on the existence of an optimal income tax schedule

The Review of Economic Studies, Ltd.

On the Existence of an Optimal Income Tax ScheduleAuthor(s): Mamoru KanekoSource: The Review of Economic Studies, Vol. 48, No. 4 (Oct., 1981), pp. 633-642Published by: Oxford University PressStable URL: http://www.jstor.org/stable/2297202 .

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Review of Economic Studies (1981) XLVIII, 633-642 0034-6257/81/00550633$02.00

? 1981 The Society for Economic Analysis Limited

On the Existence of an Optimal

Income Tax Schedule MAMORU KANEKO

University of Tsukuba and Yale University

1. INTRODUCTION

Since the pioneering work of Mirrlees (1971), many authors have considered the non-linear optimal taxation problem in models with two kinds of goods-labour and consumption- in terms of the variational method or the Pontryagin maximum principle. That is, several necessary conditions have been derived and investigated assuming the existence of an optimal tax schedule. These works, however, have been done without any verification of the existence of an optimal tax schedule. This problem is not a simple maximization problem but a compound maximization problem, i.e. it has individuals' utility maximization and government's social welfare maximization depending upon each other. So, the existence of an optimal tax schedule is not clear either mathematically or economically. On the other hand, Kaneko (1979) proved the existence of an optimal tax schedule for the class of progressive (convex) ones, which is narrower than that employed in the preceeding works. Although the progressiveness of tax schedules is an important condition to consider in an optimal taxation problem, it should be checked whether or not this condition can be derived in a more general theory which permits non-progressive tax schedules.' Thus we have two general problems-one is the existence of an optimal tax schedule in a class which permits non-progressive (non-convex) tax schedules and the other one is the progressiveness (in some sense) of the optimal tax schedules. Here we consider only the existence problem.

In this paper, we model the optimal taxation problem in the context of general equilibrium analysis. We consider an economy with a finite number of individuals, a finite number of firms and a government, where a finite number of consumption goods and one public good are produced by the firms and the government respectively using labour as input. An optimal tax schedule means one which gives a competitive equilibrium with the maximum value of a social welfare function. Thus the model of this paper is much wider than those of the preceding works, though this paper concerns only the existence of an optimal tax schedule in an economy with a finite number of individuals. The existence problem in an economy with a continuum of individuals is important for the further consideration of the shape of an optimal tax schedule, because tax rates or disposable incomes of rich individuals can be considered naturally in such a model.2 The technique given in this paper, however, can not be applied to the existence problem in such a case. This author continues to investigate the continuum case.

This paper is written as follows. In Section 2, we describe the model and define the optimal taxation problem. In Section 3, we provide one important theorem and prove the existence of an optimal tax schedule.

2. TAX SCHEDULE AND COMPETITIVE EQUILIBRIUM

We consider an economy consisting of a finite number of individuals, a finite number of firms and an agent called "government". Let N = {1, 2,. . ., n} be the set of all individuals.

633



634 REVIEW OF ECONOMIC STUDIES

We assume that leisure, c-kinds of consumption goods and one public good enter the individuals utility functions Ui(t, x, Q) (i E N), where t denotes leisure time, x = (x1,.. ., xc) a level of consumption goods and Q a level of the public good supplied by the government. Every U' is defined on the consumption set Y [0, L] x E"i+', where L > 0 is the initial endowment of leisure time and Ec++l the non-negative orthant of the c + 1-dimensional Euclidean space EC+ . We assume:

Assumption A. For all i e N, U'(t, x, Q) is monotonically increasing, continuous and quasi-concave function of (t, x, Q).

The following argument can be directly applicable to the case of more than one public goods. But we discuss the economy with one public good for notational simplicity.

Each individual i owns a labour production function f'(h). That is, if he works for h-hours, he can provide a quantity f'(h) of service called "labour". For example, labour may be measured in terms of the unit of manpower/hour. We assume:

Assumption B. For all i e N, f'(h) is a continuous and concave function of h E [0, L] with f'(0) = 0 and f'(h)> 0 for some h > 0.

We assume that all the individuals are endowed with no consumption goods. The consumption goods are produced by firms. There are m firms in the economy. Each firm j(j = 1, . . . , m) has a production set Z' c E x Ec. We also denote z' e Z' by (z4, zc) = (zi, z.. , Z). We make the following assumptions:

Assumption C. Zj c E- x Ec for allj= 1, . . ., m.

