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Externalities and PG
MWG- Chapter 11
Chapter 11 MWG: Externalities and Public Good
Simple Bilateral Externality
When external e¤ects are present, CE are not PO. Assume:
1 Two consumers i = 1, 22 The actions of these consumers do not a¤ect prices p 2 RL
3 wi Consumers i�s wealth4 Ui (x1i , ..., xLi , h)5 ∂U2
∂h 6= 0, consumer 1�s choice of h a¤ects consumer 2�swell-being (externality)
Each consumer i derived utility function over the level ofh:
vi (p,wi , h) = maxxi�0
ui (xi , h)
s.t p � xi � wi
Chapter 11 MWG: Externalities and Public Good
Simple Bilateral Externality
We shall assume that the consumer�s ut. function takes aquasilinear form:vi (�) = φi (p, h) + wiwe can rewrite φi (h) and assume φ00i (�) < 0
Competitive Equilibrium: each of the two consumers maximizeher utility limited only by her wealth and P
φ01(h�) � 0, with equality if h� > 0
Interior solution : φ01(h�) = 0
Pareto Optimal Allocation: the optimal level of h mustmaximize the JOIN surplus of the 2 consumers
max φ1(h) + φ2(h)
FOC : φ01(ho ) � �φ02(h
o )
Interior solution : φ01(ho ) = �φ02(h
o )
Chapter 11 MWG: Externalities and Public Good
Simple Bilateral Externality
Considers (ho , h�) >> 0. If φ02(�) < 0 (so h generatesnegative ext). Then, we have φ01(h
o ) = �φ02(ho ) > 0,
because φ01(ho ) is decreasing and φ01(h
o ) = 0! ho < h�
h
Φ’2(h)
Φ’1(h)
h*ho
Chapter 11 MWG: Externalities and Public Good
Traditional Solutions to the Externality Problem
Quotas and Taxes
Suppose negative externality ho < h�
1 Government intervention to achieve e¢ ciency is the directcontrol of the externality: Mandate h = ho
2 Tax on the externality-generating activity
Pigouvian Tax: (A) th = �φ02(ho ) > 0
Consumer 1 then choose the level of h that solves:
maxh�0
φ1(h)� th � h
(B) FOC : φ01(h) � th (with equality if h > 0)
Given that: th = �φ02(ho ), h = ho satis�es (B). In addition,
φ001 (h) < 0, ho must be unique to solution (A)
Chapter 11 MWG: Externalities and Public Good
Traditional Solutions to the Externality Problem
Quotas and Taxes
h
Φ’2(h)
Φ’1(h)
ho
th=Φ’2(h)
Chapter 11 MWG: Externalities and Public Good
Fostering Bargaining over externalities: enforceable property
Assume: Property Rights with regard to theexternality-generating activity
Assign the right to an externality-free environment toconsumer 2Consumer 1 is unable to produce externality without Consumer2�s permissionAssume consumer 2 makes consumer 1 a take-it-or-leave-ito¤er demanding a payment TConsumer 1 accepts i¤ φ1(h)� T � φ1(0)
Consumer 2 will choose her o¤er (h,T ) to solve
Maxh�0,T
φ2(h) + T
s.t φ1(h)� T � φ1(0)
Maxh�0
φ2(h) + φ1(h)� φ1(0)
FOC : φ02(h) + φ01(h) = 0
ho = φ01(h) = �φ02(h)
Chapter 11 MWG: Externalities and Public Good
Fostering Bargaining over externalities: enforceable property
Summary
Consumer 1 has the right to generate as much as theexternality she wants
In the absence of any agreement, consumer 1 will generate h�
Consumer 2 will need to o¤er T < 0 to have h < h�
Consumer 1 will agree to have h i¤: φ1(h)� T � φ1(h�)
Consumer 2 will choose her o¤er (h,T ) to solve
Maxh�0,T
φ2(h) + T
s.t φ1(h)� T � φ1(h�)
Maxh�0
φ2(h) + φ1(h)� φ1(h�)
FOC : φ02(h) + φ01(h) = 0
ho = φ01(h) = �φ02(h)
Chapter 11 MWG: Externalities and Public Good
Fostering Bargaining over externalities: enforceable property
Consumer 1 pays φ1(h)� φ1(0) > 0! to be allowed to setho > 0
Consumer 1 receives φ1(ho )� φ1(h) < 0 for setting ho < h�
Coase Theorem: If trade of the externality can occurthen bargaining will lead to an e¢ cient outcome nomatter how PR are allocated.
