THE OPTIMAL PROGRESSIVE INCOME TAX

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M. Kaneko 1 The optimalprogressive income tax. 195 taxation and states the existence of an optimal tax function under relatively weak assumptions. The proof of ...
Mathematical Social Sciences 3 (1982) 193-222 North-Holland Publishing Company

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THE OPTIMAL PROGRESSIVE INCOME TAX* THE EXISTENCE AND THE LIMIT TAX RATES Mamoru KANEKO hstitute of Socio-Econ-ric

Planning, University of Tsukuha, Sakuramura,

Iharaki-ken 305,

Japan

Communicated by K.H. Kim Received 30 October 1981 The purpose of this paper is to consider the problem of optimal income taxation in the domain of a progressive (convex) income tax function. This paper proves the existence of an optimal tax function and proves that the optimal marginal and average tax rates tend asymptotically ro 100 percent as the income level becomes arbitrarily high. Keywords: Progressive income tax; Nash social welfare function;

optima1 income tax.

1. Introduction 1.1

It is commonly recognized that there exists a conflict between equity and efficienty in income taxation. Progressive income taxation is a device to enhance the degree of equity. A highly progressive income taxation, however, obstructs labor incentives of individuals with high productivities or earning abilities, and so, it decreases the degree of efficiency. Therefore optimal income taxation should be determined to harmonize these contrary norms. To discuss this problem Mirrlees provided a model of optimal income taxation in his pioneering work (1971). Since Mirrlees (1971), many authors have been considering this problem in variations of Mirrlees’ model. There are two directions of studies from a technical point of view. One is the approach initiated by Mirrless himself, where income tax ftinctions of any forms are allowed. In this approach the method of the calculus of variations or Pontryagin’s Maximum Principle is used. Another one is the approach initiated by Sheshinski (1972), where only linear tax functions are allowed. The first approach is superior to the second to consider the progressiveness of an optimal income tax function, *An earlier version of this paper was presented in the 1979 Meeting of the Japan Association of Economics and Econometrics. The research in this paper was partially sponsored by the Office of Naval Research Contract Number NOOO14-77-0518.

0165-4896/82/0000-0000/$02.75

0 1982 North-Holland

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though there are many problems arising in the optimal taxation which can be discussed in the domain of linear tax functions without losing the essence. The first approach, however, has a serious defect as Mirrlees himself pointed out. It is the lack of mathematical rigorousness. That is, although several necessary conditions for an optimal tax function are discussed under the assumption of the existence of it, the existence has never been verified in the works of this approach. The differentiability of several variables derived from an optimal tax function is also doubtful.’ Furthermore, if we allow nonconvex tax functions, there exists multiple equilibria for each tax function with different welfare levels in general. Therefore the objective function of government becomes a set-valued function. This means that the usual maximal principle or the variational method is not applicable to the problem in this case (see Kaneko, 1981b). The second approach does not have such a weak point, though we can not fully consider the progressiveness of an optimal income tax function because of the assumption of linearity. 1.2 This paper considers the problem of optimal income taxation in the domain of progressive (convex) tax functions in an economy with a continuum of individuals.2 Of course, the progressive tax functions include the linear and proportional tax functions. The model we provide is a variation of Mirrlees’ model, while we take a public good - government’s service - into account explicitly. One of the main results of this paper is to prove the existence of an optimal income tax function in the model. The merit of our approach is tlcPbe able not only to discuss rigorously the existence of an optimal income tax function but also to consider its shape to a certain extent, e.g., progressiveness in a different sense broader than convexity. Since convexity of an optimal tax function is imposed as an assumption, we can not discuss it in our model. But convexity is no more than one concept of progressiveness in a weak and local sense. We prove that the optimal marginal and average tax rates tend asymptotically to 100 percent as income level becomes arbitrarily high. This result can be interpreted as progressiveness in a stronger sense than convexity. Another important point of this paper is to employ the Nash social welfare function as the welfare criterion, i.e., government% objective function in the model. The Nash social welfare function has been recently discussed by many authors, e.g., Kaneko and Nakamura (1979a 1979b), Kaneko (1980, 198la), Brock (1979), Wagner (1980), Kim and Roush 1.,1981)and others. One purpose of this paper is an explicit application of this concept to the income taxation problem. This paper is written as follows. Section F presents a model of optimal income j Mirrlees (1971, 4) pointed this out. 2Kaneko (1981b) established the existence tax schedule w&h a finite num!ber of individuals but the functions. But the problem tax rates for people can not be treated appropriately in a model with a finite number of individuals. For purpose, the model with a continuum

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taxation and states the existence of an optimal tax function under relatively weak assumptions. The proof of it is given in Section 4. Section 3 provides the limit theorem that the optimal marginal and average tax rates tend asymptotically to 100 percent as income level becomes arbitrarily high. This theorem is proved in Section 5 under stronger assumptions than those used in the proof of the existence theorem.

2. Model and problem 2. I (N, .#, p) is a measure space of all individuals, where N is the set of all individuals, .ilga o-algebra of subsets of N and p a measure on 2 with 0 0 is the initial endowment of leisure time and Ef the nonnegative orthant of the 2-dimensional Euclidean space E*. We assume the following. Assumption A. For all i e N, Ui(a, b, Q) is a strictly increasing and strict& quasiconcave function on P- ((a, b, Q) E Y: a ~0 and b >O), with Ui(a, 0, Q) = Ui(O,0,O) for ali (a, 0, Q). Assumption B. For all i E N, Ui(a, b, Q) is a continuous function on Y. Assumption C. For each (a, b, Q) E Y, Ui(a, 6, Q) is a measurable function of i. Assumption D. For each i EN, U ‘(a,6, Q) is bounded, Ui(a, b, Q) s A@for all (a, b, Q) E X3

i.e., for

some M i

It is not necessary to explain the strict monotonicity and the strict quasi-concavity in Assumption A. The assumption that Ui(a, 0, Q) = Ui(O,O,0) says that if individual i’s consumption level is zero, then his utility level is equal to that of the worst state (0, 0,O).This would not be a strange condition. Furthermore, it may be natural to assume that Ui(O,b, Q) = Ui(O,0,O) for all (0, 6, Q), but since we do not need this assumption, we do not assume it. Assumption B is a technical condition. Assumption C is also a technical condition, which is familiar in the theory of market with a continuum of traders. Assumption D is the boundedness of utility functions. The boundedness is justified by St. Petersburg paradox and is thought of as a IThe boundedness from below follows Assumption A.

M. Kaneko / ?%eoptimal progressive income tax

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natural assumption (see Owen (1968, Ch. VI) or Aumann (1977)). Of course, we assume that the utility functions U @, Q) E are

Exactly speaking, when a(i) and j?(i) are measurable functions of i with cr(i) > 0 for all i E N, we can employ i (i EN).

