Chapter Thirty-Three
Externalities
外部效应
Structure
Definition
Consumption externality
Production externality
The tragedy of commons
Externalities
An externality is a cost or a benefit
imposed upon someone by actions
taken by others,The cost or benefit is
thus generated externally to that
somebody.
An externally imposed benefit is a
positive externality.
An externally imposed cost is a
negative externality.
Examples of Negative Externalities
Air pollution.
Water pollution.
Loud parties next door.
Traffic congestion.
Second-hand cigarette smoke.
Examples of Positive Externalities
A well-maintained property next door
that raises the market value of your
property.
A pleasant cologne or scent worn by
the person seated next to you.
Improved driving habits that reduce
accident risks.
Education.
Externalities and Efficiency
Crucially,an externality impacts a
third party; i.e,somebody who is not
a participant in the activity that
produces the external cost or benefit.
Externalities and Efficiency
Externalities cause Pareto
inefficiency; typically
–too much scarce resource is
allocated to an activity which
causes a negative externality
–too little resource is allocated to an
activity which causes a positive
externality.
Externalities and Property Rights
An externality is viewed as a purely
public commodity.
A commodity is purely public if
–it is consumed by everyone
(nonexcludability),and
–everybody consumes the entire
amount of the commodity
(nonrivalry in consumption),
E.g,a broadcast television program.
Consumption Externality
Pareto efficient amount of smoke
Inefficient equilibrium with negative
externality
Property rights and price mechanism
Quasi-linear utility and the Coase
theorem (科斯定理 )
Consider two agents,A and B,and
two commodities,money and smoke.
Both smoke and money are goods
for Agent A.
Money is a good and smoke is a bad
for Agent B.
Smoke is a purely public commodity.
Consumption Externality
Agent A is endowed with $yA.
Agent B is endowed with $yB.
Smoke intensity is measured on a
scale from 0 (no smoke) to 1
(maximum concentration).
Consumption Externality
OA
1
0
Smoke
mAyA
Money and smoke are
both goods for Agent A.
Consumption Externality
OA
1
0
Smoke
mAyA
Money and smoke are
both goods for Agent A.
Consumption Externality
OB
1
0
Smoke
mByB
Money is a good and smoke
is a bad for Agent B.
Consumption Externality
OB
1
0
Smoke
mB yB
Money is a good and smoke
is a bad for Agent B.
Consumption Externality
Pareto Efficient Allocations
What are the efficient allocations of
smoke and money?
OA
1
0
Smoke
mAyA OB
1
0
Smoke
mB yB
Pareto Efficient Allocations
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
Pareto Efficient Allocations
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
Pareto Efficient Allocations
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
Pareto Efficient Allocations
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
Efficient
allocations
Pareto Efficient Allocations
Inefficiency & Negative Externalities
Suppose there is no means by which
money can be exchanged for
changes in smoke level.
What then is Agent A’s most
preferred allocation?
Is this allocation efficient?
Inefficiency & Negative Externalities
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
Efficient
allocations
Inefficiency & Negative Externalities
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
Efficient
allocations
A’s choices
Inefficiency & Negative Externalities
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
Efficient
allocations
A’s most
preferred choice
is inefficient
Inefficiency & Negative Externalities
Continue to suppose there is no
means by which money can be
exchanged for changes in smoke
level.
What is Agent B’s most preferred
allocation?
Is this allocation efficient?
Inefficiency & Negative Externalities
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
Efficient
allocations
B’s choices
Inefficiency & Negative Externalities
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
Efficient
allocations
B’s most
preferred choice
Inefficiency & Negative Externalities
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
Efficient
allocations
B’s most
preferred choice
is inefficient
Inefficiency & Negative Externalities
So if A and B cannot trade money for
changes in smoke intensity,then the
outcome is inefficient.
Either there is too much smoke (A’s
most preferred choice) or there is too
little smoke (B’s choice).
Externalities and Property Rights
Ronald Coase’s insight is that most
externality problems are due to an
inadequate specification of property
rights and,consequently,an
absence of markets in which trade
can be used to internalize external
costs or benefits.
Externalities and Property Rights
Causing a producer of an externality
to bear the full external cost or to
enjoy the full external benefit is
called internalizing the externality(外部效应内部化 ).
Externalities and Property Rights
Neither Agent A nor Agent B owns
the air in their room.
What happens if this property right is
created and is assigned to one of
them?
Externalities and Property Rights
Suppose Agent B is assigned
ownership of the air in the room.
Agent B can now sell,rights to
smoke”.
Will there be any smoking?
If so,how much smoking and what
will be the price for this amount of
smoke?
Externalities and Property Rights
Let p(sA) be the price paid by Agent
A to Agent B in order to create a
smoke intensity of sA.
Externalities and Property Rights
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
Externalities and Property Rights
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
Externalities and Property Rights
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
p(sA)
sA
Externalities and Property Rights
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
p(sA)
Both agents
gain and
there is a
positive
amount of
smoking.
sA
Externalities and Property Rights
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
p(sA)
sA
Establishing
a market for
trading rights
to smoke
causes an
efficient
allocation to
be achieved.
Externalities and Property Rights
Suppose instead that Agent A is
assigned the ownership of the air in
the room.
Agent B can now pay Agent A to
reduce the smoke intensity.
How much smoking will there be?
How much money will Agent B pay to
Agent A?
