Pollution taxes when firms choose technologies.

1. Introduction

In the past several years, considerable emphasis has been placed on stimulating the adoption of "clean" technologies by polluting firms (National Research Council 1996; Hart 1997). In the United States, an Environmental Technology Initiative was established in 1993 to promote the development and adoption of innovative environmental technologies. Internationally, development and adoption of clean technologies is viewed as instrumental in reducing greenhouse gas emissions from both developed and developing countries (Watson, Zinyowera, and Moss 1996).

This emphasis on technology has been accompanied by increased use of taxes to limit industrial pollution and achieve environmental goals. Most European and many Asian countries now use pollution taxes to control a range of air and water pollutants. Such taxes have long been advocated by economists and have been studied extensively in the literature on externalities and environmental policy design. A primary focus of this literature has been on examining the properties of pollution taxes in a range of settings that capture, at least partially, the complexities of real-world policy choices. Complexities considered include asymmetric information about firms' control costs, cheating by firms, imperfections in the product market in which firms operate, and nonconvexities in the damages from pollution. (1)

With few exceptions, neither this body of work nor the larger theoretical literature on environmental policy design explicitly considers firms' technology choices. The bulk of the literature simply assumes that a firm's costs of reducing pollution emissions are given by a smooth convex function, effectively implying that the firm has available a continuum of technologies for reducing emissions. This assumption, though analytically convenient, is not necessarily consistent with reality. Firms' options for reducing emissions are typically discrete, as is observed in the applied literature on pollution control. For example, Palmer et al. (1995) and Fullerton, McDermott, and Caulkins (1997) emphasize the discrete nature of the abatement technology choices available to electric utilities.

Accordingly, in this paper we develop and analyze a model in which a regulated firm has a discrete choice of technologies with which to reduce its emissions. The technologies differ in terms of the trade-off they embody between the fixed and marginal costs of emissions reductions. There is a "cleaner" technology that has low marginal costs but high fixed costs and a "dirtier" one for which the opposite is true. This trade-off between marginal and fixed costs is not unusual: Firms subject to environmental regulation are often faced with a choice of either modifying/replacing their existing production process with a new, inherently cleaner process or keeping their existing production process and simply adding on pollution treatment equipment. (2) Modifying or replacing a production process entails larger outlays on both equipment and retraining of personnel (Kemp 1997, pp. 11-4, 103-4), leading to higher fixed costs.

Given this discrete choice of technologies, we consider a firm's behavior when facing a regulator that has perfect information. The regulator imposes the typical linear emissions tax to achieve the socially optimal emissions level. We find that there is a range of emissions levels the regulator is unable to achieve, regardless of the magnitude of the tax it imposes. The first-best emissions level could well lie in this range; thus, the first-best outcome may not be attainable. This finding is rather surprising since it arises from a seemingly minor modification of the standard model of a regulated firm.

These results are obtained assuming the traditional sequence of play between the firm and the regulator: The regulator moves first by announcing the tax rate, and the firm then follows, choosing its technology and emissions level (Cropper and Oates 1992). Making explicit the firm's technology choice allows for an alternative sequence of play: The firm could now move first by choosing a technology, with the regulator then announcing the tax rate. In some cases, this sequence of play better reflects the actual behavior of firms and regulators. For example, Stavins (2002) points out that some U.S. oil refineries adopted lead emissions-reducing technologies before a ban on lead emissions was instituted by the U.S. Environmental Protection Agency. Similarly, Sterner (2001, chap. 2) notes that in Sweden, paper mills adopted chlorine-free bleaching technologies in anticipation of a possible tax on chlorine emissions. The tax was imposed two years later. More generally, Stavins (1998) argues that as a matter of marke t strategy, firms may have an incentive to invest in cost-saving technologies when faced with the prospect of price regulation.

Given the usual advantages of moving first, we would expect the regulator to be in a better position to achieve the first-best outcome when it, rather than the firm, moves first. However, we find that this is not necessarily true. Under some conditions, the first-best outcome can be attained only when the firm moves first. Thus, allowing the firm to move first can be socially desirable, bringing forth the possibility that firms should be encouraged to take the lead in regulatory settings.

As noted previously, the theoretical literature on environmental policy design has focused little attention on firms' technology choices. The primary exception is a small but growing literature that examines the incentives for innovation provided by alternative policy instruments for controlling pollution. Recent contributions to this literature include those by Biglaiser and Horowitz (1995), Laffont and Tirole (1996), Parry (1998), and Denicolo (1999). (3) This literature differs from our work in at least one of two respects. First, it considers innovation rather than adoption, and second, like most of the theoretical literature, it assumes that firms have a continuum of technology choices. An exception is a paper by Requate (1995) that explicitly considers technology adoption when firms have discrete technology choices. However, he does not identify the results we present here. He focuses on the ability of an emissions tax to induce the optimal pattern of technology adoption when there are multiple firms an d the output market effects of technology adoption are recognized.

A further difference between our work and the existing literature is that the latter restricts attention to the traditional sequence of play--the regulator leading and the firm following. (4) This is also true of an early, and now classic, paper on externalities by Turvey (1963) that is closely related to our work. Turvey considers a setting in which an agent has a choice between two activities (or technologies) both of which generate an externality. The activities yield different marginal benefits to the agent and impose different marginal damages on society. Turvey shows that a Pigouvian tax does not achieve the first-best outcome in this setting. Turvey's model differs from ours in two respects: (i) We assume that the two technologies differ only in terms of abatement costs and not marginal damages, and (ii) we include fixed costs, which are important to technology adoption, whereas Turvey ignores them. Turvey's assumption that marginal damages differ for the two activities is critical to his results. It c an be shown that if marginal damages were the same, a Pigouvian tax would achieve the first-best outcome in his model. (5)

Both in Turvey's model and in ours, the regulator can unambiguously achieve the first-best outcome if it is able to impose a suitable nonlinear tax or dictate the technology the firm uses. We restrict attention to taxes that are linear in emissions (i.e., a constant amount is charged per unit of pollutant emitted) because the emissions taxes in place are typically of this form (Sterner 2001, chap. 5; Stavins 2002). The reliance on simple linear taxes is due in large part to the administrative difficulties in implementing more complex tax schemes (Sterner 2001, chap. 3).

