文档介绍：Ec2723, Asset Pricing I
Class Notes, Fall 2003
Choice Under Uncertainty
John Y. Campbell1
First draft: July 30, 2003
This version: September 11, 2003
1 Department of Economics, Littauer Center, Harvard University, Cambridge MA 02138, USA.
Email john_******@.
ontrastbetween:
Ordinal utility Υ(x) is invariant to monotonic transformations, so Υ(x) is equiv-
• alent to Θ(Υ(x)) for any strictly increasing Θ.
Cardinal utility Ψ(x) is invariant to positive affine (aka linear) transformations,
• so Ψ(x) is equivalent to a + bΨ(x) for any b>0.
In finance we rely heavily on von Neumann-Morgenstern utility theory which
says that choice over lotteries, satisfying certain axioms, implies maximization of the
expectation of a cardinal utility function, defined over es.
Sketch of von Neumann-Morgenstern theory
Define states s = 1... es xs in a set X. Probabilities ps of the
different es then define lotteries. When S =3, we can draw probabilities in
2dimensions(sincep3 = 1 p1 p2). We get the “Machina triangle”.
−−
We define pound lottery as one which determines which primitive lottery we
are given. For example pound lottery L mightgiveuslotteryLa with proba-
bility α, and lottery Lb with probability (1 α).ThenL has the same probabilities
esasαLa +(1 α)Lb. −
−
We define a preference ordering over lotteries. A person is indifferent between
lotteries La and Lb, La Lb,iff La º Lb and Lb La.
∼º º
Continuity axiom: For all La, Lb, Lc . La Lb Lc, there exists a scalar
α[0, 1] . º º
∈ Lb αLa +(1 α)Lc.
∼−
This implies the existence of a preference functional defined over lotteries, .
indifference curves on the Machina triangle.
Independence axiom: La Lb αLa +(1 α)Lc αLb +(1 α)Lc.
º ⇒−º −
This implies that the preference functional is linear in probabilities (indifference
curves on the Machina triangle are straight lines), which means that we can define a
1
scalar us for each e xs .
XS XS
a b a b
L L psus