1. business and industrial
2. company
3. bankruptcy

Natural Resources as Capital:
Theory and Policy1
Larry Karp
May 2015, copyright
1 This
manuscript bene…tted from comments by Yang Xie, Huayong Zhi, Zhen
Sun, Sangeeta Bansal, and Karolina Ryszka. It also bene…tted from the test-runs
provided by the 2013 and 2014 classes in EEP 102.
ii
Contents
1 Resource economics in the Anthropocene
1
2 Preliminaries
2.1 Arbitrage . . . . . . . . . . . . . . .
2.2 Comparative statics . . . . . . . . . .
2.3 Elasticities . . . . . . . . . . . . . . .
2.4 Competition and monopoly . . . . .
2.5 Discounting . . . . . . . . . . . . . .
2.6 Welfare . . . . . . . . . . . . . . . .
2.7 Summary . . . . . . . . . . . . . . .
2.8 Terms, study questions, and exercises
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3 Nonrenewable resources
3.1 The competitive equilibrium . . . . .
3.2 Monopoly . . . . . . . . . . . . . . .
3.3 Comparative statics . . . . . . . . . .
3.4 Summary . . . . . . . . . . . . . . .
3.5 Terms, study questions, and exercises
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4 Additional tools
4.1 A more general cost function . . . . . . . . . . . . . . . .
4.2 The perturbation method . . . . . . . . . . . . . . . . .
4.2.1 It is optimal to use all of the resource . . . . . . .
4.2.2 It is optimal to leave some of the resource behind
4.2.3 Solving for the equilibrium . . . . . . . . . . . . .
4.3 Rent . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Examples . . . . . . . . . . . . . . . . . . . . . .
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . .
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iii
iv
CONTENTS
4.5 Terms, study questions, and exercises . . . . . . . . . . . . . . 77
5 The
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
Hotelling model
The Euler equation (Hotelling rule) . .
Rent and Hotelling . . . . . . . . . . .
Lagrange multipliers and shadow prices
Resources and asset prices . . . . . . .
The order of extraction of deposits . .
Monopoly . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . .
Terms, study questions, and exercises .
6 Empirics and Hotelling
6.1 Hotelling and prices . . . . . . . . . .
6.2 Non-constant costs . . . . . . . . . .
6.3 Extending the model . . . . . . . . .
6.4 Summary . . . . . . . . . . . . . . .
6.5 Terms, study questions and exercises
7 Completing the solution
7.1 The change in price . . . . . . . . . .
7.2 The transversality condition . . . . .
7.3 The algorithm . . . . . . . . . . . . .
7.4 Leaving some resource in the ground
7.5 Summary . . . . . . . . . . . . . . .
7.6 Terms, study questions, and exercises
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8 Backstop technology
8.1 The backstop model . . . . . . . . . . . .
8.2 Constant extraction costs . . . . . . . . .
8.3 More general cost functions . . . . . . . .
8.3.1 Costs depend on extraction but not
8.3.2 Stock-dependent costs . . . . . . .
8.4 Summary . . . . . . . . . . . . . . . . . .
8.5 Terms, study questions and exercises . . .
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CONTENTS
v
9 The
9.1
9.2
9.3
9.4
“Green Paradox”
The approach . . . . . . . . . . . . . . .
Cumulative extraction . . . . . . . . . .
Extraction pro…le . . . . . . . . . . . . .
Why does the extraction pro…le matter?
9.4.1 Catastrophic changes . . . . . . .
9.4.2 Rapid changes . . . . . . . . . . .
9.4.3 Convex damages . . . . . . . . .
9.5 Assessment of the Paradox . . . . . . . .
9.5.1 Other investment decisions . . . .
9.5.2 The importance of rent . . . . . .
9.5.3 The importance of elasticities . .
9.5.4 Strategic behavior . . . . . . . . .
9.6 Summary . . . . . . . . . . . . . . . . .
9.7 Terms, study questions, and exercises . .
10 Policy in a second best world
10.1 Second best policies and targeting . .
10.2 Monopoly + pollution . . . . . . . .
10.3 Collective action and lobbying . . . .
10.4 Policy complements and substitutes .
10.5 Summary . . . . . . . . . . . . . . .
10.6 Terms, study questions and exercises,
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11 Taxes: an introduction
11.1 Types of taxes . . . . . . . . . . . . . .
11.2 Tax incidence and tax equivalence . . .
11.3 An application to environmental policy
11.4 Tax equivalence . . . . . . . . . . . . .
11.5 Approximating tax incidence . . . . . .
11.6 The leaky bucket: deadweight cost . .
11.7 Taxes and cap & trade . . . . . . . . .
11.8 Summary . . . . . . . . . . . . . . . .
11.9 Terms, study questions and exercises .
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139
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183
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12 Taxes: nonrenewable resources
211
12.1 Resource taxes in the real world . . . . . . . . . . . . . . . . . 212
12.2 The logic of resource taxes . . . . . . . . . . . . . . . . . . . . 215
vi
CONTENTS
12.3 An example . . . . . . . . . . . . . . . . . . .
12.3.1 Calculating the price trajectory . . . .
12.3.2 The price trajectories . . . . . . . . . .
12.3.3 Tax incidence . . . . . . . . . . . . . .
12.3.4 Welfare changes . . . . . . . . . . . . .
12.3.5 Consumer welfare in a dynamic setting
12.4 Summary . . . . . . . . . . . . . . . . . . . .
12.5 Terms, study questions, and exercises . . . . .
13 Property rights and regulation
13.1 Overview of property rights . . . . .
13.2 The Coase Theorem . . . . . . . . .
13.3 Regulation of …sheries . . . . . . . .
13.3.1 Over-capitalization . . . . . .
13.3.2 Property rights and regulation
13.4 Subsidies to …sheries . . . . . . . . .
13.5 Summary . . . . . . . . . . . . . . .
13.6 Terms, study questions and exercises
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14 Renewable resources: tools
14.1 Growth dynamics . . . . . . . . . . . . . . . . . . .
14.2 Harvest and steady states . . . . . . . . . . . . . .
14.3 Stability . . . . . . . . . . . . . . . . . . . . . . . .
14.3.1 Discrete time versus continuous time models
14.3.2 Stability in continuous time . . . . . . . . .
14.3.3 Stability in the …shing model . . . . . . . .
14.4 Maximum sustainable yield . . . . . . . . . . . . .
14.5 Summary . . . . . . . . . . . . . . . . . . . . . . .
14.6 Terms, study questions, and exercises . . . . . . . .
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257
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15 The
15.1
15.2
15.3
15.4
15.5
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open access …shery
Stock-independent costs . . . . . . .
Stock-dependent costs . . . . . . . .
Policy applications . . . . . . . . . .
Summary . . . . . . . . . . . . . . .
Terms, study questions, and exercises
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271
272
274
278
284
285
CONTENTS
vii
16 The sole-owner …shery
16.1 The Euler equation for the sole owner . . . . . . . .
16.1.1 Intuition for the Euler equation . . . . . . .
16.1.2 Rent . . . . . . . . . . . . . . . . . . . . . .
16.2 Policy . . . . . . . . . . . . . . . . . . . . . . . . .
16.2.1 Optimal policy under a single market failure
16.2.2 Optimal policy under two market failures . .
16.2.3 The rent falls as the stock increases . . . . .
16.2.4 Empirical challenges . . . . . . . . . . . . .
16.3 The steady state . . . . . . . . . . . . . . . . . . .
16.3.1 Harvest costs independent of stock . . . . .
16.3.2 Harvest costs depend on the stock . . . . . .
16.3.3 Empirical evidence . . . . . . . . . . . . . .
16.4 Summary . . . . . . . . . . . . . . . . . . . . . . .
16.5 Terms, study questions, and exercises . . . . . . . .
17 Dynamic analysis
17.1 The continuous time limit . . . . . . . . . . .
17.2 Harvest rules for stock-independent costs . . .
17.2.1 Tax policy implications of Tables 1 and
17.2.2 Con…rming Table 2 . . . . . . . . . . .
17.3 Harvest rules for stock dependent costs . . . .
17.3.1 Tax policy . . . . . . . . . . . . . . . .
17.3.2 The phase portrait . . . . . . . . . . .
17.4 Summary . . . . . . . . . . . . . . . . . . . .
17.5 Terms, study questions, and exercises . . . . .
18 Sustainability (Under construction)
18.1 Concepts of sustainability . . . . . . . .
18.2 The Hartwick rule . . . . . . . . . . . . .
18.3 Discounting and intergenerational equity
18.4 Economics and climate policy . . . . . .
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A Math Review
337
B Comparative statics
345
C More complicated perturbations
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viii
CONTENTS
D The full solution in the backstop model
351
E Geometry of the Theory of the Second Best
355
F Algebra of taxes
F.1 Algebraic veri…cation of tax equivalence . . . . .
F.2 The open economy . . . . . . . . . . . . . . . .
F.3 Approximating tax incidence . . . . . . . . . . .
F.4 Approximating deadweight loss and tax revenue
G Continuous versus discrete time
G.1 The continuous time limit . . . .
G.2 Stability in discrete time . . . . .
G.2.1 Explanation of Table 1 . .
G.2.2 Applying the stability test
G.3 Exercises . . . . . . . . . . . . . .
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H Bioeconomic equilibrium
381
I
385
The Euler equation for the sole owner …shery
J Dynamics of the sole owner …shery
389
J.1 Derivation of equation 17.2 . . . . . . . . . . . . . . . . . . . . 389
J.2 The di¤erential equation for harvest . . . . . . . . . . . . . . . 390
J.3 Finding the full solution . . . . . . . . . . . . . . . . . . . . . 392
Afterword
395
Preface
Please send suggestions for how this manuscript can be improved (e.g. changes
in emphasis or the addition or deletion of topics), and let me know about
mistakes and typos: [email protected]. For the online version of this book,
click here.
The book is designed for upper division undergraduates. Together with
the appendices, it is also suitable for a masters level course. Prerequisites
include an intermediate micro-economics course and a grounding in calculus. The presentation uses derivatives, and in a few cases partial and total
derivatives. Appendix A reviews the required mathematical tools.
The text covers standard resource economics topics, including the Hotelling
model for nonrenewable resources, and renewable resource models such as
…sheries. The distinction between natural resource and environmental economics has blurred and become less useful over the decades. This book
re‡ects that evolution by including some topics that also …t in an environmental economics text. For example, the problem of climate change involves
resource stocks, and therefore falls under the rubric of natural resources. Environmental externalities drive the problem, so the topic also …ts in an environmental economics text. It is not productive to pigeonhole such problems
into either resource economics or environmental economics.
Two important ideas run through this book. First, resources are a type
of natural capital; their management is an investment problem, requiring
forward-looking behavior, and thus requiring dynamics. Second, our interest
in natural resources stems largely from the prevalence of market failures,
notably imperfect property rights, in resource markets. The book emphasizes
skills and intuition needed to think sensibly about dynamic models, and
about regulation in the presence of market failures.
ix
x
PREFACE
Standard topics Chapters 3 –5 and 8 cover nonrenewable resources.
Chapters 3 and 4 study the two-period model, …rst in the simplest setting
and then including stock-dependent costs. Chapter 4 explains the role of resource scarcity and stock-dependent extraction costs in determining resource
rent. This chapter shows how to obtain the optimality condition using the
“perturbation method”; Chapter 5 adapts that method to the T period
setting to obtain the Euler equation, known in this context as the Hotelling
condition. All of the dynamic analysis uses the perturbation method, the discrete time calculus of variations. This approach enables students to perform
constrained optimization almost without being aware of it, and is simpler
and more intuitive than the method of Lagrange. We use the two-period
optimality condition to write the T -period optimality condition merely by
replacing time subscripts. Chapter 5 also discusses the idea of the shadow
value of a resource stock. Chapter 8 presents the backstop model. Backstops
are economically important; this material gives students practice in working
with the Hotelling model and prepares them for a subsequent policy-focused
chapter.
Chapters 13 – 17 study renewable resources. We emphasize …sheries
because these provide a concrete setting, and they illustrate most of the issues found in other renewable resources. Chapter 13 de…nes and provides
historical perspective on di¤erent types of property rights, and then de…nes
and illustrates the Coase Theorem. The rest of the chapter uses real world
examples and a one-period analytic model to describe the di¢ culties and
the unintended consequences arising from …shery regulation. It discusses
attempts to establish property rights in …sheries, as an alternative or a complement to regulation.
Chapter 14 introduces the concepts needed to study renewable resources,
including the growth function, steady states, (local) stability, and maximum
sustainable yield. Chapter 15 discusses the open access …shery, showing
how taxes a¤ect the equilibrium outcome when harvest costs are stock independent and in the more realistic case where costs depend on the stock
size. This chapter also shows why the e¤ect of regulation may be sensitive
to the initial stock size at the time regulation is introduced. Chapter 16
discusses the sole owner …shery, deriving the Euler equation and illustrating
the e¤ect of policy under one or more market failure. This chapter examines
the steady state. Chapter 17 provides more advanced material needed to
analyze the sole owner …shery outside the steady states, building up to the
phase portrait.
PREFACE
xi
Chapter 18 explains concepts of sustainability and then discusses the
Hartwick rule. The chapter de…nes the social discount rate, explains its
relation to the discount rate used in all previous chapters, and discusses its
role in intertermporal cost bene…t analysis. An overview of economists’
approach to the problem of climate policy ties together these issues. Closing
the circle, readers are reminded that the climate is a type of natural capital
for which there are weak or non-existent property rights; the market will not
e¢ ciently solve the investment problem for this capital stock.
These ten chapters (3 –5, 8, 13 –18) cover the nuts and bolts of resource
economics, and could stand alone as a mini-course.
Less standard topics Chapter 2 reviews topics in micro-economics
needed to study resource economics. These topics include the concept
of arbitrage, the use of elasticities, the relation between competitive and
monopoly equilibria, and the use of discounting. Chapter 6 explains what
it means to empirically test the Hotelling model, and summarizes research
results. A numerical example in Chapter 7 shows how to use the transversality condition and the Euler equation to obtain the equilibrium extraction
and price trajectory for a nonrenewable resource. Although not essential for
subsequent chapters, this material gives students practice in assembling different pieces of information needed to solve a model that, a few weeks earlier,
would have been entirely mysterious to them; it also gives them practice in
using an algorithm.
Chapters 9 –12 introduce policy problems. Chapter 9 uses the Hotelling
model to examine the “Green Paradox”, an important topic in climate policy.
In addition to its intrinsic interest, this material gives students practice in
using the Hotelling model, and more generally illustrates the use of models to
study policy questions. The material promotes critical thinking by discussing
limitations of the green paradox model.
Chapter 10 provides the foundation for policy analysis when market failures are important. It explains and illustrates the Theory of the Second Best
and the Principle of Targeting, and discusses the importance of political lobbying and the distinction between policy complements and substitutes. In
order to present this material simply, examples in the chapter use static environmental problems, instead of stock-dependent natural resource problems.
Chapter 13 uses the concepts developed here, again in a one-period setting.
Chapter 15 then develops these concepts in a dynamic setting.
xii
PREFACE
Taxes and other market-based instruments are becoming increasingly important regulatory tools. Chapter 11 introduces the principles of taxation
in a static framework. The material includes the de…nition of di¤erent types
of taxes, the meaning and circumstances of tax equivalence, and the meaning and measurement of tax incidence and deadweight loss. These basic
ideas provide a conceptual framework for estimating the fraction of permits
in a cap and trade scheme that need to be grandfathered in order to compensate …rms for the cost of regulation. Chapter 11 is essential for understanding
taxes in the dynamic natural resource setting, the topic of Chapter 12. That
chapter provides an overview of actual taxation (and subsidy) or fossil fuels,
and then explains how to synthesize the Hotelling model with the information on taxes studied in the static setting. A numerical example illustrates
this synthesis.
The opportunity cost of including nonstandard topics is that other important issues, such as forestry, water, and habitat protection, are not covered.
The objective of this book is to give students a strong foundation in resource
economics, enabling them to acquire and assess information about natural
resource topics not covered in the book.
Chapter 1
Resource economics in the
Anthropocene
Mounting evidence shows that we are using resources unsustainably, destroying habitat, increasing the rate of species extinction, damaging supplies of
water and other essentials, and creating climate-related risks. The power of
human ingenuity, operating through markets, also provides examples where
we have delayed or overcome impending resource scarcity. This book o¤ers
a framework for analyzing resource use, providing tools that can contribute
to improved stewardship.
Natural resources are a type of capital: natural, as opposed to man-made
capital. Resource use potentially changes the stock of this capital, and is
a type of investment decision. Change is thus a key feature of natural resources, requiring a dynamic perspective. Natural resources, like other types
of capital, provide services that a¤ect human well-being. In some cases,
as with burning oil or eating …sh, we consume those services by consuming
a part of the resource. In other cases, we consume natural resource services indirectly. Wetlands provide …ltration services, reducing the cost of
clean water. Transforming wetlands into farms or cities changes the ‡ow
of these services. Other resource stocks, such as bees and bats, critical to
agricultural production, provide indirect but essential services. Diminishing
those stocks, by reducing habitat or otherwise changing the ecosystem, alters
future pollination services.
These examples are anthropocentric, attributing value to natural resources
only because they provide services to humans. Species or wilderness areas
may have intrinsic value apart from any e¤ect, however indirect, they have
1
2 CHAPTER 1. RESOURCE ECONOMICS IN THE ANTHROPOCENE
on human welfare. Regardless of whether one begins with a purely anthropocentric view or a more spiritual/philosophical perspective, resource value
depends on the remaining stocks: oil, …sh, wetlands, pollinators, wilderness
etc. These stocks a¤ect the current and future ‡ows of services. The anthropocentric view, putting humans at the center of the narrative, can lead
to more e¤ective remedies. The protection of forests or …sheries is more
likely achieved if the people surrounding these resources have a stake in their
protection. Ecotourism in nature preserves can give local residents an incentive to respect the preserve; using the dung of elephants and rhinos to
create paper products for sale abroad gives people an incentive to protect
the animals when they stray from the preserves. Ignoring the human aspect
of resource problems leads to ine¢ cient policy.
In common usage, “capital” refers to man-made productive inputs, such
as machinery, or the monetary value of those inputs. A broader de…nition
treats capital as anything that yields a ‡ow of services. For example, education augments our stock of human capital, because it makes us more productive, or otherwise enhances our lives in the future. Natural resources fall
under this broader de…nition of capital. A …rm’s decision about purchasing
additional machinery, or an individual’s decision about acquiring more education, are investment decisions, and thus “forward looking”; they depend on
beliefs about their future consequences. The decisions are “dynamic”rather
than “static”because their temporal aspect is central to the decision-making
process.
Natural capital, like machinery, obeys laws of physics: trees age, machines rust. The fact that the two types of capital inhabit the same physical universe connects resource economics to the broader …eld of economics.
However, natural and man-made capital di¤er in the severity of the market
failures that a¤ect them. An “externality”, a type of market failure, arises
when a person does not take into account all of the consequences of their action. Unregulated pollution or excessive use of a resource stock are leading
examples of externalities. Externalities and other market failures are central
to the study of resource economics, whereas they are usually peripheral in a
general micro economics course.
Changes in resource stocks occur either intentionally, via the exercise of
property rights, or accidently, arising from externalities. A mine owner understands that extracting a unit of ore today means that this material is not
available to extract and sell in the future; in this case, the change in the stock
of ore is intentional. Where property rights to resources do not exist, indi-
3
vidual decisions in modern societies are often made without regard to their
e¤ect on resource stocks. Farmers choose levels of pesticide and fertilizer in
order to increase their pro…ts. Some of these inputs enter waterways, where
they damage the publicly owned ecosystem, as has occurred in the Everglades
and in the Gulf of Mexico. Individual farms have negligible e¤ect on the
aggregate outcomes, and individual farmers have no (sel…sh) interest in those
outcomes, so it is rational for them to ignore these consequences. In these
cases, society chooses how the publicly owned resources are used. Often the
choice, made by default, is to do little to protect the resource.
Rapid changes in resource stocks, and the expected change of future
stocks, make resource economics an especially important …eld of study. The
2005 Millennium Ecosystem Assessment reports that the recent speed and
extent of human-induced change in ecosystems is greater than the world
has previously seen. Some of these changes, such as those associated with
the expansion of agricultural production, contribute to current human wellbeing. However, many of the changes degrade resources, eroding natural
capital and threatening future well-being. The last half-century saw a …fth
of the world’s coral reefs lost, and another …fth degraded; a third of mangrove forests have been lost; water use from rivers and lakes has doubled;
nitrogen ‡ows entering ecosystems have doubled, and phosphorus ‡ows have
tripled. Human actions have increased the rate of species extinction as much
as 1000-fold, relative to rates found in the geological record. Estimates of
mammal, bird and amphibian species threatened with extinction range from
10 –30%. Tropical forests and many …sheries are in decline.
The Millennium Assessment evaluates 24 di¤erent types of ecosystem
services and concludes that 60% of these are degraded or threatened with
degradation. The loss in natural capital may lead to abrupt changes, as has
occurred with sudden ‡ips in water quality (eutrophication) and the rapid
emergence of new diseases. Moreover, these damages tend to disproportionately harm the poor and most vulnerable. Barring major policy changes or
technological developments, ecosystems will likely face increasing pressure.
Standard measures of wealth, including Gross National Product, ignore
these changes in natural capital. Attempts to account for natural capital,
show that 45% of the 136 countries studied by the World Bank are depleting
their wealth even as income measures increase. These countries are living
o¤ natural capital.
Climate change may pose the single greatest danger to future well-being.
Climate change is likely to exacerbate the types of problems already seen,
4 CHAPTER 1. RESOURCE ECONOMICS IN THE ANTHROPOCENE
such as loss of species, the spread of diseases, and increased water shortages. It may also lead to new problems, including rising sea levels, increased
frequency of severe weather events, and decreased agricultural productivity.
The costs of these changes will depend on uncertain relations between stocks
of greenhouse gasses and changes in temperature, ocean acidity, and the sealevel, and on the uncertain relation between these variables and economic and
ecological consequences (e.g. decreased agricultural productivity and species
loss). The future stocks of greenhouse gasses depend on future emissions,
which depend on uncertain changes in policy and technology.
In recognition of man’s ability to fundamentally alter the earth’s ecosystem, many scientists refer to the current geological period as the Epoch of the
Anthropocene (“New Human Epoch”). Proposals for this Epoch’s starting
date range from the early industrial age to the middle of the 20th century.
The view that current resource use will create large costs to future generations leads to resource pessimism. Thomas Malthus, an early resource
pessimist, claimed that if unchecked by war, disease, or starvation, human
population tends to rise faster than food production. He concluded that population eventually outstrips food supplies, until starvation, war, or disease
brings them back into balance.
This description was quite accurate for most of human history, but events
since Malthus wrote in 1800 have contradicted both parts of his argument.
Many countries have seen a demographic shift associated with higher income,
leading to stabilization or decreases, not increases, in population. In poor
societies, children are a form of investment for old age. In rich societies,
children do not provide the primary support for their aged parents, and the
cost of raising children is high. These factors encourage smaller family sizes
with rising income. Technological innovations have increased agricultural
productivity and reduced the cost of transporting and storing food. Population and food security have both increased. Most recent famines were caused
not by the absolute lack of food, but by its unequal distribution, exacerbated
by political dysfunction.
Resource pessimism spurred the development of resource economics. In
1931 Harold Hotelling produced one of the cornerstones of the …eld, responding to the pessimists of his time. He wrote
Contemplation of the world’s disappearing supplies of ... exhaustible assets has led to demands for regulation of their exploitation. The feeling that these products are now too cheap
5
for the good of future generations, that they are being sel…shly
exploited at too rapid a rate, and that in consequence of their excessive cheapness they are being produced and consumed wastefully has given rise to the conservation movement.
Hotelling studied the use of natural resources in an idealized market with
perfect property rights, where a rational owner takes the …nite resource supply into account. In this setting, prices signal scarcity, in‡uencing decisions about extraction, exploration, and the development of alternate energy
sources and new technologies. More generally, prices can induce fundamental changes in human behavior, such as family size. Hotelling’s contribution
faded as resource pessimism diminished.
The 1960’s saw a resurgence in resource pessimism, exempli…ed by Paul
Ehrlich’s The Population Bomb. Ehrlich, a biologist, observing rapid increases in the use of natural resources in the 1960s, and accustomed to working with mechanistic models of insect populations, predicted imminent and
catastrophic resource scarcity. Julian Simon, an economist, thought that
the market would take care of scarcity, as higher prices encouraged exploration, discovery, production and conservation. Simon proposed, and Ehrlich
accepted, a bet that over a decade the in‡ation-adjusted price of a basket
of …ve minerals would fall. This period of time seemed long enough to
test Ehrlich’s forecast of imminent scarcity. Simon won the bet, but Ehrlich
never accepted that he was fundamentally wrong, arguing that he had merely
underestimated the rapacity of man’s resource extraction. He believed that
his prediction of scarcity was wrong only in the timing.
The Ehrlich-Simon bet involved prices of metals for which there are well
established property rights. These metal prices have been volatile over the
past century, and in one respect Ehrlich was simply unlucky. Had he picked
a di¤erent decade in the 20th century, he would have had a better than even
chance of winning the bet. The price of this basket of metals fell dramatically
during the economic upheaval following World War I, and in most subsequent
decades rose gradually. The volatility of metal prices has been largely due
to macro economic cycles (recessions or booms) or political events (wars or
boycotts) unrelated to scarcity. The theory developed in subsequent chapters explains why, putting aside these reasons for price volatility, modest but
not spectacular price increases might be expected. The basis for this theory
is that resource owners can “arbitrage over time”, advancing or postponing their sales in order to take advantage of expected price changes. This
6 CHAPTER 1. RESOURCE ECONOMICS IN THE ANTHROPOCENE
arbitrage tends to lead to modest expected price increases.
Making the wager depend on these metal prices was, in some respects, an
odd choice for both the resource optimist (Simon) and pessimist (Ehrlich). It
is an odd choice for Simon, the economist, because economic theory predicts
modest price increases, not decreases. It is (at least with the bene…t of
hindsight) an odd choice for Ehrlich, the ecologist, because it involves the
category of resources for which market forces are most likely to “work well”:
those having strong property rights.
Five years later, Ehrlich proposed a di¤erent bet, one involving changes
in, among other measures, CO2 concentrations, temperature, tropical forest
area, and rice and wheat stocks per person, rather than commodity prices.
Simon rejected Ehrlich’s o¤er, and countered with a wager involving direct
measures of human well-being, including life expectancy, leisure time, and
purchasing power. Ehrlich declined that o¤er. Some of the components
of these di¤ering proposals re‡ect di¤erent views about the relation between
mankind and natural resources. The resource pessimist begins with the
premise that human welfare is inextricably linked to resource stocks; their
degradation (e.g. as measured by increased pollution or lower …sh stocks)
must eventually lower human welfare. For example, society may be well o¤
during the period it uses …sh or forest resources intensively, degrading their
stocks. If resource use is unsustainable, resource extraction must eventually
fall, and with it, human well-being. This overuse is more likely for resources
where property rights are weak. The resource optimist, in contrast, starts
from the premise that society will be able to …nd new resource stocks (new
sources of fossil fuels) or alternatives to those resources (solar power instead
of fossil fuels). Both camps present plausible scenarios that support their
position.
The resource pessimism of the 1960’s led to renewed interest in resource
economics during the 1970s and 1980s. The dominant strand of this literature extends Hotelling’s earlier work, using the paradigm of rational agents
operating with secure property rights. However, there has also been increased recognition of market failures, especially externalities associated with
missing markets and weak property rights. Modern resource economics provides a powerful lens through which to study natural resources precisely
because it takes market failures seriously. The discipline provides a counterweight to pessimists’tendency to understate society’s ability to respond to
market signals, while also providing a remedy to the excessively optimistic
belief that markets, by themselves, will solve resource problems.
7
Agriculture and …sheries illustrate the power of markets and the problems
arising from market imperfections. In both cases, markets have unleashed
productivity gains leading to abundance. But these gains occur in the
presence of market failures, giving rise to problems that threaten (in the case
of agriculture) or actually (in the case of …sheries) o¤set the initial gains.
Neither markets nor natural forces will automatically solve these problems
without regulation.
Box 1.1 The impossibility of extinction (?). In the mid 1800s, driftnet
herring …shermen ask for regulation to restrict the use of “longlines”,
which they claimed damaged …sh stocks and reduced catches. Many
scientists, believing that the self-correcting power of nature would
take care of any temporary problems, resisted those requests. The
in‡uential scienti…c philosopher Thomas Henry Huxley, a member
of British …shing commissions charged with investigating the complaints, explained in 1883 why the requests were unscienti…c, and
merely designed to impede technological progress: “Any tendency to
over-…shing will meet with its natural check in the diminution of the
supply,... this check will always come into operation long before anything like permanent exhaustion has occurred.”
Others disagreed. Maine’s …shery commissioner Edwin Gould stated
in 1892 “Its the same old story. The bu¤alo is gone; the whale is
disappearing; the seal …shery is threatened with destruction. Fish
need protection.”
Market-mediated increases in agricultural productivity since the 1960s
made it possible to feed twice the population with slightly more than a 10%
increase in farmed land, reducing or eliminating the threat of starvation for
hundreds of millions of people. Those changes were associated with increased
pesticide and fertilizer use that threaten waterways, increased and likely unsustainable use of water, and increased loss of habitat. Human ingenuity,
operating through markets, helped to alleviate the problem of insu¢ cient
food. But they achieved this progress in a world with imperfect and missing
markets, giving rise to signi…cant externalities. Increased demand for …sh
has resulted in increased harvest and reduced …sh stocks, leading to relative scarcity and higher prices. Responding to these market signals, …shers
developed and adopted new technologies, hugely increasing their ability to
catch …sh. These gains have often been short-lived, as the increased harvest
8 CHAPTER 1. RESOURCE ECONOMICS IN THE ANTHROPOCENE
degrades …sh stocks, ultimately lowering harvest.
Markets have been essential in “disproving”the resource-pessimists thus
far. Markets are powerful in part because they are self-organizing. They
require a legal and institutional framework that creates respect for private
property and con…dence in contracts; they often require regulation, but not
detailed governmental management. However, the bene…cial changes assisted by markets, occurring in the context of market imperfections, may in
the longer run validate the resource-pessimists. Where market imperfections are severe, markets are unlikely to solve, and may exacerbate resource
and environmental problems.
There are genuine technological and demographic obstacles to solving
resource problems. However, the main obstacles are probably political.
Remedies to improve resource problems usually create winners and losers,
with the losers often in a better position to defend their interests. For
example, e¤ective climate policy will decrease fossil fuel owners’wealth; these
groups have substantial political in‡uence in many countries.
Clearer thinking will not dispel these objective obstacles to reform. However, clearer thinking and more precise language can provide a basis for negotiations between people with di¤erent values and perspectives. People might
disagree on a conservation measure, but it is counter-productive to base the
disagreement on one’s identi…cation as an economist or an environmentalist.
Economists understand the importance of market failures; environmentalists
understand tradeo¤s and markets’ potential in helping to correct a problem e¢ ciently. Too often debate becomes a re‡exive exchange of slogans.
Resource economics can help to solve problems by providing a common language and an analytic framework, creating the possibility of moving beyond
ideology.
Resource economics also helps in understanding that institutional reform,
such as the creation of property rights, is often an e¤ective remedy to problems. Some people distrust property rights because they (correctly) see these
as the basis for markets, and they (incorrectly) think that markets are responsible for the resource problem. Resource economics teaches that many
problems are due not to markets, but to market failures.
Disagreements about resource-based problems tend to be easier to resolve where the problems are local or national rather than global, and where
changes occur quickly (but not irreversibly), rather than unfolding slowly.
The local or national context makes the horse-trading needed to compensate
losers easier. The rapid speed of change makes the problem more obvious,
9
and makes the potential bene…ts of remedying the problem, and the costs
of failing to do so, more pressing for the people who need to engage in this
horse-trading. The most serious contemporary resource-based problems are
global and unfold over long periods of time, relative to individuals’lifetime,
and certainly relative to the political cycle.
A prominent international treaty, the Montreal Protocol, succeeded in
halting and reversing the global problem of ozone depletion. The rapid
increase in the ozone hole over the southern hemisphere made the problem
hard to ignore, and the availability of low-cost alternatives to ozone-depleting
substances made it fairly cheap to …x.
International negotiations on other global problems, prominently climate
change, have been notably unsuccessful. It is di¢ cult to summon the political will to make the international transfers needed to compensate nations
that would be, or think they would be, better o¤ without an agreement. The
most serious e¤ects of climate change likely impact future generations, who
have no direct representation in current negotiations.
An overview of the book
Two points made above set the stage for the rest of this book. (i) Markets
have the potential to ease environmental and resource constraints, contributing hugely to the increase in human welfare. (ii) Many problems arise from
market failures, such as externalities associated with pollution; those market failures may diminish or even reverse the bene…cial e¤ects arising from
markets that function well. A corollary to these claims is that regulation
that harnesses the power of market forces, or the establishment of property
rights, may make it easier to solve resource problems. Nevertheless, those
regulations and institutional changes require political intervention; they do
not arise spontaneously from market forces. This book develops and uses a
theoretical apparatus that can contribute to coherent analysis of these issues.
Theory makes it possible to intelligently evaluate the facts of speci…c cases,
in pursuit of better policy prescriptions.
The pedagogic challenge arises from the fact that resources are a type of
capital, requiring a dynamic setting in which agents are forward looking. In
a static setting, …rms’and regulators’decisions depend on current prices and
(for example) pollution. In the dynamic resource setting, a …rm’s decision
on how much of the resource to extract and sell in a period depend on the
price in that period, and the …rm’s beliefs about future prices. A regulator’s
10CHAPTER 1. RESOURCE ECONOMICS IN THE ANTHROPOCENE
(optimal) policy depends on beliefs about future actions, and these depend
on future prices. This di¤erence between the static and dynamic setting is
central in our presentation of resource economics.
The …rst half of the book provides the foundation for studying nonrenewable resources, such as coal or oil. This foundation requires a review
of some aspects of microeconomic theory, and the development of methods
needed to study dynamic markets. We apply these methods to the resource
problem, emphasizing perfectly competitive markets with no externalities or
other distortions.
This material identi…es incentives that are important
in determining resource use, and helps the reader understand the potential
for markets, when they work well. The second half of the book applies
these tools, emphasizing situations where market failures create a rationale
for policy intervention. We consider both the possibility that policies ameliorate the market failure, and the possibility that policy is harmful due to
unintended consequences. We also move from nonrenewable to renewable
resources, making it possible to show how di¤erent systems of property rights
and policies alter resource levels at di¤erent time scales.
Terms and concepts
Epoch of the Anthropocene, renewable versus nonrenewable resource, eutrophication, market failure, externality, resource pessimist/optimist
Sources
The United Nations’ (2005) Millennium Ecosytem Assessment reports an
international group of scholars’assessment of recent environmental changes,
their consequences on human well-being, and likely scenarios for future changes.
Alix-Garcia et al (2009) discuss the payment of environmental services in
agriculture.
Elizabeth Kolbert The Sixth Extinction: an Unnatural History documents species extinction.
Duncan Foley (2006) Adam’s Fallacy discusses Thomas Malthus and
other important economists.
Scott Barrett Environment and Statescraft (2003) discusses the di¢ culty
of creating e¤ective global environmental agreements.
Gregory Clark A Farewell to Alms provides a long run historical perspective on Mathus’ideas.
11
Paul Sabin, “The Bet” examines the tension between resource optimists
and pessimists, in particular between Ehrlich and Simon.
The World Bank’s (2014) Little Green Data Book provides statistics on
green national accounting.
Robert Solow (1974) provides the quote from Hotelling (1931).
Partha Dasgupta (2001) Human Well-Being and the Natural Environment
and develops the concept of natural resources as a type of capital.
Edward Barbier Capitalizing on Nature: Ecosystems as Natural Assets
extends this concept to ecosytems, or ecological capital, e.g. wetlands, forests,
and watersheds.
The quote from Huxley in Box 1.1 is from Kurlansky (1998), and the
quote from Gould in from Bolster (2015).
12CHAPTER 1. RESOURCE ECONOMICS IN THE ANTHROPOCENE
Chapter 2
Preliminaries
Objectives
To prepare students to study resource economics.
Information and skills
Understand the meaning of arbitrage.
Understand the distinction between exogenous and endogenous variables; analyze the e¤ect of an exogenous change on an endogenous
outcome (comparative statics).
Know the de…nition and purpose of elasticities, and be able to calculate
them.
Understand the relation between a competitive and a monopoly outcome.
Know the de…nition and purpose of a discount rate and a discount
factor; use them to calculate present values and to compare pro…t or
cost streams.
Know the basics of welfare economics, in particular the fact that in the
absence of market failures, a competitive equilibrium is e¢ cient.
This chapter reviews and supplements the micro-economic foundation
needed for natural resource economics. In the familiar static setting, a
13
14
CHAPTER 2. PRELIMINARIES
competitive equilibrium occurs when many price-taking …rms choose output
to maximize their pro…ts. A competitive equilibrium in the resource setting
involves a time-path of output, not merely output at a single point in time.
Resource-owning …rms begin with an initial stock of the natural resource, and
decide how much to supply in (typically) many periods, not just in a single
period. For a non-renewable resource such as coal or oil, the cumulative
extraction over the …rm’s planning horizon cannot exceed the …rm’s initial
stock level. For renewable resources such as …sh, where a natural growth
rate can o¤set harvest, the …rm faces a more complex constraint. For both
types of resources, the …rm solves a constrained optimization problem. We
use the solution to that problem to analyze the competitive equilibrium.
In each period, the …rm obtains revenue and incurs costs. A dollar
today is not the same as a dollar ten years from now; they are apples and
oranges. A sensible description of the …rm’s optimization problem requires
that revenue and costs in di¤erent periods are commensurable. Discounting
achieves this objective.
The necessary conditions for optimality to the …rm’s problem involve
techniques developed in subsequent chapters. We interpret these optimality conditions as “intertemporal arbitrage conditions”. Before introducing
the idea of intertemporal arbitrage, it helps to discuss the meaning and the
implication of plain vanilla “spatial”arbitrage. “Arbitrage”is the practice
of taking advantage of di¤erences, often price di¤erences, in di¤erent markets. Moving a commodity from one place to another is the most familiar
form or arbitrage. This spatial arbitrage, and “intertemporal arbitrage”,
selling a commodity in one period instead of another, can be studied using similar methods. Intertemporal arbitrage is the basis for understanding
equilibria in resource markets, and spatial arbitrage provides the foundation
for understanding intertemporal arbitrage.
Models serve a variety of purposes, one of which is to understand how
changes in a piece of data, or an assumption, a¤ect an equilibrium outcome.
The piece of data or assumption can often be characterized by assigning a
value to a parameter in the model. These “exogenous”parameters are determined “outside the model”. The model helps us discover or understand an
“endogenous outcome”, determined “inside the model” For example, the cost
of shipping goods from one location to another a¤ects the quantity shipped.
We may want to know how changes in shipping costs alter the equilibrium
amount shipped. In this example, the shipping cost is an exogenous parameter, and the amount shipped is an endogenous outcome. The question “How
2.1. ARBITRAGE
15
does a change in the shipping costs alter the equilibrium amount shipped?”
is a comparative statics question. This chapter provides the foundation for
answering these kinds of questions.
Economics involves the relation between di¤erent things, such as price
and quantity. “Elasticities”provide a “unit free”measure of these types of
relations; “unit free”means, for example, that the relation does not depend
on whether we measure prices in dollars per pound or Euros per kilo.
We emphasize competitive equilibria. However, many important resource
markets, including markets for petroleum, diamonds, and aluminium (produced using bauxite, a natural resource) are not, or have not always been,
competitive. We therefore supplement the study of competitive markets by
considering the case of monopoly. Few, if any resource markets are literally monopolistic, unless one adopts an extremely narrow de…nition of the
commodity (e.g. diamonds from a particular region, instead of diamonds in
general). The monopoly model is useful as a limiting case, against which to
compare perfect competition. Many actual resource markets lie somewhere
on the continuum between these two. We use the elasticity of demand to
relate the equilibrium conditions under perfect competition and monopoly.
Two welfare theorems explain the relation between a competitive equilibrium and the outcome of a social planner who maximizes social welfare.
This relation is essential for understanding the circumstances under which a
competitive equilibrium is e¢ cient. It also provides the basis for a discussion
of market failures.
2.1
Arbitrage
Objectives and skills
Understand arbitrage and graphically represent and analyze the “noarbitrage”condition.
Much of the intuition for later results rests on the idea of “arbitrage over
time”. This idea is closely related to the more familiar idea of arbitrage
over space, reviewed here. Suppose that there is a …xed stock of tea in
China, 10 units. The inverse demand for tea in China is pChina = 20 q China ,
where q China is the quantity consumed in China. A demand function gives
quantity as a function of price, and the inverse demand function gives price
16
CHAPTER 2. PRELIMINARIES
price
20
China demand
15
US demand
10
5
0
0
1
2
3
4
5
6
7
8
9
10
11
12
tea
Figure 2.1: Solid line shows demand for tea in China. Dashed line shows
demand for tea in US. Dotted line shows price received by Chinese exporter
with 20% ttransportation costs.
as a function of quantity. The inverse demand for tea in the U.S. is pU.S. =
18 q U.S. .
Figure 2.1 shows the inverse demand in China (solid line) and in the
U.S. (dashed line). Moving left to right on the horizontal axis represents
an increase in consumption in China, and a corresponding decrease in the
U.S., because total supply is …xed at 10 by assumption. The U.S. demand
function is therefore read “right to left”; for example, the point 3 on the
tea axis means that China consumes 3 units and the U.S. consumes 7 units.
Except where noted otherwise, we assume that the equilibrium is “interior”,
meaning (here) that some tea is sold in both countries; at a “boundary”
equilibrium, all of the tea is sold in a single country.
If there is no cost of transportation, and if sales are positive in both
countries, then price in a competitive equilibrium must be equal in the two
countries. This equality represents a “no-arbitrage”condition. “Arbitrage”
refers to the practice of taking advantage of price di¤erences (possibly in
di¤erent locations), to make bene…cial trades. There are no arbitrage opportunities when no further pro…table trades are possible. If the price in
China was lower than the U.S. price, an agent with zero transportation costs
could make money by buying a unit of the good in China and selling it in
the U.S.. If the price in the U.S. was lower, an agent would ship tea from
the U.S. to China. There are no remaining opportunities for arbitrage if
and only if the price is the same in the two locations. In the competitive
equilibrium, using the demand functions above, China consumes 6 units, the
U.S. consumes 4 units, and the equilibrium price, 14, is the same in both
countries.
2.1. ARBITRAGE
17
It is easy to become confused about causation in a competitive equilibrium. The competitive exporter in our tea example does not move tea from
one location to another in order to cause the prices in the two locations to
be the same. This competitive exporter takes prices as given. Prices in the
two locations are the same because of the exporter’s ability to move goods
from one location to another. If the price in China was lower than the U.S.
price, the exporter would purchase tea in China to export to the U.S.. Her
purchases increase the demand in China, causing price to rise there. Selling the tea in the U.S. increases the supply there, causing the U.S. price to
fall. A single exporter may move too few goods to a¤ect prices, but if all
exporters have the same incentives, their collective actions a¤ect prices. The
outcome is the same regardless of whether there are many small exporters,
each of which is an insigni…cant part of the market, or whether there is a
single exporter who takes price as given.
Transportation costs are important in the real world. These costs can
be expressed on a per unit basis (some number of dollars per unit) or on
an ad valorem basis (some percent of the value). In the latter case, costs
are sometimes called “iceberg costs”; it is as if a certain fraction of the
value melts in moving the good from one place to another. The transport
costs include the physical cost of transportation, and ancillary costs such
as the cost of setting up distribution networks, acquiring information about
prices in di¤erent locations, and insurance. We represent these iceberg
costs using b (for “berg”). Expressed as a percent (rather than a fraction),
the transportation costs are b 100%. If b = 0:3, then an exporter faces
transportation costs of 30% of the purchase cost. If the price of a unit is
pChina in China, the cost of selling the unit in the U.S. is (1 + b) pChina : an
exporter buys the good for pChina and spends bpChina in order to transport
the good, for a total cost of (1 + b) pChina . In a competitive equilibrium with
positive sales in both countries, (1 + b) pChina = pU.S. , or
pChina =
1
pU.S. :
(1 + b)
(2.1)
Equation 2.1 is the no-arbitrage condition in the presence of iceberg transportation costs, b
0. If b > 0, then equation 2.1 implies that the U.S.
price exceeds the China price in a competitive equilibrium.
The dotted line in Figure 2.1 shows the U.S. demand, adjusted for 30%
transport costs (b = 0:3). A point on this dotted line gives the amount that
an exporter receives, at a given level of U.S. consumption, after the exporter
18
CHAPTER 2. PRELIMINARIES
pays transportation costs. The equilibrium sales occurs at the intersection
of the solid line and this dotted line. At this quantity, the price in China
equals the price, net of transportation costs, that an exporter receives for
U.S. sales. For our example, this equilibrium occurs where China consumes
7.83 units, at price 12.17; the U.S. consumes the remaining 2.17 units. The
U.S. transportation-inclusive price, a point on the U.S. demand function (the
dashed line), is 1:3 12:17 = 15:83. Net of transportation costs, the exporter
receives 12:17, a point on the dotted line.
In the examples above, “the law of one price” holds: the price, adjusted
for transport costs , is the same in both locations. This “law” describes a
tendency. Arbitrage requires that people have information about prices in
di¤erent locations. If information about price di¤erences is imperfect, then
arbitrage would still create a tendency for prices to move together, but we
would not expect them to be exactly equal. The potential to pro…t from
price di¤erences gives people an incentive to acquire information. It is not
necessary that the individual selling tea in China know the price in the U.S..
There may be a chain of people in a range of markets between the China
and the U.S.. Each of these knows the price in neighboring markets, so they
know whether it is pro…table to move tea east or west.
Markets “aggregate” information; the middlemen move the commodity
from where the price is low to where it is high. The actions of these middlemen reveal information about price di¤erences, making it unnecessary for
the seller in China to incur the cost to learn about the price in the U.S.,
thus lowering a component of transportation costs. The middlemen also
require payment for their e¤ort, thus increasing transport costs. Modern
technology leads to the cheaper ‡ow of information, reducing transportation
costs, making it more likely that the forces of arbitrage cause prices to move
together. Apps make it possible to instantly compare prices in di¤erent
stores. Farmers in developing countries use cell phones to learn about price
di¤erences in di¤erent markets.
2.2
Comparative statics
Objectives and skills
Understand the distinction between endogenous and exogenous variables; understand the meaning of a comparative statics question.
2.2. COMPARATIVE STATICS
19
Be able to use an equilibrium condition to conduct comparative statics
analysis.
This section uses an example to illustrate the meaning of a comparative statics experiment (or question), and to show how that question can be
answered. Using the tea example from Section 2.1, consider the comparative statics question “How do transportation costs a¤ect the equilibrium
prices and quantities in the two markets?” This question is simple enough
to answer without the use of a model: higher transportation costs decrease
exports from China to the U.S., increasing supply and decreasing the equilibrium price in China, and having the opposite e¤ect in the U.S. Figure
2.1 illustrates the graphical approach to answering this question. Mathematics helps for questions that are too complicated to answer using informal
reasoning or graphs.
The …rst step in addressing a comparative statics question is to be clear
about which variable(s) is (are) exogenous (= given, or determined outside
the model), and which are endogenous (= determined by the model). In this
example, the transportation parameter, b, is exogenous, and the prices and
quantities in the two markets are endogenous.
The second step is to identify the equilibrium condition that determines
the endogenous variables. In this case, the no-arbitrage condition 2.1 is
the equilibrium condition. Substituting the demand functions for the two
countries, pchina = 20 q China and pU.S. = 18 q U.S. and using the constraint
that q U.S. = 10 q China , we can write this equilibrium condition as
1
18
10 q China
20 q China =
| {z } 1 + b
|
{z
} :
China
China
L q
=
R q
;b :
(2.2)
Once we know q China it is straightforward to …nd q U.S. and the two prices, so
we consider only the comparative statics of q China with respect to b.
Using the explicit solution Equation 2.2 gives q China as an implicit function of b. Because of its linearity, this equation can be easily solved to yield
the explicit equation.
20b + 12
q China =
:
b+2
20
CHAPTER 2. PRELIMINARIES
Using the quotient rule, we obtain the derivative:
28
dq China
=
>0
db
(b + 2)2
(2.3)
As already noted, an increase in transport costs increases equilibrium sales
in China.
Using the implicit relation In cases where the equilibrium condition is
too complicated to solve explicitly, we can still obtain information merely by
using the equilibrium condition, which gives the endogenous variable as an
implicit function of the exogenous parameters. The second line of equation
2.2 shows that the left side is denoted as L q China and the right side as
R q China ; b . An exogenous change in b alters the right side without having a direct e¤ect on the left side. In order for the equality L q China =
R q China ; b to continue to hold after the change in b, there must be a compensating change in q China . The change in R, denoted dR, must equal the
change in L, denoted dL. These changes are called the “di¤erentials” of R
and L. Equilibrium requires
dL q China
= dR q China ; b :
(2.4)
Using the de…nition of a di¤erential and the rules of partial derivatives (Appendix A), equation 2.4 yields the comparative statics expression (Appendix
B)
q China + 8
dq China
= 2
> 0:
(2.5)
db
b + 3b + 2
Comparing the two methods Equations 2.3 and 2.5 both show how
q China responds to a change in transport costs, b. The right side of equation
2.5 involves the unknown value q China , whereas the right side of equation 2.3
involves only numbers and the exogenous parameter b.
In this respect,
the comparative statics expression 2.3 is more informative than equation 2.5.
The approach using di¤erentials is useful when it is di¢ cult to solve the
equilibrium condition to obtain the explicit expression for the endogenous
variable. Equation 2.5 tells us only that an increase in b increases sales in
China. Often we use models to obtain “qualitative”rather than “quantitative”information, e.g. we care more about the direction than the magnitude
of the change.
2.3. ELASTICITIES
2.3
21
Elasticities
Objectives and skills
Calculate elasticities and understand that they are “unit free”.
Economists frequently use elasticities instead of slopes (derivatives) to
express the relation between two variables, e.g. between price and quantity
demanded. The slope of the demand function tells us the number of units
of change in quantity demanded for a one unit change in price. The elasticity of demand with respect to price tells us the percent change in quantity
demanded for a one percent change in price. Unless the meaning is clear
from the context, we have to specify the elasticity of something (here, quantity demanded) with respect to something else (here, price). In general, the
value of an elasticity depends on the price at which it is evaluated.
denotes the rate
The symbol dQ denotes the change in Q. The ratio dQ
Q
100 is the
of change; multiplying by 100 converts a rate to a percent, so dQ
Q
percent change in Q. The percent change in Q for each one percent change
in P is the ratio
dQ
dQ
100
dQ P
Q
Q
:
= dp =
dP
dP Q
100
P
p
The …rst equality follows from cancelling the 100’s, and the second follows
from rearranging the ratio of the ratios1 .
The demand function Q = D (P ) gives the relation between quantity
demanded and price, Q and P . The elasticity of Q with respect to P ,
denoted , is de…ned as
=
dQ P
=
dP Q
D0 (P )
P
:
D (P )
We follow the convention of including the negative sign in the de…nition,
making the elasticity a positive value. If we want to evaluate this elasticity
at a particular price, say Po , we express it as
(Po ) =
a
D0 (Po )
Po
:
D (Po )
This rearrangement uses the rule cb = ab dc = ac db . For our problem, a corresponds to
d
dQ and c corresponds to dP . We can manipulate these variables in the same way that
we manipulate a; b; c; d.
1
22
CHAPTER 2. PRELIMINARIES
Suppose that we measure quantity in pounds and prices in dollars per
pound, and the demand function is
Q=3
5P:
The inverse demand function corresponding to this demand function is
3 Q
:
P =
5
The slope of the inverse demand function is 15 . The elasticity of demand
with respect to price, evaluated at P = 0:5 is
0:5
dQ P
=5
= 5:
dp Q
3 5(0:5)
Box 2.1 Elasticities are unit free. For the example above, suppose
that we want to express the price in pesos rather than dollars, and
1
suppose that there are 10 pesos per dollar; one peso is 10
dollars.
Using upper case P to denote the number of dollars, and lower case
p
. The demand
p to denote the number of pesos, p = 10P , so P = 10
function is
1
Q = 3 5P = 3 5 p:
10
The …rst equality gives demand as a function of dollars, and the second
gives demand as a function of pesos. The inverse demand, expressed
in pesos instead of dollars, is
p=
3
Q
1
2
=6
2Q:
The change in units from dollars per pound to pesos per pound
changes the slope of the inverse demand function from 15 to 2 and
changes the vertical (price) intercept from 35 to 6. The elasticity of
demand with respect to price, evaluated at a price of 0.5 dollars, or 5
pesos, is
5
dQ p
1
=
=5
dp Q
2 3 12 5
Changing the units –in this case from dollars per pound to pesos per
pound – changes the appearance of the demand function, but does
not change the elasticity.
2.4. COMPETITION AND MONOPOLY
23
Economists frequently use elasticities instead of slopes, or derivatives,
because the elasticity, unlike the slope, is “unit free”.2 For example, the
elasticity does not change if we change units from pounds to kilos, or measure
prices in pesos rather than dollars (Box 2.1). In contrast, the derivative does
change if we change these units.
2.4
Competition and monopoly
Objectives and skills
Be able to write the objectives and the equilibrium conditions for a
competitive industry and a monopoly.
Understand the similarities and the di¤erences between these two market structures and the corresponding equilibrium conditions.
Understand the meaning of “marginal revenue”, and be able to write
it using the elasticity of demand.
We consider both perfect competition and monopoly. The competitive
…rm takes the market price as given, and the monopoly understands that the
price responds to sales. In order for the equilibrium prices in the two settings
to be comparable, we assume that the inverse demand function, p (Q), and
the industry cost function, c (Q), are the same in the two types of markets;
Q is aggregate sales.
The industry and …rm cost functions What does it mean to say that
an industry, consisting of many …rms, has a particular cost function? The
simplest way to think of this is to imagine that the industry consists of a large
number, n, of factories. Under monopoly, a single …rm owns all factories;
under the competitive structure, each …rm own a single factory. Suppose
that the cost of producing q in a single factory is c~ (q). All …rms in the
competitive industry are identical, making it is reasonable to assume that
2
Units are important. Columbus underestimated the diameter of the world partly as
a consequence of confusion over units. The common belief that Napoleon was quite short
(the “little man complex”) resulted from confusing English with French units; in fact, he
was slightly taller than the average man of his era. The 1999 Mars Climate Orbiter was
lost due to miscommunication about units between Lockheed Martin and NASA.
24
CHAPTER 2. PRELIMINARIES
they all produce the same quantity, n1 ’th of industry quantity, so nq = Q.
If each factory produces q = Qn , then the cost in each …rm is c~ (q) = c~ Qn .
The total industry cost is then n~
c Qn . We can de…ne the industry cost as
c (Q) = n~
c
Q
n
:
Taking the derivative of both sides of this equation, with respect to Q, using
the chain rule, gives
c0 (Q)
d~
c dQ
d~
c1
d~
c
dc
=n Q n =n
=
dQ
dq n
dq
d n dn
c~0 (q)
(2.6)
Equation 2.6 states that the marginal cost of the industry (the expression
on the left) equals the marginal cost of the …rm (or factory), the last expression. This relation holds for any number of …rms, n, provided that q = Qn .
This formulation enables us to compare the competitive and the monopoly
market structures. In making this comparison, it is essential that we recognize that the cost of producing an arbitrary amount does not change with
the market structure. A change in the market structure alters the equilibrium amount produced, but not the technology and therefore not the relation
between costs and output.
We …nd the competitive and the monopoly equilibria by solving maximization problems. Both types of …rms want to maximize pro…ts, and
they both have the same cost structure and face the same demand function.
They di¤er because competitive …rms take price as exogenous, whereas the
monopoly understands that price is endogenous.
The competitive equilibrium A representative …rm chooses its output
to maximize pro…ts, taking price as given. This …rm’s optimization problem
is:
max pq c~ (q) :
q
The …rst order condition sets the derivative of the objective (pro…ts), with
respect to the choice variable, q, equal to 0:
d (pq
c~ (q))
dq
=p
set
c~0 (q) = 0 ) p = c~0 (q) :
2.4. COMPETITION AND MONOPOLY
25
Using the relation in equation 2.6, we can replace c~0 (q) with c0 (Q); recognizing that the price depends on aggregate sales, we can replace p with the
inverse demand function p (Q). With these two substitutions, we rewrite
the optimality condition for the representative …rm as
price = industry marginal cost: p (Q) = c0 (Q) :
(2.7)
The monopoly equilibrium The monopoly recognizes that the price,
p (Q), depends on its sales. Its pro…ts, revenue minus costs, equal p (Q) Q
c (Q). The monopoly’s optimization problem is
max p (Q) Q
Q
c (Q) :
The …rst order condition for the monopoly is3
1
(Q)
Marginal revenue = Marginal cost: p (Q) 1
Box 2.: Derivation of equation 2.8.
pro…t maximization is
d(p(Q)Q c(Q))
dQ
= c0 (Q) .
(2.8)
The …rst order condition for
= p0 (Q) Q + p
set
c0 (Q) = 0
) p0 (Q) Q + p = c0 (Q) ;
which states that marginal revenue, p0 (Q) Q+p, equals marginal cost,
c0 (Q). We can write marginal revenue (denoted M R (Q)) using the
elasticity of demand, :
M R (Q)
d[p(Q)Q]
dQ
p 1+
3
1
dq p
dp Q
= p0 (Q) Q + p = p 1 +
= p (Q) 1
1
(Q)
dp Q
dQ p
:
Recognizing that the equilibrium price and quantity lie on the demand function, we
have p = p (Q), so we can write the elasticity of demand as (p (Q)), or (with some abuse
of notation) as (Q).
26
CHAPTER 2. PRELIMINARIES
Comparing the two equilibria The equilibrium conditions for the competitive industry and the monopoly are
Competition: p (Q) = c0 (Q)
Monopoly p (Q) 1
1
(Q)
= c0 (Q) :
Both of these equations involve the inverse demand function, p (Q), and
the industry marginal cost function, c0 (Q). The competitive equilibrium
condition sets price equal to the industry marginal cost, and the monopoly
condition sets marginal revenue equal to industry marginal cost. Once we
have the necessary condition for a competitive equilibrium, we can obtain
the necessary condition for a monopoly equilibrium merely by replacing p
with p 1 1 .4
The more elastic is the demand function that the monopoly faces (the
larger is ), the less opportunity the monopoly has to exercise market power:
it understands that any e¤ort to raise the market price requires a large reduction in sales, and a corresponding fall in revenue. As the market demand
becomes in…nitely elastic, i.e. as ! 1 (“ goes to in…nity”), the monopoly
looses all market power, and behaves like a competitive …rm. The monopoly
never produces where < 1; at such a point, marginal revenue is negative.
An example of calculating and comparing the equilibria In general,
the demand elasticity, , is a function of sales. The “constant elasticity
1
of demand function”, where p (Q) = AQ , is the exception. Here, the
elasticity of demand is a parameter, not a function. With this demand
function and the cost function c (Q) = 2b Q2 , we can solve the competitive
equilibrium condition (price = marginal cost)
AQ
4
1
A
= bQ ) = Q
b
set
+1
)Q=
A
b
1+
)p=A
A
b
1
1+
:
Early in the study of arithmetic we learn that the order in which operations are
performed matters: (3 + 4) 7 = 49 6= 3 + (4 7) = 31. The order of operations also
matters in carrying out economic calculations.
For the monopoly, we …rst replace “price” with the inverse demand function, and then
we take the derivative of pro…t with respect to sales. For the competitive …rm, we …rst take
the derivative of pro…t with respect to sales, taking price as given, and then substitute the
inverse demand function into the …rst order condition to obtain an equation in quantity.
The order of these two steps is critical.
2.4. COMPETITION AND MONOPOLY
27
price ratio 1.8
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1.0
1
2
3
4
5
elasticity
Figure 2.2: Ratio of monopoly to competitive price
For the monopoly problem, assume that > 1 (so marginal revenue is positive). Solving the monopoly equilibrium condition (marginal revenue =
marginal cost) gives
p (Q) 1
1
(Q)
)Q=
= AQ
A(1
b
1
)
1
1
1
set
= bQ )
1+
)p=A
A(1
b
A(1
1
)
b
1
)
=Q
+1
1
1+
:
Figure 2.2 graphs the ratio of the monopoly to the competitive equilibrium price. For low elasticity of demand ( close to 1) monopoly power
is substantial, and the monopoly price is much larger than the competitive
price. For large elasticity of demand, the monopoly has little market power,
and the monopoly and competitive prices are similar.
Optimization and equilibrium In most economic contexts, an equilibrium occurs where all agents simultaneously solve their optimization problems: no one wants to move unilaterally away from an equilibrium. At a
competitive equilibrium consumers are maximizing utility, resulting in quantity demanded being on the demand function, and producers are maximizing
pro…ts, resulting in quantity supplied being on the supply function. Markets
clear, so supply equals demand.
Models help to clarify complex situations, but do not literally describe behavior. If people behave completely randomly, then optimization-based eco-
28
CHAPTER 2. PRELIMINARIES
nomic models are useless. If people attempt to pursue their self-interest, and
behave with a modicum of rationality, these models are informative. Apart
from Dr. Spock, everyone behaves irrationally in some contexts. However,
managers of …rms who are consistently irrational are not likely to survive
long in the market place. Shoppers may not buy food that is good for them,
but they buy the food that they want, and in that respect they act in their
self interest.
Equilibrium is also an abstraction. Competitive …rms’optimal production level depends on the price that they expect to receive. Their expectations may be incorrect. If they have already committed a certain quantity
to the market, but the price is lower than they expected (perhaps because
demand is lower than they expected), the price-quantity point is below the
supply curve. Markets are unlikely to be exactly in equilibrium, but people
respond to their mistakes, and those responses likely move a market towards equilibrium. If …rms …nd that the price has systematically been lower
than their marginal cost, they have an incentive to decrease quantity. The
reduction in quantity leads to a higher price, moving the outcome toward
equilibrium.
Optimality and no-arbitrage conditions The …rst order condition to
an optimization problem is a particular type of no-arbitrage condition. To
illustrate this relation, consider the price-taking tea seller who has 10 units
of tea in China and faces iceberg transport costs b in moving the tea to the
U.S. Using the constraint q U.S. = 10 q China , this seller obtains pro…ts
= pChina q China + pU.S 10
q China
bpChina 10
q China :
The …rst order condition (at an interior equilibrium) is
d
= pChina
dq China
set
pU.S + bpChina = 0:
Rearranging the last equation gives the no arbitrage condition 2.1, repeated
here:
1
pChina =
pU.S. :
(1 + b)
Optimality conditions can be interpreted as no-arbitrage conditions, although
the exact interpretation varies with the context.
2.5. DISCOUNTING
2.5
29
Discounting
Objectives and skills
Understand the rationale for and the implementation of discounting.
Be able to use discount factors to calculate the present value of a
“stream”(= sequence) of future costs or payments.
Understand the relation between the magnitude of a discount rate and
the length of a period over which discounting occurs.
A box of tea in China and a box of tea in the U.S. are not the same
commodity if it is costly to move the box from one location to the other.
Similarly, a dollar ten years from now is not the same as a dollar today,
because there is an opportunity cost, the foregone investment opportunity, of
receiving the dollar later rather than earlier. Most of the resource problems
studied in this book involve decisions of when to extract a resource, e.g.
when to take a unit of oil out of the ground, or when to harvest a unit
of …sh. Deciding when to extract requires comparing the …rm’s value of
extracting at di¤erent points in time. This comparison involves discounting.
Pick a period of arbitrary length, say one year. Suppose that the most
pro…table riskless investment available pays a return of r after one year. We
call r the discount rate. If a person at period 0 invests z dollars for one year
in an asset that returns the rate r, then at the end of the year the person
has z (1 + r). The person with this investment opportunity is indi¤erent
between receiving $1 at the beginning of the next year and $z in the current
1
. We de…ne
period, if and only if z (1 + r) = 1, i.e. if and only if z = 1+r
1
= 1+r as the discount factor. This person is indi¤erent between receiving
(for example) $43.60 at the beginning of next period, or $43.60
at the
beginning of the current period. Multiplying an amount received one year in
the future by , produces the “present value” of the future receipt. Today,
the “present value”of $43.60 at the beginning of next year equals $43.60
.
A person is indi¤erent between receiving one dollar at the end of two years
and $z today if and only if (1 + r) (1 + r) z = 1, i.e. if and only if z = 2 .
Thus, 2 is the present value today of a dollar in two years. Similarly, n is
the present value today of a dollar n years from now.
30
CHAPTER 2. PRELIMINARIES
Suppose that a …rm obtains pro…ts t during periods t = 0; 1; 2:::T . The
present discounted stream of pro…ts equals
T
X
t
t:
t=0
In the special case where t = , a constant, we can simplify this sum using
the formula for a geometric series:
1
X
t
=
t=0
1
(2.9)
:
This formula implies
T
X
t=0
t
=
1
X
t=0
t
T +1
1
X
t=0
t
=
T +1
1
1
:
(2.10)
Relation between the discount rate and the length of a period The
numerical value of the discount rate, and thus of the discount factor, depends
on the length of a period of time. For example, suppose that a period lasts
for ten years, and an asset held for one ten-year period pays a return of r~.
Thus, one dollar invested in this asset for three periods (30 years) returns
(1 + r~)3 . We say that this return is “compounded” every decade. We
can convert this decadal return to an annual return by choosing the annual
discount rate, r to satisfy (1 + r)10 = (1 + r~). Taking logs of both sides and
r)
. If r~ = 0:8, then r = 0:061; this asset
simplifying gives ln (1 + r) = ln(1+~
10
pays an 80% return over a decade, or a 6.1% return compounded annually.
(We multiply by 100 to translate a rate into a percentage.)
Given the decadal and annual discount rates, the corresponding decadal
1
1
and annual discount factors are ~ = 1+~
and = 1+r
. We could have found
r
the relation between the two discount factors …rst, and used this result to
compute the annual discount rate. The present value of one dollar in one
decade is ~. With annual compounding, the present value of one dollar after
1
10 years is 10 . Setting 10 = ~ and taking logs gives ln = 10
ln ~.
Example: the levelized cost of electricity Electricity can be produced
using coal, natural gas, nuclear power or renewable resources. Each of
these power sources has di¤erent production methods, e.g. di¤erent types
2.5. DISCOUNTING
31
of coal powered or nuclear powered plants, and di¤erent types of renewable
resources, including wind, solar, and hydro. Di¤erent methods of electricity
generation require di¤erent cost streams and produce di¤erent amounts of
power. Nuclear-powered plants are expensive to build and require decommissioning, but have low fuel costs. Fossil fuel plants have relatively low
investment and decommissioning costs, but high fuel costs.
The “levelized cost” of electricity (LCOE) provides a basis for comparison across di¤erent power sources. Calculating the levelized cost of a power
source requires estimates of the year-t capital cost, Ct , variable cost, Vt (including fuel and maintenance), the amount of energy produced, Et , and the
lifetime (including construction and decommissioning time) of the project,
n + 1 years. The formula for LCOE is
Pn t
(C + Vt )
Pn tt
:
(2.11)
LCOE = t=0
Et
t=0
Although the LCOE provides a straightforward means of comparing costs
of di¤erent sources, it usually leaves out potentially important considerations.
For example, a new power source such as wind or solar may require signi…cant
network upgrades in order to bring the power to market. Fossil fuels have
health and climate-related externalities. Nuclear power creates the risk of
rare but catastrophic events. Incorporating these and other considerations
requires additional data. Table 2.1 shows estimates for the U.S. for several
power sources.
Conventional
Coal
96
Wind
IGCC
116
Wind
o¤shore
66
Advanced
Nuclear
96
Solar PV2
Hydro
Natural Gas
CCC
80
204
130
85
Table 2.1 Estimated LCOE (2012$/MWh) for plants entering service in
2019. *Integrated Coal-Gasi…cation Combined Cycle. ** Conventional
Combined Cycle (U.S. Energy Information Administration, 2014)
The example in Table 2.2 shows that comparison of LCOEs can be sensitive to the discount rate. Type A generation method is expensive to
construct but cheap to run and lasts a long time. Type B is cheap to build,
expensive to run, and has a shorter lifetime. They both produce the same
amount of energy per year (one unit).
32
CHAPTER 2. PRELIMINARIES
ratio 1.3
1.2
1.1
1.0
0.9
0.8
0
1
2
3
4
Figure 2.3: Ratio of LCOE
capital cost
(billion $)
Type A
Type B
annual
operating cost
0.05
0.12
2
0.3
5
r (%)
Type A
:
Type B
lifetime
45 years
30 years
construction
time
5 years
2 years
Table 2.2: Data for two di¤erent types of power plants
A
LCOE =
2+
P50
t=5
P50
t=5
t
(0:05)
t1
B
and LCOE =
0:3 +
P32
t
(0:12)
t=2
P32 t
1
t=2
Figure 2.3 shows the ratio of the two levelized costs as a function of the
discount rate. The levelized costs are equal (the ratio is 1) for r = 2:7%;
Type A is 10% cheaper at r = 2% and 17% more expensive at r = 4%.
When “money is cheap” (the interest rate is low), it is economical to use
the method that has large up-front costs but lower costs overall (Type A).
However, when the interest rate is high, it is economical to use Type B, which
has lower initial costs but higher undiscounted total costs.
Example: the social cost of carbon The “social cost of carbon”(SCC)
is de…ned as the present discounted stream of damages due to a unit of carbon emissions. It plays an important role in climate economics, in particular
in estimates of the optimal carbon tax. The SCC depends on: the relation
between a unit of emissions today and future carbon stocks; the relation
between carbon stocks and temperature changes; the relation between temperature changes and economic damages; and the discount rate. Higher
2.6. WELFARE
33
discount rates (lower discount factors) place less weight on future damages,
and therefore lead to a lower SCC.
The Environmental Protection Agency (EPA) provides estimates of the
SCC for di¤erent years and under di¤erent discount rates. It estimates, for
the year 2015, a SCC of $12 per metric ton of CO2 if the discount rate is 5%,
and a SCC of $61 if the discount rate is 2.5%.
The estimates of SCC involve complex models, but a simple model illustrates the basic idea, and in particular the role of discounting. For this
model, suppose that one metric ton of CO2 creates d dollars of economic
damage, via temperature changes, and that carbon decays at a constant rate
.5 With these assumptions, one unit of emissions today increases the carbon stock t periods from now by (1
)t and creates d (1
)t dollars of
damage in period t. The present discounted value of the stream of damages
due to this unit of emissions equals
SCC =
1
X
d (1
)t
t
t=0
The last equality uses from formula 2.9.
(the larger is ), the larger is the SCC.
2.6
=
1
d
(1
)
:
The smaller is the discount rate
Welfare
Objectives and skills
Understand the meaning of “Pareto e¢ cient”.
Be familiar with the two Fundamental Welfare Theorems, and know
the conditions under which a competitive outcome is e¢ cient.
Resource economics studies the allocation of a natural resource over time.
Under certain conditions, the allocation under competitive markets is Pareto
e¢ cient, meaning that there does not exist another allocation of the resource
that makes at least one agent better o¤, without making any agent worse
o¤. “Pareto e¢ cient” has a technical meaning; it does not express a value
5
Carbon does not literally decay. CO2 is emitted to the atmosphere, and over time
some of it moves to di¤erent oceanic and terrestrial “reservoirs”. The model of constant
decay is one of the simplest ways to approximate carbon leaving the atmosphere.
34
CHAPTER 2. PRELIMINARIES
judgement. “E¢ cient” does not mean “good”. The simplest example of
Pareto e¢ ciency involves two people who both like to consume a product
that has …xed supply, one unit. These two people, Jiangfeng and Mary,
get utility only from their own consumption. A feasible allocation gives
Jiangfeng z
0 and Mary w
0 units of the good, with z + w
1. An
e¢ cient allocation makes sure that all of the good is consumed; any outcome
with z + w = 1 is e¢ cient. Giving all of the good to Jiangfeng or all of it
to Mary are both e¢ cient, although neither allocation seems particularly
ethical. In this sense, e¢ ciency is a rather weak criterion. For example,
we might prefer the equal but ine¢ cient allocation, z = w = 0:4999 to the
e¢ cient but ethically questionable allocation w = 1, z = 0.
The …rst part of this book emphasizes the competitive equilibrium without market failures, occasionally discussing the monopoly equilibrium. Firms
own natural resource stocks and choose how much to extract and sell in each
period. In order to make statements about the social welfare arising from
the …rms’allocation, we have to specify a “welfare criterion”, i.e. we have to
say exactly how one measures welfare. Once we have this criterion, we can
imagine a social planner who maximizes it, choosing how much to extract and
sell in each period. We can then compare the outcome under competition
or monopoly with the outcome under this social planner.
We work exclusively with partial equilibrium models: those that involve
a single market, e.g. the market for a particular resource. We take as
given all “outside” considerations that might in‡uence this market. For
example, if the resource is petroleum, the partial equilibrium model seeks to
explain petroleum prices and quantities, over time, taking as given: levels
of income (which a¤ect demand for petroleum); prices of factors used to
produce petroleum (labor, machinery); prices of substitutes (natural gas)
and complements (cars); and technology (drilling techniques). With a partial
equilibrium model, consumer and producer surplus are reasonable measures
of consumer and producer welfare, and their sum is a reasonable measure
of social welfare in a period. We take the social welfare function to be
the discounted sum (over time) of welfare in each period. This criterion,
known as “discounted utilitarianism”, is used in most resource models. Our
…ctitious social planner is a discounted utilitarian.
If markets are competitive, and certain conditions hold, then a competitive equilibrium is Pareto e¢ cient, using the criterion of discounted social
surplus: the competitive equilibrium maximizes the present discounted sum
of consumer + producer surplus. In this case, there is no role for government
2.6. WELFARE
35
intervention if we care only about e¢ ciency, not about equity. No taxes or
lump sum transfers are needed in order to insure that the competitive equilibrium leads to the same prices and quantities as under the planner.
The equivalence between the outcomes under competition and under the
social planner requires that markets are “complete” and that there are no
unpriced externalities. Markets are complete if there is a market for every
type of transaction that people would like to make. For example, if someone
would like to buy insurance and someone else is willing to sell insurance, then
“complete markets”requires that there is actually an insurance market that
makes their exchange possible. An “unpriced externality” is a consequence
of the market transaction not fully re‡ected in the price of the good. For
example, the price of fossil fuels does not include the environmental damage
arising from extracting and using the fuels.
In the real world, markets are not complete and externalities are significant, weakening the practical importance of the theoretical possibility that
the competitive equilibrium is Pareto e¢ cient. Nevertheless, it is worth understanding this possibility: it is central to modern economics, and provides
a good place to begin (but not to end) the analysis of resource markets. The
result is known as
First Fundamental Welfare Theorem: Suppose that markets are
complete, there are no unpriced externalities, and agents are
price-takers. Then any competitive equilibrium is Pareto e¢ cient.
A second theorem provides conditions under which a particular Pareto ef…cient outcome can result from a competitive equilibrium. We say that a set
of transfers and taxes “supports”a particular “outcome X”in a competitive
equilibrium if, in the presence of those transfers and taxes, the competitive equilibrium is identical to (e.g. has the same prices and quantities) as
“outcome X”. The second theorem is
Second Fundamental Welfare theorem: Provided that a technical requirement (“convexity”6 ) is satis…ed, any Pareto e¢ cient
6
For example, constant and decreasing returns to scale technologies are “convex”. A
technology with increasing returns to scale is not convex. Constant returns to scale means
that doubling inputs doubles output. Decreasing returns to scale means that doubling
inputs less than doubles output. Increasing returns to scale means that doubling inputs
more than doubles output.
36
CHAPTER 2. PRELIMINARIES
equilibrium can be supported as a competitive equilibrium using
taxes and lump sum transfers.
These two theorems provide the basis for attributing welfare properties
to a competitive equilibrium. The …rst theorem provides conditions under
which the competitive equilibrium is e¢ cient. With complete, competitive
markets where there are no externalities, the First Fundamental Welfare Theorem assures us that the competitive equilibrium is e¢ cient. The second
theorem tell us (under “convexity”) that we can obtain any other e¢ cient
outcome, as a competitive equilibrium, by using appropriate taxes and transfers.
2.7
Summary
An example of arbitrage over space illustrates the meaning of arbitrage.
Many of the main ideas in this book are based on arbitrage over time: instead
of selling the commodity in one country rather than another, the …rm sells it
at one point in time rather than another. Understanding spatial arbitrage
makes it easy to understand intertemporal arbitrage.
Economic models help to determine how a change in an exogenous parameter changes an endogenous variable. This kind of question is known as a
comparative statics question. Casual reasoning or graphical methods su¢ ce
to answer very simple comparative statics questions. In more complicated
cases, we use mathematics, beginning with an equilibrium condition (e.g.,
supply equals demand). One approach uses this condition to …nd an explicit expression for the endogenous variable, as a function of the exogenous
variables. An alternative uses the di¤erential of the equilibrium condition.
We de…ned elasticities, and illustrated the de…nition using the elasticity
of demand with respect to price. It is important to be able to calculate an
elasticity, and to understand why it is unit free.
In order to compare perfect competition and monopoly, we want to “hold
everything else constant”, apart from the market structure. In this context,
we require that the demand and cost functions (not their levels) are the
same for both market structures. We can think of the industry consisting
of many factories. In the competitive environment, each …rm owns one of
these factories, and in the monopoly, a single …rm owns all factories.
Both the monopoly and the representative …rm want to maximize pro…ts.
For the price-taking competitive …rms, the equilibrium condition is “price
2.8. TERMS, STUDY QUESTIONS, AND EXERCISES
37
equals marginal cost”. The monopoly understands that its sales a¤ect the
price; the monopoly marginal revenue equals p (1 1= ), where is the elasticity of demand. Provided that is …nite, the monopoly sells less than the
competitive industry. As demand becomes more elastic, the monopoly has
less market power.
We use the discount factor to compare money (e.g. pro…ts) received
in di¤erent periods. The discount factor is = 1= (1 + r), where r is the
discount rate, equal to the highest riskless return available to the agent. The
discount factor converts future values into present values. The magnitude
of r, and thus of , depend on the length of a period. We can transform
the magnitudes of ; r corresponding to a particular length of period, e.g. 10
years, into the magnitudes that correspond to some other length of period,
e.g. one year.
An outcome, such as the allocation of a product across individuals, geographical regions, or time, is said to be Pareto e¢ cient if there is no reallocation that makes some agent better o¤ without making any agent worse
o¤. The two fundamental welfare theorems describe the relation between a
competitive equilibrium and the outcome under a social planner. The …rst
of these theorems provides conditions under which a competitive equilibrium
is Pareto e¢ cient. The second provides conditions under which any Pareto
e¢ cient equilibrium can be supported as a competitive equilibrium by means
of taxes or income transfers.
2.8
Terms, study questions, and exercises
Terms and concepts
Demand function, inverse demand function, arbitrage, no-arbitrage condition, interior equilibrium, boundary equilibrium, iceberg transportation costs,
endogenous and exogenous variable, law of one price, implicit function, explicit function, comparative statics, di¤erential, …rst order condition, marginal revenue, order of operations, discount function, discount factor, opportunity cost, compounded, capital cost, operating cost, decommissioning
cost, levelized cost, partial equilibrium, externality, complete markets, e¢ cient, Pareto e¢ cient, consumer and producer surplus, feasible, discounted
utilitarianism.
38
CHAPTER 2. PRELIMINARIES
Study questions
1. You should be able to use the type of …gure in Section 2.1 to illustrate
the e¤ect of a change on demand in one country, or a change in available
supply, or in transportation costs, on the equilibrium allocation of sales
across country.
2. Given inverse demand functions in the two countries, available supply,
and the transport costs, you should be able to write down the equilibrium condition. Taking the di¤erential of this condition, you should
be able to write a comparative statics expression showing the e¤ect of
a change in an exogenous variable on an endogenous variable. You
should be clear about the distinction between exogenous and endogenous variables.
3. You should know what an elasticity it, how to calculate it, and what it
means to say that the elasticity is unit free.
4. Given an industry cost function and an inverse demand, you should
be able to write down the equilibrium conditions that determine sales
under competition and under monopoly.
5. You should know the relation between a discount rate and a discount
factor, and understand what they are used for. There is no need to
memorize the formula for the sum of a geometric series, but if given
this formula, you should be able to use it to calculate (or show how to
calculate) the present discounted stream of payo¤s. For example, you
should be able to work through an example like the …rst one in the text
that compares the cost of two methods of electricity generation.
6. You should understand the meaning and be able to describe the two
Fundamental Welfare theorems.
Exercises
1. When quantities are measured in pounds and prices in dollars, the
demand function is Q = 3 5P . (a) What is the elasticity of demand,
evaluated at P = 1? (b) Express the same relation between demand
and price, using di¤erent units, q and p: q are in units of kilos, and p
in pesos. There are 3 pesos per dollar. (c) What numerical value of p
2.8. TERMS, STUDY QUESTIONS, AND EXERCISES
39
corresponds to P = 1? (d) Using the new units, q and p, express the
elasticity of demand with respect to price, evaluated at the value of p
you obtained from part c. (d) What is the point of this exercise?
