Stochastic pollution, costly sanctions, and optimality of emission permit banking

Journal of Environmental Economics and Management 45 (2003) 546–568

Stochastic pollution, costly sanctions, and optimality ofemission permit banking

Robert Innes

Department Agricultural and Resource Economics, University of Arizona, Tucson, AZ 85721, USA

Received 27 April 2001; revised 10 January 2002

Abstract

This paper identifies a new economic motive for pollution regulations that allow polluting firms to bankand borrow emission permits over time. When aggregate pollution is stochastic, an intertemporal permittrading regime can provide firms with efficient incentives for pollution abatement without the need forcostly government enforcement actions that would otherwise be required.r 2003 Elsevier Science (USA). All rights reserved.

Keywords: Environmental regulation; Emission permit banking; Stochastic pollution

1. Introduction

In recent times, marketable emission permit systems, both in the US and internationally, havespecified a variety of different rules that allow for some ‘‘banking’’ or ‘‘borrowing’’ of emissionallowances over time. For example, the 1990 Clean Air Act Amendments allow banking of sulfurdioxide permits. California’s Low-Emission Vehicle Program allows firms to bank permits from 1year to the next at a 50 percent discount [10]. US fuel economy regulations allow automobileproducers to save or borrow permits for up to three years [11]. Under the Framework Conventionon Climate Change, permit banking is allowed on a one-for-one basis and borrowing is allowed ata discount.A potentially important motive for a firm to bank emission permits is that it has unexpectedly

low emissions in one particular period because (for example) the firm has unexpectedly lowproduction or unexpectedly superior equipment performance. Conversely, firms may want toborrow permits from the future when they have unexpectedly high emissions. At an aggregate(market-wide or emission basin) level, actual emissions can also be unexpectedly low or high. Such

E-mail address: [email protected] (R. Innes).

0095-0696/03/$ - see front matter r 2003 Elsevier Science (USA). All rights reserved.

doi:10.1016/S0095-0696(02)00021-9

uncertainty can lead to excess demand or excess supply for current-period emission permits that,absent intertemporal trading opportunities, can necessarily yield regulatory violations and anassociated enforcement action that is costly to both the government and regulated firms.The specific question explored in this paper is this: In the presence of such uncertainty, can—and

how can—intertemporal trading economize on the enforcement costs that the government mustbear in order to provide regulated firms with a desired deterrent to pollution? Such an inquiry willenable us to understand the potential economic gains from enacting intertemporal marketableemission permit approaches, as well as how one may want to design such systems in practice.A recent and growing academic literature describes how to design efficient intertemporal

trading rules in non-stochastic environments. For example, the pioneering work of Kling andRubin [10], Cronshaw and Kruse [4], Rubin [15], and Rubin and Kling [16] describes tradingbehavior, attendant efficiency effects, and optimal design of intertemporal permit markets whenpollution is certain. Contrary to prior work, Leiby and Rubin [11] consider both stock and flowpollutants. In this literature, however, optimality can be achieved with either a sequence of single-period emission permit markets in which the total number of permits issued each period isoptimally chosen, or a suitably designed intertemporal permit trading program. A notableexception to this rule is a recent paper by Yates and Cronshaw [20]. These authors argue thatintertemporal emission trading can sometimes (but not always) be strictly superior to period-by-period permit markets when the government has imperfect information about aggregateabatement costs. In essence, when aggregate marginal abatement costs are high, intertemporaltrading opportunities enable firms to mitigate these costs by borrowing emission permits fromfuture periods; conversely, when aggregate marginal abatement costs are low, intertemporaltrading enables firms to take advantage of attendant abatement economies by abating morepollution, and banking excess emission permits.1

In the present paper, in contrast, the government knows the distribution of abatement costfunctions in the population of polluting firms, although it does not know any particular firm’s

costs of pollution prevention. The government’s resulting cognizance of aggregate abatementcosts vitiates the potential motive for intertemporal trading identified by Yates and Cronshaw[20]. The analysis here focuses instead on the implications of stochastic emissions and the simplestof possible enforcement costs—administrative and other costs of penalizing firms—for optimalpollution regulation.2 In this setting, a new efficiency motive for intertemporal emission permittrading is identified.

1See also an interesting paper by Requate [14]. In a model with regulatory uncertainty about aggregate abatement

costs (but non-stochastic emissions), Requate [14] identifies potential benefits of emission banking similar to those of

Yates and Cronshaw [20], while also exploring implications of pollutant stock effects for the sign and size of these

benefits.2 In doing so, this paper builds upon a number of literatures, most importantly those on incomplete enforcement (e.g.,

[1,5,7,12,13,18]) and the regulation of stochastic pollution (e.g., see [2,3,8]). In contrast to the present inquiry, these

literatures do not study the merits of intertemporal emission trading and also, in general, focus on a fixed (exogenous)

regime of fines for regulatory violations. Because the objective of the present paper is to compare optimal single-period

emission regulations with intertemporal trading counterparts, the government must choose its regime of regulatory fines

in this paper. However, by positing the simplest possible administrative (enforcement) costs of sanctions—and not

modeling government costs of auditing (monitoring) firms—this paper ignores many of the optimal enforcement design

issues that occupy prior work.

R. Innes / Journal of Environmental Economics and Management 45 (2003) 546–568 547

2. The model

Consider N risk neutral polluting firms in a two-period model with three dates. In period one, firmsmake production and abatement choices ex ante (at time 0), which yield stochastic emissions ex post(at time 1). In period two (at time 2), production and abatement choices are again made, yielding non-stochastic emissions. (Results can and will be generalized to allow for repeated periods with stochasticemissions; see Section 6 below.) Formally, let us denote period t production for firm i by yit, andperiod t pollution abatement by firm i by ait. Firm i’s production and abatement choices give rise toperiod t profit (absent pollution taxes or fines) of pit(yit,ait), where pit is concave and decreasing in ait.A firm’s period 1 production and abatement yield the stochastic emissions/pollution,

ei1 ¼ ei1ðyi1; ai1; eiÞ;

where ei is a random variable with support ½%e; %e�;3 and ei1 is increasing in yi1 (output leads to more

pollution), decreasing in ai1 (abatement decreases pollution), and increasing in ei. Similarly, a firm’s(non-stochastic) period 2 emissions are ei2¼ei2(yi2,ai2) (where ei2 also rises with yi2 and falls with ai2).Firms are ‘‘small’’ in the sense that any individual firm views regulatory and emission permit marketoutcomes as exogenous to its own choices. However, time 1 emissions are imperfectly correlated

across firms, so that total time 1 pollution (by all firms), E1 ¼PN

i¼1 ei1 is stochastic.4

Pollution causes damage to the public. Damage is assumed (for simplicity) to depend on totalpollution by all firms

Dt ¼ DtðEtÞ; where Et ¼XN

i¼1eit ¼ total pollution at time t;

with aggregate pollution increasingly damaging at the margin, Dt0()40 and Dt

00X0.

Period t profits and damages are measured in period t dollars. The intertemporal discountfactor is denoted by do1.

3. The first-best

Before turning to the regulatory choice problem, consider the problem of a benevolent socialplanner who has perfect information and, thus, can freely maximize social welfare by choice of{yit,ait} for t¼1,2. The period 1 (time 0) choice problem maximizes the difference between totalprofits and total expected damages as follows:5

maxfyi1;ai1g

XN

i¼1pi1ðyi1; ai1Þ E D1

XN

i¼1ei1ðyi1; ai1; eiÞ

!( ); ð1Þ

3 In principle, ei may be vector valued. For example, we will later consider a general class of linear emission risk

functions in which ei is a two-dimensional vector, with corresponding two-dimensional support bounds.4With a large number of ‘‘small’’ firms (as assumed here), aggregate pollution at time 1 would be (approximately)

deterministic if firms had completely uncorrelated emissions. For reasons that will become clear in a moment, this case

is of little interest for this analysis.5For simplicity, we consider markets in which output and input prices are exogenous, so that we can ignore consumer

surplus and profits in other sectors. As is well known, this abstraction comes at no cost in generality when markets are

competitive.

