Comparing specific and ad valorem Pigouvian taxes and output quotas.

I. Introduction

When the production or use of a good creates an external cost, an unregulated market produces a socially-inefficient quantity of output. One remedy for this inefficiency is to impose a Pigouvian tax equal to the marginal external damage associated with the good. In many circumstances, privately optimizing agents facing a Pigouvian tax make decisions that are consistent with the maximization of social welfare.

Various authors have studied the effects of imposing Pigouvian taxes on commodities such as carbon-based fuels [8; 24], products containing ozone-depleting chemicals [4], fertilizer [18], pesticides [30(gasoline [10], virgin materials [13], alcohol [27; 28] and cigarettes 123]. A Pigouvian tax is more efficient the closer is the link between the traded quantity of the good and its total external damage. Yet even when external damage varies according to how the good is produced or used, a Pigouvian tax remains a useful policy option if administrative costs are reduced by taxing at the point of the commodity's sale, rather than at the point of its production or use.

A number of complications hinder attempts to set Pigouvian taxes [4; 10]. One pervasive problem is that policy-makers may wish to impose taxes or other controls when they are uncertain about the demand or supply of the relevant commodity, and thus also uncertain about the marginal social costs of reducing its output. For example, a mixture of taxes and quantity restrictions is used to limit output of products containing ozone-depleting chemicals [4], yet future demand for these products is uncertain because of imperfect information about the prices and availability of substitutes.(1)

Researchers have examined a number of issues involving externality control under uncercertainty, focusing mainly on enforcement and on the choice of policy instruments. When choosing a control policy under ex ante uncertainty, a key problem is that a policy that is invariant to realized market conditions is unlikely to achieve an efficient outcome ex post. The government could alter its policy as it learns more over time, but the political and administrative difficulties of this strategy are well known. Repeated policy adjustments also may lead to strategic responses by polluters [9]. Consequently, researchers often follow Weitzman [35] by comparing the expected welfare losses created when policies are fixed ex ante.

Yet studies of this type have not compared the relative expected efficiencies of the two basic forms in which a Pigouvian commodity tax can be imposed -- specific (the per-unit tax is a fixed amount) and ad valorem (the tax is a percentage of the good's price). The distinction between tax forms is relevant for several reasons. Governments impose both types of tax, and academic studies have estimated the optimal value of a Pigouvian tax in both forms.(2) Most importantly, while the two tax forms are equivalent in a competitive market with no uncertainty,(3) they generally produce differing outcomes (for all market structures) when consumer demand or firm cost is uncertain.(4) The difference arises for a straightforward reason: the per-unit equivalent value of a fixed ad valorem tax generally depends on realized market parameters; the value of a fixed specific tax does not.

Only two papers to date have considered the specific-ad valorem distinction in the context of Pigouvian taxation. Koenig [21] compares administered prices, a quota system, and a tax system in which both a specific and an ad valorem tax are applied to the externality-generating good. By using two taxes (and linear functions), Koenig designs a tax system that responds perfectly to either demand or supply uncertainty. Koenig [20] also assumes that regulators use both specific and ad valorem output taxes when examining a related question -- the benefits (under uncertainty) of allowing regulators to impose output taxes along with a direct effluent tax. One weakness in Koenig's valuable contributions may be the generality of the tax systems he analyzes. We know of no actual tax system in which both specific and ad valorem taxes are simultaneously applied to the same good; the more common approach is to impose either a specific or an ad valorem tax. In this "either/or" case, Koenig's results provide little insight about the relative expected welfare performance of externality-control policies. Important differences in the performance of specific and ad valorem taxes can be clarified by considering the two types of taxes separately. Moreover, like most research in this area, Koenig considers only competitive markets.

While there has been little consideration of differences between specific and ad valorem forms of Pigouvian taxation, the two tax forms have been compared in other contexts. Two main lines of research appear in the literature. The first approach focuses on noncompetitive markets, and often concludes that an ad valorem tax can raise a fixed amount of revenue more efficiently than can a specific tax [12; 32]. The second approach investigates how the two tax forms affect markets in which product quality is endogenous [5; 19]. In addition, Fraser [15] considers how the tax forms affect a price-taking firm producing a homogeneous product under uncertain market conditions. His analysis emphasizes the firm's attitudes toward risk-bearing, but does not consider overall welfare effects.

This paper compares the specific and ad valorem forms of Pigouvian taxation under market uncertainty. In contrast to previous research [1; 14; 20; 21; 35] it employs partial-equilibrium models of both competition and monopoly. It emphasizes the behavior-modifying goal of taxation by assuming that the government has no revenue requirement (implicitly, tax revenue is rebated to consumers).(5) Uncertainty is modeled by assuming that a random vertical shift can affect either the demand or supply curve. To link results with prior research on externality control under uncertainty, and because actual controls may consist of quantity regulations, the paper also compares the two taxes to an optimal quota on output.(6) In this context, neither of the two tax forms nor the quota is unambiguously welfare superior when market conditions are uncertain. Rather, the welfare comparison among the three policies hinges on the slopes of the demand, supply and marginal damage functions, as well as on the amount of uncertainty.(7)

Certain market characteristics that alter the rankings of the three policies can, however, be identified. Consider first the case of demand uncertainty in a competitive market. An unexpectedly high level of demand causes an increase in both equilibrium quantity and price (assuming upward-sloping supply), and thus raises the per-unit size of a given ad valorem tax. This automatic adjustment in the size of the tax can be welfare-improving if marginal damage also rises with output. Consequently, an ad valorem tax system becomes more attractive relative to a specific tax system as the marginal damage function becomes steeper. Once the marginal damage function becomes sufficiently steep, however, the quota dominates both tax forms, as would be expected based on the work of Weitzman [35] and Koenig [21]. In other words, all else equal, a small (moderate) (large) slope of marginal damage suggests that the welfare-maximizing policy is a specific tax (ad valorem tax) (quota).

Three other results are established. First, it is only for the ad valorem tax that the optimal policy design under uncertainty differs from the design that is optimal when the expected demand curve holds with certainty. Second, an increase in uncertainty expands the range of parameter values over which the ad valorem tax is preferred. Third, the ad valorem tax is the only policy that is never dominated by both other policies.

The above results extend, at least qualitatively, to the case of demand uncertainty in a monopoly market. One new result is derived by comparing the monopoly and competitive outcomes: other things equal, the existence of monopoly power expands the range of marginal-damage slopes over which taxation in general (and specific taxation in particular) is the preferred policy. This result contrasts markedly with efficiency comparisons of specific and ad valorem taxes as revenue instruments.

