INVESTMENT AND CONSUMPTION 0 The purchase and sale of insurance on the lives of others will be viewed s a subset of the productive opportunities. Each person will be assumed to give rise to a separate investment opportunity, the return of which is ependent of the returns of all other opportunities 2.2. The Utility Function. The amount spent on consumption in period j will be designated c. As indicated, c, is a decision variable; in order to give it economic meaning, we require it to be nonnegative. We now postulate that the individuals preference ordering at the beginning of period m, conditioned on the event that death occurs in period k 2 m, is representable by a numerical utility function Umk. This utility function is de fined on the Cartesian product of all possible consumption programs(cm, ... ck and the amount of his estate ak+ at the end of period k; thus, the utility function is independent of the opportunities faced by the individual. We assume in this paper that the conditional utility function Umk has the form (11)Um&cm c,xk+1)=-1 ∑(量)00+…m-a m,k=1 (m8≤k) Implicit in this form is the assumption that preferences are independent over time We shall call u(c) the one-period utility function of consumption and g(')the utility function of bequests. The constant aj>0(ao= 1)is the patience factor linking the (one-period) utility functions of periods j and j+1 given that the individual will be alive at decision point j+l.When ai<1(a,2 1)we shall say that impatience (patience) prevails in period j with respect to period 3+1. Similarly, the constant &; expresses the relative weight attached to bequests by the individual at decision point 3, given that death will occur in period 3. Since ar, and a; are constants, we note that the rate of patience, while dependent on time, is independent of the overall level of utility(see [10]). We also postulate that the individual obeys the von Neumann-Morgenstern postulates [15]; accordingly, the individuals objective is to maximize the expected utility attainable from consumption over his life-time and the estate remaining and bequeathed upon his death. We also assume that the individual always prefers more consumption to less in any period, i.e., that u(c)is monotone increasing, and that the bequest function g(ar)is non-decreasing Finally, we assume that the individual is risk averse with respect to con- plies that u(c) is strictly concave, and that u(c) and g(a') are twice differentiable The notation developed in the previous section is summarized below before e proceed to construct our basic model z We assume, however, that the continuity postulate has been modified in such a way as to permit unbounded utility functions In congruence with this premise, we assume that the functions(11)are cardinal This content downloaded from 202.115.118.13 on Wed, II Sep 2013 02: 34: 55 AMINVESTMENT AND CO.NSUMPTION 447 where, by assumption, (10) tn = 0. The purchase and sale of insurance on the lives of others will be viewed as a subset of the productive opportunities. Each person will be assumed to give rise to a separate investment opportunity, the return of which is independent of the returns of all other opportunities. 2.2. The Utility Function. The amount spent on consumption in period j will be designated Cj. As indicated, cj is a decision variable; in order to give it economic meaning, we require it to be nonnegative. We now postulate that the individual's preference ordering at the beginning of period m, conditioned on the event that death occurs in period k ? m, is representable by a numerical utility function Umk. This utility function is de￾fined on the Cartesian product of all possible consumption programs (cm, ... , Ck) and the amount of his estate xk+1 at the end of period k; thus, the utility function is independent of the opportunities faced by the individual. We assume in this paper that the conditional utility function Umk has the form (11) Umk(Cm, * * Ck, Xk+?) = aE au)(cj) + am a ak-lOkg(Xk+1) am-1 j=m \irm-1 m, k-1, *,n(m < k) . Implicit in this form is the assumption that preferences are independent over time. We shall call u(c) the one-period utility function of consumption and g(x') the utility function of bequests. The constant aj > 0 (ao 1) is the patience factor linking the (one-period) utility functions of periods j and j + 1 given that the individual will be alive at decision point j + 1. When a. < 1 (aj ? 1) we shall say that impatience (patience) prevails in period j with respect to period j + 1. Similarly, the constant sj expresses the relative weight attached to bequests by the individual at decision point j, given that death will occur in period j. Since aj and dj are constants, we note that the rate of patience, while dependent on time, is independent of the overall level of utility (see [10]). We also postulate that the individual obeys the von Neumann-Morgenstern postulates [15];2 accordingly, the individual's objective is to maximize the expected utility attainable from consumption over his life-time and the estate remaining and bequeathed upon his death.3 We also assume that the individual always prefers more consumption to less in any period, i.e., that u(c) is monotone increasing, and that the bequest function g(x') is non-decreasing. Finally, we assume that the individual is risk averse with respect to con￾sumption, which implies that u(c) is strictly concave, and that u(c) and g(x') are twice differentiable. The notation developed in the previous section is summarized below before we proceed to construct our basic model: 2 We assume, however, that the continuity postulate has been modified in such a way as to permit unbounded utility functions. 3 In congruence with this premise, we assume that the functions (11) are cardinal. 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