R. Innes / Journal of Environmental Economics and Management 45 (2003) 546–568548

where E is an expectation operator over the random variables, (e1,...,eN). First-order optimalityconditions for this problem (assuming interior solutions) are

yi1 : @pi1ðyi1; ai1Þ=yi1 EfD01ðE1Þ½@ei1ðyi1; ai1; eiÞ=@yi1�g ¼ 0; ð2Þ

ai1 : @pi1ðyi1; ai1Þ=ai1 EfD01ðE1Þ½@ei1ðyi1; ai1; eiÞ=@ai1�g ¼ 0: ð3Þ

Let us denote the solution to this problem by fyi1; ai1g; which yield the stochastic optimal

aggregate pollution level, E1 ¼

PNi¼1 ei1ðyi1; ai1; eiÞ:

Similarly, the period 2 (time 2) choice problem is

maxfyi2;ai2g

XN

i¼1pi2ðyi2; ai2Þ D2

XN

i¼1ei2ðyi2; ai2Þ

!; ð4Þ

which yields the solutions, fyi2; ai2; e

i2 ¼ ei2ðy

i2; ai2Þg; the optimal level of total period 2 pollution,

E2 ¼

PNi¼1 ei2; and period 2 marginal pollution damage, d

2 ¼ D02ðE

2Þ:

4. The regulator’s problem

In truth, the regulator does not observe a firm’s type. Hence, achieving a first-best by directlyregulating firms’ production and abatement choices is not possible. In fact, let us suppose that theregulator only observes each firm’s actual ex post emissions (costlessly) and nothing else.Moreover, because our principal concern is to understand the relative merits and optimal designof emission permit banking programs, this analysis focuses on two alternative regulatory regimes:(1) a sequence of single-period emission permit programs, with taxes/fines for emission violationsand no allowance for banking or borrowing of emission permits between periods; and (2) anemission permit program that allows banking and/or borrowing of emission permits betweenperiods. The policy instruments available to the government under these two regimes are identicalwith one exception—the permit banking policy specifies an intertemporal trading ratio at whichfirms can exchange emission permits between periods.In the first (single-period permit market) regime, the government issues each firm a certain

number of emission permits at the beginning of each period. Each permit entitles its owner to emitone unit of pollution tax-free. Permits can be freely traded, but are useless in the subsequentperiod. After time 1 trading occurs, a firm can be fined (taxed) if its emissions exceed its permitholdings (as in [9]). The government specifies a period 1 regime of taxes/fines a priori (before time0), which takes the form f(v), where v denotes the size of a firm’s violation (i.e., the amount bywhich its emissions, ei1, exceed its permit holdings), f(v)¼0 for vp0 (firms cannot be fined if theydo not ‘‘violate’’), and f() is assumed to be monotone non-decreasing (larger violations cannot befined less) with a bounded first derivative. Note that this regime combines non-linear pollutiontaxes with a single-period permit market. For example, consider the case of constant marginaldamages, D1¼d1P1; in this case, the Pigovian tax is a constant per-unit emission tax equal to d1.This Pigovian tax is simply a special case of the regulatory regime considered here, with zeropermits issued and a linear fine, f(v)=d1v.The Pigovian tax regime is instructive because, for the case of constant marginal damages, this

tax will achieve the first-best when there are no costs of penalization (i.e., no costs of levying fines or


taxes). Let us suppose, however, that there are costs of penalization. Specifically, whenever a positivefine (or tax) is levied on a firm, administrative and other related costs of collecting the fine areincurred. These administrative costs are likely to increase with the level of the fine (due to increasingefforts of firms to avoid larger sanctions); indeed, it is likely that, as they become larger, fines will beincreasingly costly to levy at the margin (see [5]). For purposes of this analysis, however, all thatmatters is that there are positive administrative costs of levying positive fines/taxes. In the presence ofsuch costs, a Pigovian tax regime, by requiring that penalization costs be borne, will not achieve afirst-best. In fact, we will see that, in general, no period-by-period regulation can achieve a first-best.The same cannot be said for an emission banking program, however. In this regime, the

government chooses: (1) the total number of period 1 (time 0) emission permits to issue, %Q1; (2)

the number of period 2 (time 2) permits to issue, %Q2; and (3) the rate of exchange between period 1and period 2 permits, r (where r is the number of period 2 emission permits that a firm obtains

when it banks one period 1 permit). %Q1 is chosen at the start of period 1 (before firms’ period 1

decisions unfold), and %Q2 need not be chosen until the start of period 2. For expository purposes,it is first assumed that r can be chosen ex post (at time 1) and, hence, can depend upon ex postaggregate emissions, r(E1); the analysis then turns to a restriction that r be a constant that ischosen ex ante—as is almost certainly the case in practice.The adjustment in the permit endowment each period is a function of the two-period model

structure posited here at the outset; when turning to a multi-period generalization (in Section 7below), we will see that permit endowments can be adjusted infrequently without upsetting thelogic of the arguments that follow. In practice, of course, permit endowments (and associatedemission standards) are revised periodically over time, but not at every point in time. However, asis standard in the emission-banking literature, it is important that the intertemporal tradingration, r, be tailored to each period.The logic of this intertemporal permit trading environment can be related quite simply to

counterparts in non-stochastic environments (e.g., [10]). In the latter settings, the government usesa time-specific intertemporal trading ratio to determine the desired relationship between emissionpermit prices at different times (as needed to reflect relative marginal costs of pollution); theoverall level of permit prices (and aggregate pollution) is then determined by a one-time (lumpsum) permit endowment. The difference here is that stochastic emissions give rise to stochasticpermit bankings and borrowings. Hence, a one-time permit endowment could give rise topersistent permit surpluses or shortages; to avoid such outcomes, periodic (but infrequent)adjustments in permit endowments are needed to pin down the level of permit prices. As in non-stochastic analogs, a time-specific intertemporal trading ratio drives the relationship betweenemission permit prices across time.For our two period structure, we now turn to the pitfalls of period-by-period regulation when

sanctions are costly to impose.

5. Period-by-period regulation

With the stochastic pollution and costly sanctions modeled here, a period-by-period regulationcannot, in general, achieve a first-best. In essence, the reason is this. With stochastic emissions,


emission permits can sometimes be in surplus as of time 1—when there are more permits thanneeded to cover actual time 1 pollution (because, ex post, aggregate emissions can turn out to belower than the fixed level of emission permits issued at time 0). In such cases, permits have zerovalue in time 1; no fines are paid; and firms thus pay no ‘‘price’’ for their emissions. Of course, thiscannot always be true if firms are to engage in positive (and costly) pollution abatement ex ante(at time 0). Rather, there must sometimes be positive emission permit prices at time 1 that providesome positive benefit of ex ante pollution reductions—namely, saved emission permit costs oravoided fines for pollution violations. Hence, emission permits must sometimes be in shortage attime 1. In such cases, moreover, permits have value only to the extent that they enable firms toavoid fines. In sum, fines must be paid in order to deter emissions directly and/or to provide valueto permits—value that also serves to price (and thereby deter) pollution. Because fines are costlyto impose—and efficiency calls for positive emission deterrence—a first-best cannot be achieved.In other words, first-best emission deterrence requires that fines be imposed, but because sanctionsare costly to assess, fines cannot be imposed in a first-best.To develop this result, note first that if there is no aggregate emission uncertainty, then a first-

best can be achieved in period-by-period regulation, with the following permit (and fine)regulation: issue E

1 pollution permits ex ante (which is now non-stochastic) and fine all violations

at the marginal rate d1 ¼ D0

1ðE1Þ: This policy yields a non-stochastic time 1 permit price of

p1 ¼ d1 ; and permit market clearing that yields no violations by any firm—and hence, no fines or

penalization costs. To understand this conclusion, recall that firms’ actual emissions are assumedto be costlessly (and accurately) observed, as are their emission permit holdings. Firms aretherefore subject to fines on their true violations, the excess of their emissions over their permitallowances. Now, if the non-stochastic aggregate emissions are less than E

1 (with some firms

abating more pollution than required by a first-best), then the foregoing policy would yieldsurplus permits—and thus, a zero permit price and no emission violations. Anticipating a zeropermit price, however, firms would pollute more than first-best, implying (by contradiction) thataggregate emissions must be at least E