Assumption D. Z' is a closed convex cone for all i = 1, . . . , m.

Assumption E. There are =(z , z ) e Z zm = (zrm, zM) e Zm such that -=1 z >0,butthere areno z' , . .' , zm EZm such that ETm 1 z> 0.

Assumption F. There are ei>0 and 82>0 for any zEX'1mZj and k1, k2 (1 k1, k2 c) with zkl > 0 such that z -le 1e + k2 2 E lZ.

Here E_ is the set of all non-positive real numbers and Zc >0 means Zk >0 for all k = 1, .. ., c and z?_ 0 means z-0 but z ? 0, and ek is the unit vector in E x Ec in the k-direction (all entries zero except the k-th, 1).

Assumption C means that labour can not be produced by the firms. We would not need to explain Assumption D, but this is a crucial assumption in this paper. Assumption E means that the economy is productive as a whole in the capital-theory sense, and the impossibility of free production. Assumption F means that when the economy produces a positive amount of a consumption good, it can decrease an amount of input or can produce a positive amount of any other good by decreasing the production level of the consumption good.

Each firm j has a profit-share vector d' = (d, ... , d'l) with d '-0 for all i E N and Ei.ENdl = 1.

The government produces and supplies the public good using labour and consumption goods as inputs. The government has a production set Z? c E x EC x E. We denote, by zo = (Zo, z%, z?+1), a typical element in Z?. z0 and Zo mean amount of labour and the consumption goods used as inputs, respectively, and z?+1 an amount of the public good produced. We assume:

Assumption G. Z? c E_ x E_ x E+ and 0 e Z?.



KANEKO OPTIMAL INCOME TAX 635

Assumption H. Z0 is a closed convex set.

Assumption I. There is no E Z? such that zo 0.

Here E'I is the non-positive orthant of EC. Assumption G means that labour cannot be produced and the public good cannot

be used as inputs. Further it excludes the possibility that the government produces consumption goods. We do not need to explain Assumption H. Assumption I implies the impossibility of free production.

We are now in a position to define our optimal taxation problem. A tax function T is a function from the interval [0, M] to E which satisfies

T(y) is a continuous and non-decreasing function of y with

T(y) y for all y E [0, M].

Here ,M is a positive real number with M ?-maxi maxhf'(h). We assume that the government imposes taxes on the individuals' incomes measured in terms of labour. That is, a tax function T implies that when an individual i works for h-hours and earns income f'(h), he must pay an income tax T(f'(h)) to the government. Hence we assumed that T(y) ? y for all y e [0, M]. We denote, by X, the set of all tax functions. For notational simplicity, we may denote, by Tf(h) or Tf, the composite function Tof of T and f in the following.

Kaneko (1979) assumed convexity and (1) on tax functions. The significance of convexity (progressiveness) in optimal taxation theory is clear and we do not need discussion of it here. Mathematically, however, convexity is a kind of uniformness condition and a strong one. Disposing of convexity permits a great deal of variety on tax functions. But we will show in the next section that we can restrict our consideration to a certain narrower class than 9, which plays the same role as convexity in the existence proof of Kaneko (1979) in the domain of progressive tax functions.

We say that ((t', x', Q), . . . ,(tn, xn, Q), zo, z,. . . , zm) is an allocation iff

(t',x', Q)e Y for all ieN, ziEZj for allj=0, 1,.. .,m (2)

and zO,i =

Y.j=j f l(L - t) +E =0 zo == O13

j=1 jx -Emoi= =0?. (4)

It is easily verified that the set of all allocations is a compact set under our assumptions. The economy works as follows. The government plans to employ a tax function

T e and a production schedule z0 E Z0. We call (T, z0) a tax schedule and denote, by 9, the set of all tax schedules. The government announces the tax schedule employed to the individuals. Under the tax schedule, the individuals and the firms behave as price takers, and the prices of consumption goods and labour are determined by the market mechanism (but not the government). Thus we get the following definition.

