Journal of Public Economics 6 (1976) 37-54. 0 North-Holland
OPTIMAL
Publishing Company
TAXATION
An introduction to the literature Agnar SANDMO* Norwegian School of Economics and Business Administration, Bergen, Norway Revised version received September 1975 This paper attempts to give an introductory survey of recent contributions to the literature on optimal taxation. It covers the basic theorems on optimal commodity taxation and discusses the insights that the theory provides into the structure of an optimal tax system. No systematic study is made of the possibilities for application, but the theory is surveyed with a view towards its implications for actual tax policy.
1. Introduction Among a subset of economists the term ‘optimal taxation’ has come to acquire a meaning which is not obvious to economists who have not been following modern developments in public finance and welfare economics. Clearly, one could think of at least three different criteria for ‘optimality’ of the tax system. First, one could argue that a good tax system is one which minimizes the resource cost involved in assessing, collecting and paying the taxes. This is frequently a rather dominant concern of tax administrators, although they typically emphasize the costs incurred by the tax collectors and tend to neglect those borne by the firms and consumers who pay the taxes. Second, one could evaluate alternative tax systems in terms of justice or fairness. This would seem to be the line of thought which it is most natural for the ordinary taxpayer to follow, although his concept of justice may not be very precise and may include considerations which the economist would prefer to group under a different heading. Third, tax systems can be ranked according to the criterion of economic efficiency, and this was the original point of departure for the economic theory of optimal taxation; the optimal tax system is the one which minimizes the aggregate deadweight loss for any given tax revenue or level of public expenditure. The theory has then gradually been extended to take account of distributional *This paper was presented to the Conference on the Economics of Taxation, sponsored by the International Seminar in Public Economics, which was held at the Abbaye de Royaumont, France, January IS-20,1975. I appreciate the helpful comments received by the participants in the seminar, in particular Peter Diamond, Martin Feldstein and Serge-Christophe Kohn, and by Tony Atkiin, Arne Gabrieken and Victor Norman.
38
A. Sandmo, Optimal taxation
considerations. As regards the costs of administration, these have so far not been satisfactorily integrated in the theory, which of course to some extent limits its relevance for discussions of actual tax policy and tax reform. From an efficiency point of view an ideal tax system is one which is consistent with a Pareto optimal allocation of resources. The classical solution to the problem is to advocate lump-sum taxes, which are clearly neutral with respect to all marginal evaluations made by consumers and producers, but this is not a very helpful conclusion for the public finance economist. Although lump-sum taxes can be envisaged in the context of a once-and-for-all levy, it is much more difficult to imagine such taxes as a permanent system. If the public sector levies lump-sum taxes each year in such a way that the elasticity of the tax payment with respect to the taxpayer’s income exceeds one everywhere, taxpayers will soon discover that they do in fact have a progressive income tax system and adjust their actions accordingly. Therefore, it is hard to resist the conclusion that lump-sum taxation is a bad assumption both from a descriptive and a normative point of view. However, even if lump-sum taxes are ruled out, there are still taxes which are consistent with Pareto optimality. There is for example a large literature on the conditions under which a profits tax will be neutral with respect to production and investment decisions. Moreover, it has been argued since Pigou (1920) that indirect taxes can be used to improve the efficiency of the market allocation of resources in the presence of externalities. Thus, taxation need not be distortionary by the standard of Pareto optimality. But it seems definitely sensible to admit the unrealism of the assumption that the public sector can raise all its tax revenue from neutral or Pigovian taxes, and once we admit this we face the second-best problem of making the best of a necessarily distortionary tax system. This is the problem with which the optimal tax literature is mainly concerned. The treatment of the problem in the literature has an interesting and rather curious history. This has been well described by Baumol and Bradford (1970), so that there is no reason to go into details here. Although the early history of the subject goes back at least to 19th century writers on public utilities, the tist analytical formulation and solution of the problem appears in the celebrated article by Ramsey (1927). Ramsey gives credit to Pigou for suggesting the problem, and Pigou himself gave a very good, although simplified, treatment of it in his book on public iinance (1947). In spite of its exposure to the profession the analysis seems to have fallen into oblivion for many years.’ It was hardly mentioned in textbooks on public finance, nor did it have any impact on the analysis of the welfare economics of the second best which began with the article by Lipsey and Lancaster (1956-57); in fact, these authors do not even refer to ‘However,note should be made of the paper by Samuelson (1951), which is unfortunately still unpublished, and of Corlett and Hague (1953-54). whose important pioneering contribution is discussed in section 4 below.
