simple pigovian taxes vs. emission eees to control negative

8/10/2019 Simple Pigovian Taxes vs. Emission Eees to Control Negative

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SIMPLE PIGOVIAN TAXES VS. EMISSION EEES TO CONTROL NEGATIV

EXTERNA LITIES: A PEDAGOGICAL NOTE

by Robert S. Main*

Abstract

Many economics texts introduce their analysis of negative externalities by examining a tax on the

output of polluting firms, sometimes called a simp le Pigovian tax, often pointing out that taxing

pollution directly is superior to taxing output and proceeding to discuss an emission fee as an alterna-

tive. They do not show how and why an emission fee is more efficient than an output tax. This note

presents a numerical example allowing comparison of the welfare effects of the two approaches, as

well as showing why simply reducing the pollution intensity of polluters' output would be inferior to

an emission fee.

eywords Simple Pigovian tax. Emission fee. Pollution, Efficiency, Command and control

I. Introduction

Almost all principles of economics texts, public

finance texts, environmental economics texts and

intermediate microeconomics texts deal with the

economics of extemalities. (Representative Princi-

ples texts: Mankiw (2007) and Frank/Bemanke

(2007).

Public Finance texts: Anderson (2003),

Gmber (2007), Holcombe (2006), Hyman (2008),

Steineman, et al (2005), Stiglitz (2000), and

Ulbrich (2003). Environmental texts: Callan and

Thomas (2000), Kahn (2005), Keohane/Olmstead

(2007) and Tietenberg (2003). Intermediate Micro-

economics texts: Besanko/Braeutigan (2005) and

Waldman (2004).) The discussion of negative pro-

duction extemalities usually begins by pointing out

that the social cost of producing goods that entail

negative extemalities is higher than the private

cost. The authors then point out that the resulting

exce ss output can be corrected by imposing a

taxoften called a Pigovian taxon the output of

the offending product. Many texts continue by

pointing out that taxing pollution directly, instead

of output, can be more efficient. However, the

texts do not show or fully explain why one ap-

proach is more efficient than the other. This note

offers a straightforward numerical example to il-

lustrate the efficiency gains from taxing pollution

directly rather than taxing the output of pollu

generating goods. The example also illustrates

simply ordering firms to reduce the pollution

tensity of their products is less efficient than ta

pollution direcdy. The issue of how to deal

negative extemalities is important, because the

ferent approaches discussed in the paper can

significantly different welfare implications.

II. Current Textbook Treatment

Economics textbooks often introduce the

cept of negative extemalities using an exampl

which the production of each unit of a good re

in a certain amount of extemal damage. (See

example, Mankiw (2007, 206-7), Frank and

nanke (2007, 349-51), Tietenberg

(2003,

6

Kahn (2005, 47-8), Keohane and Olmstead (2

67- 70 ) , Callan and Thomas (2000, 86-95),

Besanko/Braeutigan (2005, 638-6 47).) The ex

ple usually shows the competitive market su

curve (Marginal Private Cost) the demand c

(Marginal Private Benefitequal to Marginal

cial Benefit, if there is no extemality in consu

tion) and Marginal Social Cost, equal to

vertical sum of the MPC and the Marginal Ext

Damage (MED). The socially optimal quantit

where MSC intersects MSB, but a compet

Department of Economics, Law and Finance, College of Business, BuUer University, Indianap


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the competitive market produces too much

Pigovian tax equal to the level of MED. (If

careful graphical

and Kahn (2005, 47-8).)

Some texts point out that taxing the output of

is fully efficient only if the amount

ient to tax the output of firms. Rather, the pol-

t uses the word Pigovian to describe the direct

to connect the two stories.

n

The xample

Imagine a competitive industry producing good

10 units of a pollutant. (Call this amount E.) Each

unit of the pollutant that is emitted causes a dam-

age of $7.50. (Call this amount f ) There is no

extemality associated with the consumption of the

good. Then the total extemal damage (TED)

caused by the production of the good is: f E-x

where x is the total amount of the good produced

by the industry, and the damage per unit is f E, or

75,

in this example.

