43 Optimal Taxation- An Introduction to the Literature

8/8/2019 43 Optimal Taxation- An Introduction to the Literature

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Journal of Public Economics 6 (1976) 37-54. 0 North-Holland Publishing Company

OPTIMAL TAXATIONAn introduction to the literature

Agnar SANDMO*Norw egian School of Economics and Business Administration, Bergen, Norw ay

Revised version received September 1975This paper attempts to give an introductory survey of recent contributions to the literature onoptimal taxation. It covers the basic theorems on optimal commodity taxation and discussesthe insights that the theory provides into the structure of an optimal tax system. No systematicstudy is made of the possibilities for application, but the theory is surveyed with a view towardsits implications for actual tax policy.

1. IntroductionAmong a subset of economists the term optimal taxation has come to acquirea meaning which is not obvious to economists who have not been following

modern developments in public finance and welfare economics. Clearly, onecould 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 theresource cost involved in assessing, collecting and paying the taxes. This isfrequently a rather dominant concern of tax administrators, although theytypically emphasize the costs incurred by the tax collectors and tend to neglectthose borne by the firms and consumers who pay the taxes. Second, one couldevaluate alternative tax systems in terms of justice or fairness. This would seemto be the line of thought which it is most natural for the ordinary taxpayer tofollow, although his concept of justice may not be very precise and may includeconsiderations which the economist would prefer to group under a differentheading. Third, tax systems can be ranked according to the criterion of economicefficiency, and this was the original point of departure for the economic theoryof optimal taxation; the optimal tax system is the one which minimizes theaggregate 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 bythe 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 inthe seminar, in particular Peter Diamond, Martin Feldstein and Serge-Christophe Kohn,and by Tony Atkiin, Arne Gabrieken and Victor Norman.


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38 A. Sandmo, Optimal taxation

considerations. As regards the costs of administration, these have so far not beensatisfactorily integrated in the theory, which of course to some extent limits itsrelevance for discussions of actual tax policy and tax reform.

From an efficiency point of view an ideal tax system is one which is consistentwith a Pareto optimal allocation of resources. The classical solution to theproblem is to advocate lump-sum taxes, which are clearly neutral with respectto all marginal evaluations made by consumers and producers, but this is not avery helpful conclusion for the public finance economist. Although lump-sumtaxes can be envisaged in the context of a once-and-for-all levy, it is much moredifficult to imagine such taxes as a permanent system. If the public sector levieslump-sum taxes each year in such a way that the elasticity of the tax paymentwith respect to the taxpayers income exceeds one everywhere, taxpayers willsoon discover that they do in fact have a progressive income tax system andadjust their actions accordingly. Therefore, it is hard to resist the conclusionthat lump-sum taxation is a bad assumption both from a descriptive and anormative point of view.

However, even if lump-sum taxes are ruled out, there are still taxes which areconsistent with Pareto optimality. There is for example a large literature on theconditions under which a profits tax will be neutral with respect to production andinvestment decisions. Moreover, it has been argued since Pigou (1920) thatindirect taxes can be used to improve the efficiency of the market allocation ofresources in the presence of externalities. Thus, taxation need not be distortionaryby the standard of Pareto optimality. But it seems definitely sensible to admit theunrealism of the assumption that the public sector can raise all its tax revenuefrom neutral or Pigovian taxes, and once we admit this we face the second-bestproblem of making the best of a necessarily distortionary tax system. This is theproblem with which the optimal tax literature is mainly concerned.

The treatment of the problem in the literature has an interesting and rathercurious 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 ofthe subject goes back at least to 19th century writers on public utilities, the tistanalytical formulation and solution of the problem appears in the celebratedarticle by Ramsey (1927). Ramsey gives credit to Pigou for suggesting theproblem, and Pigou himself gave a very good, although simplified, treatmentof it in his book on public iinance (1947). In spite of its exposure to the professionthe analysis seems to have fallen into oblivion for many years. It was hardlymentioned in textbooks on public finance, nor did it have any impact on theanalysis of the welfare economics of the second best which began with the articleby 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 unfortunatelystill unpublished, and of Corlett and Hague (1953-54). whose important pioneering contributionis discussed in section 4 below.


