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Microeconomics II
MWG, Ch. 11 Externalities and Public Goods
Alzahra University Department of Economics
Hamid Kordbacheh
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Externalities
• FFWT any competitive equilibrium is Pareto optimal. i.e. markets allocate resources efficiently.
• SFWT (given suitable convexity assumptions) any Pareto optimal allocation can be supported as a competitive equil.
• What happens if the behavior of some agent affects the welfare of others?
• When external effects are present, CE is still PO, as long as the effects are transmitted via prices, markets are efficient.
• market failures: violation of welfare theorems assumptions in which markets fail to deliver optimal results.
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11.B: A simple Bilateral Externalities
• Bilateral Externality vs Multilateral Externality
• Externalities can be produced by consumers as well as firms.
o Consumption side: noise pollution
o Production side: Chemical plant’s discharges reducing fishery’s catch
• Externalities can be positive or negative
• Public Good -special kind of externality
o Non-rivalry in consumption: National defense & flood control
o Non-excludable: lighthouse
Basic Problem – Market failure or lack of market (no price)
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Definition 11.B.1: An Externality is present whenever the wellbeing of a
consumer or the production possibilities of a firm are directly affected by the
actions of another agent in the economy.
• Directly Affected - not through prices
Viner (1931)
• Pecuniary externalities: actions affecting prices;
• Non-pecuniary (true) externalities actions not affecting prices (that's
what we're studying)
Fishery’s productivity affected by emissions from oil refinery.
Fishery’s profitability affected by price of oil.
Neighbor's flower garden
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Simple Bilateral Externality
A two-agent partial equilibrium model
• 2 consumers (can think of 2 producers or 1 of each)
• L traded goods
• 𝒘𝐢 consumers I’s wealth
• Consumer i’s utility function
𝑢𝑖 𝒙𝐢, ℎ
𝒙𝐢= (𝑥𝑖1, 𝑥𝑖2,…𝑥𝑖𝐿)
h: amount of externality
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Simple Bilateral Externality
• Consumer 1 chooses externality h, assume h 0, h
• Consumer 2 takes the externality so 𝜕𝑢−𝑖
𝜕ℎ≠ 0
The Consumer derived their utility function on the level of h
𝑣𝑖 𝑤𝑖 , 𝑝, ℎ = max𝑥𝑖>0
𝑢 𝒙𝒊, ℎ
s.t. 𝑝𝑥𝑖 ≤ 𝑤𝑖
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Equilibrium Choices are Not Efficient
• Given the quasi-linear preferences the consumer’s indirect utility function
takes the form
𝑣𝑖 𝑤𝑖 , 𝑝, ℎ = 𝑥1𝑖 + 𝑔(𝑥−1𝑖 , ℎ)
We know that 𝑥𝑖1= 𝑤𝑖 − 𝑝𝑥−1𝑖(𝑝, ℎ), then
𝑣𝑖 𝑤𝑖 , 𝑝, ℎ = 𝑤𝑖 − 𝑝𝑥−1𝑖 𝑝, ℎ + 𝑔 𝑥𝑖2, 𝑥𝑖3,…𝑥𝑖𝐿
or
𝑣𝑖 𝑤𝑖 , 𝑝, ℎ = 𝜙𝑖(𝑝, ℎ) + 𝑤𝑖
It can be written
𝜙𝑖(𝑝, ℎ)=𝜙𝑖(ℎ)
What is 𝜙𝑖(ℎ)?
Assume 𝜙′𝑖(ℎ) ≤ 0 and, 𝜙"𝑖 ℎ < 0
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Equilibrium Choices are Not Efficient
• How will consumer 1 choose h?
• Efficient outcome
𝜙′𝑖(ℎ) ≤ 0 with equality if ℎ > 0
Figure 11.B.1 shows this results
• Is there any problem with this result?
• What is the socially optimal level of h?
