Lecture Externalities General Equilibrium

8/13/2019 Lecture Externalities General Equilibrium

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Theory of Externalities

General Equilibrium Model

Consider an economy with

x i two consumption goods indexed by i = 1,2 consumed by representativeconsumer

two production goods indexed by i = 1,2 , produced by two firms indexedy i k

by k = 1,2 , where superscripts denote identity of firm and subscripts

denote type of good

There are two externalities affecting firm 2 :

one generated by consumers consumption of good 1 , x 1

one generated by production of good 1 by firm 1 , y 11

Illustrative example: pollution of a river by city inhabitants (municipalsewage) and a firm (industrial effluents) that affect adownstream waterusing firm


3/23

Consumer

U ( ) is differentiable, increasing and strictlyU x x ( , )1 2quasiconcave

initial endowment( , ) 1 2

Producers

firm 1 's production technology, where is firm 1 'sy f y 11 1

21= ( ) y 11

output, is differentiable and concave and is firmf 1 y 21

1 's input (negative output) of good 2

firm 2 's production technology, where is firm 2 'sy f y y x 22 2

12

11

1= ( , , ) y 22

output, is differentiable and concave, is firm 2 'sf 2 y 12

input (negative output) of good 1 , is firm 1 's outputy 11

of good 1 and x 1 is consumers consumption of good 1

= inputs = negative outputsy y 21 12,

Since inputs have negative sign

f

y

f

y

1

21

2

120 0,

Hence, firm 1 's production of good 1 and consumers consumption of good1 both enter firm 2 's production function, generating two negative externalities.


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Pareto Optimal Allocation

{ }max ( , )

, , , , ,

( )

( , , )

U x x

x x y y y y

y y x

y y x

y f y

y f y y x

1 2

1 2 11

21

22

12

11

12

1 1

21

22

2 2

1

1 1

2

1

22 2

12

11

1

0

0

0

0

s.t.scarcity contraints

technological constraints

+ + + +

+ +

We can ignore the inequality constraints because of the convexity and concavityassumptions these assumptions are sufficient conditions for interior solutions .

The Lagrangian can be written as

{ }( ) ( )( ) ( )

max

, , , , ,

( , )

( ) ( , , )x x y y y y

L U x x y y x y y x

y f y y f y y x 1 2 11

21

22

12

1 2 1 11

12

1 1 2 21

22

2 2

1 11 1

21

2 22 2

12

11

1

= + + + + + + + + + +

(1)

L

x

U

x

f

x 1 11 2

2

1

0= + =

(2)

L

x

U

x 2 2 2 0= =

(3)

L

y

f

y 11 1 1 2

2

11 0= + =

(4)

L

y

f

y 21 2 1

1

21 0= + =

(5) L

y 22 2 2 0= =

(6)

L

y

f

y 12 1 2

2

12 0= + =


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Solution

Eliminate . Express efficiency conditions in terms of . 1 2and

1

2

Efficient consumption

Rewrite equations (1) and (2). Substitute from equation (5): 2 2=

(1) 1 1 2

2

1= +U

x

f

x

(2)

2 2=

U

x

Substituting equation (2) into equation (1) and dividing yields:

(7)

1

2

1 2

2

1

2

1

2

2

1

=+

= +

U

x

U

x

f

x U

x

U

x U

x

f

x

where the RHS of equation (7) is the social M RS 1,2 . The social M RS 1,2 takesinto account all effects of consumption activities (marginal benefits to consumeras well as marginal external costs imposed on others as a result of consumptionactivity).

The social M RS 1,2 must take into account that substituting one unit of good1 for good 2 affects the production of good 2 by and therefore the

f

x

2

1

individuals utility level is changed by .

U

x

f

x 2

2

1


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Efficient production for firm 1

Rewrite equations (3) and (4). Substitute from equation (5): 2 2=

(3) 1 1 22

11=

f

y

(4) 2 11

21=

f

y

Substitute equation (4) into equation (3). Dividing the equations yields:

(8)

1

2

1

21

2

11

1

2

1

1

2

1

2

11

11=

+=

f

y

f

y

f

y

f

y

f

y

The RHS of equation (8) is the social M RT 2,1 . By using one additional unit

of good 2 as an input, firm 1 produces and consequently affects the

f

y

1

21

production of good 2 by .

f

y

f

y

2

1

1

1

2

1


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Efficient production for firm 2

Rewrite equations (5) and (6):

(5) 2 2=

(6) 1 22

12=

f

y

Dividing equation (6) by equation (5) yields:

(9)

1

2

2

12=

f

y

N ote that firm 2 's production activity does not generate an externality,

hence the RHS of equation (9), the social M RT 1,2 , equals the private M RT 1,2 .


