Externalities and PG - WordPress.com · Chapter 11 MWG: Externalities and Public Good Traditional Solutions to the Externality Problem Quotas and Taxes Suppose negative externality

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



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 )



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


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)


Traditional Solutions to the Externality Problem

Quotas and Taxes

h

Φ’2(h)

Φ’1(h)

ho

th=Φ’2(h)


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)



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)



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.


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


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


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



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



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



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


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


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)


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


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.


Quota or Tax Performs betters?

Quota or Tax performs betters?

It depends!

Quota h = h� Maximizes aggregate surplus for all θ


Quota or Tax Performs betters?

1 Tax t = t� maximizes aggregate surplus for all θ

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Externalities and PG - WordPress.com · Chapter 11 MWG: Externalities and Public Good Traditional Solutions to the Externality Problem Quotas and Taxes Suppose negative externality