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Pareto Optimal Allocation
The socially optimal level of h must maximize the JOIN surplus of the 2
consumers
maxℎ≥0
𝜙1(ℎ) + 𝜙2(ℎ)
FOC
𝜙′1(ℎ𝑜) + 𝜙′2(ℎ
𝑜) ≤ 0
For interior solution
𝜙′1(ℎ
𝑜) = − 𝜙′2(ℎ𝑜)
Result: ℎ∗ > ℎ𝑜
Figure 11.B.1 shows this results
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Equilibrium Choices are Not Efficient
• Three important points that come out of this results
• Externalities are not necessarily eliminated at the Pareto optimal solution
o When dose this happen?
• What would be the result if we have positive externalities
• What would happen if we relax the assumption of a quasilinear utility
functions?
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Traditional Solutions to the Externality Problem
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Quotas
• Suppose negative externality ℎ𝑜 < ℎ∗
• Social planer sets maximum of ℎ = ℎ𝑜
• Emitter solves
max0<ℎ<ℎ𝑜
𝜙1(ℎ)
We know ℎ𝑜 < ℎ∗, so polluter will do as much as he can
Perfect information Requirements:
• Policy maker needs to compute ℎ𝑜
• So he needs to compute 𝜙𝑖(ℎ)
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• Taxes
• Pigouvian taxation: Imposing tax on the externality-generating activity
• Taxes consumer 1 for producing h ; sets per unit tax
𝑡ℎ = −𝜙′2 ℎ𝑜 > 0
• What does this mean?
• Emitter solves
maxℎ>0
𝜙1 ℎ − 𝑡ℎℎ
FOC 𝜙′1 ℎ𝑜 ≤ 𝑡ℎ with equality if ℎ𝑜 > 0
Figure 11.B.2
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Two questions:
• How much tax must we impose in case that the negative externality is very
substantial (and ℎ𝑜=0)
• How can the previous discussion be extended to positive externality
Subsidies
• Policy maker sets the unit of subsidy 𝑠ℎ = −𝑡ℎ= 𝜙′2 ℎ𝑜 > 0
Where 𝑆 ≡ 𝑠(ℎ∗−h)
Emitter solve
maxℎ>0
𝜙1 ℎ +𝑠ℎ(ℎ∗−h)
FOC 𝜙′1 ℎ𝑜 + 𝑠ℎ ≤ 0 with equality if ℎ𝑜 > 0
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Some important points about Pigouvian taxation:
• The Pigouvian view:
o Assumes ethical standpoints, and relies on social attitudes or norms to
determine the direction of an externality.
o Emphasizes an externality generator and a victim
• The Pigouvian tax charges a tax on the externality-generating activity but
not on the output that generated such pollution
o What would happen if the output was taxed?
o When does the tax on output lead to the same result?
• The quota and the Pigouvian tax are equally effective under complete
information
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Fostering bargaining over externalities
Coase/Bargaining Solution
• Coase/Bargaining Solution: Coase’s famous paper (The Problem of
Social Cost ,Coase 1960), was a direct response to Pigou’s argument
• The key features of Coasean paradigm:
o Emphasizing reciprocity
o Relying on property right based on social attitudes and norms
o Free market alternative (possibility of private bargaining) to the
Pigouvian idea of explicit intervention in response to a “market failure”
o Bargaining between two parties results in Pareto efficient outcome
(irrespective of who has property rights).
o Irrelevance Theorem ( the neutrality proposition): the initial allocation of property rights does not affect the final allocation of resources
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The key conditions
• Small numbers of agents involved
• Perfect information among the agents
• Assigned property right to externality otherwise No Big Deal
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Example: two agents
• Still looking at negative externality
Case 1: 2 Has Property Right
• Assign right to externality-free environment to consumer 2.
