Second-Best Pollution Taxes and the Structure of Preferences.

Helmuth Cremer [*]

Firouz Gahvari [+]

Norbert Ladoux [++]

We characterize optimal taxes on polluting and nonpolluting goods in Ramsey and Mirrlees second-best environments. The polluting good tax differs from the Pigouvian tax by Ramsey terms in the first and by Stiglitz/Mirrlees plus another adjustment term in the second. These terms can be positive, negative, or zero. If preferences are weakly separable in public and private goods, with the private good subutility weakly separable in labor and produced goods, nonpolluting goods are taxed uniformly and the concept of a tax differential between polluting and nonpolluting goods is well defined. The differential is then less than the Pigouvian tax in the Ramsey framework, but it can be greater, equal to, or smaller than the Pigouvian tax in the Mirrlees second best. In Mirrlees second best, if preferences are separable in labor supply and other goods, the second-best tax differential is identical to the Pigouvian tax.

1. Introduction

The Pigouvian prescription for correcting an externality is to levy a tax on it equal to its marginal social damage. This is a first-best remedy, which may have to be modified in the second best. Sandmo (1975) made this point in a pioneering work more than two decades ago. He characterized second-best taxes in an economy where there are other distortionary taxes in the system. More recently, a number of authors have refined and extended Sandmo's results in a number of ways. [1] On the policy front, many European countries are considering carbon taxes, or taxes on energy consumption, to fight global warming. This is aimed at curtailing [CO.sub.2] emissions, from fossil-fuel combustion, deemed as the most important of greenhouse gases. Other counties like Australia, New Zealand, Sweden, Netherlands, and Finland already have carbon taxes in place. Such a tax has also been proposed and discussed in the United States (Poterba [1991, 1993] discusses various aspects of this tax at length). [2]

An important policy question in this context is to determine how second-best considerations change the level of the optimal tax on a polluting good. This question was first studied by Bovenberg and van der Ploeg (1994) and Bovenberg and de Mooij (1994). The latter paper concluded that "in the presence of preexisting distortionary taxes, the optimal pollution tax typically lies below the Pigouvian tax ... " (p. 1085). As stated, however, this inquiry can take many different meanings. What these authors have in mind resembles Atkinson and Stern's (1974) discussion of the rule for, versus the level of, optimal provision of public goods in the second best. The first-best rule for levying a tax on a polluting good is to set it equal to its marginal social damage. This rule will be modified in the second best. Now suppose we were to set the tax on the basis of this modified rule. Would the resulting tax be lower or higher than that which would have resulted had we followed the first-best conventional rule? [3]

Posed this way, one sees immediately that one cannot in general provide a clearcut, or even a meaningful, answer to this question. The first-best rule calls for a tax that is solely corrective. The second-best tax rule, on the other hand, embodies both corrective and optimal tax objectives. In a Ramsey tax setting, for example, the tax structure may be described as "Ramsey-plus-Pigou." [4] It thus seems rather odd that one is able to draw a clearcut conclusion regarding the size differential between the second-best and the Pigouvian tax. Indeed, one wonders what exactly one learns by comparing these two conceptually different taxes.

A second unanswered question arises in the context of the "normalization debate" that followed Bovenberg and de Mooij's contribution. Fullerton (1997) and Schob (1997) challenged the validity of Bovenberg and de Mooij's claim and argued that it is due to their particular normalization in which the tax on the nonpolluting good is set equal to zero. They pointed out that the correct statement of Bovenberg and de Mooij's result should have referred to the tax differential between polluting and nonpolluting goods (and not the tax on the polluting good per se). Fullerton (1997, p. 248) also quotes Bovenberg as writing, "To avoid confusion, we probably should have said that optimal tax differentiation is less than the Pigouvian rule would suggest." Now when there is one polluting good and one nonpolluting good (as the participants in this debate have assumed), the concept of a tax differential between the polluting and nonpolluting good is clear enough. However, one good per category is only a simplification. There must generally be at least many nonpolluting goods. One must then wonder how the tax differential is defined. Can we meaningfully speak of one tax differential when there are many goods per category?

The first aim of the current paper is to clear up these unanswered questions. To this end, following Bovenberg and van der Ploeg (1994), Bovenberg and de Mooij (1994), Fullerton (1997), and Schob (1997), the article at first examines the optimal tax problem in a traditional Ramsey setting. In contrast with these authors and to shed light on the question of a single tax differential, we allow for m nonpolluting goods. For simplicity, however, we keep the one-polluting-good formulation. We characterize the optimal tax rates on polluting and nonpolluting goods for a general specification of preferences and show that the second-best tax on the polluting good generally reflects both optimal tax considerations as well as Pigouvian concerns. The presence of optimal tax considerations implies that one cannot draw general conclusions regarding the relative size of the second-best polluting tax versus the Pigouvian tax. Nor can one speak of one tax differential.

We then follow Bovenberg and de Mooij and assume that preferences are weakly separable in public and private goods, with the private good subutility being weakly separable in labor and produced goods. We will argue that this assumption provides the key to answering both of our questions. We prove that these preferences imply that all nonpolluting goods must be taxed at a uniform proportionate rate. This property makes the concept of one tax differential meaningful, thus answering our second question. These preferences also imply that optimal tax considerations do not call for differential tax treatment of polluting and nonpolluting goods. That is, the tax differential reflects solely externality-correcting objectives. The separability assumption thus purges the second-best tax differential from its optimal tax objectives, making its comparison with the Pigouvian tax (set on the basis of the first-best rule) to be meaningful. This answers the first question. We point out that these results are closely linked t o Sandmo's (1974) earlier result concerning uniform commodity taxes in the traditional Ramsey tax framework (and in the absence of externalities). Finally, we formally prove that the tax differential between polluting and nonpolluting goods must be less than the Pigouvian tax regardless of the tax normalization rule (setting the income tax rate or the tax on nonpolluting goods equal to zero).

The second aim of the current article is to argue that the validity of Bovenberg and de Mooij's claim is very much dependent on the particular second-best environment they posit. Theirs is a one-consumer economy in which lump-sum taxation is ruled out. There are of course many types of second-best distortions. The least arbitrary, we believe, is that of the more modern optimal tax theory a la Mirrlees (1971). This theory allows for individuals to be heterogeneous and justifies the absence of first-best taxes by the existence of informational asymmetries between tax authorities and taxpayers. We thus recast and reexamine the issues raised by Bovenberg and de Mooij in Mirrlees' setting. An important feature of our analysis is that we allow for nonlinear income taxation. The optimal income tax literature typically assumes that incomes are publicly observable. This allows income to be taxed nonlinearly (as well as linearly). Consequently, there are no informational grounds for restricting income taxes to be linea r.

We consider a model with heterogeneous agents. There are H types of agents who differ in earning abilities and/or tastes. This follows Cremer, Gahvari, and Ladoux (1998). However, our specification of preferences is more general than theirs. We impose no particular restrictions on preferences initially, except for the customary quasi-concavity and differentiability. Additionally, we consider a number of different separability restrictions. Cremer, Gahvari, and Ladoux had assumed, at the outset, that preferences are additive in emissions and goods. [5]

We give a characterization for the optimal commodity taxes (on polluting and nonpolluting goods) and argue that the second-best tax on the polluting good reflects optimal tax objectives as well as Pigouvian concerns. This dual role prevents one from drawing general conclusions regarding the relative size of the second-best tax and the Pigouvian tax. We define the concept of the optimal environmental levy by the additional terms due to environmental considerations that appear in the optimal tax formula for the polluting good (and are absent in the tax formulas for nonpolluting goods) and show that it is independent of individual types. This is a different concept from the tax differential between polluting and nonpolluting goods (which includes both environmental and optimal tax objectives and may also depend on individual types). We show that the environmental levy thus defined may exceed, fall short of, or be equal to the Pigouvian tax.

Next we consider the implications of different preference structures for optimal pollution taxes. We prove that, if preferences are separable in labor supply and all other goods, the environmental levy will be equal to the Pigouvian tax and to the tax differential between polluting and nonpolluting goods. Specifically, we prove that polluting goods should be taxed only for externality-correcting purposes and at a rate equal to the full Pigouvian tax. Finally, we study preferences that are weakly separable in public and private goods, with the private good subutility being weakly separable in labor and produced goods. We show that, in contrast with the second-best Ramsey environment, the second-best pollution tax (which in this case is equal to the environmental levy) can now be greater, equal to, or smaller than the Pigouvian tax. [6] The relationship depends crucially on the interaction of environmental quality and labor supply. When the environmental quality and labor supply are complementary and individual s have identical tastes, we show that the optimal tax exceeds the Pigouvian tax. On the other hand, when the two goods are substitutes, the optimal tax will be less than the Pigouvian tax. The reason for this result is that the complementarity/substitutability relationship between environmental quality and labor supply allows the government to weaken otherwise binding self-selection constraints through departures from Pigouvian taxes.

2. The Second-Best Ramsey Setting

Consider an economy with N identical individuals each endowed with one unit of time. They have preferences defined over n "clean" goods, C = ([C.sub.1], [C.sub.2], ..., [C.sub.m]), a "dirty" good, D, labor supply, L, a public good, G, and environmental quality, E. The production technology is linear. Preferences are represented by

U = U(C,D,L,G,E), (1)

where U is strictly quasi-concave, twice continuously differentiable, and strictly increasing in C D, G, E and decreasing in L. The link between environmental quality and consumption of the dirty output is

E = e(ND), (2)

with e'(*) [less than] 0. This specification is more general than that in the cited literature, where only one clean good is considered. This allows a better understanding of the structure of second-best taxes.

Define the Pigouvian tax to measure "the direct [marginal] social damage of pollution" (Bovenberg and van der Ploeg 1994, p. 361), or in Fullerton's (1997, p. 246) terminology, "the [private] dollar cost of environmental damage per unit of the dirty output." This is the definition also used by Bovenberg and de Mooij (1994). We have the following.

DEFINITION 1. Let [alpha] denote the marginal utility of income, which is the same for everyone, and [U.sub.E] denote the marginal utility of environmental quality. The Pigouvian tax, [tau], is defined (in the first as well as the second best) as the private dollar cost of environmental damage per unit of dirty output. Formally,

[tau] [equivalent] -Ne'[U.sub.E]/[alpha]. (3)

Bovenberg and de Mooij's claim is that "in the presence of preexisting distortionary taxes, the optimal pollution tax typically lies below the Pigouvian tax" as measured by [tau].