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A. Sandmo, Optimal taxation 39

the Ramsey-Pigou analysis.? Among French economists the subject receivedmore attention; important analysis was contributed by Boiteux (1951, 1956) andmany further developments were made by Kahn (1969, 1970). 3 Around 1970there began a general revival of interest in the subject, with publication of articlesby Baumol and Bradford (1970), Lerner (1970), Dixit (1970) and Diamond andMirrlees (1971); of these, the Diamond-Mirrlees article in particular representsa major generalization and extension of the Ramsey formulation. The field nowseems well established as one of considerable interest both from a theoreticaland a practical point of view, and future textbooks in public economics willsurely come to devote space to it both in their chapters on taxation and on publicutility pricing.

The present paper attempts to provide an introductory survey of the fieldwhich is intelligible to the nonspecialist, and at the same time to evaluate therelevance 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 incompletein many respects. Of the analytical detail, only the minimum which is necessaryto gain some real insight into the subject is presented, and the discussion of policyimplications are also rather sketchy and unsystematic.Section 2 introduces the basic theory of optimal commodity taxation. Section 3analyzes the question of the possible uniformity of the optimal tax structure,and section 4 presents some formulas for simplified cases. The discussion isextended in section 5 to take account of production and supply conditions andin section 6 to incorporate redistributional objectives of taxation. Section 7discusses briefly some additional problems in commodity taxation that areraised by public goods, externalities, international trade, public utilities and bythe introduction of income taxes. The final section is an attempt to evaluatebriefly the contribution that the optimal tax literature has made so far to thepractical aspects of public economics.

The paper attempts mainly to give an introductory survey of the field ratherthan of the literature itself, and it is therefore inevitable, although regrettable,that some important contributions have gone unmentioned: a more completecoverage of the literature would necessarily have implied less attention toanalytical detail.2. Optimal commodity taxation: The simplest case

We shall start with the very simplest model imaginable, given that the inherentcomplexities of the problem are not entirely to be lost. Suppose that there are

ZThe analytical approach of Lipsey and Lancaster may have prevented them from discover-ing the similarity between their theory and that of Ramsey and Boiteux. In fact, there can beshown to exist a duality relationship between the two sets of formulations; this has been studiedin detail by Bronsard (1971).3These contributions have later been incorporated into Kolms treatise on pubhc economics(1971a,b).


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40 A. Sandmo, Optimal taxationm + 1 commodities in the economy, the first of which is labour (to be denotedcommodity 0) and the remaining m commodities are consumer goods. Thelatter are subject to indirect taxation, and we imagine that the public sector hasa fixed tax revenue constraint, which says that a given amount - expressed inunits 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 thisconstraint as

~ t,Xi = T,i=l (1)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 producerprices are given; this assumption has been shown to be equivalent in terms of itsimplications to the more general assumption of constant returns to scale. Thisthen means that the problem of selecting a tax structure is equivalent to choosinga structure of consumer prices. We now make the further drastically simplifyingassumption that the consumer side of the economy can be treated as if there wereonly one consumer. Taken literally, this assumption is of course quite uninterest-ing, so we need to be careful about the possible economic interpretation of thisas if assumption. We shall return to this question later on. For the moment wejust postulate the existence of a social utility function,

u = wo, Xl,. * ., x,), (2)satisfying the usual concavity properties of consumer theory. Our problem isthen to choose a tax structure (tI, . . ., t,) - or, equivalently, a consumer pricestructure (PI, . . ., I,,,) -which satisfies (1) and maximizes (2) subject to thisconstraint. We can formulate this problem in terms of a Lagrange function,

L = lJ(XO,Xl,. . ., X)+P(i~~iXT)and we obtain the necessary conditions for a constrained maximum of U bysetting the partial derivatives of L with respect to the tax rates equal to zero4 :

~o~-~+p(~;i~+xk)=O,= l,...,m.These conditions can be simplified once we take account of the optimizing

4Note tha t Xx,/W, = ax&t,. It is convenient to write the derivativesof demand functionsin terms of prices rather than as functions of the taxes.