A. Sandmo, Optimal taxation
39
the Ramsey-Pigou analysis.? Among French economists the subject received more attention; important analysis was contributed by Boiteux (1951, 1956) and many further developments were made by Kahn (1969, 1970). 3 Around 1970 there began a general revival of interest in the subject, with publication of articles by Baumol and Bradford (1970), Lerner (1970), Dixit (1970) and Diamond and Mirrlees (1971); of these, the Diamond-Mirrlees article in particular represents a major generalization and extension of the Ramsey formulation. The field now seems well established as one of considerable interest both from a theoretical and a practical point of view, and future textbooks in public economics will surely come to devote space to it both in their chapters on taxation and on public utility pricing. The present paper attempts to provide an introductory survey of the field which is intelligible to the nonspecialist, and at the same time to evaluate the relevance of the theoretical results for economic policy in the field of taxation. This is quite a lot for one paper, and the treatment is necessarily incomplete in many respects. Of the analytical detail, only the minimum which is necessary to gain some real insight into the subject is presented, and the discussion of policy implications are also rather sketchy and unsystematic. Section 2 introduces the basic theory of optimal commodity taxation. Section 3 analyzes the question of the possible uniformity of the optimal tax structure, and section 4 presents some formulas for simplified cases. The discussion is extended in section 5 to take account of production and supply conditions and in section 6 to incorporate redistributional objectives of taxation. Section 7 discusses briefly some additional problems in commodity taxation that are raised by public goods, externalities, international trade, public utilities and by the introduction of income taxes. The final section is an attempt to evaluate briefly the contribution that the optimal tax literature has made so far to the practical aspects of public economics. The paper attempts mainly to give an introductory survey of the field rather than of the literature itself, and it is therefore inevitable, although regrettable, that some important contributions have gone unmentioned: a more complete coverage of the literature would necessarily have implied less attention to analytical detail. 2. Optimal commodity taxation: The simplest case We shall start with the very simplest model imaginable, given that the inherent complexities of the problem are not entirely to be lost. Suppose that there are ZThe analytical approach of Lipsey and Lancaster may have prevented ing the similarity between their theory and that of Ramsey and Boiteux. shown to exist a duality relationship between the two sets of formulations; in detail by Bronsard (1971). 3These contributions have later been incorporated into Kolm’s treatise (1971a,b).
them from discoverIn fact, there can be this has been studied on pubhc economics
A. Sandmo, Optimal taxation
40
m + 1 commodities in the economy, the first of which is labour (to be denoted commodity 0) and the remaining m commodities are consumer goods. The latter are subject to indirect taxation, and we imagine that the public sector has a fixed tax revenue constraint, which says that a given amount - expressed in units of labour, which serves as the numeraire - has to be collected in taxes. Letting I, be the tax on commodity i and xi its quantity, we can write this constraint as
(1)
~ t,Xi = T, i=l
where ti is defined as the difference between the price paid by the consumer (Pi) and that which is received by the producer (pi). Let us assume that producer prices are given; this assumption has been shown to be equivalent in terms of its implications to the more general assumption of constant returns to scale. This then means that the problem of selecting a tax structure is equivalent to choosing a structure of consumer prices. We now make the further drastically simplifying assumption that the consumer side of the economy can be treated as if there were only one consumer. Taken literally, this assumption is of course quite uninteresting, so we need to be careful about the possible economic interpretation of this ‘as if’ assumption. We shall return to this question later on. For the moment we just postulate the existence of a social utility function,
u = wo, Xl,.
* .,
(2)
x,),
satisfying the usual concavity properties of consumer theory. Our problem is then to choose a tax structure (tI, . . ., t,) - or, equivalently, a consumer price structure (PI, . . ., I’,,,) -which satisfies (1) and maximizes (2) subject to this constraint. We can formulate this problem in terms of a Lagrange function, L = lJ(XO,Xl,.
