Taxes vs. Quotasor Taxes vs. Upper Bounds?

Nicholas Brozovic,∗

David L. Sunding, and David Zilberman†

November 2, 2004

Abstract

We compare taxes and quotas where a regulator and a non-strategic firmhave asymmetric information about a pollution-producing activity. In previ-ous studies, optimal quotas are generally assumed to bind with probabilityone. We analyze the conditions under which a quota that bindswith prob-ability less than one is optimal. A quota that may be slack canbe targetedtowards some firms whilst allowing others to operate unconstrained. Failureto consider optimal quotas that may be slack overestimates the advantage oftaxes over quotas, even in situations where the marginal benefit function ismuch steeper than the marginal cost function.

Keywords: Pollution control; Taxes and quotas; Asymmetric informa-tion

JEL classification:D81; H23; L51

∗Corresponding author; University of Illinois at Urbana-Champaign, Department of Agricul-tural and Consumer Economics, 307 Mumford Hall, Urbana, IL 61801; emailnbroz@uiuc.edu;tel 217-333 6194; fax 217-333 2312

†Both at University of California at Berkeley, Department ofAgricultural and Resource Eco-nomics, 207 Giannini Hall, Berkeley, CA 94720

1 Introduction

Many pollution problems are characterized by uncertainty on the part of the regu-

lator. This may be because polluting firms have better information than regulators

on their emissions or abatement costs, so that there is asymmetric information.

Alternatively, the damages caused by pollution and the pollution process itself

may be subject to uncertainty. Price instruments and quantity instruments are

commonly used to control pollution. With a price instrumentthe regulator uses a

tax, on pollution or on a suitable proxy, and firms equate the marginal benefits of

their productive activity to this tax. Under a tax with asymmetric information, the

aggregate level of output is uncertain. Conversely, with a quantity instrument and

asymmetric information, the regulator uses a quota to obtain a certain aggregate

level of output; the marginal benefits of productive activity are uncertain.

If the regulator is able to use contingent regulations that differentiate each

type of firm or possible state of nature, first-best taxes and quotas have identical

welfare outcomes. If there is asymmetric information and first-best regulation

is not feasible, the welfare effects of taxes and quotas willvary. An extensive

literature in the tradition of Weitzman [11] considers the choice between taxes

and quotas under asymmetric information (e.g. Roberts and Spence [9]; Laffont

[5]; Malcomson [6]; Nichols [7]; Kolstad [4]; Stavins [10];Hoel and Karp [2]).

These papers assume, implicitly or explicitly, that quantity regulation involves

the dictation of a specific level of activity that is strictlyadhered to. The chosen

quota will thus influence the decisions of the regulated group in all states of na-

ture, and binds with probability one. However, quotas oftenimply an upper bound

on regulated activities, because quota recipients have theoption of not taking full

advantage of their allocation. Hence, the assumption that optimal quotas bind with

probability one is unduly restrictive. Indeed, in practicemost quantity regulations

are implemented as upper bounds: for example, environmental regulators are gen-

1

erally unconcerned if a firm emits less pollution than it is legally entitled to, and

there is no penalty for driving below the speed limit on a highway. Moreover,

if an industry is very heterogeneous in terms of either its productive or polluting

capacities, it is unlikely that firm response to a uniform quota will be identical

in all possible states of nature. This paper analyzes the implications of allowing

regulated groups to treat quotas as upper bounds – that is to say, allowing quotas

to be slack with positive probability.

We are aware of only a few previous studies that consider the possibility that

quotas may not bind with probability one. Hochman and Zilberman [1] compare

the welfare impacts of taxes and quotas using fixed proportions production and

pollution functions. Because of this functional form, firm adjustment to regula-

tion only occurs at the extensive margin, and individual production units cannot

adjust at the intensive margin in response to regulation. Thus, under each kind of

regulation, firms either continue to operate at full capacity, or shut down. Wu and

Babcock [12] consider firm adjustment at both intensive and extensive margins

for the problem of second-best regulation under heterogeneity. Although they do

recognize the possibility that quotas may not bind with probability one, they do

not analyze either the potential optimality or the implications of this. Karp and

Costello [3], using a dynamic model with asymmetric information, show that a

regulator may use a slack quota to gain information with which to better target fu-

ture quotas. However, in their model, the optimal one-period quota still binds with

probability one, and use of a slack quota leads to loss of surplus in that period.

Finally, in a numerical simulation of alternative policiesto mitigate global climate

change, Pizer [8] shows that for realistic cost and benefit parameters, second-best

quotas do not bind in all cases. For the particular application he considers, Pizer

calculates that a failure to account correctly for slack quotas overestimates by a

factor of five the advantage of taxes over quotas, yielding anestimated gain of $10

2

billion in 2010, rather than the correct figure of $2.2 billion.

Our objective is to compare taxes and quotas in a static framework when the

regulator may use a quota that is slack with positive probability. Setting a quota

that may be slack allows targeting of the regulation on a subset of possible firms.

