Emissions standards and ambient environmental quality standards with stochastic environmental services

Emissions Standards and Ambient

Environmental Quality Standards with

Stochastic Environmental Services

Stephen F. Hamilton∗

Cal Poly San Luis Obispo

Till Requate†

University of Kiel

May 8, 2012

Abstract

Many important environmental policies involve some combination of

emission controls and ambient environmental quality standards, for in-

stance 2 emissions are capped under Title IV of the U.S. Clean Air

Act Amendments while ambient 2 concentrations are limited under

National Ambient Air Quality Standards (NAAQS). This paper examines

the relative performance of emissions standards and ambient standards

when the natural environment provides stochastic environmental services

for assimilating pollution. For receiving media characterized by greater

dispersion in the distribution of environmental services, the optimal emis-

sions policy becomes more stringent, whereas the optimal ambient policy

generally becomes more lax. In terms of economic performance, emissions

policies are superior to ambient policies for relatively non-toxic pollutants,

whereas ambient standards welfare dominate emissions standards for suf-

ficiently toxic pollutants. In the case of combined policies that jointly

implement emissions standards and ambient standards, we show that the

optimal level of each standard relaxes relative to its counterpart in a uni-

lateral policy, allowing for greater emissions levels and higher pollution

concentrations in the environmental medium.

JEL Classification: D62; Q38; Q50

Keywords: Environmental policy; ambient standards; emissions standards

∗Correspondence to: S. Hamilton, Department of Economics, Orfalea College of Business,California Polytechnic State University, San Luis Obispo, CA 93407. Voice: (805) 756-2555,

Fax: (805) 756-1473, email: [email protected]. We would like to thank Robert Innes,

David Sunding, Cyrus Ramezani and seminar participants at UC Berkeley, UC Davis, and

the University of Kiel for helpful comments.†Department of Economics, University of Kiel, Olshausenstrasse 40, 24118 Kiel, Germany,

email: [email protected].

1 Introduction

Most industrial countries adopt environmental policies that involve some com-

bination of emissions controls and ambient environmental quality standards.

Ambient standards, which set limits on allowable pollution concentrations in

receiving environments, have been implemented since the U.S. Rivers and Har-

bors Act of 1899 and they currently serve as the backbone of U.S. environmental

policy in the National Ambient Air Quality Standards (NAAQS) of the Clean

Air Act and in the Water Quality Standards (WQS) of the Clean Water Act.1

Yet, despite the long-standing use of ambient standards to control environmen-

tal externalities, the trend in environmental policy over the last several decades

has been towards the use of emissions controls, typically in the form of “cap

and trade” programs.

The recent gravitation of environmental policy towards emissions instru-

ments in many cases has led to overlapping policies that combine ambient stan-

dards with emissions standards. For example, the U.S. cap and trade programs

for sulfur dioxide (2) and for nitrogen oxides () currently operate in

conjunction with ambient standards on 2 and concentrations under

NAAQS.2 Given the long-standing prevalence of ambient standards in U.S.

environmental policy, it is surprising to note that the economic performance of

ambient environmental quality standards and emissions standards have not been

examined in settings where the natural environment has a stochastic ability to

1NAAQS set allowable pollution concentrations for six so-called “criteria pollutants” —

Carbon Monoxide (), Nitrogen oxides (), Ozone (3), Lead (), Sulfur Dioxide

(2), and Particulate matter (10 and 25)— and specify both an acceptable annual

mean concentration and a maximum concentration in a given interval of time, generally the

second highest 24-hour period each year. For more details on the requirements of the U.S.

Clean Air Act, see Lave and Omenn (1981) and Liroff (1986).2The NAAQS primary standard for 2 establishes a 1-hour standard of 75 ppb on ambient

2 concentrations. The cap and trade system for 2 currently limits annual emissions

among electric utilities to 8.95 million tons under Title IV of the Clean Air Act Amendments

of 1990.

1

assimilate pollution.

In this paper we consider the relative performance of optimal emissions stan-

dards and optimal ambient environmental quality standards when emissions and

environmental services jointly produce ambient environmental quality. Specifi-

cally, we model ambient pollution concentrations as the product of emissions and

a stochastic environmental input, for instance when pollution is released into

receiving water with variable streamflow. We examine the relative performance

of emissions standards and ambient standards under circumstances in which en-

vironmental policy cannot be fully customized in a system of state-contingent

pollution controls.

Our framework has several interpretations. One interpretation is that pol-

luting firms have superior information on the extent of environmental services

available to assimilate their pollution while environmental services are stochas-

tic from the perspective of the regulator. A second interpretation is that the

regulator has full information regarding environmental services, but the extent

of environmental services available at a given time varies temporally and spa-

tially in the receiving media at a rate that makes fully adoptive policies infea-

sible. Our policy comparison thus encompasses ambient standards of the form

underlying NAAQS and WQS that stipulate uniform limits on pollution concen-

trations across a diverse set of regional airsheds and water bodies in the U.S. as

well as emissions standards of the form underlying U.S. cap-and-trade programs

for 2 and of the Clean Air Act and pollution discharge permits of the

Clean Water Act.

We organize the paper by first examining the properties of the optimal am-

bient standard and optimal emissions standard for each policy levied in iso-

lation. When damage functions are convex in pollution concentrations, we

2

demonstrate that the optimal emissions standard becomes more stringent (i.e.,

mandates lower emissions) when the receiving media is characterized by greater

dispersion in environmental services; however, the optimal ambient standard

can either become more stringent (require lower pollution concentrations) or

more lax. We provide sufficient conditions for the ambient standard to relax

in response to greater dispersion in environmental services. This allows us to

characterize how the optimal uniform ambient standard is amended when the

level of environmental services is drawn from a distribution with greater disper-

sion of receiving conditions, for instance in air basins such as Los Angeles that

are prone to temperature inversion.