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behaviour of consumers, who maximize the utility function (2) subject to thebudget constraint

(5)

This way of writing the budget constraint is easy to understand if we think oflabour supply as being measured negatively; (5) then says simply that earningsmust be equal to expenditure. Note in particular that there is no exogenousincome which is not related to factor supply, nor are there any lump-sum taxesor 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 p."'+pi=* apk f t.%+,t-1 apk = 0, k = 1,. . ., m. (4)

But from the budget constraint (5) we have that

f piz+Xk ,i=Oso that (4) can be written as

-2x,+/d (5 t axii=l i&+Xk > =Osand finally asCondition (7) provides the starting point for a discussion of what kind of rulesfor commodity taxation can be derived from the analysis. The first such rule isthe one first derived by Ramsey (1927). Since we can write the Slutsky equation asax,= --_x,

a p k$if Sik, i, k = 1, . . ., m,


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where I is income 5 and sik is the substitution effect, we can substitute from thisinto (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+ f t ax,fEl ivy 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 theconsumer is compensated so as to stay on the same indifference curve. Since theright-hand side is constant, i.e. independent of k, it follows that this proportionatereduction of demand should be the same for all commodities.6

This result is particularly valuable when contrasted with the idea that indirecttaxation 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 atthis stage that an optimal allocation is defined in terms of quantities, not in termsof prices, and that a proportional reduction of all prices in terms of the numerairehas no obvious claim to be considered as optimal.

Nevertheless, the Ramsey rule is hardly of great significance as a guide topractical tax policy. As it stands, it is valid only for an arbitrarily small taxrevenue. Going back to eq. (7), we see easily that we could have a different andmore interesting version of the Ramsey result if it were true that

axi axk-=-)ap, a p i i, k = 1, . , ;

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 usfrom utilizing the Slutsky equation, however, for we are simply using the income derivative tocharacterize 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 forT > 0, we must have Y = (d-p)/@ < 0. Intuitively this means that if consumers were to begiven an exogenous (lump-sum) increase in income, and if this amount were then to be taxedaway from them by means of indirect taxation, they would suffer a net loss. This is of courseanother restatement of the superiority of lump-sum taxation.


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which implies that the Ramsey result of uniform proportional reduction ofdemand would be true without the restrictive assumption of zero tax revenue.Then question then becomes: when is (10) true? Going back to the Slutskyequation (8), and taking account of the symmetry of the substitution effects,we see that (10) implies

I ax, z a x ,- * - - -*-3xl 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 thefall in demand resulting from taxation should be along the line OQ of equalproportionate reduction. Note that this also implies uniform taxation, i.e. nochange of relative prices within the group of taxed goods.

This suggests that deviations from the rule of uniform proportional reductionmust be sought in unequal elasticities of income, and an interesting result to thiseffect 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, sinceit does not adequately take into account the existence of the third commodity, labour, which isthe numeraire good.


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44 A. Sandmo, Optimal taxationterms we have that

ax, ax, ax, ax;-zap, aP,+xi ZDXk;iT*Substituting into (7) and rearranging terms, we obtain

If the proportionate change in demand for commodity k resulting from ahypothetical change in exogeneous income is higher on the average (using taxpayments as weights) than for the other taxed commodities, then this implies alarger than average proportionate reduction of demand.

This result lends itself nicely to an intuitive interpretation. Tax increases haveboth income and substitution effects, and the income effects are analogous tothe changes that would have resulted if the revenue had been raised by lump-sumtaxes. Since the latter effects are nondistortionary, so are the pure income effectsand we should therefore reduce the demand most for the commodities wherethese effects dominate.