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Pareto Optimal Allocation

1

2

1

2

2

11

21

2

11

2

12

1 2 2 1 1 2

1= + = =

= = =

U

x U

x

f

x f

y

f

y

f

y

relative

scarcity

prices

social M RS social M RT social M RT , , ,

Key results

optimal organization of economy does not necessarily require totalelimination of externalities, even when they are negative consumptionand production good 1 affects production of good 2 negatively butoptimality does not require eliminating production of good 1

in general, zero pollution is not optimal

external costs (or direct effects of one agents actions upon others) must beinternalized in consumption and production decisions


9/23

Competitive Equilibrium

Consumers decision problem

max,

( , ) ( , )x x

U x x B p p p p p x p x 1 2

1 2 1 1 2 2 1 2 1 1 2 2s.t. = + + = +

where is the value of the consumers initial endowment andp p 1 1 2 2 + are industrial profits( , )p p 1 2

max

,

( , ) ( )

x x

L U x x + B p x p x

1 2

1 2 1 1 2 2=

L

x

U

x p

1 11 0= =

L

x

U

x p

2 22 0= =

Eliminating 8 yields:

p

p

U

x U

x

1

2

1

2

=

relative prices = private M RS 1 2,


10/23

Firm 1 's decision problem

max ( )

y

p y p y y f y

21

11 1

12 2

111 1

21 = + =s.t.

max ( )y

p f y p y

21

11

121

2 21 = +

1

21 1

1

21 20

y p

f

y p

= + =

p p

f y

121

21

=

p p f

y

1

21

21

1=


11/23


max ( , , )

y

p f y y x p y

12

22

212

11

1 1 12 = +

2

12 2

2

12 1 0y

p f

y p = + =

p p

f y

212

12

=

p

p

f

y 1

2

2

12=


12/23

Competitive Equilibrium

A competitive equilibrium is a vector of prices (p 1 ,p 2 ) and an allocation such that( )x x y y y y 1 2 11 21 22 12, , , , ,

firms profits and consumers utility are maximized (firstordernecessary and sufficient conditions are satisfied)

supply and demand is equalized in both markets

At a competitive equilibrium, selfinterest maximization leads each agent toequate his/her private marginal rate of substitution or transformation to the priceratio, resulting in the equalization of private rates:

p

p

U

x U

x f

y

f

y

relative prices

private M RS private M RT private M RT

1

2

1

2

1

21

2

12

12 2 1 1 2

1= = =

= = =

, , ,


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Comparing Competitive Equilibrium and Pareto Optimum

In contrast, a Pareto optimal allocation requires equalization of social

marginal rates of substitution and transformation to relative scarcity or shadowprices.

divergence of private and social valuations

economic decisions too decentralized at competitive equilibrium

agents generating negative externality will produce/consume too much

prices of negative externalitygenerating production or consumption goodsare too low (since full social costs of economic activity not reflected in

market prices)

In general, CE with externalities not PO!


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Decentralized Solutions

Optimal Taxation

Is there a tax structure that will sustain a competitive equilibrium as aPareto optimum? Can we find a set of taxes for polluters and victim damage orcompensation payments that will induce consumers and firms to choose Paretooptimal levels of economic activities?

Define: t c garbage tax per unit of good 1 consumed

t f emission tax per unit of good 1 produced

Jdamage compensation per unit of good 2 produced

Assume taxes collected are redistributed to consumer as lumpsum transfer T .


( )max,

( , ) ( , ) ( )x x

L U x x p p T p p p t x p x c

1 2

1 2 1 1 2 2 1 2 1 1 2 2= + + + + +

L

x

U

x p t c

1 11 0= + =( )

L

x

U

x p

2 22 0= =


p

p

U

x U

x

t

p c 1

2

1

2

2=


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max ( ) ( )

y

p t f y p y f

21

11

121

2 21 = +

1

21 1

1

21 2 0y

p t f

y p f = + =( )

p p

f y

t f 1

2

1

21= +

p

p f y

t

p f 1

21

21

2

1= +


16/23


Damage compensation rates depend on victims activity level. For example,if exogenous shift in demand leads to an increase in output, damage caused by

firm 1 and consumers will necessarily increase and hence compensation paymentmust increase.

max ( ) ( , , )y

p f y y x p y

12

22

212

11

1 1 12 = + +

2

12 2

2

12 1 0y p

f

y p = + + =( )

p p

f y

212

12

=

p

p

f

y p

f

y 1

2

2

12

2

2

12=


17/23

Competitive Equilibrium with Taxes and Compensation

p

p

U x U

x

t

p f

y

t

p p

f

y

t p f

x t p

f

y

c f

c f

1

2

1

2

21

21

2 2

2

12

2

2

12

2

11

11

0

= = + = +

= = =

* * *

set of taxes and compensation payments which will sustain a{ }t t c f * * *, , competitive equilibrium which is Pareto optimal

optimal taxes are standard Pigouvian taxes taxes are set equal to the valueof the marginal damage of economic activities evaluated at the Paretooptimal level of these activities taxes are based on a polluters payprinciple

Pigouvian taxes are asymmetric no subsidization or compensation of victims necessary to sustain Pareto optimum provided t c and t f set optimally

market prices are now equal to shadow prices in social planners problem


18/23

Creation of markets by specifying property rights

Social planner wishes to establish a complete set of competitive marketswhich incorporates externalities must assign property rights.