We also consider the relative economic performance of emissions standards

and ambient standards as environmental policy instruments. We find emissions

standards to be superior to ambient standards for relatively non-toxic pollu-

tants, while ambient policies welfare dominate emissions policies for sufficiently

harmful pollutants. The reason is that emissions policy equalizes marginal

abatement cost across polluters, which economizes on compliance costs. Nev-

ertheless, uniform emissions policies that align marginal abatement cost among

polluters can lead to “hot spots” with highly damaging pollution concentrations

in receiving media that provide little in the way of environmental services. In-

deed, it is precisely this shortcoming of emissions policy —the inability to predict

where traded emissions ultimately are released— that prompted the U.S. EPA to

limit 2 and emissions trading across state lines under the Cross-State

Air Pollution Rule (CSAPR) of 2011.

We examine the design of optimal “over-lapping” policies that incorporate

emissions standards in markets with pre-existing ambient standards. We show

that policies that jointly implement ambient standards and emissions standards

3

involve less stringent standards relative to a unilateral environmental policy

of either type. Thus, when an emissions standard is introduced in a market

with a pre-existing ambient standard, the optimal emissions standard allows

for greater pollution levels than would be desirable under a unilateral emissions

policy and the optimal ambient standard relaxes to allow for higher pollution

concentrations in the receiving medium than would be optimal absent an over-

lapping emissions standard.

Our analysis of emissions standards and ambient standards with stochastic

environmental services is related to previous work following Weitzman (1974)

that considers taxes and standards on emissions when firms have stochastic

benefits (or costs). Beavis and Walker (1983), Beavis and Dobbs (1987), and

Innes (2003), which are the papers closest to ours, consider circumstances in

which firms respond to environmental policy by making production or abate-

ment decisions that result in stochastic pollution levels. Our paper departs

from this literature by modeling ambient environmental conditions as the prod-

uct of firm’s decisions on emissions levels and nature’s draw of environmental

services. A distinguishing feature of our framework is that firms can observe

the level of environmental services being provided and adjust their pollution

loads in response to prevailing receiving conditions, for instance by exploiting

periods of high streamflow to release effluent into receiving water. The ability

of firms to adjust to spatial and temporal variation in environmental services

under ambient standards is a neglected element of pollution control policy that

is particularly germane in light of recent empirical evidence by Henderson (1996)

that U.S. firms vary their emissions levels with environmental conditions under

NAAQS.

Our analysis is also related to the literature on relative standards. Hochman

4

and Zilberman (1978), Harford and Ogura (1983) and Helfand (1991) examine

standards that limit emissions per unit of output in competitive markets, and

compare the performance of these policies to emissions policies. Ebert (1998)

extends the analysis to oligopoly markets; however, these models involve de-

terministic damages and their policy comparison does not encompass ambient

standards. To the best of our knowledge, our analysis is the first to model the

relationship between emissions and ambient pollution concentrations in settings

where pollution creates stochastic environmental damages.

The remainder of the paper is structured as follows. In the next section we

construct a simple model framed around a polluting firm that releases emissions

into a receiving medium with stochastic environmental services. We character-

ize the socially optimal resource allocation in Section 3 and consider the relative

performance of unilateral ambient standards and unilateral emissions standards

in Section 4. In Section 5, we examine combined policies that jointly imple-

ment ambient standards and emissions standards, and then conclude the paper

with a brief discussion of policy implications for pollution control in settings

that are characterized in principle by both spatial and temporal variation in

environmental services.

2 The Model

Consider a polluting entity, which may be a firm, a community waste-disposal

facility, or a collection of firms and facilities that deposits pollution into an en-

vironmental medium. The polluting entity (hereafter “firm”) faces opportunity

costs for reducing emissions, , below an unregulated emissions level, max, and

we represent these costs by a smoothly increasing and convex abatement cost

function −(), where 0 and 0 for max and, by definition,

5

(max) = 0.

We refer to − as the marginal abatement cost function. We assume that

lim→max() = 0, lim→max () 0 and

lim→0

()

−() 2 (1)

Condition (1) is a mild regularity condition that limits the convexity of the

abatement cost function as emissions approach zero. In a pollution market,

− represents the firm’s inverse demand function for pollution permits, and

the more stringent condition that lim→0()

−() 1 would be required for

marginal revenue to remain positive in a permit market as the firm’s emissions

approach zero. It is straightforward to verify that condition (1) holds for

quadratic abatement cost functions.

Emissions produce social damage in the environmental medium. The social

damage from emissions depends on ambient environmental quality, which is the

product of emissions and environmental services. In practice, ambient environ-

mental quality levels can be determined by the interaction between pollution

emissions and a potentially large number of stochastic environmental variables;

however, for analytic convenience we confine our attention to circumstances in

which it is possible to characterize the prevailing state of the environmental

medium with a single stochastic variable drawn from a known distribution of

environmental services, as in the case of receiving water that varies in volume

according to rainfall inputs. For expedience, we refer to the distribution of

environmental services in the environmental medium as a stochastic “receiving

distribution”. The receiving distribution may be interpreted as capturing ei-

ther temporal variation in pollution assimilative capacity at a single location or

spatial variation in pollution receptivity at a single point in time.

6

We represent the distribution of environmental services in the receiving

medium by the random parameter and relate the ambient pollution con-

centration to the level of emissions as = . One interpretation of is a

production function that relates inputs of emissions and environmental services

to ambient pollution concentrations. Pollution concentrations are higher (and

social damages are greater) for a given level of emissions when the provision

of environmental services is characterized by larger values of in the receiving

distribution. We assume that is distributed according to a density function

with compact support [ ], where ≥ 0.3 Accordingly we write the

cumulative distribution function as () =R ()

Environmental damage is given by the function ( ), which depends on

the ambient pollution concentration, , and a damage parameter 0 that

represents the toxicity of the pollutant. We assume social damages are increasing

and convex in the ambient pollution level, ( ) 0 and( ) ≥ 0, andthat an increase in the damage parameter increases both the damage and the

marginal damage from an increment in pollution concentrations, ( ) 0

and ( ) 0.