• Initial state ℎ = 0
• Consumer 1 cannot produce externality without Consumer 2’s permission
• Bargain – agents will bargain to reach an agreement over (h,T) ; if no agreement is reached the default value is (0,0)
• Bargaining Power - this is independent of the property right and reliant on the ability of negotiation.
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• Assume consumer 2 has all the bargaining power so he can make a take-
it-or-leave-it offer, (h, T ) to consumer 1 demanding a payment T
• Consumer 1 accepts iff 𝜙1 ℎ − 𝑇 ≥ 𝜙1(0)
• Consumer 2 will pick the offer (h,T) to solve
maxℎ≥0
𝜙2 h + T
𝑠. 𝑡 𝜙1 ℎ − 𝑇 ≥ 𝜙1 0
The constraint will be binding in
maxℎ≥0
𝜙2 h + 𝜙1 ℎ − 𝜙1 0
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Efficient Choice
𝜙′2 ℎ + 𝜙′1(ℎ) ≤ 0 with equality if ℎ > 0
• This coincides with that solving the social planner’s problem
• i.e. ℎ = ℎ0 with 𝑇 = 𝜙1 ℎ0 − 𝜙1 0 > 0
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Case 2: 1 Has Property Right
• Assign right to the polluter (consumer 1) to generate as much as the
externality she wants
• In the absence of any agreement, consumer 1 will generate ℎ∗
• Assume consumer 2 has all the bargaining power so he can make a take-
it-or-leave-it offer, (h, T ) to consumer 1 for a payment T
• Indeed, consumer 2 can pay $T the consumer 1 in exchange of a lower
level of pollution
• Consumer 1 accepts iff 𝜙1 ℎ + 𝑇 ≥ 𝜙1(ℎ∗)
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• Consumer 2 will choose the offer (h,T) to solve
maxℎ≥0
𝜙2 ℎ − T
𝑠. 𝑡 𝜙1 ℎ + 𝑇 ≥ 𝜙1 ℎ∗
The constraint will be binding in
maxℎ≥0
𝜙2 h + 𝜙1 ℎ −𝜙1 ℎ∗
Efficient Choice
𝜙′2 ℎ + 𝜙′1(ℎ) ≤ 0
• Again this coincides with that solving the social planner’s problem
• i.e. ℎ = ℎ0 with 𝑇 = 𝜙1 ℎ0 − 𝜙1 ℎ∗ > 0
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Results
Case 1: 2 Has Property Right
• Consumer 1 pays 𝑇 = 𝜙1 ℎ0 − 𝜙1 0 > 0 to be allowed to set ℎ0> 0
Case 2: 1 Has Property Right
• Consumer 2 pays 𝑇 = 𝜙1 ℎ0 − 𝜙1 ℎ∗ > 0 for setting ℎ0 < ℎ∗
Coase paradigm : If trade of the externality can occur then
bargaining will lead to an efficient outcome no matter how PR are
allocated.
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Although the initial allocation of PR does not affect the level of the ext., it
affects the wealth distribution of the two agents.
Case 1: 2 Has Property Right
• 1 must pays 𝑇 = 𝜙1 ℎ0 − 𝜙1 0 to 2
Then consumer 2’s utility is 𝜙2 ℎ0 + T and then consumer 1’s utility is
𝜙1 ℎ0 − T= 𝜙1 0
Hence, consumer 2’s utility is higher than that of consumer 1 if
𝜙2 ℎ0 + 𝜙1 ℎ0 > 2𝜙1 0
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Coase/Bargaining Solution
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Case 2: 1 Has Property Right
• 2 must pays 𝑇 = 𝜙1 ℎ∗ − 𝜙1 ℎ0 to 1
Then consumer 1’s utility is 𝜙1 ℎ0 + T = 𝜙1 ℎ∗ but consumer 2’s utility is
𝜙2 ℎ0 − T
Hence, 1’s utility is bigger than that of consumer 2 if
2𝜙1 ℎ∗ > 𝜙1 ℎ0 + 𝜙2 ℎ0
• Therefore, when the agent has bargaining power has a total utility higher
the average of welfare at the Pareto optimum, and vice versa.
