Economics of Global Warming: How to Decentralize the Social Optimum in

Economics of Global Warming:
How to Decentralize the Social Optimum in
the Presence of an Oil Rent ?
Institut de l’Environnement
Project ’Dynamiques des Territoires et D´eveloppement Durable’
Universit´e de Corse - Campus Caraman - 7, avenue Jean Nicoli
In this paper, I study the optimal climate policy in the presence of an oil rent.
Previous papers show that, in the long run, because of the Hotelling’s rent, the
optimal tax must decrease. However, as the full depletion of the polluting non
renewable resource may be suboptimal, it is an equilibrium condition for the resource
market, if the price is strictly positive. In this situation, an optimal tax would
imply the disappearance of the hotelling rent. Thus, the interaction between global
warming and oil depletion can be broken, and the tax can be increasing.
Key words: Hotelling rent, Global warming, Pigovian tax, Economic growth
JEL Classification: H30, Q32, Q40
This paper studies the optimal climate policy in the presence of an oil rent.
This study is done by combining the features of the cumulative pollution
models and those of the non renewable resources models.
Since the seminal papers by Keeler et al. (1971), Plourde (1972) and Forster
(1973), a large literature in environmental economics has focused on the problem of cumulative pollution. Concerns about climate change explain this inEmail address: [email protected] (Antoine BELGODERE).
The author is grateful to Sjak Smulders, Flora Bellone, Dominique Prunetti and
Sauveur Giannoni for their precious comments.
terest, as greenhouse effect is caused by the accumulation of carbon in the
atmosphere. Optimal control models indicate that an increasing Pigovian tax
should be raised on the use of polluting goods, such as fossil fuels. This result
matches the intuition. As pollution accumulates, the marginal external cost
of polluting is increasing over time, for a given amount of polluting emissions.
In a situation where pollution is a flow, the Pigovian tax, at each period,
must equate the marginal external cost for the period. But when pollution
accumulates into a stock, the Pigovian tax must take into account not only
the current external cost of pollution, but also the actual value of the future
external cost caused by accumulation. In these conditions, the shadow price
of the pollution grows over time, and asymptotically reaches a steady-state
value. This result implies that, if an optimal policy were to be enforced by an
international agreement on climate change, one should expect the price of the
fossil fuels, including a Pigovian tax, to grow over time.
Actually, such kind of expectations already exists in the public’s mind, but
the main cause is not the global warming. Fossil fuels are a non-renewable
resource. Hotelling (1931) showed that, at the market equilibrium, the price
of a nonrenewable resource must grow according to a rate that equates the
discount rate. This result directly follows from 1) a non-arbitrage condition
and 2) the fact that, by definition, the stock of a non-renewable resource can
only decrease. A stock of non-renewable resource is an asset, and must yield
the same rate of return to its owner than any other asset. The only way a
stock of non-renewable resource can yield the same return as other assets, is
that its price grows at a rate that equates this common rate of return. Indeed,
the limits of the oil stocks, that are rather low from a human scale, may allow
to predict an increasing trend of oil prices for the next decades 2 .
As cumulative pollution models advocate an increasing Pigovian tax on fossil
fuels while nonrenewable resources models predict an increase in the price of
fossil fuels, one can ask if a model integrating both features could make the
market equilibrium closer to the social optimum. Indeed, the increasing rent
can be expected to have the same effect as a Pigovian tax, i.e. to incite users
of fossil fuels to save energy and to adopt renewable energy sources, which are
assumed not to increase the carbon concentration in the atmosphere.
This idea has led to a debate in the early 90’s, engaged by Sinclair (1992)
with an article whose provocative title was: ’High does nothing and rising is
worse: Carbon taxes should be kept declining to cut harmful emissions’. As
the title indicates, his main results are 1) the optimal path of a tax is defined
only by its rate of growth, and not by its initial condition that is neutral,
and 2) this rate of growth must be negative 3 . As we will see in more detail,
Although recent pressures on oil prices can be explained, to a large extend, by
short-run phenomena.