. .,
X”)+P(i~~iX’T) 3
and we obtain the necessary conditions for a constrained maximum of U by setting the partial derivatives of L with respect to the tax rates equal to zero4 :
~o~-~+p(~;i~+xk)=O, k=
l,...,m.
These conditions can be simplified once we take account of the optimizing 4Note that Xx,/W, = ax&t,. It is convenientto write the derivativesof demand functions in terms of pricesrather than as functionsof the taxes.
A. Sandmo, Optimal taxation
41
behaviour of consumers, who maximize the utility function (2) subject to the budget constraint (5) This way of writing the budget constraint is easy to understand if we think of labour supply as being measured negatively; (5) then says simply that earnings must be equal to expenditure. Note in particular that there is no exogenous income which is not related to factor supply, nor are there any lump-sum taxes or subsidies. The optimum conditions for consumers take the form
Ui_APi= 0,
i = 0, 1, . . ., m,
(6)
where Ui = au/&, . Substituting from (6) into (4) we obtain
A f
f
p."'+p
‘apk
i=*
=
t.%+,
t-1
0,
‘apk
k = 1,. . ., m.
(4’)
But from the budget constraint (5) we have that
f i=O
piz+Xk =
0,
so that (4’) can be written as
-2x,+/d
5 t axi
(
i=l
i&+Xk
>
=Os
and finally as
Condition (7) provides the starting point for a discussion of what kind of rules for commodity taxation can be derived from the analysis. The first such rule is the one first derived by Ramsey (1927). Since we can write the Slutsky equation as
ax, -= apk
--_x,
$if Sik,
i, k = 1, . . ., m,
A. Sandmo, Optimal taxation
42
where I is income 5 and sik is the substitution effect, we can substitute from this into (7) to obtain
We can now utilize the fact that the substitution effects are symmetric (i.e. Sik = ski) to rewrite the condition as xk=v+
ax,
f t ivy fEl
k= l,...,
m.
The left-hand side of this equation can be interpreted as the relative decrease in demand for commodity k following on the tax change, provided that the consumer is compensated so as to stay on the same indifference curve. Since the right-hand side is constant, i.e. independent of k, it follows that this proportionate reduction of demand should be the same for all commodities.6 This result is particularly valuable when contrasted with the idea that indirect taxation at uniform rates is obviously best from an efficiency point of view. The uniformity issue will be discussed later, but it is well to remind the reader at this stage that an optimal allocation is defined in terms of quantities, not in terms of prices, and that a proportional reduction of all prices in terms of the numeraire has no obvious claim to be considered as optimal. Nevertheless, the Ramsey rule is hardly of great significance as a guide to practical tax policy. As it stands, it is valid only for an arbitrarily small tax revenue. Going back to eq. (7), we see easily that we could have a different and more interesting version of the Ramsey result if it were true that
axi -=-)
ap,
axk
api
i, k =
1, ...,m;
for we could then rewrite (7) as
(i$ti~)/xk=v~
k=
l,...,m,
(11)
5Actually, there is no exogenously given income in this model. This does not prevent us from utilizing the Slutsky equation, however, for we are simply using the income derivative to characterize the consumption indifference map. 6 -p is the marginal social value of an increase in T. Since this is negative, no account being taken in the present model of the uses of T, it follows that I( > 0. It can be shown that for T > 0, we must have Y = (d-p)/@ < 0. Intuitively this means that if consumers were to be given an exogenous (lump-sum) increase in income, and if this amount were then to be taxed away from them by means of indirect taxation, they would suffer a net loss. This is of course another restatement of the superiority of lump-sum taxation.
A. Sandmo, Optimal taxation
43
which implies that the Ramsey result of uniform proportional reduction of demand would be true without the restrictive assumption of zero tax revenue. Then question then becomes: when is (10) true? Going back to the Slutsky equation (8), and taking account of the symmetry of the substitution effects, we see that (10) implies
z ax, I ax, -*- -*-3 xl az X, az
i, k = 1, . . ., m,
i.e. equal income elasticities for all taxed goods.