Results differ dramatically, however, when competitive supply or monopoly marginal cost is uncertain. An unexpected supply increase, for example, causes a reduction in the per-unit size of the ad valorem tax, which runs counter to the desired outcome when marginal damage is nondecreasing. Consequently, the specific tax always dominates the ad valorem tax, while the slopes of demand, supply and marginal damage again determine the welfare comparison between the specific tax and the quota.

The paper proceeds as follows. Section II describes the model and compares the two tax forms in the case of demand uncertainty in a competitive market. Section III adapts the model to a monopoly setting, while section IV briefly considers supply uncertainty. Section V discusses policy implications and conclusions.

II. Demand Uncertainty: Competition

Consider first the case of demand uncertainty in a competitive market. To facilitate computation of welfare differences between policies, we follow virtually all previous authors [1; 14; 20; 21; 35] and assume that the marginal benefit and cost functions are linear. The demand function is given by [P.sub.d] = a + [Alpha] - bQ, where [P.sub.d] denotes demand price, Q denotes market quantity, a, b [is greater than] 0, and [Alpha] is a mean zero random intercept shift with variance [[Sigma].sup.2], So that E [Alpha] = 0 and E [[Alpha].sup.2] = [[Sigma].sup.2]. The supply price is [P.sub.s] = c + d Q, where c, d [is greater than] 0. Marginal external damage is a function of industry output: G = e + f Q, where e, f [is greater than] 0.(8) Finally, to insure that positive output is always efficient, [Alpha] is bounded by the condition [Alpha] [is greater than] - (a - c - e).

If a welfare-maximizing regulator could set policy after demand is revealed, an efficient quota or tax would be chosen to maximize the sum of consumer plus producer surplus plus tax revenue minus total external damage.(9) For a particular value of [Alpha], the welfare-maximizing output [Q.sup.*] is found by equating marginal damage to the marginal cost of controlling output, which in this context equals marginal foregone consumption benefit. Solving G([Q.sup.*]) = [P.sub.d] ([Q.sup.*]) - [P.sub.s]([Q.sup.*]) for the (ex post) optimal output yields [Q.sup.*] = (a + [Alpha] - c - e)/(b + d + f). This outcome can be achieved by setting Q* as a quota or by setting an appropriate tax. The optimal ex post specific tax equals marginal external damage at the efficient output G([Q.sup.*]) = [f (a + [Alpha] - c) + e(b + d)]/(b + d + f); the optimal ex post ad valorem tax rate equals the ratio of marginal external damage to consumer price G([Q.sup.*])/[P.sub.d] ([Q.sup.*]) = [f (a + [Alpha] - c) + e (b + d)]/[(a + [Alpha]) (d + f) + b(c + e)]. In all cases, the resulting welfare is [W.sup.*] = (1/2)(b + d + f)[Q.sup.*2].

In contrast, we assume the regulator must set policy before demand is known, but production occurs after [Alpha] is revealed. The regulator seeks to maximize expected welfare given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where E denotes the expectations operator and [W.sub.i] measures welfare associated with quantity [Q.sub.i], which in turn denotes the quantity traded under a quota (i = q), specific tax (i = s) or ad valorem tax (i = a). The expected loss arising from setting policy with imperfect information is proportional to

[L.sub.i] = 2E([W.sup.*] - [W.sub.i]) = [(b + d + f) E([Q.sub.i] - [([Q.sup.*]).sup.2],

where [L.sub.i] denotes (twice the) expected welfare loss from quantity [Q.sub.i], i = q, s, a. As shown, the expected welfare loss (which can be viewed graphically as the expected size of a welfare-loss triangle) is proportional to the mean square difference between actual and ex post optimal output.

The Optimal Quota

The optimal quota, or ex ante optimal output, is the fixed quantity with minimum mean square deviation from [Q.sup.*]. Clearly, this quantity must equal expected optimal output:

(1) [Q.sub.q] = E [Q.sup.*] = (a - c - e)/(b + d + f) = [Q.sup.*] [[Epsilon].sub.q],

where [[Epsilon].sub.q] = - [Alpha]/(b + d + f) measures the deviation from ex post optimal output. The quota achieves the ex post optimum on average because it equates the expectations of marginal damage and marginal control cost, but of course yields an expected welfare loss relative to a policy based on perfect information:

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The expected loss rises with the amount of uncertainty (measured by [[Sigma].sup.2]), but falls as the (absolute) slope of demand, supply or marginal damage rises. The greater the slopes of these curves, the less optimal output responds to demand shifts, and thus the less it deviates from its expected value.

The Optimal Specific Tax

Like the quota, the optimal specific tax maximizes expected welfare by balancing expectations of marginal control cost and marginal damage, but after accounting for the market reaction to the tax. The market quantity in the presence of a specific tax is (a + [Alpha] - c - t)/(b + d). The optimal specific tax is

[t.sub.s] = [f(a - c) + e(b + d)]/(b + d + f) = e + f E [Q.sup.*] = E [G.sup.*],

where [G.sup.*] equals marginal damage at the ex post optimal output. Thus, [t.sub.s] equals the expectation of the specific tax that maximizes welfare ex post. The resulting output is [Q.sub.s] = [Q.sup.*] + [[element of].sub.s], where [[element of].sub.s] = - [f/(b + d)][[element of].sub.q] measures the deviation of quantity from the ex post optimum. The specific tax achieves the ex post optimal output on average (in the sense that E [Q.sub.s] = E [Q.sup.*]), but again results in an expected welfare loss relative to a policy based on perfect information:

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

An important special case arises if f = 0: [t.sub.s] = e and [Q.sub.s] = [Q.sup.*]. Naturally, a specific tax set equal to a constant marginal damage will maximize welfare for any realization of demand. Otherwise, the expected loss from a specific tax rises with the amount of uncertainty and with the slope of marginal damage, but falls as the (absolute) slope of demand or supply increases. This pattern arises because an increase in the slope of marginal damage has no effect on how the market quantity adjusts to a demand shift; such an increase, however, reduces the extent to which the optimal quantity responds to a demand shift. In consequence, an increase in the slope of marginal damage widens the divergence between market and optimal quantities. Holding f constant, increases in the absolute slopes of demand or supply narrow the divergence.

