optimal Taxation and Public Production I: Production...

optimal Taxation and Public ProductionI: Production Efficiency

By PETER A. DIAMOND AND JAMES A. MIRRLEES*

Theories of optimal production in aplanned economy have usually assumedthat the tas system can allow the govern-ment to achieve any desired redistributionof property.^ On the other hand, somerecent discussions of public investmentcriteria have tended to ignore taxation asa complementary method of controllingthe economy.^ Although lump sum trans-fers of the kind required for full optimality'are not feasible today, commodity andincome taxes can certainly be used to in-crease welfare.^ We shall therefore examinethe maximization of social welfare using

* The authors are at Massachusetts Institute ofTechnology and Nuffield College, Oxford, respectively.During some of the work, Diamond was at ChurchillCollege, Cambridge and Nuffield College, Oxford andMirrlees was at M.I.T. Earlier versions of this paperwere given at Econometric Society winter meetings atWashington and Blarlcum, 1967, at tie UniversitySocial Science Council Conference, Kampala, Uganda,December 1968, and to the Game Theory and Math-ematical Economics Seminar, Hebrew University,Jerusalem. The authors wish to thank M.A.H. Demp-ster, D. K. Foley, P. A. Samuelson, K. Shell, andparticipants in these seminars for helpful discussionson this subject, and referees for valuable comments.Diamond was supported in part by tlie National ScienceFoundation under grant GS 1585. The authors bear soleresponsibOity for opinions and errors,

* For a discussion of this literature, see Abram Berg-son.

^ For a survey of this literature, see .\lan Prest andRalph Turvey,

* We wish to distinguish here between lump sumtaxes, which may vary from individual to individualwhile being unaffected by the individual's behavior,and poll taxes which are the same for all individuals, orperhaps for all individuals within several large groups,distinguished perhaps by age, sex, or region.

* For another study of the general equilibrium im-pact of taxation, which does not explore the optimalityquestion, see Gerard Debreu (1954).

both taxes and public production as con-trol variables. In doing so, we intend tobring together the theories of taxation,public investment, and welfare economics.

There are two main results of the study:the demonstration of the desirability ofaggregate production efficiency in a widevariety of circumstances provided thattaxes are set at the optimal level; and anexamination of that optimal tax structure.It is widely known that aggregate produc-tion efficiency is desired as one part ofachieving a Pareto optimum. It is alsowidely known that when the desiredPareto optimum cannot be achieved, ag-gregate production efficiency may not bedesirable. Our conclusion differs from theseresults in that production efficiency isdesirable although a full Pareto optimumis not achieved. In the optimum position,the presence of commodity taxes impliesthat marginal rates of substitution are notequal to marginal rates of transformation.Furthermore, the absence of lump sumtaxes implies that the income distributionis not the best that can be conceived. Yet,the presence of optimal commodity taxeswill be shown to imply the desirability ofaggregate production efficiency.

This result is similar to that derived byMarcel Boiteux, although he considered aneconomy where lump sum redistributionsof income were possible. Boiteux also ex-amined the optimal tax structure that wasnecessary for this result. The optimal taxstructure for the case of a single consumer(or equivalently with lump sum redistribu-tion) has also been examined by Frank

DIAMOND AND MIRRLEES: OPTIMAL TAXATION

Ramsey and Paul Samuelson.^ Our resultsmove beyond theirs in considering theproblem of income redistribution togetherwith that of raising revenue. Even in theabsence of government revenue require-ments, if lump sum redistribution is im-possible, the government will want to useits excise tax powers to improve incomedistribution. It will subsidize and taxdifferent goods so as to alter individualreal incomes. The optimal redistributionby this method occurs when there is abalance between the equity improvementsand the efficiency losses from further taxa-tion.

The general situation we want to discussis an economy in which there are manyconsumers, public and private production,public consumption, and many differentkinds of feasible tax instruments. Wethink that it is easier to understand theproblem if we present the analysis first fora single consumer, no public consumption,and only commodity taxation, althoughthis case has little intrinsic interest. Themain point of the paper is that the anal-ysis of this special case carries over in themain to the general case.

The first two sections are devoted to thisspecial case. In the first, the situation isportrayed geometrically (for a two-com-modity world with no private production);in the second, production efficiency andconditions for the optimal taxes are derivedby application of the calculus. The use ofthe calculus here and elsewhere is notperfectly rigorous for the usual reasons.These issues are taken up in Section IV.In the third section, we extend the analysisof production to an economy with manyconsumers, elucidating precise conditionsunder which production efficiency is desir-able (and presenting certain exceptions).

* For a detailed history of analysis of this problem,see William Baumol and David Bradford. A summaryand discussion of the work of Boiteux has been given byJacques D r ^ .

Section IV provides a rigorous statementof the theorems. In the fifth section, wediscuss briefly certain applications andextensions of the basic efficiency result.

A following paper, referred to here asDiamond-Mirrlees II, will appear in theJune 1971 Review. In it we will examinethe optimality rules for commodity taxes,for other taxes including income taxes,and for public consumption. We will alsogive a rigorous statement of conditionsunder which the first-order conditions ob-tained (heuristically) below are indeednecessary conditions.

I. One-Consumer Economy—Geometric Analysis

We begin by considering an economywith a single, price-taking consumer andtwo commodities. We assume, for themoment, that aU production possibilitiesare controlled by the government. Whilethere is no scope for redistribution ofincome in this economy, the governmentmight need to raise revenue to cover lossesif there are increasing returns to scale orif there are fixed expenditures (such asdefense) and constant returns to scale.Alternatively, the technology might ex-hibit decreasing returns to scale, facing thegovernment with the problem of disposingof a surplus if all transactions are carriedout at market prices. The optimal solutionto either raising or disposing of revenueis well known. A poll tax or subsidy, as thecase may be, will permit the hiring of theneeded resources and permit the economyto achieve a Pareto optimum, which, in aone-consumer economy, is equivalent tothe maximization of the consumer's utility.While this is a reasonable possibility in aone-consumer economy, lump sum taxesvarying from individual to individual donot seem feasible in a much larger econo-my. An identical problem of distributinga surplus among many people arises if it isdesired to improve income distribution.

