1. art and entertainment
2. visual art and design
3. painting

EC202 Microeconomic Principles II: Lecture 10
Leonardo Felli
NAB.LG.08
19 March 2015
Private Good
I
A good is rival if consumption by an agent reduces the
possibilities of consumption by the other agents.
I
A good is subject to exclusion if you have to pay to consume
the good.
Definition (Private Goods)
A private good is both rival and subject to exclusion.
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Public Goods
Definition (Public Goods)
A public good is nonrival. A pure public good is both nonrival and
not subject to exclusion.
I
A pure public good example is national defense
I
A good that is subject to exclusion, but nonrival is patented
research
I
A good that is not subject to exclusion, but rival is free
parking
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The Free Rider Problem
I
Free riders are economic agents that consume more than their
fair share of a public resource,
I
They are also economic agents that bear less than a fair share
of the costs of its production.
I
Free riding is considered to be a problem when it leads to:
I
the under-production of a public good (and thus to Pareto
inefficiency),
I
the excessive use of a common resource.
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The Free Rider Problem (cont’d)
I
This problem was first formalized by Wicksell in 1896
I
Though discussions about the under-provision of public goods
date back to Adam Smith’s ”Wealth of Nations” in 1776
I
The 1st solution to the problem was proposed by Lindahl in
1919
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A Simple Economy with Pure Public Goods
Consider the following two goods economy:
I
Let n denote the number of consumers (i denotes any one of
them).
I
Let x denote the private good (an aggregate of all private
goods).
I
Let z denote the pure public good.
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Technology
I
Good z is produced using good x according to a production
function:
z = f (x)
I
The production function f is monotonic, hence can be
expressed as the function g :
x = g (z) = f −1 (z)
I
Endowments of the two goods are
(ωx , ωz ) = (X , 0)
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Preferences and Feasibility
I
Preferences of consumer i are given by
U i (xi , zi )
I
The feasibility constraint for the private good requires that:
n
X
xi ≤ X − x
i=1
I
Since good z is nonrival, feasibility only requires that:
zi ≤ z for any i ∈ {1, ..., n}
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EC202 Microeconomic Principles II
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Pareto Optimality with Pure Public Goods
I
Let αi denote the Pareto weight on the utility of player i
I
The Pareto optimal allocation is the solution to the problem:
max
x,z,xi ,zi
s.t.
n
X
αi U i (xi , zi )
i=1
z = f (x)
n
X
xi ≤ X − x
i=1
zi ≤ z
Leonardo Felli (LSE)
∀i ∈ {1, ..., n}
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Pareto Optimality with Pure Public Goods (cont’d)
I
Clearly since utility is monotonic, both feasibility constraints
will be binding hence the problem becomes:
max
xi ,z
s.t.
n
X
i=1
n
X
αi U i (xi , z)
xi = X − g (z)
(λ)
i=1
I
The first order conditions are:
n
X
αi Uxi (xi , z) = λ
(xi )
αi Uzi (xi , z) = λ g 0 (z)
(z)
i=1
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EC202 Microeconomic Principles II
19 March 2015
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Pareto Optimality with Pure Public Goods (cont’d)
I
Solving the two equations above yield the Pareto Optimality
conditions:
n
X
Uzi (xi , z)
1
= g 0 (z) = 0
i
Ux (xi , z)
f (x)
i=1
I
Where we use the rule:
df −1 (z)
1
= 0
dz
f (x)
I
These conditions are known as the Bowen-Lindhal-Samuelson
conditions and imply that the Social Marginal Rate of
Substitution equals the Marginal Rate of Transformation.
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EC202 Microeconomic Principles II
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Private Contribution/Subscription Equilibrium
Consider an economy in which individuals decide how much to
contribute to the production of the public good:
I
Denote Xi the resources of player i.
I
Denote by di the resources that individual i is willing to
donate for the production of z.
I
The budget constraint for player i requires that:
xi + di = Xi
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EC202 Microeconomic Principles II
19 March 2015
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Private Contribution Equilibrium (cont’d)
I
The amount produced of public good z therefore satisfies:
!
n
X
z =f
di
i=1
I
The best reply for consumer i is then to choose his
contribution di given the contributions of all remaining
consumers d−i :
max U i (xi , z)
xi ,di
s.t.
z =f
n
X
!
di
i=1
xi + di = Xi
Leonardo Felli (LSE)
EC202 Microeconomic Principles II
19 March 2015
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Private Contribution Equilibrium (cont’d)
I
Since the constraints are binding, the problem of consumer i
simplifies to:
!!
n
X
max U i Xi − di , f
di
si
I
i=1
The first order conditions are then:
Uzi (xi , z)
1
= 0 Pn
i
Ux (xi , z)
f ( i=1 di )
I
To be compared to BLS:
n
X
U i (xi , z)
z
Uxi (xi , z)
i=1
Leonardo Felli (LSE)
=
1
f 0 (x)
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Private Contribution Equilibrium (cont’d)
I
The Nash equilibrium of the private contribution game is such
that consumer i’s Private Marginal Rate of Substitution
equals the Marginal Rate of Transformation.
