STOCHASTIC

or if
s"(s'
MULTIPERIOD RISK PREFERENCE 51
0 and | s" | < min \ (1 - s')\ s' -o\ s'\: • o \ l
A + e A + e
where s" = s"(A), s' = s'(A), then Un is an increasingly relative risk
averse function.
Proof. The proof is exactly the same as for Theorem 2.
Note that, as indicated in the remarks following Theorem 2, the hypotheses
of Theorem 4 imply that rf, will be nondecreasing whenever the
functions c„ (and consequently the functions sn) are linear.
Having developed sufficiency conditions for the functions i/„ to exhibit
decreasing absolute and increasing relative risk aversion, it is now appropriate
to turn to a discussion of the uses of these measures in studying
wealth elasticities. It is rather unfortunate that, since the measure of
absolute risk aversion is preserved under more general assumptions than
is the measure of relative risk aversion, the former's ability to characterize
the sensitivity of the optimal policy precisely is less strong than that of the
latter.
4. WEALTH ELASTICITIES AND RISK PREFERENCE
This section discusses how the concavity measures studied in the context
of the multiperiod decision model of Section 3 can be employed to develop
properties of the policy functions c-„ and s„ . The present work provides
some results additional to those in Refs. [2, 14], thus further illustrating
the usefulness of the Arrovv-Pratt concavity measures.
The first of these extensions is concerned with developing benchmark
measures of the elasticity of demand for risky investment.
LEMMA 8. If both c„ and z„ exhibit constant relative risk aversion equal
to unity 1 " the wealth elasticity of demand for risky investment is also equal
to unity.
Proof By Eq. (3.8),
By constant relative risk aversion, .
•v' -