Voluntary Pollution Abatement when Emissions are Not Monitored

William S. Neilson Department of Economics Texas A&M University and Geum Soo Kim Department of Economics Hoseo University

February 2005

Abstract: We analyze an infinitely-repeated game between a regulator and a polluting firm in which the regulator does not monitor emissions but does monitor ambient environmental quality and threatens to impose a period of command-and-control regulation if the environment deteriorates too far. We show that the firm pollutes less during unregulated episodes than it would if it were never regulated at all, and identify circumstances in which threats are the regulator’s preferred policy instrument. The model can be used to explain why firms abide by voluntary environmental agreements and how firms behave when they are not yet subject to regulation. Keywords: Voluntary environmental agreements, overcompliance, environmental regulation, ambient environmental quality, trigger strategies JEL code: Q58

Neilson thanks the Private Enterprise Research Center for financial support.

1. Introduction The existence of overcompliance with environmental regulations raises an important issue, namely why do polluting firms bear the cost of reducing pollution more than they have to? 1 Several answers have been proposed. Salop and Scheffman (1983) suggest that firms in an oligopoly reduce pollution in the hopes of fostering stricter regulations in order to give them a competitive advantage over their rivals (see also Maxwell, Lyon, and Hackett, 2000; and Puller, 2004). Arora and Gangopadhyay (1995) construct a duopoly model in which voluntary overcompliance shifts outward the demand for the firm’s product, allowing it to increase revenue. While Khanna and Damon’s (1999) empirical study of the EPA’s 33/50 Program finds that participation is driven in part by expected gains from public recognition and technical assistance, it is also driven by the threat of harsh future regulation.2 Harrington (1988) constructs a model in which unregulated firms reduce emissions to avoid future regulation.3 The regulator places a firm into one of two groups, either an unregulated group or a heavily regulated group. A firm in the favored group cuts back its emissions to avoid being placed in the disfavored group.4 Bose (1995) constructs a model in which firms choose their emissions first and then the regulator chooses the enforcement parameters second, and finds that firms reduce their emissions in order to obtain less-stringent future enforcement parameters.

1

Khanna and Damon (1999) find that participants in the EPA’s voluntary 33/50 Program do reduce emissions. 2 Goodin (1986) and Segerson and Miceli (1998) note that the threat of regulation underlies many successful voluntary environmental agreements, including the 33/50 Program and the Dutch National Environmental Policy Plan. Further empirical evidence on regulatory threats can be found in Hartman, Huq, and Wheeler (1997). 3 Segerson and Miceli (1999) also look at the ability of legislative threats to influence participation in a voluntary agreement, but unlike Harrington, they do not make compliance with the voluntary agreement endogenous, and instead assume that the firm automatically abides by any agreement. 4 Raymond (1999) shows that Harrington’s results are not robust under asymmetric information.

2

In this paper we explore a different mechanism through which the threat of future regulation induces firms to voluntarily reduce emissions.5 In our model, the regulator does not directly monitor the firm’s emissions but it does monitor ambient environmental quality, which is stochastic but related to the level of the firm’s emissions. If the ambient environmental quality ever drops below some trigger level, the regulator initiates a command-and-control regime.

At the beginning of the game, the regulator sets

parameters that relate to the harshness of the regulation and impact the profitability of the firm, as well as the length of the command-and-control regime and the ambient environmental quality level that triggers it. The firm then sets its emission level each period to maximize the sum of expected discounted profits, and therefore faces a tradeoff between emitting more and earning higher profit in the current period and cutting back emissions to avoid future regulation. This framework is reminiscent of Green and Porter’s (1984) model of collusion in a repeated game with imperfect monitoring.6 In their model rival output is unobservable but the market price is an imperfect signal of market output. They establish the existence of an equilibrium in which firms produce a collusive level of output, but if the price falls below an endogenously-determined threshold level they expand output in a temporary punishment phase. In our model, the regulator first commits to the parameters governing the harshness and length of the punishment phase as well as the trigger level of the ambient environmental quality, and the firm chooses an emissions level as a best response. Our model differs from the Green and Porter model in two respects. First, in their model the players are oligopolistic firms, and so all players move every period, 5

The threat of future regulation has also been studied in the industrial organization literature. See, for example, Glazer and McMillan (1992) for theoretical work and Stango (2003) for recent empirical work. 6 See also Radner (1986).