V ‘(a,6, Q) = ar(i)U ‘(a,b, Q) + /3(i)

Vi E N, (a, b, Q) E Y,

(1)

as the same utility functions. Hence we can obtain utility functions Vi (i EN) satisfying (2) by certain positive linear transformations:

Assumption El,

For any e > 0, there is a 6 > 0 such that U ‘(e, e, 0) - Gi(O,0,O)> 6

for all i EN. This assumption is a kind of uniformness of utility functions. This assumption is not necessarily preserved for arbitarty positive linear transformations. In the following we allow only the following positive linear transformations. For some e>O and IM>O, E~~(i)dkf

V~EN.

(3)

That is, when u(i) satisfies (3), the new Vi obtained from Ui which satisfies (2) and Assumption E.4 The following lemma shall be necessary to define the Nash social welfare function. The proof is standard and so we omit it. Lemma 2.1. Let (a(i), b(i), Q(i)) be a measurablefunction of i such that(a(i), b(i), Q(i)) E Yfor all i EN. Then U “(a(i),b(i), Q(i)) is a measurablefunction of i. 2.2 Each individual i EN owns a kborproduction functionf ‘(x). That is, if he works for time X, he can provide a quantity of laborf ‘(x). For simplicity, we assume that f’(x) coincides with the quantity of the consumption good produced by his labor f j(x), being independent of the other individuals’ labors? We call f i the null labor production function iff f’(x) = 0 for all x E [0, L). We assume that if f i is nonnull, then f i satisfies the following. 4Assumptions A, B and C are oreserved for arbitrary positive linear transformations. . Vor example, !abor can be measured in terms of the unit of man-power/hour.

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F. There is an L’ (0 < L’c L) such that f’(x)

is monotonicah’y increasing on [O,Li] and is nonincreasing on [Li, L] with f ‘(0) = 0. Assumption C.

f’(x) is a continuous and concave function.

Furthermore, we assume the following. Assumption H.

For each x E [O,L], f’(x) is a measurable function of i.

Assumption I. j,,, f

‘(L’) dp < += and for some co > 0, co I Li s L - co for all i E N, where Li is attached appropriately to every i whose f i is null?

Assumption A means that there is an interior point Li at which i’s labor productivity is saturated, i.e., his marginal productivity is zero. Assumption G is a standard condition. Assumption H is just a technical condition. Assumption I is the integrability of f’(t’) and a kind of uniformness of Li. The government produces the public good using the consumption good as input. We assume that the public godd is measured in terms of the consumption good needed to produce it. In other words, the cost function of the public good is C(Q) = Q. The government’s expenditure coincides with Q when it plans TVsclpply Q-amount of the public good. The government’s revenue is determined by a tax function and a level of the public good.’ 2.3 Now we are in a position to discuss the problem of optimal taxation. A tax function T is a real-valued function on the set of nonnegative real numbers E, which satisfies T(y) is a monotonically nondecreasing and convex function such that T(y) s y for all y E E, .8s9

(4)

We denote, by .$ the set of all tax functions. (See Fig. 1.)

ah is assured by Assumptions A, G and H that Liandfi(x(i)) are measurable functions of i, where x(i) is a measurable function with x(i) F (0, L] for all i E N. ’ Im and Kaneko (1981) showed that when cost functions of public goods are linearized by measuring the public goods in terms of the costs themselves, ratio equilibrium is invariant for such a trxsformation but not Lindahl equilibrium. The equilibrium concept of this paper has the same property, i.e., is invariant for the linearization of the cost function of the public good. Hence our assumption C(Q) = Q does not lose any generality. sThis condition implies that T(y) is a continuous function. 9Monotonicity is not essential because we proceed the following discussion replacing it by the continuity.

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Fig. 1.

A tax function T(y) means that when an individual i works for time x and earns income y = f ‘(x), he must pay income tax T(y) = T(f j(x)) to the government. Hence T must satisfy y ,( T(y) for all y E E,. Note that negative taxes, i.e., subsidies are allowed. We take income redistribution directly into account besides the supply of the public good. Convexity of tax functions means just the progressiveness of income taxation in a local and weak sense. This paper allows only progressive (convex) income tax functions. Of course, Y includes the proportional and linear tax functions. Here we should give a comment on this formulation. Seade (1977), Ordover and Phelps (1979) and others showed that the optimal marginal tax rate is zero at the maximal income level, which implies that the optimal tax function is not convex. They assumed, however, that there exists the maximal income group, and they derived the above result from this assumption. In our model we do not assume the existence of the maximal income group but rather we do the nonexistence of it. Our assumption is much better to corlsider the optimal tax rate for ‘rich’ people. In our formulation, rich people are described as ones with arbitrarily high incomes. If the existence of the maximal income group is assumed, it represents only ‘relatively’ rich people in the model. Furthermore we shall prove in Theorem 3.1 that the k>ptimalmarginal and average tax rates tend asymptotically to 100 percent as income level becomes arbitrarily high under certain assumptions. This result can be interpreted as progressiveness in a different sense broader than convexity. This would justify the assumption of progressiveness (convexity) against the criticism from Seade and others’ result.

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Let us suppose that the government employs a tax function T and decides to supply Q-amount of the public good. When each individual i e N decides to work for time x(i), his gross income is fix(i) =f$x(i)) and his disposable income is f k(i) - T(fl(x(i))). In the following, we denote T(f ‘(.x(i)))by Tf k(i) for notational simplicity. In this case the government’s revenue is j,,, Tf ‘x(i) dp. Since Q is the government’s expenditure the following must hold,

Qs[,Tf

(5)

‘x(i)&

With above explanation in mind we provide the following definitions. Every individual i maximizes his utility function under the assumption that T and Q are fixed. That is, individual i chooses x(i) E [0, L] such that Ui(L -x(i), f ‘x(i) - Tf k(i), Q) z r U ‘(L - x, f’(x) - Tf ‘(x), Q> Vx E [0, L].

(6)

In general, x(i) is not uniquely determined by (6). Let X(i) be the set of all solutions for (6). Then the following lemma holds. Lemma 2.2. X(i) is a singleton if and only lff ‘(x) - Tf ‘(x) > 0 for some x E X(i). The proof is straightforward. We will choose one function x(i) from X(i) by x(i) =

x if X(i) is singleton and X(i) = (x}, t 0 otherwise.

(7)

Since f’(x) - Tf ‘(x) = 0 and U ‘(L - x, f’(x) - Tf ‘(x), Q) = 0 for all x E X(i) if X(i) is not singleton, the above selection by (7) would be natural. We shall employ this selection as a representative of X(i) in the following. We call (x(i))iEN the labor (time) supply schedule for (T, Q) iff each x(i) satisfies (6) and (7). Lemma 2.3. The labor sup& schedule (x(i))i. and is a measurablefunction of i.

