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Proof of this is left as an exercise.
We will now consider necessary and sufficient conditions on preference relations
for an expected utility representation to exist.
Theorem 5.5.1 If X contains only a finite number of possible values, then the
substitution and Archimidean axioms are necessary and sufficient for a preference
relation to have an expected utility representation.
Proof We will just sketch the proof that the axioms imply the existence of an
expected utility representation; the proof of the converse is left as an exercise.
For full details, see ?.
Since X is finite, and unless the consumer is indifferent among all possible choices,
there must exist maximal and minimal sure things, say p+ and p- respectively.
By the substitution axiom, and a simple inductive argument, these are maximal
and minimal in L as well as in X . (If X is not finite, then an inductive argument
can no longer be used and the Sure Thing Principle is required.)
From the Archimedean axiom, it can be deduced that for every other lottery, p,
Ü
there exists a unique V (p) such that
Ü
p
Ü Ü Ü
Revised: December 2, 1998
96 5.5. THE EXPECTED UTILITY PARADIGM
It is easily seen that V represents .
Linearity can be shown as follows:
Üthen
We leave it as an exercise to deduce from the axioms that if x
Ü Ü
Ü
Àx •" (1 - À) z
Ü Ü
Define z a" Àx •" (1 - À) ù.
Ü Ü
Then, using the definitions of V (x) and V (ù),
Ü
z
Ü Ü
Ü Ü
= (ÀV (x) + (1 - À) V (ù)) p+ •" (À (1 - V (x)) + (1 - À) (1 - V (ù))) p-
Ü Ü
It follows that
V (Àx •" (1 - À) ù) = ÀV (x) + (1 - À) V (ù) .
Ü Ü
This shows linearity for compound lotteries with only two possible outcomes: by
an inductive argument, every lottery can be reduced recursively to a two-outcome
lottery when there are only a finite number of possible outcomes altogether.
Q.E.D.
Theorem 5.5.2 For more general L, to these conditions must be added some tech-
nical conditions and the Sure Thing Principle.
Proof We will not consider the proof of this more general theorem. It can be
found in ?.
Q.E.D.
Note that expected utility depends only on the distribution function of the con-
sumption plan.
Two consumption plans having very different consumption patterns across states
of nature but the same probability distribution give the same utility. E.g. if wet
days and dry days are equally likely, then an expected utility maximiser is indiffer-
ent between any consumption plan and the plan formed by switching consumption
between wet and dry days.
The basic objects of choice under expected utility are not consumption plans but
classes of consumption plans with the same cumulative distribution function.
Chapter 6 will consider the problem of portfolio choice in considerable depth.
This chapter, however, must continue with some basic analysis of the choice be-
tween one riskfree and one risky asset, following ?.
Such an example is sufficient to show several things:
1. There is no guarantee that the portfolio choice problem has any finite or
unique solution unless the expected utility function is concave.
2. probably local risk neutrality and stuff like that too.
Revised: December 2, 1998
CHAPTER 5. CHOICE UNDER UNCERTAINTY 97
5.6 Jensen s Inequality and Siegel s Paradox
Theorem 5.6.1 (Jensen s Inequality) The expected value of a (strictly) concave
function of a random variable is (strictly) less than the same concave function of
the expected value of the random variable.
Ü Ü
E u W d" u E W when u is concave
Similarly, the expected value of a (strictly) convex function of a random variable
is (strictly) greater than the same convex function of the expected value of the
random variable.
Proof There are three ways of motivating this result, but only one provides a fully
general and rigorous proof. Without loss of generality, consider the concave case.
1. One can reinterpret the defining inequality (3.2.1) in terms of a discrete
Ü
random vector x taking on the value x with probability À and x with prob-
ability 1 - À:
"x = x " X, À " (0, 1)
f (Àx + (1 - À)x ) e" Àf(x) + (1 - À)f(x ), (5.6.1)
which just says that
Ü Ü
f (E [x]) e" E [f (x)] . (5.6.2)
An inductive argument can be used to extend the result to all discrete r.v.s
with a finite number of possible values, but runs into problems if the number
of possible values is either countably or uncountably infinite.
2. Using a similar approach to that used with the Taylor series expansion in [ Pobierz całość w formacie PDF ]