Microeconomics II MWG, Ch. 11 Externalities and Public Goods
Transcript of Microeconomics II MWG, Ch. 11 Externalities and Public Goods
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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
β’ FFWT any competitive equilibrium is Pareto optimal. i.e. markets allocate resources efficiently.
β’ SFWT (given suitable convexity assumptions) any Pareto optimal allocation can be supported as a competitive equil.
β’ What happens if the behavior of some agent affects the welfare of others?
β’ When external effects are present, CE is still PO, as long as the effects are transmitted via prices, markets are efficient.
β’ 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
β’ Bilateral Externality vs Multilateral Externality
β’ Externalities can be produced by consumers as well as firms.
o Consumption side: noise pollution
o Production side: Chemical plantβs discharges reducing fisheryβs catch
β’ Externalities can be positive or negative
β’ Public Good -special kind of externality
o Non-rivalry in consumption: National defense & flood control
o Non-excludable: lighthouse
Basic Problem β 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.
β’ Directly Affected - not through prices
Viner (1931)
β’ Pecuniary externalities: actions affecting prices;
β’ Non-pecuniary (true) externalities actions not affecting prices (that's
what we're studying)
Fisheryβs productivity affected by emissions from oil refinery.
Fisheryβs profitability affected by price of oil.
Neighbor's flower garden
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Simple Bilateral Externality
A two-agent partial equilibrium model
β’ 2 consumers (can think of 2 producers or 1 of each)
β’ L traded goods
β’ ππ’ consumers Iβs wealth
β’ Consumer iβs utility function
π’π ππ’, β
ππ’= (π₯π1, π₯π2,β¦π₯ππΏ)
h: amount of externality
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Simple Bilateral Externality
β’ Consumer 1 chooses externality h, assume h 0, h
β’ Consumer 2 takes the externality so ππ’βπ
πββ 0
The Consumer derived their utility function on the level of h
π£π π€π , π, β = maxπ₯π>0
π’ ππ, β
s.t. ππ₯π β€ π€π
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Equilibrium Choices are Not Efficient
β’ Given the quasi-linear preferences the consumerβs indirect utility function
takes the form
π£π π€π , π, β = π₯1π + π(π₯β1π , β)
We know that π₯π1= π€π β ππ₯β1π(π, β), then
π£π π€π , π, β = π€π β ππ₯β1π π, β + π π₯π2, π₯π3,β¦π₯ππΏ
or
π£π π€π , π, β = ππ(π, β) + π€π
It can be written
ππ(π, β)=ππ(β)
What is ππ(β)?
Assume πβ²π(β) β€ 0 and, π"π β < 0
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Equilibrium Choices are Not Efficient
β’ How will consumer 1 choose h?
β’ Efficient outcome
πβ²π(β) β€ 0 with equality if β > 0
Figure 11.B.1 shows this results
β’ Is there any problem with this result?
β’ 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
maxββ₯0
π1(β) + π2(β)
FOC
πβ²1(βπ) + πβ²2(β
π) β€ 0
For interior solution
πβ²1(β
π) = β πβ²2(βπ)
Result: ββ > βπ
Figure 11.B.1 shows this results
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Equilibrium Choices are Not Efficient
β’ Three important points that come out of this results
β’ Externalities are not necessarily eliminated at the Pareto optimal solution
o When dose this happen?
β’ What would be the result if we have positive externalities
β’ 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
β’ Suppose negative externality βπ < ββ
β’ Social planer sets maximum of β = βπ
β’ Emitter solves
max0<β<βπ
π1(β)
We know βπ < ββ, so polluter will do as much as he can
Perfect information Requirements:
β’ Policy maker needs to compute βπ
β’ So he needs to compute ππ(β)
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β’ Taxes
β’ Pigouvian taxation: Imposing tax on the externality-generating activity
β’ Taxes consumer 1 for producing h ; sets per unit tax
π‘β = βπβ²2 βπ > 0
β’ What does this mean?
β’ Emitter solves
maxβ>0
π1 β β π‘ββ
FOC πβ²1 βπ β€ π‘β with equality if βπ > 0
Figure 11.B.2
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Two questions:
β’ How much tax must we impose in case that the negative externality is very
substantial (and βπ=0)
β’ How can the previous discussion be extended to positive externality
Subsidies
β’ Policy maker sets the unit of subsidy π β = βπ‘β= πβ²2 βπ > 0
Where π β‘ π (βββh)
Emitter solve
maxβ>0
π1 β +π β(βββh)
FOC πβ²1 βπ + π β β€ 0 with equality if βπ > 0
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Some important points about Pigouvian taxation:
β’ 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
β’ 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?
