Consider a static, one period model of the macroeconomy. There is a representative household. The household can choose how much to consume and how much to work. There is no saving since the model is static. The household problem is: max Ct,Nt U = ln Ct + t ln (1 Nt) s.t.

Ct = wtNt + Dt

Here Nt is labor, Ct consumption, Dt a dividend received from ownership of the firm, and wt is the real wage. t is an exogenous parameter governing the disutility from work. A firm produces output according to the production technology: Yt = AtNt .

The firms dividend is: Dt = Yt wtNt .

The firms objective is to pick Nt to maximize Dt : max Nt Dt = AtNt wtNt .

(a) Use calculus to derive a first order necessary condition characterizing optimal behavior by the household.

(b) Use calculus to derive a first order necessary condition characterizing optimal behavior by the firm.

(c) Given your previous answers, what will be true about Dt in equilibrium?

(d) Given previous answers, derive the aggregate resource constraint.

(e) Use previous answers to derive an expression for equilibrium Yt as a function of exogenous variables, At and t . Verify that Yt is increasing in At and decreasing in t .

(f) Re-do the previous parts, but with a utility specification that is non-separable in consumption and leisure: U = ln [Ct + t ln (1 Nt)] .