N

for (3; Q) is uniquely determined

The proof is standard and so we will omit it. We call (T, Q) a feasible fax schedule iff (5) holds for (x(i)),,,v satisfying (6) and (7). Further if Q = J, Tf ‘x(i) dp, then we call (T, Q) an equilibrium tax srctedule. We denote, by g, the set of all feasible tax schedules. We call T a feasible taxfunction iff for some Q E E+, (T, 0) is a feasible tax schedule. Since (N, 9J,pu)is a non-atomic measure space, the influence of each individual’s choice on the government’s revenue is negligible. Hence an equilibrium tax schedule

Ad. Kaneko / The optimal progressive income tax

(T, Q) determines a Nash equilibrium (x(i&N, -x(i), f ‘x(i) -

Ui

Tfk(i),

i.e., for all i E N

Tf k(i) dp

1

-x,f ‘(x)- Tf ‘(x),

u’

. J

1 >

N

Tfjx(

j) dp + T4.fi(xM(i))

N- {i}

vx E

>

[O, 1)

and Q = IN Tf ‘x(i) dp =

I Nti

Tfjx(j)G

+

TlfiWccW.

{i}

This is the reason why we call such a (z g rn equilibrium tax schedule. Let 7) be the function such that To(y) = 0 for all y E E, . Then (TO,0) is always a feasible dax schedule and an equilibrium tax schedule. (To,0) means that the government neither imposes any tax nor supplies the public good. We call (To,0) the trivial tax schedule. This is a feasible behavior of the government. Hence we have the following. Proposition 2.4. There exist a feasible tax schedule and an equilibrium tax schedule.

Let (xo(i))iEN be the labor supply schedule for (To,0). Then we assume the following. Assumption 3

This assumption means that individuals work for their consumptions if the government neither imposes any tax nor supplies the public good. The following proposition states that if T is a feasible tax function, it is always possible to achieve the equation of the government’s revenue and expenditure. Proposition 2.5. For every feasible tax function z there exists a Q E E, such that (r Q) is an equilibrium tax schedule.

Furthermore it is not difficult to verify the following proposition. Proposition 2.6. Let T be a feasible tax function. Then: (i) Jf T(0) < 0, then T(y) > 0 for some y E E,. (ii) If d-T F

then ?IyO)> 0.

1 yo=



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Note. Since T is convex and continuous with y ,> T(y) for all y E E,, T has the derivatives on the left and the right d-T/dy and d+T/dy with d-T

--00. We call T* = (T*, Q*) E 3 satisfying (10) an optimal tax schedule. The purpose of this paper is to investigate the optimal tax schedules. The first result of this paper is the existence ol”an optimal income tax schedule. The proof of Theorem 2.7 will be provided in Section 4. IOIn Kaneko and Nakamura (1979a), the necessity of this setting is discussed briefly (see Kaneko and Nakamura, 1979a, Section 4).

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Theorem 2.7 (Existence Theorem). Under Assumptions A-J, there exists an optimal tax schedule r* = (T*, Q-j3 which is an equilibrium tax schedule. 2.5 The concept of optimal tax schedule defined in the above can be interpreted as a noncooIperative equilibrium point of a game in extensive form in which the government and the individuals appear as players. The game is formulated as follows. First, the government decides and announces a tax schedule (T, Q) to the individuals. Then every individual independently decides his labor time x(i). The game tree is drawn in Fig. 2. The individuals’ payoffs are the utilities from (T, Q) and x(i)%. If the government’s revenue is smaller than it’s expenditure Q, then it suffers a punishment P, which is sufficiently large. The government’s payoff is the Nash social welfare function -6P,

(11)

where 6 = 1 if the revenue is smaller than Q and 6 = 0 otherwise. Now we have a game ;n extensive form. Let r* be an optimal tax schedule and let (x*(i))iEN be the labor supply schedule for T+. Then (r *, (x*(i))iEN) is a subgame perfect equilibrium point of the game. 1J 2.6 In this subsection, we provide a simple but important necessary condition for an

Fig. 2. JJFor definition of subgame perfect equilibrium,

see Selten (1975).

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or$mal tax schedule. For any T E .;‘;the minimal disposable income for T is given as -T(O) because of (4). Lek T(0) = 0. Then the individuals with the null labor production function can earn no disposable income. They must ‘live’ under no consumption. Is this tax function optimal? The answer is ‘no’ under the condition that there exist such individuals with a positive measure. Proposition 2.8. Under Assumptions A-J and p( { i E N: f i is null )) > 0, if (T *,Q *)I is an optimal tax schedule, then the minimal disposable income for T* is positive. Proof. Let T*(O) = 0. Then iffi is null, his consumption must be zero. Let (x*(i))i Eh be the labor supply schedule for (T*, Q*). By Assumption A, U’(L -x*(i), f ‘x*(i) T*f’x*(x), Q*) = Ui(O, 0,O) = 0 and all i such that f i is null. Since ,u({i EN: f’ is

null}) > 0,

s

log

U’(L -x*(i),

f ‘x*(i) - T*f ‘x*(i), Q*)dp = -oo.

N

This contradicts (10). This proposition depends crucially upon the structure of the Nash social welfare function. If we employ another social welfare function, e.g., the utilitarian or the maximin, then the above proposition could be true under ad hoc assumptions.

3. The limit marginal and average income tax rates 3.1

In the previous section we have shown the existence of an optimal income tax schedule. The next task is to investigate the shape of the optimal income tax schedule. In this section we consider the limit optimal marginal and average tax rates when income level becomes arbitrarily high. The result which we prove is the theorem that both the optimal marginal and average tax rates tend asymptotically to 100 percent. To prove this theorem we need new assumptions and replace some of the assumptions of the previous section by stronger orres. Initially we make the following assumptions. Assumption A’. For all i E N, U ‘(a,6, Q) is a strictly increasing, strictly concave and continuously differentiable function on p with Ui(a, 0, Q) = U '(0, 0,O) for aN (a,4 Qb t Assumption K. For all i E N, Ui(a, 6, Q) is weakly separable with respect to (a, 6) and Q, i.e., there are functions g i and hi such that U ‘(a, 6, Q) = h ‘(g ‘(a,b), Q> for all (a, b, Ql E Y-

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Assumption

L. For each Q 2 0, U’(L -- Li, b, Q) converges uniformly to Mp

supb Ui(L - Li, 6, Q) as b + 00, i.e., for any e > 0, there is a b0 such that Ui(L -L$b,Q)zMi-&foraNbzboandaNiEN. Assumption

M. For each QZ 0, 1 u; wb

u;

(L-Lib,,)

converges uniformly to 0 as b + 00, i.e., for any 6 > 0, there is a bOsuch that --1 u; b u;

ce

Vbrbj,

ieN.

