prevention and treatment in environmental policy design

Ž .JOURNAL OF ENVIRONMENTAL ECONOMICS AND MANAGEMENT 33, 196]213 1997ARTICLE NO. EE970983

Prevention and Treatment in Environmental Policy Design*

James Barrett and Kathleen Segerson

Department of Economics, Uni ersity of Connecticut, Storrs, Connecticut 06269

Received November 10, 1995; revised September 19, 1996

In many environmental contexts, the damages that result from some externality-generatingactivity can be reduced either through prevention prior to the occurrence of the event ortreatment after that occurrence. The existing literature has focused on the use of prevention.

w xOnly Polinsky and Shavell 11 consider explicitly the use of mitigation, but their analysis islimited to the case of Pareto efficiency. In reality, policymakers often pursue other objectives,such as minimizing damages subject to a budget constraint or minimizing the cost ofmaintaining a certain safety level. This paper provides an analytical framework for examiningthe choice of prevention vs treatment in the context of these alternative behavioral assump-tions. It considers first the case where the effectiveness of treatment is known at the time thatthe allocation decision must be made. We then extend the analysis to the case where theeffectiveness of treatment is uncertain and develop a framework for addressing the tradeoffbetween prevention and treatment under a ‘‘safety’’ rule. In both cases, we examine thefactors that influence the allocation of resources between prevention and treatment and showthat these factors can affect the second best allocations of resources in ways that are contraryto the effect under the Pareto efficient allocation. Q 1997 Academic Press

I. INTRODUCTION

Many production and consumption activities can lead to unintended side effectsthat are stochastic, i.e, uncertain at the time that the activity is undertaken.Examples of stochastic damages include contamination of groundwater or soils as aresult of activities such as agricultural production or waste disposal and damagesfrom spills or other unintended releases of toxic substances and hazardous wastes.Other examples are the illnesses that can result from the consumption of contami-

Ž .nated foods e.g., eggs with salmonella , diseases from the use of hazardousŽ .products e.g., tobacco, contaminated intravenous needles , and illness from expo-

Ž .sure to infectious environments e.g., houses with inadequate sanitation .Much attention has been devoted in the economics literature to evaluation of

policies designed to reduce stochastic damages through increased prevention.1

However, relatively little attention has been given to a second approach toŽreducing damages, namely, through treatment once the event e.g., the contamina-

.tion or illness has occurred. In contrast to prevention, which is undertaken ex ante,

* The authors gratefully acknowledge the helpful comments of two anonymous referees and financialsupport from the Food Marketing Policy Center, University of Connecticut.

1 w xFor an overview, see Shavell 13 .

1960095-0696r97 $25.00Copyright Q 1997 by Academic PressAll rights of reproduction in any form reserved.

PREVENTION AND TREATMENT 197

or before the event occurs, treatment is an ex post response.2 For example, thedamages from groundwater contamination can be reduced either by preventing

Žcontamination in the first place or by cleaning contaminated aquifers where.possible or by tapping replacement water supplies. Similarly, damages from

exposure to contaminated soils or toxic releases can be reduced either by prevent-ing the contamination or release or by removing andror treating contaminatedsoils or isolating the contaminated area. The damages from illness can be reduced

Žeither by reducing the likelihood of contracting the illness through, e.g., improved.sanitation or reduced use of hazardous products or by treating victims with

medical services once an illness has occurred.When both prevention and treatment are possible, decisions must be made

about how much to invest in prevention of events that can cause damages versusinvestment in treatment to reduce damages if and when they occur. However, asnoted above, to date research relating to the control of stochastic damages hasfocused on prevention, with little attention given to the possibility of treatment orto the question of how scarce resources should be allocated between these twoalternative means of reducing damages. A notable exception is Polinsky and

w x 3Shavell 11 . Their model examines the choice of accident prevention and damageŽmitigation in the production of some good. Oil spills and toxic waste leakages are

.cited as examples. Their focus is on the incentive effects of different liability rules.They conclude that a strict liability rule will induce the firm to employ the optimallevels of care and mitigation.

2 It is useful to be clear from the start about the distinction between what we are calling treatmentand the concepts of victim care and mitigation. In a general model of stochastic externalities, the

Ž .probability of an ‘‘accident,’’ P, can be affected by prevention undertaken by either the injurer A or1Ž . Ž .the victim A , i.e., P s P A , A . Similarly, the total damages from an accident occurring, D, can be2 1 2

Ž .affected by prevention as well as by response or treatment undertaken by either the injurer T or the1Ž . Ž .victim T , i.e., D s D A , A , T , T . Since our interest is in defining efficient choices rather than2 1 2 1 2

analyzing the incentives of injurers and victims, we do not distinguish between those actions undertakenby injurers and those undertaken by victims. For our purposes, the important distinction betweenprevention and treatment is that the former is chosen ex ante, i.e., before an accident occurs, while thelatter is chosen ex post, i.e., after the accident has occurred. Of course, when there are compoundprobabilities, the dividing line between prevention and treatment is less clear cut. For example, in thecontext of groundwater contamination, if expected damages depend on both the probability that anaquifer becomes contaminated and on the probability that the contaminated water is consumed, then

Ž .efforts to clean the aquifer e.g., through pumping out the contaminated water would be viewed as exante relative to the consumption of the water and ex post relative to the contamination of the aquiferitself. However, for our purposes, this type of ambiguity in the definition of prevention vs treatment isunimportant.