Fig. 1
This can be illustrated diagrammatically for the case of two taxed goods (fig. I).’ The indifference map for the two taxed goods is homothetic, and the fall in demand resulting from taxation should be along the line OQ of equal proportionate reduction. Note that this also implies uniform taxation, i.e. no change of relative prices within the group of taxed goods. This suggests that deviations from the rule of uniform proportional reduction must be sought in unequal elasticities of income, and an interesting result to this effect can in fact be derived. From (8) and the symmetry of the substitution ?The reader is warned that the diagram must be interpreted with more than usual care, since it does not adequately take into account the existence of the third commodity, labour, which is the numeraire good.
44
A. Sandmo, Optimal taxation
terms we have that
ax, ax,
-z
ap,
aP,+xi
ax,
ax; ZDXk ;iT*
Substituting into (7) and rearranging terms, we obtain
If the proportionate change in demand for commodity k resulting from a hypothetical change in exogeneous income is higher on the average (using tax payments as weights) than for the other taxed commodities, then this implies a larger than average proportionate reduction of demand. This result lends itself nicely to an intuitive interpretation. Tax increases have both income and substitution effects, and the income effects are analogous to the changes that would have resulted if the revenue had been raised by lump-sum taxes. Since the latter effects are nondistortionary, so are the pure income effects and we should therefore reduce the demand most for the commodities where these effects dominate. 3. The uniformity issue We have already remarked that the rule of uniform taxation - i.e. taxation of all commodities at equal percentage rates - has no obvious claim to optimality. Yet in the example shown in fig. 1 above, this rule did after all turn out to be optimal. It is certainly of great interest both theoretically and practically to study the conditions under which this result holds.’ One might perhaps think that if only the set of taxable commodities were extended so as to include labour, then uniform taxation would turn out to be optimal, since this would mean that no relative prices in the system would be changed, as compared with the pre-tax Pareto optimal equilibrium. But this is wrong, for the simple reason that such a tax structure would result in zero tax revenue. Let Bi = ti/Pi, i.e. the tax rate as a percentage of the consumer price. Total tax revenue is then T=
F t$Pixi = 0 i=O f Pixi,
i=O
where the last equality follows from the uniformity assumption. But from the *This section is based on Sandmo (1974); see also Atkinson and Stiglitz (1972).
A. Sandmo, Optimal taxation
45
budget constraint (5) this expression is necessarily zero. Keeping all relative prices constant when consumer goods are taxed implies subsidizing labour supply at the same rate, and subsidies and taxes must necessarily cancel each other. The possible optimality of uniform taxation cannot be established in this way. It has long been realized that if there exists a commodity which is completely inelastic in demand, not only with respect to its own but to all prices, then this commodity is an ideal object of taxation from an efficiency point of view. Suppose now that labour is completely inelastic in supply. Ideally, we would have liked to choose labour as the only taxed commodity; this would have meant a change in the relative price of labour and consumer goods, but no change in the relative prices among the consumer goods themselves. It then becomes clear that if labour is not taxed, we can achieve exactly the same result by taxing all consumer goods at the same rate, so that this is a case where uniform taxation is optimal. The other case in which uniform consumer goods taxation is optimal can be identified by referring again to fig. 1. Here uniform taxation is optimal because the income elasticities for both taxed goods are the same; the indifference map is homothetic. However, the argument is incomplete because the indifference map must be understood as drawn for a given supply of labour, while this supply will in reality change with the structure of prices. But if the indifference map is in effect invariant with respect to changes in the labour supply, the argument obviously holds. This implies that we have to add an assumption of utility separability between consumption and labour to that of homotheticity in the consumption indifference map; under these conditions we have again that uniform taxation is optimal Thus, although there do exist interesting cases in which uniform taxation is optimal, these must definitely be considered as exceptions. In the general case it is not easy to see the structure of taxation which follows from the general optimality conditions. There are some special cases, however, in which this is possible, and to these we now turn. 4. Elasticity formulae Let us assume that all cross derivatives of the demand functions vanish as between the taxed goods. Conditions (7) are then simplified to read
k=
l,...,m,
where &,& is the (direct) price elasticity of demand. This is the well-known inverse elasticity rule, which has also been derived from partial consumer surplus
A. Sanaho, Optimal taxation
46
analysis, e.g. by Hicks (1947). The idea behind the rule of imposing the highest tax rates on the commodities with the lowest price elasticities of demand is of course to minimize the deviations from the nondistortive, pre-tax allocation. Elasticity formulae become very complicated in the general case and provide little intuitive insight into the structure of taxation. One particular case which it may be useful to consider is a three-good model, containing labour and two taxed goods, This case was first considered by Corlett and Hague (1953-54).’ Eq. (9) then becomes @ll
+t,s12
t,s,,+t2s,*
=
-
KXl,
=
- KX2.