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while in our model the players are the regulator and the regulated firm, and the regulator only moves at the beginning of the game. Second, our model adds an additional variable, the harshness of the punishment phase. The results show that (i) the firm does not overcomply in the absence of a regulatory threat, and (ii) threatening a harsher command-and-control regime leads to more overcompliance. Furthermore, it is possible to identify conditions under which the regulator prefers voluntary overcompliance through threats to the optimal traditional command-and-control regime, as well as conditions under which it prefers voluntary overcompliance through threats to never regulating. Even though some of these findings are not terribly surprising, they constitute an important addition to the literature, in particular the literature on voluntary environmental agreements.7 Although the model does not explicitly account for negotiations between the regulator and the firm, the equilibrium of the game can still be thought of as a voluntary agreement to reduce emissions, and this model provides an explanation, missing from the literature, for why firms abide by voluntary environmental agreements. The paper is organized as follows. Section 2 discusses the assumptions required for our model. Section 3 sets up the model, and Section 4 derives and discusses the firm’s equilibrium strategy. Section 5 explores conditions under which the regulator prefers to induce voluntary overcompliance through threats to other regulatory regimes. Section 6 offers a conclusion and suggestions for further research.

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This is not, of course, the only issue of interest. Other issues involve the role of the negotiation process (e.g. Amacher and Malik, 1996; and Manzini and Mariotti, 2003), the efficiency of the agreement (e.g. Wu and Babcock, 1999; and Lyon and Maxwell, 2003), and the propensity of firms in voluntary agreements to truthfully self-report their emissions (e.g. Malik, 1993; Kaplow and Shavell, 1994; and Livernois and McKenna, 1999).

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2. Assumptions We consider an infinite-period two-player game. One player, the firm, decides how much pollution to emit each period in order to maximize the sum of its expected discounted profits. The other player, the regulator, decides how to regulate the firm in order to maximize a welfare function which depends on the sum of expected discounted environmental damages, the sum of expected discounted regulatory costs, and the sum of the firm’s expected discounted profits. To construct the model, we make three major assumptions about the natural environment. First, the firm’s emissions have only a transitory effect on environmental quality. More specifically, the ambient environmental quality depends on emissions in the current period but not on emissions in past periods. This rules out situations in which pollutants build up over time, but may apply to noise pollution, air pollution in a localized area, or water pollution in a flowing water source.

The second major

assumption is that the ambient environmental quality in a given period is stochastic but also dependent on the firm’s emissions in that period. Ambient air quality, for example, depends not only on pollutant emissions but also on such things as wind, humidity, and temperature. The third assumption is that the environment is stationary in the sense that the same random process that governs ambient environmental quality in period t1 also governs it in period t2. We restrict attention to strategies that allow for only two regulatory regimes. In one regime the regulator adopts command-and-control regulation by setting and enforcing a standard. In the other regime the regulator does not enforce any standard but monitors ambient environmental quality. We operationalize the restriction by assuming

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that the regulator commits to the values of three parameters before the start of the game. Those parameters are the level of ambient environmental quality below which a command-and-control phase is triggered, the length of the command-and-control phase, which may be infinite, and a parameter governing the strictness of the command-andcontrol phase. In keeping with this, we call a period in which the regulator enforces a standard an enforcement period, and a period in which it does not enforce a standard a non-enforcement period. At first glance, the assumptions that the regulator can commit to a regulatory regime and the firm knows what the regulator has committed to may seem farfetched. We contend that they are not. First, the results still hold, at least for early periods, if the firm believes that the regulator uses a trigger strategy and that command-and-control regulation, if it occurs, will have a fixed harshness and duration. These restricted beliefs provide an alternative interpretation of our model in which the regulator manipulates the firm’s beliefs to get it to alter its emissions instead of by changing the harshness or duration of the threatened command-and-control regime.8