3 Other studies have discussed certain aspects of prevention]treatment decisions, but none fullyaddress the ex ante and ex post nature of the variables. For example, using a first-order Markov process,

w xHeffley 5 develops a decision rule for allocating expenditures between prevention and treatment in thecontext of health care. However, in his model, the choices regarding prevention and treatment are madesimultaneously, so that the level of prevention taken has no effect on the level of treatment needed, and

w xthe ex ante vs ex post nature of the problem is not fully examined. Raucher 12 presents a theoreticalframework for measuring the costs and benefits of groundwater protection. However, the modelassumes that the government always chooses the ‘‘economically appropriate’’ response to incidences ofcontamination so that treatment decisions are taken as given, leaving no need to choose between thetwo. This implicitly assumes that there is no constraint on the level of resources, so that any level ofremediation can be used with any level of preventative action.

BARRETT AND SEGERSON198

The criterion for resource allocation used by Polinsky and Shavell is the standardeconomic criterion of Pareto optimality or the minimization of total social costs,4

where those costs include the costs of prevention and treatment and the residualdamages from contamination. However, in many cases, Pareto optimality may notadequately describe a policymaker’s objective. There may be cases where the desirefor efficiency is superseded by other factors. For example, when resources arescarce, health and environmental agencies may not be able to achieve the Paretooptimal solution; instead, the objective may be to minimize total social costssubject to a budget constraint and to distribute limited resources efficientlybetween prevention and treatment.

Alternatively, policymakers may explicitly reject the balancing of costs andbenefits required by Pareto optimality and instead want to guarantee some mini-mum level of safety at least cost. This approach can be seen in current practice atSuperfund cleanup sites. The Comprehensive Environmental Response, Compen-

Ž .sation and Liability Act CERCLA prohibits a balancing of benefits and costs insetting target risk levels. Similarly, the Superfund Amendments and Reauthoriza-

Ž .tion Act SARA requires that sites be cleaned until they are in compliance withfederal and state statutes such as the Safe Drinking Water Act, which specifymaximum contamination levels. Empirical evidence suggests that Environmental

Ž .Protection Agency EPA decisions are, in fact, consistent with the statutes. Guptaw xet al. 3 found that the EPA does not consider costs in setting target risk. Thus, at

least in the context of Superfund cleanups, EPA decisions appear to be based onan objective of ensuring a given target risk level at minimum cost. A similarapproach has been used to evaluate different means of attaining minimum ground-

Ž w x. Žwater quality Lichtenberg et al. 8 and worker safety standards Harper andw x. 5Zilberman 4 .

This paper considers optimal investment in prevention and treatment under twoalternative criteria that might characterize a policymaker’s objective function.Specifically, we consider the case where a budget-constrained policymaker seeks tominimize damages subject to a budget or a resource constraint and the case wherea policymaker seeks to minimize the cost of ensuring that expected damages do notexceed some pre-determined threshold. The basic model can be viewed as an

w xextension of the model in Polinsky and Shavell 11 that allows for other behavioralobjectives.6 In addition, we present an extension of the basic model that allows foran additional source of uncertainty, namely, uncertainty about how effectivetreatment will be in reducing damages, an issue that is likely to be important in

4 Ž .Given linear utility functions constant marginal utilities and the availability of exogenous wealthtransfer mechanisms, Pareto optimality is equivalent to aggregate wealth maximization, which is, in turn

w x w xequivalent to cost minimization if costs are separable. See Miceli and Segerson 9 and Shavell 14 for athorough discussion of these issues.

5 w xFor other examples of safety standards in the economics literature, see Baumol and Oates 1, 2 .Additionally, maximum damage constraints are the basis of much work in marketable pollution permit

w x w xschemes; see, for example, Montgomery 10 and Krupnick et al. 6 .6 Because the focus of this paper is on allocations made by a benevolent dictator seeking to achieve

the different criteria, we do not consider the role of alternative policies in inducing desirable behavior,as Polinsky and Shavell do. An evaluation of policies depends on the incentives faced by individualparties, which will vary across the particular contexts noted above since these contexts differ with regardto the extent to which the effects of prevention and treatment decisions are internalized. Considerationof incentive issues is the subject of another paper, which must be more context specific.


determining the appropriate extent of reliance on the treatment alternative. Thew xextension builds on an approach proposed by Lichtenberg and Zilberman 7 .

The paper is organized as follows. In Section II, we present the basic model andexamine the relationship between the Pareto optimal allocation and the optimalallocations under the two alternative criteria. In Section III, we examine the role of

Žexogenous factors such as the magnitude of interim damages i.e., damages that.would occur prior to any treatment that may be undertaken and the likelihood of

the contamination or illness occurring, given a level of prevention. In particular, weconsider how changes in these factors affect optimal decisions regarding preventionand treatment under the alternative criteria. Section IV extends the analysis toallow for the fact that there might be uncertainty about how effective treatmentwill be in reducing damages. It examines the effect of an increase in uncertainty onthe optimal levels of prevention and treatment under the alternative criteria.

