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ECO-5340 Decision Making Under Uncertainty Name_____________________________________

Problem Set 3. Risk Aversion

Spring, 2018

DUE DATE IS WEDNESDAY, FEB 28th AT 6.20 PM

USE THE HOMEWORK BOX, IN THE DOOR OF MY OFFICE, TO TURN IN YOUR HOMEWORK

PLEASE WRITE YOUR ANSWERS IN THE SPACES PROVIDED

1. Consider three individuals with the following VN-M utility functions:

u(W) = W , v(W) = W2 , and r(W) = ln W, where W denotes wealth and ln the natural log.

1.1 Take the first and second derivatives of the utility functions, and then calculate the absolute risk aversion

index of every individual. Write your calculations in the spaces provided in Table 1.

Table 1.

Individual First derivative Second derivative Absolute risk aversion

u(W) = W u(W) = u(W) = Au =

v(W) = 2 W v(W) = v(W) = Av =

r(W) = ln W r(W) = r(W) = Ar =

1.2 The individual u is _________________ (risk averse, risk neutral, risk lover) because the second derivate of

his utility function is _________________(negative, zero, positive)

1.3 The individual r is _________________ (risk averse, risk neutral, risk lover) because the second derivate of

her utility function is _________________(negative, zero, positive)

1.4 Individual u is _________________(more risk averse than, less risk averse than, has the same degree of risk

aversion of) v because his absolute risk aversion measure is ______________(greater than / lower than / equal

to) that of v.

1.5 Individual u is _________________(more risk averse than, less risk averse than, has the same degree of risk

aversion of) r because his absolute risk aversion measure is ______________(greater than / lower than / equal

to) that of r.

2. Consider the risky prospect X = (4, 16, 25; ¼, ½, ¼), and the individuals defined in problem 1 above.

2.1 Calculate the utility of having the expected value of the risky prospect X with probability 1 (column 2), and

the expected utility of prospect X for every individual. Write your result in the corresponding space of Table 2.

Table 2.

Individual Utility of the expected value of the

prospect X

Expected utility of the risky prospect X.

u(W) = W u(EX) = Eu(X) =

v(W) = 2 W v(EX) = Ev(X) =

r(W) = ln W r(EX) = Er(X) =

2.2 Individual u prefers __________________ (facing the prospect X / having the expected value of the lottery

X) because u(EX) is _________________ (greater/smaller) than Eu(X). This result is_______________

(expected/unexpected) given that the individual is __________________(risk averse/risk neutral/risk lover)

2.3 Individual r prefers __________________ (facing the prospect X / having the expected value of the lottery

X) because r(EX) is _________________ (greater/smaller) than Er(X). This result is_______________

(expected/unexpected) given that the individual is __________________(risk averse/risk neutral/risk lover)

3. Now consider the risky prospect X, as specified in problem 2, and the individuals described in problem 1.

3.1 Find the risk premium (column 2) and the Arrow-Pratt approximation of the risk premium (column 3) that

corresponds to each individual and the prospect X. Write your results in Table 3.

Table 3.

Individual Risk premium Arrow-Pratt approximation

u(W) = W (X, u) = (X, u) 

v(W) = 2 W (X, v) = (X, v) 

r(W) = ln W (X, r) = (X, r) 

3.3 The risk premium of individual v is ________________ (higher than /lower than / equal to) the risk

premium of individual u. This result _____________ (is/is not) consistent with the fact that the absolute

measure of risk aversion of v is _______________(higher than / lower than /equal to) that of individual u.

3.4 The risk premium of individual u is ________________ (higher/lower) than the risk premium of individual

r. This result _____________ (is/is not) consistent with the fact that the absolute measure of risk aversion of u is

_______________(higher / lower) than that of individual r.