In this problem, you will see why the ‘‘Equity Premium Puzzle’’ described in Application 4.5 really is  puzzle. Suppose that a person with $100,000 to invest believes that stocks will have a real return over the next year of 7 percent. He or she also believes that bonds will have a real return of 2 percent over the next year. This person believes (probably contrary to fact) that the real return on bonds is certain—an investment in bonds will definitely yield 2 percent. For stocks, however, he or she believes that there is a 50 percent chance that stocks will yield 16 percent, but also a 50 percent chance they will yield 2 percent. Hence stocks are viewed as being much riskier than bonds.

 

a. Calculate the certainty equivalent yield for stocks using the three utility functions in Problem 4.6. What do you conclude about whether this person will invest the $100,000 in stocks or bonds?

 

b. The most risk-averse utility function economists usually ever encounter is ^. If your scientific calculator is up to the task, calculate the certainty equivalent yield for stocks with this utility function. What do you conclude?

 

Problem 4.6

 

Sometimes economists speak of the certainty equivalent of a risky stream of income. This problem asks you to compute the certainty equivalent of a risky bet that promises a 50-50 chance of winning or losing $5,000 for someone with a starting income of $50,000. We know that a certain income of somewhat less than $50,000 will provide the same expected utility as will taking this bet. You are asked to calculate precisely the certain income (that is, the certainty equivalent income) that provides the same utility as does this bet for three simple utility functions:

 

 

 

What do you conclude about these utility functions by comparing these three cases?