Definition 1. y=(p, (,xQ). tn Xn,), z,z2,.. .,zm) is said to be a competitive equilibrium under a tax schedule r = (T, z0) iff

P= (Po,Pc) = (po,pj, . .. ,pc)EE+c (5)

with P> 0 and ((tl, x Q), (tn, xn, Q), z1,... , zm) is an allocation,

Pcx =-po(l-T)[f1(L-t1) +> i-ldipz'/po] for all i EN, (6) for all i E N, U1(t', xi, Q) ' U1(t, x, Q) for all (t, x) such that

pCx _po(l - T)[f (L - t) +?zn1 d lpz'/pol, (7)




for all-,.. .m,

pz'i_pz for all z eZ. (8) We call T = (T, z0) a feasible tax schedule iff there exists a competitive equilibrium under r which satisfies

-p(zo, z c) -Po g-1 T[f1(L - t ) + }.m1 d i'pz '/po]. (9) We denote, by 5f, the set of all feasible tax schedules and by C(ir) the set of all competitive equilibria under r with (9).

Condition (5) guarantees a positive price of labour and the coincidence of total demands and total supplies of consumption goods and labour. Condition (6) is the individuals' budget constraint and (7) is the individuals' utility maximization under the budget constraint. Condition (8) is the firms' profit maximization. Condition (9) means that the government's expenditure for the production of the public good does not exceed revenue.

Let (TO, 0) be the trivial tax schedule, i.e. T0(y) = 0 for all y E [0, M] and zo = 0. That is, the government does nothing. If there exists a competitive equilibrium under (TO, 0), then (2.9) is automatically satisfied by it. The definition of competitive equilibrium under the trivial tax schedule (TO, 0) is reduced to the standard one, so the existence of a competitive equilibrium in this case can be proved, slightly modifying the standard existence proof, e.g. Nikaido (1970). Thus we get the following theorem.

Theorem 1. There exists a feasible tax schedule T, i.e. "f # 0.

Under our assumptions, we can easily prove:

Lemma 1. p >0 for any competitive equilibrium (p, (t', x', Q),...(tn,x,Q), zl z') eCOO) and any -r Yf.

Since Zj is a convex cone for all j= 1,... ,m, firm i's profit pzi is always zero in equilibrium for all j = 1, ... m. In the following, we will use this fact and Lemma 1 without any remark.

The government employs a social welfare function such that

1'=1 G'[U'(-)], (10)

where G' is a monotonically increasing and continuous function on [U'(L, 0, 0), +X). For example, if G [ Ui (.)] = log [ Ui (*) - U1(0, 0, 0)] for all i E N, then this welfare function is the Nash social welfare function of Kaneko and Nakamura (1979). When the government employs a feasible tax schedule r = (T, zo) and so a competitive equilibrium y = (p, (t', x1, Q), ... , (tn. xn, Q), z, ... , zm) in C(Tr) results in the economy, the value of the social welfare function is represented as

WQr, -y) = st.= G1[U'(t', xi, Q)].

The government tries to maximize the social welfare function over the feasible tax schedules.

Definition 2. "r e 9f is said to be an optimal tax schedule iff for some 5 E C(e )

W(1r Y) = SUp"SY, W(,r, ).(11

Ye C(r)

The purpose of this paper is to prove the existence of an optimal tax schedule. Although the meaning of an optimal tax schedule is clear, the definition deserves a special critical comment and another note:




Remark 1. Let us remember the role of the government in this economy. The government has tax functions and production schedules as the controllable variables. When the government employs and announces a tax schedule (T, zo), the environment of the economy is determined. Under the environment the participants, i.e. the individuals and firms adopt plans independently to maximize their utilities and profits. Then a competitive equilibrium is determined by the market mechanism but not the government. If there are multiple-equilibria, i.e. C(T, zo) consists of more than one equilibrium, then we cannot give a unique answer to the question which equilibrium occurs. We can answer only that C(T, z?) corresponds to a tax schedule (T, zQ). It is natural to assume that the government cannot select one equilibrium from C(T, zo) but that it can manipulate only T and zo. This situation is like a "game against nature". See Luce and Raiffa (1957, Chapter 13). In this case, the government plays against the market mechanism. There are many solution concepts for a "game against nature" (see Luce and Raiffa (1957, Chapter 13)). Our definition "max max" (an optimal tax schedule satisfies W(, 9) = max,yfEmaxyEC(,) W(ir, y)) is the most optimistic criterion. The opposite criterion, the most pessimistic one is "max min", i.e. we define an optimal tax schedule T by

W(', 9) = maxEsyf min,Ec(,) W(r, y). (12)

This definition means that an optimal tax schedule r maximizes the guaranteed level of social welfare minyeC(,) W(r, y). Therefore this definition would be safer than the former. But this author has not succeeded in proving the existence of an optimal tax schedule using the "max min" criterion. If the correspondence of equilibria to r was always lower-semi-continuous, the existence would be proved without much difficulty, but unfortunately, lower-semi-continuity does not necessarily hold (see Hildenbrand (1974, Section 2.3)).

