Qffice of Tax Analysis

U.S. Treasury Department

Washington, D. C. 20220 

Issued March, 1976 

The Optimal Taxation of 

Commodities and Income 

David F. Bradford 

U.S. Treasury Department 

Harvey S. Rosen 

Princeton University 

OTA Paper 8 December 1975 
TABLE OF CONTENTS 

I. Some History of Thought 

11. Optimal Commodity Taxation 

III. Optimal Income Taxation 

IV. Concluding Remarks 

Page 

1 

4 

10 

14 

n 
The last few years have seen a resurgence of interest in 

the old question of how best to raise tax revenue. Roughly 

speaking, two different problems have been studied. The first 

is to find a set of commodity taxes that is optimal given certain 

efficiency and (sometimes) equity considerations. In a second 

strain of the literature, it is assumed that the revenue system 

is based upon income rather than commodity taxation, and the 

problem is to determine the optimal degree of progressivity 

(or regressivity).-11, 21 

The principal motivation of some writers in the optimal 

taxation literature seems to be the discovery of fairly simple 

rules which policy makers actually can implement. Others are 

more interested in theoretical exploration of the implications 

of alternative economic assumptions than in developing usable 

policy recommendations. Practically all the contributions, 

however, have been quite mathematical and thus inaccessible to 

many practitioners in the public finance area. The purpose of 

this essay is to discuss in a nontechnical way the methodology 

and principal conclusions of the optimal taxation literature.-31 

In section I we present briefly the history of thought 

on optimal taxation. Sections I1 and I11 discuss the optimal 

commodity and income tax literatures, respectively. We conclude 

with some observations on the accomplishments of the optimal 

taxation research and on some open questions. 
-2-

I. SOME HISTORY OF THOUGHT 

The debate over the properties of a good tax system goes 

back hundreds of years. Some of the discussion may seem at once 

cynical and amusing to contemporary economists. One eighteenth 

century writer considered a good tax to be one that was easy to 

disguise. (Jones, p. 93) Similarly, the French statesman Colbert 

argued 'I. . . the art of taxation is the art of plucking the 

goose so as to get the largest possible amount of feathers with 

the least possible squealing." (Armitage-Smith,p. 36) 

However, the striking aspect of the old literature is its 

concern with the same efficiency and equity issues discussed 

today. It was viewed as desirable that tax induced distortions 

be kept at a minimum: "Taxation should interfere as little as 

possible with the processes of industry." (Armitage-Smith,p. 55) 

The effect of taxes on work incentives was a concern for politicians 

as well as economists. Gladstone opined that the income tax did 

t1 . . . more than any other tax to demoralize and corrupt the 

people." (Wells, p. 516) 

Equity was also a major issue, and the fairness of progressivity was hotly debated. Adam Smith believed that individuals should pay taxes ". . . in proportion to their respective abilities, that is in proportion to the revenue enjoyed." 

(Stamp, p. 29) Mill characterized a graduated tax as 'I. . . a 
-3-

graduated robbery," (Stamp, p. 38) while others thought that 

progressivity had a firm scientific basis in the theory of 

diminishing marginal utility. (Stamp, p. 40) . 

As far as formal theorizing is concerned, the history of 

optimal commodity taxation is rich and long, while that of 

optimal income taxation is surprisingly thin, if not short. 

This is no doubt because the Commodity tax problem is formally 

equivalent to the problem of pricing policy in nationalized, 

increasing returns enterprise. Recent work is in a tradition 

dating at least from J. Dupuit, writing in the middle of the 

nineteenth century. However, the most famous forerunner is 

Frank Ramsey, who derived the proposition that (second-best) 

optimal commodity taxes cause an equi-proportionate contraction 

in quantities of all commodities, in a paper published in 1927. 