The demand for the good (the Marginal Private

Benefit, which in this case is also the Marginal

Social Benefit) is given by:

= 10 0 - 2 X

1

The function showing the cost of production (total

private cost) incorporates the notion that the amount

of pollutant emitted per unit of output can be re-

duced, at some cost. In accordance with the assump-

tions of our example, assume that the least costly

way to produce good X is to emit 10 units of pollut-

ant per unit of X produced. The cost function then

reflects both the cost of producing more output, hold-

ing emissions per unit constant, and the cost of de-

creasing emissions per unit, holding output constant.

2

where j is a positive parameter, EQ is 10 in this

example, and E is the amount of emissions per uni

of output that firms actually produce. Since j is

assumed to be positive, firms minimize their cos

of producing any given quantity of output by emit

ting E =

EQ

units of pollutant per unit of X pro

duced, if there is no tax on pollution, and no

regulation of pollution. Thus, if the firms in this

example were unregulated, they would emit 10 units

of pollutant per unit of X produced.

Since the supply curve in an unregulated com-

petitive market is the Marginal Private Cost

and since firms minimize their cost by emitting

E = Eo of pollutant per unit of output, the relevan

TPC = X- and the MPC = 2 x.

nregulated quilibrium

Given the information provided, it is a simple

matter to calculate the market equilibrium quantity

and price, along with the total extemal damage


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The results are as follows:

p = 50, X = 25, CS = 625, PS = 625, TED =

75 25 = 1875, NSB = -6 2 5 , and DW L = 703.125.

DWL is anived at by calculating the area between

the MSC curve and the MSB (demand) curve over

the difference in quantity between the unregulated

equilibrium quantity and the socially efficient quan-

tity. The socially efficient quantity, given that each

unit of output results in 10 units of pollution, each

resulting in a damag e of 7.50, is the quantity at

which the Marginal Social Cost (equal to MPC

75) intersects the demand curve. In this exa

that quantity is 6.25 units, so the

DW L = 0.5 (25 - 6.25) (125 - 50) = 703

(See Fig. 1)

2. Simple Pigovian Tax

Since each unit of the good prod uced results

extem al dam age of 75 , a simple Pigovian ta

125

87.5

75

50

12.5

MSC = MPC + MED

Simple Pigovian

Tax = 75 per unit

MPC

D = MPB

MSB

6.25 25


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of $75 perunit corrects theextemality

Itresultsin amarket equilibrium priceof

aquantityof6.25 units. At that quantity,

= 75 -hMPC= $87.50= P = MSB.At

CS = PS =$39.0625, while Tax

=Total Extemal Damage=$468.75. Net

=CS -I- PS -I- Tax Revenue- Total

=

$78.125. There

is no

Dead-

theequilibrium is at theinter-

of MSC and MSB, and the increase in

theimposition of the tax is

was thesizeof theDWL in the

It would bepossible to

intotal extemal damage exceeds the reduction

andproducer surpluses by$703.125.

1)

This is the standard story one might expect to see

a textbook treatment of aPigovian taxthatis

to correct anextemality.' Thestory allows

to see anumber of important points:

inthe presenceofa negative production extemal-

anunregulated competitive market will produce

ofthe good, because theproducersof

notreflect) all of thecostsofproducingthe

2) ifthe extemal costs are broughttobear on

(by

imposing

a

tax equal

to

oneither buyersor

themarketcan beinduced toproducethe

3)usually,theefficient quantity

isnot zerorather,it is the

atwhich the last unit produced hasavalue

to thefull (private

of

it.Authors often note that thisis asim-

of the tme story, since it implicitly

the

amount

of

pollutant emitted

per

ofthe good produced cannot be changed. (See,

etal (2005, 214-16), and

saythat, because

ofpollution emissionscan bechanged,

inprinciple,todeal with the problem

is to tax thepollutantdi-

notext-

Ihave found attemptstomodel how costsof

theamount of pollutant

per

unit

is

reduced. Nor does any textbook

amodel orexample that allows thecom-

all authors (forexample, seeAnderson, Gmber

Hyman, Ulbrich, andWaldman) simply moveon

to thedi.scussion of how a firm would react to

an emissions fee and other approaches to controlling

emissions (suchasemission standardsandtradabl

emission permits).

