1. Savage State-Dependent Expected Utility Lecture IV

2. Savage’s state dependent expected utility Savage’s subjective expected utility theory takes as the object of choice state dependent outcomes.

3. Security 1Security 2

4. Writing the expected utility of the gamble out, we have What is changing? The probability or the states?

5. Savage proved a similar result to the von Neumann-Morgenstern model–if agents’ preferences obey certain axioms they have an expected utlity representation through both the probabilities of each outcome and the utility function. Specifically, the probabilities in the expected value framework may be endogenous.

6. Axiomatization of State– Dependent Expected Utility Start with some notation. In the state–preference format we speak of the utility function “mapping” from state-space into preferences:

7. Next, we want to introduce the notation of replacement when the return in state s is replaced with y.

8. We are interested in making the comparison Note that c,d  R S or are two vectors of outcomes or payoffs while y,w  R are two scalar incomes.

9. For example, take an investment

10. The independence axiom for state- dependent expected utility requires that Intuitively, the ordering remains the same if we replace on of the states with a common payoff.

11. Theorem 8.5.1 Assume that there are at least three states, S  3. Utility function u has a state-dependent expected utility representation iff it obeys the independence axiom.

12. Cardinal Utility Systems A second article: Friedman and L.J. Savage (1952) “The Expected-Utility Hypothesis and the Measurability of Utility” Journal of Polictical Economy 60(6): 463–74 followed the Friedman and Savage article we analyzed last class. This article responded to a comment by William Baumol that raised questions about the cardinality of utility measures.

13. Most of our current microeconomic theory is based on ordinal utility measurement where the actual levels of the utility are not important only the order or relative rank. Cardinality assigns importance to the actual numbers.

14. The Cardinal Coordinate Independence axiom is a stronger version of the independence axiom discussed above. Proposition 8.6.1 Cardional coordinate independence implies independence. Theorem 8.6.2 Utility function u has a state- independent expected utility representation iff it obeys the cardinal coordinate independence axiom.