Optimal Taxation in the Presence of Externalities

Agnar Sandmo

The Swedish Journal of Economics, Vol. 77, No. 1, Public Finance: Allocation and Distribution.(Mar., 1975), pp. 86-98.

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http://www.jstor.orgFri Sep 7 11:47:26 2007

OPTIMAL TAXATION IN THE PRESENCE OF EXTERNALITIES

Agnar Sandmo*

Sorwegian School oi Econonlics and Business Administration, Bergen, Sorway

Summary

This paper attempts to integrate the theory of optimal taxation with the analysis of t,he use of indirect taxation to counteract negative external effects (Pigovian taxes). X first-best solution to the problem of the optimal tax on an externality- generating good is contrasted with the case where the government also needs other, distortionary taxes in order to satisfy its revenue requirements. The main result is that the Pigovian principle holds in a modified form in the latter case as w-ell. The problem of the distributional impact of taxation is also studied for the special case of individuals with identical preferences and a utilitarian social wel- fare function.

1. Introduction

Since the idea was first introduced by Pigou (1920)economists have generally

accepted that when externalities are present, indirect taxation can be used as a tool for correcting inefficiencies in the competitive allocation of resources. In particular, this is true if the externality in question is of the public good (or "bad") variety, so that negotiations between the parties concerned can be effectively ruled out. The effect of the tax in this case is to confront the generator of the externality with a price reflecting the damage (or benefit, as the case may be) which his production or consumptioll of the commodity in question inflicts on 0thers.l

There is already considerable literature on the applicability of the Pigovian tax solution. This question has recently come t o the foreground in connection with the choice of policies aimed a t protecting the natural environment. However, practical difficulties of implementation are not the direct concern of this paper. Instead, we consider the theoretical problem of determining op- timal taxes when externalities of the public good type are present.

The existing literature on this topic suffers from several shortcon~ings.

* I am indebted to Victor D. Korman, who has read the n~anuscript and given helpful comments. The advice of a referee is also appreciated.

For a critical evaluation of Pigou's own analysis and of ~ t sapplicability to the case where the parties can negotiate, see Coase (1960) and the discussion oi Coase's arguments by Bauinol (19i2).

Optimal taxation in the presence oj externalities 87

First, most of it is partial equilibrium in nature and does not contain satis- factory treatment of the benefits resulting from control of externalities. Second, the analyses do not take into account the second-best problems which arise as soon as it is realized that Pigoviail taxes only constitute part of a more comprel~ensive system of commodity taxes. Thus, the form in which optimality conditions are stated must frequently be taken to imply that the proceeds from Pigovian taxes are distributed to consumers in a lump-sum manner and that the net tax requirement of the public sector is zero. These assumptions are surely unattractive. Third, distributional aspects of optimal taxes are ignored.

The first of these objections, although probably insufficiently emphasized, requires only fairly straightfor\+ard analysis. The second, however, raises more interesting problems. Taxes on commodities that do not involve external effects introduce distortions of their oxn and raise the familiar problem of second-best theory, i.e. since the remaining conditions for Pareto optimality are not satisfied. the imposition of a tax on the externality-generating com- modity reflecting the marginal social damage may no longer be optimal. This point was made in a slightly different context by Buchanan (1969),who argued that the Pigovian tax may in fact reduce efficiency if the externality-generat- ing commoditjr is produced byv a monopolist. Buchanan seems to draw the conclusion that the Pigovian tax solution is only applicable under ideal com- petitive conditions. Finally, distributional aspects of optimal taxation are of particular interest in the context of environmental externalities, where there has been a good deal of concern about the distributional impact of economic policies.

These three points will be taken up in turn in this paper. The "first-best" model is introduced in Section 2, while Section 3 analyzes the second-best problem of the choice of an optimal set of commodity taxes under a budgetary constraint for the public sector. In this section the distributional problem is assumed away; it is taken up in Section 4. Section 5 contains some conclud- ing remarks.

2. The Structure of the Model and the First-Best Solution

The basic model to be used in this paper is a very simple one; given the com- plexity of the problem, it is probably the simplest model imaginable. Thus, we assume that consumers have identical preferences. This assumption is useful when one wishes to ignore the problem of distribution as well as when one wants to come to grips with it. We also treat the production side of the economy in an aggregate fashion, assuming a simple linear production struc- ture.