Here we have written K for the right-hand side of (9). These equations can be solved to yield t, = --K
Xl sz2 -
X2Sl2 >
S11S22-3
t2=
--Ic
X2%1
:2
-x1s21
SllS22
42
We can rewrite these expressions in terms of ad valorem tax rates and compensated elasticities as follows : 8, =
1
x1x2
--K
(c722-(712)
=
-G~22--%2h
w22-~:2plpz
o2 =
1
=%rri
-K
-cr21) = - K’(cQi-02J.
W22-42PlP2
Here the compensated elasticity uki = Ski(PI/Xk)(i, k, = 1,2). It follows from the theory of consumption that sllsz 2 - sf2 > 0. It also follows that ~lo+~li+~lz
=
0
=
cr2,+cr21+a22.
Substituting for crl2 and a,, in the expressions for the tax rates, we obtain
4 = -“‘(“11+a,,+o,fJ,
(14)
02
(15)
=
--‘tfJ,l+~22+Q20),
9An early and apparently neglected reference is Meade (1955). The model has also been studied by Diamond and Mirrlees (1971, part II) and by Andersen (1971).
A. Sandmo, Optimal taxation
41
and it follows that
The consumer good which is to be taxed at the highest rate is the one with the lowest compensated cross-elasticity with labour. This implies that a consumer good which is a complement with iabour (substitute for leisure) should be taxed at a lower rate than one which is a substitute for labour (complementary with leisure). The economic rationale of this rule is clearly that since we have barred ourselves from taxing leisure, we can do it indirectly by taxing the commodities that are complementary with leisure.
5. Production and supply Our assumption of given producer prices has led us to rules of optimal taxation which are independent of the conditions of production and supply. It was shown by Diamond and Mirrlees (1971) that these rules continue to be valid in the more general case of constant returns to scale. This is an important result which may not be intuitively obvious, since marginal cost under constant returns is constant only in a partial equilibrium and not in a general equilibrium sense. However, this raises the question of how the rules will have to be changed in the case of nonconstant returns to scale. We shall focus here on the case of decreasing returns, since increasing returns to scale raises some particularly difficult questions which it would be impossible to discuss at all adequately in the present context. Analytically, the most direct and easy approach to this issue is just to add a production constraint to our maximization problem. Derivatives of the supply functions will then enter the optimal tax formulae along with the demand derivatives. This does not, of course, make the tax structure any easier to evaluate than before, and to get some feeling for the implications of the analysis we have to consider special cases. If we make the assumption that all crosselasticities of both demand and supply functions vanish, it can be shown that, as an approximation, the optimal tax rate for any one commodity is proportional to the sum of the inverses of the demand and supply elasticities; this is an extension of our previous rule (13). It can be left to the reader as an exercise to extend our previous analyses to the case of production along these lines; this has also been done by Dixit (1970). However, this approach raises some awkward questions. How can we work with an aggregate production constraint if there are decreasing returns to scale? And if we disaggregate the model to the level of the individual firm, does not this open up the possibility that firms should be taxed at different rates? The latter result obviously implies production inefficiency, i.e. a movement away from
48
A. Sam&no, Optimal taxation
the production possibility curve, but in a second-best world it clearly cannot be ruled out. One should also take into account that with decreasing returns, profits will exist in general equilibrium, and that one should therefore consider profit taxes as components of an optimal tax structure. By our assumption of given producer prices equal to marginal costs, we have in fact assumed productive efficiency. Assuming constant returns to scale, Diamond and Mirrlees (1971) showed that productive efficiency was indeed desirable. The argument is very simple and is presented particularly clearly in Mirrlees (1972). The essential point turns out to be that consumer welfare does not depend directly on producer prices; it is then optimal for producers to maximize profit at prices that imply production efficiency, which again implies that they must face the