I
Clearly this differs from BLS the reason is that individual
agents only care about their private benefits from their
investment in the public good.
I
Under general conditions the private contribution equilibrium
leads to the production of too little public good.
I
In other words, there exists a free rider problem.
Leonardo Felli (LSE)
EC202 Microeconomic Principles II
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Lindahl Equilibrium
In 1919 Lindahl (a Swedish economist) proposed the following
solution to the free rider problem:
I
Assume that personal prices can be established
I
Let pi be the price paid by consumer i
I
The producer of the public good thus receives a price:
p=
n
X
pi
i=1
I
The producer chooses his output given the price to maximize
profits:
max p z − g (z)
z
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EC202 Microeconomic Principles II
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Lindahl Equilibrium (cont’d)
I
Thus he produces until marginal costs are equal to the price p:
p = g 0 (z)
I
Consumer i, on the other hand, chooses xi and zi to maximize
utility given his price pi :
max U i (xi , zi )
xi ,zi
s.t.
Leonardo Felli (LSE)
xi + pi zi = Xi
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Lindahl Equilibrium (cont’d)
I
The FOC require that he equalizes his Marginal Rate of
Substitution to the price pi :
Uzi (xi , zi )
= pi
Uxi (xi , zi )
I
Market clearing in the public good sector requires that the
individual demand of any consumer be equal to the supply:
zi (p1 , . . . , pn ) = z(p1 , . . . , pn ) for any i ∈ {1, ..., n}
Leonardo Felli (LSE)
EC202 Microeconomic Principles II
19 March 2015
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Lindahl Equilibrium (cont’d)
Adding up for any i condition
Uzi (xi , zi )
= pi
Uxi (xi , zi )
And using condition
p = g 0 (z)
We obtain that the BLS condition for Pareto optimality is satisfied:
n
X
U i (xi , zi )
z
Uxi (xi , zi )
i=1
Leonardo Felli (LSE)
=
n
X
pi = p = g 0 (z) =
i=1
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1
f 0 (x)
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Lindahl Equilibrium (cont’d)
I
The advantage of such implementation is that it restores
Pareto optimality.
I
The disadvantage is that it requires the existence of n
”micromarkets” in which a sole consumer buys the public
good at his personalized price.
I
Thus it is a hard solution to implement if consumers
valuations for the public good are private information, since
everyone has an incentive to underestimate his demand in
order to pay a lower price.
Leonardo Felli (LSE)
EC202 Microeconomic Principles II
19 March 2015
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Personalized Taxation
If consumers can be taxed for the consumption of the public good:
I
The budget constraint of any consumer i becomes:
xi + ti (zi ) = Xi
I
Consumers equalize MRS to the marginal cost of the public
good:
Uzi (xi , zi )
= ti0 (zi )
Uxi (xi , zi )
I
Setting tax rates equal to the Lindahl prices yields Pareto
optimality:
ti (zi ) = pi zi
Leonardo Felli (LSE)
EC202 Microeconomic Principles II
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Pros and Cons
I
This approach has the same advantages and disadvantages
than the Lindahl equilibrium (it works if the government has
detailed information about the preferences in the population)
I
Recently devised mechanisms do not require the regulator to
be informed about the tastes of consumers and still restore
Pareto optimality.
Leonardo Felli (LSE)
EC202 Microeconomic Principles II
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Planning Procedures and Pivotal Mechanisms
I
These may be Planning Procedures (Malinvaud, Dreze and
Poussin, 1971): dynamic revelation mechanisms that
converges to Pareto optimality
I
They may also be Pivotal Mechanisms (Vickrey, Clarke,
Groves 1971): static revelation mechanisms that implements
any Pareto optimal allocation.
I
In both cases the regulator is uninformed about preferences of
consumers, he implements the Pareto optimal allocation
giving individual consumer the incentive to reveal their true
valuation for the public good.
Leonardo Felli (LSE)
EC202 Microeconomic Principles II
19 March 2015
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Property of Public Goods
I
It is often asserted that because of the free rider problem
public goods should be provided by the public sector (e.g.
police, defense, justice system)
I
Such claim was first made by Adam Smith in the ”Wealth of
Nations”
I
Several classical authors (Mill & Samuelson) illustrated the
principle through the lighthouse example (a pure public good)
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EC202 Microeconomic Principles II
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Coasian Solution
I
In 1974 Coase contested such argument pointing out that the
free rider problem was simply a problem of positive
externalities.
