Suppose that Natashas utility function is given by u 10 whe

Suppose that Natasha\'s utility function is given by u = 10, where l represents annual income n thousands of dollars. Is Natasha risk loving, risk neutral, or risk averse? Explain. O A. She is risk neutral because her utility function exhibits constant marginal utility. B. She is risk averse because her utility function exhibits diminishing marginal utility C. She is risk loving because her utility function exhibits increasing marginal utility. Suppose that Natasha is currently earning an income of $40,000 (I- 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? Natasha should not take the new job because her expected utility of 19.852 is less thanher current utility. (Round expected utility to three decimal places.) Suppose for some reason Natasha takes the new job. Would she 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?) Natasha would be willing to pay $198 for insurance. (Round your answer to the nearest penny)

Solution

U(I) = (10 I)^0.5    I = annual income in thousand of dollars.

(a) Suppose that Natasha has $10,000 , it implies $10 thousand of dollars and is offered a gamble with $1,000 gain with 50% probability and a $1,000 loss with 50% probability. So, her total utiltiy is = [(10)(10)] ^0.5 = 10.

Her expected utility , EU = (0.5) [(10)(9)]^0.5 + (0.5)[(10)(11)]^0.5

= (0.5)(90)^0.5 + (0.5)(110)^0.5

= 0.5(9.48) + 0.5(10.48)

= 4.74 +5.24 = 9.98 <10 .

It implies that Natasha is risk averse , she would avoid the gamble. Therefore, she is risk-averse because her utility function exhibits diminishing marginal utility.

Suppose she is currently income of $40,000 and can earn with certainity . 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.

Then, her utility of her current salary = [(10)(40)]^0.5 = (400)^0.5 = 20.

Expected utility of the new job = (0.6)[(10)(44)]^0.5 + (0.4)[(10)(33)]^0.5

= 0.6 (440)^0.5 + (0.4)(330)^0.5

= 0.6(20.97) + 0.4(18.16)

= 12.582 + 7.264

= 19.85

She should not take the new job because her expected utility is less than her current utility .

Suppose for some reason Natasha takes the new job . Then Natasha would be willing to pay the risk premium equal to the difference between $40,000 and the utility of gamble to ensure that she obtains a level of utility equal to 20. Utility of gamble = 19.85 as we calculated earlier. Substituting this into the utility function we get,

19.85 = (10 I)^0.5

I = 39,410. Thus, she would be willing to pay for insurance equal to the risk premium (40,000 -39,410) = $590.