We consider the choice of abatement technology to be exogenous to firms.

This case accords with environmental polices that combine ambient standards

with separate policy controls in the form of technological requirements on pol-

luters, for instance the Best Available Control Technology requirements of the

U.S. Clean Water Act. In practice, abatement technology is often fixed at the

time short-run abatement decisions are made, as would be the case when pol-

luters make infrequent investments in abatement equipment and emit pollution

3Our analysis of continuous states of nature applies to the case of river pollution, where the

environmental damage from a unit of pollution with a given biochemical oxygen demand varies

continuously with streamflow in the receiving water, and to the case of urban air pollutants,

where ambient pollution concentrations are a decreasing function of the atmospheric lapse

rate.

7

into receiving media that continuously vary in environmental services.

3 The Social Optimum

The social objective is to minimize social cost, , which we take to be the sum

of abatement cost and environmental damage. Given a particular state of the

environmental medium, , social cost can be written in terms of the emissions

level,

( ) = () +( ) (2)

or equivalently in terms of the ambient pollution level,

( ) = () +( ) (3)

In the socially optimal resource allocation, firms observe the state of nature

and then align their abatement effort with prevailing environmental conditions

in a manner to minimize social cost in equation (2), or equivalently equation (3).

Minimizing social cost with respect to the emissions level gives the first-order

necessary condition

−() = ( ) (4)

Condition (4) sets marginal abatement cost equal to state-contingent marginal

damages. For the case of a single firm polluting into a receiving medium

that varies temporally in environmental services, the firm makes continuous

adjustments in emissions levels to align the benefits and costs of pollution on

the margin in the social optimum. For the case of homogeneous firms that

emit pollution in regions with a spatially distributed environmental services,

the social optimum involves selecting unequal standards across firms according

to the environmental services available in each region.

8

Let ∗( ) denote the solution to equation (4). Making use of the implicit

function theorem, it is straightforward to verify that ∗( ) is decreasing in

both and . The optimal level of emissions decreases for environmental states

associated with higher social damages and for pollutants with greater toxicity.

The optimal ambient pollution level, ∗( ) = ∗( ), also decreases

in ; however, the optimal ambient pollution concentration can either increase

or decrease in , depending on the elasticity of the marginal abatement cost

function. Specifically, letting = −()() 0 denote the elasticity

of the marginal abatement cost function, we have:

Proposition 1. ∗( ) is decreasing (increasing) in when ≤ ()1.

Under usual circumstances, this condition on the elasticity of marginal abate-

ment cost is only locally satisfied, so that Proposition 1 describes how the opti-

mal ambient pollution level responds to small changes of . The ambient envi-

ronment can either become “cleaner” or “dirtier” in the socially optimal resource

allocation when the degree of environmental services in the receiving medium

facilitates higher damages from a given input of pollution.

In response to an increment in environmental services that raises damages

( 0), the optimal ambient pollution concentration standard “relaxes” (∗

increases) whenever the marginal abatement cost function is elastic (1 ).

The reason is that pollution concentrations rise at a unit rate in emissions, while

marginal abatement cost rises at more than a unit rate in emissions when the

marginal abatement cost function is elastic. Under circumstances in which the

marginal abatement cost function is unit elastic ( = 1), the optimal ambient

standard is independent of the level of environmental services in the receiving

medium.

The socially optimal resource allocation can be decentralized by a state-

9

contingent emissions policy. To see this, suppose the regulator can select a tax

schedule on emissions, (), that varies according to realized states of nature .

Under this regulation, the compliance cost of the firm in the abatement stage

is given by

( ()) = () + ()

which is minimized when the firm selects an emissions level that solves

−() = () (5)

By inspection of (4) and (5), this policy results in the optimal state-contingent

ambient environmental quality level whenever the tax in each state of nature is

set equal to the realized marginal damage; that is, when () = ( ).4

But environmental policies that vary according to spatial and temporal vari-

ation in environmental services are rarely observed in practice, and one reason

for this is that continuously varying policies are difficult to implement. Instead,

most environmental policies levied in industrial countries involve some combi-

nation of two forms of policy that do not vary with the level of environmental

services: () uniform ambient standards, which impose limits on the maximum

allowable pollution concentration in the receiving medium, = ; and ()

uniform emissions standards, which sets a maximum allowable emissions level,

= .

4 Emissions Policies vs. Ambient Policies

In this section we consider emissions standards that levy a non-varying cap on

emissions and ambient standards that limit allowable pollution concentrations

in the receiving medium. These policies conform with emissions standards

4Alternatively, the regulator could issue state-contingent tradable pollution allowances in

the amount ∗( ), which would imply a competitive permit price of () = ( )

in state of nature .

10

in cap-and-trade systems and ambient standards that limit allowable pollution

concentrations in U.S. airsheds and waterways under NAAQS of the Clean Air

Act and WQS of the Clean Water Act.

The information structure of our policy framework is as follows. The regu-

lator knows the distribution of the states of nature at the time the emission

standard is determined, but either does not know the particular draw of envi-

ronmental services received by the firm or is not able to adjust the policy in

response to spatial or temporal variation in environmental services.

4.1 Emissions Standards

Consider first the case of a uniform emissions standard, = . In the

interesting case in which the emissions standard is set below the unregulated

emissions level of the firm, the abatement decision involves selecting an emissions

level that exactly meets the standard. The regulator’s problem is therefore to

minimize expected social costs

() +

Z

( ) ()

which has the first-order condition

− () =

Z

( ) () (6)

Condition (6) states that marginal abatement cost be set equal to the expected

marginal damage under a uniform emissions standard of units.