The Optimal Tax Problem

The representative consumer maximizes utility subject to the budget constraint 

[[[sigma].sup.m].sub.i=1] [q.sub.i][C.sub.i] + [q.sub.d]D = [w.sub.n]L - T, (4)

where [q.sub.1] and [q.sub.d] are the consumer prices of [C.sub.i] (i = 1, 2, ..., m) and D, [w.sub.n] is the net of tax wage, T the lump-sum tax, if any. Denote the unit tax rates on [C.sub.i], D, and L by [t.sub.i], [t.sub.d], and [t.sub.w]. Normalize the producer prices of C and D at one and denote the gross of tax wage by w. The following relationships hold: [q.sub.i] = 1 + [t.sub.i] for all i = 1, 2, ..., m, [q.sub.d] = 1 + [t.sub.d], and [w.sub.n] = w(1 - [t.sub.w]). The maximization problem yields the demand functions for C and D and the supply function for L. The individual's indirect utility function can then be defined as

v = v(q, [q.sub.d], [w.sub.n], T, G, E)

[equivalent] U(C(q, [q.sub.d], [w.sub.n], T, G, E), D(q, [q.sub.d], [w.sub.n], T, G, E), L(q, [q.sub.d], [w.sub.n], T, G, E), G, E), (5)

where q is the vector of clean good prices.

To determine the optimal tax rates, the indirect utility function (Eqn. 5) is maximized with respect to the available tax instruments and subject to the government's budget constraint

[[[sigma].sup.m].sub.i=1] [t.sub.i][C.sub.i] + [t.sub.d]D + [t.sub.w]wL + T = [p.sub.g]G/N, (6)

where [p.sub.g] denotes the producer price of G. Note that the government is not optimizing over G. This corresponds to the standard assumption of a fixed revenue constraint in the optimal taxation literature. [7] Note also that, since the demand and the labor supply functions are homogeneous of degree zero in T and q, [q.sub.d], [w.sub.n], one of the consumer prices (including the wage) can be normalized to one. Equivalently, one of the tax rates can be set to zero.

Introduce

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

and define

[gamma] [equivalent] -Ne'/1 - Ne'[delta]D/[delta]E[[[[sigma].sup.m].sub.i = 1] [t.sub.i][delta][C.sub.i]/[delta]E + ([t.sub.d] + Ne'[U.sub.E]/[micro])[delta]D/[delta]E + [t.sub.w] w[delta]L/[delta]E]], (8)

where [micro] is the shadow cost of raising one dollar of tax revenue (the Lagrange multiplier associated with the government's budget constraint). Next denote the matrix derived from A by deleting its third row and column (associated with [C.sub.1]) by [A.sub.C]. Similarly, define [A.sub.L] by deleting the second row and column of A (associated with L). Further, let [[gamma].sub.C] denote the expression obtained for [gamma] when [t.sub.1] = 0 and [[gamma].sub.L] the expression when [t.sub.w] = 0. In Appendix A, we prove that

[MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII] (9)

[MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII] (10)

The system of equations in Equation 9 characterizes the optimal tax rates when [t.sub.1] is normalized at zero, and the system of equations in Equation 10 characterizes the optimal tax rates for the normalization [t.sub.w] = 0. The first corresponds to the normalization rule followed by Bovenberg and de Mooij (1994) and Bovenberg and van der Ploeg (1994), who set the tax on their single clean good equal to zero. The second is the alternative discussed by Fullerton (1997) and Schob (1997).

First-Best Taxation

In Appendix A, we show that, if the lump-sum tax instrument T is available, the only taxes that will be used are T and

[t.sub.d] = -Ne'[U.sub.E]/[micro]. (11)

All other taxes are equal to zero. Note also that, in this case, [micro] = [alpha]. Consequently, the first-best tax is equal to the Pigouvian tax as the latter was defined.

Second-Best Taxation

The systems of Equations 9 and 10 characterize the second-best rule for taxation of the polluting good. Note that these are the same characterizations that one gets in the absence of externalities except that, instead of [t.sub.d] + Ne'[U.sub.E]/[micro] - [[gamma].sub.C] and [t.sub.d] + Ne'[U.sub.E]/[micro] - [[gamma].sub.L], we would have only [t.sub.d]. The equations show that, regardless of the normalization rule adopted, the optimal tax on the dirty output, [t.sub.d], reflects optimal tax objectives (Ramsey terms) as well as Pigouvian corrections (Pigouvian terms). This mix of terms makes it generally impossible to derive clearcut results concerning the relative size of [t.sub.d] and [tau].

We summarize these results in Proposition 1.

PROPOSITION 1. Assume individuals are identical, taxes are linear, lump-sum taxation is infeasible, and preferences are represented by U = U(C, D, L, G, E). Then

(i) Optimal taxes are characterized by Equation 9 if [t.sub.1] is normalized at zero and by Equation 10 if [t.sub.w] is normalized at zero.

(ii) Environmental concerns affect the second-best tax rule formulas via the additional expression, -- Ne'[U.sub.E]/[micro] + [[gamma].sub.c] or - Ne' [U.sub.E]/[micro] + [[gamma].sup.L].

(iii) No clearcut conclusions can be made concerning the relative size of the optimal tax on the polluting good and the Pigouvian tax, regardless of the normalization rule adopted.

(iv) The concept of tax differential between polluting and nonpolluting goods is not well defined.

Items (iii) and (iv) in Proposition 1 are the formal statements of the two unanswered questions we mentioned in the Introduction in relation to the debate on taxation of polluting goods within the Ramsey framework. To surmount these problems, one will have to impose suitable restrictions on preferences. Below, we will show that the separability restrictions adopted by the contributors to this literature have done precisely this.

Nested Separability of Preferences

Assume preferences are weakly separable in private and public goods (including environmental quality), with the private good subutility being separable in leisure and produced goods. Preferences are thus represented by U = U(H(Q(C, D), L), G, E), (12)

where Q(., .) is assumed homothetic.

Now consider the normalization rule of [t.sub.1] = 0 followed by Bovenberg and de Mooij and Bovenberg and van der Ploeg. Appendix A shows that, given the preference structure in Equation 12, the second-best environmental tax on clean and dirty goods have the following characterization:

[t.sub.s] = 0, s = 1, 2, ..., m, and [t.sub.d] = -Ne'[U.sub.E]/[micro] (13)

This is accompanied by a wage tax and a zero tax on all the clean goods. The structure of the environmental levy is thus precisely the one derived by these authors for their setup with one clean good.

There are two interesting features of this result. First, the second-best tax on D now reflects only externality-correcting objectives. It is thus legitimate to refer to it as an environmental levy and compare it with the Pigouvian tax, [tau]. Note also that [t.sub.d] is identical in form to the formula for the first-best tax. [8] Second, clean goods must be taxed uniformly. The tax is zero here because of the normalization rule. However, the uniformity result is not due to any particular normalization; it always holds (see Appendix A.) It is important to realize that these features are entirely due to the particular specification of preferences. In particular, the uniform tax on clean goods is a reincarnation of an earlier result on uniform taxation by Sandmo (1974). He has shown, in a traditional Ramsey setting without externality, that, when preferences are weakly separable in goods and labor supply and when the subutility in goods is homothetic, optimal second-best taxes are uniform. Of course, we now al so have an additional tax on the dirty good. But the separability assumptions also imply that this tax differs from the tax on clean goods only by an externality correcting term.

Note also that the particular characterization of the second-best environmental levy as given by Equation 13 shows that Bovenberg and de Mooij's (1994) discussion of the relationship between [t.sub.d] and [tau] in the first and in the second best can be reduced to one of a comparison between [micro] and [alpha]. Introduce [eta] [equivalent] [mirco]/[alpha] for the marginal cost of public funds. The comparison can be stated in terms of the size of [eta] in these two environments. The observation that [eta] is equal to one in the first best and greater than one in the second best will then be sufficient to prove that [t.sub.d] is equal to [tau] in the first best and smaller than [tau] in the second best, as claimed. The role that the assumption of an upward-sloping labor supply function plays in Bovenberg and de Mooij's is precisely to ensure that [eta] [greater than] 1 in the second best. [9]

The Normalization Issue

Now consider setting [t.sub.w], instead of [t.sub.1], equal to zero. Appendix A shows that, with this normalization too, the optimal tax rate on clean goods is uniform. Furthermore, it proves that the following relationship holds between the optimal tax rates on clean and dirty goods:

[t.sub.s]/[q.sub.s] = [t.sub.d] - (-Ne'[U.sub.E]/[micro])/[q.sub.d], s = 1, 2, ..., m. (14)

Equation 14 is illuminating in terms of the uniformity results it suggests. First, the tax rates on clean goods are proportionately uniform. Second, this uniform rate is also equal to the optimal tax rate, appropriately adjusted, on the dirty good. Note that the tax on the dirty good, [t.sub.d], incorporates a correction term for externality, namely, -Ne'[U.sub.E]/[micro]. Equation 14 shows that, if we subtract this corrective term out from [t.sub.d], the remaining part should be set, as a proportion of the consumer price of the dirty output, equal to the tax rate on clean goods (as a percentage of their consumer prices).

There are two sources behind the uniformity result. The first is due to the separability of preferences in environmental quality, E, and other goods. This ensures that a change in E will leave the demand for clean and dirty goods unaffected. Consequently, a tax that affects E through changing the dirty good, D, will not impact the clean goods, [C.sub.s]'s. Nor will it have a second-round effect on D. The second source is the separability of preferences in labor supply and other goods. Sandmo (1974) has proved, in a Ramsey world without externality, that this type of preferences, when the subutility of goods is homothetic, implies optimal commodity taxes are proportionately uniform. The uniformity result is thus an extension of Sandmo's (1974) to situations with consumption externalities.

Finally, note that one may reinterpret Equation 14 in another way. Denote the uniform tax rate by t. We will have

[t.sub.d] - t = (1 + t)(-Ne' [U.sub.E]/[micro]).

This leads to a comparison between the differential between the tax on the dirty and on the clean according to [10]

[t.sub.d] - t [less than] [tau] [Left/Right Arrow] [micro]/[alpha] [greater than] 1 + t.

We summarize these results in Proposition 2.

PROPOSITION 2. Assume individuals are identical, taxes are linear, lump-sum taxation is infeasible, and preferences are represented by U = U(H(Q(C, D), L), G, E) Then

(i) Second-best taxes on clean goods, C, are uniformly proportionate. This implies that the concept of a tax differential between polluting and nonpolluting goods is well defined.