(L-Lib,,,)

=auVaQ. In the above assumption U[ = NP/i#a, ui = a&lab and V;: Although Assumption A’ is stronger than Assumption A, it is not yet so strong. Assumption K means that the individual choice of leisure and consumption is not affected by the level of the public good. Since we already take income redistribution into account by tax functions, our public good is very pure, e.g., national defense, fire fighting, etc. Hence it would be natural to assume that the level of the public good does not affect the individual choice. Assumption L is a kind of uniform bound dness, which is stronger than Assumption D and (2). When the utility func?ions Ui can be transformed into an identical one, this assumption is, of course, true. Assumption M means that although the marginal rate of substitution of leisure and consumption may tend to infinity as b + 00, the order is smaller than that of b. If this is not true, it happens that the marginal rate of substitution at (t - Li, 6, Q) has the magnitude of the same order or a greater order than b, which would be implausible. The utility functions which are represented as g’(a,b)=a=+bfl

VbzsomebandO-;crJ
0, then ,u({i E N: n(i)Lk S}) > 0, andfor any E > 0 there is a /9 > 0 such that

lim 030 p({i E N: n(i)L’> a+ &})/p({i E N: n(i)Li > a}) = #?.I2 I*u denotes the usual Lebesgue measure on E+ .

E f Kancko

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L

Fig. 3.

The function n(i) assigns to each individual i his marginal productivity of labor. Assumption F’ means that f i (i EN) is approximated by a piecewise linear function (Fig. 3). This implies Assumptions G and H. Assumption N means that the distribution of the abilities n(i) overspreads everywhere of E,, and that the density of n(i) converges asymptotically to 0 but the speed of the convergence ic not too rapid. When Li is a constant, i.e., Li = Lo, this limit property is satisfied by the Pareto distributions but not by the normal distributions. Therefore it is fair to say that this assumption restricts the applicability of our theory to a certain extent. We are now in a position to state the main result of this section. The proof of the following theorem is provided in Section 5. Theorem 3.1 (The Limit Tax Rates Theorem). Let (T*, Q*) be an optimal income tax schedule. Then it holds under Assumptions A’-N that d-T*(Y)

lim,_+,

d+T*W =

=lim

_V’O

dY

dy

1

(12)

and lim,,,

,Te(y)=l Y



(13)

Theorem 3.1 says that both the optimal marginal and average tax rates tend to 100 percent as income level tend to infinity. It should be, however, noted that the disposable income y- T*(y) is always a monotonically nondecreasing function of y. (See Fig. 1.)

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3.2 The result (13) follows immediately from (12) (see Subsection 5.1). Here we should give an intuitive sketch of the exact proof of the result (12) to help us to understand the essence of the proof because it is long and complicated. Suppose that the marginal tax rate does not tend to 1 as y --)00. In this case, the disposable income y- T+(y) tends toinfinity with the sameorder as that ofy, i.e., lim(y- T*(y))/y>O. Then, if an individual i with very high ability works for time Li, then his disposable income n(i)Li- T*n(i)L’ can be also very large. This observation and Assumption M imply that individuals with sufficiently high abilities work for Lib. In this case, if the government increases the marginal tax rate a little in a range of sufficiently high incomes, then every individual i with an income in the range still works for L’. That is, this change doqs not obstruct the labor incentives. Then the government’s revenue increases and so, the government can increase the level of the public good. This increase in thle level of the public good makes the utility levels of individuals with low incomes rise because the tax function in the range they face does not change. Even if thle utility levels of the individuals with high incomes may decline, the decrements are smaller than the increments of the utility levels of the individuals with low incomes because of the boundedness of utility functions (Assumptions D and L). Therefore the government can increase the social welfare by changing a little the ma,ginal tax rate in the range of sufficiently high incomes. This is a contradiction to the optimality of (T*, Q*). Thus we have the result (12) of Theorem 3.1. 3.3 Theoren 3.1 is quite different from the results of the preceding studies. Mirrlees (1971) showed many possibilities of the optimal marginal tax rate but concluded that they are much less than 100 percent. Furthermore Sheshinski (1972), Fair (1971), Feldstein (1973), Kesselman (1976) and others discussed the same problem and provide different upper-bounds of the optimal marginal tax rate. But they are also smaller than 100 percent. The apparent difference between our model and their models is that we take a public good into account explicitly which the others do not. Another difference is in welfare criteria. However, these are not very crucial. The main reason for the difference is that we made the assumption of boundedness on the utility functions which has not been done in the other papers. The boundedness is a natural assumption, which is justified by St. Petersburg paradox. This fact can be observed in comparison between utility increments of poor people and utility decrements of rich people in the sketch of the previous subsection. , If only linear tax functions are allowed, then Proposition 2.6(ii) states that the 100 percent marginal tax rate is impossible. This is one reason for the difference between our result and that of Sheshinski (1972) in which linear tax functions are only allowed. The results (12) and (13) can be regarded as progressiveness in a different sense

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broader than convexity. In this sense we can say that our model does not lose the essence of the optimal taxation problem assuming convexity. 3.4

When T* satisfies (12) and (13) of Theorem 3.1, there are two possibilities, i.e., lim(y - T*(y)) = +a

(14)

lim(y - T*(y))< +a.

(1%

and If (15) is true, the disposable income has the upper bound. In this case even if an individual i with very high ability works for L’, his disposable income can not exceed the upper bound. Therefore he would not work for a long time. That is, the tax function obstructs the labor incentives of individuals with high abilities. Therefore we conjecture that (14) holds but not (15). Regretfully, the author has not succeeded in proving this conjecture. This is an open problem of great importance.

4. Proof of Theorem 2.7 4.1 From (2) there exists an M > 0 such that U ‘(a,b, Q) I M for all (a, b, Q) E Y and i E N. Hence W(T) =

N

lOi3W -x,(i), f ‘x,(i) - Tf ‘x,(i), Q) dp

is bounded from above. That is,

sup, E

.i

W(7) < +a.

(16)

Furthermore the following lemma holds. Lemma 4.1. There exists a feasible tax schedule T with W(r) > --oo. Proof. Let us consider the sequence {TV) such that TV(y) = y/v for all y EE, and all V 2 1. Let (Xv(i))jEN be the labor supply schedule for (T”,0). It is easily verified that x,(i) converges pointwise to x0(i), where (xo(i))i, N is the labor supply schedule for the trivial tax schedule (To,0). By Assumption I and Lebesgue’s dominated convergence theorem we have lim

s N

x,(i)dp

=

? 3 N

x&l dw

SincejNx0(i) dp > 0 by Assumption J, there is an integer v. such that j,%x,,,(i) dp > 0.

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It follows from this and (7) that j,,,T”of’x,,(i) dp > 0. Let us consider another sequence {r8) such that rs( y) = Y/V*- l/s for ally E E, and all s 2 1. Let (&(& ,,, be the labor supply schedule for (1”: 0), theu it is also easy to see that $Ji) converges pointwise to x,,,li) as s -+ 00. By Assump’cion I and Lebesgue’s dominated convergence theorem we have a

3Ji)dp

lim

= sN

N

x,(i) dcr.

Therefore there is an integer SOsuch that 1, ~sofi.fso(i) dp > 0. This means that (TsqO) is a feasible tax schedule. Since the minimal disposable income -F”o(O) is positive, by Assumptions A, E and I we have

wpq0) 2

s

log

u’(L -L\

-rso(0),

O)dp > --oo.

