STOCHASTIC

MULTIPERIOD RISK PREFERENCE 47
(ii) s = A. In this case UN(A) = const + zN(A); hence #•„' is
determined by r2', and the sign of rz' is nonpositive by Lemma 4.
(iii) J e (0, A). It is useful first to establish some preliminary results.
By hypothesis and by definition of c and s,
Therefore,
-vN'(c) + z„'(s) = 0. (3.5)
UN'(A) = u„'(c)c' + zK'(s)s' = !>.v'(c)(l — s') + zN'(s)s',
so that by Eqs. (3.4) and (3.5),
Furthermore,
UN\A) = vN'(c) = zA.'(5). (3.6)
U'^A)-=i"N(c)c' = z"N(s)s'. (3.7)
Finally, total differentation of (3.5) gives
s' = r;.(c)/[t;(f) + -;(^)] e (o, i) (3.8)
by the strict concavity of rv and wN and by the fact that strict concavity
is preserved under expectation operations. 10
Accordingly,
rv(A) = rv(c)c' = r,(s)s', (3.9)
and it is desired to show that ru'(A) < 0. Now s' e (0, 1) by Eq. (3.8), and
rv, r3 are positive functions by strict concavity of vK and zK . Let [^ , S.,]
be any interval on which s' is monotone. Then on [flx, B2] either s', c\ or
both must be nonincreasing. Suppose s' is nonincreasing. Then
r v{A) = rz(s) s' must be nonincreasing since it is a product of nonnegative,
nonincreasing functions. If c' is nonincreasing, the same result obtains
by taking r^A) = r„(c) c and by noting that Eq. (3.8) implies c to be a
nondecreasing function of A.
The process may be continued for all intervals on which s' is monotone,
thus completing the first part of the proof.
The fact that UK will be strictly concave is proved in Ref. [7], where it is
shown that if g(x, y) is a (strictly) concave function and if C is a convex
set, then/(x) = max{»(x, y)} is a (strictly) concave function. Q.E.D.
yeC
THEOREM 2. Suppose that the functions DN and wN of Eqs. (3.3) and (3.4)
are strictly concave, and that the functions r,.* and rs* are nondecreasing
functions. Let e > 0. If' s"(s' — a) < 0, or ifs"(s' — a) > 0 and
, „ . „ . ( (1 — s')\ s' — a I s' \s' — a I )
i 5" K= mm P -f- , —'-—-
( A + e A + e )
10 This last fact is quite well-known. See, e.g., Ref. [5].
PART V. DYNAMIC MODELS