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7 Externality（外部性）

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7.1 Introduction

externality: outside the price system

pollution from a factory, envy felt by individuals

inefficiency

control of externalities is practically important

global warming（地球温暖化）

destruction of ozone layer（オゾン層の破壊）

government intervention ---> internalization（内部化） Pigouvian taxes（ピグー税）

tradable licenses（取引可能な許可証）

Coase theorem（コースの定理）

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7.2 Externalities Defined

Definition 4 (Externality 外部性) An

externality is present whenever some

economic agent’s welfare is directly affected

by the action of another agent in the economy.

directly = exclude effects mediated by prices

(pecuniary externality 金銭的外部性)

production externality

consumption externality

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consumption x={x1,x2,...,xH}

production y={y1,y2,...,ym}

Uh=Uh(x,y), h=1,...,H

Yj=Yj(x,y), j=1,...,m

utility fns and production sets are possibly

dependent on the entire arrays

actions of the agents in the economy will no

longer be independent（独立） or determined solely

by prices

competition will generally not achieve efficiency in

an economy with externalities

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7.3 Market Inefficiency presence of externalities

Pareto-inefficient = never max welfare

incorrect quantities of goods in the market

two-consumer, two-good economy

utility fns: U1=x1+u1(z1)+v1(z

2)

U2=x2+u2(z2)+v2(z

1)

positive externality if vh(･) is increasing

negative externality if vh(･) is decreasing

assume: supply of x <--- endowment wh z=x

z is produced from x by a competitive industry

normalize: price of x =1, so in eq. price of z =1

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max decision of consumers

u’h(zh)=1, h=1,2

private marginal benefit = private marginal cost

xh+zh=wh, h=1,2

x1+z1+x2+z2=w1+w2

Pareto-efficient allocation

max U1+U2=[x1+u1(z1)+v1(z

2)]+[x2+u2(z2)+v2(z

1)]

s.t. w1+w2-x1-z1-x2-z20

solution: u’1(z1)+v’2(z

1)=1

u’2(z2)+v’1(z

2)=1

externalities affect the optimal allocation（最適配分）

v’h>0: positive externality, v’h<0: negative

individual consumer’s maximization

max U1=x1+u1(z1)+v1(z

2) s.t. x1+z1=w1

max U2=x2+u2(z2)+v2(z

1) s.t. x2+z2=w2

substitute the constraints into utility functions

U1=w1-z1+u1(z1)+v1(z

2)

U2=w2-z2+u2(z2)+v2(z

1)

dU1/dz1 =-1+u1’(z1)=0 ∴ u1’(z

1)=1

dU2/dz2 =-1+u2’(z2)=0 ∴ u2’(z

2)=1

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Pareto efficiency

max W=U1+U2=x1+x2+u1(z1)+u2(z

2)+v1(z2)+v2(z

1)

s.t. x1+x2+z1+z2=w1+w2

substitute the constraints into welfare function

W=w1+w2 -z1-z2 +u1(z1)+u2(z

2)+v2(z1)+v1(z

2)

∂W/∂z1 =-1+u1’(z1)+v2’(z

1)=0

∴ u1’(z1)+v2’(z

1)=1

∂W/∂z2 =-1+u2’(z2)+v1’(z

2)=0

∴ u2’(z2)+v1’(z

2)=1

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Another way to find Pareto efficiency max U1=x1+u1(z

1)+v1(z2)

s.t. x2+u2(z2)+v2(z

1)≥U2, x1+x2+z1+z2=w1+w2

substitute the constraints into welfare function

L=x1+u1(z1)+v1(z

2)+λ[x2+u2(z2)+v2(z

1)≥U2]

+μ[w1+w2-x1-x2-z1-z2 ]

∂L/∂x1 =1-μ=0 ∴ μ=1 ∂L/∂λ=0

∂L/∂x2 = λ-μ=0 ∴ λ=μ=1 ∂L/∂μ=0

∂L/∂z1 =u1’(z1)+λv2’(z

1)-μ=0 ∴ u1’(z1)+v2’(z

1)=1

∂L/∂z2 =λu2’(z2)+v1’(z

2)-μ=0 ∴ u2’(z2)+v1’(z

2)=1

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marginal utilities at the optimum

positive externality: below those in market outcome

negative: above those in market outcome

market outcome is not Pareto-efficient

in general

positive: more of z is consumed at the optimum

than under the market outcome

negative: converse

Fig. 7.1

market outcome: PMB=MC

Pareto-efficient outcome: SMB=MC

SMB= uh’(zh)+ )('~

h

hzv

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MC

SMB( )