2. How does an increase in transportation costs a¤ect the location of the
dotted line in Figure 2.1, and how does this change alter the equilibrium
price and the allocation of tea between the two countries?
3. How does an increase in the available supply (e.g. from 10 units to 12
units of tea) change the appearance of Figure 2.1, and how does this
change in supply alter the equilibrium quantities and prices in the two
countries?
4. Calculate (Q) for the demand function Q = a hp and then for the
demand function Q = ap h , where h is a positive number. Graph the
two elasticities as a function of Q.
5. Suppose that a monopoly owns the ten units of tea in China. There is
no transportation costs (b = 0). Using the inverse demand functions
in Section 2.1, …nd monopoly sales two markets.
6. Consider the monopoly in the previous question. Suppose that iceberg transportation costs are b. Using the equilibrium condition for
the monopoly, …nd the comparative statics of the monopoly’s sales in
China, with respect to b. (Use the method that does not rely on …nding
an explicit solution to …rst period sales as a function of b.)
7. Suppose that the industry has the cost function c (Q) = 2Q + 32 Q2 .
This industry consists of n …rms, each with cost function c (q). Find
c (q). Hint: “Guess”that the single …rm’s cost function is of the form
c Qn , to
c~ (q) = aq + 2b q 2 . Then use the requirement that c (Q) = n~
write
!
2
3 2
Q b Q
2Q + Q = n a +
:
2
n
2 n
This relation must hold for all Q (not just a particular Q), so we can
“equate coe¢ cients”of Q and Q2 to …nd the values of a and b.
8. A plant that supplies 1 unit of electricity per year, costs $1 billion to
build, lasts 25 years, and has an annual operating cost of $0.2 billion; it
40
CHAPTER 2. PRELIMINARIES
costs $0.1 billion to decommission the plant at the end of its lifetime (25
years). (Assume that the construction costs and the operating costs
are paid at the beginning of the period, and that the decommissioning
cost is paid at the end of the life of the plant.) The annual discount rate
is r. Write the formula for the present value of the cost of providing 1
unit of electricity for 100 years, including the decommissioning costs.
9. A monopoly faces the demand function q = p 1:2 and has production
costs c (q) = 2b q 2 . Find the comparative statics of equilibrium price,
with respect to the cost parameter, b.
10. A person plans to save $1 for 20 years. They can invest at an annual
rate of 10% (r = 0:1). This investment opportunity “compounds
annually” (meaning that they receive interest payments at the end of
each year). A second investment opportunity pays a return of r~ 100%,
compounded every decade. (After one decade, the investment of one
dollar yields 1 + r~.) For what value of r~ is the person indi¤erent
between these two investments? (Assume that there is no chance that
the person wants to cash in the investment before the 20 year period.)
Explain the rationale behind your calculation.
11. (*) This exercise illustrates the First Fundamental welfare theorem.
Inverse demand equals p (q) and total cost is 12 cq 2 . (a) Write the
condition for equilibrium in a competitive market. (b) For a linear
demand function, draw the graphs whose intersection determines the
competitive equilibrium. Using this graph, identify consumer and
producer surplus. (c) Explain in words why consumer surplus equals
Z q
p (w) dw p (q) q:
0
(d) Write the expression for producer surplus, equal to revenue minus
costs. (e) De…ne social surplus, S (q), as the sum of consumer and
producer surplus. Write the expression for social surplus. Using
Leibniz’s rule (see math appendix) write the …rst order condition for
maximizing social surplus, by choice of q. (f) Compare this …rst order
condition with the equilibrium condition under competition. Explain
why this comparison implies that the competitive equilibrium and the
solution to the social planner’s problem (maximizing social surplus)
2.8. TERMS, STUDY QUESTIONS, AND EXERCISES
41
are identical. What does this have to do with the First Fundamental
Welfare Theorem?
Sources
The U.S. Energy Information Administration (2014) presents and explains
estimates of the Levelized Cost of Electricity.
The Interagency Working Group (2013) presents estimates of the Social
Cost of Carbon.
Amartya Sen On Ethics and Economics (1987) New York, Blackwell provides a background on welfare economics.
Aker (2010) provides evidence of the relation between cell phones and
agricultural markets.
Glaeser and Kohlhase (2004) provide estimates of transport costs, and
discuss their role in international trade.
Prominent estimates of market power in resource markets include: Hnyilicza and Pindyck (1976), Stollery (1985), Pindyck (1987), Ellis and Halvorsen
(2001), Cerda (2007).
42
CHAPTER 2. PRELIMINARIES
Chapter 3
Nonrenewable resources
Objectives
Ability to analyze equilibria under competition and monopoly for the
two-period model of nonrenewable resources.
Information and skills
Translate the techniques and intuition from the “trade in tea” model
to the nonrenewable resource setting.
Understand the relation between transport costs in the trade model
and the discount factor in the resource setting.
Derive and interpret an equilibrium condition and analyze it using
graphical methods, for both competition and monopoly.
Do comparative statics with respect to extraction costs and the discount factor.
A two-period model provides much of the intuition needed to understand
equilibrium in a nonrenewable resources market. We use graphical methods
to analyze the equilibrium under competition or monopoly when …rms are
unable to save any resource beyond the second period. We emphasize the
case where the initial stock is small enough, relative to demand, that …rms
want to exhaust the resource during this time.
Arbitrage provides the basis for the intuition in resource models, but here
we speak of arbitrage over time, instead of over space. After discussing the
43
44
CHAPTER 3. NONRENEWABLE RESOURCES
competitive and the monopoly models, we explain how to answer comparative
statics questions of the following sort: How does a change in a demand or a
cost parameter a¤ect the equilibrium level of …rst-period sales?
A sales trajectory is the sequence of sales, and a price trajectory is the
associated sequence of prices. In the two-period setting, each of these sequences contains only two elements.
3.1
The competitive equilibrium
Objectives and skills
Write the objective and the constraints for a competitive …rm.
Obtain and interpret the “no-intertemporal” arbitrage (equilibrium)
condition under competition, and analyze it graphically.
Understand the e¤ect of constant average extraction costs on the equilibrium sales and price trajectories.
Chapter 2.1 considered the allocation of a …xed quantity of tea over two
countries, in the presence of iceberg transportation costs. Here, a …xed stock
of the resource replaces tea, two periods replace the two countries, and the
discount factor replaces the iceberg transportation costs.
We require a bit of notation. The …rst period is denoted t = 0, and
the second period, t = 1. A price-taking …rm has discount factor , faces
prices p0 and p1 in periods 0 and 1, must pay extraction cost c for each unit
extracted, and has a …xed stock of the resource, x units. In this model,
marginal extraction costs = average extraction costs = c, a constant. The
trade example did not include production costs. Except for this di¤erence,
the trade and the resource models are exactly the same; we merely call things
by di¤erent names. Not surprisingly, the intuition for the equilibrium in the
two models is also the same.
Let y be sales in period 0. Assuming that all of the resource is sold, x y
equals period-1 sales. At an interior equilibrium, extraction is positive in
both periods: x > y > 0. The …rm wants to maximize the sum of present
value pro…ts in the two periods:
competitive
(y; p0 ; p1 ) = (p0
c) y + (p1
c) (x
y) :
(3.1)
3.1. THE COMPETITIVE EQUILIBRIUM
45
Multiplying period-1 pro…ts by the discount factor, , gives the present value
of period-1 pro…ts. The derivative of (y; p0 ; p1 ) with respect to y is
d
competitive
dy
= (p0
c) y
(p1
c) :
The …rm prefers to sell all of its stock in period 0 ( d
c o m p e titive
dy
d
> 0) if
c o m p e titive
(p0 c) > (p1 c). It prefers to sell all of its stock in period 1 (
<
dy
0) if (p0 c) < (p1 c). In order for the …rm to sell a positive quantity
in both periods, as we assume, it must be indi¤erent about when to sell the
stock. This indi¤erence requires that the present value of period t = 1 price
minus marginal cost equals the value of price minus marginal cost in period
t = 0:
d competitive
= 0 if and only if (p0 c) = (p1 c) :
(3.2)
dy
The second equation is a “no-intertemporal arbitrage” condition; it holds
in an interior competitive equilibrium. The equation states that the …rm
cannot increase its pro…ts by moving sales from one period to another.
The competitive resource owner, just like the competitive exporter in the
trade example, takes prices as given. These prices adjust in response to
the amount of supply brought to market. Actions of an individual resource
owner, just like the actions of an individual exporter, have negligible a¤ect
on the price. However, all resource owners (in our model) have the same
costs and discount factor, so they have the same incentives. Therefore, we
can proceed as if there is a “representative …rm” that takes price as given,
and owns all of the stock in the industry. The price responds to changes in
this representative …rm’s supply.
Figure 3.1 shows the market when extraction costs are c = 0 and the
inverse demand in both periods is p = 20 y. Sales in period 0 equal
y. Assuming that the initial stock is x = 10, period-1 sales equal 10 y.
The solid line shows the demand function in period 0: as y increases, the
equilibrium price falls. The dashed line shows the demand function in period
1: as y increases, period 1 sales, 10 y, fall, so the price in that period rises.
1
, equals
If the discount rate is 0 (r = 0) then the discount factor, = 1+r
1. In this scenario, …rms allocate the stock evenly between the two periods,
and the price in each period equals 15. With zero discount rate, the …rm is
indi¤erent between selling in periods 0 or 1 if and only if the prices in the
two periods are equal.
46
CHAPTER 3. NONRENEWABLE RESOURCES
$
20
first period demand
second period demand
15
10
5
0
0
1
2
3
4
5
6
Figure 3.1: Demand in period 0 (solid).
Present value of period 1 price (dotted).
7
8
9
10
11
12
y
Demand in period 1 (dashed).
The dotted line shows the present value (in period 0) of the period-1 price
1
= 0:77. The equilibrium occurs
if the discount rate is r = 0:3, so = 1:3
where the present value of price is the same in both periods, i.e. where
the solid and the dotted lines intersect. With our demand function and
discount factor, equilibrium sales in period 0 equal 6: 96 and the price is
13: 04. Period-1 sales equal 3:04 and the price is 16: 96. The equilibrium
period-1 price is a point on the dashed curve, above the intersection of the
solid and the dotted curve. Discounting the future makes future revenue less
valuable from the standpoint of the …rm in period 0. The …rm therefore has
an incentive to sell more in period 0. As the …rm reallocates sales in this
manner, the period-0 price falls, and the period-1 price rises. Equilibrium
is restored when the present value of prices in the two periods are equal.
The price-taking representative …rm does not shift sales from the future
to the present (i.e. from period 1 to period 0) with the intention of causing
price to change. If, following an increase of r from 0 to 0.3, the …rm did not
reallocate sales, then the present value of a unit of sales in period 0 remains
1
=
at p0 = 15 and the present value of a unit of sales in period 1 is 15 1:3
11: 5 < 15. In this case, the …rm has an opportunity for intertemporal
arbitrage. Prices adjust as the …rm moves sales from period 1 to period
0, until, at equilibrium, there are no further opportunities for intertemporal
arbitrage
Figure 3.2 illustrates the model with constant average (= marginal) costs
c = 4 (instead of c = 0 as above). The solid and dashed lines show price
minus cost, instead of price, in the two periods. With zero discount rate,
3.1. THE COMPETITIVE EQUILIBRIUM
47
$
20
15
10
5
0
0
1
2
3
4
5
6
7
8
9
10
11
12
y
Figure 3.2: Demand - cost in period 0 (solid). Demand - cost in period 1
(dashed). Present value of period 1 price - cost (dotted).
sales are again allocated evenly between the two periods, and the price in
both periods is again p = 15, so price - costs = 11. In the absence of
discounting, the cost increase reduces the …rm’s pro…ts, but has no e¤ect on
consumers.
The dotted line in Figure 3.2 shows the present value of period-1 price
minus extraction costs, for a discount factor = 0:77. In this case, the
equilibrium occurs where the present value of price minus extraction costs
are equal in the two periods, at the intersection of the solid and the dotted
lines. Here, period-0 sales equal 6: 43 and the period-0 price is 13: 57. With
discounting, higher extraction costs cause the …rm to move production from
period 0 to the period 1. This reallocation causes period-0 price to rise and
period-1 price to fall. With discounting, the higher extraction costs lowers
consumer surplus in period 0, and increases consumer surplus in period 1.
=1
= 0:77
c = 0 p0 = 15 p0 = 13:04
c = 4 p0 = 15 p0 = 13:57
Table 3.1: Period-0 price for di¤erent discount factors and cost
Table 1 summarizes the e¤ects of discounting and extraction costs on
period-0 price. A higher extraction cost (raising costs from c = 0 to c = 4)
has no e¤ect on period-0 price in the absence of discounting, but leads to
a reallocation of sales “from the present to the future” (i.e. from period 0
to period 1) under discounting. From the perspective of the …rm in period
48
CHAPTER 3. NONRENEWABLE RESOURCES
0, a one unit increase in costs increases the average and marginal extraction
cost today by one unit, and increases the present value of cost in the next
period costs by only . Other things equal, higher extraction costs make it
less attractive to extract the resource. However, discounting diminishes the
period-1 cost-driven disincentive to sell, relative to the period-0 disincentive.
Therefore, in the presence of discounting, …rms respond to a higher cost by
reducing period-0 sales and increasing period-1 sales.
3.2
Monopoly
Objectives and skills
Write down the objective and constraints of a monopoly resource owner.
Obtain and understand the equilibrium condition for the monopoly.
Use graphical methods to illustrate the relation between exogenous
parameters and the monopoly equilibrium.
Compare the monopoly and the competitive outcomes (e.g. period 0
sales and price in the two cases).
Understand the statement “the monopoly is the conservationist’s friend”.
The monopoly, like the competitive …rm, wants to maximize the present
discounted value of pro…ts, given by expression 3.1. However, the monopoly
recognizes that its sales a¤ect the price, whereas the competitive …rm takes
price as given. The monopoly understands that period 0 price is the function
p (y) and period 1 price is the function p (x y). The present discounted
stream of monopoly pro…ts equals
monopoly
(y) = (p (y)
c) y + (p (x
y0 )
c) (x
y) :
(3.3)
We can …nd the equilibrium condition for the monopoly by using the …rst
order condition to the problem of maximizing monopoly (y). A simpler approach, discussed in Chapter 2.4, …nds this equilibrium condition by adapting
the equilibrium condition for the competitive industry, replacing price with
marginal revenue. Marginal revenue, M R, is
M R (Q) = p (Q) 1
1
(Q)
, with
(Q)
dQ p
;
dp Q
3.2. MONOPOLY
$
49
18
16
14
12
10
8
6
4
2
0
0
1
2
3
4
5
6
7
8
9
10
11
12
y
Figure 3.3: Solid curves show …rst and second period demand minus marginal
cost as a function of …rst period sales. Dashed curves show marginal revenue
minus marginal cost curves corresponding to these demand curves.
= 1.
where is the price elasticity of demand. Again denoting period-0 sales by y
(instead of Q) and period-1 sales by x y, we have the equilibrium condition
for the monopoly
M R (y)
c = [M R (x
y)
c] :
(3.4)
Figure 3.3 shows the period-0 and period-1 demand functions, p0 = 20 y
and p1 = 20 (10 y), minus marginal cost, c = 4, (the two solid lines) as
a function of period-0 sales, y. The dashed lines beneath those two demand
functions show the marginal pro…t (marginal revenue minus marginal cost)
corresponding to those two demand functions. In the absence of discounting
( = 1) optimality for the monopoly requires that marginal pro…t in the two
periods are equal.
Absent discounting, the monopoly sells the same amount in both periods,
so y = 5, exactly as in the competitive equilibrium with no discounting. In
this market, moving from a competitive market to a monopoly has no e¤ect
on the outcome. This result is due to the fact that the stock of resource is
…xed, together with the assumption that the world lasts only two periods, and
the assumption that the discount rate is 0 ( = 1). In this setting, neither
the monopoly nor the competitive …rm has any incentive to save the resource
beyond period 1. Below we show that if r > 0 (so < 1) the monopoly
50
CHAPTER 3. NONRENEWABLE RESOURCES
sells less in period 0 than competitive …rms; the monopoly saves more of the
resource for the future, period 1, and in that sense is the “conservationist’s
friend”. First, however, we consider the simpler case where = 1 in a bit
more detail.
Digression: Comparing competition and monopoly in the static
setting To emphasize that there is nothing peculiar about the possibility
that the monopoly and competitive …rm might choose the same level of sales,
consider an even simpler, one-period model. In this model, the …rm (either a
monopoly or a representative competitive …rm) can produce up to 10 units at
constant costs 4. Production beyond that level is not feasible; equivalently,
the marginal cost of production becomes in…nite at y = 10. In the …rst
scenario, the inverse demand function is p = 30 y, and in the second
scenario it is p = 15 y. Figures 3.4 and 3.5 show the demand functions
in these two cases (the solid lines), and the corresponding marginal revenue
curves (the dashed lines). In both cases the marginal production cost is 4
for y < 10 and in…nite for y > 10.
In the high-demand scenario (Figure 3.4), the constraint y 10 is binding
for both the monopoly and the competitive …rm, so both …rms produce y =
10. In this case, the price is also the same under the competitive …rm or
the monopoly. In the low-demand scenario (Figure 3.5), the constraint is
binding for the competitive …rm, which produces y = 10. Here, the marginal
revenue curve (which lies below the demand curve) equals marginal cost at
y = 5:5. The monopoly produces less than competitive …rms, and receives
a higher price.
There is nothing special about the possibility that a monopoly and a
competitive …rm might produce at the same level. The outcome depends on
the relation between the point at which the cost function becomes vertical
(y = 10 in this example) and the demand and marginal revenue functions.
The monopoly is the conservationists’friend We return to the twoperiod resource model. Under our assumption that both the monopoly
and competitive industry extract the same cumulative quantity over two
periods, the two market structures lead to the same allocation of the resource
if there is no discounting and if extraction costs are either 0 or constant
(as in our example): both sell half the aggregate quantity in each period.
With discounting (or with non-constant average extraction costs, studied in
3.2. MONOPOLY
51
$
30
20
10
0
0
1
2
3
4
5
6
7
8
9
10
11
12
y
Figure 3.4: The solid line shows the demand function p = 30 y; and the
dashed line is the marginal revenue function corresponding to this demand
function. Marginal costs are constant at 4 for y < 10 and in…nite for y > 10:
$
20
15
10
5
0
0
1
2
3
4
5
6
7
8
9
10
11
12
y
Figure 3.5: The solid line shows the demand function p = 15 y; and the
dashed line is the marginal revenue function corresponding to this demand
function. Marginal costs are constant at 4 for y < 10 and in…nite for y > 10:
52
CHAPTER 3. NONRENEWABLE RESOURCES
$
20
15
10
5
0
0
1
2
3
4
5
6
7
8
9
10
11
12
y
Figure 3.6: The solid curves show the present value (with = 0:5) demand
functions in the two periods, minus c = 4, and the dashed lines show the
present value marginal revenue curves, minus c = 4. First period sales in
the competitive equilibrium = 7.33; …rst period sales under the monopoly =
5.33. “The monopoly is the conservationist’s friend.”
Chapter 4), the allocations in the two equilibria are, in general, di¤erent.
For “most” demand and cost functions, including those used in this book,
the monopoly sells less in period 0, compared to the competitive industry.
The monopoly saves more of the resource for the future, compared to the
competitive industry: “the monopoly is the conservationists’friend”. This
result is not completely general: for some combinations of demand and cost
functions (not included in this book) period-0 monopoly sales are greater
than period-0 competitive sales.
For = 0:5, Figure 3.6 shows the present value of price minus marginal
revenue minus cost, illustrating that (in this setting) the monopoly is the
conservationist’s friend. The intersection of the solid lines (present value of
price minus marginal costs) identi…es the period-0 sales under competition,
y = 7:33. The intersection of the dashed lines (present value of marginal revenue minus marginal costs) identi…es the period-0 sales under the monopoly,
y = 5:33.
A decrease in the discount factor from = 1 to = 0:5 increases period-0
sales under both market structures, but the increase is greater in the competitive market. (Compare Figures 3.3 and 3.6.) A decrease in shifts down
and ‡attens both of the period-1 curves in Figure 3.6, but the marginal revenue curve is steeper than the inverse demand function, causing the point of
intersection on the marginal revenue curves to move further to the left..
3.3. COMPARATIVE STATICS
3.3
53
Comparative statics
Objectives and skills
Reinforce the distinction between exogenous and endogenous variables,
and the distinction between an explicit and implicit relation between
variables.
Use calculus, and both the explicit and the implicit relations, to answer
a comparative statics question.
This section provides more practice in working through the comparative
statics of a model. Following the procedure outlined in Chapter 2.2, we …rst
…nd an explicit expression for the endogenous variable of interest (period-0
sales) as a function of model parameters, and take derivatives to …nd comparative static expressions. The alternative approach proceeds by totally
di¤erentiating the equilibrium condition with respect to the endogenous and
the exogenous variables of interest, and rearranging the resulting di¤erential
in order to obtain an expression for the derivative of the endogenous variable
with respect to the exogenous variable.
We can sometimes, but not always, determine the sign of the comparative
static expressions. In cases where we cannot determine the sign, we have
to be careful in how we describe our ignorance. If we say that we cannot
determine the sign, we mean just that. If, however, we say that the sign of
the expression is “ambiguous” we mean that for some parameter values the
sign is positive, and for others it is negative. These two statements mean
di¤erent things.
In one of our examples, we can determine the sign of the comparative
static expression when we use the explicit solution, but not when we use the
alternative based on the di¤erential of the equilibrium condition. Because
the two approaches to …nding the comparative statics are both correct, their
answers must be mutually consistent. However, one approach gives more
information than the other.
The endogenous variables in this model are the prices and quantities in
the two periods. The exogenous variables are c and . A slightly richer
model replaces the numerical values in the demand function with symbols,
replacing p = 20 y with p = a by, and replaces the initial stock, 10, with
a symbol, x. For that model, the equilibrium condition in the competitive
54
CHAPTER 3. NONRENEWABLE RESOURCES
model is
(a
by
c) = (a
b (x
y)
(3.5)
c) :
Comparative statics via the explicit relation Because of its linearity,
we can solve this equation to obtain an explicit expression for …rst period
sales, as a function of the model parameters:
y=
1
(a
b+b
c + (c
a + bx)) :
We can answer comparative statics questions by di¤erentiating this expression with respect to model parameters. For example,
dy
1
=
dc
b+b
0 and
dy
1
1
= 2
(a
db
b +1
c)
0:
(3.6)
The “choke price”, de…ned as the price at which demand falls to 0, is a in
this model. In order for …rms to extract the resource, it must be the case
that a > c. Therefore, the two comparative statics inequalities are “strict”
(< instead of =) for < 1. We already showed graphically that when < 1
an increase in extraction costs shifts extraction from the …rst to the second
period. A larger value of b makes the inverse demand function steeper, i.e.
it reduces demand at any price. The second comparative statics expression
shows that this decrease in demand also reduces …rst period sales.
Comparative statics via the implicit relation For even slightly more
complex models, it would not be possible to obtain explicit comparative
statics expressions. Chapter 2.2 illustrates how to …nd comparative static
expressions using only the equilibrium condition, not an explicit solution to
the model. Following the procedure outlined there, we denote the left side
of the equilibrium condition, equation 3.5, as L (y; ) = (a by c) and the
right side as R (y; ) = (a b (x y) c). The “ ” notation is shorthand
for all of the exogenous variables, the parameters of the model: a; b; c; x; .
With this notation, we rewrite the equilibrium condition, equation 3.5 as
L (y; ) = R (y; ) :
Equilibrium requires that a change in an exogenous parameter, such as
the demand slope b, be o¤set by a change in the endogenous variable, period
0 supply, y: dL = dR. Evaluating this equality implies the comparative
3.3. COMPARATIVE STATICS
55
statics expression (Box 3.1)
x + (1 + ) y
:
b (1 + )
dy
=
db
(3.7)
The denominator of the right side of this equation is negative, but without
additional information we cannot determine whether the numerator is positive or negative. The most that we can say, using only the information in
equation 3.7, is that dy
< 0 if and only if
x + (1 + ) y > 0, i.e. if and
db
only if y > 1+ x.
Box 3.1: Derivation of equation 3.7. We totally di¤erentiate the
equilibrium condition, equation 3.5, with respect to y and b, to write
dL =
@L
@R
@R
@L
dy +
db =
dy +
db = dR:
@y
@b
@y
@b
Rearrange the di¤erentials on the two sides of the equality to write
@L
@y
@R
@y
dy =
@R
@b
@L
@b
dy
=
db )
db
@R
@b
@L
@y
@L
@b
:
@R
@y
@R
=
@b
(x
y) :
To evaluate this expression we use
@L
=
@y
b;
@R
@L
= b,
=
@y
@b
y;
Putting these results together, we have
dy
=
db
@R
@b
@L
@y
@L
@b
@R
@y
=
(x
y)
b
( y)
b
Simplifying the right side of this equation produces equation 3.7.
Comparing the two approaches to comparative statics It is instructive to compare the derivatives dy
in equations 3.6 and 3.7. Both of them
db
are correct, but the former contains more information; it tells us that dy
< 0,
db
whereas equation 3.7 gives us only a condition ( y > 1+ x) under which
dy
< 0. The fact that the two approaches yield di¤erent amounts of infordb
mation is not surprising, because the …rst approach begins with more informa-
56
CHAPTER 3. NONRENEWABLE RESOURCES
tion: it uses the explicit expression for y as a function of model parameters.
In contrast, the second approach uses only the equilibrium condition. More
information is preferred to less, so in this sense the …rst approach is better
than the second. But bear in mind that the …rst approach is not always
available to us, because many models are too complicated to yield explicit
solutions for the endogenous variables.
3.4
Summary
Equilibrium in a nonrenewable resource market has many of the characteristics of the equilibrium in the previous trade example. The no-arbitrage
condition in the trade model requires that transportation costs account for
any price di¤erence between the two locations. In the two-period nonrenewable resource setting, …rms can reallocate sales from one period to another.
Intertemporal reallocation in the resource setting corresponds to transportation across space in the trade setting. An intertemporal no-arbitrage condition requires that the present value of the marginal return from selling a
good is the same in both periods. The discount factor, used to convert
a future receipt into its present value equivalent, plays a role analogous to
transportation costs in the trade model.
We obtained the equilibrium condition for a monopoly by taking the equilibrium condition for a competitive …rm, and replacing price with marginal
revenue, where marginal revenue = p (1 1= ), and is the price elasticity
of demand. With constant marginal extraction costs and no discounting
(r = 0, so = 1), in our example both types of …rms sell the same amount,
half of the available stock, in period 0. Under discounting, ( < 1), the
monopoly sells less in the …rst period than the competitive …rm. Here, the
monopoly is the conservationist’s friend.
Using graphical methods, we showed that for < 1, increasing extraction
costs leads to a fall in period-0 sales under both competition or monopoly.
Increasing costs decreases the sales incentive both periods; but because of
discounting, the incentive for period-1 sales (= extraction) falls by less than
does the incentive for period-0 sales.
An example illustrated two calculus-based ways to conduct comparative
statics: by taking derivatives of an explicit solution, or by using the di¤erential of the equilibrium condition. Both approaches are correct, but the
comparative statics expressions that they produce have di¤erent appearances,
3.5. TERMS, STUDY QUESTIONS, AND EXERCISES
57
and possibly contain di¤erent amounts of information. For many models of
interest, it is not possible to obtain explicit expressions for the endogenous
variables; in those cases, we can use the method based on the di¤erential.
3.5
Terms, study questions, and exercises
Terms and concepts Extraction costs, intertemporal arbitrage, trajectory, “monopoly is the conservationist’s friend”, choke price.
Study questions For these questions, use the linear inverse demand function, p = 10 y.
1. In the two-period setting, with discount factor < 1, use a …gure to
describe the e¤ect of an increase in extraction costs, from C = 0 to
C = 2, on the equilibrium price and sales trajectory. Provide the
economic explanation for this change.
2. In a two period setting with linear demand and constant average extraction costs C use two …gures to illustrate the equilibrium sales trajectory
under competition and under monopoly for the two cases where (a) the
discount factor is = 1 and (b) the discount factor is less than 1.
Explain the e¤ect of discounting.
3. Answer questions 1 and 2 algebraically, using the equilibrium conditions
under competition and monopoly.
Exercises
1. Find the …rst order condition to the problem of maximizing monopoly (y)
and show that this …rst order condition is identical to equation 3.4.
2. We assumed that the initial stock of the resource is exhausted in two
periods, i.e., cumulative extraction equals the initial stock. Continue
to assume that the resource has no value after period 1; perhaps the
world ends, or perhaps the demand for this resource disappears. (a)
Using the demand function p = 20 y and constant extraction costs
c = 5, …nd the critical level of the initial stock, xc (a number) such
that a competitive equilibrium exhausts the resource if and only if the
58
CHAPTER 3. NONRENEWABLE RESOURCES
initial stock, x, satis…es x
xc . (b) If x
xc , what is true of price
in the two periods? (c) Does the critical value depend on ? Does
it depend on c? Explain. (d) Now suppose that instead of ending
after two period, the model last for three periods. Does this change in
assumption alter your answer to part (a)? Explain. (Note: you do not
need to …nd the critical value corresponding to the three-period model
to answer this question. A bit of economic reasoning is su¢ cient.)
3. Suppose that c = 0 and demand is constant elasticity, p (y) = Ay .
(a) Write the equilibrium conditions for the competitive …rm and the
monopoly in this case. (b) In order for the monopoly equilibrium condition to be sensible, what restriction must be imposed on ? Provide
the economic explanation for this restriction. (c) Compare the level of
…rst period sales under competition and under monopoly.
4. The …rst approach to comparative statics (based on the explicit solution
< 0
for y as a function of model parameters) demonstrates that dy
db
(for
< 1), and the second approach (based on the di¤erential of
the equilibrium condition) demonstrates that dy
< 0 if and only if
db
y > 1+ x. Based on these two (correct) results, what must be true
about the relation between y and x for the two results to be mutually
consistent?
5. Consider the two-period model with inverse demand p = a bq, constant average extraction cost c, initial stock x, and discount factor .
Period 0 extraction is y, the endogenous variable. Suppose that in the
competitive equilibrium extraction is positive in both periods, and all
of the resource is used. Using the equilibrium condition for the competitive …rm and the “general approach” to comparative statics (i.e.,
not relying on …nding an explicit solution to the equilibrium value),
.
…nd dy
dc
6. In the text we assumed that extraction is positive in both periods.
Using the demand and cost assumptions in the example in Chapter 3.1,
…nd the critical discount factor, denoted , such that second period
sales in the competitive equilibrium are 0 if < . Provide the
economic intuition.
7. Consider the inverse demand function p = y , where 0 < < 1. Marginal extraction costs are constant, at c, and the initial stock is x. (a)
3.5. TERMS, STUDY QUESTIONS, AND EXERCISES
59
What is the interpretation of ? (b) Write the equilibrium conditions
for the competitive industry and for the monopoly. (c) Find the comparative statics expression for ddy in the monopoly market. resources.
60
CHAPTER 3. NONRENEWABLE RESOURCES
Chapter 4
Additional tools
Objectives
Work with a general (stock-dependent) cost function; use the “perturbation method” to obtain equilibrium conditions; and express these
conditions using “rent”.
Information and skills
Understand the rationale for using a stock-dependent extraction cost
function, and be able to work with a particular cost function.
Write down a …rm’s objective function and constraints.
Derive and interpret the optimality condition to this problem, for the
two cases where the resource constraint is binding or is not binding.
Understand the logic of the perturbation method, and apply it in the
two-period setting.
Understand the meaning of “rent” in the resource setting, and use it
to express the optimality (equilibrium) condition.
Understand the relation between rent in period 0 and in period 1.
We build on the previous chapter, introducing: (i) a more general cost
function, (ii) the “perturbation method”, and (iii) the concept of rent. The
constant-average-cost model in Chapter 3 provides intuition, but obscures
61
62
CHAPTER 4. ADDITIONAL TOOLS
important features of many resource settings. A more general cost function
captures these features. By dealing with this generalization at the outset, we
can present the subsequent material in a uni…ed and concise manner, without
repeating the discussion for each important special case.
The “perturbation method” provides a quick way to obtain the equilibrium condition in resource models. The idea behind this method, if not the
term, will be familiar to many readers. Imagine the …rm beginning with a
“candidate” for an optimal plan, e.g. selling 53% in period 0 and 47% in
period 1. The …rm can test whether this candidate is optimal by “perturbing”it, moving a small (in…nitesimal) amount of sales from one period to the
other. If this perturbation increases the …rm’s present discounted value of
pro…ts, the original candidate was not optimal. It the perturbation decreases
the …rm’s pro…ts, then using the “opposite” perturbation, e.g. moving sales
from period 0 to period 1, instead of from period 1 to period 0, would increase pro…ts. Thus, if the perturbation either increases or decreases pro…ts,
the candidate is not optimal. In order for the candidate to be optimal, an
in…nitesimal perturbation must have “zero …rst order e¤ect” on the payo¤.
This statement means that the derivative of the payo¤, with respect to the
perturbation, evaluated at a zero perturbation, is zero.
We also discuss the concept of rent in a resource setting. Although “rent”
is a common word, it has a particular meaning in economics, and a still more
particular meaning in resource economics. It provides a convenient way to
express the equilibrium conditions.
This chapter considers the competitive equilibrium. We obtain the equilibrium condition for the monopoly using the competitive equilibrium condition, merely replacing “price”with “marginal revenue”.
4.1
A more general cost function
Objectives and skills
Understand the reasons for allowing average extraction costs to depend
on the stock and the extraction level.
Understand the relation between parameter values and the characteristics of cost for an example.
The distinction between stock and ‡ow variables is central to resource
economics. A stock variable is measured in units of quantity, e.g. billions of
4.1. A MORE GENERAL COST FUNCTION
63
barrels of oil, or tons of coal, or number of …sh, gigatons of carbon. The units
of measurement do not depend on units of time. The number of tons of coal
might, of course, change over time, but the statement that we have x tons of
coal today does not depend on whether we measure time in months or years.
In contrast, the units of measurement of ‡ow variables do depend on units of
time. For example, the statement “This well produces 1000 barrels of oil”is
meaningless unless we know whether it produces this number of barrels per
hour, day, or week. The variable xt denotes the stock of a resource, with
the subscript identifying time, or the period number. The variable yt is a
‡ow variable, denoting extraction during a period. Thus, if a period lasts
for one year, and are quantity units are tons, then xt is in units of tons and
and yt is in units of tons per year.
The constant average cost function used in Chapter 3 has two restrictive
assumptions: (i) Marginal extraction costs do not depend on the size of the
remaining resource stock, and (ii) Marginal extraction costs do not depend
on the rate of extraction.
In reality, marginal (and average) extraction costs typically increase as
the size of the remaining stock falls. This relation likely holds at both the
level of the individual mine or well, and at the economy-wide level. For
example, at the individual level, shallow and relatively inexpensive wells are
adequate to extract oil or water when the stock of oil in a …eld or water in an
aquifer is high. As the stocks diminish, it becomes necessary to dig deeper
and more expensive wells to continue extraction. At the economy-wide level,
di¤erent deposits have di¤erent extraction costs. Because it is (generally)
e¢ cient to extract from the cheaper deposits …rst, extraction costs increase
as the size of the remaining economy-wide stock falls. People began mining
coal from seams that lay close to the ground; early oil deposits could be
scooped up with little e¤ort. As society exhausted these cheap deposits,
it became economical to remove mountaintops to obtain coal and to exploit
deep-water deposits to extract oil. In these cases, extraction costs rose as
the remaining economy-wide supply of the resource declined.
The assumption that the rate of extraction does not a¤ect average and
marginal cost implies, for example, that total extraction costs double in a
period if we double the amount extracted. In many circumstances, average
and marginal costs increase with the rate of extraction.
It is important not to confuse stock-dependent extraction costs with increasing average and marginal costs. The former causes average or marginal
costs to rise over time, as the stock falls; the latter causes higher extraction
64
CHAPTER 4. ADDITIONAL TOOLS
within a period to increase these costs. Average and marginal extraction
costs might increase as the stock falls, or as extraction increases, or for both
reasons. These two types of cost-related considerations are distinct. To
accommodate both of these features, we need a cost function of the form
c (x; y), with the following characteristics
h
i
c(x;y)
@
y
@ 2 c (x; y)
@c (x; y)
0,
0,
0:
@x
@y
@y 2
The …rst inequality states that a higher stock either lowers costs or (in the
case of equality) leaves them unchanged. The second states that higher
extraction either increases average costs or leaves them unchanged. The
third states that higher extraction either increases marginal costs or leaves
them unchanged.
A parametric example helps to make this cost function more concrete:
y 1+ ;
Parametric example: c (x; y) = C ( + x)
(4.1)
where C; ; ; and are non-negative parameters. Table 1 shows the relation
between parameter values and marginal costs.
parameter values
C=0
C > 0;
>0
and > 0
C > 0; = 0
and = 0
C > 0; = 0;
and > 0
C > 0; > 0;
and = 0
cost function
0
C (x + )
y 1+
marginal cost
0
C (1 + )(x + )
Cy
C
Cy 1+
C (1 + ) y
C (x + )
y
C (x + )
y
marginal extraction costs are:
zero
increasing in extraction,
decreasing in stock
independent of both
extraction and stock
increasing in extraction,
independent of stock
independent of extraction,
decreasing in the stock
Table 4.1: Relation between parameter values and marginal extraction costs
If the stock is zero, it is not possible to extract anything: extraction costs
are in…nite if the stock is zero. For = = 0, c (x; y) = Cy for x > 0 and
c (x; y) = 1 if y > 0 and x = 0. We adopt the common usage, referring the
cost function Cy as stock-independent; in fact, this cost function is stockindependent only for x > 0.
4.2. THE PERTURBATION METHOD
4.2
65
The perturbation method
Objectives and skills
Write down the …rm’s objectives and constraints, based on a statement
of the problem.
Write down and interpret the …rst order condition (= optimality condition) for this problem, both in the case where the resource constraint
is binding and where it is not binding (= “slack”).
Implement two ways of obtaining the optimality condition: (i) eliminating the constraint by substitution, and (ii) using the perturbation
method.
Given demand and cost functions and all parameters of the model, solve
for the equilibrium.
This section uses two approaches to derive the necessary condition for
optimality (the “equilibrium condition”) in the two-period competitive market. The …rst approach begins by (i) substituting in the constraint, (ii) then
taking the derivative of the present discounted value of pro…ts, with respect
to period-0 sales, and (iii) …nally, replacing the price in each period (which
the …rm takes as exogenous) with the inverse demand function. The resulting …rst order condition is the equilibrium condition for this market. The
second approach uses the perturbation method. The perturbation method
has no advantages, relative to the more familiar …rst approach, in this simple
setting. However, the perturbation method is useful for models with many
periods, so we introduce it in the two-peiod setting.
The initial stock of the resource, at the beginning of period 0, is x0 . A
candidate consists of feasible extraction levels in the two periods, y0 and y1 .
These must satisfy the resource constraint and the non-negativity constraints:
0
y0
x0 , 0
y1
x1 = x0
y0 and x1
y1
0:
Extraction cannot be negative, and cannot exceed available stock; the available stock in period 1 equals the initial stock minus the amount that was
extracted in period 0. The …nal inequality states that the ending stock,
66
CHAPTER 4. ADDITIONAL TOOLS
after period-1 extraction, must be non-negative. If extraction is positive in
both periods, and all of the resource is used, the constraints imply
y0 .
y1 = x1 = x0
(4.2)
Even in this two-period setting, there are some circumstances where it is not
optimal to use all of the resource; in that case, we have y1 < x1 instead of
y1 = x1 . We consider these two cases separately.
We present the equilibrium for competitive …rms. As discussed in Chapters 2 and 3, we can obtain the equilibrium condition for the monopoly,
merely by replacing price in each period with marginal revenue in that period.
4.2.1
It is optimal to use all of the resource
In this section, we assume that in equilibrium y1 = x1 , i.e. the resource
constraint is binding. The present discounted value of total pro…t for the
price-taking …rm is
p0 y 0
c (x0 ; y0 ) + [p1 y1
(4.3)
c (x1 ; y1 )] :
Eliminating the constraint by substitution
Probably the most natural way to obtain the equilibrium condition for this
model is to substitute the constraints 4.2 into the objective, to write the
present discounted value of pro…ts as
(y0 ) = p0 y0
c (x0 ; y0 ) + [p1 (x0
y0 )
c (x0
The …rst order condition to the problem of maximizing
y0 ; x0
y0 )] :
(y0 ) is
d (y0 ) set
= 0:
dy0
This …rst order condition implies (Box 4.1 ):
2
@c (x0 ; y0 )
@c (x1 ; y1 )
p0
= 4 p1
@y0
@y1
3
@c (x1 ; y1 ) 5
@x
|
{z 1 }
(4.4)
4.2. THE PERTURBATION METHOD
67
The optimal decision balances the gain from additional extraction in period
0 (the left side of equation 4.4) with the loss from lower extraction and lower
stock in period 1 (the right side of the equation). The left side is the familiar
“price minus marginal cost”, the increase in period-0 pro…ts from extracting
one more unit in that period. The right side is the present value of two
terms, the underlined and the “under-bracketed” terms. The underlined
term equals price minus marginal cost in period 1, the loss arising from
having one less unit to sell. The under-bracketed term is the cost increase
due to a reduction in the stock at the beginning of period 1, resulting from
the higher period-0 extraction.
1 ;y1 )
= 0; in this special
If costs are independent of the stock, then @c(x
@x1
case, the under-bracketed term vanishes and equation 4.4 then states that
the present value of marginal pro…t is equal in periods 0 and 1.
Box 4.1 Derivation of equation 4.4 The …rst order condition for the
competitive …rm’s maximization problem is
h
i
d (y0 )
@c(x0 ;y0 )
=
p
+
0
dy0
@y0
h
1
p1 dy
dy0
@c(x1 ;y1 ) dx1
@x1
dy0
@c(x1 ;y1 ) dy1
@y1
dy0
i
set
= 0:
The second square brackets on the right side uses the chain rule. For
1 ;y1 )
picks up
example, period-1 costs depend on the period-1 stock; @c(x
@x1
this dependence. From the …rst constraint in equation 4.2, the period1
= 1. Similarly,
1 stock depends on period-0 extraction, via dx
dy0
dy1
= 1.
dy0
We can use these two equalities to write the …rst order condition as
d (y0 )
= p0
dy0
@c (x0 ; y0 )
+
@y0
p1 +
@c (x1 ; y1 ) @c (x1 ; y1 ) set
+
= 0:
@x1
@y1
Rearranging this condition gives equation 4.4.
The alternative: perturbation method
The approach used above to derive equation 4.4 is straightforward in the twoperiod setting, but becomes cumbersome in the many-period problem. With
that problem in mind, we consider the method of perturbation. We begin
with a candidate, y0 and y1 , and the associated period-1 stock, x1 = x0 y0 .
68
CHAPTER 4. ADDITIONAL TOOLS
The assumption that it is optimal to consume all of the resource, means
that any candidate worth considering sets y1 = x1 ; the ending stock is 0.
Expression 4.3 shows the payo¤ associated with this candidate.
Suppose that we “perturb”this candidate by changing period-0 extraction
by a small (positive or negative) amount, ", the “perturbation parameter”.
Because, (by assumption) it is optimal to consume all of the resource, a
change in period-0 extraction of " requires an o¤setting change in period-1
extraction of ". For example, a perturbation with " < 0 means that we
extract a bit less in period 0 and a bit more in period 1. The “gain” from
this perturbation equals
g ("; y0 ; x1 ; y1 ) = p0 (y0 + ")
c (x0 ; y0 + ")
(4.5)
+ [p1 (y1
")
c (x1
"; y1
")] :
For " = 0, g ("; y0 ; x1 ; y1 ) equals the payo¤ under the candidate, shown in
expression 4.3
If the candidate is optimal, then a perturbation causes zero …rst order
change to the payo¤. This statement means that at an optimum, it must be
the case that
dg ("; y0 ; x1 ; y1 )
= 0:
d"
j"=0
(4.6)
Evaluating this derivative (Box 4.2 )produces the same …rst order condition
that we obtained above, equation 4.4. The perturbation method is a di¤erent
route to the same goal.
4.2. THE PERTURBATION METHOD
69
Box 4.2 Evaluating derivative in equation 4.6. Here we are interested
in the e¤ect on the payo¤ of changing ", given the candidate y0 ; x1 ; y1 .
The period-1 cost function, c (x1 "; y1 "), involves " in two places,
so we need to take the total derivative to evaluate the e¤ect of " on
this function. To evaluate this derivative, we use the fact that
d (x1 ")
d (y1 ")
=
=
d"
d"
1:
Using the chain rule and these equalities, the total derivative of period1 costs, with respect to ", evaluated at " = 0 is:
dc (x1
"; y1
d"
")
=
j"=0
@c (x1 ; y1 ) @c (x1 ; y1 )
+
@x1
@y1
:
Using this result, we have
dg(";y0 ;x1 ;y1 )
d"
j"=0
p0
@c(x0 ;y0 )
@y0
h
p1
=
@c(x1 ;y1 )
@x1
+
@c(x1 ;y1 )
@y1
i
:
Setting this derivative equal to 0 and rearranging, yields the …rst order
condition 4.4.
4.2.2
It is optimal to leave some of the resource behind
Depending on the relation between demand and costs, it might not pay to
extract all of the resource. Here we consider the situation where, in equilibrium, y1 < x1 : it is optimal to not exhaust the resource. In this case, the
resource constraint is “slack”instead of “binding”.
If exhausting the resource in period 1 would cause the …rm’s marginal
pro…t to become negative, then the …rm chooses not exhaust the resource.
The …rm’s period-1 pro…ts equal p1 y1 c (x1 ; y1 ), so its marginal pro…t, price
minus marginal cost, is
@c (x1 ; y1 )
p1
:
@y1
The …rm wants to continue extracting as long as marginal pro…t is positive,
but it never extracts to the level that causes marginal pro…t to be negative.
If the …rm does not want to exhaust the stock (set y1 = x1 ) then it must
70
CHAPTER 4. ADDITIONAL TOOLS
be because doing so would drive marginal pro…ts negative, i.e. it must be
the case that
@c (x1 ; y1 )
p1
< 0,
@y1
jy1 =x1
If this inequality holds, then the …rm extracts up to the point where period-1
marginal pro…t is 0:
@c (x1 ; y1 )
= 0.
(4.7)
p1
@y1
If this equality holds, then an argument similar to that which produces equation 4.4 yields the optimality condition.
p0
@c (x0 ; y0 )
=
@y0
@c (x1 ; y1 )
:
@x1
(4.8)
Note that substituting equation 4.7 into equation 4.4 produces equation 4.8.
Thus, the situation where the resource constraint is slack is a special case of
the situation where the constraint is binding.
4.2.3
Solving for the equilibrium
Thus far, we have written the …rm’s equilibrium condition in terms of the
prices in the two periods. Firms take these prices as exogenous, but they
are determined by equilibrium behavior. (Prices are “endogenous to the
model” – not to the …rm.) Given speci…c demand and cost functions, we
have enough information to actually solve for the equilibrium. For example,
suppose that demand is linear, p = a by, and cost is given by equation
4.1. We can replace p0 and p1 in equation 4.4 with a by0 and a by1 , so
equation 4.4 now involves the three unknowns, y0 , y1 , and x1 .
We have to consider two cases. If the resource is exhausted, then the
…rst order condition, equation 4.4, and the two constraints y1 = x0 y0 = x1
comprise three equations in the three unknowns. We can (in principle) solve
these three equations to …nd a candidate.
We then have to check to see whether the solution is actually a competitive
equilibrium. If price minus marginal costs is greater than or equal to 0 in
both periods, then we have the correct solution. However, if price minus
marginal cost is negative in either period, the solution to the three equations
is not actually an equilibrium. Instead, it must be optimal not to exhaust
the resource, y1 < x1 .
4.3. RENT
71
If we have found that it is optimal to not exhaust the resource, then
we use equations 4.7 and 4.8, together with the de…nition x1 = x0 y0 to
1 ;y1 )
…nd the three unknowns, y0 , y1 , and x1 . By assumption, @c(x
0,
@x1
so any solution to equation 4.8 implies that period-0 price minus marginal
cost is non-negative (typically, positive). Thus, we obtain the competitive
equilibrium when the resource is not exhausted.
4.3
Rent
Objectives and skills
Know the meaning of rent and write the optimality condition using
rent.
Understand the relation between rent in the two periods.
Understand that period-0 rent depends on whether the resource is exhausted, and on whether extraction costs depend on the stock.
In economics, “rent”is the payment to a factor of production that exceeds
the amount necessary to make that factor available. The classic example
is rent to unimproved land, used in production. Because the land is in
limited supply, it receives a payment. The payment is not needed in order
to create the land –it already exists, regardless of the payment. However,
the limited supply means that other potential users are willing to bid for the
land; the highest value use determines the rent in a competitive market. In
the resource setting, the limited initial stock, x0 , gives rise to resource rent.
Very little productive land is “unimproved”. Usually, a previous investment increased the land’s productivity by, for example, removing trees and
rocks. Once these improvements have been made, they are sunk. Nevertheless, they receive a payment for the same reason that the unimproved land
does: their limited supply. The payment occurs after the sunk investment,
and the improvements continue to exist regardless of whether the payment
is actually made. Because the actual payments are not necessary for the
continued existence of the improvements, they resemble rent. However, the
improvements were made with the anticipation of the payments, so the payments are not precisely rent. For this reason, payments resulting from a
sunk investment are known as “quasi-rent”.
72
CHAPTER 4. ADDITIONAL TOOLS
Most natural resource stocks become available only after signi…cant investments in exploration and development. Thus, the payments in excess of
extraction costs, arising from the sale of resources, are the sum of rents and
quasi-rents. For the most part (until Chapter 12) we ignore this distinction,
and refer the resource rent merely as “rent”.
In competitive markets (resource) rent is de…ned as the di¤erence between
price and marginal extraction costs. Denoting rent in period t as Rt , we have
R0 = p0
@c (x0 ; y0 )
and R1 = p1
@y0
@c (x1 ; y1 )
:
@y1
We can use this de…nition to write the optimality conditions in the two
cases where the resource is exhausted or is not exhausted, using a single
equation. If the resource is exhausted, then R1
0; except for knife-edge
cases, the inequality is strict. If the resource is not exhausted, then R1 = 0;
see equation 4.7 and the de…nition of R1 . We can write the equilibrium
conditions 4.4 and 4.8 as
@c (x1 ; y1 )
:
(4.9)
R0 =
R1
@x1
Equation 4.9 states that period-0 rent equals the present value of the
sum of two terms: period-1 rent, plus the cost reduction due to having higher
period-1 stock. Either of the terms on the right side could be zero or positive
(but never negative). These two terms capture the two reasons that period-0
rent is (typically) positive:
1. We will run out of the resource. Extracting one more unit today means
that we have one less unit to extract in the future. That extra unit of
potential future extraction is valuable if and only if R1 > 0. If, instead,
R1 = 0, then we will not run out of the resource, thus eliminating one
of the reasons that period-0 rent is positive.
2. Extraction of an extra unit today makes future extraction more expensive. For example, we will have to start extracting from a more
1 ;y1 )
= 0 (extraction
expensive mine in the future. If, however @c(x
@x1
cost does not depend on stock) this reason for positive period-0 rent
also vanishes.
Under monopoly, we de…ne rent as marginal revenue (instead of price) minus marginal cost. With this modi…cation, equation 4.9 gives the monopoly
equilibrium condition.
4.3. RENT
4.3.1
73
Examples
To examine the two components of rent, we specialize our parametric cost
function, equation 4.1, setting = 0 to obtain C ( + x) y. For given x,
this function has constant average and marginal extraction costs C ( + x) .
An increase in stock changes costs by
@C ( + x)
@x
y
=
this relation holds with equality if
C ( + x)
1
y
0;
= 0 and with inequality if
> 0.
Stock-independent costs
We begin with the simplest case, = 0, where extraction costs do not depend
on the stock. Here, equation 4.9 simpli…es to R0 = R1 : the present value
of rent is the same in both periods. If it is optimal to consume all of the
resource then R1 > 0. If it is optimal to not exhaust the resource, then
we have seen that R1 = 0; using R0 = R1 we conclude that in this case,
period-0 rent is also zero.
Figure 4.1 illustrates these two possibilities, using C = 4, = 0:77 and
x0 = 10. (Review the discussion of Figure 3.1 if the meaning of Figure
4.1 is unclear.) The solid lines in this …gure show price minus marginal
cost with high demand (p = 20 y) and the dashed lines show price minus
marginal cost with low demand (p = 7 y). The high-demand (solid) lines
intersect at y0 = 6:4, where rent (R0 = p0 4) is positive, at 9.6. Period-0
extraction is 6.4 and period-1 extraction is 3.6. The resource is exhausted in
the competitive equilibrium. The low-demand (dashed) lines intersect below
the y (horizontal) axis. This point of intersection corresponds to p0 4 < 0.
A competitive …rm does not lose money on sales, so extraction would stop
where 7 y 4 = 0, i.e. at y = 3. In this equilibrium, rent is 0 in both
periods. The condition R0 = R1 still holds, because 0 = 0. This example
illustrates the fact:
1 ;y1 )
= 0 (as it is in our example with = 0), then the
If @c(x
@x1
only source of rent is the fact that cumulative supply is limited.
Rent is positive in this case if and only if the resource constraint
is binding, i.e. if competitive …rms extract all of the …xed supply
over the life of the program (here, two periods).
74
CHAPTER 4. ADDITIONAL TOOLS
$
18
16
14
12
10
8
6
4
2
0
1
2
3
4
5
6
7
8
9
10
11
-2
12
y
Figure 4.1: With demand p = 20 y (solid lines), the resource is exhausted.
With demand p = 7 y, the resource is not exhausted.
Stock-dependent costs
We now consider the case where > 0, so extraction costs depend on the
stock. In this situation, the second term on the right side of equation 4.9 is
positive, so …rst period rent is also positive. We use this example to illustrate
the possibility that:
With stock-dependent extraction costs: (i) period-0 rent is positive even if the resource constraint is not binding, and (ii) period-0
sales might be either greater or less than period-1 sales.
In order to illustrate claim (i), we choose parameter values so that the resource constraint is not binding; therefore, period-1 rent is zero. For this
example, C = 7, = 0, = 0:77, x0 = 10; = 0:1 and the inverse demand
is p = 10 y. Here, the cost function is Cx y. If y0 is extracted in period
0, then period-1 costs are C (10 y0 ) y1 . The assumption that period-1
rent is 0 implies
R1 = 10
y1
7 (10
y0 )
= 0 ) y1 = 10
7 (10
y0 )
:
(4.10)
Using this relation and the optimality condition, some calculation (Box 4.3)
shows that period-0 extraction is y0 = 4:12, period-1 extraction is y1 = 4:14,
and aggregate extraction equals 8: 26 < 10. It is optimal to leave 10 8:26 =
1: 74 units of resource in the ground.
4.4. SUMMARY
75
Box 4.3 Finding extraction levels. Using R1 = 0 and equations 4.9
and 4.10, we obtain the equilibrium condition
R0 = 10
y0
7 (10)
0 + C (10
=
y0 )
0 + C (10
1
10
7 (10
y0 )
1
y1 =
y0 )
The de…nition of rent gives the …rst equality. The second equality is the equilibrium condition: period-0 rent equals the present
value of period-1 rent (here equal to 0), plus the added cost due
@c(x1 ;y1 )
to the diminished second period stock,
(here equal to
@x1
1
y1 ). The third equality uses equation 4.10. It
C (10 y0 )
can be con…rmed (numerically) that the solution to this equation is
y0 = 4:12, where rent is R0 = 0:318. The stock available at the beginning of period 1 is 10 4:12 = 5: 88, so the average (= marginal) extraction cost in the second period is 7 (10 y0 ) = 7 (5: 88) = 5:86.
With these extraction costs, period-1 rent is R1 = 10 y1 5:86. It
is optimal to extract up to the point where period-1 rent is 0, which
implies y1 = 4:14. Aggregate extraction is 4:12 + 4:14 =
The statements “rent is zero in period 1” and “the resource is not exhausted”are equivalent: one statement is true if and only if the other statement is also true. With stock-dependent extraction costs, period-0 rent is
positive even if period-1 rent is zero, simply because higher period-0 extraction increases period-1 costs. The …rm extracts less in period 0 than the
level that equates price and marginal costs, in order to reduce next-period
costs. In this example (but not in general), period-1 extraction is greater
than period-0 extraction, so the equilibrium period-1 price is lower than the
equilibrium period-0 price.
4.4
Summary
Extraction costs may depend on the remaining resource stock, and the marginal extraction costs might increase with the level of extraction (the “extraction rate”). We introduced a parametric cost function that has these
features. We used both the method of substitution and the perturbation
method to obtain the necessary condition for optimality in a two-period
nonrenewable resource problem.
76
CHAPTER 4. ADDITIONAL TOOLS
If demand is low relative to extraction costs, it might be optimal not to
exhaust the resource. We therefore have to consider the possibility that the
resource is exhausted, and also the possibility that some resource remains
after period t = 1 (the second and last period). If the resource is exhausted,
then the resource constraint means that extraction of an additional unit at
t = 0 requires an o¤setting reduction in extraction at t = 1. If the resource
constraint is slack, it is not necessary to make this o¤setting change at t = 1.
In the resource setting with competitive …rms, rent is de…ned as price
minus marginal cost. Under monopoly, rent is de…ned as marginal revenue
minus marginal cost. Recognizing this di¤erence in the de…nition of rent under a competitive …rm and under a monopoly, we can express the optimality
condition for both markets in the same manner:
Rent in period 0 equals the present value of the rent in period
1 plus the cost increase due to a marginal reduction in period-1
stock. In mathematical notation, this condition is
R0 =
R1
@c (x1 ; y1 )
@x1
:
The …rm never extracts where rent is negative; rent is either
strictly positive or it is zero.
We can use this fact, together with the optimality condition, to solve for
the equilibrium, given speci…c functional forms and parameter values for costs
and demand. To do this, we …rst solve the problem under the assumption
that the resource is exhausted in two periods. That assumption implies
y1 = x1 = x0 y0 . These two equations, plus the optimality condition,
comprise three equations in the three unknowns, y0 , x1 , and y1 . If, at the
solution to these three equations, rent is nonnegative in both periods, we
have the solution. If rent in either period is negative, then this proposed
solution is incorrect, and it must be the case that y1 < x1 , i.e. the resource
is not exhausted. In that case, period-1 rent is 0, and we have
R0 =
@c (x1 ; y1 )
@x1
and R1 = 0:
These two equations, together with the de…nition x1 = x0
equations that we solve to obtain the three unknowns.
y0 , give the three
4.5. TERMS, STUDY QUESTIONS, AND EXERCISES
77
When costs depend on the resource stock, we saw by example that there
are di¤erent possibilities for the trajectory of extraction and price. In the
“usual” case, or at least the one that we emphasize, extraction falls and
price rises over time (from period 0 to period 1). But it is also possible that
period-1 sales are higher (and the price lower) than in period 0.
4.5
Terms, study questions, and exercises
Terms and concepts
Binding constraint, slack constraint, stock-dependent and stock-independent
costs, perturbation, rent, quasi-rent.
Study questions
1. Given an inverse demand function p(y), an extraction cost function
c (x; y), a discount factor , and an initial stock x0 : (i) Write down
the competitive …rm’s objective (= payo¤) and constraints for the twoperiod problem. (ii) What assumption does this …rm make regarding
price in the two periods? (iii) Write down and interpret the optimality
condition (= …rst order condition) for the …rm. (Explain what the
various terms in the equation mean.) (iv) Write down the de…nition
of rent, and then restate the optimality condition in terms of rent in
the two periods.
2. (i) In this two-period problem, what does it mean to say that the
resource constraint is not binding? (ii) If the resource constraint is
not binding, what is the value of period-1 rent? (iii) If the resource
constraint is not binding, what is the value of period-0 rent? (iv) What
does your answer to part (iii) tell you about the components of period0 rent? (In answering parts iii and iv of this question you need to
discuss the two situations where extraction costs are independent of,
or depend on, the stock.)
3. Using the objective (= payo¤) for the …rm in question 1, and the assumption that the resource constraint is binding, describe how you
can derive the optimality condition, …rst by eliminating the constraint,
and second by the perturbation method. It is not necessary to take
derivatives or do any calculation; just describe the steps.
78
CHAPTER 4. ADDITIONAL TOOLS
2 cost functions
cost
0.4
0.3
0.2
0.1
0.0
0
1
2
3
4
5
6
7
y
1 1:pdf
Figure 4.2: Graphs of two cost functions, as functions of extraction, y, for
…xed stock, x.
4. Using the information provided in Chapter 2.4 and the competitive optimality condition, equation 4.4, write down and interpret the monopoly’s
optimality condition.
Exercises
1. This chapter considers only the competitive equilibrium. (a) For a general inverse demand function, and the parametric cost function, write
down the monopoly’s optimization problem in the two-period setting.
(b) Under the assumption that the monopoly exhausts the resource,
write down the equilibrium condition for the monopoly. (Hint: Review Chapter 2.4, especially the last subsection. (c) Say in words
(“interpret”) this equilibrium condition.
2. Figure 4.2 graphs two cost functions of the form C ( + x) y 1+ , with
C > 0 and > 0. (a) What can you conclude from these graphs about
the parameter in these two functions? (b) What can you conclude
about ? Explain the basis for your answers.
3. (a) For the parametric cost function c (x; y) = C ( + x)
mine the four partial derivatives:
@c (x; y) @c (x; y) @c (x; y) @c (x; y)
,
,
,
.
@C
@
@
@
y
+1
, deter-
4.5. TERMS, STUDY QUESTIONS, AND EXERCISES
79
(You may want to consult Appendix A to review a particular rule of
derivatives.) (b) Say in words what each of these partial derivatives
mean. (This is a one-liner.)
4. Replace the general cost function used in the …rst order condition equation 4.4 with the parametric example given in equation 4.1. Next,
rewrite the equation, specializing by setting = 0. Explain in words
the meaning of this equation.
5. Section 4.2.2 claims that the …rm never extracts to a level at which
marginal pro…t is negative. Explain, in a way that a non-economist
will understand, why this claim is true.
6. In our two-period setting, the gain function when it is optimal to consume all of the resource is
g ("; y0 ; x1 ; y1 ) = p0 (y0 + ") c (x0 ; y0 + ")+ [p1 (y1
")
c (x1
"; y1
and the gain function when it is optimal not to consume all of the
resource (so that the resource constraint x1 0 is not binding) is
g ("; y0 ; x1 ; y1 ) = p0 (y0 + ")
c (x0 ; y0 + ") + [p1 y1
c (x1
"; y1 )] :
Identify the di¤erence between these two functions, and provide the
economic explanation for this di¤erence.
7. (a) With stock-dependent resource costs, explain why period-0 rent is
positive even if it is not optimal to exhaust the resource (so that the
constraint x1 0 is not binding). (b) Explain why period-1 rent must
be zero in this circumstance. (c) In this circumstance, what happens
to rent over time (i.e. from period 0 to period 1)? This question has
a two-word answer.
8. Under the assumption that the monopoly exhausts the resource in
this two-period setting, write down the equilibrium condition for the
monopoly that faces inverse demand p = 20 y.
9. (Rent and quasi-rent for agricultural land.) Suppose that there is a
…xed stock of unimproved land, L = 10. The value of marginal product
of this land per year is 20 q, where q is the amount of land that is
rented. (a) What is the equilibrium annual rental rate for land? (b)
")] ;
80
CHAPTER 4. ADDITIONAL TOOLS
How much would someone with an annual discount rate of r be willing
to pay for this land? Recall equation 2.9. (The price of land is the
amount that someone pays to buy the land; the rent is the amount
they pay to use it for a period of time, e.g. one year.) (c) Suppose
that if the land is improved, its value of marginal product increases by
2, to 22 q. What is the equilibrium annual rental rate if all land is
improved? (d) What is the equilibrium price of improved land? (e)
Suppose that the cost of this improvement is a one-time expense of 10.
Assuming that the improvement (like the land) lasts forever, what is
the critical value of r at which the landowner is indi¤erent between
leaving (all of) the land in its unimproved state, and improving it?
10. (*) Use the inverse demand function p = 20 y and the cost function
Cx y, to write the equilibrium condition for a competitive market in
the two-period setting. (i) By using the di¤erential of this equilib0
.
rium condition, write down the comparative statics expression for dy
d
(You have to identify the endogenous and the exogenous variable of
interest in this exercise, in order to know the relevant di¤erential.) (ii)
0
. (ii) State in words the
Then specialize this expression to …nd dy
d j =0
meaning of these expressions, and provide the economic intuition for
them.
Sources
Livernois and Uhler (1987) study the role of extraction costs in nonrenewable
resource models.
Chermak and Patrick (1995), Ellis and Halvorsen (2002), and Stollery
(1983) provide empirical estimates of extraction costs.
Chapter 5
The Hotelling model
Objectives
Interpret and use the optimality condition for the T -period problem.
Information and skills
Understand the relation between the two-period and the T -period problems, and between their optimality conditions.
Write down the objective, the constraints, and the Euler equation for
the competitive …rm in the T -period problem.
Understand the relation between rent in any two periods, and the role
of extraction costs in determining the rent.
Explain the relation between price in two adjacent periods.
Understand the meaning of the “shadow value”of a resource.
Understand the meaning of a Lagrange multiplier in a constrained optimization problem.
Understand the relation between the Euler equation and an “asset pricing equation”.
Show that …rms exhaust cheaper deposits before beginning to extract
from more expensive deposits.
81
82
CHAPTER 5. THE HOTELLING MODEL
The intuition developed in the two-period setting survives when the resource can be used an arbitrary number of periods, T 1. The perturbation
method produces the optimality condition, known as the Euler equation in
general settings, and the Hotelling rule in the resource setting. The de…nition of rent provides a succinct means of expressing this rule, which has a
close parallel to an asset pricing equation used in investment models.
Rent can be interpreted as the “shadow value”of the resource, the amount
that a competitive resource owner would pay for one more unit of the resource
in the ground. It can also be interpreted as the opportunity cost of extracting
the resource. We use the Hotelling rule to show that it is optimal to exhaust
mines with relatively low extraction costs, before beginning to extract from
more expensive mines.
Most of our examples involve physical exhaustion of the resource; in these
cases, extraction continues until there is zero resource stock left. However, if
extraction costs increase as the resource stock falls, it may be uneconomical
to extract all possible stocks. In that case, the stock is economically but not
physically exhausted. Most natural resources have increasing costs, and future physical exhaustion is more likely the exception than the rule. At some
point, coal will become more expensive than other energy sources, particularly when one factors in the externality cost associated with climate change;
coal deposits are unlikely ever to be exhausted. Keeping the atmospheric
temperature change below 2o , considered by some to be the maximum safe
threshold, will require leaving over 50% of known fossil fuel stocks below the
ground. For most fossil fuel resources, economic exhaustion will occur before
physical exhaustion.
5.1
The Euler equation (Hotelling rule)
Objectives and skills
Write the competitive …rm’s objective and constraints.
Write and interpret the necessary condition (the Euler equation) to this
problem.
Understand how the perturbation method produces this condition.
We discuss the necessary condition for a …xed length of the program, T +1,
leaving the determination of T to Chapter 7. The competitive …rm wants
5.1. THE EULER EQUATION (HOTELLING RULE)
83
to maximize the present discounted sum of pro…ts, subject to the resource
constraint. The …rm’s optimization problem is:
max
xt+1 = xt
PT
t
t=0
(pt yt
c (xt ; yt ))
subject to
(5.1)
yt , with x0 given, xt
0 and yt
0 for all t.
The …rst order (necessary) condition for this problem is known as the
Euler equation; in the nonrenewable resource setting, it is also called the
Hotelling rule. The equation is
pt
2
@c (xt ; yt )
= 4pt+1
@yt
@c (xt+1 ; y+1 )
@yt+1
3
@c (xt+1 ; yt+1 ) 5
:
@xt+1
|
{z
}
(5.2)
Compare equations 4.4 (for the two-period problem, where T = 1) and 5.2
(for general T ), shows that the two are identical, except for the time subscripts. Equation 5.2 must hold for t = 0; 1; 2:::T 1, i.e. for all pairs of
adjacent periods when extraction is positive.
Equations 4.4 and 5.2 have the same interpretation. If the …rm sells
one more unit in period t and makes an o¤setting reduction of one unit in
period t + 1, it receives a marginal gain in period t and incurs a marginal
loss in period t + 1. The marginal gain in period t (the left side of equation
5.2) equals the increased pro…t, price minus marginal cost, due to the one
unit increase in sales. The marginal loss in period t + 1 is the sum of the
two terms on the right side of equation 5.2. The underlined term equals
the reduced pro…t due to reduced sales, price minus marginal cost in period
t + 1; the under-bracketed term equals increased cost due to the lower stock
in period t + 1.
Derivation of equation 5.2
By de…nition, T is the last period during which extraction is positive, so
extraction at T + 1 is zero. In the class of problems we consider, extraction
is also positive at earlier times: yt > 0 for all t < T . This fact means that
we can make small changes (perturbations) in any of the yt ’s, and o¤setting
changes in other yt ’s, without violating the non-negativity constraints on
84
CHAPTER 5. THE HOTELLING MODEL
extraction, or on the stocks. A perturbation is “admissible” if it does not
violate these constraints.
Recall that a “candidate” is a series of extraction and stock levels that
satisfy the non-negativity constraints. At the optimum, any admissible perturbation of the candidate yields zero …rst order change in the payo¤. In
the two-period setting, only “one-step”perturbations, where we make a small
change in period-0 extraction and an o¤setting change in period-1 extraction,
are possible. In a multiperiod setting, in contrast, in…nitely many types of
perturbations are possible. For example, we can reduce extraction by " in
period t, make no change in period t + 1, and increase extraction by "=3 in
each of the subsequent three periods. To test the optimality of a particular candidate, we have to be sure that no admissible perturbation, however
complicated, creates a …rst order change in the payo¤. With in…nitely many
possible perturbations, that sounds like a di¢ cult job. However, the task
turns out to be simple, because any admissible perturbation, no matter how
complicated, can be broken down to a series of “one-step” perturbations
(Appendix C).
Therefore, we can check whether a candidate is optimal by considering
only the one-step perturbations a¤ecting pairs of adjacent periods during
which extraction is positive. Let t be any period less than T , so that extraction is positive in periods t and t + 1. Because we are considering one-step
perturbations that a¤ect only these two periods, we only have to check that
the perturbation has zero …rst order e¤ect on the combined payo¤s during
these two periods. The combined payo¤ in these two periods, under the
perturbation is
g ("; yt ; xt ; yt+1 ) =
t
[(pt (yt + ")
c (xt ; yt + "))
(5.3)
+ (pt+1 (yt+1
")
c (xt+1
"; yt+1
"))]:
This gain function and the gain function from the two-period problem, equation 4.5, are the same, except for the time subscripts (and the fact that t
multiplies the right side of equation 5.3). In the two-period setting, we noted
that an optimal candidate has to satisfy the …rst order condition 4.6. The
necessary condition in the T -period setting is exactly the same, except for
the time subscripts:
dg ("; yt ; xt+1 ; yt+1 )
= 0:
d"
j"=0
5.2. RENT AND HOTELLING
85
Evaluating this derivative (repeating the steps that lead to equation 4.4)
produces the Euler equation 5.2.
5.2
Rent and Hotelling
Skills and objectives
Rewrite the Euler equation using the de…nition of rent.
Explain the relation between rent in period t and in any other period
(not merely the next period).
Understand how and why this relation depends on whether extraction
cost depends on the stock.
Explain the relation between rent (and price) in adjacent periods for
the case where average extraction cost is constant, C.
Using the de…nition of rent when …rms are competitive,
Rt = pt
@c (xt ; yt )
;
@yt
(5.4)
we can write the Euler equation (Hotelling rule) more compactly, as
Rt =
Rt+1
@c (xt+1 ; yt+1 )
:
@xt+1
(5.5)
Constant marginal costs If marginal costs are constant (i.e. if c (x; y) =
Cy), the Hotelling rule simpli…es to
Rt = Rt+1 or pt
C = (pt+1
C) :
(5.6)
The …rst equation states that the present value of rent is the same in any
two adjacent periods. A stronger result states that the present value of rent
is the same in any two periods where extraction is positive:
Rt =
j
Rt+j :
(5.7)
The Hotelling rule states that the …rm cannot increase its payo¤ by moving a unit of extraction between adjacent periods, t and t + 1; this is a
86
CHAPTER 5. THE HOTELLING MODEL
“no-intertemporal arbitrage condition”. Equation 5.7 is more general: it
states that the …rm cannot increase its payo¤ by moving extraction between
any two periods where extraction is positive (not merely between any two
adjacent periods). The intuition for this relation is that a …rm can sell the
marginal unit in period t, invest the marginal pro…t (Rt ) for j periods and
obtain the return (1 + r)j Rt ; alternatively, the …rm can delay extraction of
this marginal unit until period t + j, at which time it earns Rt+j . If there
are no opportunities for intertemporal arbitrage, the …rm must be indi¤erent
between these two options, i.e.
(1 + r)j Rt = Rt+j ) Rt =
1
Rt+j =
(1 + r)j
j
Rt+j :
(5.8)
With constant marginal = average extraction cost, rent has a particularly
simple interpretation. The value of the mine equals the initial rent times
the initial stock:
Value of mine =
T
X
t=0
t
(pt
C) yt =
T
X
t
Rt yt = R0
t=0
T
X
yt = R0 x0 : (5.9)
t=0
The …rst equality is a de…nition: the value of the mine equals the present
discounted stream of pro…ts from extraction. The second equality uses
the de…nition of rent, equation 5.4. The third equality uses equation 5.7
to replace t Rt with P
R0 . The fourth equality uses the stock constraint:
aggregate extraction ( Tt=0 yt ) equals the initial stock (x0 ).
We can determine the rate of change of price or rent by (i) multiplying
both sides of the two equations 5.6 by 1 + r, (ii) using = (1 + r) 1 , and
(iii) rearranging, to obtain
pt+1 pt
Rt+1 Rt
= r or
=r
Rt
pt
rC
:
pt
(5.10)
The …rst of these two equations says that rent rises at the rate of interest.
, i.e. price rises
The second says that price rises the rate of interest minus rC
pt
at less than the rate of interest if C > 0. For C = 0, the second equation
simpli…es to
pt+1 pt
= r;
(5.11)
pt
which states that in a competitive equilibrium where extraction is costless,
price rises at the rate of interest: the competitive …rm is indi¤erent between
5.2. RENT AND HOTELLING
87
selling in any two periods if and only if the present value of the price is
the same in the two periods. For C > 0, equilibrium price rises more slowly
than the interest rate. In Chapter 3, we noted that constant extraction costs
cause the …rm to delay extraction, causing the initial price to be higher, and
the later price to be lower than would have been the case for C = 0. Thus,
C > 0 reduces p1p0p0 in the two-period setting. Equation 5.11 shows that in
the T -period setting, a positive C lowers the rate of change of price at every
point in time.
Box 5.1 The trade analogy. Chapter 2 uses a two-country trade
example. Suppose instead that there are n + 1 countries in a row,
labelled 0; 1; 2; ::n, with iceberg costs b between any two countries.
In order to move tea from one country to another, it is necessary to
move it through the intermediate countries. Initially all of the tea is
in Country 0. If tea is sold in two adjacent countries, i and i + 1, then
1
pi+1 (review equation 2.1). If tea
spatial arbitrage requires pi = 1+b
is sold in countries i and i + j, with j 1, spatial arbitrage requires
1 j i+j
pi = 1+b
p . This equation is the spatial analog of equation 5.8.
Stock dependent extraction costs If costs depend on the resource stock
t ;yt )
> 0) the “no intertemporal arbitrage” condition implies that the
( @c(x
@xt
rent in period t equals the sum of two terms: (i) the present discounted value
of rent in a future period t + j (the underlined term on the right side of
equation 5.12) and (ii) the present discounted stream of cost savings due to
the extra unit of stock in the ground, from the current period, t, until period
t + j (the under-bracketed term in this equation):
Rt =
j
Rt+j
|
j
X
i=1
i @c (xt+i ; yt+i )
{z
@xt+i
}
:
(5.12)
If the …rm reduces extraction by one unit in period t and increases extraction
by one unit in period t + j, it looses Rt in the current period and gains Rt+j
in period t + j. The time-t present value of the additional pro…t at t + j
is j Rt+j . The under-bracketed term equals the time-t present value of the
cost savings arising from the higher stock between periods t + 1 to t + j.
Equation 5.12 shows that current rent depends on rent and extraction
costs in future periods. Rent is a “forward-looking variable”, because it
depends on prices and costs in the future.
88
CHAPTER 5. THE HOTELLING MODEL
Rent can be interpreted as the opportunity cost of extracting the resource:
the loss from extracting the marginal unit now rather than at some other
time. Current extraction reduces future pro…ts, creating an opportunity cost.
t ;yt )
. By rearranging the
The standard marginal extraction cost is simply @c(x
@yt
de…nition of rent, we have
pt =
@c (xt ; yt )
+ Rt :
@yt
This equation states that in a competitive equilibrium, price equals “full”
marginal extraction cost, where “full”means that it includes both the standard marginal cost and the opportunity cost (= rent).
5.3
Lagrange multipliers and shadow prices
Objectives and skills
Understand the meaning of the “shadow value”of a resource.
Know that the perturbation method used above yields the same optimality condition as the Method of Lagrange.
Recognize that the Lagrange multiplier can be interpreted as the shadow
value of the resource.
There does not exist a market that enables a resource owner to buy an
additional unit of stock in the ground (“in situ”). If there were such a market,
how much would the resource owner be willing to pay for this additional unit?
The answer is known as the “shadow price” of the resource; the modi…er
“shadow” is a way of recognizing that this market is hypothetical. The
shadow price at time t equals the equilibrium rent at that time, Rt .
The equality between rent and the shadow value of the resource holds
in general, but it is particularly obvious in the case of constant marginal
extraction costs. Here, equation 5.9 shows that the value of the mine is
R0 x0 . The increase in this value, due to the increase in the stock, x0 , is the
rent, R0 . The mine owner would be willing to pay R0 for one more unit of
the resource at time 0.
We used the perturbation method to obtain the equilibrium condition
in the competitive resource market. The method of Lagrange provides
5.4. RESOURCES AND ASSET PRICES
89
an alternative derivation. The …rm’s problem contains the T constraints
xt+1 = xt yt for t = 0; 1:::T 1. To each of these constraints we assign a
variable knows as the Lagrange multiplier. Having one more unit of the stock
“relaxes” the constraint, i.e. makes it less severe. The Lagrange multiplier
associated with a particular constraint equals the amount by which a “relaxation”in that constraint would increase the present discounted stream of
pro…ts. In the resource setting, the Lagrange multiplier associated with the
constraint in a particular period equals the amount that the resource owner
would pay for a marginal increase in the stock of the resource in that period:
the Lagrange multiplier equals the shadow price of the resource, which equals
the rent.
If someone o¤ered to sell the competitive owner a unit of the stock in situ
at time t, the owner would be willing to pay exactly Rt (= the rent = the
shadow value = the Lagrange multiplier) for that unit. This claim is not
self-evident, because it might seem that an owner who receives one more unit
of the resource would not simply extract that unit in the current period, and
thus earn Rt . Instead, the owner could extract some of the extra unit in the
current period, and some of it later. With that reasoning, it appears that
the amount the owner would pay for the marginal unit might di¤er from Rt .
This conjecture is false because of the no-intertemporal-arbitrage condition:
the owner has no desire to reallocate extraction over time. An owner who
acquires one extra unit is indi¤erent between extracting it now, and earning
Rt , or extracting it later. In either case, the present value of the owner’s
additional pro…t is Rt . The owner would therefore pay Rt for a marginal
unit of the resource in situ.
5.4
Resources and asset prices
Objectives and skills
Understand the relation between the basic asset pricing equation and
the Hotelling rule.
Competitive equilibria eliminate opportunities for intertemporal arbitrage, both for natural resources and for other types of assets such as shares
in companies. To illustrate that intertemporal arbitrage is important in
many disparate contexts, we discuss the relation between the Hotelling rule
and an asset pricing equation from …nancial economics.
90
CHAPTER 5. THE HOTELLING MODEL
By multiplying both sides of equation 5.5 by
the result, we can write the Hotelling rule as
rRt = Rt+1
Rt
1
= 1 + r and rearranging
@c (xt+1 ; yt+1 )
:
@xt+1
(5.13)
Now consider the equilibrium price of an asset such as shares in a company. There is no risk in our model, and a person can borrow a dollar for one
year at the interest rate r. In this perfect information world, people know
that next period, t + 1, the price of the asset will be Pt+1 , and they know
that there will be a dividend on the stock of Dt+1 . What is the equilibrium
price of this asset today, in period t?
If the price of the stock at the beginning of period t is Pt , a person who
can borrow at annual rate r can borrow Pt at the beginning of the period
and buy a unit of the stock. They must repay (1 + r) Pt at the beginning of
the next period. If they collect the dividend paid at the beginning of period
t + 1, and sell the stock, they collect Pt+1 + Dt+1 . Because the person has
not used their own money to carry out this transaction (and thus incurred
no opportunity cost), they must make 0 pro…ts, which implies
Pt+1 + Dt+1
revenue
Rearranging this equality gives:
rPt = Pt+1
(1 + r) Pt = 0:
| {z }
costs
Pt + Dt+1 :
(5.14)
Equation 5.14 is a no-arbitrage condition: it means that a person cannot
make money merely by making riskless purchases and sales. The left side
is the yearly cost of borrowing enough money to buy one unit of the stock.
