Externalities

© Allen C. Goodman 2009

Ideal Market Processes are desirable if …

• We accept the value judgment that “personal wants of individuals should guide the use of society’s resources.”

• Three structural characteristics are necessary:– All markets are competitive.

– All participants are fully informed.

– All valuable assets can be individually owned and managed without violating the competition assumption.

• If these hold, government’s best role involves determining an income distribution, providing rules of property and exchange and enforcing competition.

Markets

If markets behave properly, COST of item equals the PRICE that buyers are willing to pay.

Value to consumer = Value to producer

With competition, in the short run, the firm produces to where:

MC = MR = P

Value of resources in production = MR = Value to consumer

We can do a little bit of geometry to show this:

Typical Firm Diagram

Mkt. Firm

Firm qEconomy Q

PP*

D

S

MC

AC

MR

Pareto Efficiency

Context of trade.

One can’t make oneself better off, without making someone else worse off.

We usually do this with an exchange Edgeworth Box.

Abner

Belinda

Abner’s Preferences

Belinda’s Preferences

Pareto Efficiency

Start at Point A.

Is this an Equilibrium?

Abner

Belinda

No, they can trade

Belinda can be better off.

ASo can Abner.

B

Pareto Efficiency

We can plot similar points, which we recognize as a “contract curve”

Abner

Belinda

And so on.

A

B

Pareto Efficiency

We must recognize that point X is Pareto Optimal.

Abner

Belinda

So is point Y.

A

B

X

Y

Utility Possibility Frontier

We can plot Abner’s utility against Belinda’s Utility.

Why do we draw it this way?

Abner’s Utility

Bel

inda

’s U

t ili

ty

X

Y

What if we want a perfectly egalitarian society?

Does equal utility mean equal allocations?

So, are markets always great?

• Externality – A cost or benefit in production or consumption that does NOT accrue to the producer or the consumer of the commodity.

• No single person can own or manage air or water.

• Consider a person who wants to heat a house with a wood fire.

1. More wood more heat.

2. W/ more heat, willingness to pay for additional heat .

3. More wood and more heat more smoke

Heat and smokeIndividual sees price of wood as

P1.

Compares price to marginal benefit (demand curve).

Individual purchases quantity A of wood.

BUT…

Heat, smoke

$

MC

MSC

D = WTP

P1

A

Wood Smoke.

Assume that more burning more smoke.

We get MSC curve B

Heat and smoke

• If we go past B the marginal benefits are:

Heat, smoke

$

MC

MSC

D = WTP

P1

A

• Wood Smoke.

• Assume that more burning more smoke.

• We get MSC curve

B

Inc.Ben.Inc.

Costs• If we go past B the marginal

costs are:

Heat and smoke• If we go past B we get

societal losses.

Heat, smoke

$

MC

MSC

D = WTP

P0

P1

AB

Inc.Ben.

Losses

• This is a NEGATIVE externality.

• How to remedy?

• A tax of P0 – P1.

• Called a Pigovian Tax, after, Arthur Pigou early 20th century economist

Tax

Heat and smoke• Tax of P0 – P1.

Heat, smoke

$

MC

MSC

D = WTP

P0

P1

AB

Inc.Ben.

Losses

Tax

• Has nothing (necessarily) to do with cleaning up the air.

• We must set up a market for a resource that no one specifically owns.

• Think of it as taking revenues and refunding it back to population.

• Who gains? Who loses?

A general problem – the LakeExternalities Equations

n industrial firmsYi = outputPi = pricexi units of labor at wage W

Production Function+ + +

Yi = Yi (zi, xi, q), where:

zi = waste dischargesq = quality of lakeL = assimilative capacity of Lake - - - +q = Q (z1, z2, ..., zn, L)

Society’s ObjectiveSocietal Objective:Max U = Pi Yi (xi, zi, q) - W xi - C (L) - [q - Q (z1, z2, ..., zn, L)]

Pi is the willingness to pay (related to utility of goods).PiYi is the amount spent (related to utility of goods). is the valuation of the extra unit of environmental quality.

First Order Conditions: U / xi = Pi Yi

xi - W = 0. (a) U / zi = Pi Yizi + Qzi = 0 (b) U / q = Pj Yj

q - = 0 (c) U / L = QL - C' = 0 (d)

Public Good

Society’s ObjectiveFirst Order Conditions:

U / xi = Pi Yixi - W = 0. (a)

U / zi = Pi Yizi + Qzi = 0 (b) U / q = Pj Yj

q - = 0 (c) U / L = QL - C' = 0 (d)

For Firm 1:P1 Y1

x1 = WP1 Y1

z1 = - Qz1

P1 Y1q =

Eq'm:P1 Y1

z1 = [P1 Y1q] [- Qz1]

z1

$P1 Y1

z1

[P1 Y1q] [- Qz1]

z*1

AmountCollected

Society’s ObjectiveFirst Order Conditions:

U / xi = Pi Yixi - W = 0. (a)

U / zi = Pi Yizi + Qzi = 0 (b) U / q = Pj Yj

q - = 0 (c) U / L = QL - C' = 0 (d)

For Firm 1:P1 Y1

x1 = WP1 Y1

z1 = - Qz1

P1 Y1q =

z1

$P1 Y1

z1

[P1 Y1q] [- Qz1]

z*1

For Society:P1 Y1

x1 = WP1 Y1

z1 = - Qz1

Pj Yjq =

Optimum: P1 Y1z1 =

[P1 Y1q + 2,n Pj Yj

q ] [- Qz1] > [P1 Y1

q] [- Qz1]

[P1 Y1q + 2,n Pj Yj

q ] [- Qz1] > [P1 Y1q] [- Qz1]

z*1

TAX

So …

• Societal optimum dictates that each firm produce less than in an autarkical system.

