STOCHASTIC

MULTIPERIOD RISK PREFERENCE 41
models which have recently appeared, preservation of decreasing absolute
and increasing relative risk aversion under maximization and expectation
operations is considered only for the special cases in which these measures
are constant (see, e.g., Refs. [6, 8, 13]), or else in terms of examples (such
as in Ref. [9]). On the other hand, when the measures are not assumed to
be constant, their properties have been examined in the context of singleperiod
decision models, as is the case in the classic papers [2,12], and in
the extensions developed in Ref. [14]. This paper's results thus contribute
to integration of the directions that investigations involving the
Arrow-Pratt measures have taken.
The outline of the remaining sections of the paper is as follows. In
Section 2 some new properties of risk aversion measures are developed,
with emphasis being given to the behavior of the risk aversion measures
under expectation operations. In the paper's third section, the preservation
of both decreasing absolute and increasing relative risk aversion under
maximization is considered in the context of the consumption-investment
decision problem already outlined. The fourth section considers some
extensions of the measures to the analysis of relations between consumption-investment
decisions and wealth. One of the characteristics of the
present approach is to consider measures of risk aversion somewhat more
generally as concavity measures, and to use the measures for comparisons
of the concavity properties of the additive utility functions.
2. SOME PROPERTIES OF RISK AVERSION MEASURES
This section is concerned primarily with studying the behavior of
absolute and relative risk aversion measures under expectation operations.
For the most part, this section develops results which will be employed in
Section 3 to study the consumer's multiperiod consumption-investment
decisions. However, the first two lemmas are not employed in the sequel;
they are included for the sake of their general interest.
DEFINITION. If/is a thrice-differentiable function defined on the real
line, then /'is said to exhibit decreasing absolute risk aversion iff r/ < 0,
where rf = —/"//'•
LEMMA 1. Iff is a function defined on the integers 1,2,..., and iff can be
written in the form
/(/;) = £ A""",
(=0
where k e (0, 1) and g(t) is an increasing function for all t > 0, then g(t) is
concave for all t ^Oiffr, is decreasing.
502 PART V. DYNAMIC MODELS