(ii) Normalize the uniform tax on clean goods, t, at zero. The second-best tax on D reflects only externality-correcting objectives with the characterization [t.sub.d] = -Ne' [U.sub.E]/[micro]. Moreover, if labor supply is upward sloping, [micro] [greater than] [alpha] [Right Arrow] [t.sub.d] [less than] [tau], where [tau] [equivalent] -Ne'[U.sub.E]/[alpha].

(iii) Normalize the wage tax at zero. The second-best taxes on D and C are related according to ([t.sub.d] - t)/(1 + t) = -Ne' [U.sub.E]/[micro]. Moreover, we have [t.sub.d] - t [less than] [tau] [Left/Right Arrow] [micro]/[alpha] [greater than] 1 + t.

3. Second-Best a la Mirrlees

Consider the setup of the previous section. However, instead of postulating a representative consumer, assume that individuals differ in earning abilities and/or tastes. Specifically, assume there exists a continuum of individuals who can be grouped into H types. A person of type j has [w.sup.j] as his wage and

[[omega].sup.j] = U(C, D, L, G, E, [[theta].sup.j]), j = 1, 2, ..., H, (15)

where [[theta].sup.j] is a taste parameter. Normalize the population size at one and denote the proportion of persons of type j(j = 1, 2, ..., H) in the population by [[pi].sup.j].

It is the standard assumption in the optimal tax literature that an individual's type (i.e., earning ability) and labor input, L, are not observable by the government. His before-tax income, I = wL, on the other hand, is. This rules out first-best taxation of types as a policy instrument while allowing nonlinear taxation of income. It is then convenient to introduce a type-specific utility function describing preferences over the observables according to

[u.sup.j](C, D, I, G, E) [equivalent] U(C, D, I/[w.sup.j], G, E, [[theta].sup.j]). (16)

Note also that, with the H-group specification, the link between environmental quality and consumption of the dirty output is

E = e([[[sigma].sup.H].sub.j = 1] [[tau].sup.j][D.sup.j]). (17)

Given this framework, we set out to examine the properties of second-best environmental levies on the dirty good as part of an optimal tax scheme. In deriving the properties of optimal taxes, we follow the framework used by Stiglitz (1982, 1987) to discuss Pareto-efficient self-selection tax structures. (Of course, in his setup, there is no externality). This framework recognizes that the reason for the existence of distortionary taxes is the limitation on the type of information available to the government. If the government had information on individuals' types, it could simply impose first-best (differential) lump-sum taxes on everyone. In that case, the optimal tax on the dirty good would be the Pigouvian one. However, in its choice of tax structures, the government must recognize the limitations to its information. If it wishes a certain group of individuals to pay a certain tax, it must make sure that the intended group can do no better for itself by behaving differently. Otherwise, the group will simpl y adjust its behavior to obtain the allocation it prefers most and, as a result, pay a different amount of tax. An immediate implication of this observation is that the government's action is subject to the individuals' self-selection constraints.

To derive the (constrained) Pareto-efficient tax structures, we consider the standard equivalent problem of the government first choosing optimal allocations subject to resource balance, self-selection, and the relevant nonnegativity constraints. Having derived optimal allocations, we then describe the properties of the optimal environmental levies as part of the tax structures that can implement them.

Note that our procedure does not impose any restrictions on the nature of the implementing commodity taxes. As such, it allows for nonlinear taxes. Whether or not the government has the required information to levy nonlinear commodity taxes depends on the structure of public information in the economy. We ignore this question here. We shall see below that the type of preferences assumed by Bovenberg and de Mooij and Bovenberg and van der Ploeg make nonlinear commodity taxes redundant. In other words, we will not use them even if we are allowing for them [11]

Optimal Allocations

Denote the utility level of a j-type individual by [u.sup.j] when he chooses the allocation intended for him and by [u.sup.jk] when he chooses a k-type person's bundle. Thus denote, for all j [not equal to] k = 1, 2, ..., H,

[u.sup.j] = [u.sup.j] ([C.sup.j], [D.sup.j], [I.sup.j], G, E), (18a)

[u.sup.jk] = [u.sup.j] ([C.sup.k], [D.sup.k], [I.sup.k], G, E). (18b)

One can then describe the set of Pareto-efficient allocations as follows. Maximize

[[[sigma].sup.H].sub.j=1] [[gamma].sup.j][u.sup.j] (19)

with respect to [C.sup.j], [D.sup.j], and [I.sup.j], subject to the resource constraint

[[[sigma].sup.H].sub.j=1] [[pi].sup.j] ([I.sup.j] - [[sigma].sub.i] [[C.sup.j].sub.i] - [D.sup.j]) [greater than or equal to] [p.sub.g] G, (20)

the self-selection constraints

[u.sub.j] [greater than or equal to] [u.sup.jk], j [not equal to] k; j, k = 1, 2, ..., H (21)

and the production function for environmental quality given by Equation 17. Parameters [[gamma].sup.j] are positive constants with the normalization [[sigma].sub.j] [[gamma].sup.j]= 1. Note that, as in the representative consumer case, the government is not optimizing over G.

Denote the (nonnegative) Lagrangian multipliers associated with the resource constraint (Eqn. 20) by [micro] and with the self-selection constraint (Eqn. 21) by [[lambda].sup.jk]. Further, denote the partial derivatives of [u.sup.j] and of [u.sup.kj] with respect to [[C.sup.j].sub.i], [D.sup.i], E by [[u.sup.j].sub.i], [[u.sup.j].sub.D], [[u.sup.j].sub.E], and [[u.sup.kj].sub.i], [[u.sup.kj].sub.D], [[u.sup.kj].sub.E]. In Appendix B, by manipulating the first-order conditions of the government's problem, we show that the optimal allocations satisfy the following relationships for all j [not equal to] k = 1, 2, ..., H and all i, s = 1, 2, ..., m:

[[u.sup.j].sub.s]/[[u.sup.j].sub.i] = 1 + [[sigma].sub.k[not equal to]j] [[lambda].sup.kj] [[u.sup.kj].sub.s]/[micro][[pi].sup.j]/1 + [[sigma].sub.k[not equal to]j] [[lambda].sup.kj] [[u.sup.kj].sub.i]/[micro][[pi].sup.j], (22)

[[u.sup.j].sub.D]/[[u.sup.j].sub.i] = 1 + [[sigma].sub.k[not equal to]j] [[lambda].sup.kj] [[u.sup.kj].sub.D]/[micro][[pi].sup.j]/1 + [[sigma].sub.k[not equal to]j] [[lambda].sup.kj] [[u.sup.kj].sub.i]/[micro][[pi].sup.j] - [[[sigma].sub.j] ([[gamma].sup.j] + [[sigma].sub.k[not equal to]j] [[lamba].sup.jk]) [[u.sup.j].sub.E]/[micro] - [[sigma].sub.j] [[sigma].sub.k[not equal to]j] [[lambda].sup.kj] [[u.sup.kj].sub.E]/[micro]]e'/1 + [[sigma].sub.k[not equal to]j] [[lambda].sup.kj] [[u.sup.kj].sub.i]/[micro][[pi].sup.j]. (23)

Note that in deriving this characterization we do not rely on any single-crossing property and the characterization is valid regardless of which self-selection constraint binds.

Equations 22 and 23 serve as the basis for characterizing the second-best taxes on clean and dirty goods. We will undertake this exercise in the next section, where we will also discuss the concept of the Pigouvian tax and how it may be compared with the second-best environmental levy. To see the connection between second-best taxes and Equations 22 and 23, we note that the left-hand sides of these equations are the familiar marginal rates of substitution between [C.sub.s] and [C.sub.i] and between D and [C.sub.i] for an individual of type j. Moreover, a j-type person who purchases goods in the market treats E as given [12] and sets these marginal rates of substitution equal to their relative consumer prices. Similarly, [[u.sup.kj].sub.s]/[[u.sup.kj].sub.i] and [[u.sup.kj].sub.D]/[[u.sup.kj].sub.i] are the marginal rates of substitution between [C.sub.s] and [C.sub.i] and between D and [C.sub.i] for a k-type person who chooses the allocation intended for a j-type (i.e., a k type who mimics a j type). Note als o that, in implementing the constrained Pareto-efficient allocations, the government will always have one degree of freedom in setting the tax rates facing individual j. One can thus set, without any loss of generality, the tax rate on [C.sub.1] equal to zero for all individuals, that is, use [C.sub.1] as the numeraire. Proposition 3, proved in Appendix B, gives a characterization of the various tax rates.

PROPOSITION 3. Assume individuals are grouped into H different types whose preferences are represented by Equation 15. Individual j must face (marginal) tax rates on [C.sub.s], s = 2, 3, ..., m, and on D equal to [[t.sup.j].sub.s] = [[u.sup.j].sub.s]/[[u.sup.j].sub.1] - 1 and [[t.sup.j].sub.d] = [[u.sup.j].sub.d]/[[u.sup.j].sub.1] - 1 as characterized by

[[t.sup.j].sub.s] = -[[sigma].sub.k[not equal to]j] [[lambda].sup.kj] [[u.sup.kj].sub.1]/[micro][[pi].sup.j] ([[u.sup.j].sub.s]/[[u.sup.j].sub.1] - [[u.sup.kj].sub.s]/[[u.sup.kj].sub.1], (24)

[[t.sup.j].sub.d] = -[[sigma].sub.k[not equal to]j] [[lambda].sup.kj] [[u.sup.kj].sub.1]/[micro][[pi].sup.j] ([[u.sup.j].sub.D]/[[u.sup.j].sub.1] - [[u.sup.kj].sub.D]/[[u.sup.kj].sub.1] + [[sigma].sub.j] -[[gamma].sup.j]e'[[u.sup.j].sub.E]/[micro] + [[sigma].sub.j] [[sigma].sub.k[not equal to]j] -[[lambda].sup.jk]e'([[u.sup.j].sub.E] - [[u.sup.jk].sub.E])/[micro]. (25)

4. Pigouvian Tax Comparisons

Bovenberg and de Mooij's claim as stated has no obvious parallel in an economy with heterogeneous agents. The point is that, with heterogeneous individuals, there does not exist one Pigouvian tax because the concept is defined for a one-consumer economy. Recall that this is typically defined as "the dollar cost of environmental damage per unit of dirty output" (Fullerton, 1997, p. 246). In a many-consumer economy, the dollar cost of social marginal damage differs across individuals. This is true even in the first best if the social welfare function assigns different weights to different individuals. Put simply, one is prompted to ask, "To whose dollar cost of environmental damage should the tax be set equal?"