N

This lemma and (16) imply that there exists a sequence {rS} = {(TS,Qs)) such that (TS,Qs) ~9 Vs and { W(rs)} is a nondecreasing sequence with lim W(rs) = sup W(r).

(17)

Since Q 4 j,,, f’(L’) dp < +a, for all (T, Q) E g by Assumptions F and I, every Qs (s=l, . ) belongs to a compact interval. Hence there is a convergent subsequence ( Qsy) of ( Qs}. Since {(TS: es’)} also satisfies (17), we can assume without, loss of generalit that (Qs} itself converges to Q*. The purpose of Subsections 4.1 and 4.2 is to show that we can choose a subsequence of {TS} which converges in a certain sense. Lemma 4.2. infs TS(0) > --oo. Proof. Suppose inf, TS(0) = -00. Let {(X,(i))i.N} be a sequence of labor supply schedule for rs (s= 1, . . . ), i.e., each x,(i) satisfies (6) and (7) for tS. Then it is clear by Assumption F that xs(i)sLi

Vs, &N.

(18)

Let T5(y) = y + T’(O) for all y E E,. Of course these rs (s = 1, . . . , ) satisfy (4), i.e., Fs E CTfor all s. Since TS is convex and TS(y) I y for all y E E,, we have P(y) 5 r”(y)

Vy E E,.

WI

(See Fig. 1.) It follows from (18) and (19) that sN

T”fi(Li) dp 1

T Efk,(i) dp. s N

The left-hand term of this inequality is rewritten as

s

(f i(Li) + TS(0)) dp =

N

sN

f ‘(L’) dp + TS(0)p(N).

M. Kaneko 1 The optimal progressive income ta.v

209

By Assumption I, there is an s for which this value is negative. Since (T”, Q”) is a feasible tax schedule, it holds that

Q-’s
0, let 6 = e, and so it holds that for all y, y’ E [O,k] and all tEC[O,k],

ly -y’l I6 =$If(y) - f(y’)) Se. This means that C[O, k] is equicontinuous. Next we show that C[Q k] is a closed set. Let {P} be a convergent sequence of functions in C[O, k]. Let to = lim,,, P. Then it is easily verified that 1’ is continuous, convex and nondecreasing. Since tS is convex with d-P(y)/dy Ik5 1, we have ts(k -E) - t”(k) 51 -&

KS, V&>O.

M. Kaneko / The optimalprogressive income tax

Hence we

have

~“w-e--ow) =

ve>()

1

.

-&

This implies d-t”( y)/dy Ik5 1. Thus we have shown that to E C[O, k]. Let tS be the restriction of TS on [0,1) (s = 1, . . ., ). Then it follows from Lemma 44 that (P} has a convergent subsequence {ru). Let T Isbe the original tax function on E, in (TS} corresponding to f-Is. Then {T Is} converges uniformly on [0,11. Let f Is be the restriction of Tls on [0,2] (s = 1, . . . , ). In the same way let {f&} be a subsequence of (f Is) such that {t’zs) converges. Let T2s be the original tax function on E+ in ( Ts) corresponding to f‘2s. Then ( Tzs) converges uniformly on [0,2]. Continuing this process we get an array of sequences of the form (TS) = (T’, T2, . . . . ), . . . . ),

(TIS) = (T”,T’2,

(T&) = (T2’, T22, . . . . ), (T3”) = (T3’, T32, . . . . ),

in which each sequence is a subsequence of the one directly above it, and for each k the sequence ( Tk”) has the property that (T”) converges uniformly on (0, k]. If we define r *,T2, . . . , by r1 = T1’, r2= T22, r3 = Tj3, . . . . then the sequence (rs) is the sequence ( rs) is the ‘diagonal’ subsequence of ( TS). It is clear from this construction that for any k, (rS) converges uniformly on [0, k]. Hence ( rs) also converges pointwise everywhere on E,. We can define the function T* on E, by T*(y) = lim,,,

r”(y)for

allyEE+.

(21)

Lemma 4.5. T+ E .K

Proof. Since each rs is convex and nondecreasing with rs( y) s y for all y E E,, it holds that for any yl, y2 (0 r yl~yz) and a! E [0, 11, f%h

+ (I-

f3Y’) =

Mu2)

f?Y2)9

= afb)

WY’)

+

=Y1

(1 -

dYY2h

t’s*

‘*Vehave, by (21),

T*(ayl + (1 -ah) WYI)

s TYy2)

5 and

aT*trl) + (1 - a)T*trzh T*tu’) 5~‘.

Let ( Qs) be the subsequence of ( Qs) corresponding to ( Ts). This ( Qs) also converges to Q*. Hence we have shown the following lemma.

M. Kaneko / The optimal progressive income tax

211

Lemma 4.6. (P) = ((TS, Q”)} has a subsequence {(r”, OS)) which satisfies (i) lim,,, Qs = Q*z 0, lim,,, rs( y) = T*(y) for ally E E, and T+ E J: (ii) (r”) converges uniformly to T+ on [0, k) for any k. 4.3 The purpose of this subsection is to prove that (T*, Q*) is a feasible tax scheduk and W(P) = W(T*, Q*) = SUPS,,. W(r),

(22)

which means that r+ = (T*, Q*) is an optimal tax schedule. That is, if we show the feasibility and (22), we complete the proof of the existence of an optimal tax schedule. To show the feasibility it is sufficient to prove

Q* s SN W’x*(i) dcc,

(23)

where (x+(i))i,N is the labor time supplies for (T*, Q*). Lemma 4.7. Let ((r: OS)} be the sequence given in Lemma 4.6 and let (&(i)); Eh7be the labor supply schedule for (F”, 0”) (s = 1, . . . , ). Then ($(i)) converges to x*(i) for each i EN. For the proof see Appendix A. Let individual ie N be arbitrarily fixed. By the continuity of f I, we have lim_ f’&(i) =.f’x*(i). Since 0 5 f i.2s(i)5 f ‘(L’) for all s, we have limS4, Tsf’x,(i) == T*f ‘x*(i), because lr”fiZs(i) - T*f’x*(i)l

5 ITsfigs +o

-

T*f%Ji)l

+ IT*f’&(i) - Ttf’x*(i))

(s+oo)

by Lemma 4.6(ii).13 Since rsf i2s(i) 5 f ‘(I,‘) for all s and all i E IV, and since f ‘(L’) is integrable by Assumption I, we have, by Lebesgue’s dominated convergence theorem,

lim,,,

s

T*f ‘x+(i) dp.

r”f%,(i) dp =

N

sN

Since (r: Qs) is feasible, it holds that &” = IN Tsf’&(i)dN

VS .