The Optimal Ad Valorem Tax

When an ad valorem tax is imposed, equilibrium market output is [(a + [Alpha])(1 - t) - c]/[b(1 - t) + d]. The appendix shows that the optimal tax under uncertainty is

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [Delta] [Q.sub.a]/[Delta] [t.sub.a] = - [Psi]/[[b(1 - [t.sub.a]) + d].sup.2], [Psi] = (a + [Alpha])d + bc, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] equals demand price at the ex post optimum. When [t.sub.a] is imposed and a particular [Alpha] is realized, output is [Q.sub.a] = [Q.sub.a] + [[element of].sub.a], where

[[elements of].sub.a] = (cf - de)[(ad + bc)[Alpha] - d[[Sigma].sup.2]] /[(b + d + f)E [[Psi].sup.2]]

measures the deviation from the ex post optimum quantity, and where E [[Psi].sup.2] = [(ad + bc).sup.2] + [d.sup.2] [[Sigma].sup.2]. The expected welfare loss is one-half of

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The expected welfare loss rises with the amount of uncertainty, and is smaller the closer is cf to de. Note also that whether [L.sub.a] rises, stays constant, or falls as f rises depends on whether f is greater than, equal to or less than de/c.

As it does in the last two displayed formulae, the term cf - de plays a critical role throughout this section of the paper; we thus consider this term in detail. Define [[Eta].sub.g] = f Q/(e + fQ) to be the elasticity of marginal damage with respect to changes in output, and [[Eta].sub.s] = dQ/(c + dQ) to be the elasticity of supply price similarly defined. It is simple to show that the sign of cf - de equals the sign of [[Eta].sub.g] - [[Eta].sub.s]. Thus, cf [is greater than] de (for example) implies that a change in output produces a change in marginal external damage that is proportionally larger than the change in marginal private cost.(10)

Several points about the optimal ad valorem tax policy deserve further discussion. Begin by noting that when an ad valorem tax is in place, the optimal policy with uncertain demand does not equal the policy that would be optimal if the expected demand curve held with certainty. For both a quota and a specific tax, in contrast, the optimal policy under uncertainty would also be optimal if expected demand obtained for sure. With certain demand ([[Sigma].sup.2] = 0), the optimal ad valorem tax is

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If [??.sub.a] is imposed in the uncertainty case, expected output equals the expected ex post efficient output, E [??.sub.a] = E [Q.sub.*].(11) When cf [not equal to] de and d [not equal to] 0, however, expected welfare is maximized by imposing [t.sub.a] [not equal to] [??.sub.a]. The direction in which [t.sub.a] differs from [??.sub.a] is easy to identify; equations (4) and (6) can be used to show that the sign of [t.sub.a] - [??.sub.a] matches that of cf - de (which, as noted above, in turn matches the sign of [[Eta].sub.g] - [[Eta].sub.s]). Imposing [t.sub.a] thus produces an output that is expected to differ from E [Q.sup.*]; in particular,

(7) E [[element of].sub.a] = - (cf - de)d[[Sigma].sup.2]/[(b + d + f)E [[Psi].sup.2]].

As would be expected, assuming that supply slopes upward, the sign of cf - de determines whether E [Q.sub.a] is less than or greater than E [Q.sub.*].

The explanation for why [t.sub.a] [not equal to] [??.sub.a] hinges on the fact that an ad valorem tax can be seen to have two distinct effects on market behavior. Namely, the tax affects both the expected market output and the manner in which the market output reacts to realized conditions. The first of the effects is obvious from the fact that E [Q.sub.a] = [a(1 - [t.sub.a]) - c]/[b(1 - [t.sub.a]) + d]. The second effect arises because, with upward-sloping supply, a high realization of [Alpha] increases the per-unit size of any given ad valorem tax. This effect in turn dampens the reaction of output to a demand increase. The higher the rate at which the ad valorem tax is imposed, the more significant is this dampening effect: [Delta] [Q.sub.a]/[Delta] [Alpha] [is greater than] 0, but

(8) [[Delta].sup.2] [Q.sub.a]/[Delta] [Alpha] [Delta] [t.sub.a] = - d/[[(b1 - [t.sub.a) + d).sub.2]]. [is less than or equal to] 0.

A higher tax rate thus both reduces expected market output and (for d [is greater than] 0) reduces variation in actual output about the mean. The manner in which the optimal tax balances these two effects can be understood by comparing outcomes under positive, negative and zero values for cf - de, given d [is greater than] 0, and finally considering the case of d = 0.

In the presence of an ad valorem tax, the market reacts to realized demand in a way that equalizes the marginal foregone net consumption benefit [P.sub.d] - [P.sub.s] with the implicit unit tax rate [tP.sub.d]. The market of course ignores marginal damage. Assume that supply is upward-sloping and begin by considering the case in which [[Eta].sub.g] [is greater than] [[Eta].sub.s] (equivalently, cf [is greater than] de). In this situation, a demand shock causes a change in marginal damage that is proportionally larger than the change it causes either in market price or in the implicit unit tax rate (which equals marginal foregone benefit). These observations imply that the efficient output changes by a smaller amount than does the market output. The market "overreacts" to demand shocks; as a result, expected welfare falls. In such a situation, it is best to restrict the adjustments in market output [35]. A welfare-maximizing government therefore wishes to dampen the output reaction to a demand shock; in view of equation (8), it accomplishes this goal by setting a relatively high tax rate.

The optimal ad valorem tax [t.sub.a] thus exceeds [??.sub.a] whenever marginal damage is more output elastic than is supply price.(12) The advantage of the higher tax rate lies in reducing damaging fluctuations in market output. The disadvantage is that it causes expected output to fall short of the efficient level, E [Q.sub.a] [is less than] E [Q.sup.*] when cf [is greater than] de. The optimal tax maximizes expected welfare by balancing these two effects. Note also that controlling market reactions becomes more important the larger is the expected variation in demand, and thus the optimal tax rises with the amount of uncertainty: [Delta] [t.sub.a]/[Delta] [[Sigma].sup.2] [is greater than] 0 when [[Eta].sub.g] [is greater than] [[Eta].sub.s].

The argument is reversed when marginal damage is less output elastic than is supply price; i.e., [t.sub.a] [is less than] [??.sub.a] when [[Eta].sub.g] [is less than] [[Eta].sub.s] (or cf [is less than] de). In this case, a demand shock causes a change in marginal damage that is proportionally smaller than the change in the implicit unit tax or in marginal foregone benefit. A demand shock thus changes the efficient level of output by relatively more than it changes the market output. Consequently, it is best to give the market greater latitude to adjust to realized conditions; this goal is met by setting [t.sub.a] [is less than] [??.sub.a]. Of course, the optimal tax rate again balances the gain from altering how [Q.sub.a] responds to [Alpha] against the loss from allowing E [Q.sub.a] to exceed E [Q.sub.*]. The relative benefit of imposing [t.sub.a] [not equal to] [??.sub.a] again increases with the size of the expected variation in demand, so that [Delta] [t.sub.a]/[Delta] [[Sigma].sup.2] [is less than] 0 when [[Eta].sub.g] [is less than] [[Eta].sub.s].