10 THE AMERICAN ECONOMIC REVIEW

good 2

good 1

FlGUEZ 1

Thus we shall consider the use of com-modity taxes when lump sum taxes are notpermitted to the government, not for theintrinsic interest of this question in a one-consumer economy, but as an introductionto the many-consumer case. Furthermorewe shall hold constant the governmentexpenditure pattern which directly affectsconsumer utility. Thus we can ignore it,since the utility function already reflectsits impact. The addition of choice forpublic consumption will be considered inDiamond-Mirrlees II.

Assuming free disposal, the technologi-cal constraint on the planner is that thegovernment supply be on or under theproduction frontier. Such a constraint isshown by the shaded area in Figure 1.Let us measure on the axes the quantitiessupplied to the consumer. Thus, the out-put being produced (good 2) is measuredpositively, while the input (good 1) ismeasured negatively. The case drawn isthe familiar one of decreasing returns toscale. If the government needed a fixedbundle of resources, for national defensesay, then the production possibility fron-tier (describing the potential transactionswith the consumer) would not pass through

the origin. With constant returns to scalethis might appear as in Figure 2, where aunits of good 1 are needed for defense. (Itis perhaps convenient to think of good 1 aslabor and good 2 as a consumption good.)

In a totally planned economy, where theplanner selects a fixed consumption bundle(including labor to be supplied) for eachconsumer, the planner would have no fur-ther constraint and could choose anypoint that was technologically feasible.Again, this is not implausible for theplanner in a one-consumer economy, butbecomes so as the number of householdsgrows. A more realistic assumption, then,is to assume that the planner can only dealwith consumers through the market place,hiring labor and selling the consumer good.Assume further that the planner is con-strained to charge uniform prices. Theplanner must now set the price of theconsumer good relative to the wage (or in-versely the real wage), and is constrainedto transactions which the consumer is will-ing to undertake at some relative price.The locus of consumption bundles whichthe consumer is willing to achieve by tradefrom the origin is the offer curve or price-consumption locus. It represents the

good 2

0 good I

FlGUSE 2

DIAMOND AND MIRRLEES: OPTIMAL TAXATION U

bundles of goods that the consumer wouldpurchase at different possible price ratios.Figure 3 contains an example of an offercurve with several hypothetical budgetHnes and the corresponding indifferencecurves drawn in. The planner thus has twoconstraints: he must choose a point whichis both technologically feasible and anequilibrium bundle from the point of viewof the consumer. Combining these twoconstraints, the range of consumptionbundles which are both feasible and poten-tial consumer equilibria is shown as theheavy line in Figure 4.

We can state these two constraintsalgebraically. Let us denote by z= (si, . . . , Sn) the vector of governmentsupply. The production constraint is thenwritten

(1) G(z) ^ 0, or, equivalently,

The constraint that the government sup-

good 2

good 2

FIGURE 3

FIGURE 4

ply equal the consumer demand for someprice can be written in vector notation

(2) x{q) = z,

where x={xi, . . . , a;™) is the vector ofconsumer demands and q={gi, • • • , ?«)is the vector of prices faced by the con-sumer.

Now consider the government's objec-tives. Since the consumer's equilibriumposition is determined by the prices hefaces, we can, in the usual circumstances,describe the objective function as a func-tion of prices, say v{q). The problem is tochoose q so as to

(3) Maximize v{q)

subject to G{xig)) ^ 0

This simply formulated problem is thefocus of attention of the paper and cantake on a variety of interpretations. Thereader may note that the consideration ofmany consumers does not alter the formof this problem. This is a major advantage


good 2

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of using prices rather than quantities asthe focus of the analysis.

Let us consider the case where theplanner seeks to maximize the same func-tion of consumption as the consumer'sutility function. The welfare function issaid to be individualistic, or to respect indi-vidual preferences, since welfare can bewritten as a function of individual utility.Returning to Figure 3 we see that theconsumer moves to higher indifferencecurves as he proceeds along the ofFer curveaway from the origin. Thus, in Figure 4 wewish to move as far along 00* as possible,subject to the constraint of the shadedproduction possibility set. The optimalpoint is therefore A, where the offer curveand the production frontier intersect.

The prices which will induce the con-sumer to purchase the optimal consump-tion bundle are defined by the budget lineOA. In Figure 5 we show the optimal pointand the implied budget line, and indiffer-

ence curve / / . All the points above / / andin the shaded production set are Pareto-superior to A and technologically feasible,but not attainable by market transactionswithout lump sum transfers. For contrast,in Figure 6, we show the Pareto optimalpoint, B, and the implied budget line, andindifference curve / ' / ' , which will permitdecentralization. In the case drawn, theconsumer's budget line does not passthrough the origin; this represents his pay-ment of a lump sum tax to cover govern-ment expenditures in excess of profits fromproduction.

We see that the optimal point is on theproduction possibility frontier of theeconomy, not inside it. This importantproperty of the optimum can easily be seento carry over to the case of many com-modities, but still one consumer. Withmany commodities, the offer curve is aunion of loci, each of which is obtained byholding the prices of all but one commodityconstant and varying the price of that onecommodity. Doing this for each com-

good 2

DIAMOND AND MIRRLEES: OPTIMAL TAXATION 13

modity, and for all possible configurationsof prices for the other commodities, gener-ates all the loci. The offer cur\'e is the unionof such loci. On each locus, the point whichis also on the production frontier is betterthan the other points on the locus. Thus,any point which is not on the productionfrontier is dominated by some point whichis on the frontier. Therefore, the optimalpoint is one of the points on the frontier.The implications of this result will beseen more clearly below, when we considerboth public and private production. Forthis result to carry over to the case ofmany consumers requires one further, mildassumption which will be discussed in thethird section. First, we treat the one con-sumer economy algebraically, ^\'ith bothpublic and private production, showing bycalculus the desirability of aggregate pro-duction efficiency, and obtaining the opti-mal relationship between consumer pricesand the slope of the production possibili-ties. This relationship defines the optimaltax structure.