Second, there may be

circumstances under which the regulator’s actions are heavily restricted. For example, other firms in the same industry but different regions might already be regulated, and if regulation is triggered for the firm in question, it stands to reason that the details of the command-and-control regime would not differ too greatly from the firm’s counterparts in different regions. Also, even though the regulator may not be able to commit to an ending time for the command-and-control regime, the model is able to accommodate permanent regulation once it is triggered. 8

Given that the results still hold when the firm assigns probability one to a single harshness/duration pair, continuity arguments suggest that the results continue to hold when probability mass is spread over a range of harshness/duration pairs, although the mathematics become much more complicated.

6

While the firm knows the distribution of the random factors that determine ambient environmental quality, it cannot observe any of them (except its own emissions level) in a given period until after it has committed to its emissions level for that period. Given that the environmental quality variable is stationary and that the regulator commits to its strategy before the first period, it then makes sense to restrict attention to stationary strategies for the regulated firm. Consequently, we restrict attention to strategies that specify a level of emissions for the firm for non-enforcement periods and a level of emissions for enforcement periods.

3. The model In a non-enforcement period t a firm emits effluent xt generating profit π(xt), where π is assumed to be strictly increasing and concave. The emissions impact the ambient environmental quality qt given by qt = q(xt,θt), where θt is the realization of an i.i.d. random variable θ~t with distribution function F and density function f, which are common knowledge. The function q is decreasing in its first argument, so that increased emissions lead to lower environmental quality. Assume without loss of generality that q increases with θ. It is further assumed that q is invertible with respect to θ so that θ = ~

k(x,q) is the realization of the random variable θ that yields ambient environmental quality level q when the firm emits x. It follows that k increases in x, since when the firm raises its emissions environmental quality falls, and to keep environmental quality fixed θ must increase as well. Similarly, k increases in q, since achieving a higher environmental

7

quality level when emissions remain the same requires a higher realization of the random variable. The environmental quality is below some level q0 when the firm emits xt if θt ≤ k(xt,q0), and the probability of such an event is given by F(k(x,q0)). Zero is the lower bound for environmental quality so that F(k(x,0)) = f(k(x,0)) = 0 for all x, and f’(k(x,0)) > 0 for all x. A command-and-control regime is described by a regulatory parameter r, which may reflect such measures as the standard that is set by the regulator, the monitoring probability, or a parameter of the fine schedule should a violation be detected. When in a command-and-control regime with regulatory parameter r, the firm’s maximal per-period profits are πR(r) and its corresponding emissions level is xR(r). We abstract away from the details of how much the firm emits during enforcement periods because the problem is well-understood in the literature. It suffices to assume that πR decreases in r, so that increases in r reflect more restrictive regulatory regimes. It is assumed that for all r, π(xR(r)) > πR(r), that is, the profit from emitting xR in an enforcement period is less than the profit from producing the same level of emissions in a non-enforcement period. There are two reasons why this would hold. The first is that the firm bears costs from being regulated that occur in addition to the costs of reducing emissions.9 The second is that firms often benefit from voluntarily reducing

9

Glachant (1999) documents the cost savings from a voluntary environmental agreement, and his findings imply that for a given level of emissions, costs are lower under non-enforcement periods than under enforcement periods.