In addition, we assume that the regulator is unable to dictate the technology the firm uses. This could be due to legislative restrictions on the nature or extent of regulatory intervention in the firm's decisions. In the United States, this is true under existing policies even when environmental regulations are technology based. For example, the Clean Water Act calls for emissions standards that are based on "best available technologies." Yet firms are free to choose other control technologies or other means (such as process or input changes) to comply with the standards (Freeman 1990).

The rest of the paper is organized as follows. In the next section we present our basic model. In section 3, we consider the traditional setting in which the regulator moves first. The alternative setting in which the firm moves first is examined in section 4. A comparison of the results for these two settings leads to a discussion of first-mover advantages in section 5. The final section contains our conclusions.

2. Basic Model and Assumptions

Before describing our model, it would be appropriate to comment on our use of the term "technology." We use the term in its vernacular sense to refer to a specific option available to a firm for reducing its pollutant emissions. An example of such an option is installation of waste treatment equipment. The term could be used instead in a more formal sense to refer to the entire set of options available for reducing emissions. Rather than referring to technologies, we would then refer to abatement options within an existing technology. Our choice of terminology is not critical to our analysis. What is critical is our assumption that there is a discrete set of abatement options rather than the conventionally assumed continuum of options.

Firm

The firm's technology choice is motivated by a tax on its pollutant emissions x. The firm can use one of two technologies to reduce (or abate) its emissions; we label these technology 1 and technology 2. For each technology, the total costs of abating emissions are separated into fixed costs and variable costs: (6)

[C.sub.i](x) [equivalent to] [K.sub.i] + [V.sub.i](x). (1)

The variable cost function, [V.sub.i](*), is strictly convex, with

0 < [V.sub.i](x) < [infinity], -[infinity] < [V'.sub.i](x) < 0 for x [member of] [0, [X.sub.i]); [V.sub.i](x) = 0, [V'.sub.i](x) = 0

for x [member of] [[X.sub.i], [infinity]), (2)

where [X.sub.i] is the firm's maximum emissions level when using technology i. Note that the firm's variable abatement costs fall to zero when emitting this amount.

The technologies differ in that technology 1 has lower marginal abatement costs than technology 2:

-[V'.sub.1](x) < -[V'.sub.2](x) for x [member of] [0, [X.sub.2]). (3)

This, together with Equation 2, implies that technology 1 has lower maximum emissions than technology 2: [X.sub.1] < [X.sub.2]. The inequality in Equation 3 also implies that technology 1 has lower variable abatement costs than technology 2: [V.sub.1](x) < [V.sub.2](x) for x [member of] [0, [X.sub.2]). Technology l's drawback is that it has higher fixed costs:

[K.sub.1] > [K.sub.2]. (4)

We can think of technology 1 as being a "cleaner" technology, such as a new, less polluting production process, and technology 2 as being a "dirtier" technology, such as pollution treatment equipment that is added to the firm's existing production process. (7)

The differences in fixed and variable costs for the two technologies are assumed to be such that the cleaner technology is cheaper at low emissions levels, in particular, at zero emissions:

[C.sub.1](0) [equivalent to] [V.sub.1](0) < [K.sub.2](0) [equivalent to] [C.sub.2](0). (5)

The dirtier technology is cheaper at high emissions level, in particular, at the maximum level, [X.sub.2]: [C.sub.1]([X.sub.2]) [equivalent to] [K.sub.1] > [K.sub.2] [equivalent to] [C.sub.2]([X.sub.2]) given Equation 4.

For a given technology, the firm's total costs are obtained by adding the firm's emissions tax payments to its abatement costs. Letting t denote the emissions tax rate, the firm's total costs are

[C.sub.i](x) + tx. (6)

The emissions level that minimizes these costs is denoted [x.sup.*.sub.i](t). If this emissions level is positive, it satisfies the following first-order condition (of the firm):

-[V.sub.i]([x.sup.*.sub.i]) = t. (7)

If the tax rate is sufficiently high, [x.sup.*.sub.i](t) will equal zero--the firm will choose to stop emitting the pollutant. Let [[tau].sub.i] denote the lowest tax rate for which this is true. The assumptions about variable costs in Equations 2 and 3 imply that [[tau].sub.i] is unique and higher for the dirtier technology:

[x.sup.*.sub.i](t) = 0 for t [greater than or equal to] [[tau].sub.i], [[tau].sub.2] > [[tau].sub.1].

The assumptions about variable costs also imply that for any tax in the interval [0, [[tau].sub.2]), the firm emits strictly more of the pollutant when it employs the dirtier technology (see Appendix A for a proof):

[x.sup.*.sub.2](t) > [x.sup.*.sub.1](t) for t [member of] [0, [[tau].sub.2]). (8)

Finally, as is readily established, for a given technology, the firm's emissions fall as the tax rate rises:

d[x.sup.*.sub.i](t)/dt {< 0 if t < [[tau].sub.i]/= 0 if t [greater than or equal to] [[tau].sub.i].

Hence, the firm's emissions are at their maximum level when the tax is set to zero, [x.sup.*.sub.i](0) = [X.sub.i].

The firm's indirect total cost function can be obtained by substituting its emissions choice function, [x.sup.*.sub.i](t), into Equation 6:

[TC.sub.i](t) [equivalent to] [C.sub.i]([x.sup.*.sub.i](t)) + t[x.sup.*.sub.i].

Application of the envelope theorem reveals that this function is strictly concave and strictly increasing when t < [[tau].sub.i] and is linear with a zero slope when t [greater than or equal to] [[tau].sub.i]:

[TC'.sub.i](t) [equivalent to] [x.sup.*.sub.i](t) {> 0 if t < [[tau].sub.i]

= 0 if t [greater than or equal to] [[tau].sub.i],

[TC".sub.i](t) [equivalent to] d[x.sup.*.sub.i]/dt {< 0 if t < [[tau].sub.i]

Figure 1 depicts the indirect cost function associated with each technology. The hashed segments of the two curves constitute the lower envelope of costs; this gives the firm's minimum total costs as a function of the tax rate. Figure 1 indicates that the following relationship holds:

= 0 if t [greter than or equal to] [[tau].sub.i]. (9)

LEMMA 1. The firm's minimum total costs are equal for the two technologies at the unique tax rate t. For tax rates above t, total costs are lower for the cleaner technology, and for tax rates below t, they are lower for the dirtier technology:

[TC.sub.1](t) >/< [TC.sub.2](t) [left and right arrow] t </> t. (10)

The uniqueness of t is demonstrated in Appendix B. Examining Figure 1, it can be seen that the position of t is influenced by the relative magnitude of fixed costs. The larger is the difference in fixed costs for the two technologies, the larger is the value of t.