Externalities and Property Rights
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
Externalities and Property Rights
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
Externalities and Property Rights
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
sB
p(sB)
Externalities and Property Rights
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
p(sB)
Both agents
gain and
there is a
reduced
amount of
smoking.
sB
Externalities and Property Rights
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
p(sB)
Establishing
a market for
trading rights
to reduce
smoke
causes an
efficient
allocation to
be achieved.
sB
Externalities and Property Rights
Notice that the
–agent given the property right is
better off than at her own most
preferred allocation in the absence
of the property right.
–amount of smoking that occurs in
equilibrium depends upon which
agent is assigned the property
right.
Externalities and Property Rights
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
p(sB)p(sA)
sAsB
sB
sA
Externalities and Property Rights
Is there a case in which the same
amount of smoking occurs in
equilibrium no matter which agent is
assigned ownership of the air in the
room?
Externalities and Property Rights
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
p(sB)p(sA)
sA = sB
Externalities and Property Rights
OA
1
0
Smoke
OB
1
0
Smoke
yA yB
p(sB)p(sA)
sA = sB
For both agents,the MRS is constant as
money changes,for given smoke intensity.
Externalities and Property Rights
OA
1
0
Smoke
OB
1
0
Smoke
yA yB
p(sB)p(sA)
sA = sB
So,for both agents,preferences must be
quasilinear in money; U(m,s) = m + f(s).
Coase’s Theorem
Coase’s Theorem is,If all agents’
preferences are quasilinear in
money,then the efficient level of the
externality generating commodity is
produced no matter which agent is
assigned the property right.
Production Externalities
A steel mill produces jointly steel
and pollution.
The pollution adversely affects a
nearby fishery.
Both firms are price-takers.
pS is the market price of steel.
pF is the market price of fish.
Issues of Interest
Independent firms
Merger and internalization
Non-merger solutions
–Property rights and price
mechanism
–Coase theorem
Production Externalities
cS(s,x) is the steel firm’s cost of
producing s units of steel jointly with
x units of pollution.
If the steel firm does not face any of
the external costs of its pollution
production then its profit function is
and the firm’s problem is to
s s ss x p s c s x(,) (,)
Production Externalitiesm a x (,) (,).
,s x s s ss x p s c s x
The first-order profit-maximization
conditions are
Production Externalitiesm a x (,) (,).
,s x s s ss x p s c s x
The first-order profit-maximization
conditions arep c s x
ss
s
(,)0
c s x
x
s (,),
and
Production Externalities
p c s xss s(,)states that the steel firm
should produce the output level of steel
for which price = marginal production cost.
Production Externalities
p c s xss s(,)states that the steel firm
should produce the output level of steel
for which price = marginal production cost.?
c s x
x
s (,)
is the rate at which the firm’s
internal production cost goes down as the
pollution level rises
Production Externalities
p c s xss s(,)states that the steel firm
should produce the output level of steel
for which price = marginal production cost.?
c s x
x
s (,)
is the rate at which the firm’s
internal production cost goes down as the
pollution level rises,so
c s x
x
s (,)
is the marginal cost to the
firm of pollution reduction.
Production Externalities
c s xxs (,)is the marginal cost to the
firm of pollution reduction.
What is the marginal benefit to the steel
firm from reducing pollution?
Production Externalities
c s xxs (,)is the marginal cost to the
firm of pollution reduction.
What is the marginal benefit to the steel
firm from reducing pollution?
Zero,since the firm does not face its
external cost.
Hence the steel firm chooses the pollution
level for which
c s x
x
s (,),0
Production Externalities
0 2 4( ).x
and the first-order profit-maximization
conditions are12 2? s
( ) 2 ( 4 ),mM C x xand
E.g,suppose cS(s,x) = s2 + (x - 4)2 and
pS = 12,Then
Production Externalities
p ss12 2,determines the profit-max.
output level of steel; s* = 6.
Production Externalities
p ss12 2,determines the profit-max.
output level of steel; s* = 6.
2 4( )xis the marginal cost to the firm
from pollution reduction,Since it gets
no benefit from this it sets x* = 4,
Production Externalities
p ss12 2,determines the profit-max.
output level of steel; s* = 6.
2 4( )xis the marginal cost to the firm
from pollution reduction,Since it gets
no benefit from this it sets x* = 4,
s s x s s x( *,*) * * ( * )
( )
$36,
12 4
12 6 6 4 4
2 2
2 2The steel firm’s maximum profit level is
thus
Production Externalities
The cost to the fishery of catching f
units of fish when the steel mill emits
x units of pollution is cF(f,x),Given f,
cF(f,x) increases with x; i.e,the steel
firm inflicts a negative externality on
the fishery.
Production Externalities
The cost to the fishery of catching f
units of fish when the steel mill emits
x units of pollution is cF(f,x),Given f,
cF(f,x) increases with x; i.e,the steel
firm inflicts a negative externality on
the fishery.
The fishery’s profit function is
so the fishery’s problem is to
F F Ff x p f c f x( ; ) ( ; )
Production Externalities
m a x ( ; ) ( ; ).f F F Ff x p f c f x
The first-order profit-maximization
condition is
Production Externalities
m a x ( ; ) ( ; ).f F F Ff x p f c f x
The first-order profit-maximization
condition is p
c f x
fF
F
( ; ),
Production Externalities
m a x ( ; ) ( ; ).f F F Ff x p f c f x
The first-order profit-maximization
condition is p
c f x
fF
F
( ; ),
Higher pollution raises the fishery’s
marginal production cost and lowers both
its output level and its profit,This is the
external cost of the pollution.
Production Externalities
E.g,suppose cF(f;x) = f2 + xf and pF = 10.
