STOCHASTIC

MULTIPERIOD RISK PREFERENCE 43
The next two lemmas are concerned with the behavior of the absolute
risk aversion measure under expectation operations. In these lemmas, as
well as in the remainder of the paper, only situations in which the expectations
exist and are finite will be considered.
LEMMA 3. (Pratt). Let g, h be functions such that r„ and rh are decreasing,
and let f = /, g -f tji; tt , to > 0. Then rf is a decreasing function.
Proof. The proof consists of showing that r/ ••".; 0; see Ref. [12, p. 132].
LEMMA 4. Let f(x) = E',g(k -f ?*)}, where I is a random variable
defined on any interval (C/-). Then if g exhibits decreasing absolute risk
aversion, f exhibits decreasing absolute risk aversion.
Proof For any fixed value of £, say c0, the function g(k -j- C0x)
exhibits decreasing absolute risk aversion. Lemma 3 states that the nonnegative
sum of decreasingly absolute risk averse functions is decreasingly
absolute risk averse. Then since expectation can be regarded as a nonnegative
summation operation whether £ is discrete or continuous, it
follows that/ exhibits decreasing absolute risk aversion.
Some results concerning the behavior of the relative risk aversion
measure under expectation operations will next be considered.
LEMMA 5. If g is a strictly increasing, strictly concave function which
exhibits constant relative risk aversion, and ifk > 0, then rh* is increasing,
where h(x) = g(x + k).
Proof. Note that rh*(x) =-• -xh"(x)/h'(x) = -xg"(x + k)jg'(x + k).
Since r,*(x) = m > 0, rg(x) = mix, and r„(x 4- k) = m!(x + k). But
*•»*(•*) = xr A x + k) = xm/(x + k), an increasing function of x.
COROLLARY. If k < 0, g(x + k) exhibits decreasing relative risk
aversion.
Lemma 5 does not imply that E{ g(x + />)} exhibits constant relative
risk aversion when E(p) — 0, as is shown by considering
(i)ln(jc+l) + tt)ln(*-2),
where ln(.v) denotes the natural logarithm of .Y. The measure of relative
risk aversion for this function is (3.Y 3 — 6x 2 + 9X)/(3Y 3 — 6x 2 — 3x -f 6),
a decreasing function.
EXAMPLE. The nonnegative sum of increasingly relative risk averse
functions is not necessarily increasingly relative risk averse.
504 PART V. DYNAMIC MODELS