100 people each have certain incomes of $500. Each has the utility function , where U is utility and I is income Suppose there is an investment opportunity that would require an up-front cost of $ 500 and would yield gross income of $1500 or $0 with equal probability next year. Suppose i=0, so the present value of $1 next year is $1. Notice that this project if undertaken would raise the economy's expected income by $250, since it has expected value of $750 and costs $500. Would any one of the people be inclined to make this investment on his or her own? Why or why not? Find how the investment would affect her certainty-equivalent income. (If the investment is not undertaken, she has certainty-equivalent- because certain - income of $500.)

Suppose two people shared the costs and the benefits of the investment, each putting up $250 in return for an equal chance of $750 or $0 added to the $250 of income each would have net of the cost of making the investment. Would each want to make this happen? Would it raise or lower the certainty equivalent income of each; by how much?

Suppose finally that each of $100 people put up $5 each in return for a payment of $150 if the project goes well or nothing if it fails. What happens to the certainty-equivalent income of each person? What is the total increase in certainty-equivalent income that the investment produces in this case?

Can you see, with this problem, the importance of a financial system able to facilitate risk-sharing?