The right side is the sum of capital gains (the change in the price) and the
dividend. If, contrary to equation 5.14, we had rPt < (Pt+1 Pt ) + Dt+1 ,
then people can make money by borrowing at the riskless rate, buying the
asset and then selling it after a year. Here, the asset is too cheap: there
is excess demand for it. If rPt > (Pt+1 Pt ) + Dt+1 , then the asset is too
expensive, and no one wants to hold it: there is excess supply. Equilibrium
therefore requires that equation 5.14 hold.
;xt+1 )
To compare equations 5.13 and 5.14, recall that @c(yt+1
0, because
@x
a higher stock decreases or leaves unchanged extraction costs; this term corresponds to the dividend, Dt in equation 5.14. It equals the bene…t that
5.5. THE ORDER OF EXTRACTION OF DEPOSITS
91
the resource owner obtains (here, from lower costs) due to holding the asset.
The asset price, P , corresponds to R, the resource rent.
In the asset price equation, if the dividend is 0, then the price of the stock
must rise at the rate of interest in equilibrium: the capital gain due to the
change in the asset price must equal the opportunity cost of holding the stock,
rPt . A positive dividend makes investors willing to hold the stock at lower
capital gains. The same reasoning explains why, in the resource setting,
stock dependent extraction costs cause the equilibrium rent to increase at
less than the rate of interest. The stock-dependent extraction costs play
the same role in the resource setting as the dividend does in the asset price
equation.
5.5
The order of extraction of deposits
Objectives and skills
Use the Euler equation to demonstrate that it is optimal to exhaust a
cheaper deposit before beginning to use a more expensive deposit.
We have assumed that there is a single deposit, for which extraction costs
can be de…ned using a function, c (x; y). We can think of this model as an
approximation of a more realistic situation where there are many di¤erent
deposits, having di¤erent extraction costs. In a competitive equilibrium,
the cheaper deposits are exhausted before beginning to extract from more
expensive deposits.
Suppose, for example, that there are three di¤erent deposits, with stock
size xa = 3, xb = 7, and xd = 2, having associated constant average (=
marginal) extraction costs C a = 4, C b = 5:5, C d = 7. Figure 5.1 shows
the average cost function for this example, as a function of remaining stock.
The graph is a step function, because the extraction costs are constant while
a particular deposit is being mined. Costs jump up once that deposit is exhausted, and it becomes necessary to begin mining a more expensive deposit.
If instead of there being only three mines, with signi…cantly di¤erent costs,
there were many mines, with only small cost di¤erences between the most
similar mines, then the …gure would approach a smooth curve, showing costs
decreasing in remaining stock.
In order to show that, in a competitive equilibrium, cheaper deposits
are exhausted before …rms begin to extract from more expensive deposits, it
92
CHAPTER 5. THE HOTELLING MODEL
$7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
x
Figure 5.1: Extraction costs as a function of remaining stock, when there are
three deposits, with individual stocks 3, 7 and 2, and corresponding constant
marginal extraction costs, 4, 5.5., and 7
is su¢ cient to consider the case where there are two mines, with constant
extraction costs C a < C b . For the sake of exposition, we assume that one
competitive …rm owns the low-cost mine and another …rm owns the high-cost
mine.
The Euler equation must hold for both …rms. In particular, for any two
adjacent periods, t and t + 1, during which a …rm is extracting, equation
5.10 must hold, with C replaced by C a or C b (depending on which …rm is
extracting). There might be a single period when extraction from both
deposits occurs. For example, during the last period when the low cost
…rm extracts, its remaining stock may be insu¢ cient to satisfy demand at
the equilibrium price. In this case, there is a single period when both …rms
extract a positive amount. Extraction of high-cost deposits begins only after
or during the last period when extraction from the low-cost deposit occurs.
To verify this claim, consider any two adjacent periods, t and t+1, during
which the low-cost …rm is extracting. We need to show that the high-cost
…rm does not want to extract in t, the …rst of these two periods. If the
high-cost …rm did want to extract in period t, then the claim would not be
true, because in that case extraction of high-cost deposits occurs in a period
prior to the period when the low-cost deposits are exhausted.
Given that the low-cost …rm extracts in both t and t + 1, equation 5.10
implies
(pt
C a ) = (pt+1
C a) :
(5.15)
5.6. MONOPOLY
93
Some manipulations (Box 5.2) produce
Cb <
pt
pt+1
Cb .
(5.16)
This inequality implies that the high-cost …rm strictly prefers to extract
nothing in period t. If this …rm were to extract a unit in period t, it would
earn pt C b . It could earn strictly higher present value pro…ts by holding on
to this unit and then selling it in period t + 1. Therefore, it is not optimal
for the high-cost …rm to sell anything in period t.
Box 5.2 Derivation of inequality 5.16. Adding C a
of equation 5.15 produces
C b = (pt+1
pt
C a) + C a
C b to both sides
C b:
Add and subtract C b to the right side of this equation and then
simplify the result to obtain
Cb =
pt
Because 1
=
r
1+r
pt+1
C b + (1
> 0 and C a
(1
) Ca
Cb :
(5.17)
C b < 0, we have
) Ca
C b < 0:
This inequality and equality 5.17 establish inequality 5.16.
5.6
Monopoly
Objectives and skills
Use previous results to write down and interpret the optimality condition for the monopoly.
We obtain the optimality condition under monopoly by using equation 5.2
and replacing price with marginal revenue, M Rt = pt (1 1= (pt )), where
(pt ) is the elasticity of demand evaluated at price pt . The Euler equation
for the monopoly is
pt 1
h
pt+1 1
1
(pt+1 )
1
(pt )
@c(xt ;yt )
@yt
@c(xt+1 ;y+1 )
@yt+1
=
@c(xt+1 ;yt+1 )
@xt+1
i
(5.18)
:
94
CHAPTER 5. THE HOTELLING MODEL
For the special case where extraction is costless and the elasticity of demand
1
is constant (inverse demand is p = y , and is a constant), equation 5.18
simpli…es to equation 5.11. In this special case, the monopoly and the
competitive industry have the same price and sales path.
More generally, the monopoly and competitive outcomes di¤er. If extraction is costless but demand becomes more elastic with higher prices (as
occurs for the linear demand function and many others) then monopoly price
rises more slowly than the rate of interest. In this situation, the initial
monopoly price exceeds the initial competitive price, so monopoly sales are
initially lower than competitive sales. Here again, “the monopoly is the
conservationist’s friend”.
De…ning rent for the monopoly as marginal revenue minus marginal cost
(instead of price minus marginal cost, as under competition), we can write
the monopoly’s optimality condition as in equation 5.5. The interpretation
of this equation is the same as under competition, provided that we keep in
mind that the de…nition of rent under monopoly di¤ers from the de…nition
under competition.
5.7
Summary
The perturbation method is as easy to use for the T -period problem as for
the two-period problem. It leads to a necessary condition for optimality
(equivalently, an equilibrium condition) that expresses current price and costs
as a function of next-period price and costs. This equation is known as the
Euler equation; in the resource setting, it is also known as the Hotelling rule.
We can express this equilibrium condition in terms of rent.
The equilibrium market price and the rent in a period depend on future
prices (or rent) and costs. Rent is a forward-looking variable. The Hotelling
rule states that rent in an arbitrary period equals the present value of rent in
the next period, plus the cost increase due to a marginal reduction in stock.
For the special case where extraction costs are independent of the stock, the
Hotelling rule states that the present value of rent is equal in any two periods
where extraction is positive. If extraction costs are zero, the Hotelling rule
states that price rises at the rate of interest. For positive constant average
extraction costs, price rises more slowly than the rate of interest.
The Hotelling rule has a close analog in investment theory, where the
asset pricing equation states that the opportunity cost of buying an asset
5.8. TERMS, STUDY QUESTIONS, AND EXERCISES
95
must equal the capital gains plus the dividend from owning the asset. By
re-labelling the resource rent as the asset price, and the cost increase due to
a lower stock as the dividend, the Hotelling rule becomes identical to this
asset pricing equation.
We also discussed the following two points:
Rent in a period equals the amount that the resource owner would
pay in that period for an extra unit of resource in the ground. Rent
equals the shadow price for this hypothetical transaction, which equals
the Lagrange multiplier associated with the resource constraint in this
period.
It is optimal to exhaust cheaper deposits before beginning to use more
expensive deposits. We con…rmed this result for the case of mines with
di¤erent constant extraction costs.
5.8
Terms, study questions, and exercises
Terms and concepts
Euler equation, Hotelling rule, asset price, capital gains, dividend, Lagrange
multiplier, shadow value, in situ.
Study questions
1. (a) For a general inverse demand function p (y) and the parametric cost
function in equation 4.1, write down the competitive …rm’s objective
and constraints. (b) Write down and interpret the Euler equation for
this problem. (c) Without actually performing calculations, describe
the steps of the perturbation method used to obtain this necessary
condition.
2. (a) What is the de…nition of “rent”in the renewable resource problem
for the competitive …rm? (b) Use this de…nition to rewrite the Euler
equation for the problem described in question #1. (c) Write down
the relation between rent in period t and in period t + j (j 1), and
interpret this equation. In particular explain the di¤erence between the
case where extraction costs depend on the stock and where extraction
cost does not depend on the stock.
96
CHAPTER 5. THE HOTELLING MODEL
Exercises
1. Write the Euler equation for a monopoly facing the demand function
p = 20 7y with the cost function c (x; y) and discount factor .
2. Con…rm, using a proof by induction, that if extraction costs do not
depend on stocks, the present value of rent is the same in any two
periods where extraction is positive. (The Euler equation only applies
if extraction is positive.) This proof requires that you demonstrate
that for any integer j = 0; 1; 2; 3; 4::: Rt = j Rt+j is true. In case this
is your …rst inductive proof, here is how to proceed. Step 1. First,
show that the claim is true for j = 0. (This is a one-liner.) Step
2. Now consider the hypothesis that the present value of rent is the
same in periods t and t + j, for some integer j (as distinct from for
all integers). That is, your hypothesis is that Rt = j Rt+j for some
integer j. Using the Euler equation, show that if this hypothesis is
true, then Rt = j+1 Rt+j for that particular integer j These two steps
complete the inductive proof. The idea is that Step 1 shows that the
claim (Rt = j Rt+j ) is true for j = 0. Step 2 shows that if the claim
is true for j = 0 (which we know is true), then it is also true for j = 1.
Invoking step 2 again, it follows if the claim is true for j = 1 (as indeed
it is) then it is also true for j = 2..... and so on.
3. Fill in the missing algebraic steps that lead from equation 5.5 to 5.10
when extraction costs are constant.
4. Begin with the Hotelling rule, equation 5.5.
condition implies
Rt+1 =
Rt+2
(a) Explain why this
@c (xt+2 ; yt+2 )
@xt+2
(This is a one-liner.) (b) Use this fact to write Rt as a function of Rt+2
and @c(x;y)
in periods t + 1 and t + 2. (c) Repeat this step to write Rt
@x
as a function of Rt+3 and @c(x;y)
in periods t + 1, t + 2, and t + 3. (d)
@x
Generalize your formula to write Rt as a function of Rt+j and @c(x;y)
in
@x
periods t + 1, t + 2, ....to t + j for integers j 1. (To avoid having to
t ;yt )
carry around a lot of unnecessary notation, de…ne ht = @c(x
. With
@xt
this de…nition, write Rt as a function of Rt+j and ht in periods t + 1,
t + 2, ....to t + j (e) Provide an economic explanation for this formula.
5.8. TERMS, STUDY QUESTIONS, AND EXERCISES
97
5. Suppose that extraction costs are constant, c (x; y) = Cy. (a) How
does an increase in C change the equilibrium rate of change of price
in a competitive equilibrium? [This is a one-liner if you examine
the text carefully.] (b) How does an increase in C a¤ect the initial
equilibrium price, p0 , in a competitive equilibrium. [This question calls
for intelligent speculation, not algebra.]
Sources
Hotelling (1931) is the classic paper in the economics of nonrenewable resources.
Solow (1974) and Gaudet (2007) provide time-lapse views of the role of
the Hotelling model in resource economics.
Dasgupta and Heal (1979), Fisher (1981) and Conrad (2010) are three
excellent graduate texts covering the nonrenewable resource model. Berck
and Helfand (xx), Hartwick and Olewiler (1986), and Tietenberg (2006),
provide undergraduate-level treatment of this material.
98
CHAPTER 5. THE HOTELLING MODEL
Chapter 6
Empirics and Hotelling
Objectives
Understand what it means to test the Hotelling model, and the practical
di¢ culties of performing such a test.
Information and skills
Summarize the chief implications of the Hotelling model.
Have an overview of historical price patterns for several natural resources.
Understand some of the ways that people test the Hotelling model, why
data limitations complicate the e¤ort.
A theory, like a map, involves a trade-o¤ between realism and tractability.
A map of the world that consists of a circle is too abstract to be of any use. A
map of the world containing all of the details of the world is equally useless.1
As with maps, economic models should be judged on how useful they are in
helping us to understand the world, not how faithfully and completely they
describe it. The Hotelling model, the basis of the economics of nonrenewable
resources, rests on a fact and an assumption. The fact is that resource stocks
1
Jorge Luis Borges’ (very) short story “On Exactitude in Science" tells the tale of
an empire in which cartography becomes so precise that the empire creates a map of its
territory on a 1–1 scale. Later generations decided that this achievement was useless as
a map, although beggars use scraps of it to cover themselves.
99
100
CHAPTER 6. EMPIRICS AND HOTELLING
are …nite, so extraction costs must eventually rise. The assumption is that
forward looking resource owners attempt to maximize the present discounted
stream of pro…ts.
If average cost is constant, the model predicts that price minus this constant rises at the rate of interest. Price data contradict this version of the
Hotelling theory. Data limitations make it di¢ cult to test more sophisticated versions of the theory, which describe the evolution of rent (= price
minus marginal cost). We can observe price, but it is hard to estimate
marginal cost, and therefore di¢ cult to assess whether rent actually evolves
in the manner the theory predicts. We are therefore unable to empirically
validate the foundation for the …eld of nonrenewable resource economics.
Why does it matter whether the theory is empirically (approximately)
“true”? Concern about long run sustainability is an important motivation
for studying resource economics. Resource-pessimists worry that we will
run out of essential resources. Resource-optimists think that prices respond
to and impending shortages, causing markets to create alternatives. This
optimism rests on the belief that market outcomes are determined by rational
pro…t maximizing competitive agents, i.e. that the Hotelling model describes
resource markets.
If the theory is correct, then the First Fundamental Theorem of Welfare
Economics (Chapter 2.6) implies that the competitive equilibrium is e¢ cient.
In this case, the fact that nonrenewable natural resources are …nite does not,
in itself, create a basis for government intervention. There may, of course, be
distinct motivations for intervention, e.g. concerns about equity or market
failures, just as arise in many other markets. If, however, nonrenewable
resource markets are broadly inconsistent with even sophisticated versions of
the Hotelling model, then outcomes in these markets are not e¢ cient, and
there is no basis for ascribing any particular welfare properties to them. In
particular, the theory provides no basis for resource-optimism.
In addition, resource economists use the Hotelling model to think through
the consequences of policy changes. For example, it is widely accepted that
extracting and using fossil fuels contribute to climate change, a type of market failure. Taxing these fuels or subsidizing alternatives alters market
outcomes, potentially changing the market failure. We cannot conduct actual policy experiments, similar to the types of experiments that laboratory
scientists perform. We do not have one group of economies that serve as
6.1. HOTELLING AND PRICES
101
the “control group”and another that serve as the “treated group”.2 Therefore, economists use mathematical models, an alternative to bona…de (and
infeasible) experiments to assess the likely e¤ect of policy. For applications
involving fossil fuels, the Hotelling model is key. By comparing a model
outcome in the absence of a policy, and second outcome under a particular
policy, we have at least some basis for evaluating that policy. Chapter 9
illustrates this procedure. If the underlying model is fundamentally wrong,
then this kind of experiment is not informative. Therefore, the validity of
the Hotelling theory is important for practical, not just for academic reasons.
Given the uncertain empirical foundation of the Hotelling model, why do
resource economists rely so heavily on it? One answer is that we do not
have a better alternative.3 Also, the pro…t maximizing assumption of the
Hotelling model seems at least as true for natural resources as for other types
of capital. If there were a more valuable use of the asset, markets should be
able to identify it at least as well as a planner could. It would be absurd
to take literally the deterministic Hotelling model, in which agents perfectly
forecast future prices. But it is unlikely that resource owners –or owners of
other types of capital –ignore the future in deciding how to use their asset.
Investors make mistakes, but investors who systematically make mistakes are
eventually culled from the herd.
6.1
Hotelling and prices
Objectives and skills
Have an overview of the time trajectories of prices for major resources.
The simplest version of the Hotelling model, with zero extraction costs,
implies that price rises at the rate of interest. With constant average extraction cost, C > 0, prices rise more slowly than the rate of interest.
2
To test the e¢ cacy of a new drug, researchers compare the outcome of people who
receive the drug (the treated group) with those who receive a placebo (the control group).
If we could conduct experiments with economies, the economies subject to the policy of
interest would be in the treated group, and those without the policy would be in the
control group. Of course this kind of experimentation is not feasible.
3
Alternatives matter. How can you tell if someone is an economist? Ask them how
their husband/wife/partner is. An economist will answer “Compared to what?”
102
CHAPTER 6. EMPIRICS AND HOTELLING
250
200
150
100
50
0
00
10 20 30 40 50 60 70 80
ALUMINUM
ALUMINUM_T
80
70
60
50
40
30
20
10
1880 1900 1920 1940 1960 1980
COPPER
COPPER_T
1880 1900 1920 1940 1960 1980
COAL
COAL_T
100
25
200
180
160
140
120
100
80
60
80
20
60
15
40
10
20
5
1880
1880 1900 1920 1940 1960 1980
LEAD
LEAD_T
1900 1920 1940 1960
IRON
IRON_T
200
160
120
80
40
14
12
10
8
6
4
2
20
30
40 50 60 70 80
NICKEL
NICKEL_T
90
16
14
12
10
8
6
4
2
0
0
20 25 30 35 40 45 50 55 60 65 70 75 80 85 90
GAS
GAS_T
1200
1000
800
600
400
200
1880 1900 1920 1940 1960 1980
PETROLEUM
PETROLEUM_T
0
1880 1900 1920 1940 1960 1980
SILVER
SILVER_T
Figure 6.1: Real commodity prices. (Lee, List, and Strazicich, 2005)
Figure 6.1 shows the pro…les of real prices (nominal prices adjusted for
in‡ation) for nine commodities. The dotted lines show the time trends …tted
to this price data. The points of discontinuity in the dotted lines capture
abrupt changes in the price trajectory. For most of these commodities there
are long periods during which the price falls, and at least one abrupt change
the price trajectory.
We need only price data to test the Hotelling model under the assumption
of constant costs. In light of Figure 6.1, it is not surprising that most research
…nds that this version of the Hotelling model is not consistent with observed
prices. We therefore consider versions of the model with non-constant costs,
and then we consider more fundamental changes to the model or the testing
procedure.
6.2
Non-constant costs
Objectives and skills
Understand why it is di¢ cult to test the Hotelling model with nonconstant marginal extraction costs.
6.3. EXTENDING THE MODEL
103
Lack of data makes it di¢ cult to test the Hotelling model with nonconstant costs. The case where costs depend on extraction level, but not on
remaining stocks, illustrates the problem. Here, the Hotelling rule is
pt
dc (yt )
=
dy
pt+1
dc (yt+1 )
dy
:
(6.1)
The problem is that we do not observe marginal costs. If we assume that
costs equal a + h2 y y, then marginal cost equals a + hy. Given data on
prices and sales, we can substitute a + hy for marginal costs into equation
6.1 and estimate the parameters ; a; h. If, for example, the estimate of is
implausible, either unreasonably small or large, the researcher concludes that
the model does not …t the data. This procedure involves a joint hypothesis:
the hypothesis that the no-arbitrage condition, equation 6.1, is a reasonable
description of behavior, and the hypothesis that the cost function a + h2 y y
is a reasonable description of the actual cost function. If the statistical tests
reject our joint hypothesis, we do not know whether the rejection was due to
the failure of one or both parts of the hypothesis.
If we had good data on costs (in addition to prices and quantities), then
we could estimate a ‡exible functional form (e.g. a higher order polynomial)
for the cost function, and have a reasonable level of con…dence in the resulting estimate of marginal cost. In that case, we would be closer to testing
the single hypothesis that we actually are interested in, the one involving
behavior. We rarely have good cost data.
Low quality cost data limits many …elds of empirical economics, not just
resource economics. A common procedure in other …elds uses a …rm’s optimality condition, together with information about factor prices, to estimate marginal costs. Here, however, the empirical objective is to determine
whether the optimality condition, the Hotelling rule, describes …rms’behavior. It is not possible to both assume that the Hotelling rule holds and also
to test whether it holds.
6.3
Extending the model
Objectives and skills
Recognize that extensions of the model lead to a better …t with data,
but increased di¢ culty of empirical testing.
104
CHAPTER 6. EMPIRICS AND HOTELLING
Understand the use of proxies in estimation.
The theory presented in Chapter 5 omits many features of the real world,
including: (i) the discovery of new stocks of the resource; (ii) unanticipated
changes in demand due, for example, to changing macro-economic conditions
or the discovery of alternatives to the resource; (iii) changes in extraction
costs due to changes in technology or regulation, (iv) and general uncertainty.
The literature on resource economics includes all of these features. We
consider the empirical implications of some of these extensions.
Stock changes We took the level of the resource stock in the initial
period as …xed, and assumed that the stock falls over time, with extraction.
In reality, …rms invest in …nding and developing new resource stocks, with
uncertain success. The level of risk and …rms’attitude toward risk in‡uence
their decisions about extracting known reserves and about investing in the
search to …nd undiscovered reserves. The theory presented above ignores the
discovery and development of new reserves, and all risk.
Some extensions of the Hotelling model recognize that the exploration
decision is endogenous. A simpler model takes the exploration decision as
given, and treats the timing and the magnitude of new reserves as random
variables. Rational …rms would factor this randomness into their extraction
decisions. Resource owners understand that large new discoveries create
competition for previously existing deposits, lowering the value of those deposits. The new discovery therefore causes the rent on existing deposits
to fall; recall that rent equals the “shadow price” of the stock, de…ned as
the amount that a …rm would be willing to pay for a marginal unit of the
stock. Rent equals price minus marginal cost. Because the discovery has
no direct e¤ect on extraction costs of previously existing deposits, the fall in
rent implies a fall in price. Thus, in a model with random discoveries, price
rises between discoveries, but falls at the time of a large new discovery. In
this scenario, the price path is saw-toothed, rising for a time and then falling
at the times of new discoveries.
6.3. EXTENDING THE MODEL
105
Box 6.1 Royal Dutch Shell and the Kulluck. Royal Dutch Shell’s
experience with Arctic drilling illustrates the complexity of oil exploration. In 2005 Shell announced that it had previously overstated its
proven reserves by more than 20%, resulting in an overnight drop of
10% in the value of company. (“Proven reserves”are stocks that are
highly likely to exist.) In an e¤ort to increase proven reserves, Shell
ramped up its Arctic exploration, buying drilling leases and drilling
equipment, including a large rig named the Kulluck. In 2012 it received permission to begin exploration and towed the Kulluck into
position. By the end of the year and a $6 billion investment, the Kulluck was destroyed in a storm, without having succeeded in drilling a
well. Shell paused its exploration e¤orts in 2013 was set to continue
them in 2014. In August 2014 oil was $100/barrel, but it had fallen
to $55/barrel by the end of the year. “Unconventional” sources of
oil, such as in the Arctic, require a price of about $70/barrel to be
economical.
Changes in demand Unanticipated changes in demand that are expected to persist for a substantial period of time can also change the price
path. For example, the Great Recession beginning in 2008 saw a reduction
in aggregate demand, including a reduction in demand for many resources.
Unexpectedly strong growth in China from 1990 – 2010 increased demand
for natural resources. The discovery of alternatives, e.g. the 18th century
replacement of petroleum for whale oil, may also change the demand for a
resource.
If a change in the economic environment causes …rms to expect that future
demand will be weaker than they had previously believed, then they revise
downward their estimate of the value of their resource stock. This downward
revision in rent requires a reduction in price, just as occurs following discovery
of a large new deposit. Thus, over a period when …rms are revising downward
their projections of future demand, and thus revising downward their belief
about the current value of a marginal unit of the stock, the equilibrium price
rises more slowly than the theory predicts, and might even be falling. The
deterministic (perfect information, no surprises) model ignores unanticipated
changes in demand, although those random events are an important feature
of the real world and could explain observed price falls.
106
CHAPTER 6. EMPIRICS AND HOTELLING
Cost changes The simple Hotelling model ignores changes in costs,
apart from those associated with changes in stock or extraction rates. In
fact, the extraction cost function might shift up or down over time. Downward shifts are associated with price rises that are lower than the standard
model predicts, or even negative. Upward shifts lead to faster price rises
than the model predicts. Technological advances over the last 20 years
made horizontal drilling both practical and relatively inexpensive, making it
cheaper to extract from current deposits, and possible to develop resource
deposits that were previously inaccessible. Other changes, such as stricter
environmental or labor rules, increase extraction costs. For example, in projecting extraction costs for new reserves, Royal Dutch Shell in 2014 included
a (still nonexistent) carbon tax of $40/Mt C02 , or approximately $17/barrel
(using the estimate 0.43 metric tons CO2 /barrel). A higher tax corresponds
to a higher cost.
Consider a model in which average costs are constant with respect to
both extraction rates and stock levels, but falling over time; we replace the
constant C with
a
;
(6.2)
C (t) = C0 +
1 + ft
where C0 , a, and f are positive. The Euler equation is
pt
C0 +
a
1 + ft
=
pt+1
C0 +
a
1 + f (t + 1)
:
(6.3)
Figure 6.2 shows the graphs of price (solid), rent (dashed), and marginal
cost (dotted) under the cost function in equation 6.2.4 Falling costs put
downward pressure on the equilibrium price, just as with a standard good.
Discounting the future puts upward pressure on the price, just as in the
nonrenewable resource model with constant costs, C. Initially, the cost
e¤ect is more powerful, so the equilibrium price falls. However, the cost
decreases diminish over time, because costs never fall below C0 . Eventually,
the e¤ect of discounting becomes more powerful, and the equilibrium price
rises, just as in the model with constant C. Rent (price minus marginal
4
This …gure corresponds to the continuous time version of the discrete time Euler
equation 6.3. We use the discrete time model in order to derive equations and provide
intuition.
The graphs of equilibrium outcomes of the discrete time model are step
functions. The continuous time analogs are easier to graph, and because they are smooth,
also easier to interpret. Figure 6.2 uses a = 30, Co = 10, f = 0:5, and the discount rate
r = 0:03.
6.3. EXTENDING THE MODEL
$
107
60
50
40
30
20
10
0
2
4
6
8
10
12
14
16
18
20
time
Figure 6.2: Equilibrium price (solid), rent (dashed) and marginal cost (dotted).
cost) rises at the constant rate, r, and marginal costs steadily decline to
C0 . Adding frequent small shocks and occasional large discoveries causes
the graph in Figure 6.2 to become saw-toothed and bumpy, making it more
closely resemble the graphs of actual time series shown in Figure 6.1.
General uncertainty A rich literature studies the role of uncertainty
about new discoveries, changes in demand and extraction costs, and other
features of nonrenewable resource markets. Uncertainty alters the expected
time path of rents, and thus of prices, even during periods when a large shock
does not occur.
Uncertainty complicates the already formidable data problem of testing
the deterministic Hotelling model, so researchers use proxies, de…ned as variables that can be observed and which are closely related to the unobserved
variable of interest: resource rent. For example, in forestry the “stumpage
price”is the price paid to a landowner for the right to harvest timber. Oldgrowth timber is a nonrenewable resource, and the stumpage price of this
timber is a good proxy for rent. Researchers using this proxy …nd moderate
support for the Hotelling model.
The stock price of a mining company is a proxy for the rent associated
with the resource that the company mines. Both the observable stock price
and the unobservable resource rent are forward-looking variables that re‡ect
the expected value of the stream of future pro…ts from owning one more unit
108
CHAPTER 6. EMPIRICS AND HOTELLING
of an asset. For rent, the asset consists of the resource. For the company
stock, the asset is a composite of the resource, the mining equipment that
the company owns, and many other factors, including intangibles such as
reputation. These two assets are not exactly the same, so the company stock
price is not a perfect measure of the resource rent. However, because the
resource is a large part of the company assets, the two are closely related.
The 2005 fall in Shell’s stock price following the downward revision in its
proven reserves illustrates this relation. (Box 6.1)
Chapter 5.4 summarizes a model of the equilibrium change in a company’s
stock price under certainty. Under uncertainty, the equilibrium expected
change in the stock price depends on the correlation between the price of
the company and the price of a diversi…ed portfolio of stocks (a “market
portfolio”). If the two are negatively correlated, then the company tends
to do well when the market does poorly. In this case, owning stock in the
company provides a hedge against market risk. A hedge provides an o¤set
to the risk associated with a particular activity, here the investment in the
market. Owning stock in this company is valuable partly because it reduces
an investor’s risk. This reduction in risk is a bene…t of owning the stock,
and in that respect, is analogous to the dividend discussed in Chapter 5.4.
This “pseudo-dividend”, much like the real dividend, lowers the expected
capital gains needed to make investors willing to hold the mining asset. (Box
6.2)
Researchers …nd that the stock price of companies owning copper mines
is negatively correlated with the return on a market portfolio. The theory
sketched in the previous paragraph implies that this negative correlation
lowers the equilibrium rate of increase of the mining company’s stock price.
If the observed stock price is a good proxy for the unobserved rent, then
evidence that one variable is consistent with theory provides some evidence
that the other is as well. Using data on copper mining companies, this
research “fails to reject”the Hotelling model.5
In general, the market return might have positive, negative, or zero correlation with resource prices. Therefore, in a Hotelling setting, uncertainty
might either increase or decrease the expected change in prices. Resource
5
If a statistical test …nds that a hypothesis is consistent with observations, we say
that the test “fails to reject” the hypothesis. We do not say that the test “accepts” the
hypothesis, because there are usually many other hypotheses that are also consistent with
the observed outcome. It does not make sense to accept all of these mutually exclusive
hypotheses.
6.3. EXTENDING THE MODEL
109
prices respond to both supply and demand shocks. A positive demand shock
(e.g. unexpectedly high consumer demand) and a negative supply shock (e.g.
a disruption in supply due to war or political turmoil) both increase the resource price. These two types of shocks may be associated with di¤erent
e¤ects on the market return; the supply shock is likely associated with a low
return on the market portfolio, and the demand shock with a high return.
In this case, resource prices tend to be positively correlated with the market
return if the demand shocks overwhelm the supply shocks, negatively correlated in the opposite case, and uncorrelated if the two types of shocks are
approximately equally important. Empirical evidence …nds that oil prices
are uncorrelated with the market return, even though oil prices are negatively correlated with future economic growth. Large increases in oil prices
preceded most of the post-World War II economic downturns.
Box 6.2 The Capital Asset Pricing Model. Consider a world in which
people can buy a risk-free asset (e.g. government bonds) that pays a
return rf . They can also invest in the stock market that has a risky
return, the random variable r~m with expectation E r~m = rm . An
investor’s opportunity cost of investing in the stock market is rf (the
riskless rate), so their expected reward (the “market premium”) from
investing in the stock market is rm rf . The investor can also buy a
company that owns and mines a nonrenewable resource, e.g. copper,
that has a risky return of r~c and an expected return rc . The “beta”
of the mining asset is de…ned as
=
covariance (~
rm ; r~c )
:
variance (~
rm )
Under a number of assumptions, including that all investors are rational and risk averse and have the same information and no borrowing or lending constraints, the Capital Asset Pricing Model (CAPM)
shows that the equilibrium expected return to the copper asset must
be rc = rf + (rm rf ). This equation states that if the copper
asset is negatively correlated with the market (the beta is negative),
then investors are willing to purchase the copper asset at an expected
return lower than the expected market return. They accept this expected loss because the copper asset provides a hedge against market
uncertainty.
110
6.4
CHAPTER 6. EMPIRICS AND HOTELLING
Summary
Observed prices are not consistent with the deterministic Hotelling model
under constant average costs. Adding real-world features, including nonconstant marginal extraction costs, exploration, random shocks to stock,
demand or cost, or systematic time-varying changes to costs, can generate
simulated price paths that resemble observed price paths. Lack of data
makes it di¢ cult to directly test models that incorporate these kinds of realistic features. Because we cannot observe rents, they have to be estimated
using estimates of marginal costs. Any empirical test of the Hotelling model
using such constructed data thus involves the test of joint hypotheses. One
part of the hypothesis concerns the Hotelling model, and the other part involves the estimates of marginal costs. A few studies use proxies, such as
stumpage fees for timber or the stock price of companies that mine resources,
to indirectly test the Hotelling model. Those indirect tests provide modest
support for the Hotelling model.
Occam’s razor describes a preference for the simpler of any two models
that do equally well in explaining observations. Arguably, a model based on a
succession of static equilibria with exogenously changing supply and demand
functions is simpler than the resource model based on dynamic optimization.
Consider a model of a standard good (not a nonrenewable resource), for
which costs initially fall faster than demand increases; subsequently, demand
rises faster than costs fall. In addition, the supply and demand functions are
both subject to random shocks. That model would also produce a bumpy
U-shaped trajectory of prices, simply by equating supply and demand in each
period. Thus, we could explain observed price patterns for nonrenewable
resources using a model that has nothing to do with nonrenewable resources.
However, the static model assumes that …rms simply ignore the future.
The deterministic Hotelling model, in which resource owners can exactly predict future prices, is not intended to be taken literally, but provides a starting
point for studying forward-looking behavior. The …nite supply and stockdependent extraction costs are the distinguishing features of nonrenewable
resources, and are the source of rent. For some resources (perhaps coal), rent
arising from scarcity is not important, and rent arising from stock-related extraction costs may or may not be important. For other commodities, such as
oil, rent is substantial. Low cost oil can be extracted for less than $20 a barrel. When the price is multiples of that, as it has been for long stretches of
time, the di¤erence can be attributed to rent (and sometimes to the exercise
6.5. TERMS, STUDY QUESTIONS AND EXERCISES
111
of market power). The theory of nonrenewable resources studied here applies
only to commodities where rent is a major component of market price.
Many factors apart from the …nite supply and stock-dependent costs
(their distinguishing features) are important in determining equilibria in nonrenewable markets. Non-resource-based commodities share most of those
factors, including changes that alter costs or change demand. Often those
other factors dominate the distinguishing features of nonrenewable resources,
leading to outcomes that are not consistent with the Hotelling theory. What
is the point in developing a theory that focuses on the “distinguishing features” of the nonrenewable resources, when for long periods of time other
factors are more important in determining market outcomes? The answer
is that we want to strike a balance between a “theory of everything” and a
description of the particular.
A theory of everything includes all of the aspects of the world likely to
be important in (at least) a large class of markets, including technological
changes, unexpected demand shifts, risk. A theory that includes all of
these features would be too complicated to produce insights, or to use for
policy analysis. In contrast, a description of the particular would choose
a particular nonrenewable resource, or even a particular resource during a
particular period of history or a particular geographic area, and study that
market in detail. That kind of description would provide limited insight
about other resources.
6.5
Terms, study questions and exercises
Terms and concepts
Real versus nominal prices, joint hypothesis, control group, treated group,
maintained hypothesis, proven reserves, proxies, hedge, market portfolio,
market return, CAPM, stumpage, Occam’s razor
Study questions
1. (a) Describe the simplest version of the Hotelling model’s predictions
about change in prices over time. (b) Describe important features
of price series for actual nonrenewable resources, and contrast these
to the predictions of the simplest Hotelling model. (c) What does it
mean to empirically “test” the Hotelling model? Discuss some of the
112
CHAPTER 6. EMPIRICS AND HOTELLING
di¢ culties in conducting such a test. (Your answer should develop the
idea that there are many versions of the Hotelling model, not just the
simplest one. Your answer should also discuss the fact that a test of
the Hotelling model is almost certainly a test of a “joint hypothesis”,
and explain why this matters.)
Exercises
1. (a) Beginning with the equilibrium condition (or de…nition),
p (y)
@c (x; y)
= R;
@y
what is the relation between a change in R and a change in y? Justify
your answer and provide the economic explanation. (b) Based on your
answer to part (a), what is the relation between a change in R and a
change in p?
2. Derive the Euler equation 6.3.
3. Suppose that average extraction cost is constant, C, and the period t
inverse demand function is
pt =
a
a0 + a1 t
yt ;
where a0 ; a1 ; and a are positive known parameters.
(This type of
model is called “non-stationary” because there is an exogenous timedependent change –here, a change in the demand function.) (a) State
in words what this demand function states in symbols. (b) Consider
a competitive …rm facing this demand function. Refer to Chapter
5 to write the Euler equation for this model (no derivation needed).
(c) Explain what this equilibrium condition implies about the rate of
change in prices. Provide an economic explanation. (d) This problem
and the previous one show that nonstationary costs and nonstationary
demand have di¤erent e¤ects on the appearance of the Euler equation.
Identify this di¤erence, and provide an economic explanation for it. (e)
The text discusses the e¤ect, on the evolution of rent, of unanticipated
changes in demand. Parts (b) and (c) of this question ask you to
consider the e¤ect of anticipated changes in demand. Summarize in a
6.5. TERMS, STUDY QUESTIONS AND EXERCISES
113
sentence of two the di¤erence in the e¤ect, on the evolution of rent, of
unanticipated versus anticipated changes in demand. (f) Provide an
economic explanation for this di¤erence.
4. Using the parametric example in equation 4.1, discuss the equilibrium
price e¤ect of unanticipated cost reductions. To answer this question,
you have to consider the various ways in which costs might fall; review
your answer to Exercise 2 in Chapter 4.
5. Suppose that the return from investing in the market is a random
variable r~m with expectation E r~m = rm and variance 2m ; the return
from investing in a copper mining company is a random variable r~c
and an expected return rc and variance 2c . The correlation coe¢ cient
between the two assets is < 0. A person buys one unit of stock in
the market and a units in the copper company. The return on this
portfolio is the random variable rm + arc . (a) What is the mean and
variance of the return on this portfolio? (b) What is the value of a that
drives the variance on the portfolio to zero? [Hint: look up the formula
for the mean and variance of the weighted sum of random variables, and
the formula that shows the relation between the correlation coe¢ cient
and the covariance of two random variables.)
Sources
This chapter relies heavily on the surveys by Livernois (2009) and Slade
and Thille (2009). Kronenberg (2008) and Krautkraemer (1998) also provide
useful surveys.
Heal and Barrow (1980), Berck and Roberts (1996), Pindyck (1999), and
Lee, List and Strazicich (2006) test the Hotelling model using price data.
Miller and Upton (1985) for oil and gas, Slade and Thille (1997) for
copper, and Livernois, Thille, and Zhang (2006) for old-growth timber use
proxies to test the Hotelling model.
Halvorsen and Smith (1991) estimate the interest rate under the hypothesis that the Hotelling condition holds.
Pindyck (1980) pioneered the large literature on uncertainty and nonrenewable resources.
Farzin (1995) discusses the impact of technological change on resource
scarcity.
114
CHAPTER 6. EMPIRICS AND HOTELLING
Hamilton (2011) studies the relation between oil prices and economic
activity.
Kilian (2009) discusses the distinction between oil demand and supply
shocks.
Fama and French (2004) explain the Capital Asset Pricing Model and
discuss its empirical signi…cance.
Reuters (2014) reports Shell’s assumption of carbon tax in projecting
future extraction costs.
Funk (2014) describes Shell’s experience with the Kulluck.
Chapter 7
Completing the solution
Objectives
Derive the optimality condition for the last period during which extraction is positive, and use this condition to determine the complete
solution to the resource extraction problem.
Information and skills
Use the logic of intertemporal arbitrage to determine the transversality
condition that holds during the last period of extraction.
Use this condition, together with the Euler equation and the resource
stock constraint, to determine the number of periods (T + 1) during
which extraction is positive.
Use T , the Euler equation, and the stock constraint, to determine the
level of extraction and the price in each period (the “full solution”).
Obtain the transversality condition for the case where the resource is
economically but not physically exhausted.
The analysis in Chapter 5, using only the Euler equation, sheds light on
some of the qualitative properties of a nonrenewable resource market. The
reader still does not have enough information to …nd the level of extraction
and the corresponding price in each period. Up to this point, we took T ,
the …nal date of extraction, as given, without explaining how that value is
115
116
CHAPTER 7. COMPLETING THE SOLUTION
determined, or how to use it to compute the equilibrium price and extraction
trajectories.
Leaving things at this stage is unsatisfactory for two reasons. First, the
model remains a black box. Readers cannot fully understand the equilibrium
without knowing the missing information. Working through an example in
detail removes some of the mysteries. Second, there are occasions when we
actually do want to be able to calculate the full equilibrium. We know that
the competitive equilibrium maximizes the present discounted value of the
sum of producer and consumer welfare. If we want to know how a resource
should be consumed to maximize this payo¤, we have to …nd the “complete
solution”to the problem. This chapter …lls in the missing pieces. We derive
and discuss most of the new results using a numerical example.
In the static setting, competitive …rms base their sales decision in a period on the price in only that period. In that setting we can calculate the
equilibrium in a period using only the single period demand and the cost
function, together with the …rst order condition (Section 2.4).
In the resource setting, in contrast, the equilibrium price and quantity
in a particular period depend on the equilibrium prices in all subsequent
periods. In making their current sales decision, a …rm compares the bene…t
of selling the marginal unit in that period or delaying extraction and selling it
in a subsequent period. That comparison requires that …rms know not only
the current market price, but also know (or form beliefs about) subsequent
prices. In the resource setting (but not in the static setting), agents are
forward looking. The Euler equation shows the equilibrium relation between
prices in adjacent periods. In order to use this equation to determine the
trajectory of prices, we need an “anchor”, the price in a particular period.
The transversality condition provides that anchor.
Box 7.1 Crossing the ravine Crossing a ravine provides a metaphor
to the nonrenewable resource problem. Mathematics corresponds to
the rope bridge across this ravine. The price and quantity in each
period correspond to the rungs of the bridge, and the Euler equation
corresponds to the rope that connects one rung to the next, thereby
indirectly connecting the …rst and the last rung. The initial condition,
the beginning stock, anchors this bridge on one side of the ravine, and
the transversality condition is the anchor on the other side. Without
all of these parts, the rope bridge is useless.
7.1. THE CHANGE IN PRICE
7.1
117
The change in price
Objectives and skills
Use the Euler equation to …nd a relation between price in an arbitrary
period, and price in period T .
We use an example with constant average extraction costs, C, and discount factor . The constraints and optimality condition are:
constraints: xt+1 = xt
yt , xt
Euler equation: pt
0, x0 given
0, yt
C = [pt+1
(7.1)
C]
The full solution to the …rm’s optimization problem is a sequence of extraction levels, y0 ; y1 ; y2 :::yT and corresponding prices. We need the value of
T to obtain this sequence. Recall our convention that the initial period is
period 0: the number of periods during which extraction is positive is T + 1,
not T .
The Euler equation implies the following relation between the price in
period T and T n (n periods from the date of …nal extraction):
pT
n
=
n
(pT
(7.2)
C) + C:
Box 7.1 Derivation of equation 7.2 We can write the Euler equation
for di¤erent periods:
pT
pT
1
C = (pT
3
C = (pT
2
C)
pT
C) :::::::::pT
C = (pT
2
n
C)
1
C = (pT
C) :
n+1
Substituting the …rst of this sequence of equations into the right side
of the second equation, and adding C to both sides, gives
pT
2
= ( (pT
C) + C
C) + C = pT
Substituting this expression for pT
and adding C to both sides, gives
pT
3
= ( ( (pT
2
2
=
2
(pT
C) + C
into the third of the sequence,
C))) + C =
3
(pT
C) + C
Repeating this process n times, produces equation 7.2.
118
7.2
CHAPTER 7. COMPLETING THE SOLUTION
The transversality condition
Objectives and skills
Provide the logic and interpretation of the transversality condition.
With constant average extraction costs, it is optimal to eventually extract
all of the resource, so xT +1 = 0. This equality implies that yT = xT : in the
last period during which extraction is positive, all of the remaining resource
is extracted. Thus, we need only consider candidates for which yT = xT and
yT +1 = 0. Hereafter, we also assume that demand is linear: p = a by, with
a > C.
For times t < T , where extraction is positive at both time t and at t + 1,
it is possible to make a small increase or decrease in time t extraction, and
make an o¤setting change in the subsequent period (t + 1). In contrast, at
time T , current extraction is positive and extraction in the next period is
0: yT > 0 = yT +1 . It is possible to make a small decrease in yT and an
o¤setting increase in yT +1 , but (because negative extraction and a negative
stock are infeasible) it is not possible to make a small increase in yT and an
o¤setting decrease in yT +1 . Therefore, in order to test the candidate at time
T , we need to consider only perturbations that decrease yT . Consequently,
the optimality condition at time T , known as the transversality condition is
[pT
C]
[a
C] :
(7.3)
Inequality 7.3 has a straightforward interpretation. Under the candidate
trajectory, period T is the last date at which extraction is positive. Therefore, under this candidate, the price in period T + 1 is a b0 = a. A feasible
perturbation reduces period T extraction, moving the marginal unit to period T + 1. The cost to the …rm of this perturbation is the marginal loss
in pro…t at time T , the left side of inequality 7.3. The present value of
the increased T + 1 pro…ts is [a C]. Inequality 7.3 states that the loss
exceeds the gain. The …rm does not want to delay extracting the …nal unit:
it prefers to extract the …nal unit at time T .
Because extraction is positive at T , (yT > 0), we know that pT < a.
Combining this inequality and the transversality condition gives
a > pT
[a
C] + C:
(7.4)
7.3. THE ALGORITHM
7.3
119
The algorithm
Objectives and skills
Be able to put together the pieces (the constraints, the Euler equation,
and the transversality condition) to …nd the equilibrium price and extraction level in all periods.
We obtain T by using information at the end of the problem to obtain
information in earlier periods, i.e. by “working backwards”. We then “work
forwards” to complete the solution. Equations 7.1, 7.2, and 7.4 enable us
to …nd the complete solution. The algorithm consists of the following six
steps:
Step 1: Combine equations 7.2 and 7.4 to …nd inequalities that the
equilibrium price must satisfy:
n
(a
C) + C > pT
n
n
=
(pT
n
C) + C
([ [a
C] + C]
C) + C
or more simply
n
(a
C) + C > pT
n+1
n
(a
(7.5)
C) + C
Step 2 Use inequality 7.5 and the inverse demand function p = a
to obtain bounds on extraction in period T n:
n
(a
C) + C > a
a+
n (a
C)+C
b
byT
< yT
n+1
n
a+
(a
n+1 (a
n
C) + C )
C)+C
by
(7.6)
b
Step 3 For n = 0; 1; 2:::; sum each of the three parts of the inequality
7.6 from periods T n to T to …nd bounds on cumulative extraction over
these periods:
n
X
i=0
a+
i
(a
C) + C
b
<
n
X
i=0
yT
i
n
X
i=0
a+
i+1
(a
b
C) + C
(7.7)
120
CHAPTER 7. COMPLETING THE SOLUTION
Step 4 Compare the initial level of the stock, x0 , with the left and the
right sides of equation 7.7, for di¤erent values of n, to identify the value of
T.
An example illustrates this procedure. Let a = 10, b = 1, C = 2, and
= 0:9. Table 7.1 contains values of the Left and the Right sides of equation
7.7 for n = 0; 1; 2 and a …nal column showing the relation between the initial
stock and the equilibrium value of T .
For example, the row corresponding to n = 2 states that cumulative
extraction during the …nal three periods of the program is greater than 2.32
and less than or equal to 4.488. If 2:32 < x0
4:488, then T = 2. If
x0 > 4:488, we need more rows of the table to determine the equilibrium
value of T . If x0 2:32 then we know that T = 0 or T = 1.
n Left side Right side The value of T
0 0
0.8
T = 0 if 0 < x0 0:8
1 0.8
2.32
T = 1 if 0:8 < x0 2:32
2 2.32
4.488
T = 2 if 2:32 < x0 4:488
Table 7.1: Left and Right sides of equation 7.7, and the relation between x0
and T .
Step 5 We use the optimal value of T , the Euler Equation, and the
initial stock level, to obtain an equation for the unknown period-T level of
extraction, yT . We solve that equation to …nd yT .
For our linear example, pT = a byT and pT n = a byT n . Substituting
these relations into equation 7.2 gives
a
byT
yT
n
n
=
=
Summing from 0 to n we have
n
n
X
X
a
yT i =
i=0
i=0
n
(a
a (
n (a
( i (a
byT
byT
b
byT
b
C) + C )
C)+C)
:
C) + C)
set
= x0
(7.8)
To continue the numerical example above, suppose that x0 = 2:34. For
this value, we know from Table 7.1 that T = 2 (= n). For this value of n,
equation 7.8 simpli…es to
n
X
set
yT i = 2: 71yT + 2: 32 = 2:34 ) yT = 0:0074:
i=0
7.4. LEAVING SOME RESOURCE IN THE GROUND
121
Step 6 Using yT and the inverse demand function, we obtain pT . This
information and equation 7.2 gives us the prices and the quantities in earlier
periods, pt and yt .
For the example above, we have yT = 0:0074 so pT = 9: 992 6, rounded
to 9:993. Table 6.2 uses equation 7.2 to obtain the price and extraction in
each period: the complete solution to the model.
t
T
T 1
T 2
Table 7.2: Price and
7.4
pt
yt
9:993 0:0074
9:19 0:807
8:47 1:53
extraction in periods before T
Leaving some resource in the ground
Objectives and skills
Derive and interpret the Transversality condition when extraction costs
depend on the stock.
With stock-independent extraction costs, it is optimal to physically exhaust the resource. With stock-dependent costs, extraction might become
so expensive that it is uneconomical to physically exhaust the resource. In
this situation, the resource is “economically” (rather than physically) exhausted. Specializing the parametric cost function in equation 4.1 by setting
= = 0, and > 0, costs are c (x; y) = Cx y, and marginal cost equals
Cx . Marginal costs become large as x approaches 0. For the inverse
demand function p = 20 y, the choke prices is p = 20. As long as price
is greater than marginal extraction cost, it is optimal to continue extracting.
When extraction stops, at T + 1, price equals the choke price, 20. Because
it is optimal not to extract in period T + 1 (by de…nition of T ), we know that
1
1
C
C
. Therefore, as long as xt > 20
, it is optimal to
20 Cx , or x
20
continue extraction.
Economic
exhaustion occurs as soon as the stock reaches the critical level
1
C
. As the stock approaches this level, marginal costs approach the price,
20
so rent approaches 0. An additional unit of the resource in the ground is so
expensive to extract, that it is worth almost nothing (rent is close to 0) when
122
CHAPTER 7. COMPLETING THE SOLUTION
x is slightly above this critical level; at x less than or equal to the critical
level, the rent is 0 and the resource is economically exhausted.
7.5
Summary
In static models, a competitive …rm’s equilibrium sales depends only on the
price in that period and on the …rm’s cost function. In the resource setting,
in contrast, a …rm’s equilibrium sales depends on the current price, and on
the …rm’s beliefs about all future prices. These links arise because the …rm
has to compare the bene…t of extracting a marginal unit today and extracting
that unit in the future. In order to make that comparison, the …rm has to
form beliefs about future prices. There is no uncertainty in the models that
we use, so …rms correctly anticipate future prices.
The fact that …rms in the resource setting are forward looking complicates
the problem of solving for the equilibrium. We have to understand how the
di¤erent periods …t together. The Euler equation links price and sales in the
current period with price and sales in the next periods; the latter are linked
to the price and sales in the subsequent period. In this way, the price and
sales in a particular period are linked to the prices and sales in all subsequent
periods.
In the T period model, the number of periods during which the …rm extracts the resource, T + 1, is endogenous. Using an example, we showed how
to piece together the necessary conditions for optimality to …nd the complete
solution. The necessary conditions consist of (i) the Euler equation, (ii) a
transversality condition, and (iii) the resource constraint. The transversality condition to the problem states that at an optimum, the resource owner
prefers to extract the …nal unit in period T instead of holding on to it until
period T + 1.
If extraction costs depend on the resource stock, it might be optimal to
leave some of the resource in the ground. In that case, both the price and the
marginal extraction costs rise toward the choke price, and rent approaches 0
as the stock approaches the level below which extraction is not economical.
7.6. TERMS, STUDY QUESTIONS, AND EXERCISES
7.6
123
Terms, study questions, and exercises
Terms and concepts
Transversality condition, algorithm, economic exhaustion.
Study questions
1. Suppose that marginal extraction costs are constant, C, inverse demand
is p = 20 y, the discount factor is , and the initial stock is x0 . Your
job is to explain to a computer programer how to write an algorithm
that …nds the competitive equilibrium for this model. You need to
describe the algorithm clearly enough so that this person can code it.
(Do not do any calculations: that is the computer’s job.)
Exercises
1. Use an inductive proof to establish equation 7.2. We begin the inductive chain by showing that the equation is true for the smallest possible
value of n, here, n = 0. This demonstration requires only noting that
for n = 0, pT = 0 (pT C) + C = pT , which is obviously true. Then
show that if the equation holds for arbitrary n (our “hypothesis”) it
must also hold for “the next n”, n + 1. To accomplish this, use the
Euler equation and the “hypothesis”.
2. Using the methods and the example from Chapter 7.3 …nd the inequalities on the initial stock for which it is optimal to exhaust the resource
in 4 periods (T = 3).
3. Consider the linear example in this chapter, with all of the same parameter values: p = 10 y, C = 2, = 0:9, x0 = 2:34. Suppose that
the mine owner is a monopoly instead of a competitive …rm. Find the
complete solution for the monopoly. (Mimic the steps used for the
competitive …rm, making the one change arising from the di¤erence
between competition and monopoly.)
124
CHAPTER 7. COMPLETING THE SOLUTION
Chapter 8
Backstop technology
Objectives
Apply the methods and insights developed above to examine the e¤ect
of a backstop technology on equilibrium extraction of a nonrenewable
resource.
Information and skills
Know the meaning of and the empirical importance of backstop technologies.
Explain why the existence of the backstop a¤ects equilibrium extraction
even before the backstop is used.
Use the transversality condition to determine the e¤ect of a backstop
technology on equilibrium resource extraction.
Understand how extraction costs a¤ects the simultaneous use of the
resource and the backstop.
Understand how cumulative extraction depends on extraction costs.
A “backstop”technology provides an alternative to the nonrenewable resource. Solar and wind power, and other methods of generating electricity,
are backstop technologies for fossil fuels. We assume that the backstop is
a perfect substitute for the resource, can be produced at a constant average
cost, b, and can be supplied without limit. In contrast, the natural resource
125
126
CHAPTER 8. BACKSTOP TECHNOLOGY
has a …nite potential supply, given by the stock level. We want to know how
this backstop a¤ects resource use. As in the previous models, a representative …rm owns the resource. The backstop is available to the economy at
large; any …rm –not just resource owners –can use it.
The potential to use the backstop a¤ects the equilibrium price and resource extraction paths, even during periods when the backstop is not actually used. The equilibrium level of production and price in a period depend
on future prices. The backstop has a direct e¤ect on future prices in periods
when the backstop is used. Thus, the backstop has an indirect, but still
important, e¤ect on price and resource extraction even in periods when the
backstop is not actually used.
This insight is true generally. Anything that changes future resource
prices –or, in an uncertain world, people’s beliefs about these future prices –
changes current extraction decisions. In the static model, competitive supply
in a period depends on the price in that period. In the nonrenewable resource
setting, however, current supply depends on current and future prices.
8.1
The backstop model
Objectives and skills
Understand the components of the simplest model of a backstop technology.
As above, we use yt to denote resource use in period t. We need two
new pieces of notation. Denote zt as the amount of the alternative produced
using the backstop technology in period t, and wt = yt + zt as supply of the
resource plus the backstop good. In some settings, e.g. with fossil fuels
and renewable power, it is natural to think of the good as energy. Fossil
fuels and renewable power are physically di¤erent, but we can convert both
to common units of energy.
The assumption that the resource (e.g. oil) and the alternative produced
using the backstop technology (e.g., solar power) are perfect substitutes
means that the price in any period depends only on the sum of the resource
and the backstop good brought to market, wt . In any period where both
the resource and the backstop good are produced and consumed, the price of
the two goods must be equal. If they were imperfect substitutes, we would
have to consider the di¤erent prices for the two products.
8.1. THE BACKSTOP MODEL
127
If society is using the backstop, then price = marginal cost. Therefore,
z > 0 ) p = b:
(8.1)
The inverse demand function, p (w), equals the price consumers pay for the
quantity w.
Chapters 5 and 7 provide the tools needed to analyze the model of a
backstop technology. Chapter 5 derives the Euler equation, the condition
that relates prices and costs in any two adjacent period when extraction
is positive. Chapter 7 shows how to use the transversality condition to
…nd the …nal date of extraction, and then to use that information to …nd
the complete description of the competitive equilibrium. Here we use the
constant marginal cost model, c (x; y) = Cy, to show how the presence of a
backstop a¤ects equilibrium resource extraction. We then consider matters
under more general extraction costs.
This model of the backstop technology misses important features of realworld markets. Just as coal is not a perfect substitute for oil, a low-carbon
alternative such as solar power is not a perfect substitute for a fossil fuel.
Solar power creates an “intermittency” problem: the power goes o¤ when
the sun goes down. Oil and coal do not su¤er from this problem. Many
machines that can run on fossil fuels have not been adapted to run on solar or
wind power. However, solar-generated electricity is the same as oil-generated
electricity.
Our model uses a constant marginal cost for the backstop, but actual costs
might either rise of fall with production levels or with cumulative production.
These costs rise if there are decreasing returns to scale. For example, producing solar power uses land. The most economical locations are likely to
be used …rst, making subsequent solar farms more expensive. Economies of
scale or learning-by-doing might o¤set those cost increases. The unit cost of
producing solar power in large scale solar farms might be lower than the unit
costs of producing this power on houses and buildings, resulting in economies
of scale. The history of new technologies shows that learning-by-doing tends
to make costs fall over time, with experience. The cost of solar power is estimated to have fallen by a factor of 50 between 1976 and 2010. Our model
overlooks these features in order to illuminate some of the signi…cant features
arising from the interaction of backstop and nonrenewable resource markets.
128
8.2
CHAPTER 8. BACKSTOP TECHNOLOGY
Constant extraction costs
Objectives and skills
Identify and interpret the transversality condition in the backstop model
with constant average extraction costs.
Explain how and why the presence of the backstop changes the equilibrium price and extraction trajectories.
The simplest case involves constant average extraction costs, c (x; y) =
Cy, where C < b. If C b, the resource is worthless. We also assume that
b is less than the choke price; otherwise, the backstop is worthless. The
price-taking mine-owner wants to maximize
1
X
t
[pt yt
Cyt ] ;
t=0
and the price-taking producer of the backstop wants to maximize
1
X
t
[pt zt
bzt ] :
t=0
Both of these representative …rms take price as given, so they act as if their
individual decisions have no e¤ect on the other …rm. Therefore, we can
combine these two representative …rms into a single, larger representative
…rm that chooses both resource extraction and sale of the backstop. This
…rm wants to maximize the discounted sum of pro…ts from both the resource
and the backstop:
1
X
t
[pt
(yt + zt )
Cyt
bzt ] :
(8.2)
t=0
Apart from one minor di¤erence, and di¤erent notation, this problem is
the same as the problem discussed in Chapter 5.5, where we had mines with
di¤erent extraction costs. In that chapter, we denoted the extraction costs
as C a and C b . Here we denote them as C and b. The minor di¤erence is
that here, the “expensive mine” is actually a backstop that, by assumption
can be supplied in unlimited quantities at constant costs, b. In Chapter 5.5,
both the cheaper and the more expensive mines have …nite stocks.
8.2. CONSTANT EXTRACTION COSTS
129
Chapter 5.5 shows that it is never optimal to begin to extract from the
more expensive mine while there remains available stock in the cheaper mine.
Either the cheaper mine is exhausted before extraction begins from the more
expensive mine, or there is a single period during which extraction occurs
from both mines. In that case, in previous periods the cheaper mine satis…es the entire market, and in later periods, when the cheaper mine has
been exhausted, the more expensive mine supplies the entire market. The
same result occurs here, if we replace “the more expensive mine” with “the
backstop technology”.
The trajectory consists of periods during which the resource is used, followed by in…nitely many periods when only the backstop is used; there may
be a single “transitional” period during which both the resource and the
backstop are used. The transversality condition and the Euler equation
make it possible to …nd the equilibrium. We begin with the transversality
condition. As above, T is the last date during which extraction of the
natural resource is positive. If, in period T , the remaining resource stock
is not su¢ cient to supply the entire market, then the backstop is also used
in that period; in this case, the price in period T is b. If, in period T , the
remaining resource stock leads to a price below b, then the resource supplies
the entire market, and the backstop is not used until the next period, T + 1.
We have to consider both of these possibilities.
The transversality condition, i.e. the …rm’s optimality condition in period
T is
pT C
[b C] :
(8.3)
Compare this equation with 7.3. Here we use the fact that in period T + 1,
when the backstop is being used, the price equals the backstop cost, b (not the
choke price). The logic used to derive equation 8.3 is the same as the logic
behind equation 7.3. Appendix D provides a numerical example that uses the
Euler equation and the transversality condition, and the algorithm described
in Chapter 7.3, to …nd the equilibrium price and extraction trajectories.
A graphical example Figure 8.1 shows the graph of price and quantity
for a nonrenewable resource with demand function D = 10p 2 and the parameters b = 5, C = 2, = 0:95 and the initial stock x0 = 301. As with
Figure 6.2, Figure 8.1 shows the graphs for the continuous time analog of the
discrete time model. While the resource is being used (up to time = 150),
extraction is falling and the price is rising. With = 0:95, it is sensible to
130
CHAPTER 8. BACKSTOP TECHNOLOGY
5
price
4
3
2
quantity
1
0
0
10
20
30
40
50
60
70
80
90 100 110 120 130 140 150 160 170
time
Figure 8.1: Resource price and quantity with a backstop
think of the unit of time as approximately a year, i.e., the discount rate is
about 5%. The vertical axis represents both price and quantity, explaining
the absence of a label on that axis. Price and quantity have di¤erent units,
so the comparison of the levels of the two graphs (e.g. the crossing point)
has no interest.
The …gure illustrates the role of the backstop. The initial stock of the
resource is so high that it takes 150 years to exhaust it. The rent, at the
initial time, is p0 C = 150 (b C) 0 (because (0:95)150 is a small number).
The initial price is imperceptibly greater than C = 2. Over time, as the
exhaustion date approaches, the rent, and thus also the price, rises; quantity
falls. At the time of exhaustion, the price has reached the backstop b = 5.
Resource sales falls from a level su¢ cient to satisfy the market at a price
approximately equal to b, to zero. Immediately after this date of exhaustion,
the backstop supplies the entire market, and price remains constant at b = 5.
Thus, total supply and the price change continuously over time; the resource
supply changes continuously until the date of exhaustion, at which point the
supply falls abruptly from a positive level to 0. At that point in time, the
backstop provides the entire market, at price p = b.
8.3. MORE GENERAL COST FUNCTIONS
131
Box 8.1 Continuous versus discrete time The continuous and the
discrete time models are qualitatively similar, except for the transitional period when the backstop is …rst used. In the discrete time
setting, both the resource and the backstop may supply the market
during this transitional period. As we move toward a continuous
time model, the length of each period, including the transitional period, becomes smaller. In the continuous time limit, the resource
and the backstop are never used simultaneously. Because Figure 8.1
shows the continuous time setting, the extraction rate falls from a
positive level an instant before the transitional time T , to zero.
8.3
More general cost functions
Objectives and skills
Understand how the backstop changes the price and extraction trajectory when average extraction costs depend on either the level of
extraction or on the resource stock.
The constant average (= marginal) cost function in the previous section
is adequate for explaining the basic features of the backstop model, but it
has two empirically false implications. First, the model implies that, with
the exception of a single transitional period, the resource and the backstop
are never used in same period. When marginal costs increase with extraction rates, the resource and the backstop might be used simultaneously in
many periods. Second, the constant-cost model implies that the resource
is physically exhausted before the backstop starts being used. When costs
increase as the stock of resource falls, it may not be economical to physically exhaust the resource. In the interest of clarity, we consider these two
features separately.
8.3.1
Costs depend on extraction but not stock
Here we assume that marginal costs depend on the rate of extraction, but
not the stock: c = c (y), with c0 (y) > 0 and c00 (y) > 0; for example, c (y) =
Cy 1+ . Using the same argument that led to equation 8.2, we obtain the
objective of the representative mine owner with extraction costs c (y) and
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CHAPTER 8. BACKSTOP TECHNOLOGY
access to the backstop:
1
X
t
[pt (yt + zt )
c (yt )
bzt ] :
t=0
In this setting, there may be many periods during which both the resource
and the backstop are used (y > 0 and z > 0). Thus, it is no longer true,
in general, that society uses the backstop only when the resource is about to
be, or has been, exhausted. Once …rms begin to use the backstop, they do
not stop using it.
In the model here, where extraction costs are convex (marginal extraction
costs increase with the extraction level) there can be either one or two distinct
phases of resource extraction. In early periods, provided that the initial stock
is su¢ ciently large, it is optimal to extract only the resource. During this
phase, the Euler equation holds:
pt
c0 (yt ) = (pt+1
c0 (yt )) :
(8.4)
Along this part of the trajectory, price rises to maintain equality of the
present value of rent (price - marginal cost) across periods.
Whenever production of the backstop is positive (z > 0) the condition
that price equals marginal cost implies pt = b. Firms do not stop using
the backstop, once they have begun using it. Therefore, if zt > 0, then
pt = pt+1 = pt+2 = :::: = b. Putting these equalities together with the
Euler equation 8.4, it follows that if the …rm uses both the resource and the
backstop in adjacent periods, then
b
c0 (yt ) = [b
c0 (yt+1 )] :
(8.5)
At some time, say T1 , it becomes optimal to begin using the backstop,
while continuing to extract the resource. During this phase, the price remains
constant at p = b, the cost of the backstop, but extraction falls (so c0 falls)
in order to main equality of the present value of rent across periods. At a
later time, T2
T1 the resource is exhausted; thereafter, the sole source of
supply is the backstop. The trajectory might consist of these two stages of
resource extraction, or it might be optimal to begin using both the resource
and the backstop, and then move to using only the backstop.
Equation 8.5 is a no-arbitrage condition. Because the backstop is being
used in both periods t and t+1, the price in both periods equals b. If the …rm
8.3. MORE GENERAL COST FUNCTIONS
133
extracts one more unit of the resource in period t, it incurs the additional
cost c0 (yt ). However, because the price remains at b, the perturbation does
not alter total (resource + backstop) sales or revenue: a unit increase in
extraction requires a one unit reduction of backstop production. The net
cost reduction due to extracting an additional unit of the resource in t and
reducing backstop production by one unit is therefore b c0 (yt ).
The …rm must eventually reduce resource extraction, because the stock is
…nite. On an e¢ cient trajectory, there are no remaining arbitrage opportunities: the …rm is indi¤erent about the timing of the o¤setting reduction in
future extraction, required by the original perturbation. If the …rm reduces
next period extraction by one unit, it reduces next period cost by c0 (yt+1 ),
but incurs the additional cost due to the one unit increase in backstop production. The net increase in future cost is therefore b c0 (yt+1 ), and the
present value of that additional cost equals the right side of equation 8.5.
Along an optimal extraction path, the …rm has no desire to reallocate the
resource use: there are no opportunities for intertemporal arbitrage.
8.3.2
Stock-dependent costs
We now consider the case where the …rm’s extraction cost is c (x; y) =
C ( + x) y, i.e. average extraction cost depends on the resource stock,
but not on the rate of extraction: costs are lower when the resource stock
is higher. We assume that b < C ( + 0) , which implies that it is not
optimal to physically exhaust the resource. This assumption implies that
there is a positive level of x that solves C ( + xmin ) = b; the solution is
1
> 0.
It is never optimal to extract when the stock is
xmin = Cb
below this level, because at such levels it is cheaper to produce the good
using the backstop. If the initial stock of the resource is x0 , then the “economically viable”stock, i.e. the amount that will eventually be extracted, is
approximately x0 xmin .1
In this model, the resource is not physically exhausted, but for low stocks
1
Why “approximately”instead of “exactly”? We assumed that extraction costs depend
on the stock at the beginning of the period. Thus, for example, if the stock is slightly
above xmin , it might be optimal to extract to a level slightly below xmin . However, if the
current stock is below xmin , further extraction is uneconomical. This complication does not
arise in a continous time model, where the economically viable stock is exactly x0 xmin :
Provided that the length of each period is reasonably small, say a year or so, the discrete
time model and the continuous time model are similar.
134
CHAPTER 8. BACKSTOP TECHNOLOGY
$
7
6
extraction cost
5
4
3
backstop cost = 2
2
1
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
stock
Figure 8.2: A lower backstop cost, b, increases the minimum economically
viable stock.
it is not economically viable to extract more. Coal is one of the many
examples of such a resource. Even apart from issues related to climate
change, we will not use all of the coal on the planet, simply because at some
point extraction costs exceed the cost of an alternative. Figure 8.2 shows
a graph of stock-dependent average costs, the solid curve, and a backstop
cost of 2 at the dashed line. In this example, extraction does not occur
if the stock is below xmin
13. The cost of extracting stocks below this
level exceeds the production cost using the backstop. It is always optimal
to drive the resource at least to the critical level xmin . A trajectory that
ceases extraction when the stock is above this critical level “leaves money
on the table” (valuable resource in the ground). If the backstop cost falls
from b = 2 to b = 1, the dashed line shifts to the dotted line, leading to an
increase in the intersection, xmin , and a decrease in the economically viable
stock, x0 xmin .
As with constant average extraction cost, there is an initial phase during
which the backstop is not used, and an in…nitely long phase during which
only the backstop is used. There is at most a single period when both the
resource and the backstop are used. In the model with constant average
costs equal to C, a constant (Section 8.2) we assumed at the outset that
C < b. The corresponding assumption in the model with stock-dependent
costs C ( + x) , is that at the initial stock, x0 , C ( + x0 ) < b. If this
assumption does not hold, the resource is again worthless, because there is
an alternative with equal or lower costs.
8.4. SUMMARY
8.4
135
Summary
The presence of the backstop substitute for the resource a¤ects the entire
trajectory of the resource price, even during periods when the backstop is
not being used. This dependence re‡ects the fact that a …rm’s resource extraction decision is an investment problem. Extraction in a period depends
on the relation between price in that period and in all subsequent periods.
The backstop model drives home an important point: when the commodity is a resource, the supply of the resource brought to market in a period
depends on the entire trajectory of current and (anticipated) future prices.
In contrast, in the familiar static model, it is often reasonable to model the
supply as a function the current price.
Where average extraction cost is independent of the extraction level, we
have the extreme result that there is at most a single transitional period
during which both the resource and the backstop are used. In all other periods, only one energy source is used. If the marginal extraction cost increases
with extraction, society might use both sources of energy simultaneously over
many periods.
Variations of this model include the possibility that: the resource and the
backstop are imperfect substitutes; the marginal backstop production costs
rises with output (decreasing returns to scale) or falls with output (increasing
returns to scale); and that the cost falls with cumulative output (learning by
doing). In these cases, the backstop and the resource might also be used
simultaneously for many periods.
8.5
Terms, study questions and exercises
Terms and concepts
Backstop technology, economically viable stock, proof by contradiction, inductive proof.
Study questions
1. Explain why the presence of the backstop technology a¤ects the competitive equilibrium even in periods before the backstop is actually
used.
136
CHAPTER 8. BACKSTOP TECHNOLOGY
2. Di¤erent assumptions about extraction costs have di¤erent implications concerning the simultaneous use of the natural resource and the
backstop. (a) Suppose that average (and marginal) extraction costs
are constant. Describe the extraction pro…le of the resource, in relation
to the production pro…le of the backstop. In particular, under what if
any conditions are the two used in the same period? (b) Suppose that
extraction costs are increasing in the rate of extraction. Describe the
extraction pro…le of the resource, in relation to the production pro…le
of the backstop. In particular, under what if any conditions are the
two used in the same period? (c) Explain the source of the di¤erence
in parts (a) and (b).
Exercises
1. Use a proof by contradiction to establish equation 8.1. [Hint: To use
this type of proof, write the hypothesis that states the “opposite” of
what we want to prove, and derive a contradiction. Here, the hypothesis is “in some period when z > 0, p 6= b.” This hypothesis is the
“opposite”of equation 8.1, the statement that we want to verify. Then
show that the hypothesis must be false, by considering individually the
two possibilities,“z > 0 and p < b” and then “z > 0 and p > b”. The
proof must explain why neither of these two statements can be true in
a competitive equilibrium. Consequently, equation 8.1 must be true.]
2. In the constant cost model, explain why the resource is worthless if
C b. What is the resource rent in this case?
3. (a) In Chapter 8.3.1, why do we assume that c0 (0) < b? (b) Explain
why the assumption c0 (0) < b implies that it is optimal to exhaust the
resource.
4. Chapter 8.3.1 claims that once …rms begin using the backstop, they do
not stop using it. Verify this claim, using a proof by contradiction.
See the hint for problem 1 above.
Sources
Timilsina et al. (2011) review the evolution and the current status of solar
power.
8.5. TERMS, STUDY QUESTIONS AND EXERCISES
137
Heal (1974) is an early paper on natural resources and a backstop.
Dasgupta and Heal (1974) study the case where the backstop becomes
available at an uncertain time.
Tsur and Zemel (2003) consider R&D investment that lowers the cost of
the backstop.
138
CHAPTER 8. BACKSTOP TECHNOLOGY
Chapter 9
The “Green Paradox”
Objectives
Use the nonrenewable resource model to analyze the e¤ect of a lower
backstop price on extraction trajectories and cumulative extraction of
fossil fuels, and to consider possible climate-related consequences.
Information and skills
Understand how a lower backstop price a¤ects cumulative extraction
and/or the extraction pro…le.
Explain why both of these changes might have climate-related consequences.
Be familiar with the terms “industrial policy” and “green industrial
policy”.
Be able to synthesize this information to describe the “Green Paradox”.
The “Green Paradox”is an unconvential choice for a textbook on resource
economics. We include the material for three reasons. First, the topic is
intrinsically important. Second, it provides an example of a situation where
well-intentioned policies can back…re, a possibility that arises in many other
contexts. Chapter 10 provides a more general perspective on this issue;
the current chapter sets the stage for the general discussion by considering
a speci…c example in detail. Third, the material shows how the Hotelling
model can be used to illuminate a policy question. The policy conclusions
139
140
CHAPTER 9. THE “GREEN PARADOX”
are straightforward once readers understand the Hotelling model, but without that background the conclusions might be hard to understand or seem
counterintuitive. Economists build formal models, like the Hotelling model,
in order to better understand real world concerns. This chapter illustrates
that process.
Burning fossil fuels increases atmospheric stocks of greenhouse gasses.
Evidence from climate science shows that these higher stocks will e¤ect the
world’s climate, possibly leading to serious environmental and economic consequences. This risk has spurred interest in the development of “green”
alternative energy sources, such as solar and wind power, which emit little
or no carbon. Much of the political discussion concerns the use of public
policy to reduce the cost of providing these alternatives.
We consider a particular policy, a subsidy that encourages …rms to undertake research that decreases the cost, b, of a green backstop energy source. A
cheaper backstop provides economic bene…ts during periods when it is used.
The cheaper backstop also reduces the fossil fuel price trajectory prior to the
backstop being used, bene…tting energy consumers. However, if fossil fuel
consumption was already socially excessive, the lower price of fossil fuels can
lower aggregate welfare. The possibility that an apparently bene…cial change
(the lower-cost backstop) harms society is known as the Green Paradox.
Other forms of this paradox build on the same general idea. The key
feature in all of these cases is that resource owners anticipate a change that
directly e¤ects the market in the future. Possible changes include future
carbon taxes or the future availability of substitutes to fossil fuels. Those
future changes reduce future consumption of fossil fuels, bene…tting the climate. But they induce changes in current behavior that might harm the
climate. When these kinds of o¤setting changes occur, their net e¤ect on
the climate may be ambiguous.
In an economy without market failures (and given the convexity condition discussed in Chapter 2.6), cheaper energy increases social welfare. Although the lower backstop cost bene…ts society in this perfect world, the
Fundamental Welfare Theorems imply that there is no need to subsidize
green alternatives here. We do not consider providing public subsidies to
competitive computer manufacturers, even though their innovations bene…t
consumers. In this perfect world, the market rewards the innovators by exactly the amount needed to induce them to undertake the socially optimal
level of innovation.
141
However, we are interested in public policy because market failures are
prevalent. A monopoly might want to produce either too much or too little
innovation, relative to the socially optimal level. Competitive …rms might
face research spillovers, a positive externality where one …rm’s innovation
bene…ts other …rms. The innovator does not take those bene…ts into consideration and therefore undertakes too little research.
These kinds of market failures are potentially important, but our focus
is the pollution externality associated with fossil fuels. Greenhouse gasses
such as carbon dioxide are a classic example of a public bad. In the absence
of “win-win opportunities” (Box 9.1) it is expensive for a country to reduce
emissions. Those reductions lower atmospheric stocks of greenhouse gasses,
reducing climate-related damages and creating global bene…ts. The individual country making the sacri…ce to reduce emissions obtains only a small
share of these global bene…ts. Due to this market failure, countries have
inadequate incentive to reduce emissions. There is scope for public policy
at the global level.
Does a seemingly bene…cial policy, such as a subsidy that lowers the cost
of the backstop, actually help to correct this externality, or does it make
things worse? The Green Paradox provides examples where a policy of
this sort makes matters worse. The subsidy has two types of e¤ects on the
externality. First, the subsidy tends to reduce cumulative extraction of fossil
fuels over the life of the resource, improving the climate problem. Second,
the subsidy alters the extraction path, increasing extraction early on and
decreasing extraction later. This “tilting”of the extraction path can worsen
the climate problem. Either of these e¤ects might dominate, so we cannot
presume that the lower backstop price bene…ts the climate.
142
CHAPTER 9. THE “GREEN PARADOX”
Box 9.1 Win-win opportunities With “win-win” opportunities, the
unilateral reduction of CO2 emissions can bene…t a country. Reducing C02 tends to also reduce local pollutants such as SO2 and
Total Suspended Particles (TSP), generating local health bene…ts.
These kinds of “co-bene…ts” have been documented for China, and
the Obama administration used these bene…ts as an additional justi…cation for environmental rules announced in the summer of 2014.
If the co-bene…ts are large relative to the cost of C02 reductions, the
country can gain from the reductions even without taking into account
the global bene…t of lower atmospheric carbon stocks: a “win-win”.
Using sequestered carbon to improve soil creates or reduce oil extraction costs create other win-win possibilities.
9.1
The approach
Objectives and skills
Recognize that the Green Paradox does not rely on fact that investment
is costly.
Rather than speak generally of improvements in green technology, it is
useful to have a speci…c model. The simplest of these describes the green
technology using the production costs, b, which we again take as a constant,
as in Chapter 8. With this model, an improvement in technology corresponds
to a reduction in b, to b0 < b.
Improving technology of any hue usually involves costs. Suppose that
society must invest I (b0 ; b) > 0 (“I” for “investment costs”) to reduce the
backstop cost from b to b0 . The factors of production, e.g. scientists and
lab space used to improve the technology, are costly. Proponents of green
subsidies argue that the social bene…ts, arising from reduced climate-related
damages, justify these investment costs. To focus on the Green Paradox,
we ignore the investment costs. That is, we ask, “Even if I (b0 ; b) = 0,
does society want the lower backstop costs?” This question is similar to
the comparative static questions that we introduced in Chapter 2.2, except
that here we ask how changing the parameter b alters the entire pro…le of
extraction (not merely extraction at a single point in time) and thereby alters
climate-related damages.
9.2. CUMULATIVE EXTRACTION
143
A large literature discusses the merits of “Industrial Policy”, governmental attempts to promote speci…c industries. All of the arguments for and
against this type of government intervention also apply to green industrial
policy. The Green Paradox applies uniquely to green industrial policy, raising the possibility that society might not want the better technology even if
it were free.
9.2
Cumulative extraction
Objectives and skills
Recognize that with stock-dependent costs, lowering the backstop lowers cumulative extraction, making the Green Paradox less likely.
Chapter 8 considers one case where extraction costs increase as the remaining stock of the resource falls (costs equal C ( + x) y) and two cases
where extraction costs are independent of the remaining stock (costs equal
Cy, and costs equal Cy 1+ ). In the …rst case, we noted that zero extraction is optimal for stocks less than or equal to xmin , the solution to
C ( + xmin ) = b. In this case, cumulative extraction, over the life of the
resource, is x0 xmin .
A reduction in the backstop price, e.g. from b to a lower level b0 , increases
the critical stock level below which it is not worth extracting the resource.
Figure 8.2 illustrates a case where reduction in the cost of the backstop from
b = 2 to b = 1 shifts the dashed line to the dotted line. This downward
shift increases the critical stock below which it does not pay to extract, from
about 13 to 21, thus reducing cumulative extraction by 8 units.