3. The uniformity issueWe 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 beoptimal. It is certainly of great interest both theoretically and practically to studythe conditions under which this result holds.

One might perhaps think that if only the set of taxable commodities wereextended so as to include labour, then uniform taxation would turn out to beoptimal, since this would mean that no relative prices in the system would bechanged, as compared with the pre-tax Pareto optimal equilibrium. But this iswrong, for the simple reason that such a tax structure would result in zero taxrevenue. 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 f Pixi,i=O i=Owhere the last equality follows from the uniformity assumption. But from the

*This section is based on Sandmo (1974); see also Atkinson and Stiglitz (1972).


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budget constraint (5) this expression is necessarily zero. Keeping all relativeprices constant when consumer goods are taxed implies subsidizing laboursupply at the same rate, and subsidies and taxes must necessarily cancel eachother. The possible optimality of uniform taxation cannot be established in thisway.

It has long been realized that if there exists a commodity which is completelyinelastic in demand, not only with respect to its own but to all prices, then thiscommodity is an ideal object of taxation from an efficiency point of view.Suppose now that labour is completely inelastic in supply. Ideally, we would haveliked to choose labour as the only taxed commodity; this would have meant achange in the relative price of labour and consumer goods, but no change in therelative prices among the consumer goods themselves. It then becomes clear thatif labour is not taxed, we can achieve exactly the same result by taxing all con-sumer goods at the same rate, so that this is a case where uniform taxation isoptimal.

The other case in which uniform consumer goods taxation is optimal can beidentified by referring again to fig. 1. Here uniform taxation is optimal becausethe income elasticities for both taxed goods are the same; the indifference mapis homothetic. However, the argument is incomplete because the indifferencemap must be understood as drawn for a given supply of labour, while thissupply will in reality change with the structure of prices. But if the indifferencemap is in effect invariant with respect to changes in the labour supply, theargument obviously holds. This implies that we have to add an assumption ofutility separability between consumption and labour to that of homotheticity inthe consumption indifference map; under these conditions we have again thatuniform taxation is optimal

Thus, although there do exist interesting cases in which uniform taxation isoptimal, these must definitely be considered as exceptions. In the general case it isnot easy to see the structure of taxation which follows from the generaloptimality conditions. There are some special cases, however, in which this ispossible, and to these we now turn.

4. Elasticity formulaeLet 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 &,&s the (direct) price elasticity of demand. This is the well-known inverseelasticity rule, which has also been derived from partial consumer surplus


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46 A. Sanaho, Optimal taxation

analysis, e.g. by Hicks (1947). The idea behind the rule of imposing the highesttax rates on the commodities with the lowest price elasticities of demand is ofcourse to minimize the deviations from the nondistortive, pre-tax allocation.

Elasticity formulae become very complicated in the general case and providelittle intuitive insight into the structure of taxation. One particular case which itmay be useful to consider is a three-good model, containing labour and twotaxed goods, This case was first considered by Corlett and Hague (1953-54).Eq. (9) then becomes

@ll +t ,s12 = - KXl,

t,s,,+t2s,* = - KX2.Here we have written K for the right-hand side of (9). These equations can besolved to yield

t, = --K Xl sz - X2Sl2 >S11S22-3 :2

t2= --Ic X2%1 -x1s21SllS22 42

We can rewrite these expressions in terms of ad valorem tax rates and com-pensated elasticities as follows :

8, = --K 1 x1x2w22-~:2plpz

(c722-(712) = -G~22--%2h

o2 = -K 1 =%rri -cr21) = - K(cQi 02J.W22-42PlP2

Here the compensated elasticity uki = Ski(PI/Xk) i, k , = 1,2). It follows fromthe theory of consumption that sllsz 2 - sf2 > 0. It also follows that

~lo+~li+~lz = 0 = cr2,+cr21+a22.