3

A Tax on Pollution Emissions

Wecan use themodel introduced aboveto ex

amine how a tax on pollution emissions would

work. Wealready know that thedamage caused

per unitofemissionsisrepresentedbythe parame

ter/($7.50, in this example).Ifa firmisassessed

taxof$7.50perunitofpollutant emitted, itwil

reduceE(the amountofpollutant emittedperuni

of output)aslongas thesaving inemission taxe

paid exceeds the increase in production cos

entailed thereby.Itwill stop reducingEwhenth

reduction in taxes due to reducing E by 1uni

equals the increase in production cost due to reduc

ingE by1 unit. The savingintaxes equalsT x (o

$7.50

X

inthis case), while the increaseinprodu

tion cost (the negative

of

the partial derivative

o

TPC with respecttoE)is:

2-j-(Eo-E)-x.

3

The optimal amountof E isthus whereT = 2

j

Q

-E). (Thex'scancel.) The optimalE forfirm

to emit isthus E* = 10 - (T/(2j)). Tokeepthe

computations simple,

I

selected

a

value

of j =

0.536, resultingin avalueofE*of3.0. Therefore

in my example, firms faced with an emissionfeeo

$7.50 per unitofpollutant they emit would choos

to reduce the emissionsperunitofoutput from1

to3. Of course, this results in higher totaland

marginal production costs

for

firms. Given

a re

duction of emissions from 10 perunit to 3 pe

unit, themarginal costofproducing more outpu

(the partial derivative ofTPC with respect to x

becomes

MPCx =

(0.536)-(7^)-2-x-t-26.264 (4

Recall, however, that the firms are paying the gov

emment $7.50perunitofpollutant emitted. Sinc


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the firms are paying $22.50 to the govemment, per

unit of output produced, in addition to the produc-

tion cost. W hen that co st is included , the height of

the supply curve becomes 2 x + 26.264 + 22.5 =

2 X + 48.764. This is the full MC (including tax)

that firms incur to prod uce X . It is the ind ustry

supply curve, in the presence of the emissions fee,

and the intersection of this curve with the demand

curve gives the market equilibrium.

Given the parameters of our example, this equi-

librium is as follows: P = $74.382, output =

12.809, TSB = TPB = $1116.829, TD = Tax

Reven ue = $28 8.56 1, Total Private (production)

Cost = $500.12 7, NSB = CS + PS + Tax Reve-

nue - TD = $32 8.141 1. (See Fig. 2)

4 Com parison of an Em ission Fee with

Simple Pigovian Tax

Note that the output of the good, consu

surplus, producer surplus and net social be

are all greater than they were whe n the unr

lated amoun t of pollution per unit of good X

used to determine the simple Pigovian tax. W

no govemment intervention, the least costly

to produce X was to emit 10 units of pollutan

unit of X p roduc ed. That resulted in an exte

dam age of $75 per unit of X produced. If

govemment responds by imposing a $75 tax

each unit produced, firms have no incentiv

reduce the amount of pollutant they emit, per

1

74 382

63 132

5

48 764

26 264

Equilibrium

with emission

fee

MSC = MPC + MED = MPC + T =

26.264 + 2-x + 22.5 = 48.764 + 2-x

MPC - 26.264 + 2-x

(when E is reduced from 10 to 3)

Equilibrium with command

and control regulation

S = M PC with no

regulation

Equilibrium with

no regulation

D = MPB =

MSB

12.809 18.434 25 50


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X produced. The equilibrium in the market is

the price per unitof X is $75 higher

the marginal cost of production. This occurs

is 6.25,price is$87.50,and the mar-

of

production

is

$12.50.

By way of

is

taxed directly,

via an

fee of

$7.50

per

unit

of

pollution, firms

to

reduce

E

from

10 per

unit

X to

per

unit

of X,

since

the

marginal cost

to

firms

is less than $7.50 per unit

for emission levels higher than 3 per

of X. When the simple Pigovian tax was

all of the correction for the pollution

had to come in the form of reduced

of X. When the emission fee was

of the correction camein the form

the emissions per unitof X. The rest

in the

form

of

reduced output.