Regulator

Turning to the regulator, we make the usual assumption that the regulator minimizes social costs. For a given technology, social costs consist of the firm's abatement costs and the damages from its emissions, which are captured by a smooth, strictly increasing, convex function, D(x). (8) Following convention, the tax payments made by the firm are excluded from social costs because they are assumed to be transfers. Thus, social costs are given by

[SC.sub.i](x) [equivalent to] [C.sub.i](x) + D(x). (11)

The strict convexity of [C.sub.i](*), together with the convexity of D(*), implies that [SC.sub.i](*) is strictly convex for each technology. This in turn implies that, for each technology, there is a unique emissions level that minimizes social costs:

[s.sup.*.sub.i] [equivalent to] argmin [SC.sub.i](x).

To ensure that this emissions level is strictly positive for both technologies, we assume that D'(0) < [absolutely value of [V'.sub.1](0)]. Then, [s.sup.*.sub.i] is the emissions level at which marginal abatement costs are equal to marginal damages: -[V'.sub.i]([s.sup.*.sub.i]) = D'([s.sup.*.sub.i]), i = 1, 2. This emissions level must be lower for the cleaner technology given its lower marginal abatement costs (see Appendix C):

0 < [s.sup.*.sub.1] < [s.sup.*.sub.2].

To render our problem interesting, minimum social costs, [SC.sub.i](s), must differ for the two technologies. We will assume that they are lower for the cleaner technology:

[SC.sub.1]([s.sup.*.sub.1]) < [SC.sub.2] ([s.sup.*.sub.2]).

Thus, the "first-best" outcome would occur when the firm adopts the cleaner technology and emits [s.sup.*.sub.1]. We will write this outcome as {l, [s.sup.*.sub.1]}. The outcome at which the firm adopts the dirtier technology and emits [s.sup.*.sub.2] will be referred to as the "second-best" outcome and written as {2, [s.sup.*.sub.2]}. It is, of course, just one of many possible second-best outcomes. We will encounter another possible second-best outcome later.

Since the firm's costs are most conveniently expressed as a function of the tax rate, it is useful to restate social costs in terms of the tax rate and to identify the tax rates corresponding to [s.sup.*.sub.1] and [s.sup.*.sub.2]. This is accomplished by substituting the firm's emissions choice function, [x.sup.*.sub.i](t), for x in Equation 11. Because of the monotonicity of [x.sup.*.sub.i](t), there is a unique tax rate, [t.sup.*.sub.i], which induces the firm to emit [s.sup.*.sub.i]: [x.sup.*.sub.i]([t.sup.*.sub.i]) = [s.sup.*.sub.i], i = 1, 2. Accordingly, the first-best and second-best outcomes can be written in terms of tax rates as {1, [t.sup.*.sub.1]} and {2, [t.sup.*.sub.2]}, respectively.

The relationship between [s.sup.*.sub.1] and [s.sup.*.sub.2] and [t.sup.*.sub.1] and [t.sup.*.sub.2] can be characterized with the aid of Figure 2, which depicts marginal damages and marginal abatement costs for the two technologies. The socially optimal emissions level for each technology is given by the intersection of the marginal damage curve and the relevant marginal abatement cost curve. The emissions level chosen by the firm is given by the intersection of the tax rate and the relevant marginal abatement cost curve (see Eqn. 7). Inspecting the figure, one can verify that the following relationships hold between [t.sup.*.sub.1] and [t.sup.*.sub.2]:

LEMMA 2. Given [s.sup.*.sub.1] < [s.sup.*.sub.2], the relative magnitude of [t.sup.*.sub.1] and [t.sup.*.sub.2] depends on the curvature of the damage function. If the damage function is strictly convex (D" > 0), [t.sup.*.sub.1] < [t.sup.*.sub.2]; and if it is linear (D" = 0), [t.sup.*.sub.1] = [t.sup.*.sub.2].

3. Game with Regulator Moving First

We are now ready to begin our analysis of the interaction between the firm and the regulator. As noted earlier, the traditional assumption in the literature on environmental regulation is that the regulator moves first. When using an emissions tax, it does so by announcing the tax rate, cognizant of the firm's response to the tax. The firm then follows, choosing its emissions level. We begin by considering this traditional sequence of play. The setting we examine differs from the traditional one only in that the firm must choose both its emissions level and its technology. As is usual, we analyze the game backwards, starting with the firm's response.

Firm's Best Response

The firm's best response to the announced tax rate is easily derived from Lemma 1. When the tax rate imposed by the regulator is greater than t, the firm chooses the cleaner technology, and when it is smaller than t, the firm chooses the dirtier technology. Thus, the firm's best response, in terms of its technology choice, is simply

i = {1 if t [greater than or equal to] t

2 if t < t. (12)

In writing Equation 12, we assume, for simplicity, that the firm picks the cleaner technology (which is the technology preferred by the regulator) when it is indifferent between the two. Having chosen a technology, the firm's emissions level is given by the associated emissions choice function, [x.sup.*.sub.i](t), where i = 1 or 2.

Figure 2 helps in understanding the firm's best response. Let [x.sup.*.sub.i] denote the firm's emissions level when it employs technology i and faces a tax of x, that is, [x.sup.*.sub.i] = [x.sup.*.sub.i](t). The best response in Equation 12 implies that when the tax rate chosen by the regulator is in the interval [0, t), the firm's emissions lie in the interval [[X.sub.2], [x.sup.*.sub.2]). If the tax rate is greater than or equal to t that is, t [member of] [t, [[tau].sub.2]], the firm's emissions are in the interval [[x.sup.*.sub.1], 0].