By choice of units, we can set one unit of extraction equal to one unit of
emissions, so the reduction of cumulative extraction implies an equal reduction of cumulative emissions. One short ton of subbituminous coal contains
about 3700 pounds of C02 . We can de…ne a “unit of coal” to equal a short
ton and a “unit of C02 ”to equal 3700 pounds, so that one unit of coal equals
one unit of C02 .
With stock-dependent extraction costs, a lower backstop cost reduces cumulative emissions over the life of the resource. Insofar as climate-related
damages arise because of cumulative emissions, the lower backstop cost reduces climate-related damages. Stock-dependent extraction costs therefore
militate against the Green Paradox. If extraction costs are independent
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CHAPTER 9. THE “GREEN PARADOX”
extraction
low backstop
1.0
0.8
0.6
high backstop
0.4
0.2
0.0
0
10
20
30
40
50
60
70
80
90
100
110
time
Figure 9.1: Extraction pro…les under high and low backstop costs
of the stock, the reduction in backstop costs has no e¤ect on cumulative
extraction and thus no e¤ect on cumulative emissions.
9.3
Extraction pro…le
Objectives and skills
Understand the e¤ect of a reduction in backstop cost on the extraction
pro…le.
Climate damage may depend on the shape of the extraction trajectory
(the pro…le), not just cumulative extraction. In order to make this point
as simply as possible, we consider the case where average extraction costs
are constant at C < b0 < b; here, cumulative extraction is the same for
both backstop costs. The assumption of constant average costs thus allows
us to isolate the e¤ect of the backstop cost on the extraction pro…le. A
more descriptive model, with stock dependent costs, would show how a lower
backstop cost changes both cumulative extraction and the extraction pro…le.
The intuition from such a model merely combines, without increasing, the
intuition that we obtain by considering separately the e¤ects of cumulative
extraction and the extraction pro…le.
A reduction in the backstop cost reduces future resource prices, thereby
reducing the rent in earlier periods. This reduction in rent corresponds to a
9.3. EXTRACTION PROFILE
145
decrease in the …rm’s opportunity cost of selling the resource. A reduction in
the opportunity cost, like the reduction in any kind of (marginal) cost, leads
to an increase in equilibrium sales. Therefore, a reduction in the backstop
costs leads to higher sales during the time that these are positive. Because
the resource stock is …nite, it is not possible to increase sales at every point
in time. Therefore, the extraction paths (showing extraction as a function
of time) under a low and under a high backstop cost, must cross at some
point in time.
Figure 9.1 shows extraction pro…les corresponding to a high and a low
backstop costs. The important features of the …gure are: (i) early in the
program (t < 41 for this example) extraction is higher under the low backstop
cost, and (ii) exhaustion occurs earlier under the low backstop costs. At
T 0 = 41, the resource is exhausted under the low backstop cost; exhaustion
occurs at T = 100 under the high backstop cost. The extraction paths cross
at T 0 , but not before.
The lower backstop cost “tilts the extraction trajectory toward the present”
(small t). Provided that the reduction in backstop costs is not large enough
to make the resource stock worthless, the timing of the cost reduction may
be unimportant in a deterministic setting. In the example above, it does
not matter whether the cost reduction occurs right away, at t = 0, or if it is
anticipated to occur at t = 40. In either of those cases, the low-cost backstop
is not used until t = 41.
Box 9.2 Description of Figure 9.1. The extraction pro…les in Figure 9.1 are generated using a continuous time model with constant
average extraction costs, C = 5, = 0:95 (a discount rate of about
5%), demand D = 10p 1:3 , and an initial stock of x0 = 46. The high
backstop cost, b = 100, leads to the solid curve of extraction, and the
low backstop cost, b0 = 6, leads to the dashed extraction trajectory.
Because b0 > C, in both cases it is optimal to exhaust the resource;
rent is positive. The curves are smooth, except for the points of
discontinuity, because of the continuous time setting; in the discrete
time setting, where the time index is integer-valued, the graphs would
be step functions rather than smooth curves. (See the discussion of
Figure 8.1.)
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CHAPTER 9. THE “GREEN PARADOX”
Box 9.3 The logic behind Figure 9.1. The Euler equation does not
depend on the backstop price. In the continuous time setting, the
price at the last instant of extraction, time T , equals the backstop
price. The graphs of the price trajectories, under the two backstop
prices, cannot cross: if they were to intersect at any point, they would
have to coincide at every point when extraction is positive, simply
because both paths satisfy the same Euler equation. However, if the
two price trajectories coincide, then it is not possible for both to result
in exhaustion of the resource, because the two price trajectories terminate at di¤erent prices, equal to the backstop cost in the two cases.
Consequently, the two price trajectories never cross. Therefore, one
path must lie strictly below the other, during times when extraction
is positive under both. Because the price path corresponding to low
backstop cost ends at a lower price, it must be lower during the entire
time that extraction is positive. Therefore, extraction must be higher
while it is positive, and exhaustion occurs sooner, under the low-cost
backstop, as Figure 9.1 shows.
9.4
Why does the extraction pro…le matter?
Objectives and skills
Understand three reasons why the change in extraction pro…le, associated with the reduction in backstop cost, might increase climate-related
damages.
Scienti…c evidence points strongly to the conclusion that higher atmospheric
stocks of greenhouse gasses contribute to climate change. Climate change
may create substantial social costs, including more serious epidemics, rising
sea level, and increased frequency and severity of storms and droughts. Climate change may also lead to rapid and large-scale extinction of species, with
unpredictable ecological, and ultimately social, consequences; temperature
and precipitation changes might decrease agricultural productivity, worsening food insecurity; these changes may also induce massive human migration, worsening social con‡ict. Higher cumulative extraction –thus higher
atmospheric stocks of greenhouse gasses –worsens the climate problem. We
9.4. WHY DOES THE EXTRACTION PROFILE MATTER?
147
have seen that a subsidy that reduces backstop costs cannot increase cumulative emissions. There can be no Green Paradox through that channel, leaving only the possibility that tilting the extraction pro…le toward the present
worsens climate change.
There are many reasons why such a tilt is likely to make things worse.
We consider three of these reasons: the tilt might make it more likely that
we cross a threshold that triggers a catastrophe, such as rapid melting of
the Antarctic ice sheet; the tilt might speed the rate of climate change, and
society is worse o¤ when change occurs more quickly; the tilt creates a higher
maximum stock level, and costs may be nonlinear in the stock. Figures
9.2 – 9.4 are based on the same assumptions used to generate Figure 9.1.
That …gure shows two extraction pro…les that lead to the same cumulative
extraction, but the dashed pro…le, corresponding to the low backstop cost,
is tilted toward the present: extraction is initially higher with the lower
backstop cost, but is eventually lower.
9.4.1
Catastrophic changes
Figure 9.2 illustrates the possibility of crossing a threshold that triggers a
catastrophe. The two extraction pro…les in Figure 9.1 have di¤erent e¤ects
on the stock of carbon. The distinction between stock and ‡ow variables is
critical. In the setting here, emissions is a ‡ow variable. It is measured in
(for example) tons of carbon per year. The stock variable in this setting is
the amount of carbon in the atmosphere. It is measured in (for example)
tons of carbon. The ‡ow variable is measured in units of quantity per unit
of time, whereas the stock variable is measured in units of quantity.
A t = 0, the beginning of the program, the level of the stock is given.
Historical emissions, prior to t = 0, determine this initial condition. Some
of the carbon entering the atmosphere migrates to other reservoirs, including
the ocean and biomass. Although not literally correct, climate economists
often describe this migration as a decay of the stock, and for purpose of
exposition use a constant decay rate. The emissions generated by fossil fuel
use increases the stock of atmospheric carbon and decay reduces the stock.
Early in the program, when emissions are high, the stock increases. Later
in the program, when emissions are low, the natural decay rate of the stock
dominates, and the stock falls.
Because the initial stock level is historically determined, it is the same for
both extraction paths. However, after the initial period, the stock trajectory
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CHAPTER 9. THE “GREEN PARADOX”
stock 40
low backstop
high backstop
30
20
10
0
10
20
30
40
50
60
70
80
90
100
time
Figure 9.2: The stock trajectories corresponding to the two extraction trajectories. Solid curve corresponds to high-cost backstop and dashed curve
corresponds to low-cost backstop.
depends on the extraction (= emissions) trajectory. Because extraction is
initially higher under the low-cost backstop, the stock grows more quickly in
that scenario, relative to the high cost backstop scenario. Figure 9.2 shows
the stock trajectories corresponding to the two extraction pro…les in Figure
9.1. For approximately the …rst 70 years, the (dashed) stock trajectory
under the extraction path corresponding to the low backstop cost lies above
the (solid) trajectory corresponding to the high backstop cost.
Figure 9.2 also shows the ‡at dotted line at a stock of 38. If, for example,
a stock above 38 triggers a catastrophe, then that catastrophe occurs in about
40 years under the low-backstop-cost trajectory, but never occurs under the
high-backstop-cost trajectory. If the catastrophe is su¢ ciently severe, then
the future economic bene…ts arising from eventual availability of the low-cost
backstop do not compensate society for the fact that this low-cost backstop
“causes” the catastrophe. The lower cost backstop does not literally cause
the catastrophe: the accumulation of stocks does that. But the lower cost
backstop changes the competitive equilibrium extraction trajectory, thereby
changing the stock trajectory, thereby triggering the catastrophe.
The model of the “carbon cycle” Figure 9.2 is drawn using the assumption that the stock of carbon decays at a constant rate: the time derivative
of the stock, S(t), for this …gure is
dS (t)
= y (t)
dt
S (t) ;
(9.1)
9.4. WHY DOES THE EXTRACTION PROFILE MATTER?
149
where y (t) is emissions (= extraction) at time t and > 0 is the constant
decay rate. For this model, the stock rises ( dS(t)
> 0) when y (t) > S (t)
dt
and the stock falls when this inequality is reversed.
This climate model is simple to work with, and therefore often used in policy models where the goal is to obtain insight, instead of making quantitative
policy recommendations, e.g. levels of the optimal carbon tax. The process
that governs changes in atmospheric carbon (or more generally, greenhouse
gas) stocks is much more complicated. In particular, a constant decay rate
does not accurately describe the e¤ect of emissions on the stock. In addition, GHG stocks likely cause damages indirectly, via the e¤ect of the stocks
on temperature or precipitation, instead of directly. The inertia in the climate system causes global average temperature and other climate variables
to respond to changed GHG stocks with a delay. Therefore, the climateimpact of current emissions might increase over decades, before eventually
diminishing.1
In addition to this delay, climate scientists have identi…ed a number of
positive feedback e¤ects that might cause stocks to increase even if anthropogenic emissions were close to zero. For example, higher temperatures
caused by higher stocks of greenhouse gasses might melt permafrost, releasing additional greenhouse gasses. Our model of catastrophes provides a
simple way to think about this possibility. There may be a threshold level
of atmospheric stocks that triggers such an event. However, the actual
dynamics are much more complicated.
Figure 9.2 illustrates a possibility, but it does not establish that a particular outcome is likely. The …gure does not show the unit of measurement
of the stock variable, partly to defuse the danger that readers give it undue
weight. Its key feature is that the maximum stock level under the low-cost
backstop (the dashed trajectory) is above the maximum stock level under
the high-cost backstop (the solid trajectory). Provided that the probability
of catastrophe increases with the maximum stock level, this model (together
with parameter assumptions) implies that the lower backstop cost increases
the probability of catastrophe. This result is due to the fact that the initial
emissions pro…le is higher under the low-cost backstop.
1
Prominent climate-economics models, e.g. DICE, due to Nordhaus (2008), use climate
components in which the major e¤ect on temperature occurs …ve or six decades after the
release of emissions. Recent evidence by Ricke and Caldeira (2014) suggests that the
major e¤ect occurs within the decade of emissions release.
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CHAPTER 9. THE “GREEN PARADOX”
damages 45
40
35
low backstop
30
high backstop
25
20
0
10
20
30
40
time
Figure 9.3: Damages are related to the stock and the speed of change in
the stock. Solid curve corresponds to high-cost backstop and dashed curve
corresponds to low-cost backstop.
Box 9.4 The half life of the stock The half life of the stock equals the
amount of time it takes half of a given stock to decay. With constant
decay rate , e t of a unit that is emitted at time 0 remains at time
t. Setting e t = 0:5 and solving for t, produces the half life of the
stock, ln 0:5 . If the half life is between 100 and 200 years, and if we
pick the unit of time to equal one year, then 0:0035 < < 0:007.
9.4.2
Rapid changes
Even in the absence of catastrophic events, tilting the extraction trajectory
toward the present may harm society. If change occurs su¢ ciently slowly,
society may be able to adapt to it with moderate costs. Over the very long
run, society replaces most infrastructure. Climate-related change alters the
speed at which this replacement must occur. If we know, for example, that
rising sea levels will make some highways and bridges obsolete in 150 years,
then we can divert investment away from maintaining these highways and
bridges, and toward building more resilient substitutes. If we have to replace
this infrastructure within the next 50 years, then we may be forced to write
o¤ much of the current infrastructure that would, absent rising sea levels,
still be useful for decades.
As a simple way of modeling this dependence of climate-related costs
on the speed of change, denote the stock at time t as S (t) and the speed
. Suppose that total
of change in the stock, the time derivative, as dS(t)
dt
9.4. WHY DOES THE EXTRACTION PROFILE MATTER?
151
damages depend linearly on the stock, and are convex increasing in the speed
of change:
2
dS (t)
Damages = S (t) + 10
.
dt
With this formulation, marginal damages increase with the speed of change
of the stock. From Figure 9.2, it is evident that the stock initially increases
more rapidly in the dashed trajectory: its slope – the time derivative – is
greater. During the early part of the program, the stock levels are similar,
because the stock levels are exactly the same at the initial time, and the
stock levels change continuously with time. Therefore, early in the program,
damages must be higher in the trajectory corresponding to the low cost
backstop.
Figure 9.3 shows the graphs of damages in two cases, as a function of
time. Damages corresponding to the low-cost backstop are higher early in
the program, and only slightly lower late in the trajectory (not shown); the
present discounted value (not shown) of damages are also much higher under
the low-cost backstop.
9.4.3
Convex damages
In the previous example, where the ‡ow of damage is linear in the stock of
atmospheric carbon, doubling S (t) S (0) doubles damage. Damages may
be convex in the stock. For example, the …rst 10% increase in these stocks
may cause negligible additional damage; the second 10% increase may cause
only small additional damages; but the fourth or …fth 10% increase may
cause substantially higher damages. Convex damages mean that marginal
damages are higher, the higher is the stock. This convexity might arise from
the relation between atmospheric stocks and temperature change (e.g. due
to feedbacks) or from the relation between temperature change and damages.
To illustrate the e¤ect of this kind of nonconvexity, suppose that damages
equal S (t) + 12 S (t)2 . Figure 9.4 shows the graphs of damages, over time,
under the high-cost backstop (solid) and the low-cost backstop (dashed). At
the beginning of the program, damages in the two scenarios are exactly the
same, because in this model damages depend only on the level of the stock;
the initial stock, given by historical emissions, is the same under the low- or
high-cost backstop. However, as Figure 9.2 shows, the stock becomes much
higher in the low-cost backstop scenario. This increase makes damages much
152
CHAPTER 9. THE “GREEN PARADOX”
damages 900
800
low backstop
700
600
high backstop
500
400
300
200
100
0
10
20
30
40
50
60
70
80
90
100
time
Figure 9.4: Damages are convex (quadratic) in the stock. Solid curve corresponds to high-cost backstop and dashed curve corresponds to low-cost
backstop.
larger, early in the program, in the low-cost backstop scenario.
9.5
Assessment of the Paradox
Objectives and skills
Understand some of the nuances of the Green Paradox.
We explained the Green Paradox in the context of industrial policies
(“green subsidies”) that lower backstop costs. The same logic applies to
phased-in carbon taxes, i.e. taxes that begin low and rise over time. Both
of these policies lower future producer prices of fossil fuels, and therefore
tend to lower current prices, increasing current extraction. Green subsidies
and future carbon taxes are politically more palatable than policies that
discourage current fossil fuel use. Because of their greater political appeal,
and the resulting higher likelihood that they will be implemented, it is worth
asking whether such policies have unintended consequences.
Firms’ current sales decisions, and thus the equilibrium price of a nonrenewable resource in the current period, depends on …rms’expectations of
future prices. A lower expected future producer price decreases the scarcity
rent, making it less attractive to store the resource rather than sell (and consume) it today. Thus, lower expected future prices increase the availability
9.5. ASSESSMENT OF THE PARADOX
153
of supply in the current period, lowering the current equilibrium price and
increasing current consumption.
The Green Paradox exempli…es the constructive role that theory can play
in informing policy, and also illustrates how easy it is to hijack theory to
promote a particular agenda. Theory works best when it is simple enough
to communicate easily. That simpli…cation almost always requires focusing
on a small set of issues to the exclusion of others. Once the theory has been
understood in the simple setting, it is important to recognize its limitations.
9.5.1
Other investment decisions
The Green Paradox is usually studied in the Hotelling setting where fossil
fuel extraction is the only investment decision explicitly modeled. That
treatment ignores other investment decisions, including those related to the
development of substitutes for fossil fuels or adaptations to anticipated policy, and those related to the discovery and development of new stocks of
fossil fuels. When we recognize that businesses solve a host of investment
problems, of which resource extraction is only one, the Paradox appears in a
di¤erent light. Instead of providing a strong basis for rejecting green industrial policy, it reminds us that green industrial policy might have unintended
consequences.
Green alternatives and current adaptation
The immediate elimination of carbon emissions would be prohibitively expensive, but there is considerable uncertainty about the cost of moderate
emissions reductions. Resolving that uncertainty requires research and development in green alternatives, which likely requires a combination of current
R&D subsidies and future carbon taxes. Current subsidies reduce current
investment costs, and the anticipation of future carbon taxes increase the expectation of the future pro…tability of current investment. Current carbon
taxes increase the current pro…tability of low-carbon alternatives, but not
their future pro…tability. Investment incentives depend on the anticipation
of future pro…tability, because the fruits of current investment are available
only in the future.
The Green Paradox focuses on the current response of resource owners
to future changes in the market. However, resource users also have an
incentive to adapt early to future changes. Consequently, anticipated future
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CHAPTER 9. THE “GREEN PARADOX”
policy a¤ects both current supply and demand. For example, the U.S. Acid
Rain program was phased in over a decade, so coal producers and consumers
were aware of future sulfur emissions constraints. This noti…cation reduced
the rent on high-sulfur coal, inducing owners to increase sales prior to the
restrictions coming in to force; this is the supply e¤ect examined by the
Green Paradox. Power plants, the major consumers, recognized that the
policy would make future emissions expensive. Businesses replace capital as
it wears out; their replacement decisions depend on their expectation of future
market conditions. The future implementation of the sulfur emissions policy
gave power plants an incentive to replace aging capital stock with cleaner
technology. Thus, the announcement of the future constraints increased
current supply of dirty coal, but reduced near-term demand for that coal,
leading to statistically insigni…cant e¤ect on equilibrium consumption.
New sources of fossil fuels
The discovery and development of new deposits, such as tar sands deposits
in Canada and oil o¤ the coast of Brazil, usually involves substantial investment costs, including the costs of infrastructure needed to bring the oil to
market. A fundamental rule of economic logic is to ignore sunk costs. The
decision whether to develop the new deposits depends on the magnitude of
the investment costs relative to potential pro…ts. A green policy that lowers future resource prices, lowering future pro…ts, can change the investment
calculation. However, once …rms have incurred the investment costs, the
subsequent extraction pro…le does not depend on the now-sunk costs.
The Keystone pipeline would bring oil from the Canadian tar sands
to U.S. re…neries, and thence to world markets. Extracting and re…ning
these deposits creates higher carbon emissions per unit of energy produced,
compared to other petroleum deposits. Climate-change activists devoted
substantial e¤ort to in‡uence U.S. policymakers to reject permits for this
pipeline. Some of this opposition was due to concern about local environmental a¤ects arising from possible leaks in the pipeline. Some of the opposition was for symbolic reasons, to show that the danger of climate change is
great enough to justify derailing a project of national importance to Canada.
Delaying tactics can sometimes achieve a strategic goal. In 2012, with
high oil prices, the pipeline looked like a solid business proposition. After a
40% decrease in oil prices, the economic viability of the project is less certain.
A further decline in oil prices might change the oil companies’calculation,
9.5. ASSESSMENT OF THE PARADOX
155
even if they obtain permission to build. Green industrial policy can increase
uncertainty about the value of major new exploration and development efforts, delaying and possibly stopping these e¤orts. However, oil prices have
historically been volatile (Figure 6.1) even before green industrial policy.
Oil producers are accustomed to this volatility; the uncertainty about future
green industrial policy (regulatory risk) is just one additional source of risk.
As noted in Chapter 6.3, Shell already (in 2014) builds in a carbon tax to
the cost of production, in anticipation that this tax will eventually arise.
Divestment from fossil fuel companies
Climate change activists encourage universities, pension funds and other investment funds to divest from fossil fuel companies on the grounds of social
responsibility. These activists draw parallels with the divestment from South
Africa during the apartheid regime. By 2015, several universities (including Stanford) and cities (including Seattle, San Francisco and Portland) had
begun to divest from coal companies. Other funds have divested from coal
on business grounds. In 2014 Norway’s Government Pension Fund Global
(GPFG), the worlds largest sovereign fund, divested from 114 companies,
including 32 coal companies and several oil sands producers. GPFG stated
that these companies where exposed to unacceptable levels of regulatory risk
or other changes that would reduce future demand.
To date, these divestments are too small to a¤ect fossil fuel companies’
cost of capital or their investment decisions. However, if green industrial
policy is large enough to a¤ect current extraction decisions via the change in
resource rent, then it is also large enough to alter incentives to invest in the
discovery and development of new reserves, and to a¤ect the share price of
fossil fuel companies.
9.5.2
The importance of rent
The Paradox is relevant only for resources that have a substantial component of rent in their price. As noted in Chapter 6, rent is a signi…cant
component of the price of oil, but a much smaller component of the price
of coal. Resource-based commodities that have only a small component
of rent are similar to standard commodities. The Green Paradox has only
slight relevance for such commodities, but so does the theory of nonrenewable
resources.
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CHAPTER 9. THE “GREEN PARADOX”
The Green Paradox concerns policies that directly a¤ect markets in the
future, and only indirectly a¤ect current markets. Some green policies have
direct e¤ects on current markets. For example, renewable fuel portfolio
standards impose a minimum requirement on the fraction of energy produced
using green technologies. Current subsidies to solar or wind energy increase
the demand for green energy sources today, not merely in the future. These
sources are substitutes for fossil fuels, so the portfolio standards and the
subsidies decrease the current consumption of fossil fuels. These policies do
not lead to a Green Paradox.
9.5.3
The importance of elasticities
The signi…cance of the Green Paradox depends on the price elasticities of
supply and demand for fossil fuels. At least in the short run, demand is quite
inelastic. With a low elasticity of demand, a lower current price transfers
income from fossil fuel owners to fuel consumers, and having modest e¤ects
on current consumption.
The Green Paradox is based on the assumption that a downward revision
of beliefs about future energy prices would lead to a signi…cant increase in
current supply. Technical constraints may limit the supply response, at least
in the short run. Many resource …rms operate at or near capacity, and
therefore have limited ability to quickly increase their supply. It may also
be costly for them to shut down operations, lowering their ‡exibility to reduce
supply. These considerations tend to reduce short run supply elasticities,
reducing the signi…cance of the Green Paradox.
9.5.4
Strategic behavior
The Paradox depends on the behavior of oil exporters, in particular OPEC,
the oil cartel. OPEC is less powerful than a monopoly, because it faces a
competitive fringe, but it is more sophisticated than the textbook monopoly
because it understands that the demand function is not exogenous. The
demand function in any period is predetermined by past events. Some past
investments in infrastructure (e.g. highways) increase the current demand for
fossil fuels, and other investments (e.g. development of the ethanol industry)
decrease that demand. Because these investments have already occurred,
the demand function (not the quantity demanded) at a point in time is
predetermined.
9.5. ASSESSMENT OF THE PARADOX
157
OPEC observed that its oil embargo of the early 1970s changed behavior
in importing countries, leading to increased conservation and the development of alternative supplies. OPEC wants to increase its rents, but it
understands that the best way of doing that is not to extract every cent of
consumer surplus available in the near term. OPEC’s long-term strategy
includes maintaining a reliable and reasonably priced source of petroleum,
to discourage changes in behavior or the development of alternative sources
that would reduce its future demand.
Green policies that reduce future demand can have ambiguous e¤ects
on current OPEC strategic behavior. One possibility is that the presence
of these policies makes OPEC redouble its e¤orts to create a reliable and
reasonably priced source of oil, in order to counteract the e¤ect of the policies.
The green policies encourage the development of green technologies, and
OPEC may decide to try to o¤set that encouragement. How might OPEC
go about achieving this goal?
OPEC might think that the developers of the green technologies base
their beliefs about future energy prices chie‡y on current energy prices.
With this view, OPEC could o¤set the subsidies to green technologies,
discouraging their development, by reducing current price. In that
case, OPEC’s strategic response causes an even larger reduction in
current prices, and thus a larger increase in current consumption than
the standard Green Paradox suggests.
Alternatively, OPEC might think that the developers of green technologies understand that future energy prices will depend on future
stocks. With that view, OPEC could save more of its resource stock,
in order to make credible its commitment to relatively low future prices.
This commitment to low future prices requires a reduction in current
extraction, thus working against the Green Paradox.
A third possibility is that OPEC decides that e¤orts to discourage green
substitutes for fossil fuel are doomed, thus diminishing its incentive to
maintain stable and reasonable prices. If those e¤orts contributed
to relatively low fossil fuel prices, then the reduction in those e¤orts
increases current fossil fuel prices, working against the Green Paradox.
That is, OPEC might decide that it is rational to exercise market power
to the full extent possible, without concerning itself with the e¤ect that
high current prices have on the future demand function.
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9.6
CHAPTER 9. THE “GREEN PARADOX”
Summary
The Green Paradox illustrates the possibility that well-intentioned policies
can back…re. The paradox potentially applies to policies that directly a¤ect
future energy markets, e.g. carbon taxes that begin in the future, or subsidyinduced reductions in the costs of backstop technologies that will be used
in the future. These policies directly a¤ect future demand for fossil fuels.
Because of the dynamic linkages in resource markets, those future prices
a¤ect resource owners’current supply decisions.
Any policy that reduces future demand, makes it less attractive for resource owners to hold on to their stock. This kind of policy therefore tilts the
extraction trajectory toward the present, increasing current extraction and
reducing extraction in some future periods. If extraction costs depend on the
remaining resource stock, these policies also reduce cumulative extraction.
Lower cumulative extraction, and the associated reduction in cumulative
carbon emissions, bene…t the climate. Tilting the extraction pro…le toward
the present is likely to harm the climate. The higher earlier extraction likely
increases the peak stock of atmospheric carbon, increasing the probability
of a “catastrophe”. The tilted extraction pro…le also increases the speed of
change of atmospheric stock. Society may be worse o¤, the more rapid this
change occurs. Finally, if marginal damages are increasing in the level of
the stock, the tilted extraction pro…le is likely to increase damages. The net
e¤ect of policies that lower future demand for fossil fuels therefore depends on
the balance between the e¤ects of lower cumulative extraction and of higher
earlier extraction.
The Green Paradox is valuable as a caution to policymakers, but practical
considerations may limit its importance. The Green Paradox emphasizes the
resource owners’incentives. Consideration of investment in resource exploration and development, and taking into account the externalities associated
with investment in green alternatives, can overturn the paradoxical result.
9.7
Terms, study questions, and exercises
Terms and concepts
Research spillovers, business as usual, green industrial policy, green paradox,
win-win opportunities, climate threshold, catastrophic change, stock and ‡ow
9.7. TERMS, STUDY QUESTIONS, AND EXERCISES
159
variables, decay rate, half-life of a stock, convex damages, predetermined
versus exogenous.
Study questions
1. (a) State the meaning of the Green Paradox in the context considered
in this chapter (where green industrial policy reduces the cost of a
low fossil renewable alternative to fossil fuels). (b) Discuss some of
the reasons that the paradox might occur in the case where fossil fuel
marginal extraction costs are constant. Your answer should include
both a description of how, and an explanation of why, the industrial
policy changes the extraction pro…le. It should also include a discussion
of how and why this change in extraction pro…le might change climaterelated damage. (c) Suppose now that extraction costs increase as
the remaining resource stock falls. How and why does this di¤erent
assumption about extraction costs a¤ect the likelihood that the green
paradox occurs?
2. Explain why the consideration of investment decisions other than the
resource extraction decision might make the Green Paradox less likely.
Exercises
1. Following the sketch in Box 9.3, provide a detailed argument showing
that the extraction trajectory under the low-cost backstop lies above
the trajectory under the high-cost backstop during the time that both
are positive. Explain why this fact implies that exhaustion occurs
earlier under the low-cost backstop.
Sources
The DICE model due to Nordhaus (2008) is probably the most widely used
model that “integrates”the economy and a climate cycle
Ricke and Caldeira (2014) provide evidence showing that major e¤ect of
emissions occurs within the …rst decade of emissions release.
Sinn (2008) (elaborated in Sinn 2012) is an early study of the Green
Paradox.
Hoel (2008 and 2012) studies the role of extraction costs and demand
characteristics.
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CHAPTER 9. THE “GREEN PARADOX”
Gerlagh (2011) distinguishes between a “weak”and “strong”paradox.
van der Ploeg and Withagen (2012) provide an in-depth analysis of the
paradox.
van der Werf and Di Maria (2012) survey the literature.
Pittel, van der Ploeg and Withagen’s (2014) edited volume brings together recent contributions.
Winter (2014) studies the Green Paradox in the presence of climate feedbacks.
Ellerman and Montero (1998) examine the e¤ect of future emissions constraints on earlier sulfur emissions.
Di Maria et al. (2013) discuss the Green Paradox in the context of the
U.S. Acid Rain Program.
Alberini et al. (2011) provide estimates of the elasticity of demand for
electricity.
Vennemo et al. (2006) estimate the health bene…ts to China of reducing
CO2 emissions.
Lal (2004) discusses a win-win possibility involving carbon sequestration
and soil enhancement.
Karp and Stevenson (2012) discuss green industrial policy.
Chapter 10
Policy in a second best world
Objectives
Understand the basics of designing policy to correct market failures,
and understand why well-intentioned policies can back…re when there
are multiple market failures.
Information and skills
Explain and apply the Theory of the Second Best and the Principle of
Targeting.
Calculate and illustrate graphically the Pigouvian tax, for competitive
markets.
Describe a “collective action problem”.
Understand the relation between the optimal tax under a monopoly
and a competitive industry.
Understand why two policies might be either complements or substitutes.
The study of economics usually emphasizes perfectly competitive markets
where there are no distortions, where the equilibrium is e¢ cient (Chapter
2.6). E¢ ciency means that no one can be made better o¤ without making someone else worse o¤; the e¢ cient equilibrium might not be ethically
161
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CHAPTER 10. POLICY IN A SECOND BEST WORLD
appealing. Market imperfections such as monopoly or externalities, if discussed at all, are usually relegated to a second tier of importance. A negative
externality is an unpriced harm, or cost, borne by some part of society other
than the …rm or individual that creates the harm. A positive externality is
an unpriced bene…t.
The emphasis on competitive markets is logical because we can make
general statements about these, but not about markets with imperfections.
To paraphrase Tolstoy: Perfect markets are all alike; every imperfect market
is imperfect in its own way. Theory aims to uncover general truths, not to
describe particular cases. In addition, the study of perfect markets yields
valuable and counter-intuitive insights. For example, the theory of comparative advantage explains why trade can make all participants better o¤,
even if they have very di¤erent levels of development. The Hotelling model
explains why scarcity per se is not a rationale for government intervention.
Finally, for many markets, the perfectly competitive paradigm is reasonably
accurate.
However, the emphasis on perfectly competitive markets sometimes seems
like an elaborate justi…cation of Dr. Pangloss’claim that “Everything is for
the best in this best of all possible worlds”. In fact, market failures are
important, especially in natural resource settings. Economics can help us
understand how market failures a¤ect outcomes, and how policy can remedy
those failures.
A “second best world”is a world in which there are two or more market
failures. For example, there may be research spillovers, causing …rms to
have inadequate incentive to undertake R&D, and at the same time a zero
price on pollution, causing …rms to pollute excessively. There are obviously
more than two market failures in our world: we inhabit a second best world.
Much economic theory (and the …rst part of this book) ignores market failures
altogether. Economic analysis of market failures (e.g. research spillovers),
often uses models in which the particular failure of interest is the only one
that exists. This procedure provides focus, but it can be misleading if two
or more market failures interact in important ways.
A collection of results having certain features in common, known as the
“theory of the second best” (TOSB), sheds light on policy in the presence
of multiple market failures. A “…rst best” policy corrects a distortion or
achieves an objective (e.g. raising government revenue) in the most e¢ cient
manner possible. We say that a policy is “second best” whenever it is not
…rst best. Typically it is di¢ cult to rank all of the possible policies; we
10.1. SECOND BEST POLICIES AND TARGETING
163
might not even know which to include. Therefore, we may be unable to say
whether a particular policy is 4’th or 17’th best, but able to say that it is
not …rst best. We merely say that it is second best.
The TOSB warns us against applying, in a second best world, the intuition obtained from the theory of perfect markets. A policy intervention
that seems likely to improve welfare might make matters worse. In less extreme circumstances, a policy intervention might improve matters, but in
the process might create unnecessary collateral damage. We also discuss an
idea closely related to the TOSB: the Principle of Targeting. This principle
provides commonsensical but still valuable advice about policy.
The insights developed in this chapter are relevant to a broad range of
both resource and non-resource markets. We discuss these insights in a
fairly general but simple context, in order to make them clear and to emphasize their broad applicability. This material is important for the subsequent
study of market failures and policy intervention in natural resource markets.
Chapter 9 contains our …rst sustained discussion of market failures, related to
climate change. Many other resource markets have similar market failures.
Imperfect or non-existent property rights to …sh stocks or groundwater create externalities analogous to those associated with climate change. Those
failures lead to overharvesting of …sh or excessive pumping of groundwater,
much as the lack of property rights to the atmosphere leads to excessive
emissions of greenhouse gasses.
10.1
Second best policies and targeting
Objectives and skills
Have an intuitive understanding of the Theory of the Second Best and
the Principle of Targeting.
Di¤erent types of policies might alleviate a particular social, economic, or
environmental problem. The choice of policy depends on political and social
considerations. In democracies, and under most other types of governance,
no single social planner makes the policy decision based on his or her view of
what is best. Even though this social planner is a …ction, there is value in
trying to determine the policy that would be chosen by a planner who wants
to maximize social welfare. That information provides a benchmark against
which we can compare policies that are actually used.
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CHAPTER 10. POLICY IN A SECOND BEST WORLD
As noted in Chapter 2.6, the idea of social welfare is itself problematic. In
our partial equilibrium setting without market failures and without taxes, we
take social welfare to be the present discounted stream of the sum of producer
and consumer surplus. A competitive equilibrium maximizes this measure
of welfare; it leads to the same outcome as the …ctitious social planner.
Here we are interested in market failures, so we need a broader de…nition
of welfare. If, for example, the market failure arises from an unpriced externality such as pollution, we have to include the social cost of pollution and
the …scal cost or bene…t of policies that attempt to remedy it. Subsidizing
…rms to encourage reductions in pollution requires generating tax revenue, in
order to …nance the subsidy, and therefore involves a …scal cost. A pollution
tax, in contrast, generates tax revenue, making it possible to reduce taxes
elsewhere in the economy, and therefore involves a …scal bene…t. These
costs or bene…ts are components of social welfare, along with producer and
consumer surplus and the costs or bene…ts arising from the externality.
The …rst best policy maximizes social welfare; second best policies might
increase social welfare, but they do so imperfectly, creating collateral damage
or unnecessary costs.
Examples: Trade restrictions and industrial policy Some activists
promote trade restrictions as ways of achieving environmental or resource
objectives. Trade may increase environmentally destructive production, as
occurred with shrimp harvesting that kills turtles. In the 1990s the U.S.
imposed a trade restriction to decrease, or at least redirect, U.S. shrimp
imports, hoping to decrease turtle mortality. This and other trade restrictions might have environmental bene…ts, but trade restrictions are seldom
the optimal type of policy to achieve these goals.
Turtle mortality was a consequence but not the goal of shrimp harvesting.
The policy objective was to decrease turtle mortality, not to decrease trade.
An e¢ cient policy “targets”the environmental/resource objective. The U.S.
trade restriction led to an international dispute that was resolved by the
World Trade Organization (WTO). Although the WTO accepted that the
U.S. had the right to use policies for the purpose of protecting international
resources such as turtles, it also found that the U.S. policy contravened WTO
law because it restricted trade unnecessarily. The dispute was resolved when
the U.S. dropped its trade restriction but required exporting countries to use
nets with “turtle excluding devices”that protected the turtles.
10.2. MONOPOLY + POLLUTION
165
The Green Paradox provides another example of a policy that may be
poorly targeted to an objective. The policy goal is to reduce carbon emissions. Low cost and low carbon alternatives to fossil fuels might help to
achieve that goal, but green industrial policy that promotes these alternatives potentially change the extraction pro…le in a way that harms the climate system. The net e¤ect of the green industrial policy might be positive.
However, …rst best policies, such as emissions taxes or cap and trade, directly
target the environmental objective of reducing carbon emissions.
The Principle of Targeting These examples illustrate the Principle of
Targeting (POT). This principle makes an intuitive recommendation. When
faced with a market failure (i.e., a “distortion” such as an unpriced externality), it is best to use a policy that “targets” that distortion as closely as
possible. In some respects, this principle (like much of economics!) is a
tautology. It might be interpreted as enjoining the policy maker to use the
best policy – hardly advice that anyone needs. However, the principle actually contains some subtle wisdom. Many policies in‡ict collateral damage
in correcting a distortion, or cause people to reduce a distortion (e.g. lower
pollution) ine¢ ciently. The POT reminds us to be aware of this collateral
damage or ine¢ ciency, and to try to avoid it.
In most cases, the application of the POT is straightforward. It is necessary to clearly identify the objective or the problem, and to distinguish
features that cause the problem from those that are associated with it. In
the trade example, the problem is not trade, but that turtles are killed in
catching shrimp. In the Green Paradox example, the problem is carbon
emissions, not an excessively high backstop cost. The POT tells us that
the e¢ cient policy alters …shers’harvesting techniques in the …rst case, and
reduces carbon emissions in the second. It is worth being able to identify the
e¢ cient policy, even if social or political considerations prevent its adoption.
10.2
Monopoly + pollution
Objectives and skills
Use graphical methods to compare output under a social planner, a
competitive …rm, and a monopoly, in the presence of an externality.
Show how a tax alters output under competition and monopoly.
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CHAPTER 10. POLICY IN A SECOND BEST WORLD
p
20
18
16
14
12
10
8
B
D
6
4
C
2
A
0
1
-2
2
3
4
5
6
7
8
q
-4
Figure 10.1: A: the competitive equilibrium with no pollution tax. B: the
competitive equilibrium with the optimal (Pigouvian) tax t = 6. C: the
monopoly equilibrium with no pollution tax. D: the monopoly equilibrium
with the non-optimal tax t = 6
Illustrate the possibility that a tax that induces competitive …rms
to produce at the socially optimal level might lower welfare under a
monopoly.
A basic idea of the TOSB is a that an ill-advised policy, designed to
alleviate some problem, might make another problem worse. The net e¤ect
of the policy might be either harmful or bene…cial. One of the most famous,
and easily illustrated, example of this possibility arises in a monopoly setting
where production creates a negative externality, pollution.
Figure 10.1 shows the inverse demand function p = 20 3q, the solid
line, and the corresponding marginal revenue curve, M R = 20 6q, the
dashed line. Average and marginal costs are constant at 2. Each unit of
production (or consumption) creates $6 of environmental damages. In this
example, pollution is proportional to output, and social costs are proportional
to pollution. The private cost of production here is 2 and the social cost of
production, which includes environmental damages, is 2 + 6 = 8.
Reserving t for the time index, we use to represent a unit tax. When a
policy, such as a tax, can be either positive or negative, we sometimes refer
to it as a “tax/subsidy”. More commonly, we call it a tax, recognizing that
a negative tax is a subsidy.
An untaxed competitive industry sets price equal to marginal cost and
produces at point A. The monopoly sets marginal revenue equal to marginal
cost, and produces at point C. The optimal tax for the competitive industry,
10.3. COLLECTIVE ACTION AND LOBBYING
167
also known as the Pigouvian tax, is = 6. The socially optimal level of
production and the price occur at point B. The tax = 6 “supports” (or
“induces”) the socially optimal outcome: when competitive …rms face this
tax, they produce at the socially optimal level. In general, the optimal
tax causes …rms to face the social cost of production; it causes …rms to
“internalize”the cost of pollution.
If the monopoly where charged the same tax, = 6, it’s tax-inclusive
cost of production also equals the social cost. The monopoly facing = 6
produces at point D. Absent the tax, the monopoly produces too little,
relative to the socially optimal level: point C lies to the left of point B. The
tax causes the monopoly to reduce output even more, lowering social welfare:
the tax that is optimal in a competitive setting ( = 6) lowers social welfare
if imposed on the monopoly.
This example illustrates the basic point of the TOSB: a policy that improves matters in one circumstance might make things worse in another circumstance. In the competitive setting, there is a single distortion, arising
from pollution. In the monopoly setting, there are two distortions, one arising from pollution and the second arising from the exercise of market power.
A tax that …xes the …rst distortion makes the second one worse. For our
example, the net e¤ect of the policy that is optimal under competition, lowers welfare under monopoly. For this example, the optimal policy under a
monopoly is a subsidy (a negative tax).
A caveat Our example uses two assumptions: (i) a …xed relation between
output and pollution, and (ii) a …xed relation between pollution and social
marginal cost. Assumption (i), which means that the only way to reduce
pollution is to reduce output, is not realistic; usually it is possible to reduce pollution without reducing output, by using a more costly production
method. Despite its lack of plausibility, we adopt this assumption here and
later in the text because it makes it simple to deliver the basic message.
Assumption (ii) also makes the message simple to deliver, but it can easily
be dropped.
10.3
Collective action and lobbying
Objective and skills
Use a payo¤ matrix to identify a noncooperative Nash equilibrium.
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CHAPTER 10. POLICY IN A SECOND BEST WORLD
Illustrate the collective action problem that arises with lobbying.
Understanding the collective action problem helps in making sense of
political outcomes. A collective action is a costly action taken by a group, for
the bene…t of the group. The “problem”is that people prefer other members
of their group to incur the costs, while they share the bene…ts. In some
cases, society imposes a solution to this problem by forcing group members
to contribute, provided that a su¢ ciently large fraction of the group have
voted in favor of the action. For example, the U.S. Agricultural Marketing
Agreement Act of 1937 authorizes “marketing orders”for some commodities.
Producers are obliged to participate in these marketing orders, which can
require minimal quality levels or limited production (in order to maintain
high prices). About half the U.S. states require workers to pay union dues
to a legally recognized union, on the ground that all workers bene…t from
union representation of their workplace. “Right to work”laws remove that
requirement, making union membership, and thus union dues, voluntary.
Real world policies emerge from a political process, not as the dictate of
a benevolent social planner. Political lobbying or naked corruption a¤ects
these outcomes. The Sunlight Foundation estimates that in the U.S. between
2007 –2012, 200 companies spent $5.8 billion in lobbying and campaign contributions, and received $4.4 trillion in federal support or contracts: $760
for each dollar contributed. Changes in U.S. law, notably Citizens United,
make it easier to use money to in‡uence outcomes. In 2014, Transparency
International (www.transparency.org) ranked the U.S. as the 17th least corrupt out of 175 countries. Not all lobbying is corruption, but lobbying uses
money and connections to in‡uence outcomes.
An example illustrates lobbying and the collective action problem. Society at large may bene…t from reduced pollution, but this reduction imposes
costs on some groups. Here we take the extreme case where a group of …rms
incur all of the costs and consumers obtain all of the bene…ts of a particular
emissions policy. Both groups can lobby to in‡uence the probability that
this policy is implemented, but lobbying is expensive. Both consumers and
…rms face a “collective action problem” in deciding whether to lobby; the
collective action is the lobbying e¤ort.
Suppose that the particular emissions policy increases consumer welfare
by 100 units and reduces …rm welfare by 50 units, yielding a net bene…t to
society of 50 units. Absent lobbying e¤ort, the probability that the policy
is implemented equals 0.5, so the expected bene…t to society (the probability
10.3. COLLECTIVE ACTION AND LOBBYING
169
that the bene…t occurs times the level of the bene…t if it does occur) is 25.
If only one group spends 10 units on lobbying, that group has its preferred
outcome with certainty. If both groups spend 10 units on lobbying, their
e¤orts cancel each other, leaving unchanged the probability that the policy
is implemented, but wasting 20 units of welfare.
Table 1 shows the payo¤ matrix if each group is represented by a single
agent who can enforce the decision whether to lobby.
consumersn …rms
…rms lobby …rms do not lobby
consumers lobby
(40; 35)
(90; 50)
consumers do not lobby (0; 10)
(50; 25)
Table 101: Payo¤ matrix for the lobbying game. First element in an ordered pair
shows consumers’expected payo¤, second element shows …rms’expected payo¤.
The …rst element of each ordered pair shows consumers’expected payo¤ for a
combination of actions, and the second element shows producers’payo¤. For
example, if both groups lobby, consumers’expected payo¤ is 0:5 (100) 10 =
40, and …rms’ expected payo¤ is 0:5 (50) 10 = 35. This game is an
example of the Prisoners’ Dilemma. Both groups are better o¤ if they
forswear lobbying, but if one group decides not to lobby, the other group
wants to lobby. The only (Nash) equilibrium in this game is for both groups
to lobby.
This payo¤ matrix assumes that each group has solved the collective
action problem: each acts as a uni…ed agent, i.e. it has delegated authority
to a single agent who decides, on the group’s behalf, whether to lobby. This
assumption is unrealistic; there may be millions of consumers and only a few
…rms. The bene…ts of lobbying are more dispersed for consumers, and each
consumer faces a great deal of uncertainty about whether other consumers
will contribute to the lobbying e¤ort. Producer groups are legal for the
purpose of representing the industry’s interests to legislators, and they can
help to coordinate the individual …rm’s lobbying contributions. In this
setting, …rms are more likely than consumers to solve their collective action
problem.
For this example, society’s expected payo¤ is 25 if neither group solves
the collective action problem, so that neither lobbies. It is 5 if both groups
solve the collective action problem, so that both lobby. Society’s payo¤ is
-10 if only …rms solve the collective action problem. A group’s ability to
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CHAPTER 10. POLICY IN A SECOND BEST WORLD
solve its collective action problem may or may not bene…t society. In this
example, society’s payo¤ is highest if only consumers solve their collective
action problem, and it is lowest if only producers solve their collection action
problem –the opposite of what would be likely to occur in this situation.
Renewable Fuel Standard In 2005 the U.S. introduced a Renewable Fuel
Standard (RFS), requiring annual minimum consumption levels of di¤erent
types biofuels; 2007 legislation increased these levels. The Environmental
Protection Agency (EPA) implements the policy by estimating gasoline demand in the next year and dividing annual targets for the di¤erent biofuels
by the estimated gasoline consumption, to obtain a ratio i for biofuel i.
Gasoline producers are required to use i gallons of biofuel i for each gallon
of gasoline they produce. These producers therefore face a blending constraint that increases their cost of production, because the biofuels are more
expensive than gasoline.
Proponents of this policy justify it chie‡y using an “infant industry”
argument. The idea is that biofuels will eventually be important both as
low carbon alternatives to fossil fuels, and as alternatives to foreign sources of
petroleum. Because the current state of technology and infrastructure would
not enable this industry to survive under market conditions, government
policy is needed to protect this “infant”until it grows into a mature industry.
Infant industry arguments go back at least to the early 1800s, when they
were used to justify trade restrictions. Many opponents of the RFS begin as
skeptics of the infant industry argument in general, because of experiences
where infants fail to mature. In addition, the applicability of the infant
industry argument is questionable in this case, because the RFS has chie‡y
promoted the production of corn-based ethanol, for which the technology was
already mature.1
The RFS emphasis on corn-based ethanol has two major disadvantages,
in addition to having low potential to encourage new technology. First, it
leads to a small, and by some estimates non-existent reduction in carbon
emissions. Second, it diverts a major food crop from food to fuel, likely
increasing food prices and possibly worsening food insecurity in some parts
1
After 2015, ethanol produced using cellulosic material, including the inedible part of
corn and special crops such as switchgrass, is scheduled to become more important in
the RFS. Cellulosic biofuels rely on an immature technology, where government support
can potentially lead to large improvements. However, the RFS’s support for corn-based
ethanol does nothing to promote the development of cellulosic biofuels.
10.4. POLICY COMPLEMENTS AND SUBSTITUTES
171
of the world. The policy has also encouraged farmers to cultivate marginal
land that would otherwise likely have been left fallow under a conservation
program.
Recent evidence estimates that carbon reductions achieved using the RFS
are about three times as costly as the reduction that could have been achieved
under an e¢ cient policy such as an emissions tax or cap and trade. An
important consequence of the RFS was to provide large transfers from the
general public (in the form of higher fuel prices) to corn growers, likely with
little environmental or technological bene…t. The RFS was estimated to
increase U.S. fuel costs by $10 billion per year. Why did the U.S. government
implement an ine¢ cient policy instead of an e¢ cient policy?
Holland et al. (2015) estimate the transfers resulting from the RFS at
the congressional district level, and simulate the transfers that would have
resulted from the cap and trade proposal under the Waxman-Markey bill in
2009. A provision in Waxman-Markey required that fuels eligible for the
RFS produce greater carbon reduction than currently achieved by corn-based
ethanol. Thus, Waxman-Markey would have reduced the transfers that corn
producers receive under the RFS.
Not surprisingly, representatives tend to vote their constituents’ interest. Those from districts that bene…t under the RFS were more likely to
oppose Waxman-Markey, and those from districts that would have bene…ted
under Waxman-Markey were more likely to vote in favor of it. Representatives from districts bene…tting from RFS, which would have been harmed by
Waxman-Markey, also received greater campaign contributions from groups
opposing the bill. Although the cap and trade policy is more e¢ cient than
the RFS, the gains from the latter are concentrated in a small number of
districts, whereas the bene…ts of the former were more widely dispersed (as
in the numerical example above). It was therefore easier to generate e¤ective
lobbying opposing Waxman-Markey than to generate lobbying in its support.
10.4
Policy complements and substitutes
Objectives and skills
Understand why a tax and a subsidy have the same e¤ect on a polluter’s
incentives.
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CHAPTER 10. POLICY IN A SECOND BEST WORLD
Understand the political and the economic disadvantages of a subsidy,
compared to a tax.
Understand the arguments for and against subsidizing R&D in “green
technologies”.
Determine the socially optimal pollution tax when marginal costs increase with the level of pollution.
Determine whether two policies are complements or substitutes.
The previous section illustrates the possibility that a socially bene…cial
policy might be blocked in a political equilibrium, and provides an example
where this actually occurred. Policies typically have di¤erent e¤ects on
di¤erent groups of producers and consumers. The success of these groups
in the political competition to implement or block a policy may have little
relation to the social merits of a policy. E¢ cient policies might be politically
infeasible. In this case, other feasible policies might seem like reasonable
alternatives, even if they are less e¤ective or more costly. We continue to
assume that the social goal is to reduce excessive pollution. The numerical
example in Section 10.3 illustrates why a pollution tax might not be a political
equilibrium. What are the alternative policies?
Subsidizing pollution reduction
The most obvious alternative is to subsidize …rms if they reduce pollution
instead of taxing them if they create pollution. A …rm has the same incentive
to reduce pollution if it is taxed $1 for each unit of pollution, or given a
subsidy of $1 for each unit that it reduces pollution (“abates”) relative to
some baseline. (“Abatement”is de…ned as the reduction of pollution.) With
the tax, a unit of pollution creates a direct cost to …rms; with the subsidy,
a unit of pollution creates an opportunity cost to …rms. These two types of
costs have the same e¤ect on the …rm’s incentives, so (in principle) the two
policies achieve the same level of pollution.
There are both political and economic impediments to using the subsidy
instead of the tax. The political impediment is that the subsidy imposes a
cost on taxpayers; it involves a transfer from general tax revenue to a speci…c
group of …rms. Even if the reduced pollution is worth this cost, politicians
10.4. POLICY COMPLEMENTS AND SUBSTITUTES
173
may have a hard time explaining to voters why they should tax themselves
to pay …rms not to do something that is socially harmful.
The economic impediment is that raising revenue by means of taxes typically creates additional costs, above the direct distributional e¤ect of taking
income away from a group. These additional costs are called deadweight
costs, a subject discussed in Chapter 11. The deadweight costs may be
20 - 30% of the revenue raised. The distinction between a transfer and a
deadweight cost is important. If the government takes $1 from Mary to give
to Jiangfeng, this is merely a transfer; it might make Mary unhappy, but it
makes Jiangfeng happy, and is not a cost to the economy. However, if the
government has to take $1.25 from Mary in order to give $1 to Jiangfeng,
there is a $0.25 cost to the economy. This situation is sometimes described
by saying that the government makes transfers using a leaky bucket.
Due to our assumption of a …xed relation between pollution and output,
the only way to reduce one is to reduce the other. Therefore, there is no
deadweight cost in taxing the polluting sector. However, taxing other sectors
in order to …nance a subsidy for the polluting sector typically does create a
deadweight cost.
As an illustration, suppose that the deadweight cost associated with general taxes (outside the polluting industry) is 25% of revenue raised by a tax.
In that case, …nancing a $1 subsidy to this polluting industry requires rasing $1 of tax revenue elsewhere, creating a deadweight cost of $0.25. If,
instead, the polluting industry is taxed $1, and that tax revenue transferred
to general funds, then other taxes can be reduced by $1.00, saving society
the deadweight cost $0.25. Here, replacing a $1 pollution tax with a $1
abatement subsidy increases social costs by $0.50. This example illustrates
why taxing pollution is likely more e¢ cient than subsidizing abatement.
Subsidizing R&D in “green technologies”
If pollution taxes are politically infeasible because of the collective action
problem, and subsidies to reduce pollution are both politically and economically problematic, it is natural to seek alternatives. Green industrial policy,
such as a subsidy to develop and implement low-carbon alternatives to fossil
fuels, is often proposed as an alternative to a carbon tax or a cap and trade
policy. The American Enterprise Institute and the Brookings Institute, a
conservative and a liberal think tank, respectively, both endorsed research
and development (R&D) subsidies for green technology as an alternative
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CHAPTER 10. POLICY IN A SECOND BEST WORLD
to carbon taxes. These green policies subsidize …rms for doing something
socially useful (developing new technologies) instead of for refraining from
doing something socially harmful (polluting), and therefore encounter less
political resistance than a pollution subsidy.
These green policies still involve subsidies, and give rise to deadweight
costs, as discussed above. They also invite the criticism that the government does a poor job of picking winners.2 Countering these objections, proponents of green industrial policy argue that the subsidies address market
failures other than excessive pollution, such as cross-…rm research spillovers
(a positive externality) that leads to insu¢ cient investment in green R&D.
Regardless of the merits of these opposing arguments, it is reasonable to
ask whether green policies are really an alternative to, i.e. a substitute for
emissions taxes, or if instead they should be regarded as a complement to
emissions taxes. The terms complements and substitutes are familiar from
demand analysis. Beer and pretzels are probably complements, because
eating pretzels increases our marginal utility of beer. A big screen TV and
a sports car are perhaps substitutes, because our marginal utility of driving
the hot car may be lower if we have the option of watching the fantastic
TV screen. Two policies are said to be substitutes if the implementation
of one makes the other less valuable to society; they are complements if the
implementation of one makes the other more valuable.
Chapter 9 shows that green industrial policies might aggravate the pollution problem. Here we develop the closely related idea, that instead of
being a substitute for a carbon tax, green industrial policy might make the
carbon tax more rather than less vital to society. In this case, the policies
are said to be complements.
Competitive resource …rms’equilibrium condition is
pt
@c (xt ; yt )
@c (xt ; yt )
= Rt ; or pt =
+ Rt ;
@y
@y
(10.1)
where, as in previous chapters, @c(x@yt ;yt ) is extraction costs and Rt is the
…rms’period-t rent. Recall that Rt is an opportunity cost, the amount lost
by extracting a unit of resource in period t instead of at some other time.
Thus, @c(x@yt ;yt ) + Rt is the “combined” marginal extraction cost, including
both the standard marginal cost and the opportunity cost. Chapter 5.2
2
For example, subsidies to Solyndra, a manufacturer of components to solar panels,
cost U.S. taxpayers $500 million. Solyndra went bankrupt in 2011.
10.4. POLICY COMPLEMENTS AND SUBSTITUTES
175
shows that Rt equals the …rms’present discounted value of future rents, plus
any cost reduction due to a higher stock. Chapter 9 explains why green
industrial policy is likely to reduce future rents, thus reducing the …rms’
current (period-t) rent, Rt .
Increasing marginal costs of pollution
As with the example in Chapter 10.2, we assume that one unit of production (here, extraction) creates one unit of pollution: the only way to reduce
pollution is to make a corresponding reduction in output. In the previous
example, we took the marginal damage resulting from pollution to be a constant. Here we consider the case where the marginal damage increases with
the level of extraction. If extraction (= emissions) is y, then in this example,
damages equal y 2 , so marginal damages equal 2y.
Where marginal damage are constant, the Pigouvian tax equals that constant. Here, however, where marginal damages depend on the level of extraction, the Pigouvian tax also depends on the level of extraction. In this
case we …nd the Pigouvian tax by:
1. Identifying the socially optimal level of extraction, the level that equates
price to social marginal costs (= marginal extraction costs, plus the opportunity cost, Rt , plus marginal damage, 2yt .)
2. Once we have found socially optimal level of output, denoted yt , we
set the Pigouvian tax equal to the marginal external cost, 2yt for this
example.
The e¤ect of rent on the Pigouvian tax
Figure 10.2 illustrates the case where the …rm has constant marginal extraction costs, @c(x@yt ;yt ) = C.3 Suppose that at a point in time the private
combined marginal cost is C + Rt = 10, shown by the solid ‡at line in the
…gure. For our example, the social marginal cost equals the private marginal
3
This …gure relies on two assumptions: green industrial policies reduce rent, and marginal damages are increasing in emissions. The …rst assumption is plausible, but in the
context of climate change, the second is not. Climate damages are associated with the
pollution stock, not with emissions in a single period. This example is useful for illustrating the di¤erence between policy complements and substitutes, but it is not “realistic”for
the climate application.
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CHAPTER 10. POLICY IN A SECOND BEST WORLD
cost plus the (external) environmental marginal damage, 10 + 2y. If rent
falls by …ve units (e.g. due to the introduction of a green industrial policy
that reduces the cost of a backstop), from Rt to Rt0 = Rt 5, the private
combined marginal cost, and also the social marginal cost, falls by 5 units.
The upwardly sloping solid curve in Figure 10.2 is the graph of 2y+C +Rt ,
the social marginal cost, equal to marginal damage plus the private cost (the
cost of extraction, C, plus the opportunity cost, R). A …ve-unit reduction
in rent causes this combined social marginal cost to shift from the positively
sloped solid line to the dashed line. This reduction in social cost increases
the socially optimal level of production from the intersection shown at point
A, to the intersection shown at point B.
The reduction in rent decreases the combined social cost of extraction,
thereby increasing the socially optimal level of extraction. Because marginal
damages increase with the level of extraction, the optimal level of marginal
damages increase with the fall in resource rent. As a consequence, the
optimal tax also increases with the reduction in rent. This result might
seem counter-intuitive, but it holds quite generally: where marginal damages
increase with the level of extraction (or in other contexts, with the level of
production), a decrease in …rms’ marginal cost leads to an increase in the
optimal tax. When …rms’marginal cost falls, they tend to produce more,
and the higher production leads to higher marginal damages and a higher
optimal tax.
The Pigouvian tax induces the …rm to face the social (instead of the
private) cost of production, thereby giving …rms the correct incentive to
produce at the socially optimal level. If the …rm faces a constant tax
and has private costs (inclusive of opportunity cost, its rent) equal to 10,
its private cost equals social costs if and only if the tax equals the vertical
distance from point A to 10. The Pigouvian tax thus equals this distance,
which we denote A . If the …rm’s rent falls so that its rent-inclusive private
cost now equals 5, the Pigouvian tax equals the vertical distance from point
B to the ‡at dashed line; we denote this tax as B . It is apparent from
the …gure, and easy to demonstrate using either geometry or algebra, that
B
> A . The decrease in rent increases the optimal tax. In this case, an
emissions tax and a green industrial policy that reduces rent are complements,
not substitutes.
10.5. SUMMARY
p
177
25
20
A
15
B
10
5
0
0
2
4
6
8
10
12
14
16
18
q
Figure 10.2: External marginal costs = 2q. Solid positively sloped curve
shows social marginal costs when private marginal cost C +Rt = 10. Socially
optimal production occurs at A. If private marginal costs fall to C + Rt0 = 5,
socially optimal production occurs at B. When private costs fall, the optimal
tax increases.
Relation between this example and the TOSB
This tax example illustrates the general possibility that policies that appear
to be either unrelated or substitutes might be, on closer examination, complements. The TOSB is based on a related idea: connections that are not
apparent may nevertheless be important. The tax example shows that green
industrial policy, instead of making carbon taxes less important, might instead make them more important. The TOSB warns that in the presence
of two or more distortions, or market failures, correcting only one of those
distortions might exacerbate the other distortion to such an extent that welfare falls. Of course, if it is possible to correct all of the distortions, doing
so increases welfare.
10.5
Summary
This chapter introduces the notion of a second-best policy or an outcome:
one that is not optimal, or “…rst best”. The Principle of Targeting recognizes
the importance of carefully matching policies and objectives. A Pigouvian
tax causes competitive …rms to internalize what would otherwise be an ex-
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CHAPTER 10. POLICY IN A SECOND BEST WORLD
ternality for the …rm. For example, a Pigouvian tax causes …rms to take
into account the social cost of pollution when making its production decisions. Two policies are said to be substitutes if the implementation of one
policy makes the other policy less important, or decreases the optimal level
of the other policy; they are complements if the implementation of one policy
makes the other more important, or increases the optimal level of the other
policy.
The theory of the second best (TOSB) and the Principle of Targeting
(POT) are deceptively simple ideas, with important economic implications.
The TOSB reminds us that in economies, “the hip bone is connected to the
shoulder bone”, although perhaps not directly. Simply because markets
connect apparently unconnected outcomes, a policy that reduces one market failure may, in the presence of a second market failure, actually lower
welfare. We illustrated this result using an example of a monopoly who
produces pollution, and showed that moving from the zero emissions tax to
the Pigouvian tax might either increase or decrease social welfare. This
possibility is not surprising: a policy that is optimal in one setting (perfect
competition) might be harmful in another (monopoly). The POT reminds
us to beware of collateral damage in setting policy, and to attempt to select
policies that do not in‡ict such damage.
There is sometimes the risk of examples that are intended to illuminate a
theoretical issue being interpreted as policy prescription. This chapter uses a
particular component of green industrial policy –subsidizing the development
of low cost alternatives to fossil fuels –as a means of illustrating the TOSB
and of explaining the di¤erence between policies that are complements or
substitutes. Our example does not put green industrial policy in a favorable
light. However, the example was not designed to provide a balanced view of
green industrial policy. The TOSB can also be used to construct arguments
in support of green industrial policy. For example, the development of some
green technologies may provide positive spillovers that encourage still better
technologies, making it economically attractive to phase out fossil fuel use
early enough to avert serious climate damage. Economic theory can help
identify the right questions to ask about a policy. Except in rather extreme
cases, theory cannot provide de…nitive judgment of policy. Where it does
appear to provide a de…nitive judgement, the reader should question whether
theory has been inappropriately applied.
10.6. TERMS, STUDY QUESTIONS AND EXERCISES,
10.6
179
Terms, study questions and exercises,
Terms and concepts
Theory of the second best, Principle of Targeting, Pigouvian tax, a tax “supports” or “induces” an outcome, internalize an externality, collective action
problem, payo¤ matrix, Nash equilibrium, Prisoners Dilemma, Renewable
Fuels Standard, deadweight loss, abatement, increasing marginal pollution
costs, policy complements and substitutes.
Study questions
1. Use a graphical example (and a static model) to show that the Pigouvian tax that corrects a production-related externality (e.g. pollution) in a competitive setting might lower social welfare if applied to a
monopoly.
2. (a) Consider a static model. Suppose that inverse demand is 10 q,
…rms are competitive with constant average = marginal costs C, and
pollution-related damages (arising from output) are q + 12 q 2 , with
> 0 What is marginal damage? (b) What is the socially optimal
level of production and consumer price, and what is the Pigouvian
tax? (c) How does the Pigouvian tax change with ? (d) Provide the
economic explanation of this relation.
3. (a) Continue with the model in question #2. Suppose that there is a
policy that reduces C, e.g. by making production more e¢ cient. How
does this reduction in C alter the Pigouvian tax that you identi…ed in
question #2b? (b) Are the two policies (the pollution tax and the
policy that reduces C) complements are substitutes? (c) How would
the answer to part (b) have changed if = 0? Explain.
4. You are in a conversation with someone who correctly states that, in
a particular market, international trade increases production in poorer
countries with weaker environmental standards, thereby increasing a
global pollutant (i.e. a pollutant that causes worldwide damage, not
just damage in the location where production occurs). The person
claims that a trade ban is a good remedy for this problem. Regardless
of your actual views, use concepts from this chapter to argue against
this person’s proposal.
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CHAPTER 10. POLICY IN A SECOND BEST WORLD
5. Ignoring the possibility of the Green Paradox, what are the principle
arguments (actually used in the policy debate) for an against a subsidy
that makes it cheaper to reduce pollution?
Exercises
1. Suppose that consumers of the product bear the cost of pollution. The
model then contains three types of agents: the …rm, consumers, and
taxpayers. Taxpayers bene…t from tax revenue and they do not like
having to pay the cost of subsidies. Inverse demand is p = 20 3q,
private average = marginal cost is 2, and environmental damage per
unit of output is 6. A monopoly chooses the level of sales. Using a
…gure like Figure 10.1 identify graphically (by shading in appropriate
areas) the change in welfare of the three types of agent when a regulator
imposes a unit tax of = 6.
2. For this example, identify (graphically) the socially optimal tax/subsidy
under the monopoly. That is, identify the tax/subsidy that induces
the monopoly to produce at the socially optimal level.
3. Change the example in Exercise 1 to p = 20 0:4q. (Replace the slope
3 by 0:4) Other parameter values are unchanged. (a) Does this change
make demand more or less elastic? (b) Find the optimal pollution tax
for the competitive …rm. (c) Find the optimal pollution tax under
the monopoly. (d) Provide an economic explanation for the relation
between the optimal tax and the slope of the inverse demand function,
under both a competitive …rm and a monopoly.
4. Suppose that (private) constant marginal costs is c, each unit of pollution creates d dollars of social cost (external to the …rm), and the
demand function is demand = Q (p). (a) What is the optimal pollution tax for the competitive industry? (b) Use the formula for marginal
revenue (a function of price and elasticity of demand) to …nd the equation for the optimal tax (or subsidy) under a monopoly. (c) Now
suppose that Q = p with > 1. Use your formula from part (b)
to …nd the optimal tax/subsidy under the monopoly. (d) Provide an
economic explanation for the relation between the optimal tax/subsidy
under the monopoly and .
10.6. TERMS, STUDY QUESTIONS AND EXERCISES,
181
5. Suppose that demand is p = 20 q, and marginal environmental damages equal 2 + q, where the parameter
0. Suppose that green
industrial policy causes private costs, de…ned as @c(x@yt ;yt ) + Rt , to fall
from 10 to 5. (a) Discuss the economic interpretation of the parameter
. In particular, explain the di¤erence in the model with = 0 and
with > 0. (b) Find the socially optimal level of production when
private costs equal 10. Find the optimal (Pigouvian) tax in this case.
(Both of these are functions of .) (c) Now …nd the optimal production level and the Pigouvian tax when private marginal costs fall to 5.
(d) Write the di¤erence in the Pigouvian tax, in these two settings, as
a function of . (e) Describe and explain the e¤ect of on the change
in the Pigouvian tax arising from the change in private costs.
6. Inverse demand is p = 20 3q, private average = marginal cost is 2,
and environmental damage per unit of output is 6. There is a 10%
deadweight cost of raising public funds. This cost is symmetric in
the following sense: the social value of raising $1 of tax revenue is
$1.10, and the social cost of raising $1 in revenue to pay for a subsidy
(or anything else) is also $1.10. Explain the qualitative e¤ect, on the
optimal policies under competition and monopoly, of the social cost of
raising public funds.
Sources
Lipsey and Lancaster (1956) is the classic article on TOSB.
Okum (1975) is credited with the “leaky bucket”metaphor.
Fowlie (2009) and Holland (2012) provide recent applications of the TOSB
related to environmental regultion.
Tritch (2015) discusses the statistics on U.S. lobbying provided by the
Sunlight Foundation.
Auerbach and Hines (2002) survey the literature on taxation and e¢ ciency.
Prakash and Potoski (2007) provide examples of collective action in environmental contexts.
Winter (2014) shows that carbon taxes and green industrial policy are
likely to be policy complements.
Leonhardt 2010 describes the political popularity of green industrial policy.
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CHAPTER 10. POLICY IN A SECOND BEST WORLD
Holland et al. (2015) study the political economy connections between
the RFS and the Waxman - Markey bill.
Bryce (2015) provides the estimated cost to motorists ($10 billion/year)
of the RFS.
Chapter 11
Taxes: an introduction
Objectives
Understand how taxes a¤ect equilibrium outcomes and welfare in a
static setting.
Information and skills
Know the de…nitions of unit and ad valorem taxes.
Be able to use graphs and mathematics to determine the e¤ect of taxes
on equilibrium prices.
Understand the de…nition of tax incidence and be able to explain why a
tax on consumers or on producers are “equivalent”in a closed economy.
Understand why this equivalence does not hold in an open economy.
Be able to illustrate graphically the relation between tax incidence and
supply and demand elasticities.
Be able to use graphs to identify tax-induced changes in consumer and
producer surplus, and to identify the deadweight cost of a tax.
Understand the meaning of a “cap and trade”policy, and know the relation between the equilibrium price of permits, and a tax that induces
the same level of pollution as the cap.
183
184
CHAPTER 11. TAXES: AN INTRODUCTION
Before examining the e¤ect of taxes in the resource setting, readers should
understand their e¤ect in the simpler static setting, the topic of this chapter.
By changing prices, taxes change consumer and producer behavior. Tobacco
taxes discourages smoking, and carbon taxes would reduce carbon emissions.
The static model of a competitive economy has a downward sloping demand curve and an upward sloping marginal cost (= supply) curve. The
intersection of these two curves determines the equilibrium price and quantity. Provided that neither the demand nor the supply curve is ‡at, producers and consumers both obtain surplus in this equilibrium. A tax, levied
on either …rms or consumers, changes the equilibrium price and quantity,
changing consumers’ and producers’ levels of surplus. The revenue raised
from a tax makes it possible to reduce other taxes while funding the same
level of government expenditures.
There are three types of agents in our model: consumer welfare depends
on consumer surplus; producer welfare depends on producer surplus; and
taxpayer welfare depends on tax revenue. Many people belong to two or
even all three of these groups. The sum of consumer surplus, producer
surplus, and tax revenue is our measure of social welfare.
Understanding the e¤ect of taxes in the familiar static model of a competitive …rm is worthwhile for its own sake, and also necessary for understanding
the e¤ect of taxes applied to natural resources, studied in Chapter 12. We
emphasize competitive markets, dealing only in passing with monopolistic
markets. We also emphasize the case where the economy is closed: there
is no international trade in the taxed commodity. This assumption means
that domestic supply equals domestic demand.1
11.1
Types of taxes
Objectives and skills
Introduce and de…ne a unit tax, an ad valorem tax, and a pro…ts tax.
1
Appendix F contains technical material related to this chapter, including: algebraic
veri…cation of tax equivalence in the closed economy; an example showing that tax equivalence does not hold in an open economy; details on the approximation of tax incidence,
deadweight loss, and tax revenue, and details related to the material on cap and trade.
11.2. TAX INCIDENCE AND TAX EQUIVALENCE
185
A unit tax, denoted , is measured in units of currency, such as dollars
or Euros. If producers receive $p per unit sold, and the government imposes
a tax of = $6 per unit, consumers pay p + 6 per unit. The unit tax is $6.
An ad valorem tax, denoted , is measured as a rate. If producers receive
$p per unit sold, and the government imposes an ad valorem tax of = 0:2,
consumers pay (1 + 0:2) p. The ad valorem tax is a fraction; we multiply
by 100 to express the tax as a percent.
There is a simple relation between the unit and the ad valorem tax. If
producers receive the price p and one group of consumers pays a unit tax
and another group of consumers pays an ad valorem tax , the two groups
pay the same price if p + = (1 + ) p. Thus, two taxes yield the same
consumer price if and only if = p. We can work with whichever type
of tax we want, and easily translate one type of tax into another. For the
monopoly, the relation between a unit and an ad valorem tax is slightly more
complicated. Exercises ask students to investigate taxes under a monopoly.
A pro…ts tax takes a percent of pro…ts. Exercises ask students to investigate the e¤ect of a pro…ts tax.
11.2
Tax incidence and tax equivalence
Objectives and skills
Understand the meaning of tax incidence and tax equivalence.
Ability to explain intuitively why producer and consumer taxes are
equivalent in a closed economy.
We assume that people are “rational”, in the sense that their willingness
to buy a commodity depends on the price they pays, not the precise manner
in which the price is calculated. For example, an informed consumer is just
as likely to buy a commodity priced at $1.10 “out the door”as a commodity
marked at $1.00 that requires payment of a 10% sales tax at the cash register.
In both cases, the …nal price equals $1.10. A …eld known as behavioral
economics shows that people sometimes react di¤erently to these two settings,
and in that sense are not rational.
It might seem that it should matter whether a tax is levied on consumers
or producers (on which group has the “statutory obligation” of paying the
tax). However, in a closed economy, the equilibrium price and quantity,
186
CHAPTER 11. TAXES: AN INTRODUCTION
and thus the consumer and producer surplus and the tax revenue, are the
same regardless of whether the tax is levied on consumers or producers.
The consumer and producer taxes are “equivalent”. Both producers and
consumers “e¤ectively”end up paying part of the tax.
What does it mean to say that one agent, e.g. producers, “e¤ectively”
pays part of the tax that is levied on the other agent, e.g. consumers?
Suppose that in the absence of a tax, the equilibrium price is $12 and the
supply and demand is 100 units. Suppose that a tax of $2 per unit is
imposed on consumers. Does the imposition of this tax mean that the
price consumers pay rises to $12+$2=$14? In general, the answer is “no”.
The tax does increase the price that consumers pay, but (in general) this
higher price decreases the amount that they demand. In order for producers
to want to decrease the amount that they supply, the price that producers
receive must fall. The increase in consumer price, as a percent (or fraction)
of the tax is called the consumer incidence of the tax, and the decrease in
producer price, as a percent (or fraction) of the tax is the producer incidence.
Suppose that the $2 tax causes the tax-inclusive price that consumers face
to rise from $12 to an equilibrium of $13.50. This higher price reduces their
demand. In order for supply to fall by enough to maintain supply equal to
demand, the price that producers receive must have fallen. In this example,
it must have fallen to $11.50, because the tax drives a $2 wedge between the
consumer (tax-inclusive) price and the producer price. The di¤erence in the
tax-inclusive price paid by consumers and the price received by producers is
$13.50-$11.50=$2, the amount of the unit tax. Consumers “e¤ectively”pay
the share
1:5
13:5 12
=
= 0:75;
tax
2
or 75% of the tax, and producers “e¤ectively”pay the remaining 25% of the
tax. In this example, the tax incidence on consumers is 75% and the tax
incidence on producers is 25%.
In general, the tax incidence depends on the elasticities of supply and
demand, a relation we discuss below. However, the tax incidence and the
equilibrium quantity and price do not depend on whether the tax is directly
levied on consumers or producers. This equivalence between the producer
and consumer taxes arises simply because, in a closed economy (no international trade) supply equals demand.
11.3. AN APPLICATION TO ENVIRONMENTAL POLICY
11.3
187
An application to environmental policy
Objectives and skills
Introduce the “Polluter Pays Principle”.
Understand the relation between this Principle and the tax equivalence
result.
The equivalence of producer and consumer taxes is potentially important
to environmental policy; it implies that it may not matter whether an externality is corrected using a tax on production or on consumption. That
equivalence appears to undercut the Polluter Pays Principle, which is based
on the idea that it is important which agent –the polluter or the pollutee –
directly pays for the damage caused by pollution. If we think of consumers
as being a proxy for the agent that su¤ers from the pollution, the principle
seems to imply that it would matter whether the tax is levied on consumption
or production. However, the equivalence between these two taxes implies
that it is immaterial whether a consumer or producer tax is used. In this
(limited) sense, the Polluter Pays Principle sets up a meaningless distinction.
In order to make the point that the Polluter Pays Principle is sometimes
irrelevant, consider the case where each unit of production creates $2 worth
of environmental damage. This damage is external to the …rm, meaning
that the …rm does not take the damage into consideration in making its
production decisions. We also assume that the environmental damage is an
inevitable consequence of production. No abatement technology is available,
so production and pollution are equivalent: society cannot have one without
the other. (See the Caveat at the end of Chapter 10.2.)
The optimal policy causes …rms to internalize this environmental cost,
just as they internalize costs of production associated with hiring capital and
labor. One means of achieving this internalization is to charge producers a
$2 per unit tax on output. However, in view of the equivalence of a producer
and consumer tax, we achieve the same goal by charging consumers a $2 per
unit tax on consumption. The incidence of the two taxes is the same and
they have the same e¤ects on welfare, i.e. on the level of environmental
damage, tax revenue, and consumer and producer surplus. It does not
matter whether the polluter (producers) or the “pollutee” (consumers, as
proxies for society) faces the statutory obligation to pay the tax.
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CHAPTER 11. TAXES: AN INTRODUCTION
It also does not matter which agent is responsible for the environmental damage. In some cases (e.g. driving), a major source of environmental
damage arises from consumption of the good (cars) rather than production.
Suppose that production causes no pollution, but that each unit of consumption causes $2 worth of environmental damage and, as in the previous
example, there are no opportunities for abatement. In this case, targeting
pollution and targeting consumption are equivalent. The optimal policy
charges consumers a consumption tax equal to the marginal cost of pollution. In view of the equivalence between producer and consumer taxes for
non-traded goods, we obtain the same outcome by imposing the statutory
tax obligation on producers.
11.4
Tax equivalence
Objectives and skills
Use graphs to show how a tax causes a “shift”in demand or supply.
Use graphs to determine tax incidence and the e¤ect of a tax on price
and output.
Use graphs to show the equivalence of consumer and producer taxes.
This section shows that for “rational” agents (as de…ned above), consumer and producer taxes are equivalent for non-traded goods. Figure 11.1
shows a supply and demand curve (the heavy line) in the absence of taxes.
Consumers and producers face the same price; the equilibrium price is p .
Once we introduce a tax, the consumer and producer prices are di¤erent,
so we can no longer use the same axis to measure both prices. We have to be
clear about what the vertical axis now measures. Suppose that we introduce
a consumer unit tax of . We continue to let the vertical axis be the price
that producers receive and we continue to denote the producer price by p.
Therefore, the tax does not alter the location of the supply curve.
The tax causes the consumer price to be p + . The original demand
function under the tax, the heavy downward sloping line, shows the relation
between quantity demanded and the price that consumers pay. However,
under the tax, we decided to use the vertical axis to represent the price that
producers receive. Since the price that consumers pay and the price that
11.4. TAX EQUIVALENCE
189
Figure 11.1: Upward sloping line: supply function. Heavy downward sloping
line: demand function with no tax. Light downward sloping line: demand
function with tax. The tax equals the distance k bd k, labelled f:
producers receive are not the same when a tax is imposed, we cannot use the
original demand function (the heavy line) to read o¤ the quantity demanded
for an arbitrary producer price. The di¢ culty is that supply is a function
of p and demand is a function of p + , and we cannot let one axis represent
both of these values. This di¢ culty is easily resolved.
Box 11.1 A tax shifts the demand function. Pick an arbitrary point,
e.g. b, on the original demand function in Figure 11.1. Suppose that
point b corresponds to q = 10 and p = $6. If consumers have to pay a
tax $ , then they would be willing to buy 10 units if the producer price
is $(6
); in that case, consumers’tax-inclusive price is $6. Here,
where we graph consumer demand as a function of producer price,
the point q = 10, p = 6
is a point on “new” (dashed) demand
function, under the tax. This point is units below the point on the
“original” (solid) demand function.. This procedure is the same for
any point on the original demand curve. Thus, the consumer tax
shifts the original demand down by units.
We want to have the supply and demand curves on the same graph, in
order to be able to determine the equilibrium by …nding the intersection
between these two graphs. What does the demand function look like, when
we hold the tax …xed and consider the quantity demanded to be a function
190
CHAPTER 11. TAXES: AN INTRODUCTION
of the producer price, p? The answer is that the tax shifts down the demand
curve by the magnitude , resulting in the light dashed demand function
shown in Figure 11.1. This “new” demand function contains exactly the
same information as the original demand function; it merely shows demand as
a function of the producer price rather than the consumer price, recognizing
that the consumer price is p + . The unit tax shifts the demand function
down by units: the vertical distance between the original demand function
and the demand function under the tax, is .
