Suppose that Natasha’s utility function is given by ( I ) = 1 10 I , where I represents annual
income in thousands of dollars.
a. 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 $1000 gain with 50% probability and a $1000 loss with 50% probability.
The utility of her current income of $10,000 is u(10) 10(10) 10. Her expected
utility with the gamble is:
EU (0.5)( 10(11)) (0.5)( 10(9)) 9.987 10.
She would avoid the gamble. If she were risk neutral, she would be indifferent
between the $10,000 and the gamble, and if she were risk loving, she would prefer
the gamble.
You can also see that she is risk averse by noting that the square root function
increases at a decreasing rate (the second derivative is negative), implying
diminishing marginal utility.
b. Suppose that Natasha is currently earning an income of $40,000 (I 40) and can 40) and can
earn that income next year with certainty. She is offered a chance to take a new
job that offers a 0.6 probability of earning $44,000 and a 0.4 probability of
earning $33,000. Should she take the new job?
The utility of her current salary is 10(40) 20. The expected utility of the new job
is
EU (0.6)( 10(44)) (0.4)( 10(33)) 19.85,
which is less than 20. Therefore, she should not take the job. You can also determine
that Natasha should reject the job by noting that the expected value of the new job is
only $39,600, which is less than her current salary. Since she is risk averse, she
should never accept a risky salary with a lower expected value than her current
certain salary.
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?)
This question assumes that Natasha takes the new job (for some unexplained reason).
Her expected salary is 0.6(44,000) 0.4(33,000) $39,600. The risk premium is the
amount Natasha would be willing to pay so that she receives the expected salary for
certain rather than the risky salary in her new job. In (b) we determined that her new
job has an expected utility of 19.85. We need to find the certain salary that gives
Natasha the same utility of 19.85, so we want to find I such that u(I) 19.85. Using
her utility function, we want to solve the following equation: 10I 19.85. Squaring
both sides, 10I 394.0225, and I 39.402. So Natasha would be equally happy
with a certain salary of $39,402 or the uncertain salary with an expected value of
$39,600. Her risk premium is $39,600 39,402 $198. Natasha would be willing to
pay $198 to guarantee her income would be $39,600 for certain and eliminate the
risk associated with her new job.
income in thousands of dollars.
a. 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 $1000 gain with 50% probability and a $1000 loss with 50% probability.
The utility of her current income of $10,000 is u(10) 10(10) 10. Her expected
utility with the gamble is:
EU (0.5)( 10(11)) (0.5)( 10(9)) 9.987 10.
She would avoid the gamble. If she were risk neutral, she would be indifferent
between the $10,000 and the gamble, and if she were risk loving, she would prefer
the gamble.
You can also see that she is risk averse by noting that the square root function
increases at a decreasing rate (the second derivative is negative), implying
diminishing marginal utility.
b. Suppose that Natasha is currently earning an income of $40,000 (I 40) and can 40) and can
earn that income next year with certainty. She is offered a chance to take a new
job that offers a 0.6 probability of earning $44,000 and a 0.4 probability of
earning $33,000. Should she take the new job?
The utility of her current salary is 10(40) 20. The expected utility of the new job
is
EU (0.6)( 10(44)) (0.4)( 10(33)) 19.85,
which is less than 20. Therefore, she should not take the job. You can also determine
that Natasha should reject the job by noting that the expected value of the new job is
only $39,600, which is less than her current salary. Since she is risk averse, she
should never accept a risky salary with a lower expected value than her current
certain salary.
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?)
This question assumes that Natasha takes the new job (for some unexplained reason).
Her expected salary is 0.6(44,000) 0.4(33,000) $39,600. The risk premium is the
amount Natasha would be willing to pay so that she receives the expected salary for
certain rather than the risky salary in her new job. In (b) we determined that her new
job has an expected utility of 19.85. We need to find the certain salary that gives
Natasha the same utility of 19.85, so we want to find I such that u(I) 19.85. Using
her utility function, we want to solve the following equation: 10I 19.85. Squaring
both sides, 10I 394.0225, and I 39.402. So Natasha would be equally happy
with a certain salary of $39,402 or the uncertain salary with an expected value of
$39,600. Her risk premium is $39,600 39,402 $198. Natasha would be willing to
pay $198 to guarantee her income would be $39,600 for certain and eliminate the
risk associated with her new job.