There are n consumers in the economy and m +1 consumer goods. Let r , be the amount consumed of good i by the individual consumer and X , the

Swed. J . of Economics 1975

total amount consumed and produced in the economy. We assume now that x, denotes hours worked and that leisure can be written as 1-xo. I t is also assumed that the consumption of good m creates a negative externality which is a function of the total consumption of that good X,. We can then write the utility function of the representative consumer as

This function is assumed to be strictly concave1 and to satisfy the usual dif- ferentiability requirements with the following sign assumptions for the first- order partial derivatives:

u i > 0 ( i = O , l , ...,m), u,+,<O (2)

This treatment of the externality is of course highly stylized, although it does seem to capture some essential aspects of the concern over environmental externalities. Traffic congestion, air and water pollution resulting from the production of certain commodities, etc. can all be interpreted as examples of this general description.

The production side of the economy is summarized in a fixed-coefficients transformation function

The fixed-coefficients assumption is obviously a strong one, but a t least in the present context it is less restrictive than it may seem (see footnote 6 below). It should be observed that only the total amount of labor enters in (3); all individuals are perfect substitutes for each other in production.

The welfare function is taken to be the utilitarian unweighted sum of utilities. Since each individual has the same preferences and the same pro- ductivity, the optimum will obviously be characterized by full equality, and we can write Xi=nx,for all i. To characterize the properties of an optimal allocation the Lagrangian can now be written

Setting the partial derivatives of L equal to zero, me can write the optimality conditions as

1 While guaranteeing the sufficiency of the first-order maximum conditions, the assump- tion of strict concavity is stronger than necessary for the main results in Sections 2 and 3, which require only quasi-concavity. However, when we come to distributional problems we utilize arguments based on decreasing marginal utility of income, which imply strict concavity. For simplicity, the stronger assumption is made from the start.

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Optimal taxation in the presence of externalities 89

Eq. (5)says that for the first m -1 consumer goods the marginal rate of sub- stitution between any one good and leisure should be equal to their marginal rate of transformation. Eq. (6) says that, for the mth good, its marginal rate of substitution for leisure should not be equated directly to the marginal rate of transformation; the former should be multiplied by a factor involving the sum of the marginal rates of substitution between the mth good as a private and as a public good.

To interpret these conditions in terms of a competitive equilibrium we de- fine a consumer price vector P=(Po , PI, ...,P,) and a producer price vector p =(po, pl, ...,p,). Prices may be normalized by choosing labor as our numdraire, setting Po=po=1. In a competitive equilibrium consumers maxi- mize utility subject to a budget constraint involving the prices P , while firms maximize profit,, taking the price vector p as given. Thus, the consumer's optimum is found as the maximum of (1)subject to the budget constraint

where S is the amount of lump-sum transfers received from the public sector. When maximizing utility subject to this constraint the individual consumer, being very small compared to the market, will ignore the effect of a small change in his own consumption on the total quantity of good m consumed, effectively assuming that the derivative aXm/axm=0.l This is simply the usual competitive assumption.

We can now reformulate the first-order conditions for the consumer as

ui- = p i . ( i = l , ...,m). u0

Firms maximize profit subject to the production constraint (3) and the price vector p. Obviously, the only structure of producer prices which is com- patible with competitive assumptions is

pi =a i (i= 1, ...,m). (9)

This says that producer prices should be equal to marginal costs measured in labor units.

We now ask, what is the nature of the competitive price system that will lead to an efficient allocation of resources? Comparing (8) and (9) with (5) and (6) we see that efficiency requires

However, the externality problem would remain, although in a slightly different form, if the individual were to assume (rightly) that aXm/axm= 1. The reason for this, of course, is that the individual only takes into account the externality effect on himself, while continuing to ignore the effects of his own consumption on others.

Swed. J . of Economics 1975

Thus, efficiency results when there is a price structure with equality between consumer and producer prices for the first m - 1 goods and an inequality for the ,nth externality-creating good. The optimal rate of the Pigovian tax may now be defined as

since (1 -8,) is the optimal ratio between the producer and the consumer price for good r r ~ .For this solution to make sense we must obviously assume that 8, < 1 or

so that our local optimalitp conditions also describe a global optimum. Conditions (10) and (11) provide a general equilibrium solution to the prob-

lem of the optimal Pigovian tax; from (12) we see that the optimal tax rate reflects the marginal "social damage" as the sum of the marginal rates of substitution between good 77t as a private and as a public good. But the under- lying assumptions about the public sector are still unsatisfactory. Its only task is to correct for externalities by imposing Pigovian taxes and distribute the tax revenue through lump-sum payments. We now turn to a model which sets more realistic constraints for the public sector.