2𝜙1 ℎ∗ > 𝜙1 ℎ0 + 𝜙2 ℎ0 >2𝜙1 0
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Figure 11.B.3: The final distribution of utilities under different PR and BP
• At point a consumer 2 has the property right ( h = 0)
• Therefore, the take-it-or-leave-it offer leads to point f in the first case and
point e in the second case
• At point b consumer 1 has the property right
• The , the take-it-or-leave-it offer leads to point d in the first case and point
c in the second case
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Some important points
The Key assumptions of Coase theorem:
• Property rights must be perfectly defined.
• Property rights must be perfectly enforced,
• The polluter must know the cost of the externality for the affected agents
• The affected agents must know the polluter’s profit function
Questions on Coase theorem
• Are these assumptions practical?
• Isn’t Coase theory itself a blackboard economics?
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Externalities as missing markets
Missing Market definition
• Lack of coordination
• Technology
• Transaction costs
• Trust or information
An alternative view to externalities:
• Externalities are a commodity which lacks a market.
• We can simply show that, if externalities were a traded commodity, the
produced level of that coincides with the Pareto optimal level ℎ = ℎ0.
• Suppose a well defined property rights, and a competitive market for the
right
• 𝑝ℎ: the price of one unit of externality-generating activity
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Externalities as missing markets
• consumer 1 (the emitter) chooses how many polluting rights to buy
maxℎ1≥0
𝜙1 ℎ1 − 𝑝ℎℎ1
F.O.C: 𝜙′1 ℎ1 < 𝑝ℎ with equality if ℎ1 >0
• Similarly, the individual affected decides how many polluting rights to sell,
maxℎ2≥0
𝜙2 ℎ2 + 𝑝ℎℎ2
• F.O.C: 𝜙′2 ℎ2 + 𝑝ℎ < 0 with equality if ℎ2 >0
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Externalities as missing markets
• the competitive market for polluting rights must clear
ℎ1 = ℎ2 = ℎ∗∗
𝜙′1 ℎ∗∗ ≤ 𝑝ℎ ≤ −𝜙′2 ℎ∗∗
Or simply
𝜙′1 ℎ∗∗ ≤ -𝜙′2 ℎ∗∗ with equality if ℎ∗∗ >0
• Interestingly, this condition coincides with the F.O.C under the Pareto
optimal level of the externality ℎ0. i.e.
𝑝∗ℎ = 𝜙′1 ℎ0 = − 𝜙′2 ℎ0
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11.C. Public Goods
• DEFN 11.C.1: A public good is a commodity for which use of a unit of the
good by one agent does not preclude its use by other agents.
Distinction
• Non-Excludable: public goods usual known non-excludable, but in Mas-
Colell they can be either excludable or non-excludable;
• Non-Rivalrous: consumption of additional units of the good involves zero
social marginal costs of production
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The taxonomy of four different types of goods
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Rivalrous Non-rivalrous
Excludable Private Good Club Good
Non-excludable Common property
resource
Public Good
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Conditions for Pareto Optimality
Model
• I consumers,
• One public good x
• L traded private goods
Assume:
• quasi-linear utility function over L private goods and the public good
• 𝑢 𝐱𝐢, 𝑥 = 𝑥𝑖1 + 𝑢 (𝑥𝑖2, 𝑥𝑖3,…𝑥𝑖𝐿) + 𝜙𝑖(𝑥)
• 𝐱𝐢 = (𝑥𝑖1, 𝑥𝑖2,…𝑥𝑖𝐿)
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• 𝑥: amount of public good which is actually a good (𝜙′𝑖 𝑥 > 0)
• 𝜙𝑖(𝑥) is concave (𝜙′′𝑖 𝑥 < 0)
• level of consumption of x has no effect on prices of the private goods
• Cost to produce public good is C(𝑞)
• C(𝑞) are convex in q
• q : amount of public good produced
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Pareto Optimal Solution
The partial equilibrium model
The social planner maximizes aggregate surplus,
max𝑞≥0
𝜙𝑖(𝑞)𝐼1 -C(q)
F.O.C: Samuelson rule
𝜙′𝑖(𝑞𝑜)𝐼
𝑖=1 > C′(𝑞𝑜) with equality if 𝑞 >0
• The social planner increases the provision of a public good until that the
sum of the consumers’ marginal benefit ( or marginal social benefit) is
equal to its marginal cost.