Substituting for crl 2 and a,, in the expressions for the tax rates, we obtain4 = -(11+a,,+o,fJ, (14)02 = --t fJ,l+~22+Q20), (15)

9An early and apparently neglected reference is Meade (1955). The model has also beenstudied by Diamond and Mirrlees (1971, part II) and by Andersen (1971).


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and it follows that

The consumer good which is to be taxed at the highest rate is the one with thelowest compensated cross-elasticity with labour. This implies that a consumergood which is a complement with iabour (substitute for leisure) should be taxedat a lower rate than one which is a substitute for labour (complementary withleisure). The economic rationale of this rule is clearly that since we have barredourselves from taxing leisure, we can do it indirectly by taxing the commoditiesthat are complementary with leisure.

5. Production and supplyOur assumption of given producer prices has led us to rules of optimal taxa-

tion which are independent of the conditions of production and supply. It wasshown by Diamond and Mirrlees (1971) that these rules continue to be valid inthe more general case of constant returns to scale. This is an important resultwhich may not be intuitively obvious, since marginal cost under constant returnsis 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 thecase of nonconstant returns to scale. We shall focus here on the case of decreasingreturns, since increasing returns to scale raises some particularly difficultquestions which it would be impossible to discuss at all adequately in the presentcontext.

Analytically, the most direct and easy approach to this issue is just to add aproduction constraint to our maximization problem. Derivatives of the supplyfunctions will then enter the optimal tax formulae along with the demandderivatives. This does not, of course, make the tax structure any easier toevaluate than before, and to get some feeling for the implications of the analysiswe have to consider special cases. If we make the assumption that all cross-elasticities of both demand and supply functions vanish, it can be shown that,as an approximation, the optimal tax rate for any one commodity is proportionalto the sum of the inverses of the demand and supply elasticities; this is anextension of our previous rule (13). It can be left to the reader as an exercise toextend our previous analyses to the case of production along these lines; this hasalso been done by Dixit (1970).

However, this approach raises some awkward questions. How can we workwith 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 notthis open up the possibility that firms should be taxed at different rates? Thelatter result obviously implies production inefficiency, i.e. a movement away from


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48 A. Sam&no, Optimal taxation

the production possibility curve, but in a second-best world it clearly cannot beruled out. One should also take into account that with decreasing returns, profitswill exist in general equilibrium, and that one should therefore consider profittaxes as components of an optimal tax structure.

By our assumption of given producer prices equal to marginal costs, we havein fact assumed productive efficiency. Assuming constant returns to scale,Diamond and Mirrlees (1971) showed that productive efficiency was indeeddesirable. The argument is very simple and is presented particularly clearly inMirrlees (1972). The essential point turns out to be that consumer welfare doesnot depend directly on producer prices; it is then optimal for producers tomaximize profit at prices that imply production efficiency, which again impliesthat they must face the same price vector. It was brought out by Stiglitz andDasgupta (1971) that the important assumption here is not that of constantreturns per se but of zero profits in equilibrium. Thus, they showed that thedesirability of productive efficiency continues to hold under decreasing returnsto scale, provided that the government imposes a 100 percent profits tax on allproducers.

We shall not go any further into this line of argument, which easily becomesquite involved. However, we note that with the possibility of differential taxationof producers we must also take into account that producers may find it profitableto merge or dissolve firms for tax reasons. This might make it both difficult andcostly 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 proportionaltax. Perhaps, therefore, one should not worry too much about the exceptions tothe rule that productive efficiency is desirable; the administrative and informa-tional costs of deviating from the rule might easily be too high for it to be aninteresting alternative.

6. Distributional considerationsSo far we have been assuming a one-consumer economy, or, alternatively,

that the preferences of the community can be represented by social indifferencecurves. This can evidently only be a first approximation. Social indifferencecurves exist if lump-sum transfers are constantly being used in the backgroundto keep the income distribution optimal according to some individualisticwelfare function; but in the optimal tax literature such transfers are ruled outby assumption. Social indifference curves also exist if all individuals have identicaland homothetic indifference maps; but in that case we can safely ignore distribu-tional issues, since nothing in the way of redistribution can be achieved bycommodity taxation anyway.l

For a fuller discussion of social indifference curves, see Samuelson (1956).


