446 NILS HAKANSSON satisfy the following conditions for all j and all finite Bi; such that 0 i20 for all i>s, and 0,j+0 for at least one i.( 5) is known as the"no-easy-money condition for the case when the lending rate equals the borrowing rate [8]. This condition states that no combination of productive investment opportunities exists in any period which provides, with probability 1, a return at least as high as the(borrowing)rate of interest; no combination of short sales is available in which the probability is zero that a loss will exceed the (lending)rate of interest; and no combination of productive investments made from the proceeds of any short sale can guarantee against loss In some variants of the basic model the individual has the opportunity to purchase term insurance on his own life and to sell (purchase)term insurance on the lives of others in each period. Let t20 denote the premium paid by the individual at the j-th decision point for life insurance on his own life during period 3. If the individual dies during this period, which by(1)has probability pii of happening, we assume that his estate will receive ti/pi; at the end of period 3; if he is alive at decision point 3+1, he will receive nothing Since in this contract the mathematical expectation of the"return equals the cost we shall say that the insurance is available at a"fair" rate We assume that insurance is issued only when pjj<l, i.e. at decision points We shall allow the possibility of contracting in advance for purchases of insurance on the individuals own life. Such an arrangement will be called an insurance contract. The unexpired portion of such a contract at decision point 3 will be denoted (ti, ti+i,., tw-i, where tu/pek is the amount of insurance the individual will keep in force in period k given that he is alive at the ke-th decision point (when the premium tk is paid) For convenience we define 7=2+ We also assume that t≤x+B, where B, denotes the maximum an individual may borrow at the j-th decision point on the security of his non-capital income stream and his insurance contract. Since no insurance can be issued at the nth decision point, it is clear that (8) Bw=yn/r. and that t计+1+B This content downloaded from 202.115.118.13 on Wed, II Sep 2013 02: 34: 55 AM446 NILS H. HAKANSSON satisfy the following conditions: (4) O? jij <oo, 2, M; j = 1, n Mi (5) Pr E (jj - rj)Oij < 0 > 0 for all j and all finite Oij such that Oij > 0 for all i > Sj and Oiji 0 for at least one i. (5) is known as the "no-easy-money" condition for the case when the lending rate equals the borrowing rate [8]. This condition states that no combination of productive investment opportunities exists in any period which provides, with probability 1, a return at least as high as the (borrowing) rate of interest; no combination of short sales is available in which the probability￾is zero that a loss will exceed the (lending) rate of interest; and no combination of productive investments made from the proceeds of any short sale can guarantee against loss. In some variants of the basic model the individual has the opportunity to purchase term insurance on his own life and to sell (purchase) term insurance on the lives of others in each period. Let tj ? 0 denote the premium paid by the individual at the j-th decision point for life insurance on his own life during period j. If the individual dies during this period, which by (1) has probability pjj of happening, we assume that his estate will receive tj/pjj at the end of period j; if he is alive at decision point j + 1, he will receive nothing. Since in this contract the mathematical expectation of the "return"' equals the cost, we shall say that the insurance is available at a "fair" rate. We assume that insurance is issued only when pjj < 1, i.e., at decision points. I, * **, n-1. We shall allow the possibility of contracting in advance for purchases of insurance on the individual's own life. Such an arrangement will be called an insurance contract. The unexpired portion of such a contract at decision point j will be denoted (tj, tj+?, * *, tn-1), where tklPkk is the amount of insurance the individual will keep in force in period k given that he is alive. at the k-th decision point (when the premium tk is paid). For convenience we define (6) Tj-tj+ ti+ ... + tn-i j-1,...,n-1. rj ri ... rn-2 We also assume that (7) tj < xj +Bj, j-1, * l ,n￾where Bj denotes the maximum an individual may borrow at the j-th decision point on the security of his non-capital income stream and his insurance contract. Since no insurance can be issued at the n-th decision point, it is, clear that (8) Bn = yn/rn, and that (9) Bj = min { r__+ ' _ jtj B, } = X-1 ri ri~~~I 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