The external cost inflicted on the fishery
by the steel firm is xf,Since the fishery
has no control over x it must take the steel
firm’s choice of x as a given,The fishery’s
profit function is thus
F f x f f xf( ; )10 2
Production Externalities
Given x,the first-order profit-maximization
condition is
F f x f f xf( ; )10 2
10 2f x,
Production Externalities
Given x,the first-order profit-maximization
condition is
So,given a pollution level x inflicted upon
it,the fishery’s profit-maximizing output
level is f
x*,5
2
F f x f f xf( ; )10 2
10 2f x,
Production Externalities
Given x,the first-order profit-maximization
condition is
So,given a pollution level x inflicted upon
it,the fishery’s profit-maximizing output
level is
F f x f f xf( ; )10 2
Notice that the fishery produces less,and
earns less profit,as the steel firm’s
pollution level increases.
f x*,5 2
10 2f x,
Production Externalities
The steel firm,ignoring its
external cost inflicted upon the fishery,
chooses x* = 4,so the fishery’s
profit-maximizing output level given the
steel firm’s choice of pollution level is
f* = 3,giving the fishery a maximum
profit level of
.9$343310
*xf*f*f10)x*;f(
2
2
F
Notice that the external cost is $12.
f x*,5 2
Production Externalities
Are these choices by the two firms
efficient?
When the steel firm ignores the
external costs of its choices,the sum
of the two firm’s profits is $36 + $9 =
$45.
Is $45 the largest possible total profit
that can be achieved?
Merger and Internalization
Suppose the two firms merge to
become one,What is the highest
profit this new firm can achieve?
Merger and Internalization
Suppose the two firms merge to
become one,What is the highest
profit this new firm can achieve?
What choices of s,f and x maximize
the new firm’s profit?
m s f x s f s x f xf(,,) ( ),12 10 42 2 2
Merger and Internalization
m s f x s f s x f xf(,,) ( ),12 10 42 2 2
The first-order profit-maximization
conditions are
m
m
m
s
s
f
f x
x
x f
12 2 0
10 2 0
2 4 0
.
( ),
The solution is
s
f
x
m
m
m
6
4
2.
Merger and Internalization
m m m m
m m m m m m m
s f x
s f s x f x f
(,,)
( )
( )
$48,
12 10 4
12 6 10 4 6 2 4 4 2 4
2
2
2
2 2 2And the merged firm’s maximum profitlevel is
This exceeds $45,the sum of the non-
merged firms.
Merger and Internalization
Merger has improved efficiency.
On its own,the steel firm produced x*
= 4 units of pollution.
Within the merged firm,pollution
production is only xm = 2 units.
So merger has caused both an
improvement in efficiency and less
pollution production,Why?
Merger and Internalization
s s x s s x(,) ( )12 42 2The steel firm’s profit function is
so the marginal cost of producing x units
of pollution is MC x xs ( ) ( )2 4
When it does not have to face the
external costs of its pollution,the steel
firm increases pollution until this marginal
cost is zero; hence x* = 4.
Merger and Internalization
In the merged firm the profit function is? m s f x s f s x f xf(,,) ( ),12 10 42 2 2
The marginal cost of pollution is thusMC x fm
x( ) ( )2 4
Merger and Internalization
In the merged firm the profit function is? m s f x s f s x f xf(,,) ( ),12 10 42 2 2
The marginal cost of pollution isMC x fm
x( ) ( )2 42 4( ) ( ).x MC xs
Merger and Internalization
In the merged firm the profit function is? m s f x s f s x f xf(,,) ( ),12 10 42 2 2
The marginal cost of pollution isMC x fm
x( ) ( )2 42 4( ) ( ).x MC xs
The merged firm’s marginal pollution cost
is larger because it faces the full cost of
its own pollution through increased costs
of production in the fishery,so less
pollution is produced by the merged firm.
Merger and Internalization
But why is the merged firm’s
pollution level of xm = 2 efficient?
Merger and Internalization
But why is the merged firm’s
pollution level of xm = 2 efficient?
The external cost inflicted on the
fishery is xf,so the marginal external
pollution cost is MC fxE?,
Merger and Internalization
But why is the merged firm’s
pollution level of xm = 2 efficient?
The external cost inflicted on the
fishery is xf,so the marginal external
pollution cost is
The steel firm’s marginal cost of
reducing pollution is
MC fxE?,
( ) 2 ( 4 ),mM C x x
Merger and Internalization
But why is the merged firm’s
pollution level of xm = 2 efficient?
The external cost inflicted on the
fishery is xf,so the marginal external
pollution cost is
The steel firm’s marginal cost of
reducing pollution is
Efficiency requires
MC fxE?,
( ) 2 ( 4 ),mM C x x
( ) 2 ( 4 ),EmxM C M C x f x
Merger and Internalization
.ExM C f?
( ) 2 ( 4 ),mM C x x
42 x
Merger and Internalization
Merger therefore internalizes an
externality and induces economic
efficiency.
How else might internalization be
caused so that efficiency can be
achieved?
Coase and Production Externalities
Coase argues that the externality
exists because neither the steel firm
nor the fishery owns the water being
polluted.
Suppose the property right to the
water is created and assigned to one
of the firms,Does this induce
efficiency?
Coase and Production Externalities
Suppose the fishery owns the water.
Then it can sell pollution rights,in a
competitive market,at $px each.
The fishery’s profit function becomes?
F f xf x p f f xf p x(,),2
Coase and Production Externalities
Suppose the fishery owns the water.
Then it can sell pollution rights,in a
competitive market,at $px each.
