Question

6. Lindsey checks her voice mail and is pleased to find that she has two messages from families wanting her to babysit for 5 hours on Friday night. She has sat for one family, the

Paradines, before and knows that they pay $10 an hour. She knows the second family, the Welches, but has never babysat for them before. From what she has heard, she thinks there is a 75% chance the Welches pay $12 an hour, and a 25% chance that they pay only $5 an hour. Because everyone knows each other very well, Lindsey knows that she will have to take the job from whichever family she calls first (that is, she can’t bail on the Welches if she discovers they are cheapskates). Other than the money, she values each babysitting job the same (all the kids are angels).

a) Assuming that Lindsey is risk neutral with respect to money, which option should she choose?

b) Now assume Lindsey’s utility curve for cash wealth is ln(w). She has just spent all her money, so has none. Based on this, should she sit for the Welches or the Paradines?

c) What would be the least amount of money per hour that the Paradines could pay her before she would prefer to switch to the Welches?

d) Lindsey needs the babysitting money to pay for gas. She is planning to drive to the coast with a friend on Saturday, and her tank is running low. On Saturday morning, Lindsey drives to her favorite gas station in Seattle and discovers that gas costs $3.20 per gallon. She has enough gas to make it to Tacoma, where she knows of a station that sometimes has cheaper gas. She estimates that there is a 50% chance that the Tacoma station will be charging $3.40 per gallon, and a 50% chance that it will be charging $3.10 per gallon. She decides that she would rather drive to Tacoma to get gas (she knows that the prices go up once you get past Tacoma, so she will definitely get gas there). Explain why we should not be surprised that she made this decision.

Answer 1

a) Witj certainty, Lindsey gets $10 an hour. With uncertainty, she expects to get 0.75* 12 + 0.25*5 = 9 + 1.25 = 10.25

Since 10.25 > 10 and Lindsey is risk neutral, she would prefer to work with uncertainty i.e Welches.

b) If her utility for money is ln(w) which is a concave function. Risk averse individuals have concave utility functions. Risk averse people would go for even lesser amount but with certainty. In his case, Lindsey has spent all her money and the utility derived only from the income she earns from babysitting, he would go for Paradines as that is a certain payment of $10/ hr.

c) utility with certainty = ln(10)= 2.30

Utility with uncertainty = 0.75* ln(12) + 0.25 ln (5)= 0.75*2.48 + 0.25*1.60 = 1.86 + 0.4 = 2.26

She gets a utility of 2.26 from Welches and hence if Paradines had to offer the least pay , they should give the certainty equivalent of 2.26

Ln(w) = 2.26

Taking exponent both sides

W= e^2.26

w= 9.58 is the least amount Paradines could offer before Lindsey switches to Welches

d) with certainty, cost of gas= 3.20

With uncertainty, cost of gas= 0.50*3.40 + 0.50*3.10= 3.25

Since she is risk neutral, she would take the risk and go to Tacoma and expect a lesser price than 3.20