3 Schulze (1973), Hoel (1978) and Forster (1980) already dealt with the interaction
point 1) is explained by the nil price-elasticity of the resource supply. Point
2) is due to the fact that, under several assumptions, the resource stock is
asymptotically exhausted. In this situation, the only effect of an environmental policy is to transfer pollution from the present to the future. At least two
arguments advocate this transfer. First, future damages are perceived to be
less harmful than present ones because future utility is discounted. Second,
while the use of the resource becomes asymptotically nil, the stock of pollution decreases after reaching a top, because of the natural assimilation. Thus,
the external cost of carbon emissions becomes also asymptotically nil. Several
articles were published in response to Sinclair’s (Ulph and Ulph (1994), Sinclair (1994), Hoel and Kverndokk (1996), Farzin and Tahvonen (1996), among
others). Tahvonen (1997) provides a major contribution to this debate. He establishes, in a very general framework, that 11 different kinds of paths can be
obtained, depending on the assumptions about the use of backstop technology
and the decay process. Schou (2002) takes into account the incentive to invest
in knowledge, and finds that no tax is needed to implement the optimum. But
Grimaud and Roug´e (2005) show that Schou’s result holds for a Cobb-Douglas
utility specification, but is not always valid for more general specifications.
This article adds to the debate by describing a regime where the climate policy
makes the Hotelling’s rent to disappear. The argument presented here is very
similar to the intuitive one made on Berck and Roberts (1996) (p68), but is
presented in a complete macroeconomic framework. With no Hotelling’s rent,
the link between oil depletion and climate change is broken. In this context, an
increasing Pigovian tax is needed. This result is based on two key assumptions:
1) the presence of a backstop technology, and 2) there is some irreversibility
in pollution.
The paper is organized as follows: section 2 presents the model and characterize
the optimal path, section 3 investigates in which way the social optimum can
be decentralized, section 4 discusses some issues involved in this model and
then section 5 concludes.
Presentation of the model and analysis of the optimal path
The model
The representative consumer of the economy has an instantaneous utility function U which depends on consumption C(t) and on the stock of pollution M(t),
between oil depletion and cumulative pollution, but did not focus on the specific
questions discussed here.
with 4
> 0 and
< 0. U is assumed to be strictly concave.
M is assumed to follow
M˙ = ζE
with ζ ∈ [0; 1] being a parameter and E the rate of resource extraction. It is
assumed here that the stock of pollution is expressed in the same unit as the
stock of resource. In the case of global warming, this unit can be the ton of
carbon. Every unit of resource extracted will be released into the nature. A
fraction (1 − ζ) is absorbed by the ecosystem (afforestation, oceans,...), and a
fraction ζ accumulates into the pollution stock.
Notice that no decay process is assumed here. This choice needs to be justified. Our point concerns the cases where full asymptotic depletion of a nonrenewable resource is not optimal. With a linear decay function, which is often
assumed in the literature, and a non-renewable polluting resource, the stock
of pollution will be asymptotically nil. So will be its external cost. In this
situation, only an excessive private cost of extraction could explain a partial resource depletion. More complex decay processes, such as that used by
Forster (1975) and Withagen and Toman (1998), become nil after a given
threshold attained by the stock of pollution. In Farzin and Tahvonen (1996),
carbon accumulates into 4 stocks, 3 of which have a strictly positive linear
rate of decay, while the fourth has a nil rate of decay. These features account
for the limits in the absorptive capacity of the environment. In this case, the
stock of pollution can, on an optimal path, satisfy lim M (t) > 0 . However,
for simplicity, I will assume that decay is nil.
The stock of non-renewable resource evolves according to:
S˙ = −E
With this formulation, for M(0) and S(0) given, M(t) can be expressed as a
function of S(t).
M (t) = M (0) + ζ [S(0) − S(t)]
It follows that M can be replaced by S in the utility function 5 . We then have
> 0 and ∂∂SU2 < 0.
When no confusion is possible, time references are removed.
Similar modelings are used in Schulze (1973), Hoel (1978), Forster (1980) and
Sinclair (1994).
In these conditions, our model becomes a cake eating model with a stock effect,
close to Krautkraemer (1985).
The final output Y is produced using capital K(t) 6 and an energy input N(t).