3. The Second-Best Problem

As in most of the recent contributions to the theory of optimal commodity taxation1 we approach the optimization problem by way of the dual. Thus, let us define the consumer's indirect utility function as

where x is the consumption rector. The derivative of r: with respect to the kt11 price is then

Substituting from the first-order conditions we obtain

See e.g. Diamond 6:Mirrlees (1971), Stiglitz & Desgupta (1971), Dixit (1974) and Sand- mo (1974).

Swed. J . of Econotnics l 9 i 5

Optimal faxation in the presence of externalifies 91

av ax, rn ax.--- - A - + A 2 Pi - +apk i = l apk

But from the budget constraint (7) we have that

so that we get

We can now formulate the second-best optimization model. We omit an explicit and detailed treatment of the expenditure side of the public sector's budget and simply assume that the public sector requires a given amount of tax revenue, T. As the required tax revenue is in la,bor units1 we might inter- pret it as, e.g. the amount of labor required to keep up a pre-determined level of national defense. Unit taxes are defined as t , = P , -p,. The government's budget constraint can then be written as

The optimization problem can be formulated as maximization of the sum of the indirect utility functions with respect to consumer prices subject to the budget constraint (16).The Lagrangian becomes

and the first-order conditions are

These are the fundamental optimality conditions. In general it is not pos- sible to solve for the optimal taxes explicitly, since the quantities demanded depend on the taxes. However, it is illuminating to solve for the t , from (IS), although the resulting expressions still only give us implicit solutions.

The coefficient matrix of the t i is the transpose of the Jacobian matrix of the demand functions for the taxed goods:

The choice of labor as the untaxed numdraire good is of course arbitrary, and the reader is free to reinterpret the model so as to include labor among the taxed goods. However, the income tax considered is then only a proportional one; more complicated income tax schedules would require substantial extensions of the analysis.

Swed. J . of Economics 1975

Let J be the determinant of J* and J , the cofactor of the element in the ith row, kth column of J.Assuming J* to be non-singular we can then use Cramer's rule to solve for t, from (18):

This expression can be rewritten as

According to some well-known theorems about the expansion of determinants we have that

m

t2% -

0 for k + m = api J for k = m .

Rewriting (20) in terms of the tax rates Ok(=tk/Pk),and defining p= --il/P, we can then conclude that the optimal tax structure has the following form:

t xi J ik

e k = i i - p l [ - i & ] for k + ~

C xi J i m

m = - p ) [ - i l i i - j + p [ - n ~ ~ ~ .

Two main conclusions follow from this analysis. The first is that the optimal tax structure is characterized by what might

be called an additivity property; the marginal social damage of commodity m enters the tax formula for that commodity additively, and does not enter the tax formulas for the other commodities, regardless of the pattern of com- plementarity and substitutability. Thus, the fact that a commodity involves a negative externality is not in it'self an argument for taxing other commodities which are complementary with it, nor for subsidizing substitutes. This result has obvious relevance for economic policy and is not evident from the view- point of t'he theory of second best.

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Optimal taxation in the presence of externalities 93

This conclusion should be interpreted with some caution, however. It does not imply that the marginal social damage is the same under a system of commodity taxation as in the first-best situation described in Section 2. In general, both relative prices and incomes of consumers will differ between the two situations and so, therefore, will the marginal social damage. Thus, our conclusion is no more than a statement about the terms contained in the optimal tax formulas.

The second implication is that the optimal tax rate on the externality- creating commodity is a weighted a,verage of two terms, of which the second is the marginal social damage of commodity m. The first term, similar to the corresponding terms in the formulas for the remaining commodities, is com- posed of the efficiency terms familiar from the theory of optimal taxation. To see that this is indeed so, consider the case of independent demands, so that ax,/aPk=O for j +k. The determinant of the transposed Jacobian is then simply equal to the product of the diagonal terms, and formulas (22) and (23) reduce to

(24) is the inverse elasticity formula familiar from the literature and says that the highest tax rates should be levied on commodities with inelastic demand.' Eq. (25)shows that the optimal tax rate for the externality-creating commodity is a weighted average of the inverse elasticity and the marginal social damage. (We assume that E~ <O for a11 k.)