8

emissions, such as with voluntary environmental agreements, through such mechanisms as public recognition and government-provided technical and financial assistance.10 The regulator’s strategy is restricted to the following class. Before the first period the regulator commits to the values of three parameters, r, qR, and T. The first period is a non-enforcement period, meaning that the regulator does not enforce any standard. If the ambient environmental quality in period t falls below qR, the regulator begins a command-and-control phase with parameter r that lasts for a total of T periods beginning in period t + 1 and lasting through period t + T. Because of the random nature of ambient ~

environmental quality, regulation is triggered in the next period if θ t ≤ k ( xt , q R ) . The regulator may choose T = ∞, which has the interpretation that command-and-control regulation lasts forever once it starts.11 In period t + T + 1, if it is finite, the regulator once again stops enforcing the standard. The regulator may also choose qR = ∞, which has the interpretation that every non-enforcement period is followed by an enforcement period. If qR and T are both infinite, command-and-control regulation begins in period 2 and lasts for the remainder of the game. Consider a non-enforcement period t. The firm chooses x to maximize

T  V ( x ) = π ( x ) + 1 − F ( k ( x, qR ) ) β V ( x ) + F ( k ( x, qR ) )  β sπ R ( r ) + β T +1V ( x )  ,  s =1 

∑

(1)

where β is the firm’s discount factor. This recursive representation reflects the fact that the firm is restricted to stationary strategies, which in turn follows from the stationarity 10

Khanna and Damon (1999) find that participation in voluntary environmental agreements is often driven by expected gains from public recognition and technical assistance. Indeed, for reasons of credibility the regulator might only have the choice of T = ∞.

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assumptions of the game. The firm’s expected payoff in period t is its profit in period t, which is the first term, plus its expected discounted profit if command-and-control regulation is not triggered by the ambient environmental quality in period t, plus its expected discounted profit if regulation is triggered. In this last term the firm receives per-period profit of πR(r) for each of the next T periods and then it receives its expected discounted profit from a non-enforcement period. An enforcement period is triggered by period-t emissions if qt ≤ qR, which occurs with probability F(k(x,qR)). Rearranging (1) yields

V ( x) =

π R (r) π ( x) − π R (r) + . 1 − β 1 − β + ( β − β T +1 ) F ( k ( x, qR ))

(2)

The firm is guaranteed at least πR(r) each period forever, as shown by the first term in (2). The second term measures the additional amount the firm earns during non-enforcement periods. The firm chooses the emissions level to maximize its payoff, and denote the resulting value by x*(r,qR,T).

We write V*(r,qR,T) to denote the maximized value,

V*(r,qR,T) = V(x*(r,qR,T)). We assume throughout that V is strictly concave so that the second-order condition for a maximum is satisfied. The regulator’s objective function is

W(r,qR,T) = V*(r,qR,T) − C*(r,qR,T) − D*(r,qR,T),

10

(3)

where V* is the firm’s maximized expected discounted profit, C* is the expected discounted regulatory cost associated with the command-and-control regime (when they occur), and D* is the expected discounted environmental damages from pollution, all when the firm chooses its profit-maximizing emissions level x*(r,qR,T). The regulatory cost in an enforcement period is cR(r), which increases in r, and the regulatory cost in a non-enforcement period is zero. Consequently, the sum of the expected discounted regulatory costs is

T  C = F k ( x* , qR )  β s cR ( r ) + β T +1C  ,  s =1 

(

)∑

(4)

which can be solved for C to yield

C * ( r , qR , T ) =

cR ( r ) cR ( r ) . − 1 − β 1 − β + ( β − β T +1 ) F ( k ( x * , qR ))

(5)

The first term is the cost of permanent regulation with parameter r, and the second term is the cost savings from not regulating every period. The environmental damage depends on the amount of emissions made by the firm, which equal xR in enforcement periods and x*(r,qR,T) in non-enforcement periods. The single-period damage cost is given by the increasing, convex function ∆(xt). Letting ∆R(r) denote the damage in an enforcement period, the sum of the expected discounted damage costs is

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T  D = ∆ ( x* ) + 1 − F k ( x * , qR )  β D + F k ( x* , qR )  β s ∆ R ( r ) + β T +1 D  .    s =1 

(

)

(

)∑

(6)

Rearranging yields

D * ( r , qR , T ) =

∆ R (r ) ∆( x* ) − ∆ R ( r ) + . 1 − β 1 − β + ( β − β T +1 ) F ( k ( x* , qR ))