Subsequent development has consisted of refinement and rediscovery of Ramsey's result. Important landmarks since then 

include Samuelsonls 1952 U.S. Treasury memorandum (unfortunately 

never published, but widely circulated informally) and Marcel 

Boiteux's elegant treatment of the regulation of public monopolies 

which are subject to budget constraints. In almost all of the 

work the predominant question was the same: how can we raise 

a specified amount of revenue (or finance a specified program 
-4-

of expenditures), using commodity taxes, in such a way as to 

minimize deadweight loss. While distributional issues were 

generally acknowledged, the focus of attention was on the 

efficiency question, 

In thinking about income taxation the early contributions 

tended to lean in the other direction, to the point that Edgeworth, 

in the first important attempt to derive a tax schedule in an 

optimizing framework, ignored efficiency altogether. He pointed 

out that if all individuals have identical declining marginal 

utility of income schedules and the government's goal is to 

collect its revenue with the minimum aggregate loss of utility, 

then the appropriate policy is to level off income from the top. 

Edgeworth's analysis does not take into account the probable 

efficiency effects of the confiscatory tax rates. Perhaps 

because the conclusion made so little sense, there seems to 

have been no further attempt to derive income tax characteristics 

from an explicit optimizing problem until very recently. Richard 

Musgrave in his well known text reviewed and clarified this 

and other criteria (equal absolute or relative utility sacrifice --

interestingly, not derivable from a utilization maximizing framework) which might be used in determining equitable tax shares, 

but he did not in that context formally introduce the equityefficiency trade off. Integrating these aspects has been the 

principal objective of the optimal income tax theorists of the 

1970's. 
-5-

11. -OPTIMAL COMMODITY TAXATION 

Since the cited literature contains many and varied derivations of the principal theorems of optimal commodity taxes, we 

shall not carry out detailed proofs here. We can point out, 

however, some of the variations in the way the problem is posed. 

Most commonly a revenue constraint is taken as a starting point, 

together with an assumption that the government must use per unit 

commodity taxes. (Thus lump-sum taxes are excluded.) If xi is 

the quantity of the ith good purchased by the household sector 

from the production sector (xi is negative if the households 

are net sellers, as in the case of the commodity "leisure") , and 

Ti is the per unit tax, the revenue constraint is 

(1) C Ti~i= R, 

where R is the required revenue level. 

The taxes are the difference between the prices, pi, received by 

producers and Pi, paid by the consumers. 

(2) Ti = Pi - Pi 

It is frequently assumed that producer prices are fixed, so that 

by setting taxes we set consumer prices and hence consumer welfare. 

The problem is then to make the choice of taxes in such a way as 

to maximize the resulting consumer welfare. Another way of 

describing the objective is to obtain the required revenue with 
-6-

minimum excess burden or deadweight loss to consumers -- the cost 

in inefficiency which is in addition to the value of output 

necessarily foregone to meet the government's requirements. 

Samuelson uses a somewhat different formulation of the problem 

and one which has the virtue of emphasizing the resource releasing 

function of taxes. He begins with the assumption that the 

government wishes to obtain a vector g = (81 . . . gn) of 

quantities of each of the n commodities. Assuming constant 

returns to scale (and hence no profits under competition) the 

behavior of firms will be governed by the producer price vector 

p = (ply . . . pn) while the demands and welfare of the households 

will be determined by consumer prices P = (PI, . . . Pn). At 

p the firms will supply the vector y(p) = (y1, . . . yn) of net 

outputs; at P the households will demand x(P) = (XI, . . . +). 

The trick of feasibility is to choose P and p so that 

(3) Y(P> - X(P> = g, 

that is the amounts produced less the amounts demanded by the 

household just equal the government's requirements. The problem 

of -optimality is to pick from among the pairs of consumer and 

producer price vectors satisfying (3) one which maximizes consumer 

welfare. To work out the problem it is generally easier to go 

behind the producer supply relationships. Thus if F(y) = 0 

implicitly defines the transformation frontier of the economy, 

the constraint (3) might be expressed by 
-7-

(4) F(x(P)+g) = 0 

Fi(x(P)+g) = api, i = 1 . . . n, 

where the subscripted conditions are those associated with 

producer profit maximization. 

