STOCHASTIC

distribution with respect to which the given random variables form a submartingale
or martingale sequence; see Exercise V-CR-19 for elementary
properties of martingales. Equivalently, one of the random variables must
have the same distribution as the other plus nonpositive "noise." Exercise
ME-12 presents another approach to the proof of this result. Exercise ME-13
shows how dominance for random variables having unequal means can be
reduced to the equal mean case. Exercise ME-15 asks for an algorithm to
construct the random noise variable mentioned above.
II. Measures of Risk Aversion
The Pratt paper shows how to relate risk preferences of an individual to
specific properties of his utility function. Pratt introduces two important riskaversion
indices which are often called Arrow-Pratt risk-aversion indices
because they were independently studied by Arrow [see Arrow (1965, 1971)]
and Pratt. These are (1) the absolute risk-aversion index, and (2) the relative
risk-aversion index. Since an individual's preference orderings are invariant
under positive linear transformations of the utility functions, any meaningful
risk-aversion measures must also be invariant under such transformations.
The absolute and relative risk-aversion indices both satisfy this requirement.
Pratt shows that a natural restriction on the utility function is that its absolute
risk-aversion index be a nonincreasing function. This assumption implies that
the insurance premium an investor is willing to pay to cover a fixed risk is a
nonincreasing function of wealth. Several useful properties of such utility
functions are presented. First, the property of nonincreasing absolute risk
aversion is shown to remain valid for positive linear transformations of the
wealth level. Composition of one such function with another is shown to yield
a function of the same type. Finally, the set of such functions is shown to be
convex. These properties can be utilized to generate a large number of utility
functions having nonincreasing absolute risk aversion by starting with a small
sample of such functions. Pratt also discusses investor behavior with respect
to risks which are proportional to wealth. He shows that the assumption of
nondecreasing relative risk aversion is equivalent to the requirement that the
investor's proportional insurance premium increases with wealth.
Additional material regarding the risk-aversion measures and their application
is found in the exercises. Exercise CR-10 shows that risky assets are not
inferior goods if the absolute risk-aversion index is nonincreasing. Exercise
CR-24 considers the case when the risky asset has a two-point distribution.
Exercise CR-25 shows how the results apply to simple insurance and foreign
exchange problems. Exercise ME-19 is concerned with plunging (avoidance
of a secure asset) in portfolio selection. Exercise ME-21 presents a more
84 PART II QUALITATIVE ECONOMIC RESULTS