The inequality in Equation 8 implies that [x.sup.*.sub.2] > [x.sup.*.sub.1]; that is, when facing a tax of t, the firm's emissions are higher when it employs the dirtier technology rather than the cleaner one. This, together with the previous observations, implies that there is a discontinuity in emissions in the vicinity of t: specifically, the firm cannot be induced to emit in the interval ([x.sup.*.sub.1], [x.sup.*.sub.2]]. As we shall see, this discontinuity is significant. Note, however, that it only exists ex ante: Once the firm has adopted a technology, the firm's emissions become a continuous function of the tax rate, and all emissions levels can be attained with a suitably chosen tax.

Regulator's Choice of Tax Rate and Nash Equilibria

We now turn to the regulator's decision problem. The solutions to this problem yield the Nash equilibria of the game. The problem is one of choosing the tax rate that minimizes social costs, subject to the firm's best response:

[min.sub.t] [SC.sub.i]([x.sup.*.sub.i]) s.t. (12). (13)

Given Equation 12, it is easily seen that the first-best outcome {1, [t.sup.*.sub.1]} is the solution to this problem if and only if [t.sup.*.sub.1] [greater than or equal to] t. Referring to Figure 2, in this case, by announcing a tax of [t.sup.*.sub.1], the regulator would induce the firm to choose the cleaner technology and emit [s.sup.*.sub.1] = [x.sup.*.sub.1] ([t.sup.*.sub.1]). If [t.sup.*.sub.1] < t, the first-best outcome cannot be achieved because the firm would not choose the cleaner technology in response to a tax of [t.sup.*.sub.1]; it would choose the dirtier technology instead. In this case, the second-best outcome {2, [t.sup.*.sub.2]} could be the solution to Equation 13. This would be true only if [t.sup.*.sub.2] < t. Given Equation 12, the firm would then choose the dirtier technology in response to the tax rate of [t.sup.*.sub.2]. As explained in the following, even though the second-best outcome is attainable in th0is case, it may not be the solution to Equation 13 because it may be dominat ed by another feasible outcome. Finally, if [t.sup.*.sub.2] [greater than or equal to] t, the second-best outcome is unattainable because the firm would pick the cleaner technology when faced with [t.sup.*.sub.2].

Lemma 2 indicates that if the damage function is strictly convex, then [t.sup.*.sub.1] < [t.sup.*.sub.2]. If t lies between these two tax rates, that is, [t.sup.*.sub.1] < t [less than or equal to] [t.sup.*.sub.2], the previous observations imply, rather strikingly, that neither the first-best nor the second-best outcome can be attained. This possibility is illustrated in Figure 2: The first-best outcome is unattainable because [t.sup.*.sub.2] < t, and the second-best outcome is unattainable because [t.sup.*.sub.2] > t. We cannot rule out the possibility of t lying in the interval specified since t is determined by the firm's total costs, whereas [t.sup.*.sub.1] and [t.sup.*.sub.2] are determined by marginal abatement costs and marginal damages. (9)

For this case, the obvious question is, What is the equilibrium of the game when neither the first-best nor the second-best outcome is attainable? As shown in Appendix D, the equilibrium is a corner solution, {1, t}, at which the regulator announces a tax of t. The firm responds by picking the cleaner technology, which is the technology preferred by the regulator, and emitting [x.sup.*.sub.1] = [x.sup.*.sub.1](t). (Recall that t is the lowest tax rate that induces the firm to pick the cleaner technology.) This corner solution can be viewed as another second-best outcome; however, to avoid confusion, we shall restrict use of the term second-best outcome to refer to {2, [t.sup.*.sub.2]}.

The regulator's inability to achieve either the first-best or second-best outcome occurs only if the damage function is strictly convex. With a linear damage function, [t.sup.*.sub.1] = [t.sup.*.sub.2] = D' and either the first-best or the second-best outcome is achievable, depending on the relative magnitude of t and D'. If t [less than or equal to] D', the first-best outcome is the equilibrium of the game since t [less than or equal to] [t.sup.*.sub.1] holds then. If t > D', the first-best outcome is not attainable; however, the second-best outcome is because t > [t.sup.*.sub.2] holds then.

Although it is tempting to conclude that the second-best outcome is the equilibrium of the game when it is attainable (and the first-best outcome is not), this is, in fact, not necessarily true, regardless of the curvature of the damage function. Instead, the corner optimum {1, t} may be the equilibrium. This possibility arises because social costs could be lower at the corner optimum than at the second-best one, that is, [SC.sub.1]([x.sup.*.sub.1]) < [SC.sub.2]([x.sup.*.sub.2]([t.sup.*.sub.2])) could hold. Using the definition of t, the difference in social costs can be written as

[SC.sub.1]([x.sup.*.sub.1]) - [SC.sub.2]([x.sup.*.sub.2]([t.sup.*.sub.2])) [equivalent to] [[SC.sub.2]([x.sup.*.sub.1]) - [SC.sub.2]([x.sup.*.sub.2]([t.sup.*.sub.2]))] + [[C.sub.1]([x.sup.*.sub.1]) - [C.sub.2]([x.sup.*.sub.1])]. (14)

The first term on the right-hand side of Equation 14 is positive since [t.sup.*.sub.2] [equivalent to] argmin [SC.sub.2]([x.sup.*.sub.2](t)), whereas the second term is negative, as shown in Appendix E. The second term captures the savings in abatement costs from having the firm adopt the cleaner technology. Equation 14 indicates that if these savings are sufficiently large, social costs are lower at the corner optimum than at the second-best one. This is consistent with intuition. The regulator would impose a tax t > [t.sup.*.sub.1], which induces the firm to emit less than [s.sup.*.sub.1], only if there are large savings in abatement costs from having the firm adopt the cleaner technology.

We summarize the previous results in the following proposition.