β’ The quota and the Pigouvian tax are equally effective under complete
information
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Fostering bargaining over externalities
Coase/Bargaining Solution
β’ Coase/Bargaining Solution: Coaseβs famous paper (The Problem of
Social Cost ,Coase 1960), was a direct response to Pigouβs argument
β’ 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 βmarket failureβ
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
β’ Small numbers of agents involved
β’ Perfect information among the agents
β’ Assigned property right to externality otherwise No Big Deal
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Example: two agents
β’ Still looking at negative externality
Case 1: 2 Has Property Right
β’ Assign right to externality-free environment to consumer 2.
β’ Initial state β = 0
β’ Consumer 1 cannot produce externality without Consumer 2βs permission
β’ Bargain β agents will bargain to reach an agreement over (h,T) ; if no agreement is reached the default value is (0,0)
β’ Bargaining Power - this is independent of the property right and reliant on the ability of negotiation.
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β’ 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
β’ Consumer 1 accepts iff π1 β β π β₯ π1(0)
β’ Consumer 2 will pick the offer (h,T) to solve
maxββ₯0
π2 h + T
π . π‘ π1 β β π β₯ π1 0
The constraint will be binding in
maxββ₯0
π2 h + π1 β β π1 0
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Efficient Choice
πβ²2 β + πβ²1(β) β€ 0 with equality if β > 0
β’ This coincides with that solving the social plannerβs problem
β’ i.e. β = β0 with π = π1 β0 β π1 0 > 0
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Case 2: 1 Has Property Right
β’ Assign right to the polluter (consumer 1) to generate as much as the
externality she wants
β’ In the absence of any agreement, consumer 1 will generate ββ
β’ 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
β’ Indeed, consumer 2 can pay $T the consumer 1 in exchange of a lower
level of pollution
β’ Consumer 1 accepts iff π1 β + π β₯ π1(ββ)
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β’ Consumer 2 will choose the offer (h,T) to solve
maxββ₯0
π2 β β T
π . π‘ π1 β + π β₯ π1 ββ
The constraint will be binding in
maxββ₯0
π2 h + π1 β βπ1 ββ
Efficient Choice
πβ²2 β + πβ²1(β) β€ 0
β’ Again this coincides with that solving the social plannerβs problem
β’ i.e. β = β0 with π = π1 β0 β π1 ββ > 0
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Results
Case 1: 2 Has Property Right
β’ Consumer 1 pays π = π1 β0 β π1 0 > 0 to be allowed to set β0> 0
Case 2: 1 Has Property Right
β’ Consumer 2 pays π = π1 β0 β π1 ββ > 0 for setting β0 < ββ
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
β’ 1 must pays π = π1 β0 β π1 0 to 2
Then consumer 2βs utility is π2 β0 + T and then consumer 1βs utility is
π1 β0 β T= π1 0
Hence, consumer 2βs utility is higher than that of consumer 1 if
π2 β0 + π1 β0 > 2π1 0
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Coase/Bargaining Solution
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Case 2: 1 Has Property Right
β’ 2 must pays π = π1 ββ β π1 β0 to 1
Then consumer 1βs utility is π1 β0 + T = π1 ββ but consumer 2βs utility is
π2 β0 β T
Hence, 1βs utility is bigger than that of consumer 2 if
2π1 ββ > π1 β0 + π2 β0
β’ Therefore, when the agent has bargaining power has a total utility higher
the average of welfare at the Pareto optimum, and vice versa.
2π1 ββ > π1 β0 + π2 β0 >2π1 0
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Figure 11.B.3: The final distribution of utilities under different PR and BP
β’ At point a consumer 2 has the property right ( h = 0)
β’ Therefore, the take-it-or-leave-it offer leads to point f in the first case and
point e in the second case
β’ At point b consumer 1 has the property right
β’ 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:
β’ Property rights must be perfectly defined.
β’ Property rights must be perfectly enforced,
β’ The polluter must know the cost of the externality for the affected agents
β’ The affected agents must know the polluterβs profit function
Questions on Coase theorem
β’ Are these assumptions practical?
β’ Isnβt Coase theory itself a blackboard economics?
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Externalities as missing markets
Missing Market definition
β’ Lack of coordination
β’ Technology
β’ Transaction costs
β’ Trust or information
An alternative view to externalities:
β’ Externalities are a commodity which lacks a market.
β’ We can simply show that, if externalities were a traded commodity, the
produced level of that coincides with the Pareto optimal level β = β0.
