4. (22 points) MIU Model: Consider the representative household in the Money-in-the- Utility model, who chooses a path of consumption, leisure, and nominal money balances over an infinite horizon, {c+s,lt+s, Metso, to maximize the following objective func- tion: 1-a Mits V = 8 = - (.()" --... ct+s Pt+s + o ln(lt+s) s=0 where 1 > a > 0 and 0 > 0 are exogenous preference parameters, and B is a discount factor. The household faces the following nominal budget constraint each period: Ptct + At + M = (1 + it)At-1+ M_1 + Wint where Pt is the aggregate price level, At is nominal wealth, it is the nominal interest rate, W+ is the nominal wage rate, and nt is labor supply. The household also faces a unitary time endowment which holds each period so that 1 = lt + nt. The representative firm chooses labor and capital to produce output. In nominal terms: max {nt+s,kt+s+1} Profit = > (1+ir+s) **{Pt+s (ktontto) Petsinvets We+s4=+8} +s s=0 where 0) a > 0 and 0 > 0 are exogenous preference parameters, and B is a discount factor. The household faces the following nominal budget constraint each period: Ptct + At + M = (1 + it)At-1+ M_1 + Wint where Pt is the aggregate price level, At is nominal wealth, it is the nominal interest rate, W+ is the nominal wage rate, and nt is labor supply. The household also faces a unitary time endowment which holds each period so that 1 = lt + nt. The representative firm chooses labor and capital to produce output. In nominal terms: max {nt+s,kt+s+1} Profit = > (1+ir+s) **{Pt+s (ktontto) Petsinvets We+s4=+8} +s s=0 where 0)