1 ; hence, pollution permits will not be in surplus, and the

value of a permit, in avoiding a marginal fine, will be p1 ¼ d1 : Marginal pollution, by requiring

either the purchase of an additional permit or the payment of the marginal fine, thus bears theprice d

1 ; which in turn elicits first-best emissions—E1 in aggregate. In sum, permits exactly cover

elicited pollution, and no fines are paid.6

6 If emission permits could be issued ex post (at time 1), this logic implies that a first-best could be achieved, in

general, with period-by-period regulation. The government could simply issue emission permits sufficient to cover

actual ex post aggregate pollution, P1. This policy is ruled out here—with permit endowments and fines that are

invariant to P1—for a number of reasons. First, while such a dependence is unrealistic on its face, it also denies, by fiat,

the regulatory issues implicitly raised by stochastic pollution, essentially turning a stochastic problem into a non-

stochastic one by focusing on ex post regulation. For this reason, a premise of non-stochastic permit endowments and

fines—like mine—is standard in the literature on stochastic pollution regulation (e.g., [19,17]), including its one

application to emission banking [20]. Secondly, under the ex post permit endowment policy described above, firms

would have a powerful incentive to collude in the production of maximal emissions levels that keep their collective (and

individual) abatement costs at a minimum and, yet, are met with minimal (zero) collective costs of emission permit

purchases and regulatory fines. Third, the premise of non-stochastic (period 1) permit endowments and fines is imposed

in both regulatory regimes under investigation in this paper and thus—given its consistency with observed practice—

provides a natural baseline for comparison of the two alternatives. Although explicit ex post (time 1) regulation is ruled

out, a referee points out that, by allowing ex post (time 2) adjustment in period 2 permit endowments and linking the


Aggregate emission uncertainty is thus necessary for the failure of period-by-period regulation.For simplicity at this juncture, it is useful to model such uncertainty with the following (rathermild) regularity restriction.

Assumption 1. The vector e (=(e1,..., eN)), has strictly positive support on ON � ½%e; %e� �

½e; %e�yx½%e; %e�:

Assumption 1 implies, for example, that first-best aggregate emissions have positive support on

[E1min, E1

max], where Emin1 �

PNi¼1 ei1ðy

i1; ai1;%eÞo

PNi¼1 ei1ðy

i1; ai1; %eÞ � Emax

1 :In view of this aggregate uncertainty, a policy regime with no tradable emission permits either

levies positive taxes/fines with positive probability or, if not, does not elicit first-best pollutionabatement (because pollution bears no penalty); either way, a first-best is not achieved.

Lemma 1. A period-by-period regime of pure taxes/fines (with no tradable emission permits) cannot

achieve a first-best.7

Turning to a period-by-period policy with tradable permits, we begin with each firm’s time 1permit market trade and attendant violation choice problem, given a time 1 emission permit pricep1, a fine function f(v), emissions ei1 and permit endowments of %qi1:

8 The firm will choose itsviolation size to minimize the sum of fines to be paid and costs of requisite permit trades, asfollows:

Cðff ðvÞg; p1; ei1; %qi1Þ ¼ minvff ðvÞ þ p1½ei1 %qi1 v�g s:t: vX0; ð5Þ

where ei1 %qi1 is the violation size before any time 1 permit trades, and hence, ½ei1 %qi1 v� is thenumber of time 1 permit purchases (or sales, if negative) needed to achieve a violation of size v.The solution to (5) will be denoted v*(). We assume (for obvious reasons) that firms cannot havenegative violations (vX0). Implicit in problem (5) is also a premise that firms can ‘‘borrow’’violations from one another (violating by more than their own emissions). In practice, firms maybe able to form an emission ‘‘bubble’’ with another firm (by merging, for example); a ‘‘bubble’’permits the firms to aggregate their emissions and permits, and split the difference whencalculating each firm’s violation. However, if firms cannot ‘‘bubble’’ their emissions, their choiceproblem in (5) will be subject to the additional constraint, vpei1. Both cases are considered inwhat follows.

(footnote continued)

two periods by intertemporal trade, this paper’s proposed intertemporal permit trading regime implicitly achieves a type

of ex post regulation. Indeed, the purpose of this paper is to show how, by exploiting natural temporal separations in

regulation policy (between distinct periods of pollution production and not within periods), the benefits of ex post

regulation can be obtained.7See the appendix for a proof of Lemma 1. Note that Lemma 1 does not imply that, with tradable permits, a period-

by-period regulation can achieve a first-best. Indeed, the ensuing analysis shows that, in general, the converse is true.8Because firms are endowed with emission permits at time 0, permits could be traded at time 0, as well as time 1.

However, because there is no incentive for time 0 trade in this model, we abstract from this possibility without loss.


Stepping back to time 0, firms choose their production/abatement plans, taking the distributionof the equilibrium time 1 permit price (p1) as parametric

maxyi1;ai1

pi1ðyi1; ai1Þ þ E Cðff ðvÞg; p1; ei1ðyi1; ai1; eiÞ; %qi1Þf g: ð6Þ

In view of these choice problems, several observations imply useful necessary conditions for aperiod-by-period regulation to yield a first-best:

Observation 1. If the time 1 permit price is zero (p1=0), the firm’s time 1 cost of pollution (C in(5)) is also zero. Therefore, in view of problem (6), a necessary condition for positive pollutiondeterrence—as required for a first-best (by Eqs. (2) and (3))—is that p1 be positive with positiveprobability.

Observation 2. The time 1 permit price will be positive only if there is not an excess supply ofemission permits,

p140 ) D � E1 %Q1 ¼X

i

vðÞX0: ð7Þ

Observation 3. Whenever there are positive violations (v*()40), as there must be when there isexcess demand for permits (D40), marginal fines must be at least as high as the permit price(f0(v*())Xp1) in order for the violation to be optimal (by problem (5), with or without an upperbound constraint on violations, vpei1).

Observations 1 and 2 imply that, under a putative first-best policy, there must be some set ofemission realizations such that (i) there is positive excess demand for permits (in order for positivepermit prices to prevail with positive probability); (ii) the permit price is positive; and (iii) no finesare actually imposed. Together, these conditions require that there be positive violations thatproduce zero fines, but yield positive marginal fines (by Observation 3):

(%v40 : f ðvÞ ¼ 0 8vA½0; %v�; f 0ð%vÞXp140:

With a monotone fine function, these properties also imply positive fines for all violations above

%v : f ðvÞ40 8v4%v:Now, in view of this rule, consider cases in which firms can ‘‘bubble’’ their emissions.

Collectively, the N firms can then achieve ‘‘free violations’’ equal to N %v (the ‘‘free violation’’ perfirm, %v; times the number of firms). Hence, in order for the emission permit price to be positive,total excess emissions—the excess of actual pollution over the supply of pollution permits—mustbe at least as high as N %v :

p140 ) free violations ¼N %vpD ¼ E1 %Q1 ¼ actual aggregate violation:

Moreover, if the maximum amount of free violation ðN %vÞ is less than the aggregate violation (D),positive fines must be imposed (because some firms must then violate by more than %vÞ: To avoidthe imposition of positive fines with positive probability, the maximum free violation musttherefore be no lower than the maximum possible aggregate violation:

N %v ¼ Dmax � Emax1 %Q1 ¼ maximum aggregate violation: ð8Þ


However, condition (8) is easily seen to imply that a first-best is not achieved. In particular, Eq. (8)implies that the maximum free violation ðN %vÞ strictly higher than the actual aggregate violationwhenever the aggregate violation is less than maximal. In such instances—which (by Assumption1) occur almost surely—the pollution permit price is zero. Clearly, with a zero permit priceprevailing with virtual certainty, the incentive for pollution abatement is less than first-best. Insum, when emission ‘‘bubbles’’ are possible, necessary conditions for a period-by-period policy tosupport a first-best imply that a first-best is not supported; the implied contradiction establishesthat no period-by-period policy can achieve a first-best.Similar logic applies when firms are ex ante identical, but cannot ‘‘bubble’’ their emissions. However,

when firms are not identical—and cannot ‘‘bubble’’—this logic is not sufficient. In this case, positivepermit prices can prevail with positive probability even when the maximum possible ‘‘free violation’’equals the maximum possible aggregate violation (the analog to Eq. (8)).9 The contradiction implied inthe foregoing argument is thus voided. Nevertheless, with no emission ‘‘bubbles’’ allowed, it can beshown that a period-by-period regulation generally provides heterogeneous firms with heterogeneousemission prices that are inconsistent with the first-best calculus of Eqs. (1)–(3).10

These observations can be summarized as follows (see the appendix for proof.)