Chapter 11 MWG: Externalities and Public Good
Multilateral Externalities
Assumptions
Agents who su¤ers externalities are di¤erent than those whogenerates
Generators of ext: Firms
Experiencing ext: Consumers
Partial equilibrium approach: Given price P of L tradablegoods
J �rms generate the externality
πj (hj ) derived pro�t function over the level of the externality
I consumers, who have quasilinear utility function
φi (ehi ) consumer i�s utility over the amount of ext. eh
Negative externality: φ0i (�) < 0, φ00i (�) < 0,π00j (�) < 0
Chapter 11 MWG: Externalities and Public Good
Non-Depletable Externalities
The externality experienced by each consumer is ∑jhj (The
total amount of the externality produced by the �rm)At any CE, each �rm will wish to set the h�j satisfying
πj (h�j ) � 0 (with equality if h�j > 0)
In contrast, any PO allocation involves (ho1 , ..., hoj )
Max(h1,...,hj )�0
I
∑i=1
φi (∑jhj ) +
J
∑j=1
πj (hj )
FOC :I
∑i=1
φ0i (∑jhoj ) � �π0j (h
oj )
In the case of a Non-Depletable Externality, a market-basedsolution would require personalized markets for the externality,as in Lindhal eq. concept.
In contrast, given adequate information, the government canachieve optimality using quotas or taxes
Chapter 11 MWG: Externalities and Public Good
Private Information and Second-Best Solutions
Presence of asymmetric information!
Generators of ext: Firms
Experiencing ext: Consumers
φ(h, η) consumer�s derived utility
π(h, θ) derived pro�t function θ 2 R
η and θ are privately observed
The ex-ante likelihoods (prob. distribution) of various valuesof η and θ are publicly known
η and θ are independently distributed
φ(h, η) and π(h, θ) are strictly concave in h for any givenvalue of θ and η
Chapter 11 MWG: Externalities and Public Good
Private Information and Second-Best Solutions
Clarke�s example: Sausage Company
Two types of individuals: a¤ected (sensitive nose) and una¤ected
Two types of �rms: e¢ cent and ine¢ cient
Chapter 11 MWG: Externalities and Public Good
Private Information and Second-Best Solutions
Presence of asymmetric information
Measurement of �rm�s bene�ts: b(θ) = π(h, θ)� π(0, θ) > 0
Measurement of consumer�s cost from h:c(η) = φ(0, η)� φ(h, η) > 0
G (B) and F (C ) distribution functions of these two variablesinduced by the underlying probability distribution of η and θ
density functions g(b) and f (c)
In the absence of an agreement h = 0
Any arrangement that guarantees PO outcomes, the �rmshould allow to set h = h whenever b > c
Chapter 11 MWG: Externalities and Public Good
Private Information and Second-Best Solutions
Decentralized Bargaining
h? when consumer cost is c
1 Firms will agree to pay T i¤ b � T2 Consumers knows that if she demands a payment of T , theprob. that the �rm accepts is equal the prob. thatb � T ! (1� G (T ))
MaxT(1� G (T ))(T � c)
Solution : T �c > c
Chapter 11 MWG: Externalities and Public Good
Quotas and Taxes
The aggregate surplus from ext: φ(h, η) + π(h, θ)
Firm : maxh�0
π(h, θ)
st h � bhOptimal Choice : ho (bh, θ)
The e¤ect of the quota is to make h less sensitive to η and θthan is required by optimality. Firms will be insensitive to η
Chapter 11 MWG: Externalities and Public Good
Quotas and Taxes
The loss in aggregate surplus arising under the quota for typesη and θ is given by:
φ(hq(bh, θ), η) + π(hq(bh, θ), θ)� φ(ho (θ, η), η)� π(ho (θ, η), θ)
=
hq (bh,θ)Zho (θ,η)
( ∂π(h,θ)∂h ,
∂φ(h,η)∂h )dh
The loss in aggregate surplus under a quota for types (θ, η)
Chapter 11 MWG: Externalities and Public Good
TAX ON THE FIRM OF t UNITS
Externalities
Firm:maxh�0
π(h, θ)� t � h, Optimal Choice : ht (t, θ)
The loss in aggregate surplus arising under the tax for types ηand θ is given by:
φ(ht (t, θ), η) + π(ht (t, θ)), θ)� φ(ho (θ, η), η)� π(ho (θ, η), θ)
=
ht (t ,θ)Zho (θ,η)
( ∂π(h,θ)∂h ,
∂φ(h,η)∂h )dh
But now assuming that a tax is set a t = � ∂φ(ho (θ,η),η)∂h
Chapter 11 MWG: Externalities and Public Good
TAX ON THE FIRM OF t UNITS
Externalities
the loss in aggregate surplus under a tax for types (θ, η)
Note that quotas and taxes, the level of externality isresponsive to changes in Mg bene�ts but not to changes inthe Mg cost of the consumer.
Chapter 11 MWG: Externalities and Public Good
Quota or Tax Performs betters?
Quota or Tax performs betters?
It depends!
Quota h = h� Maximizes aggregate surplus for all θ
Chapter 11 MWG: Externalities and Public Good
Quota or Tax Performs betters?
1 Tax t = t� maximizes aggregate surplus for all θ