For the sake of concreteness, we discuss the model and the results in the contextof damages that can result from unintentional releases of hazardous wastes.However, as noted above, the model is sufficiently general to capture the basicelements of decisions regarding optimal prevention and treatment in other contextsŽ .e.g., agricultural externalities and health care as well.

II. AN OVERVIEW OF THE MODEL

Consider a production process that creates hazardous wastes as a by-product.The producer can undertake some degree of prevention in disposal of the wastesthat reduces the probability of these wastes entering the environment. If theenvironment does become contaminated, remedial measures can be taken to cleanthe environment. Closing off the site, excavating and treating contaminated soil,and counterpumping contaminated groundwater are examples of measures that canbe used alone or in combination to treat a potentially dangerous hazardous wastesite. Additionally, the effectiveness of prevention and of treatment are initiallyassumed to be known with certainty.7 It is also assumed that both the producer andthe victims are risk neutral or that actuarially fair insurance is available and that alldamages are monetarily compensable.

Pareto Optimality

This section will explore the familiar efficiency criterion of maximizing oneindividual’s expected utility subject to a constraint on the minimum level of theother individual’s expected utility. Given risk neutrality and an income redistribu-tion mechanism, Pareto optimality is equivalent to minimizing total expected socialcosts:

Minimize A q P A T q P A D T , 1Ž . Ž . Ž . Ž .A , T

where A is the amount of prevention taken by the producer, with units chosen sothat the price of A is one; P is the probability of contamination, with P9 - 0 andP0 ) 0; T is the expenditure on treatment of the contaminated site, with unitschosen so that the price of T is one; and D is the damages suffered, with D9 - 0

7 The effect of uncertainty with respect to the effectiveness of treatment is examined in Section IV.


and D0 ) 0. We assume that the producer has already decided how much of theproduct to produce. Thus, we do not model the production decision.8 In addition,Ž .1 assumes that damages are affected only by treatment and not by the amount ofprevention undertaken.

The first-order conditions for T and A are

yD9 s 1 2Ž .and

yP9 ? T q D s 1, 3Ž . Ž .Ž .respectively. These simply require that the expected marginal benefit of each is

equal to its marginal cost. The marginal benefit of treatment is simply thereduction in damages associated with an increase in treatment. The expectedmarginal benefit of prevention is the reduced probability that treatment will beneeded and that damages will be suffered. Note that under this criterion the

Ž .optimal level of T is determined solely by 2 and is independent of the probabilityof contamination. Thus, the treatment decision is separable from the preventiondecision.

In Fig. 1, the point EC* represents the minimum level of total expected costsŽ . Ž . Ž .EC s A q P A T q P A D T that could be incurred by society. Higher levels of

Žtotal expected costs are represented by ovular iso-expected cost curves drawn as

8 In a more general model where production is endogenous, the probability that contaminationoccurs could depend on the amount of wastes produced, inter alia. It is also possible to model thesechoices per unit of production, which would also abstract from the production decision.

FIGURE 1


.circles for simplicity spreading out from EC*. The further from EC* an iso-expected cost curve lies, the higher the level of expected costs it represents.

Ž .The solution under the Pareto criterion A*, T* yields EC* and the associatedŽ . Ž .level of expected damages P A* D T* represented by the iso-expected damage

curve D*. It should be noted that at EC* the slope of the iso-expected damageŽ . Žcurve is equal to the slope of the iso-expenditure curve, R s A q P A T not

. Ž . Ž .drawn . This can be demonstrated by combining Eq. 2 and 3 to obtain themarginal condition

1 q P9T P9Ds . 4Ž .

P PD9

Thus, at the Pareto optimal levels, expenditure is minimized given the associatedlevel of expected damages. Likewise, expected damages are minimized given thelevel of expected expenditure. It is possible to draw an expansion path through all

Ž .the points where the slope of an iso-expected damage curve, y P9DrPD9 , isŽŽ . .equal to the slope of its associated iso-expenditure curve, y 1 q P9T rP . As

shown below, the efficient levels of prevention and treatment under the second twocriteria lie on this expansion path as well.

Maximum Damage Constraint

We now replace the Pareto optimality criterion with an alternative criterion.Here, the objective is to minimize total expected expenditure subject to a con-straint on the level of damages expected to be suffered by the victim. The problemis to

Minimize A q P A TŽ .A , T 5Ž .

s.t. P A D T F D.Ž . Ž .

Ž .It can be easily shown that 5 is equivalent to minimizing expected social costssubject to a maximum damage constraint, i.e.,9

Minimize A q P A T q P A D TŽ . Ž . Ž .A , T 6Ž .

s.t. P A D T F D.Ž . Ž .