Firms that are targeted in this way will operate closer to thesocial first-best level

of production. Conversely, firms for which the quota is slackwill operate uncon-

strained and will be producing further from the social first-best level. There are

thus two opposite effects from allowing a quota to be slack: firm targeting leads

to an increase in net surplus, whereas unconstrained firm operation leads to a de-

crease in net surplus. In order to allow comparison with existing work, we derive

analytical results using the same assumptions about functional forms as previous

literature. We derive conditions under which an optimal quota may be slack, and

compare taxes and quotas under these conditions. We show that if there is enough

heterogenity in the regulated industry, the optimal quota must be slack with posi-

tive probability. Moreover, our analysis suggests that previous studies have over-

estimated the relative advantage of taxes over quotas. The ability to use a quota

that may be slack means that quotas are preferred to taxes over a wide range of

parameter values where previous studies would indicate theopposite result.

2 The model

We begin by describing the functional forms and informationasymmetry used in

the general model. We then present optimality conditions for second-best regula-

tion with taxes and quotas.

3

2.1 Elements of the model

We assume a representative firm which has a production technology and a pollu-

tion technology. The production technology captures the quasi-rents of a given

level of production activity by the firm. The pollution technology captures the ex-

ternality costs of the firm’s activity. We assume that the production and pollution

technologies are independent attributes of the representative firm.

The regulator has full information about the firm’s pollution technology, but

is uncertain of the firm’s production technology.1 For simplicity, we assume that

there are two possible production technologies, low-productivity (L) and high-

productivity (H). The representative firm is anL-type with probabilityθ and an

H-type with probability1−θ. Our major results hold for continuous distributions

of production technology, but the proofs are more difficult to interpret.

The firm uses a scalar input,xi, in the production of a numeraire good, where

i ∈ {L, H} and denotes whether the firm is anL-type or anH-type. Its quasi-

rents net of input prices are given by the production function fi(xi). Additionally,

we assume thatfi(xi) ≥ 0, f ′i(xi) > 0, f ′′

i (xi) < 0, fi(0) = 0, and thatf ′L(xi) ≤

f ′H(xi) for all positive values ofxi.

Input use by each firm causes a negative externality. The damage caused by

using xi units of input is given by the functiong(xi), where we assume that

g(xi) ≥ (0), g′(xi) > 0, g′′(xi) > 0, and g(0) = 0. We also assume that

f ′L(0) > g′(0), so that it is socially desirable for the firm to continue to oper-

ate at some positive level even if it is a low-productivity type. This assumption

implies that the representative firm will never shut down as aresult of regulation,

and adjustment will take place at the intensive margin only.

The expected net surplus is given by the sum of quasi-rents and damages

1Although it is possible to consider uncertainty by the regulator over both the pollution andproduction technologies, it is well known that if uncertainty in the pollution and production tech-nologies is uncorrelated, then it is sufficient to consider only one source of uncertainty [10] [11].

4

caused by production taken over realizations of the production technology:

Ei [fi(xi) − g(xi)] (1)

In the absence of any regulation, the firm will maximize quasi-rents by us-

ing an input levelxci such thatf ′

i(xci) = 0, so thatxc

L ≤ xcH . This means that

without regulation, if the firm is a high-productivity type,it will on average pro-

duce more pollution than if it is a low-productivity type. Becauseg′′(xi) > 0, a

high-productivity firm will also produce more pollution perunit of input than a

low-productivity firm.

The firm does not internalize the damage caused by its production and net

surplus may be improved by regulation. In a first-best setting, the regulator can

maximize net surplus by using a regulation – either a tax or a quota – that is con-

tingent on the firm’s production technology. Such contingent taxes and quotas will

attain the same surplus level. However, contingent regulation is often impossible

for two reasons. First, the regulator may be unable to ascertain firm type. Second,

even if the regulator has full information, reasons of politics or perceived equity

may prevent the government from using contingent regulation. In many settings

of heterogeneity or uncertainty, the regulator is constrained to use a single, uni-

form, instrument. Under such conditions, the welfare effects of second-best price

and quantity instruments will, in general, differ.

2.2 Optimal second-best regulation

The regulator seeks a uniform policy instrument,M∗, which will maximize net

surplus, defined by

M∗ = arg maxM∈ℜ+

Ei [fi(xi(M)) − g(xi(M))] (2)

5

In equation (2),xi(M) is the input use decision by a firm of productivity type

i facing a policy instrument given byM . The first-order condition corresponding

to equation (2) is

Ei [x′

i(M∗) (f ′

i(xi(M∗)) − g′(xi(M

∗)))] = 0 (3)

The marginal response of ani-type firm to a unit increase in the policy in-

strumentM is given byx′i(M). In this paper, we consider the choice of policy

instrument between a second-best tax and a second-best quota.

