STOCHASTIC

MIND-EXPANDING EXERCISES
1. Suppose that u(w) is concave where wealth w = £j_! £(xt = u(x b ) if x" > 0 for all / and x, b = 0 for some t;
(iii) duldx, > 0; and (iv) u is twice continuously differentiable. We say that « is additive if
there exist T functions u,(x,) such that u(x) = J«,W; ordinally additive if there exists
an F, F > 0 such that F(u(x)) = ^u,(xt); and log additive if there exist constants a and
b > 0 such that \og[a + bu(x)] = X H t(*t)-
(a) Illustrate some common utility functions that are additive, ordinally additive, and
log additive, respectively.
Let X" and Y" represent two T-dimensional consumption paths and let y„e[0,l].
A lottery ticket La denoted by (ya,X",Y°) provides consumptions X" and Y° with probabilities
y„ and 1 — ya, respectively. Note that u(La) = y„u(X a ) + {\ — y„)u(Y°). Two lottery
tickets La and Lb are a pair of ^-standard lottery tickets if y„ = yb and x° = xt b for a given t.
(b) Interpret the concept of a /-standard lottery.
An individual's preferences are said to satisfy the strong additivity axiom if his preference
between two f-standard lottery tickets in a given pair is independent of the level of x, for
all pairs of /-standard lottery tickets and all /.
(c) Interpret the strong additivity assumption.
(d) Show that an individual's preferences satisfy the strong additivity axiom if and only
if u is additive.
We say that two simple lottery tickets are a pair of /-normal lottery tickets if y„ = yb = i
and x," = y," = x, b = y," = z, for a given /.
(e) Interpret the concept of a /-normal lottery ticket.
MIND-EXPANDING EXERCISES 67