PROPOSITION 1. In the traditional regulatory setting in which the regulator moves first, the first-best and second-best outcomes may be unattainable when the damage function is strictly convex. (i) The first-best outcome {1, [t.sup.*.sub.1]} is the unique (Nash) equilibrium if and only if [t.sup.*.sub.1] [greater than or equal to] t. (ii) The corner optimum {1, t} is the unique equilibrium if [t.sup.*.sub.1] < t and t < [t.sup.*.sub.2]. (iii) If [t.sup.*.sub.1] < t and t [greater than equal to] [t.sup.*.sub.2], one of two equilibria arises depending on the savings in abatement costs associated with the cleaner technology. If the savings are sufficiently large, the corner optimum is the equilibrium; otherwise, the second-best outcome is the equilibrium.

The potential inability of the regulator to achieve the first-best or second-best outcomes, and the existence of the corner optimum, arise from the ex ante discontinuity in emissions identified earlier. Figure 2 shows that the magnitude of this discontinuity, [absolutely value of [x.sup.*.sub.2] - [x.sup.*.sub.1]], depends on the difference in marginal abatement costs for the two technologies. We can see from the figure that the discontinuity would increase in magnitude if the difference in marginal abatement costs were larger. This suggests that the likelihood of the first-best outcome being unattainable increases when the technologies available to the firm have very different marginal abatement costs since the interval ([x.sup.*.sub.1], [x.sup.*.sub.2]) would then increase in length.

The underlying cause of the discontinuity in emissions is nothing other than the tax payments made by the firm. These payments obviously influence the firm's choice of technology. However, to the regulator and society, the payments are transfers that should not affect the firm's technology choice. Thus, the tax payments create a divergence between the firm's technology preferences and those of the regulator. This is reflected by the presence of the abatement cost term in the corner optimum condition (Eqn. 14). If the firm considered only abatement costs when making its technology choice, the term would vanish because the firm would on its own choose the technology that offered lower abatement costs. (10)

4. Game with Firm Moving First

We have assumed thus far that the regulator is able to commit to the tax rate it announces, thereby ensuring its ability to move first. This commitment assumption is quite innocuous in the typical model of environmental regulation because there is little scope for the firm to move first: The firm only chooses its emissions level, and this is presumed to be continuously variable with no fixed/sunk costs incurred. In our model, however, because of the firm's choice of technologies, the commitment assumption is important. Without it, the corner solution identified previously would not be subgame perfect because the regulator would choose to lower the tax from t to [t.sup.*.sub.1] once the firm had adopted the cleaner technology. This would benefit the regulator since [t.sup.*.sub.1] [equivalent to] argmin [SC.sub.2]([x.sup.*.sub.1](t)) and would benefit the firm since the firm's costs fall with a lower tax.

The ability of the regulator to commit to a tax depends on the political constraints it faces and on the nature of the process by which regulations are designed and implemented. In the United States, the latter process is quite involved, making it difficult for the regulator to change a tax rate once it is promulgated. But in other settings, commitment may be more problematic. Accordingly, in this section we consider a game in which the regulator is able to commit to using a tax but not to a specific tax rate. In this alternative game, the firm moves first by adopting one of the two technologies; the regulator then announces the tax rate. (11) The credibility of the firm's commitment to its choice of technology could be questioned, but this commitment is unlikely to be problematic since commitment is commonly assured when capital is installed, as occurs here when the firm adopts a technology (see Spence 1977; Dixit 1980).

The firm's problem now is to choose the technology that minimizes the sum of its abatement costs and tax payments, [TC.sub.i](t), taking into account the best response of the regulator. The regulator's best response is straightforward: It will choose a tax rate of [t.sup.*.sub.1] if the firm adopts the cleaner technology and a tax rate of [t.sup.*.sub.2] if the firm adopts the dirtier technology, thereby achieving the first-best outcome and second-best outcome, respectively. (12)

Formally, the firm's decision problem is

[min.sub.i] [TC.sub.i](t) s.t. t = {[t.sup.*.sub.1] if i = 1

[t.sup.*.sub.2] if i = 2. (15)

Now, because the regulator issues the tax after the firm has chosen a technology, the ex ante discontinuity in emissions observed in the traditional sequence of play does not arise. However, characterizing the solution to Equation 15 is more difficult because our assumptions do not predetermine the relative magnitude of [TC.sub.1]([t.sup.*.sub.1]) and [TC.sub.2]([t.sup.*.sub.2]). Nonetheless, we can derive the following proposition:

PROPOSITION 2. When the firm moves first, the only possible (Nash) equilibria are the first-best and second-best outcomes. (i) The first-best outcome is the unique equilibrium if and only if (a) [t.sup.*.sub.2] [greater than or equal to] t or (b) [t.sup.*.sub.2] < t and [t.sup.*.sub.2] is sufficiently large, specifically, [t.sup.*.sub.2] [greater than or equal to] [t.sup.*.sub.1] + [delta], where [delta] is defined by the equality [TC.sub.1]([t.sup.*.sub.1]) = [TC.sub.2]([t.sup.*.sub.1] + [delta]). (ii) The second-best outcome is the unique equilibrium if and only if [t.sup.*.sub.2] < t and [t.sup.*.sub.2] is sufficiently small, specifically, [t.sup.*.sub.2] < [t.sup.*.sub.1] + [delta].

These results can be explained with the aid of Figure 1. (A formal proof of the proposition is provided in Appendix F.) The first condition in part (i) of the proposition ([t.sup.*.sub.2] [greater than or equal to] t) together with Lemma 2 implies that [t.sup.*.sub.1] [less than or equal to] [t.sup.*.sub.2] > t. In this case, the firm cannot be better off choosing the dirtier technology because it would then face a tax rate of [t.sup.*.sub.2]. Not only is this tax rate higher than the rate the firm would face if it chose the cleaner technology ([t.sup.*.sub.1]), but it is also a rate for which the cleaner technology yields lower costs since [t.sup.*.sub.2] > t. Thus, the following inequalities hold: [TC.sub.1]([t.sup.*.sub.1]) [less than or equal to] [TC.sub.1]([t.sup.*.sub.2]) < [TC.sub.2]([t.sup.*.sub.2]); these imply that the firm is better off choosing the cleaner technology.