Notice the difference between the outcome of the emissions cap in equation

(6) and the first-best outcome for emissions in expression (4). Under a uniform

emissions standard, the level of the standard depends on expected marginal

damage in the environmental medium rather than on the actual marginal dam-

age under a particular realization of . Let ∗ denote the solution to (6) and

11

let (∗ ) denote expected social cost under the optimal uniform emissions

standard.

It is straightforward to show that regulation of a more toxic pollutant in-

volves a more stringent emission standard. Formally, making use of the implicit

function theorem on (6) gives ∗ 0

Next, consider how the dispersion of environmental services in the receiving

distribution impacts the optimal choice of the emission standard. We can derive

the following result:

Proposition 2. Let and be two distributions with supports [( ) ( )]

and [() ()], and let ∗ ( ) and ∗ () denote the optimal emis-

sions standards with respect to and , respectively. IfZ ( )

( )

( ) ()

Z ()

()

( )() (7)

for all , then ∗ ( ) ∗ ().

Proposition 2 states that if the expected marginal damage under distribution

is smaller than the expected marginal damage under distribution , then the

optimal emissions standard allows a greater level of emissions under than

under . The proof, formally given in the appendix, follows immediately from

the first-order condition for the optimal emissions standard (6).

It is possible to express this result in terms of second-order stochastic dom-

inance as follows:

Corollary 1 Let and be two distributions of where second-order

stochastically dominates .5 If 2 + ≥ 0, then ∗ ( )

∗ ().5A distribution is defined to second-order stochastically dominate a distribution if

(()) ≥ (()) for all concave functions , where (·) is the expectation operatorunder distribution = .

12

The regularity condition 2 + ≥ 0 is a mild restriction that

requires the damage function to be sufficiently convex. It is satisfied for power

functions of the form ( ) = with ≥ 1, but may be violated formore complex functional forms. Corollary 1 implies that for two distributions

and with the same mean but a greater dispersion under , the emission

standard under involves a lower emissions level than under .

4.2 Ambient Standards

Next consider an ambient standard that limits permissible pollution concen-

trations in the receiving medium. For a given level of an ambient standard,

= , the firm is allowed to emit any level of pollution that abides by

the standard in the sense that ≤ for any state of nature . Given an

ambient standard of , one of three outcomes must occur. For sufficiently

large values of , the standard never binds, and in this case the firm remains

free to pollute at the unregulated level, = max, in all states of nature. For

smaller values of that limit emissions for at least some draws of environmen-

tal services, it is possible that the ambient standard binds on emissions levels

in states of nature with high pollution damages, but does not bind on emissions

levels in states of nature with low pollution damages. In this case there exists a

state of nature = () ∈ [ ] such that for ≥ the firm must choose

an emissions level that complies with the ambient standard, = , while

for , the firm remains free to emit pollution at the unregulated level,

= max. Finally, if is sufficiently stringent that it binds for all ∈[ ] we have = . For () we obtain () = max which

implies that 0() = 1max 0.

For expedience, we consider the case in which a single firm emits into a

receiving medium with an uncertain level of environmental services; however,

13

our analysis generalizes directly to the case of multiple firms polluting a col-

lective receiving medium. To see this, consider the case in which a group of

firms = 1 emit pollution into a shared environmental medium. Firm

emits the quantity , where =P

=1 is the total pollution level of the

firms. The optimal environmental policy is analogous to the one we consider

below provided that the ambient standard ≤ is satisfied collectively by

all firms utilizing the medium. To ensure this is the case, a mechanism involv-

ing collective punishment is required, and the general form of this regulation is

well-known in the literature on non-point source pollution.6

As in the case of an emissions standard, the regulator knows the distribution

of the states of nature at the time the ambient environmental quality standard

is determined, but cannot subsequently adjust the standard in response to the

particular draw of environmental services. The regulator’s objective function

is to minimize expected social cost, which is given by

( ) =

Z

()

µ

¶ () +( )[1− (())]

+

Z ()

(max ) () (8)

The social cost function under an ambient standard has three terms. The

first two terms represent abatement cost and social damages in the binding

region ( ≥ ) in which the firm selects the emission level = to meet

the ambient standard of . The third term represents social damages from

pollution in the non-binding region where the polluter selects = max. The

interval over which these damages are expressed collapses to zero as ()→ .

6One way to collectively punish firms is to levy a Pigouvian tax on the group, as suggested

by Meran and Schwalbe (1988) and Segerson (1989), where each firm pays a tax proportional

to the ambient pollution concentration in excess of the standard; that is, = [ − ] if

and zero otherwise. Such a tax can be chosen sufficiently high to deter violation.

Another way to collectively punish firms is to design a random punishment mechanism that

imposes a lump-sum payment on a single firm, as in Xepapadeas (1991).

14

The regulator’s first-order condition with respect to is given by7

−Z

()

µ

¶1

() = ( )[1− (())]

8 (9)

The optimal uniform ambient standard that satisfies condition (9) equates

expected marginal abatement cost with marginal damage in the regulated tail

of the receiving distribution. For states of nature in which the ambient standard

binds, which occur with probability 1 − (()), marginal social damage is

constant at the level of the ambient standard, ( ), so that marginal social

damage in the regulated portion of the receiving distribution is ( )[1 − (())]. Marginal abatement cost varies with environmental services in the

upper tail of the receiving distribution as the firm expends greater resources

to comply with the ambient standard for larger draws of . Pollution levels

remain unregulated at = max for all other levels of environmental services.

Notice that the manner in which the ambient standard integrates over bind-

ing states of nature is a mirror image of the emissions standard. The opti-

mal ambient standard equates marginal social damage with expected marginal

abatement cost, whereas the optimal emissions standard equates expected mar-

ginal social damage with marginal abatement cost.