same price vector. It was brought out by Stiglitz and Dasgupta (1971) that the important assumption here is not that of constant returns per se but of zero profits in equilibrium. Thus, they showed that the desirability of productive efficiency continues to hold under decreasing returns to scale, provided that the government imposes a 100 percent profits tax on all producers. We shall not go any further into this line of argument, which easily becomes quite involved. However, we note that with the possibility of differential taxation of producers we must also take into account that producers may find it profitable to merge or dissolve firms for tax reasons. This might make it both difficult and costly to operate such a tax system and is presumably one of the reasons why, for example, the corporate profits tax is almost always a genera1 proportional tax. Perhaps, therefore, one should not worry too much about the exceptions to the rule that productive efficiency is desirable; the administrative and informational costs of deviating from the rule might easily be too high for it to be an interesting alternative. 6. Distributional considerations So far we have been assuming a one-consumer economy, or, alternatively, that the preferences of the community can be represented by social indifference curves. This can evidently only be a first approximation. Social indifference curves exist if lump-sum transfers are constantly being used ‘in the background’ to keep the income distribution optimal according to some individualistic welfare function; but in the optimal tax literature such transfers are ruled out by assumption. Social indifference curves also exist if all individuals have identical and homothetic indifference maps; but in that case we can safely ignore distributional issues, since nothing in the way of redistribution can be achieved by commodity taxation anyway.l’ “‘For a fuller discussion of social indifference curves, see Samuelson (1956).
A. Sandmo, Optimal taxation
49
The possibility of a conflict between our efficiency rules and distributional objectives becomes evident when we consider the practical implications of the inverse elasticity rule. We tend to think of commodities with numerically low price elasticities as being necessities, and of those with high elasticities as luxuries. The rule then implies that necessities should be taxed at higher rates than luxuries. Atkinson and Stiglitz (1972) have performed some illustrative calculations on empirical data which confirm this ; they come out with tax rates on food which are two to three times higher than those on consumer durables. There is clearly a need for a modification of the analysis to give some scope for distributional considerations. Let us now assume that individuals are heterogenous both with respect to preferences and productivities. The utility function of individual j is Uj
=
Uj(XjO,
. .
.)
Xj,),
j
=
1,. . ., n,
(17)
and the welfare function, which is assumed to belong to the individualistic Bergson-Samuelson family, is
w=
W(u,, . . ., u,),
wj >
0.
(18)
We can now obtain a generalization of the analysis in section 2 by maximizing (18) with respect to consumer prices subject to the government’s budget constraint. There is no need to go through the whole analysis again and we shall just give the main result for the case of independent demands,‘l in which we obtain a generalization of the inverse elasticity formula:
The ad valorem tax rate is still proportional to the inverse of the market demand elasticity, but the proportionality factor has a more complicated form. Given the market demand elasticity, the tax rate on good k should be higher, the lower is consumers’ average social marginal utility of income, when weighted by their consumption of good k. In the special case where all individuals have the same strictly concave utility function and the welfare function is the unweighted sum of individual utilities, we can draw the stronger conclusion that “It may be worth noting that to derive the generalized inverse elasticity formula, it is only necessary to assume that the cross-derivatives of the market demand functions vanish. Theoretically at least, one could imagine cases where two commodities are substitutes for some consumers and complements for others, with zero cross-effect in the aggregate, so that this assumption is weaker than that of demand independence for all consumers.