The second condition in part (i) of the proposition ([t.sup.*.sub.2] < t and [t.sup.*.sub.2] [greater than or equal to] [t.sup.*.sub.1] + [delta]) indicates that the firm also prefers the cleaner technology when [t.sup.*.sub.2] < t, provided [t.sup.*.sub.2] is sufficiently larger than [t.sup.*.sub.1]. If [t.sup.*.sub.2] is not that much larger than [t.sup.*.sub.1], then, as part (ii) of the proposition indicates, the firm prefers the dirtier technology.

5. First-Mover Advantages and the Desirability of Commitment

A comparison of Propositions 1 and 2 yields some interesting conclusions about the merits of moving first. It is not difficult to see that the firm always benefits by moving first: By doing so, it can determine whether the first-best or second-best outcome is realized, choosing the one that entails lower costs. Although there is the added possibility of the corner optimum {1, t} when the regulator moves first, this outcome is inferior to the first-best one from the firm's perspective because it entails higher costs to the firm, given that t > [t.sup.*.sub.1] holds at the corner optimum.

The benefit to the regulator of moving first is less clear. According to Proposition 1, when the regulator leads, the first-best outcome is achieved with a tax if and only if [t.sup.*.sub.1] [greater than or equal to t. Proposition 2 reveals that this condition does not guarantee that the first-best outcome is achieved when the firm leads. But Proposition 2 also indicates that with the firm leading the first-best outcome could be achieved when [t.sup.*.sub.1] < t. From part (i) of the proposition, this is unambiguously true when [t.sup.*.sub.2] [greater than or equal to] t, and it could be true even when [t.sup.*.sub.2] < t. Thus, the regulator may be better able to achieve the first-best outcome if the firm moves first. Figure 2 depicts such a situation. Recall that for the case depicted in the figure, neither the first-best nor the second-best outcome is attainable when the regulator moves first. However, when the firm moves first, the first-best outcome is the equilibrium since [t.sup.*.sub.2] > t.

We therefore have the following corollary:

COROLLARY 1. (i) The regulator may be better off when the firm moves first; in particular, when [t.sup.*.sub.1] < t [less than or equal to] [t.sup.*.sub.2], the first-best outcome is the equilibrium when the firm leads but not when the regulator leads. (ii) The firm, on the other hand, is always better off moving first.

6. Conclusions

We set out to evaluate pollution taxes when a regulated firm must choose among discrete pollution abatement technologies. The firm's choice of technology makes possible two sequences of play: the traditional one in which the regulator moves first, committing to a tax rate before the firm adopts a technology, and an alternative one in which the firm moves first by adopting a technology.

The results we obtain raise issues that are potentially important in the application of pollution taxation, issues that have not been identified in the existing literature. For both sequences of play, we find that the regulator may not be able to achieve the first-best outcome. The underlying reason for this is that the firm's tax payments cause a divergence between the technology preferences of the firm and those of the regulator (or society): The firm takes into account its tax payments when choosing a technology, but from the regulator's (and society's) perspective, these payments should not influence the choice of technology because they are just transfers. As a result, the firm may choose the wrong technology from the regulator's perspective.

Not surprisingly, the conditions under which the regulator is able to achieve the first-best outcome differ for the two sequences of play. But they differ in an unexpected manner. We would expect the regulator to be better able to achieve the first-best outcome if it moves first. However, we find that, in some cases, the first-best outcome can be achieved only if the firm moves first. This result suggests that the regulator may be better off forgoing whatever ability it has to commit to a tax rate and should instead allow or even encourage firms to choose technologies before the tax rate is announced. The regulator could then, after observing the pattern of technology adoption, promulgate the appropriate tax. Indeed, as noted in the introduction, such behavior has been observed in a few settings. However, in most discussions and applications of pollution taxes, the possibility of allowing or encouraging firms to adopt abatement technologies in advance of regulation is simply not considered. Our results sugges t it should be, particularly in cases where differences in marginal abatement costs between technologies are very large. The discontinuity in emissions when the regulator moves first is severe in such cases, with a wide interval of emissions being unattainable.

A question of some interest is whether our findings have any bearing on the debate over the choice of policy instruments. A well-known duality exists between an emissions tax and an emissions standard when the regulator has perfect information (as we assume here): Any outcome achievable with a standard can also be achieved with a suitably chosen tax. We can show that this duality fails to hold in our model. The discontinuity in emissions that arises with a tax in the traditional sequence of play does not arise with a standard, for the simple reason that with a standard, the regulator chooses an emissions level, and the firm then chooses the technology that enables it to meet this level at the lowest cost. More important, we can show that when the regulator moves first, the first-best outcome is always attainable with a standard. This is not true when the firm moves first; in this case, the advantage of a standard over a tax is less clear.

We have assumed for simplicity that the firm has a choice between two technologies. An obvious question is whether the ex ante discontinuity we identify is mitigated when the number of technologies available to the firm increases. Intuition might suggest that the magnitude of the discontinuity diminishes as the number of technologies increases. We can show, however, that this is not necessarily true. As the number of technologies rises but remains finite, the number of discontinuities (or gaps in emissions) rises, and the aggregate length of the discontinuities could increase. It is only when there is a continuum of technologies that discontinuities cease to arise. However, as noted earlier, the empirical relevance of a continuum of technologies is questionable.

Appendix A

Proof that [x.sup.*.sub.2](t) > [x.sup.*.sub.1](t) for t [epsilon] [0, [[tau].sub.2]).

For t [epsilon] [0, [[tau].sub.1]), - [V'.sub.1]([x.sup.*.sub.1](t)) = t = -[ [V'.sub.2]([x.sup.*.sub.2](t)). Given Equation 3 and the strict convexity of [V.sub.i](.), it follows that [x.sup.*.sub.2](t) > [x.sup.*.sub.1](t) for t [epsilon] [0, [[tau].sub.1]). For t [epsilon] [[[tau].sub.1], [[tau].sub.2]), [x.sup.*.sub.1](t) = 0 and [x.sup.*.sub.2](t) > 0, so [x.sup.*.sub.2](t) > [x.sup.*.sub.1](t) holds trivially.

Appendix B

Proof of existence and uniqueness of t.