II. One-Consumer Economy—Algebraic Analysis

We assume constant returns to scale inthe private production sector and thepresence of competitive conditions there.In equilibrium there are, therefore, noprofits. (This is a critical assumption forthe efficiency analysis.) We also assume,for the present, that the only taxes used bythe government are commodity taxes.^Consumer prices, q, therefore determinethe choices available to the consumer, andwe may write the welfare function as afunction of consumer prices, v{q). Noticethat this covers the case where the govern-ment's assessment of welfare does notcoincide with the consumer's utility, al-

• This assumption is made solely for simplicity. InDiamond-Mirrlees II, tbe general principles will be seento carry over with additional taxes, including a pro-gressive income tax.

though depending on what he consumes.In the special case where social preferencescoincide with those of the single consumer,his utility may be taken to measurewelfare. Then we have

(4) v{q) = uixiq))

We shall not use this special form for v{q)in the analysis below until we come toevaluate the tax structure explicitly. Untilthat point, the analysis applies also towelfare functions that are not individual-istic. For later use let us express the deriva-tives of V in this special case. WritingVk = dv/dqk, Ui = du/dXi, and using (4), wehave

(5)dqt

where a is a positive constant (i.e., inde-pendent of k), the marginal utility ofincome. Equation (5) follows from thebudget constraint,

(6) Z qix, = 0,

which on differentiation with respect toqk yields

(V)

Since utility-maximization implies thatUi=aqi, (5) now follows from (7).

Production

Let us denote the vector of prices facedby private producers by p = ipi, . . . , pn)-Because of taxes, /, these may differ fromthe prices faced by consumers: qi = pi-\-ti( j= l , . . . , n). y=iyi, . . . , >'») is thevector of commodities privately supplied(inputs will thus appear as negative sup-plies) , and we write the private productionconstraint,

(8) yi = f(y2,..., yn)

Notice that we assume equality in the


production constraint, that is, that pro-duction is efficient in the private sector.This follows from profit maximization ifthere are no zero prices. We assume tha t /is a differentiate function, and thaty.FÔ ( i= l , . . . , n). Then, profit maxi-mization means that

(9)Pi= -

where/i denotes the derivative of/ withrespect to y^ Also, by the assumption ofconstant returns to scale, maximizedprofits are zero in equilibrium:

(10) — 0

So that we may conveniently employcalculus, we shall assume that the govern-ment production constraint, (1), is satisfiedwith an equality rather than an inequality:

(11) S l = g{Z2, . . . , Z . )

Thus we do not give the government theoption of inefiicient government produc-tion. Rather, we shift our attention toaggregate production efficiency. Efficiencywill be present if marginal rates of trans-formation are the same in publicly andprivately controlled production. It willthen be seen quite easily that the assump-tion of efficiency in the public sector isjustified.

Walras^ Law

We have chosen an objective functionand expressed the government's produc-tion constraint above. To complete theformulation of the maximization problem,it remains to add the requirement that theeconomy be in equilibrium. The conditionsthat all markets clear can be stated interms of the vectors x, y, and 2.

(12) x{q) = y^ z

The reader may be puzzled that at noplace in this formulation has a budget

constraint been introduced for the govern-ment. (Other readers may be puzzled byour failure to include only «—I marketsin our market clearance equations. Theseare aspects of the same phenomenon.)Walras' Law implies that if all economicagents satisfy their budget constraints andall markets but one are in equilibrium, thenthe last market is also in equilibrium. Italso implies that when all markets clearand all economic agents but one are ontheir budget constraints, then the lasteconomic agent is on his budget constraint.In setting up our problem, we have as-sumed that the household and the privatefirms are on their budget constraints. Thus,if we assume that all markets clear, thiswill imply that the government is satisfy-ing its budget constraint,' which we canexpress as

= Z] îî + ^ PiZi

Alternatively, if we consider the govern-ment budget balance as one of the con-straints, then it is only necessary to im-pose market clearance in «—1 of themarkets.

In this model we can make two pricenormalizations, one for each price struc-ture. Since both consumer demand andfirm supply are homogeneous of degreezero in their respective prices, changingeither price level without altering relativeprices leaves the equilibrium unchanged.As normalizations let us assume.

(14) = 1, = 0

It may seem surprising that it does notmatter whether the government can taxgood one. But the reader should rememberthe budget balance of the consumer. Sincethere are no lump sum transfers to the

^ In an intertemporal interpretation of this model,the government budget is in balance over the horizon ofthe model, not year by year.


consumer, net expenditures are zero. Thus,levying a tax at a fixed proportional rateon all consumer transactions results in norevenue. (It should be noticed that apositive tax rate applied to a good sup-plied by the consumer is in effect a subsidyand results in a loss of revenue to thegovernment.)

Welfare Maximization

We can now state the maximizationproblem. In the statement we shall use thetwo sets of prices as control variables. Itwould be a more natural approach to usethe taxes which the government actuallycontrols as decision variables. However,once we have determined the optimal pand g vectors we have determined theoptimal taxes. Using taxes as decision vari-ables complicates the mathematical formu-lation and leads to a control problem sincethe tax vector may not uniquely determineequilibrium.

Rather than calculate the first-orderconditions from the formulation spelledout above, we shall alter the problem tosimplify the derivation. We have to choose

( 1 5 ) 92, • • • ) qn, P2, • • • , pi, Sl , . - • » Zn

to maximize v{q) subject to

Xi{q) — yi — Zi — yJ {i — i, z, . . . , nj,

where y maximizes X P^y^ subject to

and

Since the choice of producer prices canbe used to obtain any desired behavior onthe part of private producers, we can useany vector y consistent with the produc-tion constraint (8). Producer prices arethen determined by equation (9). Usingthe equations

we reduce the constraints in (15) to the

single constraint

We have therefore simplified the problem(15) to:

(16) Choose q2, . . . , qn, Z2, . . . , s™

to maximize v{q) subject to

•^1(9) ~ f(^2(q) — Za, . . . , Xniq) — Zn)

Forming a Lagrangian expression from(16), with multiplier X,

L = v{q) -~ \[xi{q)

(17) -fiX2-Z2,...,X^-Zn)

we can differentiate with respect to qt:

(18) \dqk i=a dqk /

Making use of the equations (9) for pro-ducer prices, this can be written

(19)

k = 2, 3, . . . , n

Differentiating L with respect to Zk wehave

(20) - gk) =0

Provided that X is unequal to zero (i.e.,that there is a social cost to a marginalneed for additional resources), equation(20) implies equal marginal rates of trans-formation in public and private productionand thus aggregate production efficiencyas was argued above. The assumption thatX?^0 needs justification. This is providedby the rigorous arguments of Sections IIIand IV.