Because the emphasis is generally on efficiency, a typical 

approach is to assume there to be only one consumer (hence no 

distribution problem). Thus the objective might be to choose 

P and p to 

(5) Maximize U(x(P)) 

subject to (4) and to 

(6) Ui(x(P)) = yPi, i = 1, . , . n 

Conditions (6) are the familiar first order implications of the 

household's optimization. More often an indirect utility function 

V(P, 0) = U(x(P)) is used, (the zero argument draws attention to 

the assumption of no transfer income) as the derivations become 

very simple when use is made of "Roy's Identity" 

av av 

(7) -- -xi(P) . ­ ap aM 

where aV/aM is the derivative of the indirect utility function 

with respect to budget level. 

Putting these pieces together in any of several sequences 

leads to the famous Ramsey result on optimal commodity taxation: 

= @xk, k = 1, . . ., n 
-- 

- - 

-8-

where Sik is the Slutsky coefficient, the derivative of the demand 

for the ith good with respect to the kth price, other prices and 

utility being held constant, and 6 is independent of k. The 

left hand side gives an estimate of the change in demand for 

the kth good which would occur if the taxes were removed. Hence 

(8) says that the proportional change in demand (thus estimated) 

should be the same for all comodities the Ramsey result. 

Condition (8) can also be expressed in terms of elasticities. 

Probably the most familiar "optimal tax" result is the form which 

applies when the off-diagonal elasticities are zero. In this 

case the first order conditions associated with (5) lead to the 

"inverse elasticity rule": 

- (9) tr 

6 , r = 1, . . ., my 

where tr -

- T /P , the percentage or -ad valorem rate of tax, 6 

rr 

is a constant, and E,, is the elasticity of the ordinary (uncompensated) demand function for the kth good. This formula has 

certainly been of importance in forming economists' intuitions 

on tax and price regulatory ques.tions. In underlies the notion 

of charging according to "what the traffic can bear" in transportation, for example, and is the basis for the acceptance on 

efficiency grounds of high taxes on tobacco and alcohol, the 

demand for which is presumed price inelastic. 
-9-

One of the important contributions of the optimal commodity 

tax literature, indeed, has been to reconcile economists' sometimes opposing intuitions. For example, the intuition that prices 

should be set at marginal cost, so that producers' and consumers' 

price vectors are at least proportional, is seen to be correct 

under the assumption^ that (a) distributional objectives are 

otherwise achieved and (b) sufficient revenue can be raised. 

If there is insufficient profitability in the economy (as, for 

example, in the case of constant returns to scale production 

technologies with competition) the second of these conditions 

cannot be met. &/ Where prices must deviate from marginal costs, 

the inverse elasticity rule is appealing, but we see that it will 

be strictly appropriate only under the rather strong assumptions 

of independent commodity demands. 

Another application of the analysis is to the presumptive 

case for direct over indirect taxation. The classic Hotelling 

argument for marginal cost pricing seemed to support the conclusion that an "income tax" will involve no efficiency cost. 

When it was recognized, however, that the "income" of the tax 

system is not the "budget level" of the elementary theory of 

consumer demand, but rather the product of a certain price, 

the wage, and a demanded quantity (net purchase) of leisure, 

the apparent a priori advantage of an income tax was lost. 

The analyses of Corlett and Hague, Little and Friedman to this 

effect all are applications of the theory of optimal commodity 

taxation, as is neatly shown by Sandmo. 
- 10 -

While the extensive subsequent work has shown how difficult 

it is to sustain any simple rules for commodity taxation, the 

result of the spreading awareness of this work has been to make 

economists think about tax questions in a new way and to hasten 

the search for rules which are reasonably robust. 

For example, as Sliglitz and Atkinson point out, optimal 

tax analysis makes it clear that there is no a priori assurance 

that the income tax is the single best instrument for income 

redistribution. "Commodity taxes", such as housing subsidies 

or food stamps, might contribute to an optimal program. Boskin 

notes that, in view of the differences in the observed elasticities of household supply of the two types of labor (husband 

labor and wife labor), it is probably efficient to tax these 

"commodities" at different rates. Feldstein (1975) uses the 

same basic approach to examine the choice between "tax expenditures" and direct expenditure methods of achieving an increase 

in a specified activity. 