The equivalence of the two problems implies that minimizing expected expendituresubject to an expected damage constraint is, in fact, a constrained version of the

Ž .first-best Pareto optimal problem in 1 . It therefore yields second-best levels ofprevention and treatment that will in general result in a higher level of expectedtotal social costs than the first-best Pareto optimal levels. Thus, while choosing tomaintain a minimum level of safety, or maximum level of expected damages, willreduce one component of total expected social costs, namely, expected damagesŽ . Ž .P A D T , it will necessarily increase the other component, expected expenditure

Ž .A q P A T , by at least as much or more.

9 Ž . Ž .5 and 6 can be shown to be equivalent by comparing their Lagrangians.


Ž . 10If the constraint in 5 holds with equality, the first-order conditions for theoptimal values of A and T under the damage-constrained problem are

D y PD s 0, 7Ž .1 q P9T y lP9D s 0, and 8Ž .

1 y lD9 s 0, 9Ž .

Ž . Ž .where l is the Lagrangian multiplier on the constraint. The conditions 8 and 9Ž . Ž .are clearly similar to the conditions 2 and 3 from the Pareto criterion and would

Ž . Ž .be identical if l were equal to y1. Combining 8 and 9 yields the marginalcondition

1 q P9T P9Ds , 10Ž .

P PD9

Ž .which simply states that in equilibrium the slope of the iso-expenditure curve lhsŽ .is equal to the slope of the iso-damage curve rhs . This marginal condition was

Ž .also implied by the first-order conditions for the Pareto criterion 4 . Thus, asnoted above, Pareto optimality implies that expected expenditure is minimizedsubject to the associated level of expected damages, although minimizing expectedexpenditure subject to a given level of expected damages does not imply Paretooptimality. In this sense, Pareto optimality is a stronger criterion than is theefficiency criterion used in the damage-constrained problem. Note also that underthe damage-constrained criterion the decision regarding the level of treatment isno longer separable from the prevention decision. The constraint implies that anyincrease in prevention or treatment requires an offsetting decrease in the other,forcing them to be traded off for one another at a rate that keeps expecteddamages constant.

In Fig. 1 the level of expected damages is constrained to be D, which we assumeŽ .is lower than D* so that the constraint in 5 is binding. Graphically, the efficient

Ž .levels of A and T are the levels that are on the lowest i.e., inner mostŽ D D.iso-expected cost curve given the constraint. The solution is A , T , which

requires that R D resources be allocated to the problem and results in expectedtotal costs of EC D. An expansion path can be drawn to show the differentequilibrium points associated with different levels of the constraint. This expansionpath is simply the upper half of the expansion path shown in Fig. 1.

Resource Constraint

In this case, the objective is to minimize expected damages subject to aconstraint on the amount of resources that can be spent on prevention andtreatment. Note that at the time the expenditure on prevention is made, it is notknown whether or not contamination will occur and thus whether or not treatment

10 Ž . Ž .If the constraint is not binding, 6 reduces to 1 .


will be required. We therefore specify the resource constraint in terms of theactual expenditure on prevention plus the expected expenditure on treatment.11

The problem is then to

Minimize P A D TŽ . Ž .A , T 11Ž .

s.t. A q P A T F R .Ž .

This problem is simply the dual of the damage-constrained problem consideredŽ .above. Again, it can be shown that 11 is equivalent to

Minimize A q P A T q P A D TŽ . Ž . Ž .A , T 12Ž .

s.t. A q P A T F R .Ž .

Thus, as with the damage-constrained problem, minimizing expected damagessubject to a resource or budget constraint is equivalent to a constrained version ofthe Pareto optimality problem. It also yields second-best levels of prevention andtreatment that generate higher expected total social costs than occur under Paretooptimality. While the expected expenditure on prevention and treatment will besmaller, the decrease in this component of social costs will be more than offset bythe resulting increase in the expected damages.

If the constraint holds with equality, the first-order conditions for the resource-constrained problem are

R y A y PT s 0, 13Ž .P9D y g ? 1 q P9T s 0, and 14Ž . Ž .

D9 y g s 0, 15Ž .

where g is the Lagrangian multiplier on the constraint. Again, this would beŽ .identical to the Pareto solution if the multiplier were equal to y1. Combining 14

Ž .and 15 again yields the marginal condition

1 q P9T P9Ds , 16Ž .

P PD9

Ž . Ž . Ž .which is equivalent to 10 . This reflects the fact that 5 and 11 are duals of eachother. Thus, in order to minimize expected damages subject to an expectedexpenditure constraint, it is necessary to minimize expected expenditure subject tothe corresponding expected damage constraint. As with the damage-constrainedproblem, the optimal level of treatment is not independent of the level ofprevention.

In Fig. 1, the amount of resources available is constrained to be less than orŽ .equal to R which must be below R* if the constraint is binding . The solution is

Ž R R. RA , T , which results in expected damages of D and leaves society with total

11 An alternative is to specify a constraint that requires the sum of actual expenditure on bothŽ .A q T to be less than or equal to the resources available. However, under such a constraint,resources would be ‘‘wasted,’’ i.e., the constraint would not be binding, if no contamination occurredand thus no expenditure on treatment was needed.


expected costs of EC R. An expansion path can again be drawn; in this case it is thelower half of the expansion path in Fig. 1.