Two energy sources are available. The first is the polluting non-renewable
resource E(t). The second is an infinitely but costly available non-polluting
resource A(t). So:
N (t) = E(t) + A(t)
Perfect substitution is assumed here 7 . This is justified by the fact that what
are principally expecting energy purchasers are joules. No distinction can be
made between joules produced by different energy sources. Of course, complementary services are also required, such as storage and transportation. The
substitution is imperfect when one takes into account these features of energy
demand. The differences between energy sources from the point of view of
these complementary services will be taken into account by the convex cost
function of the renewable energy. If oil and wind are not perfect substitutes
for a car driver, this is not because the joules produced by the wind are different from those produced by the oil. This is because the joules must be stored
and transported. To store and transport wind energy in the same way as oil,
one has to transform it into hydrogen, which is is costly. To reflect this, we
introduced q(A), the amount of final good devoted to produce A. We assume
q 0 (A) ≥ 0, q 0 (0) = 0, and q 00 (A) > 0.
> 0,
We assume ∂N
according to:
> 0 and the strict concavity of Y. The capital evolves
K˙ = Y − C − δK − q(A)
Where δ is a depreciation parameter.
Social optimum
Characterization of the optimal path
The social planner maximizes 0∞ U e−ρt dt, where ρ is the social discount rate,
under (1), (2) and E, C, A ≥ 0. C, E and A are the control variables. The
By seek of simplicity, we will not take into account any other input, such as labor.
Thus, K(t) must be thought of as an aggregate of all the inputs but the energy.
7 Such as in Tahvonen (1997) among others.
Hamiltonian associated with this problem is:
H = U − λE + µ [Y − C − δK − q(A)]
Where λ and µ are the co-state variables associated, respectively, with S and
K. The maximum principle 8 give the following conditions 9 :
UC = µ
YE ≤ λ/µ E ≥ 0 E (µYE − λ) = 0
YA = q 0 (A)
λ˙ = ρλ − US
µ˙ = µ(ρ + δ − YK )
lim e−ρt λS = 0
lim e−ρt µK = 0
The assumptions made about UC and q(A) allow to look for an interior solution
for C and A, but not for E. (3) states that the marginal utility of consumption
must equate the shadow price of the capital. In order to interpret (4) when
YE = λ/µ 10 , one can use (3) in order to write UC YE = λ. Along an optimal
path, UC YE is the marginal utility of the extractions. Indeed, if an extra unit of
resource is extracted, then the output will increase by YE . Along an optimal
path, this marginal output can be indifferently affected to consumption or
to the capital accumulation. If it is consumed, then the utility will increase
by UC YE . Thus, (4) states that the marginal utility of the extractions must
equate the shadow price of the stock of resource. (5) states that the marginal
productivity of the renewable resource must equate its marginal cost. (6) is the
modified Hotelling’s rule. It can be interpreted as a non-arbitrage condition.
An asset whose value is λ, must yield, in term of utility and from a social point
of view, the same as if this value was placed at a rate ρ. What it yields is US ,
the marginal utility of the stock of resource, which corresponds to the avoided
˙ which is the gain in capital. (7) is the classical Ramseypollution, plus λ,
Keynes rule. (8) and (9) are the transversality conditions. Eliminating the costate variables in (3)-(7) provides the two following equations, that describe
These conditions are both necessary and sufficient thanks to the conditions on U
and Y.
9 I denote by M the derivative of the variable M with respect to N.
10 That is to say when the non-negativity constraint on E is not binding.
the optimal path:
d + Yc = ρ −
d =ρ+δ−Y
where a letter with a hat is the growth rate of the variable. (10) and (11) are
benchmarks that will be compared to the equilibrium path. The elimination
of the co-state variables is necessary to make this comparison, because the
co-state variables are not identically equal to the corresponding prices in the
market equilibrium.
An analysis of the whole optimal path could be provided, probably with some
difficulty. Indeed, if the economy starts with a low stock of capital, then one
expects the marginal productivity of the resource to be low. But, if the economy also starts with a low stock of pollution (i.e. a hight S), then the marginal
external cost of using the resource is low. While the low stock of capital advocates a low rate of extraction, the low pollution stock allows a hight rate of
extraction. As the economy grows, one expects both stocks (i.e. capital and
pollution stock) to increase, which have an ambiguous effect on the evolution
of E(t). In particular, a path in which the non negativity constraint on E is
binding for a period, and then non binding for a latter period cannot be excluded. However, the main purpose of this paper, is to focus on the behavior of
the system when t tends toward infinity, because the difficulty to decentralize
the optimum with a tax arises from asymptotic conditions. That is why the
next sub-section deals with the steady state of the model.