Let ∗ denote the solution to equation (9) and let (∗ ) denote ex-

pected social cost under the optimal uniform ambient standard. Given that

= max at the boundary of the regulated region, the solution ∗ defines the

truncation point in the environmental medium where the regulation goes into

effect, ∗ = (∗) =∗max

. The optimal outcome under a uniform ambient

standard that solves equation (9) differs from the first-best outcome in expres-

sion (4) in that a single regulated pollution concentration is chosen to equate

7For a derivation see the Appendix.8For the remainder of the paper we assume the second-order condition is satisfied.

15

marginal social damage with expected marginal abatement cost under a uniform

ambient pollution standard, whereas the socially optimal resource allocation se-

lects a different ambient pollution concentration in each state of nature to align

state-contingent benefits and costs on the margin.

The optimal ambient standard, ∗, in (9) depends critically on the toxicity

of the pollutant, . Making use of the implicit function theorem on (9) and

the second-order condition for the regulator’s problem, it is straightforward to

show that the optimal ambient policy satisfies ∗ 0. For more damag-

ing pollutants, the optimal uniform ambient standard requires lower pollution

concentrations to be maintained in the environmental medium. Moreover, as

rises, the optimal uniform ambient standard binds more frequently across states

of nature, which implies that decreases towards . Thus, as the toxicity level

of the pollutant rises, the ambient standard adjusts smoothly from a regime in

which the ambient standard binds only in some states of nature to a regime in

which the ambient standard is always binding.

Next consider how a change in the character of the receiving distribution

impacts the optimal ambient standard. For this purpose we consider two dis-

tributions and where involves greater dispersion in environmental services

than . We also limit our attention to cases where the states of nature for

which the ambient standard is non-binding under receive more weight than

under , as would be the case when the distribution of is a transformation of

that shifts weight to both tails. For this case we can show that the optimal

ambient standard allows higher pollution concentrations under distribution

than under distribution . Formally:

Proposition 3. Let and be two distributions with supports [( ) ( )]

and [() ()], and let ∗( ) and ∗() denote the optimal ambient

16

standards with respect to and , respectively. Suppose there exists an

ambient standard ≥ max{∗( ) ∗()} such that for all ≤ :Z ( )

()

[−

µ

¶1

] ()

Z ()

()

[−

µ

¶1

]() (10)

and

(()) ≤ (()) (11)

Then ∗( ) ∗().

Proposition 3 states that the optimal ambient standard on pollution con-

centrations relaxes (∗ increases) for distributions of associated with higher

expected marginal abatement cost. Notice that condition (10) parallels condi-

tion (7). Condition (11) requires that in the probability mass across all states

of nature for which the standard is non-binding under distribution is no larger

than the respective probability mass under . This condition is satisfied with

equality when the pollutant is sufficiently toxic ( is sufficiently “large”) that the

policy binds for all ∈ [ ]. In cases where (11) is satisfied with inequality,the condition ensures that the truncation point in the environmental medium,

, “lops off” at least as great a proportion of the frequency distribution under

than under .

Again we reformulate the our result in terms of second-order stochastic dom-

inance.

Corollary 2 Let and be two distributions of where second-order

stochastically dominates . If Ψ() ≡ −

³

´1is convex in on

the support of and and if (11) holds in the relevant range of ambient

standards then ∗( ) ∗().

In contrast to Proposition 2, where the convexity of the integrand ( )

followed from the convexity of the damage function, an additional condition is

17

needed here: Greater dispersion of environmental services in the receiving dis-

tribution relaxes the ambient standard whenever expected marginal abatement

cost, Ψ(), is convex in the domain of and . This condition is satisfied for

hyperbolic abatement costs of the form () = − , with 0 for all 0,

and for quadratic abatement cost functions of the form () = ( − )22

for sufficiently large (i.e. 3()).

It should be noted that Corollary 2 provides only a sufficient condition, not

a necessary one, for the ambient standard to relax when moving from to

. For quadratic abatement cost functions and constant density functions with

() = [−+2] for ∈ [− + ], the corollary holds even though

Ψ(·) is not globally convex. In this case, an increase in corresponds to a meanpreserving spread in , and it is straightforward to show that ∗ increases as

increases for all 0.

To understand the implication of Proposition 3 and Corollary 2, take for

example the case of ambient air quality regulations in Los Angeles and Bakers-

field, California. These regions are similar in terms of annual average ozone

concentrations, but Los Angeles is characterized by substantially greater vari-

ation in daily and hourly ozone concentrations (due, in part, to more frequent

temperature inversion). If the expected marginal abatement cost of complying

with a given ambient standard is higher in Los Angeles than in Bakersfield, then

the optimal ambient standard would involve higher ozone concentrations in Los

Angeles than in Bakersfield. Proposition 3 and its Corollary provide sufficient

conditions for this to be true.

4.3 Policy Comparison

It is useful to draw some general implications on the comparative performance

of unilateral policies of each type levied in isolation. Ambient standards, which

18

moderate damages at the upper tail of the distribution, have the advantage

of providing better matches between pollution loads and receiving conditions.

Emissions polices, which equalize abatement costs on the margin across all draws

from the receiving distribution, minimize total abatement costs across states of

nature, but fail to exploit matches between emissions levels and prevailing envi-

ronmental conditions. For this reason, when environmental damage functions

are highly convex and for relatively toxic pollutants ( “large”), ambient po-

lices tend to outperform emissions policies. We formally state this outcome as

follows.

Proposition 4:

i) For sufficiently large, (∗ ) (∗ ).

ii) With constant marginal damage, there exists an interval of damage parame-

ters [ ] and an environmental support [] such that (∗ )

(∗ ) for ∈ [ ]

Proof See the appendix.