In what follows, we consider one possible interpretation of the Pigouvian tax when individuals are heterogeneous. It corresponds to Bovenberg and van der Ploeg's (1994) definition in their setup with many but identical consumers. The definition is motivated by Samuelson's condition for optimal provision of public goods. [13] The idea is to define the Pigouvian tax as the sum of the individuals' marginal rates of substitution between environmental quality, per unit reduction in the dirty output, and a private good used as the numeraire. Equivalently, this is (the negative of) the sum of the individuals' marginal rates of substitution between environmental damage, per unit of the dirty output, and the numeraire. Given our normalization that [C.sub.1] is the numeraire, [[u.sup.j].sub.1] will also show the marginal utility of income for individual j, which we denote by [[alpha].sup.j] We have the following definition. [14]

DEFINITION 2. The Pigouvian tax is the sum of the private dollar costs of the environmental damage per unit of the dirty output across all individuals. Formally,

[tau] [equivalent] [[sigma].sub.j] -[[pi].sup.j]e'[[u.sup.j].sub.E]/[[alpha].sup.j]. (26)

First-Best Taxation

The characterization of second-best allocations in Equations 22 and 23 includes the characterization of the first best as a special case. At first-best allocations, self-selection is not a constraint on the government's problem. Thus, set [[lampda].sup.kj] = 0 for all k [not equal to] j in Equations 22 and 23. We will have, for all I, s = 1, 2, ..., m and j = 1, 2, ..., H,

[[u.sup.j].sub.s]/[[u.sup.j].sub.i] = 1, (27a)

[[u.sup.j].sub.D]/[[u.sup.j].sub.i] = 1 + [[sigma].sub.j] -[[lambda].sup.j]e'[[u.sup.j].sub.E]/[micro]. (27b)

Recall that optimization in the market entails equality of marginal rates of substitution with relative consumer prices. It then follows from Equation 27b that the implementation of the first-best allocation requires no distortionary taxes on [[C.sup.j].sub.i]'s as expected, but a tax on [D.sup.j] relative to [[C.sup.j].sub.i] (j = 1, 2, ..., H) equal to

[t.sub.d] = [[sigma].sub.j] -[[gamma].sup.j]e'[[u.sup.j].sub.E]/[micro]. (28)

Note that the right-hand side of Equation 28 shows the social cost of environmental damage associated with any person's purchase of a last unit of D in terms of dollar cost to the government. Note also that the tax is the same for everyone because the nature of the externality is such that only the aggregate size of D matters.

To determine the relationship between [t.sub.d] as characterized by Equation 28 and the Pigouvian tax as defined by Equation 26, note that, in the first best, we have [[pi].sup.j]/[[alpha].sup.j] = [[gamma].sup.j]/[micro] for j = 1, 2, ..., H (see Equation B3 in Appendix B). Consequently, the two expressions are identical, that is,

[t.sub.d] = [tau].

Second-Best Taxation

Introduce

[delta] [equivalent] [[sigma].sub.j] [[[sigma].sub.k[not equal to]j] [[lambda].sup.kj]e'[[u.sup.kj].sub.1]/[micro]([[u.sup.j].sub.E]/[[u.s up.j].sub.1] - [[u.sup.kj].sub.E]/[[u.sup.kj].sub.1] (29)

Appendix B shows that Equation 25 which characterizes [[t.sup.j].sub.d], can alternatively be written as

[[t.sup.j].sub.d] = - [[sigma].sub.k[not equal to]j] [[lambda].sup.kj][[u.sup.kj].sub.1]/[micro][[pi].sup.j]([[u.sup.j].su b.D]/[[u.sup.j].sub.1] - [[u.sup.kj].sub.D]/[[u.sup.kj].sub.1]) + [tau] + [delta]. (30)

In the second best, a subset of [[lamda].sup.kj]'s are nonzero. It thus follows from the characterization of [[t.sup.j].sub.d] in Equation 30 that, in contrast with first best, the optimal tax now differs from the Pigouvian tax, [tau]. The tax has three distinct components, one that is type specific and two that are the same for all types.

The type-specific term in [[t.sup.j].sub.d] has an anologue appearing in the tax formula for nonpolluting goods as shown in Equation 24. This is the Stiglitz/Mirrlees term indicating the distortion on j-type's consumption (in this case, of good D) in order to relax the self-selection constraints for types k [not equal to] j. This term is generally nonzero unless preferences are of special forms. The other two components of [[t.sup.j].sub.d] do not appear in the tax formula for nonpolluting goods. One is the Pigouvian tax, [tau], as defined earlier. The other term, [delta], has the following interpretation. It reflects the distortion on each type's consumption of D in order to change E (the environmental quality) and relax any possible binding self-selection constraint with respect to E. [15]

The preceding discussion tells us that the optimal tax on D may distort D from what it would be with just the Pigouvian tax, [tau] for two reasons--one related to j-type's consumption of D and the other related to every type's impact on E through each type's consumption of D. This also tells us that, because the sign and the size of the Stiglitz/Mirrlees term varies across goods, the concept of a tax differential (between polluting and nonpolluting goods) is again not well defined. Nevertheless, because [tau] + [delta] appear only in [[t.sup.j].sub.d] and not in the tax formula for nonpolluting goods, it will be useful to focus on these two terms separately. [16] This suggests the following terminology.

DEFINITION 3. We call the additional terms due to environmental considerations that enter in the tax formula determining [[t.sup.j].sub.d] (and are absent in the tax formulas for nonpolluting goods) the optimal environmental levy. This is equal to [tau] + [delta] with [tau] given by Equation 26 and [delta] by Equation 29.

The optimal environmental levy thus defined exceeds, is equal to, or falls short of the Pigouvian tax as [delta] happens to be positive, zero, or negative. To see what determines the sign of [delta], let us assume that everyone has the same tastes (i.e., [[theta].sup.j] = [[theta].sup.k] for all k, j = 1, 2, ..., H) and that the optimum is characterized by redistribution from high- to low-wage individuals. This implies that, if [[lamda].sup.kj] [greater than] 0, then [w.sup.k] [greater than] [w.sup.j]. Now [[u.sup.j].sub.E]/[[u.sup.j].sub.1] denotes the marginal rate of substitution between the environmental quality (E) and a numeraire good ([C.sub.1]) for individual j and [[u.sup.kj].sub.E]/[[u.sup.kj].sub.1] shows the same marginal rate of substitution for k at the consumption bundle intended for j (the mimicker). Given [w.sup.k] [greater than] [w.sup.j], the mimicker will have to work less to earn the same income level (and consume the same consumption bundle) as j. It follows that which type has a higher or a lower marginal rate of substitution between E and [C.sub.1] depends on whether E and labor are complements or substitutes. If E is complementary to labor, the lower labor supply by k also implies that k will have a lower marginal rate of substitution between E and [C.sub.1], (values E less). Under this circumstance, [[u.sup.j].sub.E]/[[u.sup.j].sub.1] - [[u.sup.kj].sub.E]/[[u.sup.kj].sub.1] will be positive. This results in [delta] [greater than] 0 and an optimal environmental levy that exceeds the Pigouvian tax. In the same manner, if E and L are substitutes, then [[u.sup.j].sub.E]/[[u.sup.j].sub.1] - [[u.sup.kj].sub.E]/[[u.sup.kj].sub.1] is negative. Consequently, [delta] [less than] 0 also and the optimal environmental levy will fall short of the Pigouvian tax. Finally, it is also possible that environmental quality and labor are neither substitutes nor complements. Under this circumstance, [[u.sup.j].sub.E]/[[u.sup.j].sub.1] = [[u.sup.kj].sub.E]/[[u.sup.kj].sub.1]. We will then have [delta] = 0 and a n optimal environmental levy identical to the Pigouvian tax.

The intuition behind the preceding result is as follows. Suppose the tax system is optimized subject to the constraint that the tax on the polluting good for type j is equal to his Stiglitz/Mirrlees term plus the Pigouvian tax. An increase in the environmental quality associated with this pattern of taxation then allows for more redistribution from higher to lower wage persons, if environmental quality is complementary to labor supply. [17] This is achieved by increasing the polluting good tax for all types and thus lowering aggregate emissions and increasing E. To see how the increase in E helps, consider a high-wage person who misrepresents his type (the mimicker). He earns the low-wage person's income but works less. The marginal rate of substitution between environmental quality and a numeraire good for the mimicker thus differs from the low-wage person's marginal rate of substitution (at the latter's consumption bundle) essentially because of their different labor supply. With the mimicker supplying less labor and with complementarity, he values environmental quality less (than the low-wage type). Consequently, the increase in E is worth less to the mimicker than to the low-wage type. The required gain to the truthful high-wage type, to keep the self-selection constraint binding for him, will thus also be less than the gain to the low-wage type; that is, we can have further redistribution from high- to low-wage types. A similar argument shows why the optimal environmental levy must fall short of the Pigouvian tax if environmental quality and labor are substitutes.

We summarize our results thus far in Proposition 4.

PROPOSITION 4. Assume preferences are represented by Equation 15. Then

(i) Environmental concerns do not change the rule for determining optimal taxes on nonpolluting goods; they consist of Stiglitz/Mirrlees terms only as characterized by Equation 24.

(ii) The rule for determining the optimal tax on the polluting good changes. The tax will have three components: the Stiglitz/Mirrlees term, the Pigouvian tax term [tau], defined in Equation 26, and an additional term [delta], defined by Equation 29.

(iii) We have [delta] [Left/Right Arrow] 0, so that the optimal environmental levy [tau] + [delta] may exceed, fall short of, or be equal to the Pigouvian tax [tau].

It is clear from Proposition 4 and our previous discussion that the ambiguity in the signs of Stiglitz/Mirrlees term and [delta] do not allow one to draw any general conclusions about the relative size of the optimal tax on the polluting good and the Pigouvian tax. Indeed, the optimal tax on the polluting good may very well differ according to the individual types (i.e., be nonlinear). Nor will it be possible to talk meaningfully of a tax differential between polluting and nonpolluting goods. To allow such inferences, one must impose certain restrictions on preferences. Below, we will discuss two such restrictions. One, separability in labor, ensures that both Stiglitz/Mirrlees term and [delta] will vanish so that the optimal tax on the polluting good is Pigouvian. The other, nested separability, implies that the Stiglitz/Mirrlees term is zero so that the optimal tax differs from the Pigouvian tax only in [delta].

