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 AM450 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