Proposition 1. If firms can ‘‘bubble’’ their emissions or are ex ante identical,11 then there is noperiod-by-period regulation that can achieve a first-best. If firms are not identical ex ante, and cannot

‘‘bubble’’ their emissions, there are primitive economic conditions under which no period-by-periodregulation can achieve a first-best.12

Proposition 1 establishes the initial claim that, under a variety of circumstances, a first-bestcannot prevail under period-by-period regulation.

6. A first-best emission banking program

We will now see that the government can achieve a first-best with an appropriate choice of theemission banking policy parameters, combined with fines that are sufficiently large that they deter

9With ‘‘bubbles’’ not allowed, total available ‘‘free violations’’ becomeP

i minfei1ð:Þ%vg; and actual violations, as

before, are D ¼P

i ei1ð:Þ %Q1: Hence, if a firm i’s emissions fall (because its ei is lower)—and its maximum free

violation equals its actual emissions (because ei1ð:Þo%vÞ—total free and actual violations fall by the same amount. As a

result, even when the maximum level of free violation,P

i minfei1ð:; %eÞ%vg; equals the maximum possible aggregate

violation, Dmax, there can be a non-degenerate set of e realizations that produce positive emission permit prices—

because total free violations remain equal to actual violations and marginal violations are costly ðf 0ð%vÞ40Þ:10Without emission ‘‘bubbles,’’ a firm faces a positive ‘‘emission price’’ only when its actual emission (ei1()) are higher

than the government’s stipulated ‘‘free violation’’ ð%vÞ—so that it actually needs to use pollution permits in order to

avoid positive sanctions. Firms that have higher optimal emissions levels (with higher probability) thus pay a positive

‘‘emission price’’ with higher probability than do firms with lower optimal emission levels. As a result, the symmetric

emission prices required for a first-best are not achieved.11Firms that are ‘‘identical ex ante’’ have identical profit and emission functions, and have identically (but not

independently) distributed emission risks (ei).12Among the primitive economic conditions under which period-by-period regulation necessarily fails is that

emissions functions take the linear risk form of Assumption 2. (See proof of Proposition 1.)


any violations and thereby avoid any actual penalization. The essential advantage of an emissionbanking program is this: Whether the number of emission permits issued initially (at time 0) ishigher or lower than actual ex post emissions (at time 1), firms pay a price for a higher emissionlevel. When a firm has higher ex post emissions than it has permits to cover them—so that it has apermit shortfall—it must pay for each unit of excess emission by either buying another permit or‘‘borrowing’’ future (next period) permits; ‘‘borrowing’’ will, in turn, require the firm to ‘‘repay’’from either its next-period permit endowment or additional next-period permit purchases.Similarly, when a firm has lower ex post emissions than its permits allow—so that it has apermit surplus—it can either sell the excess permits or ‘‘bank’’ them for next-period sale or use.Either way, a firm pays a positive ‘‘price’’ for additional emissions: higher permit costs inthe case of a permit shortfall, and reduced permit sales (or bankings) in the case of a permitsurplus. Importantly, this positive ‘‘emission price’’—the putative object of environmentalregulation—prevails even when firms never actually pay any regulatory fines and the governmentthus bears no administrative costs of prospective sanctions. With period-by-period regulation, incontrast, regulatory fines must actually be paid in order to provide firms with a deterrent topollution.Turning to the formal analysis, note that in period 2 (time 2), a first-best requires total

emissions of E2*, and no fines (so that no penalization costs are borne). With net banked permits of

B1 (from time 1, in period 2 emission units), the government can achieve this outcome by issuingE2*–B1 permits in time 2, and threatening fines for positive violations that are sufficiently large to

ensure that no violations occur (and hence, no fines are actually levied).13 We assume, of course,that these stochastic time 2 permit issues are allocated to firms in a lump-sum way; that is, thefirms’ allocations do not depend upon individual firm behavior or performance. Followingstandard arguments, equilibrium in the resulting period 2 permit market will yield a permit price(in period 2 dollars),

p2 ¼ d2 ð¼ D0

2ðE2ÞÞ: ð9Þ

Turning now to period 1, let us first consider how the price of a time 1 emission permit, p1, mustbe related to the time 2 permit price, p2. Because these permits are exchangeable at the rate r, we

13Although this threat could also be made in a period-by-period policy regime, doing so cannot lead to a first-best (as

shown above). Absent an ability to bank or borrow permits—the premise underpinning period-by-period regulation—

an absence of any regulatory violations requires that permits be available for the maximum possible amount of

pollution. However, permits then have value—and emissions are thus priced—only when the maximum amount of

pollution occurs, which is with probability of measure zero; hence, with bounded marginal fines, firms will face a zero

price for pollution ex ante and will abate less than is optimal; indeed, they will not abate at all. In sum, if a period-by-

period policy is to yield no regulatory violations, it must issue permits for the maximum possible amount of pollution

when there is no abatement; firms, in turn, will choose not to abate. In a period-by-period regulation (with aggregate

emission uncertainty), the provision of some positive incentive for ex ante pollution abatement requires that there

sometimes be aggregate permit shortfalls (when aggregate pollution turns out, ex post, to be higher than permit

endowments); moreover, these shortfalls must give rise to positive regulatory fines—due to positive regulatory

violations—in order for the permit shortfalls to be priced and thus provide the requisite emission deterrence. In

contrast, the intertemporal regulatory policy at issue here implicitly enables firms to have permit shortfalls (or

surpluses) without any regulatory violation. Firms simply borrow future permits to cover current period shortfalls, or

bank current surplus permits; either way, the shortfalls (or surpluses) are ‘‘priced’’—and thus, the underpinning

pollution is also priced—even though no regulatory violation occurs and no regulatory fine is actually levied.


have the following necessary relationship in order to avoid arbitrage opportunities:

p1ðE1Þ ¼ drðE1Þp2; ð10Þwhere r(E1) is the intertemporal permit trading ratio specified by the government at time 1. Forexample, if p1odr()p2, then it would be profitable to buy time 1 permits, bank them, and sell themat time 2, a strategy that would yield a discounted (time 1 dollar) profit per-permit equal to thepositive difference between (i) the discounted sale proceeds, dr()p2, and (ii) the cost of the time 1permit purchased, p1. Conversely, if p1odr()p2, then it would be profitable to borrow r time 2permits and sell the resulting time 1 emission entitlement, yielding a profit equal top1dr()p240.14 Because firms will exploit arbitrage opportunities such as these, Eq. (10)describes the time 1 permit market equilibrium price, which will be determined by the time 2permit price (as given in (9)) and the chosen intertemporal exchange rate r().Now, in view of the permit price relationship in (10), firms are indifferent between banking

permits vs. selling them, or borrowing permits vs. buying them. Firm i’s period 1 (time 0) choiceproblem is to maximize the sum of profits from operations (pi1) and expected profits fromrequisite time 1 permit transactions, as follows:

maxyi1;ai1

pi1ðyi1; ai1Þ þ EfdrðE1Þp2½ %qi1 ei1ðyi1; ai1; eiÞ�g: ð11Þ

In time 1, firms (i) sell off or bank any period 1 permits that they have, over and above what theyrequire to cover their emissions, and (ii) buy or borrow permits that they need to cover their actualemissions, over and above their permit endowments (qi1). Given the pricing relationship in (10), thesetime 1 strategies yield the expected (time 0) profits and losses described by the second term in (11).Solving problem (11) yields the following optimality conditions:

yi1 : @pi1ðyi1; ai1Þ=yi1 EfdrðE1Þp2½@ei1ðyi1; ai1; eiÞ=@yi1�g ¼ 0; ð12Þ

ai1 : @pi1ðyi1; ai1Þ=ai1 EfdrðE1Þp2½@ei1ðyi1; ai1; eiÞ=@ai1�g ¼ 0: ð13ÞIn order for the emission banking program to elicit first-best outcomes, conditions (12) and (13)

must be satisfied at {yi1* ,ai1

* } (as implicitly defined by conditions (2) and (3)). Comparing the twosets of first-order conditions (and recalling Eq. (9)), it is easily seen that the two coincide when

drðE1Þp2 ¼ D01ðE1Þ 3 rðE1Þ ¼ D0

1ðE1Þ=dd2 : ð14Þ

In sum, we have:

Proposition 2. A first-best is achieved by an emission banking program that (i) specifies the

intertemporal trading ratio in (14), (ii) adjusts the quantity of time 2 permits, in view of time 1bankings and borrowings, so that the net supply at time 2 equals the optimal period 2 level of

aggregate emissions, E2*, and (iii) promises sufficiently stiff penalties to emissions violations

(emissions above levels covered by permits) that violations do not occur.15

14We implicitly assume that the intertemporal permit exchange rate r is the same for bankings as it is for borrowings.As we will see, a first-best can be achieved with this implied restriction, thus implying no economic motive for

differential (borrowing vs. banking) trading ratios.15 In order to ensure that firms always pursue ‘‘legal’’ permit trading strategies to cover their emissions—rather than

‘‘illegal’’ violation strategies—marginal fines for violations must be set to exceed the maximum permit price that

government policies support, D10(E1

max).


7. First-best emission banking: generalizations and discussion

7.1. (A) Can the optimal intertemporal trading ratio be constant?

Clearly, the most unreasonable property of Proposition 2’s solution is that r varies withrealized ex post aggregate pollution, E1. Can the intertemporal trading ratio, r, be a constant—and thus set a priori (before actual period 1 emissions, E1, are known)—without upsetting thefirst-best? Under plausible conditions, the answer is ‘‘yes.’’ For example, when there are constantmarginal damages in period 1 (D1

0¼d1), then the optimal trading ratio in (14) is also a constant,r¼d1/dd2

*.With increasing marginal damages (D1

0040) and a constant intertemporal trading ratio (r),firms cannot simply be confronted with the ex post marginal harm from their pollution (becausethe implicit pollution tax, drp2, is non-stochastic, whereas the true harm D1

0(E1) is stochastic).Rather, one must inquire as to whether there is a fixed r that confronts firms with the societal costfrom their emission production on average, and in doing so, provides desired marginal incentivesfor choices of output and abatement. Formally, note that the societal cost of the emissionproduction from marginal output and marginal abatement, respectively, are

EfD01ðÞ½@ei1ðÞ=@yi1�g ¼ EfD0

1gEf@ei1ðÞ=@yi1g þ CovfD01; @ei1ðÞ=@yi1g

¼ EfD01gEf@ei1ðÞ=@yi1g½1þ riyðÞ�; ð15aÞ

EfD01ðÞ½@ei1ðÞ=@ai1�g ¼ EfD0

1ðÞgEf@ei1ðÞ=@ai1g þ CovfD01; @ei1ðÞ=@ai1g

¼ EfD01gEf@ei1ðÞ=@ai1g½1þ riaðÞ�; ð15bÞ

where ‘‘Cov’’ denotes the covariance operator, and

rix � normalized covariance between D01 and ½@ei1=@xi1�

¼CovfD01; @ei1=@xi1g=½EfD0

1gEf@ei1=@xi1g� for xAfy; ag:

Eq. (15) reveals how average marginal emission production from output and abatement—E{qei1()/qyi1} and E{qei1()/qai1}—need to be ‘‘taxed’’ in order to confront firms with the averageharm that they cause by their choices at the margin. Specifically, the required tax inflates (ordeflates) the average marginal damage from pollution—E{D1

0}—by the normalized covariancebetween marginal damage (D1

0) and marginal emission production (qei1()/qyi1 and qei1()/qai1). Forexample, if the covariance is positive, then average marginal emission production causes a higherlevel of average harm than measured by expected marginal damage (E{D1

0})—and thus requires ahigher tax.Now under an intertemporal permit trading regime with constant r, note from Eqs. (12) and

(13) that the private cost of emission production from marginal output and marginal abatementare, respectively,

drp2Ef@ei1ðÞ=@yi1g and drp2Ef@ei1ðÞ=@ai1g: ð150Þ

That is, the implicit tax on average marginal emission production is drp2—a constant that appliesto both output and abatement, and to all firms.


Comparing Eqs. (15) and (150), it is clear that the constant tax implied by the intertemporalpermit regulation (from Eq. (150)) can, in all cases, replicate the tax required for first-bestproduction/abatement incentives (from Eq. (15)) when and only when the normalized covariancebetween marginal emission production and marginal damage are the same, at the optimum, foroutput, abatement, and all firms:

riy ¼ ria ¼ r for all i; at the optimum: ð16ÞMoreover, when Eq. (16) holds, first-best outcomes are supported by setting the followingconstant trading rate (combined with optimal time 2 regulation, p2¼d2

*):

d ¼ EfD01ðE

1Þg½1þ r�=rd2 : ð17Þ

Although one can find emission-production technologies that do not satisfy condition (16), ageneral class of linear risk emission functions do satisfy this requirement:

Assumption 2. Let

eit ¼ eitðyit; aitÞyit þ Zit;

where yit ¼%yt þ git; Zit ¼

%Z

tþ jit; and (git, jit) are zero mean ‘‘idiosynchratic risk’’ random

variables that are independent of all other firms’ counterparts (gjt and jjt, jai)) and the ‘‘systematic

risk’’ random variables,%yt and

%Z

t:

As required by Eq. (16), Assumption 2 implies (at the optimum)16

riy ¼ ria ¼ r ¼ CovfD01ðE

1Þ; yg=½EfygEfD01ðE

1Þg�40: ð18ÞIn essence, linear risk emission functions yield a ‘‘price’’ for amerage emissions, E{D1

0(E1*)ei1()},

that is proportional to average emissions, with a proportional constant that does not depend uponany individual firm choices or technologies,

EfD01ðÞei1ðÞg ¼ EfD0

1ðÞ%ygei1ðÞ:

As a result, a constant ‘‘tax’’ on average emissions—as implied by an intertemporal trading regimewith constant r—can confront firms with the true societal price for their emissions.Note finally that, under Assumption 2, the optimal intertemporal trading ratio in Eq. (17) is

higher than the expected value of its random counterpart in Eq. (14) (given that r40 by Eq. (18)).Because individual firm emissions are positively correlated with aggregate emissions (E1), they arealso positively correlated with marginal pollution damage (D0(E1)). Hence, when a firm’semissions are high, the per-unit societal cost of the emissions (D0(E1)) tends to be high. Andconversely, when a firm’s emissions are low, their per-unit social cost tends to be low. Now if afirm is confronted with just the average per-unit cost of emissions, E{D0()}, then emissions willtend to be over-priced when emissions are low (because (D0(E1)) then tends to be lower than itsaverage, E{D0(E1)}) and under-priced when emissions are high. However, the latter effectdominates; because under-pricing occurs when emissions are higher—and hence, there are moreemissions that are under-priced than are over-priced—emissions will be under-priced on average.

16To obtain (18), I use the following relationships: (i) E{qei1()/qyi1}=qei1* ()/qyi1E{y} (likewise for ai1),

(ii) Cov{D10, qei1()/qyi1}=Cov{D1

0,yi}[qei1* ()/qyi1] (likewise for ai1), and (iii) Cov{D1

0,yi}=Cov{D10,y}.


To counter this effect, the emission permit price needs to be raised above the average marginaldamage, E{D0}. This can be done by raising the cost of using an emission permit in the currentperiod, rather than a future period—that is, by raising the rate at which current period permitscan be converted into future period permits (r).