Separability in Labor

The previous subsection argued that, if environmental quality and labor supply are neither complements nor substitutes, the environmental levy will be identical to the Pigouvian tax. It turns out that this case has very important implications for the whole structure of second-best taxes. We thus study this possibility in more detail.

Assume preferences are weakly separable in labor and all other goods are specified by

[omega.sup.i] = U(F(C, D, G, E),f(L, [[theta].sup.j])), j = 1, 2, ..., H. (31)

These preferences imply that, at the same consumption bundles, j- and k-type persons have identical marginal rates of substitution for every pair of goods, including the environmental quality. Specifically, we will have

[[u.sup.j].sub.s]/[[u.sup.j].sub.i] = [[u.sup.kj].sub.s]/[[u.sup.kj].sub.i], [[u.sup.j].sub.D]/[[u.sup.j].sub.i] = [[u.sup.kj].sub.D]/[[u.sup.kj].sub.i], [[u.sup.j].sub.E]/[[u.sup.j].sub.i] = [[u.sup.kj].sub.E]/[[u.sup.kj].sub.i]. (32)

The implications of these equalities for the structure of second-best taxes are drastic. Substituting from Equation 32 in Equations 24 and 30, it immediately follows that second-best taxes call for the nonpolluting goods to go tax free, with a tax on the polluting good equal to

[t.sub.d] = [tau].

This means that commodity taxes are not useful instruments for optimal taxation. Polluting goods should be taxed only for externality-correcting purposes and at a rate equal to the full Pigouvian tax. This also means that the optimal tax on the polluting good is now equal to the

Pigouvian tax, the tax differential between polluting and nonpolluting goods, and the optimal environmental levy.

This result is in keeping with an earlier result by Boadway and Keen (1993), who show, given the same labor separability assumption as here, that distortionary taxation does not obviate the Samuelson condition for optimal provision of public goods as long as the government has the ability to levy a general income tax. It is also related to Kaplow (1996), who makes a similar argument to Boadway and Keen's. The result is also a generalization of Atkinson and Stiglitz's (1976) result on the nonusefulness of differential commodity taxes in the presence of a general income tax. [18]

Nested Separability

Assume that preferences are weakly separable in private and public goods, with the private good subutility being separable in labor and produced goods. This is specified by

[[omega].sup.j] = U(F(H(C, D), Q(L, [[theta].sup.j])), f(G, E, [[theta].sup.j])), j = 1, 2,...,H. (33)

The separability of the private good subutility in labor and produced goods implies that, at the same consumption bundles, j- and k-type persons have identical marginal rates of substitution for every pair of produced goods. Specifically, we have

[[u.sup.j].sub.s]/[[u.sup.j].sub.i] = [[u.sup.kj].sub.s]/[[u.sup.kj].sub.i], [[u.sup.j].sub.D]/[[u.sup.j].sub.j] = [[u.sup.kj].sub.D]/[[u.sup.kj].sub.i] (34)

Substituting these equalities into Equations 24 and 30 also has interesting implications for our tax structure. The nonpolluting goods must again go untaxed, while the polluting goods should be taxed according to

[t.sub.d] = [tau] + [delta].

The optimal tax on the polluting good is independent of the individual type. It is also equal to the environmental levy and the tax differential between polluting and nonpolluting goods.

Bovenberg and de Mooij's Claim

The nested separability is the preference structure that Bovenberg and de Mooij (1994) postulate in their Ramsey second-best analysis of optimal polluting good taxes. The validity of their claim for the Mirrlees second best must thus be examined under this preference structure. Now, as shown in Equation 35, the optimal tax in this case differs from the Pigouvian tax by [delta]. Moreover, [delta] can take positive, negative, or zero values depending on (if tastes are identical) the complementarity, substitutability, or independence of environmental quality and labor supply. Consequently, [t.sub.d] may exceed, fall short of, or be equal to [tau].

The preceding result tells us that Bovenberg and de Mooij's claim, that the second-best environmental levy is necessarily less than the Pigouvian tax, does not carry over to the second-best setting a la Mirrlees. In fact, the simplest way to recognize this is to note that the preference structure (Eqn. 33) includes preferences that are represented by

[[omega].sup.j] = U(H(C,D) + Q(G,E) + f(L, [[theta].sup.j])), j = 1,2,...,H. (36)

This corresponds to the case where environmental quality and labor are neither substitutes nor complements. Under this circumstance, [[u.sup.j].sub.E]/[[u.sup.j].sub.i] = [[u.sup.kj].sub.E]/[[u.sup.kj].sub.i]. In turn, this implies that [delta] = 0, so that, from Equation 35, we have [t.sub.d] = [tau]; that is, the second-best tax is identical to the Pigouvian tax. The implication of this for Bovenberg and de Mooij's claim is plain. It tells us that the environmental levy, rather than being smaller than the Pigouvian tax, is identical to it. [19]

A Simpler Characterization

If preferences are of the type specified by Equation 33, [t.sub.d] will have a very simple characterization. To see this, note that there are two separability assumptions in Equation 33, each having a particular implication for the tax structure. We have already noted that the separability of the private good subutility in labor and produced goods results in the properties given by Equation 34. Second, we should point out that the separability of environmental quality from private goods implies that the last expression in the right-hand side of Equation 25 will be zero. This result is stated in the following lemma and proved in Appendix B.

LEMMA 1. Assume preferences are weakly separable in public and private goods as in [[omega].sup.j] = U(F(C,D,L, [[theta].sup.j]), f(G,E, [[theta].sup.j])), j = 1,2,...,H. Then

[[lambda].sup.jk] [greater than] 0 [Right Arrow] [[u.sup.j].sub.E] - [[u.sup.jk].sub.E] = 0.

Put together, the properties in Equation 34 and Lemma 1 in conjunction with Equation 25 imply a second-best tax structure for [t.sub.d] of the form

[t.sub.d] = [[sigma].sub.j] -[[gamma].sup.j]e'[[u.sup.j].sub.E]/[micro] (37)

The separability thus implies that the optimal polluting-good tax will have the same characterization in the second best as in the first best. However, in the first best, we also had [[pi].sup.j]/[[alpha].sup.j] = [[gamma].sup.j]/[micro], so the tax was equal to the Pigouvian tax. This latter property does not hold here, resulting in a relationship between the two as given by Equation 35.

We summarize these results in Proposition 5.

PROPOSITION 5. Define the Pigouvian tax [tau] by Equation 26 and the optimal environmental levy by [tau] + [delta], with [delta] given by Equation 29.

(i) Assume preferences are weakly separable in labor and other goods as in Equation 31. Then the Stiglitz/Mirrlees terms and [delta] vanish so that the second-best tax on the polluting good, [t.sub.d], is equal to the Pigouvian tax, [tau].

(ii) Assume preferences are weakly separable in private and public goods, with the private good subutility being separable in labor and produced goods, as in Equation 33. Then the second-best tax on the polluting good is characterized by

[t.sub.d] = [[sigma].sub.j] -[[gamma].sup.j]e'[[u.sup.j].sub.E]/[micro] = [tau] + [delta].

The tax can be greater than, equal to, or smaller than the Pigouvian tax.

5. Conclusion

This paper has studied the problem of taxation of polluting goods in two different second-best environments: Ramsey and Mirrlees. By treating the issue in a consistent and unified way, it has attempted to achieve three objectives. First, it has shown that, in both settings, the tax on the polluting good generally reflects both optimal tax and externality-correcting considerations. This dual role implies that one cannot draw general results regarding the relative size of second-best polluting and Pigouvian taxes.

Second, the paper has traced the nature of the relationship between second-best and Pigouvian taxes to the consumers' preferences. In the context of the traditional Ramsey setting, Bovenberg and de Mooij (1994, p. 1085) had claimed that "in the presence of preexisting distortionary taxes, the optimal pollution tax typically lies below the Pigouvian tax...." This result was subsequently questioned by Fullerton (1997) and Schob (1997), who argued that the claim rests on a particular tax-normalization rule: It holds if the tax on nonpolluting goods is set equal to zero but not if the income tax rate is normalized at zero. They pointed out that the correct statement of Bovenberg and de Mooij's result should have referred to the tax differential between polluting and nonpolluting goods (and not the tax on the polluting good per se). The current paper has formally demonstrated the validity of this claim in a model where there is more than one nonpolluting good. By allowing for more than one nonpolluting good, our t reatment has shed light on the importance and the relevance of the separability restrictions on preferences required for this result. Preferences that are weakly separable in public and private goods, with the private good subutility being weakly separable in labor and produced goods, have two important implications. First, they call for a uniform proportionate tax rate on all nonpolluting goods. This property makes the concept of tax differential between polluting and nonpolluting goods meaningful. Second, they imply that optimal tax considerations do not call for differential tax treatment of these goods. This makes it possible to compare the size of the tax differential with the corresponding Pigouvian tax.

The third contribution of the article has been to study the relationship between the optimal tax on the polluting good and the Pigouvian tax in a second-best environment a la Mirrlees (1971). This setting allows for individuals to be heterogeneous and justifies the absence of first-best taxes by the existence of informational asymmetries between tax authorities and taxpayers. Defining the optimal environmental levy by the additional expressions due to environmental considerations that appear in the optimal tax formula for the polluting good (and are absent in the tax formulas for nonpolluting goods), it has shown that the optimal environmental levy may exceed, fall short of, or be equal to the Pigouvian tax. Next, assuming that preferences are separable in labor supply and all other goods, it has proved that the optimal environmental levy is now equal to the Pigouvian tax and to the tax differential between polluting and nonpolluting goods. Specifically, it has proved that polluting goods should be taxed only for externality-correcting purposes and at a rate equal to the full Pigouvian tax. Finally, following the specification for the second-best Ramsey setting, the article has considered preferences that are weakly separable in public and private goods, with the private good subutility being weakly separable in labor and produced goods. It has shown that the optimal tax on the polluting good (which in this case is equal to the environmental levy) can now be greater, equal to, or smaller than the Pigouvian tax. When the environmental quality and labor supply are complementary (and preferences are identical), the optimal tax exceeds the Pigouvian tax. On the other hand, when the two are substitutes, the optimal tax will be less than the Pigouvian tax.

(*.) IDEI and GREMAQ, University of Toulouse and Institut Universitaire de France, Toulouse, France.

(+.) Department of Economics, University of Illinois at Urbana-Champaign, 313 David Kinley Hall, Urbana, IL 61801, USA; E-mail [e-mail]; corresponding author.

