1. ( (from Macho-Stadler and Perez-Castrillo) For several months not a drop of rain has fallen in Bilbao, and daily water usage restrictions have had to be enforced. All classic methods of solving the problem have failed, and the mayor is searching for drastic and imaginative solutions.One of the mayor's aides tells him about a sorcerer from India who is able to make it rain.Naturally, the mayor is unconvinced, and is afraid the sorcerer will turn out to be a fake. Therefore, the mayor is trying to find a contract to offer the sorcerer such that he will accept only if he is authentic.

Everyone knows that a phony sorcerer has no power over the rain.If the mayor contracted with a fake, the probability of rain would remain unchanged.It is currently estimated that the probability of rain in the next week (which is the time required to adequately test the powers of a sorcerer) is 2/100. An authentic sorcerer, in spite of having powers, is not infallible, and will increase the probability of rain, within a week, to 20/100.

All sorcerers are risk averse, with utility functions given by u(y) = y^0.5 , whereydenotes income.No authentic sorcerer will work unless the contract gives him at least U(lower bar) = 10; a fake, on the other hand, needs only U(lower bar) = 1.The mayor of Bilbao, who is risk neutral, is concerned with designing a contract which would be accepted only by an authentic sorcerer, since the political cost of being publicly ridiculed, should anyone discover that a fake had been hired for the job, is prohibitively high.

Suppose the mayor offered a contract promising to pay a sorcerer yr if it rains, and y0 if there is no rain.

1. What constraints must the payments in the contract satisfy?Why?
2. Derive the optimal contract (given the mayor has all the bargaining power).
3. Calculate the costs of this contract compared to that offered under symmetric information.