Consider an economy described by the Real Business Cycles model where at any time t, the aggregate output is given by

Yt=Bktl1t

where (0,1), l is labor input and k is capital input. In turn, B is a fixed number, B>0. There is a continuum of mass one of agents who own everything in the economy. The agents derive utility from consumption and leisure. They have the following expected lifetime utility

U=+t=0(11+)t[u(ct)+ln(1lt)],

where ,>0, c is consumption, l is labor supply, and u(ct) is the instantaneous utility from consumption at time t. The instantaneous utility function is increasing and concave u'(ct)>0, and u''(ct).The agents face a per-period budget constraint

ct+it=wtlt+rtkt,

where wt is the wage rate paid for a unit of labor and rt is the return paid on a unit of capital k. In turn, i are investments that create new capital in each period

kt+1=it+(1)kt

where (0,1) is the depreciation rate.

a. (3 points) Derive the demand for labor and capital and use the demand functions to show that the budget constraint reduces to the national accounts identity.

b. (7 points) Derive labor supply and the optimal rule describing savings and consumption (Euler Equation).

c. (5 points) Write down the system of equations that solve for the equilibrium of the model. Write down also the variables which need to be solved.

d. (5 points) Assume that

u(ct)=c1t11

with >0 and derive the values of c and k in the steady-state as functions of model parameters.