13 In Section 13.2, we discussed the concept of the risk premium of a lottery and a risk-averse decision maker. In many situations, we would like to measure the degree of risk aversion associated with a utility function, and how a decision maker’s risk aversion depends on his or her wealth.

In this problem, we develop Pratt’s measure of absolute risk aversion. Consider Ivana, who has initial wealth W and utility function u(w) for final wealth position w. She has placed money in a small investment. The investment will increase her wealth by a random amount X, with E(X) 0.

We want to investigate how the risk premium of X depends on W. Let RP(W, X) be the risk premium associated with investment X if the decision maker’s wealth is W.

a Explain why RP(W, X) satisfies the following equation:

E(Utility for wealth level of W X)

utility of wealth level [W RP(W, X)]

b Perform a second-order Taylor series expansion on E(utility for wealth level of W X) about W.

c Perform a first-order Taylor series expansion on utility of wealth level [W RP(W, X)] about W.

d Equating the answers in

(b) and

(c) (disregard the remainder terms), show that

e Pratt’s measure of absolute risk aversion at wealth level W, called ARA(W), is defined to be twice the amount of risk premium per unit of variance when a decision maker is faced with a small lottery that has a zero expected value. Use your answer in part

(d) to explain why

f If ARA(W) is an increasing function of W, then u(w)
is said to exhibit increasing risk aversion, and if ARA(W)
is a decreasing function of W, then u(w) exhibits decreasing risk aversion. Is increasing or decreasing risk aversion more consistent with most people’s behavior?
Determine whether the following utility functions exhibit increasing or decreasing risk aversion:

Transcribed Image Text:

RP(W, X) -var(X)u"(W) 2u'(W)