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 AM
INVESTMENT 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 defined 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 consumption, 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. 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
448 NILS H. HAKANSSON Puy probability of death in period j(j s n), pj probability of death in period 3(2m), given that the individual is alive at the beginning of period m capital position at decision point estate at the end of period j-l, given that death occurs in period y, non-capital income received at the end of period 3 if the individual is alive at the beginning of period 3 Y present value at decision point j of the potential non-capital income tream rs-1 interest rate in period j, s amount lent at decision point 3 M; number of investment opportunities in period 3, number of investment opportunities which may be sold short in period 3 net proceeds realized at the end of period j from each unit invested in opportunity i,i=2,., M,, at the beginning of period j, Fs joint distribution function of B2; ,.. AM it zi amount invested in opportunity i, i=l,., Mi, at the beginning of t insurance premium paid at the beginning of period 3 for insurance in period t contractual insurance premium payable at the beginning of period 3 if individual is alive at that point, T present value at decision point j of potential premiums t;, tj+l,., tn-1 amount of consumption in period 3 one-period utility function of consumption utility function of bequests a, patience factor linking periods j and j+I if the individual remains alive at the end of period j patience factor linking periods j and 3+l if the individual passes 3. DERIVATION OF THE BASIC MODEL Te shall now identify the relation which determines the amount of capital (debt)on hand at each decision point in terms of the amount on hand at the previous decision point. This leads to the pair of difference equations: =∑2计+721+y x+=∑1+T1+y+tP √=1 where This content downloaded from 202.115.118.13 on Wed, II Sep 2013 02: 34: 55 AM
448 NILS H. HAKANSSON pj5 probability of death in period j(j ? n), Pmi probability of death in period j(2 m), given that the individual is alive at the beginning of period m, Xj capital position at decision point j, Xi estate at the end of period j - 1, given that death occurs in period j-l, yi non-capital income received at the end of period j if the individual is alive at the beginning of period j, Y3 present value at decision point j of the potential non-capital income stream, rj- 1 interest rate in period j, Z1j amount lent at decision point j, M6 number of investment opportunities in period j, Si number of investment opportunities which may be sold short in period j, 13ij net proceeds realized at the end of period j from each unit invested in opportunity i, i = 2, ***, Mj, at the beginning of period j, Fj joint distribution function of 2,j, *, 19M6j, zi6 amount invested in opportunity i, i = 1, ***, Mj, at the beginning of period j, tj insurance premium paid at the beginning of period j for insurance in period j, ti contractual insurance premium payable at the beginning of period j if individual is alive at that point, T3 present value at decision point j of potential premiums t6, t6+1, *.* *, cj amount of consumption in period j, Umk utility function at the beginning of period mn of consumption and bequests given that the individual passes away in period k ? m, u one-period utility function of consumption, g utility function of bequests, Ja, patience factor linking periods j and j + 1 if the individual remains alive at the end of period j, ai patience factor linking periods j and j + 1 if the individual passes away in period j. 3. DERIVATION OF THE BASIC MODEL We shall now identify the relation which determines the amount of capital (debt) on hand at each decision point in terms of the amount on hand at the previous decision point. This leads to the pair of difference equations: Mj (12) xj+l = E fiijzij + rjzlj + yj, -,*** n -1 i=2 and M j (13) xj+1 = X Iijzij + rjzlj + yj + tjlpjj, j- = * *, n i=2 where 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
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 AM
INVESTMENT 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
NILS H. HAKANSSON 22)(x1)=max{t(c)+Epg(x3+)+a(1-p)f+12x+ 1,……,n8 Letting a≡P b;≡a(1-p;), (22) may be written more concisely as f,(ai)= max u(ci)+ Elaig(a;+1)+ b fi+(ecj+iJI We shall now attempt to obtain the solutions to(25) for certain classes of the functions u(c)under different sets of assumptions concerning the bequest function g(a)and the availability of insurance More specifically, we shall consider the class of functions u(c) such that u(e) satisfies one (or more)of the functional equations u(ey)= v(ew(y), u(ay)=v(e)+ wly for c20. The functional equations(26) and(27)in three unknowns the set of equations usually referred to as the generalized Cauchy or Pexider's equations. That subset of their solutions, which is increasing and strictly concave in u, is given (leaving out v and (28) 0<7<1 Model I 7>0 Model il ule= le Model III Note that since u(c)is a cardinal utility function, the solutions(28)-(30)also include every solution 21+ 22u(c) to(26)and(27) where a, and 22>0 are In [9, it was also noted that (28)-(30)is the solution to the differential cv"(c)+r(c)=07>0 Thus,(28)-( 80) are also the only monotone increasing and strictly concave utility functions for which the proportional risk aversion index q*(c)≡-cw"c)/u'(c is a positive constant 4. NO BEQUEST MOTIVE, NO INSURANCE We shall first consider the simplest case, namely that in which there is no bequest motive and no insurance is available. The absence of a bequest motive implies that 6;g(x3+)=0, This content downloaded from 202.115.118.13 on Wed, II Sep 2013 02: 34: 55 AM
450 NILS H. HAKANSSON (22) fj(xj) = max {u(cj) + E[6jpjig(x'+1) + aj(1 -pjj)fj+,(xj+,)]l ji=,**, n. Letting (23) aj3 5jp and (24) b aj(l a - pjj) (22) may be written more concisely as (25) fj(xj) = max {u(cj) + E[aig(xj'+) + bjfj_1(xj+1)]}, j 1, ***, n We shall now attempt to obtain the solutions to (25) for certain classes of the functions u(c) under different sets of assumptions concerning the bequest function g(x') and the availability of insurance. More specifically, we shall consider the class of functions u(c) such that u(c) satisfies one (or more) of the functional equations (26) u(xy) = v(x)w(y), (27) u(xy) = v(x) + w(y), for c 2 0. The functional equations (26) and (27) in three unknowns belong to the set of equations usually referred to as the generalized Cauchy equations or Pexider's equations. That subset of their solutions, which is monotone increasing and strictly concave in u, is given (leaving out v and w) by [9]: (28) u(c) = cy 0 < r < 1 Model I (29) u(c) =-c-' r > 0 Model II (30) u(c) = log e Model III . Note that since u(c) is a cardinal utility function, the solutions (28)-(30) also include every solution 21 + 22u(c) to (26) and (27) where 21 and 22 > 0 are constants, if simultaneously, g(x') is represented by ;,2g(X'). In [9], it was also noted that (28)-(30) is the solution to the differential equation (31) cu"(c) + yu'(c) = 0 r > 0 . Thus, (28)-(30) are also the only monotone increasing and strictly concave utility functions for which the proportional risk aversion index (32) q*(c) -cu"(c)/u'(c) is a positive constant. 4. NO BEQUEST MOTIVE, NO INSURANCE We shall first consider the simplest case, namely that in which there is no bequest motive and no insurance is available. The absence of a bequest motive implies that (33) aig(xi+)=, j1,* ** ,n . 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