III. COMPARATIVE-STATIC RESULTS

Having characterized the efficient levels of prevention and treatment underthree alternative efficiency criteria, we turn next to a consideration of some of thefactors that affect those choices. The first is the level of interim damages. Sincetime often passes between the incidence of contamination and the beginning oftreatment, some damages may be suffered in this interim. The severity of theseinterim damages can be expected to alter the efficient levels of prevention andtreatment. For example, one might expect that the greater the interim damageswould be, the more we would want to invest in prevention rather than treatment.The model confirms this intuition in that it calls for an increase in prevention, butthe effects on treatment are less clear and depend on which criterion is used.

ŽSecond, the likelihood that contamination would occur given some level of.prevention is expected to affect the choice of prevention and treatment as well.

Specifically, one might expect that the more likely contamination is, the more wewould want to invest in preventing it. However, the model does not necessarilysupport this intuition, and the results again are sensitive to the criterion examined.This section will examine how these two factors affect the efficient levels ofprevention and treatment. Table I summarizes the comparative static results forthis section.

Interim Damages

The effect of interim damages can be seen by letting

˜ ˆD T s D q D T , 17Ž . Ž . Ž .

TABLE IŽ .Comparative Static Results When P ) 0 Section IIIa A

Damage- ResourceDamage- substitution substitution Resource- Total

shift effect effect effect shift effect effect

Interimdamages

Pareto optimality nra nra nra nra AT unchanged

Damage-constrained A, T A, T x nra nra A, T ?Resource-constrained nra A, T x nra nra A, T x

Probabilityof accident

Pareto optimality nra nra nra nra AxT unchanged

Damage-constrained A, T Ax, T A?, T ? nra A?, T Resource-constrained nra Ax, T A?, T ? Ax, T x Ax, T ?


˜where D is interim damages. This formulation assumes that the interim damagessimply shift the damage function without altering the effectiveness of treatment inreducing damages.

˜Ž . Ž . Ž .From 2 and 3 , we derive the comparative-static results A*r D G 0 and˜Ž . T*r D s 0. These results follow clearly from the first-order conditions. An

increase in interim damages increases the damages suffered if contaminationoccurs, so that the marginal benefit of reducing the probability of an accidentincreases, and the equilibrium level of prevention therefore rises. Interim damageshave no effect on the marginal benefit of treatment, so that there is no effect onthe equilibrium treatment level.

The effect of interim damages in the damage constrained model can be deter-Ž . Ž .mined from 7 ] 9 . Given that the objective function is convex in A and T , it

D ˜ D ˜Ž . Ž .can easily be shown that A r D G 0 and T r D is indeterminate. Thus,under this efficiency criterion an increase in interim damages has the expectedresult of increasing investment in prevention. However, the effect on treatment isindeterminate.

This indeterminacy results from two different effects that occur when interimdamages increase. The first is a ‘‘damage-shift effect.’’ The increase in interimdamages shifts the iso-expected damage curve outward, requiring an increase inexpenditures to keep the constraint satisfied. The second effect is a ‘‘damage-sub-stitution effect.’’ The increase in interim damages causes the iso-expected damagecurve to become steeper. This occurs because reducing the probability of contami-nation through increased prevention now reduces the probability of suffering agreater level of damages than before. It would take a greater amount of treatmentto match the reduction in expected damages, thereby increasing the marginal rateof substitution and making the iso-expected damage curve steeper. Both of theseeffects tend to increase prevention. However, the effects work in opposite direc-tions on treatment; the damage-shift effect tends to increase treatment, and thedamage-substitution effect tends to decrease it. Hence, the effect on treatment isindeterminate.

Ž . Ž .In the resource-constrained case, assuming convexity, 13 ] 15 imply thatR ˜ R ˜Ž . Ž . A r D G 0 and T r D F 0. In this case, the comparative static results are

consistent with a priori intuition. The increase in interim damages does not shiftthe constraint, so that there is no damage-shift effect. However, as before, there isa damage-substitution effect, which causes the iso-expected damage curves tobecome steeper. However, the fact that expenditures can not go up in this casemeans that the level of expected damages must rise. This decreases treatment andincreases prevention.

Change in Probability of an Accident

One would expect the probability of contamination occurring to depend on otherfactors in addition to the amount of prevention taken. In the context of hazardouswastes, several exogenous parameters can be expected to play a role in theprobability of contamination. For example, weather could affect the probability ofcontamination as excessive rain may increase the likelihood of improperly disposedwastes escaping from a lagoon into surface or groundwaters. This section examineshow an exogenous increase in the probability of contamination affects the choiceof prevention and treatment.


The effect of an exogenous increase in the probability of contamination can beseen by letting

P s P A , a , 18Ž . Ž .

where a is an exogenous shift parameter. We define a such that the partialŽ . Ž .derivative, P , is positive. Under Pareto optimality, 2 and 3 imply that the signa

Ž .of A*ra is the opposite of the sign of the cross-partial P and thata AŽ . T*ra s 0. Again, the results are fairly intuitive. Because a only affects theprobability of an accident and treatment is chosen without regard to the probabil-ity, a change in a will not affect the choice of treatment. The effect of a change ina on the choice of prevention depends on the sign of P . If P is negative, thena A a Athe increase in a has made prevention more effective in reducing the probabilityof an accident, and more prevention should be used to take advantage of this. IfP is positive, then prevention has become less effective, and less should be used.a AIf P is equal to zero, then the increase in a has had no effect on thea Aeffectiveness of prevention, and the equilibrium level of prevention does notchange. For simplicity of exposition, the results in Table I are based on thearbitrary assumption that P is positive, although there is no a priori reason whya Athis should be so. Where possible, we have included explanations of how theseresults would be affected if this assumption did not hold.