The fishery’s profit function becomes
Given pf and px,how many fish and
how many rights does the fishery
wish to produce? (Notice that x is
now a choice variable for the fishery.)
F f xf x p f f xf p x(,),2
Coase and Production Externalities
F
f
F
x
f
p f x
x
f p
2 0
0
F f xf x p f f xf p x(,),2
The profit-maximum conditions are
Coase and Production Externalities
F
f
F
x
f
p f x
x
f p
2 0
0
F f xf x p f f xf p x(,),2
The profit-maximum conditions are
and these give f px p px
S f x
*
*,
2(fish supply)(pollution
right supply)
Coase and Production Externalities
The steel firm must buy one right for
every unit of pollution it emits so its
profit function becomes
Given pf and px,how much steel does
the steel firm want to produce and
how many rights does it wish to buy?
S s xs x p s s x p x(,) ( ),2 24
Coase and Production Externalities
S
s
S
x
s
p s
x
x p
2 0
2 4 0( )
S s xs x p s s x p x(,) ( ),2 24
The profit-maximum conditions are
Coase and Production Externalities
S
s
S
x
s
p s
x
x p
2 0
2 4 0( )
S s xs x p s s x p x(,) ( ),2 24
The profit-maximum conditions are
and these give
s
p
x
p
s
D
x
*
*,
2
4
2
(steel supply)
(pollution
right demand)
Coase and Production Externalities
In a competitive market for pollution rights
the price px must adjust to clear the market
so,at equilibrium,x p p p x
D x f x S* *.4 2 2
Coase and Production Externalities
In a competitive market for pollution rights
the price px must adjust to clear the market
so,at equilibrium,x p p p x
D x f x S* *.4 2 2
The market-clearing price for pollution
rights is thus p
p
x f?
2 8
3
Coase and Production Externalities
In a competitive market for pollution rights
the price px must adjust to clear the market
so,at equilibrium,x p p p x
D x f x S* *.4 2 2
The market-clearing price for pollution
rights is thus p
p
x f?
2 8
3
and the equilibrium quantity of rights
traded is x x
p
D S f* *,
16
3
Coase and Production Externalities
s p f p x x ps x D S f* ; * ; * * ;2 16 3
p px f2 83,
Coase and Production Externalities
s p f p x x ps x D S f* ; * ; * * ;2 16 3
p px f2 83,
So if ps = 12 and pf = 10 then
s f x x pD S x* ; * ; * * ;,6 4 2 4
This is the efficient outcome.
Coase and Production Externalities
Q,Would it matter if the property right to
the water had instead been assigned to
the steel firm?
A,No,Profit is linear,and therefore
quasi-linear,in money so Coase’s
Theorem states that the same efficient
allocation is achieved whichever of the
firms was assigned the property right,
(And the asset owner gets richer.)
The Tragedy of the Commons
公地悲剧
Consider a grazing area owned,in
common” by all members of a village.
Villagers graze cows on the common.
When c cows are grazed,total milk
production is f(c),where f’>0 and f”<0.
How should the villagers graze their
cows so as to maximize their overall
income?
The Tragedy of the Commons
c
Milk
f(c)
The Tragedy of the Commons
Make the price of milk $1 and let the
relative cost of grazing a cow be $pc,
Then the profit function for the entire
village is
and the village’s problem is to
( ) ( )c f c p cc
m a x ( ) ( ),c cc f c p c0?
The Tragedy of the Commonsm a x ( ) ( ),
c cc f c p c0?
The income-maximizing number of cows
to graze,c*,satisfies
f c p c( )
i.e,the marginal income gain from the
last cow grazed must equal the marginal
cost of grazing it.
The Tragedy of the Commons
c
Milk
f(c)
pcc
slope =
f’(c*)
c*
slope
= pc
The Tragedy of the Commons
c
Milk
f(c)
pcc
slope =
f’(c*)
c*
slope
= pc
Maximal incomef(c*)
The Tragedy of the Commons
For c = c*,the average gain per cow
grazed is
because f’ > 0 and f” < 0.
( *)
*
( *) *
*
( *)
*
c
c
f c p c
c
f c
c p
c c 0
The Tragedy of the Commons
c
Milk
f(c)
pcc
slope =
f’(c*)
c*
f c
c p c
( *)
*?
f(c*)
The Tragedy of the Commons
For c = c*,the average gain per cow
grazed is
because f’ > 0 and f” < 0,So the
economic profit from introducing one
more cow is positive.
Since nobody owns the common,entry
is not restricted.
( *)
*
( *) *
*
( *)
*
c
c
f c p c
c
f c
c p
c c 0
The Tragedy of the Commons
Entry continues until the economic
profit of grazing another cow is zero;
that is,until? ( ) ( ) ( )
.cc f c p cc f cc pc c 0
The Tragedy of the Commons
c
Milk
f(c)
pcc
slope =
f’(c*)
c*
f c
c p c
(? )
f(c*)
c
The Tragedy of the Commons
c
Milk
f(c)
pcc
slope =
f’(c*)
c*
f c
c p c
(? )
f(c*)
The commons are over-grazed,tragically.
c
The Tragedy of the Commons
The reason for the tragedy is that
when a villager adds one more cow
his income rises (by f(c)/c - pc) but
every other villager’s income falls.
The villager who adds the extra cow
takes no account of the cost inflicted
upon the rest of the village.
The Tragedy of the Commons
Modern-day,tragedies of the
commons” include
–over-fishing the high seas
–over-logging forests on public
lands
–over-intensive use of public parks.