2.2.1 Second-best taxes

The regulator may choose to use a uniform tax as the second-best policy instru-

ment. In this case,xi(t) is the input use decision by ani-type firm facing a per-unit

input tax oft. We assume that tax revenues are recycled in a non-distorting fash-

ion and thus net out of equation (2). The marginal response ofan i-type firm to

a unit increase in the tax may then be written asx′i(t). Given a tax oft, the firm

will choose an input use so thatf ′i(xi(t)) = t, equating the value of its marginal

product and the per-unit tax.2 With the second-best tax, high-productivity (H-

type) firms will continue to use more inputs than low productivity (L-type) firms,

so thatxL(t) ≤ xH(t). As before, a firm that isH-type will on average produce

more pollution and more pollution per unit of input than a firmthat isL-type.

2We assume that the second-best tax does not induce the low-productivity firm to shut down.This would occur if the optimal tax were larger than the quasi-rent for the low-productivity firmfor the first unit of production. If this were the case, the regulator could target the tax to the high-productivity firm, and increase net surplus, even though expected production would decrease. Fora qualitative analysis of firm shut down under second-best taxes, see Wu and Babcock [12]. In thereal world, quotas that are slack on some firms are common, butpollution taxes are not generallylarge enough to lead to firm closure. For this reason, in this paper, we assume that second-besttaxes do not lead to shut down.

6

2.2.2 Second-best quotas

Alternatively, the regulator may address the production externality using a quota

on input use. In this case, the input use decision by a firm of productivity type

i facing a quota ofQ may be written asxi(Q), and its change in input-use in

response to a marginal increase in the quota is given byx′i(Q). If the quota binds

with probability one – that is to say, if it binds on both theL-type and theH-type

firm – thenxL(Q) = xH(Q) = Q, x′L(Q) = x′

H(Q) = 1 and first-order condition

(3) simplifies to

Ei [f′

i(Q∗

1)] = g′(Q∗

1) (4)

Equation (4) states that if it is known that the quota will bind with probability

one (denoted by the subscript onQ∗), then the optimal quota equates the expected

benefits for both firm types and the costs of pollution. This isa well-known result

(e.g. Weitzman [11]; Laffont [5]; Stavins [10]).

However, ifL-type andH-type production technologies are different enough,

then the quota may not bind with probability one. In particular, the quota may

bind on anH-type firm, but may be large enough that it does not bind on an

L-type firm. In this case, input use and marginal input use are given by

xH(Q) = Q

x′

H(Q) = 1

xL(Q) =

xcL if f ′

L(Q) ≤ 0

Q if f ′L(Q) > 0

x′

L(Q) =

0 if f ′L(Q) ≤ 0

1 if f ′L(Q) > 0

(5)

7

If the quota does not bind on anL-type firm, it follows that it is operating at

its unconstrained optimum, and increasing the quota further will have no effect on

input use. First-order condition (3) simplifies to

f ′

H(Q∗

θ) = g′(Q∗

θ) (6)

In this case, the regulator ignores the possibility that thefirm is a low-productivity

type entirely, and equates the marginal benefits of production and marginal pollu-

tion damages for the high-productivity firm only – even though with probability

θ, the quota will be slack (denoted by the subscript onQ∗). This possibility is

illustrated using linear marginal production and pollution functions in Figure 1.

What are the advantages to the regulator from setting a quotathat does not bind

with probability one? By specifically targeting the quota atthe high-productivity

firm, there is a probability of1 − θ that there will be no loss of surplus relative to

the first-best contingent regulation. The benefits of such targeting may be larger

than the additional damages caused by allowing a low-productivity firm to operate

unconstrained, with probabilityθ.

The literature comparing taxes and quotas under conditionsof asymmetric

information is extremely large. However, in ranking second-best taxes and quotas,

existing analyses assume that quotas bind with probabilityone (corresponding to

equation (4) above) and ignore the possibility that an optimal quota need not bind

in all states of nature (corresponding to (5) and (6)). As will be shown below, the

assumption that quotas are never slack underestimates the efficiency of second-

best quotas in comparison to second-best taxes.

Moreover, for a set of production functionsfi(·) and a pollution functiong(·),

it is possible that there exist solutionsQ∗1 andQ∗

θ that are admissible solutions to

first-order conditions (4) and (6) respectively. In this case the net surplus function

has two local maxima. However, the assumption of only two production tech-

8

nologies allows the support of the net surplus function to bepartitioned into two

sets. Each portion of the net surplus function then containsa unique, well-behaved

maximum. If multiple local maxima exist, the choice of a global maximum will

depend on the functional forms of the production and damage functions. Simi-

larly, if production technologies are continuous rather than discrete, multiple local

maxima may still exist that satisfy the relevant first-orderconditions.3 We limit

our analysis to a two-part discrete distribution: with continuous distributions, the

general results still hold, but analytical solutions are intractable.

3 Analytical example

The model shows that a regulator can theoretically choose anoptimal quota that

does not bind with probability one. However, in order for this result to have

practical implications, two things need to be demonstrated. First, there must be

a significant range of relevant conditions under which the optimal quota does not

bind in all states of nature. Second, consideration of such quotas must alter the

choice between taxes and quotas under at least some of these conditions.

