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 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 do this? Would it raise or lower the certainty-equivalent income of each; by how much?

Suppose finally that each of $100 people put $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?