(++.) IDEI and GREMAQ, University of Toulouse, Toulouse, France.

We thank Jonathan Hamilton and two anonymous referees for helpful comments.

Received February 2000; accepted January 2001.

(1.) See, among others, Bovenberg and van der Ploeg (1994), Bovenberg and de Mooij (1994), Bovenberg and Goulder (1996), Kaplow (1996), Fullerton (1997), Mayeres and Proost (1997), Pirttila and Tuomala (1997), Schob (1997), and Cremer, Gahvari, and Ladoux (1998).

(2.) As actual taxes go, Fullerton (1996) explains that each environmental tax in the U.S. is "intended to collect from a particular industry thought to be responsible for that pollution problem, but non represents a good example of an incentive-based tax designed to discourage the polluting activity itself" (p. 33). Moreover, the taxes that do in practice discourage fossil-fuel emissions (e.g., federal excise taxes on gasoline, tires, coal, and transportation) have been designed with other purposes in mind. Barthold (1994) contains an interesting discussion of the U.S. environmental taxes.

(3.) This may be interpreted as a rule question in that one is asking if the conventional Pigouvian rule is overestimating or underestimating the marginal social damage of emissions in the second best. Metcalf (2000) examines Bovenberg and de Mooij's claim in light of Atkinson and Stern's (1974) discussion.

(4.) A reviewer even wonders if "it makes any sense to refer to anything as the 'Pigouvian tax' [in a second-best setting].

(5.) Pareto-efficient tax structures with a finite number of types have also been studied by Brito et al. (1990).

(6.) This statement is based on Bovenberg and de Mooij's (1994) original normalization of setting the tax on (one of the) nonpolluting goods equal to zero. With nonlinear income taxes, this is the natural normalization rule. It can, of course, also be reinterpreted as a tax differential.

(7.) This is the assumption adopted by Bovenberg and de Mooij (1994), Bovenberg and van der Ploeg (1994), and Fullerton (1997).

(8.) Of course, the equality of formulas does not imply equality of values. This is because both [micro] and e'[U.sub.E] assume different values in the first and the second best.

(9.) One can show that, in this setup,

[eta] = 1/1 - (W[t.sub.w]/L)([partial]L/[partial][w.sub.n]).

Consequently, [eta] [greater than] 1 [Left/Right Arrow][partial]L/[partial][W.sub.n] [greater than] 0.

(10.) Comparing a consumption tax to an equivalent wage tax, indicates that [alpha] (1 + t) under a consumption tax equals [alpha] under a wage tax. This tells us that [tau] under a consumption tax is equal to [tau](1 + t) under a wage tax. This also explains why the tax differential is -Ne'[U.sub.E]/[micro] under a wage tax and (-Ne'[U.sub.E]/[micro])(1 + t) under a consumption tax.

(11.) For example, it is reasonable to assume that there is public information on individual purchases of polluting goods such as electricity, thus allowing the government to levy nonlinear taxes on electricity. On the other hand, for most other goods, it seems more reasonable to assume that the available information is on anonymous purchases (i.e., on aggregate sales and not on who buys how much). Under this circumstance, the government can only levy linear commodity taxes. Cremer, Gahvari, and Ladoux (1998) study this question in some detail.

(12.) Treating E as given in a natural consequence of the continuum of individuals' setup of the article.

(13.) As Kaplow (1996) has put it, "After all, correcting externalities may be viewed as a public good" (p. 523).

(14.) This definition differs from Cremer, Gahvari, and Ladoux's, who define the Pigouvian tax, and the marginal social environmental damage, as -[[sigma].sub.j] [[gamma].sup.j]e'[[u.sup.j].sub.E]/[micro].

(15.) This term is the anologue of Boadway and Keen's (1993) adjustment term to the Samuelson's condition for optimal provision of a public good when the public good is financed by a nonlinear income tax rather than first-best lump-sum taxes.

(16.) Observe that the presence of [tau] + [delta] will generally affect the value of the common terms that appear in the tax formulas for polluting and nonpolluting goods as well and with it the values of all the tax rates.

(17.) This is precisely the argument of Cremer and Gahvari (1997), who call for public provision of goods that are complements to labor supply.

(18.) Boadway and Keen (1993) do not address the question of Pigouvian taxes, They are concerned solely with the optimal provision of public goods in the presence of nonlinear income taxes. Moreover, they consider an economy with two types of persons who differ only in earning abilities. Our setting is more general than theirs in two respects: We allow for H types as well as variation in tastes. Kaplow (1996) too is primarily concerned with the question of optimal provision of public goods; but he addresses the Pigouvian tax problem as well. He does not explicitly show the equivalence of the second-best levy with the Pigouvian tax in a framework with self-selection constraints. Instead, he shows that, if one adjusts the income tax schedule to offset the distributive effects of a proposed environmental levy, the effect of such a policy on total tax revenues will be positive if and only if it is efficient to impose the proposed levy.

(19.) Recall that, in the representative-consumer case, the separability assumptions were not sufficient to guarantee that the optimal environmental levy would be less than the Pigouvian tax. A further assumption of an upward-sloping labor supply was also required. This assumption will not buy us much in the present context. In particular, note that the preference structure (Eqn. 36) is consistent with having upward-sloping labor supply functions. An example of this is U = [C.sup.0.5] + [D.sup.0.5] + [(1 - L).sup.0.5] + [G.sup.0.5] + [E.sup.0.5].

(20.) With one extra degree of freedom in setting the tax rates, one of the commodity tax instruments (including the wage tax) can be set equal to zero. The equation corresponding to the tax that is normalized at zero must then be deleted from the set of first-order conditions. In the second best, the equation corresponding to T is also deleted, as lump-taxation is infeasible.

References

Atkinson, A. B., and N. H. Stern. 1974. Pigou, taxation and public goods. Review of Economic Studies 41:119-28.

Atkinson, A. B., and J. E. Stiglitz. 1976. The design of tax structure: Direct versus indirect taxation. Journal of Public Economics 6:55-75.

Barthold, T. A. 1994. Issues in the design of environmental taxes. Journal of Economics Perspectives 8:133-51.

Boadway, R., and M. Keen. 1993. Public goods, self-selection and optimal income taxation. International Economic Review 34:463-78.

Bovenberg, A. L., and R. A. de Mooij. 1994. Environmental levies and distortionary taxation. American Economic Review 84:1085-9.

Bovenberg, A. L., and L. Goulder. 1996. Optimal environmental taxation in the presence of other taxes: General equilibrium analyses. American Economic Review 86:985-1000.

Bovenberg, A. L., and F. van der Ploeg. 1994. Environmental policy, public finance and the labor market in a second-best world. Journal of Public Economics 55:349-90.

Brito, D. J., J. H. Hamilton, S. M. Slutsky, and J. E. Stiglitz. 1990. Pareto efficient tax structures. Oxford Economic Papers 42:61-77.

Cremer, H., and F. Gahvari. 1997. In-kind transfers, self-selection and optimal tax policy. European Economic Review 41:97-114.

Cremer, H., F. Gahvari, and N. Ladoux. 1998. Externalities and optimal taxation. Journal of Public Economics 70:343-64.

Fullerton, D. 1996. Why have separate environmental taxes? In Tax policy and the economy, volume 10, edited by J. M. Poterba. Cambridge, MA: NBER and The MIT Press, pp. 33-70.

Fullerton, D. 1997. Environmental levies and distortionary taxation: Comment. American Economic Review 87:245-51.

Kaplow, L. 1996. The optimal supply of public goods and the distortionary cost of taxation. National Tax Journal 49:513-33.

Mayeres, I., and S. Proost. 1997. Optimal tax and public investment rules for congestion type of externalities, Scandinavian Journal of Economics 99:261-79.

Metcalf, G. E. 2000. Environmental levies and distortionary taxation: Pigou, taxation, and pollution. Mimeo.

Mirrlees, J. A. 1971. An exploration in the theory of optimal income taxation. Review of Economic Studies 38:175-208.

Pirttila, J., and M. Tuomala. 1997. Income tax, commodity tax and environmental policy. International Tax and Public Finance 4:379-93.

Poterba, J. M. 1991. Tax policy to combat global warming: On designing a carbon tax. In Global warming: Economic policy responses, edited by R. Dornbusch and J. M. Poterba. Cambridge, MA: MIT Press, pp. 71-98.

Poterba, J. M. 1993. Global warming policy: A public finance perspective. Journal of Economic Perspectives 7:47-63.

Sandmo, A. 1974. A note on the structure of optimal taxation. American Economic Review 64:701-6.

Sandmo, A. 1975. Optimal taxation in the presence of externalities. Swedish Journal of Economics 77:86-98.

Schob, R. 1997. Environmental taxes and pre-existing distortions: The normalization trap. International Tax and Public Finance 4:167-76.

Stiglitz, J. E. 1982. Self-selection and Pareto efficient taxation, Journal of Public Economics 17:213-40.

Stiglitz, J. E. 1987. Pareto efficient and optimal taxation and the new welfare economics. In Handbook of public economics 2, edited by A. Auerbach and M. Feldstein. Amsterdam: North-Holland, pp. 991-1042.