–urban traffic congestion,
Externalities
外部效应
Structure
Definition
Consumption externality
Production externality
The tragedy of commons
Externalities
An externality is a cost or a benefit
imposed upon someone by actions
taken by others,The cost or benefit is
thus generated externally to that
somebody.
An externally imposed benefit is a
positive externality.
An externally imposed cost is a
negative externality.
Examples of Negative Externalities
Air pollution.
Water pollution.
Loud parties next door.
Traffic congestion.
Second-hand cigarette smoke.
Examples of Positive Externalities
A well-maintained property next door
that raises the market value of your
property.
A pleasant cologne or scent worn by
the person seated next to you.
Improved driving habits that reduce
accident risks.
Education.
Externalities and Efficiency
Crucially,an externality impacts a
third party; i.e,somebody who is not
a participant in the activity that
produces the external cost or benefit.
Externalities and Efficiency
Externalities cause Pareto
inefficiency; typically
–too much scarce resource is
allocated to an activity which
causes a negative externality
–too little resource is allocated to an
activity which causes a positive
externality.
Externalities and Property Rights
An externality is viewed as a purely
public commodity.
A commodity is purely public if
–it is consumed by everyone
(nonexcludability),and
–everybody consumes the entire
amount of the commodity
(nonrivalry in consumption),
E.g,a broadcast television program.
Consumption Externality
Pareto efficient amount of smoke
Inefficient equilibrium with negative
externality
Property rights and price mechanism
Quasi-linear utility and the Coase
theorem (科斯定理 )
Consider two agents,A and B,and
two commodities,money and smoke.
Both smoke and money are goods
for Agent A.
Money is a good and smoke is a bad
for Agent B.
Smoke is a purely public commodity.
Consumption Externality
Agent A is endowed with $yA.
Agent B is endowed with $yB.
Smoke intensity is measured on a
scale from 0 (no smoke) to 1
(maximum concentration).
Consumption Externality
OA
1
0
Smoke
mAyA
Money and smoke are
both goods for Agent A.
Consumption Externality
OA
1
0
Smoke
mAyA
Money and smoke are
both goods for Agent A.
Consumption Externality
OB
1
0
Smoke
mByB
Money is a good and smoke
is a bad for Agent B.
Consumption Externality
OB
1
0
Smoke
mB yB
Money is a good and smoke
is a bad for Agent B.
Consumption Externality
Pareto Efficient Allocations
What are the efficient allocations of
smoke and money?
OA
1
0
Smoke
mAyA OB
1
0
Smoke
mB yB
Pareto Efficient Allocations
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
Pareto Efficient Allocations
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
Pareto Efficient Allocations
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
Pareto Efficient Allocations
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
Efficient
allocations
Pareto Efficient Allocations
Inefficiency & Negative Externalities
Suppose there is no means by which
money can be exchanged for
changes in smoke level.
What then is Agent A’s most
preferred allocation?
Is this allocation efficient?
Inefficiency & Negative Externalities
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
Efficient
allocations
Inefficiency & Negative Externalities
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
Efficient
allocations
A’s choices
Inefficiency & Negative Externalities
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
Efficient
allocations
A’s most
preferred choice
is inefficient
Inefficiency & Negative Externalities
Continue to suppose there is no
means by which money can be
exchanged for changes in smoke
level.
What is Agent B’s most preferred
allocation?
Is this allocation efficient?
Inefficiency & Negative Externalities
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
Efficient
allocations
B’s choices
Inefficiency & Negative Externalities
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
Efficient
allocations
B’s most
preferred choice
Inefficiency & Negative Externalities
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
Efficient
allocations
B’s most
preferred choice
is inefficient
Inefficiency & Negative Externalities
So if A and B cannot trade money for
changes in smoke intensity,then the
outcome is inefficient.
Either there is too much smoke (A’s
most preferred choice) or there is too
little smoke (B’s choice).
Externalities and Property Rights
Ronald Coase’s insight is that most
externality problems are due to an
inadequate specification of property
rights and,consequently,an
absence of markets in which trade
can be used to internalize external
costs or benefits.
Externalities and Property Rights
Causing a producer of an externality
to bear the full external cost or to
enjoy the full external benefit is
called internalizing the externality(外部效应内部化 ).
Externalities and Property Rights
Neither Agent A nor Agent B owns
the air in their room.
What happens if this property right is
created and is assigned to one of
them?
Externalities and Property Rights
Suppose Agent B is assigned
ownership of the air in the room.
Agent B can now sell,rights to
smoke”.
Will there be any smoking?
If so,how much smoking and what
will be the price for this amount of
smoke?
Externalities and Property Rights
Let p(sA) be the price paid by Agent
A to Agent B in order to create a
smoke intensity of sA.
Externalities and Property Rights
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
Externalities and Property Rights
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
Externalities and Property Rights
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
p(sA)
sA
Externalities and Property Rights
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
p(sA)
Both agents
gain and
there is a
positive
amount of
smoking.
sA
Externalities and Property Rights
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
p(sA)
sA
Establishing
a market for
trading rights
to smoke
causes an
efficient
allocation to
be achieved.
Externalities and Property Rights
Suppose instead that Agent A is
assigned the ownership of the air in
the room.
Agent B can now pay Agent A to
reduce the smoke intensity.
How much smoking will there be?
How much money will Agent B pay to
Agent A?