Remark 2. We have chosen labour as numeraire, and so we have assumed that the government imposes taxes upon the individuals' incomes measured in terms of labour. But it is also possible to choose any other good as numeraire. Or, when the government does not measure the individuals' incomes by any particular good, the argument of this paper remains true if we define the unit of incomes appropriately. For example, we always normalize the price vector p such that c = 1 and assume that incomes are measured in terms of the value defined by p, i.e. when individual i works for h-hours, his gross income and disposable income are pof'(L - h) and pof'(L - h) - T[pof'(L - h)]. But since the individuals can get incomes only by selling their labour in our economy, there is no substantial difference among these alternative settings.

3. THE RESTRICTION THEOREM AND THE EXISTENCE THEOREM

Before we state and show the existence of an optimal tax schedule, we need to state the theorem which makes it possible to restrict our consideration within a narrower class, Jo, than J. Here Jo designates the set of all tax functions T e J having the following property: T(y1) - T(y2) c1 for all Yi, Y2 E [0, M] with yl # Y2 (13)

Y1-Y2

A tax function T = Jo has the property that its marginal tax rate is not greater than 100% everywhere.

Let (T1, z0) and (T2, z0) be feasible tax schedules. Then (T1, z0) is said to be equivalent to (T2, zo) iff C(T1, zo) = C(T2, zo). The equivalence of (T1, zo) and (T2, zo) means that even if the government employs either, the same equilibria can be expected to result. Of course, this relation is an "equivalence relation" on Yf.




Theorem 2. (The Restriction Theorem). For any feasible tax schedule (T, z?), there exists a tax function To E9 0 such that (T, z 0) is equivalent to (To, z 0).

Theorem 2 says that for every feasible tax schedule, we can curve the tax function so that the new tax function has the same equilibria but has marginal tax rates not greater than 100%. Hence this theorem ensures that we can restrict our consideration to Jo instead of J.

This theorem can be explained intuitively as follows. Suppose T is not in Jo. Then there are points y, and Y2 in [0, M] such that (T(y1) - T(y2))/(y1 - Y2)> 1 and yl > Y2. This is equivalent to y, - T(y1) < Y2 - T(y2). Any individual i does not choose labour time h1 with yi = f1(h1), because he can get more net income Y2- T(y2) than y1- T(y1), working for the less time h2 with Y2 = f1(h2). So, we can flatten out T without any effect on individuals' behaviour so that a new tax function belongs to Jo. A precise proof will be given in the Appendix.

We are now in a position to state the main result of this paper.

Theorem 3. (The Existence Theorem). There exists an optimal tax schedule.

Let 9o = {(T, z0) E 9f : T e Jo}. Then it follows from Theorem 2 that

supC(T) W(r, y) = supy,EC(T) W(r, y). (14) TEYO ~~~~T EEff

Since U' and G' are continuous functions for all i E N and the set of all allocations is a compact set, we have sup W(r, y) < +oo. Hence there is a sequence {(rS, ys)} = {((TS, zos), (ps (tls, xs QS) s, (tns, xns, QS) zls zms))} such that ys e C(-rs) and rss e 90 for all

s _ 1 and lims,c W(r5, W ) = sup W(r, y). The outline of the following proof is that we can choose a convergence subsequence {(rs', es )} from {(rs, yS)} and then the limit point (r*, Y*) really attains (14).

Lemma 2. infs Ts(0)>-cxo.

Proof. Suppose infs Ts(0) = -oo. Since each Ts belongs to J9o, Ts(y) _ Ts(0) + y for all y e [0, M] and all s. Hence there is an so such that Ts(y) Ts(0) +M <0 for all y E [0, M] and all s ?s0. This means that for large s, the government's revenue is negative. This contradicts the supposition that ys E C(rs) for all s.