If we had introduced several public


production sectors, each described by aconstraint like (11), we should have ob-tained an equation of the form (20) foreach sector. Thus marginal rates of trans-formation in all public sectors should beequal, since they are all to be equal to theprivate marginal rates of transformation.This argument—which we only sketchhere, since the conclusion will be provedmore directly in the next section—justifiesour assumption that there should be pro-duction efficiency in the public sector.

Optimal Tax Structure

The relations (19) determine the optimaltax structure, since they show how pro-ducer and consumer prices should be re-lated. These equations show that con-sumer prices should be at a level such thatfurther increases in any price result in anincrease in social welfare, Vky which is thesame ratio, X, to the cost of satisfying thechange in demand arising from the priceincrease. Reintroducing taxes explicitlyinto the problem we can obtain an alterna-tive interpretation for the first-order con-ditions.

Since Xi is a function of p-\-t,

(p is held constant in this latter deriva-tive.) Consequently, the optimal tax struc-ture, (19), can be rewritten:

.f— dXi d --(21) v.-\Zpi~- = '>^~~ZPiX<

h dt

Since Y, piXi='Zl qiX— X) tiXi= — XI t,Xi(by the consumer's budget constraint (6)),we have

(22)

This last set of equations asserts the

proportionality of the marginal utility of achange in the price of a commodity to thechange in tax revenue resulting from achange in the corresponding tax rate, cal-culated at constant producer prices. Likethe first-order conditions for the optimumin standard welfare economics, our first-order conditions are expressions in con-stant prices. The tax administrator, likethe production planner, need not be con-cerned with the response of prices togovernment action when looking at thefirst-order conditions.

If we now make the further assumptionthat the welfare function is individualistic,we can use equation (5) to replace vt. Thefirst-order conditions then become

(23)

Thus for all commodities the ratio ofmarginal tax revenue from an increase inthe tax on that commodity to the quantityof the commodity is a constant. This formof the first-order conditions has the ad-vantage of showing the information neededto test whether a tax structure is optimal.The amount of information does not seemexcessive relative to the data and knowl-edge which a planner in an advancedcountry should have.

The statements of the first-order condi-tions thus far do not directly indicate thesize of the tax rates required, nor the im-pact upon demand that the optimal taxrates would have. In his pioneering studyof optimal tax structure, Frank Ramseymanipulated the first-order conditions soas to shed light on the latter question. Heemployed the concept of demand curvescalculated at a constant marginal utilityof income. Paul Samuelson reformulatedthis using the more familiar demandcurves calculated at a constant level ofutility. We shall return to this question inDiamond-Mirrlees II.


III. Production Effidenqr in theMany-Consumer Economy

We have remarked already that manyof the results carry over directly to aneconomy of many consumers, even whenlump sum taxation is excluded. We noticeat once that the device of expressing wel-fare as a function of the prices, q, faced byconsumers can be used perfectly well.Explicitly, we assume that there are Hhouseholds, with utility and demand func-tions «'• and x'̂ (/?=1, 2, . . . ,H) . If, as wemay generally suppose, in the absence ofexternalities from producers to consumers,social welfare can be expressed as afunction of the consumption of thevarious consumers in the economy,U{x^, x'^, . . . , x"), it may also be written

(24) V{q) = U{x\q), x'{q), . . . , x^C?)),

where we assume that there are no lumpsum incomes or transfers that would beinfiuenced by producer prices or govern-ment policy. In the case where social wel-fare depends only on individual utilityand there are no externalities, we can write

(25) V(q) = W[uK^Kq)),

where W is presumed to be strictly in-creasing in each of its arguments.

Using this indirect welfare function, wecan carry out the analysis already pre-sented for the one-consumer economy, andconclude in the same way that aggregateproduction efhciency is desirable. For thatargument to be correct, we must confirmthat the Lagrange multiplier X is not zero.Rather than attempt to do this directly, weshall present a different argument for thedesirability of production efficiency. A fur-ther condition will be required to secureour conclusion. In considering this prob-lem, we shall concentrate on the case whereall production is under government con-

trol. The desirability of production effi-ciency in this case will be seen to imply thesame conclusion when there is also aprivate sector (provided that privateproducers are price takers, and profits, ifany, are transferred to the government).Assume then (as we did in Section I) thatall production takes place in the publicsector: our problem is to find g that will

(26) Maximize V(q),

subject to G{X{q)) ^ 0,

where we define X{q) = J2h x'^iq) as aggre-gate demand at prices g. We shall alsoexpress the production constraint a littlemore generally by saying that X{q) is tobelong to the production set G, the set oftechnologically feasible production plans.(Thus the letter G denotes both the pro-duction set, and also the function that canbe used to describe it; but we shall hardlyever use the fuJtction G explicitly).

Suppose we establish that, at the opti-mum for problem (26), production isefficient. Consider an economy with thesame technolgical possibilities, partly un-der the control of private, competitiveproducers. The government can induceprivate firms to produce any efficient netoutput bundle by suitable choice of pro-ducer prices p. In particular, it can obtainthe production plan that would be optimalif the government controlled all produc-tion. The choice of p does not affect con-sumer demands or welfare, since pureprofit arising from decreasing returns toscale go to the government, and since, anycommodity taxes being possible, g can bechosen independently of p. Thus, if thesolution to (26) is efficient, the same equi-librium can be achieved when some pro-duction is under private control, and isoptimal in that case too. Proof that pro-duction efficiency is desirable in the "spe-cial" case (26) therefore implies that pro-


duction efficiency is desirable in the moregeneral case.