When f/e = d/c, the marginal damage and supply curves mirror one another, so that marginal damage and supply price have equal output elasticities: [[Eta].sub.g] [not equal to] [[Eta].sub.s]. Marginal damage is thus a fixed proportion of marginal social cost: G/(G + [P.sub.s]) = f/(d + f). It follows that an ad valorem tax set equal to this proportion maximizes welfare for any realization of demand. Substitution of cf = de into (4) and (6) indeed yields [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; the resulting output is [Q.sub.a] = [Q.sub.*] and thus [L.sub.a] = 0. When [[Eta].sub.g] = [[Eta].sub.s] and tax rate f/(d + f) is imposed, demand shocks do not cause market output to differ from optimal output, and thus the optimal tax policy is independent of [[Sigma].sup.2]: [Delta] [t.sub.a]/[Delta] [[Sigma].sup.2] = 0 when [[Eta].sub.g] = [[Eta].sub.s].

Finally, consider the case of perfectly elastic supply, so that [[Eta].sub.s] = 0. An upward-sloping marginal damage curve produces cf [is greater than] de; the reasoning above would thus suggest that the optimal tax should exceed [??.sub.a]. Notice, however, from equation (8) that when d = 0, the tax rate has no effect on market reaction to demand shocks. This fact arises because a horizontal supply curve means that the per-unit value of an ad valorem tax is invariant to demand shifts. There is thus no advantage to increasing the tax above the level that equates expectations of market and optimal output; substituting d = 0 into (4) produces [t.sub.a] = [??.sub.a]. When d = 0, the optimal tax is independent of [[Sigma].sup.2] and results in both E [[element of].sub.a] = 0 and [[Epsilon].sub.a] = [[Epsilon]].sub.s]; the optimal ad valorem and specific taxes produce identical market outcomes. This is not surprising; with supply price fixed, the two tax forms are equivalent.

Welfare Comparisons

The expected welfares produced by the three alternative policies can be directly compared. Equations (2) and (3) make clear that the specific tax yields higher (lower) expected welfare than does the quota for all values of f [is less than] ([is greater than]) [f.sub.1] where [f.sub.1] = b + d. If marginal damage responds less to an output change than does the marginal cost of restricting output, the specific tax is preferred to the quota; if marginal damage responds more, the quota is preferred. This outcome mirrors Koenig's [21] result and would be expected based on earlier work [1; 14; 35] which establishes, for example, that emissions charges achieve higher (lower) expected welfare than do emissions quotas as the slope of the marginal damage of emissions is less (greater) than the slope of the marginal cost of controlling emissions. In the presence of demand uncertainty, however, the comparison of quotas and specific taxes is largely irrelevant, because the ad valorem tax is never dominated by both of the other policies.

To compare the ad valorem tax to the quota, note that equations (2) and (5) imply that the sign of [L.sub.a] - [L.sub.q] equals the sign of

(9) [(cf - de).sup.2] - E [[Psi].sup.2]

When f = 0, expression (9) is negative, so that [L.sub.a] [is less than] [L.sub.q] and the ad valorem tax is preferred to the quota. As f increases, [L.sub.a] - [L.sub.q] declines until reaching a minimum at f = (de/c); further increases in f raise [L.sub.a] - [L.sub.q]. Expression (9) thus has one root that must exceed de/c. Setting the expression equal to zero and applying the quadratic formula reveals that this root is

(10) [f.sub.2] = (de/c)[1 + [(E [[Psi].sup.2]).sup.1/2]/de].

The ad valorem tax is preferred to the quota for all values of f [is less than] [f.sub.2], while the quota is preferred for all f [is greater than] [f.sub.2].

Turning to the comparison of the two tax forms, equations (3) and (5) imply that the sign of [L.sub.a] - [L.sub.s] equals the sign of

(11) [[(cf - de).sup.2]/E [[Psi].sup.2] - [[f/(b + d)].sup.2].

An important special case occurs under perfectly elastic supply: [L.sub.a] = [L.sub.s] when d = 0. As discussed previously, the two tax forms are equivalent in this case. If d [is greater than] 0, the slope of the marginal damage function again plays a key role in the welfare comparison. Notice that expression (11) is a decreasing (for f [is greater than or equal to] 0), concave (for f [is greater than] 0) function of f with a maximum at f = 0. This observation, along with the fact that [L.sub.a] - [L.sub.s] [is less than] 0 at f = de/c, implies that [L.sub.a] - [L.sub.s] has one root that must be less than de/c, namely

(12) [f.sub.3] = (de/c)[c(b + d)/[c(b + d) + [(E [[Psi].sup.2]).sup.1/2]]].

The specific tax is preferred to the ad valorem tax for all values off [is less than] [f.sub.3]; the ad valorem tax is preferred for all f [is greater than] [f.sub.3]. Three special cases are apparent: (a) if cf = de, the ad valorem tax maximizes welfare and thus is preferred to the specific tax; (b) if e = 0 (marginal damage proportional to output), the ad valorem tax is again preferred; (c) if f = 0 (marginal damage constant), the specific tax maximizes welfare and thus is preferred.

More generally, comparing (10) and (12) shows that (when d [is greater than] 0), [f.sub.2] [is greater than] [f.sub.3].(13) This inequality allows us to summarize the policy implications of changes in the slope of marginal damage. If marginal damage is relatively unresponsive to output (f [is less than] [f.sub.3]), then [L.sub.s] [is less than] [L.sub.a] [is less than] [L.sub.q]; the specific tax achieves the highest expected welfare, the quota the lowest. If marginal damage is moderately responsive to output ([f.sub.3] [is less than] f [is less than] [f.sub.2]), then [L.sub.a] [is less than] [L.sub.s] and [L.sub.a] [is less than] [L.sub.q]; the ad valorem tax achieves the highest expected welfare. Finally, if marginal damage is quite responsive to output ([f.sub.2] [is less than] f), then [L.sub.q] [is less than] [L.sub.a] [is less than] [L.sub.s]; the quota achieves the highest expected welfare, the specific tax the lowest. In other words, a small output elasticity of marginal damage favors the specific tax, while a moderate elasticity favors the ad valorem tax, and a large elasticity favors the quota.

The intuition for these results is clear. When the marginal damage curve is relatively flat, marginal damage is nearly constant over wide fluctuations in output, though the marginal cost of foregone consumption may vary. These conditions favor one of the tax policies, which allow the market quantity to react to unexpected realizations of demand. Between the two taxes, the specific form is preferred when marginal damage is quite flat, because the constant specific tax approximately equals the nearly constant marginal damage. But if marginal damage is moderately responsive to output changes, actual marginal damage may differ significantly from a constant specific tax. In this case, the ability of the ad valorem tax to adjust automatically to a realization of demand and thus to restrict the market's reaction to a demand shift becomes relatively more important. When marginal damage becomes quite responsive to output changes, however, optimal output does not deviate far from the expected optimal output or quota, while market responses may; thus the quota is the preferred policy.