7.2. (B) Multi-period stochastic emissions

Clearly, positing non-stochastic period 2 emissions is a simplifying device, not a realistic one.Let us now extend the analysis to allow for multiple periods with stochastic emissions in eachperiod; in doing so, let us focus on the case of linear emission risk (Assumption 2) and constant(but period-specific) intertemporal permit trading ratios, rt.In the general multi-period setting, each period (like period one in the foregoing analysis) has

an ex ante (time 0) and ex post (time 1) structure. Denoting the period t (time 1) emission permitprice by pt, intertemporal trading opportunities imply the following no-arbitrage condition (theanalog to Eq. (10)):17

pt ¼ dtrtEfptþ1g; ð19Þ

where dt is the period t (to period (t+1)) discount factor. The no-arbitrage condition (19) alsoimplies that individual firms are always indifferent between (i) buying (selling) current periodemission permits in order to make up (liquidate) permit deficits (surpluses) and (ii) borrowing(banking) permits from (to) a future period. Hence, firm i’s period t choice problem can be statedas follows:

maxyit;ait

pitðyit; aitÞ þ Efpt½ %qit eitðyit; ait; eitÞ�g; ð20Þ

where %qit is firm i’s period t permit endowment (the sum of banked (less borrowed) permitallowances and new period t permit endowments), and Eq. (20) maximizes firm i’s period t profitsfrom operations (pit()) and expected permit transactions. Comparing problem (20)’s optimalityconditions to their first-best counterparts (cf., Eqs. (2) and (3)) yields the following necessary andsufficient conditions for first-best choices, ðyit; a

itÞ; to be made:

Efpt½@eitðÞ=@yit�g ¼EfD0tðE

t Þ½@eitðÞ=@yit�g;Efpt½@eitðÞ=@ait�g ¼EfD0

tðEt Þ½@eitðÞ=@ait�g; ð21Þ

17Eq. (19), as expressed, does not follow from the absence of riskless arbitrage per se because pt+1 can, in principle,

be uncertain as of period t. However, when allowing for ex ante (time 0) permit trades in all periods, the absence of

riskless arbitrage implies an equivalent relationship in which the non-stochastic time 0 (period t+1) permit price takes

the place of its expected (time 1) counterpart (see the expanded version of this paper, available from the author upon

request). Moreover, under the optimal policies described below, permit prices (pt+1) are non-stochastic and, thus,

Eq. (19) does indeed follow from the absence of riskless arbitrage. Most importantly, Eq. (19) follows from the presence

of risk neutral permit traders, which is a premise of this inquiry. The latter observation becomes important when

information about damage functions, emission risks, and abatement costs is acquired over time; as discussed below (see

‘‘Damage Uncertainty’’), such information will yield optimal updating of policies and behavior, with attendant

revisions—and associated a priori uncertainty—in optimal permit prices.


with eit() evaluated at ðyit; aitÞ: Moreover, by Assumption 2 and the logic of Section 7(A) above,

condition (21) will be satisfied when

pt ¼ Kt � EfD0tðE

t Þg þ ½CovfD0tðE

t Þ;%ytg=Ef

%ytg�: ð22Þ

Comparing Eqs. (22) and (19), we see that the intertemporal trading ratios needed to support thepermit pricing of Eq. (22) are

rt ¼ Kt=½dtKtþ1�: ð23Þ

Beyond implementing the trading ratio of Eq. (23), the government must regulate emissionpermit supplies in order to support the optimal permit price levels of Eq. (22). There are a widevariety of ways in which this might be done. For example, the government could issue newemission permits in each period t equal to the average optimal emission level less total net permitbankings from the prior period (Bt),

18

%Qt ¼ period t permit issues ¼ EfEt g Bt:

Alternately, the government could issue permits for the minimum possible optimal emissionlevel,

%Qt ¼ Emint �

Xi

ei1ðyi1;a

i1;

%eÞ;

and periodically auction additional permits in order to offset accumulated permit borrowings. Or,if auctions are costly and the government only wants to alter its per-period permit allowancesinfrequently, it could do so to adjust for accumulated permit surpluses or shortfalls (relative toactual emissions)—surpluses (shortfalls) that translate into permit bankings (or borrowings). For

example, if the per-period permit allowance ð %Qt ¼ %QÞ is adjusted once every T periods and B1

represents accumulated net permit bankings at the start of a T-period interval, then %Q could be setso as to equate current-period-equivalent permit issues with current-period-equivalent optimalemissions, less accumulated net bankings:

%Q ¼XT

t¼1EðE

t Þ=Yt1k¼0

rt

( ) B1

" #=XT

t¼1

Yt1k¼0

rt

( )124

35; with r0 � 1:

To see how these policies can support first-best outcomes, consider first a finite horizon oflength NT, with stochastic emissions in each and every period (including the final period NT).Note that, with any finite horizon, the final period regulation necessarily takes a single periodform, which cannot be first-best (by Proposition 1); at this juncture, there is simply no subsequentperiod with which one could construct an intertemporal regulatory policy. Hence, in terms of thebenefits and optimal structure of intertemporal (vs. period-by-period) regulation, it is only

18 In order to ensure that permit issues are always non-negative under this policy, the following requirement must be

satsified: Emint1XEfE

t1g EfEt g: (This condition ensures that, with maximum possible bankings from period (t1)—

Bmaxt ¼ EfE

t1g Emint1 —period t permit issues are non-negative.) This requirement is clearly a very modest one,

stipulating that optimal reductions in average pollution levels (from period (t1) to period t) not be exceedingly drastic.


meaningful to talk about the intervening time (before the final period). To do so, let us supposethe following: In the final (NT) period’s ex ante time 0, let the government commit to buy and/orsell emission permits at the fixed price KNT.

19 By inductive logic, the policies described above thensupport the desired permit prices (qt¼Kt), and attendant first-best outcomes, throughout theintervening interval from period 1 through period (NT1).20 Now, if one goes to an infinitehorizon, by letting N go to infinity, then the ‘‘final period’’ inefficiency is postponed indefinitely,giving us a first-best throughout time.

7.3. (C) Strategic behavior

Under the policies described above, higher levels of emission permit borrowing—or lower levelsof permit banking—can give rise to the issuance of more permits in future periods. A potentialconcern is that, anticipating these effects, firms may over-pollute in order to reap attendantbenefits in raised future permit endowments. However, so long as the issuance of permits toindividual firms is not tied to their history of pollution or permit banking/borrowing behavior,this concern is vitiated by this paper’s premise that actors in pollution markets are ‘‘small.’’Because new permit issues depend only upon aggregate permit bankings and borrowings—whichare not affected by the actions of any ‘‘small’’ individual firm—no such firm can ascribe a newpermit issue benefit to a strategy of over-pollution or excess permit borrowing.Strategic behavior does become an issue when actors in pollution permit markets are ‘‘large.’’

Although a full treatment of imperfectly competitive permit market participants is beyond thescope of this paper, note that when large actors over-pollute, they can anticipate increases inaggregate permit borrowings and/or reductions in aggregate permit bankings, which will in turnraise aggregate future permit endowments—and their own share of these endowments, even if thisshare is otherwise decoupled from individual decisions.21 To counter the resulting over-pollutionincentive, the government may want to set different intertemporal trading rates (r) for permitbankings and permit borrowings. By setting a higher trading rate for permit borrowings (i.e.,allowing borrowings only at a discount), a firm’s incentive to raise its emissions, in order to raiseaggregate borrowings, will be reduced. For example, suppose that there is one large actor and acompetitive fringe in the permit market (the case considered by Hahn [6]). To preserve optimal

19Because this final period commitment is ex ante, it does not violate this paper’s premise that explicit ex post

regulation be ruled out (note 6). Moreover, because this commitment implicitly clears the permit market of outstanding

borrowings and bankings, it leaves the final period regulatory policy open.20By this logic, the intertemporal trading ratios of Eq. (23) will support an optimum, even without any of the

proposed adjustments in permit endowments to reflect accumulated bankings/borrowings. However, this is not the case

if there are perceived bounds on the government’s final period commitment; for example, it may be perceived that the

government will expose itself to no more than a fixed (upper bound) level of permit purchase costs, and will be willing to

sell only a fixed (upper bound) number of additional pollution permits. Such bounds will motivate clearing the permit

markets periodically with permit endowment adjustments, as proposed, in order to ensure that the bounds are never

breached.21Note that this incentive is quite different than that identified by Hahn [6]. Hahn argues that a large actor, endowed

with more emissions permits than it will optimally use, will over-pollute in order to capture monopoly rents in the

permit market. Here, in contrast, dynamic arbitrage relationships prevent a large actor from manipulating market

permit prices in this way, so long as the government can pin down an optimal permit price in some future period (with

standby authority to buy/sell permits, for instance).


incentives for pollution abatement by the competitive fringe, the elevated ‘‘borrowing r’’ will needto be combined with a lowered ‘‘banking r’’ so as to preserve the average trading rate.22 For thelarge firm, the elevated ‘‘borrowing r,’’ even when combined with the lowered ‘‘banking r,’’ candeter excess pollution because such excess leads to a higher probability of aggregate borrowing(precisely its intent) and, therefore, a greater weight on the ‘‘borrowing r.’’ Imperfectlycompetitive players in an intertemporal emission permit market may thus provide an economicmotive for the sorts of distinctions that governments have made in practice between the bankingand borrowing of pollution permits—namely, more liberal treatment of the former and morerestrictive treatment of the latter.