Appendix A

The Ramsey Tax Problem

Form the Lagrangian,

[delta] = v(q, [q.sub.d], [w.sub.n], T, G, E) + [micro] [[[[sigma].sup.m].sub.i=1] [t.sub.i][C.sub.i] + [t.sub.d]D + [t.sub.w]wL + T - [p.sub.g] G/N]. (A1)

The first-order conditions with respect to all the tax instruments are [20]

[partial][delta]/[partial][q.sub.s] = dv/d[q.sub.s] + [micro] ([C.sub.s] + [[sigma].sub.i] [t.sub.i] [dC.sub.i]/[dq.sub.s] + [t.sub.d] dD/[dq.sub.s] + [t.sub.w]W dL/[dq.sub.s] = 0, s = 1, 2, ..., m, (A2)

[partial][delta]/[partial][q.sub.d] = dv/[dq.sub.d] + [micro] (D + [[sigma].sub.i] [t.sub.i] [dC.sub.i]/[dq.sub.d] + [t.sub.d] dD/[dq.sub.d] + [t.sub.w]W dL/[dq.sub.d] = 0, (A3)

[partial][delta]/[partial][w.sub.n] = dv/[dw.sub.n] + [micro] (-L + [[sigma].sub.i] [t.sub.i] [dC.sub.i]/[dw.sub.n] + [t.sub.d] dD/[dw.sub.n] + [t.sub.w]W dL/[dw.sub.n]) = 0, (A4)

[partial][delta]/[partial]T = dv/dT + [micro] (1 + [[sigma].sub.i] [t.sub.i] [dC.sub.i]/dT + [t.sub.d] dD/dT + [t.sub.w]W dL/dT) = 0, (A5)

The derivatives that appear in the above first-order conditions are total in that they reflect both the direct impact of a change in a price (or income) as well as the indirect effect induced through a change in E. One can thus derive the expressions for the derivatives of v according to

dv/[dq.sub.s] = [partial]v/[partial][q.sub.s] + [v.sub.E] dE/[dq.sub.s] = -[alpha][C.sub.s] + [U.sub.E]Ne' dD/[dq.sub.s], (A6)

where we have also made use of Roy's identity, with [alpha] (= -[partial]v/[partial]T) denoting the marginal utility of income vE] = [U.sub.E], and Equation 2. Using Equation A6 and similar expressions for dv/[dq.sub.d], and dv/[dw.sub.n], and dv/dT, we may rewrite Equations A2-A5 as

(1 - [alpha]/[micro])[C.sub.s] + [[sigma].sub.i] [t.sub.i] [dC.sub.i]/[dq.sub.s] + ([t.sub.d] + Ne'[U.sub.E]/[micro])dD/[dq.sub.s] + [t.sub.w]W dL/[dq.sub.s] = 0, s = 1, 2, ..., m, (A7)

(1 - [alpha]/[micro])D + [[sigma].sub.i] [t.sub.i] [dC.sub.i]/[dq.sub.d] + ([t.sub.d] + Ne'[U.sub.E]/[micro])dD/[dq.sub.d] + [t.sub.w]W dL/[dq.sub.d] = 0, (A8)

(1 - [alpha]/[micro])(-L) + [[sigma].sub.i] [t.sub.i] [dC.sub.i]/[dw.sub.n] + ([t.sub.d] + Ne'[U.sub.E]/[micro]) dD/[dw.sub.n] + [t.sub.w]W dL/[dw.sub.n] = 0, (A9)

(1 - [alpha]/[micro]) + [[sigma].sub.i] [t.sub.i] [dC.sub.i]/dT + ([t.sub.d] + Ne'[U.sub.E]/[micro]) dD/dT + [t.sub.w]W dL/dT = 0, (A10)

Next, simplify the derivatives in Equations A7-A10 according to [dC.sub.i]/[dq.sub.s] = [partial][C.sub.i]/[partial][q.sub.s] + [partial][C.sub.i]/[partial]E dE/[dq.sub.s], (A11)

with similar expressions for the other derivatives. Note also that, from differentiating Equation 2, one obtains

dE/[dq.sub.s] = Ne' [partial]D/[partial][q.sub.s]/1 - Ne' [partial]D/[partial]E, (A12)

and similarly for dE/[dq.sub.d], dE/[dw.sub.n], and dE/dT. Equations A7-A10 are then simplified to

(1 - [alpha]/[micro]) [C.sub.s] + [[sigma].sub.i] [t.sub.i] [partial][C.sub.i]/[partial][q.sub.s] + ([t.sub.d] + Ne'[U.sub.E]/[micro] - [gamma])[partial]D/[partial][q.sub.s] + [t.sub.w] w[partial]L/[partial][q.sub.s] = 0, (A13)

(1 - [alpha]/[micro]) D + [[sigma].sub.i] [t.sub.i][partial][C.sub.i]/[partial][q.sub.d] + ([t.sub.d] + Ne' [U.sub.E]/[micro] - [gamma]) [partial]D/[partial][q.sub.d] + [t.sub.w] w[partial]L/[partial][q.sub.d] = 0, (A14)

(1 - [alpha]/[micro]) (-L) + [[sigma].sub.i] [t.sub.i] [partial][C.sub.i]/[partial][w.sub.n] + ([t.sub.d] + Ne' [U.sub.E]/[micro] - [gamma]) [partial]D/[partial][w.sub.n] + [t.sub.w]w[partial]L/[partial][w.sub.n] = 0, (A15)

(1 - [alpha]/[micro]) + [[sigma].sub.i] [t.sub.i] [partial][C.sub.i]/[partial]T + ([t.sub.d] + Ne' [U.sub.E]/[micro] - [gamma]) [partial]D/[partial]T + [t.sub.w]w[partial]L/[partial]T = 0, (A16)

where [gamma] is defined in Equation 8 in the text. Rewriting Equations A13-A15 in matrix notation, once normalizing [t.sub.1] = 0 and once [t.sub.w] = 0 and using the definitions of A, [A.sub.c], and [A.sub.L], yields the system of Equations 9 and 10 in the text.

First-Best Taxation

Denote the compensated demand and supply functions with a tilde over the corresponding variable. Through the Slutsky decompositions, one can then simplify Equations A13-A15 to

([t.sub.d] + Ne' [U.sub.E]/[micro] - [gamma]) [partial]D/[partial][q.sub.s] + [t.sub.w] w[partial]L/[partial][q.sub.s] + [[sigma].sub.i] [t.sub.i] [partial][C.sub.i]/[partial][q.sub.s] = [C.sub.s] [phi], s = 1, 2, . . ., m, (A17)

([t.sub.d] + Ne' [U.sub.E]/[micro] - [gamma]) [partial]D/[partial][q.sub.d] + [t.sub.w]w[partial]L/[partial][q.sub.d] + [[sigma].sub.i] [t.sub.i] [partial][C.sub.i]/[partial][q.sub.d] = D[phi], (A18)

([t.sub.d] + Ne' [U.sub.E]/[micro] - [gamma]) [partial]D/[partial][w.sub.n] + [t.sub.w]w[partial]L/[partial][w.sub.n] + [[sigma].sub.i] [t.sub.i] [partial][C.sub.i]/[partial][w.sub.n] = (-L)[phi], (A19)

where [phi] is defined as

[phi] = - (1 - [alpha]/[micro]) - [[sigma].sub.i] [t.sub.i][partial][C.sub.i]/[partial]T - ([t.sub.d] + Ne' [U.sub.E]/[micro] - [gamma]) [partial]D/[partial]T - [t.sub.w]w[partial]L/[partial]T. (A20)

Let [[phi].sub.c] denote the expression obtained for [phi] when [t.sub.1] = 0 and [[phi].sub.L] the expression when [t.sub.w] = 0. Introduce A to be the Slutsky matrix corresponding to A, that is, the (i, j)th element of A is the compensated version of the (I, j)th element of A. Further, denote the matrix derived from A by deleting its third row and column (associated with [C.sub.1]) by [A.sub.c]. Similarly, define [A.sub.L] by deleting the second row and column of A (associated with L). Rewriting Equations A17-0A19 in matrix notation, once normalizing [t.sub.1] = 0 and once [t.sub.w] = 0, yields

[MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII], and (A21)

[MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII] (A22)

The system of Equations A21 and A22 yield another characterization for the optimal tax rates for each of the normalizations [t.sub.1] = 0 and [t.sub.w] = 0.

Now, in the first best, T is available and Equation Al6 holds so that [phi]= [[phi].sub.C] = [[phi].sub.L] = 0. It then follows from Equations A21 and A22 and the definitions of [gamma], [[gamma].sub.C], and [[gamma].sub.L] that, for either normalization, the only taxes to be used are T and [t.sub.d] = -Ne' [U.sub.E]/[micro]. All other taxes are equal to zero.

Derivation of Equation 14

Set [t.sub.w] = 0. With weakly separable preferences in E and all other goods, we have the demand functions for [C.sub.s] and D as well as the supply function for L to be independent of E. It then follows from Equation 8 that [gamma] = 0. Substituting 0 for [gamma] in Equations Al3 and A14 results in

(1 - [alpha]/[micro]) [C.sub.s] + [[sigma].sub.i] [t.sub.1] [partial][C.sub.i]/[partial][q.sub.s] + ([t.sub.d] + Ne' [U.sub.E]/[micro]) [partial]D/[partial][q.sub.s] = 0, S = 1, 2, . . ., m, (A23)

(1 - [alpha]/[micro])D + [[sigma].sub.i] [t.sub.i] [partial][C.sub.i]/[partial][q.sub.d] + ([t.sub.d] + Ne' [U.sub.E]/[micro]) [partial]D/[partial][q.sub.d] = 0. (A24)

Second, assume that the private goods subutility is weakly separable in L and (C, D), with the subutility of (C, D) being homothetic. Sandmo (1974) has shown that these assumptions imply

[partial]L/[partial][q.sub.s] = [delta][C.sub.s], s = 1, 2, . . . , m, (A25)

[partial]L/[partial][q.sub.d] = [delta]D, (A26)

where [delta] is some function of the arguments of the utility function U. Now partially differentiating the individual's budget constraint with respect to [q.sub.s] and [q.sub.d] yields

[C.sub.s] + [[sigma].sub.i] [q.sub.i] [partial][C.sub.i]/[partial][q.sub.s] + [q.sub.d][partial]D/[partial][q.sub.s] = w [partial]L/[partial][q.sub.s], S = 1, 2, . . . , m, (A27)

D + [[sigma].sub.i] [q.sub.i] [partial][C.sub.i]/[partial][q.sub.d] + [q.sub.d] [partial]D/[partial][q.sub.d] = w [partial]L/[partial][q.sub.d], (A28)

Substitute from Equations A25 and A26 into Equations A27 and A28 and solve for [C.sub.s] and D. Substituting the resulting expressions in Equations A23 and A24 yields, for all s = 1, 2, . . . , m,

[[[sigma].sup.m].sub.i=1] (1 - [alpha]/[micro]/1 - w[delta] [q.sub.i] - [t.sub.1]) [partial][C.sub.i]/[partial][q.sub.s] + [1 - [alpha]/[micro]/1 - w [delta] [q.sub.d] - ([t.sub.d] + Ne' [U.sub.E]/[micro])] [partial]D/[partial][q.sub.s] = 0. (A29)

[[[sigma].sup.m].sub.i=1] (1 - [alpha]/[micro]/1 - w [delta][q.sub.i] - [t.sub.i]) [partial][C.sub.i]/[partial][q.sub.d] + [1 - [alpha]/[micro]/1 - w[delta] [q.sub.d] ([t.sub.d] + Ne' [U.sub.E]/[micro])] [partial]D/[partial][q.sub.d] = 0. (A30)

Assuming the matrix associated with Equations A29 and A30 is nonsingular, the solution to these equations is given by

