Problem 4. Consider an exchange economy with two states. There are two agents with the same utility function UC) = ln(c)

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Problem 4 Consider An Exchange Economy With Two States There Are Two Agents With The Same Utility Function Uc Ln C 1 (156.1 KiB) Viewed 31 times

Problem 4. Consider an exchange economy with two states. There are two agents with the same utility function UC) = ln(c). Assume that their utility function is in the group of N-M utility. State 1 has a proba- bility of 7. 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 e in state 1 and en in state 2. Arrow-Debreu state prices are denoted by q1 and 22. (a) Write down agents' optimization problems and show that 91 7 Y2 Y1 92 1 – (what is yi and y2?). Assume that qı + 92 = 1 and solve for the state prices (what is qı and q2?). Hint: Recall the simple algebraic fact that ( = = 8 = = (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 Qi and Q2. If the two states are equally likely, show that the price of the risky asset is 5Q1 +4Q2 P2 = 4Q1 + 502 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.