Ž . Ž .Under the damage-constrained case, 7 ] 9 imply that the sign of bothŽ D . Ž D . A ra and T ra depend on the cross-partial derivative, P . If P isa A a A

Ž D . Ž D .positive, then T ra G 0. The sign of A ra depends not only on the signŽ D .of P , but also on its size. If P is positive and large enough, A ra F 0;a A a A

Ž D .otherwise, A ra G 0.When a increases, three different effects can be identified. The first is the

damage-shift effect. As was the case with interim damages, the increase in a shiftsthe iso-expected damage curve outward so that more resources are required tomaintain the same level of expected damages. This effect increases both preventionand treatment. At the same time, the damage-substitution effect tilts the iso-ex-pected damage curve so that it becomes flatter. This also tends to increasetreatment but tends to decrease prevention. The third effect is a ‘‘resource-sub-stitution effect.’’ The change in a rotates the iso-expenditure curve so that the rateof substitution between prevention and treatment changes. In the absence of aconstraint on expected damages, the increase in a would cause the iso-expenditure

Žcurve to tilt inward, resulting in a higher level of expected damages a lower.iso-expected damage curve . However, the expected damage constraint does not

allow that, so that the iso-expenditure curve merely rotates, rather than tiltinginward, remaining on the same iso-expected damage curve. The direction of theresource-substitution effect depends on the size of P . If P is positive anda A a Asufficiently large, the iso-expenditure curve will get steeper, and the resource-sub-stitution effect will add to the damage-substitution effect’s tendency to decrease

Ž .prevention and increase treatment. If P is small or negative , the iso-expendi-a Ature curve gets flatter as it rotates, tending to decrease treatment and increaseprevention. If this flattening effect were large enough, then the combination of theresource-substitution effect and the damage-shift effect would dominate the dam-age-substitution effect’s tendency to decrease prevention, and prevention wouldincrease. Otherwise, the damage-substitution effect would dominate, and AD willdecrease.


As in the damage-constrained problem, under the resource-constrained crite-rion, the signs of the comparative statics depend on the size and sign of the

Ž R . Ž R .cross-partial P . If it is positive and large, A ra F 0 and T ra F 0. Ifa AŽ R . Ž R .it is positive and small or negative, A ra F 0 still but T ra G 0. Again,

three effects can be identified. The first is the resource-substitution effect. Theincrease in a causes the iso-expenditure curve to tilt inward, and this time there isno maximum damage constraint to keep it from doing so. Again, however, it isimpossible to tell whether the tilt causes the iso-expenditure curve to becomesteeper or flatter without making additional assumptions about P . Unlike thea Aprevious case, because the level of expected damages is not constrained, theresource-substitution effect is accompanied by a resource-shift effect. This isanalogous to the income effect in standard microeconomic theory, as the increasein P raises the expected price of treatment. The resulting decrease in purchasingpower makes both prevention and treatment less affordable, so that they both tendto decrease. Since treatment is now more likely to be needed, the same actuallevels of prevention and treatment would result in a higher expected level ofexpenditures, so that a reduction in both is required to satisfy the resourceconstraint. The combination of these two effects results in the new iso-expenditurecurve. At the same time, the damage-substitution effect flattens the iso-expecteddamage curves. This adds to the resource-shift effect’s tendency to reduce preven-tion, but moves treatment in the opposite direction, tending to increase it. If theresource-substitution effect is large enough, when combined with the resource-shifteffect, it will dominate the damage-substitution effect’s tendency to increasetreatment, and treatment will fall. If not, the damage-substitution effect willdominate, and T R will increase.

IV. UNCERTAINTY

Many elements of the hazardous waste problem are characterized by uncer-tainty. One important source of uncertainty is the effectiveness of the means usedto reduce the expected damages caused by contamination of soil, air, and surfaceand groundwaters. Using a method first introduced by Lichtenberg and Zilbermanw x w x7 and further developed by Lichtenberg et al. 8 , this section examines howuncertainty regarding the effectiveness of treatment in reducing damages affectsthe efficient allocation of resources between prevention and treatment. This‘‘safety rule’’ approach allows for ‘‘uncertainty-compensated trade-offs betweenrisk and social cost . . . reflecting both uncertainty about risk and decision-makers’

Ž w x.preferences regarding that uncertainty’’ Lichtenberg et al. 8 . Under this ap-proach, the policymaker sets a risk standard and a safety margin, r, expressed as apercentage. The object is to ensure that the risk standard is violated no more than1 y r percent of the time at a minimum cost to society.