Let K = infs Ts(0). We define C[0, M] by

C[0, M] = {t: t is a continuous non-decreasing function with (t(y1) - t(y2))/(y1 - Y2) _ 1 for all yi, Y2(Y1 # Y2) and K _ t(O) 0 O}. (15)

Lemma 3. C[0, M] is a compact set with respect to the topology of uniform convergence.

Proof. By Ascoli's theorem (Simmons (1963, Section 25, Theorem C)), it is sufficient to show that C[0, M] is closed, bounded and equicontinuous.

It is easily verified that C[0, M] is closed and bounded. By (15) it holds that for all t EC[0, M],

IYl - Y2_ 8 implies jt(y1) - t(y2)1 86

This means that C[0, M] is equicontinuous. ||




We can normalize {pS} without loss of generality such that each ps belongs to P-{p eE +c: Ek=oPk = 1}. Let A be the set of all allocations. Then the sequence {(rs, es)} is in C[0, M] x P x A. Clearly C[0, M] x P x A is a compact set, so there is a subse ence {(rs, Sy`)} of {(S, yS)I which converges to (r*, y*) = ((T* zo), (p* (t , xl , Q*) . . , (t" , x" , Q*), z',... zm*)) in the sense of the product topology.

We assume for notational simplicity that {(rs, ys)} itself converges to (r*, y*). Then since U' and G1 are continuous functions for all i e N, we have

SUp W(Y ) = lims,oo ZiEN Gi[U (ts xs QS)] = ZiEN G1[UJ(ti*, x*, Q*)]

Further 'it holds that T* e C[O, M] c Jo. Hence it is sufficient to show that y* is a competitive equilibrium under r* with (9).

Lemma 4. p* > 0.

Proof. Let p =0. By Assumption E, there are z' eZ',. . . , zm EZm such that Z> > 0. Then for some so, ps O Em zj >0 for all s _so. That is, there i> a j such

that pSzI > 0. Then psz)s _ psz j > 0 by (8).. This is impossible.

Suppose:

fi(L-ti*)-T*fi(L-ti )=O for all ieN. (16)

Then it is easily verified that if p* > 0, then lims,w xis = 0 for all i E N because of Lemma 4 and (16). Let pC =0. Then since ps =1 zcs converges to 0 by the compactness of the set of allocations, pS oF.m 1 = po oE z +P-c Y> z c =0 for all s implies lims,w EX> z's = 0, because of Lemma 4. This implies X7L1 zic = 0 for all j = 1,... , m by Assumption E, and so x' =0 for all i e N. Finally let us consider the case where

=0 and Pk2>O for some k1, k2 (1-kl, k2 _ c). In this case, if E =z > 0, then there are > 0 and 82 > 0 by Assumption F such that EL zi _ Elek +e2ek2 in YmJZ. Hence pS(fil zi -sle 1+ 2ek2) -*P 2s2 >>0, which is a contradiction to the fact that maxzipsz = 0 for allj= 1, . . ., m and all s. Thus we have shown that El7 z= 0 for all j. Hence it follows that x'* =0 for all i eN. Thus, it is always true that xi* =0 for all i e N.

Since U1 is continuous for all i E N, it follows from the above argument that lim,,. Ui(tis, xis, QS) = Ui(ti*, 0, Q*) for all i E N. If t'* <L, then for a sufficiently small E > 0, there is an so such that U'(L, 0, 0*)-s > U1(t's, x's, QS) for all s -s0. Further there is an s1 such that U1(L, 0, QS)> U1(L, 0, Q*)-E for all s -'s. These imply U1(L, 0, QS)> Ul(tls, x's, QS) for all s -max (so, sl), which is a contradiction because (L, 0) always satisfies his budget. Hence we have t1 = L for all i E N. This also implies Q* = 0. Hence we have

SUp W(r, Y) = EieN G'[U'(L, 0, 0)].

