STOCHASTIC

that the property of nonincreasing Arrow-Pratt absolute risk aversion is
preserved under rather general conditions. For intertemporaily independent
risky returns, if the single-period utilities for consumption and the utility for
terminal bequests exhibit nonincreasing absolute risk aversion, then all the
induced utility functions for wealth also have this property. Mathematically,
this result arises from the fact that convex combinations of functions having
decreasing absolute risk aversion also have this property (see the Pratt paper,
Part II). Thus, the decreasing absolute risk-aversion property is preserved
under the expected-value operation, and is further preserved under maximization,
as Neave shows. The paper specifically includes the possibility of endpoint
as well as interior solutions in the presentation.
Neave also considers the behavior of the Arrow-Pratt relative risk-aversion
index of induced utility for wealth. He presents some sufficient conditions under
which the property of nondecreasing relative risk aversion is preserved. Since
this property is not, in general, preserved under convex combinations, results
can be obtained only under special conditions. Unfortunately, these conditions
are not easily interpreted, and involve the specific values of the optimal
decision variables as well as the properties of the utility functions. In general,
however, nondecreasing relative risk aversion obtains for sufficiently large
values of wealth. In Exercise ME-6 the reader is asked to attempt to widen
Neave's classes of utility functions that generate desirable risk-aversion
properties for the induced utility functions. Neave also relates properties of
the relative risk-aversion measures to wealth elasticity of demand for risky
investment. Arrow (1971) performed a similar study in the context of choice
between a risk-free and a risky asset, with no consumption. In the present case,
the choice is between consumption and risky investment, with no risk-free
asset. Neave shows that if the utilities for consumption and terminal wealth
in any period exhibit constant relative risk aversion equal to unity, then the
wealth elasticity for risky investment in that period is also equal to unity.
He further shows that the wealth elasticity of risky investment exceeds unity
if and only if the relative risk-aversion index of expected utility for wealth is
less than or equal to that of utility for consumption in the relevant period.
The Samuelson paper studies the optimal consumption-investment problem
for an investor whose utility for consumption over time is a discounted sum
of single-period utilities, with the latter being constant over time and exhibiting
constant relative risk aversion (power-law functions or logarithmic functions).
Samuelson assumes that the investor possesses in period t an initial wealth
w, which can be consumed or invested in two assets. One of the assets is safe,
with constant known rate of return r per period, while the other asset is risky,
with known probability distribution of return z, in period /. The zt are assumed
to be intertemporaily independent and identically distributed. Samuelson thus
generalizes Phelps' model to include portfolio choice as well as consumption.
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