Behavioral eco-lect 4: Choice under risk and uncertainty part 2

Expected utility

Solution to st-petersburg paradox (Bernouilli, 1738): people maximize expected utility, not expected value

EU(L) = p1 u(C1) + … + pn u(Cn) = ∑

Question: How much is an EU-maximizer with u(x) = ln(x) willing to pay to play the game?

Suppose u(x) = ln(x), then
EU = ½ln(2) + ¼ln(4) + … = 2ln(2) = ln(4) ≈ 1.39

Certainty equivalent

Your certainty equivalent CE of a lottery L is the amount for sure that you find as good as the lottery L:

Every lottery has its own certainty equivalent, which depends on the preferences of the decision maker

consider lottery L=(p:A, 1-p:B)

certainty equivalent to elicit utility

Required assumption: expected utility holds.

N.B. If a person does not satisfy expected utility, the method used to measure utility in tutorial B is not accurate.

Certainty equivalents can be used to measure utility functions

Risk attitudes in general

(100% : EV(L)) ≻ L <=> riskaverse

(100% : EV(L)) ~ L <=> risk neutral

(100% : EV(L)) ≺ L <=> risk seeking (= risk prone)

Risk attitudes in general cont.

The definitions of risk attitudes have the following implications for the certainty equivalents of lotteries:

CE(L) = EV(L) <=> risk neutral

CE(L) > EV(L) <=> risk seeking (=risk prone)

CE(L) < EV(L) <=> risk averse

Risk attitudes under expected utility

Under expected utility we have:

Linear utility <=> risk neutral

Convex utility <=> risk seeking (=risk prone)

Concave utility <=> risk averse

Expected utility and rationality

Standard economics uses expected utility.

NB: utility u in expected utility is cardinal.

Expected utility is considered rational (=normative) by many people.