Without specific functional forms, analytical comparisonsof second-best taxes

and quotas are not possible. We illustrate the basic resultsof the model using the

same assumptions about functional forms as Weitzman [11], namely that quadratic

approximations to the production and damage functions are adequate. Thus, the

production and damage functions have the following forms:

fi(x) ≈ f ′

i(x)x +f ′′

i (x)

2x2

3A necessary, but not sufficient, condition for the existenceof multiple solutions for the choiceof second-best quota is that as the quota increases, the expected marginal product for the subsetof firm types for which the quota binds increases over some interval. Discrete distributions withmore than two possible values may thus have more than two solutions.

9

g(x) ≈ g′(x)x +g′′(x)

2x2 (7)

By following Weitzman’s functional forms, the effect of allowing quotas to be

slack can be compared directly with the results of previous work. Additionally, as

in Weitzman’s study, we assume thatf ′′L(x) = f ′′

H(x) = f ′′ and thatg′′(x) = g′′.

These assumptions imply that the marginal production functions for each firm

type have the same slope.4 It also follows that a firm of typei will have a constant

response to a marginal increase in the per unit tax, so thatx′i(t) = 1/f ′′. Hence,

equation (3) can be simplified tot∗ = Ei [g′(xi(t

∗))], which is the optimal uniform

tax.

3.1 Conditions under which the optimal quota may not bind

With these assumptions about the functional forms of the production and damage

functions, it is possible to obtain analytical expressionsfor the conditions under

which the optimal quota may be slack, or there exist two choices for an optimal

quota. In order to simplify interpretation of the analytical results, we introduce

several further parameters. First, recall that the input use Q∗1 corresponds to the

quota chosen by the regulator if it is assumed that the optimal quota will always

bind with probability one. Note that the input useQ∗1 is defined even if it is

not an admissible solution to first-order condition (4). Figure 1 illustrates this

possibility. The input useQ∗1 forms a convenient baseline for comparison. Define

the parameterγ1 = f ′H(Q∗

1) − f ′L(Q∗

1), so thatγ1 > 0 and is the difference in

marginal products between anH-type and anL-type firm.

Recalling that the probabilities of the firm being low-productivity and high-

productivity areθ and 1 − θ respectively, the variance off ′i(Q

∗1) is defined as

4Relaxing this assumption adds significantly to the complexity of the problem, and has beenanalyzed by Malcomson [6].

10

σ2 = Ei [f′i(Q

∗1)

2] − (Ei [f′i(Q

∗1)])

2 = θ(1 − θ)γ21 . The productθ(1 − θ) is a

measure of the skewness of the distribution of production technologies. It attains

a maximum value when there is an equal probability that the firm is either type.

The parameterγ2 is the difference in marginal products between the two pos-

sible types of firm divided by the marginal product of a high productivity firm and

is defined asγ2 = γ1/f′H(Q∗

1). Note that the parametersγ1 andγ2 are defined

even when the quotaQ∗1 may not be an admissible solution to (4). A value of

γ2 < 1 implies that at the quotaQ∗1, a low productivity firm will have a positive

marginal product. A value ofγ2 > 1 implies thatQ∗1 > xc

L, so that at a hypo-

thetical production level ofQ∗1, the low productivity firm would have a negative

marginal product. Note also that there is a maximum value ofγ2 that is possi-

ble for the problem of interest. By a geometric argument, it is impossible forγ2

to take a value greater than1/θ as this is not consistent with a marginal damage

function that has a positive value atQ∗1.

Finally, the parameterβ is defined as the ratio of the elasticity of the expected

marginal productivity with respect to input use to the elasticity of marginal pollu-

tion damage with respect to input use. Thus, noting that by definition Ei [f′i(Q

∗1)] =

g′(Q∗1), β is given by the expressionβ = − (Q∗

1 · f′′/Ei [f

′i(Q

∗1)]) / (Q∗

1 · g′′/g′(Q∗

1)) =

−f ′′/g′′, which is the negative of the ratio of the slope of the marginal product to

the slope of the marginal pollution damage. The parameterβ is always positive,

and it captures the relative trade-off between increased production and increased

pollution as input use increases. For the functional forms used, a value ofβ of

more (less) than one implies that the marginal production function is steeper (shal-

lower) than the marginal damage function.

These parameters can be used to determine the conditions under which an

optimal quota that is slack with positive probability exists:

Lemma 1 A quota that binds with probability 1 is an admissible solution to first-

11

order condition (4) ifγ2 ≤ 1.

Lemma 2 A quota that binds with probability(1−θ) is an admissible solution to

first-order condition (6) ifγ2 ≥1+β

1+(1+θ)β.

Note that because 1+β

1+(1+θ)β< 1, Lemmas 1 and 2 imply that it if 1+β

1+(1+θ)β≤

γ2 < 1, there will be two different quotas that are potential solutions to problem

(2). Conditions under which is one preferred to the other will be analyzed below.

