Problem 3. Consider an exchange economy with two states. There are two agents with the same utility function U(c) : ln(c). Assume that their utility function is in the group of vNM utility. State 1 has a proba bility of 71'. The agents are endowed with the units of the consumption good at each state. Their endowments across themselves and across states are not necessarily equal. Total endowment of this consumption good is el in state 1 and 92 in state 2. Arrow-Debreu state prices are denoted by q1 and q2. (a) Write down agents' optimization problems and show that a _ W (a) 12 1 '11" 9'1 (what is yl and yg?). Assume that q1 + qg : 1 and solve for the state prices (what is ql and 192?). Hint: Recall the simple algebraic fact that % = g 2 :13. (b) Suppose there are two types of asset in the economy. A riskless asset (asset 1) pays off 1 (unit of the consumption good) in each state and has market price of P1 = 1. The risky asset (asset 2) pays off 0.5 in state 1 and 2 in state 2. Aggregate supplies of the two assets are Q1 and Q2. If the two states are equally likely, show that the price of the risky asset is _ 5621 + 4Q2 _ 4Q1 + 5Q2 Hint: Note that in this case state-contingent consumption of the agents are assured, in equilibrium, through their holdings of the two assets. To solve the problem you will need to use the results of part (a). You do not need to set up another optimization problem. P2