Problem Set #3
Micro II – Spring Term 2010
1. Suppose that Natasha’s utility function is given by
, where I represents annual
income in thousands of dollars.
Is Natasha risk loving, risk neutral, or risk averse? Explain.
Natasha is risk averse. To show this, assume that she has $10,000 and is
offered a gamble of a $1,000 gain with 50 percent probability and a $1,000 loss
with 50 percent probability. Her utility of $10,000 is 10, (
10). Her expected utility is:
) + (0.5)(110
) = 9.987 < 10.
She would avoid the gamble. If she were risk neutral, she would be indifferent
between the $10,000 and the gamble; whereas, if she were risk loving, she
would prefer the gamble.
You can also see that she is risk averse by noting that the second derivative is
negative, implying diminishing marginal utility.
b.Suppose that Natasha is currently earning an income of $40,000 (
= 40) and
can earn that income next year with certainty. She is offered a chance to take a
new job that offers a .6 probability of earning $44,000, and a .4 probability of
earning $33,000. Should she take the new job? The utility of her current salary
, which is 20. The expected utility of the new job is
) + (0.4)(330
) = 19.85,
which is less than 20. Therefore, she should not take the job.
c. In (b), would Natasha be willing to buy insurance to protect against the variable
income associated with the new job? If so, how much would she be willing to pay for
that insurance? (Hint: What is the risk premium?)
Assuming that she takes the new job, Natasha would be willing to pay a risk
premium equal to the difference between $40,000 and the utility of the gamble
so as to ensure that she obtains a level of utility equal to 20. We know the