Let rT = (TO, 0) be the trivial tax schedule and y0 = (pO, (t10, x10, 0), . . ., (tn?, x no, 0), z,... ,zM) e C(r0). The existence of such a i0 has been stated in the preceding paragraph of Theorem 1. It is clear that UL(tio, xi, 0) _ U1(L, 0, 0) for all i E N. Hence we have

W(Cr, y0) = sup W(r, y). (17)

In the following we consider the case where

fi(L - ti*) - T*fi(L - ti*) >O for some i eN. (18)

Lemma 5. p > 0.

Proof. Let p* =0 for some k (1'k5c). Then it is easily verified that xks-+O (s-*oc) for all i with f'(L-t'*)-Tfl(L-ti )>0. This contradicts the impossibility of free production. 11




It is easily verified that y* satisfies (5), (6) and (8) under i*. We show that 'y* satisfies (7). Suppose that there is a (to, x0) such that Ui(to, x0, Q*) > Ui(ti*, xi, Q*) and pcx0 c p * (1 - T*)f i(L - to). Now let us show (1 - T*)ff(L - to) > 0. On the contrary, let (1- T*)fi(L - t?) =0. Then x? =0 because p*>O. Hence U'(L, 0, Q*)-> U'(t, 0, Q*) > U(ti*, x, Q*) Since {(tWs, x s, QS)} converges to (t'*, x'l, Q*), there is an so such that U1(L, 0, Qs) > Ui(tis, xis, QS) for all s 2 so. But (L, 0) satisfies i's budget constraint under (Ts, zs). This is a contradiction. Hence (1- T*)fi(L - to)>0. Then there is another (t, I) in a neighbourhood of (to, xo) by the continuity of Ui and Lemma 5 such that U'(t, x, Q*)> Ui(ti*, x, Q*) and p x<p (1-T*)fi(L -). Since {TS} converges uniformly to T* and {pS} converges to p*, there is an s, such that pscX< p0(1 - Ts)f'(L-T) for all s 's1. Since (t's, x's, QS) converges to (t' , x' , Q*), there is an S2 for some E >0 such that U'(tJ, x, 0Q) - > U' (tls, x's, QS) for all s_iS2. Since {QS} converges to Q*, there is an S3 such that U'(t, x, QS) _ U'(T, X, Q*) - e for all S -53.

Hence it holds that U1(t, x, QS) > Ui(tis, x's, QS) and pS S <p(1- TS)fi(L-T) for all s ' max (S1, S2, S3). This contradicts the fact that yS is a competitive equilibrium under rs for all s ?1. Thus we have proved (7).

Finally we show that y* satisfies (9) under r*. Since {ly} and {z0'} converges to -y* and z?0, and since {Ts} converges uniformly to T*, {Tsfi(L - t's)} converges to T*fl(L - ti*) for all i E N. Hence the condition

pS(Zos z?) c p- o ?i= Tsfi(L-tis) for all s ?

1

implies -p*(zo*, z?c)-Po* 1 T*fi(L-t*).

APPENDIX

Proof of Theorem 2. Let (T, z0) be any feasible tax schedule. Let c(y) be the net income function corresponding to T, i.e. c(y) = y - T(y) for all y e [0, M]. Using this function c(y), we define co(y) and To(y) by

co(y) = maxo=yj-_ c(y1) for all y E [0, M], (A.1) To(y) = y - co(y) for all y E [0, M]. (A.2)

Lemma 6. (i) co is a non-decreasing continuous function with property (13). (ii) To(y) is a nondecreasing continuous function with property (13) and To(y) ? y for all ye[0,M], i.e. Toe-o.

Proof. It is clear that co(y) is a non-decreasing continuous function. Since T(y) is non-decreasing, we have, for YI > Y2,

C(y1)-C(Y2) = y,- T(y1)-(y2- T(y2))

= Yl- Y2 - (T(yi) - T(y2)) Y - Y2, which is (13). It follows from this and (A.1) that co(y) also satisfies (13). It is clear that To(y) is a continuous function. Since co(y) is nondecreasing and satisfies (13), we have, for Y1 > Y2,

TD(y1) - To(y2) = Y1 - cO(y1) - (Y2 - CO(Y))

= Y- Y2 - (cO(y) - cO(y2)) _ - Y2, and

To(y1) - To(y2) = Yl - Y2 - (co(yl) - Co(Y2))A 0O

Since To(O) = T(O), we have, by the above inequality, To(y) c y for all y E [0, MI. 1I




We define the set I(T, To) for T and To by

I(T, To) = {y E [0, M]: T(y) = To(y)}. (A.3)

Lemma 7. (i). If y = (p, (tl,x,Q), ..., (tn, xn, Q), z,.. .,zn) is a competitive equilibrium under (T, z?), then f i(L - t1) E I (T, To) for all i E N. (ii). If y is a competitive equilibrium under (To, z?), then f'(L - t') E I (T, To) for all i E N.

