Chapter 9

A farmer's tomato crop is wilting, and he must decide whether to water it. If he waters the tomatoes, or if it rains, the crop will yield $\mathrm{\$}1,000$ in profits; but if the tomatoes get no water, they will yield only $\mathrm{\$}500.$ Operation of the farmer's irrigation system costs $\mathrm{\$}100.$ The farmer seeks to maximize expected profits from tomato sales. a. If the farmer believes there is a 50 percent chance of rain, should he water? b. What is the maximum amount the farmer would pay to get information from an itinerant weather forecaster who can predict rain with 100 percent accuracy? c. How would your answer to part (b) change if the forecaster were only 75 percentaccurate?

In Problem $8.5,$ Ms. Fogg was quite willing to buy insurance against a 25 percent chance of losing $\mathrm{\$}1,000$ of her cash on her around-the-world trip. Suppose that people who buy such insurance tend to become more careless with their cash and that their probability of losing $\mathrm{\$}1,000$ rises to 30 percent. What is the actuarially fair insurance premium in this situation? Will Ms. Fogg buy insurance now? (Note: This problem and Problem 9.3 illustrate moral hazard.)

Problem 8.4 examined a cost-sharing health insurance policy and showed that risk-averse individuals would prefer full coverage. Suppose, however, that people who buy cost-sharing policies take better care of their own health so that the loss suffered when they are ill is reduced from $\mathrm{\$}10,000$ to $\mathrm{\$}7,000.$ Now what would be the actuarial fair price of a cost-sharing policy? Is it possible that some individuals might prefer the cost-sharing policy to complete coverage? What would determine whether an individual had such preferences? (A graphical approach to this problem should suffice.)

Blue-eyed people are more likely to lose their expensive watches than are brown-eyed people. Specifically, there is an 80 percent probability that a blue-eyed individual will lose a $\mathrm{\$}1,000$ watch during a year, but only a 20 percent probability that a brown-eyed person will. Blue-eyed and brown-eyed people are equally represented in the population. a. If an insurance company assumes blue-eyed and brown-eyed people are equally likely to buy watch-loss insurance, what will the actuarially fair insurance premium be? b. If blue-eyed and brown-eyed people have logarithmic utility-of-wealth functions and current wealths of $\mathrm{\$}10,000$ each, will these individuals buy watch insurance at the premium calculated in part (a)? c. Given your results from part (b), will the insurance premiums be correctly computed? What should the premium be? What will the utility for each type of person be? d. Suppose that an insurance company charged different premiums for blue-eyed and brown-eyed people. How would these individuals' maximum utilities compare to those computed in parts (b) and (c)? (This problem is an example of adverse selection in insurance.

Suppose there are two types of workers, high-ability workers and low-ability workers. Workers' wages are determined by their ability- high ability workers earn $\mathrm{\$}50,000$ per year, lowability workers earn $\mathrm{\$}30,000.$ Firms cannot measure workers' abilities but they can observe whether a worker has a high school diploma. Workers' utility depends on the difference between their wages and the costs they incur in obtaining a diploma. a. If the cost of obtaining a high school diploma is the same for high-ability and low-ability workers, can there be a separating equilibrium in this situation in which high-ability workers get high-wage jobs and low-ability workers get low wages? b. What is the maximum amount that a high-ability worker would pay to obtain a high school diploma? Why must a diploma cost more than this for a low-ability person if having a diploma is to permit employers to identify high-ability workers?

Suppose Molly Jock wishes to purchase a high-definition television to watch the Olympic Greco-Roman wrestling competition. Her current income is $\mathrm{\$}20,000,$ and she knows where she can buy the television she wants for $\mathrm{\$}2,000$. She has heard the rumor that the same set can be bought at Crazy Eddie's (recently out of bankruptcy) for $\mathrm{\$}1,700,$ but is unsure if the rumor is true. Suppose this individual's utility is given by \[\text { utility }=\ln (Y),\] where $Y$ is her income after buying the television. a. What is Molly's utility if she buys from the location she knows? b. What is Molly's utility if Crazy Eddie's really does offer the lower price? c. Suppose Molly believes there is a $50-50$ chance that Crazy Eddie does offer the lowerpriced television, but it will cost her $\mathrm{\$}100$ to drive to the discount store to find out for sure (the store is far away and has had its phone disconnected). Is it worth it to her to invest the money in the trip?

Suppose an individual knows that the prices of a particular color TV have a uniform distribution between $\mathrm{\$}300$ and $\mathrm{\$}400$. The individual sets out to obtain price quotes by phone. a. Calculate the expected minimum price paid if this individual calls $n$ stores for price quotes. b. Show that the expected price paid declines with $n$, but at a diminishing rate. c. Suppose phone calls cost $\mathrm{\$}2$ in terms of time and effort. How many calls should this individual make in order to maximize his or her gain from search?

In some cases individuals may care about the date at which the uncertainty they face is resolved. Suppose, for example, that an individual knows that his or her consumption will be 10 units today $\left(C_1\right)$ but that tomorrow's consumption $\left(C_2\right)$ will be either 10 or $2.5,$ depending on whether a coin comes up heads or tails. Suppose also that the individual's utility function has the simple Cobb-Douglas form \[U\left(C_{1}, C_{2}\right)=\sqrt{C_{1} C_{2}}.\] a. If an individual cares only about the expected value of utility, will it matter whether the coin is flipped just before day 1 or just before day 2 ? Explain. b. More generally, suppose that the individual's expected utility depends on the timing of the coin flip. Specifically, assume that \[ \text { expected utility }=E_{1}\left[\left(E_{2}\left\{U\left(C_{1}, C_{2}\right)\right\}\right)^{\alpha}\right] \] where $E_1$ represents expectations taken at the start of day $1,E_2$ represents expectations at the start of day $2,$ and $\alpha$ represents a parameter that indicates timing preferences. Show that if $\alpha =1,$ the individual is indifferent about when the coin is flipped. c. Show that if $\alpha =2$, the individual will prefer early resolution of the uncertainty-that is, flipping the coin at the start of day 1 d. Show that if $\alpha =.5,$ the individual will prefer later resolution of the uncertainty (flipping at the start of day 2 ). e. Explain your results intuitively and indicate their relevance for information theory. (Note: This problem is an illustration of "resolution seeking" and "resolution-averse" behavior. See D. M. Kreps and E. L. Porteus, "Temporal Resolution of Uncertainty and Dynamic Choice Theory," Econometrica [January $1978]:185-200$.)