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A MathematicalNote * Wilfried M. A. Pauwels The purpose of this note is twofold. First, we want to propose an alternative and simpler derivation of the results obtained [above] by E.S. Phelps. Secondly, we want to apply these results to a continuous version of the earlier model on game-equilibrium growth examined by E. S. Phelps and R. A. Pollak.1 In section I we examine the general properties of the game-equilibrium consumption function using the Hamilton-Jacobi partial differential equation. In section II we apply these results to a model which uses the same pro duction function and utility function as Phelps and Pollak. Finally, in section III, we derive the same results as in section II via the calculation of the "second-best" optimum as proposed by Phelps and Pollak. I. THE GAME-EQUILIBRIUM CONSUMPTION FUNCfION Using the same notation as in Phelps, we can formulate the present generation's problem as follows: Find a consumption function c(t), to ~ t ~ tl, so as to maximize (1)·The author is indebted to E. S. Phelps with whom he had the opportunity to discuss this note. t E. S. Phelps and R. A. Pollak, "On Second-Best National Saving and Game-Equilibrium Growth," Review ofEconomic Studies 35 (April, 1968). 107 108 • ALTRUISM, MORALITY, AND ECONOMIC THEORY subject to k(t) = g(k) - c(t) ; k(to) = ko > 0 (2) It is assumed that the present generation lives from to till tl, at which time it bequeaths k(tl) of capital per capita to the next generation. The utility to the present generation of this bequest is represented by V [k(tl)]. 0 can be interpreted, either as 0 =U(c), where c =c(k) =g(k) (see section I of Phelps), or as 0 = 0 (see section II of Phelps). Let j[k(t),t] (3) where c(t), to « t « tl, is the optimal consumption policy of the present generation. j [k(t),t] must then satisfy the following Hamilton-Jacobi partial differential equation2 : aj[k(t),t] . jaj[k(t),t] a + maximum 1 ak() [g(k) - c(t)] + t c(t) t U[c(t)] e-pt - ue-pt} 0 subject to Maximization of the expression in brackets in (4) gives U[ ()] -pt = aj[k(t),t] c t e ak(t) From (5), we then obtain for the terminal time t =tl aj[k(tl )] ak(tl) (4) (5) (6) (7) 2See, e.g., E. B. Lee and L. Markus, Foundations of Optimal Control Theory (New York: John Wiley & Sons, Inc., 1968), pp. 340-60. A Mathematical Note. 109 Let us now specify V [k(tl)] as V[k(tl)] = [) r[U[c(k(t»] - 0] e-ptdt t, Then aV[k(tl)] = [) jd[U[c(k(t»] - 0] -ptd 3k(tl) t, dk(id e t = k(~l) [e-Ptd[U[C(k(t»] - 0] = pV[k(tl)] - [)[U[c(k(td)] - 0] e-pt , k(tl) assuming the convergence condition lim e-pt [U[c(k(t»] - 0) = 0 ~ holds. Using (7), we then obtain U[c(td) = pePt , V[k(~l») - [) [U[C(k(tl») - 0] k(tl) (8) (9) (10) (11) Let us now make the assumption that the lifetime of each generation, tl - to, tends to zero so that at each moment of time there is a generation that disappears. The implication of this assumption is that (7), and hence (11), must hold for all t, to ~ t ~ 00. We can then write and r,l[ ()] -pt = aV[k(t)] u c t e ak(t) u' [c(t») peptV[k(t») - [) [U[c(k(t») - ii) k(t) (12) (13) By the game-equilibrium property, we also require c(t) =c(k(t», so that, from (13) and (2), we obtain 110 • ALTRUISM, MORALITY, AND ECONOMIC THEORY u'[c(k(t»] = pePtV[k(t)] - 0 [U[c(k(t»] - 0] g(k(t» - c(k(t» (14) This is the basic condition characterizing the game-equilibrium consumption function. Taking the derivative of (14) with respect to k(t), and making use of (12), we obtain dc(k(t» dk(t) u' [c(k(t»] [g'(k(t» - p] iJ [c(k(t»] (1 - 0) - U" [c(k(t»] [g(k(t» - c(k(t»] which characterizes the slope of the consumption function. (15) If we interpret 0 as 0 = U(c), where C= c(k) = g(k), we can gain some insight about the asymptote k by evaluating (14...


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