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Microeconomics II
MWG, Ch. 11 Externalities and Public Goods
Alzahra University Department of Economics
Hamid Kordbacheh
2
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.
3 Chapter 11 MWG: Externalities and Public Goods,
Hamid Kordbacheh, Alzahra university, Iran
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)
4 Chapter 11 MWG: Externalities and Public Goods,
Hamid Kordbacheh, Alzahra university, Iran
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
5 Chapter 11 MWG: Externalities and Public Goods,
Hamid Kordbacheh, Alzahra university, Iran
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
Chapter 11 MWG: Externalities and Public Goods, Hamid Kordbacheh, Alzahra university, Iran
6
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
Chapter 11 MWG: Externalities and Public Goods, Hamid Kordbacheh, Alzahra university, Iran
7
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
8 Chapter 11 MWG: Externalities and Public Goods,
Hamid Kordbacheh, Alzahra university, Iran
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?
9 Chapter 11 MWG: Externalities and Public Goods,
Hamid Kordbacheh, Alzahra university, Iran
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
10 Chapter 11 MWG: Externalities and Public Goods,
Hamid Kordbacheh, Alzahra university, Iran
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?
11 Chapter 11 MWG: Externalities and Public Goods,
Hamid Kordbacheh, Alzahra university, Iran
Traditional Solutions to the Externality Problem
12 Chapter 11 MWG: Externalities and Public Goods,
Hamid Kordbacheh, Alzahra university, Iran
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)
• 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
Chapter 11 MWG: Externalities and Public Goods, Hamid Kordbacheh, Alzahra university, Iran
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
Chapter 11 MWG: Externalities and Public Goods, Hamid Kordbacheh, Alzahra university, Iran
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
Chapter 11 MWG: Externalities and Public Goods, Hamid Kordbacheh, Alzahra university, Iran
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
Chapter 11 MWG: Externalities and Public Goods, Hamid Kordbacheh, Alzahra university, Iran
The key conditions
• Small numbers of agents involved
• Perfect information among the agents
• Assigned property right to externality otherwise No Big Deal
17 Chapter 11 MWG: Externalities and Public Goods,
Hamid Kordbacheh, Alzahra university, Iran
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.
18 Chapter 11 MWG: Externalities and Public Goods,
Hamid Kordbacheh, Alzahra university, Iran
• 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
19 Chapter 11 MWG: Externalities and Public Goods,
Hamid Kordbacheh, Alzahra university, Iran
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
Chapter 11 MWG: Externalities and Public Goods, Hamid Kordbacheh, Alzahra university, Iran
20
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∗)
Chapter 11 MWG: Externalities and Public Goods, Hamid Kordbacheh, Alzahra university, Iran
21
• 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
22 Chapter 11 MWG: Externalities and Public Goods,
Hamid Kordbacheh, Alzahra university, Iran
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.
23 Chapter 11 MWG: Externalities and Public Goods,
Hamid Kordbacheh, Alzahra university, Iran
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
Chapter 11 MWG: Externalities and Public Goods, Hamid Kordbacheh, Alzahra university, Iran
24
Coase/Bargaining Solution
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
Chapter 11 MWG: Externalities and Public Goods, Hamid Kordbacheh, Alzahra university, Iran
25
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
Chapter 11 MWG: Externalities and Public Goods, Hamid Kordbacheh, Alzahra university, Iran
26
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?
Chapter 11 MWG: Externalities and Public Goods, Hamid Kordbacheh, Alzahra university, Iran
27
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
Chapter 11 MWG: Externalities and Public Goods, Hamid Kordbacheh, Alzahra university, Iran
28
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
Chapter 11 MWG: Externalities and Public Goods, Hamid Kordbacheh, Alzahra university, Iran
29
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
Chapter 11 MWG: Externalities and Public Goods, Hamid Kordbacheh, Alzahra university, Iran
30
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
Chapter 11 MWG: Externalities and Public Goods, Hamid Kordbacheh, Alzahra university, Iran
31
The taxonomy of four different types of goods
Chapter 11 MWG: Externalities and Public Goods, Hamid Kordbacheh, Alzahra university, Iran
32
Rivalrous Non-rivalrous
Excludable Private Good Club Good
Non-excludable Common property
resource
Public Good
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)
Chapter 11 MWG: Externalities and Public Goods, Hamid Kordbacheh, Alzahra university, Iran
33
• 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
Chapter 11 MWG: Externalities and Public Goods, Hamid Kordbacheh, Alzahra university, Iran
34
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.
Chapter 11 MWG: Externalities and Public Goods, Hamid Kordbacheh, Alzahra university, Iran
35
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
Chapter 11 MWG: Externalities and Public Goods, Hamid Kordbacheh, Alzahra university, Iran
36
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
Chapter 11 MWG: Externalities and Public Goods, Hamid Kordbacheh, Alzahra university, Iran
37
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
Chapter 11 MWG: Externalities and Public Goods, Hamid Kordbacheh, Alzahra university, Iran
38
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.
Chapter 11 MWG: Externalities and Public Goods, Hamid Kordbacheh, Alzahra university, Iran
39
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)
Chapter 11 MWG: Externalities and Public Goods, Hamid Kordbacheh, Alzahra university, Iran
40
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
Chapter 11 MWG: Externalities and Public Goods, Hamid Kordbacheh, Alzahra university, Iran
41
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
Chapter 11 MWG: Externalities and Public Goods, Hamid Kordbacheh, Alzahra university, Iran
42
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
Chapter 11 MWG: Externalities and Public Goods, Hamid Kordbacheh, Alzahra university, Iran
43
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!
Chapter 11 MWG: Externalities and Public Goods, Hamid Kordbacheh, Alzahra university, Iran
44
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
Chapter 11 MWG: Externalities and Public Goods, Hamid Kordbacheh, Alzahra university, Iran
45
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
Chapter 11 MWG: Externalities and Public Goods, Hamid Kordbacheh, Alzahra university, Iran
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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
Chapter 11 MWG: Externalities and Public Goods, Hamid Kordbacheh, Alzahra university, Iran
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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
Chapter 11 MWG: Externalities and Public Goods, Hamid Kordbacheh, Alzahra university, Iran
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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
Chapter 11 MWG: Externalities and Public Goods, Hamid Kordbacheh, Alzahra university, Iran
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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
Chapter 11 MWG: Externalities and Public Goods, Hamid Kordbacheh, Alzahra university, Iran
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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
Chapter 11 MWG: Externalities and Public Goods, Hamid Kordbacheh, Alzahra university, Iran
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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)
Chapter 11 MWG: Externalities and Public Goods, Hamid Kordbacheh, Alzahra university, Iran
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