STOCHASTIC

(b) Show that the only utility functions that have the property that their preference
orderings are independent of wealth are u(w) = aw or u(w) = be° w where a, b, and c are
real numbers. [Hint: Differentiate the equation in (a) with respect to w and /?, combine
these, and solve the resulting differential equation with da(f))/d/l = or ^ o.]
6. Suppose that an investor's utility function is u(w) = w 3 .
(a) Construct a fair gamble, an actuarily unfair insurance situation, and a wealth level
for which the investor simultaneously wishes to gamble and purchase insurance.
For the example used in (a), show that at different wealth levels the investor is
(b) unwilling to purchase the given insurance, and
(c) unwilling to take the given gamble.
(d) Is there a wealth level at which the investor is unwilling to purchase the given
insurance or to take the given gamble?
(e) Is the result of (d) general in the sense that its answer remains the same for all gambles
that illustrate (a)?
Illustrate the wealth ranges and sets of gambles for which the investor is
(f) a risk averter, or
(g) a risk taker.
Show that on E, u is
(h) quasi-concave, and
(i) not pseudo-concave.
7. (Jensen's inequality) Suppose A