STOCHASTIC

(c) Interpret (b).
(d) Illustrate some problem types for which such utility functions would arise naturally.
(e) Recall from the Hanoch-Levy paper that X is preferred to Y by risk averters, say
Y~$ X, if Eu(x) g Eu(Y) for all concave nondecreasing utility functions u. Under what
conditions are the statements Y~$ X and Y~< X equivalent ? [Hint: Investigate the case
in which Fand G cross.]
(f) Find a counter example to the following statement. If Y ^ X, that is, X is preferred
to Y by all risk lovers, then X~? Y; that is, Yis preferred to JTby all risk averters.
6. Consider the class of utility functions U3 defined on a finite interval [a, b] such that u is
three times continuously differentiable and u' > 0, u" g 0, and «'" 3: 0.
(a) Referring to the paper by Pratt, show that the absolute risk-aversion index may be
nondecreasing or nonincreasing for ue U3.
(b) Interpret the result in (a).
Recall from the Brumelle and Vickson paper that the concept of third-degree stochastic
dominance Y "^ X is defined by the condition
f u(x)[dF(x)-dG(x)\ g 0 for all u s U3,
where F and G are the cumulative distribution functions of X and Y, respectively.
(c) Show that Y