Variables in steady state
In order to search the steady-state, we introduce the following specifications
of U, Y and q:
U (C, S) =
S 1−
C 1−σ
Y (K, E) = ΩK α (A + E)1−α
q(A) = 0.5βA2
Where σ > 0, > 0, α ∈]0; 1[, Λ > 0, Ω > 0 and β > 0 are parameters.
˙ = µ∗
˙ = K∗
˙ = 0. I note with an
I am looking for a steady-state, where λ∗
˙ = S∗
asterisk the steady-state values. The co-state variables are now reintroduced
for computational purpose. Simple calculations give:
E∗ = 0
A∗ = ψχ α−1
K∗ = ψχ α−1
Y ∗ = Ωψχ α−1
C∗ = ψχ α−1 ζ
ρ α−1
S∗ =
(1 − α)Ωχ α−1
ψχ ζ
µ∗ = ψχ α−1 ζ
λ∗ = ψχ α−1 ζ
Where ψ ≡
− 1
(1 − α)Ωχ α−1
> 0, χ ≡
> 0 and ζ ≡ Ω − δχ−1 −
> 0.
= ∞. It means
With this specification, S∗ > 0. That is because lim ∂U
S→0 ∂S
that there is a threshold in the pollution stock, from which pollution has
¯ = M (0) +
catastrophic consequences. It is only for simplicity that I chose M
S(0) as particular value for this threshold. The qualitative result of this model
would be the same with lim ∂U
= ∞ with φ ≥ 0.
From a comparative statics point of view, one can remark that S*, the asymptotic remaining oil stock, is negatively related to β and positively related to Λ.
The interpretation is rather trivial. The costlier are renewable resources, the
more oil will be used. The more weight is put on the environment in the utility
function, the less oil will be used. Surprisingly, A* does not depend on Λ. This
is because, on steady state, no oil is used, and then, using renewable resource
is not related, on the long run, to a trade off between clean and polluting
energy, but only to a cost-advantage analysis of the use of energy.
It can be shown 11 that the steady state is a saddle path for certain values of
the parameters, among which the set of reasonable values given in table 1 12 .
In this case, the variables tend asymptotically to their steady-state values. In
particular, I can state:
lim S = S∗ > 0
See appendix A for the demonstration.
See appendix B for a justification of these values.
3.22 × 10−11
Table 1
Parameters values
This equation is important for our purpose, as it states that the stock of
resource will not be fully exhausted along an optimal path. (12) will be compared to the equilibrium condition for the resource market in the next section.
Furthermore, transversality conditions (8) and (9) are fulfilled, because µ, λ,
S and K tend asymptotically to a constant steady-state value.
Implementation of the optimum
Agents’ behavior
The economy is composed of 3 representative agents, plus the social planner:
a representative household, a representative firm in the final good sector, and
representative firm that manages the stock of nonrenewable resource.
The household
The household
chooses a consumption path in order to maximize his intertemR
poral utility 0∞ U e−ρt dt. The constraint he faces is the evolution of its assets
X, given by X˙ = rX + T − C, where r is the interest rate, and T are transfers
from the government. X is a mix of equities in the firm of the final sector and
of equities of the firm that manages the stock of resource. As risk is absent
in this model, in the market equilibrium both assets yield the same rate of
return r. This is a standard Ramsey model, from which one gets:
d =ρ−r
The final good sector
In the final good sector, the firm maximizes its profit π = Y − (r + δ)K −
P τ E −q(A), where P is the price of the resource, the final good being taken as
the numeraire, and τ is an ad-valorem tax on extraction plus one. This yields:
YE = P τ ⇔ YcE = Pb + τb
YK − δ = r
YA = q 0 (A)
The tax revenue and the profit 13 are redistributed lump-sum to the consumer:
T = P E(τ − 1) + π, which does not affect the consumer’s behavior.
The resource sector
The stock owner maximizes the sum of the actual value of its future rent
P Ee−
under (1). As perfect competition prevails, P does not depend on E, from the
firm’s point of view. In equilibrium, the two following equalities hold:
P˙ = rP
lim e−
P (t)S(t) = 0
(17) is the Hotelling ’r-rule’, and (18) is the classical equilibrium condition
that states that the value of the exceeding demand must be nil. Indeed, from
anRinter-temporal point of view, the resource supply is S(0), and the demand
is 0∞ E(t)dt. Notice that
lim S(t) = S(0) −
So, lim S(t) that appears on (18) is the opposite of the excess demand.