A further implication of these results is that the ad valorem tax is never dominated by both other policies. If the market supply curve is perfectly elastic the two tax forms are welfare equivalent, but otherwise either the specific tax or the quota must achieve lower expected welfare than does the ad valorem tax.

Apart from the importance of the slope of marginal damage, the amount of uncertainty can also have an important effect on the expected-welfare comparisons. If d = 0, then changes in [[Sigma].sup.2] leave the welfare comparisons unaltered. For d [is greater than] 0, however, increases in [[Sigma.]sup.2] reduce [f.sub.3] and increase [f.sub.2], thus widening the range in which the ad valorem tax is preferred. Put differently, there is some critical value of [[Sigma].sup.2] such that for fixed values of other parameters (and d, f [is greater than] 0), larger values of [[Sigma].sup.2] insure that the ad valorem tax achieves the highest expected welfare.(14) The advantage of the ad valorem tax is that it responds to realized market conditions; this advantage is most attractive when the amount of uncertainty is greatest.

One disadvantage of the ad valorem tax policy is that computing the optimal tax rate in equation (4) requires more information than computing the specific tax or quota; to set these latter policies optimally, one needs to know only the value of expected demand. It is therefore useful to examine the welfare effects of imposing the tax rate [t.sub.a] given in equation (6). Briefly, (a) there is an intermediate range of marginal damage slopes where the simpler ad valorem tax policy achieves higher expected welfare than both the optimal specific tax and quota; (b) the simpler ad valorem tax policy is never dominated by both the specific tax and quota (but of course never dominates the optimal ad valorem tax); (c) changes in the amount of uncertainty do not affect the welfare achieved by the simpler ad valorem tax relative to the specific tax or quota.(15)

Finally, we note that the adjustment in the per-unit size of the ad valorem tax as output increases bears some similarity to the penalty function proposed by Roberts and Spence [31]. Their scheme consists of a fixed quantity of traceable emission licenses together with a subsidy to firms emitting less, and a charge to firms emitting more, than licensed amounts. Setting the unit charge higher than the unit subsidy creates a kinked, linear penalty function which more closely approximates a convex damage function than the linear penalty function associated with a unit emissions tax alone. Roberts and Spence note that this policy could yield significant efficiency gains over either licenses or emissions taxes used separately, provided that marginal damage increases sharply with emissions or that there is substantial uncertainty about marginal control cost.

In a similar manner, the ad valorem tax yields efficiency gains relative to either the specific tax or quota if marginal damages are increasing and there is uncertainty about demand. While Roberts and Spence focus on controlling emissions, the results derived here pertain to controlling output. In this case, setting a single ad valorem tax rate is far simpler than implementing the three-part policy described by Roberts and Spence.

III. Demand Uncertainty: Monopoly

In this section, we study how the existence of market power affects the comparisons among the three alternative policies. For simplicity, we assume that the market power takes the form of an unregulated monopoly. It is widely known that market power has implications in situations related to the one analyzed here. For example, monopoly power affects both the level of an optimal Pigouvian tax [2; 22] and the welfare comparison between revenue-raising specific and ad valorem taxes [32]. The differences between the competitive and monopoly results in our context arise from the fact that, all else equal, monopoly output responds less to a demand shock than does competitive output.

When a monopoly is assumed, C = c + dQ is used to denote the monopolist's marginal cost rather than industry supply. Marginal damage and demand are specified as they were above, with marginal revenue given by R = a + [Alpha] - [Beta] Q, where [Beta] = 2b. The optimal quantity remains [Q.sup.*], while the unregulated monopolist produces [Q.sup.m] = (a + [Alpha] - c)/([Beta] + d). Because the monopolist equates marginal revenue and marginal cost, [Q.sup.m] [is greater than] [Q.sup.*] if marginal revenue exceeds marginal cost at the optimum, or [R.sup.*] [is greater than] [C.sup.*]. Equivalently, the monopolist produces socially excessive output if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Intuitively, this condition means that the distortion caused by market power is smaller at the margin than is the distortion caused by external damage.

With perfect information, the ex post optimum can be achieved by setting [Q.sup.*] as a quota or indirectly by setting a specific tax of [R.sup.*] - [C.sup.*] = [[Phi](a + [Alpha] - c)+ e([Beta] + d)]/(b + d + f), or an ad valorem tax rate of [[R.sup.*] - [C.sup.*]]/[R.sup.*] = [[Phi](a + [Alpha] - c) + e([Beta] + d)]/[(d + [Phi])(a + [Alpha]) + [Beta](c + e)], where [Phi] = f - b. Since the monopolist's marginal revenue is less than price, both (ex post) optimal taxes are lower under monopoly than are the corresponding taxes under competition. Indeed, both taxes would be negative (subsidies) if [R.sup.*] [is less than] [C.sup.*]. Of course, these results occur because the monopolist's tendency to restrict output implies that market quantity exceeds [Q.sup.*] by less (if at all) than does market quantity in a competitive market with equal parameter values.

The Optimal Ex Ante Policies

Turning now to the case of policies set before demand is revealed, we make the simplifying assumption that the welfare-maximizing government policy always restricts output.(16) In such a situation, the optimal quota remains the same as in the competitive case, and the same welfare loss occurs. Market power is irrelevant to the outcome because the quota does not allow any market reaction to changes in demand. The optimal taxes, however, depend on market reactions and thus differ from those under competition.

When a specific tax is imposed on a monopoly, the profit-maximizing output is (a + [Alpha] - c - [Tau])/([Beta] + d). The optimal specific tax equates expectations of marginal revenue and marginal cost at the ex post optimum:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus [[Tau].sub.s] equals the expectation of the optimal ex post specific tax. Alternatively, [[.Tau].sub.s] equals the optimal tax rate in the absence of output distortions, less the optimal output subsidy. The output produced under [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] measures the deviation from the ex post optimal output. Thus, the specific tax achieves the optimum on average [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] but results in an expected welfare loss of (one-half of)

(13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In this situation, a demand shift has differing effects on market output and optimal output unless f = b. When this equality holds, marginal damage, which is the difference between marginal private cost and marginal social cost, changes at the same rate as does the difference between price and marginal revenue. Thus, the monopoly's own pattern of output choice exactly offsets the change in marginal damage due to a demand shift.(17) When the monopoly's behavior offsets the variable part of marginal damage, the optimal specific tax can offset the fixed part.(18) The optimal specific tax is thus guaranteed to achieve the ex post optimum not when f = 0 (as was the case for competition), but rather when f = b. Otherwise, the expected welfare loss rises widh the amount of uncertainty and falls or rises as the slope of marginal damage increases depending on whether f is less or greater than b.