This result is highly intuitive. For a given support [ ], the variance of

the damage distribution increases as rises, and this favors the use of ambient

standards to control the formation of severely-damaging pollution concentra-

tions. Under an ambient standard, abatement costs depend only on the partic-

ular outcome for , and are unrelated to the degree of toxicity of the pollutant.

Since abatement costs do not depend on while expected damage is increas-

ing in , it follows that there must be some level of toxicity for which policies

targeted to control damages outperform policies designed to reduce abatement

costs.

19

5 Combined Policies

Most industrial countries implement some form of ambient air and water quality

standards. As emissions policies continue to increase in popularity, an impor-

tant environmental policy issue is the jointly optimal mix of ambient standards

and emissions standards. Ambient standards serve to control damages in the

upper tail of the receiving distribution, but leave polluters unregulated for all

other draws of environmental services. Nevertheless, even when pollution con-

centrations are in attainment with the ambient standard, emissions produce

environmental damages, so that a combined policy provides scope for emissions

policy to address the external cost of pollution when environmental services are

sufficient to satisfy the ambient standard.

We again consider circumstances in which cannot be observed directly by

the regulator, but that the prevailing ambient pollution concentration, , can

be observed. The regulator sets a combination of an emission standard and

an ambient standard . Thus the policy is set such that the firms’ emissions

level must satisfy

≤ min{

} ≤ (12)

The intersection of the two standards defines a switching point at the state of

nature ≡ such that for states of nature ≤ the emissions standard is

binding, while for states of nature the ambient standard is binding. As

increases beneath , pollution concentrations rise in the environmental medium

at the regulated emissions level until the ambient standard eventually is met;

thereafter, for further increases in , the ambient standard is binding and the

firm must continually adjust emissions downward for to maintain ambient

environmental quality at the regulated threshold, .

20

Environmental policies that jointly implement both ambient standards and

emissions standards can be characterized either by the choice of and , or

by the selection of one of the two standards and the switching state of nature ,

where ≡ We make the following observation on the switching point:

Proposition 5 The optimal switching point in a combined policy, , satisfies

≥ .

The intuition for this result is straightforward. Given that states of nature

with relatively low pollution concentrations are now regulated by the emissions

standard in a combined policy, emissions levels are below max at the boundary

point of the receiving distribution; that is ≤ max. As a result, marginal

abatement cost is positive at the switching point between the two forms of

regulation, raising abatement costs in the regulated portion of the receiving

distribution. The ambient standard, accordingly, must relax; an outcome that is

necessary to reconcile expected marginal abatement cost with marginal damage

in the regulated tail of the receiving distribution.

We are now ready to characterize the optimal combined policy. Since ≡ or = the regulator’s problem can be reduced to the selection of

and to minimize

( ) = () ()+

Z

() ()+

Z

µ

¶ ()+( )[1− ()]

(13)

21

The regulator’s first-order necessary condition with respect to and are9

−() () =

Z

( ) () (14)

Z

−

µ

¶1

() = ( )[1− ()] (15)

Let (∗ ∗) denote the solution to conditions (14) and (15). The optimal

ambient standard is then given by ∗ = ∗∗ . By setting

∗ = ∗

∗ in (15)

we can interpret (15) as the first-order condition with respect to .

Before we interpret these conditions it is instructive to investigate whether

or not the switching point is bounded away from . We obtain the following

result:

Proposition 6 The optimal switching point in a combined policy satisfies

.

Propositions 5 and 6 together imply that a corner solution does not arise in

the combined policy. That is, the optimal combined policy never reduces to a

pure emissions standard or to a pure ambient standard. We summarize this

finding as:

Corollary 3 An optimal combined policy involves the joint use of both types

of standard. The emission standard binds for low levels of , while the

ambient standard binds for high levels of .

An implication of Corollary 3 is that combined policies perform strictly bet-

ter than unilateral polices of either type.

Next consider the optimality conditions (14) and (15). Replacing with

9For a derivation of these conditions see the appendix.

22

in (15), we obtain

Z

−

µ

¶1

() = ( )[1− ()] (16)

Together, conditions (14) and (16) completely characterize the optimal com-

bined policy, ∗ and ∗ . Notice that conditions (14) and (16) represent the

same essential trade-offs as conditions (9) and (6) that describe the optimal uni-

lateral standards, but with expectations defined over the binding region for the

policy in each respective support. Condition (14) equates marginal abatement

cost with expected marginal damage over states of nature bound by the emis-

sions standard while condition (16) equates expected marginal abatement cost

with marginal damage over states of nature bound by the ambient standard.

Applying the implicit function theorem to (14) and (16) we can verify that

0 (see the appendix). Since ≤ , it therefore follows

that ∗ ∗ and ∗ ≤ ∗. The reason is that under (14) the relevant support

of the distribution determining the expected marginal damage of emissions is

censored from above. The most damaging states of nature in the receiving dis-

tribution are no longer relevant for the assessment of expected marginal damage

in a combined policy, because outcomes in these states are already controlled by

the ambient standard, and this allows the optimal emissions standard to relax.

Under condition (16) the relevant support of the distribution determining the

expected marginal abatement cost similarly is censored from below compared

to a unilateral ambient policy. The ambient standard must (at least weakly)

relax, permitting higher pollution concentrations in the environmental medium

than in the case of a unilateral ambient standard.

We summarize these results as follows:

Proposition 7. Relative to the outcome under a unilateral emissions or ambi-

23

ent policy, a combined policy:

i) levies an ambient standard that allows higher pollution concentrations, ≥;

ii) levies an emissions standard that allows higher emissions, ; and

iii) results in lower social cost.

6 Policy Implications

On June 2, 2010 the U.S. EPA tightened the NAAQS primary standard for

2 from the original ambient standards set in 1971 and simultaneously im-

posed dramatic restrictions in interstate trading of 2 and emissions

that subsequently led to the Cross-State Air Pollution Rule (CSAPR) of 2011.

