There are two farmers, A and B, in a valley. There are two states, ?ood and no ?ood, and each occurs with probability 1/2. Each farmer has a von Neumann-Morgenstern utility function with the utility of an amount, x, of the crop in any one state being ln(x). The harvest of farmer A equals 10 if there is no ?ood and equals 5 if there is a ?ood. The harvest of farmer B equals 10 whether there is a ?ood or not.

(a) Compute an Arrow-Debreu equilibrium with the price of the crop being 1 if there is no ?ood. Now suppose that a dam can be built that would prevent the ?ood. That is, if the dam were built, the crop of each farmer would be   10 in both states. The dam would cost a certain amount, say, t , of crop payable after the dam was built and after the harvest, in either state.

(b) Suppose there is no insurance and, hence, no Arrow-Debreu markets, so that each farmer's consumption equals his or her crop in either state. How much would each farmer be willing to pay for the dam, assuming that she or he pays for it alone?

(c) Suppose there is insurance, so that the farmers would consume their allocation under the Arrow-Debreu equilibrium. How much would each farmer be willing to pay for the dam, assuming that he or she pays for it alone? For each farmer, does insurance increase or decrease willingness to pay for the dam?