• Remedy, again, would be a tax.

• Once again, a situation where ownership is not well-defined and one’s actions affect others.

Coase TheoremThe output mix of an economy is identical, irrespective of the

assignment of property rights, as long as there are zero transactions costs.

Does this mean that we don’t have to do pollution taxes, that the market will take care of things?

Some argue that it’s not really a theorem.

It does set out the importance of transactions costs.

Let’s analyze.

Externalities and the Coase Theorem

X F L K Y

Y G L K

L L L

K K K

x x

y y

x y

x y

( , , )

( , )Production of Y decreasesproduction of X, or FY < 0.

+ + -

If we maximize U (X, Y) we get:

U F L K Y G L K

U F L K G L L K K G L L K K

x x y y

x x x x x x

[ ( , , ), ( , )]

[ ( , , ( , )), ( , )]

Planning Optimum

U F L K Y G L K

U F L K G L L K K G L L K K

x x y y

x x x x x x

[ ( , , ), ( , )]

[ ( , , ( , )), ( , )]

If we maximize U (X, Y) we get:

If we maximize U (X, Y) w.r.t. Lx and Kx, we get:

U

U

F

GF

F

GFY

X

L

LY

L

LY (*)

Does a market get us there?

Planning optimum Market Optimum

U

U

F

GF

F

GFY

X

L

LY

K

KY (*)

Does a market get us there?

If firms maximize conventionally, we get:

p F p G w

p F p G rX L Y L

X K Y K

F

F

G

G

w

rL

K

L

K

F

G

F

G

p

pL

L

K

K

Y

X

So?

U

U

F

GF

F

GFY

X

L

LY

K

KY (*)

U

U

p

p

F

G

F

GY

X

Y

X

L

L

K

K

(**)

Society’s optimum

Market optimum

Since FY < 0, pY/pX is too low by that factor. Y is underpriced.

Coase TheoremThe output mix of an economy is identical, irrespective of the

assignment of property rights, as long as there are zero transactions costs.

Suppose that the firm producing Y owns the right to use water for pollution (e.g. waste disposal). For a price q, it will sell these rights to producers of X.

Profits for the firm producing X are:

X

( )

( , , ) ( ) (***)

0

X X X X X X

X Y

Y Y T Tickets

p F L K Y wL rK q Y Y

p F qY

Reduced by paying to pollute

Coase Theorem

Y Y Y Y Y Y

Y L

Y K

p G L K wL rK q Y Y

Lp q G w

Kp q G r

( , ) ( )

( )

( )

Y

Y

Y

Y

0

0

Y Y T

p F L K Y wL rK q Y Y

p F q

X X X X X X

X Y

( , , ) ( ) (***)

YX 0

We know that q = -pXFY

α 1 gets to Y;Like the iceberg model

Coase Theorem

Y

Y

Y

Y

( ) 0

( ) 0

Y L

Y K

p q G wL

p q G rK

X

Y p F qX Y 0

We know that q = -pXFY

F

GF

F

GF

p

pL

LY

K

KY

Y

X

If 1, this looks like (*)

Change the ownership - X owns

Y Y Y Y Y Y

Y L

Y K

p G L K wL rK qY

Lp q G w

Kp q G r

( , )

( )

( )

Y

Y

Y

Y

0

0

X X X X X X

X Y

p F L K Y wL rK qY

p F q

( , , ) (****)

YX 0

We know that q = -pXFY/

If Y owns

If 1, this looks like (*)

F

GF

F

GF

p

pL

LY

K

KY

Y

X

If X owns

If 1, this looks like (*)

F

G

F F

G

F p

pL

L

Y K

K

Y Y

X

If = 1 We are at a Pareto optimum We are at same P O.

If is close to 1 We may be Pareto superior We are not necessarily at same place.Where we are depends on ownership of prop. rights.

Remarks

• These are efficiency arguments.

• Clearly, equity depends on who owns the rights.

• We are looking at one-consumer economy. If firm owners have different utility functions, the price-output mixes may differ depending on who has property rights.

If X holds, Y pays this muchIf Y holds, X pays this much

Graphically

T = Tx + Ty

q

Y’s supply (if Y holds)X’s demand (if Y holds)

-pxFY Py -r/GK = Py -w/GL

T*

If X holds, Y pays this muchIf Y holds, X pays this much

But, with transactions costs

T = Tx + Ty

q

Y’s supply (if Y holds)X’s demand (if Y holds)

-pxFY Py -r/GK = Py -w/GL

T*

Y gets this muchq

X gets this muchq

The equilibria are not the same!