- The Indeterminancy of Game-Equilibrium Growth in the Absence of an Ethic
- Russell Sage Foundation
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The Indeterminacy of Game-Equilibrium Growth in the Absence ofon Ethic* Edmund S. Phelps A recent paper by Robert Pollack and myselfl essays the optimal saving problem when, by reason of (limited) selfishness on the part of each generation, future generations will not consume and save the capital they inherit in the proportions that the current generation would like them to do. This situation poses for each generation a "second best" problem or "sub-optimization" problem-neither term is wholly appropriate-whose solution depends upon the assumptions made by each generation about future saving behavior. If each generation expects future generations to behave as it would behave in their situations, then there may result a kind of game-equilibrium growth path which is self-warranting: Every generation acts in such a way as to validate earlier assumptions as to how it will act and, given these assumptions, no generation acting alone can increase its estimated overall utility-despite the fact that cooperative action among the generations, were it enforceable, could produce an improvement in every generation's utility. That paper postulated a discrete-time production process in which the·This paper is a reVISion of a discussion paper dated May 1968 and presented at the Econometric Society Congress in September 1970. This version corrects the admissible range of the asymptotic capital-labor ratio and it elaborates the potential role of an ethic in rescuing the determinacy of the growth path. In the minds of some, my dynamic-programming analysis makes excessive demands on intuition. I am grateful therefore to Dr. Pauwels for recasting the argument in terms of differential game analysis. See his Mathematical Note which follows this chapter. I E. S. Phelps and R. A. Pollak, "On Second-Best National Saving and Game-Equilibrium Growth," Review ofEconomic Studies 3S (April 1968). 87 88 • ALTRUISM, MORALITY, AND ECONOMIC THEORY output-capital ratio is constant and labor is inessential and unproductive. The population is constant in size and is completely replaced each period. The present contribution postulates an exponentially growing population and continuous-time production under possibly variable proportions and constant returns to scale along the lines of the one-good Solow-Swan neoclassical growth model.2 Every generation is born directly into the labor force and dies with its boots on. There is no overlap-no births intervene during a generation's tenure. I shall study only the limiting case in which every generation's tenure shrinks to zero. In any finite length of time, then, infinitely many generations hold sway in a continuum, one after the other, each for an infinitely short time. It is hoped that there is heuristic value in such a model. I. BOUNDED SUSTAINABLE CONSUMPTION PER CAPITA AND UNBOUNDED RATE OF UTILITY The production-and-growth equations are LIL = r = constant> 0 f(k) = F(k, 1) = F(K, L) IL, k = KIL with the following specifications in the first model: with [(0) ;;;. 0, [' (k) >0, [" (k) r, f' (00) = 0 k =f(k) - rk - c = g(k) - c, g(O) ;;;. 0, g' (0) >0, gil(k) O k(b + 6) = x;;;'O k = g(k) - c The solution is of the form c(t) = h(t; kb, x), b~t~b+6 The utility-maximizing bequest must thus satisfy aUb ( 6 ) , b+6. '{ l ak(b + 6) = Vb [k(b + 6») + f u h [t; kb, k(b+ll) f b (5) (6) ah ak(b +ll) .dt (7) Letting cdenote the average per capita consumption rate over the interval 6, we can, for small 6, approximate the latter integral by 6u' (c) ak(~c+ 6) (8) 90 • ALTRUISM, MORALITY, AND ECONOMIC THEORY which, using the further approximation, can then be written 6.u'(c) I 6. I U (C) - = - u (c). 6. (9) Hence utility maximization equates the marginal utility of average consumption to the marginal utility of the bequest. In the limit, as 6. + 0 and overlapping vanishes, utility maximization by any generation requires consuming so as to equate the marginal utility of consumption to the marginal utility of the bequest, (10) the latter being completely predetermined in the limiting case. The marginal utility for the present generation of the capital it bequeaths depends upon the value the present generation assigns to future consumptions and the disposition of capital for consumption by future generations. Let each generation make the. assumption that all generations infinitely far into the future will consume according to some unknown, to-be-calculated, stationary and continuous consumption function...

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