3.2 Ranking of second-best policies

It is not sufficient to show that an optimal quota may be slack in some states of

nature. We must also demonstrate that a failure to consider such a possibility can

lead to incorrect rankings of second-best policy instruments. Define the net sur-

plus under a second-best tax aswt, that under a second-best quota that binds with

probability one aswq1, and that under a second-best quota that is slack with prob-

ability θ aswqθ. In Weitzman’s original analysis – with quotas assumed to bind

with probability one – the choice between a second-best tax and a second-best

quota simplified to a comparison of the steepness of the marginal production and

marginal damage functions. We begin our analysis by briefly rederiving Weitz-

man’s result using our notation. The difference in net surplus between a uniform

tax and quota that binds with probability one is given by

wt −wq1 = θ

∫ Q∗

1

xL(t∗)(f ′

L(z) − g′(z)) dz + (1− θ)∫ xH(t∗)

Q∗

1

(f ′

H(z) − g′(z)) dz (8)

The first and second terms on the right-hand side of equation (8) give the ex-

pected advantage of a tax over a quota for theL- andH-type firms, respectively.

Under our assumption of quadratic production and damage functions, the differ-

ence in net surplus is given by

12

wt − wq1 =

σ2(f ′′2 − g′′2)

−2f ′′2(f ′′ − g′′)=

σ2(β2 − 1)

−2β2(f ′′ − g′′)(9)

This is equivalent to the well-known expression derived by Weitzman [11]

that states the comparative advantage of a tax over a quota interms of the relative

slopes of the marginal production and damage functions. Equation (9) implies that

if the parameterβ is greater than one, then a tax will yield a higher net surplusthan

a quota that binds with probability one. This is equivalent to the statement that the

marginal production function is steeper than the marginal damage function.

Lemma 3 For quadratic production and damage functions, a uniform tax is a

mean-preserving spread of input use relative to a uniform quota that binds with

probability one, so thatxL(t∗) < Q∗1 < xH(t∗).

Becausef ′′ < 0 andg′′ > 0, Jensen’s inequality and Lemma 3 imply that a

uniform tax has higher expected output levels and higher expected damages than a

uniform quota ofQ∗1. An intuitive explanation for the relative slope criterionthen

follows easily from Lemma 3: if the tax is a mean-preserving spread of input use

relative to the binding quota, and the marginal production function is steeper than

the marginal damage function, then the gains from increasedproduction under a

tax relative to a quota are greater than the increased damages.

If it is assumed thatγ2 > (1 + β)/ (1 + (1 + θ)β), so that an admissible

solution to first order condition (6),Q∗θ, exists, then the difference in net surplus

between a tax and a quota that binds with probability less than one is given by

wt−wqθ = θ

∫ xc

L

xL(t∗)(f ′

L(z) − g′(z)) dz+(1−θ)∫ xH(t∗)

Q∗

θ

(f ′

H(z) − g′(z)) dz (10)

The first term on the right hand side of equation (10) is the difference in net

13

surplus between a tax and unconstrained operation for low productivity firms un-

der the quotaQ∗θ. This term may be positive or negative, as the gains from in-

creasing aggregate output for a low productivity firm may be more or less than the

concurrent increase in aggregate pollution. The second term on the right hand side

of equation (10) is the difference in surplus between the taxand quota for the sub-

set of high productivity firms. Because the quotaQ∗θ is targeted specifically to the

high productivity firm, this term is always negative. As shown in the Appendix,

equation (10) may be rewritten as

wt − wqθ =

σ2((1 − θ)β2 − 1 + ∆)

−2β2(f ′′ − g′′)(11)

In (11) the parameter∆ ≡ 1−γ2

γ22

(1 + β)(

1−γ2

1−θ(1 + β) + 2γ2β

)

. Finally, for

the case where bothQ∗1 andQ∗

θ are admissible solutions for the optimal quota (i.e.

1+β

1+(1+θ)β≤ γ2 < 1), the difference in net surplus between these two quotas is

simply the difference between expressions (8) and (10):

wq1 − wq

θ =∫ xc

L

Q∗

1

Ei [f′

i(z) − g′(z)] dz + (1 − θ)∫ Q∗

θ

xc

L

(f ′

H(z) − g′(z)) dz (12)

The first term on the right hand side of equation (12) is alwaysnegative and

represents the loss in surplus from allowing the unconstrained operation of a low

productivity firm and higher input use (calculated over the range fromQ∗1 to xc

L)

for a high productivity firm. The second term on the right handside of equation

(12) is positive and represents the expected gains from targeting a quota just on a

high productivity firm. From expressions (9) and (11), we obtain:

wq1 − wq

θ = (wt − wqθ) − (wt − wq

1) =σ2(∆ − θβ2)

−2β2(f ′′ − g′′)(13)

