STOCHASTIC

592 NILS H. HAKANSSON
Pratt [11] notes that (16H18) are the only monotone increasing and strictly
concave utility functions for which the relative risk aversion index
is a positive constant and that (19) is the only monotone increasing and strictly
concave utility function for which the absolute risk aversion index
is a positive constant. 6
THEOREM 1: Let u{c), a, y, r, {/?,}, and Ybe defined as in Section 2. Then, whenever
u(c) is one of the functions (16H18) and ky < 1/a in Model I, a solution to (11)
subject to (12)-(14) exists for x > — Y and is given by
(22)
(23)
(24)
(25)
f(x) = Au(x + Y) + C,
c*(x) = B(x + Y),
z\(x) = (1 - B)(l - v*)(x + Y) --
Y,
z'(x) = (1 - B)v](x + Y)
where the constants u* (u* = £" 2 v]) and k are given by
(26) k = £r«(E'(ft-r)i>: + r)]
subject to
(27) D.-SSO, ifS,
and
(28) Prl^(pt-r)vi + r^o\ = \,
and the constants A, B, and C are given by
(i) in the case of Models /-//,
A = (1 -(a/cy) 1 '"-")'- 1 ,
(29) B= 1 -(ofcy) 1 " 1 "".
C = 0;
(i = 2,...,M)
6 The underlying mathematical reason why solutions are obtained in closed form (Theorems 1 and 2)
for the utility functions (16H19) is that these functions are also the only (monotone increasing and
strictly concave utility function) solutions (see [8]) to the functional equations u(xy) = v(x)w(y),
u(xy) = v(x) + w{y), u(x + y) = v(x)w{y), and u(x + y) = v(x) + w(y)t which are known as the
generalized Cauchy equations [1, p. 141].
530 PART V. DYNAMIC MODELS