When an ad valorem tax is imposed on a monopoly, the profit-maximizing output is [(a + [Alpha]) (1 - [Tau]) - c]/[[Beta](1 - [Tau]) + d]. The optimal ad valorem tax is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denotes the monopoly reaction to variations in the ad valorem tax rate and where [micro] = (a + [Alpha])d + [Beta] c. The resulting output is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

measures the deviation from the ex post optimal output, and where E [[micro].sup2] = [(ad + [Beta] c)[sup.2] + [d.sup.2] [[Sigma].sup.2]].

The ad valorem tax achieves the ex post optimum on average if d = 0 and for sure if c [Phi] = de, or f = b + (de/c). This equality holds when marginal damage rises fast enough to mirror the firm's marginal private cost after accounting for the monopoly's tendency to restrict output. As was the case for ad valorem taxation under competition, the optimal tax does not generally equal the expected value of the ex post optimal tax, and it thus produces an expected output chat does not equal the expected-optimal output. In fact, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is greater or less than E [Q.sup.*] depending on whether f is less or greater than b + (de)/c. The expected welfare loss is (one-half of)

(14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The expected loss rises with the amount of uncertainty and is smaller the closer is c [Phi] to de. It falls or rises as f increases depending on whether f [is less than] ([is greater than]) b + (de/c).

The optimal values for both taxes, the resulting output levels, and the expected losses are qualitatively similar under competition and monopoly. Quantitative differences arise because of the need to account for the divergence between marginal revenue and the marginal social benefit of consumption. Since R [is less than] [P.sub.d], profit-maximizing output is lower under monopoly than it is under competition for any realization of demand. Holding parameter values constant, therefore, both (ex ante) optimal taxes are also lower under monopoly. More importantly, R [is less than] [P.sub.d] implies that monopoly output responds less to a demand shock than does competitive output; this fact has important implications for welfare comparisons among the policies.

Welfare Comparisons

Equations (2) and (13) indicate that the specific tax achieves higher (lower) expected welfare than the quota whenever [Phi] [is less than] ([is greater than]) [[Phi.sub.1], where [[Phi].sub.1] = [Beta] + d, or equivalently when f [is less than] ([is greater than]) [Beta] +b+d. All else equal, marginal damage must rise more sharply under monopoly than under competition before the quota outperforms the specific tax. As in the case of competition, however, this comparison is largely irrelevant because the ad valorem tax is never dominated by both other policies at once.

Using equations (2) and (14) to compare the quota and ad valorem tax reveals that the ad valorem tax achieves higher (lower) expected welfare whenever [Phi] [is less than] ([is greater than]) [[Phi].sub.2], where

[[Phi].sub.2] = (de/c)[1 + (E [[micro.sup.2]) [sup.1/2]/de].

Comparing the two tax forms, equations (13) and (14) show that the specific tax achieves higher (lower) expected welfare than the ad valorem tax whenever [Phi] [is less than] ([is greater than]) [[Phi].sub.3], where

[[Phi].sub.3] = (de/c)[c([Beta] + d)/[c([Beta] + d) + (E [[micro].sup.2]) [sup.1/2]]].

As in the competitive case, the two taxes are equivalent if d = 0, but for d [is greater than] 0, [[Phi].sub.2] [is greater than] [[Phi]sub.3]. The qualitative policy implications under monopoly thus mirror those obtained under competition. A relatively flat marginal damage curve favors the specific tax, while a moderately sloped curve favors the ad valorem tax, and a steep curve favors the quota. The ad valorem tax is never dominated by both other policies at once, and its relative welfare performance improves as uncertainty increases.

The critical values for f computed under monopoly can be compared with those computed under competition. Doing so reveals that the critical values for monopoly, [[Phi].sub.2] + b and [[Phi],.sub.3] + b, exceed the corresponding values for competition, [f.sub.2] end [f.sub.3]. Thus, the range of marginal damage slopes for which the quota is the most preferred policy is smaller under monopoly than it is under competition, while the range of marginal damage slopes for which the specific tax is preferred to the ad valorem tax is larger under monopoly.(19)

These results hinge on the major difference between competition and monopoly in this context; namely, that for given parameter values, the existence of monopoly decreases the extent to which output responds to demand shocks. In turn, this effect limits the harm that can result from the output changes permitted by a tax. This harm is potentially more severe when the marginal damage curve is steep; under these conditions, therefore, the existence of a monopoly reduces the need to rely on a quantity restriction rather than a tax. Likewise, the monopolist's muted reaction to a demand shock reduces the policy-maker's need to rely on the self-adjusting nature of the ad valorem tax.

The result that market power strengthens the case for specific taxation contrasts markedly with analyses of specific and ad valorem taxes as revenue instruments under imperfect competition. In those analyses (assuming homogeneous goods), market power produces results that are "strikingly unambiguous in favouring ad valorem taxation" [12, 366].(20)

IV. Uncertain Supply

The welfare comparisons among the three policies differ when supply, rather than demand, is subject to random shocks, largely because of the behavior of the ad valorem tax. Consider a competitive market again and denote demand and supply respectively as [P.sub.d] = a - bQ and [P.sub.s] = c + [Gamma] + dQ, where [Gamma] denotes a mean zero random intercept shift with a - c - [Gamma] - e [is greater than] 0; the remainder of the model is specified as in section II.(21)

Results for the quota and specific tax parallel those obtained under demand uncertainty. The optimal quota equals the expectation of ex post optimal output and does not allow the traded quantity to adjust to realizations of [Gamma]. The optimal specific tax equals expected marginal damage at the optimum, and on average attains expected optimal output. Because the tax is fixed, the traded quantity rises and falls with increases or decreases in supply.

The welfare comparison between the quota and specific tax again hinges on the responsiveness of marginal damage relative to that of the marginal cost of reducing output. Whether it is the location of the demand or the supply curve that is random, the specific tax achieves higher expected welfare than the quota wherever f [is less than] ([is greater than]) (b + d).

In contrast to the quota and specific tax, the ad valorem tax performs quite differently when supply, rather than demand, is random. Specifically, the automatic adjustment in the size of the ad valorem tax works at odds with the desired outcome. As supply increases, marginal damage increases (or stays constant if f = 0), while the per-unit size of the ad valorem tax decreases. This perverse pattern of adjustment causes the ad valorem tax policy to attain lower expected welfare than the specific tax policy for all parameter values.