A natural question in view of the interpretation of the 

income tax as a commodity tax is whether taxation of labor only 

(i.e., uniform taxation of commodities) is appropriate. Not 

surprisingly, the answer is that it will be appropriate when 

labor is inelastically supplied. Sandmo shows that this in 

turn will follow if utility is separable between leisure and 
-- 

- 11 -

all other goods and homogeneous in those goods. Intuitively 

this separability means that further efficiency cannot be gained 

by differential taxation of goods that are "re1ated"to leisure. 

Several writers have noted an important consequence when this 

result is reinterpreted in an intertemporal context. If utility 

is a function of consumption and leisure at different dates and 

separability obtains, then no taxes on interest income should be 

levied consumption is the appropriate tax base. This simply 

illustrates the challenge, implicit in the optimal tax approach, 

to the widespread acceptance of taxation on the basis of HaigSimons income which has been emphasized by Feldstein. 

While an "income tax'' can be regarded as a tax on the sale 

of labor (negative net purchase of leisure), there is a feature 

of actual income taxes which is slighted by such a point of view: 

it is institutionally feasible to assess taxes at different rates 

on different individuals; in particular progressive taxation of 

earnings is possible. Depending upon the allowable features, 

the possibility arises of, in effect, duplicating a lump-sum 

tax by a (regressive) income tax structure. When distributional 

considerations are introduced this is not terribly useful; 

however, the fact that tax rates can vary from household to 

household makes the income tax, and such related taxes as the 

expenditure tax, the principal instruments for distribution 

objectives. We now turn to the studies which consider the 

trade-off between such distributional objectives and economic 

efficiency. 
- 12 -

111. OPTIMAL INCOME TAXATION 

As we noted in Section I, the problem of optimal income 

taxation has a long history in economics. However, most of the 

recent literature stems from a paper published by James Mirrlees 

in 1971. A natural way to organize our discussion, then, is to 

summarize Mirrlees' techniques and conclusions, and then view 

the ensuing literature as an attempt to explain and modify some 

of his results. 

In Mirrlees' model, society is composed of individuals 

who have identical atemporal utility functions in after-tax 

income and leisure. Individuals differ only in their earnings 

abilities (wage per hour). The government must collect an exogenously determined amount of tax revenue. The problem is to 

find an income tax schedule (tax function) which maximizes the 

sum21 of individuals' utilities subject to this revenue constraint. 

Using the tools of the calculus of variations to solve 

the constrained maximization problem, Mirrlees finds that the 

optimal tax function exhibits marginal tax rates between zero 

and one, and that when it is operative, part of the population 

does not work. Although these results may seem weak, they are 

really quite remarkable given the absence of specific frrnctional 

forms for the key relationships in the problem. 
- 13 -

In order to get more specific results, more specific assumptions must be built into the analysis. Mirrlees assumes that the 

utility functions are Cobb-Douglas, and considers both lognormal 

and Pareto distributions of earnings abilities. With these 

assumptions, the following results emerge: a) the optimal tax 

function is approximately linear with a negative intercept, and 

b) the optimal tax function is characterized by 'low' marginal 

tax rates which fall somewhat with income. (Atkinson's interpolations of Mirrlees ' results indicate rates in the neighborhood of 20 percent). 

Mirrless was surprised at how low the marginal tax rates 

were: '' . . . I must confess that I had expected the rigorous 

analysis of income-taxation in the utilitarian manner to provide 

an argument for high tax rates. It has not done so." A study 

by Fair in the same year also generated fairly low implied 

marginal tax rates. Apparently, those who read the Mirrlees 

paper also found the low marginal tax rates counter-intuitive, 

for much of the literature appears to be an attempt to explain 

them. 