50
A. Sat&no, Optimal taxation
the tax rate should be higher, the more strongly the consumption of the commodity is concentrated among high-income individuals.’ 2 It is of course difficult to claim much in the way of immediate applicability for the formula (19) and for the more general conditions for optimal taxation in the case of many individuals. What we may claim for the analysis seems mainly to be that it gives some insights into the nature of the compromise between equity and efficiency considerations that will have to be made in practice. These insights are not so general as to be empty of all empirical content. One may note that if two goods have the same proportionality factor-if, in the words of Feldstein (1972a), they have the same distributional characteristic - then it is still true, in the case of independent demands, that the ratio of their tax rates should be equal to the inverse ratio of their price elasticities. One might also attempt to develop the distributional characteristic into a more operational measure; this is also done by Feldstein (1972a,b). 7. Further problems in commodity taxation With the recent growth of the literature on optimal taxation, the grounds for considering it as a special field within the general area of public economics are weakening. This is not only because an increasing number of economists are becoming familiar with this type of analysis, but also because the theory is rapidly becoming integrated with other well-established subfields of public economics and general economic analysis. It is obviously impossible here to go into all of these in any detail, but they should at least be mentioned. (I) Public goods and taxes. It was argued by Pigou (1947) that distortionary taxation introduces an additional cost factor in calculations of the optimal supply of public goods. Consequently, he argued, one would want to curtail the supply of public goods compared to the first-best rule of making marginal benefit equal to marginal cost. This conclusion has been analyzed in the contributions of Stiglitz and Dasgupta (1971) and Atkinson and Stern (1974). Their work represents a generalization of the analysis of previous sections, in that the public sector’s tax revenue requirement is derived from the cost of supplying public goods. It turns out that Pigou’s conjecture is not generally correct. Under firstbest conditions, the correct benefit measure, assuming an optimal distribution of income, is the sum of the marginal rates of substitution. With distortionary taxation, the correct benefit measure may exceed the first-best measure if the public good is complementary with taxed goods or if taxation releases income effects which increase the demand for the taxed goods; the latter case would be of importance if taxed goods are inferior. Atkinson and Stern also point out
1*A more generaltreatment of distributionalissues,attemptingto generalizethe Ramseyrule (9) is contained in the articles by Mirrks (1975) and Diamond (1975).
51
A. Sam&no, Optimal taxation
that the answer to the question of the correct benefit measure does not in itself provide a solution to the over- or under-supply problem. This is more difficult, and there does not seem to be available a simple and general solution. (2) Externalities. If externalities exist we know that taxes and subsidies can be used to improve the competitive allocation of resources and in fact make it Pareto optimal. However, the standard analysis of Pigovian taxes assumes, more or less implicitly, that the public sector either needs no additional tax revenue or that it distributes the proceeds from Pigovian taxes in a lump-sum manner. If neither of these conditions hold, we again have a second-best problem where the government must simultaneously employ taxes which improve and taxes which distort the allocation of resources. Sandmo (1975) analyzes the optimal combination of such taxes for the case of a negative consumption externality and concludes that the marginal social damage should only be reflected in the tax on the externality-creating commodity, regardless of the pattern of complementarity and substitutability; exceptions to this rule have been discussed by Green and Sheshinski (1974). Another issue in this area concerns the choice of optimal taxes or subsidies when the government is constrained to tax uniformly generators of externalities whom it really would have been optimal to tax at different rates. This problem has been investigated by Kolm (1971b) and Diamond (1973). (3) International trade. The theory of the optimal tariff has some important similarities with the theory of optimal commodity taxation, and it seems a natural undertaking to try to integrate the two bodies of literature. This has been attempted in articles by Boadway, Maital and Prachowny (1973) and Dasgupta and Stiglitz (1974). The problem here has some similarities with one which arises in the analysis of externalities; we wish to derive rules for optimal tariffs (which improve the allocation of resources) and optimal commodity taxes (which distort it) simultaneously. The two articles also discuss optimality criteria for public goods and relate the criteria to the problem of the use of international prices in domestic cost-benefit analyses.r3 (4) Public utility pricing. The theory of optimal commodity taxation can be reinterpreted as a theory of public utility pricing, and the latter furnishes the frame of reference for many contributions to the subject; in particular, this is true of the work of French economists like Boiteux (1956) and Kolm (1971a, b). This interpretation is perhaps a more natural one as long as economic efficiency is taken as the sole criterion; one could imagine the government as ordering public utilities to set their prices according to efficiency criteria on the assumpIsFor an application of optimal tax theory to the problems integration, see Kolm (1969b).
of international
economic
52
tion that the government policies.