The specific tax thus always dominates the ad valorem tax under supply uncertainty and is preferred to the quota for relatively flat marginal damage functions. Since the quota outperforms the specific tax when marginal damage is steep, the ad valorem tax can be dominated by both other policies at once. These results imply that a detailed welfare comparison of the quota and ad valorem tax under supply uncertainty is largely irrelevant, and we simply note that the ad valorem tax may achieve higher expected welfare than the quota when the marginal damage curve is relatively flat.

V. Conclusion

This paper has shown that a Pigouvian commodity tax imposed in specific form generally leads to a different market outcome than does a tax imposed in ad valorem form, even in a competitive market. Two important exceptions to this conclusion hold when demand uncertainty is paired with constant marginal production costs or when market conditions are fully certain. When market conditions are uncertain, the two taxes also produce different outcomes than does an output quota.

In the case of demand uncertainty in either a competitive or monopoly market with rising marginal costs, the model produces some expected results--a marginal damage curve that is quite steep favors a quota system, while a curve that is quite flat curve favors a specific tax. In addition, we have shown that there always exists an intermediate range of marginal damage slopes over which the ad valorem tax is preferred. Furthermore, an increase in uncertainty widens the range of marginal damage slopes over which the ad valorem tax is preferred. Finally, these conditions imply that the ad valorem tax is always strictly preferred to at least one of the other policies.

Given demand and marginal cost parameters, the existence of market power strengthens the case for the tax policies relative to quantity controls. Furthermore, market power favors the specific tax relative to the ad valorem tax; in particular, the range of marginal damage slopes for which the specific tax is preferred is larger under monopoly than it is under competition. This result contrasts sharply with conclusions reached from comparing the efficiencies of the two taxes as revenue-raising devices.

The above results suggest that regulators who are relatively uncertain about future demand, as might be the case for products containing ozone-depleting chemicals, may find ad valorem taxation to be an attractive policy option. An ad valorem tax attains the highest expected welfare under some conditions; it also avoids the possibility of producing the lowest possible expected welfare, a chance that exists with both the specific tax and the quota. When the marginal damage curve is steep, an ad valorem tax does not risk the large losses caused by the great output flexibility present with a specific tax; when marginal damage is flat, an ad valorem tax does not risk the large losses caused by the rigidity of a quota. The fact that the relative performance of the ad valorem tax is best when the amount of uncertainty is greatest may also be attractive to policy makers.

If the major uncertainty in a market concerns the level of supply, however, the ad valorem tax appears to be a poor policy choice. Under supply uncertainty, the specific tax dominates the ad valorem tax in all circumstances. The specific tax also dominates the quota when the marginal damage curve is flat relative to the supply and demand curves; otherwise the quota is preferred.

Our results also indicate that computing the optimal ad valorem tax rate may be rather complicated, since (unlike the specific tax or quota) it differs from the policy that is optimal when the expected outcome holds for sure, even assuming linear functions. In light of this complication, we have also considered welfare effects of imposing the ad valorem tax that would be optimal under certainty. Like the optimal ad valorem tax, this simpler tax policy dominates the specific tax and quota over an intermediate range of slopes of marginal damage, and is never dominated by both the specific tax and quota.

The policies considered in this paper are less direct than are controls imposed when a good is produced or when it is used. Commodity taxes and quotas can, however, be important elements of externality regulation if damage is closely linked to output, or if administrative costs are reduced by imposing controls when a good is sold. Because market parameters will rarely be known with certainty, and adjusting policies over time is costly, the conclusions of this paper are relevant to the choice of control instrument. Of course, policy design also depends on a number of factors not considered above, such as political and administrative feasibility, reaction to inflation or to expected growth or decline in an industry, and revenue yield. To the extent that governments are motivated by both externality control and revenue raising concerns when imposing some commodity taxes, the analysis in this paper is complementary to comparisons of commodity tax forms in the context of a Ramsey problem. This paper has shown, however, that the comparison between tax forms in an uncertain world depends in part on factors not previously appearing in the literature.

Appendix

This appendix illustrates the derivation of the optimal ad valorem tax rates found in sections II and III of the text. Inspection of the expression for expected welfare loss [L.sub.i] given in section II reveals that the optimal tax may be found by minimizing the mean square difference between market and ex post optimal quantities. Thus, choose t to minimize E [([Q.sub.a] - [Q.sup.*]) [sup.2]], where [Q.sub.a] = [Q.sub.a] (t, [Alpha]) denotes the competitive market quantity with the ad valorem tax in place and [Q.sup.*] = [Q.sup.*] ([Alpha]) denotes the ex post optimal quantity. The first-order condition can be written as

(A.1) E [([Q.sub.a] - [Q.sup.*]) [Delta] [Q.sub.a]/ [Delta] t] = 0

The equilibrium quantity [Q.sub.a] must also satisfy

(A.2) (1 - t)[P.sub.d]([Q.sub.a]) - [P.sub.s]([Q.sub.a]) = 0.

Solve equation (A.2) for [Q.sub.a] and differentiate to obtain [Delta] [Q.sub.a]/ [Delta] t. The resulting two expressions and the solution for [Q.sup.*] (all of which are presented in section II of the text) are [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [Q.sup.*] = (a + [Alpha] - c - e)/[D.sup.*], where [D.sub.a] = b(1 - t) + d and [D.sup.*] = b+d+f . Substitute these three expressions into the first-order condition (A.1) and take expectations to obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where E [Q.sub.a] = [a(1 - t) - c]/[D.sub.a], E [Q.sup.*] = (a - c - e)/ [D.sup.*], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Multiplying through by [D.sup.*] [D.sub.a] yields

(A.3) -t[a(d + f) + b(c + e)] + f (a - c) +e(b + d) + [d [[Sigma].sup.2]/(ad + bc)] [(f - t(d + f)] = 0.

Solving expression (A.3) for t yields the formula for the optimal ad valorem tax [t.sub.a] given in equation (4) of the text.

For the corresponding tax rate in a monopoly market, simply replace expression (A.2) with

(A.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denotes the monopoly equilibrium quantity with the ad valorem tax r in place. Solve equation (A.4) for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and differentiate to find [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (both expressions are presented in the text of section III). Substitute these along with the expression for [Q.sup.*] into (A.1) and solve for the optimal tax rate as in the competitive case. We thank Thomas D. Crocker and an anonymous referee for helpful comments on previous versions of this research.

(1.) Similarly, Poterba [29] and others have emphasized that the estimated costs of reducing carbon dioxide emissions are very uncertain. For alcohol and cigarettes, future demand is unknown because of difficulty in forecasting the adoption or impact of other policies to reduce consumption, such as restrictions on public use, increases in minimum age requirements, information campaigns or moral suasion. To illustrate how the empirical treatment of regulations governing smoking in public can affect demand estimates, see Wasserman et al. [34] and Grossman [16]. (2.) See, for example, Pogue and Sgontz [28] for computations of optimal specific taxes, and Phelps [27] for ad valorem taxes.

