When old, the household consumes his savings:

co,t+1  ≤ (1 + Rt+1) st                                                 (3)

where Rt+1  is the gross return on capital in t+1. Note that we assume here that capital does not depreciate, i.e. δ = 0. Cohort size at t is given by Nt . Population grows at rate n.

(a) Derive expressions for the household’s optimal consumption when young (cy(∗),t ), optimal consumption when old (co(∗),t+1), and optimal savings st(∗) .

Assume that output in this economy is produced by a firm which lives forever and combines labor Lt  and capital Kt  to produce output according to

Yt  = At F(Kt ,Lt ) = At Kt(α)Lt(1)−α (4)

The firm hires labor at wage rate wt  and capital at interest rate Rt .

(b) Solve the firm’s profit maximization problem to derive its optimal labor demand Lt(D) and capital demand Kt(D) .

(c) Determine the equilibrium on the labor market. That is, derive Lt(∗)  and w . The capital stock evolves according to

Kt+1  = st Nt                                            (5)

(d) Determine the equilibrium on the capital market. That is, derive K and Rt(∗) . (e) Show that in equilibrium, the goods market clears: Yt  = Ct + It .

(f) Find an expression that links the future capital stock per worker kt+1  ≡ to the current capital stock per worker kt   ≡  N(K)t(t)    and the exogenous variables of the model (α,β,A,n).

(g) Solve for the capital stock per worker and output per worker in the steady state of the economy.

(h) Determine what happens to output per worker in steady state if households become more patient, i.e. if β increases. Provide some intuition.

(i) Assume that you have two countries, A and B .  Suppose that in steady state, income per capita in country A is three times higher than income in country B. It holds that βA  = 0.7, βB  = 0.8, and αA  = αB  = 1/3.  Further assume that nA  = nB .  Assuming that AB  = 1, determine the value of AA  that rationalizes this income difference.

Question 3 (20 points)

This question is similar to Question 2 with the only difference that we assume a different lifetime utility function for households. More specifically, lifetime utility for a household born at t is now given by

U = (cy,t )0.5 + β(c0,t+1)0.5 ,   β ∈ [0, 1]                                     (6)

As before, cy,t  denotes consumption when young, and c0,t+1  denotes consumption when old. The household faces two flow budget constraints. When young, the households inelastically supplies one unit of labor and earns wage rate wt :

cy,t + st  ≤ wt                                           (7)

When old, the household consumes his savings:

co,t+1  ≤ (1 + Rt+1) st                          (8)

where Rt+1  is gross the return on capital in t+1. Note that we assume here that capital does not depreciate, i.e. δ = 0. Cohort size at t is given by Nt . Population grows at rate n.

(a) Derive an expression for the household’s optimal savings st(∗) .  Note how it differs from the expression for savings you obtained in part (a) of Question 2.

Assume that output in this economy is produced by a firm which lives forever and combines labor Lt  and capital Kt  to produce output according to

Yt  = At F(Kt ,Lt ) = At Kt(α)Lt(1)−α (9)

The firm hires labor at wage rate wt  and capital at interest rate Rt .

(b) Determine the equilibrium on the labor market. That is, derive Lt(∗)  and w . Is the labor market equilibrium different from the equilibrium you found in Question 2?

The capital stock evolves according to

Kt+1  = st Nt                                                  (10)

(c) Sketch the equilibrium on the capital market, i.e. provide a graph of capital supply and capital demand in a space where capital Kt  is on the x-axis and the interest rate Rt  is on the y-axis. [Note:  You are not required to solve for the equilibrium analytically here as this would be quite challenging algebraically.]

(d) Describe how the capital market equilibrium you depicted in part (c) is different from the capital market equilibrium in Question 2.

(e) What happens to equilibrium capital when households become more patient, i.e.  if β increases? Is the effect on equilibrium capital smaller or greater compared to the effect of an increase in β on equilibrium capital in Question 2? Provide some intuition. [Note: You do not need to provide any formal analysis here, intuition based on your results in part (c) and (d) suffices.]

Question 4 (20 points)

The central equation governing the dynamics of capital per worker in the Solow model is given by:

kt+1  = kt(α)

Output per worker is given by

yt  = At kt(α)                                                         (12)

(a) Solve for expressions for steady state capital and output per worker as functions of At and other parameters.

Consider a country with β = 0.9, α = , n = 0 and At  = 1.

(b) Calculate the steady state level of output per worker for this country.