The intersection of the original supply function and the new demand
function occurs at the price p1 ; this is the equilibrium producer price in the
presence of the tax. The equilibrium quantity is obtained by reading o¤ the
quantity associated with this price, q1 . The equilibrium consumer price is
p1 + . Note that
p1 < p < p1 + :
The tax incidence for producers in this example is p p1 100 and the tax
incidence for consumers is p1 + p 100. We see that both tax incidences lie
between 0 and 100%, and they sum to 100%:
p
p1
100 +
p1 +
p
100 = 100:
Figure 11.2 shows the e¤ect of a unit tax when producers face the statutory tax obligation. In this graph we let the vertical axis represent the price
that consumers pay. The price that producers receive is the consumer price
minus the tax. Using the same reasoning that was used to explain Figure
11.1, we see that the producer tax shifts the supply curve (as a function of
the consumer rather than the producer price) up by the amount . For example, if producers are willing to supply 10 units when they receive a price
of $6, once they have to pay the tax they would need to receive 6 + to
persuade them to supply 10 units.
In the absence of the tax, the equilibrium price is p as before. When
producers pay the tax, the equilibrium price that consumers pay increases to
p2 = p1 + , but producers’after-tax revenue on each unit sold is p2
= p1 .
If we were to superimpose Figure 11.2 onto Figure 11.1, the resulting …gure
would show that the equilibrium quantity and the tax-inclusive consumer and
producer prices are the same, regardless of whether producers or consumers
face the statutory obligation. Tax equivalence does not hold in an open
economy (one with international trade).
11.5. APPROXIMATING TAX INCIDENCE
p
191
tax
P1+t
P*
p1
Q
Figure 11.2: Heavy upward sloping line: supply function. Light upward
sloping line: supply function under tax. Downward sloping line: demand
function.
11.5
Approximating tax incidence
Objectives and skills
Introduce the idea of a an approximation of tax incidence.
Show the e¤ect of supply and demand elasticities on tax incidence.
Calculating the exact tax incidence requires that we …nd the equilibrium
price in the absence of the tax, and the equilibrium consumer (or producer
price) under the tax, and compare the two. However, we can use supply and
demand elasticities to approximate the tax incidence for small taxes. We
use the de…nitions of elasticities
elasticity of supply
elasticity of demand
=
=
dS(p) p
dp S
dD(p) p
;
dpc D
(11.1)
where it is understood that the elasticities are evaluated at the equilibrium
price in the absence of a tax. Using these de…nitions and the condition that
supply equals demand, we obtain an expression for the change in equilibrium
price due to a change in the tax, starting from a zero tax:
dp
=
d
+
:
(11.2)
Equation 11.2 is an example of the type of comparative static expression
introduced in Chapter 1.
192
CHAPTER 11. TAXES: AN INTRODUCTION
Using equation 11.2 and the elasticity de…nitions, we obtain the approximations for the producer and consumer tax incidence (see Appendix F):
producers’approx. tax incidence:
+
100
consumers’approx. tax incidence:
+
100:
(11.3)
These approximations are valid only in the neighborhood of a 0 tax: the
formulae are “close to being accurate”if the tax is small (close to 0).
By their de…nition, the two tax incidences must sum to 100%. Therefore,
if we know the tax incidence for one agent, we subtract that number from
100% to …nd the tax incidence for the other agent. Smaller values of mean
that demand is less elastic, which translates into a steeper inverse demand
function. As demand becomes less elastic, the consumer incidence of the
tax increases. If demand is completely inelastic ( = 0) then the inverse
demand function is vertical. In that case, the consumer incidence of the
tax is 100%. Similarly, a lower elasticity of supply corresponds to a steeper
supply function and a larger producer incidence of the tax.
To understand the relation between elasticities and tax incidence, the
student should draw a supply and demand curve, pick a tax, and identify the
tax incidences. Then change one of these curves by rotating it around the
no-tax equilibrium. A steeper curve corresponds to a reduction in elasticity,
and a ‡atter curve corresponds to an increase in elasticity. From this …gure,
it is easy to see how a more elastic supply or demand (i.e. a ‡atter supply
or demand) alters the tax incidence.
11.6
The leaky bucket: deadweight cost
Objectives and skills
Identify graphically the deadweight cost of a tax.
Approximate the deadweight cost using elasticities of supply and demand.
Understand why a greater elasticity of supply or demand increases the
deadweight cost of a tax.
Approximate the tax revenue using supply and demand elasticities.
11.6. THE LEAKY BUCKET: DEADWEIGHT COST
193
Use the approximations of deadweight cost and of tax revenue to obtain
an approximation of the deadweight cost per unit of tax revenue.
Understand the di¤erence between partial and general equilibrium analysis.
Understand the di¤erence between short and long run elasticities, and
the resulting “time-consistency”problem.
As long as the elasticities of supply and demand are both positive and
…nite, both consumers and producers bear some of the incidence of the tax.
The tax causes consumers’ after-tax price to rise, and producers’ after-tax
price to fall. Thus, the tax reduces both consumer and producer surplus.
In Figure 11.1, the area of the trapezoid abcp measures the loss in consumer
surplus due to the tax, and the area of the trapezoid p cde is the loss in
producer surplus. Tax revenue equals the area of rectangle abde. Social
welfare is the sum of producer and consumer surplus and tax revenues. The
reduction in social welfare, due to moving from a zero tax to a positive
tax, equals the reduction consumer surplus, plus the reduction in producer
surplus, minus the increase in the tax revenue. In Figure 11.1, this net loss
equals the area of the triangle bce. This triangle is society’s deadweight cost
of the tax.
In the case of linear supply and demand functions, the deadweight loss
is literally a triangle (known as the “Harberger triangle”), and its area can
be easily measured. However, we use the linear functions only for ease of
viewing. For general supply and demand functions, we can use the elasticities
to obtain an approximation of the deadweight loss, denoted DW L:
DW L
q
1
2 + p
2
:
(11.4)
The approximation uses estimates of the supply and demand elasticities,
observation of quantity and price, and the magnitude of the tax.
The formula also illustrates an important and general result in economics.
The deadweight loss is approximately proportional to the square of the tax.
Consequently, the derivative of the (approximation of) deadweight loss is
proportional to the tax. In particular, this derivative, evaluated at a zero
tax, is 0: a zero tax minimizes the deadweight loss.
194
CHAPTER 11. TAXES: AN INTRODUCTION
DWL
9
8
7
6
5
4
3
2
1
-3
-2
-1
0
1
2
3
tax
Figure 11.3: The graph of the approximation of deadweight loss.
In discussing the fundamental welfare theorems in Chapter 2.6, we noted
that a competitive equilibrium maximizes social welfare, when the assumptions of the theorems hold. Thus, a tax, which moves the outcome away
from a competitive equilibrium, must lower social welfare. The deadweight
loss increases with the magnitude of the tax, but it increases faster than the
tax. Figure 11.3 illustrates this relation. This fact means that it is e¢ cient
to use a broad tax basis. For example, suppose that in one scenario the
government uses a tax = 2 in each of 10 markets; in each markets sales
equal 4, so tax collection equals 10 2 4 = 80. In another scenario it
doubles the tax base from 10 to 20 commodities and lowers the tax on each
from = 2 to = 1, collecting the same amount of revenue, 20 1 4 =
80. If + pq is the same for all of these commodities, the DWL falls from
10 12 + pq 22 to 20 12 + pq 12 , a decrease of 50%.
Extremely small taxes cause essentially no deadweight loss, but as the tax
increases, the deadweight loss increases rapidly. Doubling a tax increases the
approximate deadweight loss by a factor of 22 = 4. Tripling tax increases the
approximate deadweight loss by a factor of 32 = 9. Therefore, eliminating
an extremely small tax leads to a negligible increase in social welfare. But
reducing a tax that is not extremely small, may lead to a substantial increase
in social welfare. Notice also that the deadweight loss is symmetric around
0: a small subsidy creates the same deadweight loss as does a small tax.
This analysis suggests two “rules” for optimal tax policy. In order to
raise a given (arbitrary) amount of tax revenue:
(1) It is better to tax goods that have lower elasticity of supply
or demand. (2) It is better to have a broad tax base (i.e. tax
many instead of few goods).
11.6. THE LEAKY BUCKET: DEADWEIGHT COST
195
A third rule is so obvious that it does not require analysis: It is better to tax
“bads”, such as pollution, rather than “goods”, such as labor or investment.
Many taxes ignore some or all of these rules.
The deadweight cost is expressed in the same monetary unit as prices,
e.g. dollars. An alternative (unit-free) measure shows the deadweight loss
per unit of the tax revenue. For example, a ratio .2 (20%) means that
the deadweight loss is 20% of tax revenue. Chapter 10.4 discusses this
measure in a policy setting. Denote p and q as the zero-tax equilibrium
price and quantity. We can divide the approximation of deadweight loss by
an approximation of tax revenue, to obtain an estimate of the deadweight
cost per unit of tax revenue:
DWL
tax revenue
1
2 q
q
+ p
q
+ p
(11.5)
Figure 11.4 graphs the right side of equation 11.5 for two arbitrary values
of q and for a particular normalization.2 Figure 11.3 shows that the deadweight loss rises “very slowly” for small taxes (i.e. it rises with the square
of the tax). Figure 11.4, in contrast, shows that the deadweight loss per
unit of tax revenue rises “quite quickly” with the tax. A small tax leads
to very little deadweight loss, but it also leads to very little tax revenue.
Increasing the tax increases both the deadweight loss and the tax revenue,
but the deadweight cost rises more rapidly than does tax revenue.
In the context of environmental economics, we are principally interested in
taxes or subsidies as a means of correcting a market failure such as a pollution
externality, not as a means of raising tax revenue. Properly implemented,
those environmental taxes increase social welfare. Many taxes are designed
to raise tax revenue. The tax revenue is (we assume here) socially useful, but
raising it creates a deadweight cost. That cost might eliminate a signi…cant
fraction of the potential social bene…t of each dollar of tax revenue. The
revenue raised by taxing a “bad” such as pollution, makes it possible to
reduce other taxes whose purpose is to raise revenue, thereby lowering the
deadweight loss associated with those taxes.
2
Because of the arbitrary choicea of q, the …gure has no speci…c empirical content, but
it illustrates how “easy” it is to obtain ratios of 0.2 – 0.3, i.e. where the deadweight loss
reaches 20% –30% of tax revenue.
196
CHAPTER 11. TAXES: AN INTRODUCTION
ratio
0.5
0.4
0.3
0.2
0.1
-4
-3
-2
-1
1
2
3
4
tax
-0.1
Figure 11.4: Approximate social cost per unit of tax revenue, using normalization (achieved by a particular choice of units) + pq = 1 and the arbitrary
values q = 10 (solid curve) and q = 8 (dashed curve).
Product markets and general equilibrium analysis The discussion
above considers taxes for produced goods. Taxes are also applied to “factors
of production”, such as land, labor and capital. The e¤ects of commodity
and factor taxes are similar, both having an incidence and a deadweight
cost. For example, a tax applied to labor potentially alters the supply of
labor, changing the equilibrium wage, a¤ecting both people who supply labor
and those who purchase it, and creating a deadweight loss. The factor-tax
incidence and deadweight loss depend on elasticities, just as with a product
tax.
The partial equilibrium analysis examines a single market. There, tax
incidence depends entirely on the relative supply and demand elasticity of the
commodity or the factor, and the deadweight loss depends on the product of
the supply and demand elasticities. If the elasticity of supply is zero, then
producers or factor owners bear the entire incidence, and the deadweight
loss is zero. In this case, the tax shifts income from producers or factor
owners to taxpayers, but causes no e¢ ciency loss. Unimproved land is the
classic example of a factor whose supply is perfectly inelastic. Henry George,
a 19th century political economist, proposed a “single tax” on unimproved
land; this tax causes no economic loss, and falls entirely on people who do
not create wealth.
A general equilibrium setting recognizes that markets for di¤erent products or factors are interlinked, and that it is seldom possible to alter one
market without altering others. For example, if landowners are also farmers, the land tax lowers their income and wealth. The tax-induced reduction
11.6. THE LEAKY BUCKET: DEADWEIGHT COST
197
in income may cause farmers to work harder, changing the supply of labor,
thus changing the level and equilibrium price of the output. The tax-induced
reduction in farmers’ wealth might induce them to rebuild their wealth by
accumulating more capital. The higher stock of capital increases the marginal productivity labor, increasing the equilibrium wage and lowering the
return to capital. The changes in these factor prices causes the incidence to
be shifted to their owners. If the revenue from the land tax is given to workers, their higher income might cause them to supply less labor, increasing
the equilibrium wage. Even ignoring these complications, there are practical
di¢ culties in implementing the single tax, whose rationale depends on the
zero elasticity of supply of unimproved land. Virtually all productive land,
however, has been improved in some manner, often by previous generations.
The indirect (or “general equilibrium”) changes are too complicated to
summarize in a simple formula. They are sometimes studied using numerical
“computable general equilibrium” (CGE) models. Table 1 reports CGEbased estimates of the deadweight loss of various U.S. taxes. The property
tax has the lowest deadweight loss, consistent with the low elasticity of supply
of land. The fact that most taxes create a deadweight loss means that there
is a social cost to raising government revenue. Prominent estimates are
that this social cost is at least 20% of tax revenue. Opponents of expensive
government programs sometimes invoke this cost to explain that the actual
cost of the program exceeds the budgetary cost.
income
payroll
consumer
property
capital
output
tax
tax
sales tax
tax
tax
tax
50
38
26
18
66
21
Table 11.1 Estimates of percent deadweight loss of U.S. taxes
Time consistency Putting aside the general equilibrium complications,
the partial equilibrium analysis has a simple and powerful policy message:
the deadweight cost associated with taxes is lower, the lower is the elasticity
of supply or demand associated with the taxed product or factor. The
di¤erence between short and long run elasticities complicates this message.
In order to make this point, suppose that the government wants to minimize the deadweight loss, subject to the constraint that it raise a certain
amount of tax revenue in each period. The government has two policy
instruments, a tax on capital and a tax on labor. Investment takes time;
the current stock of capital depends on previous, not on current investment.
198
CHAPTER 11. TAXES: AN INTRODUCTION
This fact causes the supply of capital to be quite inelastic in the short run,so
current capital taxes create little deadweight loss. This observation implies
that most of the revenue in the current period should be raised using a capital tax. However, the future stock of capital depends on current investment,
which depends on beliefs about future capital taxes. This fact causes the
future stock of capital to be quite elastic, militating against the use of a
capital tax in the future.
If the policymaker today can make a binding commitment to the time
pro…le of taxes, she would like to raise most of current revenue using a capital
tax, but promise to use a low future capital tax. The current labor tax can
therefore be low, but the future labor tax must be high, in order to raise the
required amount of revenue in each period.
The time consistency problem is that “once the future arrives, it has become the present”. The policymaker in the future has the same incentives
as the policymaker today. She has the temptation to renege on her predecessor’s commitment to use a low capital tax, in order to raise the required
amount of revenue with little deadweight cost. This is the time consistency
problem: the planner in the future does not want to implement the program
that her predecessor would like to see used. If today’s planner cannot bind
her successors to carry out the program that she wants them to use, that
program is not time-consistent. Investors would not believe that they will
in fact face low capital taxes in the future, and this understanding informs
their current investment decisions.
11.7
Taxes and cap & trade
Objectives and skills
Understand the basic ingredients of a cap and trade policy.
Be able to explain why the equilibrium price of permits, under cap
and trade, depends on the number of permits (the “cap”) but not on
whether permits are given or auctioned to …rms.
Understand that corresponding to every “cap” in the cap and trade
policy, there is a “quota-equivalent emissions tax” that leads to the
same level of pollution, and the same distribution of pollution across
…rms.
11.7. TAXES AND CAP & TRADE
199
Understand how to use tax incidence to estimate the fraction of permits
that have to be “grandfathered”in order for a cap and trade policy not
to reduce industry pro…ts.
Emissions taxes and cap & trade are market-based environmental policies,
in contrast to “command and control”policies that reduce emissions by mandating certain types of technology or production methods. Most economists
think that market-based policies reduce emissions more cheaply than command and control policies. By the end of 2014 there were almost 50 carbon
markets worldwide, the largest being the The European Union’s Emissions
Trading Scheme. Southern California’s trading scheme for NOx , RECLAIM,
has operated since 1994. The (never passed) 2009 Waxman-Markey bill envisioned setting up a U.S. market for carbon emissions. Environmental reform
requires people to change their behavior, and usually imposes some cost on
the economy. Cap and trade can reduce the industry cost by “grandfathering” permits, i.e. giving rather than selling businesses a certain number of
emissions permits.
Our discussion of taxes emphasizes that regardless of who has the statutory obligation to pay the tax, in general both consumers and producers
bear some incidence of the tax. Both consumers and producers also typically bear some of the incidence of any other environmental policy, including
cap and trade. The Waxman-Markey proposal gave businesses a declining (over time) share of the permit allowance. When does grandfathering
merely cushion businesses from a loss of pro…ts, and when does it amount to
a give-away?
The answer to this kind of question depends on the speci…cs of the provision, but the logic of the answer is straightforward. To explain this logic,
imagine that we were interested in slightly di¤erent question. Suppose we
pick a tax, and ask what fraction of the tax revenue …rms would have to be
given, in order to make them as well o¤ as they are without the tax. Referring to Figure 11.1, we see that the tax causes the …rms to lose the amount of
surplus equal to the areas edf p + dce. If …rms were given tax revenue equal
to edf p , plus a bit more to make up for their surplus loss dce, they would
p
be as well o¤ as without the tax. The area edf p equals the fraction edf
edba
of the tax revenue. Because the length of the rectangles in the numerator
and the denominator of this fraction are equal, the ratio equals the ratio of
the heights of the two rectangles. This ratio equals pp1 +tp1 , which equals the
producer incidence of the tax. In conclusion, we see that if …rms are given a
200
CHAPTER 11. TAXES: AN INTRODUCTION
share of the tax revenue equal to their incidence of the tax, plus a bit more
(to make up for the missing triangle dce), then they are as well o¤ as before
the tax.
To apply this logic to cap and trade, instead of taxes, it is necessary
to understand the relation between the two types of policies. We explain
how a cap and trade system works, and why the equilibrium level of each
…rm’s emissions do not depend on whether …rms are given permits or have
to buy them. We then explain the sense in which a cap and trade policy is
equivalent to an emissions tax. We use that equivalence, together with the
logic above, to approximate the fraction of permits that …rms would have to
be given, in order to make them just as well o¤ under cap and trade as they
are in the absence of regulation.
The basic ingredients of cap and trade. The regulator chooses the
cap on emissions, denoted Z. The many competitive …rms are able to
buy and sell permits. This buying and selling is the “trade” part of the
cap and trade policy. Each of these …rms takes the price of an emissions
permit as given. Denote the equilibrium price of permits as pe (Z). This
relation recognizes that (as in all markets) the equilibrium price depends on
the supply. Here, the supply is a number, Z. Firms might be given some
permits (“grandfathered”) or no permits at the beginning of the program.
Due to the (assumed) …xed relation between output and emissions, by
choice of units we can set one unit of output to equal one unit of emissions.
For example, if one unit of output creates 2.5 kilograms of pollution, we can
measure pollution in units of 2.5 kilograms.
Claim #1: The permit price and emissions are independent of
the allocation of permits The equilibrium permit price depends on the
aggregate number of permits, Z. However, if …rms are price taking and
pro…t maximizing, and if the transactions cost in buying and selling permits
is low, then …rms’ pollution levels are independent of the distribution of
allowances, e.g. whether …rms are given or sold the permits. To verify this
claim, we show that each …rm’s demand for permits is independent of its own
allocation. Consider an arbitrary …rm that is given an allowance A (possibly
equal to zero). This price-taking faces the output price, p, and the permit
price, pe , and wants to maximize pro…ts:
pq
c (q) + pe (A q) :
| {z }
11.7. TAXES AND CAP & TRADE
201
The underlined term equals the …rm’s revenue from selling the good minus its
cost of production; the under-bracketed term equals the …rm’s pro…ts from
selling (if A > q) or its costs of buying (if A < q) permits.
The …rst order condition to the …rm’s problem states that price equals
marginal cost. Marginal cost here equals the sum of the “usual” marginal
dc
), and the cost of buying an emissions permit, pe . The …rst order
cost ( dq
condition
dc
p=
+ pe ;
(11.6)
dq
does not depend on its permit allocation, A. Firms’ decision about how
much to produce, and thus about how many permits to use, does not depend
on the allocation of permits.
A …rm that buys permits has to pay pe for the additional permit needed to
produce an additional unit. A …rm that sells permits incurs an opportunity
cost pe in using an additional permit: by using that permit it is no longer
able to sell it. Thus, regardless of whether the …rm is a net buyer or seller
of permits, it incurs the cost pe of using an additional unit. Recent research
(Box 11.2) …nds empirical support for Claim 1.
Box 11.2 The empirical validity of Claim 1 If allowances were randomly assigned to …rms, then Claim 1 implies that …rms’allowances
are uncorrelated with their emissions. In practice, allowances are not
randomly assigned; with grandfathering, …rms that emitted more in
the past receive higher allowances. These are the same …rms that are
more likely to have high future emissions, thus creating a mechanical
(and therefore uninformative) positive correlation between allowances
and emissions. California’s RECLAIM emissions trading program
randomly assigned …rms to di¤erent permit allocation cycles: …rms
in the …rst cycle receive their annual allowance at the beginning, and
…rms in the second cycle receive their allowance half-way through the
calendar year. Moreover, …rms in the second cycle tended to receive a
higher allowance than those in the …rst cycle. Similar …rms randomly
assigned to di¤erent groups therefore receive, on average di¤erent allowances. Using this design feature (and “instrumental variables”),
statistical tests provide empirical support for Claim 1.
Claim #2: There exists a quota-equivalent emissions tax To simplify the exposition, we assume that all …rms have the same cost function,
202
CHAPTER 11. TAXES: AN INTRODUCTION
$
10
9
p(Q)
8
dC/dQ
7
6
5
4
3
2
permit price
1
0
0
1
2
3
4
5
Q= Z
Figure 11.5: Solid lines: inverse demand and marginal cost. Dashed curve:
the equilibrium permit price, pe , is the vertical di¤erence between inverse
demand and marginal cost.
so that we can use the representative …rm model. As in Chapter 2.4, we
denote the cost function for the representative …rm as c (Q). Using the fact
that Q = Z (because one unit of output produces one unit of emissions),
equation 11.6 implies
p (Q)jQ=Z =
dC (Q)
+ pe (Z) or
dQ jQ=Z
p (Q)
dC (Q)
dQ
= pe (Z) :
jQ=Z
(11.7)
The second equation shows that the equilibrium permit price equals the
di¤erence between the inverse demand function, p (Q), and the marginal
cost function, dC(Q)
.
dQ
Figure 11.5 shows linear demand and marginal cost curves, and the equilibrium permit price (dashed line) as the vertical di¤erence between the two.
For this example, the equilibrium quantity (= emissions) absent regulation
is 3:33. If the regulator chooses Z
3:33, then the regulation is vacuous,
and the permit price is zero. But for Z < 3:33, the emissions constraint is
binding, and the permit price is positive. Every value of Z below the unregulated “Business as usual”level (3.33) corresponds to a di¤erent equilibrium
permit price.
If the representative …rm faces a tax , the equilibrium condition (price
equals “usual”marginal cost plus the tax) is
p (Z) =
dC (Q)
+ :
dQ
11.7. TAXES AND CAP & TRADE
203
Comparing this equation to the …rst equation in 11.7 shows that the tax
= pe (Z) induces the competitive industry to produce at the same level as
under the cap and trade policy with cap Z.
Claim #3: There is a simple formula for compensating …rms A
cap and trade policy with cap Z determines the amount of emissions. We
showed that there is an equivalent tax that leads to the same amount of
emissions. Under a cap and trade policy the regulator can reduce the cost
to the …rms by giving them (for free) some permits, instead of auctioning all
of the permits. What fraction of permits would the regulator have to give
…rms, to make them (almost) as well o¤ under regulation as under Business
as Usual?
This question has a simple answer in our setting. If the regulator auctions
all of the permits (gives none to the …rms) then from the standpoint of …rms,
it is exactly as if they face the tax = pe (Z). Figure 11.1 shows that the
…rms’ loss in surplus, due to a tax (or to being forced to buy all of its
emissions permits at the price p^ (Z) = ) is the area p cde = edf p + dce. If
…rms are given (instead of being forced to buy) the fraction + of permits,
then the value of this gift is + times the potential tax revenue. (Review
equation 11.3.) This value equals the area of the rectangle p f de. Firms’
net loss equals their loss in producer surplus minus the value of the gift, the
area of the triangle f cd. This triangle is the small correction that is needed
to make …rms whole. The fraction + is (typically) much less than 1, so
even with the correction, it would be necessary to give …rms only a fraction
of the permits, to compensate them for the regulation.
Lessons and caveats In considering industry’s request for the free allocation of permits, regulators need to remember a fundamental economic fact.
A regulation that increases …rms’ costs typically reduces their incentive to
supply the good that they produce, thereby increasing the market price of
that good. The higher market price provides an automatic cushion for …rms
facing the cost increase caused by stricter regulation. The requirement that
…rms reduce emissions, together with the free distribution of all emissions
permits, might increase …rms’pro…ts relative to the business as usual level.
In that case, it is possible to give …rms fewer permits than they actually use
(forcing them to buy the rest), and still fully compensate the …rms for lost
pro…ts due to the restriction on emissions.
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CHAPTER 11. TAXES: AN INTRODUCTION
All of this may sound too good to be true, so it is worth repeating two
assumptions that underlie the argument. The most important of these is
that the economy is closed, i.e. there is no international trade. In a closed
economy, supply equals demand. If an open economy imposes cap and
trade, the regulation may have little or no e¤ect on the price at which the
country can buy or sell the commodity. To the extent that the price does not
adjust, the market gives producers no automatic cushion to the cost increase
resulting from regulation. In that situation, …rms bear the entire incidence
of the regulation; they are worse o¤ even if they are given all of the permits.
The second assumption is that pollution is proportional to output, i.e.
there is no way to lower pollution except by lowering output. This assumption is not realistic, because …rms typically can choose di¤erent production
methods, with di¤erent ratios of emissions to output. The …rst two of the
claims above remain true in a model with variable emissions/output ratio:
(1) Firms’choice of emissions is independent of the allocation of permits (i.e.
whether …rms have to buy or are given permits), and (2) For every quota
Z there is an emissions tax that leads to the same level of emissions. The
third claim needs to be modi…ed if the emissions/output ratio is not a …xed;
the formula for compensating …rms becomes more complex.
11.8
Summary
The unit and ad valorem taxes are di¤erent ways of expressing a tax. For
each unit tax, there is an ad valorem tax that leads to exactly the same
outcome. These taxes change equilibrium price and quantity and they create
a deadweight cost to society. In contrast, the constant pro…ts tax merely
transfers surplus from producers to taxpayers, having no e¤ect on consumers’
welfare or on aggregate social welfare. The deadweight cost of a tax equals
the net loss in social welfare arising from the tax: the reduction in the sum
of consumer and producer surplus, minus the tax revenue.
The consumer incidence of the tax equals the increase in the price that
consumers pay, as a percent of the tax. The producer incidence equals the
reduction in the price that producers receive, as a percent of the tax. These
two incidences sum to 100%. In a closed economy (no international trade)
it does not matter whether the tax is imposed on consumers or producers;
the two taxes are equivalent. This fact implies that in some settings, the
Polluter Pays Principle is vacuous: regardless of whether the polluter or the
11.8. SUMMARY
205
pollutee directly pays, their actual cost is the same.
A simple formula relates approximate consumer and producer incidence
to supply and demand elasticities. Another approximation shows the deadweight loss of a tax. We also provided an estimate of the deadweight loss
of a tax, per unit of tax revenue. The deadweight loss of a tax rises is approximately proportional to the square of the tax. Therefore, reductions in
small taxes typically lead to small decreases in deadweight loss. In contrast,
reductions in large taxes result in large decreases in deadweight loss. Small
taxes lead to small deadweight loss, and also small tax revenue. Increases in
the tax increase both the deadweight loss and the tax revenue (at least up to
a certain tax level). However, the deadweight loss rises more rapidly than
does tax revenue. The deadweight loss might create a sizeable reduction in
the potential social bene…t arising from tax revenue.
The two basic “rules” of optimal taxation are that it is better to tax
commodities or factors for which the elasticity of supply or demand is small
(so that the deadweight loss is small), and it is better to have a broad tax
base (so that a given amount of revenue can be raised using small taxes in
each sector.
In the environmental context, we are primarily interested in taxes as a
means of correcting externalities such as pollution. These taxes remedy
market failures; they do not create social costs. However, the purpose of
many taxes, levied on “goods rather than bads”, is to raise tax revenue for
social or political purposes. To the extent that taxes on “bads”can replace
taxes on “goods”, the former not only correct market failures, but also reduce
the deadweight cost of revenue-raising taxes.
We used this machinery to examine a recent policy question: Under a cap
and trade system, what fraction of permits would …rms have to be given, in
order to leave them as well o¤ under the policy as they were before regulation?
We addressed this question using a particular model, in which there is a …xed
relation between emissions and output. In that setting, we explained and
illustrated the equivalence of an emissions tax and a cap and trade policy.
In particular, if a policy sets a cap of Z and allows trade, the equilibrium
permit price is p^ (Z). The equivalent emissions tax, = p^ (Z), results in
the same level of emissions, Z. If …rms are given the fraction of permits,
equal to the tax incidence they face under the tax = p^ (Z), plus a small
correction to account for their share of the deadweight cost of taxes, then the
cap and trade policy has approximately zero e¤ect on producer surplus. In
that case, the cap and trade policy can be used to reduce pollution without
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CHAPTER 11. TAXES: AN INTRODUCTION
harming …rms. Giving away instead of selling pollution permits is a loss
to the treasury, hurting taxpayers in general but bene…tting the regulated
…rms.
The e¤ect of taxes in a general equilibrium setting is more complicated
than in the partial equilibrium setting we emphasize. In particular, general
equilibrium relations can shift the tax incidence in subtle ways. In a dynamic
setting, involving investment, the short run elasticity of supply (or demand)
typically di¤er from their long run analogs. This di¤erence complicates the
problem of designing tax policy, and can lead to the time inconsistency of
optimal policy.
11.9
Terms, study questions and exercises
New terms or concepts
Unit tax, ad valorem tax, consumer and producer tax incidence, rational
buyers, behavioral economics, approximation of tax incidence, closed and
open economies, Polluter Pays Principle, tax equivalence, Harberger triangle, deadweight loss (or cost) of taxes, approximation of deadweight loss,
approximation of tax revenue, factor prices, general equilibrium e¤ects, computable general equilibrium (CGE) model, time consistency, cap and trade,
equivalence of a cap and trade and a tax policy, grandfathering.
Study questions
1. (a) What does it mean to say that a producer and a consumer tax
are “equivalent” in a closed economy? (Your answer should include
a de…nition of the term “incidence”. (b) Use either a graphical or a
numerical example to illustrate this equivalence in a closed economy.
(c) Using either a numerical or a graphical example, show that this
equivalence breaks down in an open economy.
2. (a) Use a graphical example to show how the consumer and producer
tax incidences (in a closed economy) depend on the relative steepness
of the supply and the demand functions at the equilibrium price. (b)
Using this example, explain how the approximation of consumer tax
incidence depends on the demand elasticity relative to the supply elasticity (i.e. the ratio of the two elasticities) evaluated at the no-tax
11.9. TERMS, STUDY QUESTIONS AND EXERCISES
207
equilibrium. [Begin by drawing a supply demand function, picking a
tax, and identifying the tax incidences. Then rotate one of the curves
around the no-tax equilibrium, making it much steeper (= less elastic) or much ‡atter (= more elastic) and show graphically how the
incidences change.]
3. Suppose that a regulator imposes a producer tax in a closed economy.
(a) Use the concept of producer tax incidence to approximate the fraction of the tax revenue that would have to be turned over to producers
to make them almost as well o¤ under the tax + transfer as they were
before the tax. (b) Use the concept of consumer tax incidence to approximate the fraction of the tax revenue that would have to be turned
over to consumers to make them almost as well o¤ under the tax +
transfer as they were before the tax. (c) Is it possible, by means of
transferring the tax revenue (associated only with this particular tax),
to make both producers and consumers exactly as well o¤ under the
tax + transfer as they were before the tax? Explain
4. (a) Describe a cap and trade policy. (Explain how it works.) (b)
Explain what it means to auction permits. (c) Explain why …rms’
equilibrium level of pollution does not depend on whether permits are
given to the …rm or auctioned.
5. (a) Explain what is meant by the claim that a pollution tax and a cap
and trade policy are equivalent. (b) Explain why the claim is true (in
the particular setting we used). (c) Suppose that instead of using a cap
and trade, a regulator uses a “cap and no trade”policy, in which …rms
are allocated pollution permits but not allowed to trade them. Is a
pollution tax equivalent to a “cap and no trade policy”? Explain. For
example, if you claim that the two policies are still equivalent, without
trade, you should justify that conclusion. If you claim that the two
policies are di¤erent, with respect to some signi…cant outcome, you
should identify and explain the di¤erence.
Exercises
Assume for all questions that the economy is closed.
1. Consider a monopoly, in a static setting, with constant marginal cost
c. If the monopoly receives price p and consumers pay a unit tax
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CHAPTER 11. TAXES: AN INTRODUCTION
, consumers’ tax-inclusive prices is pc = p + ; if instead they pay
the ad valorem tax , they pay the tax-inclusive price pc = p (1 + ).
The demand function is q = a pc . (a) Write down the monopoly’s
maximization problem and …rst order condition in the two scenarios,
with a unit and an ad valorem tax. (b) Solve these two …rst order
conditions to …nd the equilibrium consumer price in these two cases.
(In one case this price is a function of and in the other case it is a
function of .) (c) Use part b to …nd the relation between and
such that if the taxes satisfy this relation, the consumer price is the
same under either tax.
2. (a) Draw a linear demand function and a linear marginal cost function. Use this …gure to identify (graphically) consumer and producer
incidence of a unit tax, , in a competitive equilibrium. (b) Reproduce the …gure you drew from part (a), except now make the supply
function steeper at the zero-tax competitive equilibrium. Identify the
consumer and producer tax incidence and compare these to your answer in part (a). (c) Explain the relation between your answer to part
(b) and equation 11.3. (d) Reproduce the …gure that you drew from
part (a). A regulator uses a unit tax . Use this new …gure to compare
the consumer tax incidence in a competitive equilibrium and under a
monopoly.
3. (a) Using the approximation of deadweight loss in equation 11.4, show
that deadweight cost increases with either the elasticity of demand or
the elasticity of supply. (Hint: take a derivative.) For this question,
you are holding the tax and the zero-tax equilibrium quantity and price
constant, and considering the e¤ect of making either the demand or the
supply function ‡atter (more elastic) at this equilibrium. (b) Provide
an economic explanation for the relation you showed in part (a).
4. A pro…t tax is usually expressed in ad valorem terms. If …rms have profits py c (y) and pay a pro…t tax , their after-tax pro…t is (1
) [py c (y)].
(a) How, if at all, does a pro…ts tax a¤ect the equilibrium price in a
competitive equilibrium? (b) What, if anything, does a pro…t tax affect in a competitive equilibrium? (Make a list of the features of the
competitive equilibrium that we care about, and ask which if any of
those features are altered by the pro…ts tax.) (c) Now answer parts
(a) and (b), replacing the competitive …rm with a monopoly.
11.9. TERMS, STUDY QUESTIONS AND EXERCISES
209
5. A monopoly has constant costs, c. Consumers, facing price pc , demand
q = a pc units of the good; so the inverse demand is pc = a q. (a)
Write down the monopoly pro…ts, as a function of its sales, q, in the two
cases where consumers pay the unit tax , and then when the monopoly
pays the unit tax . (b) Compare the pro…t function in these two cases.
Based on this comparison, does the monopoly equilibrium depend on
which agent (consumers or the monopoly) directly pays the tax?
Sources
Gentry (2007) reviews evidence that labor bears a signi…cant share of the
incidence of corporate taxes.
Feldstein (1977) discusses the general equilibrium e¤ects of a land tax.
Diewert et al. (1998) and Conover (2010) review estimates of tax incidence; Table 11.1 is based on Conover.
Judd (1985) discusses the time path of capital taxes, and Karp and Lee
(2003) discuss the time-inconsistency of the optimal program.
The World Bank (2014) surveys carbon markets across the world.
Fowlie, Holland and Mansur (2012) document the success of the RECLAIM market for NOx .
Fowlie and Perlo¤ (2013) …nd support for the hypothesis that emissions
levels do not depend on permit allocations (Claim 1 and Box 11.2).
McAusland (2003 and 2008) compares environmental taxes in open and
closed economies.
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CHAPTER 11. TAXES: AN INTRODUCTION
Chapter 12
Taxes: nonrenewable resources
Objectives
Be able to synthesize the material on nonrenewable resources and on
static taxes to understand the role of taxes applied to nonrenewable
resources.
Information and skills
Have an overview of actual fossil fuel taxes.
Understand the time consistency problem arising from quasi-rent.
Understand how taxes alter a …rm’s extraction incentives.
Be able to analyze the equilibrium e¤ects of di¤erent (e.g. constant
versus increasing) tax pro…les .
Understand how a tax-induced change in the price trajectory maps into
a trajectory of consumer and producer tax incidence.
Be able to explain why, in the resource setting, consumer tax incidence
is negative over some periods.
Understand and be able to describe the measures of the present discounted ‡ow of consumer surplus/producer surplus/tax revenue.
211
212
CHAPTER 12. TAXES: NONRENEWABLE RESOURCES
With static markets, consumers’ and producers’ decisions depend only
on the current price. For natural resources markets, …rms’ current sales
decisions depend on their expectations of future prices, and on the relation
between those prices and the current price. Some results from static tax
analysis carry over to the dynamic setting. For example, in a closed economy
the incidence of the tax is the same regardless of whether it is levied on
consumers or producers, and for every unit tax, there is an equivalent ad
valorem tax. Therefore, there is no loss in generality in assuming that
producers rather that consumers have the statutory obligation to pay a unit
tax.
In other respects, taxes might have qualitatively di¤erent e¤ects in a static
setting and in a dynamic setting with natural resources. In a static setting,
taxes reduce equilibrium supply. In the resource setting, a tax reallocates
supply across periods, but might have no e¤ect on cumulative supply. The
tax-induced change in the timing of sales can have important welfare e¤ects.
12.1
Resource taxes in the real world
Natural resources, particularly fossil fuels, are an important part of the world
economy, so it is not surprising that governments derive substantial revenue
from their taxation. Between 2005 – 2010 the (mostly rich) 24 countries
in the Organization for Economic Cooperation and Development (OECD)
raised about $850 billion per year on petroleum taxes, including goods and
services taxes and value added taxes. These taxes served a variety of purposes, including internalization of environmental externalities, road building
and maintenance, and general government expenses. For large oil producing
countries, government receipts from the hydrocarbon sector was a large fraction of total government revenue (2000 - 2007 data): 72% for Saudi Arabia,
48% from Venezuela, and 22% for Russia.
Despite the large contribution of resource taxes to government co¤ers,
the important story is that in many rich countries the oil sector receives signi…cant implicit subsidies in the form of tax deductions. Just as a tax on
producers is an implicit tax on consumers (Chapter 11), so a producer subsidy is an implicit consumption subsidy. Other countries, principally middle
income and developing countries that export fossil fuels, directly subsidize
domestic fuel consumption by maintaining a domestic price lower than the
world price. For both groups of countries, these policies subsidize fuel con-
12.1. RESOURCE TAXES IN THE REAL WORLD
213
sumption, creating signi…cant distortions.
In 2009, leaders of the G20 (a group of wealthy countries) committed
to “rationalize and phase out over the medium term ine¢ cient fossil fuel
subsidies that encourage wasteful consumption”. A group of international
organizations, including the OECD and World Bank, estimated the scope
of energy subsidies and made suggestions for their reduction. Their report
identi…ed 250 individual mechanisms that support fossil fuel production in
the OECD countries, having an aggregate value of USD $ 45 -75 billion per
year over 2005 - 2010; 54% of this subsidy went to petroleum, 24% to coal,
and 22% to natural gas. In the U.S., tax breaks provide implicit fossil
fuel subsidies of about $4 billion per year. The three most important of
these are: write-o¤s for intangible drilling costs, a domestic manufacturing
tax deduction, and a percentage depletion allowance for oil and gas wells.
These tax breaks transfer income from taxpayers to resource owners, without
promoting the development of new technologies.
A group of 37 emerging and developing countries provide direct domestic
fuel consumption subsidies, maintaining domestic prices below international
prices. This group accounts for over half of world fossil fuel consumption
in 2010. With a domestic price of pd and a world price of pw , the per unit
subsidy is pw pd . The nation loses pw pd times the amount of subsidized
consumption from selling fuel at the low domestic price instead of the higher
world price. Most of the countries maintained a stable domestic price, while
the world price ‡uctuated, causing the per unit subsidy to also ‡uctuate. The
cost of the subsidies to their domestic treasuries amounted to $409 billion in
2010 and $300 billion in 2009. Oil received 47% of total, and the average
subsidy was 23% of the world price. The subsidy rates were highest among
oil and gas exporters in Middle East, North Africa and Central Asia.
A common justi…cation for fuel subsidies is that they bene…t the poor,
providing them with access to energy services. However, only 8% of the
$409 billion subsidy in 2010 went to poorest 20% of the population. If
the subsidy had been eliminated, and the fuel sold at world price, and each
person then given an equal share of the proceeds, the poorest 20% would
have received approximately twice as much as they did under the subsidy.
This approximation probably overstates the ine¢ ciency of the subsidy as a
means of helping the poor. First, if all of this fuel had been diverted from
the domestic to the world market, it would have lowered the world price,
thus lowering the gain from the sales. Second, given the levels of domestic
corruption, it is unlikely that the poor would have received an equal share
214
CHAPTER 12. TAXES: NONRENEWABLE RESOURCES
of the increased revenue resulting from removal of the subsidies. Despite
these two quali…cations, fuel subsidies – like most commodity subsidies –
are an ine¢ cient way to help the poor. There is modest evidence that the
magnitude of these subsidies has been decreasing. They fell from about 1.8%
of government budgets in 2004 to 1.3% in 2010.
Absent reforms, 2011 estimates project that these fossil fuel subsidies
would reach $660 billion per year by 2020. The elimination of consumption subsidies was estimated to reduce 2020 fuel demand by 4.1% and CO2
emissions by 4.7%.
A 2015 International Monetary Fund (IMF) study updates estimates of
the magnitude and economic cost of fossil fuel subsidies, and also includes
unpriced externalities. For example if the cost of supplying a unit of the
good, inclusive of rent, is 10, and the externality cost is 2, the e¢ cient price
is 12. If the externality is not taxed, and in addition a subsidy reduces the
cost to consumer by 1, then consumers pay 9. In this example, the total
subsidy per unit equals 3. For fossil fuels, the externality cost is larger than
the direct subsidy. Local health e¤ects, not climate change, accounts for the
bulk of this externality cost. The IMF study estimates that global energy
subsidies (including the externality cost) amounted to about $5 trillion, or 6%
of world Gross Domestic Product (GDP) in 2013; removal of these subsidies
would have raised almost $3 trillion in government revenue, and would have
increased global GDP by more than 2%. For comparison, estimates of the
increase in welfare due to major trade liberalization are typically only a
fraction of a percent of GDP.
Fossil fuel subsidies are ine¢ cient for at least four reasons. Most importantly, they subsidize a commodity that should, because of environmental
externalities, be taxed. The subsidies also violate both of the two “rules”
of optimal taxation discussed in Chapter 11.6. The …rst rule states that,
for the purpose of raising government revenue, goods with inelastic supply
and/or demand should be the most highly taxed. Fossil fuels likely fall into
that category, but in practice they are subsidized. The second rule states
that governments should seek to have a broad tax base, so that taxes on each
sector can be low. Subsidizing rather than taxing the fossil fuel sector ‡ies in
the face of this advice, requiring higher taxes in other sectors to …nance the
fossil fuel subsidies. Finally, commodity subsidies are an ine¢ cient means of
making transfers to the poor. Political power, not economic logic, explains
the tax and subsidy policies used in a wide range of fossil fuel markets, for
both importers and exporters countries, and for rich and developing nations.
12.2. THE LOGIC OF RESOURCE TAXES
12.2
215
The logic of resource taxes
Just as in the static setting, taxes (and subsidies) drive a wedge between consumer and producer price. Typically, they alter people’s decisions, creating
an e¢ ciency cost, the deadweight loss. If either the elasticity of supply or
demand are zero, then the tax (or subsidy) does not alter behavior, and it
creates zero deadweight loss. The deadweight cost of a tax is low when either
of these elasticities is low. This fact is the basis for much of the theory of
taxes; it helps to explain the popularity of the 19th century recommendation
of a “single tax” on unimproved land, a factor of production with inelastic
supply.
Taxes applied to natural resources can alter both the timing of extraction
and cumulative extraction. To focus on the former, we consider the case
where …rms have constant average (and marginal) extraction cost C. We
then discuss the relation between taxes and investment.
Taxes and the timing of extraction: special cases
(First special case) Suppose that …rms have constant average production
costs C and the government introduces a positive constant unit tax, . From
the standpoint of the …rm, it is as if its costs increased from C to C 0 = C + .
We saw in Chapter 3.3 that a higher extraction cost causes …rms to shift
production from the current period to future periods. The intuition for this
result is that discounting reduces the present value of costs that are incurred
in the future. If a …rm moves one unit of extraction from the current period
to t periods in the future, its current tax liability falls by $ and the present
value of its future tax liability increases by t < . Shifting extraction
from the current time to the future lowers the present value of the …rm’s tax
liability. Therefore, the constant tax causes the …rm to delay extraction.
In a static model, if …rms have constant production costs, then their
elasticity of supply, , is in…nite. In this case, using equation 11.3, the ratio
= 0: …rms incur
of producer tax incidence to consumer tax incidence,
none of the incidence of the tax. That result has a simple explanation.
With constant production costs, …rms earn zero producer surplus. They
cannot possibly earn less than zero surplus, so imposition of the tax leaves
their surplus unchanged at zero. Because the tax has no e¤ect on the price
that producers receive, consumers’price rises by the magnitude of the tax:
consumers bear the entire tax incidence.
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CHAPTER 12. TAXES: NONRENEWABLE RESOURCES
The resource model with constant extraction costs resembles the static
model, but there is an important di¤erence. In the resource model, extraction costs are constant up to the point where the resource is exhausted, but
beyond that point, extraction costs are in…nite, i.e. the elasticity of supply
becomes zero: …rms cannot supply the resource when they have no more of
the stock. The …nite supply of the resource creates rent. In the resource
model, we do not have a formula for measuring tax incidence as simple as
those in equation 11.3. However, from the discussion above, we see that a
constant tax decreases supply, increasing the equilibrium price early in the
program, and has the opposite e¤ect later in the program. Thus, the consumer incidence of the tax early in the program is positive (just as in the
static model), but late in the program it is negative.
(Second special case) Now we assume that (i) the tax increases at the
discount rate, so that the tax in period t is t = (1 + r)t 0 , where 0 is a
positive constant; and (ii) the tax does not become large enough to make the
…rm want to cease production while it has remaining resource stock. Let T
be the …nal date of extraction under a zero tax. Under the tax, the taxinclusive unit cost for the …rm in period T is C + (1 + r)T 0 . Assumption
(ii) states that C + (1 + r)T 0 is less than the choke price.
Consider the perturbation that reduces period 0 extraction by one unit,
and makes an o¤setting one unit increase in period t extraction. The reduction in current extraction reduces the present value of the …rm’s tax liability
by 0 ; the increase in period t extraction increases the present value of the
…rm’s tax liability by t t = t (1 + r)t 0 = 0 . In this case, moving a
unit of extraction from the current period to a future period has no e¤ect
on the present value of the …rm’s tax liability. This tax has no e¤ect on the
extraction pro…le or on consumer prices. The consumer incidence is 0; …rms
bear the entire tax incidence.
The present value of the tax receipts is
T
X
t=0
t
(1 + r)
t
0 yt
=
0
T
X
yt =
0 x0
t=0
This tax transfers the rent 0 x0 from producers to taxpayers, without creating
a distortion. Equation 5.9 shows that under constant extraction costs and
zero tax, the value of the …rm is R0 x0 . Therefore, under the tax considered
here, the after-tax value of the …rm is (R0
0 ) x0 , where R0 is the pre-tax
initial rent. By setting the initial tax, 0 , close to R0 , the government can
12.2. THE LOGIC OF RESOURCE TAXES
217
extract nearly all of the rent from the resource owner.
Investment
Rent is the return to a factor of production in …xed supply; quasi-rent is
the return to a previous, now “sunk”, investment. Chapter 4.3 notes that
most natural resource stocks require signi…cant investment in exploration and
development. Thus, the di¤erence between the price that a …rm receives and
its marginal cost of extracting the resource, is actually the sum of rent and
quasi-rent. We referred to this sum as “rent” merely in the interest of
simplicity. Now, for the purpose of studying tax policy, the distinction is
important.
For example, suppose that the initial rent + quasi-rent for a mine with
constant extraction costs is R0 = 10 and the initial stock is x0 = 10, so the
value of this mine (using equation 5.9) is 100. Suppose, in addition, that
each mine costs 5 to develop, and on average a …rm must develop …ve mines to
…nd one that is successful. We assume that the development cost 5 includes
the cost of capital and that the …rm can test many potential mines at the
same time, understanding that on average 20% of them will be successful.1
It is important to account for all of the unsuccessful mines in estimating
the development cost of a successful mine. In this example, the expected
investment cost for a successful mine is 25. Here, the rent on the mine is 75
and the quasi-rent is 25.
Prior to the exploration and development, the government might announce a tax that begins, at the time of initial extraction, at 7:5, and rises
at the rate of interest. With this tax, the government captures all of the
rent, but leaves …rms with the quasi-rent. In this case, …rms break even,
and are willing to undertake the investment. (As a practical matter, it
would be important to leave …rms with some rent, in order to induce them
to undertake the risk; we ignore that complication, in assuming that …rms
are risk neutral.) Once a successful mine is in operation, the government
might be tempted to renege on its announcement, and to impose an initial
tax of 0 = 10, which increases at the rate of interest. This tax transfers all
of the rent + quasi-rent from the …rm to taxpayers. The …rm in this case
loses all of its sunk investment costs. This temptation to renege illustrates
the potential time-inconsistency of optimal plans.
1
For example, if …rms have to invest I at time 0 to learn, after three years, whether
3
the mine is successful, then I (1 + r) = 5.
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CHAPTER 12. TAXES: NONRENEWABLE RESOURCES
The real-world importance of this time inconsistency varies with the setting. If there are many cycles of investment and extraction, then the government has an incentive to adhere to its promises in order to maintain its
reputation. By behaving opportunistically, this government obtains some
short run bene…ts, in the form of higher tax revenue, but it discourages future investment: it kills the goose that lays the golden eggs. This time
inconsistency is therefore more important for a single large project, e.g. a
one-time development of o¤shore oil deposits. (All of these considerations
also apply to non-resource related infrastructure projects, e.g. building a harbor or developing a transportation network.) In this case, the government’s
temptation to renege may be greater if the major investors are foreigners; domestic investors might be better able to defend their interests in the political
process that determines the tax.
For large one-o¤ foreign-sourced investment projects, time inconsistency
can be a major issue. Without some protection, the foreign investors would
be unwilling to undertake the project, or would require a large risk premium
for doing so. This reluctance to invest is an example of the “holdup problem”, which occurs whenever an action that is mutually bene…cial (here,
investment) leaves one party exposed to the risk of opportunistic actions
(here, con…scatory tax) by the other.
The OECD and other international organizations attempted, during the
1980s and 1990s, to negotiate a Multilateral Agreement on Investment to
resolve this hold-up problem. The hope was that this agreement would encourage international investment, in much the same way that the World
Trade Organization promotes international trade. Largely due to resistance
from developing countries, this e¤ort was unsuccessful. However, there are
currently thousands of Bilateral Investment Treaties (BITs), most of which
involve one rich and one developing country; the U.S. is party to over 40
BITs.
The parties of these treaties are countries, not private …rms, but many
of the treaties have an “investor-to-state”provision. This provision permits
a private investor originating in one signatory (usually the rich country)
to sue, in an international court, the government of the other signatory
(usually the developing country) for violation of the treaty. A primary
purpose of the treaties is to protect an investor against overt con…scation of
their investment, but many treaties also provide protection against measures
that are “tantamount to expropriation”, such as “con…scatory taxes”or even
post-investment changes to environmental rules. Business interests regard
12.3. AN EXAMPLE
219
tax
7
6
5
4
3
2
1
0
1
2
Figure 12.1: The tax pro…le for
(dashed).
3
4
5
6
7
8
9
= 0:05 (solid),
10
11
12
t
= 0:5 (dotted) and
= 0:8
the investor-to-state provision of these treaties as essential, because they
are not con…dent that their own government would act in their interests.
For example, foreign policy considerations might make the U.S. government
reluctant to invoke a treaty in protection of a U.S. investor. The investor,
in contrast, has no such qualms about exercising the treaty rights.
The treaties provide a self-enforcing mechanism, a credible commitment
against opportunistic behavior. However, they also limit a country’s ability
to exercise what are known as “police powers” (e.g. environmental regulation) and to respond to contingencies not foreseen at the time of investment
(e.g. a major recession). Although business groups strongly support these
treaties, some NGOs think that they harm developing country interests.
12.3
An example
A numerical example illustrates the e¤ect on resource markets of a particular
time-varying tax, (t) = b (t + 1) , with b
0 and 1 >
0. At time
t = 0, the tax equals is (0) = b. The rate of increase of the tax is
d (t)
dt
(t)
=
t+1
:
The rate of increase begins (when t = 0) at and falls steadily to 0 at t
increases. The tax increases over time, approaching 1 as t ! 1, but the
tax increases at a decreasing rate; after a while, the tax increases very slowly.
The e¤ect of the tax on the incentive to extract is intuitive. If is close
to 0, so the tax grows slowly, then the situation is similar to the constant tax
220
CHAPTER 12. TAXES: NONRENEWABLE RESOURCES
case. We saw (Chapter 12.2) that in this case the tax creates an incentive
to delay extraction, as a means of decreasing the present value of the tax
liability. If is large, the tax grows quickly. Here, the …rm has an incentive
to accelerate extraction so that more of its sales incur the relatively low
current taxes instead of high future taxes, reducing the …rm’s present value
tax liability.
Figure 12.1 shows the tax pro…le for three values of for b = 1. These
tax begin at (0) = 1 and approach in…nity as t becomes large. With
= 0:05, the tax is approximately 1.12 after 12 periods and with = 0:8
the tax reaches 7.6 after 12 periods. Larger values of imply a more rapid
increase in the tax.
Box 12.1 The e¤ect of the tax on the timing of extraction. By extracting one less unit in period 0 and instead extracting that unit
in period t, the present value of the tax liability falls by b and int
creases by t b (t + 1) , for a net decrease of b
b (t + 1) . A bit of
computation establishes that
d ( t b (t + 1)
dt
b)
> 0 if and only if ln (1 + r) >
t+1
:
For small r, ln (1 + r)
r. Using this approximation, we see that
if r > t+1 , the tax creates an incentive to delay extraction. This
inequality holds for all t if = 0, the case we already considered in
Chapter 12.2. However, if is large, then the inequality does not hold
early in the program (when t is small); in that case, the tax creates
an incentive for the …rm to extract earlier rather than later.
12.3.1
Calculating the price trajectory
For our example, demand is 10 p, the constant average cost is C = 1,
the initial stock is x0 = 20, and annual discount rate is 4%. We …nd
the equilibrium price trajectory (for a given tax pro…le) and then graph
the consumer and producer incidence corresponding to that tax trajectory.
Consumers pay the price p (t), so the producer’s after-tax price is p (t)
(t).
The rent at at time t is p (t)
(t) C.
The Euler equation requires that the present value of rent is the same at
each instant when extraction is positive. In particular, the present value of
12.3. AN EXAMPLE
221
rent at time t equals the present value of rent at the …nal time, T :
t
(p (t)
C
(t)) =
T
(p (T )
C
(T )) :
(12.1)
The resource is (by assumption) exhausted, i.e. cumulative sales equal the
initial stock, x0 :
T
X
(10
p (t)) = x0 = 20:
(12.2)
t=0
The amount extracted at time t equals the amount consumed, which equals
10 p (t). Therefore, the left side of equation 12.2 equals the amount
extracted during the program, which equals the initial stock; for our example,
it is not optimal to stop extracting while some resource remains in the ground.
Chapter 7 shows that it is possible to solve these equations (together
with the transversality condition) to …nd the complete solution: the number
of periods, T , and the level of sales and the price in each period. As we noted
before, it is easier to solve (but more di¢ cult to explain) the continuous time
version of this problem. In addition, the smooth graphs in the continuous
time setting are easier to interpret than the step functions (or points) that
arise in the discrete time model. We therefore present all results in this
section using a continuous time model.
Box 12. The continuous time model The Euler equation in the continuous time setting implies equation 12.1. To obtain the continuous
time version of equation 12.2, we replace the summation with an integral, to write
Z
T
(10
p (t)) dt = 20:
(12.3)
0
At any time after T , when the resource has been exhausted, the price
equals the level that drives demand to 0, so p (t) = 10 for t > T . If
it were the case that p (T ) < 10, there would be an upward jump in
the price at time T . That jump would imply that by delaying extraction an in…nitesimal amount of time, the resource owner increases
the present discounted value of pro…ts; the jump therefore violates the
no-intertemporal-arbitrage condition. Consequently, p (T ) = 10, the
choke price, in the continuous time model.
222
CHAPTER 12. TAXES: NONRENEWABLE RESOURCES
p
11
10
9
8
7
6
5
4
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
t
Figure 12.2: The equilibrium consumer price trajectory under the tax =
(t + 1)0:05 (solid); under the tax = (t + 1)0:8 (dashed); and price under zero
tax, = 0 (dotted). Demand = 10 p, constant extraction costs c = 1,
initial stock x0 = 20, and discount rate r = 0:04. Graphs are shown for
continuous time model.
12.3.2
The price trajectories
Using p (T ) = 10, we can solve equations 12.1 and 12.3 numerically for the
three taxes. For the zero tax, = 0, we obtain the terminal time T = 11:2.
For the slowly growing tax, = (t + 1)0:05 , the terminal time is only slightly
greater, T = 11:5. The rapidly growing tax, = (t + 1)0:8 , gives producers a
strong incentive to extract early, while the tax is still relatively low, leading to
a much earlier exhaustion time, T = 7:5. Figure 12.2 shows the equilibrium
price trajectories under these three tax pro…les. Each trajectory reaches the
choke price at the time of exhaustion.
The equilibrium price trajectories under the zero tax (dotted) and under
the slowly increasing tax, = (t + 1)0:05 , (solid) are almost indistinguishable.
The positive tax encourages …rms to delay extraction, which tends to increase
the initial price; but the fact that the tax pro…le is increasing encourages …rms
to move extraction forward in time, which tends to reduce the initial price.
When the tax rises slowly, these two e¤ects almost cancel each other, but
the net e¤ect is to slightly increase the initial consumer price, from 6.72 to
6.86, and to slightly delay the exhaustion date. (Compare the solid and
the dotted graphs.) The steeply rising tax trajectory gives …rms a much
stronger incentive to extract early, while the tax is still relatively low. The
12.3. AN EXAMPLE
incidence
223
120
100
80 producer
incidence
60
40
20
consumer incidence
0
1
2
3
4
5
6
7
8
9
10
11
t
-20
Figure 12.3: Consumer and producer incidence of tax under the slowly increasing tax
steeply rising tax therefore leads to a substantially lower initial price and
earlier exhaustion. (Compare the dashed and the solid graphs.)
12.3.3
Tax incidence
Figure 12.3 graphs the producer and the consumer tax incidences, as a function of time, under the slowly growing tax. As in Chapter 11.2, the consumer
incidence is de…ned as the di¤erence in the consumer price with and without
the tax, divided by the tax, times 100 (to convert to a percent). The consumer and producer tax incidences sum to 100 by de…nition, so we obtain the
producer incidence by subtracting the consumer incidence from 100. The
consumer incidence begins at about 15% but over time falls to a negative
level. The price paths corresponding to the zero and the slowly growing
tax shown in Figure 12.2 (the dotted and solid curves, respectively) cross at
about t = 6:5. At later dates, the tax reduces the equilibrium consumer
price, so the consumer incidence is negative there. Over this region, the
producer incidence is greater than 100%.
Figure 12.4 graphs the consumer and producer tax incidence over time
under the rapidly growing tax. This tax lowers the equilibrium consumer
price for t < 4:5, so over this region, the consumer incidence is negative, and
the producer incidence exceeds 100%. At later dates, the tax incidence for
both consumers and producers lies between 0 and 100%.
These last two …gures show that even in an extremely simple model, the
224
CHAPTER 12. TAXES: NONRENEWABLE RESOURCES
incidence
300
200
producer incidence
100
0
1
2
3
4
Consumer incidence
5
6
7
8
9
10
11
12
t
-100
-200
Figure 12.4: Consumer and producer incidence of tax under the rapidly
increasing tax
trajectories of tax incidence can be quite complicated. The tax causes a
reallocation of supply over time, but no change in aggregate supply (equal
to the initial stock) over the life of the resource. Thus, there must be some
periods during which the tax lowers the equilibrium consumer price. Over
those periods, the consumer tax incidence is negative, and the producer tax
incidence exceeds 100%. In contrast, in a static competitive model both the
consumer and producer tax incidences lie between 0 and 100%.
Box 12.3 The pro…ts tax Exercise 2 in Chapter 11 explores the e¤ect
of a constant pro…ts tax in a static model. That exercise shows that
a constant pro…ts tax, , extracts rent from producers, but has no
e¤ect on equilibrium price or quantity. This result also holds for
resource markets. The constant pro…ts tax reduces pro…ts in each
period proportionally, and does not alter the …rm’s incentives about
when to produce. A time varying pro…ts tax, (t), does alter the
…rm’s incentives, and therefore a¤ects consumers in addition to …rms.
12.3.4
Welfare changes
If the price in period t is p, consumer surplus (CS (t)), producer pro…t
(P S (t)), and tax revenue (G (t), for “government”) in that period equal,
12.3. AN EXAMPLE
225
respectively,
CS =
R 10
p
(10
q) dq = 50
P S = (p
(t)
10p + 12 p2
1) (10
G = (t) (10
(12.4)
p)
p) :
The equilibrium p changes over time, so the functions CS; P S; G also change
over time. In static models, we are accustomed to thinking about a component of social welfare (e.g. consumer or producer surplus or tax revenue) as
a single number. In dynamic models, it is important to think of these payo¤s
as being trajectories.
We de…ne an agent’s welfare as the present discounted value of their
stream of single period payo¤s. Welfare for consumers, producers, and
taxpayers equal, respectively,
Z T
Z T
Z T
t
t
t
G (t) dt:
(12.5)
P S (t) dt,
CS (t) dt,
0
0
0
The sum of these three integrals equals social welfare. Table 1 shows welfare
for consumers, producers, and taxpayers, and the sum of these three welfare
measures (social welfare) under the zero tax and for both the slowly and the
rapidly growing tax. The table also shows the percent change in welfare for
consumers, producers, and society as a whole, in moving from the 0 tax to
either of the two positive taxes.
consumer
welfare
zero tax
20.2
“slow tax” 19
% change
-5.9%
“fast tax” 33.1
% change
+64%
producer
welfare
114.4
97.2
-15%
50.1
-56.2%
taxpayer
welfare
0
18.3
NA
47
NA
social
welfare
134.6
134.5
-0.07%
130.2
-3.3%
DWL
tax rev
5. 5 10
3
9. 4 10
2
Table 12.1: Agents’welfare and % change in welfare under the zero tax,
the “slow tax” (t) = (t + 1)0:05 , and the “fast tax” (t) = (t + 1)0:8 . “%
change”is relative to = 0.
Just as in the static model, the slowly growing tax reduces consumer
and producer welfare, here by 6% and 15%, respectively. The increase in
tax revenue (from 0 to 18.3) almost o¤sets those two reductions, so social
226
CHAPTER 12. TAXES: NONRENEWABLE RESOURCES
welfare under this tax falls by less than a tenth of 1%. Social welfare falls
by over 3% under the rapidly growing tax. We previously observed this
pattern in the static model. In discussing Figure 11.3, we noted that any
non-zero tax leads to deadweight loss, and the deadweight loss (or at least
its approximation) increases with the square of tax. The deadweight loss
is negligible at a small tax, and increases with the magnitude of the tax.
The social loss per unit of tax revenue (both expressed as present discounted
sums) equals 134:618:3134:5 = 0:0055 for the slowly growing tax, and 134:647130:3 =
0:094 for the rapidly growing tax. To convert to percentage, multiply by
100. For example, the social cost as a percent of tax revenue for the slowly
growing tax is 0.55%, i.e., about one half of one percent. For both of these
examples, the social loss per unit of tax revenue is negligible.
In the static setting, the tax (weakly) increases the equilibrium price
that consumers face, and thus lowers consumer welfare. In contrast, in the
nonrenewable resource setting, the tax increases price in some periods and
lowers prices in other periods. Depending on the magnitude and the timing
of the changes, consumer welfare might either increase or decrease. Figure
12.2 shows that the rapidly growing tax leads to a fall in consumer price early
in the trajectory, and an increase in the consumer price late in the trajectory.
The tax thus increases the ‡ow of consumer surplus early in the trajectory
and lowers the ‡ow of consumer surplus late in the trajectory. Because of
discounting, the early increases count for more, in terms of present discounted
value, than do the later reductions. Thus, the rapidly growing tax increases
the present discounted value of the stream of consumer surplus. This tax
bene…ts consumers and leads to a large increase in tax revenue (from 0 to
47), but it also causes a large (56%) decrease in producer welfare. The tax
decreases aggregate social welfare by 3.3%.
12.3.5
Consumer welfare in a dynamic setting
We proceeded as if using the present discounted stream of surplus to measure
consumers’and producers’welfare is so obvious as not to require comment.
Although this procedure is standard in economics, it has a serious shortcoming when applied to consumers. We mention the problem here, and take it
up more carefully in Chapter xx. Suppose that we have in mind a resource
that will last 200 years. Clearly, the consumers alive today are not the same
people who will be consuming in 100 years or more.
What does it mean to say that “consumer welfare” increases under the
12.4. SUMMARY
227
rapidly growing tax? The individuals currently consuming are clearly better
o¤, because the tax causes them to face a lower price. However, the consumers late in the period are worse o¤, because the tax causes them to face
a higher price, or possibly leads to exhaustion of the resource before they
are born. The ethical objection to our welfare measure is that it adds up
the utility of people who are alive at di¤erent points in time. Moreover, it
does so in a way that privileges those currently living, because the welfare
measure discounts the surplus of future generations of consumers.
Our measure of social surplus at a point in time also adds up the surplus
of possibly di¤erent people, consumers and producers. That aggregation is
ethically less questionable, for two reasons. First, consumers and producers
may or may not be di¤erent people, whereas individuals living today and in
100 years are certainly di¤erent people. Second, consumers and producers
currently living have at least the potential for in‡uencing current government
policy that a¤ects their well-being. People living in the future have no direct
voice in the current political process.
12.4
Summary
The theory of optimal taxation suggests that fossil fuels should be relatively
heavily taxed, in order to o¤set the negative externality associated with their
consumption, and to take advantage of their relatively low elasticity of supply. In practice, both rich and developing nations, and both importers and
exporters subsidize consumption of fossil fuels. Political constraints impede
international attempts to move toward more rational resource policies.
Although the equilibrium e¤ects of taxes applied to nonrenewable resources is more complicated than in the static model, the methods developed
in earlier chapters make the outline of the analysis straightforward. A constant tax on extraction encourages …rms to delay extraction, raising consumer
prices early in the program and lowering prices later in the program. A tax
that increases at the rate of interest has no e¤ect on the extraction trajectory, because the present value of this tax is constant. A tax that increases
rapidly over time increases …rms’ incentive to extract early, before the tax
is high. This e¤ect tends to o¤set the incentive of a constant tax to delay
extraction. If the tax increases very slowly, then the “delay incentive”is the
stronger of the two. However, if the tax increases rapidly, producers respond
to the tax by moving the extraction pro…le forward in time. Thus, a rapidly
228
CHAPTER 12. TAXES: NONRENEWABLE RESOURCES
increasing tax lowers price early in the program and increases later prices.
Taxes (other than the tax with constant present value) lead to a reallocation of supply over time. Therefore, there are some periods when the tax
leads to an increase in supply, relative to the no-tax case; for those periods,
the consumer incidence of the tax is negative, and the producer incidence of
the tax exceeds 100%. These periods of negative consumer tax incidence
tend to occur late in the program, if the tax is constant or increases slowly;
those periods tend to occur early in the period if the tax increases rapidly.
The zero-tax competitive equilibrium maximizes social surplus, so in the
setting here a tax of any nature reduces social surplus. In the static competitive setting, both consumers and producers bear some of the incidence
of the tax. The tax therefore decreases the welfare of both agents. In
the resource setting with stock-independent extraction costs, the tax shifts
production from one period to another, without altering cumulative production. In this case, the tax must lower price and therefore increase consumer
surplus in some periods. The tax might either increase or decrease the
present discounted value of the stream of consumer surplus, as our examples
showed. However, the tax reduces the producer price (equal to the price
consumers pay minus the unit tax) in every period. Therefore the producer
incidence is positive in every period. In addition, the tax lowers producer
pro…t (equal to the producer price minus extraction costs, times sales) in
every period. Therefore, the tax necessarily lowers the present discounted
stream of producer pro…t.
We emphasize the relation between taxes and extraction decisions, usually
keeping the investment decision in the background. However, investment
is important in resource markets, creating “quasi-rents”, the return to a
previous investment. Price minus marginal costs, which we usually refer to
as “resource rent”is in fact the sum of genuine rent and quasi-rent. The fact
that investment is sunk at the time of extraction creates a time consistency or
holdup problem for policy makers. This problem is likely most severe in the
case of large, one-o¤ foreign investments. Bilateral investment treaties, with
an investor-to-state provision, is an attempt at solving this hold-up problem.
12.5. TERMS, STUDY QUESTIONS, AND EXERCISES
12.5
229
Terms, study questions, and exercises
Terms and concepts
Tax trajectories, hold-up problem, rate of change of a tax (or of anything
else), trajectories of producer and consumer tax incidence, intertemporal
welfare (the integral, or the sum, of the discounted stream of consumer or
producer welfare).
Study questions
For all these questions, assume that the average = marginal extraction cost
is constant with respect to extraction and independent of the stock of the
resource.
1. (a) Explain how a constant tax alters the competitive nonrenewable
resource owner’s incentives to extract, and thus how the tax a¤ects
the equilibrium price trajectory. (b) Now explain how an increasing
tax trajectory alters the competitive resource owner’s incentives, and
thereby alters the equilibrium price trajectory. (Your answer to both
parts should make clear how the taxes a¤ect the relative advantage of
extracting at one point instead of another.) (c) Use your answers to
parts (a) and (b) to explain why the three tax pro…les shown in Figure
12.1 give rise to the three price pro…les shown in Figure 12.2.
2. Using Figure 12.2, sketch the consumer and producer tax incidence over
time, for the slowly growing and the rapidly growing tax. (Figures in
the text actually show those graphs. You should see whether you can
produce the sketches using only Figure 12.2 and then compare your
answers with the …gures in the text. Your sketches will not get the
magnitudes correct, but they should correctly show the intervals of time
where an incidence is negative, positive and less than 100%, or greater
than 100%. If you do it carefully, you can also see where the incidences
are increasing or decreasing over time. The point of this question is
to see whether you REALLY know what tax incidence means.
3. (a) Explain why, in the static setting, a tax always reduces consumer
surplus. (b) Explain why, in the resource setting, the tax must increase consumer surplus at some points in time. (c) Explain why your
230
CHAPTER 12. TAXES: NONRENEWABLE RESOURCES
answer to part (b) implies that a tax might either increase of decrease
the present discount stream of consumer surplus. (d) Is there any
objection to using the present discounted stream of consumer surplus
as a measure of consumer welfare?
Exercises
1. Suppose that the government uses a pro…ts tax, (t), instead of a unit
tax. This pro…ts tax equals (t) = 1 exp ( t) with > 0. (a)
Sketch two graphs of this tax, as function of time, on the same …gure,
for a small and a large value of . (b) Following the logic in the text for
the increasing unit tax, brie‡y explain the e¤ect of an increasing pro…ts
tax on the equilibrium extraction pro…le. (You have to think about
how the increasing pro…ts tax a¤ects the …rm’s incentive to extract the
resource. (c) Compare the equilibrium e¤ect (on the extraction pro…le)
of a constant unit tax and a constant pro…ts tax. What explains this
di¤erence between the two types of constant taxes?
2. You need functional forms and parameter values to calculate the taxincidence graphs shown in Figures 12.3 and 12.4, but you can infer their
general shape by inspection of Figure 12.2, and from the de…nition of
“tax incidence”. Explain how to make this inference.
3. The text considers the e¤ect of a tax in a model with stock-independent
extraction costs. How might stock dependent extraction costs alter
the trajectories of tax incidence. (This question calls for intelligent
speculation, not calculation.)
4. Suppose that …rms have constant extraction costs and face a time varying pro…ts tax, (t) = (0:05) t . Under this tax, if extraction in period
t is y, a …rm’s after-tax pro…t is (1
(t)) (p (y) y C). (a) On the
same …gure, graph (t) (as a function of t) for < 1 and also for
> 1. (b) Using intelligent speculation (not calculation), explain how
these two pro…ts tax a¤ects the equilibrium price and extraction paths.
(Figures will make your explanations clearer.) Provide the economic
logic for your answer.
5. Justify the claim that the functions in equation 12.4 do indeed equal
consumer and producer welfare and tax revenue.
12.5. TERMS, STUDY QUESTIONS, AND EXERCISES
231
Sources
Sinclair (1992) points out the e¤ect of a rising carbon tax on the incentive
to extract fossil fuels.
Boadway and Keen (2009) survey the theory of resource taxation.
Chapter 12.1 is based largely on the IEA, OPEC, OECD and World Bank
Joint Report (2011).
The IMF paper by Coady et al (2015) provides estimates of the cost
(including environmental costs) of resource subsidies.
Aldy (2013) discusses the …scal implications of eliminating U.S. fossil fuel
subsidies.
Daubanes and Andrade de Sa (2014) consider the role of resource taxation
when the discovery and development of new deposits is costly.
Lund (2009) reviews the literature on resource taxation under uncertainty.
Bohn and Deacon (2000) discuss investment risk with natural resources.
Aisbett et al. (2010) discuss investment risk and bilateral investment
treaties.
232
CHAPTER 12. TAXES: NONRENEWABLE RESOURCES
Chapter 13
Property rights and regulation
Objectives
Understand how the establishment of property rights alters the problem
of second-best regulation.
Information and skills
A basic knowledge of the consequences of and the evolution of property
rights.
Familiarity with the Coase Theorem, and understanding its relevance
to policy in the presence of externalities.
An overview of the history of …shery regulation, and understanding
why regulation has led to overcapitalization in the sector
Understanding how individual quotas establish property rights, and the
e¤ects of these quotas.
We have emphasized competitive equilibria in nonrenewable resource markets (e.g., oil, coal) with perfect property rights. This chapter sets the stage
for a discussion of renewable resources (e.g. …sh, groundwater, forests, the
climate), emphasizing the role of imperfect property rights. We begin with
a general discussion of property rights and then show that the emergence
of an e¢ cient outcome depends on the existence but not on the allocation
of property rights (i.e., who possesses the property rights). This result is
known as the Coase Theorem.
233
234
CHAPTER 13. PROPERTY RIGHTS AND REGULATION
We then discuss the role of property rights and regulation in …sheries.
Fisheries provide a natural focus for this discussion, because: they are economically important; they are plagued by imperfect property rights; there
is a large body of research devoted to their study; and the insight gained
from studying …sheries is applicable to many other resource problems, e.g.
water and forests. Due to over…shing, loss of habitat, and climate change,
30% of the world’s …sheries are at risk of population collapse. Fisheries support nations’ well-being through direct employment in …shing, processing,
and ancillary services amounting to hundreds of billions of dollars annually.
Fish provide nearly 3 billion people with 15 percent of their animal protein needs; including post-catch activities and workers’dependents, marine
…sheries support nearly 8% of the world’s population.
13.1
Overview of property rights
Objectives and skills
Know the characteristics of the three leading types of property rights,
and understand that none of them eliminate the need for regulation.
A spectrum of social arrangements govern the use of natural resources.
The three leading modes are private property, common property, and open
access. With private property, an individual or a well-de…ned group of
individuals (e.g. a company) owns the asset and determines how it is used.
Under common property, use of the asset is limited to a certain group of
people, e.g. those living in a town or an area; community members pursuing
their individual self-interest, instead of a single agent, decide how to use the
asset. Anyone is free to use an open access resource. A farm owned by an
individual, family, or corporation, is private property. The owner decides
how to use the farm. A …eld on which anyone in the village can graze
their cows, but from which those outside the village are excluded, is common
property. A …eld that anyone can use is open access.
This taxonomy identi…es di¤erent types of ownership structure, but in
practice the boundaries between them are often blurred. Labor, health, and
environmental laws govern working conditions and pesticide use on privately
owned land, limiting the exercise of private property rights. In Britain,
common law allows anyone to use paths across privately owned farmland,
13.1. OVERVIEW OF PROPERTY RIGHTS
235
provided that they do not create a nuisance, such as leaving gates open, further limiting private property rights; in Norway, people are allowed to enter
uncultivated private property to pick berries. Common property dilutes
property rights, but social norms often limit community members’actions.
Anyone in the village may be allowed to use the village common, but they
might be restricted to grazing one cow, not ten. Even for an asset that is
nominally open access, members of a group might exert pressure to restrict
outsiders. Local surfers at some California public beaches make it uncomfortable or dangerous for outsiders to surf.
As these examples suggest, there is a continuum of types of property
rights, not three neat types. De jure property rights describe the legal
status of property, and de facto property rights describe the actual property
rights. De jure and de facto property rights might coincide or di¤er. In
the sur…ng example, the de jure property rights are open access, but to the
extent that the local surfers are successful in excluding outsiders, the de facto
property rights more closely resemble common property.
Property rights to a resource may change over time, often responding to
migration and increased trade, and often accompanied by social upheaval.
The enclosure movement in the UK, converting village commons to
private property, began in the 13th century and was formalized by
acts of parliament in the 18th and 19th century. These enclosures
increased agricultural productivity and dispossessed a large part of the
rural population.
In the late 19th and early 20th century Igbo groups in Nigeria converted
palm trees from private to common property in response to increased
palm oil trade. Trade increased the value of the palm trees, increasing
the need to protect them from over-harvesting. In this case (but not
in general) monitoring and enforcement costs needed to protect the
resource were lower under common property.
The 1924 White Act in Alaska, later incorporated into the state’s constitution, abrogated aboriginal community rights to the salmon …shery.
The Act also disallowed private ownership, insuring that non-residents
would not control the resource. The Act was rationalized by the claim
that the resource is owned by the nation, not the communities previously managing it. Open access replaced e¤ective common property
management, leading to resource degradation and requiring regulation.
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CHAPTER 13. PROPERTY RIGHTS AND REGULATION
A long-running dispute in the U.S. tests the limits of private property rights. The legal doctrine of “regulatory taking” seeks to de…ne
zoning and environmental rules that diminish the value of property as
“takings”, requiring compensation under the Fifth Amendment to the
U.S. constitution. The doctrine’s objective is to weaken governments’
police powers (e.g. environmental regulation), strengthening the rights
of private property owners.1 Periodic legal disputes in coastal states,
including California and Texas, test landowners’ability to impede access to beaches, or to insist on public investment (e.g. sea walls) that
maintains the value of private property.
Private property diminishes or eliminates some externalities that are
prevalent under common property or open access. Grazing an additional
cow on a …eld creates bene…ts for the cow’s owner. If the cow competes with
other animals for fodder, it creates a negative externality for other owners,
much in the way that an additional driver contributes to road congestion.
The overgrazing also damages the …eld, lowering its long run productivity,
thus lowering both short and long run community welfare. This outcome
is known as the “tragedy of the commons”. Private owners internalize the
congestion created by the additional cow, and are therefore less likely to
overgraze the …eld. To the extent that private property avoids the tragedy
of the commons, it improves resource management.2
An emerging body of research shows that many societies have successfully managed common property natural resources, avoiding the tragedy of
the commons. Common property management requires widespread agreement on the rules of use, and mechanisms for monitoring and enforcement of
the rules. Private property also requires adjudication of disputes, and monitoring and enforcement of laws. Stable conditions and homogenous users
1
Supreme Court rulings have largely rea¢ rmed these police powers, undermining the
Doctrine of Regulatory Takings. Bilateral Investment Treaties (Chapter 12.2) requiring compensation for regulation that is “tantamount to expropriation” strengthens the
Doctrine in the sphere of international, rather than US domestic law.
