The Review of Economic Studies, Ltd.

Distributional Goals and Optimal GrowthAuthor(s): Owen StanleySource: The Review of Economic Studies, Vol. 45, No. 2 (Jun., 1978), pp. 389-390Published by: Oxford University PressStable URL: http://www.jstor.org/stable/2297355 .

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Distributional Goals and Optimal Growth OWEN STANLEY

Australian National University

This paper considers a more general form of a problem discussed previously by K. Hamada [1]. He was concerned with the income transfer from workers to capitalists which was optimal from the workers' point of view. We shall be concerned with the optimality of such transfers and the stationary states when the relative importance of workers' and capitalists' consumption is chosen.

Workers are assumed to consume all of their incomes and capitalists consume all of theirs net of a fixed rate of savings, fi. The government uses a redistribution tax, x, as its control. Capitalists receive profits equal to the marginal product of capital and since workers receive the rest of income

cW =f-f'k+x ...(1)

cp = (1 - f(f 'k -x) ... .(2)

where cp is capitalists' consumption, cw is workers' consumption and k is the capital-labour ratio; f is a neoclassical production function with constant returns to scale and

f '(O) = 0, f '(k) > O, f"(k) < O. ...(3) The redistribution tax can be no greater than profits, when it is imposed on capitalists, and no greater than wages when it is imposed on workers. Thus it is constrained in the following way

f'k ? x ? -f+f'k. .. (4)

Since only capitalists save the rate of change of capital stock is

k = f(f'k-x)-Ak ... (5) where )i is the rate of depreciation.

The government has two arguments in its utility function, the present value of workers' consumption and the present value of capitalists' consumption. It weights each of these according to its political colour and uses an index a E [0, 1] such that a = 1 when workers' consumption only is important and capitalists' consumption is of no importance. If the converse is true then a =0. Thus the government's goal is to

maximize [aI cwe tdt+(1-a) cpe .(6) x(t) 0 O

where p is the rate of discount applied by the government, subject to (4), (5) and k(0) = ko. Then since

CW-=f(k)-l- '(k+Ak)

C= 1(1-f)(k+Ak) (6) becomes:

{ 1 {cxflf(k) + [(1 -x)(1- -x(k + Ak)}dt 0

or, after integrating by parts: 00 -

{cxff (k) + (, + p)k[(l - )(1 - f)-a] d t. ... (7)

389

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390 REVIEW OF ECONOMIC STUDIES

Since the maximand is a strictly concave function of k alone, the optimal approach to the Golden Age capital-labour ratio will be the most rapid [2] and will involve suppressing workers' income after tax to the lowest level regardless of a, if k is less than the optimal capital-labour ratio.

Provided that the optimal capital-labour ratio k* is k* ? k, where k is the maximum sustainable capital-labour ratio, then k* is given by

f (k*)= +P [ _ (1)(1O] . . (8)

and so k* declines as the importance of capitalists' consumption is increased in the objective function. At J, f(k) = (i/Of)k, c - = 0 and a = a

(l-(P+)) = i (9)

(P + ,- ff '()) + G( l-(P+ Thus a<6. has no effect on the terminal capital-labour ratio.

Consumption enjoyed by workers and capitalists, and the redistribution tax at the Golden Age are as follows. From (5) the redistribution tax is

X_-a P- #1 - )((p + A) k* ... 10)

and so there will be redistribution of income from workers to capitalists if

cL< (l-f5)(P+A)L).. G

(1 - (p +A) ...(11)

The tax will be zero if both sides of (11) are equal, otherwise capitalists will subsidize the workers. Capitalists' consumption is

cp = 1

(t ? k* ..(12)

and workers' consumption is

cw =f(k*)-A k*. *.(13)

Thus it has been shown that the government's attitude to workers' and capitalists' consumption affects only the Golden Age capital-labour ratio and that any policy requiring an increase in the capital-labour ratio always involves suppressing workers' consumption to the minimum.

First version received February 1976; final version accepted April 1977 (Eds.). I am indebted to Professor J. D. Pitchford and N. Vousden for helpful comments on this paper.

REFERENCES [1] Hamada, K. " On the Optimal Transfer and Income Distribution in a Growing Economy ", Review of

Economic Studies, 34 (1967). [2] Spence, M. and Starrett, D. " Most Rapid Approach Paths in Accumulation Problems ", International

Economic Review, 16 (2) (1975).

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