Chapter 6: Utility for Money

Question 1

A utility function U is continuous if and only if (context is gambles)

Answer 1

for every gamble and prize such that delta_prize not ~ gamble, then for some \epsilon>0, all points near x are not sim to gamble.

Question 2

Concavity (for gambles)

Answer 2

delta_{E\pi}\succeq \pi

Question 3

Certainty equivalent

Answer 3

x such that delta_x ~ pi

Question 4

risk premia

Answer 4

E\pi - C(\pi). C(\pi) exists by continuity. It is unique when U is strictly increasing.

Question 5

Coefficient of risk Aversion

Answer 5

• curvature

Question 6

Coefficient of relative risk aversion

Answer 6

• outcome * curvature.

Question 7

First order dominance

Answer 7

1) pi \succeq \rho for nondecreasing U; 2) CDF of pi is always less than rho on domain; 3) X_\pi = X_\rho + Y which is strictly positive.

Question 8

Second order dominance (strong or weaker)

Answer 8

weaker.

1) \pi \succeq \rho for nondecreasing concave U;
2) Integral of CDF of pi less than \rho.
3) X_\pi = X_\rho + Y which has positive value condition on X_\pi.

Question 9

Induced Preferences for Income asssumptions

Answer 9

Suppose mixture space theorem holds (vNM) and U is continuous.

Question 10

Induced Preferences for Income Results

Answer 10

Preferences over lotteries of income satisfy mixture space, are continuous, strictly increasing in y, and are concave w.r.t. y if U is concave w.r.t. x.