Suppose that social planner assigns firm 2 the right to an unpolluted river. The initial assignment of property rights creates two pollution rights markets:

Consumer must purchase the right to pollute from firm 2 :

price of right for consumer to pollute one unitp 112

Firm 1 must also purchase the right to pollute from firm 2:

price of right for firm 1 to pollute one unitq 112


19/23


max

,

( , ) ( , )

x x

U x x p p p p p x p x p x

x x 1 2

1 2 1 1 2 2 1 2 1 1 112

112

2 2

112 1

s.t. + + = + +

=

where , the quantity of rights demanded by consumer, must equal x 1 ,x 112

the number of units of pollution created.

( )max,

( , ) ( )

x x

L U x x B p p x p x

1 2

1 2 1 112

1 2 2= + +

L

x

U

x p p

1 11 1

12 0= + =( )

L

x

U

x p

2 22 0= =


p

p

U

x U

x

p

p 1

2

1

2

112

2=


20/23


max ( )

y

p y p y q y y f y

y y 21

11 1

12 2

1112

112

11 1

21

112

11

= + =

=

s.t.

where is the quantity of pollution rights demanded by firm 1 and isy 112 y 1

1

firm 1 's level of pollution. N ote that, institutionally, firm 1 is constrained topurchase as many rights as it creates units of pollution.

max ( ) ( )y

p q f y p y 21

11 1

12 121

2 21

= +

1

21 1 1

121

21 2 0y

p q f

y p = + =( )

p p f y

q 1 21

21

112= +

p

p f

y

q

p 1

21

21

112

2

1= +


21/23


Firm 2 will supply quantities of pollution rights at prices ,$ , $y x 112

112 q p 1

12112,

respectively.

max

, $ $$ $ ( , $ , $ )

y y x

p y p y q y p x y f y y x

12

112

112

22 2

21 1

2112

112

112

112

22 2

12

112

112 = + + =+ s.t.

max

, $ $( , $ , $ ) $ $

y y x

p f y y x p y q y p x

12

112

112

22

212

112

112

1 12

112

112

112

112 = + ++

2

12 2

2

12 1 1 2

2

12

1

2

2

120y

p f

y p p p

f

y

p

p

f

y = + = = =

2

112 2

2

112 1

12112

2

2

112

112

2

2

1120$ $ $ $y

p f

y q q p

f

y

q

p

f

y = + = = =

2

112 2

2

112 1

12112

2

2

112

112

2

2

1120$ $ $ $x

p f

x p p p

f

x

p

p

f

x = + = = =

N ote that firm 2 will sell the right to pollute so as to equate the marketvalue of the property right ( M B of selling the right) to the value of the damagecreated by selling the right ( M C of selling the right).


22/23

Competitive equilibrium with marketable pollution permits

p p

U

x U

x

p p f

y

q p

f y

1

2

1

2

112

21

21

112

2

2

12

1= = + =

Definition of competitive equilibrium requires marketclearing in bothpollution permit markets:

$

$x x x y y y

1

12

1

12

1

112

112

11= == =

From firm 2 's profitmaximization problem and marketclearing conditionswe have that

p p f

x p

f

x t

q p f

y p

f

y t

c

f

112

2

2

112 2

2

112

112

2

2

112 2

2

112

=

=

permitprices

=marginal damage

at Pareto optimumPigouvian

taxes

= =

= =

=

$

$

*

*

Substituting marketclearing permit prices into competitive equilibriumconditions will yield conditions for Pareto optimal allocation.

by defining property rights, social planner can create missing markets

can sustain a competitive equilibrium as Pareto optimum with a completeset of markets!


23/23

Marketable Permits versus Taxes

two decentralized solutions to externality problem

price set by central authority in tax regime price determined by market forces in marketable permit system

economic agents find optimal price of pollution in decentralized,atomistic economy invisible hand works provided there is acomplete set of markets

both solutions force economic agents to fully internalize costs of actions

solutions have identical allocative consequences but different distributionalconsequences

Caveats

models assume externalities the only source of institutional failure

other possible sources of institutional failure

market failure transaction costs administrative costs may differ strategic behaviour trading restrictions

regulatory failure other distortionary policies may be in place such as

subsidization of externalitygenerating activity incomplete information regulator must know MB and MD

curves exactly to set policies efficiently incomplete enforcement models assumes full compliance highly unlikely regulator has the resources and knowledge todesign perfect expost governance structure

regulatory capture

global failure

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Lecture Externalities General Equilibrium