STOCHASTIC

An individual's preferences are said to satisfy the weak additivity axiom if his preference
between two /-normal lottery tickets in a given pair is independent of the level of z, for all
pairs of f-normal lottery tickets and all t.
(f) Interpret the weak additivity axiom.
(g) Show that if an individual's preferences satisfy the weak additivity axiom, then u is
ordinally additive.
(h) Show that an individual's preferences satisfy the weak additivity axiom if and only
if u is additive or log additive.
Pollak (1967) summarizes his findings as follows:
The strong additivity axiom is satisfied if and only if the von Neumann-Morgenstern
utility function is the sum of functions each of which depends on the level of consumption
in a single period. The weak additivity axiom is satisfied in this case and is also
satisfied if the von Neumann-Morgenstern utility function can be written as the
product of functions each of which depends on the level of consumption in a single
period. Furthermore, these are the only cases in which the weak additivity axiom is
satisfied.
The assumption that an individual's von Neumann-Morgenstern utility function
is ordinally additive is a severe restriction on his preferences; it is equivalent to the
assumption that his ordinal utility function is additive. The assumption that his
von Neumann-Morgenstern utility function is additive or log additive implies a
more severe restriction on preferences, namely, that they satisfy the weak additivity
axiom. The assumption that his von Neumann-Morgenstern utility function is
additive implies a still more severe restriction, namely, the strong additivity axiom.
Simplifications resulting from special assumptions about the form of the von
Neumann-Morgenstern utility function are paid for in lost generality and applicability.
By expressing these assumptions in terms of preferences, it may be easier to assess
their cost.
4. Suppose/(x) is positive and is defined on the open convex set C in E".
(a) Show that /is convex on C if log /is convex on C.
(b) Show that /is pseudo-concave on C if log /is concave on C.
5. Suppose/, gi,...,g„: R"-* R, where/is pseudo-concave and the gj are quasi-concave.
Show that x* solves {max/(*)|g,(x)jg0, i=l,...,«} if and only if it solves the problem
with a linear objective function {maxV/(x*)'x|0i(x)=:O, /= 1,...,«}.
6. Consider the problem
Z(a,b) = min/(a,x),
s.t. g(x) Si, x £ 0,
where x e E", b e E m , g: E" -» E m , a e A