Suppose there are two equally probable states, s1 and s2, of the world and there are two consumers, A and B. Suppose further there are two goods, barley and wheat, which are perfect substitutes in consumption. Specifically, if c is the total amount of barley and wheat consumed by a consumer, then the consumer's utility is ln(1 + c) in either state. Consumer A has initially a stock of 1 unit of barley, consumer B has initially a stock of 1 unit of wheat; there is no production. In state s1, all the wheat is destroyed and none of the barley; in state s2, all the barley is destroyed and none of the wheat. Since the two goods are perfect substitutes (the marginal rate of substitution is constant and equal to 1), we can think of the situation as having just one consumption good with A endowed with 1 unit in state s1 and B endowed with 1 unit in state s2. Two types of contingent claims can be created: delivery of 1 unit of the consumption good in state s1 and delivery of 1 unit of the good in state s2. Consumers A and B are price-takers.
a) Find the competitive equilibrium of the market for the contingent claims.
b) Suppose a perfect signal is publicly available. Find the new competitive equilibrium.
c) Are consumers A and B better off with the public signal than without it? Explain why or why not.
d) Suppose now s1 occurs with probability 1/3 and s2 occurs with probability 2/3. Redo parts a)-c).