INVESTMENT AND CONSUMPTION (14) by direct application of the definitions given in Section 2. 1. The first terms of(12)and (13)represent the proceeds from productive investments the e secon terms the pay ment of the debt or the proceeds from savings, the third terms the non-capital income received, and the fourth term in (13)the proceeds from life insurance Inserting (14)into(12)and (13) we obtain (5)2+=(56=)十(--)+,=1,…n-1 (16)+1=∑(1-r)2;+;m--t)++tp,了=1,……, The restriction that only the first S, opportunities may be sold short in period (17) 0,=S+1,…,M must hold while the assumption that all borrowing must be fully secured implies that a must satisfy the condition Pr{m≥0}=1 j=2, y()it follows that there is an upper limit on consumption in period jj=1, 3+Bi-tj which, since c:20, must be non-negative in order that a feasible solution exist in period 3. We shall now define f(ai as the maximum expected utility attainable b the individual over his remaining life-time, as of the beginning of period j on the condition that he is alive at that point and that his capital is w Utilizing(1)and(11), we may write this definition formally ∫x)≡ max elpis(e,x)+p,+1U,;+(e,,x +pin(c max Elu(ci)+pj dig(3+1)+.2 pika jiu(cj+1) +p,+100;+g(x3+2 ……+pn By the principle of optimality,(20) may be written, using (1) The principle of optimality states that an optimal strategy has the property that whatever the initial state and the initial decision are the remaining decisions must onstitute an optimal strategy with regard to the state resulting from the first decision. See [2,(83)1 This content downloaded from 202.115.118.13 on Wed, II Sep 2013 02: 34: 55 AMINVESTMENT AND CONSUMPTION 449 Mj (14) Zl; = xj-c;-tj- Zij, j=1,***, n i=2 by direct application of the definitions given in Section 2.1. The first terms of (12) and (13) represent the proceeds from productive investments, the second terms the payment of the debt or the proceeds from savings, the third terms the non-capital income received, and the fourth term in (13) the proceeds from life insurance. Inserting (14) into (12) and (13) we obtain Maj (15) xj+1 = , (Aii -rj)zij + ri(x - cj- ti) + yj, j=, n*** n-1 i=2 and Mj (16) x3 +l= E (~ij- rj)zij + ri(x - cj- tj) + yj + tjlpjj, j=1***, n. i=2 The restriction that only the first Sj opportunities may be sold short in period j implies that (17) Zij 2 0 , i = Sj + 1, * ,Mj, j = 1, *** must hold while the assumption that all borrowing must be fully secured implies that x; must satisfy the condition (18) Pr{x >2o}=1, j=2, *- ,n+1. By (5) it follows that there is an upper limit on consumption in period j,j=1, j * ,n, given by (19) xj + Bj -t, jz=1, n which, since Cj ? 0, must be non-negative in order that a feasible solution exist in period j. We shall now define fi(xx) as the maximum expected utility attainable by the individual over his remaining life-time, as of the beginning of period j, on the condition that he is alive at that point and that his capital is xj. Utilizing (1) and (11), we may write this definition formally: fj(xi) max E[pjj Ujj(cj, x>+1) + Pj,j+lUj,j+I(Ci, Cj+l, X42) + * + P.Un(C, ** C, xn+l)] j = 1, ***, n - max E[u(cj) + pjj3jg(xj+i) + E Pikaju(cj+i) (20) k=j+l + pj,j+jayjj+jg(x4+2) + > Pikajaj+IU(Cj+2) k=j+2 + + Pinaj *.. an-1Jng(X$n+1)] j = 1, n By the principle of optimality, (20) may be written, using (1):4 4 The principle of optimality states that an optimal strategy has the property that whatever the initial state and the initial decision are, the remaining decisions must constitute an optimal strategy with regard to the state resulting from the first decision. See [2, (83)]. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:34:55 AM All use subject to JSTOR Terms and Conditions