# There are many points of overlap between the fundamental

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4 Now, let us make explicit the assumptions about knowledge of n. Assume that at period 2, the parent knows the actual realization of n; but that in period 1, the parent does not (although the parent does know the distribution of n). In period 2, then, we optimize on c 2, given K, which has been determined by the choice of c 1, and given n. This choice determines a second-period payoff (the expression in curly brackets) as a function of n and c 1. Take the expectation of this payoff with respect to n; we then have 2 functions of c 1, whose weighted sum is to be maximized. To simplify the later discussion, it is convenient to state a general lemma about functions of this kind, which we will use twice in the following analysis. Lemma. Let (c) [c 1 /(1 )] P[(K c) 1 /(1 )], where P is a positive constant. Then (c) is maximized at c [Q/(1 Q)]K, where Q is defined by Q P, and the maximum value of is P(1 Q) K 1 /(1 ). The result follows straightforwardly from setting the derivative equal to zero (note that is a concave function) and then substituting. We now proceed as before. Conjecture that V K AK 1 / 1, for some A, then show that we can find a value of A such that Eq. 14 is satisfied. Consider first the second-stage optimization, where n is known and c 2 is chosen. Note that V(K ) An 1 (K c 2 ) 1 /(1 ), so that, from Eq. 14, c 2 maximizes, 2 c 2, n c 2 1 / 1 P 2 K c 2 1 / 1, where P 2 A n. Define B by B A [15] Then, if we define Q 2 in terms of P 2 as in the Lemma, we see that, Then, and Q 2 B/n. [16] c 2 BK / B n, [17] K K / B n. [18] From the second part of the Lemma, we find that, max c2 2 c 2, n A n 1 B/n K 1 / 1 A B n K 1 / 1. [19] Now we proceed to the determination of c 1. Because n is unknown at time 1, we substitute the expected value of Eq. 19 for the expression in curly brackets in Eq. 14. Let 1 c 1 c 1 1 / 1 E n max 2 c 2, n. Substitute from Eq. 19 and then substitute for K from Eq. 12. where 1 c 1 c 1 1 / 1 P 1 K c 1 1 / 1, P 1 A 1 E B n. Define C by C A 1. [20] For any positive random variable, X, we can define the -mean of X by X E X 1/. Then, if we define Q 1 in terms of P 1 as in the Lemma, we see that, Hence, by the Lemma, and Q 1 C/ B n. c 1 C/ B n C K. max 1 c 1 P 1 1 Q 1 K 1 / 1 A 1 B n C K 1 / 1. Because this has to equal V(K) by the Bellman equation, we conclude that 1 B n C 1. [21] We can express C in terms of B and so reduce Eq. 21 to an equation in one unknown. Divide Eq. 20 by Eq. 15 and then raise both sides to the power, 1/ : C 1 1/ B. [22] Then Eq. 21 becomes, after some simplification, 1/ 1 1/ B n B 1. [23] The left-hand side is a strictly increasing function of B and approaches infinity as B approaches infinity. Hence, Eq. 23 has a unique solution if the left-hand side is 1 atb 0, i.e., if 1 1/ n) 1. [24] From Eqs. 21 and 22 and the formula for c 1, we can write c 1 1 1/ CK 1/ BK. We can then derive the law of motion of the capital stock of each individual by substituting into Eq. 12 and then Eq. 18. K 1 1/ B / B n K. [25] This is the initial capital of each child when the parent s capital is K. Once again, it leads to the conclusion that the asymptotic distribution of wealth is log-normal. Discussion It is not surprising that fundamental problems in economics have strong analogies to similar problems in evolutionary ecology. In both realms, individuals must trade off current and future discounted benefits, to themselves and to others. Evolution favors those strategies that increase payoffs, and the solutions evolution favors broadly should be represented in the ways individuals make decisions, for example about current consumption versus savings for the future and legacies for offspring. Fundamental in both arenas is how uncertainty affects decisions, and when it selects for higher discounting of future benefits. We explore this issue through a generalization of standard consumption models to uncertain environments, providing a framework that we hope will serve as a springboard for further studies. 4of5 cgi doi pnas Arrow and Levin

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