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The possibility of a conflict between our efficiency rules and distributionalobjectives becomes evident when we consider the practical implications of theinverse elasticity rule. We tend to think of commodities with numerically lowprice 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 onempirical data which confirm this ; they come out with tax rates on food whichare two to three times higher than those on consumer durables. There is clearlya need for a modification of the analysis to give some scope for distributionalconsiderations.

Let us now assume that individuals are heterogenous both with respect topreferences and productivities. The utility function of individual j is

U = Uj(XjO, . . .) Xj, j = 1,. . ., n, (17)and the welfare function, which is assumed to belong to the individualisticBergson-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 governments budget con-straint. There is no need to go through the whole analysis again and we shalljust give the main result for the case of independent demands,l in which weobtain a generalization of the inverse elasticity formula:

The ad valorem tax rate is still proportional to the inverse of the marketdemand elasticity, but the proportionality factor has a more complicated form.Given the market demand elasticity, the tax rate on good k should be higher, thelower is consumers average social marginal utility of income, when weightedby their consumption of good k . In the special case where all individuals havethe same strictly concave utility function and the welfare function is the un-weighted 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 onlynecessary 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 someconsumers and complements for others, with zero cross-effect in the aggregate, so that thisassumption is weaker than that of demand independence for all consumers.


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50 A. Sat&no, Optimal taxationthe tax rate should be higher, the more strongly the consumption of the com-modity is concentrated among high-income individuals. 2It is of course difficult to claim much in the way of immediate applicabilityfor the formula (19) and for the more general conditions for optimal taxation inthe case of many individuals. What we may claim for the analysis seems mainlyto be that it gives some insights into the nature of the compromise betweenequity and efficiency considerations that will have to be made in practice. Theseinsights are not so general as to be empty of all empirical content. One may notethat if two goods have the same proportionality factor-if, in the words ofFeldstein (1972a), they have the same distributional characteristic - then it isstill true, in the case of independent demands, that the ratio of their tax ratesshould be equal to the inverse ratio of their price elasticities. One might alsoattempt to develop the distributional characteristic into a more operationalmeasure; this is also done by Feldstein (1972a,b).

7. Further problems in commodity taxationWith 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 areweakening. This is not only because an increasing number of economists arebecoming familiar with this type of analysis, but also because the theory israpidly becoming integrated with other well-established subfields of publiceconomics and general economic analysis. It is obviously impossible here to gointo 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 distortionarytaxation introduces an additional cost factor in calculations of the optimal supplyof public goods. Consequently, he argued, one would want to curtail the supplyof public goods compared to the first-best rule of making marginal benefit equalto marginal cost. This conclusion has been analyzed in the contributions ofStiglitz and Dasgupta (1971) and Atkinson and Stern (1974). Their workrepresents a generalization of the analysis of previous sections, in that the publicsectors tax revenue requirement is derived from the cost of supplying publicgoods. It turns out that Pigous conjecture is not generally correct. Under first-best conditions, the correct benefit measure, assuming an optimal distribution ofincome, is the sum of the marginal rates of substitution. With distortionarytaxation, the correct benefit measure may exceed the first-best measure if thepublic good is complementary with taxed goods or if taxation releases incomeeffects which increase the demand for the taxed goods; the latter case would beof importance if taxed goods are inferior. Atkinson and Stern also point out

1*A more general reatment of distributional ssues, ttempting o generalize he Ramsey ule(9) is contained in the articlesby Mirrks (1975) and Diamond (1975).