PBM

SMB( )

zh

marginal

benefit

and cost 0'~ h

v

0'~ h

v

hz hz

)ˆ(')('0'~h

h

h

hhzuzuv

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7.4 Externality Examples

7.4.1 River Pollution（汚染）

example of Louis Gevers

two firms: located along the same river

upstream（上流） firm u pollutes the river

reduces the production of downstream（下流） firm d

both produce the same output

sell at the constant unit price of 1 (same revenue)

inputs: labor (market price w) and water (free)

technologies: Fu(Lu), Fd(Ld,Lu), ∂Fd/∂Lu<0

profit: pi=Fi(･)-wLi i=u, d

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equilibrium 0u0d

total stock of labor is allocated between u and d

labor input of u: from the left, d: from the right

u’s profit max: upper part of the diagram ---> Lu*

given Lu* d’s profit max: lower part ---> Ld*

profits: pu, pd

inefficiency of competitive equilibrium

small reduction in L to u

no effect on pu since choice is optimal ∂pu/∂Lu=0

outward shift of d’s production function ---> pd↑

this change raises the aggregate profit（集計利潤）

externality ---> u uses too much of labor

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upstream

cost

revenue

downstream

cost

revenue

0u 0dw

w

Lu*,Ld*

p d

p u

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7.4.2 Traffic Jams（交通渋滞）

externalities imposed by drivers on each other

N: commuters（通勤者） choice of commuting by train or by car

train: 40 minutes

car: commuting time increases as the number of car

users increases ---> externality

minimization of transportation time（輸送時間）

equilibrium

travel time by car = travel time by train ---> 40%

optimum = max of aggregate time saving（時間の節約）

20% ---> overusing of cars

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minutes

commuting maximum

time saving car

train

% car users0 20 40

20

40

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7.4.3 Pecuniary Externality（金銭的外部性） set of students: economist or lawyer

being economist is great when there are few

economists, but not when there are many

price competition

students choose the profession with the best

earning prospects（所得見込み）

pecuniary externality: decision to become a

economist lowers all other economists’ incomes

equilibrium: % of economists

earnings of economist = earnings of lawyer

laissez-faire（自由放任） equilibrium ---> efficient

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income of

lawyers

income of

economists

lawyer

economist

E 1000

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7.4.4 The Rat Race Problem contest for relative position (George Akerlof)

students work too hard when the ranking is given

intense competition for a promotion in workplace

classroom example （相対評価）

performance is judged in relative term (not absolute)

advantage over other students can only be gained

by working harder than they do

all must work harder ---> all extra effort cancels out

inefficient rat race（出世争い）

if all agree less work, the same grades are obtained

agreement of less work is not self-supporting

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low high

low1/2

1/2

1-c

0

high0

1-c

1/2 -c

1/2 -c

player 1

player 2

cost of effort c, 0<c<½

Prisoners’ Dilemma（囚人のジレンマ）

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performance-enhancing drugs by athletes

no effective drug regulation（薬物規制）

---> enhance performance by using anabolic steroid

steroid（ステロイド） ---> increase the chance to win

unregulated contests: more and more drugs

---> substantial risks (cancer, health problems)

voluntary restraint is not an effective solution

public intervention（公的介入） of strict drug testing

rat race problem <--- relative（相対的） position

electoral competition ---> too much advertising

ban on cigarette advertising

---> beneficial to cigarette companies

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7.4.5 The Tragedy of the Commons（共有地の悲劇）

common right of access to a resource

inefficiency <--- divergence between individual

and social incentives

fishermen’s problem

lake used by fishermen from a village in its bank

fishermen do not own boats but can rent them for

daily use at a cost c

B boats ---> F(B): number of fish caught by each

boat, decreasing in B

w: wage (opportunity cost) p=1: price of fish

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equilibrium number of boats

p =F(B*)-c=w

optimal number of boats for the community: Bo

max{B} B[F(B)-c-w]