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Inefficiency of private provision of public goods
• Show that market provision of PG (i.e. competitive, price taking
equilibrium) is inefficient
• Assume a market exists for the public good
• Market-Clearing - at p* public good produced (supplied) equals public
good consumed
• Total amount of PG purchased 𝑥 = 𝑥𝑖𝐼𝑖=1
• No Exclusion: 𝑥∗𝑘𝐼𝑘≠𝑖 optimal purchases of public good all other
consumers
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At a competitive equilibrium:
1. Consumers maximize utility
each consumer i’s purchase of the public good 𝑥∗𝑖 must satisfy
max𝑥𝑖≥0
𝜙𝑖 𝑥𝑖 + 𝑥∗𝑘𝐼𝑘≠𝑖 −𝐼
1 𝑝∗𝑥𝑖
F.O.C
𝜙′𝑖 𝑥∗𝑖 + 𝑥∗𝑘
𝐼𝑘≠𝑖 ≤ 𝑝∗ with equality if 𝑥∗𝑖 >0
or
𝜙′𝑖 𝑥∗ ≤ 𝑝∗ with equality if 𝑥∗ >0
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2. Firms maximize profit:
The firm producing the public good must solve the fallowing PMP
max𝑞≥0
(𝑝∗𝑞 − 𝐶(𝑞))
F.O.C
𝑝∗ − 𝐶′(𝑞∗) ≤ 0 with equality if 𝑞∗ >0
At a competitive equilibrium 𝑞∗ = 𝑥∗
𝜙′𝑖 𝑞∗ = 𝐶′(𝑞∗) if 𝑞∗ >0 and 𝜙′𝑖 𝑞
∗ < 𝐶′(𝑞∗) if 𝑞∗= 0
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Compare:
Pareto Efficient Solution 𝜙′𝑖(𝑞𝑜)𝐼
𝑖=1 = C′(𝑞𝑜)
Market Provision 𝜙′𝑖 𝑞∗ = 𝐶′(𝑞∗)
Conclusion: when people make voluntary contributions, the market will
provide too little of the public good, 𝑞𝑜 > 𝑞∗
• This fact can be understood in terms of positive externalities
• Free rider problem.
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Free rider problem.
• Suppose
𝜙′1 𝑥 < 𝜙′2 𝑥 < ⋯ < 𝜙′𝐼 𝑥
In this case, condition 𝜙′𝑖 𝑥∗ ≤ 𝑝∗ can hold for at most one consumer.
This consumer must be consumer I, who values it the most (on the margin).
(why?)