One concern was the maximand of Mirrlees' problem, an 

unweighted sum of individual utilities, which implies that a 

'util' to a rich individual adds as much to social welfare as 

a 'util' to a poor individual. To what extent would more 

egalitarian results (i.e., higher marginal tax rates) emerge 
- 14 -

if a social welfare function were used which weighted the utilities 

of the rich less than those of the poor? Atkinson and Feldstein 

(1973) consider social welfare functions of the form: 

(10) w = ( cui) l/U uc1 

Clearly, when u = 1, welfare (W) is the simple sum of utilities 

(Ui) . When u is less than 1, however, it can be shown that 

a given increment to the utility of a low utility individual adds 

more to W than if awarded to a high utility individual. It 

should be noted, however, that the specifications of the social 

welfare function and the individual utility functions are not 

really independent of each other. We could, for example, specify 

the utility of the ith individual to be Ut, 6’ and then write 

social welfare as the arithmetic sum of these utilities. 

Atkinson focuses attention on the case in which u approaches 

minus infinity. Under such circumstances, maximizing W is 

equivalent to maximizing the utility of the worst off individual 

in society: the maximin case.l’ This case has received considerable attention due to philosopher John Rawls’ argument that 

it is particularly compelling as an ethical criterion. (A number 

of criticisms are suggested by Klevorik.) 

Atkinson uses a Rawlsian social welfare function in a model 

with a linear income tax, no net government revenue requirement 

(i.e., taxation for redistribution only), and a Pareto distribution of skills in the economy. He finds that optimal marginal 

tax rates range between 30 and 45 percent. Thus, one solution 
-- 

- 15 -

to the mystery of Mirrlees' low marginal tax rates is his formulation of the objectives of the government. Social welfare functions which are more egalitarian than the classical utilitarian 

variety may yield higher marginal rates. 

Another potential explanation for Mirrlees' results is the 

Cobb-Douglas a/ssumption concerning the form of individuals ' 

utility functions. Stern has investigated this possibility by 

assuming that individuals have constant elasticity of substitution (CES) utility functions in leisure and income. Using results 

on the elasticity of labor supply from the econometric literature,-8/ 

he finds that an elasticity of substitution of 0.4 is more realistic 

than 1.0. el When a variant of Mirrlees' problem is solved using 

CES utility functions with this lower elasticity of substitution, 

the optimal marginal tax rates are substantially higher without 

appeal to a more egalitarian social welfare function. 

So far, it has been assumed that there is one type of 

labor, and individuals differ only in their ability to perform 

it. Feldstein (1973) investigates the importance of this 

assumption by analyzing a two person society consisting of a 

skilled and unskilled worker whose wages are endogenously 

determined, He finds that relaxing the exogenous determination 

of wages has no major impact on optimal marginal tax rates, and 

as in the Mirrlees article, they are still 'low'. Even for the 

maximin case Feldstein finds a marginal tax rate of only 45 

percent (assuming Cobb-Douglas utility functions). 
- 16 -

Reexamination of the social welfare function suggests 

another possible explanation for the low tax rates typically 

generated by optimal income tax studies. Out intuition about 

optimal income taxation may perhaps be conditioned on social 

objective functions which are not utilitarian-individualistic. 

For example, the presence in the social welfare function of a 

variable parameterizing the ‘aesthetics ’ of the income distribution would lead to more egalitarian results.-1” Similarly, 

Feldstein (forthcoming) has shown that if interdependent utility 

functions are allowed for, very high marginal tax rates may 

be appropriate. 

We now turn to a limitation of the Mirrlees model which 

is just beginning to receive attention, its atemporal setting. 

The appropriate taxation of capital income is one of the most 

controversial aspects of the tax system, yet the studies cited 

above for the most part ignore it. Ordover and Phelps examine 

the optimal mix of taxes on two factors of production (capital 

and labor) in a one sector neo-classical growth model.-11/ 

Their model is very general, and therefore no results on tax 

rates emerge which can be compared to those discussed above. 