(7)

The first term is the sum of the damages that would occur if every period were an enforcement period with regulatory parameter r, and the second term captures the additional environmental damages that occur during non-enforcement periods. Substituting (2), (5), and (7) into (3) yields

W ( r , qR , T ) =

π R ( r ) − cR ( r ) − ∆ R ( r ) (π ( x* ) − π R ( r )) + cR ( r ) − ( ∆ ( x* ) − ∆ R ( r )) + . 1− β 1 − β + ( β − β T +1 ) F ( k ( x* , qR ))

(8)

The first term is the value that the regulator’s objective function obtains if the regulator institutes a standard command-and-control regime with regulatory parameter r beginning in the first period. The second term is the addition to the objective function from using regulatory threats instead. There are two potential sources of gain, namely increased firm profit in non-enforcement periods and the avoidance of regulatory costs in nonenforcement periods, and one potential loss, the increased environmental damages during non-enforcement periods. Consequently, the second term in (8) depicts the tradeoff faced by the regulator when it chooses its policy parameters.

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A subgame perfect equilibrium of the game is a combination of r, qR, and T which maximizes W, combined with the firm’s best response in non-enforcement periods, x*(r,qR,T). In equilibrium there will be non-enforcement phases and command-and~ control phases, with the latter occurring if the realization of θ is below k(x*(r,qR,T),qR).

To ensure that some regulation is desirable, so that the problem is interesting, we impose the following restriction. Let xˆ be the level of emissions that maximizes the firm’s single-period non-enforcement profit function π, so that xˆ is the amount of effluent emitted each period by a permanently-unregulated firm. In the same vein, let rˆ denote the value of the regulatory parameter that maximizes the regulator’s single-period objective function, πR(r) – cR(r) – ∆R(r). We assume throughout the remainder of the paper that

∆ ( xˆ ) − ∆ R ( rˆ) > π ( xˆ ) − π R ( rˆ) + cR ( rˆ) .

(9)

The left-hand-side is the benefit to the regulator from imposing (single-period) optimal command-and-control regulation on an otherwise-unregulated firm, with the benefit arising from the reduction in pollution damage. The right-hand-side is the cost to the regulator from imposing (single-period) optimal regulation, with the costs coming from reduced profit and from regulatory costs. Condition (9) states that in a static setting the regulator prefers some regulation to none.

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4. The firm’s behavior At the beginning of the game the regulator sets r, the strictness of the regulation during command-and-control phases, qR, the ambient environmental quality level that triggers enforcement phases, and T, the length of enforcement phases. The first period is a non-enforcement period, and so the firm chooses its emissions level x to maximize its payoff function V given in (2). The first-order condition is

V '( x ) =

π '( x ) [π ( x ) − π R ( r )]( β − β T +1 ) f ( k ( x, qR )) ∂k L

−

L2

⋅

∂x

= 0,

(10)

where L = 1 − β + (β − βT+1)F(k(x,qR)) > 0, which is the denominator of the last term in (2). The first-order condition simplifies to

π '( x ) =

[π ( x ) − π R ( r )] ∂k ( β − β T +1 ) f ( k ( x, qR )) . T +1 ∂x 1 − β + ( β − β ) F ( k ( x, qR ))

(11)

The left-hand term is the additional profit the firm makes in a single non-enforcement period by increasing emissions by a (marginal) unit. The right-hand term is the expected loss in profit caused by the increased likelihood of triggering a command-and-control episode. Before getting to our first result, differentiate (11) with respect to r and rearrange to get dx * 1 dπ R ∂k =− ( β − β T +1 ) f ≤ 0. dr V " dr ∂x

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(12)

The second-order condition for a maximum guarantees that V” < 0.

Also, by

construction, dπR/dr < 0 and ∂k/∂x > 0. Since f is a (non-negative) density function, it follows that dx*/dr ≤ 0, so that the firm emits less when regulation is more restrictive during command-and-control episodes. An examination of equation (11) leads to our first result.

