STOCHASTIC

wealth, then his utility function must be a logarithmic or power function
(Exercise CR-17). Exercise ME-3 investigates the behavioral significance of
assuming that a utility function defined over several commodities is separable
or that its logarithm is separable.
The calculation of an allocation vector that maximizes expected utility
may proceed via a stochastic programming or nonlinear programming
algorithm as discussed in Part III. However, when the utility function is
concave one can easily determine bounds on the maximum value of expected
utility (Exercise ME-1). The lower bound is a consequence of Jensen's
inequality (Exercise CR-7) which states that the expected value of a concave
function is never exceeded by the value of the function evaluated at its mean.
Some sharper Jensen-like bounds applicable under more restrictive assumptions
appear in Ben-Tal and Hochman (1972) and Ben-Tal, Huang, and
Ziemba (1974). Exercise CR-11 introduces the notion of a certainty equivalent,
that is, a fixed vector whose utility equals the expected utility.
The expected utility approach provides a natural framework for the analysis
of financial decision problems. However, there are several alternative
approaches which are of particular interest in certain instances. Most notable
are the approaches that trade off risk and return, and various safety-first and
related chance-constrained formulations. These approaches and their
relations with the expected utility approach are considered in some detail in
Part III. See also Exercise CR-18 for an approach that augments a decisionmaker's
returns and costs in such a way that gambling and insurance conclusions
that normally hinge on the shape of the utility function may be
obtained for linear utility functions.
Fishburn's article provides a succinct but brief presentation of an expected
utility theory. The mathematical level in this article is perhaps higher than in
nearly all of the other material in this book. For this reason some readers
may wish to consult the less general but more lucid developments by Arrow
(1971), Jensen (1967a, b), and Pratt et al. (1964). A much fuller treatment
(and comparison) of many expected utility theories may be found in the work
of Fishburn (1970). See also Fishburn's paper (1968) for a concise survey of
the broad area of utility theory.
II. Convexity and the Kuhn-Tucker Conditions
Many of the functions involved in financial optimization problems are
convex (or concave, the negative of a convex function). Convex functions
are defined on convex sets, which are sets such that the entire closed line
segment joining any two points in the set is also in the set. Exercise ME-15
outlines some important properties of convex sets. Convex functions have
4 PART I MATHEMATICAL TOOLS