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A. Sam&no, Optimal taxation 51

that the answer to the question of the correct benefit measure does not in itselfprovide 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 canbe used to improve the competitive allocation of resources and in fact make itPareto optimal. However, the standard analysis of Pigovian taxes assumes, moreor less implicitly, that the public sector either needs no additional tax revenue orthat 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 wherethe government must simultaneously employ taxes which improve and taxeswhich distort the allocation of resources. Sandmo (1975) analyzes the optimalcombination of such taxes for the case of a negative consumption externalityand concludes that the marginal social damage should only be reflected in thetax on the externality-creating commodity, regardless of the pattern of comple-mentarity and substitutability; exceptions to this rule have been discussed byGreen and Sheshinski (1974).

Another issue in this area concerns the choice of optimal taxes or subsidieswhen the government is constrained to tax uniformly generators of externalitieswhom it really would have been optimal to tax at different rates. This problemhas been investigated by Kolm (1971b) and Diamond (1973).

(3) International trade. The theory of the optimal tariff has some importantsimilarities with the theory of optimal commodity taxation, and it seems anatural undertaking to try to integrate the two bodies of literature. This hasbeen attempted in articles by Boadway, Maital and Prachowny (1973) andDasgupta and Stiglitz (1974). The problem here has some similarities with onewhich arises in the analysis of externalities; we wish to derive rules for optimaltariffs (which improve the allocation of resources) and optimal commoditytaxes (which distort it) simultaneously. The two articles also discuss optimalitycriteria for public goods and relate the criteria to the problem of the use ofinternational prices in domestic cost-benefit analyses.r3

(4) Public utility pricing. The theory of optimal commodity taxation can bereinterpreted as a theory of public utility pricing, and the latter furnishes theframe of reference for many contributions to the subject; in particular, this istrue 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 efficiencyis taken as the sole criterion; one could imagine the government as orderingpublic utilities to set their prices according to efficiency criteria on the assump-

IsFor an application of optimal tax theory to the problems of international economicintegration, see Kolm (1969b).


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tion that the government itself will determine appropriate redistributionalpolicies.

(5) Income taxation. Many of the ideas discussed so far are of courseapplicable to income taxation as well as to commodity taxation. Thus, we shouldexpect the efficiency loss associated with an income tax to be larger the moreelastic is the labour supply with respect to the wage rate. If we consider thedegree of progression in the tax schedule, it would obviously be important toknow how the elasticity of labour supply varies with income. If people withhigher income are also characterized by elastic labour supply, this would act asa brake on the degree of progression that one might otherwise prefer from aredistributional point of view. Nevertheless, income taxation has some peculiarfeatures which are difficult to incorporate in the previous framework. Spaceprevents a full treatment of this topic here; the reader is referred to the articlesby Mirrlees (1971) and Sheshinski (1972) as well as to the survey in Atkinson(1973).

8. Concludiug remarksThere can be no doubt that the recent developments in the analysis of optimal

taxation have brought welfare economics closer to the realities of economicpolicy. We know how to model optimization problems in the public sector withfairly realistic assumptions about the set of policy tools available. New insightshave been gained into the efficiency aspects of taxation, and we can probablyalso claim to have obtained a better understanding of the tradeoff betweenequity and efficiency.

The theory obviously has its limitations. It is at its best in yielding rules forthe optimal structuring of a given tax system and has less to contribute to thediscussion of major problems of tax reform, which typically involves the choicebetween alternative tax systems. A difficulty with the extension of the theory tocover these global problems is that the costs of administration have not beenincorporated into the theory;i4 this is one aspect of the neglect of transactionscosts in the theory of general equilibrium. The incorporation of costs of adminis-tration is an extremely complicated task, and it remains to be seen whether aformally more complete theory can still yield conclusions which are interestingand meaningful from the point of view of implementation.

However, this raises the question of whether optimum tax formulae can haveany claim to be taken seriously, given that they abstract from such centralconcerns 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 avery abstract level.15For an answer which is mainly in the negative, see Hahn (1973).