Proposition 1. Let xˆ be the level of emissions that maximizes the firm’s single-period non-enforcement profit function, π. In the absence of a threat (i.e. r is set so that xR = xˆ , or T = 0, or qR = 0), the firm chooses x * = xˆ . If the regulator sets r high enough that xR < xˆ , sets qR so that F (k ( xˆ , q R )) > 0 , and sets T ≥ 1 then the firm chooses x * ∈ ( x R , xˆ ) .

Proof. In the absence of a threat, it follows from (10) that V’(x) = π’(x)/L, and xˆ maximizes π.

To prove the second statement, by construction π ' ( xˆ ) = 0 and

π ( xˆ ) − π R > 0 . Consequently, when the left-hand side of (11) is zero the right-hand side is strictly positive and when the right-hand side of (11) is zero the left-hand side is strictly positive. The value x* that satisfies (11) must therefore be between xR and xˆ .

□

Proposition 1 contains two results. First, the firm reduces emissions from its unregulated optimum during non-enforcement periods, but it does not reduce them as much as it would during enforcement periods. In other words, the threat of command-and-control regulation leads firms to reduce emissions. Second, the absence of a threat leads to no emissions reduction. Thus, voluntary environmental agreements between the firm and the regulator can only lead to equilibrium cutbacks in emissions if the agreement is backed up by the threat of regulation.

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5. The regulator’s decision

The regulator correctly forecasts the firm’s response to its decisions when it chooses the values of r, qR, and T to maximize W. By setting r high enough to get xR < xˆ the regulator can get the firm to reduce emissions as part of the voluntary agreement, and it can use its three parameters as tools for fine-tuning the firm’s emissions.

The

appropriate question, though, is whether or not the regulator prefers the voluntary agreement over traditional command-and-control regulation. To address this question we restrict attention to situations in which the regulator can only commit to permanent regulation, that is, T = ∞. Given that regulation is permanent once it begins, it is natural to also assume that the regulator can only commit to r = rˆ , the optimal regulatory parameter in a one-period version of the game. The reason is that in the first enforcement period the regulator’s welfare function becomes [πR(r) – cR(r) – ∆R(r)]/[1 – β], and rˆ maximizes πR – cR – ∆R. Consequently, in the first enforcement period the regulator would choose to set r = rˆ if it could, and it is natural to assume that it cannot commit to any other value of the regulatory parameter but this one. With the restrictions on r and T, the firm’s profit function becomes

V ( x) =

π R ( rˆ) π ( x ) − π R ( rˆ) + . 1 − β 1 − β [1 − F ( k ( x, qR ))]

(13)

The regulator’s objective function in (8) can be reduced to

W ( qR ) =

π R ( rˆ) − cR ( rˆ) − ∆ R ( rˆ) [π ( x* ) − π R ( rˆ)] + cR ( rˆ) − [ ∆( x* ) − ∆ R ( rˆ)] + . 1− β 1 − β [1 − F ( k ( x* , qR ))]

16

(14)

Now consider what happens when the regulator institutes command-and-control regulation beginning in the first period. It would set the regulatory parameter at r = rˆ , and the objective function takes the value

WR =

π R ( rˆ) − cR ( rˆ) − ∆ R ( rˆ) . 1− β

(15)

We wish to explore conditions under which regulatory threats are the optimal policy for the regulator. This entails finding conditions under which the regulator prefers to threaten regulation in the first period rather than actually impose regulation in the first period, and conditions under which it prefers to threaten regulation in the first period rather than leave the firm completely unregulated. To begin, subtract WR from W(qR) to get

W ( qR ) − WR =

[π ( x* ) − π R ( rˆ)] + cR ( rˆ) − [ ∆( x * ) − ∆ R ( rˆ)] . 1 − β + β F ( k ( x* , qR ))

(16)

Since the denominator is positive, the difference is positive as long as

cR ( rˆ) + π ( x* ) − π R ( rˆ) > ∆( x* ) − ∆ R ( rˆ).

