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,{ct+s, lt+s, M dt+s}∞s=0, to maximize the following objective function:

Macroeconomic TheoryProblem Set 111. 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,{ct+s, lt+s, M dt+s}∞s=0, to maximize the following objective function:

V =∞∑s=0βs (α ln ct+s + (1 − α) ln(Mt+s/Pt+s) + σ ln lt+s)where β is a discount factor.

The household faces the following nominal budget constraint each period: Ptct + At + M Dt = (1 + it)At−1 + M Dt−1 + Wtntwhere Pt is the aggregate price level,

At is nominal wealth, it is the nominal interest rate, Wt is the nominal wage rate, and it is the labor supply. The household also faces a unitary time endowment which holds each period