The existence of t can be established as follows. When t = 0, [TC.sub.2](0) [equivalent to] [K.sub.2] < [K.sub.1] [equivalent to] [TC.sub.1](0) given Equations 2 and 4. When t = [[tau].sub.2], [TC.sub.2]([[tau].sub.2] [equivalent to] [C.sub.2](0) > [C.sub.1](0) [equivalent to] [TC.sub.1]([[tau].sub.2] given Equation 5. It follows that [TC.sub.1](t) = [TC.sub.2](t) must hold for some t [epsilon] (0, [[tau].sub.2]) because of the continuity of the two total cost functions. The uniqueness of t can be establish by noting that, given Equations 8 and 9, [TC'.sub.2](t) [equivalent to] [x.sup.*.sub.2](t) > [x.sup.*.sub.1](t) [equivalent to] [TC'.sub.1](t) for t [epsilon] [t, [[tau].sub.2]). This implies [TC.sub.2](t) > [TC.sub.1](t) must hold for t [epsilon] (t, [[tau].sub.2]).

Appendix C

Proof that 0 < [s.sup.*.sub.1] < [s.sup.*.sub.2].

The assumption that D'(0) < \[V'.sub.1](0)\ implies [s.sup.*.sub.i] > 0, i = 1, 2, while the assumption D'(.) > 0 and [V'.sub.i]([X.sub.i]) = 0 imply [s.sup.*.sub.i] < [X.sub.i], i = 1, 2. Thus, the [s.sup.*.sub.i] are interior optima that satisfy the first-order conditions: [SC'.sub.1]([s.sup.*.sub.1]) [equivalent to] [V'.sub.1])(s.sup.*.sub.1] + D'([s.sup.*.sub.1]) = 0 = D'(s.sup.*.sub.2]) + [V'.sub.2]([s.sup.*.sub.2] [equivalent to] [SC'.sub.2]([s.sup.*.sub.2]). The relationship [s.sup.*.sub.1] < [s.sup.*.sub.2] then follows from Equation 3 and the strict convexity of [SC.sub.i](.) since they imply that [SC'.sub.1](.) > [SC'.sub.2](.).

Appendix D

Proof of existence of the corner equilibrium {1, t}.

We can demonstrate that the corner solution {1, t} is the equilibrium when neither the first-best nor the second-best outcome is attainable by making use of the strict convexity of [SC.sub.i](.). The latter implies that deviation from t raise social costs by moving the firm's emissions further away from either [s.sup.*.sub.1] or [s.sup.*.sub.2]. First, consider a higher tax, t + [epsilon]. This would induce the firm to adopt the cleaner technology and emit some amount ([x.sup.*.sub.1] - [epsilon]) < [x.sup.*.sub.1]. Since [s.sup.*.sub.1] [epsilon] ([x.sup.*.sub.1], [x.sup.*.sub.2]), \[s.sup.*.sub.1] - ([x.sup.*.sub.1]) - [epsilon])\ must be greater than \[s.sup.*.sub.1] - [x.sup.*.sub.1]\; that is, ([x.sup.*.sub.1] - [epsilon]) must be farther away from [s.sup.*.sub.1] than is [x.sup.*.sub.1]; therefore, [SC.sub.1]([x.sup.*.sub.1]) < [SC.sub.1](x.sup.*.sub.1] - [epsilon] must hold, implying that social costs are lower when t = t and the firm adopt the cleaner technology. Now consider a lower tax, t - [epsilon]. This would induce the firm to adopt the dirtier technology and emit some amount ([x.sup.*.sub.2] + [epsilon]) > [x.sup.*.sub.2]. Since [s.sup.*.sub.2] [epsilon] ([s.sup.*.sub.1], [x.sup.*.sub.2]), \([x.sup.*.sub.2] + [epsilon]) - [s.sup.*.sub.2]\ must be greater than \[x.sup.*.sub.2] - [s.sup.*.sub.2]\; therefore, [SC.sub.2]([x.sup.*.sub.2] < [SC.sub.1](x.sup.*.sub.2] + [epsilon]) must hold, implying that social costs are lower when t = t and the firm adopts the dirtier technology.

Appendix E

Proof that [C.sub.1]([x.sup.*.sub.1]) - [C.sub.2]([x.sup.*.sub.1]) < 0.

We first need to define and characterize the emissions level at which the firm's abatement costs are equal for the two technologies:

[C.sub.1](x) [less than/greater than] [C.sub.2](x) [right arrow/left arrow] x [less than/greater than] x. (A1)

The existence and uniqueness of x can be established as follows. The assumption in Equation 5 implies that technology 2 is more costly when x = 0, whereas Equations 2 and 4 imply that technology 2 is less costly when x = [X.sub.2]. Therefore, given the continuity of the cost functions, [C.sub.1](.) = [C.sub.2](.) must hold for some x [epsilon] (0, [X.sub.2]). The uniqueness of x follows from Equation 3. It implies that [C'.sub.1](.) > [C'.sub.2](.) for x [element of] (x, [X.sub.2]); therefore, [C.sub.1](.) > [C.sub.2](.) must hold for x [element of] (x, [X.sub.2]).

We now establish that x [element of] ([x.sup.*.sub.1], [x.sup.*.sub.2]). This is most easily done by contradiction. There are two alternatives to consider: (i) x [greater than or equal to] [x.sup.*.sub.2](>[x.sup.*.sub.1]) and (ii) x [less than or equal to] [x.sup.*.sub.1] (<[x.sup.*.sub.2]). (i) If x [greater than or equal to] [x.sup.*.sub.2], then, given Equation A1, we would be able to write [C.sub.1] ([x.sup.*.sub.2]) + [tx.sup.*.sub.2] [less than or equal to] [C.sub.2] ([x.sup.*.sub.2]) + [tx.sup.*.sub.2]. This inequality implies [C.sub.1]([x.sup.*.sub.1]) + [tx.sup.*.sub.1] [equivalent to] [TC.sub.1](t) < [TC.sub.2](t) [equivalent to] [C.sub.2]([x.sup.*.sub.2]) + [tx.sup.*.sub.2] since [x.sup.*.sub.1] argmax [TC.sub.1](t), but this contradicts the definition of t. (ii) The second alternative, x [less than or equal to] [x.sup.*.sub.1], can be ruled out in a similar manner.