β’ Suppose a well defined property rights, and a competitive market for the
right
β’ πβ: the price of one unit of externality-generating activity
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Externalities as missing markets
β’ consumer 1 (the emitter) chooses how many polluting rights to buy
maxβ1β₯0
π1 β1 β πββ1
F.O.C: πβ²1 β1 < πβ with equality if β1 >0
β’ Similarly, the individual affected decides how many polluting rights to sell,
maxβ2β₯0
π2 β2 + πββ2
β’ F.O.C: πβ²2 β2 + πβ < 0 with equality if β2 >0
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Externalities as missing markets
β’ the competitive market for polluting rights must clear
β1 = β2 = βββ
πβ²1 βββ β€ πβ β€ βπβ²2 βββ
Or simply
πβ²1 βββ β€ -πβ²2 βββ with equality if βββ >0
β’ Interestingly, this condition coincides with the F.O.C under the Pareto
optimal level of the externality β0. i.e.
πββ = πβ²1 β0 = β πβ²2 β0
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11.C. Public Goods
β’ 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
β’ Non-Excludable: public goods usual known non-excludable, but in Mas-
Colell they can be either excludable or non-excludable;
β’ 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
β’ I consumers,
β’ One public good x
β’ L traded private goods
Assume:
β’ quasi-linear utility function over L private goods and the public good
β’ π’ π±π’, π₯ = π₯π1 + π’ (π₯π2, π₯π3,β¦π₯ππΏ) + ππ(π₯)
β’ π±π’ = (π₯π1, π₯π2,β¦π₯ππΏ)
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β’ π₯: amount of public good which is actually a good (πβ²π π₯ > 0)
β’ ππ(π₯) is concave (πβ²β²π π₯ < 0)
β’ level of consumption of x has no effect on prices of the private goods
β’ Cost to produce public good is C(π)
β’ C(π) are convex in q
β’ q : amount of public good produced
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Pareto Optimal Solution
The partial equilibrium model
The social planner maximizes aggregate surplus,
maxπβ₯0
ππ(π)πΌ1 -C(q)
F.O.C: Samuelson rule
πβ²π(ππ)πΌ
π=1 > Cβ²(ππ) with equality if π >0
β’ The social planner increases the provision of a public good until that the
sum of the consumersβ marginal benefit ( or marginal social benefit) is
equal to its marginal cost.
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Inefficiency of private provision of public goods
β’ Show that market provision of PG (i.e. competitive, price taking
equilibrium) is inefficient
β’ Assume a market exists for the public good
β’ Market-Clearing - at p* public good produced (supplied) equals public
good consumed
β’ Total amount of PG purchased π₯ = π₯ππΌπ=1
β’ No Exclusion: π₯βππΌπβ π optimal purchases of public good all other
consumers
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At a competitive equilibrium:
1. Consumers maximize utility
each consumer iβs purchase of the public good π₯βπ must satisfy
maxπ₯πβ₯0
ππ π₯π + π₯βππΌπβ π βπΌ
1 πβπ₯π
F.O.C
πβ²π π₯βπ + π₯βπ
πΌπβ π β€ πβ with equality if π₯βπ >0
or
πβ²π π₯β β€ πβ with equality if π₯β >0
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2. Firms maximize profit:
The firm producing the public good must solve the fallowing PMP
maxπβ₯0
(πβπ β πΆ(π))
F.O.C
πβ β πΆβ²(πβ) β€ 0 with equality if πβ >0
At a competitive equilibrium πβ = π₯β
πβ²π πβ = πΆβ²(πβ) if πβ >0 and πβ²π π
β < πΆβ²(πβ) if πβ= 0
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Compare:
Pareto Efficient Solution πβ²π(ππ)πΌ
π=1 = Cβ²(ππ)
Market Provision πβ²π πβ = πΆβ²(πβ)
Conclusion: when people make voluntary contributions, the market will
provide too little of the public good, ππ > πβ
β’ This fact can be understood in terms of positive externalities
β’ Free rider problem.
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Free rider problem.
β’ Suppose
πβ²1 π₯ < πβ²2 π₯ < β― < πβ²πΌ π₯
In this case, condition πβ²π π₯β β€ πβ 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
β’ Regulation
β’ Price-based interventions: compulsory participation (Taxation, Tying)
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Suppose there are I consumers each with benefit function ππ π₯
Using the Pigouvian taxation we can implement the optimal consumption xO by setting the per unit subsidy to each consumer equal to
π π = πβ²π(π₯π)
πβ π
Because
maxπ₯πβ₯0
ππ π₯π + π₯π
πΌ
πβ π
+ π ππ₯π β πβπ₯π
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The necessary and sufficient first-order condition for this problem is
πβ²π π₯π + π₯π
πΌ
πβ π
+ π π = π
Substituting in the above subsidy and combining with the market-clearing
condition
πβ²π π₯ π + π₯ π
πΌ
πβ π
+ πβ²π(π₯π)
πβ π
= π
πβ²π π + πβ²βπ π
π β€ πβ² π
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Lindahl Equilibrium
A market-based solution to public goods
Firm charges each consumer πββ
In the demand side
maxπ₯πβ₯0
ππ π₯π β ππββπ₯π
FOC πβ²π π₯ββ β€ ππ
ββwith equality if π₯ββ > 0
In the supply side
maxπβ₯0
ππββ π β πΆ(π))
FOC ππββ β πΆβ² πββ β€ 0 with equality if πββ > 0
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Lindahl Equilibrium
Market clearing condition
πβ²π πββ = πΆβ²(πββ) for an interior solution
Thus πββ = ππ
β’ The right kind of market can result in the Pareto optimal allocation, even in
the public goods case
Problems -
β’ Need power to exclude
β’ Price taking consumer even they are the only buyers of a particular good
β’ Discriminating (needs perfect information)
β’ Consumer has to believe that to consume the good have to purchase it!