Becauseσ2/(−2β2(f ′′ − g′′)) > 0, the ranking of each policy instrument can

14

be determined by looking at the term of the numerator in parentheses in (9), (11),

and (13). Table 1 summarizes the results for ranking of second-best taxes and quo-

tas by net surplus. The expressions for ranking the net surplus of taxes and quotas

shown in Table 1 are more complex than Weitzman’s original relative slopes crite-

rion, even though the functional forms used in this analysisare identical. With the

exception of the ranking of a tax and a quota when the quota binds with probabil-

ity one, comparisons do not lead to clear intuition. In situations where an optimal

quota may be slack, or there are two possible quotas, a simplerule to rank the

different policies is not apparent. However, qualitative relationships between the

surplus rankings can be determined by graphing the relevantanalytical expres-

sions. The panels of Figure 2 show the optimal choice of second-best policy for

different values ofθ in the parameter space{β, γ2}. We have chosen values ofθ

of 0.25, 0.5, and 0.75 to represent situations where the firm is more likely to be

anL-type, equally likely to beL- andH-type, and more likely to be anH-type,

respectively. Note that the horizontal axis of each panel uses a logarithmic scale;

in the middle of each panel(log β) = 0, corresponding toβ = 1, so that the

marginal product and marginal damage functions are equallysteep. The lower

range of the horizontal axis, -1, corresponds to a marginal damage function that

is ten times steeper than the marginal product function. Similarly, the upper range

of the horizontal axis, -1, corresponds to a marginal product function that is ten

times steeper than the marginal damage function. Note also that the vertical axis

on each panel is different, showing the range of feasible values ofγ2, given by

(0, 1/θ), for each value ofθ used.

Four qualitative results emerge from Figure 2. Although it is beyond the scope

of this paper to prove these results analytically, we can provide some intuition for

them. The results are:

Result (i)As the degree of heterogeneity increases, quotas that are slack with

15

positive probability become the preferred instrument.

Result (ii)As the probability that the firm is a high productivity type increases,

the parameter space over which slack quotas are stable, and preferred, increases.

Result (iii) For any feasible pair of values of{β, γ2}, quotas that are slack

with positive probability perform better as the probability that the firm is a low

productivity type increases.

Result (iv)Quotas that are slack with positive probability perform better when

the slopes of the marginal product and marginal damage functions are similar.

Result (i) is fairly straightforward to explain. As the difference betweenL-

type andH-type production functions increases, the distance between Q∗1 andQ∗

θ

also increases (see Figure 1). As a result, the gains from targeting regulation to

theH-type firm will also increase. Similarly, asγ2 increases, the losses in surplus

from using a second-best tax relative to a first-best solution also increase, so that

the advantage of targeting the quota increases. There are two further observations

to note from result (i). First, quotas that are slack with positive probability may be

optimal even whenγ2 < 1, when there are two admissible choices for the optimal

quota. Second, forγ2 > 1, quotas are preferred to taxes over a very wide range of

the parameter space of{β, γ2}. In particular, Figure 2 suggests that if an optimal

quota exists that binds with probability less than one, it may achieve a higher net

surplus than a tax even when the marginal product function ismuch steeper than

the marginal damage function. This result is significantly different to the original

relative slopes criterion, and somewhat counterintuitive. It is generally assumed

that if the marginal production function is relatively shallow, then a uniform tax is

a good approximation to the damage function and thus minimizes surplus losses

relative to first-best regulation (Roberts and Spence [9] develop this intuition in a

clear fashion). However, our analysis shows that even though this is the case, a

higher net surplus may be achieved by using a quota, if the expected gains from

16

targeting high productivity firms exceed the expected losses from unconstrained

operation of low-productivity firms.

Result (ii) follows directly from the geometry of the regulation problem (Fig-

ure 1). All else equal, asθ decreases, it is more likely that a quota that is slack

exists, and that targeting such a quota to high productivityfirms will be optimal.

Result (ii) is a consequence of Lemma 1.

Result (iii) may appear to contradict result (ii). It statesthat all else equal, quo-

tas that may be slack are preferred for lower values ofγ2 asθ increases. However,

there is an intuitive explanation for this result. A high productivity firm will be

penalized more, in terms of level of productive activity, when the probability that

the firm is low productivity increases. Hence, once again, the gains from targeting

high-productivity firms may outweigh the losses from allowing low-productivity

firms to operate unconstrained.

Finally, result (iv) may be explained in terms of the trade-off between two

effects. When the marginal damage function is much steeper than the marginal

production function, the losses from allowing the low-productivity firm to operate

unconstrained are large. Conversely, when the marginal production function is

much steeper than the marginal damage function, the potential gains from target-

ing the high-productivity firm are small. These two effects mean that slack quotas

are most effective when the marginal production and marginal damage functions

are of similar steepness. This result is also quite different to the original relative

slopes criterion, which states that when marginal production and marginal damage

functions are of the same steepness, taxes and quotas perform equally well.

17

4 Conclusion

The problem of choosing between taxes and quotas when there is asymmetric in-

formation between polluting firms and the regulator has beenextensively studied.

This literature implicitly assumes that the optimal quota will bind with probability

one. With the additional assumptions that production and damage functions are

quadratic and uncertainty additive, the choice between a second-best tax and a

second-best quota depends only on the relative steepness ofthe marginal produc-

tion and marginal damage functions.