The net ef-

was simultaneously to reduce the amount of

62.5 units to 38.43 units)

the amount of X produced (from

to 12.809).

NSBislarger withtheemissionfeethan withthe

taxbecause,up to apoint, reducing

eE (theamountofpollution emittedperunitof X

isless costlyto society than reducingthe

of X produced, while holding E constant.

fee

allows market participants

to

One might argue that

a

well-designed output

tax

the

same result

as an

emission

fee.

is, ifeach producerof X were confronted with

the tax on itsoutput would

the amount of pollutant that firm

perunit of X produced, then firms would

toreducetheamountof pollutant emit-

d perunitof X,justas in theemission fee case.

the

output

tax

would

in

that case actually

an emission fee. (This appears to be whatSti-

has in mind.) Real woridout-

do notappearto be calculated thatway.

at theamount of pollution actu-

per unit of output to determine the

tax.

5 .

omparing

an

Emission

ee to

ommand

and

ontrol

reduce emissions from 10 per unit of X to 3 pe

unitof X (theefficient amo unt of emission reduc

tion, given the cost function and the margina

damage per unit of pollution). In that case, the

industry supply curve would

be

given

by the mar

ginal production cost curve

(MC =

26.264

-f 2x)

The equilibrium quantity would

be

18.434,

and

the price would

be

$63.132. Since

the

efficien

quantity

is

12.809, there

is a

deadweight loss

equal to$63.281. The apparatus used in this note

allows us to see how much more efficient an

emission fee is than direct control, even if firms

do not differ in their marginal cost functions and

even if the govem ment could, somehow, deter

mine the proper amount of reduction in E fo

firms to undertake. If firms were subsidized to

reduce emissions, the deadweight loss would be

even greater. ' '

IV. Conclusion

Many authors introducetheanalysisof negative

production extemalities

by

examining

a tax on the

output

of

polluting firms, what

I

have called

a

simple Pigovian

tax.

Most then point

out

tha

taxing pollution directly

is

more efficient than

tax

ing output andproceed todiscuss anem ission fee

along with command and control regulation and

tradable permits. No attempt is made in the text

books to show how and why an emission fee i

more efficient than an output tax. This note ha

presented a simple numerical example that allow

the welfare effects of the two approaches to be

easily compared. It also allows studentsto see eas

ilywhy a govemment policy of ordering firms to

reducethepollution intensityof their output would

be less efficient than theemissionfee.

Notes

1. M ore thorough treatmentsof this storyare giv

enbyCallan/Thom as (2000, 79-8 6and 92) an

by Besanko/Braeutigan (2005, 638-47).

2.

Ho lcombe explicitly asks, W hat should b

taxed?

He then argues verbally and graphi

cally that taxing pollution directly results in a

more efficient outcome.

His

Figure

4.2 (p. 74

is similar

to my

Figure

2.

However ,

he

doe


7/8

3 .

Nichols (1984, 27 -29 ) reaches similar con-

clusions using a simplified diagrammatic

approach with constant marginal costs. His

treatment does not contain an explicit algebra-

ic model showing how firms adjust the amount

of pollution emitted per unit of output in re-

sponse to a tax on emissions.

References

Anderson, John E. 2 003 .

Public Finance.

Boston:

Houghton Mifflin.

Besanko, David, and Ronald R. Braeutigam. 2005.

Microeconomics, 2 * ed. New York: Wiley.

Callan, Scott J. and Janet M. Thomas. 2000.

Envi-

ronmental Economics and Managem ent, 2 ^

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Eon Wonh: Dryden.

Erank, Roben H. and Ben S. Bemanke. 2007.

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Gruber, Jonathan. 2007.

Public Finance and P ub-

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2 ' ' ed. New Y ork: Worth.

Holcombe, Randall. 2006.Public Sector Economics.

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it h

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Wes tem.

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Cambr

MA: MIT Press.

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Wes tem.

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