Proof. We prove only (i), but can prove (ii) analogously. Suppose f1(L-t1)o I (T, To). Then T(y) > To(y), i.e. c (y) < co(y), where y = f'(L - t'). Hence there is a yi < y such that co(y) = c(y1). If he works for L - t1 hours such that f'(L - t1) = y1 and t1 > t', then the disposable income c(y1) is greater than c(y). This means that the individual can increase his utility. This is a contradiction to the supposition that y is a competitive equilibrium under (T, zo).

Lemma 8. yE EC(T, z?) if and only if yE C(To, z).

Proof. We prove that if y E C(T, zo), then y E C(TM, zo). It is sufficient to show that y satisfies (6), (7) and (9) under (TO, z0). It follows from Lemma 7(i) that y satisfies (6) and (9). Let (7) be not satisfied. Then there is a (t, x) such that f'(L - t) , I(T, TO), U'(t, x, Q)> U'(t', x', Q) and pcx < po(1 - T0)f(L-t). Since f'(L-t) = y0I(T, To), T(y) > To(y), i.e. c(y) < co(y), which implies that there is a y1 < y such that co(y) = c(y1). Let t1 be the real number such that y 1 = f '(L - t1) and t1 > t. Then f '(L - t1) - Tf '(L - t1) =

c(yj) = co(y) = fi(L - t) - T0f1(L - t). In sum, we have

fi(L-t1) E I(T, To), U1(t1, x, Q) > U1(t, x, Q) > U1(t', xi, Q)

and

PcX 'po(l - To)f1(L - t) = po(l - T)fi(L - t1).

This is a contradiction to the supposition y E C(T, z?). We can analogously prove the converse part of this lemma using Lemma 7(ii).

First version received June 1980; final version accepted April 1981 (Eds.).

The author wishes to express his thanks to Professors Yozo Ito and Toru Maruyama for valuable discussions, and to the referees of this journal for valuable comments on an earlier draft. The research in this paper was partially sponsored by the Office of Naval Research, Contract Number N00014-77-C-0518.

NOTES 1. A result of Seade (1977) appears, to a certain extent, to contradict the progressiveness of an optimal tax

schedule. Ordover and Phelps (1979) also achieved a similar conclusion in a different context. But Kaneko (1979) proved that the optimal marginal and average tax rates tend to 100% as income level becomes arbitrarily high under certain assumptions. We can interpret this result as progressiveness in a different sense broader than convexity. Anyway, the author does not think that the optimal taxation problem has been studied fully nor that we can generally answer the problem of progressiveness.

REFERENCES HILDENBRAND, W (1974) Core and Equilibria of a Large Economy (Princeton: Princeton University Press). LUCE, R. D. and RAIFFA, H. (1957) Games and Decisions (New York: John Wiley & Sons). KANEKO, M. (1979) The Optimal Progressive Income Tax-the Existence and the Limit Tax Rates (D.P.

No. 46, Institute of Socio-Economic Planning, University of Tsukuba). KANEKO, M. and NAKAMURA, K. (1979), "The Nash Social Welfare Function", Econometrica, 47,423-435. MIRRLEES, J. A. (1971), "An Exploration in the Theory of Optimum Income Taxation", Review of Economic

Studies, 38, 175-208.




NIKAIDO, H. (1970) Introduction to Sets and Mapping in Modern Economics (Amsterdam: North-Holland). ORDOVER, J. A. and PHELPS, E. S. (1979), "The Concept of Optimal Taxation in the Overlapping-

Generaions Model of Capital and Wealth", Journal of Public Economics, 12, 1-26. SIMMONS, G. F. (1963) Introduction to Topology and Modern Analysis (Tokyo: McGraw-Hill). SEADE, J. (1977), "On the Shape of Optimal Tax Schedule", Journal of Public Economics, 7, 203-235.



on the existence of an optimal income tax schedule

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