2
Anyone who has lived in a shared household already knows a great deal about the
potential bene…ts and pitfalls of common property. A person who buys or leases their
own house has private ownership, within the terms of their lease and the local laws. Private
housing is on the private property end of the property rights continuum. A squat resembles
the open access resource. No squatter has de jure property rights to the dwelling, although
a squatter or a group of squatters might be able to exclude outsiders and enforce rules,
much like the surfers on some beaches. A communal household lies between a private
house and a squat, and is a type of common property.
13.1. OVERVIEW OF PROPERTY RIGHTS
237
increase the success of common property management. A rapid change that
increases the demand for the resource, such as migration or opening to trade,
can undermine common property management.
Historical events during the last quarter century show that private property does not guarantee e¢ cient management. When the Soviet Union collapsed in 1991, reformers and their western advisors (primarily economists)
debated the right pace of privatization of state owned property. Those
supporting rapid privatization hoped that it would lead to the e¢ cient use
of natural and man-made capital, and feared that a slower pace would only
perpetuate the ine¢ ciencies and make it possible to reverse the reforms. Privatization occurred rapidly, but instead of leading it e¢ ciency it created a
class of oligarchs and an entrenched system of corruption.
Following a political upheaval, local elites might capture resources, replacing common property with private property, thus rendering the traditional
monitoring and enforcement mechanisms irrelevant. In a politically unstable environment, the new owners recognize that the next political upheaval
might replace them with another group of elites, making their property rights
insecure. The combination of current absolute but insecure property rights
is particularly likely to lead to overuse of the resource. The current owners
can use the resource as they see …t, usually to enrich themselves; the risk of
losing control of the resource gives them the incentive to attach little weight
to its future uses.
In summary, actual property rights tend to exist on a continuum that
includes the three main types as special cases. Regulation is particularly
necessary under open access, which is especially vulnerable to the tragedy
of the commons. Common property management in small communities has
(often) avoided this tragedy. As communities integrate into wider markets,
the management practices frequently break down, requiring di¤erent kinds
of regulation. Private property solves some externality problems but creates
others, and also typically requires regulation. For example, converting the
village commons to a privately owned farm gives the farmer the incentive
to manage the land e¢ ciently (e.g. to avoid erosion), but not to correct
o¤-farm externalities (e.g. pollution run-o¤). Arguably, in this example
the problem lies not with private ownership of the land, but with the lack
of property rights to the waterways that absorb the pollution. With this
view, the policy prescription is to create property rights for the waterways.
Creating those additional rights may be too expensive or politically or ethically unacceptable, in which case the policy prescription is to regulate the
238
CHAPTER 13. PROPERTY RIGHTS AND REGULATION
pollution.
13.2
The Coase Theorem
Objectives and skills
Be able to state and explain the Coase Theorem.
Be able to apply the theorem to speci…c settings.
Transactions costs include the costs of reaching and enforcing an agreement. The Coase Theorem states that if transactions costs are negligible,
and property rights are well-de…ned, agents can reach the e¢ cient outcome
regardless of the distribution (or allocation) of property rights. The basis
for this result is that rational agents will not leave money on the table if it is
costless for them to pick it up. Under the conditions of the theorem, there
is no need for a regulator, because private agents reach an e¢ cient outcome
by bargaining. The government’s only role is to insure that agents honor
their contracts, and possibly to make transfers in order to promote fairness;
e¢ ciency does not require those transfers.
In many situations where we observe regulation instead of a bargained
outcome, transactions costs are large, making it impractical for agents, bargaining amongst themselves, to achieve an e¢ cient outcome. In other situations, the property rights are not well-de…ned, leaving agents uncertain
about the payo¤s of reaching a bargain, making it more di¢ cult to reach an
agreement. Most natural resource problems, including those that arise in
…sheries, require some form of regulation.
To illustrate the Coase Theorem, suppose that the total pro…t of a …shery
depends on the number of boats operating there. With one boat, the pro…t
is 1, with two boats, the aggregate pro…t is 4, and with three boats, the
aggregate pro…t is 3. Initially, three boats, each with a separate owner,
operate in the …shery. The surplus obtained from inducing one boat to exit
equals 4 3 = 1.
In an extreme case, one person has exclusive property rights to the …shery.
That person would insist that the other two …shers leave the sector, and the
person would buy an additional boat, thus increasing industry pro…t by 1 and
reaching the optimal outcome. If all three …shers have some property rights,
and if the transactions costs are small, then they can reach an agreement
13.3. REGULATION OF FISHERIES
239
in which one of them sells her right to …sh to the other two and leaves the
sector. There are many types of bargains that they might strike, leading
to di¤erent splits of the surplus. The Coase Theorem does not identify the
particular bargain reached.
For example, if the three …shers are equally productive, they can create
a lottery that determines who leaves the sector.3 The person who leaves
receives a payment of 1 + x, their initial pro…t plus the compensation x for
leaving. The two remaining …shers split the higher pro…t, each receiving
2, and share the cost of buying out the departing …sher, for a net bene…t
. The expected payo¤ to an agent participating in a fair lottery
of 2 1+x
2
=
(where the chance of leaving the sector is 1=3) is 31 (1 + x) + 23 2 1+x
2
4
1
,
which
is
greater
than
their
payo¤
under
the
status
quo
(1).
If
x
>
,
the
3
3
person who leaves is a winner, and if the inequality is reversed, this person
is a loser; setting x = 13 insures that there are no losers. Other procedures,
not involving a lottery (e.g. arm wrestling), could also be used to determine
who leaves.
The Coase Theorem does not predict how the e¢ cient outcome is obtained (a lottery or arm wrestling) or the compensation (x in the lottery
example). It merely says that if transactions costs are small and property
rights well de…ned, rational agents will bargain and achieve an e¢ cient outcome. Di¤erences in bargaining power, perhaps due to di¤erences in outside
options (a …sher’s best alternative to remaining in the sector) or levels of
bargaining skill lead to di¤erent distributions but not di¤erent levels of the
bargaining surplus.
13.3
Regulation of …sheries
Objectives and skills
Know the basics of the recent history of …shery regulation.
Understand how, under limited regulation, …shers’ endogenous decisions limit the e¢ cacy of regulation.
Understand how assigning property rights can alleviate the regulatory
problem.
3
If the …shers are not equally productive, then an e¢ cient procedure must chose the
two most productive to remain in the …shery.
240
CHAPTER 13. PROPERTY RIGHTS AND REGULATION
Prior to the 20th century, the concept of “freedom of the seas” gave
nations sovereignty up to three miles from their coastline. Other nations’
ships were allowed to operate outside that area. In the 20th century, nations
began to claim control of larger areas, often to protect their rights to …sh
stocks. The United Nations Convention on the Law of the Sea, concluded
in 1982, replaced earlier agreements, giving nations an “exclusive economic
zone” (EEZ), e.g. to harvest …sh or extract oil, 200 miles beyond their
coastline.
The U.S. passed the Manguson-Stevens Fishery Conservation and Management Act in 1976 and amended it in 1996 and 2006, to promote optimal
use of …sheries within its EEZ. Goals included: conserving …shery resources,
enforcing international …shing agreements, developing under-used …sheries,
protecting …sh habitat, and limiting “bycatch” (…sh caught unintentionally,
while in pursuit of other types of …sh). The law established Regional Fishery
Management Councils, charged with developing Fishery Management Plans
(FMPs). These FMPs identify over…shed stocks and propose plans to restore and protect the stocks. The law requires management practices to be
based on science. For each …shery, a scienti…c panel determines the “acceptable biological catch”, and the managers then set an “annual catch limit”
(ACL), not to exceed the acceptable biological catch. The Fishery Management Councils can enforce the ACL by: restricting entry to speci…c boats or
operators (limited access); restricting …shing to certain times of the year or
certain locations; regulating …shing gear; and requiring on-board observers
to insure that boats obey regulations. A 2009 National Marine Fisheries
report states that of the 192 stocks being monitored, the percent with excessive harvest fell from 38% to 20% over the decade, and the percent with
over-…shed stocks fell from 48% to 22%.
Most of the important …sheries are concentrated in countries’ exclusive
economic zones and are regulated in some fashion; these are “limited access”
…sheries. Despite this regulation, scienti…c di¢ culties and human frailty often leads to over-…shed stocks. For most …sheries, identifying the actual stock
is a di¢ cult measurement problem. Random stock changes, due to unforeseen changes in ocean conditions and changes in stocks of predators and prey
(that impact the regulated …shery) make it di¢ cult to determine safe stock
levels, even if the stocks can be measured; imprecise measures compound the
scienti…c problem. Managers may decide to ignore scienti…c evidence if they
distrust it or are swayed by lobbying from …shers or processors. “Regulatory
capture”, a widespread phenomena, occurs when regulators are “captured”
13.3. REGULATION OF FISHERIES
241
by the groups that they are charged with regulating: human frailty.
Box 13.1 The U.S. Northwest Atlantic scallop …shery: a success story.
Scallops are caught by dragging dredges along the seabed, trapping
scallops of legal size. In 1994 the government closed 3 large banks
and scallopers responded by concentrating and subsequently exhausting other areas of the …shery. Fisherman hired University of Massachusetts biologists to conduct a stock survey, …nding that the population in the closed areas had rebounded. The industry began to
commit a fraction of its pro…ts, about $12 - $15 million per year,
to conduct surveys. Using this data, the National Marine Fishery
Services (NMFS) closes over…shed seabeds long enough to allow the
population to recover, about 3 - 5 years. This system resembles …eld
rotation in agriculture. The success of the program relied on good
data, cooperation between …shers and NMFS, leading to trust (because …shers supply the data), and a common interest among …shers
to preserve the stock.
Even without the scienti…c and political impediments, …shery regulation
is inherently di¢ cult. The problem arises from the mismatch between “targets”(or “margins”) and “instruments”(or policies) (Chapter 10). A target
(margin) is anything that a regulator would, in an ideal world, like to control.
The optimal harvest of a stock involves many considerations, in addition to
the annual catch limit. The ecology depends on the annual catch limit, but
the economic pro…ts depend on how that limit is harvested, including speci…cs
about gear, e.g. types of nets, engine size, and other boat equipment. In
principle, the Regional Fishery Management Councils have the authority to
regulate most of these features, but regulation of every important business
decision is not practical. Most regulation involves fairly simple policies. For
example, the Council can enforce the annual catch limit by restricting the
length of the season and limiting the number of boats; but the Council is
unlikely to be able to control all of the decisions that go in to …shing.
The problem is that when the regulator restricts one margin (e.g. by limiting the season length), …shers respond on another margin, (e.g. by changing
the gear they use). Regulators cannot control all margins, so regulation is
second-best. Fishers respond to regulation, acting in their individual selfinterest. However, they in‡ict externalities on others, leading to a socially
ine¢ cient outcome. This ine¢ ciency manifests as over-capitalization: too
many boats, or too much gear, chasing too few …sh. Over-capitalization
242
CHAPTER 13. PROPERTY RIGHTS AND REGULATION
reduces industry pro…ts, making a given annual catch limit more expensive
to harvest. Estimates of over-capacity vary widely, ranging from 30% to
over 200%, but there is a consensus that it is severe.
13.3.1
Over-capitalization
We illustrate the over-capitalization and the resulting loss in industry profits using a simple model with four targets: annual allowable catch, A; the
number of boats, N ; the number of days of the season, D, and the amount
of “e¤ort”per boat per day, E. E¤ort is an amalgam of all of the decisions
the …sher makes (boat size, crew, gear characteristics). Each unit of e¤ort
costs w per day, and E units of e¤ort enable a boat to catch E 0:5 …sh per
day.
In the …rst best setting, the regulator chooses the value of all four variables. To examine the forces at work in the real world, where regulators are
unable to control every margin, we consider a second best setting where the
regulator chooses the annual catch and the length of the season, A and D.
Fishers choose e¤ort, E, to maximize their pro…ts. We …rst consider the
case where N is exogenous and …xed; perhaps it is historically determined.
We then allow N to be endogenous.
Exogenous N
Individual …shers, each with a boat, choose e¤ort in order to maximize pro…t
(revenue minus cost). If the price of a unit of …sh is p, the …sher chooses E
to maximize pro…ts per day:
max pE 0:5
E
wE )
p
d (pE 0:5 wE) set
= 0 ) E = 0:5
dE
w
2
:
(13.1)
Given that the N …shers choose this level of e¤ort, the manager who wants
to set annual catch equal to A must choose the number of days of the season,
D, so that the number of …shers (N ) times the catch per day per …sher (E 0:5 )
times the number of days equals A:
NE
0:5
D=N
p
0:5
w
2
0:5
set
D =A)D=2
Aw
:
Np
The maximum feasible season length (e.g., due to weather conditions) is
, the fraction of the potential season that …shing is
S. We denote = D
S
13.3. REGULATION OF FISHERIES
allowed, and we assume that
D
S
243
< 1 i.e.,
=
2Aw
< 1:
N Sp
(13.2)
This inequality implies that, given …shers’ individually optimal choice of
e¤ort per day, the annual allowance, A, constrains the length of the season.
If, instead, inequality 13.2 is reversed (
1), then …shers can be allowed to
…sh during the maximum feasible number of days (S), without exceeding the
annual ceiling. In that case, …shers want to catch less than the limit, and
regulation is not needed.
In this second-best setting, where the regulator chooses the length of the
season and …shers choose e¤ort, total industry pro…t equals
pro…tsecond best = DN (pE 0:5
Aw
N
2N
p
p
0:5 wp
2 0:5
wE) =
w 0:5 wp
2
=
Ap
2
In the …rst best setting, the regulator chooses both e¤ort and the number
of days in order to reach the ACL, A. It is straightforward to show that
costs are minimized by setting D = S, the maximum feasible season, and
2
E = NAS (Problem #2). With these values, total pro…t in the industry is
pro…t…rst best = DN (pE 0:5
SN p
A
NS
w
A 2
NS
wE) =
= pA
A2 w
:
NS
The percent increase pro…t in moving from the second best to the …rst best
regulation is
1
Ap 1
2
pro…t…rst best pro…tsecond best
100
=
Ap
pro…tsecond best
2
2Aw
N Sp
100 = (1
) 100 > 0;
where the last equality follows from the de…nition of , and the inequality
follows from inequality 13.2.
Fishers act in their self-interest in choosing e¤ort, but they in‡ict a negative externality on the industry: as e¤ort increases, catch per day also increases, requiring that the regulator shorten the season in order to maintain
the annual catch limit. This self-interested behavior leads to 12 times the
244
CHAPTER 13. PROPERTY RIGHTS AND REGULATION
…rst best level of e¤ort. This over-capitalization decreases industry pro…ts
by the percent decrease in the length of the season. For example, if second
best regulation leads to a season only 20% as long as the …rst best level, then
industry pro…ts are only 20% of the …rst best level.
This model illustrates an empirically important phenomenon: the race to
catch …sh (and resulting over-capitalization) has lead to shorter seasons and
lower industry pro…t in many …sheries. In the early 20th century, the North
Paci…c halibut …shery operated throughout the year, leading to excessive
harvests. In 1930 the U.S. and Canada agreed to manage the …shery using
an annual quota. The management success increased pro…ts, leading to overcapitalization of the …shery. Managers responded by reducing the season
length to two months in the 1950s and to less than a week in the 1970s.
Endogenous N
Above we took the total number of boats, N , as …xed, but in the absence
of regulation, N is endogenous; it is another “target”, or “margin”. Higher
pro…ts encourage entry, changing the (endogenous) value of N unless regulation restricts entry. For example, suppose that initially the …shery is
totally unregulated, leading to depleted stocks, making it expensive to catch
…sh. The low stocks and high costs cause pro…ts in the sector to be low, discouraging entry or encouraging …shers in the sector to leave. Now suppose
that a restriction on annual catch succeeds in rebuilding the stock, making it
cheaper to catch …sh, and thus increasing pro…ts. In the absence of entry restrictions, more boats may enter, creating another type of over-capitalization.
Suppose that each boat costs $F /year to own; F is the opportunity cost
of the money tied up in owning a boat for a year. Under second best
regulation, where the length of the season adjusts in order to maintain the
. With N boats, the pro…t
catch limit, we saw above that industry pro…t is Ap
2
Ap
per boat per year is 2N . Under free entry, boats would enter the industry
until the annual opportunity cost of being in the sector equals the pro…t of
Ap
Ap 4
…shing there, which requires 2N
= F , or N = 2F
. For smaller values of N
additional boats have an incentive to enter the …shery (to obtain positive net
pro…ts), and for larger values of N existing boats have an incentive to leave
the …shery (to avoid losses).
Consider the case where the number of boats is in a second best equilib4
This calculation ignores the fact that the number of boats must be an integer.
13.3. REGULATION OF FISHERIES
245
Ap
rium (N = 2F
) and technological advances suddenly enable the regulator to
control e¤ort, determining the gear each boat uses. We showed above that
this change, at the initial level of N , increases industry pro…t by (1
) 100%,
(1 )2F 100
(1 )100
%=
% (using the expresincreasing the pro…t per boat by
N
Ap
sion for the equilibrium level of N ). This increase in pro…t encourages
additional boats to enter.
Ap
The second best equilibrium number of boats (N = 2F
) is already excessive from the standpoint of society, but at least the regulator does not have
to discourage more boats from entering: they have no desire to do so. However, successfully reducing one type of over-capitalization by reducing e¤ort
per boat creates an incentive for new boats to enter, exacerbating the other
source of overcapitalization. Getting one margin right (reducing e¤ort per
boat) causes another margin (the number of boats) to move further from the
social optimum. This kind of problem is endemic to a second best setting,
i.e. to the real world: …xing one problem can make another problem worse.
In theory, the regulator can determine the optimal number of boats and
the optimal e¤ort per boat, and then either decree that …shers accept these
levels, or impose taxes (e.g. a tax per unit of e¤ort and an entry tax) that
“induce”these levels. That level of regulation is seldom practical.
Box 13.2 The relation between N and e¤ort. In the model here,
…shers’equilibrium choice of e¤ort does not depend on the number of
boats (equation 13.1). That independence simpli…es the calculations,
but it is a consequence of functional assumptions. More generally,
equilibrium e¤ort depends on the number of boats. Regulating one
aspect of the industry a¤ects other …shing decisions. Economic agents
who are constrained in one dimension (e.g. not being able to increase
the number of boats) respond by making changes in other dimensions
(e.g. increasing the speed of boats). Over-capacity can show up in
many ways, e.g. as too many boats, too much gear, or boats that
are too big or too fast. The economy is like a water bed: applying
pressure in one location creates movement elsewhere.
13.3.2
Property rights and regulation
Fishing, like most resource extraction problems, creates an externality. Here,
the outcome when individuals pursue their self interest is not e¢ cient. The
externality leads to over-…shing and lower stocks. Because individual …shers
246
CHAPTER 13. PROPERTY RIGHTS AND REGULATION
do not own the stocks, they have little incentive to manage them for future
users. Even if the over-…shing problem can be solved by setting and enforcing
an annual catch limit, history shows that other ine¢ ciencies remain, often
leading to over-capitalization and low pro…ts.
Creating property rights is the obvious alternative to standard regulation
(such as limiting the …shing season, restricting the number of boats, and
regulating …shing gear). Problems in …sheries arise largely from the lack
of property rights, so creating these rights is the most direct remedy; recall
the Principle of Targeting from Chapter 10. Individual Quotas (IQs), or
Individually Transferable Quotas (ITQs) are the most signi…cant form of
property rights in …sheries. These measures set an annual Total Allowable
Catch (TAC) for a species and give individuals shares in that quota. An
individual with share s is entitled to harvest s times that year’s TAC of that
species. Under ITQs, owners can sell or lease (“transfer”) their shares.
Property rights-based regulation has been shown to decrease costs, increase revenue, and protect …sh stocks. However, less than 2% of the world’s
…sheries currently use property rights based regulation.
Reducing industry costs For a given TAC (or ACL), property rights
cause …shers to internalize the externalities that lead to overcapitalization.
Property rights reduce costs, increasing pro…ts. Using the model above, a
…sher with the share s of a TAC A for a …shery with price p obtains the …xed
revenue sAp. With …xed revenue, the …sher maximizes pro…ts by choosing
e¤ort and days …shed to minimize the cost of catching sA. This …sher’s
cost minimization problem is the same as that of the regulator who chooses
both e¤ort and season length (Problem #2). Thus, the …sher with property
rights chooses the socially optimal amount of e¤ort and days …shed.
If the quotas are transferable (ITQs instead of IQs), there are two additional sources of cost savings. First, some of the …shers may be more
e¢ cient. For example, the average …sher may catch E 0:5 …sh per day, but
half may catch 10% more and half 10% less with the same level of e¤ort.
Regulators are unlikely to know …shers’relative e¢ ciency, or be able to act
on that information even if they have it. Fishers probably have a better
idea of their relative e¢ ciency. Shares in the quota are worth more to the
e¢ cient …shers than to the ine¢ cient …shers, so there are opportunities for
the former to buy the latter’s’licenses. If that occurs, aggregate costs in the
…shery falls by 10% in this example.
13.3. REGULATION OF FISHERIES
247
Second, even if all …shers are equally productive, there may be too many
boats in the sector: industry pro…ts would be higher if some boats could be
persuaded to leave, as in the example in Chapter 13.2. If transactions costs
are su¢ ciently low, and …shers manage to solve the Coasian bargain, some
boats sell their quotas and leave the sector. Pro…ts under the smaller ‡eet
are higher.
The Mid-Atlantic surf clam …shery switched from restricted entry regulation to ITQs in 1990. Prior to the switch, there were 128 vessels in the
…shery. Estimates at the time claimed that the optimal number of boats
for the industry was 21 –25, and that rationalization of e¤ort could reduce
costs by 45%. By 1994, the ‡eet size had fallen to 50 vessels and costs had
fallen by 30%. The costs savings due to rationalization of ‡eet size and gear
choice likely occur over a period of time; it takes time to change.
Increasing revenue For a …xed TAC, IQs and ITQs can increase revenue,
in addition to reducing harvest costs. We showed above that partial regulation leads to too much gear and/or too many boats chasing a given number
of …sh, requiring regulators to reduce the …shing season, sometimes to a period of a week or less. Over-capitalization increases …shing costs, but it also
makes the annual harvest available during a short period of time, instead of
being spread out during the season.
If the market for fresh …sh is inelastic, a sudden increase in harvest leads
to a much lower equilibrium price. There may also be capacity constraints
in transporting fresh …sh to market. Fish processors may be able to wield
market power if …shers have to unload large catches during short periods of
time. For all of these reasons, the ex vessel price that …shers receive may be
lower when the annual catch is landed during a short interval. The lowerpriced …sh may then enter the frozen …sh market, where prices are typically
lower than for fresh …sh. Moving from regulation to IQs or ITQs eliminates
or at least reduces the race to catch …sh, causing harvest to be spread out
over the season, and increasing the average price of landings.
Before the British Columbia halibut …shery switched to ITQs in 1993,
the …shing season lasted about …ve days, and most of the catch went to the
frozen …sh market. With the more spread-out and thus steadier supply of
…sh caused by the move to ITQs, wholesalers found it pro…table to develop
marketing networks for transporting fresh …sh. Prior to the ITQs, the price
of fresh …sh fell rapidly if more than 100,000 pounds a week became available,
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CHAPTER 13. PROPERTY RIGHTS AND REGULATION
but with the development of the new marketing networks, the market could
absorb 800,000 pounds before prices dropped.
A 2003 study estimates the potential gains from introducing ITQs in
the Gulf of Mexico reef …sh …shery, accounting for both cost reductions and
revenue increases. ITQs had the potential to increase revenue by almost
50% and to reduce costs by 75%. The study also estimates that under ITQs
the equilibrium ‡eet size contains 29 – 70 vessels, compared to the actual
level of 387 vessels.
Protecting …sh stocks Standard regulation and IQs/ITQs both rely on
an annual limit (TAC) but they use di¤erent ways of enforcing that limit.
The mismatch between targets and instruments under standard regulation,
and the resulting ine¢ ciencies lead to high costs, low revenue, and …nancial
di¢ culties for …shers. Fishers have an incentive to pressure regulators to
increase the annual catch as a short-term …nancial …x. If this pressure overrides the scienti…c advice, it imperils …sh stocks. The creation of property
rights potentially eases these stresses. When …shers have property rights
to the catch, they have incentives to protect the stock. Property rightsbased regulation changes the political dynamics and likely helps to protect
…sh stocks.
The di¢ culty of measuring stocks makes it hard to know if a stock is over…shed. A common de…nition calls a …shery “collapsed” in a particular year
if the harvest in that year is less than 10% of the maximum previous harvest.
By this de…nition, 27% of the …sheries were collapsed in 2003. Fisheries
with IQ/ITQs were less likely to be collapsed, but the “selection problem”
makes it di¢ cult to tell whether this negative correlation between ITQ status
and …shery collapse implies that the property rights-based mechanism really
protects species’health. Unobserved characteristics of a …shery that decrease
its likelihood of collapse may also increase its likelihood of being “selected
into” property rights-based management. This selection problem creates a
negative correlation between ITQ status and collapse, without implying a
causal relation. Statistical methods based on “matching” can alleviate this
measurement problem. The idea is that we would like to compare collapse
status between pairs of …sheries that are alike, except for their ITQ status.
A 2008 study based on 50 years of data and over 11,000 …sheries, which
takes into account the selection problem, …nds evidence of a causal relation,
and estimates that ITQs reduce the probability of collapse in a year by about
13.4. SUBSIDIES TO FISHERIES
249
half. It also estimates that had there been a general movement to ITQs in
1970, the percent of …sheries collapsed in 2003 would have been about 9%
instead of the observed 27%.
Remaining ine¢ ciencies The property rights created by IQs and ITQs
can improve the …nancial and ecological health of a …shery, but they leave
many problems unsolved. They create particular property rights to a particular species. They may leave important externalities even for that species.
For example, increases in production costs during the season due to the
intra-seasonal decrease in stock can still create a race to catch …sh, leading
to overcapitalization.
The IQs and ITQs do not protect other species. They provide no incentive to reduce by-catch, the unintentional harvest of …sh not being targeted.
Other regulations, or the creation of additional property rights can reduce
by-catch, but these may be costly to implement and may have unintended
consequences.
A fairly new mechanism, Territorial Use Rights Fisheries (TURFs) attempt to alleviate the cross-species problem by giving groups of …shers (e.g.
coops) exclusive rights to an area, instead of to a species. TURFs are less
e¤ective if important species move in and out of the territorial area.
13.4
Subsidies to …sheries
Policy failure harms natural resources by permitting and sometimes encouraging over-harvest, damaging ecosystems. This loss in natural capital can
have serious long run social costs. The short run economic costs of these
policies appear as over-capitalized and …nancially stressed …sheries, and stagnant or diminishing supplies of high value catch. The policy targets are to
reduce catch in order to allow …sheries to recover, and to encourage rationalization and consolidation of the industry to increase pro…ts. Reducing catch
and encouraging …shers to leave the industry are both politically fraught
prospects.
Subsidizing the industry is politically easier, but ultimately counterproductive. Subsidies disguise the economic costs, enabling …shers to remain
in an unpro…table activity, worsening both the problem of over-harvest and
over-capitalization. A variety of domestic and international agencies have
documented the link between subsidies and over-…shing and over-capitalization.
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CHAPTER 13. PROPERTY RIGHTS AND REGULATION
A 2013 European Parliament study estimates annual global subsidies to the
…shing sector of $35 billion (2009 dollars). Over half of those subsidies
generate increased capacity; 22% come in the form of fuel subsidies, 20%
subsidize …shery management, and 10% subsidize ports and harbors (Table
13.1). Developed countries are responsible for most of the subsidies; 43%
originate in Asia, chie‡y Japan and China. Between 1996 –2004, the U.S.
…shing industry received over $6.4 billion in subsidies, or $713 million per
year.
type
of subsidy
source
of subsidy
fuel: 22
developed
countries: 65
management: 20
Asia: 43
Japan: 20
China: 20
ports and
harbors:10
Europe: 25
capacity
increasing: 57
North
America: 16
Table 13.1 Types and geographical sources of …shing subsidies, as a percent
of total. (Sumaila et al. 2013)
Subsidies transfer income from one group to another, here from tax payers
to …shers. Thus, the dollar value of the subsidy does not provide a measure of
the economic cost, or lost welfare. A 2009 study commissioned by the World
Bank conservatively estimates the annual global economic cost of …shery
subsidies at $50 billion. This estimate includes the costs due to reduced
stock size (which reduces harvest and increases the harvest cost) and of overcapitalization. In some cases, the value of harvest is less than the true
cost of harvest; in these cases, the industry operates at a loss, and the loss
is disguised by government subsidies.5 According to this study, half the
current number of vessels could achieve current catch. Subsidies account for
approximately 20% of …shing revenue.
5
This situation is reminiscent of many Russian and east European industries after the
collapse of the Soviet Union. These industries were not economically viable, and had been
kept alive by government subsidies; closing them down increased gross national product.
13.5. SUMMARY
251
Box 13.3 Subsidies in other sectors. Fisheries are not alone in receiving politically motivated but economically unproductive subsidies;
Chapter 12.1 discusses fossil fuel subsidies. Agriculture in many developed countries also receives large subsidies, often justi…ed as helping struggling farmers. However, endogenous changes induced by the
subsidies often undo whatever short term …nancial help the subsidies
provide. Agricultural subsidies are “capitalized”in the price of land:
the expectation that the subsidies will continue into the future make
people willing to pay more for land, raising its equilibrium price. Current land owners who sell their land, not entering farmers who must
buy the land, capture these increased rents. Subsidies, and in particular the belief that they will continue into the future, increase young
farmers’ debt (via the increase in land prices), making them more
vulnerable to future …nancial di¢ culties and more dependent on the
subsidies.
13.5
Summary
Private property, common property, and open access are the three leading
modes of property rights. Property rights to most resources lie somewhere
on the continuum that includes these three modes. De facto property rights
sometimes di¤er from de jure rights. Common property can lead to overuse of
the resource, a result known as the tragedy of the commons. Many societies
developed mechanisms that e¢ ciently manage common property resources;
integration into the global market sometimes erodes these mechanisms.
If transactions costs are small and property rights well de…ned, agents can
(plausibly) reach an e¢ cient outcome through bargaining. The Coase theorem states that in this case, the e¢ ciency of the outcome does not depend on
the distribution of bargaining power, or more generally, on the assignment of
property rights. This result implies that explicit regulation is less important
or even unnecessary, provided that transactions costs are low and property
rights are secure.
Most economically important …sheries fall within nations’Exclusive Economic Zone, and are regulated. The impracticality of regulating every facet
of …shing leaves regulators in a second best setting. Many …sheries set annual quotas, enforced using a variety of policies, notably early season closures.
The resulting race amongst …shers to catch …sh leads to over-capitalization,
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CHAPTER 13. PROPERTY RIGHTS AND REGULATION
which results in high costs and, because much of the harvest is landed during a short period of time, low revenue. Property rights-based regulation,
primarily ITQs, can lead to consolidation and rationalization of …sheries, lowering costs, increasing revenue, and via a political dynamic, increasing the
prospect for …shery health (adequate stocks). However, only a small percent of …sheries operate under ITQs. Subsidies, especially from developed
nations, provide short run bene…ts to …sheries, but exacerbate the problems
that lead to low pro…ts and over-…shing.
13.6
Terms, study questions and exercises
Terms and concepts
Open access, common property, de jure, de facto, congestion, tragedy of the
commons, Doctrine of Regulatory Taking, transactions costs, Coase Theorem, fair lottery, outside option, Law of the Sea, exclusive economic zone,
Manguson-Stevens Fishery Act, by-catch, Regional Fishery Management Councils, Fishery Management Plans, acceptable biological catch, annual catch
limit, regulatory capture, overcapitalization, Individual (Transferable) Quotas or I(T)Qs, Territorial Use Rights Fisheries (TURFs).
Study questions
1. Recently there have been a spate of disputes on airlines concerning
whether a person is entitled to recline their seat. These disputes are
essentially about who has the property rights to the several inches of
space between a seat and the one in front of it. Some airlines have
(either implicit or explicit) rules that assign property rights: a person
is entitled to recline her seat, except during meals (and of course during
landing and takeo¤). Recently these rules seem to have become vaguer,
or less well understood. Discuss the increasing occurrence of these
disputes among passengers, in light of the Coase Theorem.
Exercises
1. This exercise takes the reader through one of the classic examples of
the Coase Theorem. A factory that emits e units of pollution obtains
the total bene…t from emissions, 10e 12 e2 . A (very old fashioned)
13.6. TERMS, STUDY QUESTIONS AND EXERCISES
253
downstream laundry dries clothes outside. The pollution makes it
more expensive for the laundry to return clean clothes to its customers,
and therefore increases the laundry’s costs by 2e2 . (a) Find the socially
optimal level of pollution, i.e. the level that maximizes bene…ts minus
costs. (b) Find the emissions tax that supports this level of pollution
as a competitive equilibrium. (c) Assume that the factory has the right
to pollute as much as it wants. The laundry and the factory are able
to costlessly bargain to reduce pollution. Who pays and who gets paid
in the bargaining outcome? (d) Justify the Coasian conclusion, namely
that the outcome of an e¢ cient bargain leads to the socially optimal
level of pollution. One can establish this claim using the following proof
by contradiction: Suppose, contrary to the claim, that they reach a
bargain that entails an amount of pollution di¤erent than the socially
optimal level. (To avoid repetition, consider only the case where this
amount is greater than the socially optimal level.) Show that at such
a level, the laundry’s willingness to pay for a marginal reduction in
pollution is strictly more than the …rm would have to receive in order
for it to be willing to reduce pollution by a marginal amount. Explain
why this conclusion implies that if the agreement by the factory and
the laundry leads to more pollution than is socially optimal, then the
factory and the laundry have left money on the table. They should
resume bargaining and capture some of this waste.
2. This exercise asks the reader to obtain the formula for e¤ort under full
regulation, given in Chapter 13.3.1. Under full regulation (where the
regulator is able to choose both D and E) the total revenue is …xed
at pA, so the regulator’s objective is to minimize costs, subject to the
constraints that the limit is caught (N E 0:5 D = A) and that D not
exceed the maximum number of days, S, during which (e.g. due to
weather conditions) it is feasible to …sh (D S):
minE;D (wED) subject to N E 0:5 D = A and D
S
Proceed as follows: (i) Use the constraint involving A to solve for E.
(ii) Substitute this value of E into the minimand (the thing being
minimized, here, costs). (iii) Note that the resulting minimand is
decreasing in D. Conclude that the value of D that minimizes costs is
therefore the maximum feasible value, S. (iv) Using D = S from part
254
CHAPTER 13. PROPERTY RIGHTS AND REGULATION
(iii) in the expression you obtained from part (i) to write the optimal
level of e¤ort (per boat per day) as a function of the model parameters
3. Using the model in Chapter 13.3.1, and the formulae given there, show
that under partial regulation …shers choose 12 times as much e¤ort per
day as the regulator chooses under full regulation.
4. As in Chapter 13.3.1, suppose that cost per unit of e¤ort per boat per
day is w, the price per unit of …sh is p, the catch per boat per day is
E 0:5 , the cost per boat per year is F , and the annual allowable catch is
A. The regulator can choose the number of days of the season, D, and
can also choose the number of boats, N , via a licensing requirement.
Compare the regulator’s optimal choice of N and D in the two cases
where (a) …shers choose their individually rational level of e¤ort, and
(b) where the regulator can choose E. For part (a) proceed as follows:
(i) Recall the fact that …shers choose a particular level of E (a function),
and that given that value of E and N , the regulator chooses a particular
level of D in order to satisfy the catch constraint. (ii) Substitute these
values of D and E in the regulator’s cost minimization problem and
minimize with respect to N . For part (b) proceed as follows: (i) Write
the cost minimization problem when the regulator chooses N; E; D.
Don’t forget the …xed cost. (ii) Use your answer from question 2 to
write the optimal level of E, as a function of N and D. (iii) Substitute
this expression for E into the minimand. For any level of N note that
the minimand is decreasing in D. What does this imply about the
optimal level of D? (iv) Use your answer to the previous part to write
the minimand as only a function of N . Find the cost minimizing value
of N .
5. The answer to question 4b tells you the optimal number of boats under
full regulation. The text tells you equilibrium number of boats under full regulation. Assume that …shers are able to solve the Coasian
bargain. (a) Using this information, what is the ratio of the ‡eet size
under ITQs to the ‡eet size under free entry and partial regulation to
ITQs? (b) Using this ratio, can you determine whether the number of
boats falls or rises, in moving from the open access partial regulation
equilibrium to the …rst best? HINT: You need to use inequality 13.2.
(c) What e¤ect do the parameters F and S have on this ratio? Provide
an economic explanation for this e¤ect. (d) What is the role, in your
13.6. TERMS, STUDY QUESTIONS AND EXERCISES
255
answer to part a of the assumption that …shers are able to solve the
Coasian bargain? (How would your answer have changed if you had
not made that assumption?)
Sources
Dietz, Ostrom and Stern (2003) and Ostrom (1990 and 2007) discuss conditions under which societies successfully manage common property without
formal regulation.
Hardin (1968) introduced the term “tragedy of the commons”.
Gordon (1954) provided the …rst well known analysis of …sheries as common property resources.
Ka¢ ne (2009) documents California surfers’ e¤orts to exercise de facto
property rights on some beaches.
The Millennium Ecosystem Assessment (United Nations, 2005), Sumaila
et al. (2011) and Dyck and Sumaila (2010) provide overviews of the state of
…sheries.
Johnson and Libcap (1982) describe the replacement of common property
with open access in the Alaska salmon …shery.
Fenske (2012) documents the case of property rights for rubber trees
among the Nigerian Igbo.
The 2009 National Marine Fisheries Service report to Congress summarizes the change in regulated U.S. …sheries.
Wittenberg (2014) provides the information for Box 13.1.
Smith (2012) surveys the problems of regulating …sheries when there more
“targets”(aka “margins”) than policy variables.
Homans and Wilen (1997) show how individually rational e¤ort decisions,
and the resulting overcapitalization, lead to reductions in the length of a
…shing season.
Homans and Wilen (2005) document the e¤ect of ITQs on revenue, and
provide the example of the British Columbia halibut …shery.
Weniger and Waters (2003) estimate potential revenue gains, cost reductions, and ‡eet consolidation due to using ITQs in the Gulf of Mexico reef
…sh …shery.
Weninger (1999) provides the information in Chapter 13.3.2 on the MidAtlantic surf clam.
Abbott and Wilen (2009 and 2011) examine …shers’response to quotas
on bycatch.
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CHAPTER 13. PROPERTY RIGHTS AND REGULATION
Costello, Gaines and Lynham (2008) estimate the e¤ect of ITQs on …shery
collapse, reported in Chapter 13.3.2.
Deacon, Parker, and Costello (2013) study a situation where …shers had
the option of obtaining property rights by joining a coop.
The 2012 Symposium “Rights-based Fisheries Management”, edited by
Costello and with papers by Aranson, Deacon, and Uchida and Wilen, reviews the literature on rights-based management.
Sharp and Sumaila (2009) quantify U.S. …shery subsidies.
Sumaila et al. (2013) quantify global …shery subsidies.
Aronson, Kelleher and Willman (2009) estimate the economic cost of
global …shery subsidies.
Chapter 14
Renewable resources: tools
Objectives
Gain familiarity with the building blocks used to construct models of
renewable resources.
Skills
Understand and be able to work with a growth function.
Understand the meaning of a “harvest rule”and understand how harvest rules change the growth equation.
Know the meaning of a steady state, and understand the relation between a growth equation and a steady state.
Have an intuitive understanding of the relation between a discrete time
and continuous time model.
Understand the meaning of stability, and be able to test for it in a
continuous time model.
Understand the meaning of “maximum sustainable yield”and be able
to identify it for simple growth functions.
A few basic tools make it possible analyze a wide variety of renewable
resource models, including stock pollutants, forestry, and …sheries. In subsequent chapters we use these tools to study the open access …shery and the
sole owner …shery, and also to study regulation in both of these settings.
257
258
CHAPTER 14. RENEWABLE RESOURCES: TOOLS
Renewable resource models involve one or more “stock variables” that
(potentially) change over time. In the nonrenewable resource setting, the
stock variable equals the amount of the resource remaining in the mine; there,
any extraction decreases the stock. In the renewable resource setting, the
extraction of the stock might be o¤set by intrinsic growth. In this situation,
the stock might either decrease or increase over time.
In water economics, we might measure the stock using the level of groundwater or the amount of water in a reservoir. Consuming the water reduces
the stock, but a natural recharge (e.g. rain) can increase it. In forestry
economics, the stock might be measured using the biomass of forestry, the
number of tons of wood. Cutting down trees reduces the stock, but the
forest’s natural growth increases it. In climate economics, the stock might
be measured using the parts per million (ppm) of atmospheric CO2 . Carbon
emissions increase the stock, but some of the stock is absorbed into other
carbon reservoirs, e.g. oceans. In all of these case, the stock might increase or decrease over time, depending on the relation between the natural
growth/recharge/decay and the consumption decisions.
14.1
Growth dynamics
Objectives and skills
Understand the meaning of biomass and the growth function.
Understand the meaning of the two parameters in the logistic growth
function.
Be able to graph this function.
For the sake of speci…city, we consider …shery economics, where we measure the stock of …sh using biomass, e.g. the number of tons of …sh. Biomass
does not capture the population age and size distribution: twenty half-pound
…sh and ten one-pound …sh contribute equally to this measure. For some
purposes, the age and size distributions are important. Those features are
hard to measure and they increase model complexity. For the purpose of
understanding the basics renewable resource model, and the …shery model in
particular, biomass is a useful measure of the stock.
We denote the stock of …sh in period t as xt , so the change in the stock
is xt+1 xt . The growth function, F (x), describes the evolution of the …sh
14.1. GROWTH DYNAMICS
259
stock in the absence of harvest. Some …sh die and new …sh are born, so the
stock might increase or decrease over time. Growth (absent harvest) depends
on the stock:
xt+1 xt = F (xt ) :
Growth also depends on the possibly random stocks of predators and prey and
changes in pollution concentrations and ocean temperature. Even holding
those features …xed, there may be intrinsic randomness of the growth of x.
These types of considerations lead to a more descriptive but more complicated
model, and we ignore them.
If a period equals one year, then F (x) equals the annual growth, and
F (x)
equals the annual growth rate. The percent growth rate is F (x)
100.
x
x
The most common functional form for the growth function is the Shae¤er,
or “logistic”model:
xt
F (xt ) = xt 1
.
(14.1)
K
This model uses two parameters, > 0, and K > 0. The parameter K is
the “carrying capacity”; it measures the level of stock that can be sustained,
absent harvest. Growth is zero if xt = K or if xt = 0; the stock grows
if 0 < xt < K and it falls if xt > K. Congestion decreases the carrying
capacity, K. As the stock increases, the …sh compete for prey, and/or they
become more vulnerable to predators. This congestion limits the potential
growth of the stock. Smaller values of K correspond to greater congestion,
and thus lower potential stock sizes.
The growth rate of the stock with the logistic growth function is
xt 1
F (xt )
xt+1 xt
=
=
xt
xt
xt
xt
K
=
1
xt
:
K
(14.2)
The parameter is the “intrinsic growth rate”. In the absence of congestion
(xt = 0 or K = 1), the growth rate equals . A larger value of K implies
a higher growth rate (less congestion) for given x. For given and K,
the growth rate falls with x, so the growth rate (not growth) reaches the
maximum value, , at x = 0. The value = 0:07, for example, means that
in the absence of congestion, the stock grows at 7% per year. For x close to
0, congestion is relatively unimportant, and the growth rate is close to 7%.
However, as the stock gets larger, congestion becomes more important, until
growth ceases as x approaches the carrying capacity, K. Figure ?? shows
graphs of the logistic growth function for three di¤erent growth rates, and
K = 50. For positive stocks, a larger implies larger growth.
260
CHAPTER 14. RENEWABLE RESOURCES: TOOLS
F
0.8
0.6
gamma = 0.07
0.4
gamma = 0.03
0.2
gamnma = 0.01
0.0
10
20
30
40
50
60
x
-0.2
-0.4
-0.6
-0.8
Figure 14.1: The logistic growth function for three values of :
14.2
Harvest and steady states
Objectives and skills
Understand the meaning of a “harvest rule”and be able to graph simple
harvest rules.
Understand the meaning of “steady state”.
Using the logistic growth function and a particular harvest rule, identify
the steady states both graphically and algebraically.
The introduction of harvest, y > 0, changes the dynamics. A “harvest
rule” gives the harvest level as a function of the stock. Two examples
illustrate harvest rules: y (x) = min (x; Y ), where Y is a constant, and y (x) =
x, with > 0 a constant. For the …rst example, harvest is constant at
Y if this level is feasible, i.e. if x
Y . If the stock is less than Y , all of
it is harvested. For the second example, harvest is a constant fraction of
the stock. For example, for = 0:01, annual harvest equals one percent
of the …sh stock. It is not possible to take more than the entire stock, so
1. Figure 14.2 shows the growth functions with the zero, constant, and
the proportional harvest. With the two non-zero harvest rules, the change
in the stock is
constant harvest: xt+1
xt = x t 1
harvest proportional to stock: xt+1
xt
K
min (xt ; Y )
xt = x t 1
xt
K
xt :
(14.3)
14.3. STABILITY
261
growth
zero harvest
constant harvest
0.2
0.0
10
20
30
40
50
60
x
-0.2
-0.4
proportional harvest
-0.6
-0.8
Figure 14.2: Growth with zero harvest (heavy solid curve), with harvest
proportional to the stock (dashed), and with the stock equal to Y for x Y .
Parameter values: K = 50, = 0:03, Y = 0:1 and = 0:01
A steady state is any level of the stock at which growth minus harvest
equals 0. A stock beginning at a steady state remains there. We have three
examples of harvest rules: zero harvest, and the two rules shown in equation
14.3. We obtain the steady states in these three cases by setting the growth
minus harvest equal to 0 and solving for x:
y = 0:
x
K
x 1
=0
) x1 ; 2 f0; 50g
y = min (x; 0:1) :
x 1
x
K
min (x; 0:1) = 0
) x1 2 f0; 3:6; 46:4g
y = 0:01x:
x 1
x
K
(14.4)
0:01x = 0
) x1 2 f0; 33g :
The subscript 1 denotes that the stock is a steady state level. Where this
meaning is clear from the context, we drop the subscript. These examples
use the parameter values K = 50, = 0:03, Y = 0:1, and = 0:01.
14.3
Stability
Objectives and skills
Be able to explain what it means to say that a steady state is stable or
unstable.
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CHAPTER 14. RENEWABLE RESOURCES: TOOLS
Given a …gure showing either (i) the growth function and the harvest
rule, or (ii) the growth function minus the harvest rule: be able to
identify the steady states and to determine (for the continuous time
problem) whether a steady state is stable or unstable.
Know that the conditions for stability are slightly more complicated in
the discrete time setting, and that the continuous time analog provides
a “reasonable approximation”to the discrete time problem only if the
length of a period in the latter is small.
A steady state is said to be “stable” if the stock trajectory approaches
the steady state when the stock begins su¢ ciently close to the steady state.
(What happens in Vegas stays in Vegas.) A steady state is “unstable” if
the stock trajectory moves away from the steady state when the stock begins
close to the steady state. (What happens in Washington DC gets broadcast
to the world.)
The stability or lack of stability of a steady state provides important
information about the dynamics of the …sh stock, and is therefore important
in policy questions. For example, suppose that harvest is a constant fraction
of the stock, y = x. For the parameter values used to generate the …nal
line of equation 14.4, we see that there are two steady states, x = 0 and
x = 33:33. It matters whether the …sh stock will die out, i.e. approach
x = 0, or approach x = 33:33, which is 66% of the carrying capacity, K.
14.3.1
Discrete time versus continuous time models
We have emphasized discrete time models. The discrete time setting makes it
possible to derive the necessary condition for optimality (the Euler equation)
using only elementary calculus. The intuition is also more straightforward
in the discrete time setting. The discrete time setting also makes it easier to
describe the dynamics in the renewable resource setting (without optimization), as above. There is a cost to convenience. The cost is that the analysis,
as distinct from the description, of the discrete time dynamic system is more
complicated than for the continuous time system.
The problem arises because in the discrete time setting there is a nonnegligible change in variables (e.g. the stock of …sh) from one period to the
next, outside of a steady state. Here, the stock might jump from one interval
to another where the behavior is quite di¤erent. This possibility can lead to
14.3. STABILITY
263
chaos even for simple models like the logistic growth function. Chaos refers
to the situation where paths (trajectories) are very irregular, i.e. they do
not repeat in a …nite amount of time. These paths are very sensitive to the
initial condition, and therefore do not lend themselves to resource modeling
(at least for our purposes).
It is possible to rule out chaotic behavior by restricting parameter values,
but that still leaves special cases. For example, a steady state might be stable
(meaning that paths starting close to the steady state approach the steady
state), but the approach path might be monotonic (steadily increasing or
decreasing over time) or it might be cyclical (…rst increasing, then decreasing,
then increasing, and so on).
These possibilities can be interesting, but they require a digression that
is tangential to our study of renewable resources. In addition, in the discrete
time setting we cannot determine the stability or instability of a steady state
merely by examining a graph; we require calculation. Therefore, for the
purpose of considering stability, we consider “the continuous time analog”
to the discrete time model. Using the general growth function and harvest
rule, F (x) and y(x), we begin by replacing equation 14.3 with the more more
general relation
xt+1 xt = F (xt ) y (xt ) :
(14.5)
Instead of studying the stability of steady states of this equation, we study
the stability of steady states of the continuous time analog, the ordinary
di¤erential equation1
dxt
= F (xt ) y (xt ) :
(14.6)
dt
14.3.2
Stability in continuous time
It is simple to determine whether an interior steady state is stable in the
continuous time setting. First, we identify the steady state(s) by …nding the
solution(s) to the equation F (x) y (x) = 0, exactly as in the discrete time
setting. Figure 14.3 shows the graph of an arbitrary function F (x) y (x)
1
Appendix G explains the relation between equations 14.5 and 14.6. These two equat
tions have the same steady states: xt+1 xt = 0 if and only if dx
dt = 0. Provided that
the length of a period in the discrete time setting is su¢ ciently small, the dynamics of the
continuous and discrete systems are similar in the neighborhood of a steady state. In this
case, it is valid to study the stability of steady states in the discrete time setting using the
continuous time model.
264
CHAPTER 14. RENEWABLE RESOURCES: TOOLS
F(x)-y(x)
10
8
6
4
2
0
1
2
3
4
5
-2
6
x
-4
-6
-8
-10
= F (x) y (x). There are three steady states,
Figure 14.3: The graph of dx
dt
at x = 0:4, x = 1:4 and x = 5:2. The …rst and third are stable, and the
middle steady state is unstable
(one without a speci…c resource interpretation).
This function has three roots, i.e. three steady states, where the graph
crosses the x axis. Consider the low steady state, x = 0:4. We see that for
a value of x close to but slightly below 0.4, dx
= F (x) y (x) > 0, i.e. x is
dt
becoming larger over time. For a value of x close to but slightly above 0.4,
dx
= F (x) y (x) < 0, i.e. x is becoming smaller. Therefore, we conclude
dt
that x = 0:4 is a stable steady state: a trajectory beginning close to, but not
equal to x = 0:4 approaches the level x = 0:4. A parallel argument shows
that the middle steady state, x = 1:4, is an unstable steady state, and the
large steady state, x = 5:2, is a stable steady state.
Noticing that the slope of F (x) y (x) is negative at the two stable steady
states in Figure 14.3, and the slope is positive at the unstable steady state,
we obtain the following rule for checking stability.2
x1 is a stable steady state if and only if
d(F (x1 ) y(x1 ))
dx
x1 is an unstable steady state if and only if
14.3.3
<0
d(F (x1 ) y(x1 ))
dx
(14.7)
> 0:
Stability in the …shing model
) y(x1 ))
If d(F (x1dx
= 0, points on one side of the steady state approach the steady
state, and points on the other side move away from the steady state. This is a knife-edge
case, and we do not consider it further.
2
14.3. STABILITY
265
y
0.3
0.2
0.1
0.0
10
-0.1
20
30
40
50
x
-0.2
-0.3
-0.4
-0.5
Figure 14.4: Solid curve: the graph of the logistic curve, F (x). Dashed
curve: the graph of F (x) 0:01x. Dotted curve: the graph of F (x) 0:3:
Figure 14.4 shows the graphs of: the logistic growth function F (x) with
carrying capacity K = 50 and intrinsic growth rate = 0:03 (solid); F (x)
0:01x (dashed); and F (x) 0:3 (dotted) We can use the rule in equation 14.7
to determine whether the steady states corresponding to the three graphs in
Figure 14.4 are stable or unstable.
The points of intersection of the graphs in Figure 14.4 and the x axis
are steady states. Absent harvest, the solid curve shows that the steady
states are x = 0 and x = K = 50. Using the rule in equation 14.7, we
see that x = 0 is an unstable steady state and x = K is a stable steady
state. The dashed curve shows that under proportional harvest, y = 0:01x,
the two steady states are x = 0 and x = 33:33. Again, the low steady state
is unstable and the high steady state is stable.
The dotted curve, corresponding to constant harvest y = 0:3, has two
points of intersection with the x axis: x = 13:8 and x = 36:2. The low steady
state is unstable, and the high steady state is stable. Under constant harvest,
x = 0 is also a stable steady state, even though the graph of F (x) 0:3 does
not intersect the x axis. In the continuous time model, y is a rate. When
x > 0, y can take any …nite non-negative value, although perhaps not for
long. As soon as the stock hits x = 0, harvest must stop. If stock is small,
here lower than 13.8, constant harvest y = 0:3, exceeds natural growth, and
the stock falls. The stock heads to extinction, x = 0.
Figure 14.4 illustrates an important possibility that we will encounter
again. Under zero harvest or harvest proportional to the stock, the stock
always approaches the high steady state (x = 50 and x = 33:33 in the two
266
CHAPTER 14. RENEWABLE RESOURCES: TOOLS
harvest
y=0.3
0.3
F
0.2
y=0.01x
0.1
0.0
10
20
30
40
50
x
-0.1
Figure 14.5: The logistic growth function, and two harvest rules: constant
harvest, y = 0:3 (dotted) and harvest proportional to stock, y = 0:01x
(dotted).
examples), provided that the initial stock is positive. In contrast, under
constant harvest, beginning with a positive stock, the stock might eventually
approach either stable steady state, x = 0 or x = 36:2. The unstable steady
state, x = 13:8, is a critical stock level. For initial stocks above this critical
level, the stock approaches the high steady state, and for initial stocks below
this level, the stock approaches the low steady state, 0.
Figure 14.5 shows the logistic growth function and two harvest rules, the
constant harvest y = 0:3 and proportional harvest y = 0:01x. Despite their
di¤erent appearance, Figures 14.5 and 14.4 can both be used to identify
steady states and their stability. Figures 14.5 shows graphs of the growth
and the harvest functions in the same …gure, and Figure 14.4 shows their
di¤erence. The reader should con…rm that, as we move from left to right in
Figure 14.5:
(i) If the harvest function cuts the growth function from above,
the associated steady state is unstable.
(ii) if the harvest function cuts the growth function from below,
the associated steady state is stable.
14.4
Maximum sustainable yield
Objectives and skills
14.4. MAXIMUM SUSTAINABLE YIELD
267
Know the de…nition of Maximum Sustainable Yield (MSY) and be able
to calculate the MSY for the logistic growth function.
Understand the economic factors that determine whether optimal steady
sate harvest should be above or below MSY.
The maximum sustainable yield is the largest harvest that can be sustained in perpetuity. The maximum sustainable yield occurs where the
growth function, F (x), reaches its maximum. For logistic growth, the maximum sustainable yield occurs at the stock x = K2 , which for our example
equals 25; there, the harvest is y = F (25) = 0:375. Any constant harvest
y lower than or equal to the maximum sustainable yield can be sustained in
perpetuity.
The steady state …sh stock that maximizes sustainable harvest is the
solution to the condition dFdx(x) = 0. For the logistic growth function, this
condition is
d x 1 Kx
1
set
=
(K 2x) = 0:
dx
K
Solving this equation gives the steady state stock that maximizes the sustainable yield
K
x= :
2
Substituting this value into the growth function, x 1 Kx , gives the maximum sustainable yield
1
MSY: y = K :
4
An increase in either the intrinsic growth rate, , or the carrying capacity,
K increases the maximum sustainable.
The socially optimal steady state
Chapter 16 examines the socially optimal steady state, but even at this stage
we can use economic logic to identify the factors that determine the optimal
level. First, we need a criterion for comparing di¤erent outcomes. The
most common criterion (discounted utilitarianism) is the present discounted
value of the stream of consumer and producer surplus. (For the discussion
here we ignore externalities and taxes.)
What steady state is optimal under this criterion? By de…nition, steady
state output = consumption is highest at the MSY, so consumer surplus
268
CHAPTER 14. RENEWABLE RESOURCES: TOOLS
is highest there. Producer surplus, equal to revenue minus costs, depends
on the cost of catching …sh. If it is cheaper to catch …sh when there are
many …sh (the stock is large), increasing x above K2 reduces costs. With
stock dependent harvest costs, the increase in the stock and the consequent
decrease in the costs might increase producer surplus. (At this level of
generality we do not know whether the change actually increases producer
surplus, because the change also alters revenue.) It is easy to construct
examples where increasing the stock above K2 increases producer surplus and
also increases the sum of producer and consumer surplus.
Even in this circumstance, we cannot conclude that it is socially optimal
to have a steady state stock above K2 . With a positive discount rate, a future
bene…t (e.g. consumer or producer surplus) is less valuable than a current
bene…t. Therefore, a positive discount rate encourages society to consume
more today, leaving less for the future, and reducing the stock below the
MSY. In summary, consideration only of the consumption bene…t suggests
that the MSY is the optimal steady harvest. Recognition that harvest costs
depend on stock size suggests that the optimal steady state stock might lie
above the level of MSY. Taking into account that the future is worth less
than the present (due to discounting) suggests that society should aim for a
steady state stock below the level of MSY. At this level of generality, there
is not much more one can say. We need a model to determine the balance
of these competing forces.
14.5
Summary
Ignoring age and size distribution, we measure the stock of …sh as biomass,
e.g. the number of tons of …sh. The growth equation determines the stock
in the subsequent period as a function of the current stock. The logistic
growth function depends on two parameters, the intrinsic growth rate and
the carrying capacity K. A harvest rule, a function of the stock of …sh,
determines the harvest in a period. We emphasized two simple harvest
rules. One sets harvest equal to a constant fraction of the stock, and the
other sets harvest equal to a constant.
At a steady state, the …sh stock remains constant over time. The steady
state depends on both the growth function and the harvest rule. There may
be multiple steady states. A steady state is stable if and only if stocks that
begin su¢ ciently close to the steady state converge to that steady state. If
14.6. TERMS, STUDY QUESTIONS, AND EXERCISES
269
a trajectory beginning at any initial condition close to but not equal to the
steady state moves away from that steady state, the steady state is unstable.
We introduced the continuous time model, for the purpose of making it
easy to determine the stability or instability of a steady state. In order to
relate the discrete and the continuos time models, the reader should think of
the length of a period in our discrete time setting as being very small.
In the continuous time setting, the interior steady states and their stability, are easy to determine. If the growth function is F (x) and the harvest
= F (x) y (x). Any solution to F (x) y (x) = 0
function y (x), then dx
dt
is a steady state. That steady state is stable if the derivative, d(F (x)dx y(x)) ,
evaluated at the steady state is negative; it is unstable is this derivative is
positive. We also need to consider levels of the state at which an inequality
constraint binds. In the …shing context, the biomass cannot be negative, so
x 0. With (positive) constant harvest, we saw that the steady state x = 0
is stable.
The maximum sustainable yield equals the maximum point on the growth
function. For the logistic growth model, the maximum sustainable yield
occurs where the stock is half of its carrying capacity.
14.6
Terms, study questions, and exercises
Terms and concepts
Biomass, stock variables, annual growth rate, logistic growth (or logistic
model), carrying capacity, intrinsic growth rate, harvest rule, steady state,
chaos, stability, monotonic path, cyclical path, Maximum Sustainable Yield.
Study questions
1. Given the graph of a growth function xt+1 xt = F (x), you should
be able to identify the carrying capacity and the maximum sustainable
yield, and say in a few words what each of these mean.
2. For a di¤erential equation dx
= G (x), if you are shown a graph of G (x)
dt
you should be able to identify the steady state(s) and say which, if any
of these are stable. You should be able to explain your answer in a
couple of sentences.
270
CHAPTER 14. RENEWABLE RESOURCES: TOOLS
3. (a) Given a single …gure that shows both the graph of the growth
function F (x) and the harvest function y (x), should be able to identify
the steady states and explain (in very few words) which is stable and
which is unstable. You should be able to sketch a graph of their
= F (x) y (x), and use the rule in equation 14.7 to
di¤erence, dx
dt
con…rm that your answer to part (a) was correct.
Exercises
1. Using an argument that parallels the discussion of the low steady state
in Figure 14.3, explain why the middle steady state is unstable and
why the high steady state is stable.
2. For the logistic growth function, F (x) =
x Kx , identify the proportional harvest rule (the value of in the rule y = x) that supports
the maximum sustainable yield as a steady state. Is this steady state
stable?
3. The “skewed logistic”growth function is
F (xt ) = xt 1
xt
K
,
with > 0. For = 1 we have the logistic growth function in equation
14.1. (a) How does the magnitude of a¤ect the growth rate? (b) The
maximum sustainable yield occurs at x = K
. Derive this formula.
+1
(c) Use a software package of your choice to draw this curve for K = 50,
= 0:03, and for both = 2 and = 0:5.
Sources
Cite Shae¤er paper
Colin Clark’s book
Chapter 15
The open access …shery
Objectives
Ability to characterize the harvest rule and the evolution of biomass
under open access, and to analyze the e¤ect of speci…c policies.
Skills
Be able to determine the cost function corresponding to a particular
production function.
Determine the relation between a representative …rm cost function and
the open access “harvest rule”.
Determine the evolution of biomass under this open access harvest rule.
Understand how a tax or some other type of regulation a¤ects the
incentive to harvest, at an arbitrary level of the biomass.
Determine the e¤ect of the policy on the evolution of biomass, and on
the steady state(s).
We apply the methods developed in previous chapters to study the open
access …shery. We want to determine both the static and the dynamic e¤ects
of taxes (or some other type of regulation). The static tax e¤ect equals the
e¤ect of the tax at a point in time, for a given biomass. The dynamic tax
e¤ect equals the e¤ect of the tax on the evolution of the biomass.
271
272
CHAPTER 15. THE OPEN ACCESS FISHERY
The agents studied in the chapters on non-renewable resources were (by
assumption) sole owners of the resource. Each of those agents knows that
extracting one more unit in the current period means that they have one less
unit to extract in some future period. These agents are “forward looking”;
they take into account future prices in making current decisions.
In contrast, in the open access …shery, no individual has property rights.
Moreover, we assume that each individual is such a small part of the total
‡eet that her actions have negligible e¤ect on the aggregate catch. Under
these circumstances, it is rational for an agent to ignore the future: there is
nothing she can do about it. Here, an individual …sher chooses her current
harvest level to maximize her own current pro…ts, without regard for the
e¤ect of her decisions on other …shers or on her own future pro…ts.
Chapter 13.3 studies a static version of this scenario, where everything
happens in the same period. Dynamics are central to the …shery problem,
where the externality unfolds over time. Here we study a model in which
…shers’aggregate current harvest a¤ects the subsequent stock, thereby a¤ecting future harvest cost: there is a “dynamic”externality. We emphasized in
Chapter 13.3 that the …rst best harvest typically involves many types of decisions, and that regulators rarely have as many instruments (policy variables)
as there are targets. There we assumed that the regulator could in‡uence
harvest by choosing the length of the …shing season. Here we consider a
di¤erent policy variable, a tax on catch, known as a landing fee. We show
how this tax in‡uences the open access equilibrium.
We use two specializations of the parametric cost function from equation
4.1, c (x; y) = C ( + x) y 1+ . The …rst, with constant stock-independent
average harvest costs, sets = = = 0, so c (x; y) = Cy. The second
specialization sets = = 0 and = 1, so c (x; y) = C xy .
15.1
Stock-independent costs
Objectives and skills
Given an inverse demand function and constant harvest costs, …nd the
open access harvest rule.
Sketch the graph corresponding to this harvest rule and the logistic
growth function.
15.1. STOCK-INDEPENDENT COSTS
273
Use this graph to identify the steady states and (for the continuous time
setting) determine whether each steady state is stable or unstable.
The case of stock-independent constant harvest costs, c (x; y) = Cy, provides the simplest illustration of the open access equilibrium. The competitive supply function is in…nitely elastic (‡at) at C for y x, and perfectly
inelastic (vertical) at y = x. In the discrete time setting, it is not possible
for y to exceed x.1 We assume that …rms can adjust their output instantaneously, so under open access, pro…ts are zero in every period, and price
equals average cost.
For the inverse demand function p = a by, price = average costs requires
y = a b C . If C a, then average cost is greater than or equal to the choke
price (a) and harvest is always zero. We therefore assume C < a, where
open access harvest is positive whenever the stock is positive. Here, the
open access harvest rule is
y (x) = min x;
a
C
b
(15.1)
Chapter 14 examines the dynamics under this kind of harvest rule. Figure
15.1 reviews this material, showing the graph of a logistic growth function
and the graphs of two harvest rules, corresponding to a low and a high cost,
C (the dashed and dotted lines, respectively). For both of these costs, there
are three steady states. The zero steady state and the high steady state
(occurring to the right of the maximum sustainable yield) are both stable.
The middle steady state is unstable.
1
In the continuous time setting, harvest is a rate and can take any nonnegative value
when x > 0 (Chapter 14.3.3).
274
CHAPTER 15. THE OPEN ACCESS FISHERY
y
0.3
((a-1)/b)
0.2
0.1
((a-3)/b)
0.0
10
20
30
40
50
x
Figure 15.1: The solid curve is the graph of the logistic growth function with
= 0:03 and K = 50. The dashed line shows the open access harvest rule
for C = 1 and the dotten line shows the harvest rule for C = 3. Inverse
demand is p = a = by with a = 3:5 and b = 10.
Box 15.1 The zero pro…t assumption. The assumption that pro…t
is zero in each period is based on the implicit assumption that e¤ort
adjusts instantaneously. In fact, adjustment is likely to be sluggish,
simply because more rapid adjustment (either increases or decreases
in e¤ort) is more expensive. With this kind of adjustment cost, e¤ort
becomes a stock variable (an asset). Two types of models study this
situation. The simplest uses an ad hoc assumption that the change in
e¤ort is an exogenous function of current pro…t. For example, current
change in e¤ort is proportional to current pro…t. The second type of
model assumes that agents are forward looking with respect to their
investment decision. Both settings can lead to cyclical dynamics,
where the resource stock rises and falls over time.
15.2
Stock-dependent costs
Objectives and skills
Given a particular production function, be able to …nd the …rm’s cost
function.
Given this cost function and an inverse demand function, …nd the open
access harvest rule.
15.2. STOCK-DEPENDENT COSTS
275
Be able to sketch the graph of this harvest rule and the logistic growth
function.
Use this graph to identify the steady states, and to determine which of
these are stable and which unstable.
Be able to determine how a change in a parameter of the demand
function or the cost function alters the graph, thus altering the stock’s
dynamics.
The case of stock dependent costs is more interesting, but more complicated than the constant cost case. We therefore begin with a road map:
1. We use a production function to obtain a cost function in the open
access equilibrium.
2. We use the cost function and the market clearing condition (supply =
demand) to obtain the open-access “harvest rule”.
3. We use the open-access harvest rule to examine the dynamics of biomass, and to determine how parameters of the model (e.g. the demand
intercept) alters the dynamics.
1. From the production function to the cost function Fishing “effort”, E, can be measured in a variety of ways, including by the number of
boats and the number of person-hours per season. We treat e¤ort as an
amalgam of all of the inputs in the …shery sector. For example, one unit of
e¤ort might equal one boat, 200 hours of labor, and a particular type of net.
In a representative agent model, Et is the aggregate e¤ort in the …shery in
period t. Greater e¤ort increases harvest.
Fish are easier to catch when the stock is large, so for a given amount
of e¤ort, a larger stock increases harvest. The …shery production function
shows how e¤ort, E, and the stock, x, determine harvest, y. The simplest
production function assumes that the level of harvest per unit of e¤ort is
proportional to the size of the stock, Ey = qx, or
y = qEx:
(15.2)
The parameter q > 0 is the “catchability coe¢ cient”. A larger q means that
for a given stock size, …shers need less e¤ort to obtain a given level of harvest.
276
CHAPTER 15. THE OPEN ACCESS FISHERY
The cost per unit of e¤ort is the constant, w. If one “unit of e¤ort” equals
one boat and 200 hours of labor and a particular net, then w equals the cost
of renting the boat, paying the crew, and buying or renting the net.
We use the production function to derive the cost function, which shows
how costs depend on harvest, the stock and parameters of the model. We
can rewrite equation 15.2 as
y
= E;
qx
which shows how e¤ort depends on the level of harvest and the stock; E units
of e¤ort cost wE. Therefore, the cost of harvesting y, given the stock x, and
given parameters w and q, is
harvest cost: c (x; y) = wE =
where C
w
.
q
C
w
y = y;
qx
x
(15.3)
Harvest costs fall as the stock of …sh rises.
2. From the cost function to the open access harvest rule Individual
…shers in our model are price takers and they ignore the e¤ect of their harvest
on the subsequent stock. If each individual …sher is a small part of the
industry, as is consistent with the price-taking assumption, then it is rational
for them to act as if their own current harvest decision has negligible e¤ect
on the next-period stock, and thus on their next-period harvest cost. With
these assumptions, and the cost function in equation 15.3, the equilibrium
condition for harvest is that price equals average cost. For this model, where
average = marginal harvest costs, each …sher obtains zero pro…t (= producer
surplus) in each period (Box 15.1).
If price is less than marginal = average cost, then supply equals 0. If
supply is positive, then the zero-pro…t condition requires that price equals
average cost. This condition determines the inverse supply function
psupply (y) =
C
:
x
(15.4)
Setting inverse demand, p (y), equal to the inverse supply function in equation 15.4, gives the equilibrium condition (demand = supply, when supply is
positive)
C
p (y) = :
x
15.2. STOCK-DEPENDENT COSTS
277
With linear inverse demand p (y) = a
be solved to yield the harvest function
by, this equilibrium condition can
y (x) =
1
b
a Cx for a Cx
0 for a Cx < 0:
0
(15.5)
The second line of this equation takes into account the fact that supply =
demand = 0 if the average harvest cost, Cx , exceeds the choke price, a.
Most …sheries are regulated. Some regulations, e.g. gear restrictions,
increase harvest costs. In the model here, these types of regulations imply
an increase in the parameter C. Equation 15.5 shows that a larger C reduces
the equilibrium open access harvest rule. Taxes, or landing fees, drive a wedge
between consumer and producer prices, also altering the harvest rule.
3. Using the harvest rule to examine dynamics of the biomass
Figure 15.2 shows the graph of the logistic growth function and the open
access harvesting rule in equation 15.5 for both “low demand”, p = 3:5 10y
and “high demand”, p = 4:2 10y, the dashed and dotted curves, respectively.
The …gure also shows the three interior steady states (where x > 0), points
A; B; D, under low demand. In the high demand scenario, the only interior
steady state occurs at a low stock level, close to but slightly lower than point
A. Extinction, x = 0, is a steady state in both cases.
Consider the low demand scenario, shown by the dashed curve. Moving
from left to right, we see that the dashed curve cuts the growth function
from below at points A and D, and it cuts the growth function from above
at point B. Thus, points A and D are stable steady states, and point B is
an unstable steady state. (Refer to the …nal paragraph in Section 14.3.3 if
this claim is not clear.) The point x = 0 is an unstable equilibrium. For
su¢ ciently small but positive stock, a Cx < 0. Here, the stock is so low,
and the harvest costs so high, that the equilibrium harvest is 0. Because
growth is positive for small positive x, the …sh stock is growing in this region.
Therefore, if the initial stock is positive and below point B, the stock in the
open access …shery converges to point A. If the initial stock is above point
B, the stock converges to point D.
In the high demand scenario (the dotted curve), there are two steady
states. The stable steady state is slightly below point A; x = 0 is an
unstable steady state. Thus, in the high demand scenario, for any positive
initial stock, the stock under open access converges to a level slightly below
278
CHAPTER 15. THE OPEN ACCESS FISHERY
growth
0.4
D
B
0.3
0.2
0.1
A
0.0
10
20
30
40
50
stock
-0.1
Figure 15.2: The solid curve shows the logistic growth function with = 0:03
and K = 50. The dashed curve shows the open access harvest rule, with
a = 3:5, b = 10 and C = 5 and the three steady states, A, B, and D. The
dotted curve shows the open access harvest rule with a = 4:20, b = 10 and
C = 5.
point A.2
15.3
Policy applications
Objectives and skills
Using the …gure obtained from the previous section, describe and explain the e¤ect of a constant tax on the evolution of the biomass and
on the steady states.
Illustrate and explain the role of the initial condition (the stock at the
time the tax is imposed) on the e¤ect of the tax.
Policy a¤ects the equilibrium outcome by altering the equilibrium harvest
rule. Even this simple model is rich enough to address several types of policy
questions. We noted that policies that restrict the type of gear …shers can
use, increase C. From equations 15.1 and 15.5 we see that a larger C shifts
2
Appendix H provides a di¤erent way to visulaize the equilibrum, using the concept of
a “bioeconomic equilibrium”.
15.3. POLICY APPLICATIONS
279
down the harvest rule. Here we consider the e¤ect of taxes when average
harvest costs depend on the stock.
The fact that there are (potentially) multiple steady states in this model
means that the e¤ect of policy might be sensitive not just to the level of policy
(e.g., the magnitude of the tax), but also to the level of the stock at the time
the policy is imposed. A policy imposed when the stock is low might have
a completely di¤erent e¤ect on the outcome, than would the same policy
imposed when the stock is high. We explore this dependence under taxes.
The e¤ect of the tax on the harvest rule Suppose that producers face
the unit tax . Chapter 11 shows that (in a closed economy) a tax has
the same e¤ect regardless of whether consumers or producers have statutory
responsibility for paying it. That chapter also explains that the tax incidence
depends on relative supply and demand elasticities. In the model here,
with constant average = marginal costs in a period (i.e. for a given level of
biomass), supply is in…nitely elastic. Therefore, consumers bear the entire
incidence of the tax at a point in time (i.e. for a given level of biomass).3
Denote the price …shers receive as p, so consumers’ price is p + . The
inverse supply curve remains unchanged at psupply (y) = Cx . With linear
inverse demand, consumers facing the price p + demand the quantity y
that solves p + = a by.
The market clearing condition, supply equals demand, requires p + =
a by = Cx + . Solving the second equation for harvest gives the harvest
rule
1
C
C
0
a
for a
b
x
x
y (x) =
(15.6)
C
0 for a
< 0:
x
The second line of equation 15.6 recognizes that if the average cost plus the
C
< 0), then sales
tax, + Cx , is greater than the choke price (so that a
x
are zero. This equation shows that the harvest rule depends on the di¤erence
between the choke price and the tax, a
. A one unit decrease in a or a
one unit increase in have the same e¤ect on a
. An increase in the tax,
, has the same e¤ect on harvest as does a decrease in the demand intercept,
a. We use this fact and Figure 15.2 to study the e¤ect of a change in the
unit tax, .
3
Fishers are already receiving zero pro…t in equilibrium. A tax could not reduce their
pro…t to a negative number: …shers would leave the sector rather than experience negative
pro…ts.
280
CHAPTER 15. THE OPEN ACCESS FISHERY
growth
0.4
B'
D'
D
B
0.3
0.2
0.1
A
0.0
10
20
30
40
50
stock
-0.1
Figure 15.3: The logistic growth function (the solid graph) and three equilibrium harvest rules: the harvest rule under high demand, p = 4:2 10y for
no tax (the highest graph); the harvest rule under a unit tax = 0:35 (the
middle graph); and the harvest rule under a unit tax = 0:7 (the lowest
graph).
Figure 15.2 shows how a change in the demand intercept alters the harvest
rule, thus altering the equilibrium steady states and the dynamics of the …sh
stock. For the policy application, we hold the demand intercept …xed at
a = 4:2 (the high demand scenario) and we consider the e¤ect of changing
the tax, . We use the fact that a decrease in a or an increase in have the
same e¤ect on the harvest rule. Figure 15.3 reproduces Figure 15.2, adding
(merely for emphasis) a harvest rule laying between the original two harvest
rules. Holding a …xed at 4:2, Figure 15.3 illustrates the e¤ect of increasing
the tax from = 0, …rst to = 0:35 (so a
= 3:85), and then to = 0:7
(so a
= 3:5).
The role of the initial condition The steady states B 0 and D0 correspond
to the harvest rule for = 0:35. The low steady states for the three harvest
rules are slightly di¤erent, but they all occur so close to point A, where
xA = 1:7, as to be indistinguishable at the level of resolution contained
in the …gure. However, the intermediate and the high steady states are
appreciably di¤erent under the two positive taxes; these steady states do not
exist if = 0. The numerical values of the stock corresponding to the low,
15.3. POLICY APPLICATIONS
281
intermediate and high steady states are
xA = 1:7, xB = 15:4, xB 0 = 20, xD0 = 28:6, and xD = 33:2:
We do not distinguish between xA and xA0 because these are so close that
their di¤erence is not interesting.
The stock at the time the tax policy is imposed is the “initial condition”.
Figure 15.3 illustrates that the initial stock, – not just the policy level –
can have a signi…cant e¤ect on the stock dynamics, and particularly on the
steady state to which the stock converges. If the initial stock is less than
15.4, neither tax has an appreciable e¤ect on the steady state. The stock
converges to a point close to xA = 1:7, regardless of whether = 0 or = 0:35
or = 0:7.
If the initial stock is between 15:4 and 20, the tax = 0:35 has a negligible
e¤ect on the steady state to which the stock converges: close to xA = 1:7. In
this same circumstance, the tax = 0:7 causes the stock to converge to 33.2.
If the initial stock lies above x = 20, then the stock converges to x = 28:6
for = 0:35 and to x = 32:2 for = 0:7.
In this example, if the initial stock is “moderately small” (15:4 < x0 <
20), then = 0:35 has a negligible e¤ect on the steady state, whereas the
tax = 0:7 has a qualitative e¤ect, leading to a large increase in the steady
state. If the initial stock is “moderately large”, (x0 > 20) then both taxes
lead to a qualitatively di¤erent steady state, compared to = 0. The high
steady states under = 0:7 (xD = 33:2) and under = 0:35 (xD0 = 28:6) are
appreciably di¤erent. The zero tax leads to a very low steady state stock,
regardless of the initial condition.
Steady state welfare e¤ects of the tax Provided that the initial condition is greater than x = 20, the low tax leads to a lower steady state, but
a higher steady state harvest, compared to the high tax: point D0 is higher
(larger y) than point D in Figure 15.3. Therefore, the steady state ‡ow
of consumer surplus (at the high stable steady state) is higher under the
low tax. This claim follows from the fact that consumption = harvest is
higher at xD0 than at xD ; consequently, consumers must pay a lower price at
xD0 . In the high steady state under the high tax, xD = 33:2, so harvest =
consumption is the solution to
y = xD 1
xD
K
= 0:335.
282
CHAPTER 15. THE OPEN ACCESS FISHERY
Consumers are willing to purchase this amount if they face the tax-inclusive
price 4:2 10 0:335 = 0:853. At this price, consumer surplus (our measure
of consumer welfare) is4
Z
4:2
0:853
4:2 z
10
dz = 0:56
and tax revenue is y = 0:7 (0:853) = 0:234. Pro…ts are zero at every point,
so social welfare equals the sum of consumer surplus plus tax revenue, 0.794.
Table 1 collects these numbers at the high stable steady states under the two
taxes. It shows that steady state consumer surplus is higher, tax revenue is
lower, and their sum, social welfare, is higher under the smaller tax.
consumer
social
tax revenue
surplus
welfare
0:7
33.2
0.335
0.853 0.56
0.234
0.794
0:35
28.6
0.367
0.528 0.67
0.129
0.803
Table 15.1: High stable steady state stock, harvest, price, consumer surplus, tax
revenue, and social surplus for two taxes. K = 50; = 0:03; C = 5, inverse
demand = p = 4:2 10y
tax ( )
stock
harvest
price
Policy implications This analysis shows that the e¤ect of the tax on the
steady state payo¤ depends on the initial condition (= the value of the stock
at the time the tax is …rst imposed). For example, if the initial stock is
x0 > xD0 = 20, the smaller tax drives the stock to a steady state with a
higher payo¤, compared to the larger tax. However, if the initial stock is
15:4 < x0 < 20, the low tax drives the stock to a low steady state (close to
1.7) with a very low payo¤, whereas the high tax drives the stock to the high
state, 33.2, with the relatively high payo¤.
If the regulator actually knows the initial value of the stock, along with
all of the other parameters of the problem, then she can solve the mathematical problem to determine the optimal (constant) tax. In the real world,
the regulator does not know these things, and data limitations make their
estimation di¢ cult. It is possible to estimate the demand function, using
data on price and quantity. It is di¢ cult to get good data on costs or stocks;
4
Recall that consumer surplus equals the area “to the left” of the demand curve, between the market price, here 0.853, and the choke price, here 4.2. The integral equals
this area.
