a. b. C. This question is about Diamond-Dybvig model of banks. In that model, the economy...
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a. b. C. This question is about Diamond-Dybvig model of banks. In that model, the economy lasts three periods (t = 0, t = 1, and t = 2). There is a continuum of agents indexed by ie [0,1]. Each agent is endowed with one "unit" at t = 0. There is no endowment in t = 1 and t = 2 and there is no production or work. Agent i has the utility function U(c₁,₂)=0, Inc₁ +(1–0;) Inc₂ where 50 with probability 0.3 1 with probability 0.7 0₁ and 0; is i.i.d. across agents. There are two technologies for transferring resources across time. One unit invested in technology 1 (liquid asset) at t = 0 returns three units at t = 1 and one unit invested at t = 1 returns three unit at t = 2. One unit invested in technology 2 (illiquid asset) at t=0 yields ten units at t = 2, but only two units if the investment is "scrapped" at t= 1. Any fraction of a unit can be invested in either technology. The agent's objective function is expected utility. Solve for the agent's optimal portfolio of assets (the agent's optimal use of the two technologies) in autarky. ("Autarky" means that the agent's consumption derives from the agent's direct use of the investment technologies.) Also give the agent's consumption in each period. SET UP THE PROBLEM AND SOLVE IT. SHOW YOUR WORK TO RECEIVE CREDIT. (25 points) A bank is a coalition of young agents that forms in t = 0. The bank accepts deposits of the endowment from the members of the coalition in t = 1 and offers a deposit contract returning r₁ units if the agent withdraws their deposit at t = 1 and r₂ units if the agent withdraws their deposit at t = 2. The bank chooses r₁ and r₂ and invests deposits in the liquid and illiquid technologies. A bank is a large coalition and therefore the bank knows that 0.7 of its depositors will have a realization of equal to 1, and 0.3 of its depositors will have a realization of equal to 0. However, the bank does not observe the realization of any agent's liquidity shock. State the bank's optimization problem. Explain the objective function. Also explain each constraint in the problem. Then solve the bank's optimization problem. SET UP THE OPTIMIZATION PROBLEM AND SOLVE IT. SHOW YOUR WORK TO RECEIVE CREDIT. Finally, interpret (explain) the solution. (25 points) State the bank's deposit contract (state the values of r₁ and r₂ that the bank will offer to depositors in t = 0). Explain why these values of r; and r₂ are the optimal contract. HINT: Relate r₁ and r₂ to the values of consumption by impatient and patient depositors. Also answer the following two questions: How can we be sure that patient depositors won't pretend to be impatient depositor? How do we know that agents don't make direct use of the investment technologies in preference to depositing their endowment in a Diamond-Dybvig bank in t=0? (20 points) a. b. C. This question is about Diamond-Dybvig model of banks. In that model, the economy lasts three periods (t = 0, t = 1, and t = 2). There is a continuum of agents indexed by ie [0,1]. Each agent is endowed with one "unit" at t = 0. There is no endowment in t = 1 and t = 2 and there is no production or work. Agent i has the utility function U(c₁,₂)=0, Inc₁ +(1–0;) Inc₂ where 50 with probability 0.3 1 with probability 0.7 0₁ and 0; is i.i.d. across agents. There are two technologies for transferring resources across time. One unit invested in technology 1 (liquid asset) at t = 0 returns three units at t = 1 and one unit invested at t = 1 returns three unit at t = 2. One unit invested in technology 2 (illiquid asset) at t=0 yields ten units at t = 2, but only two units if the investment is "scrapped" at t= 1. Any fraction of a unit can be invested in either technology. The agent's objective function is expected utility. Solve for the agent's optimal portfolio of assets (the agent's optimal use of the two technologies) in autarky. ("Autarky" means that the agent's consumption derives from the agent's direct use of the investment technologies.) Also give the agent's consumption in each period. SET UP THE PROBLEM AND SOLVE IT. SHOW YOUR WORK TO RECEIVE CREDIT. (25 points) A bank is a coalition of young agents that forms in t = 0. The bank accepts deposits of the endowment from the members of the coalition in t = 1 and offers a deposit contract returning r₁ units if the agent withdraws their deposit at t = 1 and r₂ units if the agent withdraws their deposit at t = 2. The bank chooses r₁ and r₂ and invests deposits in the liquid and illiquid technologies. A bank is a large coalition and therefore the bank knows that 0.7 of its depositors will have a realization of equal to 1, and 0.3 of its depositors will have a realization of equal to 0. However, the bank does not observe the realization of any agent's liquidity shock. State the bank's optimization problem. Explain the objective function. Also explain each constraint in the problem. Then solve the bank's optimization problem. SET UP THE OPTIMIZATION PROBLEM AND SOLVE IT. SHOW YOUR WORK TO RECEIVE CREDIT. Finally, interpret (explain) the solution. (25 points) State the bank's deposit contract (state the values of r₁ and r₂ that the bank will offer to depositors in t = 0). Explain why these values of r; and r₂ are the optimal contract. HINT: Relate r₁ and r₂ to the values of consumption by impatient and patient depositors. Also answer the following two questions: How can we be sure that patient depositors won't pretend to be impatient depositor? How do we know that agents don't make direct use of the investment technologies in preference to depositing their endowment in a Diamond-Dybvig bank in t=0? (20 points)
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Related Book For
Probability And Statistics
ISBN: 9780321500465
4th Edition
Authors: Morris H. DeGroot, Mark J. Schervish
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