Externalities and Property Rights
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
Externalities and Property Rights
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
Externalities and Property Rights
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
sB
p(sB)
Externalities and Property Rights
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
p(sB)
Both agents
gain and
there is a
reduced
amount of
smoking.
sB
Externalities and Property Rights
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
p(sB)
Establishing
a market for
trading rights
to reduce
smoke
causes an
efficient
allocation to
be achieved.
sB
Externalities and Property Rights
Notice that the
–agent given the property right is
better off than at her own most
preferred allocation in the absence
of the property right.
–amount of smoking that occurs in
equilibrium depends upon which
agent is assigned the property
right.
Externalities and Property Rights
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
p(sB)p(sA)
sAsB
sB
sA
Externalities and Property Rights
Is there a case in which the same
amount of smoking occurs in
equilibrium no matter which agent is
assigned ownership of the air in the
room?
Externalities and Property Rights
OA
1
0
Smoke
mA
OB
1
0
Smoke
mB
yA yB
p(sB)p(sA)
sA = sB
Externalities and Property Rights
OA
1
0
Smoke
OB
1
0
Smoke
yA yB
p(sB)p(sA)
sA = sB
For both agents,the MRS is constant as
money changes,for given smoke intensity.
Externalities and Property Rights
OA
1
0
Smoke
OB
1
0
Smoke
yA yB
p(sB)p(sA)
sA = sB
So,for both agents,preferences must be
quasilinear in money; U(m,s) = m + f(s).
Coase’s Theorem
Coase’s Theorem is,If all agents’
preferences are quasilinear in
money,then the efficient level of the
externality generating commodity is
produced no matter which agent is
assigned the property right.
Production Externalities
A steel mill produces jointly steel
and pollution.
The pollution adversely affects a
nearby fishery.
Both firms are price-takers.
pS is the market price of steel.
pF is the market price of fish.
Issues of Interest
Independent firms
Merger and internalization
Non-merger solutions
–Property rights and price
mechanism
–Coase theorem
Production Externalities
cS(s,x) is the steel firm’s cost of
producing s units of steel jointly with
x units of pollution.
If the steel firm does not face any of
the external costs of its pollution
production then its profit function is
and the firm’s problem is to
s s ss x p s c s x(,) (,)
Production Externalitiesm a x (,) (,).
,s x s s ss x p s c s x
The first-order profit-maximization
conditions are
Production Externalitiesm a x (,) (,).
,s x s s ss x p s c s x
The first-order profit-maximization
conditions arep c s x
ss
s
(,)0
c s x
x
s (,),
and
Production Externalities
p c s xss s(,)states that the steel firm
should produce the output level of steel
for which price = marginal production cost.
Production Externalities
p c s xss s(,)states that the steel firm
should produce the output level of steel
for which price = marginal production cost.?
c s x
x
s (,)
is the rate at which the firm’s
internal production cost goes down as the
pollution level rises
Production Externalities
p c s xss s(,)states that the steel firm
should produce the output level of steel
for which price = marginal production cost.?
c s x
x
s (,)
is the rate at which the firm’s
internal production cost goes down as the
pollution level rises,so
c s x
x
s (,)
is the marginal cost to the
firm of pollution reduction.
Production Externalities
c s xxs (,)is the marginal cost to the
firm of pollution reduction.
What is the marginal benefit to the steel
firm from reducing pollution?
Production Externalities
c s xxs (,)is the marginal cost to the
firm of pollution reduction.
What is the marginal benefit to the steel
firm from reducing pollution?
Zero,since the firm does not face its
external cost.
Hence the steel firm chooses the pollution
level for which
c s x
x
s (,),0
Production Externalities
0 2 4( ).x
and the first-order profit-maximization
conditions are12 2? s
( ) 2 ( 4 ),mM C x xand
E.g,suppose cS(s,x) = s2 + (x - 4)2 and
pS = 12,Then
Production Externalities
p ss12 2,determines the profit-max.
output level of steel; s* = 6.
Production Externalities
p ss12 2,determines the profit-max.
output level of steel; s* = 6.
2 4( )xis the marginal cost to the firm
from pollution reduction,Since it gets
no benefit from this it sets x* = 4,
Production Externalities
p ss12 2,determines the profit-max.
output level of steel; s* = 6.
2 4( )xis the marginal cost to the firm
from pollution reduction,Since it gets
no benefit from this it sets x* = 4,
s s x s s x( *,*) * * ( * )
( )
$36,
12 4
12 6 6 4 4
2 2
2 2The steel firm’s maximum profit level is
thus
Production Externalities
The cost to the fishery of catching f
units of fish when the steel mill emits
x units of pollution is cF(f,x),Given f,
cF(f,x) increases with x; i.e,the steel
firm inflicts a negative externality on
the fishery.
Production Externalities
The cost to the fishery of catching f
units of fish when the steel mill emits
x units of pollution is cF(f,x),Given f,
cF(f,x) increases with x; i.e,the steel
firm inflicts a negative externality on
the fishery.
The fishery’s profit function is
so the fishery’s problem is to
F F Ff x p f c f x( ; ) ( ; )
Production Externalities
m a x ( ; ) ( ; ).f F F Ff x p f c f x
The first-order profit-maximization
condition is
Production Externalities
m a x ( ; ) ( ; ).f F F Ff x p f c f x
The first-order profit-maximization
condition is p
c f x
fF
F
( ; ),
Production Externalities
m a x ( ; ) ( ; ).f F F Ff x p f c f x
The first-order profit-maximization
condition is p
c f x
fF
F
( ; ),
Higher pollution raises the fishery’s
marginal production cost and lowers both
its output level and its profit,This is the
external cost of the pollution.
Production Externalities
E.g,suppose cF(f;x) = f2 + xf and pF = 10.