I
As such it could be solved by private bargaining: a plain
application of the Coase Theorem.
I
Coase argued: that British lighthouses were traditionally a
responsibility of a private national company that perceived a
fixed right which was discharged by any ship landing in a
British port.
I
This was not detrimental to British naval commerce since
shipowners were more conscious of paying for this service and
had more incentives to monitor that the service was rendered.
Leonardo Felli (LSE)
EC202 Microeconomic Principles II
19 March 2015
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The Importance of the Free Rider Problem
In large economies without public goods individuals have no
incentive to game prices by altering their demand
(Roberts-Postlewaite 1976)
With public goods, however, individuals have incentives to
underestimate their demand since it affects their contribution
significantly, but it affects the provision of the public good only
marginally
Leonardo Felli (LSE)
EC202 Microeconomic Principles II
19 March 2015
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People are Essentially Honest
Several empirical studies however have pointed out that the
importance of the free rider problem has been exaggerated by
economics since individuals have a tendency towards honesty
(Bohm 1972, Ledyard 1995 – survey)
The conclusions of these studies are that individuals:
I
Game prices, but less than in the private contribution game.
I
Contribute less if the game is repeated.
I
Contribute more if allowed to communicate.
Leonardo Felli (LSE)
EC202 Microeconomic Principles II
19 March 2015
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Local Public Goods
Definition (Local Public Good)
Local public goods are public goods that apply only to the
inhabitants of a particular geographical area (garbage collection,
public transport, parks)
Tiebout was the first to study their theory in 1956
The main feature of these markets is that individuals are free to
decide in which community to live
Thus individuals will move away from communities with too few or
poorly financed public goods (neighborhoods and public schools)
Tiebout showed that this process had an efficient equilibrium if
there is perfect mobility and perfect information
Leonardo Felli (LSE)
EC202 Microeconomic Principles II
19 March 2015
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Moral Hazard in Teams (Holmstr¨om 1982)
Consider now the following multi-agent model with moral hazard.
Assume that there exists two risk-neutral agents each choosing an
effort ei , i ∈ {1, 2} simultaneously and independently.
The output of their joint effort is the strictly concave function:
x(e1 + e2 ),
Leonardo Felli (LSE)
x 0 (·) > 0,
x 00 (·) < 0,
EC202 Microeconomic Principles II
x 0 (0) > 1
19 March 2015
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Partnership
Each agent’s personal cost of effort is
φ(ei ) = ei ,
∀i ∈ {1, 2}
Assume first that the agents decide to organize their activity as a
partnership.
Let si be the share of output that agent i gets in the partnership.
Agent i’s total share of surplus is then:
si x(e1 + e2 )
We assume that:
s1 + s2 = 1
Leonardo Felli (LSE)
EC202 Microeconomic Principles II
19 March 2015
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The Full-Information Case
We start from the full-information case: ei and x are observable to
all parties.
Before exerting any effort the agents negotiate and sign a
partnership contract.
The full-information partnership contract specifies:
(e1 , e2 ; s1 , s2 )
such that:
s1 + s2 = 1
Leonardo Felli (LSE)
EC202 Microeconomic Principles II
19 March 2015
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The Full-Information Case (cont’d)
The level of efforts specified in the contract are such that:
ei∗ = arg max x(e1 + e2 ) − e1 − e2 ,
∀i ∈ {1, 2}
ei
This maximization problem is strictly concave, the unique solution
is fully characterized by the necessary and sufficient FOC:
x 0 (e1∗ + e2∗ ) = 1,
∀i ∈ {1, 2}
Whatever shares agreed by the agents (s1 , s2 ) the full information
contract specifies:
(e1∗ , e2∗ ; s1∗ , s2∗ )
Leonardo Felli (LSE)
EC202 Microeconomic Principles II
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Unobservable Effort
Assume now that each agent’s effort level ei is only observable to
the agent i.
Before choosing effort the agents negotiate and sign a partnership
contract. This contract only specifies the share of surplus accruing
to every agent:
(s1 , s2 )
Given the contract each agent then chooses his effort eˆi .