[t.sub.s]/[q.sub.s] = [t.sub.d] + Ne' [U.sub.E]/[micro]/[q.sub.d] = 1 - [alph]/[micro]/1 - w[delta], s = 1, 2, . . ., m, (A31)

Derivarion of Equation 13

Denote, under the normalization [t.sub.w] = 0, the uniform tax on [C.sub.s]'s by [t.sup.*] and the optimal tax rate on D by [[t.sup.*].sub.d]. From the individual's budget constraint, it is readily seen that these taxes are equivalent to another tax system where [C.sub.s]'s are not taxed and there is a wage tax equal to 1 - 1/(l + [t.sup.*]) = [t.sup.*]/(1 + [t.sup.*]) and a tax on D equal to

[t.sub.d] = 1 + [[t.sup.*].sub.d]/1 + [t.sup.*] - 1 = [[t.sup.*].sub.d] - [t.sup.*]/1 + [t.sup.*]. (A32)

On the other hand, from Equation A31, we have

[[t.sup.*].sub.d] = [t.sup.*] - (1 + [t.sup.*]) Ne'[U.sub.E]/[micro]. (A33)

It immediately follows from Equations A32 and A33 Chat

[t.sub.d] = Ne'[U.sub.E]/[micro]. (A34)

Appendix B

Derivation of Equations 22-23

The Lagrangian expression, L for the government's problem can be written as

L = [[sigma].sub.j] [[gamma].sup.j][u.sup.j] + [micro] {[[sigma].sub.j] [[pi].sup.j] [[I.sup.j] - [[sigma].sub.j] [[C.sup.j].sub.i] - [D.sup.j] - [p.sub.g]G]} + [[sigma].sub.j] [[sigma].sub.i[not equal to]j] [[lambda].sup.jk] ([u.sup.j] - [u.sup.jk]). (B1)

Rearranging the terms, Equation B1 can be rewritten as

L = [[sigma].sub.j] ([[gamma].sup.j] + [[sigma].sub.k[not equal to]j] [[lambda].sup.jk]) [u.sup.j] + [micro] {[[sigma].sub.j] [[pi].sup.j][[I.sup.j] - [[sigma].sub.i] [[C.sup.j].sub.i] - [D.sup.j] - [p.sub.g] G]} - [[sigma].sub.j] [[sigma].sub.k[not equal to]j] [[lambda].sup.jk] [u.sup.jk]. (B2)

The first-order conditions for this problem are, for all j, k = 1, 2 ..., H and s = 1, 2 ..., m,

[partial]L/[partial][[C.sup.j].sub.s] = ([[gamma].sup.j] + [[sigma].sub.k[not equal to]j] [[lambda].sup.jk])[[u.sup.j].sub.s] - [micro][[pi].sup.j] - [[sigma].sub.k[not equal to]j] [[lambda].sup.kj] [[u.sup.kj].sub.s] = 0, (B3)

[partial]L/[partial][D.sup.j] = ([[gamma].sup.j] + [[sigma].sub.k[not equal to]j] [[lambda].sup.jk])[[u.sup.j].sub.D] - [micro][[pi].sup.j] - [[sigma].sub.k[not equal to]j] [[lambda].sup.kj] [[u.sup.kj].sub.D] + [[[sigma].sub.j] ([[gamma].sup.j] + [[sigma].sub.k[not equal to]j] [[lambda].sup.jk]) [[u.sup.j].sub.E] - [[sigma].sub.j] [[sigma].sub.k[not equal to]j] [[lambda].sup.kj] [[u.sup.kj].sub.E]] e'[[pi].sup.j] = 0. (B4)

[partial]L/[partial][I.sup.j] = ([[gamma].sup.j] + [[sigma].sub.k[not equal to]j] [[lambda].sup.jk]) [[u.sup.j].sub.i] - [micro][[pi].sup.j] - [[sigma].sub.k[not equal to]j] [[lambda].sup.kj] [[u.sup.kj].sub.i] = 0. (B5)

Dividing Equations 83 and B4 by Equation 83, written for [[C.sup.j].sub.i], yields Equations 22 and 23 in the text.

PROOF OF PROPOSITION 3. Substitute [[sigma].sub.j] [[sigma].sub.k[not equal to]j] [[lambda].sup.jk] [[u.sup.jk].sub.E] for [[sigma].sub.j] [[sigma].sub.k[not equal to]j] [[lambda].sup.kj] [[u.sup.kj].sub.E] in the bracketed expression of the numerator of the last expression in the right-hand side of Equation 23 and rearrange the terms. Then multiply the resulting equation and Equation 22 through by 1 + [[sigma].sub.k[not equal to]j] [[lambda].sup.kj][[u.sup.kj].sub.i]/[micro][[pi].sup.j] and collect the terms. We will have

[[u.sup.j].sub.s]/[[u.sup.j].sub.i] = 1 - [[sigma].sub.k[not equal to]j] [[lambda].sup.kj] [[u.sup.kj].sub.i]/[micro][[pi].sup.j] ([[u.sup.j].sub.s]/[[u.sup.j].sub.i] - [[u.sup.kj].sub.s]/[[u.sup.kj].sub.i]), (B6)

[[u.sup.j].sub.D]/[[u.sup.j].sub.i] = 1 [[sigma].sub.k[not equal to]j] [[lambda].sup.kj] [[u.sup.kj].sub.i]/[micro][[pi].sup.j] ([[u.sup.j].sub.D]/[[u.sup.j].sub.i] - [[u.sup.kj].sub.D]/[[u.sup.kj].sub.i]) + [[sigma].sub.j] -[[gamma].sup.j] e'[[u.sup.j].sub.g]/[micro] + [[sigma].sub.j] [[sigma].k[not equal to]j] -[[lambda].sup.jk] e'([[u.sup.j].sub.E] - [[u.sup.jk].sub.E])/[micro]. (B7)

Equations 24 and 25 follow immediately from Equations B6 and 87.

Derivation of Equation 30

Add and subtract [[sigma].sub.j] [[sigma].sub.k[not equal to]j] - [[lambda].sup.jk] e'([[u.sup.j].sub.E] = [[u.sup.jk].sub.E])/[micro] to [[sigma].sub.j] - [[gamma].sup.j] e'[[u.sup.j].sub.E]/[micro] and rearrange the terms. We will have

[[sigma].sub.j] -[[gamma].sup.j] e'[[u.sup.j].sub.E]/[micro] = [[sigma].sub.j] ([[gamma].sup.j] + [[sigma].sub.k[not equal to]j] [[lambda].sup.jk]) -e'[[u.sup.j].sub.i]/[micro] [[u.sup.j].sub.E]/[[u.sup.j].sub.i] - [[sigma].sub.j] [[sigma].sub.k[not equal to]j] - [[lambda].sup.jk] e' [[u.sup.jk].sub.E]/[micro] - [[sigma].sub.j] [[sigma].sub.k[not equal to]j] - [[lambda].sup.jk] e'([[u.sup.j].sub.E] - [[u.sup.jk].sub.E])/[micro]. (B8)

Next, substitute from Equation B3, ([[gamma].sup.j] + [[sigma].sub.k[not equal to]j] [[gamma].sup.jk]) [[u.sup.j].sub.i] for [micro][[pi].sup.j] + [[sigma].sub.k[not equal to]j] [[lambda].sup.kj] [[u.sup.kj].sub.i] in Equation B8 and simplify. We will get

[[sigma].sub.j] -[[gamma].sup.j] -e' [[u.sup.j].sub.E]/[micro] = [[sigma].sub.j] -[[pi].sup.j] e' [[u.sup.j].sub.E]/[[u.sup.j].sub.i] + [[sigma].sub.j] [[sigma].sub.k[not equal to]j] -[[lambda].sup.kj] e'[[u.sup.kj].sub.i]/[micro] ([[u.sup.j].sub.E]/[[u.sup.j].sub.i] - [[u.sup.kj].sub.E]/[[u.sup.kj].sub.i]) - [[sigma].sub.i] [[sigma].sub.k[not equal to]j] -[[lambda].sup.jk] e' ([[u.sup.j].sub.E] - [[u.sup.jk].sub.E])/[micro]. (B9)

Substituting from Equation B9 into Equation B7 yields

[[u.sup.j].sub.D]/[[u.sup.j].sub.i] = 1 - [[sigma].sub.k[not equal to]j] [[lambda].sup.kj] [[u.sup.kj].sub.i]/[micro][[pi].sup.j] ([[u.sup.j].sub.D]/[[u.sup.j].sub.i] - [[u.sup.kj].sub.D]/[[u.sup.kj].sub.i]) + [[sigma].sub.j] -[[pi].sup.j] e'[[u.sup.j].sub.E]/[[u.sup.j].sub.i] + [[sigma].sub.j] [[sigma].sub.k[not equal to]j] - [[lambda].sup.jk] e' [u.sup.kj].sub.i]/[micro] ([[u.sup.j].sub.E]/[[u.sup.j].sub.i] - [[u.sup.kj].sub.E]/[[u.sup.kj].sub.i]). (B10)

Equation 30 then follows immediately from (B10).

PROOF OF LEMMA 1. Given the assumed structure of preferences, we have

[u.sup.j] = U([F.sup.j]([C.sup.j], [D.sup.j], [I.sup.j]), f(G, E, [[theta].sup.j])) (B11)

[u.sup.jk] = U([F.sup.j]([C.sup.k], [D.sup.k], [I.sup.k]), f( G, E, [[theta].sup.j])), (B12)

where

[F.sup.j](C, D, I) = F(C, D, I/[w.sup.j], [[theta].sup.j]).

Now when [[lambda].sup.jk] [greater than] 0, the corresponding self-selection constraint for j binds so that [u.sup.j] = [u.sup.jk]. This in turn implies that

[F.sup.j]([C.sup.j], [D.sup.j], [I.sup.j]) = [F.sup.j]([C.sup.k], [D.sup.k], [I.sup.k]). (B13)

Next, differentiating Equations B11 and B12 with respect to E yields

[[u.sup.j].sub.E] = [U.sub.E]([F.sup.j]([C.sup.j], [D.sup.j], [I.sup.j]), f(G, E, [[theta].sup.j])), (B14)

[[u.sup.jk].sub.E] = [U.sub.E]([F.sup.j]([C.sup.k], [D.sup.k], [I.sup.k]), f(G, E, [[theta].sup.j])). (B15)

It then follows from Equations B14, B15, and B13 that [[u.sup.j].sub.E] = [[u.sup.jk].sub.E].