Remedies for the Free-Rider Problem
Government interventions
• Regulation
• Price-based interventions: compulsory participation (Taxation, Tying)
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Suppose there are I consumers each with benefit function 𝜙𝑖 𝑥
Using the Pigouvian taxation we can implement the optimal consumption xO by setting the per unit subsidy to each consumer equal to
𝑠𝑖 = 𝜙′𝑘(𝑥𝑜)
𝑘≠𝑖
Because
max𝑥𝑖≥0
𝜙𝑖 𝑥𝑖 + 𝑥𝑘
𝐼
𝑘≠𝑖
+ 𝑠𝑖𝑥𝑖 − 𝑝∗𝑥𝑖
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The necessary and sufficient first-order condition for this problem is
𝜙′𝑖 𝑥𝑖 + 𝑥𝑘
𝐼
𝑘≠𝑖
+ 𝑠𝑖 = 𝑝
Substituting in the above subsidy and combining with the market-clearing
condition
𝜙′𝑖 𝑥 𝑖 + 𝑥 𝑘
𝐼
𝑘≠𝑖
+ 𝜙′𝑘(𝑥𝑜)
𝑘≠𝑖
= 𝑝
𝜙′𝑖 𝑞 + 𝜙′−𝑖 𝑞
𝑂 ≤ 𝑐′ 𝑞
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Lindahl Equilibrium
A market-based solution to public goods
Firm charges each consumer 𝑝∗∗
In the demand side
max𝑥𝑖≥0
𝜙𝑖 𝑥𝑖 − 𝑝𝑖∗∗𝑥𝑖
FOC 𝜙′𝑖 𝑥∗∗ ≤ 𝑝𝑖
∗∗with equality if 𝑥∗∗ > 0
In the supply side
max𝑞≥0
𝑝𝑖∗∗ 𝑞 − 𝐶(𝑞))
FOC 𝑝𝑖∗∗ − 𝐶′ 𝑞∗∗ ≤ 0 with equality if 𝑞∗∗ > 0
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Lindahl Equilibrium
Market clearing condition
𝜙′𝑖 𝑞∗∗ = 𝐶′(𝑞∗∗) for an interior solution
Thus 𝑞∗∗ = 𝑞𝑜
• The right kind of market can result in the Pareto optimal allocation, even in
the public goods case
Problems -
• Need power to exclude
• Price taking consumer even they are the only buyers of a particular good
• Discriminating (needs perfect information)
• Consumer has to believe that to consume the good have to purchase it!
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Example
For a public good Assume
𝜙𝑖 𝑞 = ln𝑞
𝐶 𝑞 =𝑞2
2
a. Derive the efficient levels of x, p q
b. Derive Lindahl price
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11.D: Multilateral Externalities
• Agents who suffers externalities are different than those who generates
• Differentiate between depletable and non-Depletable externalities.
• Depletable externalities
• Partial equilibrium approach: Given price P of L tradable goods in a
competitive market
• Firms generating externality ℎ𝑗 ∈ ℝ+
• 𝜋 ℎ𝑗 : is a concave profit function over the level of the externality
• I consumers, who have quasi-linear utility function
• 𝜙𝑖 ℎ𝑖 : consumer I’s utility over the amount of depletable externalities
• Negative externality 𝜙′𝑖 ℎ𝑖 > 0, 𝜙′′𝑖 ℎ𝑖 < 0, 𝜋′′𝑗 ℎ𝑗 < 0
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• At the CE firm j (polluter) chooses the level of ℎ𝑗 that solves its PMP
maxℎ𝑗≥0
𝜋𝑗 ℎ𝑗
FOC 𝜋′𝑗(. ) ≤ 0 with equality if ℎ𝑗∗ > 0
In contrast, PO allocation involves
maxℎ 1,….,ℎ 𝐼ℎ1,….,ℎ𝑗
𝜙𝑖 ℎ 𝑖𝐼𝑖=1 + 𝜋𝑗
𝐽𝑗−1 ℎ𝑗
s.t. ℎ 𝑖 = ℎ𝑗𝐽𝑗=1
𝐼𝑖=1
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𝐹𝑂𝐶
𝜙𝑖′ ℎ 𝑖 ≤ 𝜇 with equality if ℎ 𝑖𝑜>0
𝜇 +𝜋′𝑗( ℎ𝑗𝑜) ≤ 0 with equality if ℎ𝑗
𝑜 > 0
ℎ 𝑖 = ℎ𝑗
𝐽
𝑗=1
𝐼
𝑖=1
then
𝜙𝑖 ℎ 𝑖 ≤ −𝜋′𝑗( ℎ𝑗𝑜)
• Importantly, these conditions match the same conditions at competitive
markets in Ch. 10.