The external cost inflicted on the fishery
by the steel firm is xf,Since the fishery
has no control over x it must take the steel
firm’s choice of x as a given,The fishery’s
profit function is thus
F f x f f xf( ; )10 2
Production Externalities
Given x,the first-order profit-maximization
condition is
F f x f f xf( ; )10 2
10 2f x,
Production Externalities
Given x,the first-order profit-maximization
condition is
So,given a pollution level x inflicted upon
it,the fishery’s profit-maximizing output
level is f
x*,5
2
F f x f f xf( ; )10 2
10 2f x,
Production Externalities
Given x,the first-order profit-maximization
condition is
So,given a pollution level x inflicted upon
it,the fishery’s profit-maximizing output
level is
F f x f f xf( ; )10 2
Notice that the fishery produces less,and
earns less profit,as the steel firm’s
pollution level increases.
f x*,5 2
10 2f x,
Production Externalities
The steel firm,ignoring its
external cost inflicted upon the fishery,
chooses x* = 4,so the fishery’s
profit-maximizing output level given the
steel firm’s choice of pollution level is
f* = 3,giving the fishery a maximum
profit level of
.9$343310
*xf*f*f10)x*;f(
2
2
F
Notice that the external cost is $12.
f x*,5 2
Production Externalities
Are these choices by the two firms
efficient?
When the steel firm ignores the
external costs of its choices,the sum
of the two firm’s profits is $36 + $9 =
$45.
Is $45 the largest possible total profit
that can be achieved?
Merger and Internalization
Suppose the two firms merge to
become one,What is the highest
profit this new firm can achieve?
Merger and Internalization
Suppose the two firms merge to
become one,What is the highest
profit this new firm can achieve?
What choices of s,f and x maximize
the new firm’s profit?
m s f x s f s x f xf(,,) ( ),12 10 42 2 2
Merger and Internalization
m s f x s f s x f xf(,,) ( ),12 10 42 2 2
The first-order profit-maximization
conditions are
m
m
m
s
s
f
f x
x
x f
12 2 0
10 2 0
2 4 0
.
( ),
The solution is
s
f
x
m
m
m
6
4
2.
Merger and Internalization
m m m m
m m m m m m m
s f x
s f s x f x f
(,,)
( )
( )
$48,
12 10 4
12 6 10 4 6 2 4 4 2 4
2
2
2
2 2 2And the merged firm’s maximum profitlevel is
This exceeds $45,the sum of the non-
merged firms.
Merger and Internalization
Merger has improved efficiency.
On its own,the steel firm produced x*
= 4 units of pollution.
Within the merged firm,pollution
production is only xm = 2 units.
So merger has caused both an
improvement in efficiency and less
pollution production,Why?
Merger and Internalization
s s x s s x(,) ( )12 42 2The steel firm’s profit function is
so the marginal cost of producing x units
of pollution is MC x xs ( ) ( )2 4
When it does not have to face the
external costs of its pollution,the steel
firm increases pollution until this marginal
cost is zero; hence x* = 4.
Merger and Internalization
In the merged firm the profit function is? m s f x s f s x f xf(,,) ( ),12 10 42 2 2
The marginal cost of pollution is thusMC x fm
x( ) ( )2 4
Merger and Internalization
In the merged firm the profit function is? m s f x s f s x f xf(,,) ( ),12 10 42 2 2
The marginal cost of pollution isMC x fm
x( ) ( )2 42 4( ) ( ).x MC xs
Merger and Internalization
In the merged firm the profit function is? m s f x s f s x f xf(,,) ( ),12 10 42 2 2
The marginal cost of pollution isMC x fm
x( ) ( )2 42 4( ) ( ).x MC xs
The merged firm’s marginal pollution cost
is larger because it faces the full cost of
its own pollution through increased costs
of production in the fishery,so less
pollution is produced by the merged firm.
Merger and Internalization
But why is the merged firm’s
pollution level of xm = 2 efficient?
Merger and Internalization
But why is the merged firm’s
pollution level of xm = 2 efficient?
The external cost inflicted on the
fishery is xf,so the marginal external
pollution cost is MC fxE?,
Merger and Internalization
But why is the merged firm’s
pollution level of xm = 2 efficient?
The external cost inflicted on the
fishery is xf,so the marginal external
pollution cost is
The steel firm’s marginal cost of
reducing pollution is
MC fxE?,
( ) 2 ( 4 ),mM C x x
Merger and Internalization
But why is the merged firm’s
pollution level of xm = 2 efficient?
The external cost inflicted on the
fishery is xf,so the marginal external
pollution cost is
The steel firm’s marginal cost of
reducing pollution is
Efficiency requires
MC fxE?,
( ) 2 ( 4 ),mM C x x
( ) 2 ( 4 ),EmxM C M C x f x
Merger and Internalization
.ExM C f?
( ) 2 ( 4 ),mM C x x
42 x
Merger and Internalization
Merger therefore internalizes an
externality and induces economic
efficiency.
How else might internalization be
caused so that efficiency can be
achieved?
Coase and Production Externalities
Coase argues that the externality
exists because neither the steel firm
nor the fishery owns the water being
polluted.
Suppose the property right to the
water is created and assigned to one
of the firms,Does this induce
efficiency?
Coase and Production Externalities
Suppose the fishery owns the water.
Then it can sell pollution rights,in a
competitive market,at $px each.
The fishery’s profit function becomes?
F f xf x p f f xf p x(,),2
Coase and Production Externalities
Suppose the fishery owns the water.
Then it can sell pollution rights,in a
competitive market,at $px each.
