Consider the following utility maximization problem for a household: max {Ct,at+1,bt+1,dt+1>0} Σβ to t=0 s.t. at+1 + bt+

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Consider the following utility maximization problem for a household: max {Ct,at+1,bt+1,dt+1>0} Σβ to t=0 s.t. at+1 + bt+1+ dt+1+ Ct = Ct 1 1-o - 9/4+1=1) +r 1-0 (1 + i)at +(1+ib)bt + (1 + id) a where a > 0, ß € (0, 1), y > 0, c₁ denotes consumption in period t, and at+1, bt+1, dt+1 is the amounts of capital stock, government bonds, and bank deposits held at the end of period t (and thus at the beginning of period t+1). i, iû, and id denote the constant gross rates of return on at, bt, and dt, respectively, at period t. Thus, the household faces no uncertainty.

In the utility maximization problem, qt+1 measures the "liquidity aggregate" that consists of government bonds and bank deposits. The value of this variable represents the amount of convenience provided by the two types of financial assets, which are easily convertible into cash when the household needs to make payments. The utility of this convenience is assumed to be determined by the following function form: 1 ¶t = [(1 − A) b{ + Aď{] * where A E (0, 1) and p < 1.

In the following questions, you can assume ct > 0, at+1 > 0, bt+1 > 0, and dt+1> 0 for all t at the optimum. a. Derive the first order condition for at+1. Suppose σ = 2, p=0.9, y = 0.2, and i = 0.05. Also, assume C++1 = 5. What is the value of ct implied by the first order condition for at in this case? Derive the answer with 2 decimal places. b. In addition, derive the first order conditions for bt+1 and dt+1. Suppose λ = 0.5, p = 0.5, i 0.5, p = 0.5, i = 0.05, ip = 0.02, id = 0. What is the ratio of bt+1 to dt+1 implied by the first order conditions in this case? Derive the answer with 2 decimal places.