(17)

The left-hand side is the single-period benefit from not regulating, as the regulator saves the cost cR and the firm’s profit increases by π(x*) – πR. The right-hand side is the singleperiod cost of not regulating, with damages increasing by ∆(x*) – ∆R. So, the regulator

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prefers the threat of regulation to actual regulation as long as (17) holds. This proves the following proposition.

Proposition 2. Suppose that the regulator can only commit to permanent regulation with *

r = rˆ . If there exists a value of qR such that x (qR) satisfies (17) then the regulator prefers

threatening regulation over a strict command-and-control regime beginning in the first period.

The assumption that there exists a value of qR for which (17) holds is non-trivial. Note that if qR = ∞, so that regulation is triggered after the first period no matter what the firm does, the firm’s optimal response is x1* = xˆ . Also, if qR = 0, so that regulation is never triggered, the firm also sets xt * = xˆ in every period, and consequently x*(qR) is not monotone. Furthermore, according to expression (9), condition (17) cannot hold when * ** x* = xˆ . Since x (qR) is not monotone, by continuity it achieves some minimum x with

corresponding environmental standard qR**. We can explore whether or not (17) holds when the environmental standard is set at qR**. Combining (9) and (17) yields

∆ ( xˆ ) − ∆ R ( rˆ) > π ( xˆ ) − π R ( rˆ) + cR ( rˆ) ≥ π ( x* ) − π R ( rˆ) + cR ( rˆ) > ∆ ( x* ) − ∆ R ( rˆ).

(18)

The first inequality comes from (9), the second inequality arises from the fact that xˆ maximizes π, and the third inequality comes from (17). From (18) it is clear that for both (9) and (17) to hold, per-period damages must decrease faster as x falls from xˆ than per-

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period profits do, that is, ∆( xˆ ) − ∆ ( x* ) > π ( xˆ ) − π ( x* ). This condition holds, for example, if ∆’(x) > π’(x) for all x in [ x** , xˆ ] , which is reasonable since damages are typically assumed to be convex while profits are concave with π’ near zero when x is near xˆ . The question remains, though, whether or not changes in qR can induce x* to fall enough for (17) to hold so that the results of the proposition apply. The lowest x* falls is x**, and if x* is sufficiently responsive to changes in qR, x** is sufficiently below xˆ for (17) to hold. We now turn attention to conditions under which the regulator prefers to threaten regulation in the first period rather than leave the firm completely unregulated. If the regulator never regulates, setting qR = 0, the firm sets x = xˆ , the emissions level that maximizes single-period profit. The regulator’s objective function reduces to

W (0) =

π ( xˆ ) − ∆( xˆ ) . 1− β

(19)

Differentiating W(qR) and evaluating at qR = 0 yields

W '(0) =

π '( xˆ ) − ∆ '( xˆ ) dx* ⋅ dqR 1− β

 ∂k ∂k dx*  (π ( xˆ ) − π R ( rˆ)) + cR ( rˆ) − ( ∆( xˆ ) − ∆ R ( rˆ)) − + ⋅ β f ( k ( xˆ ,0))  . 1− β  ∂qR ∂x dqR 

(20)

Since π '( xˆ ) = 0 we know that [π '( xˆ ) − ∆ '( xˆ )]/[1 − β ] < 0 . Furthermore, by assumption f(k(x,0)) = 0 for all x, and therefore the second term in (20) is zero. Further consideration of expression (20) leads to the following result.