A. Sandmo, Optimal taxation
itself will determine
appropriate
redistributional
(5) Income taxation. Many of the ideas discussed so far are of course applicable to income taxation as well as to commodity taxation. Thus, we should expect the efficiency loss associated with an income tax to be larger the more elastic is the labour supply with respect to the wage rate. If we consider the degree of progression in the tax schedule, it would obviously be important to know how the elasticity of labour supply varies with income. If people with higher income are also characterized by elastic labour supply, this would act as a brake on the degree of progression that one might otherwise prefer from a redistributional point of view. Nevertheless, income taxation has some peculiar features which are difficult to incorporate in the previous framework. Space prevents a full treatment of this topic here; the reader is referred to the articles by Mirrlees (1971) and Sheshinski (1972) as well as to the survey in Atkinson (1973). 8. Concludiug remarks
There can be no doubt that the recent developments in the analysis of optimal taxation have brought welfare economics closer to the realities of economic policy. We know how to model optimization problems in the public sector with fairly realistic assumptions about the set of policy tools available. New insights have been gained into the efficiency aspects of taxation, and we can probably also claim to have obtained a better understanding of the tradeoff between equity and efficiency. The theory obviously has its limitations. It is at its best in yielding rules for the optimal structuring of a given tax system and has less to contribute to the discussion of major problems of tax reform, which typically involves the choice between alternative tax systems. A difficulty with the extension of the theory to cover these ‘global’ problems is that the costs of administration have not been incorporated into the theory;i4 this is one aspect of the neglect of transactions costs in the theory of general equilibrium. The incorporation of costs of administration is an extremely complicated task, and it remains to be seen whether a formally more complete theory can still yield conclusions which are interesting and meaningful from the point of view of implementation. However, this raises the question of whether optimum tax formulae can have any claim to be taken seriously, given that they abstract from such central concerns as administrative costs and incomplete information.’ ’ Whatever the 14An attempt to do so is presented by Heller and Shell (1974). However, this work is still at a very abstract level. 15For an answer which is mainly in the negative, see Hahn (1973).
A. Sadno,
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final answer to this question may be, I believe that we shall probably have to reconcile ourselves to the fact that no policy model can be complete in the sense of taking account of all relevant concerns facing a policymaker. Thus, it may well be that we shall find the models of optimal taxation to be useful ones, even though we may have to supplement them with considerations which are exogenous to the models themselves. References Andersen, P.S., 1972, The optimum tax structure in a three-good, one-consumer
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Kolm, S. Ch., 197lb, Cours d’economie publique 2: Le service de masses (Dunod, Paris). Lerner, A.P., 1970, On optimal taxes with an untaxable sector, American Economic Review 60, 284-294. Lipsey, R.G. and K. Lancaster, 1953-57, The general theory of second best, Review of Economic Studies 24,11-32. Meade, J.E., 1955, Trade and welfare (Oxford University Press, London). Mirrlees, J.A., 1971, An exploration in the theory of optimum income taxation, Review of Economic Studies 38,175-208. Mirrlees, J.A., 1972, On producer taxation, Review of Economic Studies 39, 105-l 11. Mirrlees, J.A., 1975, Optimal commodity taxation in a two-class economy, Journal of Public Economics 427-33. Pigou, A.C., 1920, The economics of welfare (Macmillan, London). Pigou, A.C., 1947, A study in public finance, 3rd edition (Macmillan, London) 1st edition, 1928. Ramsey, F.P., 1927, A contribution to the theory of taxation, Economic Journal 37,47-61. Samuelson, P.A., 1951, Theory of optimal taxation, unpublished paper. Samuelson, P.A., 1956, Social indifference curves, Quarterly Journal of Economics 70, l-22. Sandmo, A., 1974, A note on the structure of optimal taxation, American Economic Review 64,701-706. Sandmo, A., 1975, Optimal taxation in the presence of externalities, Swedish Journal of Economics 77,86-98. Sheshinski, E., 1972, The optimal linear income tax, Review of Economic Studies 39,297-302. Stiglitz, J.E. and P. Dasgupta, 1971, Differential taxation, public goods, and economic efficiency, Review of Economic Studies 38,151-174.