In the real world, it is unrealistic to assume that any quota will bind with prob-

ability one. Many quotas are implemented as upper bounds: the regulated firm

may not exceed a given level of production or pollution, but there is no penalty

for overcompliance. Thus, quotas may be slack for some firms and in some states

of nature.

We extended the existing literature on the choice between taxes and quotas by

developing and analyzing the conditions under which it is optimal for a regulator

to set a quota that is slack with positive probability. When feasible, slack quotas

entail an additional trade-off for the regulator. The regulator can target the quota

towards higher productivity firms or states of nature, whichreduces the surplus

loss relative to the first-best solution. However, in doing so, the regulator also

allows firms which are less productive to operate unconstrained, increasing the

surplus loss relative to the first-best.

We provided an analytical example using quadratic production and damage

functions, and characterized the relationship between parameters that determine

the advantage of taxes or quotas. Our analysis showed that when slack quotas are

feasible, the heterogeneity of firms and the distribution ofuncertainty, as well as

the steepness of marginal production and damage functions,must be considered

when determining the ranking of taxes and quotas. In particular, we showed that

18

when there is significant heterogeneity, the only feasible quota will be slack, and

it will dominate a tax even when the marginal production function is much steeper

than the marginal benefit function – conditions under which previous studies pre-

dicted that taxes would strongly dominate quotas.

Our model has several limitations. We assume that optimal price regulation

does not cause the firm to shut down if it is a low-productivitytype. The possi-

bility that the low-productivity firm shuts down provides anadvantage for taxes

that is analogous to the potential advantage of slack quotas. Quotas that bind

with probability one are inefficient because they do not allow input use to vary

as a function of differences in production technology. Quotas that may be slack

can reduce this inefficiency by allowing input use to vary between low- and high-

productivity firms. Conversely, taxes are inefficient because they allow input use

to vary too much as a function of production technology. If the low-productivity

firm shuts down, this inefficiency can be reduced by targetingthe tax to the high-

productivity firm. Intuitively, shut down is much less likely to be a possibility

when the marginal production function is much steeper than the marginal dam-

age function. Thus, our result that quotas may be preferred to taxes even when

the marginal production function is much steeper than the marginal damage func-

tion is unlikely to be changed by extending this paper to allow firm shut down.

However, it is possible that when shut down is an admissible solution, taxes may

dominate quotas in some regions of the parameter space of{β, γ2} where the

marginal production function is shallower than the marginal damage function.

Finally, we use a binary discrete distribution to representthe regulator’s un-

certainty. In the real world, investment in alternative abatement technologies is

often lumpy, and a discrete distribution captures this lumpiness well. A discrete

distribution also allows analytical solutions for the ranking of alternative policies

and provides clear graphical intuition. Under more complicated discrete or con-

19

tinuous distributions, the major results would still hold,but characterization of the

optimal solution would be much more complex.

5 Appendix: Technical details

In this section we derive (1) conditions under which a slack quota is an admissible

solution to the quantity regulation problem, and (2) expressions for the net surplus

differences between taxes and quotas.

5.1 Existence of slack quotas

For a quota that does not bind with probability one to be an admissible solution

to first-order condition (6),Q∗θ ≥ xc

L. Equivalently, the distance fromQ∗θ to Q∗

1

should be greater than the distance fromxcL to Q∗

1. This corresponds to the condi-

tion

−θγ1

f ′′ − g′′≥ −

f ′L(Q∗

1)

f ′′(14)

Rearranging terms and noting thatγ1 = γ2f′H(Q∗

1) and thatγ2 = 1−f ′L(Q∗

1)/f′H(Q∗

1)

gives

γ2

(

1 +f ′′ − g′′

f ′′θ

)

≥f ′′ − g′′

f ′′θ(15)

Lemma 2 follows directly from this. Similarly, ifγ2 > 1, thenQ∗1 > xc

L,

implying that the low productivity firm would have a negativemarginal product.

20

This can not be an optimal solution, and so ifγ2 > 1, the optimal quota must be

slack with positive probability.

5.2 Difference in surplus between a tax and a quota

Following the functional forms in (7), and equation (8), thedifference in net sur-

plus between a tax and a quota that binds with probability oneis

wt − wq1 =

θ ((1 − θ)2γ21f

′′2 − (1 − θ)2γ21g

′′2) + (1 − θ) (θ2γ21f

′′2 − θ2γ21g

′′2)

−2f ′′2(f ′′ − g′′)

(16)

Recalling thatσ2 ≡ θ(1 − θ)γ21 , this expression simplifies to give (9). Sim-

ilarly, the difference in net surplus between a tax and a quota that binds with

probability less than one follows from (10), and is

wt−wqθ =

θ(

((1 − θ)γ1f′′ + f ′

L(Q∗1)(f

′′ − g′′))2 − (1 − θ)2γ21g

′′2)