(3.) Grossman et al. [17] review a number of studies that calculate Pigouvian taxes for cigarettes and alcohol. The assumptions that usually underlie these calculations (demand, supply, and marginal damage are known with certainty, markets are constant-cost and competitive) insure that specific and ad valorem taxes yield identical outcomes.

(4.) Ex ante specific and ad valorem taxes produce equivalent levels of output in an uncertain competitive market only if the uncertainty exclusively affects the location of demand and if marginal production cost is constant. When firms possess market power, the two tax forms produce outcomes that differ in other ways even when market conditions are certain; see Skeath and Trandel [32].

(5.) The alternative case in which Pigouvian taxes are used to help meet government revenue requirements has also received attention. For example, Ballard and Medema [3] compute the marginal cost of public funds when a small increase in government spending is financed by Pigouvian taxation, while Bovenberg and de Mooij [6] consider the impact of environmental levies on preexisting distortions and Bovenberg and Goulder [7] consider the design of optimal emissions taxes when distortionary taxes are also present. Oates [25] surveys much of the recent literature on this topic.

(6.) Comes and Sandler [11, 59] note the possibility of determining circumstances where a uniform quantitative restriction is superior to a (noncontingent) specific tax in conditions of uncertainty, but they do not undertake the determination, nor do they consider ad valorem taxes. In fact, to the best of our knowledge, there has to date been no comparison between ad valorem Pigouvian taxes and quantitative restrictions on output.

(7.) Yang and Stitt [37] recently have examined how the elasticity of supply affects optimal ad valorem and specific taxes in the context of a Ramsey problem with no uncertainty and no externalities.

(8.) The assumption that marginal damage depends on output alone simplifies the comparison of policies which restrict output. If damages depend on other variables such as emissions of a pollutant, administrative costs may still favor commodity taxes or quotas relative to policies aimed more directly at the source of external damage. Also, we assume marginal damage is known for sure, because damage uncertainty would not affect our welfare comparisons unless it were correlated with the demand shock. Stavins [33] recently considered correlated damage uncertainty.

(9.) Throughout the paper we assume that a mechanism, such as a competitive market in transferrable quotas, exists to allocate the quota efficiently among firms.

(10.) Alternatively, cf [is greater than] de also indicates that [[Eta].sub.g] [is greater than] [[Eta].sub.m] where m is used to denote the marginal social cost of production, G + [P.sub.s]. When cf [is greater than] de holds, therefore, marginal damage becomes a larger fraction of marginal social cost as output rises.

(11.) Note also that if [??.sub.a] were imposed in the uncertainty case, the expected effective unit tax rate [??.sub.a] E [P.sub.d] equals the optimal specific tax [t.sub.s].

(12.) The importance of both [[Eta].sub.g] [is greater than] [[Eta].sub.s] and [[Delta].sup.2] [Q.sub.a]/[Delta] [t.sub.a] [Delta] [Alpha] [is less than or equal to] to the result that [t.sub.a] [is greater than] [??.sub.a] can also be seen in the last term in equation (4). If [Delta] [Q.sub.a]/[Delta] [t.sub.a] [is less than] 0 was independent of [Alpha], the formula for [t.sub.a] would reduce to that for [??.sub.a]. In fact, however, when [[Eta].sub.g] [is greater than] [[Eta].sub.s], large (small) values of [Alpha] imply both that [G.sup.*] is large relative to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and that [Delta] [Q.sub.a]/[Delta] [t.sub.a] is large (small) in absolute value. Therefore, [[Eta].sub.g] [is greater than] [[Eta].sub.s], implies that [t.sub.a] [is greater than] [??.sub.a].

(13.) The critical value [f.sub.1] at which [L.sub.q] = [L.sub.s] lies between [f.sub.2] and [f.sub.3]. (14.) The ad valorem tax dominates the specific tax when [[Sigma].sup.2] [is greater than] [(b + d)/f] [sup.2][(cf - de)/d] [sup.2] - [(ad + bc)/d] [sup.2]; it dominates quota for [[Sigma].sup.2] [is greater than] [(cf - de)/d] [sup.2] - [(ad + bc)/d] [sup.2]. An implausibly large, though logically possible, variance may be required for the ad valorem tax to dominate when values of slope and intercept parameters strongly favor the specific tax or quota.

(15.) The simpler tax is preferred to the quota for all f [is less than] [f.sub.2] = (1/c)[d(a + e) + bc], and is preferred to the specific for all f [is greater than] [f.sub.3] = de(b + d)/[d(a + c) + 2bc]. It follows that (given d [Sigma] [is greater than] 0) [f.sub.2] [is greater than] [f.sub.2] [is greater than] [f.sub.1] [is greater than] [f.sub.3] [is greater than] [f.sub.3].

(16.) This assumption is broadly consistent with results of Oates and Strassman [26], which suggest that taxing polluting monopolists is likely to yield net gains in allocative efficiency.

(17.) Note that when [[Tau].sub.s] is imposed and b [is greater than] f, a positive demand shock causes the monopolist's output to fall short of the optimal output. Because the change in demand minus marginal revenue exceeds the change in social cost minus private cost, the monopolist responds to a positive [Alpha] by increasing [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by less than the increase in [Q.sup.*].

(18.) The equality f = b implies that [[Tau].sub.s] = e. When f [is less than] ([is greater than]) b, then [[Tau].sub.s], [is less than] ([is greater than]) e.

(19.) To examine outcomes under decreasing private marginal cost, consider -[Beta] [is less than] d [is less than] 0 and assume f [is greater than] b. Under these conditions, reductions in d favor the quota relative to both tax policies. The two taxes are of course equivalent when d = 0; a marginal reduction in d from zero favors the specific relative to the ad valorem tax. Indeed, when marginal cost declines slowly (0 [is greater than] d [is greater than] - 2 [Beta] c/([a.sup.2] + [[Sigma].sup.2] - [c.sup.2])) the specific tax is preferred to the ad valorem tax. If marginal cost falls sharply (-[Beta] [is less than] d [is less than] - 2[Beta] c/([a.sup.2] + [[Sigma].sup.2] - [c.sup.2])), however, then the ad valorem tax is preferred for values of [Phi] exceeding de([Beta] + d)/[c([[Beta].sub.d]) - E E [[micro].sup.2]) [sup.1/2]].

(20.) The ad valorem tax may not be unambiguously superior for a monopolist subject to rate-of-return regulation in the presence of an Averch-Johnson effect [36].

(21.) Qualitatively similar results are obtained if the policy maker is uncertain about monopoly marginal cost.

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