So, the resource market will be in equilibrium if lim S(t) = 0 or if
lim e
P (t) = 0.
(17) and (18) lead to
P (0) lim S(t) = 0
(19) is respected only for lim S(t) = 0 or for P(0)=0.
The profit is not nil because of the strict convexity of q(A).
Optimal policy tool
The optimal ad-valorem tax
(13) and (15) are identical to (11). No economic policy is needed at this stage.
(13)-(17) imply:
d + Yc = ρ + τb
Comparing (10) to (20) indicates the rate of growth that must follow the
ad-valorem tax in order to decentralizes the optimum:
τb = −
At first glance, (21) seems to confirm Sinclair’s result of a decreasing tax, only
defined by its rate of growth and not by its level. But, actually, I did not prove
that an ad-valorem following (21) enables to decentralize the optimum. Such
a tax only ensures that the variables in the decentralized economy have the
same rates of growth that in the social optimum. But in order to be identical,
two paths need not only to have the same rates of growth, but also to have one
common point. In the context of the present model, the latter condition is not
respected. Indeed, the optimal steady-state value for S, (12) does not match
the equilibrium condition (19). If P (0) > 0, then they are not compatible,
because in the market equilibrium, it implies lim S(t) = 0 6= S∗. In these
conditions, the optimum is not decentralized. If P(0)=0, then it follows from
(14) and from our assumptions about the properties of Y(.) that the firm’s
demand for E(0) will be infinite. Of course, only S(0) can be supplied, and
then S(t) = 0∀t > 0, which does not correspond to the optimum. This result
partly depends on the modeling of the tax as an ad-valorem tax. With a tax
whose amount per unit is fixed by the social planner, the optimum can be
decentralized 14 .
The optimal per-unit tax
With a per-unit tax, if the price perceived by the stock owner is nil, the
price paid by the firm, including the tax, can be strictly positive. But this
does not affect the equilibrium conditions for the resource market. Thus, if
one considers a lump sum per-unit tax, and if this tax enables to impose
lim S = S∗, the price perceived by the owner of the stock of resource is nil all
Alternatively, a constant cost of extraction could be introduced.
over the planning horizon. Let’s call θ this tax per unit of resource. As P=0,
the profit maximization in the final good sector becomes:
YE = θ ⇔ YcE = θb
In these conditions, the optimum is decentralized if:
Indeed, (6), (7), (13), (15), (22) and (23) are identical to the benchmark
(10). As the tax equates the marginal productivity of the resource, and provided K(0) < K∗, it will be continuously rising, and reach asymptotically
∂F (K∗,0+A∗)
. Thus, the necessity to maintain a strictly positive stock in the
steady-state breaks the interaction between the cumulative pollution and the
resource depletion, when the policy tool is a tax. The problem then becomes a
classical model `a la Plourde. To understand this result, let’s transpose it into
a discrete, finite-time model. Assume that, for some environmental reason, the
regulator imposes that the stock is not fully depleted at t*, which is the end
of the planning horizon. Then, E(t∗) < S(t∗), so P (t∗) = 0. In this situation,
what can be an equilibrium price at t*-1 ? Obviously, P (t ∗ −1) = 0, because
nobody will invest in an asset whose value will be nil at the next period. By
recurrence, one understands that P (t) = 0 for every t if E(t∗) < S(t∗).
The optimal subsidy
The representative household assumption does not directly allow to adopt a
distributive point of view, as Amundsen and Schob (1999) do in an international trade framework. However, one can guess that if the owners of the
resource stock are different from the rest of the population, the distributive
impact of the tax is important. This is especially true if these owners earn the
main part of their income from the resource exploitation, such as OPEC countries do. In the Amundsen and Schobs’ paper, the rent is partly transferred
to the other countries by the environmental policy. In the framework of the
present model, this rent would be fully transferred. However, another policy
tool can be used to implement the optimum, with an opposite distributive
effect: a subsidy on the remaining resource stock.
A subsidy s is paid at each point of time to the firm of the resource sector for
each unit of the stock of resource. In this context, the equilibrium conditions
in the resource market become:
rP = P˙ + s ⇔ Pb = r −
and (18), which is unchanged.
(24) is the modified Hotelling rule. Now, the investment in the resource stock
yields not only P˙ , but also s. Thus, P will grow, on the equilibrium, slower
than without subsidy. (13) is not modified by the introduction of the subsidy.