Moreover, the only social welfare function they consider is the 

maximin case. Despite these limitations, explicit attention to 

the taxation of capital income in the optimal income tax framework is an important step which will no doubt stimulate further 

research. 
- 17 -

We could continue to list additional aspects of the Mirrlees 

model which have been changed and expanded in order to determine 

121 their effects on optimal tax rates.- However, the basic thrust

of the literature should now be clear. An exogenously determined 

amount of tax revenue must be raised by income taxes on individuals whose economic choices are distorted by the presence of 

those taxes. Given technological and behavioral assumptions, 

the optimal tax schedule is that which leaves some social welfare 

function at a maximum after the tax is collected. The literature 

shows how various assumptions on these components lead to different conclusions regarding the shape of the optimal tax 

schedule. 
-- 

- 18 -

IV. CONCLUDING REMARKS 

The accomplishments of the optimal taxation research have 

been considerable. It has upset many comfortable rules of thumb 

and lent precision to many informal arguments, But there remains 

work to be done. Part of this work will, of course, consist of 

increasing the stock of variations on the basic problems for which 

solutions have been described. Another, and very important, part 

will consist in the attempt to determine quantitatively which of 

these problems best describes the actual economy to be taxed 

filling in all those empty boxes with real, estimated elasticities. 

However, work of another kind is needed to advance the 

normative power of the analysis, Normatively the optimal tax 

literature rests on a utilitarian base. It is true that the 

optimal commodity tax results, or some of them at least, can be 

cast in a form which says: if your tax system doesn't look like 

this there is a potential bargain which can be struck among your 

citizens which would make all better off. However, these bargains are complex and their possibility tends to be eliminated 

by the very assumptions that require the use of second-best 

instruments in the first place. For practical application 

implicit interpersonal utility comparisons are required. The 

optimal income tax results are also dependent on such comparisons. The missing link is a welfare function, and the question 
- 19 -

is how does one persuade a legislative or an electorate to decide 

tax questions in accordance with some particular welfare function? 

Asking the optimal tax researchers to resolve this problem is 

effectively asking them to make welfare economics persuasive, 

obviously a tall order. 

Missing from the optimal tax arguments is the idea of 

horizontal equity, the notion that ". . . people in equal positions should be treated equally." (Musgrave, 1959, p. 160) 

(Customarily, "equal positions" are defined in terms of an 

observable index of ability to pay such as income, expenditure, 

or wealth.) In none of the studies discussed above has the 

injunction to treat equals the same appeared either as a constraint in the mazimization problem, or as an argument in the 

objective function. Therefore, they will in general 131 fail 

to provide horizontal equity. In light of this, Musgrave 

(forthcoming) and others have suggested that it is inappropriate 

to characterize such schemes as 'optimal.' 

Defining horizontal equity in terms of income is inadequate 

because individuals with identical opportunity sets but different tastes will have different incomes. An alternative 

way to define equal position would be identical opportunity 

sets. However, it seems more in the spirit of the optimal 

taxation literature to define equal position in terms of 

utilities: individuals are 'the same' only if they derive 
- 20 -

identical amounts of utility from their consumption and leisure 

bundles. The choice of a criterion for horizontal equity is 

important because when tastes differ between individuals, 

different criteria may lead to different conclusions as to the 

fairness of a given tax. For example, an income tax which is 

perfectly fair according to conventional notions of horizontal 

equity hurts an 'income lover' more than a 'leisure lover.' 

Ironically, although the optimal taxation literature ignored 

horizontal equity, it has sparked new interest in the topic, and 

modified the vocabulary of the discussion. For example, the 

optimal taxation literature emphasis on efficiency has reminded 

public finance practitioners that excess burden must be taken 

into account when allocating tax burdens across individuals. 

Similarly, the concern with the impact of tax changes on utility 

has focused attention on the equity implications of the differential taxation of pecuniary and nonpecuniary forms of income. 

