INVESTMENT AND CONSUMPTION 445 the j-th decision point, given that he is alive at that point, by j. In the event the individual passes away in period j-1, the amount of his resources at the end of that period will be termed his estate and will be designated wj We assume that the individual may also be the recipient of a non-capital income stream during all or part of his life-time. If the individual is alive at decision point j, he will be paid the(finite)installment pertaining to period 3, 1320, at the end of that period; if he is not alive, he will receive nothing. In this paper, we make the fairly strong assumption that the individuals tential non-capital income stream is exogenously determined and is known in advance. It may be thought of as consisting of the income from labor, pensions, unemployment compensation, ete We postulate that the individual faces both financial and productive oppor- tunities in each period. The first of these is the opportunity to borrow or lend arbitrary amounts of money in each period at the riskless(finite)rate r-1>0 on the condition that any borrowings (including interest)must be fully secured. The amount saved at decision point j will be denoted z1i; negative aii will then indicate borrowing For cont ce, we shall define (2) Y rirj+i where Y, may be interpreted as the present value of the individual's potential non-capital income stream at the i-th decision point. The productive opportunities faced by the individual consist of the possi bility of making risky investments. Let the total number of different risky (productive) opportunities available to the individual at decision point j be Mi-l, of which the first S,-1 s M-l may be sold short. A short sale will be defined as the opposite of a long investment, that is, if the individual ells opportunity i short in the amount e, he will receive a immediately (te do with as he pleases)in return for the obligation to pay the transformed value of e at the end of the period. The net proceeds realized at the end of of that period will be denoted Bij. Thus, returns to scale are assumed to period j from each unit of capital invested in opportunity i at the beginni stochastically constant, all investments are assumed to be realized in cash at the end of each period, and taxes and conversion costs, if any, are assumed to be proportional to the amount invested The amount invested in opportunity ,讠=2,……,M, at the j-th decision point will be denoted zij, and is,as indicated earlier, a decision variable along with z1i It will be assumed that the joint distribution functions F, given by (3)Fa2,m,…,mM)≡Pr{a≤m,月3s3,…,M;≤},j=1,…… are known and independent In addition, we shall postulate that the [Bish 1 In real world situations, the individual would, of course, be forced to derive h own subjective probability distributions. Numerous descriptions of how this may be ccomplished, on the basis of postulates presupposing certain consistencies in behavior, are available in the literature; see, for example the accounts of Savage [ 14]an Marschak [1 This content downloaded from 202.115.118.13 on Wed, II Sep 2013 02: 34: 55 AMINVESTMENT AND CONSUMPTION 445 the j-th decision point, given that he is alive at that point, by x;. In the event the individual passes away in period j - 1, the amount of his resources at the end of that period will be termed his estate and will be designated x. We assume that the individual may also be the recipient of a non-capital income stream during all or part of his life-time. If the individual is alive at decision point j, he will be paid the (finite) installment pertaining to period j, yj ? 0, at the end of that period; if he is not alive, he will receive nothing. In this paper, we make the fairly strong assumption that the individual's potential non-capital income stream is exogenously determined and is known in advance. It may be thought of as consisting of the income from labor, pensions, unemployment compensation, etc. We postulate that the individual faces both financial and productive oppor￾tunities in each period. The first of these is the opportunity to borrow or lend arbitrary amounts of money in each period at the riskless (finite) rate rj- 1 > 0 on the condition that any borrowings (including interest) must be fully secured. The amount saved at decision point j will be denoted z,j; negative zlj will then indicate borrowing. For convenience, we shall define (2) yj - yj Yi+1 ... + Yn j1* ,n r3 rjrj+i ri ... rn where Yj may be interpreted as the present value of the individual's potential non-capital income stream at the j-th decision point. The productive opportunities faced by the individual consist of the possi￾bility of making risky investments. Let the total number of different risky (productive) opportunities available to the individual at decision point j be Mj- 1, of which the first Sj - 1 < Mj -- 1 may be sold short. A short sale will be defined as the opposite of a long investment, that is, if the individual sells opportunity i short in the amount 0, he will receive a immediately (to do with as he pleases) in return for the obligation to pay the transformed value of 0 at the end of the period. The net proceeds realized at the end of period j from each unit of capital invested in opportunity i at the beginning of that period will be denoted ,Bj. Thus, returns to scale are assumed to be stochastically constant, all investments are assumed to be realized in cash at the end of each period, and taxes and conversion costs, if any, are assumed to be proportional to the amount invested. The amount invested in opportunity i, i = 2, ..., Mi, at the j-th decision point will be denoted zij, and is, as indicated earlier, a decision variable along with zlj. It will be assumed that the joint distribution functions Fj given by (3) Fi(x2, X3, * * *, XMj) - Pr{j92j < X2, l3j < X3* X8, ', imjj x XMj} 'j = 1, **, n are known and independent'. In addition, we shall postulate that the {J9ij} 1 In real world situations, the individual would, of course, be forced to derive his own subjective probability distributions. Numerous descriptions of how this may be accomplished, on the basis of postulates presupposing certain consistencies in behavior, are available in the literature; see, for example, the accounts of Savage [14] and Marschak [11]. 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