Now, T can be negative 15 . The subsidy is financed by a lump-sum tax. As no
more tax is assumed, (14) gives:
YcE = Pb
(13), (15), (24) and (25) give:
d + Yc = ρ −
that is identical to the benchmark (0.9) if:
Of course, such a subsidy will create an incentive to find new resource stocks,
that is not taken into account in the present framework, where the size of the
stock is exogenously given.
Has global Warming killed the Hotelling’s rent ?
If one takes the Hotelling rule in the strict sense of an increase of the resource
price at the interest rate, then this rule is not compatible with the observed
long run trends of oil prices. Actually, empirical tests of the non renewable
resources theory rely on more complex models 16 than the genuine Hotelling
model. These models, by taking into account the presence of extraction costs
and technical progress, make compatible a non-increasing trend of prices and
the arbitrage assumption that underlies the ’r-rule’ theory.
However, these tests are not consensual about the validity of the Hotelling
’r-rule. One can advocate that the this rule holds for a given size of the known
stocks. As new stocks are discovered, a new path initiates for the rent, still
T is negative if the subvention is greater than the profit in the final good sector.
See Chermak and Patrick (2002) for a classification.
increasing, but starting from a lower level. Actually, such an explanation is
far from convincing, as it assumes very myopic expectations.
The model presented in the previous sections gives a potential explanation for
the non-increasing of the oil rent. If the owners of the oil stocks anticipate
that, on the long run, it will be socially optimal not to deplete the stocks, and
that some politic agreement will implement this optimum, then the market
equilibrium implies a non-growing price. Notice that the reason why such
expectations might exist need not to be the global warming, that was not a
big concern 25 years ago. Any other reason would produce the same effect,
such as the political wish of the countries to be energetically independent.
While there is some doubt about the existence of an increasing scarcity rent,
some oil rents actually exist, shared by oil-producing firms and oil-exporting
countries. These rents seem to be in contradiction with the model discussed
here. But the presence of oil rents does not mean that these are Hotelling
rents. They can simply be oligopoly rents. Indeed, oil extraction requires high
investments, and implies high sunk costs. That advocates an imperfectly competitive market.
Moreover, as extraction capacities, at a given point of time, are limited and
take time to expend, booms in oil demand have an impact on prices. For
technical reasons, these short-run effects can last for a rather long time. So,
big profits in the oil sector is not incompatible with a model that predicts no
Hotelling rent.
Empirical implication of the model
Slight changes in the model presented here could enable to implement an
insightful empirical test. Shortly, assume that a political decision about global
warming has to be taken at finite time tdec 17 . From 0 to tdec , the decision
is uncertain. The decision consists in choosing between the adoption or the
rejection of an environmental policy. This policy implies the hotelling rent to
disappear. The uncertainty that prevails in the first sub-period would alter
the ’r-rule’, but would not make it to disappear. The decision taken in tdec ,
by changing the state of knowledge, must change the path of the rent. In
particular, if the policy is adopted, the rent becomes nil. A simple empirical
test would consist in testing for the constancy of the rent and of its rate of
growth in the two sub-periods.
In this short, informal presentation, I assume only two sub-periods, whereas the
argument can be generalized to a higher number of sub-periods.
Scope of the paper and further research perspectives
The scope of this paper was not to give a definitive answer to the debate
about the interaction between oil depletion and climate change. The strong
assumptions that I made about the decay process and the cost of extraction
obviously don’t allow for such a claim.
In addition to the empirical test mentioned above, further researches on the
problematic discussed here should take into account the three following points:
• More complex decay processes. A nil decay process is probably not a necessary condition to make the Hotelling rent to disappear, and then to break
the link between oil depletion and global warming.
• Innovation on renewable clean energies should be taken into account. In
this context, an endogenous R&D activity would, for instance, reduce the
parameter β.
• Strategical interactions should be modeled through a dynamic game. Indeed, both the stock of carbon and the stock of knowledge on the backstop
technology have public good features.
In a model integrating both a cumulative pollution and a non renewable resource, I showed that the optimal Pigovian tax can cause the disappearance
of the Hotelling’s rent. This breaks the interaction between resource depletion
and climate change. This result is obtained by assuming no natural assimilation of the pollutant and the presence of a backstop technology.