13Note that T* is a continuous function because T*e .I:

M. Kuneko / The optimal progressive income tax

212

Hence we have Q+ = lim,_,, &” 5 lim, “Q) Tsfi&(i) dp = sN which is the feasibility of (T*, Q*).

r+f’x*(i) dp, sN

Lemma4.8. Eq. (22) holds. Proof. Since ((r: 0;‘)) is a subsequence of {(TS, Q”,} and lim,,, W(Ts, QS) = sup,, i W(r), it holds’that lim,,, W(r”, 8”) = sup,,,, W(T). For each i EN, U’(L -g,(i), f i2S(i) - rsfi.&(i),

Qs)--)

+ Ui(L -x*(i), f k*(i) - TSf ‘x+(i), Q*) as s + oo because of Assumptions A, G, Lemma 4.6 and Lemma 4.7. Since Ui is uniformly bounded by (2) and {IV@“,OS)} is nondecreasing by (17), we have, by Lebesgue’s dominated convergence theorem, I4 n log U ‘(L - &(i), f jgs(i) - rf ‘Zs(i), ii”, dp I limp,

3 N

I

c

.,N

That is, supIE ,- W(T)

log Ui(L -x*(i), f ‘x*(i) - T*f ‘x*(i), Q*)dp. =

lims+, W(r: 8”) I W(T*, Q*).

4.4 The purpose of this subsection is to show that (T*, Q*) is an equilibrium tax schedule. Suppose Q* < jN T*f ‘x*(i) dp. Lemma 4.9. Let {qs) be a sequence which is decreasing and converges to Q*. Let (%(i))i EN be the labor supply schedule for (T*, qs) (s = 1, . . . . ), then (R,(i)) converges to x*(i) for each i EN.

The proof is straightforward. Hence we have, by Lebesgue’s dominated convergence theorem, T*f ‘x*(j) dD > Q+ = lim,,, qs. limsdap T*f i&(i) dp = sN sN Therefore there is an integer s such that jN T*f iXs(i)dp > qs > Q+. This means that (T*, qs) is a feasible tax schedule. For this s it holds by Assumption A that for all Wtrictly speaking we use the following variation of Lebesgue’s theorem: Let {fS} be uniformly b@unr’edfrom above, i.e., for some M, fs(i) s M for all s and nondecreasing for all i E N. Then i%$,&(i)dCc =l,i&if,(i)dp.

M. Kaneko 1 The optimal progressive income tax

213

i E N with f ‘x+(i) - T*f ‘x*(i) > 0, iJ’(L -x*(i), f ‘x*(i) - T*f ‘x*(i), Q*) c < U’(L -x*(i), f ‘x*(i) - T*f ‘x*(i), qs) s Ui(L -R,(i), f ‘R,(i) - T*f ‘R,(i), qs).

Thus we have --oo < tV(T’, Q*) =


0 ande>O, (24)

Hence we have lim

Suppose lim,,,

d-T*

.“‘-

-

dy

=lim

y

d+T* y’oD dy p’

d-T*/dy 1.”= 1. Since T* is convex, it holds that d-T*

d+T” sT*(y+l)-T+(y)sdy dy y

This implies lim, god(T*( y + 1) - T *( sv)) = (Lemma S.l), we have lim p-+0 .m.

Y

VytzE,. p+l

1.

Using the following familiar lemma

= lim )‘+‘,,(T*(Y + 1) - T*(Y)) = 1.

Hence it is sufficient to show that lim,, ood- T*/dy l

(25)

1v = 1.

Lemma 5.1 (Komatsu 1962, Theorem 40.5). If f(x) and g(x) are defined on E+, and g(x) + 0~ (x+ =), and if (f(x + 1) -f (x))/(g(x + 1) --g(x)) -+ a (x-’ a), then f (x)/g(x) + a (x + a), where a is a real number.

M. Kaneko / The optimal progressive income tax

214

In the following we suppose lim

d--T’ = lim,,, )“= dy y

d+T* 1 1 zwQ--

1 u: n(i) uj

l-a-O----r

(L - Li, n(i)Li-

1 vi’ n(i) ui

(28)

T”n(i)Li, Q*) ’

vsr I(L - Li, n(i)Li - T%z(i)L! Q*)

1.

(29)

ht. Kaneko / The optimal progressive income tax

Proof.

It

215

follows from Assumption M that there is a bOsuch that --1 UC b vi’

(L-L’.b.Q+)

< 1 -(a+@ L

Vi&V,

br bo.

Hence we have 1 < l-(a+& Auf n(i)Li - T ‘n(i)L’ ui I(L- ti, n(i)Li- rln(i)f_l, Q*) L Vi with n(i)Li - T ‘n(i)Li 2 bo. Since n(i)L’-

T’n(i)L’ =Li n(i)

1 - T’n(i)L’ n(i)L’

>

(L -



it holds that for all i with n(i)Li - T %(i)Li 1 bO, P_1 u( n(i) uj St

I (L - Li. n(i)Li - Tln(i)Li,

Q*)

1 C; cl-(a+@. ?#(i)L’- T ‘n(i)L’ ui (L- Li,n(i)Li- rln(i)J_l, Q*)

Since lim,,i,40 (1-s

i.,,i>=

l -a-6>0g

it holds that n(i)Li - T ‘n(i)L’+ 00 as n(i) +oo. Therefore we can choose an no such that n(i)Li - T *n(i)Li 2 bO Vi with n(i) L no. It is easily verified that n(i)Li - Ts+ ‘n(i)L’ L n(i)L’ - TSn(i)Li for all s and all i. This implies n(i)Li - TSn(i)Li 2 bOfor all i w ith n(i) 2 no and all s. Hence (29) holds for this no. In the above argument, T* can be replaced by T*. Then we get (28). Lemma 5.3. Let (x+(i))iEN and (xs(i))iE N be the labor supply schedule for (T*, Q*) and (TS, Qs) (s = 1, . . . ) respectively, where Qs is any nonnegative real number. Then it holds that x*(i) =x,(i) ViEN,szno, x*(i) = L’ Vi with n(i)x*(i) p

n&,

(3(3) (31)

where no is the integer given in Lemma 5.2. Proof. Note that the labor supply schedule for a tax schedule (T, Q) does not depend upon Q by Assumption K; that is, if (xl(i))iEN and (xz(i))i,N are the labor supply schedule for tax schedules (r Q1) and (T, Qz), respectively, then xl(i) =x2(i) for all i E N.