The fishery’s profit function becomes
Given pf and px,how many fish and
how many rights does the fishery
wish to produce? (Notice that x is
now a choice variable for the fishery.)
F f xf x p f f xf p x(,),2
Coase and Production Externalities
F
f
F
x
f
p f x
x
f p
2 0
0
F f xf x p f f xf p x(,),2
The profit-maximum conditions are
Coase and Production Externalities
F
f
F
x
f
p f x
x
f p
2 0
0
F f xf x p f f xf p x(,),2
The profit-maximum conditions are
and these give f px p px
S f x
*
*,
2(fish supply)(pollution
right supply)
Coase and Production Externalities
The steel firm must buy one right for
every unit of pollution it emits so its
profit function becomes
Given pf and px,how much steel does
the steel firm want to produce and
how many rights does it wish to buy?
S s xs x p s s x p x(,) ( ),2 24
Coase and Production Externalities
S
s
S
x
s
p s
x
x p
2 0
2 4 0( )
S s xs x p s s x p x(,) ( ),2 24
The profit-maximum conditions are
Coase and Production Externalities
S
s
S
x
s
p s
x
x p
2 0
2 4 0( )
S s xs x p s s x p x(,) ( ),2 24
The profit-maximum conditions are
and these give
s
p
x
p
s
D
x
*
*,
2
4
2
(steel supply)
(pollution
right demand)
Coase and Production Externalities
In a competitive market for pollution rights
the price px must adjust to clear the market
so,at equilibrium,x p p p x
D x f x S* *.4 2 2
Coase and Production Externalities
In a competitive market for pollution rights
the price px must adjust to clear the market
so,at equilibrium,x p p p x
D x f x S* *.4 2 2
The market-clearing price for pollution
rights is thus p
p
x f?
2 8
3
Coase and Production Externalities
In a competitive market for pollution rights
the price px must adjust to clear the market
so,at equilibrium,x p p p x
D x f x S* *.4 2 2
The market-clearing price for pollution
rights is thus p
p
x f?
2 8
3
and the equilibrium quantity of rights
traded is x x
p
D S f* *,
16
3
Coase and Production Externalities
s p f p x x ps x D S f* ; * ; * * ;2 16 3
p px f2 83,
Coase and Production Externalities
s p f p x x ps x D S f* ; * ; * * ;2 16 3
p px f2 83,
So if ps = 12 and pf = 10 then
s f x x pD S x* ; * ; * * ;,6 4 2 4
This is the efficient outcome.
Coase and Production Externalities
Q,Would it matter if the property right to
the water had instead been assigned to
the steel firm?
A,No,Profit is linear,and therefore
quasi-linear,in money so Coase’s
Theorem states that the same efficient
allocation is achieved whichever of the
firms was assigned the property right,
(And the asset owner gets richer.)
The Tragedy of the Commons
公地悲剧
Consider a grazing area owned,in
common” by all members of a village.
Villagers graze cows on the common.
When c cows are grazed,total milk
production is f(c),where f’>0 and f”<0.
How should the villagers graze their
cows so as to maximize their overall
income?
The Tragedy of the Commons
c
Milk
f(c)
The Tragedy of the Commons
Make the price of milk $1 and let the
relative cost of grazing a cow be $pc,
Then the profit function for the entire
village is
and the village’s problem is to
( ) ( )c f c p cc
m a x ( ) ( ),c cc f c p c0?
The Tragedy of the Commonsm a x ( ) ( ),
c cc f c p c0?
The income-maximizing number of cows
to graze,c*,satisfies
f c p c( )
i.e,the marginal income gain from the
last cow grazed must equal the marginal
cost of grazing it.
The Tragedy of the Commons
c
Milk
f(c)
pcc
slope =
f’(c*)
c*
slope
= pc
The Tragedy of the Commons
c
Milk
f(c)
pcc
slope =
f’(c*)
c*
slope
= pc
Maximal incomef(c*)
The Tragedy of the Commons
For c = c*,the average gain per cow
grazed is
because f’ > 0 and f” < 0.
( *)
*
( *) *
*
( *)
*
c
c
f c p c
c
f c
c p
c c 0
The Tragedy of the Commons
c
Milk
f(c)
pcc
slope =
f’(c*)
c*
f c
c p c
( *)
*?
f(c*)
The Tragedy of the Commons
For c = c*,the average gain per cow
grazed is
because f’ > 0 and f” < 0,So the
economic profit from introducing one
more cow is positive.
Since nobody owns the common,entry
is not restricted.
( *)
*
( *) *
*
( *)
*
c
c
f c p c
c
f c
c p
c c 0
The Tragedy of the Commons
Entry continues until the economic
profit of grazing another cow is zero;
that is,until? ( ) ( ) ( )
.cc f c p cc f cc pc c 0
The Tragedy of the Commons
c
Milk
f(c)
pcc
slope =
f’(c*)
c*
f c
c p c
(? )
f(c*)
c
The Tragedy of the Commons
c
Milk
f(c)
pcc
slope =
f’(c*)
c*
f c
c p c
(? )
f(c*)
The commons are over-grazed,tragically.
c
The Tragedy of the Commons
The reason for the tragedy is that
when a villager adds one more cow
his income rises (by f(c)/c - pc) but
every other villager’s income falls.
The villager who adds the extra cow
takes no account of the cost inflicted
upon the rest of the village.
The Tragedy of the Commons
Modern-day,tragedies of the
commons” include
–over-fishing the high seas
–over-logging forests on public
lands
–over-intensive use of public parks.
–urban traffic congestion,