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Example
For a public good Assume
ππ π = lnπ
πΆ π =π2
2
a. Derive the efficient levels of x, p q
b. Derive Lindahl price
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11.D: Multilateral Externalities
β’ Agents who suffers externalities are different than those who generates
β’ Differentiate between depletable and non-Depletable externalities.
β’ Depletable externalities
β’ Partial equilibrium approach: Given price P of L tradable goods in a
competitive market
β’ Firms generating externality βπ β β+
β’ π βπ : is a concave profit function over the level of the externality
β’ I consumers, who have quasi-linear utility function
β’ ππ βπ : consumer Iβs utility over the amount of depletable externalities
β’ Negative externality πβ²π βπ > 0, πβ²β²π βπ < 0, πβ²β²π βπ < 0
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β’ At the CE firm j (polluter) chooses the level of βπ that solves its PMP
maxβπβ₯0
ππ βπ
FOC πβ²π(. ) β€ 0 with equality if βπβ > 0
In contrast, PO allocation involves
maxβ 1,β¦.,β πΌβ1,β¦.,βπ
ππ β ππΌπ=1 + ππ
π½πβ1 βπ
s.t. β π = βππ½π=1
πΌπ=1
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πΉππΆ
ππβ² β π β€ π with equality if β ππ>0
π +πβ²π( βππ) β€ 0 with equality if βπ
π > 0
β π = βπ
π½
π=1
πΌ
π=1
then
ππ β π β€ βπβ²π( βππ)
β’ Importantly, these conditions match the same conditions at competitive
markets in Ch. 10.
β’ Result
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NonβDepletable externalities
β’ market is normally unable to result in an efficient allocation.
β’ Assume externality is completely non-rival in consumption:
o If all J firms generate an aggregate amount of externality βππ½π=1
o Every consumer suffers an externality βππ½π=1
β’ CE: each firm increases its level of βπβ until ππ βπ
β = 0
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NonβDepletable externalities
In contrast, PO allocation involves
maxβ1,β¦.,βπ
ππ βππΌπ=1 + ππ
π½πβ1 βπ
FOC πβ²π βπππΌ
π=1 + ππβ²(βππ) β€ 0 with equality if βπ
π>0
β’ This exactly coincides with the optimality conditions for a public good
(11.C.1)
β’ Therefore, unlike in the case of depletable externalities βπβ (in CE) does not
necessarily coincide with βππ (PO)
β’ 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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NonβDepletable externalities: Methods to achieve the optimality
β’ If the regulator has adequate information over firmsβ profit functions and
consumersβ harm, it can guarantee optimality using quotas or taxes.
1. Setting quotas : β1π, β2
π, . βπ½
π
2. Taxes. π‘β = β πβ²π βπππΌ
π=1
β’ firm jβs PMP after the tax
maxβπ
ππ βπ β π‘ββπ
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F.O.C
ππβ² βππ β π‘β β€ 0 with equality if βπ
π>0
then
ππβ² βππ + πβ²π βπ
ππΌπ=1 β€ 0 with equality if βπ
π>0
β’ Which exactly coincides with the FOC that solves the social planner problem
3. Tradable Externality Permits.
β’ Using externality permits to solve the externality problem.
Assume:
β’ βπ = βππ
β’ every firm receives β π
β’ Price taking firms
β’ πββ: the permits equilibrium price
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firm jβs PMP
maxβπ
ππ βπ + πββ(β π½ β βπ)
F.O.C πβ²π βπ + πββ β€ 0 with equality if βπ
π>0
β’ If all J firms are carrying out this PMP, we need the market clearing
condition βπ = βππ
πββ = β πβ²π βπ
π
πΌ
π=1
So
πβ²π βπ β πβ²π βππ
πΌ
π=1
β€ 0
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Multilateral Externalities
β’ This exactly coincides with the F.O.C that solves the social planner
problem
β’ βπ=βππ
β’ 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, βπ. (industry profits,consumersβ damage)
Chapter 11 MWG: Externalities and Public Goods, Hamid Kordbacheh, Alzahra university, Iran
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