7.4. (D) Damage uncertainty

So far, it has been assumed that the pollution damage function, Dt(Et), is known and certain. Inpractice, there may be some uncertainty about these damages. Due to logic that is familiar fromthe taxes and standards literature, this source of uncertainty is unlikely to have a qualitativeimpact on this paper’s conclusions.To be more specific, the logic of this paper’s model suggests that, if damage uncertainty is

resolved within a given period, it will be resolved ex post (at each period’s ‘‘time 1’’). Moreover,damage uncertainty is unrelated to emission risk per se; hence, the two sources of risk canplausibly be assumed to be independent. All in the foregoing analysis now carries throughdirectly, with the regulator optimizing over an expected emission-contingent damage function,rather than the non-stochastic emission-contingent damage function posited above.23

Over time, however, there may be updating about the distribution of emission-contingentdamage functions, as well as about profit functions (abatement costs) and emission risks. Suchupdating will clearly complicate the characterization of optimal intertemporal emission tradingpolicies. For example, optimal intertemporal trading ratios ðrt Þ will reflect current (period t)

expectations about future optimal permit prices,24 and will need to be updated over time to reflectnew information. However, optimal intertemporal trading policies will nonetheless continue toenjoy the efficiency benefits identified in this paper, providing efficient pollution abatementincentives without the need for costly sanctions (and/or taxes).

8. Conclusion

When pollution is stochastic, and it is costly to impose regulatory sanctions on firms that exceedtheir emission allowances, environmental regulators can increase economic efficiency by allowingintertemporal emission permit trading. This conclusion contrasts with an extant literature on

22Because the occurrence of aggregate banking and borrowing is correlated with a firm’s emissions, the achievement

of optimal emission incentives is somewhat more complicated than suggested here.23Formally, let ex post period t damages be Dt

*(Et,ut), where ut is a random variable. Further, define the expected

marginal damages, Dt0(Et)=Eut{Dt

*(Et,ut)/qEt}, where Eut is the expectation over the damage-risk parameter ut. With ut

independent of Et and {eit}, the foregoing analysis now carries through as stated.24Note that these expectations will be quite complex, reflecting effects of possible information updating on optimal

pollution and pollution damages.


emission banking that, while characterizing important properties of optimal emission tradingregimes, also permits full economic efficiency to be achieved by a sequence of single-periodemission permit markets that allow no intertemporal permit trading.25

With stochastic pollution—and no intertemporal permit trading—single-period emissionpermit markets necessarily lead to some regulatory violations, when firms’ emissions are higherthan their permit allowances; in such instances, regulatory fines must be imposed in order forfirms to have any incentive to curb their pollution. However, intertemporal permit markets canavoid regulatory fines—and costs of their imposition—without diluting incentives for pollutionabatement. When firms have fewer emission permits than needed to cover their current periodpollution, they can ‘‘borrow’’ future permits, which imposes a cost on their excess pollution.Similarly, when firms have more permits than needed to cover their current pollution, they can‘‘bank’’ their permits to a future period, thus imparting an economic benefit to their successfulpollution abatement. In either case, costly sanctions need not be imposed in order to provide thedesired pollution deterrent.Of course, pollution permit ‘‘borrowings’’ and ‘‘bankings’’ affect future period permit supplies

in ways that, absent periodic regulatory intervention, will raise or depress future period permitprices and thereby alter future pollution abatement incentives in undesirable ways. One way tocounter these effects is for the government to periodically adjust its new emission permitallowances in view of accumulated bankings and borrowings—raising allowances to accom-modate unanticipated borrowings and, conversely, lowering allowances to reflect unanticipatedbankings. Notably, this prescription appears to be heeded in recent public debate on theregulation of greenhouse gases; prominent analysts have advocated new emission permit issues ifand when permit prices threaten to exceed a prespecified threshold.26 Implicitly, such policypositions acknowledge precisely the stochastic nature of pollution and costs of regulatorysanctions that underpin the foregoing merits of flexible intertemporal permit trading regimes.

Acknowledgments

I am indebted to Steve Hamilton, Brian Copeland, and two anonymous referees for valuablecomments on earlier drafts of this paper.

Appendix

Proof of Lemma 1. Without tradable permits, a firm’s ‘‘violation’’ v equals its emissions ei1, andits choice problem thus becomes

maxyi1;ai1

pi1ðyi1; ai1Þ þ Eff ðei1ðyi1; ai1; eiÞÞg: ðA:1Þ

In order for a first-best to be achieved, the solution to (A.1) must equal ðyi1; a

i1Þ (satisfying

Eqs. (2) and (3)) in addition, administrative costs of assessing fines/taxes/penalties must not be

25As noted in the introduction, the notable exception to this rule is Yates and Cronshaw [20], who focus on a

different potential advantage of intertemporal permit trading than is identified here.26See David Victor, ‘‘Piety at Kyoto Didn’t Cool the Planet,’’ N.Y. Times, March 23, 2001, p. A21.


borne. Because positive administrative costs are borne whenever fines are positive, fines must bezero with probability one

f ðei1ðyi1; ai1; eiÞÞ ¼ 0 with probability one: ðA:2Þ

However, in order for the solution to (A.1) also to satisfy the first-best optimality conditions, wemust have f ðei1ðyi1; ai1; eiÞÞ bounded away from zero with positive probability. Hence, with a

monotone f(), we must have f(ei1())40 with positive probability, thus violating (A.2). &

Proof of Proposition 1. To prove the Proposition, I begin by establishing further necessaryproperties of a putative first-best period-by-period policy:

Observation 4. If a period-by-period regulation achieves a first-best, then the following must betrue: (I) The regime of fines satisfies: For %v40;

f ðvÞ ¼ 0 8vA½0; %v�; f ðvÞ40 8v4%v; f ð%vÞ40:

Firms are thus entitled to ‘‘free’’ violations up to %v40: (II) When firms cannot ‘‘bubble’’ theiremissions (so that a firm’s maximum free violation is minðei1; %vÞÞ; three conditions must hold.First, in order to avoid the payment of fines (as required in a first best), violations cannot exceedthe maximum possible level of ‘‘free violations,’’

total ‘‘free violations’’ ¼X

i

minfei1ðyi1; ai1; eiÞ; %vg

XDðeÞ �X

i

ei1ðyi1; a

i1; eiÞ %Q1

¼ total ‘‘available violations’’ 8eAON : ðA:3Þ

Second, if ‘‘free violations’’ exceed ‘‘available violations,’’ emissions permits have a zero price:Xi

minfei1ðyi1; ai1; eiÞ; %vg4DðeÞ ) p1 ¼ 0: ðA:4Þ

Third, when total emissions are maximal, e ¼ emax � ð%e; %e;y; %eÞ; emission permits have a positiveprice,X

i

minfei1ðyi1; ai1; %eÞ; %vg ¼ DðemaxÞ: ðA:5Þ

(III) Similarly, when firms can ‘‘bubble’’ emissions (so that a firm’s maximum free violation is %v),

total free violations ¼ N %vXDðeÞ 8eAON ;

if N %v4DðeÞ; then q1=0; and when e ¼ emax;N %v ¼ DðemaxÞ: (IV) Whenever emission permits have apositive price ðp140Þ; p1 ¼ f 0ð%vÞ:

Proof of Observation 4. (I) By Observations 1 and 2, there must be a state of nature e such thatD(e)40, p140, and no fines are imposed under the putative first-best period-by-period regulation.By construction,

DðeÞ ¼X

i

vi ; vi � vðff ðvÞg; p1; ei1ðyi1; ai1; eiÞÞ; ðA:6Þ


where v*()¼violation demand from problem (5) (with or without the ‘‘no bubble’’ constraint,vpei1). By (A.6), vi

*40 for some i. Now define %v ¼ maxi vi 40: With no fines imposed, we must

have

f ðvÞ ¼ 0 8vA½0; %v�: ðA:7Þ

Moreover, because vi*40 solves problem (5), we have

f 0ð%vÞXp140 ðA:8Þ

by Observation 3 and p140. Finally, f ðvÞ40 8v4%v follows from (A.7), (A.8), and monotonicityof f().(II) Condition (A.3): If ‘‘available’’ (total actual) violations exceed ‘‘free violations,’’ then (by

property (I)) some firm or firms must pay positive fines, thus contradicting the premise that a first-best is obtained.