Our analysis suggests several policy implications in light of these recent rulings.

First, it is clear that efficiency gains exist in developing combined environmen-

tal policies that take into account the interaction between ambient standards

and emissions standards. The optimal environmental policy that combines the

two forms of regulation would respond to the introduction of emissions policy

by relaxing the existing ambient standard on pollution concentrations in the

receiving medium. This, in turn, would imply that the optimal ambient stan-

dard would relax in regions containing power plants subject to the 2 and

emissions reductions under the U.S. Clean Air Act Amendments relative

to regions not encompassed by the cap-and-trade program.

Second, the Cross-State Air Pollution Rule deeply erodes one of the greatest

benefits of market-based emissions policies by reducing the potential to trade

2 and across state lines. Curtailing interstate trading responds to a

shortcoming of cap and trade systems in being unable to predict the exact lo-

24

cation in which pollution is ultimately released, but it is clearly sub-optimal.

Rather than introducing frictions in pollution exchange markets through spatial

trading restrictions in CSAPR, our analysis suggests a more efficient combina-

tion of emissions standards and ambient standards would be to introduce emis-

sions allowances that provide firms with the option to exercise pollution rights

only when the prevailing ambient air quality levels are in attainment with the

ambient standard. Trading such pollution allowances across receiving media

that differ temporally and spatially in the level of environmental services would

retain the efficiency of market-based environmental policy, provided that options

markets are well-functioning.10

Our findings have implications for tailoring environmental policy to region-

ally distinct environmental media. There is often both a spatial and temporal

distribution of environmental services in receiving media for pollution. For ex-

ample, air quality in Los Angeles and Bakersfield, California are similar in terms

of annual average ozone concentrations, while Los Angeles is characterized by

substantially greater variation in daily and hourly ozone concentrations due to

periodic temperature inversion. An implication of the model is that the optimal

emissions policy would maintain a lower emissions level in Los Angeles than in

Bakersfield, while the optimal ambient standard would allow higher ozone con-

centrations in Los Angeles than in Bakersfield whenever increased dispersion

of environmental services in the receiving distribution raises expected marginal

abatement cost under the standard.

An important area for future research is to develop models that address stock

accumulation problems in environmental media. The accumulation of pollution

concentrations in environmental media can occur both over time, as in the case

10The idea of state-contingent property rights for emissions indexed to ambient standards is

related to the Pollution Offset System proposed by Krupnick, Oates, and van der Verg (1983).

25

of persistent greenhouse gases, and over space, as in the case of collective waste

disposal sites (e.g., landfills). Stock pollutants introduce a cascading effect of

pollution damages in the environmental medium, for instance pollution released

in a waste disposal network by an upstream firm increases pollution concen-

trations for downstream firms in the receiving medium. Stock accumulation

problems in correlated environmental media alters the policy implications out-

lined here by introducing a spatial (or temporal) policy gradient with higher

taxes on upstream (or early period) polluters to account for the external cost of

emissions on subsequent users of the resource. Another interesting direction for

future analysis is to formally examine combined policies of the form considered

here in a framework that accounts for compliance incentives among multiple

pollution sources along the lines developed by Montero (2008).

26

7 Appendix

This appendix contains the derivation of equations (9), (14) and (15), and the

proofs of all propositions and corollaries.

Derivation of (9): Differentiating (8) with respect to yields:

( )

=

µ

()

¶(())

0() +

Z

()

µ

¶1

()

+( )[1− (())]−( )(())0()

+(()max)(())0()

Now using () = max in the first term, we obtain ³

()

´=

(max) = 0 Using the same relation ()max = in the last term, the

last and the second-last term cancel out. Rearranging yields (9).

Derivation of equations (14) and (15): Differentiating social cost,

( ) = () () +

Z

() ()

+

Z

µ

¶ () +( )[1− ()]

with respect to yields:

( )

= ()() +()()− ()()

+

Z

µ

¶

()

+( )[1− ()]−()()

=

Z

µ

¶

() +( )[1− ()] ≤ 0

27

Notice that for = the last term is always zero, so that even at a corner

solution, = the first-order condition is satisfied with equality. Dividing

by we obtain (15).

Now differentiate social cost with respect to yields

( )

= () () +

Z

() ()

+

Z

µ

¶

() +( )[1− ()] = 0

After dividing by , the last two terms cancel out by (??). Rearranging

terms yields (14).

Proof of Proposition 1: Re-write first-order condition (4) as

−

µ

¶= ( ) (17)

Differentiating this equation with respect to and applying the implicit function

theorem, we obtain:

=−( ) +

2

³

´( ) +

1

³

´ =

³

´+ ·

³

´2( ) +

³

´ where we have used condition (17) in deriving the last equation. Recognizing

that the denominator is positive by the second-order condition completes the

proof.

Proof of Proposition 2: Since ∗ ( ) satisfies the first-order condition (6)

we obtain from that and (7):

0 = (∗ ( )) +

Z ( )

( )

(∗ ( ) ) ()

(∗ ( )) +

Z ()

()

(∗ ( ) )()

28

Thus ∗ ( ) is not optimal with respect to . Since expected marginal damage

under distribution and standard ∗ ( ) is larger than the marginal abatement

cost, social cost can be reduced by lowering ∗ . Therefore ∗ () ∗ ( ).

Proof of Corollary 1: We need to show that the integrand ( ) in

(7) is convex, implying−( ) is concave in . LetΨ() ≡ ( ).

Then it is easy to verify that Ψ00() = [2( ) + ( )],

which is positive by assumption.