− (1 − θ)θ2γ21g

′′2

−2f ′′2(f ′′ − g′′)

(17)

Expanding the quadratic term, rearranging, and noting thatf ′L(Q∗

1) = (1 −

γ2)/f′H(Q∗

1) and thatf ′H(Q∗

1) = γ1/γ2 allows this expression to be rewritten as:

wt − wqθ =

σ2(

(f ′′2 − g′′2) − θf ′′2 + 1−γ2

γ22

(f ′′ − g′′)(

1−γ2

1−θ(f ′′ − g′′) + 2γ2f

′′))

−2f ′′2(f ′′ − g′′)

(18)

Equation (11) follows directly from (18). Finally, the welfare difference be-

tween the two possible quotas,Q∗1 andQ∗

θ, assuming that both are feasible, is

given by:

21

wq1 − wq

θ =f ′2

L (Q∗1)(f

′′ − g′′)2 − (1 − θ) (θγ1f′′ − f ′

L(Q∗1)(f

′′ − g′′))2

−2f ′′2(f ′′ − g′′)(19)

Which is equivalent to subtracting (16) from (17). Consequently, (19) may be

rewritten as:

wq1 − wq

θ =σ2(

1−γ2

γ22

(f ′′ − g′′)(

1−γ2

1−θ(f ′′ − g′′) + 2γ2f

′′)

− θf ′′2)

−2f ′′2(f ′′ − g′′)(20)

Equation (13) follows directly from this.

22

References

[1] Hochman, E., and Zilberman, D. (1978), Examination of environmental poli-

cies using production and pollution microparameter distributions,Econo-

metrica, 46, 739-760.

[2] Hoel, M., and Karp, L. S. (2001), Taxes and quotas for a stock pollutant with

multiplicative uncertainty,Journal of Public Economics, 82, 91-114.

[3] Karp, L. S., and Costello, C. (2004), Dynamic taxes and quotas with learn-

ing, Journal of Economic Dynamics and Control, 28, 1661-1680.

[4] Kolstad, C. D. (1986), Empirical properties of economicincentives and

command-and-control regulations for air pollution control, Land Economics,

62, 250-268.

[5] Laffont, J. J. (1977), More on prices vs. quantities,Review of Economic

Studies, 44, 177-182.

[6] Malcomson, J. M. (1978), Prices vs. quantities: a critical note on the use of

approximations,Review of Economic Studies, 45, 203-210.

[7] Nichols, A. L. (1984),Targeting economic incentives for environmental pro-

tection(Cambridge: MIT Press).

[8] Pizer, W. A. (2002), Combining price and quantity controls to mitigate

global climate change,Journal of Public Economics, 85, 409-434.

[9] Roberts, M. J., and Spence, M. (1976), Effluent charges and licenses under

uncertainty,Journal of Public Economics, 5, 193-208.

[10] Stavins, R. N. (1996), Correlated uncertainty and policy instrument choice,

Journal of Environmental Economics and Management, 30, 218-232.

[11] Weitzman, M. L. (1974), Prices vs. quantities,Review of Economic Studies,

41, 477-491.

23

[12] Wu, J., and Babcock, B. A. (2001), Spatial heterogeneity and the choice

of instruments to control nonpoint pollution,Environmental and Resource

Economics, 18, 173-192.

24

Range ofγ2 Ranking of difference in net surplus

(0, 1] sign(wt − wq1) = sign(β2 − 1)

[

1+β

1+(1+θ)β, 1

θ

)

sign(wt − wqθ) = sign((1 − θ)β2 − 1 + ∆)

[

1+β

1+(1+θ)β, 1]

sign(wq1 − wq

θ) = sign(∆ − θβ2)

Table 1. Ranking second-best taxes and quotas.

25

-

6

x

$/x

g′(x)

f ′H(x)

f ′L(x)

Ei[f′i(x)]

xcL xc

HQ∗1 Q∗

θ

t∗

�����

@@@@@@@@@@@@@@@@@@

@@@@@@@@

@@

@@

@@

@@

@@

@@

@@

@@

@@

Figure 1. Choice of quotas when the optimal quota may be slack.

26

log β

γ 2θ = 0.25

Taxes preferredQuotas preferredBind with probability 1

Quotas preferredSlack with probability θ

A

−1 −0.5 0 0.5 1

0.5

1

1.5

2

2.5

3

3.5

4

log β

γ 2

θ = 0.5

Taxes preferredQuotas preferred

Bind with probability 1

Quotas preferred

Slack with probability θ

B

−1 −0.5 0 0.5 1

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Figures 2A and B. The choice between second-best taxes and quotas whenθ =0.25 (Panel A) andθ = 0.5 (Panel B).

27

log β

γ 2

θ = 0.75

Taxes preferredQuotas preferredBind with probability 1

Quotas preferredSlack with probability θ

C

−1 −0.5 0 0.5 1

0.2

0.4

0.6

0.8

1

1.2

Figure 2C. The choice between second-best taxes and quotas whenθ = 0.75.

28