These choices are the Nash equilibrium of the agents’ effort-choice
game:
eˆ = (ˆ
e1 , eˆ2 )
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EC202 Microeconomic Principles II
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Agents’ Effort Choice
Each agent’s best reply is then the result of the maximization of
his payoff function:
max si x(ei + e−i ) − ei
ei
The best reply is characterized by the necessary and sufficient
FOC:
si x 0 (ei + e−i ) = 1
The Nash equilibrium is then defined by the solution to the
following pair of best replies, if si > 0:
x 0 (ˆ
e1 + eˆ2 ) =
Leonardo Felli (LSE)
1
,
s1
x 0 (ˆ
e1 + eˆ2 ) =
EC202 Microeconomic Principles II
1
s2
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Impossibility of Efficient Partnership
Result (Holmstr¨om 1982)
There does not exist a partnership agreement (s1 , s2 ), where si > 0
for all i ∈ {1, 2}, that achieves the efficient outcome.
Recall that:
s1 + s2 = 1
In other words:
0 < si < 1
Leonardo Felli (LSE)
EC202 Microeconomic Principles II
19 March 2015
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Impossibility of Efficient Partnership (cont’d)
Therefore, for all partnership agreement (s1 , s2 ) we get:
x 0 (ˆ
e1 + eˆ2 ) =
1
>1
si
Recall that the full information effort is such that:
x 0 (e1∗ + e2∗ ) = 1
Hence, strict concavity of x(e1 + e2 ) implies that x 0 (e1 + e2 ) is a
decreasing function, therefore
eˆ1 + eˆ2 < e1∗ + e2∗
Leonardo Felli (LSE)
EC202 Microeconomic Principles II
19 March 2015
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Intuition of Free-Rider Problem
I
Each individual agent when choosing his effort has an
incentive to free-ride on the other agent’s effort choice.
I
The reason is that agent i does not receive full returns from
his investment eˆi .
I
He only receives returns from his investment in proportion of
his share of the partnership si hence he adjusts his investment
accordingly.
I
Can the agents do better by choosing a contract that differs
from a partnership?
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EC202 Microeconomic Principles II
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Budget Breaker
For this purpose we need to introduce a budget breaker in the
contract.
Result (Holmstr¨om 1982)
Assume that the partnership share for agent i, Si (x) is a function
of x such that:
∗
si if x ≥ x(e1∗ + e2∗ )
Si (x) =
0 if x < x(e1∗ + e2∗ )
where for all i ∈ {1, 2} si∗ is such that
si∗ x(e1∗ + e2∗ ) > ei∗
The first best effort choices (e1∗ , e2∗ ) are now a Nash equilibrium of
the agents’ game.
Leonardo Felli (LSE)
EC202 Microeconomic Principles II
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Budget Breaker (cont’d)
∗ .
If agent −i choose e−i
Agent i’s payoff is then:
I
∗ ) − e > 0, if e ≥ e ∗ hence x ≥ x ∗ ,
si∗ x(ei + e−i
i
i
i
I
−ei ≤ 0, if instead ei < ei∗ hence x < x ∗ .
The agent’s best reply is then:
ei = ei∗
Leonardo Felli (LSE)
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Budget Breaker (cont’d)
The budget breaker has the following payoff:
0 if x = x(e1∗ + e2∗ )
Π(x) =
x if x < x(e1∗ + e2∗ )
In other words his payoff is positive only if in case of deviation.
The main problem with this solution is the multiplicity of the Nash
equilibria of the agents’ subgame.
Leonardo Felli (LSE)
EC202 Microeconomic Principles II
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Multiplicity
∗ .
Assume that agents −i choose effort eˆ−i < e−i
Then if
si∗ x(e1∗ + e2∗ ) − e¯i ≥ 0
where
e¯i = (e1∗ + e2∗ − e−i )
the best reply for agent i is to choose e¯i : he feels pivotal.
This means, (¯
ei , eˆ−i ) is an (asymmetric) Nash equilibrium of the
agents’ game.
Leonardo Felli (LSE)
EC202 Microeconomic Principles II
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Inefficiency
If instead
si∗ x(e1∗ + e2∗ ) − e¯i < 0
Then the best reply for the agent i is to choose
eˆi = 0
There always exists a Nash equilibrium of the agent’s game such
that each agent chooses
ˆai = 0,
Leonardo Felli (LSE)
∀i ∈ {1, 2}
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Being Pivotal
I
The existence of an equilibrium with efficient contributions
comes from the fact that in the presence of a budget breaker
each agent feels pivotal.
I
This implies that in the absence of his contribution the
positive payoff is not awarded to anyone, him included.
I
This is why team contribution and team spirit are enhanced in
situations where there is a large prize only for the winning
team, and zero for the others.
I
This is also why in most fund-raising appeals it is critical to
establish a target and make each potential contributor feel like
in the absence of his contribution the target will not be
achieved and the general benefits will disappear from everyone.
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