1. d. 15 points Consider the Diamond-Dybvig model of bank risk developed in Chapter 13. Suppose...

 

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1. d. 15 points Consider the Diamond-Dybvig model of bank risk developed in Chapter 13. Suppose that instead of storage, fiat money provides a means to save for either one or two periods. Due to there being a constant stock of fiat money and a constant population, the rate of return on fiat money in a stationary monetary equilibrium always equals one. As usual, capital earns a return of ut - 0 = x <1 over one period and earns a return of X> 1 over two periods. Let q be the stationary level of real balances and k be the stationary level of capital that a young agent acquires. Thus, the budget constraint of the agent when young is q + k≤y. Consider next the problem of the bank. As usual, the bank offers young agents a deposit contract wherein it promises to pay c2 goods to those who arrive when middle-aged and c3 goods to those who arrive when old. Per usual, the bank's funding constraints satisfy Nc₂uNy and (1-7) Nc3 =(1-4)NXy where 0≤u≤1. In this context, the reserve requirement implies that uzy. Assuming the utility is logarithmic, the bank solves the following optimization problem: max 7 In μy +(1-7) In (1-μ) Xy 1- π subject to μεγ where I have used the budget constraints to substitute out c₂ and c3 in terms of μ. If the reserve requirement is non-binding, then the optimal value of satisfies μ>y.. Thus, we can take the derivative of expected utility with respect to u and set it equal to zero in the usual way. Show that solving this problem implies that |-л Xy 5) π.Χ 02.03 1-7 -0. Show that this in turn implies that MRS From this marginal condition and the two financing constraints, it follows that μ = . Use this last equality to show that the solutions for c₂ and c3 are c₂ = y and c = Xy. 1. d. 15 points Consider the Diamond-Dybvig model of bank risk developed in Chapter 13. Suppose that instead of storage, fiat money provides a means to save for either one or two periods. Due to there being a constant stock of fiat money and a constant population, the rate of return on fiat money in a stationary monetary equilibrium always equals one. As usual, capital earns a return of ut - 0 = x <1 over one period and earns a return of X> 1 over two periods. Let q be the stationary level of real balances and k be the stationary level of capital that a young agent acquires. Thus, the budget constraint of the agent when young is q + k≤y. Consider next the problem of the bank. As usual, the bank offers young agents a deposit contract wherein it promises to pay c2 goods to those who arrive when middle-aged and c3 goods to those who arrive when old. Per usual, the bank's funding constraints satisfy Nc₂uNy and (1-7) Nc3 =(1-4)NXy where 0≤u≤1. In this context, the reserve requirement implies that uzy. Assuming the utility is logarithmic, the bank solves the following optimization problem: max 7 In μy +(1-7) In (1-μ) Xy 1- π subject to μεγ where I have used the budget constraints to substitute out c₂ and c3 in terms of μ. If the reserve requirement is non-binding, then the optimal value of satisfies μ>y.. Thus, we can take the derivative of expected utility with respect to u and set it equal to zero in the usual way. Show that solving this problem implies that |-л Xy 5) π.Χ 02.03 1-7 -0. Show that this in turn implies that MRS From this marginal condition and the two financing constraints, it follows that μ = . Use this last equality to show that the solutions for c₂ and c3 are c₂ = y and c = Xy.