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A. Sadno, Optimal taxation 53

final answer to this question may be, I believe that we shall probably have toreconcile ourselves to the fact that no policy model can be complete in the senseof taking account of all relevant concerns facing a policymaker. Thus, it may wellbe that we shall find the models of optimal taxation to be useful ones, eventhough we may have to supplement them with considerations which areexogenous to the models themselves.

ReferencesAndersen,P.S., 1972, The optimum tax structure in a three-good, one-consumer economy,Swedish Journal of Economics 74,185-200.Atkinson, A.B., 1973, How progressive should income tax be?, in: M. Parkin, ed., Essays inmodem economics (Longmans, London) 90-109.Atkinson, A.B and J.E. Stiglitz, 1972, The structure of indirect taxation and economicefficiency, Journal of Public Economics 1,97-l 19.Atkinson, A.B. and N.H. Stern, 1974, Pigou, taxation and public goods, Review of EconomicStudies 41,119-l 28.Baumol, W.J. and D.F. Bradford, 1970, Optimal departures from marginal cost pricing,American Economic Review 60,265-283.Boadway, R., S. Maital, and M. Prachowny, 1973, Optimal tariffs, optima1 taxes and publicgoods, Journal of Public Economics 2,391-403.Boiteux, M., 1951, Le revenue distribuable et les pertes konomiques, Econometrica 19,112-133.Boiteux, M., 1956, Sur la gestion des monopoles publics astreint ii lkquilibre budghtaire,Econometrica 24,22-40.Bronsard, C., 1971, Dualit microeconomique et theorie du second best (Vander, Louvain).Corlett, W.J. and D.C. Hague, 1953-54, Complementarity and the excess burden of taxation,Review of Economic Studies 21,21-30.Dasgupta, P. and J.E. Stiglitz, 1974, Benefit-cost analysis and trade policies, Journal of PoliticalEconomy 82,1-33.Diamond, P.A., 1973, Consumption externalities and imperfectly corrective pricing, BellJournal of Economics and Management Science 4,526-538.Diamond, P.A., 1975, A many-person Ramsey tax rule, Journal of Public Economics 4,335-342.Diamond, P.A. and J.A. Mirrlees, 1971, Optimal taxation and public production I-II, AmericanEconomic Review 61,8-27,261-278.Dixit, A., 1970, On the optimum structure of commodity taxes, American Economic Review 60,295-301.Feldstein, M.S., 1972a, Distributional equity and the optima1 structure of public prices,American Economic Review 62,32-36.Feldstein, M.S., 1972b, Equity and efficiency in public sector pricing: The optimal two-parttariff, Quarterly Journal of Economics 86,175-187.Green, J. and E. Sheshinski, 1974, Direct vs. indirect remedies for externalities, unpublished

paper.Hahn, F.H., 1973, On optimum taxation, Journal of Economic Theory f&96-106.Heller, W.P. and K. Shell, 1974, On optimal taxation with costly administration, AmericanEconomic Review 64, Papers and Proceedings, 338-345.Hicks, U.K., 1947, Public finance (Cambridge University Press, Cambridge).Kolm, S. Ch., 1969a, Prixpublics optimaux (C.N.R.S., Paris).Kolm, S. Ch., 196913,De lunion douaniere sans integration monetaire, Revue dEconomiePolitique 79, 751-799.Kolm, S. Ch., 1970, La theorie des contraintes de valeur et ses applications (Dunod, Paris).Kolm, S. Ch., 1971a, Cours deconomie publique 1: LCtat et le systeme des prix (Dunod,Paris).


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54 A. Sandmo, Optimal taxationKolm, S. Ch., 197lb, Cours deconomie 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 ofEconomic 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 ofEconomic 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 PublicEconomics 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 Review64,701-706.Sandmo, A., 1975, Optimal taxation in the presence of externalities, Swedish Journal ofEconomics 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 economicefficiency, Review of Economic Studies 38,151-174.

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43 Optimal Taxation- An Introduction to the Literature

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