M. Kaneko / The optimal progressive income tax

216

Let s z no. Let i be an individual such that rs(i)x*(i) < sL. Since TS(y) = T*(y) for till y I SL by (27) and Ts is convex, x*(i) also satisfies

Ui(L -x+(i), n(i)x*(i) - Tsn(i)x*(i), Q*) L 1 Ui(L -x, n(i)x - T%(i)x, Q*) VXS Li. In fact, if this inequality is not true, there is an x0 I L’ such that n(i)xO 2 sL and

Ui(L -x0, n(i)x o - Tsn(i)xo, Q*) > > Ui(L -x*(i), n(i)x*(i) - Tsn(i)x*(i), Q*). This implies

Ui(L -x, n(i)x - Tsn(i)x, Q*) > > Ui(L -x*(i), n(i)x*(i) - Tsn(i)x*(i), Q*) for all x (x*(i) --

1 u;

n(i) ui (I_. - Li,n(i)Li-

TSn(i)Li, Q*)

I u + 6, by (27) it follows that Using d’ T’/dy (n(;)**(i)

-U; + @n(i) ( 1 -y)]i

(f.- Li, n(i)Li

>O, - Pn(i)Li,

Q*)

which is equivalent to ,

d-

ox Ui(L -x, n(i)x - Tsn(i)x, Q*)

> 0.

x=Li

This and Assumptions A’ and F’ imply x,(i) = Li. Analogously we can prove that x*(i) =Li for all i with n(i)x*(i) L noL. This is (31). Hence it holds that x,(i) = x*(i) =Li for all i with n(i)x*(i) 1 sL. Lemma 5.4. If the government employs Ts (s 2 no) instead of T*, the increment of the government’s revenue is not smaller than &(F(sL + 1)). i.e.,

Tsn(i)xs(i) id,u-

T*n(i)x+(i) dp 1 SC((F(sL+ l)), sv whereF‘(sL+l)=(iEN:n(i)LizsL+l). sN

Proof, Note that x*(i) =x,(i) for all i EN. Since

Tsn(i)x*(i) - T*n(i)x*(i) = = (a + d)(n(i)x+(i) -sL) + T*(sL) - T*n(i)x*(i) 2 0

(32)

M. Kaneko / The o~ti~?~a~ progressive income tax

217

for all i E N with n(i) L sL by (27), it holds that l

3

[T%(i) x*(i) - T*n(i)x*(i)) dp 2 0,

(33)

F@L, SL + I)

where F(sL, sL + 1) = (i E N: SL s n(i~x~(i)< sL + 1). Since T*n(i)x*(i) s &n(i) -sL) + T*(sL) for all i with n(i)x*(i) 2rsL by the definition of 4 (eq. (2~~), it holds by (27) and (31) that [Tsn(i)x*(i) - ~*~(i)x*(~)]d~ or F(sL f 1) * [(a +

2

qI F(sL-t.1) -

~)(n(i)x*(i) - sL) + T*(sL)

~a(n(i)x*(i) - sL) + T*(sL))] dp

1

I

=

~[(n~i)x~~~)-sL)] dp = &

2

G(n(i)Lj-sL)dp F(sL + 1)

F&L + 1)

Idp = &(F(sL c 1)).

(34

s F(sL + 1)

Since T~~(i)x~~i)= T~~(i)x~(i~ for all i with ~(~~x~(i)SSL by (271, we have

s s

[T%(i)x*(i) - T*~(i)x*(i?~ dp = 0. N- F(sL)

(35)

Hence it follows from (33), (34) and (35) that

[T”n(i)x*(i) - T*n(i)x~(i)~ dp =

iv

[T%(i)x*(i) - T*n(i)x*(i)] dy [T”n(i)x*(i) - T*~(i)x~(i)~ dp 2 &p(F(sL+ 1)). 5.3 In the foi~owing we assume s L no and use Lemma 5.3 without any remark, where no is the integer given in Lemma 5.2. Since @(a, b, Q) = ~~/~~ &, &,Q,> 0 for all i E N and afl (a, b, Q) E f by Assumption A’ and p({i dV: Ui(L -x*(i), ox*

- T*~~i)-~~~i), Q*) = 0)) = 0

by (IO), there is a positive 6 such that

H = {i e F(O,noL): Gi(L -x*(i), ~(i)x*~i) - T*n(i)x*(i), Q*) r 6},

M. Kaneko / The optimal progressive income tax

218

where F(0, nOL) = (i E N: 0 s n(i)x*(i) < noL}. Since (x*(i))iE N is invariant for Q by Assumption K, we can write J’(Q) = Gi(L -x*(i), n(i)x*(i) - T*n(i)x*(i), Q)

Vi E N.

(37)

It holds by Assumption A’ that for all i EN, J’(Q+ + AQ) = J’(Q*) + J”(Q*)AQ + AQ&

and t&p0

as AQ+O,

where Ji’ = (dJ’/dQ). Emma 5.5. Suppose jH J?(Q*) dp c +a. l5Let (qk) be a decreasing sequence such that lim, _,~ q&= 0. Then it holds that r 1H

&dy*Oask-,oo.

(39)

Proof. Let i be fixed. (J’(q* + q&)- Ji(Qf))/qk is the inclination of the line connecting two points (Q *,Ji(Q*)) and (Q+ + q&,Ji(Q* + q&)). Since J’(Q) is a concave function by assumption A’, this inclination is a nondecreasing function of k, i.e., (J’(Q+ + q&) - Ji(Q”))/qk s (J’(Q’ + qk + I) - Ji(Q*))/qk + I

Further it holds that (Ji(Q* + q&) - J’(Q*))/qk I dJ’(Q*)/dQ follows that

Vk.

for all k. Hence it

l&l = I(Ji(Q’ + q&) - J’(Q*))/qk - dJ’(Q*)/dQ(

is nonincreasing function of k. Since (Ji(Q* + Qk) - J’(Q*))/qk - dJ’(Q*)/dQ is an integrable function of i E H and since c& -0 as k-,oo for all ieH, we have, by Lebesgue’s dominated convergence theorem,

Since n(i)L’ - TSn(i)Li + 00 as d(i)Li --),00 and y - TS(y) I y - Ts+ l(y) for all y E E, and all s, there is an s r no by Assumption L such that mi - $(&(F(O,

noL) ~1) I G’(L - Li, n(i)Li - T%(i)t(

Vi with n(i)Li L sL,

Q*) MO)

where k$ = log Mi (M”s are given in Assumption L) and cl is the positive number such that cl = lirns*- NW (41) + 1))4(F(sU)~ If Ftis easily verifiedthat dJVdQ is a measurable function of i.

M. Kaneko

/ The optimal

progressive income tax

Note that the positiveness of cl is ensured by Assumption N. We are now in a position to evaluate the value W(Ts, Q* + &(F(sL + 1))) - W(T*, Q’) = G’(L -x*(i),

=

-I I

n(i)x*(i) - Tsn(i)x*(i), Q* + GC((F(sL + I))) dp

Gi(L -x*(i), n(i)x*(i) - T+n(i)x*(i), Q*) dp

CN

3

=

[G’( L -x*(i), n(i)x*(i) - T”n(i)x*(i), Q* + &(F(s& + 1)))

t F(O.sL) -

Gi(L -x*(i), n(i)x*(i) - T*n(i)x*(i), Q*)] dp

1

I

+

[G’(L -x*(i), n(i)x*(i) - Tsn(i)x*(i), Q* + &p(F(sL. + 1)))

*,F(sL) -

Gi(L -x*(i), n(i):-*(i) - T*n(i)x*(i), Q*)] dp.