Examples qf Inefficiency

Before considering the argument forefficiency, it is useful to consider somelimitations on that argument as demon-strated by the following examples of de-sired inefficiency. It will be recollected thata production plan is efficient if any otherfeasible production plan provides a smallernet supply of at least one commodity. Weshall use a different concept: we say that aproduction plan is weakly efficient if it ison the production frontier. It is possiblefor a production plan to be weakly efficientwithout being efficient if the productionfrontier has vertical or horizontal portions.For matters of economic importance, suchas the existence of shadow prices, weakefficiency is all that is required. It is easyto see that if all the prices corresponding toa weakly efficient production plan arepositive, the plan is in fact efficient in theusual sense.

Even with this slightly weakened con-cept of efficiency, it is not necessarily truethat, when an optimum exists, optimalproduction has to be weakly efficient. Wepresent two examples.

Example a is portrayed in Figure 7. It isa one-consumer economy where socialpreferences, as depicted in the social in-difference curve / / , do not coincide withindividual preferences. It is evident that,in the case shown, the optimal productionplan is actually in the interior of the pro-duction set.

In the second example, social preferencesdo respect household preferences, butagain optimal production lies in the inte-rior of the production set, and is thereforenot weakly effi.cient: suitable producerprices cannot be found, and the socialoptimum cannot be obtained when thereis private control of production.

Example h. There are two commoditiesand two households. One has utility func-tion x'^y, the other has utility functionxy^\ each has the nonnegative quadrant[{x, y)\x'^O, >'^0J as consumption set.The first consumer has three units of thefirst commodity initially; the second, oneunit of the second commodity. The welfarefunction is

The second commodity can be trans-formed into the first according to theproduction relation x-\-\^y^Q, (a:^0).Let the prices of the commodities be ji, q^.Then the first household's net demandsare

— 1 of the first commodity,

9i/?2 of the second commodity.

The second household has net demands

good 2

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ligz/qi) and - I

Thus, the net market demand for thecommodities is

X = \{qi/qi) - 1 and y = (qi/q^) - I

These must satisfy

x-\- lOy ^0, a; S 0

Welfare is -g2/4gi-27^i/4?2 which ismaximized when q2/qi~3\/3:_th.e corrê -sponding production vector \ /3—1, KVg— 1) is actually interior to the productionset, not on the frontier. This example hasthe unimportant peculiarity that initialendowments of the consumers are on thefrontiers of their consumption sets. Morecomplicated examples avoiding the peculi-arity have been constructed.

The Efficiency Argument

Despite these examples, the followingargument shows that optimal productionwill generally be on the production fron-tier. Suppose that the aggregate demandfunctions, X{q), are continuous. Then anysmall change in the prices, q, will notchange aggregate production requirementsby much. Therefore, if optimal productionwere in the interior of the production set,small changes in consumer prices wouldstill result in technologically feasible aggre-gate demands. Thus, if we are at the opti-mum, small changes in consumer pricescannot increase welfare. If we can arguethat, at the optimum, there exists a smallprice change which would increase V{q),we can conclude that production for theoptimum must occur on the productionfrontier. For any unsatiated single con-sumer, utility can be increased either bylowering the price of a supplied good orraising the price of a demanded good (aswe can see, algebraically, in equation (5)).With a single consumer, we need not arguefurther, provided the equilibrium involvessome trade. When there are many con-

sumers, we can be certain of increasingwelfare if we raise some consumer's utilitywithout lowering that of anyone else. Ifthere is a commodity that no consumerpurchases, but some consumer supplies(such as certain labour skills); or a good(with positive price) which no consumersupplies, but some consumer purchases(such as electricity), we could alter theprice of that commodity in such a way asto bring about an unambiguous increasein welfare. In that case, we conclude thatefficient production is required for themaximization of individualistic social wel-fare. In example b, it will be seen thatneither of the commodities is supplied, ordemanded, by both consumers. The verysimplicity of the case appears to be mis-leading.

A formal presentation of this argumentis given in the next section: these technicaldetails can be omitted without loss ofcontinuity. We conclude this section byintroducing further taxes into the dis-cussion.

First, consider the case of a poll tax (orsubsidy)—that is, a tax is paid by a house-hold on the basis of some unalterableproperty, such as its sex or age distribu-tion. Such a tax is, of course, a lump sumtax, although its availability is not, ingeneral, sufficient to enable the full opti-mum to be achieved. To fix ideas, supposethere is a single transfer, r, to be made toall households. Then welfare can be writtenV{q, T), and we are to

(27) Maximize V(q, T)

subject to X(q, r) being in G

The standard efficiency argument can beused. Let {q*, T*) be the optimum: if anysmall change m q or T would increase V,optimal production, X(^*, T*) must be onthe production frontier (assuming that Xis a continuous function). Now a pollsubsidy must make everyone better off,


unless some are already satiated, and somust a small increase in subsidy. Thus solong as a poll subsidy is possible (and itsurely is) and not every household issatiated, optimal production must be onthe frontier.

Adding further tax instruments to thegovernment's armoury in no way weakensthe efficiency conclusion. We simply notethat if there are other tax variables whichare independent of producer prices andquantities, denoted collectively by f, wecan hold them constant at their optimumvalues f*, and then apply the efficiencyargument to the problem (27) or (26),where V and X are evaluated for f=f*.

Our final conclusion is that whateverthe class of possible tax systems, if allpossible commodity taxes are available tothe government, then in general, and cer-tainly if a poll subsidy is possible, optimalproduction is weakly efficient. We wouldnot expect this conclusion to be valid ifthere were constraints on the possibilitiesof commodity taxation, or more generally,on the possible relationship between pro-ducer prices and consumer demand. Thepresence of pure profits is one example ofsuch a relationship. To show what goeswrong, suppose, by way of another ex-ample, that no commodity taxes are possi-ble, but a poll tax is possible, and that partof production is privately controlled, insuch a way that it is uniquely determinedby producer prices. Then we have tochoose a public production vector z anda poll tax T to

(28) Maximize V{p, r)

subject to X(p, T) — y{p) = s being in G,

where y{p) is the private production vectorwhen prices are p. Following the argumentused above, we consider T smaller thanr*, the optimum level, and note that

, T)>V(P*, T*). This implies that*, T)~y{p*) is not in G, and therefore

z*, the optimal s, is efficient in G. But the

argument does not imply that the aggre-gate optimal production plan, y{p*)-\-z* isefficient. Of course, in an economy whereall production is under public control,these problems do not arise. Even whensome of the qu are fixed, the efficiency argu-ment holds, for there can be no necessaryrelation between q and p.