first order necessary condition

F(Bo)-c-w+BoF’(Bo)=0

F’(Bo) < 0 ---> Bo < B*

negative externality: each fisherman is concerned

with his profit, they will reduce the quantity of fish

caught by other fishermen ---> too many boats

public intervention: tax (price-based)課税,

quota (quantity-based)数量割当

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revenue

per boat

tax

c-w

MR AR

number

of boatsBo B*

total cost

per boat

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7.4.6 Bandwagon Effect（日和見効果） how standards are adopted?

Qwerty: reduction of key-jamming problems

Dvorak’s keyboard ---> not been adopted

impossible to switch the better technology?

bandwagon effect <--- network externality

（ネットワーク外部性）

decision to use Qwerty makes it more attractive

for manufacturers to produce Qwerty keyboards,

and for others to learn Qwerty

three equilibrium: 0, 1, p* (0<p*<1, unstable)

Qwerty dominates if initial point is to the right of p*

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0 p* 1

p*

1

% Qwerty users at time t

% Qwerty

users at

time t +1

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7.5 Pigouvian Taxation

market inefficiency <--- divergence between

social and private benefits (or costs)

---> appropriate taxes or subsidies（補助金）

negative consumption externality

PMB(SMB): private (social) marginal benefit

PMC: private marginal cost

equilibrium: xm <--- intersection of PMB and PMC

efficiency ：xo <--- intersection of SMB and PMC

Pigouvian taxation: PMC’ <--- PMC + t

pay tax equal to the marginal damage

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value

xo xm quantity

PMC’

PMC

PMB

SMB

t

benefit for the

neighbor

compensation

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two-consumer two-good case

utility fns: U1=x1+u1(z1)+v1(z

2)

U2=x2+u2(z2)+v2(z

1)

social optimum: u’1(z1)+v’2(z

1)=1

u’2(z2)+v’1(z

2)=1

producer price is equal to 1

utility max: consumer price q1=u’1(z1), q2=u’2(z

2)

efficiency: q1=1- v’2(z1), q2 =1-v’1(z

2)

Pigouvian taxes or subsidies

t1=- v’2(z1), t2=- v’2(z

1) positive or negative

different tax rates ---> practical difficulty

consumer’s price

individual consumer’s maximization

max U1=x1+u1(z1)+v1(z

2) s.t. x1+q1z1=w1

max U2=x2+u2(z2)+v2(z

1) s.t. x2+q2z2=w2

substitute the constraints into utility functions

U1=w1-q1z1+u1(z

1)+v1(z2)

U2=w2-q2z2+u2(z

2)+v2(z1)

dU1/dz1 =-q1+u1’(z1)=0 ∴ u1’(z

1)=q1

dU2/dz2 =-q2+u2’(z2)=0 ∴ u2’(z

2)=q2

to attain social optimality (efficiency)

u1’(z1)=q1=1-v2’(z

1)

u2’(z2)=q2=1-v1’(z

2)

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individual tax

individual consumer’s maximization

max U1=x1+u1(z1)+v1(z

2) s.t. x1+(1+t1)z1=w1

max U2=x2+u2(z2)+v2(z

1) s.t. x2+(1+t2)z2=w2

substitute the constraints into utility functions

U1=w1-(1+t1)z1+u1(z

1)+v1(z2)

U2=w2-(1+t2)z2+u2(z

2)+v2(z1)

dU1/dz1 = -(1+t1)z1+u1’(z

1)=0 ∴ u1’(z1)=1+t1

dU2/dz2 = -(1+t2)z2+u2’(z

2)=0 ∴ u2’(z2)=1+t2

to attain social optimality (efficiency)

u1’(z1)=1+t1=1-v2’(z

1) ∴ t1=-v2’(z1)

u2’(z2)=1+t2=1-v1’(z

2) ∴ t2=-v1’(z2)