• Result
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Non−Depletable externalities
• market is normally unable to result in an efficient allocation.
• Assume externality is completely non-rival in consumption:
o If all J firms generate an aggregate amount of externality ℎ𝑗𝐽𝑗=1
o Every consumer suffers an externality ℎ𝑗𝐽𝑗=1
• CE: each firm increases its level of ℎ𝑗∗ until 𝜋𝑗 ℎ𝑗
∗ = 0
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Non−Depletable externalities
In contrast, PO allocation involves
maxℎ1,….,ℎ𝑗
𝜙𝑖 ℎ𝑖𝐼𝑖=1 + 𝜋𝑗
𝐽𝑗−1 ℎ𝑗
FOC 𝜙′𝑖 ℎ𝑖𝑜𝐼
𝑖=1 + 𝜋𝑗′(ℎ𝑗𝑜) ≤ 0 with equality if ℎ𝑗
𝑜>0
• This exactly coincides with the optimality conditions for a public good
(11.C.1)
• Therefore, unlike in the case of depletable externalities ℎ𝑗∗ (in CE) does not
necessarily coincide with ℎ𝑗𝑜 (PO)
• The free-rider problem arises in non-depletable ext. so, the equi. level of
the negative externality exceeds its optimal level
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Non−Depletable externalities: Methods to achieve the optimality
• If the regulator has adequate information over firms’ profit functions and
consumers’ harm, it can guarantee optimality using quotas or taxes.
1. Setting quotas : ℎ1𝑜, ℎ2
𝑜, . ℎ𝐽
𝑜
2. Taxes. 𝑡ℎ = − 𝜙′𝑖 ℎ𝒊𝑜𝐼
𝑖=1
• firm j’s PMP after the tax
maxℎ𝑗
𝜋𝑗 ℎ𝑗 − 𝑡ℎℎ𝑗
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F.O.C
𝜋𝑗′ ℎ𝑗𝑜 − 𝑡ℎ ≤ 0 with equality if ℎ𝑗
𝑜>0
then
𝜋𝑗′ ℎ𝑗𝑜 + 𝜙′𝑖 ℎ𝑖
𝑜𝐼𝑖=1 ≤ 0 with equality if ℎ𝑗
𝑜>0
• Which exactly coincides with the FOC that solves the social planner problem
3. Tradable Externality Permits.
• Using externality permits to solve the externality problem.
Assume:
• ℎ𝑜 = ℎ𝑗𝑜
• every firm receives ℎ 𝑗
• Price taking firms
• 𝑝ℎ∗: the permits equilibrium price
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firm j’s PMP
maxℎ𝑗
𝜋𝑗 ℎ𝑗 + 𝑝ℎ∗(ℎ 𝐽 − ℎ𝑗)
F.O.C 𝜋′𝑗 ℎ𝑗 + 𝑝ℎ∗ ≤ 0 with equality if ℎ𝑗
𝑜>0
• If all J firms are carrying out this PMP, we need the market clearing
condition ℎ𝑜 = ℎ𝑗𝑜
𝑝ℎ∗ = − 𝜙′𝑖 ℎ𝑖
𝑜
𝐼
𝑖=1
So
𝜋′𝑗 ℎ𝑗 − 𝜙′𝑖 ℎ𝑖𝑜
𝐼
𝑖=1
≤ 0
Chapter 11 MWG: Externalities and Public Goods,
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Multilateral Externalities
• This exactly coincides with the F.O.C that solves the social planner
problem
• ℎ𝑗=ℎ𝑗𝑜
• Advantage of tradable externality Permits
o Requirement of minor information is the advantage of tradable
externality permits, relative to other policy instruments, Data about the
optimal level of pollution, ℎ𝑜. (industry profits,consumers’ damage)
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