Stability analysis of the steady state in the specified model
The theorem used by Tahvonen (1991) enables to prove that a steady state
has a saddle-path stability. To use it, one has to define the so-called modified
hamiltonian dynamic system of the social planner’s problem. The three control
variables can be expressed as explicit functions of the stock variables and the
co-state variables, along an optimal path.
E = E (S, K, λ, µ) = K
µ (1 − α) Ω
A = A (S, K, λ, µ) =
!− 1
C = C (S, K, λ, µ) = µ− σ
In these conditions, the optimal path is a dynamic system with 4 differential
equations and 4 variables.
S˙ = f1 (S, K, λ, µ) =
µ (1 − α) Ω
(1 − α) µ Ω
K˙ = f2 (S, K, λ, µ) = K
#− 1
# 1−α
− µ1/σ − δ K − 1/2
β µ2
λ˙ = g1 (S, K, λ, µ) = ρ λ − Λ S −
(1 − α) µ Ω
µ˙ = g2 (S, K, λ, µ) = ρ µ − µ α Ω
# 1−α
f1 , f2 , g1 and g2 correspond to (1), (2), (6) and (7) with the control variables being replaced by E (S, K, λ, µ), A (S, K, λ, µ) and C (S, K, λ, µ). I define J (∞) as the jacobian matrix of the modified hamiltonian system of the
problem computed on steady state.
Furthermore, I define Z in the following way:
Z(t) ≡ det 
∂f1 ∂f1
∂S ∂λ
∂g1 ∂g1
∂S ∂λ
 + det 
∂f2 ∂f2
∂K ∂µ
∂g2 ∂g2
∂K ∂µ
 + 2 det 
∂f1 ∂f1
∂K ∂µ
∂g1 ∂g1
∂K ∂µ
which takes the value Z(∞) on steady-state.
Tahvonen (1991) showed that the steady state is a saddle path if the two
following conditions are respected:
det [J(∞)] > 0
Z(∞) < 0
I first show that Z(∞) is negative for a range of parameters values.
It can be computed 18 that Z(∞) has the same sign as the following expression:
(σ 2 −1)(1−α)
1−σ 2
Θ σ
ρ + δ − ΩΘ
σ 2 −1
χΘ σα
(σ 2 −1)(1−α)
ρ(1 − α)Ω ρΘ−σ (1 − α)Ωχ α−1
Θ ≡ ψχ α−1 ζ > 0
which is negative, provided:
(σ 2 −1)(1−α)
1−σ 2
Θ σ
ρ + δ − ΩΘ
σ 2 −1
χΘ σα
(σ 2 −1)(1−α)
ρ(1 − α)Ω ρΘ−σ (1 − α)Ωχ α−1
Which is true for a range of parameters values, among which those in table 1
Finding a condition on parameters in order to fulfill det [J(∞)] involves very
complex expressions 19 . However, it can be shown that a range of parameters
respect both this condition and that given by (A.2), among which those in
table 1. It should be noted that det [J(∞)] is very sensitive to the value of σ.
Calibration of the specified model
Take an aggregate stock of capital of 6’500’000 billion dollars, and an energy
consumption of 8’600’000 kt oil-equivalent, half of which being with no green18
The computation is complex, and was made with the help of a mathematical
19 The Maple file is available on demand from the author.
house effect gas emissions. Then, with the values of α, β and Ω given in table
1, the world GDP would be of 24’750 billion dollars. Gross production would
be 25’048 billion dollars, and the cost of the renewable energies would be 298
billion dollars. As inputs are paid at their marginal productivity, the capital
earning is 23’796 billion dollars, which represents a 0.961 share of the world
GDP. This latter figure is different from α because the GDP is not Y but
Y-q(A). The remaining 0.039, that is to say 964.65 billion dollars, is the oil
rent, in the case where no climate policy is enforced. These figures can be
considered as realistic ones.
ρ = 0.03, σ = 1.75 and δ = 0.1 are close to values usually used in the
growth literature. It should be noted that the saddle path stability of the
steady state heavily depends on the value of σ. In particular, det [J(∞)] is a
non-monotonous function of σ.
No empirical evidence enables to have a precise idea of the values of and Λ.
I take close to 1, which makes the part of the utility function concerning S
close to a logarithm function. I choose Λ = 1 with no possible justification.
However, it should be noted that the value of Λ does not affect the saddle-path
stability, and only affects the steady-state value of S.
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