15.3. POLICY APPLICATIONS
283
consequently, at best we can hope for ball-park estimates of the parameters
of the cost function and the growth equation, C; ; K. In addition, we have
at best a rough idea of the current biomass, the initial condition to the optimization problem. Therefore, it would be risky to recommend a tax using
an optimization problem based on the …ction that we actually know these
parameters.
However, the analysis above reveals the nature of the trade-o¤. Given
the amount of uncertainty in going from the real world to the mathematical
model, the regulator might want to build in a margin of safety by choosing a
tax that exceeds the optimal level for this toy model. The larger tax provides
protection against the possibility that we overestimated the initial biomass.
Our estimate of the initial biomass might be above the critical level where
the trajectory takes us to a high steady state; but if the actual initial stock is
below that critical level, the consequences of the tax are very di¤erent than
we expected. Because we do not actually know the precise initial stock, we
may want to include a margin of safety in choosing the tax.
Non-constant taxes Thus far we have considered only a constant tax.
We know from elementary principles that a constant tax is not …rst best,
even if we knew all of the parameters of the problem. The constant tax
is a single number. It can therefore be used to target (i.e., to select) a
single variable. We have used the steady state stock, and corresponding
payo¤, as the target of interest. However, it makes sense for the regulator to
care about the payo¤ along the trajectory en route to the steady state, and
thus to care about the trajectory –not just its endpoint (the steady state).
Thus (ignoring the uncertainty discussed above), a richer description of the
regulatory problem includes taxes in each period, or for each possible level
of the biomass.
Even if a tax has only a negligible e¤ect on the steady state to which the
stock converges, the tax may nevertheless have a signi…cant e¤ect on welfare,
by slowing the decline of the …shery. For example, if x0 is slightly lower than
15:4, the tax = 0:35 causes a 9% reduction in the initial harvest, and the tax
= 0:7 causes an 18% reduction in the initial harvest; both of these changes
are relative to harvest under = 0. Even though this di¤erence is not large
enough to keep the stock from converging to approximately the same low
level, the higher tax slows the …shery’s decline. The welfare trajectories
under the high and the low taxes lead to approximately the same low level,
284
CHAPTER 15. THE OPEN ACCESS FISHERY
but the welfare begins at a higher level and falls more slowly under the larger
tax. Therefore, for an initial condition below 15:4, the present discounted
stream of that welfare is likely higher under the larger tax.
15.4
Summary
“E¤ort” is an amalgam of inputs (boats, labor) in the …shing sector. The
production function gives harvest as a function of e¤ort and the …sh stock.
We used the production function to obtain the industry cost function. In an
open access equilibrium, where pro…ts are zero, the condition that average
cost equals price gives the industry supply function. Equating supply and
demand and solving for harvest gives the open access harvest rule: harvest
depends on the model parameters, and when costs depend on the stock, the
harvest rule also depends on the stock. With linear demand and where
harvest per unit of e¤ort is proportional to the stock, we obtain a closed
form expression for the open access harvest rule.
We used the growth function and the harvest rule to identify and classify
steady states. Our use of the continuous time model here makes it possible
to rely on graphs (e.g. Figure 15.2) not only to identify steady states, but
also to determine whether they are stable. If we had stuck with the discrete
time setting, the same …gure would identify the steady states, but checking for
stability would have required a numerical calculation. To determine stability
of a steady state in the continuous time setting, we need only determine
whether, as we move from left to right, the harvest rule cuts the growth
function from above (at an unstable steady state) or from below (at a stable
steady state).
Depending on parameter values, there might be a single interior (= positive) steady state or three interior steady states. In the former case, the
unique interior steady state is stable. In the latter case, the low and the
high interior steady states are stable, and the middle steady state is unstable.
The stock x = 0 is an unstable steady state in our models.
A unit tax on harvest shifts down the open access harvest rule, reducing
harvest for any level of the stock. A su¢ ciently high unit tax moves the
…shery from the situation where there is a single interior steady state to the
situation where there are there interior steady states. In this situation, the
tax creates a stable steady state with a high level of the stock. The e¤ect of
a tax depends on both the magnitude of the tax and on the level of the stock
15.5. TERMS, STUDY QUESTIONS, AND EXERCISES
285
at the time the tax is …rst imposed (the initial condition). We also used this
example to determine the steady state level of welfare (= consumer surplus
plus tax revenue) under di¤erent taxes.
15.5
Terms, study questions, and exercises
Terms and concepts
Myopic, catchability coe¢ cient, initial condition, steady state supply function.
Study questions
1. (a) For the case of constant average harvest cost, Cy, linear inverse
demand, p = a by, and logistic growth, obtain the open access harvest
rule, and sketch it on the same …gure as the growth function. Illustrate
how a change in C alters the steady state and the dynamics of the …sh
stock. (b) Illustrate and explain the e¤ect of a constant unit tax, ,
on the steady states. (c) Use a graph and short discussion to illustrate
the fact that the e¤ect of a given tax depends both on its magnitude
and on the stock level at the time the tax is imposed.
2. Begin with the production function showing output (harvest) y = qEx,
where E is e¤ort and x is biomass. Explain the meaning of E (perhaps
by using an example) and the meaning of the parameter q. What is
the name given to this parameter?
3. Suppose that a unit of e¤ort costs w. Derive the cost function for the
example in the previous question. Explain your derivation. (It is not
enough to merely memorize this cost function.)
4. Using the cost function in the previous question and the inverse demand
curve p = a by, derive the open access harvest rule. Be able to explain
each step. (Memorization is not su¢ cient.)
5. Write down the logistic growth function. On the same …gure, sketch
graphs of this growth function and the harvest rule obtained in the
previous question.
286
CHAPTER 15. THE OPEN ACCESS FISHERY
6. Using the …gure from the previous question, show the e¤ect of an increase in the demand slope, b, or intercept, a, on harvest rule, and on
any steady state(s). Discuss the e¤ect of this change (in a parameter
of the demand function) on the evolution of the stock. A complete
answer must explain how the parameter change can qualitatively alter
the evolution of the stock, depending on the initial condition.
7. Show how a constant tax a¤ects the open access harvest rule.
8. By means of a graph, show how an increase in the tax can alter the
steady state(s).
9. Explain why the e¤ect of the tax depends on both the magnitude of
the tax and on the initial condition of the biomass, at the time the tax
is …rst implemented.
Exercises
1. This exercise illustrates the fact that imposing restrictions typically
increases production costs. Suppose that e¤ort depends on the size
of the boat, B, and the amount of labor, L in the following manner:
E = B 0:5 L0:5 . The cost of the boat, bB is proportional to its size, with
a “price per unit size b”. The wage is w, so the labor bill is wL. Thus,
the cost of providing one unit of e¤ort is C, with
c = min (bB + wL) subject to B L1
B;L
= 1:
(a) Find the optimal labor/boat size ratio, BL and the cost of providing one unit of e¤ort„c, as a function of prices b and w (b) Suppose
that regulation doubles the required labor/boat size ratio to 2 BL .
Denote as c~, the cost of providing a unit of e¤ort under this regulation,
a function of b and w. Compare c~ and c.
2. (a) Show how a constant tax a¤ects the steady state under open access
where costs do not depend on the stock, c (x; y) = Cy. (b) Show how
this tax alters the dynamics of the …sh stock. Your answer to part (b)
needs to describe how the e¤ect of the tax on the evolution of the stock
depends on the initial condition.
15.5. TERMS, STUDY QUESTIONS, AND EXERCISES
287
3. Using Figure 15.2, show how a larger value of C alters the two harvest
rules (corresponding to low demand and high demand). Using this
information, describe the e¤ect of a larger C on the steady state(s) in
the low and high demand scenarios.
4. Using equation 15.5, provide the economic explanation for the statement that y (x) = 0 for a Cx < 0:
5. Suppose that inverse demand is p = a a0:21 y, with a > 1. (a) Show
that for this demand function, the elasticity of demand, evaluated at
p = 1, is = a 1 1 . A larger value of a implies less elastic demand. (b)
Using Figure ??, show how the set of steady states changes as demand
becomes more elastic (a decreases toward 1). Provide an economic
explanation for your observation.
6. (*) Suppose that the production function for harvest is y = (qEx)
with 0 < < 1. The cost of a unit of e¤ort is c, a constant. (a) Write
the cost function (expressing cost of harvest as a function of harvest
and the stock). Sketch the cost function as a function of harvest (for
a given stock). (b) There is free entry, so that at each point in time,
pro…ts equal 0. Use this equilibrium condition, and the linear inverse
demand function, p = a by, to write an equation that gives harvest
as an implicit function of the stock. (c) Although it is not possible
to solve this equation to obtain harvest as an explicit function of the
stock, it is simple to solve it to obtain the stock as an explicit function
of the harvest. Using this approach, graph the relation between the
harvest and the stock. It helps to pick a particular value of , e.g.
= 0:5. On the same graph, sketch the graph of the harvest rule
when = 1 (shown in Figure 15.2). (d) Transfer these two graphs
onto a …gure that also shows the growth function F , and use the result
to describe the qualitative e¤ect on the dynamics, of a decrease in
(here, from 1 to 0.5). Provide an economic explanation. Hint, part a.
Mimic the procedure used to …nd the cost function in the text. Rewrite
the production function to …nd the level of e¤ort needed to produce a
given harvest, y, at a particular stock, x. How much does this level of
e¤ort cost? The answer is the cost function.
Sources Berck and Perlo¤ for forward-looking e¤ort adjustment; earlier
papers for ad hoc adjustment rules.
288
CHAPTER 15. THE OPEN ACCESS FISHERY
Chapter 16
The sole-owner …shery
Objectives
Ability to derive and interpret the necessary condition for an equilibrium under a price-taking sole owner …shery; use this equilibrium
condition to obtain and discuss the steady state(s) and to study taxes
under open access.
Skills
Adapt the skills developed from the nonrenewable to the renewable
resource setting.
Be able to write the sole owner’s objective function and to apply the
perturbation method.
Understand how growth in the renewable resource setting enters the
no-intertemporal-arbitrage condition.
Understand the de…nition of rent and be able to use it to rewrite and
re-interpret the Euler equation.
Be able to use the optimality condition to describe optimal policy in
the presence of market failures.
Be able to use the optimality condition to identify the steady state for
the optimally controlled …shery.
289
290
CHAPTER 16. THE SOLE-OWNER FISHERY
This chapter studies the equilibrium under the price-taking sole-owner
…shery, extending earlier results on nonrenewable resources. Absent other
market failures, the outcome under the sole owner is e¢ cient. That owner
maximizes the present discounted sum of the sequence of producer + consumer surplus. There are few if any important sole owner …sheries. However,
the sole owner outcome provides a baseline against which we can compare
an open access or common property outcome. That comparison provides
insight about …shery management when property rights are imperfect.
Fisheries are intrinsically important, and the tools developed for their
study are useful in many renewable resource settings For example, excessive
greenhouse gas emissions occur because of the absence of property rights for
the atmosphere. For both the open access …shery and the climate, property
rights are weak or nonexistent. In both cases, a comparison of the outcome
without property rights, and the outcome under a social planner (or sole
owner), provides information on optimal regulation.
The sole owner …shery (typically) obtains resource rents, which may depend on the resource stock. The sole owner understands that current harvest
a¤ects future stocks, thus a¤ecting future rent. Fishers in the open-access
setting ignore the e¤ect of their harvest on future stock. Because of this difference, the former leads to socially optimal harvest, and (except for special
cases) the latter leads to excessive harvest.
We state the sole owner’s optimization problem and then derive the optimality condition, the Euler equation. The de…nition of rent leads to a more
concise statement of Euler equation. We consider a situation where the
…shery provides ecological services, e.g. as a food source for other valuable
species, and the sole owner does not internalize the value of those services.
Property rights for the particular species corrects some, but perhaps not all
market failures. The sole owner equilibrium conditions determine the steady
states, which we compare to the open access steady states.
16.1
The Euler equation for the sole owner
Objectives and skills:
State the sole owner’s optimization problem.
Write and interpret the Euler equation.
16.1. THE EULER EQUATION FOR THE SOLE OWNER
291
Express the Euler equation using rent, and interpret.
We consider two specializations of the parametric cost function. In the
…rst, average harvest costs are constant in harvest and independent of the
stock, c (x; y) = Cy. In the second, average harvest costs are constant in
the harvest and decreasing in the stock c (x; y) = Cx y, so @c(x;y)
= xC2 y < 0.
@x
The owner of the resource takes the sequence of prices, p0 , p1 , p2 ::: as given;
these prices are “endogenous to the model”, via the inverse demand function,
pt = p (yt ), but the resource owner takes the prices as exogenous. The
owner’s present discounted stream of pro…ts is
1
X
t
(pt yt
c (xt ; yt )) .
t=0
For the constant average cost speci…cation, the owner chooses the sequence of harvests, y0 ; y1 ; y2 ::: to solve
P
t
(pt C) yt
max 1
t=0
(16.1)
subject to xt+1 = xt + F (xt ) yt :
For the stock dependent average cost speci…cation, the owner’s objective and
constraints are
P
t
pt xCt yt
max 1
t=0
(16.2)
subject to xt+1 = xt + F (xt ) yt :
In general, the resource owner obtains positive rent, which depends on the
level of the stock. The owner therefore takes into account how current harvest a¤ects future stock, and future rent. In addition, with stock dependent
costs, the owner understands that a change in current harvest alters future
harvest costs, via the change in the stock.
For the sole owner facing constant harvest costs, C, the Euler equation is
pt
C=
(pt+1
C) 1 +
dF (xt+1 )
dxt+1
:
(16.3)
The Euler equation for stock dependent case is (Appendix I)
pt
C
=
xt
pt+1
C
xt+1
1+
dF (xt+1 )
dxt+1
+
C
yt+1 :
x2t+1
(16.4)
292
16.1.1
CHAPTER 16. THE SOLE-OWNER FISHERY
Intuition for the Euler equation
We emphasize the case of stock dependent harvest costs, equation 16.4. With
one important di¤erence, this necessary condition is identical to the Euler
equation 5.2 for the nonrenewable resource.1 The di¤erence is that the right
side of equation 16.4 involves
pt+1
C
xt+1
1+
dF (xt+1 )
dxt+1
;
whereas the corresponding term with nonrenewable resources (from equation
5.2) is
C
pt+1
(1 + 0) :
xt+1
t+1 )
= 0, i.e. unless the stock has
These two expressions di¤er unless dFdx(xt+1
no e¤ect on growth. Stock dependent growth is central in the renewable
resource setting in general, and to the …shery problem in particular.
A trajectory consists of a sequence of harvest and stock levels. Along
an optimal trajectory, a small change in harvest in some period, and an
o¤setting change in some other period, must lead to a zero …rst order change
in the payo¤: a perturbation does not improve the outcome. We obtain the
Euler equation by considering a particular perturbation: one that changes
harvest by a small amount in one period, and then makes an o¤setting change
to harvest in the next period, so that the subsequent stock remains on the
candidate trajectory.
The left side of equation 16.4 equals the marginal bene…t of increasing
harvest in period t, and the right side equals the marginal cost of the o¤setting change needed in the subsequent period, to return the trajectory to the
candidate path. The di¤erence between the Euler equation in the nonrenewable resource and the renewable resource settings, arising because of growth.
Growth a¤ects the change needed in period t + 1 to o¤set the change in
period t.
A “savings example” for intuition To provide intuition, we consider
a savings problem unrelated to resources. An investor earns a per period
return of r: investing a dollar at the beginning of a period produces 1 + r
1
Compare equations 5.2 and 16.4, and use the fact that for our cost function here,
t ;yt )
c (xt ; yt ) = xCt yt , so @c(x
= xCt .
@yt
16.1. THE EULER EQUATION FOR THE SOLE OWNER
293
dollars at the end of the period. This investor has a “candidate savings
plan”, a trajectory of savings (a ‡ow variable) and wealth (a stock variable)
for a certain number of periods. The savings decision corresponds to harvest
in the …shery setting, and wealth corresponds to biomass.
Suppose that the investor considers perturbing this candidate in period
t by saving one dollar less than the candidate prescribes. In order to put
her plan back on the candidate trajectory by period t + 2, she has to invest
an additional 1 + r dollars in period t + 1, over and above the amount that
her original (“pre-perturbation”) plan calls for. The extra $1 makes up for
the dollar that she took out in period t, and the extra $r makes up for the
interest that she lost by taking out that dollar.
In the nonrenewable resource setting, extracting an extra unit (above
the level speci…ed by the candidate) at t required a one unit reduction in
extraction at t + 1, in order to return the trajectory to the candidate. In the
savings example, where the stock (wealth, instead of the resource) grows on
its own, o¤setting the change at time t requires “something extra” at time
t + 1. The same type of consideration applies in the …shery setting. The
savings example is simpler than the …shery problem because in the savings
problem growth in wealth (Wt ) is a linear function of wealth (with slope equal
to 1 + r): Absent savings, wealth in the next period is Wt+1 = (1 + r) Wt .
In the …shery problem, growth in biomass is not a linear function of biomass.
Using this intuition Harvesting an extra unit in period t generates pt
additional units of revenue, and xCt additional units of cost, for a net increase
in pro…ts of pt xCt , the left side of equation 16.4. This perturbation reduces
next period pro…ts because it necessitates lower harvest, and because the
harvest that does occur is more expensive, due to the lower stock caused by
t+1 )
the perturbation. Each unit of stock contributes dFdx(xt+1
units of growth. To
o¤set the direct e¤ect of the unit of increased harvest in period t, the owner
must reduce harvest in period t + 1 by one unit; in addition, the owner must
t+1 )
reduce harvest in period t + 1 by dFdx(xt+1
to make up for the reduced growth
in period t + 1. The term
the savings example.
dF (xt+1 )
dxt+1
corresponds to the interest payment in
C
Each unit of reduced harvest in period t+1 reduces pro…ts by pt+1 xt+1
.
Therefore, the reduction in period-t + 1 pro…ts, caused by the reduced t + 1
294
CHAPTER 16. THE SOLE-OWNER FISHERY
harvest, is
pt+1
C
xt+1
1+
dF (xt+1 )
dxt+1
;
which equals the …rst term in square brackets in equation 16.4.
The higher harvest in period t also reduces period t + 1 pro…ts via a cost
C
yt+1 .
e¤ect. Harvest costs under the candidate, in period t + 1, equal xt+1
The lower t + 1 stock, caused by the period-t perturbation, leads to increased
harvest cost of
C
@ xt+1
yt+1
C
= 2 yt+1 ;
@xt+1
xt+1
which equals the second term in square brackets in equation 16.4. The
present value cost, at time t, of the additional marginal harvest at time t
equals the right side of equation 16.4.
16.1.2
Rent
We write the Euler equations, for the two types of harvest costs, using rent,
providing a more concise expression of the Euler equation. As in the nonrenewable resource setting, we de…ne rent as the di¤erence between price and
marginal cost. For our example (but not in general), marginal cost equals
average cost, so rent equals pro…t per unit of harvest.
For the case of stock independent average costs this de…nition is
Rt = pt
C
Using this de…nition, the Euler equation 16.3 in terms of rent is
Rt = Rt+1 1 +
dF (xt+1 )
dxt+1
:
(16.5)
For the case of stock dependent harvest costs, the de…nition of rent is
Rt = pt
C
:
xt
(16.6)
Substituting this expression in the Euler equation 16.4 gives
Rt =
Rt+1 1 +
dF (xt+1 )
dxt+1
+
C
yt+1 :
x2t+1
(16.7)
16.2. POLICY
295
The sole owner never sells where price is below average costs, so rent
is never negative, and in general (but not always) is positive. The open
access …shery eliminates pro…ts, driving rent to zero. Chapter 13.3.2 notes
that Individual Transferable Quotas (ITQs) create property rights, making
an open access or common property …shery more like a sole owner …shery.
The equilibrium annual lease price of an ITQ equals that amount of pro…t a
…sher can expect to obtain for the volume of harvest covered by the quota
licence. The lease price thus provides an estimate of the rent associated
with that volume of …sh. Most lease prices range from 50% to 80% of the ex
vessel …sh price (the price …shers receive). Rent accounts for a substantial
fraction of the value of …sheries with strong property rights.
16.2
Policy
Objectives and skills:
Understand why the optimal stock-dependent tax under open access
equals the rent for the agent who harvests at the …rst best level.
In the absence of externalities, the First Fundamental Welfare Theorem
implies that the price-taking sole owner harvests e¢ ciently, so there is no
role for policy. However, the sole owner outcome provides information on
the tax that induces open access …shers to harvest e¢ ciently.
Market failures can occur even under the sole owner. For example, the
biomass creates ecological services if the …sh being harvested are important
to some other …sh stock, or to the marine ecology more generally. If the
…shery owner does not receive compensation for these services, then they are
“external”to her decision problem. The owner’s failure to internalize these
bene…ts results in excessive harvest that drives the stock to too low a level,
resulting in under-provision of the ecological services. Here there is a role for
regulation even under the sole owner, and under open access, there are two
market failures. Weak property rights and the ecological externality both
cause excessive harvest.
A particular unit tax induces the open access …shery to harvest at the
socially optimal level. This tax equals the rent in the optimally managed
…shery. Absent externalities, the sole owner harvests at the optimal level,
so the tax equals the rent under this …shery. If there is an externality,
then the optimal tax equals the rent under the sole owner who is induced to
296
CHAPTER 16. THE SOLE-OWNER FISHERY
take the externalities into account. In both cases, the optimal tax induces
open access …shers to “internalize the externalities” associated with their
harvest decisions, in much the same way as the Pigouvian tax from Chapter
11 induces …rms to internalize the cost of their pollution.
16.2.1
Optimal policy under a single market failure
Here we assume that there are no externalities under the sole owner, so that
owner harvests e¢ ciently. Under open access, there is a single market failure,
arising from the absence of property rights to the …sh stock. Our goal is to
…nd a stock-dependent unit tax, a function (x), that induces open-access
…shers to harvest e¢ ciently. If open access …shers face the tax (x), they
harvest up to the point where their tax-inclusive pro…ts equal zero. The
equilibrium condition for these …shers sets price minus average harvest costs
minus the unit tax equal to zero:
p (yt )
C
xt
(xt ) = 0:
(16.8)
The period t price depends on harvest in that period, although individual
…shers take the price as given.
Comparing equations 16.6 and 16.8, we see that for all values of the stock,
xt , the levels of the price, and thus the level of the harvest, are the same under
the sole owner and under the open access with-tax, if and only if the tax is
set equal to the sole owner’s rent:
(xt ) = Rt :
(16.9)
Figure 16.1 illustrates the stock-dependent tax. The …gure shows the
graphs of p (y) Cx for x = 10 (dashed) and x = 1 (solid). The untaxed
open access harvest occurs at p (y) Cx = 0. Harvest is y = 2 when x = 1
(where the stock is low and harvest costs are relatively high) and y = 4:7
when x = 10 (where the stock is high and harvest costs are relatively low).
Under the sole owner, rent is positive at both stock levels, but the rent falls
with the stock (see below). The example in the …gure uses the assumptions
R (1) = 1:5 and R (10) = 1, illustrating that the rent is lower when the stock
is higher. The dotted curves show the graphs of p (y) Cx R (x) for the two
values of x, illustrating that the sole owner harvests less than under open
access. For this example, the e¢ ciency-inducing tax is = 1:5 at x = 1,
and the tax is = 1 at x = 10: lower stocks require higher taxes.
16.2. POLICY
297
$
8
6
4
2
0
1
-2
2
3
4
5
harvest
-4
-6
Figure 16.1: p (y) Cx for x = 10 (dashed) and x = 1 (solid). Dotted
lines show the graph of p (y) Cx R (x) for these two stock levels. In this
example, R (10) = 1 < 1:5 = R (1) :
Implementing the tax Equation 16.9 de…nes the optimal tax under open
access. The di¢ culty of implementing this tax is that in the real world we
do not have two …sheries that are identical, except that one operates under
open access the other under the sole owner. The policy problem arises under
open access, where we do not observe the sole owner’s rent. We therefore
have to estimate what the rent would be under the sole owner.
We have the de…nition of the sole owner’s rent: price minus marginal cost.
However, rent in a period depends on price, which depends on harvest, which
in general depends on the stock. (A …shery owner with a large stock is likely
to harvest at a di¤erent level than an owner with a small stock.) Denote the
sole owner’s harvest rule as y = Y (x). Given the parameters of the growth
function, inverse demand function, and cost function, and the discount factor,
numerical methods enable us to approximate the sole owner’s harvest rule.
Given the approximation of this harvest rule, we can approximate the sole
owner’s rent as
R (x) = p (Y (x))
C
(= price minus marginal cost).
x
Approximating the harvest rule, Y (x), requires the application of numerical methods that are beyond the scope of this book. However, the most
di¢ cult part of the problem is …nding reliable estimates of the parameters of
the cost and growth function, and estimates of the stock of the …sh.
298
CHAPTER 16. THE SOLE-OWNER FISHERY
16.2.2
Optimal policy under two market failures
Suppose now that the stock of …sh provides an ecological service, with the
non-market value V (xt ). This value is external to …shers: they ignore it
in choosing harvest. There is a market failure even under the sole owner.
Therefore, under open access there are two market failures: the lack of property rights and the externality due to ecological services.
To …nd the optimal tax for the open access …shery, we need to …nd the
sole owner’s rent in the case where she does receive the value of the ecological
services. For example, if the sole owner were given a payment V (xt ) in each
period, then her goals are perfectly aligned with those of the social planner.
The objective and constraints of the sole owner who receives this payment is
i
P1 t h
C
y
+
V
(x
)
p
t
t
t
t=0
xt
(16.10)
subject to xt+1 = xt + F (xt ) yt :
Comparing the maximand in the …rst line of equations 16.2 and 16.10, we
see that they are identical, apart from the addition of the lump-sum subsidy
V (xt ) in the latter.
We can repeat the steps used to obtain the Euler equation 16.4, to obtain
the Euler equation for the slightly more complicated case here, and then
rewrite that equation using the de…nition of rent to obtain
Rt =
Rt+1 1 +
dF (xt+1 )
dxt+1
+
dV (xt+1 )
C
yt+1 +
:
2
xt+1
dxt+1
(16.11)
The left sides of equations 16.7 and 16.11, showing the marginal bene…t of
harvesting one more unit in period t, are the same. The right sides, showing
the marginal next-period cost of this perturbation, di¤er only by the presence
t+1 )
of the term dVdx(xt+1
, the change in the lump-sum subsidy due to the reduced
stock.
Again, given a particular inverse demand function, p (y) ; and function
V (x), and parameters of the cost function, one can numerically approxi~ (x) instead of R (x) to
mate the rent function; we denote this function as R
recognize that the presence of the function V (x) alters the solution to the
problem. Using the same reasoning as in Chapter 16.2.1, the optimal tax
~ (x).
for the open access …shery is (x) = R
In summary, the optimal tax for the open access …shery equals the rent
(a function of the stock of …sh) in the scenario with no market failures. If
16.2. POLICY
299
the only market failure under open access is the lack of property rights, we
“merely”have to …nd the rent function, R (x), under the sole owner. If there
is an additional market failure, e.g. arising from ecological services that are
external to the resource owner, then we have to …nd the rent function that
would arise if the sole owner were induced to internalize the externality.
16.2.3
The rent falls as the stock increases
We can interpret the rent on the resource as a shadow value: the amount that
the resource owner would pay for a marginal increase in the stock (Chapter
5.3). Larger stocks decrease the sole owner’s rent for three reasons:2 (i) A
higher stock decreases harvest cost, but it does so at a decreasing rate. A
unit increase in the stock decreases harvest cost; a second unit increase in
the stock leads to an additional, but smaller decrease in the harvest cost.
(ii) Discounting reduces the value of future harvests. An additional unit
of the stock is harvested further in the future, the greater is the current
stock. Thus, the marginal value of an added unit of the stock is smaller,
the larger is the stock, because the additional unit will be used in the more
distant future. (iii) Even though the sole owner takes prices as exogenous,
they are endogenous unless the industry faces a perfectly elastic demand
function. When the demand function is downward sloping, the additional
harvest arising from the higher stock receives a lower market price. The
marginal value of the additional stock therefore falls as the stock increases.
In summary, the optimal tax for the open access …shery equals the shadow
price of the stock in the absence of market failures. This shadow price falls
with the stock, so the optimal tax also falls with the stock, as Figure 16.1
illustrates.
16.2.4
Empirical challenges
Managers are unable to observe the growth function or the cost function,
or the biological stock on which they depend. Without estimates of these
functions and the stock, the management tools described above cannot be
2
Like most categorical statements in economics, the assertion that rent falls as the stock
increases, relies on “reasonable”(or at least commonly made) assumptions. The model
here assumes a concave (e.g. logistic) growth function. If the growth function has regions
of convexity, where a small increase in the stock leads to a rapid increase in growth, the
rent might increase over some ranges of the stock.
300
CHAPTER 16. THE SOLE-OWNER FISHERY
directly used (although they might nevertheless provide insight). Here we
discuss two approaches to estimating these ingredients. The …rst approach
relies on catch and e¤ort data and simple models; the second approach uses
much more data and more complicated models.
To explain the …rst approach, consider the case where (as is true in some
…sheries) we have “panel data”of e¤ort and catch for many boats for many
years. The e¤ort data consists of characteristics of the boats and expenditures; here, “e¤ort” is not a single number. In a particular year, all of the
boats confront (approximately) the same level of the stock. Using this fact
and the panel data, we can estimate parameters relating harvest to the effort characteristics, and also estimate a “stock index”.3 This index involves
both the physical stock and economic parameters; we cannot separate these,
so we have a stock index, instead of an estimate of the actual stock. This
entanglement does not matter for the purpose of estimating the parameters
of the growth function.
The second approach is more complicated and requires more data, but
permits the estimation of more complex models involving di¤erent species
and di¤erent age or size categories within a species. Scientists collect samples
by dragging nets across …shing areas. By counting rings in the bones (e.g. in
the ear canal or the jaw), they estimate the age of individuals in the sample
in much the same way that counting rings of tree identi…es the tree’s age.
Estimation relies on a dynamic model and a measurement model. The
dynamic model consists of a system of equations that describe the evolution
of the stock(s) and age classes. Taking as given the equations’ functional
form, the goal is to estimate the parameters of the functions. In the simplest case, with a single stock variable, scientists might assume the logistic
growth function, and then estimate the two parameters of that function, the
natural growth rate and the carrying capacity. Actual applications tend to
be much more complicated. The measurement model relates the underlying
but unobservable variables of interest (e.g. stock size for di¤erent ages) to
the measured variables (e.g. age estimates of the sample).
Estimation of the unknown parameters and unknown stocks uses an iterative procedure. Beginning with a guess of the unknown values, we can
calculate what the measurements would have been, had the guess been cor3
An “index” is a measurement related to the object of interest (here, the stock), not
the object itself. For example, the economy-wide price level is a theoretical construction,
not an observable price. We use the consumer price index to measure this price level, and
then to measure in‡ation.
16.3. THE STEADY STATE
301
rect. Of course, the guess is not correct, so the calculated values di¤er from
the observed measurements. We then change our guess (using sophisticated
numerical methods) of the unknown values, in an e¤ort to make the calculated values closer to the observed measurements. We stop the iteration
when we decide that it is not possible to get a closer …t between the calculated values and the observed measures. The …nal stage of the iteration
yields estimates of the parameters and stock variables.
16.3
The steady state
Objectives and skills:
Ability to obtain and to analyze the steady state under the sole owner.
Ability to compare steady states under the sole owner and under open
access.
Comparison of the sole owner and the open access steady states tells
us something about the relation between property rights and equilibrium
outcomes. For the examples in Chapter 15, we obtained the harvest rule
under open access by solving the zero-pro…t condition, price = average cost.
There, we could determine which of the steady states is stable, and also
determine the relation between the initial condition and the steady state to
which the stock converges.
Matters are more complicated in the sole owner setting. Here, we only
have an optimality condition (the Euler equation), not an explicit harvest
rule. In this section, we only identify the sole owner steady states, and
compare these to the open access steady states. The next chapter compares
the harvest rules in the two cases. Without these harvest rules (or at least
some information about them), we do not know whether a particular steady
state is stable.
At a steady state, the harvest, the stock, and the rent are unchanging over
time. We drop the time subscripts to denote the fact that these variables
are constant in the steady state. The equation of motion of the stock is
xt+1 xt = F (xt ) yt . In the steady state, the left side of this equation is
zero; dropping the time subscripts, we write this equation, evaluated at the
steady state, as 0 = F (x) y. We write the de…nition of the steady state
.
rent as R = p(y) @c(x;y)
@y
302
CHAPTER 16. THE SOLE-OWNER FISHERY
We obtain a third equation by evaluating the Euler equation at a steady
state. This equation, unlike the previous two, depends on the cost function.
We then have three equations in three unknowns, the steady state stock,
harvest, and rent. We can solve these three equations to determine the
steady state values. We …rst consider the case of constant average extraction
costs, and then the case of stock dependent average extraction costs.
16.3.1
Harvest costs independent of stock
We begin by evaluating equation 16.5 at a steady state (mechanically, dropping the time subscripts), to obtain
R= R 1+
dF (x)
dx
:
1
Using = 1+r
, we multiply both sides of the equation by 1 + r and cancel
terms, to write
dF (x)
0=R r
:
(16.12)
dx
Equation 16.12 implies that at a steady state either R = 0 or
r=
dF (x)
:
dx
(16.13)
Equation 16.13 states at an interior steady state with R > 0, the sole
owner is indi¤erent between two investment opportunities, because they both
earn the same return. The owner can marginally increase current harvest,
and invest the additional pro…t in an asset that earns the annual return r,
or she can keep the extra unit of stock in the …shery, where it contributes
to growth, thus contributing to future harvests and future pro…ts. At an
interior optimum, the owner is indi¤erent between these two investment opportunities.
The solution to equation 16.13 depends only on the discount rate and the
growth rate of the stock, not on demand or cost. However, the rent depends
on demand (via price) and costs. We can solve equation 16.13 to …nd the
steady state “candidate”and then check to determine whether R 0; if this
inequality holds, then we have identi…ed an interior steady state. However,
if R < 0, the solution to equation 16.13 has no signi…cance.
16.3. THE STEADY STATE
y
303
0.6
0.5
e
d
0.4
C=0.4
c
b
C=1.25
0.3
f
0.2
a
C=3
0.1
0.0
0
10
20
30
40
50
x
x
Figure 16.2: For F = 0:04x 1 50
and r = 0:02 one candidate for a steady
state under the sole owner is point d, where x = 12:5 and y = :375. For
inverse demand = 5 10y, open access steady states (where rent and growth
are both zero) are points e and c for C = 0:4, points b and d for C = 1:25,
and points f and a for C = 3. These points are also steady states under the
sole owner.
We illustrate this procedure using the logistic growth function, F (x) =
. Substituting this expression into the
x 1 Kx , where dFdx(x) = 1 2x
K
right side of equation 16.13, and solving for x, we obtain the candidate steady
state stock
K
r
K
(16.14)
1
< :
x1 =
2
2
Recall that K2 is the stock level that maximizes growth, i.e. leads to Maximum
Sustainable Yield (MSY). Equation 16.14 notes that the candidate steady
state is less than K2 , so harvest is less than MSY. The equality also shows
that the candidate steady state decreases in r : a higher discount rate (greater
impatience) lowers the candidate, and faster growth increases the candidate.
Two conditions must hold in order for the solution to equation 16.13 to
be an interior steady state:
(i)The solution must be positive: x1 > 0; and (ii)R (y1 )
0:
(16.15)
First, we determine whether x1 > 0. If < r, the value of x1 given by
equation 16.14 is negative, an obvious impossibility. Thus, for < r, there
is no interior steady state with positive rent. (There might still be interior
304
CHAPTER 16. THE SOLE-OWNER FISHERY
steady states with zero rent.) If > r, then x1 > 0: the candidate passes
the …rst of the two tests in equation 16.15.
For > r we next check whether rent is non-negative at x1 . At this
steady state, harvest is y1 = F (x1 ), and the formula for rent is p (y1 ) C.
If p (y1 ) C 0, then our candidate x1 passes the second of the two tests
in equation 16.15, and is a steady state for the sole owner. If p (y1 ) C < 0,
then x1 has no signi…cance.
x
and r = 0:02.
Pursuing the logistic example, let F (x) = 0:04x 1 50
Because = 0:04 > r for this example, x1 , given by equation 16.13, is
positive: x1 = 12:5 and y1 = :375. Figure 16.2 identi…es this solution as
the tangency, point d, between the graph of the growth function and the line
with slope r = 0:02. For this example, x1 > 0, i.e. the candidate passes
the …rst test. To check whether it passes the second test, we use the inverse
demand function p (y) = 5 10y. Given the choke price 5, the …sh has
value if and only if C < 5, as we hereafter assume. Rent, evaluated at the
candidate steady state, point d, is R = 5 10(:375) C. The second test,
R 0, requires C 1:25.
Figure 16.2 shows three horizontal lines, values of harvest where R =
5 10y C = 0 (y = 510C ), for three values of C. These horizontal lines
show the harvest rule under open access. From Chapter 15.1 we know that
under open access (provided that C is less than the choke price, here 5),
there are three steady states: 0, an intermediate steady state (e; f; d for the
three values of C) and a high steady state (c; b; a for the three cases). In
addition, we know that under open access the high steady state and x = 0
are stable steady states, and the intermediate steady state is unstable.
Now consider the steady states under the sole owner. The open access
high steady states, (c; b; a) also satisfy the sole owner steady state conditions.
At these points: (i) y = F (x), so the stock is unchanging, (ii) R = 0, so
the steady state Euler equation 16.12 is satis…ed, and (iii) the de…nition of
rent is also satis…ed. Exactly the same reasoning holds at the open access
intermediate steady states, and at x = 0. Thus, for this problem, all of the
steady states under open access are also steady states under the sole owner.
If C 1:25, then the two tests in equation 16.15 are satis…ed, so point d is
also a steady state under the sole owner. If C > 1:25, d is not a steady state
under the sole owner. Under open access, d is not a steady state except in
the knife-edge case C = 1:25.
In summary, every point that is a steady state under open access is also
a steady state under the sole owner. At all of these points, rent (or harvest)
16.3. THE STEADY STATE
305
is zero. For su¢ ciently low costs (C < 1:25 in our example) there is an
additional steady state under the sole owner, that does not exist under open
access. At that steady state, the sole owner has positive rent. The next
chapter turns to the question of determining which of the sole owner steady
states is stable.
16.3.2
Harvest costs depend on the stock
Now we turn to the case where harvest costs depend on the stock. The main
insights from this material are:
For low harvest costs (small C) the sole owner steady state (with positive rent) is lower than the stock that maximizes steady state yield; for
large harvest costs, the steady state is above the stock corresponding
to maximum steady state yield.
Higher harvest costs might either increase or decrease the owner’s
steady state rent, and the steady state consumer surplus.
We use equation 16.7 to write the steady state condition for rent (merely
by dropping the time subscripts):
R=
R 1+
We simplify this equation using
cancelling terms, to write
r
=
dF (x)
dx
1
,
1+r
dF (x)
dx
+
C
y :
x2
multiplying both sides by 1 + r and
R=
C
y:
x2
We collect the steady state conditions in
F (x)
y = 0,
r
dF (x)
dx
R
C
y = 0, and R = p (y)
x2
C
x
(16.16)
The …rst two equations repeat the steady state conditions for the stock and
the rent, and the third equation repeats the de…nition of rent.
System 16.16 comprises three equations in three unknowns, x; y; and R.
All three equations must hold if x > 0, i.e. at an interior solution. Figure
306
CHAPTER 16. THE SOLE-OWNER FISHERY
y
0.5
0
0.4
20
0.3
100
0.2
200
0.1
300
0.0
0
10
20
30
40
50
x
Figure 16.3: Graphs of F and the steady state condition for rent, the upward
sloping curves. Each of those curves correspond to a value of C, shown next
to the curve. The sole owner steady state (with positive pro…ts) occurs
at the intersection of the curves. An increase in C increases the point of
intersection.
16.3 shows the graph of the growth function and the graphs of the steady
state Euler equation (the upward sloping curves) corresponding to di¤erent
x
and r =
values of C. (The …gure uses p = 10 y, F = 0:04x 1 50
0:02.) At all points on the growth function, harvest equals growth: the …rst
equation in system 16.16 is satis…ed. At all points on an upward sloping
curve (corresponding to a particular value of C), the second two equations in
the system are are satis…ed. Thus, satisfaction of all three equations occurs
(only) at a point of intersection. That intersection is the sole owner steady
state (with positive pro…ts).4
The …gure shows that an increase in C causes the graph of the steady
state Euler equation to shift to the right, increasing the point of intersection:
larger values of the cost parameter lead to larger steady states under the sole
owner. At a given level of the stock, a larger C means that costs are higher,
discouraging harvest. As the harvest falls (for a given level of stock), the
steady state stock rises.
4
Figure 16.3 and Table 16.1 show only the sole owner steady state with positive rent.
For small values of C there is a second (stable) steady state where rent is zero, just as in
the constant cost model. For large initial stocks, the stock is not a scarce resource, and
the trajectory converges to a high steady state with zero rent.
16.3. THE STEADY STATE
307
Figure 16.3 shows that the sole owner steady state might lie either to the
left or the right of the MSY stock level. Recall that harvest under the sole
owner (absent other market failures) is e¢ cient. This example demonstrates
that it might be socially optimal to maintain a stock either above or below
the MSY stock level. Discounting gives the sole owner (and the social
planner) an incentive to harvest earlier rather than later. This incentive
decreases the steady state stock. When harvest costs are sensitive to the
level of the stock, the sole owner (and the social planner) has an incentive to
build up the stock, in order to decrease future harvest costs. This incentive
increases the steady state stock. When costs are insensitive to the stock
(e.g. C = 0 or C = 20) the second incentive is small or vanishes, leaving
only the discounting incentive. Thus, for small C the steady state is below
the stock corresponding to MSY. When costs are sensitive to the stock, as
with C = 300, the second incentive dominates the …rst, leading to a stock
above the MSY.
At the steady state, harvest equals growth. Moving from left to right
along the growth function, we see that steady state harvest increases when
the stock is below the MSY level (x = 25), and harvest decreases when the
stock is above that level. Because the stock increases in C, and harvest is
nonmonotonic in the stock, there is a nonmontonic relation between C and
the steady state harvest. Table 16.1 shows sole owner (superscript “so”)
steady state values of the stock, harvest, and rent, under three values of the
cost parameter, C. The last two columns of the table show the open access
(superscript “oa”) steady state stock and harvest for those values of C.
so
so
oa
C
xso
y1
R1
xoa
y1
1
1
0
12:5 0:38 9: 6 0
0
50 18:6 0:47 1:7 5:1
0:18
300 38:2 0:36 1:8 31:47 0:47
Table 16.1 Steady state stock, harvest, and rent for the sole owner, and
stock and harvest under open access.
Price falls and consumer welfare increases as harvests increase, so an
increase in C might make consumers either better or worse o¤ in the steady
state. In the static competitive setting, in contrast, higher costs shift the
equilibrium supply function in and up, leading to a higher equilibrium price
and lower consumer surplus. Why do higher costs have di¤erent e¤ects
in the (steady state) …shery, compared to the static market? The answer
308
CHAPTER 16. THE SOLE-OWNER FISHERY
rests on two facts already discussed. First, in the …shery context, a higher
C reduces the incentive to harvest, leading to a higher steady state stock .
Second, the steady state harvest is nonmonotonic in the stock.
Steady state rent is also non-monotonic in the cost parameter. At a very
high cost (C = 300) harvest is quite low and the price correspondingly high,
tending to increase rent. Marginal costs, Cx , are also high, tending to reduce
the rent. The net e¤ect of a larger C could be to increase or decrease rent,
as Table 16.1 illustrates.
16.3.3
Empirical evidence
The sole owner (and thus the socially optimal) steady state can be below or
above the MSY stock level. The ordering depends on extraction costs, the
growth function, and the discount rate. The ordering is therefore a di¢ cult
empirical question, because it depends on functions that we can measure
imprecisely, at best.
A 2007 study of four …sheries, including slow-growing long-lived orange
roughy, …nds that the socially optimal stock level exceeds the MSY stock
level. Therefore, the steady state harvest is below the MSY. The fact
that the study includes a slow-growing …sh is important. Chapter 16.3.1
notes that a low growth rate reduces the optimal steady state stock, and
possibly leads to extinction. Because other considerations (e.g. strongly
stock-dependent harvest costs) cause the steady state to increase, the optimal
steady state depends on a balance of con‡icting forces.
A 2013 study for a di¤erent …sh stock, North Paci…c albacore, concludes
that the optimal steady state lies to the left of the level corresponding to
MSY. This study emphasizes the role of cost-reducing technology improvements. As Figure 16.3 illustrates, lower harvest costs (smaller C) reduce the
sole owner steady state.
There are important di¤erences across …sheries, regarding both growth
functions and harvest costs. As noted in Chapter 16.2.4, there are also
di¤erent ways to estimate the parameters of these functions. Models also
di¤er in important features, e.g. whether they treat technology as …xed. In
view of these di¤erences, it is not surprising that studies disagree on the
ordering of optimal stock and MSY stock.
Most actual management practices try to keep the stock at the level of
MSY. There is a plausible and a dubious argument in favor of this practice.
The plausible argument is that, lacking a strong a priori basis for thinking
16.3. THE STEADY STATE
y
309
0.5
20
100
0.4
0
0.3
0.2
200
300
0.1
0.0
0
10
20
30
40
50
x
Figure 16.4: Graphs of F and the steady state condition for rent, the upward
sloping curves. Same parameter values as in Figure 16.3 except that here
r = 0:
that the stock should be to the right or the left of this level, and in view of
the measurement di¢ culties, the MSY level is “neutral”.
The dubious argument, based on intergenerational ethics, is that a positive discount rate is unfair to future generation, because it gives them less
weight in the social welfare function. Even it one accepts this view of ethics,
it does not imply, in general, that the ethically optimal steady state occurs
at MSY. The MSY maximizes consumer surplus, but when harvest cost depends on the stock it does not maximize social welfare, the sum of consumer
and producer surplus. With stock dependent costs, low discount rates require a higher steady state stock in order to take advantage of cost reductions.
Because these stocks occur to the right of the MSY level, they correspond
to lower harvests and lower consumer surplus, but higher producer surplus,
and a higher social surplus.
Figure 16.4 illustrates this claim, reproducing the graphs in Figure 16.3,
but replacing r = 0:02 (a 2% per annum discount rate) with r = 0. For
stock-independent costs (C = 0), the optimal steady state (for r = 0) occurs
at the MSY. However, if harvest costs depend on the stock, the optimal
steady state stock always lies to right of the MSY. For any value of C
(including C = 0), a decrease in the discount rate causes the upward sloping
curves in the …gures to shift to the right, leading to a higher steady state
stock.
310
16.4
CHAPTER 16. THE SOLE-OWNER FISHERY
Summary
We derived and interpreted the Euler equation for the price-taking sole-owner
…shery, emphasizing the di¤erence between the renewable and nonrenewable
resources. We have to consider growth with a renewable resource, but not
with a nonrenewable resource. The sole owner internalizes the e¤ect of
her current harvest decisions on future stocks. Absent externalities, the
First Fundamental Welfare Theorem implies that the outcome under the sole
owner is e¢ cient. In that case, there is no e¢ ciency rationale for regulation.
Moving from open access to the sole owner solves the only market failure.
If it is not possible or politically desirable to privatize an open access
…shery (thus moving to the sole-owner scenario), the open access …shery can
be induced to harvest e¢ ciently by charging a tax per unit of harvest. The
optimal tax equals the rent under the sole owner. We also considered the
case where the stock of …sh provides ecological services that are external to
the sole owner. A subsidy or a tax can induce the sole owner to internalize
that externality, in which case the sole owner again harvests e¢ ciently. A tax
equal to the sole owner’s rent, when that owner harvests e¢ ciently, induces
the open access …shery to harvest e¢ ciently. Because this e¢ ciency-inducing
tax varies with the stock of …sh, a constant tax is not …rst best.
For the nonrenewable resource, extraction eventually ceases as the resource is exhausted. For the renewable resource, the sole owner might drive
the stock to an interior (i.e. positive) steady state, where harvest and the
stock remain constant forever; or the owner might drive the …shery to extinction. A model with constant (stock-independent) average harvest costs
illustrates these possibilities. Here, if the intrinsic growth rate is less than
the rate of interest ( < r), there is no interior steady state with positive
rent: either the owner drives the stock to extinction, or she maintains the
stock at a positive level with zero rent. If the intrinsic growth rate exceeds
the rate of interest ( > r), there is a candidate interior steady state at which
the actual growth rate equals the rate of interest. This candidate is a steady
state for the sole owner if and only if rent is greater than or equal to zero
there. In this case, the owner does not drive the stock to extinction. There
is typically another interior steady state at which rent is zero.
Stock dependent harvest costs give the sole owner an incentive to restrict
harvest in order to let the stock grow, thus reducing future harvest costs.
Thus, stock dependence tends to increase the sole owner steady state, while
discounting tends to decrease it. The sole owner steady state might lie
16.5. TERMS, STUDY QUESTIONS, AND EXERCISES
311
above or below the MSY stock level. Lower discount rates increase the sole
owner steady state. In the limiting case with zero discounting, the sole
owner steady state equals the MSY level when harvest costs do not depend
on stocks; with stock dependent harvest costs, that steady state is above the
MSY stock level.
16.5
Terms, study questions, and exercises
Terms and concepts
stock-dependent e¢ ciency-inducing tax, ecological services
Study questions
1. Given a growth function F (x), a cost function c (x; y) and discount
factor write down the sole owner’s objective function and constraints.
Without doing calculations, describe the steps needed to obtain the
Euler equation in this model.
2. If you are given the Euler equation for a particular model (with or without stock dependent harvest costs) you should be prepared to provide
an economic interpretation of this equation.
3. (a) Consider the case where marginal harvest costs equal average harvest costs. Identify the stock-dependent tax that induces the open
access industry to harvest at the same rate as the untaxed sole owner.
(Compare the equilibrium conditions under the sole owner (price –marginal costs = rent) and under open access (price - average costs = 0).
(b) Suppose instead that harvest costs are convex in harvest (so that
marginal costs exceed average costs). In order to induce the open
access …shery to harvest at the same level as the untaxed sole owner,
would you have to increase or decrease the tax you identi…ed in part
(a)? Explain.
4. For the model with constant (stock independent) average harvest costs,
C, and logistic growth F (x) = x 1 Kx , the Euler equation evaluated at the steady state is 0 = R r dFdx(x) . (a) What is the
de…nition of steady state rent? (b) Under what conditions is there an
312
CHAPTER 16. THE SOLE-OWNER FISHERY
interior steady state with positive rent? Explain. (c) If the conditions
in part (b) are not satis…ed, are there other interior steady states? If
so, describe these.
5. Consider the following two claims “(i) An open access …shery leads to
excessive harvest. (ii) It is never socially optimal to exhaust the stock.”
Discuss these two claims in light of the model described in Question 4.
State whether you agree or disagree with these claims and justify your
answer.
Exercises
1. (a) Derive the Euler equation for the sole owner price-taking …shery
when average costs are constant, independent of the stock. (b) Provide
an intuitive explanation for this equation. (Students can answer this
question by mimicking the derivation and the explanation in the text,
making changes to re‡ect the di¤erent cost function here.)
2. Write down and interpret the Euler equation for the monopoly owner of
a renewable resource facing inverse demand p (y) and constant average
harvest costs, Cy. (Hint: use the same methods that we applied to
the nonrenewable resource problem.)
3. Provide the economic intuition for the Euler equation 16.11. This
equation corresponds to the scenario where the stock provides ecological
services V (xt ), and the sole owner receives a subsidy equal to V (xt );
that subsidy internalizes the externality. You need to understand the
intuition for the simpler case (where V is absent) and then explain
why the presence of payment for the ecological services changes the
optimality condition.
4. Show that, provided that F (x) is concave, an interior steady state in
the case where costs do not depend on the stock, always lies below the
maximum sustainable yield.
5. Consider the model with constant harvest costs C and logistic growth
F (x) = x 1 Kx . Suppose that < r. Are there circumstances
where the sole owner drives the stock to a positive steady state? Explain and justify your answer. (Hint: Is there any reason to suppose
that rent is positive in this model?)
16.5. TERMS, STUDY QUESTIONS, AND EXERCISES
313
6. Using equation 16.14 for the case of stock-independent harvest costs
with logistic growth, verify that if > r, then a larger value of K or
or a smaller value of r, all increase the interior steady state. Provide an
economic (not mathematical) explanation for these results. You need
to explain how changes in these parameters changes the owner’s incentive to conserve the …sh. Think about how an increase in r changes
the current valuation of future rents. Think about how increases in K
or alter the value of a larger …sh stock.
Sources
Homans and Wilen (2005) provide the estimate of the annual lease prices for
…shing quotas, as a percent of ex vessel price of catch.
Fenechel and Abbott (2014) show how estimates of stock dynamics and
the production function can be used to estimate the gain from better management of …sh stocks.
Zhang and Smith (2011) describe and implement, for Gulf Coast reef …sh,
the …rst estimation approach discussed in Chapter 16.2.4.
http://www.nmfs.noaa.gov/stories/2012/05/05_23_12stock_assessment_101_part1.html
explains the second estimation approach in Chapter 16.2.4.
The International Scienti…c Committee for Tuna (2011) illustrates the
second estimation approach, for the case of tuna stocks.
Grafton et al. (2007) provide evidence that the socially optimal stock
exceeds the MSY stock.
Squires and Vestergaard (2013) provide evidence that increases in technical e¢ ciency cause socially optimal steady state stocks to be below the MSY
stock.
314
CHAPTER 16. THE SOLE-OWNER FISHERY
Chapter 17
Dynamic analysis
Objectives
Use the necessary conditions for optimality under the sole owner to
characterize the equilibrium harvest rule, and the evolution of the …sh
stock over time.
Skills
Use intuition and economic reasoning to analyze the case of constant
average harvest costs.
Use phase portrait analysis to analyze dynamics under stock dependent
harvest cost.
Use the comparison of harvest rules under the sole owner and under
open access to characterize the e¢ ciency-inducing tax rule for open
access.
This chapter moves beyond steady state analysis to study the evolution of
the …sh stock and harvest under the sole owner. We assume throughout the
chapter that the …shery provides no non-market (e.g. ecosystem) services,
so there are no market failures under the price taking sole owner. The sole
owner and the social planner make the same decisions. We compare the
optimally controlled stock with the stock trajectory under open access in
order to characterize the optimal tax (a landing fee)under open access.
The market failure under open access arises from the lack of property
rights. As Chapter 13.3.2 emphasizes, the …rst best remedy in this situation
315
316
CHAPTER 17. DYNAMIC ANALYSIS
(usually) involves institutional changes, e.g. the establishment of property
rights, not regulation, e.g. a landing fee or a restriction on gear. It is
important to keep this point in mind. However, only a small fraction of
…sheries are currently managed using property rights-based regulation. It is
worth understanding how other types of regulation, such as taxes, can bring
the open access …sher closer to the optimally managed sole owner.
We emphasize graphical analysis, using a parametric model. This analysis is qualitative, i.e. it provides information about the direction of change
and the relation between initial conditions and the ultimate steady state. In
one section we use numerical methods to illustrate the complete solution. As
Chapter 14.3.1 notes, the dynamics of a discrete time system is complicated
because the stock can “jump” across steady states. Without resorting to
numerical methods that are beyond the scope of this book, the alternative is
to consider the continuous time limit of the discrete state model. That is,
we proceed under the assumption that the length of each period is so small
that we can replace the (discrete time) di¤erence equations with (continuous
time) di¤erential equations. We use the continuous time system for analysis.
We consider two scenarios. In the …rst, average harvest costs are constant, equal to C. In the second, average extraction costs, Cx , depend on
the stock. The …rst scenario makes it possible to obtain results using economic reasoning, without introducing additional mathematical tools. This
approach is useful for intuition, but it disguises many subtleties, and it does
not suggest a method for analyzing more general problems. In addition,
the assumption of constant harvest costs is questionable for many resource
settings, including most …sheries. The case of stock-dependent costs Cx is
simple enough to explain most of the subtleties without a great deal of technical detail, although it does require additional mathematical tools. Those
methods are useful for a wide variety of dynamic problems.
For both of these scenarios, it is important to keep in mind that the solution to the sole owner’s problem is a “harvest rule”, giving optimal harvest
as a function of the stock; typically, we represent a harvest rule by showing
its graph. Chapter 14 uses exogenous harvest rules, ones that are not based
on theory, and do not emerge from a model. Chapter 15 derives the endogenous harvest rule under open access, by …nding the harvest, a function
of the …sh stock, that drives rent to zero. We obtained those rules merely by
solving an equation (rent = 0). We denote the endogenous harvest rule for
the sole owner’s optimization problem as y(x) (harvest as a function of the
stock). In general, we cannot …nd a closed form expression for this function.
17.1. THE CONTINUOUS TIME LIMIT
317
We rely primarily on qualitative, and occasionally on numerical analysis.
17.1
The continuous time limit
Objectives and skills
Provide an intuitive understanding of the continuous time analog of
the discrete time Euler equation.
We need one intermediate result: the continuous time version of the discrete time Euler equation. In deriving that equation, we did not specify
whether the length of a period is one year or one second. Here we assume
that the length of period in the discrete time setting is su¢ ciently small that
the continuous time limit provides a reasonable approximation.
As in Chapter 14.3, the continuous limit of the discrete time equation of
= F (x) y. The missing piece is the continuous
motion for the stock is dx
dt
time analog for the Euler equation. We use equation 16.5 for the case
of constant average harvest cost and equation 16.7 for the case of stock
dependent cost. The continuous time limit under constant average harvest
costs (Appendix J) is
dRt
= Rt r
dt
dF (xt )
dxt
:
(17.1)
and the continuous time limit under stock dependent harvest costs is
dRt
= Rt r
dt
17.2
dF (xt )
dxt
C
yt :
x2t
(17.2)
Harvest rules for stock-independent costs
Objectives and skills
Characterize the harvest rule for the sole owner with constant harvest
costs.
Compare this rule with the harvest rule under open access.
Use this comparison to characterize the optimal tax under open access.
318
CHAPTER 17. DYNAMIC ANALYSIS
y
0.6
0.5
e
d
0.4
C=0.4
c
b
C=1.25
0.3
f
0.2
a
C=3
0.1
0.0
0
10
20
30
40
50
x
Figure 17.1: Open access and sole owner steady states for F = 0:04x 1
r = 0:02, and p = 5 10y.
x
50
,
This section uses the example introduced in Chapter 16.3.1, with growth
x
function F (x) = 0:04x 1 50
, inverse demand p (y) = 5 10y, discount
rate r = 0:02, and constant average costs C, the only free parameter in this
model. By varying C, we can determine the relation between harvest costs
and the sole-owner equilibrium. The Euler equation 17.1 must hold at every
point along the sole-owner’s harvest path.
We reproduce Figure 16.2, here shown as Figure 17.1. First consider
the open-access equilibrium. The horizontal dashed lines in Figure 17.1 are
the open-access harvest rules corresponding to three values of C. These
(constant) values of y satisfy the zero-pro…t open access condition, 5 10y =
C, or y = 510C . Using the analysis in Chapter 15, the points a, b and c are
stable interior steady states, and f , d and e are unstable steady states, for
the three values of C. The origin, x = 0 is a stable steady state in all three
cases. For example, at C = 0:4, the open-access stock approaches point c
if the initial stock is greater than the horizontal coordinate of point e; if the
initial stock lies below this level, the open-access stock approaches x = 0.
The entries in Table 1 show the open access steady state, x1 , corresponding
to di¤erent values of C and di¤erent initial conditions, x0 . In writing x1 = a,
for example, we mean that x1 equals the horizontal coordinate of point a.
17.2. HARVEST RULES FOR STOCK-INDEPENDENT COSTS
319
x0 above unstable
x0 below unstable
steady state (e, d or f )
steady state (e, d or f )
C = 0:4 x1 = c
x1 = 0
C = 1:25 x1 = b
x1 = 0
C=3
x1 = a
x1 = 0
Table 1: Open-access steady state, x1 , for di¤erent initial conditions, x0
Now consider the sole-owner equilibrium. As noted above, here we cannot
explicitly determine the harvest rule for all values of x. However, we know
one important fact about this rule: the sole owner never extracts so much
that rent becomes negative. In symbols, y > 0 ) R
0. Based on
this information, and some reasoning discussed below, we can identify the
steady state that the stock approaches in the sole owner …shery, as a function
of the value of C and of the initial conditions x0 . Table 2 summarizes this
information, and Section 17.2.2 explains how we obtain it. First, we consider
the policy implications of Tables 1 and 2.1
x0 above middle
steady state (e, d or f )
C = 0:4 x1 = c
C = 1:25 x1 = b
C=3
x1 = a
Table 2: Sole owner steady state, x1 , for
17.2.1
x0 below middle
steady state (e, d or f )
x1 = d
x1 = d
x1 = f
di¤erent initial conditions, x0 :
Tax policy implications of Tables 1 and 2
The information in Tables 1 and 2 has direct and simple policy implications.
For both open access and the sole owner, the high steady state (a, b, or c,
depending on the value of the cost parameter, C) is stable. For su¢ ciently
high initial stocks, the stock approaches the same steady state level under
both property rights regimes. In contrast, x = 0 is a stable steady state
under open access, but not under the sole owner. For su¢ ciently low initial stocks, the open access …shery eventually drives the stock to extinction.
Regardless of how small the initial stock is, provided that it is positive, the
1
For C = 1:25 and C = 3, the middle steady states (d and f , respectively) are “semistable”: initial conditions to the left of these points the trajectory converges to the point,
and for initial conditions to the right, the trajectory moves away from the point.
320
CHAPTER 17. DYNAMIC ANALYSIS
stock recovers to a positive level in the sole owner …shery. This di¤erence is
key to understanding the role of tax policy under open access.
We say that the initial stock is “large” if it exceeds the middle steady
state, points e, d or f (depending on the value of C); thus, the precise
meaning of “large” depends on C. For all three values of C, if the initial
stock is large, the equilibrium is the same under open access and under the
sole owner. In these cases, the stock is large enough that the sole owner’s
rent, along the equilibrium trajectory, is 0, exactly as under open access.
Here, the resource is not scarce. There is no reason to tax harvest under
open access, when the initial stock is large, because open access does not
create a market failure in this circumstance.
For C = 0:4 or C = 1:25 (and more generally, whenever C
1:25) and
x0 “small”(i.e. below the middle steady state), open access drives the stock
to extinction. Under the sole owner, the stock approaches (the horizontal
coordinate of) point d, i.e. it remains positive. Thus, when harvest is
inexpensive (C is small) and the stock is small, it is important to tax harvest
(or otherwise the open access …shery) in order to preserve the resource stock.
The …rst best tax policy, for small stocks, varies with the level of the stock.
We previously noted (Chapter 16.2) that in a resource setting, the optimal
tax is typically stock-dependent.
We would require numerical methods to compute the optimal tax as a
function of the level of the stock, at stocks below the unstable steady state.
Optimality might be too much to ask for, but the analysis suggests some
second-best alternatives. One alternative (for C
1:25) is to simply close
down the open access …shery until the stock recovers to (the horizontal coordinate of) point d, and then maintain a constant tax that supports openaccess harvest at point d. A less extreme alternative uses a high tax to
permit recovery of low stocks, reducing the tax as the stock increases.
The analysis is a bit di¤erent when C exceeds the critical level 1:25, but
is less than 5. (If C 5, the choke price, neither the open access nor the sole
owner …shery harvest anything.) Consider the case where C = 3. Here, for
initial stocks above the unstable steady state (point f ) the stock approaches
a under both open access and under the sole owner. Just as is the case for
the lower values of C, the sole owner and the open access …shery harvest
at the same rate, and both have zero rent, provided that the initial stock is
su¢ ciently high. For stocks below point f , the open access …shery drives
the stock to extinction. The sole owner …shery, in contrast, drives the stock
to point f (not to point d, where rent would be negative).
17.2. HARVEST RULES FOR STOCK-INDEPENDENT COSTS
321
In summary, for this example, it is important to regulate an open access
…shery, e.g. by means of a tax, when the stock is low. There may be no need
to regulate the open access …shery if the stock is su¢ ciently high. Taxes can
achieve distinct, but related objectives: to alter the steady state to which the
open access stock converges, and to alter the speed at which the open access
stock declines. In general, at low levels of the stock, the optimal tax for the
open access …shery depends on the stock. It might not be practical to use
the optimal tax, but second best taxes can insure that the stock approaches
the …rst best steady state, even if the approach trajectory does not occur at
the optimal speed.
17.2.2
Con…rming Table 2
We provide details for the case C = 0:4, leaving the other two cases as
exercises. Our goal is to show that the sole owner’s harvest rule implies
the claims in the second row of Table 2. Recall that the harvest rule gives
harvest as a function of the stock of …sh. We can describe this rule using a
graph, showing harvest as function of the stock. We begin by stating seven
facts about the graph of the sole owner’s harvest rule.
1. The graph of the rule lies on or below the dashed line labelled C = 0:4
in Figure 17.1. At any point above that line, rent is negative, i.e.
harvest results in negative pro…ts. The sole owner never harvests so
much that rent becomes negative.
2. The sole owner’s harvest rule intersects x = 0 = y, where price equals
the choke price. If the stock is 0, the harvest must also be 0: you
cannot get blood out of a turnip.
3. The harvest rule is continuous. A discontinuous rule would lead to
jumps in the price trajectory (as a function of time), violating the “nointertemporal-arbitrage”condition (the Euler equation).
4. Point d is the unique interior steady state with positive rent. Chapter
16.3.1 establishes this claim.
5. At points to the left of d, r < dFdx(x) and at points to the right of d the
inequality is reversed. This claim follows from inspection of Figure
17.1.
322
CHAPTER 17. DYNAMIC ANALYSIS
6. Where x is positive and close to 0, the sole owner’s rent is positive.
This claim follows from Fact 3: for small x, price is close to the choke
price, a, and rent is close to a 0:4 > 0.
t
7. For 0 < x < d, dR
< 0. This claim uses Facts 1 and 6 to con…rm
dt
dF (x)
that Rt > 0 and r
< 0; these two inequalities and equation 17.1
dx
dRt
establish that dt < 0.
We use these facts to characterize the sole owner’s harvest rule over three
intervals of the stock: x > e, x < d, and d < x < e.
x > e: Here, the sole owner’s harvest rule is identical to the rule under
open-access: the graph of that rule equals the highest dashed horizontal line
in Figure 17.1 to the right of point e. At this constant level of harvest, the
stock approaches point c. Along that trajectory, R = 0, so the Euler equation 17.1 is always satis…ed. For this problem, the necessary conditions for
optimality are also su¢ cient, so for x greater than the horizontal coordinate
of point e, we have found the sole owner’s harvest rule.
x < d: The sole owner’s harvest rule lies below the graph of the growth
function for stocks to the left of point d. In order to con…rm this claim, we
note that there are only three possibilities for the graph of the harvest rule
over this interval: (i) it might intersect the growth function to the left of d;
(ii) it might lie strictly above the growth function to the left of d; or (iii) it
might lie strictly below the growth function to the left of d.
We can exclude possibility (i) because a point of intersection between
the harvest rule and the growth function is a steady state at which rent is
positive, contradicting Fact 4. Thus, we are left with possibilities (ii) and
(iii).
We can exclude possibility (ii), because in this scenario, harvest exceeds
growth and the stock is falling to 0 over time. As the stock approaches 0,
Fact 3 implies that harvest approaches 0, so during this part of the trajectory,
harvest is falling and rent (R = a by C) is rising. However, that conclusion
contradicts Fact 7, which states that rent is falling. Therefore, we are left
with possibility (iii): for stocks to the left of point d, the harvest rule lies
below the growth function and the stock is increasing.
17.3. HARVEST RULES FOR STOCK DEPENDENT COSTS
323
d < x < e. By Fact 4, there are no points of intersection between the
harvest rule and the growth function over this interval. We can exclude
the possibility that the harvest rule lies below the growth function over this
interval using an argument that parallels the argument in the previous paragraph. Therefore, over this interval of x, the harvest rule lies above the
growth function.
Summary Points d and e are steady states, so they are on the sole
owner’s harvest rule. We showed above that this rule is below the growth
function for x < d and above the growth function for d < x < e. Therefore,
we conclude that for 0 < x < e, point d is a stable steady state. We also
showed that for x > e, point c is a stable steady state. Thus, we con…rm
the claims in the second row of Table 2.
17.3
Harvest rules for stock dependent costs
Objectives and skills
Ability to interpret a …gure showing graphs of the open access and the
sole owner harvest rules.
Ability to use this …gure to calculate the e¢ ciency-inducing tax for the
sole owner.
Introduction to the phase portrait and its ingredients, the isoclines and
isosectors.
We carry out all of the analysis using the following parametric example
c (x; y) =
F (x) = x 1
p=a
x
K
C
y
x
with C = 5
with K = 50 and
= 0:04
(17.3)
by; a = 3:5 and b = 10; and r = 0:03:
We begin by summarizing the policy implications based on this example. Subsequent material develops the methods used to obtain those results.
There, we explain the meaning and the use of the “phase portrait”, the important new tool in this chapter. We then explain what a “full solution”
means in a model of this sort.
324
17.3.1
CHAPTER 17. DYNAMIC ANALYSIS
Tax policy
Here we explain how to interpret Figure 17.2, and the policy implications
that it contains. For the model in equation 17.3, there is a unique steady
state under the sole owner, x1 = 39:35, y1 = 0:335. We discuss only the
behavior of the …shery for x below the steady state. Figure 17.2 shows the
growth function, the heavy solid line, for x 39:35. The dotted line shows
the harvest rule under the sole owner for x below the steady state. Most of
the work involved with this analysis lies in identifying this harvest rule, i.e. in
constructing the dotted graph. For the time being, we put those di¢ culties
aside and discuss the meaning of this graph. In this example, where harvest
costs depend on the stock of …sh, the owner wants a high stock in order to
have low harvest costs. (We do not use the thin solid curve in the discussion
here, although it plays an important role in the derivation below.)
The dashed line shows the harvest rule under open access, obtained (as
always) by setting rent = 0 and solving for harvest as a function of the stock.
There are three steady states under open access, the points of intersection
between the dashed and the heavy solid curve. The middle point is unstable,
and the higher and the lower points of intersection are stable. The low
steady state, approximately x = 2, and the high steady state, slightly less
than 39:35 (the sole owner steady state), are both stable. The intermediate
steady state, approximately equal to x = 8, is unstable.
The harvest rule under the sole owner (the dotted curve) lies everywhere
below the harvest rule under open access (the dashed curve). The sole
owner always harvests less than the open access …shery. For stock levels
above approximately x = 20, the open access and the sole owner harvest
rules are nearly the same. For this range of stocks, the stock is su¢ ciently
large that the sole owner and open access …shery harvest at nearly (but not
exactly) the same level. Here, it is not important to regulate the open access
…shery.
For stocks above the open access unstable steady state (x = 8), harvest
under open access is low enough to allow the …sh stock to reach almost the
optimal steady state, x = 39:35. For 8
x < 20 there is a slight, but
appreciable di¤erence in the two harvest rules. For this range of stocks,
there is a minor role for regulation. Regulation allows the …sh stock to
recover more quickly, and produces positive rent. However, regulation has
no signi…cant long term e¤ect, because the steady states under open access
and under the sole owner are almost the same.
17.3. HARVEST RULES FOR STOCK DEPENDENT COSTS
y
325
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0
5
10
15
20
25
30
35
x
Figure 17.2: A part of the phase portrait, for stocks below the steady state
level. The higher solid curve is the x isocline (the growth function) and
the lower solid curve is the y isocline. The dashed curve is the set of
points where rent = 0. Rent is positive below this curve. The dotted line,
sandwiched between the “rent = 0” curve and the y isocline is the optimal
trajectory, giving the optimal harvest as a function of the stock.
For stocks below the open access unstable steady state, regulation is important under open access. Over this range, the stock declines to x = 2,
whereas under the sole owner, the stock eventually recovers to 39.35.
Given the two harvest rules, it is a simple matter to determine the optimal tax, for any level of the stock. An example illustrates the procedure.
Suppose that x = 7 (an arbitrary choice, merely for illustration). Reading
from the harvest rules, we see that the open access harvest is approximately
y = 0:29 and the sole owner harvest is approximately y = 0:2, yielding the
equilibrium open access price 3:5 10 (0:29) = 0:6 and the equilibrium sole
owner price 3:5 10 (0:2) = 1: 5. The tax jx=7 = 1:5 0:6 = 0:9 induces the
open access to reduce harvest y = 0:2; at that level, the market price minus
the tax equals the average harvest cost, and industry rent is zero. The tax
= 0:9 thus supports the e¢ cient level of harvest at x = 7. Because the
vertical distance between the two harvest functions changes with the level
of the stock, the optimal tax also changes with the stock. The optimal tax
is negligible for large stocks, but the tax is large at small stocks. For our
example, the optimal tax comprises 60% of the equilibrium consumer price at
x = 7. Recalling Chapter 16.2.1, the optimal tax under open access equals
326
CHAPTER 17. DYNAMIC ANALYSIS
the rent under the sole owner. Thus, for this example, rent under the sole
owner comprises about 60% of the market price when x = 7.
The comparison of Figures 17.1 and 17.2 illustrates the e¤ect of stock
dependent harvest costs. In both of these models, there are three steady
states under open access. The intermediate steady state is unstable, and the
other two are stable. The low steady state is zero under constant costs, but
positive under our example of stock-dependent harvest cost. With stockdependent harvest costs, the open access …shery drives the stock to a low
but positive level, where harvest is expensive, price is high, and consumer
surplus low.
With stock independent harvest costs, there are either three or four steady
states under the sole owner, one of which is x = 0.2 In our stock-dependent
harvest cost example, there is a single positive stable steady state, x = 39:35.
This steady state exceeds the MSY stock, x = 25. The sole owner (and the
social planner) wants to drive the stock above the level that maximizes steady
state yield, because doing so reduces harvest costs.
The tax implications of the two models are quite similar. In both, it is
unimportant to tax the open access …shery at high stock levels. At high
stock levels the optimal tax is zero under constant harvest costs, and the
optimal tax is close to zero in our example of stock-dependent harvest costs.
For high stocks, the optimal tax for the open access …shery reduces harvest
and speeds recovery of the stock to a high level, but the tax has negligible
long run e¤ect. At low stocks, the tax is important in both models; it avoids
physical extinction in one case, and economic irrelevance in the other.
2
For constant cost C = 0:4 there are four sole owner steady states, at x = 0, d, e and c;
the steady states 0 and e are unstable, and d and c are stable. For C = 3, there are three
steady states, at x = 0, f and a. The steady state 0 is unstable, and a is stable; point
f is “semi-stable”: trajectories beginning at positive stocks below this level approach f ,
but trajectories beginning above it move away, approaching a.
17.3. HARVEST RULES FOR STOCK DEPENDENT COSTS
327
Box 17.1 Back to Huxley and Gould Box 1.1 contains quotes from two
19th century …gures, one explaining why regulation is not needed, and
the other explaining why it is needed. The above analysis illustrates
the circumstances where one or the other is correct. If stocks are
above the unstable open access steady state, the open access outcome
is at least approximately socially optimal. Stock dependent costs
reinforce this tendency, by inducing …shers to reduce harvest as the
stock falls. These forces provide a kind of automatic protection, as
Huxley suggested, and there is no need for regulation. However, at
low stocks, neither the market (which limits demand) nor technology
(which limits supply by increasing costs) is adequate to protect the
stock: regulation is needed, as Gould stated.
17.3.2
The phase portrait
Equations 14.6 and 17.2 give the di¤erential equations for the stock and for
the rent under the sole owner. We can use these two equations, together
C
, to obtain a third di¤erential
with the de…nition of rent, R = p (y)
x
dy
equation, for the harvest, dt . It helps to give this di¤erential equation a
name, so we denote it as dy
= H (x; y). (Appendix J.2 explains how we
dt
…nd this function H.) The solution to these three di¤erential equations (in
x; R; y) gives the optimal paths of the stock, rent, and harvest. Apart from
the simplest problems, we cannot solve these equations analytically. Our
…rst goal is to learn as much as possible about the solution without actually
solving the equations: we seek qualitative information about the solution.
The phase portrait is the key to achieving this.
The phase portrait contains two “isoclines”. An isocline is a curve along
which the time derivative of a variable is 0. Consider the logistic growth
y. Setting this derivative equal to 0 gives
= x 1 Kx
function, dx
dt
y = x 1 Kx ; the graph of this function is the x isocline (the curve where
dx
= 0). Thus, the x isocline is simply the growth function; we are given
dt
that function as part of the statement of the problem, so no work is required
to obtain the x isocline. We can also obtain the y isocline, the curve where
dy
= H (x; y) = 0. Figure 17.3 shows the graphs of the two isoclines for our
dt
example. The intersection of these isoclines identi…es the steady state, the
point where dx
= 0 = dy
. For our example, the steady state is x = 39:35,
dt
dt
y = 0:335.
In order to understand a phase portrait, the reader has to keep in mind
328
CHAPTER 17. DYNAMIC ANALYSIS
y
0.8
0.7
A
0.6
stock constant
0.5
B
0.4
0.3
E
0.2
harvest constant
0.1
D
0.0
0
10
20
30
40
50
x
Figure 17.3: The solid curve: the x isocline (where dx
= 0; dashed curve:
dt
dy
the y isocline (where dt = 0). The two isoclines divide the plane into four
regions, A; B; C and D, known as isosectors.
that, outside the steady state, the stock and the harvest are changing over
time. Imagine that there is a third axis, labelled time, t, perpendicular to
the page, coming directly toward the reader. A point on the page represents
a particular value of x and y at t = 0. A point above the page represents a
particular value of x and y and a time t > 0.
Suppose that we start at t = 0, with some initial condition, x0 = x (0),
and we pick some initial harvest, y0 . Starting from this point, there is a
path, a curve in three dimensional space, call it (xt ; yt ; t) along which the
= F (x) y and dy
= H (x; y) are satis…ed. Now
di¤erential equations dx
dt
dt
imagine shining a light from your eyes to the page. The curve (xt ; yt ; t),
casts a shadow onto the page. We refer to this shadow as a trajectory.
The phase portrait provides information about such a trajectory (i.e., the
“shadow”); we use that information to infer facts about the behavior of the
stock and the harvest over time. To this end, we use the four “isosectors”
de…ned by the two isoclines. Figure 17.3 identi…es these four regions as
A; B; D; and E.
17.3. HARVEST RULES FOR STOCK DEPENDENT COSTS
Isosector
motion of x
motion of y
A
decreasing
(west)
increasing
(north)
B
increasing
(east)
increasing
(north)
D
increasing
(east)
decreasing
(south)
329
E
decreasing
(west)
decreasing
(south)
overall motion
north-west
north-east
south-east
south-west
of trajectory
Table 3. Direction of change of x and y and overall direction of motion of
trajectory in the four iso-sectors
Consider isosectors B and D, the region below the x isocline (the graph
of y = F (x)). For any point in either of those two isosectors, y < F (x).
= F (x) y > 0, i.e. x is increasing over
Consequently at such a point, dx
dt
time. For shorthand, we say that the trajectory is moving east (x is getting
larger). Similarly, above the x isocline, in isosectors A and E, y > F (x), so
dx
= F (x) y < 0. In these two isosectors, x is getting smaller, so we say
dt
that the trajectory is moving west. The second row of Table 3 summarizes
this information.
We can identify the direction of movement in the north-south direction
= H (x; y)
by using information about the di¤erential equation for y, dy
dt
(details in Appendix J.2). Below the y isocline (in isosectors D and E)
dy
= H (x; y) < 0, i.e. y is decreasing, so the trajectory is moving south.
dt
= H (x; y) > 0, i.e. y is
Above the y isocline (in isosectors A and B) dy
dt
increasing, so the trajectory is moving north. The third row of Table 3
summarizes this information. The fourth row puts together the previous
information, to obtain the overall direction of motion of a trajectory in each
isosector.
This qualitative information tells us that an optimal trajectory that approaches the steady state x1 from a smaller value of x (to the west of x1 ),
must lie in isosector B. Similarly, a trajectory that approaches the steady
state from a larger value of x (to the east of x1 ) lies in isosector E. To
explain and con…rm these statements, we consider the case where the optimal
trajectory approaches the steady state from below (i.e. from the west); the
situation where the optimal trajectory approaches from above is similar.
A trajectory that approaches the steady state from below cannot lie in
isosector E, because that isosector contains no stock levels less than the
steady state. Therefore, the trajectory must lie in isosectors A; B, or D. The
path cannot lie in isosector A, because trajectories there involve westward
330
CHAPTER 17. DYNAMIC ANALYSIS
movements, i.e. reductions in the stock. Therefore, the trajectory must lie
in either isosector B or D. It cannot lie in D, because from any point in D,
it would be necessary for y to increase in order to reach the steady state; but
trajectories in D move south, i.e. y falls there. Consequently, trajectories
that approach the steady state from below (with stocks lower than the steady
state level) do so in isosector E.
For this problem, regardless of the initial (positive) stock level, the optimally controlled …shery approaches the steady state. We can use this fact
to establish that for any initial stock below the steady state, the optimal trajectory lies entirely in isosector B; and for any initial stock above the steady
state, the optimal trajectory lies entirely in isosector E. For example, suppose that we begin with a stock below the steady state level. If, contrary to
our claim, the trajectory beginning with this stock lay in either isosector A
or D it would move away from the steady state.