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7.6 Licenses（許可証）

licenses ---> direct control of externality

tradable licenses ---> greatest benefit

require information on the aggregate quantity of

externality that is optimal

comparison between taxes and licenses

information for calculation: equivalent

Pigouvian taxes: preferences, production technology

licenses: total to the optimal level of externality

---> precise knowledge is needed to determine optimum

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information known when decisions are made

when all costs and benefits are known with

certainty by gvn and individuals, licenses and

taxation are equivalent in their effects

Fig 7.8: optimal xo is achieved by tax t or issuing xo

licenses

If not, no equivalence.

gvn must take decisions before the actual costs

and benefits are known for sure

timing: who chooses what and when

gvn DM --> costs and benefits --> individual DM

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pollution abatement（汚染削減） problem

PMC: PMCL, PMCH with equal prob

benefits are known for sure

gvn chooses its policy without knowing PMC

---> act on expected value PMCE ---> z*, t*

license scheme: abatement is z* for sure

tax: abatement depends on realized cost

PMCL ---> zL, PMCH ---> zH

licenses and taxation are not equivalent

taxation: PMCL ---> excess abatement

PMCH ---> underabatement（過少削減）

no answer which is better

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value

zH z* quantity

PMCH

PMCE

PMCL

SMB

t*

zL

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7.7 Internalization（内部化）Merger

forming single units out of the parties affected

private and social costs become the same

industry with production externality

internalization ---> single monopolist

---> welfare loss (greater than initial loss)

always implies construction of larger agents and a

consequent increase in market power

agents may not wish to become a single unit

c.f. household’s consumption externality

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7.8 The Coase Theorem

Theorem 3 (Coase theorem) In a competitive

economy with complete information and zero

transaction costs, the allocation of resources

will be efficient and invariant w.r.t. legal rules

of entitlement （資格）(property rights 所有権).

legal rules of entitlement or property rights

ownership

compensation payments

no need for policy intervention except prop. rights

private agreements（合意） ---> Pareto efficiency

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implication of Coase theorem

equilib. is invariant to how prop. rights are assigned

factory polluting the atmosphere of a neighboring house

firm has the right to pollute --> house pay compensation C

G: cost to the householder of the pollution

G>C ---> willing to compensate the firm, externality cease

G<C ---> externality is left to continue

polluter-pays principle（汚染者負担原則）

cost to the firm: C

compensation required by the household: G

C>G ---> firm is willing to compensate the household

C<G ---> firm stops externality

outcome is determined by the value G relative to C

differences in the final distribution of income

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value

xo xm quantity

PMC

PMB

SMB

compensation by polluter

= net loss of pollutee

net benefit

for polluter

pollutee has right

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value

xo xm quantity

PMC

PMB

SMB

compensation by pollutee

= net loss of polluter

net benefit

for pollutee

polluter has right

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practical relevance of Coase theorem

assignment of prop. rights ---> who should pay?

transaction costs（取引費用）

---> nonexistence of markets in externalities

pollution caused by car emissions（排出）

driver must purchase pollution rights from agents

affected by car emissions each time that the car is used

welfare loss due to waste of resources in organizing the

market outweigh any gain from having the market

thinness of the market

one agent on each side of the market

non-competitive behavior

bargaining between the two agents

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bargaining by two agents: polluter & pollutee

pollutee cannot observe the benefit of pollution B

but knows that it is drawn from a distribution F(B)

polluter cannot observe the cost of pollution C but

knows that it is drawn from a distribution G(C)

prop. rights to the pollutee

Pareto-efficiency ---> pollution is allowed if B>C

pollutee makes a take-it-or-leave-it offer to polluter

pollutee requires compensation T>0

polluter only accepts T if B>T ---> prob 1-F(T)

max {T} [1-F(T)][T-C] ---> T*>C

C<B<T* ---> offer is rejected ---> Pareto-inefficient

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7.9 Nonconvexity（非凸性） externality = source of nonconvexity

Fig 7.10: output is driven to zero by an externality

regardless of the level of other inputs

economy may fail to have an equilibrium with

personalized taxes

suppose firm receives a subsidy for accepting ext.

output level = zero, offer to accept ext.

---> unlimited profits

price of accepting ext. =zero ---> not accept any

demand for ext. is discontinuous

constant returns to scale（規模に関する収穫一定） all inputs including ext. <---> constant level of ext.

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output output

externality externality

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