Proof of Proposition 3: From the first-order condition for ∗( ), i.e. (9),

from (10), and from assumption (11) we obtain:

0 = (∗( ) )[1− ((∗( )))] +

Z ( )

(∗( ))

µ∗( )

¶1

()

(∗( ) )[1−((∗( )))] +

Z ()

(∗())

µ∗( )

¶1

()

Thus ∗( ) is not optimal with respect to . Since under the derivative

of the expected marginal social cost is negative, social cost can be reduced by

increasing ∗. Therefore ∗() ∗( ).

Proof of Corollary 2: Given that −

³

´1is convex on the supports

of and ,

³

´1is concave. By stochastic dominance of over we

obtainZ ( )

(( ))

[−

µ

¶1

] ()

Z ()

(())

[−

µ

¶1

]() (18)

Proof of Proposition 4: For the proof we need the following Lemma:

Lemma 1 Define Γ() = ³

´. Then for sufficiently large, Γ 0,

while for sufficiently low Γ 0.

Proof of Lemma 1: Differentiating Γ() with respect to yields

Γ = 0³

´·³−

2

´and Γ =

h00³

´+ 20

³

´i3. Now if is

29

sufficiently large, in particular if it is close to max the term 0³

´gets

arbitrarily close to zero. Since 00 0 close to max by assumption, Γ will

be positive for large. For sufficiently small, the term in brackets becomes

negative by virtue of condition (1).

Proof of Proposition 4: (i) First consider the case of a highly damaging

pollutant with a large coefficient. Differentiating the equations (9) and (6)

with respect to , it is straightforward to see that ∗ 0 and ∗ 0.

Thus a higher assessment of damage induces lower levels of emissions under both

the emissions standard and the ambient standard. Since 0 the ambient

standard is always binding in all states of the world if is sufficiently high.

Now let ∗=

∗() denote the optimal emissions standard for a given .

Next let e be chosen such that the expected damage is the same under both

the emissions standard ∗and the ambient standard e(∗()); that is,

( e(∗()) ) = (∗() ) (19)

Now observe that by Jensen’s inequality

( ) ≥ ( ) (20)

Let −1(· ) be the inverse function to (· ). Since −1(· ) is a positivemonotonic function, applying this to (19) and using (20) yields

e(∗()) = −1(( ) ) ≥ −1(( ) ) = (21)

Since (·) is decreasing in we obtain

à e(())

!≤

µ()

¶= (()) (22)

Now choose sufficiently large such that Γ( e(∗())) = ³ e(∗())

´is

concave in . Next, consider the expected benefit of the standard e and apply30

Jensen’s inequality to the function Γ( e) = ³ e

´ which is concave in fore = e(∗()). Doing so, and making use of (19) yields

à e

!+( e ) = Γ( e) +( e )

Γ( e) +( e ) =

à e

!+( e )

≤ ³∗´+ (

∗() )

where the last inequality follows from (19) and (22) and the definition of eSince e is not necessarily the optimal ambient standard with respect to , we

obtain:

{(∗() )} ≤ {( e(∗()) )} {(

∗()) )}

where ∗() is the optimal ambient standard for

Part (ii): A linear damage function is given by ( ) = . Now let

∗() denote the optimal ambient standard for , and let (∗()) denote

the emissions standard that leads to the same expected damage as ∗(), i.e.

(∗()) = {( · (∗()))} Moreover let ∗() be the optimal emissionsstandard for . By the linearity of the damage function we obtain ∗() =

{·(∗())} = ·(∗()). If is sufficiently small but bounded away fromzero, we have (∗()) max but close to max. Therefore also ∗() =

(∗()) max and ∗() max for but sufficiently close to

. Moreover for sufficiently close to the ambient standard ∗() is still

binding. Therefore, for each there exists an interval [ ] such that ∗()

is binding for all ∈ [ ]. Now from Lemma 1 we know that () is

convex in if is sufficiently close to max. Therefore Jensen’s inequality

31

yields

{ ()} () (23)

Making use of this result,

{(∗() )} = { (∗())}+ ∗ (∗()) + ∗()

= ((∗()) + {(∗())

≥ (∗()) + {∗()} = {(∗() )}

for some from some interval [ ] with 0, and a suitable interval [ ].

Proof of Proposition 5: The proof is indirect. Assume that .

Then we can rewrite (13) as

min

{() () +Z

() () +

Z

(max) ()

+

Z

µ

¶ () +()[1− ()]}

Note that in the interval [ ] the abatement cost is zero since by definition of

we have max for . Now let ∗() be the optimal emissions

standard referring to the interval [ ]. Then the social cost restricted to

interval [ ] is given by

(∗()) () +

Z

(∗()) () +

Z

(max) ()

(∗()) () +

Z

(∗()) ()

(∗()∗()) () +

Z

(∗()) ()

where ∗() is the optimal standard referring to the interval [ ]. The first

inequality holds because emission standard ∗() is extended from the interval

32

[ ] to the interval [ ]. Therefore the original policy with and ∗()

cannot have been optimal.

Proof of Proposition 6: Observe that

2( )

()2= −()

1

() +

Z

µ

¶

2 ()

+( )[1− ()]−( )()

>From this it follows that2()

()2|= = ()

[−()−( )]

0, where we have made use of (14) and the definition of ( ) as the

highest possible marginal damage. Thus = cannot yield a social cost

minimum. Since we know from Proposition 5 that ≥ , there must be a

solution ≤ with()

= 0

Proof of Proposition 7:

i) follows from Proposition 6, i.e. and the fact that satisfying (14)

is increasing in . The latter can be verified by applying the implicit function

theorem to (14), which yields

= − ()()

() +R ()

2 () 0

ii) follows from ≤ . In a similar fashion as in i) one can show

0

when applying the implicit function theorem on (16).

iii) This is obvious since the regulator cannot do worse by applying the

combined policy. She can even strictly improve welfare, because the emission

standard applies now for a lower range of and therefore can be strictly relaxed.

33

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36