IV. Theorems on Optimal Production

In this section, we explore the existenceof the optimum, and the efficiency of opti-mal production, rigorously. We rely onDebreu (1959) for the results of generalequilibrium theory that are required.

Assumptions

There are H households in the economy,each household choosing a preferred netconsumption vector x from his consump-tion set C subject to the budget constraint(^-3:^0 where q is the vector of pricescharged to consumers. (Consumption ismeasured net of initial endowment forconvenience, since the latter is unalteredin the analysis.) As usual the net demandvector X has, in general, both positive andnegative components corresponding topurchases and sales by the household.

The assumptions used below will beselected from the following list (the super-script h refers to the index of households;all assumptions, when made, hold for allh):

(a.l) C* is closed, convex, bounded be-low by a vector a>, and contains avector with every component nega-tive.

(a.2) The preference ordering is con-tinuous.

(a.3) The preference ordering is stronglyconvex. Formally, if x'^ is preferredor indifferent to x^ and 0 < f < l ,then tx'^+{i~t)x^ is strictly pre-ferred to x^.

(a.4) There is no satiation consumptionmCK

DIAMOND AND MTRRLEES: OPTIMAL TAXATION 21

Assumptions (a.l) and (a.2) guaranteethe existence of continuous utility func-tions, which we shall write w'' (see DebreuSection 4.6). Furthermore, under (a.l)-(a.3), when the demand vector x^q) isdefined, it is uniquely defined. When Cis bounded, assumptions (a.l)-(a.3) implythat x'^iq) is defined and continuous at allnon-zero nonnegative g. (See Debreu, Sec-tion 4.10.)

Let us denote aggregate demand by(q)T.{q)It is assumed that all production is con-

trolled by the government. The assump-tions on the production possibility set, G,will be taken from the following set:(b.l) Every production plan in which

nothing is produced in a positivequantity is possible: i.e., if 3^0,z is in G.

(b.2) Complete inactivity is possible:i.e., 0 is in G.

(b.3) G is closed.(b.4) There exists a vector d such that

z^d for all nonnegative z in theconvex closure of G. (i.e., theclosure of the convex hull of G)}

(b.5) G is convex.

The welfare function will be denoted byU{x^, . . . , x"). When demands are func-tions of prices only we can define the in-direct welfare function as

V{q) = U{xKq),' • • .

Similarly we can define an individual'sindirect utility function by

We shall say that the welfare function re-spects household preferences when U can bewritten

• When G is convex, this assumption is similar tothe assumption that inputs are required to obtain out-puts, but permits the government to own a vector ofinputs.

with W increasing in each argument. Weshall assume(c.l) t/ is a continuous function of

We can now state our problem as tryingto fmd q* to maximize V{q) subject toX{q) being in G. A commodity vector willbe called attainable if it is feasible and ifthere exists prices such that aggregatedemand equals the vector. The set of allsuch vectors, the attainable set, is the in-tersection of G with the set of vectors Xiq)for all nonnegative q.

Existence of an Optimum

If we assume that the attainable setis nonempty and bounded, we obtain

THEOREM 1. If assumptions (a.l)-(a.3),(b.3), and (c.l) hold, and if the attainable setis nonempty and bounded, an optimumexists.

PROOF:Consider an economy in which the con-

sumption sets are truncated by removingfrom them all points x with |l3;l|>if,where all vectors in the attainable setsatisfy ||x|l


good 2

FiGUEE 8

in G. At the same time, x' = X(q'),by the continuity of X. Thus q' is ink l -^(?) in G} , which is therefore closed.

Since the attainable set is nonempty,and prices are in any case bounded,{q\ X(q) in G] is closed, bounded, and non-empty. By the continuity of the demandfunctions, and assumption (cl ) , F is acontinuous function of q, which there-fore attains its maximum on the set{q\X{q) in G].

One criterion for the attainable set to benonempty follows immediately from theexistence of competitive equilibrium in anexchange economy:

THEOREM 2. If assumptions (a.l)-(a.4)and (b.l) hold, the attainable set is non-empty.

PROOF:See Debreu (Section 5.7) for a proof that

there exists an equilibrium for the ex-change economy with these consumers.

The equilibrium prices result in a feasibledemand.

If the production set is taken to be theset of possible production vectors net ofgovernment consumption, the assumptionthat zero production is possible is ex-cessively strong, especially for govern-ments with large military establishments.But it is easy to construct examples ofeconomies not satisfying (b.l) in whichthere is no attainable point. Consider theone-consumer economy depicted in ex-ample c shown in Figure 8.

The boundedness of the attainable setwould be implied by the boundedness ofthe consumption sets, or the boundednessof production, but the following case ismore appealing:

THEOREM 3. / / assumptions (a.l) and(b.2)-(b.4) hold, then the attainable set isbounded?

PROOF:Suppose the attainable set is not

bounded. Then there exists a sequence ofattainable vectors Xn such that \x^ is anunbounded increasing sequence of realnumbers. There exists an n' such thatll̂ Cn'll >||a||, where a is the vector employedin (b.4). Consider the sequence of vectors(lk»'ll/lF"ll)^n for «>«'. Each vector is inthe convex hull of G (being a convexcombination of the origin and a:,.)- Furtherthe sequence is bounded. Thus there is alimit point, ,̂ which is in the convexclosure of G and satisfies ||^||>||a||. Leth= X!h Q-h, where a^ are the vectorsemployed in (a.l). Then :*;„= X]A:4^E.a f = 6. Further {\xMVA)^r.^d^"'!!/!!^" 1)6. But the latter sequence ofvectors converges to zero. Thus ^ ^ 0 . Thisis a contradiction.

• The attainable set will also be bounded if (b.2)-{b.4) hold for the true production set, gross of govern-ment consumption, rather than the net production set,G. Thus the assumption that zero production is possibleis not of great consequence.