w xIn the original model by Lichtenberg and Zilberman 7 , and in subsequentw x w xapplications by Lichtenberg et al. 8 and Harper and Zilberman 4 , the random

variable of interest is the probability that a randomly selected individual willexperience some adverse health effect of a fixed magnitude; there is no possibilityfor treatment of remediation. Additionally, their models consider several parame-ters of which overall risk is a multiplicative combination. The assumption is thatthese parameters are distributed lognormally so that the log of health risk is the


sum of the logs of the parameters. In the problem considered here, the randomvariable is the level of expected damages. The distribution of this random variabledepends on the level of treatment.12 Instead of a probabilistic risk standard, thepolicymaker chooses an expected damage standard, D; actual expected damagesshould exceed D not more than 1 y r percent of the time. Rather than considerseveral sources of uncertainty, here we focus only on uncertainty regarding theeffectiveness of treatment in reducing damages. As a result, there is no need toassume a lognormal distribution, and we assume instead that the damage functionis distributed normally about some mean. Additionally, the original models andapplications assume that some parameters affect mean risk more than the varianceof risk and that the opposite is true of other parameters. Here, we assume that thevariance of damages, s 2, is exogenous and that only the mean of damages, m, isaffected by treatment.13

Formally, in the context of our model, the safety rule approach can be expressedas

Pr P A ? D T G D F 1 y r . 19Ž . Ž . Ž .� 4

Because damages are normally distributed, it is possible to express this in terms ofthe mean and standard deviation of damages:

P A m T q Z r ? s F D , 20Ž . Ž . Ž . Ž .

Ž .where Z r is the critical value of the normal distribution that is exceeded withprobability 1 y r.

For the sake of brevity, we will focus only on the unconstrained and resourceŽ .constrained problems. Assuming that 19 holds with equality, we consider two

different objective functions. The first objective is to minimize the probability thatthe given standard, D, is violated, and the other is to minimize the standardŽ .making it as stringent as possible that will be violated 1 y r percent of the time,where r is taken as given. We then examine how an exogenous change in thestandard deviation of damages, s , affects the efficient allocation of resourcesbetween prevention and treatment. Again, the results are shown to depend uponwhich criterion is used. Table II summarizes the comparative static results of thissection.

Minimizing the Damages

As mentioned above, in this formulation, the objective is to find the moststringent standard that can be expected to be satisfied a given percentage of thetime. In the absence of a resource constraint, the problem is to

Minimize D s P A m T q Z r ? s . 21Ž . Ž . Ž . Ž .A , T

12 Ž .This is equivalent to assuming D s D T , « , where « is a random variable. For simplicity, weŽ .suppress « and simply treat D T as a random variable.

13 We make this assumption in order to consider exogenous increases in the level of uncertainty.2 2Ž . 2Alternatively, we could specify s s s b , T , where b is an exogenous parameter that shifts s

given T.


TABLE IIŽ .Comparative Static Results Section IV

Unconstrained Resource-constrained

A T A TMinimize damagess 0, s 0 G 0, F 0

given confidence s s s s

level A T A TMinimize probability

s 0, s 0 s 0, s 0of exceeding s s s s

given standard

This yields the first-order conditions

P9 ? m q Zs s 0 « P9 s 0, 22Ž . Ž .Pm9 s 0 « m9 s 0. 23Ž .

These simply require that prevention and treatment be used until the probabilityand mean damages resulting from contamination can be reduced no further. Itshould be clear that this must be the case. In the absence of any cost considera-tions in the objective function or through a constraint, the marginal cost of bothprevention and treatment is zero, and they will both be employed until theirmarginal benefits are also zero. It should also be clear that the variance ofdamages has no effect on the equilibrium levels of prevention and treatment.

With a resource constraint, the problem becomes

Minimize D s P A m T q Z r ? sŽ . Ž . Ž .A , T 24Ž .

s.t. A q P A ? T F R .Ž .

Assuming that the constraint holds with equality, this yields first-order conditions

R y A y P ? T s 0, 25Ž .P9 ? m q Z ? s y l 1 q P9 ? T s 0, and 26Ž . Ž . Ž .

P ? m9 y l ? P s 0, 27Ž .where l is the Lagrangian multiplier. These yield the comparative static resultsŽ . Ž . Ars G 0 and Trs F 0.

As in the resource-constrained version of the interim damage case, only oneeffect exists. Because the change in s does not affect the relative ‘‘prices’’ ofprevention and treatment, there is no resource-substitution or resource-shift effect.Because the constraint on resources means that the iso-expenditure curve does notshift outward, there is no damage-shift effect either. The only effect that still existsis the damage-substitution effect which makes the iso-expected damage curvesteeper, so that equilibrium treatment falls and equilibrium prevention rises. Thisis consistent with intuition in that we would expect an increase in the uncertaintyregarding the usefulness of one of our tools to reduce the extent to which it isemployed. As the effectiveness of treatment becomes more uncertain, we wouldexpect that society would be more reluctant to rely on it, thus shifting resourcestoward prevention.