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Proposition 3. Suppose that the regulator can only commit to permanent regulation with *

r = rˆ . If dk/dqR = ∂k/∂qR + (∂k/∂x)(∂x /∂qR) > 0, then the regulator prefers threatening

regulation to never regulating. Proof. It is enough to show that W’(0) > 0. Since π '( xˆ ) = 0 and f(k(x,0)) = 0 for all x, it follows from (20) that W ' (0) = [∆' ( xˆ ) /(1 − β )] ⋅ [dx * / dq R ] , and therefore we can establish that W’(0) > 0 if we can show that dx*/dqR < 0 when qR = 0. Since x* maximizes V(x), the first-order condition is

V '( x ) =

[π ( x ) − π R ( rˆ)]β f ( k ( x, qR )) ∂k π '( x ) − ⋅ = 0. ∂x 1 − β + β F ( k ( x, qR )) [1 − β + β F ( k ( x, qR ))]2

(21)

From this one can compute that

 π '( x * ) β f ∂k [π ( x * ) − π R ( rˆ)]β 2 f 2 ∂k ⋅ ∂k ∂qR ∂qR ∂x dx* 1  =− ⋅ − + 2 3 dqR V "( x )  [1 − β + β F ] [1 − β + β F ]  [π ( x * ) − π R ( rˆ)]β f ' ∂k ⋅ ∂k  ∂qR ∂x  . − 2  [1 − β + β F ] 

(22)

The first two terms inside brackets sum to zero because of the first-order condition, and therefore

[π ( x* ) − π R ( rˆ)]β dx* ∂k ∂k 1 = ⋅ ⋅ f '⋅ ⋅ . 2 ∂qR ∂x dqR V "( x ) [1 − β + β F ]

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(23)

The second-order condition for a maximum guarantees that V” < 0, and by construction both ∂k/∂qR and ∂k/∂x are positive. Finally, we have assumed that f’(k(x,qR)) > 0 when qR = 0, and so dx*/dqR < 0.

□

To understand the requirement that dk/dqR > 0, recall that k(x,qR) is the realization of the random variable that yields environmental quality qR when emissions are x. If the decrease in x following an increase in qR does not fully offset the increase in qR, then dk/dqR = ∂k/∂qR + (∂k/∂x)(∂x*/∂qR) is positive, so that when qR increases a more favorable environmental shock is required to meet the constraint even after the firm reduces emissions. The requirement for the regulator to prefer threats to never regulating, then, is that any increase in the ambient environmental standard is not fully offset by cutbacks in emissions.

6. Conclusions

In this paper we have constructed a model that fits the following scenario. A regulator agrees not to regulate a firm unless the ambient environmental quality deteriorates too much, in which case it undertakes command-and-control regulation for a set period of time. As a result of the agreement, the firm cuts back its emissions to avoid the costly command-and-control regulation. This model fills a gap in the literature on voluntary environmental agreements by endogenizing the firm’s emissions decision. Existing papers assume that the firm obeys any agreement it makes. This paper, in contrast, assumes that the firm abides by an agreement only if it is in its best interest to do so.

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Besides providing a model of how firms behave once they have entered into voluntary environmental agreements, the paper also sheds light on how as-yet unregulated firms behave when they face an implicit threat of future regulation. A firm knows that if the environmental quality ever becomes bad enough the regulator will take an action. Given its beliefs about the form of this action and the level of environmental quality that would trigger it, a firm decides how much to cut back pollution to forestall impending regulation. The model relies on the regulator using an “ambient environmental standard” instead of an emissions standard. Once the ambient environmental standard is violated, for whatever reason, a command-and-control phase is triggered. This is realistic. In practice, when the ambient environmental quality deteriorates too much, political pressure for regulation mounts, and the regulator responds with a command-and-control regime. The regulator’s equilibrium strategy also requires it to initiate a command-andcontrol regime even if the deterioration in environmental quality is caused by something other than firm emissions. This, too, is realistic. For example, the Korean Ministry of Environment makes special audits of waterways after particularly heavy rains, and makes regulations stricter for a brief time if the water quality is deemed too low. The model presented here applies only to problems that are stationary, that is, problems in which discharges are transient and their impact does not accumulate over time. Many cases fit this scenario. The EPA’s voluntary 33/50 program, for example, was designed to reduce emissions of 17 airborne toxins, and because of wind patterns, precipitation, and so on, many effects of these toxins are temporary. On the other hand,

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global warming is the outcome of accumulated emissions, and further research can address the issue of voluntary agreements when the impact accumulates.

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