There remains the task of interpreting the weight factor p. Recall that 2 is the marginal utility of income (or leisure) and therefore positive. From the formulation of the Lagrangian (17) we see that t!? is the effect of an increase in the tax requirement T on social utility; /? is therefore negative. Clearly, the public sector should employ its tax revenue in such a way that the marginal social utility of income in the public sector equals the value of the marginal amount withdrawn from the private sector, which is -/I.Therefore, since ,u = -Alp, we might interpret ,u as the marginal rate of substitution between private and public income; the higher p is, the higher the marginal value of private income compared with public income, and the lower the tax require- ment, given that this is itself derived from an underlying optimization criter- ion. We also see that with increasing p, the proportionallity factor of the effi-

Supply elasticities do not enter these formulas because of our linearity assumption about the structure of production. The crucial assumption, however, is not that of fixed coefficients but of constant returns to scale; see Diamond & Mirrlees (1971). I f some commodity is in completely inelastic demand, it is the ideal tax object from the efficiency viewpoint, but this extreme case is disregarded here.

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94 A. Sandmo

ciency terms in our formulas decrease, and the marginal social damage comes to dominate the tax on good m. It is conceivable that p >1, in which case the efficiency terms become formulas for optimal subsidies. since the marginal social damage alone implies a tax on good m which creates revenue in excess of the tax requirement.

What of the case p =11 This obviously implies that we are back to the first- best solution with a tax on good m only, and at a rate reflecting the marginal social damage. This is the fortunate case where the Pigovian tax alone happens to satisfy the tax requirement exactly, so that no additional taxes or subsidies of a distortionarg kind are called for.

4. Distributional Considerations

From the viewpoint of practical economic policy, a good case can be made for ignoring distributional considerations when using taxes to correct for ex- ternal effects. Taxes on income and wealth as well as educational and social policy may all be more efficient means of achieving a socially desirable re- distribution of welfare. However, if such policies are not iu fact effective, a government committed to certain redistributional goals cannot ignore the distributional impact of taxes that are primarily designed for the purpose of "internalizing externalities".

There could be no distributional problem in the model in the preceding sections since everybody was alike. We continue to assume that all individuals have the same preferences. but we now let them have unequal productivities. The transformation function will be written as

where

n

xi= 2 xi,. (i= 1... . ,m). (27) j = l

Here ao,/aiis obviously individual j's marginal productivity in the production of good i.

I n competitive equilibrium producer prices must necessarily be such that

The indirect utility function is now1

1 The price vector encountered by consumer j is P, = ( p y , PI ,...,P,) and differs between consumers because they face different prices for the labor they supply. h-ote that since labor is still not taxed, POj= pGj .

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--

Optimal tamtion in the presence of externalities 95

and its derivative with respect to the kth price is

Letting ,if be the marginal utility of income for individual j , we proceed as in the preceding section to obtain

av, - - , i j z k j + ~ m + l ,i - .ax,

ap , apk

To find the optimal price structure we form the Lagrangian

and set its partial derivatives equal to zero:

rn ax. - i = l t i L + x b = o ( k = l ,...,m).t ~jxkj+ ( j l u m + l . j ) 5-p[zlapk I Solving for t, we obtain

rn

( k = 1, ...,m).J

Utilizing (21),the solution can be rewritten as

Here pj = -&I,!?. Perhaps the most striking feature of the solution is that the additive pro-

perty carries over to this more general case. It is still true that the marginal social damage is only an argument in the tax formula for good m.For further insights into the properties of the solution we concentrate on the case of demand independence, in which (32)and (33)become

96 A. Sandnzo

The efficiency terms of the optimal tax formulas are proportional to the aggregate demand elasticities as before. However, the proportionality factor has become some~vhat more complicated. It is a weighted average across in- dividuals of the factor (1-pj), the weights being in each case the amount of the commodity in question consumed by individual j. Now, ( I -p,) varies positively with the level of income, being low (possibly negative) for low- income individuals and high (approaching one) for high-income individuals. Thus, the proportionality factor1 takes a low (possibly negative) value if the consumption of commodity k is concentrated among low-income individuals and a high value if i t is mainly consumed by high-income people. This distri- butional modification of the inverse elasticity rule is a natural implication of our utilitarian social welfare function.