Finally, we should remark that thestrong convexity assumption, (a.3), whichwas made in Theorem 1 can be changed toconvexity without affecting the conclusion.All that is required is to replace the con-tinuous functions of the proof by uppersemi-continuous correspondences. On theother hand, one can easily construct ex-amples in which an optimum fails to existbecause of the absence of continuity.

EfficiencyThe following lemma provides two cri-

teria for optimal production to be on thefrontier of the production set. It will beused to deduce a theorem about the casewhere household preferences are respected.

LEMMA 1: Assume an optimum, q*, exists.If aggregate demand functions and the in-direct welfare function are continuous in theneighborhood of the optimal prices; and ifeither

(1) for some i, V is a strictly increas-ing function of qi in the neighborhoodof q*; or

(2) for some i with q*>0, V is astrictly decreasing function of qt in theneighborhood of q*,

then X{q*) is on the frontier ofG.

PROOF:Let li be the vector with all zero com-

ponents except the ith, which is one. Incase 1, for e sufficiently small F(^*+f/i)>V{q*). Hence X{q*-\-€li) is not in G.Letting e decrease to zero, the continuityof X shows that X{q*-) is a limit of pointsnot in G, and therefore belongs to theboundary of G. In case 2, a similar argu-ment can be made using V{q*—eli).

These conditions are weak. They are,naturally, independent of production pos-sibilities. It may also be noticed that, whenF is a differentiable function of prices, thestated conditions are equivalent to assum-ing that

(29) It is not the case that V'{q*) ^ 0

Here V'{q) is the vector of first derivativesof V with respect to prices. The equiva-lence of the conditions of the theorem and(29) is clear if we remember that

(30) V'(q)-q =dV

since V is homogeneous of degree zero in q.Therefore V ^0 if, and only \f,dV/dqk = Owhen qk>0 anddV/dqt^0 in any case.

In the following theorem, we strengthenthe assumptions in a different way: theyremain notably weak.

THEOREM 4. If (a.l)-(a.4) and (c l)hold; if social welfare respects individualpreferences; and if either

(1) for some i, Xf^O for all h, andx^i


}&'

s e l of opti t good 2

FlGUBE 9

a linear section. Then the offer curve maycoincide with the linear part of an indif-ference curve, giving a set of optima, onlyone of which is on the production frontier.As an illustration, see Figure 9.

The example suggests that we weakenthe conclusion of Theorem 4 to say thatthere exists an optimum on the frontier ofG: this generalization is indeed correct ifwe merely assume convexity of preferences.The proof follows that of Theorem 4, withupper semi-continuity of the demand cor-respondence replacing continuity of de-mand functions. ;

V. Extensions

We can summarize the efficiency resultby considering an economy with threesectors—consumers, private producers,public producers. We assumed that onlythe equilibrium position of the consumersector enters the welfare function, and thatonly market transactions take place be-tween sectors, while the government haspower to tax any intersector transaction

at any desired rate. One conclusion wasthat all sectors not containing consumersshould be viewed as a single sector, andtreated so that aggregate productionefilciency is achieved. By regrouping theparts of the economy according to thisschematic division, we can extend theefficiency result to several other problems.In each case, we indicate briefly how ap-plication of this schematic view shows therelationship of the extension to the basicmodel.

Intermediate Good Taxation

The model, as presented above, left noscope for intermediate good taxation. Ifwe separate private production possibili-ties into two (or many) sectors, we intro-duce the possibility of taxing transactionsbetween firms. In the schematic viewpresented above, we could consider a con-sumer sector and two, constant returns toscale, private production sectors. We con-clude that we want efficiency for theseprivate production possibilities taken to-gether. Therefore the optimal tax structureincludes no intermediate good taxes, sincethese would prevent efficiency. (Similarlywe conclude that government sales to firmsshould be untaxed while those to con-sumers are taxed.)

There is a straightforward interpretationof this result, which helps to explain thedesirability of production efficiency. In theabsence of profits, taxation of intermediategoods must be reflected in changes in finalgood prices. Therefore, the revenue couldhave been collected by final good taxation,causing no greater change in final goodprices and avoiding production ineffi,-ciency. This interpretation highlights thenecessity of our assumption of constantreturns to scale in privately controlledproduction.

However, it may well be desirable to taxtransactions between consumers or tocharge different taxes on producer sales to


different consumers. There are two waysin which we can consider doing this. Thecountry might be geographically parti-tioned with different consumer prices indifferent regions. Ignoring migration, andconsumers making purchases in neighbor-ing regions, our analysis can be applied todetermine taxes region by region. In gen-eral the tax structure will vary over thecountry.

Alternatively, we might consider taxa-tion on all consumer-consumer transac-tions. Here, too, we would expect to beable to increase social welfare by havingthese additional tax controls. Neither addi-tion to the available tax structure altersthe desirability of production efficiency.

Untaxable Sectors

One problem that arises with a modelconsidering taxation of all transactions isthat some transactions may not be taxable,practically or legally. An example of theformer might be subsistence agriculturewhere transactions with consumers arehard to tax while those with firms are not.If the introduction of other taxes (e.g.,on land or output) is ruled out, we canaccommodate this problem in the modelby including subsistence agriculture in theconsumer rather than producer sector (ortreating it as a second consumer sector).EfUciency would then be desired for themodern and government production sec-tors taken together; while the tax struc-ture rules would be stated in terms of de-mand derivatives of the augmented con-sumer sector rather than of just the trueconsumers.

Similarly, in an economy without taxes,a public producer subject to a budgetconstraint is unable to charge differentprices to consumers and producers. Lump-ing together the entire private sector as asingle consumer sector, we obtain the con-ditions for optimal public production of anindustry regulated in this manner. This

is the problem considered by Boiteux inthe context of costless income redistribu-tion. He also analyzed such an economywith several firms, each limited by abudget constraint.

Foreigners

It is not easy to provide a satisfactorywelfare economics for a world of manycountries. The study of world welfaremaximization is interesting, and, one mayhope, "relevant." But it has the seriouslimitation that its results can seldom beapplied to the actions of governments.However altruistic the principles on whicha government seeks to act, it has to allowfor the actions other governments maytake, based on different principles, or fordifferent reasons. (A somewhat analogousproblem arises in intertemporal welfareeconomics.) In the following two subsec-tions, we shall, in order to keep the dis-cussion brief, refer only to the case wherethe reactions of all other countries arewell-defined functions of the actions of thecountry directly considered. Thus weneglect, reluctantly, those situations thathave come to be called "game-theoretic."Also, we shall not consider the problem offormulating a social welfare function in aninternational setting.