Given that [x.sup.*.sub.1] < x, Equation A1 implies that the firm's abatement costs must be lower with the cleaner technology: [[C.sub.1]([x.sup.*.sub.1]) - [C.sub.2]([x.sup.*.sub.1])] < 0.

Appendix F

Proof of Proposition 2.

Suppose [t.sup.*.sub.2] [greater than or equal to] t. Then, given Equations 9 and 10, we can write [TC.sub.1]([t.sup.*.sub.1]) [less than or equal to] [TC.sub.1]([t.sup.*.sub.2]) [less than or equal to] [TC.sub.2]([t.sup.*.sub.2]). Therefore, the firm would pick the cleaner technology, and the regulator's best response would be to issue the tax [t.sup.*.sub.1].

Now suppose [t.sup.*.sub.2] < t. If [t.sup.*.sub.1] = [t.sup.*.sub.2], Equation 10 implies the firm is better off with the dirtier technology: [TC.sub.2] ([t.sup.*.sub.2]) = [TC.sub.2]([t.sup.*.sub.1]) < [TC.sub.1]([t.sup.*.sub.1]). Given the continuity of the total cost functions, [TC.sub.2]([t.sup.*.sub.2]) < [TC.sub.1]([t.sup.*.sub.1]) will also hold for value of [t.sup.*.sub.2] that are not much larger than [t.sup.*.sub.1]. For such values of [t.sup.*.sub.2], the firm would pick the dirtier technology and face the tax [t.sup.*.sub.2]. However, if [t.sup.*.sub.2] is much larger than [t.sup.*.sub.1], [TC.sub.2] ([t.sup.*.sub.2]) [greater than or equal to] [TC.sub.1]([t.sup.*.sub.1]) would hold, and the firm would pick the cleaner technology and face the tax [t.sup.*.sub.1]. More precisely, if [t.sup.*.sub.2] [greater than or equal to] [t.sup.*.sub.1] + [delta], where [delta] is defined by the equality [TC.sub.1]([t.sup.*.sub.1]) = [TC.sub.2]([t.sup.*.sub.1] + [delta]), the firm would pick the cleaner technology; whereas if [t.sup.*.sub.2] < [t.sup.*.sub.1] + [delta], the firm would pick the dirtier technology. Note that [t.sup.*.sub.1] + [delta] < t must hold since [t.sup.*.sub.2] must be less than t.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

Received July 2000; accepted October 2001.

(1.) Cropper and Oates (1992) survey this literature.

(2.) For example, coal-fired power plants can lower their costs of reducing sulfur-dioxide emissions by modifying their generating plants, at considerable expense, to burn low-sulfur coal (Moore 1995). Similarly, paper mills can lower the costs of reducing emissions of organic and inorganic pollutants by adopting "clean" production processes, such as one developed by Union Camp Technology that has 30% higher fixed costs than the traditional process (Virginia Department of Environmental Quality 1994).

(3.) Kemp (1997) and Jaffee, Newell, and Stavins (2002) provide a survey of much of this literature.

(4.) An exception is the paper by Biglaiser and Horowitz (1995), who consider the incentives for innovation offered by marketable permits assuming a continuum of technologies. Yao (1988) and Malik (1991) also consider the possibility of the firm moving first, but in a very different setting: one where the regulator imposes emissions standards in the face of uncertainty about either technology or damages.

(5.) It can also be shown that a set of activity-dependent taxes, with the tax for each activity given by the intersection of the relevant marginal gain and marginal damage curves, would achieve the first-best outcome in Turvey's model but not in ours (see footnote 12).

(6.) The analysis could be cast in terms of restricted profit functions, yielding identical results. This is true because the abatement cost function for each technology is given by the difference between the firm's maximal profits and its profits when emissions are restricted to some level x: [C.sub.i](x) [equivalent to] [[pi].sub.i](p, w) - [[pi].sub.i](p, w,x), where [[pi].sub.i](p, w) is an indirect profit function with vectors of output prices and input prices as arguments and [[pi].sub.i](p, w,x) is a restricted profit function.

(7.) Alternatively, technology 2 could represent the firm's existing technology. In this case, the fixed costs associated with technology 1 could be zero because they are sunk.

(8.) Damages are typically assumed to be convex in emissions. We do not present the results for a concave damage function here, but it is not difficult to verify that for such a function the results are similar to those presented for a linear damage function.

(9.) As Figure 2 shows, in terms of emissions levels, this possibility corresponds to [s.sup.*.sub.1] and [s.sup.*.sub.2] lying in the interval of unattainable emissions levels ([x.sup.*.sub.1], [x.sup.*.sub.2]].

(10.) Alternatively, the term would vanish if the regulator's objective was to minimize the sum of the firm's total costs and damages rather than the sum of abatement costs and damages.

(11.) In a recent paper, Janeba (2000) examines the consequences of a government's inability to commit to a tax policy in the context of tax competition and the investment decisions of a multinational firm. Like us, Janeba considers a setting in which the firm makes its investment decisions before the government announces its tax policy.

(12.) As noted in the introduction, to reflect actual practice, we restrict attention to taxes that are a linear function of emissions alone. It is worth noting, though, that if the regulator were able to issue and credibly commit to technology-dependent taxes, with a tax of [t.sup.*.sub.1] levied if the firm adopted the cleaner technology and a tax of [t.sup.*.sub.2] if it adopted the dirtier one, the outcome would be no different than that obtained here. In both settings, the firm would pick the technology-tax combination that entailed lower costs.

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Gregory S. Amacher (*)

Arun S. Malik (+)

(*.) College of Natural Resources, Virginia Polytechnic Institute and State University, Cheatham Hall, Room 307, Blacksburg, VA 24061, USA; E-mail [e-mail].

(+.) Department of Economics, George Washington University, Washington, DC 20052, USA; E-mail [e-mail]; corresponding author.

A version of this paper was presented at the 8th Annual Conference of the European Association of Environmental and Resource Economists. We thank Chris Snyder, Ray Prince, John Livernois, Christian Crowley, Jorge Soares, and two anonymous referees for helpful comments and discussions. We alone are responsible for any errors in the paper.