These remarks also apply to the efficiency term of the externality-generating commodity m. But distributional factors also enter into the social damage term, since each individual's marginal rate of substitution is weighted by the factor pj , which varies negatively with income. Thus, the social damage term will be high if those who suffer the most from the externality tend to have low incomes and low if they are concentrated among the high-income groups.

I n much of the literature on externalities there is a distinction between "polluters" and "pollutees". This distinction, although it may well be useful for some purposes, might be somewhat artificial-after all, the main victim of traffic congestion is the individual driver, who is also its source. In the present model there is no such sharp distinction, but when individuals differ with respect to income, it is still possible to make statements about various income classes as being "main polluters" and "main pollutees". From t'llis point of view one might consider the total tax rate on good nz. The conditions most favorable t o a high tax rate exist when we have high-income polluters and low-income pollutees, while the opposite constellation would favor a lolv tax rate. The intermediate cases are when polluters and pollutees are both located either in low-income or high-income groups; this calls for a low effi- ciency term and a high social damage term or the opposite, as the case may be.

5 . Concluding Remarks

As should be clear from the previous sections, this paper is not designed as a practical guide to the use of Pigovian taxes by economic planners. The models are extremely stylized, and the problems of estimating social damages and of

In accordance with Pestieau (1973), we might refer to the proportionality factor for good k as the distributional characteristic of that good.

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Optimal taxation in the presence of externalities 97

choosing the correct tax base, although of great practical importance, have been assumed away or ignored. Still, an exercise such as this may have some derived practical value. We have shown that the Pigovian taxation principle can be validated as pert of a more comprehensive system of indirect taxation, and we have demonstrated that it holds in a modified form even when distri- butional considerations enter as correctives to the efficiency principles of taxation. Thus, even in a world of distortionary taxation where the allocative functions of the price system cannot be separated from its effects on distribu- tion, there is scope for taxing externality-generating commodities according to the Pigovian principle. It should perhaps be stressed that our conclusions must be evaluated in relation to the framework of analysis, which is that of an optimal second-best set of commodity taxes. Thus, the conclusions cannot be expected to hold if some taxes are unalterable and have not been set ac- cording to the optimality criterion. To take an example, suppose that it is not possible to tax the externality-generating commodity. We mould then naturally expect that the allocation of resources could be improved by taxing complementary goods or subsidizing substitutes, so that the tax rates for these goods should reflect the marginal social damage. There is no inconsist- ency between this and the preceding analysis.

Some of the assumptions used are admittedly rather special, but the es- sential features of the results carry over to more general models with suitable reinterpretations. Thus, the analysis of the first-best model in Section 2 and the resulting optimal tax formula (12) can be interpreted as an analysis of the Pareto optimal tax rate in more general models. The right-hand side of (12) would then be a proper sum of marginal rates of substitution, and this formula would describe a tax rate that is optimal in the welfare sense subject to the qualification that the distribution of income is optimal according to the welfare function used. Similarly, the problem dealt with in Section 3 could be regarded as an analysis of efficient taxation when the distributional problem can somehow be solved separately, presumably by lump-sum transfers. The difficulties and paradoxes inherent in this interpretation are obvious and well known. Finally, the model incorporating distributional effects could have been made more general by the adoption of a more general social welfare function. The individual weights in the optimal tax formulas could then be interpreted as social marginal utilities of income, and there would be no need to associate marginal social "deservingness" with income level alone.

These extensions and reinterpretations are not free from serious complica- tions. This applies in particular when we wish to take distributional effects into account. In the more applied lines of welfare economics it sometimes has to be assumed that the foundations of the theory are more solid than they really are. This may not be as bad as it seems, however, since such a proce- dure may indicate more clearly the lines along which work on the foundations might most fruitfully be pursued.

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Diamond, P . A. & Mirrlees, J. A.: Optimal taxation and public production 1-11. American Ewnomic Review 61, 8-27, 261-278, 1971.

Dixit, A. K.: Welfare effects of tax and price changes. Unpublished paper, 1974.

Pestieau, P.: Equity and efficiency rules for the provision of public goods. Un- published paper, 1973.

Pigou, A. C.: The economics of welfare. Lon-don, 1920. (4th edition 1932.)

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