Figure 17.4 repeats the previous …gure, adding the dotted curve, the set
of points where rent is zero. Rent is positive below the dotted curve, where
harvest is lower and price is consequently higher than on the curve. We know
from above that if the stock begins below the steady state, it approaches the
steady state in isosector B, so the trajectory must be above the dashed
curve (a boundary to isosector B). We also know that the sole owner never
harvests where rent is negative, so the trajectory must be on or below the
dotted curve. Therefore, we conclude that if the initial stock is below the
steady state, the trajectory must be sandwiched between the dashed and the
dotted curves. We obtained this information without actually solving the
optimization problem, using only the necessary conditions for optimality and
a bit of graphical analysis. This procedure illustrates the power of the phase
portrait.
Given the resolution of Figure 17.4, it appears that the dotted and the
dashed curves are coincident around the steady state. However, if we enlarged the …gure in the neighborhood of the steady state, we would see that
the dashed curve lies below the dotted curve, and the steady state lies on
the dashed curve. For this example, rent is always positive, but it is close
to zero for …sh stocks near the steady state. For di¤erent functional forms
or parameter values, rent might be substantial along the optimal trajectory.
17.4. SUMMARY
y
331
0.8
0.7
A
0.6
stock constant
0.5
B
0.4
0.3
E
0.2
harvest constant
0.1
D
0.0
0
10
20
30
40
50
x
Figure 17.4: The dotted curve shows the combination of harvest and stock
at which rent is 0.
The full solution
Figure 17.2 contains the graph of the sole-owner’s optimal harvest rule (the
dotted curve in that …gure). To construct that graph, we need the solution
to the pair of di¤erential equations dx
= F (x) y and dy
= H (x; y) that
dt
dt
includes the point (x1 ; y1 ), the steady state. The steady state is a “boundary condition” for this mathematical problem. As noted above, except for
very few functional forms (not including our parametric example), there is
no analytic solution to this problem. However, there are numerical routines
that are straightforward to implement. The harvest rule shown in Figure
17.2 was obtained using Mupad, a feature of Scienti…cWorkplace.
17.4
Summary
Two examples, one with constant harvest costs, and the other with stockdependent harvest costs, show how to analyze the dynamics under the sole
owner. We showed how to determine the sole owner steady state(s), and to
determine which steady state the sole owner …shery approaches, as a function
of the initial condition. With constant harvest costs, this determination
requires careful economic reasoning, but no new mathematical tools. With
stock-dependent harvest costs, we require new tools. The most important
of these is the phase portrait, which has many uses in dynamic problems.
Given this information, and using the fact that the open access …shery
332
CHAPTER 17. DYNAMIC ANALYSIS
harvests up to the point where rent is zero, we were able to make qualitative
statements about the optimal tax for the open access …shery. In particular,
we learned that regulating this …shery is important if the stock is low; for
su¢ ciently high stocks, regulation of the open access …shery is unimportant.
This di¤erence illustrates the general principle that for resource problems,
the optimal tax is stock-dependent. In practice, we seldom have enough
information to determine the optimal tax. A second best alternative is to
close down the open access …shery at low stock, and leave the …shery untaxed
at high stocks. A more nuanced policy imposes low or zero taxes at high
stocks, and high taxes at low stocks.
Examples of this sort are useful for developing intuition. The pedagogic
danger of these examples is that they may make the problem of regulation
appear too simple. It might appear that all we need is a few parameters
estimates and a modest knowledge of mathematics to propose optimal policy
measures. That conclusion is too optimistic. The models studied here
are good for the big picture, but they are too simple to be directly useful
in actual policy environments. There, it may be important to consider
multiple species or multiple cohorts of a single species, and di¤erent kinds
of uncertainty, including uncertainty about functional forms and parameter
values. Nevertheless, simple models provide a good place to begin.
17.5
Terms, study questions, and exercises
Terms and concepts
Continuous time Euler equation, isocline, phase portrait
Study questions
1. Using Figure 17.1, for each of the three values of C, sketch the sole
owner’s harvest rule that is consistent with the claims in Table 2.
2. (a) Using your sketch from #1 and C = 3, pick two values of x to
the left of point f . At these two points, identify on the graph the
di¤erences in harvest under open access and under the sole owner. (b)
Explain how you would use this graph to obtain the optimal tax under
the sole owner, at these two values of x. (c) What qualitative statement
17.5. TERMS, STUDY QUESTIONS, AND EXERCISES
333
can you make about the magnitude of the optimal taxes for the two
values of x?
Exercises
1. By adapting the arguments used in Section 17.2.2, con…rm the claims
in the last two rows of Table 2.
x
, inverse demand is p (y) = 5 10y,
2. Suppose that F (x) = 0:04x 1 50
the discount rate is r = 0:02, and harvest costs are constant at C =
0:4. Suppose also that the initial condition, x0 , is below the horizontal
coordinate of point e in Figure 17.1. (a) What tax (a number) supports
an open access steady state at point d? (b) If the policymaker uses this
constant tax, does it drive the stock to point d for all initial conditions
below e? (c) For initial conditions below e (i.e., for values of x0 below
the horizontal coordinate of point e) does the optimal tax rise or fall
with higher x?
3. * For the parametric example in the Exercise 2, suppose that C = 1:25.
(a) Con…rm that for initial conditions below the unstable steady state
(f ), the sole owner drives the stock to x = 0. (b) Con…rm that along
the path to extinction, rent is positive as long as the stock is positive.
(c) Using your answer to part (b), sketch the harvest rule under the sole
owner. (Your drawing will not be accurate, but you should be able to
identify the points on the graph of the harvest rule, y(x), corresponding
to x = f and x = 0. (d) Using this sketch, how does the …rst-best
(stock-dependent) tax under open access vary with the stock, x?
4. For the model in equation 17.3, use Figure 17.2 to estimate the optimal
tax for the open access …shery, at x = 5. Explain your steps.
334
CHAPTER 17. DYNAMIC ANALYSIS
Chapter 18
Sustainability (Under
construction)
18.1
Concepts of sustainability
18.2
The Hartwick rule
18.3
Discounting and intergenerational equity
18.4
Economics and climate policy
335
336
CHAPTER 18. SUSTAINABILITY (UNDER CONSTRUCTION)
Appendices are currently under revision
Appendix A
Math Review
(Revised August 25, 2014) This appendix reviews some concepts and
results from basic calculus. It can be used as a reference during the course,
and also gives readers an idea of the level of mathematics required for the
course. The text assumes that readers have seen much this material before.
(x0 )
, is the
1 The derivative of a function, f (x) at a point x0 , written dfdx
tangent (“slope”) of a function, evaluated at a particular point, here x0 . If
f (x) = a + bx, where a; b are independent of x (and therefore constants for
df
= b; a constant.
our purposes here), then dx
More generally, however, the value of the derivative depends on x. Figure
A.1 shows the graph of f (x) = 2 + 3x 4x2 5x3 , the solid curve, the graph
2 f (x)
(x)
of g (x) = dfdx
, the dashed curve, and the graph of h (x) = dg(x)
= d dx
2 .
dx
This …gure reminds the reader that, in general, (1) a derivative is a function,
not a constant; (2) where a function reaches an extreme point (a maximum
or a minimum) the derivative of that function equals 0. If the graph of a
function has an in‡exion point, i.e. switches from being concave to convex,
the second derivative of the function equals 0 at the in‡exion point.
Another way to indicate that a function is being evaluated at a particular
point, say x = 2, uses subscripts. For example, the subscript “jx = 2”here
indicates that we evaluate the derivative of f with respect to x at x = 2:
df (x)
= 2 + 3 (2)
dx jx=2
It is worth repeating that (in general)
we called this function g (x). Similarly,
337
4 (2)2
5 (2)3 :
df (x)
is itself a function of x; above
dx
d2 f (x)
is a function of x; above we
dx2
338
APPENDIX A. MATH REVIEW
h(x)=dg/dx
10
f(x)
-1.4
-1.2
-1.0
g(x) =df/dx
-0.8
-0.6
-0.4
-0.2
0.2
0.4
0.6
0.8
1.0
x
-10
-20
Figure A.1: The solid curve shows the graph of f (x), the dashed curve
df
, and the dotted curve shows the graphof
shows the graph of g(x) = dx
2
dg
d f
h (x) = dx = dx2 .
call it h (x).
Another way to write the derivative uses the “prime sign”, 0 :
df (x)
= f 0 (x) :
dx
2. In the example above we took the derivative of a function involving an
exponent. Students should memorize the following rule: if a is a constant
(with respect to x), then
d (xa )
= axa 1 :
dx
We write that “a is a constant with respect to x", instead of merely
writing “a is a constant” because the formula above is correct even if a is a
function of other variables (not x). For the purpose of taking this derivative,
it does not matter whether a is a literally a constant or merely a constant
with “respect to x”. What matters is that a change in x does not change a.
3. Students should memorize a few of the primary rules for derivatives.
Suppose we have two functions of x, a (x) and b (x). (Note: in the previous
line we treated a as a constant. Here we treat it as a function. In general, we
are careful not to use the same symbol to mean two di¤erent things. Here,
we intentionally use the same symbol, a, to mean two di¤erent things, …rst a
339
constant and then a function. We want to encourage readers to pay attention
to de…nitions.) We can form other functions using these two functions.
If c is the sum of these two functions, then
c (x) = a (x) + b (x) and
dc
da db
=
+ .
dx
dx dx
The derivative of a sum equals the sum of a derivative. For brevity we write,
da
for example, dx
instead of da(x)
.
dx
If c is the product of the two functions, then
c (x) = a (x)
b (x) and
da
db
dc
=
b + a.
dx
dx
dx
If c is the quotient of two functions, then
c (x) =
b da a db
dc
a (x)
and
= dx 2 dx .
b (x)
dx
b
4. The chain rule enables us to take the derivative of a function of a
function. Suppose that y is a function of x and x is a function of z. Then
y is a function of z, via the e¤ect of z on x. The chain rule states
dy (x (z)) dx (z)
dy
=
:
dz
dx
dz
For example, if y = x0:3 and x = 7z, then
dy (x (z)) dx (z)
dy
=
= 0:3x0:3
dz
dx
dz
dy(x)
dx
1
= 0:3x0:3
1
7 = 0:3 (7z)
and
0:7
dx
dz
= 7, so
7.
5. Some of our functions involve two arguments, instead of one. Throughout the book we use a cost function that depends on the stock of the resource,
x, and the amount that is extracted in a period, y. We write this cost function as c (x; y). A partial derivative tells us how the value of the function
(here, costs) changes if we change just one of the variables, either x or y.
We use the symbol @ instead of d to indicate that we are interested in the
partial derivative.
We frequently illustrate concepts using the following speci…c cost function
Parametric example: c (x; y) = C ( + x)
y 1+ ;
340
APPENDIX A. MATH REVIEW
where C; ; ; and are non-negative parameters. (Note that lower case c is
a function, and upper case C is a parameter; this use of two similar symbols
to mean two di¤erent things is intentional. It is important to pay attention
to de…nitions.) The partial derivatives of this function with respect to x and
y are are
1 1+
@C(x+ ) y 1+
=
C (x + )
y ;
@x
@C(x+ ) y 1+
= (1 + ) C (x + ) y .
@y
Because we use this formulation throughout the text, the reader should be
sure to understand it at this point. For example, in taking the partial of c
with respect to x, we recognize that Cy 1+ does not depend on x. Thus,
in writing the partial of c with respect to x we treat Cy 1+ as a constant.
Although not literally a constant, this term is constant with respect to x.
In English: this term does not depend on x; therefore, changes in x do not
a¤ect this term. To drive this point home, we can write
c (x; y) = “Constant with respect to x” ( + x)
and then use the rule in item #2 above to write the partial derivative of c
with respect to x as
= “Constant with respect to x” d( +x)
dx
= “Constant with respect to x” ( ) ( + x)
1 1+
=
C ( + x)
y
@c(x;y)
@x
1
The two partial derivatives of c with respect to x and y are themselves
functions of x and y. Thus, we can di¤erentiate either of these functions,
with respect to either x or y, to obtain a higher order partial derivative. For
example,
@ 2 C(x+ )
@y 2
@
y 1+
@C(x+ )
@y
y 1+
@ [(1+ )C(x+ )
@y
1
=
=
@y
= (1 + ) C (x + ) y
and
@ 2 C(x+ ) y 1+
@y@x
@
@C(x+ )
@y
=
(1 + ) (
y 1+
=
) C (x + )
@x
y
]
;
@ [(1+ )C(x+ )
@x
1
y
]
=
y .
6. A function may depend on two variables, and each of those variables
might depend on a third variable. The total derivative tells us how much the
341
function changes for a change in this third argument. For example, suppose
that costs depend on x and y, as above, and x and y both depend on ". We
show this dependence by writing x (") and y ("). With this notation, we
write costs as c (x (") ; y (")). The total derivative of c with respect to " is
dc (x (") ; y ("))
@c dx @c dy
=
+
:
d"
@x d" @y d"
(A.1)
In writing this equation, we merely apply the chain rule twice: the …rst term
accounts for the fact that a change in " alters c via the change in x, and the
second term accounts for the fact that a change in " alters c via the change
in y. The total change in c due a change in " is the sum of these two terms.
This expression might seem complicated, but most of the applications
in this book are extremely simple. We will be interested in the case where
x = x1 " and y = y1 ", where x1 and y1 are treated as constants for the
purpose here. For these two functions, we have
dx
=
d"
1 and
dy
=
d"
(A.2)
1:
Substituting equation A.2 into equation A.1 gives the total derivative
dc (x (") ; y ("))
=
d"
@c (x1
"; y1
@x
")
@c (x1
"; y1
@y
")
:
We often want to evaluate this derivative at " = 0. In this case, we write
dc (x (") ; y ("))
=
d"
j"=0
@c (x1 ; y1 )
@x
@c (x1 ; y1 )
:
@y
Remember that the subscript “j" = 0”on the left side of this equation means
that we evaluate the derivative of c with respect to " where " = 0.
7. A function might depend on several arguments. Suppose that a
function L depends on x; y; z: L = L (x; y; z). The di¤erential of L, denoted
dL is
@L
@L
@L
dL =
dx +
dy +
dz.
@x
@y
@z
The change in L (denoted dL) equals the change in L due to the change in
x, @L
times the change in x, dx, plus the change in L due to the change in
@x
@L
y, @y , times the change in in y, dy, plus the change in L due to the change
in z, @L
, times the change in z, dz.
@z
342
APPENDIX A. MATH REVIEW
E
8
6
4
2
0
0.1
0.2
0.3
p
-2
Figure A.2: The solid graph shows the excess demand for = 3; here the
equilibrium price is p = 0:17. The dashed graph shows excess demand for
= 5. Here the equilibrium price is p = 0:15.
We use di¤erentials to perform some comparative statics exercises. An
“equilibrium condition”is (usually) an equation; this equation might be the
…rst order condition to an optimization problem, or it might be a statement
that says “supply equals demand”. The equilibrium condition determines
an “endogenous variable” (e.g., the optimal level of sales or the price that
equates supply and demand) as a function of model parameters. A comparative statics exercise asks how the endogenous variable changes as one or
more parameters of the model change.
For example, suppose that the demand is QD = p 0:6 and supply is QS =
2 + p + p0:5 , with > 0. De…ne excess demand, E (p; ), as demand minus
supply. The equilibrium price equates supply and demand, i.e. it sets excess
demand equal to 0:
E (p; ) = p
0:6
2 + p + p0:5 = 0:
For this example, we have one endogenous variable, p, and one parameter,
. We cannot solve the equilibrium price as a function of . Figure A.2
shows the graphs of excess demand for = 3 (solid) and for = 5 (dashed).
The higher value of (associated with a larger supply at every price) leads
to a lower equilibrium price.
The di¤erential of E is
dE =
@E
@E
dp +
d .
@p
@
343
As we change , the equilibrium price also changes. The equilibrium price
causes excess demand to equal 0. Therefore, equilibrium requires that dE =
0:
@E
@E
dE =
dp +
d = 0.
(A.3)
@p
@
This equation states that an exogenous change in , d , induces an endogenous change in the price, dp, in order to maintain excess demand at 0. The
partial derivatives of E are
@E
@p
=
0:6p
1:6
@E
@
0:5p
=
0:5
<0
(A.4)
p < 0:
Notice the inequalities in the two lines of equation A.4. The …rst inequality
says that an increase in p, at …xed , decreases excess demand. The second
inequality says that an increase in , at …xed p, decreases excess demand.
The endogenous price must adjust to a change in in order to keep excess
demand at 0, in order to satisfy our equilibrium condition. Inspection of
these two partial derivatives tells us that if increases (thereby lowering E),
there must be an o¤setting decrease in p, in order to maintain excess demand
at 0.
Here we can …gure out how p must change in response to a change in
simply by thinking a bit about the implication of the signs of the partial
derivatives. Many problems are too complicated for that kind of casual
reasoning to be useful. Therefore, we proceed systematically, using equations
A.3 and A.4:
0=
@E
dp
@p
+
@E
d
@
= ( 0:6p
( 0:6p
1:6
1:6
0:5p
0:5p
0:5
0:5
) dp + ( p) d )
) dp = pd :
The …rst equality repeats equation A.3; the second uses the information in
equation A.4. Rearranging this equation gives the implication in the second
line. We solve this equation, dividing both sides by d and also dividing
both sides by ( 0:6p 1:6
0:5p 0:5 ) to write
dp
=
d
( 0:6p
p
1:6
0:5p
0:5 )
:
This equation is our comparative static expression. The numerator of the
ratio on the right side is positive and the denominator is negative, so ddp < 0.
344
APPENDIX A. MATH REVIEW
Appendix B
Comparative statics
Derivation of equation 2.5 Using the de…nition of the di¤erential
and the fact that the right side of the equilibrium condition (R) depends on
both b and q China , and the left side (L) depends only on q China , we have
dL =
dR =
@L(q C h in a )
dq China
@q C h in a
@R(q C h in a ;b)
dq China
@q C h in a
+
@R(q C h in a ;b)
db:
@b
Substituting these expressions into the de…nitions of the di¤erentials, and
using the equilibrium requirement dL = dR gives
@L q China
@R q China ; b
@R q China ; b
China
China
dq
=
dq
+
db:
@q China
@q China
@b
Collecting terms gives
!
@L q China
@R q China ; b
@R q China ; b
China
dq
=
db:
@q China
@q China
@b
Rearranging this equation gives
dq China
=
db
@R(q C h in a ;b)
@b
@L(q C h in a )
@R(q C h in a ;b)
@q C h in a
@q C h in a
Using rules of di¤erentiation, we have
@L(q C h in a )
@R(q C h in a ;b)
=
1
<
0,
=
C
h
in
a
@q
@q C h in a
and
@R(q C h in a ;b)
@b
=
1
(1+b)2
18
345
10
> 0:
1
1+b
>0
q China
< 0:
346
APPENDIX B. COMPARATIVE STATICS
Substituting the expressions for the partial derivatives into the previous equation and simplifying yields equation 2.5.
Appendix C
More complicated
perturbations
For example, suppose that T = 9, and consider the e¤ect of a particular
perturbation that leaves the candidate trajectory unchanged in period 0 and
from periods 5 to 9. Suppose that this perturbation increases (relative to the
candidate) extraction by 0.2 units in period 1, by 1 unit in period 2, makes
no change to extraction in period 3, and then reduces period-4 extraction by
1.2 units. The cumulative change in extraction is 0:2 + 1 1:2 = 0, so the
period-5 stock under this perturbation is unchanged.1
Figure C.1 shows the graph of an arbitrary sequence of extraction levels,
the solid step function, a particular candidate. The dashed lines show the
extraction levels under the perturbation, described above, to this candidate.
The solid and dashed lines are coincident except in periods 1, 2, and 4,
when the perturbation is non-zero. Figure C.2 shows the graph of the stock
under the candidate shown in Figure C.1 (the solid lines) and under the
perturbation (the dashed lines). Notice that the increased extraction in
periods 1 and 2, under the perturbation, lead to lower stocks in periods 2, 3
and 4, even though the extraction in period 4 is the same under the candidate
and under the extraction.
This somewhat complicated perturbation can be broken down into three
1
The changes in this perturbation are non-in…nitesimal, whereas in deriving the optimality condition we consider only in…nitesimal changes. The procedure described here
can be made rigorous by multiplying each of the non-in…nitesimal changes by ", and letting
" become arbitrarily small but positive. With this modi…cation, each of the changes is
in…nitesimal, and the ratios of changes are as in the example.
347
348
APPENDIX C. MORE COMPLICATED PERTURBATIONS
y
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
t
Figure C.1: Solid lines show a candidate, consisting of levels of extraction
from periods 0 through 8. The dashes lines show the perturbations relative
to this candidate. The perturbation changes extraction only in periods 1,
2 and 4. The perturbation is 0 during periods when the solid and dashed
lines are coincident.
x
18
16
14
12
10
8
6
4
2
0
0
1
2
3
4
5
6
7
8
9
10
11
t
Figure C.2: The solid lines show the stock level under the candidate, beginning with stock 17.5. The dashed lines show the stock under the perturbation
illustrated in Figure C.1
349
one-step perturbations:
Perturbation 1: increase extraction by 0.2 units in period 1
and make an o¤setting reduction of 0.2 units in period 2.
Perturbation 2: increase extraction by 1.2 unit in period 2
and make an o¤setting reduction of 1.2 units in period 3.
Perturbation 3: increase extraction by 1.2 units in period 3
and make an o¤setting reduction of 1.2 units in period 4.
An exercise asks the reader to con…rm that the trajectories under the original
perturbation and under the sequence of three one-step perturbations are
exactly the same in periods 1, 2, 3 and 4. Thus, the e¤ects on the payo¤
of the original perturbation and of the three one-step perturbations are also
the same.
If our candidate is optimal, then any one of these one-step perturbations
must have zero …rst order e¤ect on the payo¤. Therefore, the e¤ect of all of
the changes combined, i.e. the e¤ect of the original three-step perturbation
also has a zero …rst order e¤ect. This example illustrates the fact that in
order to check for the optimality of the candidate, it is not necessary to
consider these complicated multi-step perturbations. It is enough to check
that there are no one-step perturbations that yield a …rst order change in
the payo¤.
350
APPENDIX C. MORE COMPLICATED PERTURBATIONS
Appendix D
The full solution in the
backstop model
An example shows how to use equation ?? to …nd the full solution, using
methods that follow Chapter 7.
Let C = 2, b = 5, = 0:9, D (p) = 10 p. For these values, (b C) +
C = 0:9 (5 2) + 2 = 4: 7. The …rst column of Table 1 shows calendar
time working backwards, from T to T 4. Corresponding to each of these
times, we use equation ?? to …nd the bounds on price in that period. With
these bounds, we …nd the range of possible extraction levels during that
period. For example, for t = T 3, the left side of equation ?? is 3 1 b +
3 1
3
C (1
) = 4: 187 and the right side of this equation is (1
)C +b 3 =
3:9683. These numbers appear in the price column corresponding to t =
T 3. The numbers in the extraction column use the demand function and
the prices in the adjacent column, and the fact that yT > 0 (by the de…nition
of T ).
The second column in Table 2, “cumulative extraction”, uses the information in the “extraction” column of the previous table to determine the
range of cumulative extraction, in the current and subsequent periods, and
denotes this value as x0 . For example, at t = T 1 there are two periods
remaining, periods T and T 1. The most that could possibly be extracted
in these two periods is 5:3 + 5:57 = 10: 87 and the least that could be consumed is 5:3 + 0 = 5: 3. (Because periods T extraction is strictly positive,
not 0, the cumulative extraction over T 1 and T must be strictly greater
than 0, not greater than or equal to 0.) We obtain the entries in the other
rows using the same logic. If, for example, the initial stock of the resource
351
352APPENDIX D. THE FULL SOLUTION IN THE BACKSTOP MODEL
satis…es 16:683 < x0 22:7174, then it takes four periods (T = 3) to exhaust
the resource. If the initial stock exceeds 28:94323, we would add additional
rows to the tables until reaching the number of periods needed to exhaust
the initial stock.
time (t = T n) price
extraction
t=T
5 pT 4:7
0 < y 5:3
t=T 1
4:7 pT 1 4: 43
5:3 y 5:57
t=T 2
4: 43 pT 2 4: 187
5:57 y 5:813
t=T 3
4: 187 pT 3 3: 968 3
5:813 y 6: 031 7
t=T 4
3: 968 3 pT 4 3: 771 47 6: 031 7 y 6: 228 53
Table 8.1 Inequalities on price, extraction, and cumulative extraction
during the remaining life of the resource, for di¤erent values of t.
Using Table 2, we can obtain the complete solution to the problem once
we know the initial stock. If x0 = 15 then Table 2 implies that the resource
is exhausted in three periods (T = 2). Choosing y0 as our unknown variable,
and using the inverse demand function, we have p0 C = 10 y0 2.
time (t = T n) “cumulative extraction”
t=T
0 < x0 5:3
t=T 1
5:3 < x0 10:87
t=T 2
10:87 < x0 < 16:683
t=T 3
16:683 < x0 22:7174
t=T 4
22:7174 < x0 28: 943 23
Table 8.2 Inequalities on price, extraction, and cumulative extraction during
the remaining life of the resource, for di¤erent values of t.
Using the Euler equation we have
10
y0
2 = 0:9 (10
y1
2) ) y1 = 8
8
y0
:
0:9
Using the Euler equation again and then substitution, we have
y2 = 8
8
y1
=8
0:9
8
8 80:9y0
:
0:9
353
Adding extraction in the three periods and setting the sum equal to the
initial stock, gives
y0 + 8
8 y0
0:9
+ 8
8
(8
8 y0
0:9
0:9
)
= 15
) y0 = 5:31 ) y1 = 5:01 ) y2 = 4:48:
The natural resource supplies the entire market during periods 0 and 1;
during these periods, supply exceeds 5 and the price is below 5, so it is
uneconomical to use the backstop. In period 2 the resource supplies 4.48
units and the backstop produces 5 4:48 = 0:52 units; the price equals b = 5.
354APPENDIX D. THE FULL SOLUTION IN THE BACKSTOP MODEL
Appendix E
Geometry of the Theory of the
Second Best
The TOSB merits a more general discussion, because of its importance in
helping us think clearly about the second best world that we inhabit. We
begin by explaining the meaning of level sets and the convexity of a set. We
then use this terminology to discuss the TOSB.
Figure ?? shows an example of a topological map. These kinds of maps
contain curves that identify the coordinates that have the same altitude.
They provide information that helps hikers not to get lost. The terrain is
relatively steep in a region where topological curves are close to each other,
and the terrain is relatively ‡at where curves are far from each other.
For this example, the origin (0; 0) is the peak of the mountain. Curves
that are closer to the origin have higher altitudes. For example, every point
on the dashed curve is higher than any point on the solid curve. Each of
these curves is a “level set”, which means that all points on a particular
curve have the same altitude. We will call the set of points that are at
the same altitude or higher than an arbitrary point as the “higher than”set
corresponding to that point. For example, every point on or inside the solid
curve is in the “higher than” set corresponding to point a, a point on the
solid curve. A set is said to be convex if the following is true: a chord drawn
between any two points in that set lies in the set. The “higher than” sets
in this …gure are not convex. For example, points a and b both lie in the
“higher than” set corresponding to point a, but the chord between points a
and b does not lie in this set.
Figure E.2 shows two indi¤erence curves for beer and pornography. Points
355
356APPENDIX E. GEOMETRY OF THE THEORY OF THE SECOND BEST
N
a
b
E
W
S
Figure E.1: A topological map. Curves closer to the center correspond to
points in space that have higher altitude: the altitude is higher at every point
on the dashed curve than the solid curve. The “higher than” sets are not
convex: a chord that connects two points on a level set, such as points a and
b does not lie in the “higher than”sets corresponding to one or both of those
sets.
on the dashed curve are preferred to points below that curve. The “preferred” set corresponding to any point consists of all combinations of beer
and pornography that yield the same or higher level of utility than that point.
In this …gure, the preferred sets are convex. For example, every point on
the chord between any two points on a level set, such as points a and b, give
the same or higher level of utility than either of those two points.
In Figure ??, the axes represent distances east or north of the top of the
mountain. In Figure E.2, the axes represent amounts of two commodities.
In Figures ?? and E.4, the axes represent the magnitude of two distortions.
The term “distortion” has both its common meaning, and a slightly more
technical meaning in the setting here. A distortion is anything that keeps
a market outcome from being e¢ cient, including market failures such as
imperfect competition and externalities. A rather mechanical interpretation
of “distortion”is helpful in some contexts.
We can think of an e¢ cient outcome as consisting of the solution to …rst
order conditions to a social planner’s problem. If the outcome consists of
n di¤erent variables, we typically require n of these …rst order conditions in
order to determine the e¢ cient outcome. For example, suppose (unlike in
357
a
c
pornography
Figure E.2: Points on the dashed indi¤erence curve are preferred to points
on the solid indi¤erence curve. The “preferred”sets are convex.
our pollution + monopoly example) it is possible to determine both output,
y, and emissions, e, independently, and that social welfare, W (q; e) depends
on these two variables. We hold all other variables in the economy …xed.
The …rst order conditions to the social planner’s problem are
@W (q ; e )
@W (q ; e )
= 0 and
= 0.
@q
@e
Suppose that a market outcome (q m ; em ) 6= (q ; e ) is not e¢ cient, i.e. it
does not solve the planner’s problem. In this case, in general it is possible
to write
@W (q m ; em )
@W (q m ; em )
= d1 and
= d2 ;
@q
@e
with either or both d1 , d2 , not equal to 0. The numbers d1 and d2 are the
manifestations of the distortions in the economy. If policy changes resulted
in d1 = d2 = 0, then the market outcome and the social planner’s solution
are identical: (q m ; em ) = (q ; e ). A policy that causes either d1 or d2 to
move closer to 0 reduces (in a mechanical sense) a distortion. The question
is: When does such a reduction also increase social welfare?
Figures ?? and E.4 show the level sets in two scenarios. Here, a point
in the plane shows the two distortions, d1 and d2 , and a level set gives the
358APPENDIX E. GEOMETRY OF THE THEORY OF THE SECOND BEST
d2
5
4
3
d2 = 2
2A
1
-5
-4
-3
-2
-1
1
2
3
4
-1
5
d1
-2
-3
-4
-5
Figure E.3: Points on level sets closer to the origen have higher welfare. The
level sets are convex.
d2
5
4
3
A
1
-5
-4
-3
-2
d2 =2
2
-1
a
1
-1
2
3
4
5
d1
-2
-3
-4
-5
Figure E.4: Points on level sets closer to the origen have higher welfare. The
preferred sets are convex
combinations of these distortions that have the same level of welfare. In
both of these …gures, the level sets are convex, and in both of them the origin
(where d1 = 0 = d2 ) is the point of highest welfare, the e¢ cient point. The
convexity assumption eliminates a slew of possibilities that complicate the
exposition. For example, convexity of these sets means that the economy
does not have increasing returns to scale. We put aside those kinds of
complications, and now discuss the signi…cance of Figures ?? and E.4.
Figure ?? is symmetric with respect to the axes, and Figure E.4 is asymmetric with respect to the axes. These two …gures correspond to di¤erent
economic stories. The symmetry of Figure ?? means that if we pick any level
of d2 , and hold that level …xed, and then ask “Given this …xed level of d2 ,
what level of d1 is optimal for the economy?”, the answer is always “d1 = 0 is
359
optimal for the economy”. Figure ?? illustrates this fact for d2 = 2. Given
that we are not able to alter this value of d2 , the policy problem is to …nd the
level of d1 that gives the highest level of welfare. The mathematical problem
is to …nd the highest level set (the one closest to the origin) that has d2 = 2.
The solution to this problem requires that the level set is tangent to the ‡at
line at d2 = 2, as at point A. At this point of tangency, d1 = 0. The reader
can verify that if we had …xed d2 at any level, and asked what level of d1 ,
maximizes welfare„the answer would always be to set d1 = 0. Similarly, if
we had …xed d1 at an arbitrary value, and asked what value of d2 maximizes
welfare, the answer is d2 = 0.
There may be two policies that potentially correct the two distortions.
Policy #1 can correct distortion #1, and Policy #2 can correct distortion
#2. Continuing our monopoly + pollution example, we can think of Policy
#1 as being a pollution tax (or subsidy, if negative) and Policy #2 as being
an antitrust policy that forces …rms to price closer to marginal cost than they
would like. In this setting, larger values of d2 mean that a …rm exercises more
market power, i.e. moves away from producing where price equals marginal
cost, towards producing where marginal revenue equals marginal cost. Here,
d1 corresponds to the di¤erence between the tax that …rms face and social
cost of emissions (6 in our earlier example). For that example, d1 = 0 means
that the tax equals the social cost of emissions; d2 > 0 means that the tax
is greater than the social cost of emissions, and d2 < 0 means that the tax
(possibly a subsidy) is less than the social cost of emissions.
If the pollution and the monopoly power occur in di¤erent sectors, then
(apart from general equilibrium complications) the two distortions are unrelated, as in Figure ??. In that case, the two policies are neither substitutes
nor complements: the …xed level of one distortion does not a¤ect the optimal
level of the other policy. In this circumstance, we can study the two policy
problems in isolation. We can try to reduce the monopoly distortion without
worrying about the magnitude of the pollution externality, and we can try to
reduce the pollution distortion without worrying about the degree of market
power.
The situation in Figure E.4 is substantively di¤erent, and is closer to the
monopoly + pollution example that we used above, where the pollution and
monopoly power both arise in the same sector. Here, if we are told that
d2 is …xed at any value other than 0, then it is optimal to set d1 to a level
other than 0. For example, we can think of the restriction d2 = 2 as the
statement that there is a given level of market power that we are not able to
360APPENDIX E. GEOMETRY OF THE THEORY OF THE SECOND BEST
in‡uence. Given that level, Figure E.4 shows that the tangency occurs at
point A, so the optimal level of d1 is negative. This means that the optimal
pollution tax, in the presence of market power, is less than the social cost of
pollution. Possibly, the optimal tax is a subsidy ( < 0) as in Figure 10.1;
the important conclusion is only that the optimal tax is less than the social
cost of pollution. (Whether the tax is positive or negative depends on the
relative magnitudes of the two distortions.)
A bit of experimentation with Figure E.4 shows that if we lower the market power distortion, (shift down the line at d2 = 2 to a lower level) the
optimal subsidy falls. If the market power distortion vanishes, then it is optimal to set the tax equal to the social cost of emissions, so that the pollution
externality also vanishes (d1 = 0). Figure E.4, together with our de…nition
of d1 and d2 imply that (for the monopoly + pollution example) environmental taxes and antitrust policy are complements: stricter enforcement of
antitrust policy (lowering d2 ) implies that the optimal pollution tax increases
(causing d1 to move toward 0). Of course, the …rst best policy combination
eliminates both distortions, putting the economy at the origin of the …gure,
the top of the mountain, i.e. the highest level of social welfare.
Appendix F
Algebra of taxes
This appendix collects technical details for the chapter on taxes.
F.1
Algebraic veri…cation of tax equivalence
Denote the producer price as ps (for supply) and the consumer price as pc
(for consumption) and write the “market price”as p. If consumers pay the
tax, the prices are ps = p and pc = ps + (producers receive the market
price and consumers pay this price plus the tax). If producers pay the tax,
ps = p
and pc = p (consumers pay the market price and producers receive
this price minus the tax). We want to con…rm that tax-inclusive prices are
the same regardless of who directly pays the tax.
If consumers pay the tax, the supply equal demand condition is
S(p) = D(p + ):
(F.1)
Let p ( ) be the (unique) price that solves this equation; this is the equilibrium producer price (a function of ) when consumers pay the tax: p (0)
is the equilibrium price when = 0. Because consumers (directly) pay the
tax, the price producers receive (the “supply price”) equals p ( ) and the
price consumers pay equals p ( ) + .
If, instead, producers directly pay the tax, the equilibrium condition is
S(p
Substitute p = p +
) = D(p):
into this equation to write equation (F:2) as
S(p ) = D(p + ):
361
(F.2)
362
APPENDIX F. ALGEBRA OF TAXES
The last equation reproduces equation (F:1) evaluated at p = p ( ), the
unique solution to that equation. Thus, the two equations (F:1) and (F:2)
lead to the same producer and consumer prices. Appendix ?? contains
further details.
F.2
The open economy
For a closed economy, domestic supply equals domestic demand: there is
no trade. For an open economy, domestic supply does not equal domestic
demand: the di¤erence equals the amount imported or exported. The tax
equivalence result above holds in a closed economy, where all sources of supply or demand are subject to the tax. An example shows that when the
economy is open, tax equivalence does not hold. With international trade,
domestically produced supply does not equal domestic demand. With trade,
the optimal policy depends on whether the environmental damage is associated with consumption or production. There, it does matter whether a
consumer or producer tax is used.
First consider the case where the economy is closed. Suppose that domestic demand is q d = 10 p and domestic supply equals q s = bp. Column
2 of Table 1 shows that the consumer and producer tax incidence does not
depend on which agent, consumers or producers, directly pays the tax. The
incidences in this column are calculated using the following steps:
1. Calculate the equilibrium price in the absence of tax by setting the
untaxed supply equal to the untaxed demand.
2. Calculate the equilibrium consumer price and producer price when one
of these agents directly pays the tax, by setting the (taxed) demand
equal to the (taxed) supply.
3. Use the tax-inclusive consumer and producer price for the two cases
(where one agent or the other directly pays the tax) and the zero-tax
price to calculate the incidences.
The third column of the table shows that in the open economy, the incidences do depend on which (domestic) agent directly pays the tax. For
example, to calculate the incidences in the open economy when consumers
directly pay the tax (regardless of the source of supply), we use essentially
F.2. THE OPEN ECONOMY
363
the same steps as above. The market clearing condition in the absence of a
tax is 10 p = (b + c) p. We solve this to …nd the zero-tax price. If consumers pay the tax, the market clearing condition is 10 (p + ) = bp + cp,
where now we understand that p is the price received by both domestic and
foreign …rms, and p + t is the consumer tax-inclusive price. We solve this
equation to …nd the equilibrium producer and consumer prices. Using the
formula for tax incidence, we obtain the expressions in the third row and
third column of Table 1. An exercise asks readers to use this procedure to
derive the formulae in the table.
closed economy
open economy
q d = 10 p
q s = bp
q d = 10 p
q s = bp and q s;f or = cp
market clearing
10
consumers
pay tax
domestic
producers
pay tax
p = bp
consumer
incidence
b
100%
1+b
market clearing
10
p = (b + c) p
consumer
incidence
producer
incidence
consumer
incidence
b
100%
1+b
producer
incidence
consumer
incidence
producer
incidence
1
100%
1+b
producer
incidence
1
100%
1+b
b+c
100%
1+b+c
1
100%
1+b+c
b
100%
1+b+c
1+c
100%
1+b+c
Table 1 consumer and producer tax incidence in closed and open economy
In an open economy, domestic supply does not equal to domestic demand.
Taxing consumers causes the market demand function to shift in, lowering the
price that both producers face and increasing the consumer’s tax-inclusive
price. Taxing only domestic supply causes the domestic supply function to
shift in, increasing the consumer price, decreasing the domestic tax-inclusive
price, and shifting supply from domestic to foreign producers. Under the
consumer tax, both the domestic and foreign producers receive the same
price. Under the (domestic) producer tax, consumers and foreign producers
face the same price, and domestic producers receive a lower after-tax price.
This example shows that although in a closed economy producer and
consumer taxes are equivalent, the two taxes are not equivalent in an open
economy. Hereafter, we return to the closed economy model.
Exercise:
364
APPENDIX F. ALGEBRA OF TAXES
1. Derive the tax incidences shown in Table 1.
F.3
Approximating tax incidence
In the closed economy, it does not matter whether consumers or producers are charged the tax. Suppose that consumers are charged the tax, so
that the equilibrium condition is equation F.1. This equation expresses the
equilibrium price as an implicit function of the tax: as the tax changes, the
equilibrium price p changes. The consumer incidence (expressed as a fraction
instead of a percent) equals
p (0)
p
+1=
+ 1:
(F.3)
0
The numerator on the left side equals the change in price that consumers
pay. We obtain the …rst equality by simplifying, i.e. using the fact that
= 1, and subtracting 0 from the denominator. We subtract 0 in order to
emphasize that both the numerator and the denominator are changes: the
numerator is the change in price, in moving from a 0 tax to a non-zero tax,
and the denominator is the change in the tax,
0. We obtain the second
equality by using the “delta notation”:
means “change in”. The next
step requires a formula for an approximation of p , which we obtain using
the fact that the derivative ddp is approximately equal to p .
Treating p = p( ) (i.e. price as a function of the tax –and dropping the
“*” to simplify notation) we can di¤erentiate both sides of the equilibrium
condition F.1 to write
dD(p + ) dp
dS(p) dp
=
+1 :
dp d
dpc
d
p ( )+
p (0)
=
p ( )
Divide both sides by the equilibrium quantity, using S = D, and multiply by
the equilibrium price p to write
dS(p) p dp
dD(p + ) p
=
dp S d
dpc
D
dp
+1 :
d
(F.4)
Because we are considering an approximation for small , we evaluate equation F.4 at = 0. Using the de…nitions in equation 11.1, and evaluating
equation F.4 at = 0, we rewrite that equation as
dp
=
d
dp
+1 :
d
F.3. APPROXIMATING TAX INCIDENCE
We can solve this equation for
dp
d
365
to obtain
dp
=
d
+
(F.5)
:
This equation shows the derivative of the equilibrium price with respect to
the tax, evaluated at a 0 tax. Notice that ddp < 0: the tax, although paid by
consumers, reduces the equilibrium price that producers receive.
We use the fact that
dp
p
d
and equations F.3 and F.5 to write the expression for the consumer incidence
as
p
dp
+1
+1=1
=
:
d
+
+
The tax incidence for producers equals
reduction in producer price
:
level of (unit) tax
Initially the tax is 0, so the level of the tax (once it is imposed) is
The producer tax incidence is
producer incidence:
p
+
0=
.
:
This expression involves
p rather than p because the de…nition of the
producer incidence involves the “price reduction”, not the “price change”. If
the price change is, for example, 3, then the reduction is 3.
Exercise:
1. Suppose that consumers are charged the tax, as above. Let the demand
function be D (p) = p with
> 1 and suppose that …rms have
constant marginal cost, c. Evaluate the consumer and producer tax
incidence under the monopoly, as a function of . Compare with
the consumer and producer tax incidence under competition, with the
same demand and cost functions. Hint Mimic the derivation above,
replacing marginal revenue with price.
366
F.4
APPENDIX F. ALGEBRA OF TAXES
Approximating deadweight loss and tax
revenue
The graphical representation of the deadweight cost of the tax is the area of
the triangle in Figure 11.1. We want an approximation for this area that
is valid even if the supply and demand functions are not linear. For this
purpose, we use the formula for the area of a triangle, 21 base height.
The height of the triangle is simply the tax, (see ). Denote the base of this
triangle as q, the change in quantity demanded. We have (by multiplying
and dividing)
q=
q
p=
p
q
p
p
qp
pq
=
q
p
p
:
(F.6)
Equation 11.2 and the de…nition of the supply elasticity imply, respectively,
the following two equations
p
dp
=
d
+
qp
pq
and
.
Inserting these formulae into equation F.6 gives the approximation
q
:
+ p
q
(F.7)
Here we used the fact that
=
0 = , because we are taking the
approximation in the neighborhood of a zero tax. This result and the formula
for the area of a triangle produces equation 11.4.
It is worth taking a moment to consider the units in which deadweight
loss is measured. Suppose that we measure quantity in kilograms and price
in dollars per kilogram. In this case, the tax is also measured in dollars per
kilogram. The units of the ratio pq are therefore
kilograms
dollars
kilogram
(kilograms)2
=
.
dollars
The units of pq t2 are therefore
(kilograms)2
dollars
dollars
kilograms
2
= dollars.
F.4. APPROXIMATING DEADWEIGHT LOSS AND TAX REVENUE367
The elasticities are already “unit free”–that is the reason that we use them.
Therefore, deadweight loss is measured in dollars (or whatever monetary unit
we begin with). This result makes sense: we want a measure of value in
monetary units, not, for example, in units of kilograms.
The tax revenue equals (q
q), where q is the pretax quantity sold
and q is the change in the quantity due to the tax. Using equation F.7,
we have
q
tax revenue: (q
q)
q
:
(F.8)
+ p
Dividing the approximation of deadweight loss by this formula and cancelling
produces equation 11.5.
We can use the approximation of tax revenue and of DWL to illustrate
the second “rule”for optimal tax policy: use a large tax base. Suppose that
the economy consists of two commodities, and for each of these (by choice of
units) + pq = 1 and in the absence of taxes, output (= consumption) is (by
assumption) q = 10. The government has to raise $20 of tax revenue. If
the government raises this revenue by taxing only one commodity, it needs
to solve
(10
1 ) = 20 )
1
(1) (2: 76)2 = 3:82:
2
2:76 ) DW L
In contrast, if the government uses a broader tax basis, it solves
2 (10
1 ) = 20 )
1:27 ) DW L
2
1
(1) (1:27)2 = 1:61:
2
For this example, doubling the tax basis leads to a 57% decrease in the total
DWL.
368
APPENDIX F. ALGEBRA OF TAXES
Appendix G
Continuous versus discrete
time
G.1
The continuous time limit
Consider the growth equation with a harvest rule y(x). Recall that a harvest
rule determines the level of harvest as a function of the stock, x. Equation
14.3 shows the discrete time dynamics for two particular harvest rules. Later
we encounter other harvest rules, so here we use the general formulation y (x).
With this harvest rule, the next-period and current-period stocks are related
according to
xt+1
xt = F (xt )
y (xt ) = [F (xt )
y (xt )] 1:
(G.1)
Multiplying F (xt ) y (xt ) by 1, as in the last equality, obviously does not
change the quantity.
We have to measure time in speci…c units. For example, it is meaningful
to say “That was three years ago,”but we would never say “That was three
ago.” We choose the unit of time to equal one year; this choice is arbitrary:
we could have chosen a unit to equal one second or one century.
There is no reason (apart from convenience) to assume that the length
of a period equals one unit of time. For some models, where events unfold
over long periods of time, it might make sense to choose the length of a
period equal to a decade; prominent climate policy models use that time
scale. For other models, where events occur quickly, we might want the
length of a period to equal one day or even an hour or less; some models
involving …nancial transactions use that kind of time scale.
369
370
APPENDIX G. CONTINUOUS VERSUS DISCRETE TIME
We use the symbol
to represent the length of a period. If a period
lasts for a decade, then
= 10 (because our unit of time is one year). If
1
a period lasts for a day, then
= 364
.
In order for our model to show
explicitly the length of a period, we can replace the number 1 wherever it
appears in equation ?? (including in the subscripts) with ; the equation
becomes
xt+
xt = [F (xt ; ) y (xt ; )] )
(G.2)
xt+
xt
= F (xt ; ) y (xt ; ) :
By introducing the parameter , we have made a subtle change in the de…nition of F (xt ) and y (xt ); these are now rates, i.e. they give growth and
harvest per unit of time (one year). To take into account this change, we
1
replace F (xt ) and y (xt ) with F (xt ; ) and y (xt ; ). If
= 364
and if
the growth per year is 0:8, and harvest per year is 0:2 , then the amount
0:8
1
) equals 364
of growth and harvest over one period (one day, for
= 364
0:2
and 364
, respectively. (The change over one day is xt+
xt = (0:8 0:2)
1
3
= (0:8 0:2) 364 = 1: 65 10 .)
Now that we explicitly recognize that growth and harvest are rates, we
no longer need to require that y
x. For example, suppose that x = 40
and y = 60. It is not possible to extract 60 units of biomass if the stock
of biomass equals only 40. However, it is certainly possible to harvest at
1
an annual rate of 60 for a short period of time. If
= 364
and y = 60,
60
then after 10 periods (= 10 days) we have extracted 364 10 = 1: 65 units of
biomass. In general, if is su¢ ciently small, then the annual harvest rate
y can be arbitrarily large without violating the non-negativity constraint on
the stock of …sh.
The last line of equation G.2 shows the ratio xt+ xt , equal to the change
1
in stock per change in time. With
= 364
, this ratio is the change in the
xt+
xt
stock per day. As ! 0, the ratio
converges to a time derivative.
We de…ne
F (x) = lim F (x; ) and y (x) = lim y (x; ) :
!0
!0
With this de…nition, the continuous time limit of the last line of equation
G.2 is
dxt
= F (xt ) y (xt ) :
(G.3)
dt
Equation G.2 is a di¤erence equation, and equation G.3 is a di¤erential
equation. They both describe how x changes over time. Hereafter, when
G.2. STABILITY IN DISCRETE TIME
371
studying stability (and for some other purposes that arise in Chapter 16) we
“jump to continuous time”. Readers interested in knowing how to think
about stability in the discrete time setting should consult Appendix G.
It is important to be clear about the relation between equations G.2 and
G.3. By construction, they have the same steady states. In other respects,
however, they may contain very di¤erent information. For example, suppose
that we have two …sh stocks; the …rst grows according to equation G.2 and
the second grows according to equation G.3. We start both stocks at the
same level, and let each evolve in the manner described by its equation of
motion. Would these two stocks evolve in the same way, i.e. would the
time-graphs of their trajectories look similar? In general, the answer is
“no”. If we want to change the length of a period (e.g. from
= 1
1
to
= 10;000;000;000 ), while keeping the trajectory qualitatively unchanged,
we have to re-calibrate the functions F (xt ) and y (xt ). However, if
is
su¢ ciently small, then trajectories arising from the continuous and discrete
time models are qualitatively similar, at least in the neighborhood of a steady
state.
G.2
Stability in discrete time
A trajectory that moves away from an unstable steady state, or toward a
stable steady state, might be changing monotonically (i.e. the path continues to increase or to decrease) or it might cycle (i.e. the path increases, then
decreases, then increases, etc.). Table 1 includes these four possibilities; we
discuss the entries of the table below. Other kinds of much more complicated behavior, including chaos, can arise in even these simple discrete time
models. We select parameter values to exclude that complexity. Roughly,
the complexity can arise in a discrete time setting because there the stock
can change by an appreciable amount from one period to another. The
stock can therefore jump from an interval in which the dynamics “behave
in a certain way” to another region where the dynamics “behave in a very
di¤erent way”. This type of complexity does not arise in continuous time
settings where the stock moves smoothly over time.
monotonic trajectory
cyclical trajectory
0
0
stable
1 < F (x1 ) y (x1 ) < 0
2 < F 0 (x1 ) y 0 (x1 ) <
0
0
0
unstable F (x1 ) y (x1 ) > 0
F (x1 ) y 0 (x1 ) < 2
1
372
APPENDIX G. CONTINUOUS VERSUS DISCRETE TIME
Table 14.1: Types of steady states
Given particular functional forms and parameter values, we can test numerically whether a steady state is stable. Consider the example where
harvest is a constant fraction of the stock, yt = xt . In this case, we can
write the second line of equation 14.3 as
xt
K
xt+1 = xt + xt 1
xt .
(G.4)
We call x0 , the level of the stock at the initial time, t = 0, the “initial
condition”. Given a numerical value of x0 and numerical values of ; ,
and K, we can solve this equation to …nd x1 . Plugging that value into the
equation we …nd x2 . Proceeding in this manner, we can determine whether
the successive values of xt approach either the high or the low steady state
(33.33 and 0 in the example above). Although this approach is easy enough
to implement, it does not provide any insight into the problem.
There is simple test for determining the stability and the adjustment
paths (monotonic or cyclical) of steady states, without actually computing
the trajectory. We apply this method in other settings, so it is worth stating
it fairly generally. Let F (x) be a growth function (e.g., logistic) and y (x)
be a particular harvest rule (e.g. a constant fraction of the stock). Given
these function, any solution to 0 = F (x) y (x) is a steady state. Given
a particular steady state (recall that there may be more than one), we can
compute the derivatives F 0 (x1 ) and y 0 (x1 ). The entries in Table 1 show
the combinations of these derivatives that result in the four types of behavior.
G.2.1
Explanation of Table 1
We appeared to pull Table 1 out of the air, but a simple graphical argument
explains it. We require one additional piece of notation. For a growth
function xt+1 xt = F (xt ) and the harvest rule y (xt ), de…ne the stock in
the next period, given xt , as g (xt )
xt+1 = xt + F (xt )
y (xt )
g (xt ) :
(G.5)
For example, the right side of equation G.4 shows the function g under the
logistic growth function and the proportional harvest rule.
Figure G.1 provides the basis for understanding Table 1. This …gure
appears to be very simple, but it requires a bit of thought to use. The
G.2. STABILITY IN DISCRETE TIME
373
Figure G.1: A stable steady state. The curve g (x) = x + F (x) y.takes the
current value of the stock, x, into the next period value. The steady state
is the intersection of g (x) and the 45o line.
horizontal axis shows the value of the stock in the “current”period, denoted
x. As above, the stock x changes according to the equation xt+1 = g (xt ) :
The heavy line in Figure G.1 is the graph of g and the 45o is the graph of
x. We use the 45o line and the graph of g for three purposes: to identify the
steady states, to determine whether each of these is stable, and also to trace
out the evolution of the stock, given an initial condition.
At the intersection of the 45o line and the graph of g (x), x = g (x).
Therefore, the intersection of the 45o line and the graph of g identi…es the
steady states. Figure G.1 shows a single point of intersection, i.e., a single
steady state, merely to keep the …gure simple. In general there could be
several steady states, as the …sh examples illustrate. In Figure G.1 it is
apparent that the derivative of g at the steady state is positive, and that in
addition, the graph of g approaches the steady state from above the 45o line,
which has a slope of 1. Therefore, 0 < g 0 (x1 ) < 1 in this …gure. Restating
this inequality using the functions F (x) and y (x), the inequality becomes
0 < 1 + F 0 (x) y 0 (x) < 1. Subtracting 1 from all parts of this inequality
produces the equivalent relation 1 < F 0 (x) y 0 (x) < 0. The upper left
entry of Table 1 states that the corresponding steady state is stable, and
that paths beginning in the neighborhood of the steady state approach it
374
APPENDIX G. CONTINUOUS VERSUS DISCRETE TIME
monotonically.
How can we reach this conclusion based only on the slope of g shown in
Figure G.1? The answer requires understanding how to interpret this …gure.
The numerical values shown on the horizontal and vertical axis of the …gure
make it easy explain this interpretation, but otherwise have no signi…cance.
Suppose that the stock in a particular period is x = 2:3, the value of the
horizontal coordinate of point a in the …gure. By de…nition of the growth
function, the stock in the next period equals g (2:3), which is the vertical
coordinate of the point above a, on the curve g. In the …gure, g (2:3) = 2:7.
The 45o line enables us to use the above information to carry on to the
next period. This line is like a mirror that re‡ects the stock from one period
to the next. It “re‡ects” a value on the vertical axis onto the horizontal
axis. The reader should draw a horizontal line from the vertical axis, at 2:7.
Where this line hits the 45o line (at point b), drop a line straight down to
the x axis. This procedure gives us the new “current period” value of the
stock, 2:7 on the horizontal axis. Given this value, we can …nd the new “next
period”stock, the point on the heavy curve above point b. The …gure shows
this value as 3:9. Proceeding in this way, we can trace out the sequence of
stocks.
Once the reader has performed these steps a few times, it becomes apparent that the procedure can be simpli…ed. We need an initial condition,
x = 2:3, in the example above, to get started. With this initial condition,
we identify the point a on the 45o curve. We then draw an arrow toward
the curve g; where the arrow hits g we draw another arrow to the 45o line.
Where that arrow hits the 45o line we draw another arrow to g. The horizontal coordinates of the points on the 45o line, a; b, c... in Figure G.1 are the
successive values of x. We can repeat this step inde…nitely, but after a few
steps it is apparent that the trajectory converges toward the steady state.
We can see that for this …gure, with a stock that begins at 2.3 (or at any
other value su¢ ciently close to the steady state), the sequence approaches
the steady state monotonically, as Table 1 states.
Exercises ask the reader to construct analogous …gures that establish the
other three entries in Table 1.
G.2.2
Applying the stability test
We begin with a graphical example that helps to develop intuition about
steady states and stability. We then proceed more formally, using the in-
G.2. STABILITY IN DISCRETE TIME
375
formation in Table 1. Figure 14.5 shows the logistic growth function with
K = 50 and = 0:03, and two harvest rules: the constant harvest y = 0:3
and the harvest rule that takes 1% of the stock in each period, y = 0:01x.
Consider …rst the constant extraction rule, shown by the dotted line. This
line intersects the growth function at two stock levels, x = 14 and x = 36.
Both of these stock levels are steady states (when y = 0:3) but the …rst is
unstable and the second is stable. At a stock slightly below x = 14, the
harvest exceeds the natural growth, F , so the stock is falling. Similarly, at
a stock slightly above x = 14, the natural growth, F , exceeds harvest, so the
stock is increasing. Therefore, at any point close to, but not exactly equal
to, the low steady state (x = 14) the stock trajectory moves away from the
low steady state. The low steady state therefore appears to be (and in fact
is) unstable. By similar reasoning, the high steady state is stable.
The upward sloping dashed line is the graph of the harvest rule y = 0:01x.
The line y = 0:01x intersects the curve F at two points, x = 0 and x = 33.
These are both steady states for this harvest rule. At a stock close to, but
slightly greater than x = 0, the natural growth exceeds the harvest, so the
stock increases; it moves away from x = 0, so that steady state is unstable.
At a point close to but slightly smaller than x = 33, the natural growth
exceeds the harvest, so the stock is increasing. At a point close to but
slightly larger than x = 33, the harvest exceeds the natural growth rate, so
the stock is falling. Thus, for stock levels close to the steady state x = 33,
the stock trajectory approaches x = 33. This steady state appears to be
(and in fact is) stable.
The de…nition of stability of a steady state involves the behavior of the
stock trajectory in the neighborhood of (i.e., close to) the steady state. In
simple cases, such as represented by …gure 14.5, we can make a stronger
statement. For the constant harvest rule, y = 0:3, beginning with any stock
level x > 14, the stock approaches the high steady state, x = 36. For initial
stock levels x < 14, the harvest always exceeds growth, so the stock is driven
to 0, which is also a stable steady state. The dynamics with the proportional
harvest rule are even simpler. Beginning with any stock greater than 0, the
stock approaches the stable steady state.
In discussing Figure 14.5, we did not use information in Table 1. Given
the simplicity of this …gure, the reader may wonder why we bothered with
that Table at all. The answer is that some of the conclusions that we drew
above, concerning Figure 14.5, happen to be correct because of our choice
of parameter values, but cannot actually be deduced from the …gure alone.
376
APPENDIX G. CONTINUOUS VERSUS DISCRETE TIME
Footnote 1, below, and an exercise provide more insight into the problem
with relying exclusively on …gures; this problem arises because the stock can
“jump around” in a discrete time setting. Graphs are useful for providing
intuition, but to be con…dent of the characteristics of a steady state, we need
the formality contained in Table 1.
The remainder of this section uses the information in Table 1 to determine
the characteristics of steady states. We …rst use Table 1 to …nd a restriction
on the growth function that insures that the carrying capacity, K, is a stable
steady state in the absence of harvest. If harvest is always zero, y (x) = 0 =
.
y 0 (x) for all x. The derivative of F (x) = x 1 Kx is F 0 (x) = 1 2x
K
2K
0
Evaluating this derivative at x = K gives F (K) = 1 K =
. Thus,
F 0 (x1 ) y 0 (x1 ) =
, for the steady state x1 = K. Substituting this
value into Table 1 gives:
monotonic trajectory cyclical trajectory
stable
1> >0
2> >1
unstable
< 0 (XX)
>2
Table 14.2: characteristics of the steady state x = K for
logistic growth with zero harvest
Table 2 states that, for the logistic growth function with zero harvest,
the carrying capacity (the steady state x = K) is stable if 2 >
> 0.
For 1 > > 0, paths beginning in the neighborhood of the steady state
approach the steady state monotonically. For 2 > > 1, paths beginning
in the neighborhood of the steady state approach the steady state cyclically.
A negative growth rate, < 0, is not sensible in this model, so we can ignore
that case; hence, the “XX”in that entry. The remaining possibility is > 2,
for which paths beginning near the steady state diverge (move away from it)
cyclically. Hereafter, we assume that 0 < < 2, so the carrying capacity is
a stable steady state.1
1
As increases, the complexity of oscillations increases. Beyond some point ( slightly
larger than 2.5) the oscillations become “chaotic”: from almost all initial conditions, they
are very irregular (they do not repeat in a …nite interval of time). Moreover, the paths are
very sensitive to the initial condition. This situation is the example, alluded to in the text,
for why inspection of graphs is insu¢ cient in some cases to determine the characteristics
of the steady state. If we graphed the logistic growth function for > 2:5, it would have
the same general appearance as the logistic growth function in Figure 14.5. Reasoning
as we did in the text, we might conclude that the x = K is stable in this case. But
the information contained in Table 2 tells us that it is in fact not stable. The point to
remember is that …gures can be deceptive in these discrete time models.
G.2. STABILITY IN DISCRETE TIME
377
We can use the same procedure to check for stability of steady states under
the proportional harvest rule, y = x, still using the logistic growth function.
We begin by …nding the steady states, by setting harvest-inclusive growth
equal to zero, F (x)
x = 0, and solving for the stock, x. There are two
). The stock must be non-negative,
steady states, x = 0 and x = K (
so the second steady state exists if and only if > . The derivative of y(x)
.
is y 0 (x) = , a constant, and the derivative of F (x) is F 0 (x) =
1 2x
K
Table 1 shows that the stability test depends on a comparison of these two
derivatives. The derivative of F is not a constant, so we need to evaluate it
at both of the steady states:
F 0 (0) =
F0
K
(
) =
1
2K (
K
)
=2
:
(G.6)
We now consider each of two steady states in turn. Using Table 1 and
the …rst line of equation G.6 produces Table 3:
monotonic trajectory cyclical trajectory
stable
1<
<0
2<
< 1
unstable
>0
< 2
Table 14.3: characteristics of the steady state x = 0 for
logistic growth with harvest proportional to the stock.
Using Table 1, the second line of equation G.6 and the fact that 2
produces Table 4:
monotonic trajectory cyclical trajectory
stable
1<
<0
2<
< 1
unstable
> 0 (XX)
< 2 (XX)
Table 14.4: characteristics of the interior steady state x = K (
for logistic growth with harvest proportional to the stock.
=
)
We place an “XX” by the two entries in the last low row of the table,
because the associated parameter values are excluded by assumption. The
fact that we are considering the interior steady state in this table implies
that this steady state exists, i.e. it is positive. That requires < , ruling
out the lower left entry. We previously assumed that < 2 (in order to
378
APPENDIX G. CONTINUOUS VERSUS DISCRETE TIME
Figure G.2: An example of an unstable steady state
exclude chaotic behavior in the absence of harvest), and because
0, it
follows that
> 2, thus ruling out the lower right entry in Table 4.
Consequently, if the carrying capacity is a stable steady state in the absence
of harvest, and if an interior steady state exists under proportional harvest,
that steady state must be stable for this model. Using this fact and Table 3,
we conclude that if an interior steady state exists, then x = 0 is an unstable
steady state. If an interior steady state does not exist (i.e. if > ), then
x = 0 is a stable steady state; it is also the only steady state here.
G.3
Exercises
1. Figure G.2 di¤ers from Figure G.1 in that the heavy curve (which
represents the growth function g (xt )) cuts the 45o line “from below”,
> 1. Pick a
as we move left to right. In particular, dg(x)
dx jsteady state
point somewhat close to the steady state (the intersection of the heavy
curve and the 45o line). Use the steps described in the discussion of
Figure G.1 to trace out the stock dynamics (the evolution of x), over
a couple of periods. This trajectory illustrates the lower left entry of
Table 1.
2. Draw two other …gures to illustrate the remaining entries of Table 1.
Each …gure should have a 45o line and a graph of g that intersects the
line. In one …gure, 1 < g 0 (x1 ) < 0 and in the other g 0 (x1 ) < 1.
Trace out the dynamics for trajectories that begin near the respective
G.3. EXERCISES
379
steady states.
3. Suppose that the harvest rate is a constant, y (for stocks x
y);
in addition y is such that there exist two interior steady states. The
growth function is logistic, F (x) = x 1 Kx . (a) Find an inequality
(depending on the parameters of the growth function) that y must
satisfy in order for there to be two interior steady states. (Hint: draw
a picture.) (b) For a y that satis…es this inequality, determine which, if
either steady state is stable. (Hint: Write the analog to equation G.4
with constant harvest:
xt
y g (xt ) :
xt+1 = xt + xt 1
K
Write the derivative dg(x)
for this expression. Use dg(x)
, expressed as a
dx
dx
function of x (not of y) to …nd a condition on x that must be satis…ed in
order for a steady state to be stable. If your calculations are accurate,
you will discover that for a constant harvest rule, a steady state is
stable for x above a critical level, and the steady state is unstable for x
below this critical level. You need to …nd this critical level. A graph
is a big help here.)
4. Figure G.3 shows a standard growth function with a very odd looking
non-monotonic harvest rule, the dashed curve. The key feature of
this harvest rule is that its slope is very large in absolute value in the
neighborhood of the two larger steady states. (a) Use this information,
together with Table 1, to determine which, if any steady states are
stable. (b) Use the information in Figure G.3 to construct a …gure
that contains the graph of g (x) = x + F (x) g (x) and a 45o line. You
do not have the functional form of y (x), so your graph of g cannot be
“accurate”. The objective is to that it has characteristics compatible
with Figure G.3. For example, your graph should have three points of
intersection between g and the 45o line. Pick an initial condition (a
points on the x axis) near each of the steady states and use your …gure
to trace out the stock trajectory.
5. Figure G.4 shows a growth function, F (x). (a) Sketch the …gure on a
piece of paper and identify: (i) the carrying capacity, (ii) the maximum
sustainable yield, and (iii) the stock level associated with the maximum
sustainable yield. (b) The dashed curve represents a particular harvest
380
APPENDIX G. CONTINUOUS VERSUS DISCRETE TIME
Figure G.3: A standard growth function, the solid curve, and an highly nonmonotonic harvest rule, the dashed line. The harvest rule is very steep in
the neighborhood of the two largest steady steady states.
Figure G.4: See Exercise 5. The solid curve is a growth function and the
dashed curve is a harvest rule.
rule. Identify each steady state(s) associated with this harvest rule
using the letters A; B; :::. For each of these steady states, what (if
anything) can you say about its stability? Provide a brief explanation.
Appendix H
Bioeconomic equilibrium
Objectives and skills
Understand the meaning of a “bioeconomic equilibrium”.
For simple cost and growth functions, be able to write down and graph
the “steady state supply function”.
Using the steady state supply function and a demand function, identify
the steady states.
Using the …ction of the “Walrasian auctioneer”, identify which steady
states are stable and which are unstable.
Here we o¤er a slightly di¤erent perspective on the open access steady
state. The zero pro…t condition, and the production function in equation
15.2, imply
C
w
= ;
(H.1)
0 = (pqx w) E ) x =
pq
p
where the last equality uses the de…nition wq = C. The production function
15.2 and the steady state condition under logistic growth (harvest equal
growth) give
x
y = qEx = x 1
:
(H.2)
K
Substituting equation H.1 into equation H.2 gives the steady state supply
function
C
C
y=
1
:
(H.3)
p
Kp
381
382
APPENDIX H. BIOECONOMIC EQUILIBRIUM
price
3
A
2
1
B
D
0
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
harvest
Figure H.1: The backward bending steady state supply function (solid) and
a linear demand curve.
The steady state supply function for harvest gives the harvest level, as
a function of the price, that is consistent with a steady state stock of the
…sh and zero pro…ts in the …shery. Figure ?? shows the supply function for
parameter values K = 50; = 0:03, and C = 5 (as in our previous examples).
The notable feature is that this supply function bends backwards. For prices
1
= 0:2, supply increases with price, and is very price-elastic (‡at). At
p < Kq
higher prices, equilibrium supply decreases with the higher price. At a
given stock, the higher price induces greater harvest, but the higher harvest
reduces the steady state stock. The net e¤ect in the steady state is that a
higher price reduces equilibrium supply over the backward bending part of
the curve.
The dashed curve in the …gure shows the linear demand curve, p = 3:5
10y. There are three “bioeconomic equilibria”, combinations of output and
price where supply equals demand and the stock is in a steady state. The
equilibria A and D, corresponding to high price and low harvest, and low
price and high harvest, are stable; the intermediate equilibrium is unstable,
just as we saw in Section 15.2. An exercise asks the reader to show how
the set of steady states changes as demand becomes more elastic.
The argument that determines stability or instability is a bit di¤erent in
the setting in this section, compared to the above. We introduce a …ctitious
“Walrasian auctioneer”. This auctioneer calls out an arbitrary price. If, at
that price, supply equals demand, the auctioneer has found an equilibrium.
383
However, if at the price the auctioneer has called out, demand exceeds supply,
then the auctioneer raises the price, in an e¤ort to bring supply and demand
into equilibrium.
Suppose that this auctioneer calls out a price slightly above the p coordinate of point B; at this slightly higher price, demand exceeds supply. In
an e¤ort to balance supply and demand, the auctioneer increases the price.
The higher price initially elicits greater supply, but that reduces future stock,
creating an even larger divergence between steady state supply and demand.
The auctioneer continues to raise the price, toward the p coordinate of point
A, where at last steady state supply equals demand. Thus, a price that
begins slightly above the p coordinate of point B moves away from that
point, so this price is unstable. Parallel arguments show that the prices
corresponding to points A and D are stable steady states.
Section 15.2 uses a graph in the stock-harvest plane, and here we use
a graph in the harvest-price plane. Both perspectives have something to
o¤er. The backward bending supply function provides a useful contrast
between familiar static models and the dynamic …sh model, but it is a bit
harder to interpret, and requires introducing the …ction of the Walrasian
auctioneer. The graphs using the harvest rule and the growth function
follow more directly from the dynamic model, and have a direct physical
interpretation.
384
APPENDIX H. BIOECONOMIC EQUILIBRIUM
Appendix I
The Euler equation for the sole
owner …shery
Objectives and skills:
Apply the perturbation method to obtain the Euler equation.
The constraints of the sole owner’s optimization problem are x 0 and
y 0. The …rst constraint implies that yt xt : it is not possible to harvest
more than the total stock of …sh. We …nd the necessary condition where
both the stock and the harvest are strictly positive, i.e. along an interior
optimum.
As in Chapter 5, we determine the necessary conditions for optimality
using the perturbation method. Natural growth, the function F (xt ), complicates the problem, but the logic is the same. We begin with a feasible
“candidate trajectory” of harvest, y0 , y1 , y2 :::; and the corresponding stock
sequence, x0 , x1 , x2 ... . We obtain the condition that must be satis…ed if no
perturbation increases the present discounted value of the payo¤. As before,
we achieve this objective by considering a particular one-step perturbation:
one that increases harvest in an arbitrary period (t) by ", and makes an o¤setting change in harvest in the next period (t+1) in order to keep unchanged
the stock in the subsequent period (t + 2). As explained in Chapter 5, we
can build a more complicated perturbation from a series of these one-step
perturbations. Thus, for the purpose of obtaining the necessary conditions,
it su¢ ces to consider this one-step perturbation.
We begin by …nding the o¤setting change needed in period t + 1, in order
to keep unchanged (relative to the unperturbed candidate) the stock in period
385
386APPENDIX I. THE EULER EQUATION FOR THE SOLE OWNER FISHERY
t + 2. If we increase harvest in period t by ", the stock in period t + 1 is
xt+1 = xt + F (xt )
(yt + ") :
This relation implies
dxt+1
= 1:
(I.1)
d"
The increased harvest in period t reduces the stock in period t + 1 from xt+1
(the level under the candidate trajectory) to xt+1 ". Under the candidate
trajectory, we plan to harvest yt+1 in period t + 1. The o¤setting change
in yt+1 , required by the fact that we increased yt by ", and by our insistence
that xt+2 be unchanged, is ("). The notation (") emphasizes that , the
change in yt+1 , depends on, ", the change in yt . Using the growth function,
we have
xt+2 = (xt+1 ") + F (xt+1 ") (yt+1 + (")) :
(I.2)
We require that the total change in xt+2 – including the changes in both
periods t and t + 1, be zero, i.e.,
dxt+2
= 0.
d"
Using this condition, and di¤erentiating both sides of equation I.2 implies
dxt+2
d"
=0=
1+
dF (xt+1 ) dxt+1
dxt+1
d"
d
d"
=
1+
d
d"
)
dF (xt+1 )
dxt+1
1
dF (xt+1 )
dxt+1
d
d"
=0)
(I.3)
:
The …rst line di¤erentiates both sides of equation I.2 with respect to ", using
the chain rule. We use equation I.1 to eliminate dxd"t+1 to obtain the equation
after the …rst “)”, and rearrange that equation to obtain the second line.
The last line of equation I.3 provides the …rst piece of information: the
required reduction in yt+1 , given that we increase yt by ", and given that we
want to keep xt+2 unchanged. A one unit increase in yt leads to a one unit
t+1 )
direct reduction in xt+1 and dFdx(xt+1
units loss in growth; the loss in growth
a¤ects xt+2 . Therefore, if we increase yt by " units, we must decrease yt+1 by
t+1 )
1 + dFdx(xt+1
" units, to o¤set both the direct e¤ect on xt+2 and the indirect
e¤ect that occurs via the reduced growth.
387
Under the candidate trajectory, periods t and t + 1 current value contribution to the total payo¤ is t times
pt
C
xt
yt +
pt+1
C
xt+1
yt+1 :
Under the perturbation, these two periods contribute
g (") =
pt
C
xt
(yt + ") +
pt+1
C
xt+1
"
t
times
yt+1 (yt+1 + ) :
If the candidate is optimal, then a perturbation must lead to a zero …rst
order change in the gain function. Using the product and the quotient rules,
we have
h
i
dg(")
C
C
C
d
y + pt+1 xt+1 d" =
= p t xt +
d" j"=0
x2t+1 t+1
h
i
dF (xt+1 )
C
C
C
p t xt +
y
pt+1 xt+1
1 + dxt+1
=0)
x2t+1 t+1
i
h
C
t+1 )
pt xCt =
pt+1 xt+1
1 + dFdx(xt+1
+ x2C yt+1
t+1
The last line is the Euler Equation 16.4.
388APPENDIX I. THE EULER EQUATION FOR THE SOLE OWNER FISHERY
Appendix J
Dynamics of the sole owner
…shery
This appendix derives the continuous time analog of the Euler equation, and
then derives the di¤erential equation for harvest.
J.1
Derivation of equation 17.2
We could have begun with a continuous time problem, and derived equation
17.2 directly. That approach is mathematically preferable, but it requires
methods that we have not discussed. Therefore, we proceed mechanically,
taking equation 16.7 as our starting point, and showing how to manipulate
it to produce the continuous time analog, equation 17.2.
We need to have in mind a unit of time. Because we want the discrete
time and the continuous time models to be “close to each other”, the unit
of time should be small. As in Chapter 14.3, we begin with the model in
which one period equals one unit of time, and then divide that period into
smaller subperiods. Mechanically, we do this by replacing the number 1, the
length of a period, with . We also need to rewrite the discount factor as
1
Using the growth equation
= 1+1 r instead of = 1+r
xt+
xt = [F (xt )
we replace F (xt ) and y (xt ) with F (xt )
dF (xt+1 )
and
dxt+1
389
y (xt )] ;
and y (xt ) . Thus, the terms
C
yt+1
x2t+1
390
APPENDIX J. DYNAMICS OF THE SOLE OWNER FISHERY
in equation 16.7 become, respectively,
dF (xt+ )
dxt+
and
C
x2t+
yt+
:
These substitutions mean that instead of having the length of a period
be one unit of time (e.g. one minute), we now have the length of a period be
units of time. With these substitutions, we rewrite equation 16.7 as
Rt =
1
Rt+
1+ r
1+
dF (xt+ )
dxt+
+
C
x2t+
yt+
(J.1)
:
Subtract Rt+ from both sides of equation J.1 and collect terms on the right
side to rewrite the result as
Rt Rt+ =
1
1+ r
1 Rt+ +
1
Rt+
1+ r
Divide both sides of this equation by
(Rt+
Rt )
=
1
1+ r
1
Rt+ +
dF (xt+ )
dxt+
(Rt+
!0
Rt )
=
1
Rt+
1+ r
dRt
and lim
!0
dt
C
x2t+
yt+
dF (xt+ )
dxt+
1
1+ r
1
+
C
x2t+
yt+
! 0, using
= lim
!0
1 1
r
1+ r
=
r
to write
dRt
=
dt
Multiplying through by
J.2
Rt r
:
to write
Now take the limit of both sides of this equation as
lim
+
dF (xt )
dxt
+
C
yt :
x2t
1 gives equation 17.2.
The di¤erential equation for harvest
Chapter 17.3.2 uses the di¤erential equation for the sole owner harvest, dy
=
dt
H (x; y). This appendix explains how we obtain this equation, i.e., how we
obtain the function H (x; y). The procedure uses the equations of motion
:
J.2. THE DIFFERENTIAL EQUATION FOR HARVEST
for the stock, the rent, and the de…nition of rent.
equations:
dx
= F (x) y
dt
dRt
dt
dF (xt )
dxt
= Rt r
Rt = p (yt )
391
We repeat these three
C
y
x2t t
(J.2)
C
xt
The …rst equation is merely the constraint of the problem, i.e. it is “data”
(given to us). The second equation is the Euler equation, expressed in terms
of rent. Both of these equations are the continuous time versions of the
discrete time model. The third equation is the de…nition of rent.
Because the third equation holds identically with respect to time (i.e., it
holds at every instant of time), we can di¤erentiate it with respect to time
to write
h
i
C
d p (yt ) xt
dyt
C dxt
dRt
=
= p0 (yt )
+ 2 yt
dt
dt
dt
xt dt
We can use the …rst two lines of equation J.2 to eliminate
write
Rt r
dF (xt )
dxt
dyt
C
C
y = p0 (yt )
+ 2 yt (F (x)
2 t
xt
dt
xt
dRt
dt
and
dxt
,
dt
to
y) :
We can now use the third line of equation J.2 to eliminate Rt , to write
p (yt )
C
xt
r
Solving this equation for
p (yt )
dy
=
dt
C
xt
dyt
C
C
y = p0 (yt )
+ 2 yt (F (x)
2 t
xt
dt
xt
dF (xt )
dxt
r
dyt
dt
y) :
gives
dF (xt )
dxt
C
y
x2t t
p0 (yt )
C
y
x2t t
(F (x)
y)
= H (xt ; yt )
The middle expression is a function of only x and y, and the model parameters. We de…ne this expression as the function H (xt ; yt ).
This function looks complicated, but for the functional forms and the
parameter values in our example, it simpli…es to
dy
500:0yx + 80:0yx2
= 0:000 02
dt
28:0x2 + 195:0x + 750:0
= H (xt ; yt ) :
x
392
APPENDIX J. DYNAMICS OF THE SOLE OWNER FISHERY
Chapter 17.3.2 uses this function to construct the y isocline, the set of points
= 0. This isocline is given by
where dy
dt
dy
=0)y=
dt
J.3
1
10x (0:08x
0:5)
0:28x2 + 1: 95x + 7: 5 :
Finding the full solution
In order to …nd the solution to the optimization problem, needed to construct
the dotted curve in Figure 17.2, we take the ratio of the di¤erential equation
for y and the di¤erential equation for x, to obtain a new di¤erential equation,
showing how y changes with changes in x:
dy
dt
dx
dt
=
dy
H (xt yt )
=
:
dx
F (x) y
The solution to this equation is a function giving the optimal harvest as a
function of the stock, the optimal “harvest rule”. Denote this function as
y = Y (x). Figure 17.2 shows the graph of Y (x), the dotted curve. Solving
the di¤erential equation to obtain the optimal harvest rule, Y (x), requires
a “boundary condition”, giving the value of y at some value of x. Our
boundary condition is given by the steady state, denoted (x1 ; y1 ). We
calculate the steady state by …nding the intersection of the x and the y
isoclines. Our boundary condition is y (x1 ) = y1 .
Some numerical algorithms encounter a problem in solving the di¤erential
dy
vanish at the
equation, because both the numerator and denominator of dx
steady state, making the ratio an indeterminate form. This problem is easily
resolved, but involves methods beyond the scope of this book. We can
linearize our original non-linear system and use the eigenvector associated
with the stable eigenvalue to replace the boundary condition (the steady
state) with a point on the “stable” eigenvector. We have to (numerically)
solve the resulting initial value problem twice, once beginning with a point
slightly below the steady state, and then beginning with a point slightly
above the steady state. Figure 17.2 shows the …rst of these two parts of the
solution.
The dotted curve in Figure 17.2 shows the graph of the optimal harvest
level, as a function of the stock, x. Harvest is positive only when the stock is
above 3. As the stock increases over time, the harvest rises. The harvest is
J.3. FINDING THE FULL SOLUTION
393
nearly constant, once the stock reaches about 20 or 25. The stock continues
to grow to its steady state, and the harvest changes very little.
394
APPENDIX J. DYNAMICS OF THE SOLE OWNER FISHERY
Afterword
The back matter often includes one or more of an index, an afterword, acknowledgements, a bibliography, a colophon, or any other similar item. In
the back matter, chapters do not produce a chapter number, but they are
entered in the table of contents. If you are not using anything in the back
matter, you can delete the back matter TeX …eld and everything that follows
it.
395
396
AFTERWORD
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