International Trade

So long as we are completely indifferentto the welfare of the rest of the world, andso long as the reactions of other countriesare well-defined, international trade simplyprovides us with additional possibilities fortransforming some goods and services intoothers. The efficiency result then impliesthat we would want to equate marginalrates of transformation between producingand importing. If there is a monopolyposition to be exploited, it should be. Ifinternational prices are unaffected by thiscountry's demand, intermediate goodsshould not be subject to a tariff, but final


good sales direct to consumers should besubject to a tariff equal to the tax on thesame sale hy a domestic producer.

Sometimes it is not possible to sell goodsto foreigners at prices different from thoseat which they are sold to domestic con-sumers, although the theory just outlinedsuggests that foreigners should be treatedlike producers. As examples, we may citetourists and commodities covered by spe-cial kinds of international agreement. Iftourism, say, is an important tradingopportunity for the country, and touristshave to be charged the same prices asdomestic consumers, this will affect theoptimal level of taxes on certain commodi-ties. The general efficiency result is notupset, however. The analysis can be per-formed by treating tourists as consumerswhose income does not affect social wel-fare.

The authors do not, of course, recom-mend indifference to the welfare of therest of the world; although it happens tomake the results somewhat neater. Inter-national trade provides the country withanother set of consumers who can tradewith it at prices different from its ownconsumers: the case (when foreign reac-tions are well-defined) is similar to thepossibility of using different consumerprices in different regions of the sameeconomy. In that case, there is no reasonwhy optimal international trade pricesshould be the same as producer prices, p,or domestic consumer prices, q.

Migration

In all that has gone before, we have beenholding constant the set of consumers inthe economy. We can introduce migrationin a straightforward manner. Social wel-fare may be a function of the consumptionof every household in the world. Changesin the consumer prices charged in thehome country cause migration in one direc-tion or another, and therefore affect wel-

fare in ways we have not previouslydiscussed (such as the effect on the in-habitants of another country of havingadditional taxpayers join them). But wecan still define an indirect welfare func-tion V{q)y so long as the reactions of therest of the world are well-defined. Similarlywe can define aggregate demand functionsX{q), but these are no longer continuous.For, when a man decides to emigrate, hiscontribution to aggregate demand changesfrom x^ to 0.̂ ° But the number of migrantsarising from a small price change may,quite reasonably, be assumed small rela-tive to the population as a whole. We cantherefore adequately approximate thissituation by considering a continuum ofconsumers. In this way we can restorecontinuity to aggregate demand, and tothe indirect welfare function. It is to heexpected, then, that production efficiencyis still desired. Since the derivatives of thedemand functions, and possihly also thederivatives of F, will be different when thepossibility of migration is allowed for, theoptimal tax structure will be changed toreflect the loss of tax revenue when nettaxpayers, for example, leave the country.While we do not wish to examine thisproblem in detail here, we believe thatthese ideas provide an interesting ap-proach to the analysis.

Consumption Externalities

The schematic view of this problemgiven above suggests that the basic struc-ture of the results, although not the specificoptimal taxes, are unchanged by compli-cations which occur wholly within theconsumer sector. Thus, if we introduceconsumption externalities that leave ag-gregate demand continuous we will stillobtain production efRcieney at the opti-mum, if we can argue that V{q) has nounconstrained local maximum for finite q,

" A similar discontinuity problem arises in the caseof tourists' decisions not to visit the country.


The conditions used above are no longersufficient for this argument since the directeffects of a price change might be offsetby the change in the pattern of externali-ties induced by the price change. Althoughwe have not examined this case in detail,there are a number of cases where argu-ments similar to those in the no-externalitycase will be valid." Furthermore it seemsquite likely to us that efficiency will bedesired in realistic settings.

Capital Market Imperfections

While some capital market imperfec-tions affecting firms are complicated todeal with, some imperfections relevantonly for consumers can be described aselements solely within the consumer sec-tor. For example, consider the constraintthat consumers can lend but not borrow.We must then rewrite consumer utilitymaximization as subject to a set of budgetconstraints for the different time periods.In the case of two periods, for example, itwould appear as

(31)

Maximize u{x^, x'^)

subject to g^x^ -\- s ^ 0

5 ^ 0

where s represents first period savings.From this consumer problem, we still haveutility and demand expressible in terms of

" We have benefited from discussions with ElishaPazner on this subject.

prices. We expect that the efficiency resultcontinues to hold. In calculating the opti-mal formula, though, it becomes necessaryto distinguish the time period of the goodin question for there are now two Lagrangemultipliers giving the marginal utility ofincome in each of the two periods. For thisconsumer we have

(32)dv

Since savings are allowed a^^a^. If theconsumer would borrow if he could, a^>a^and the optimal tax structure is altered bythis market limitation.

REFERENCESW. Baumol and D. Bradford, "Optimal De-

partures From Marginal Cost Pricing,"Amer. Econ. Rev., June 1970, 69, 265-83.

A. Bergson, "Market Socialism Revisited," / .PoUt. Econ., Oct. 1967, 75, 431-49.

M. Boiteux, "Sur la gestion des monopolespublic astreints a I'equilibre budgetaire,"Econometrica, Jan. 1956, 24, 22-40.

G. Debreu, "A Classical Tax-Subsidy Prob-lem," Econometrica, Jan. 1954, 22, 14-22.

, Theory oj Value, New York 1959.J. Dreze, "Postwar Contributions of French

Economists," Amer. Econ. Rev. Supp., June1964, 54, 1-64.

A. Prest and R. Turvey, "Cost-Benefit Analy-sis: A Survey," Ecoti. /., Dec. 1965, 75,683-735.

F. Ramsey, "A Contribution to the Theory ofTaxation," Econ. J., Mar. 1927, 37, 47-61.

P. Samuelson, "Memorandum for U.S. Trea-sury, 1951," unpublished.

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