It has been shown, for example, that if there is one type of 

ability and tastes are the same, then horizontal equity is 

satisfied even if identical individuals pay different amounts 

of tax. (See Feldstein (forthcoming)) 

In an attempt to put the discussion of horizontal equity 

and the optimal taxation literature on the same plane, Feldstein 

(forthcoming) has redefined the principle of horizontal equity 

in terms of utility rather than ability to pay.-14/ However, 

complete integration of horizontal equity into the optimal tax 
- 21 -

framework remains to be done. Perhaps this could be accomplished 

by including some measure of departure from horizontal equity as 

an argument in the social welfare function, but this approach is 

bedeviled by conceptual difficulties in measuring departures 

from horizontal equity.-151 

It may well be that horizontal equity, ancient and honorable 

criterion of tax policy though it be, is not a helpful concept. 

However, the apparent appeal of this nonoperational idea to 

practical people suggests the attractiveness of properties of 

a tax structure which are independent of the economy to which 

that structure is applied. To discover whether there are any 

such properties which significantly narrow the range of "good" 

tax structures might be a useful topic of research. 
- 22 -

-REFERENCES 

Armitage-Smith, Priciples and Methods of Taxation, London, 

1907. 

Arrow, K., ­et. al., "Capital-Labor Substitution and Economic 

Efficiency," Review of Economics and Statistics, 

August 1961. 

Atkinson, A.B., "Capital Taxes the Redistribution of Wealth 

and Individual Savings," Review of Economic Studies, 

April 1971. 

Atkinson, A.B., "How Progressive Should the Income Tax 

Be?", in Longmans, Essays on Modern Economics, London, 

1973. 

Atkinson, A.B., J. Stiglitz, "Alternative Approaches to 

the Redistribution of Income," Journel of Public 

Economics, forthcoming. 

Boiteux, M., "Sur la gestion des monopoles publics astreint 

a l'equilib budgetaire, . Ecancxhetrica (24), 1956, 

reprinted in translation by W. J. Baumol and D. F. 

Bradford, "On the Management of Public Monopolies 

Subject to Budgetary Constraints," Journal of Economic 

Theory, (3), September 1971. 

Corlett, W. J., Hague, D.C., "Complementarity and the Excess 

Burden of Taxation," Review of Economic Studies, (21), 

1953-54, pp. 21-30. 
- 23 -

Diamond, P.A., J. A. Mirlees, "Optimal Taxation and Public 

Production: 11," American Economic Review, June 1971. 

Dupuit, J., Troite theorique et pratique de la conduite 

et de la distribution des eaux, Paris, 1854. 

Edgeworth, "The Pure Theory of Taxation, Economic Journal, 

1897. 

Fair, R.C. , "The Optimal Distribution of Income, Quarterly 

Journal of Economics, 1971. 

Feldstein, M., "On the Optimal Progressivity of the Income 

Tax," -Journal of Public Economics, 1973. 

Feldstein, M., "The Theory of Tax Expenditures," Harvard 

University, mimeo, 1975. 

Feldstein, M., "On the Theory of Tax Reform," Journal of 

-Public Economics, forthcoming. 

Jones, R., The Nature and First Principle of Taxation, London, 

1914. 

Klevorick, A.K., "Discussion," American Economic Review, 

Papers and Proceedings, May 1974. 

Little, J.M.D.,"Direct vs. Indirect Taxes," Economic 

Journal, (61), pp. 577-584. 

Mirrlees, J.A., "An Exploration in the Theory of Optimum 

Income Taxation," Review of Economic Studies, 1971. 

Musgrave, R.A., The Theory of Public Finance, New York, 

1959. 
- 24 -

Musgrave, R.A., "Optimal Taxation, Equitable Taxation and 

Second-Best Taxation, Journal of Public Economics, 

forthcoming. 

Ordover, J.A., E.S.Phelps, "Linear Taxation of Wealth 

and Wages for Intragenerational Justice: Some SteadyState Cases," -American Economic Review, September 1975.

Ramsey, F.P., "A Contribution to the Theory of Taxation," 

Economic Journal, 1927. 