Condition (A.4): When emission permits only enable firms to avoid violations that are free, atthe margin, the permits cannot command a positive price.

Condition (A.5): Note the following:

Claim 1.Xi

minfei1ðyi1; ai1; eiÞ; %vg DðeÞX

Xi

minfei1ðyi1; ai1; eiÞ; %vg DðemaxÞ 8eAON :

Proof of Claim 1. The claim follows from

DðemaxÞ DðeÞ ¼X

i

½ei1ð:; %eÞ ei1ð:; eiÞ�

X

XiAA

½ei1ð:; %eÞ ei1ð:; eiÞ� þXiAB

½%v ei1ð:; eiÞ�

¼X

i

minfei1ð:; %eÞ; %vg X

i

minfei1ð:; eiÞ; %vg;

where A ¼ fi : ei1ð:; eiÞpei1ð:; %eÞp%vg; B ¼ fi : ei1ð:; %eÞ4%vXei1ð:; eiÞg; and the inequality is due toei1ð:; %eÞXei1ð:; eiÞ 8eiA½

%e; %e�; fig*fA,Bg; and %voei1ð:; %eÞ for iAB. &

Now, if (A.5) does not hold, then (by (A.3)),

minfei1ð:; %eÞ; %vg4DðemaxÞ ) minfei1ð:; eiÞ; %vg4DðeÞ 8eAON ; ðA:9Þ

where the implication follows from Claim 1. Together, (A.9) and (A.4) imply p1¼0 withprobability one, contradicting the premise that a first-best is supported (Observation 1).(III) Similar to (II).(IV) When firms cannot ‘‘bubble,’’ p140 implies (by Observation 4(II)) thatX

i

minfei1ðyi1; ai1; eiÞ; %vg ¼

Xi

ei1ðyi1; ai1; eiÞ %Q1: ðA:10Þ


With %Q140 (Lemma 1), (A.10) implies

ei01ðyi01; ai01; ei0 Þ4%v for some i0: ðA:11Þ

With p140 (and f(v)¼0 for vA½0; %v� by Observation 4(I)), firm i0 (of (A.11)) chooses a violation ofat least %v; hence, in order for firm i0 to choose a violation that solves problem (5) and involves zerofines (as is necessary for a first-best), problem (5) must be solved by v ¼ %v; which requires (by(A.11)) that p1 ¼ f 0ð%vÞ40: Similar logic applies when firms can bubble. &Appealing to these necessary conditions for a first-best to prevail, we now have:

Proposition 1. (1) If firms can ‘‘bubble’’ their emissions, there is no period-by-period regulation thatcan achieve a first-best.(2) If firms are identical ex ante, but cannot ‘‘bubble’’ their emissions, then there is no period-by-

period regulation that can achieve a first-best.(3) If firms are not identical ex ante, and cannot ‘‘bubble’’ their emissions, there are primitive

economic conditions under which no period-by-period regulation can achieve a first-best.

Proof. (1) Suppose not. Then, by Observation 4 (I) and (III), we have

N %v ¼ DðemaxÞ4DðeÞ 8eAON ; eaemax:

Hence (by Observation 4 (III)), p1¼0 almost surely (a.s.), which (by Observation 1) contradicts thepremise that a first-best is supported.(2) Suppose not. Then, by Observation 4(II) (Eq. (A.5)),

N minfeð%eÞ; %vg ¼ DðemaxÞ ¼ Neð%eÞ %Q1; ðA:12Þ

where eðoÞ � e1ðy1; a1;oÞ for oA½

%e; %e� (with i subscripts dropped due to ex ante firm symmetry).

Furthermore, with %Q140 (Lemma 1), (A.12) implies that

%voeð%eÞ ) N %v ¼ DðemaxÞ ¼ Neð%eÞ %Q1: ðA:120Þ

However, (A.120) is easily seen to imply:Xi

minfeðeiÞ; %vg4DðeÞ ¼X

i

eðeiÞ %Q1 8eAON ; eaemax: ðA:13Þ

By Observation 4(II) (Eq. (A.4)), (A.13) implies p1¼0 almost surely, thus contradicting thepremise that a first-best is supported (Observation 1).(3) Suppose the contrary, that a first-best is achieved. Then, by Observation 4 and the absence

of fines in a first-best, a firm’s time 1minimal costs of emissions (C of Eq. (5)) under the putativefirst-best policy are

CðÞ ¼ p1 %qi1 þ p1d½ei1ðÞ %v�; d �0 if ei1ðÞp%v

1 if ei1ðÞ4%v

)CðÞ ¼ p1 %qi1 þ p1½maxfei1ðÞ; %vg %v�: ðA:14Þ


Now define

S � e :X

i

minfei1ðyi1; ai1; eiÞ; %vg ¼ DðeÞ( )

¼ set of e realizations such that q140 under the putative first-best policy

s � probabilty of S:

Substituting CðÞ from (A.14) into problem (6) and differentiating gives the following firmoptimality conditions for choice of (yi1, ai1) under the putative first-best regulation:

@pi1=@yi1 sf 0ð%vÞEeijeASf½@ maxfei1ðÞ; %vg=@ei1�½@ei1ðÞ@yi1�g ¼ 0; ðA:15aÞ

@pi1=@ai1 sf 0ð%vÞEeijeASf½@ maxfei1ðÞ; %vg=@ei1�½@ei1ðÞ@ai1�g ¼ 0; ðA:15bÞwhere I have substituted for p1 ¼ f 0ð%vÞ when p140 (Observation 4(IV)), and EeieAS is a conditionalexpectation operator for ei. Comparing (A.15) to Eqs. (2) and (3) yields the following necessarycondition for the posited first-best policy to indeed support first-best choices, (yi1,ail)¼(yi1

* ,ai1* ):

EeijeASf½@ maxfei1ðÞ; %vg=@ei1�½@ei1ðÞ@yi1�g ¼ EfD01ðE1Þ½@ei1ðÞ@yi1�g=sf 0ð%vÞ; ðA:16aÞ

EeijeASf½@ maxfei1ðÞ; %vg=@ei1�½@ei1ðÞ@ai1�g ¼ EfD01ðE1Þ½@ei1ðÞ@ai1�g=sf 0ð%vÞ ðA:16bÞ

with ei1() evaluated at (yi1,ail)¼(yi1* ,ai1

* ) and E1 ¼ E1 �

Pi ei1ðyi1; a

i1; eiÞ: In general, Eq. (A.16)

cannot hold for all i when firms are not ex ante identical. For example, suppose that firms havelinear emission risk (see Assumption 2 in the text), identically (but not independently) distributedei ¼ ðy; ZÞ and heterogeneous first-best emissions:

ei1ðÞ ¼ ei1ðyi1; ai1Þyþ Z; ei1ðyi1; a

i1Þ4ej1ðyj1; aj1Þ for i4j: ðA:17Þ

Then condition (A.16) requires

Ey;ZjeASf½@ maxfei1ðyi1; ai1; :Þ; %vg=@ei1�yg

¼ fEfyg½EfD01g þ CovfD0

1; yg�g=sf 0ð%vÞ 8i: ðA:160ÞBy (A.17), the left-hand-side of (A.160) is larger when i is higher; hence, with the right-hand-side of(A.160) invariant to i, (A.160) cannot hold for all i. &

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Stochastic pollution, costly sanctions, and optimality of emission permit banking