Minimizing the Probability of Exceeding the Standard

In this scenario, the policymakers have chosen some maximum level of expecteddamages to serve as the safety standard, and the objective is to choose the levels ofprevention and treatment that minimize the probability that the standard is

Ž .violated 1 y r . In the absence of a resource constraint, the problem is to

D y P A ? m TŽ . Ž .y1Maximize r s Z , 28Ž .ž /P A ? sA , T Ž .

or, equivalently,

DMaximize y m T . 29Ž . Ž .

P AA , T Ž .

This yields the first-order conditions

2yP9DrP s 0 « P9 s 0, 30Ž .ym9 s 0. 31Ž .

Ž . Ž .As one might expect, these are equivalent to the conditions 22 and 23 . Thereasoning is identical: if resources are unlimited, prevention and treatment shouldbe used until expected damages cannot be reduced any further. Again it should beapparent that the variance of damages does not influence the equilibrium levels ofprevention and treatment.

With a resource constraint, the objective function is to

D y P A ? m TŽ . Ž .y1Maximize r s Z ž /P A ? sA , T Ž . 32Ž .s.t. A q P A ? T F R ,Ž .

or, equivalently,

DMaximize y m TŽ .

P AA , T 33Ž . Ž .s.t. A q P A ? T F R .Ž .

Ž . Ž . Ž .It is clear from 33 that under this criterion Ars s 0 and Trs s 0.Thus, with this formulation, an increase in uncertainty regarding the effectivenessof treatment has no effect on the amount of resources invested in prevention ortreatment.

These results are less intuitive at first glance. Figure 2 shows the distribution ofexpected damages in the absence of any prevention or treatment and the distribu-

Ž .tion after prevention and treatment are employed but before s increases .Prevention and treatment are used to shift the distribution of expected damages to


FIGURE 2

the left by decreasing the probability of contamination occurring and by reducingthe mean of damages if contamination does occur. This has the effect of reducingthe area of the distribution which lies above the expected damage standard, D.Figure 3 shows the left most distribution from Fig. 2 before and after s increases.The increase in s widens the distribution, so that a greater area of the distributionis above D. However, prevention and treatment can do no further good in reducingthis area, because they have already shifted the distribution as far left as possible,given the resource constraint. Thus, under this criterion, an increase in uncertaintyhas no effect on the optimal investment in prevention or treatment.

FIGURE 3


V. CONCLUSIONS

In many environmental contexts, the damages that result from some externality-generating activity can be reduced either through prevention prior to the occur-rence of the event or through mitigation after that occurrence. With the exception

w xof Polinsky and Shavell 11 , the existing literature has focused on the use ofprevention. Polinsky and Shavell consider explicitly the use of mitigation, but theiranalysis is limited to the case of Pareto efficiency. In reality, policymakers oftenpursue other objectives, such as minimizing damages subject to a budget constraint,or minimizing the cost of ensuring that damages to not exceed a certain level. Forexample, cleanup standards under CERCLA are not based on a weighing ofbenefits and costs but rather seek to ensure a given level of protection. Similarly,health care officials are often faced with limited budgets for use in trying toimprove the health status of a given population.

This paper has provided an analytical framework for examining the choice ofprevention vs treatment in the context of these alternative behavioral assumptions.It considers first the case where the effectiveness of treatment is known at the timethat the allocation decision must be made. We show that the two criteria ofminimizing damages subject to a budget constraint and minimizing the cost ofmeeting a prespecified damage constraint are both constrained versions of Paretoefficiency and hence result in second best outcomes. Nonetheless, to the extentthat they more accurately reflect the criteria used by decisionmakers in designingenvironmental policies, they are appropriate benchmarks for policy evaluation. Wethen extend the analysis to the case where the effectiveness of treatment isuncertain and develop a framework for addressing the tradeoff between preventionand treatment under a ‘‘safety’’ rule.

In both cases, we examine the factors that influence the allocation of resourcesbetween prevention and treatment. We showed that factors such as a shift in thelikelihood of contamination occurring and an increase in the level of interimdamages can affect the second best allocations of resources in ways that are

Ž .contrary to the effect under the first-best Pareto efficient allocation. For exam-ple, when the probability of an accident increases exogenously, the Pareto optimallevel of treatment does not change, while the optimal level of treatment unambigu-ously increases under the damage constraint, regardless of whether preventionincreases or decreases. Similarly, when the effectiveness of treatment is uncertain,we show that an increase in uncertainty regarding the effectiveness of treatmentdoes not change the optimal levels of prevention and treatment in the absence of abudget constraint, but given a limited budget, this can lead to an increase in theoptimal level of prevention and a decrease in the optimal level of treatment.

The characterization of the second best levels of prevention and treatment andthe factors that affect those levels developed here should provide guidance todecisionmakers charged with designing policies to reduce damages from stochastic

Ževents. Whether the context is environmental policy prevention of contamination. Ž . Žvs cleanup , health care prevention of disease vs treatment , or drugs prevention

.vs treatment of drug addiction , society is faced with the need to choose preventionand treatment levels to achieve some social objective, which may not be simply aPareto optimal balancing of benefits and costs. In such cases, the framework andanalysis presented in this paper can provide a starting point for determining how


resources should be allocated between these two alternative means of reducingsocial costs.

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