(42)

For a sufficiently large s, the second term of the right-hand side can be rewritten as follows. 7

J

[G’(L -x*(i), n(i)x*(i) - Tsn(i)x*(i), Q* + S,u(F(sL + 1)))

F(sL) -

G’(L -x*(i), n(i)x*(i) - T*n(i)x+(i), Q*)] dp L

L s F(sL1

[mi - +(bSp(F(O, noL))cl) - Mi] dp

Let us suppose that jH Ji’(Q*) d/l < +a. Since T*(y) = TS( y) for all y I SL by (27), the first term of the right-hand side of (42) can be rewritten by (36) as follows.

s

[J’(Q’ + &(F(sL + 1))) - J’(Q*)] dp =

FV-4sL.1

[J’(Q* + &(F(sL

L:

+ 1))) - J’(Q*)] dp

c kH

t&(F(sL

2 sH

+ 1))dp + ’ e; d&(F(sL I %H

+ I))

1

r‘66p(F(sL + l))p(F(O, noL))/2 + where ci = &(F(sL

+ I))

e; dptSp(F(sL + l)), I CH

W)

and his the number defined by (36). Hence it holds for a suffi-

220

M. Kuneko / The optimal progressive income tax

ciently large s that W(Ts, Q*+ Sp(F(sL + 1))) - W(T*, Q’) L , L

&(F(sL

+ l))p(F(O, n,-,L))/2 +

- 66~(F(sL))c,

,u(F(o,

\ IH

E; dp&(F(sL

+ 1)) (45)

n&)/4.

Since cl = lim,,, p(F(sL + l))/p(F(sL)) > 0 and j,, &idp +O as s + 00 by Lemma 5.4, the right-hand side of (45) becomes positive for a sufficiently large s. This is a contradiction to the optimality of (T*, Q*). When jH J”(Q+)dp = +oo, (45) can be rewritten as follows. W(TS, Q* + &(F(sL + 1))) - W(T*, Q*) 1 L ’ [J’(Q’ + Gp(F(sL + 1))) - J’(Q*)] dp I

~H6*~(~(.~~))~,p(F(0, noL))/4.

(46)

Since I,., [J’(Q* + &(F(sL + 1))) - J’(Q’)] dp G,u(F(sL + 1))

-*a ass-*00

by Lemma 5.6 and lHJ”(Q*)dp = 00,and since lim,_ p(F(sL + l))/‘(F(sL)) by Assumption N, it holds or a sufficiently large s that

= cl >O

In [J’(Q* + &(F(sL + 1))) - J’(Q’)] dp

_

b

P(F(SL)) cl (F(0, noL))/4 > O?

MF(sL + 1))

This means that the right-hand side of (46) is positive for such an s. This is a contradiction to the optimality of (T*, Q*). Lemma

5.6. Let {fs} be a seqr,ence of measurable functions on H such that

0 5 f,(i) I_&+ I (i) for all s and all i EH. If there is a measurable function f such that lim_ f,(i) = f (i) for all i E H and In f(i) dl = 00, then lim_, jH f,(i) dp = 00.

We can prove this lemma in the standard way. 163ince J$Q) is convex and monotonically increasing with respect to Q for all k H, (JYQ* + Gp(F(sL + 1))) - Ji(Q*))/Gp(F(sL+ 1)) is positive and nondecreasing with respect to s for each i.

Al. Karteh

/ The optintai progressive

irtwnte

tax

221

Appendix A. Proof of Lemma 4.7

Suppose that for some k N, (g-(i)) has a limit point x**(i) such that j%**(i) >O and T+f’x+*(i) =f’x+*(i). Then T*(y) =y for ally E E, by (4). Therefore sup W(T)= lim W(p,Qs)= - 00, which is a contradiction. This means that if for some itz N, ($(i)} has a limit point x+*(i) such that T+f’x**(i) =f’x+*(i), then f’ is null or x**(i)= 0. If f’ is null, then &(i) = 0 for all sz 1 by Assumption A and (7), which implies lim &(i) =x*(i) =O. If x**(i)=0 and f’ is nonnull, it also follows from Assumptions A, B and (7) that lim, -+- Qi) =x*(i) = 0. For any k IV such that there is no limit point x**(i)of {&(i)} with T*f'x**(i)=f’x+*(i), it is easily verified that lims,, &(i)=x*(i). References R.J. Aumann, Markets with a continuum of traders, L conometrica 32 (1963) 39-50. R.J. Aumann, The St. Petersburg Paradox: -4 discussion of some recent comments J. Econom. Theory 14 (1977) 443-445. H.W. Brcck, A game theoretic account of social justice, Theory and Decision I 1 (1979) 239-265. R.C. Fair, The optimal distribution of income, Quart. J. Econom. 85 (1971) 551-579. M. Feldstein, On the optimal progressivity of the incttme tax, J. Publ. Econom. 2 (1973) 357-376. J.N. Herstein and J. Milnor, An axiomatic approach to measurable utility, Econometrica 21 (1953) 291-297. Y. Ito and M. Kaneko, Linearization of cost functions in a public goods economy, Econom. Stud. Quart. 32 (1981). M. Kaneko and K. Nakamura, The Nash social welfare function, Econometrica 47 (1979a) 423435. M. Kaneko and K. Nakamura, Cardinalization of the Nash social welfare function, Econom. Stud. Quart. 30 (1979b) 227-233. M. Kaneko, An extension of the Nash bargaining problem and the Nash social welfare function, Theory and Decision 12 (1980) 135-148. M. Kaneko, The Nash social welfare function for a measure space of individuals, J. Math. Econom. 8 (1981a) 173-200. M. Kaneko, On the existence of an optimal income tax schedule, Rev. Econom. Stud. 48 (1981 b) 633442. J.R. Kesselman, Egalitarianism of earnings and income taxes, J. Publ. Econom. 5 (1976) 285-301. K.H. Kim and F.W. Roush, Economic planning based on social preference functions, Math. Sot. Sci. 1 (1981) 193-200. Y. Komatsu, Introduction to Analysis (Hirokawa-shoten, Tokyo, 1962), in Japanese. J.A. Mirrlees, An exploration in the theory of optimum income taxation, Rev. Econom. Stud. 38 (1971) 175-208. J.F. Nash, The bargaining problem, Econometrica 18 (1950) 155-162. J.A. Ordover and E.S. Phelps, The concept of optimal taxation in overlapping generations model of capital and wealth, J. Publ. Econom. 12 (1979) l-26. G. Owen, Game Theory (Saunder, Philadelphia, l%S). J. Seade, On the shape of optimal tax schedule, J. Publ. Econom. 7 (1977) 203-235. R. Selten, Reexamination of the perfectness concept of equilibrium points in extensive games, hernat. J. Game Theory 4 (1975) 25-55. E. Scheshinski, The optimal linear income tax, Rev. Econom. Stud. 39 (1972) 297-302. G.F. Simmons, Introduction to Topology and Modern Analysis (McGraw-Hill, New YorL 1963). J. von Neumann and 0. Morgenstern. Theory of Games and Economic Behavior (Princeton University Press, Princeton, 3rd ed., 1953). R.H. Wagner, Impartiality and equity, Theory and Decision 12 (1980) 61-73.