Rawls, J., "Some Reasons for the Maximin Criterion," American 

Economic Review Papers and Proceedings, May 1974. 

Rosen, H., "Income, Utility, and Horizontal Equity under 

the U.S, Income Tax,It Princeton University, mimeo, 

1975. 

Samuelson, P.A., "Theory of Optimal Taxation," unpublished, 

approximately 1952. 

Sandmo, A., "A Note on the Structure of Optimal Taxation," 

American - Economic Review, September 1974.

Sheshinski, E. , Income Taxation and Capital Accumulation,I' 

Garterly Journal of Economics, forthcoming, a. 

Sheshinski, E., "On the Theory of Optimal Income Taxation," 

Journal of Public Economics, forthcoming, b. 

Stamp, Josiah S., Fundamental Principles of Taxation in the 

Light of Modern Developments, London, 1921. 

Stern, N., "On the Specification of Models of Optimum 

Income Taxation, Journal of Public Economics, forthcoming. 

Wells, D , The Theory and Practice of Taxation, New York, 1900. 
- 25 -

FOOTNOTESThe authors would like to thank Roger Gordon for useful 

conversations and Jay Stuart for assistance in gathering 

material. 

1

There is some overlapping of these strains. For 

example, Atkinson and Stiglitz consider the problem of 

differential commodity taxation in the presence of an 

income tax. 

2Although we shall focus upon these problems in this 

paper, the optimal tax literature has had a somewhat wider 

scope. For example, Diamond and Mirrlees concider the problem of optimal expenditure along with taxation, and 

Atkinson considers the issue of wealth taxation. 

30ur goal is not to provide a comprehensive literature review. Consult Atkinson and Stiglitz and Sandmo for 

more biblographical material. 

4Thus, if y is the sector of net outputs of the production sector, net profits are given by p.~. Equilibruim requires that p.y=O (otherwise firms would expand 

all components of y proportionately, which is possible 

under the assumption of constant returns to sale). For a 

vector T of taxes proportional to p, say T=rp, where r is 

a scalar, the revenue raised will be T.y=rp.y=O. In other 

words, a tax on economic profit would raise no revenue. 
-- 

-- 

- 26 -

5He also considers a social welfare function of the form 

1 

Be -B(un) f(n)dn, where u n 

is the utility of the nth 

individual and f(n) is the distribution of abilities. In the 

application Mirrlees takes the cases 13 = 0 (yielding a simpl? 

sum of utilities) and 13 = 1. 

'Such a transformation changes none of the behavioral 

implications of the utility function. 

7The proof is similar to the demonstration of Arrow, 

et. al., that as the elasticity of a CES production function 

goes to zero, technology is characterized by fixed coefficients. 

8These are measures of the elasticity of hours per 

year with respect to the wage, and thus do not take into 

account other, perhaps more important dimensions of labor 

supply * 

9 

If the elasticity of substitution were zero, lump 

sum taxation would be possible. If the elasticity of substitution were infinite, no revenue could be raised. 

losuch a social welfare function would be non-paretian, 

but there is nothing to prevent a reasonable set of value 

judgments from allowing for such a possibility. 

''Sheshinski (forthcoming, a) considers taxation in a 

one sector neo-classical growth model with earned and 

unearned income taxed at the same rate. 

I2For example, Stern has suggested changing the assumptions on the underlying distribution of skills, while 

Sheshinski (forthcoming, b) focuses on a model in which 
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taxes influence human capital accumulation. 

I3It can be shown that if all individuals have identical tastes and there is only one type of ability, then 

horizontal equity will be satisfied by virtually any broadbased tax. (See Feldstein (forthcoming)). Such assumptions, 

as we have seen, are built into a number of the optimal 

tax studies. (For an exception, see Diamond and Mirrlees .) 

14"If two individuals would be equally well off (have 

the same utility level) in the absence of taxation, they 

should be equally well off if there is a tax." 

I5See Rosen for a discussion of these problems and 

some attempts to surmount them.