Assignment: Macroeconomic Theory

Question 1

Suppose we allow for installation costs, the utility function and production function are given by

U(c_t) = ln c_t                 (1)

y_t = F(k_t) = Ak^α_t         (2)

In addition, the resource constraint and capital accumulation are given by

y_t = c_t + [1 + (φi_t/2k_t)] i_t and                  (3)
Δk_t+1 = i_t - δk_t,                                      (4)
where φ ≥ 0.

(a) Write the First Order Conditions for consumption, capital, investment, and Lagrange multipliers for the optimal solution.

(b) Derive the equation for Tobin's q.

(c) Using (a) and (b), derive the Euler equation in terms of Tobin's q, capital, and consumption.

(d) Determine the steady state levels for Tobin's q, capital, investment, and consumption.

(e) Determine the effects of a permanent productivity increase (i.e. ΔA > 0).

Question 2

Suppose we allow for technological progress, the utility and production functions are given by

U(C_t) = ln C_t and                (5)
Y_t = (1 + µ)^tK^α_t N^1-α_t,       (6)

where µ > 0. In addition, capital accumulation is given by

ΔK_t+1 = I_t - δK_t.          (7)

(a) Write the First Order Conditions for consumption, capital, labor, and Lagrange multiplier for the optimal solution .

(b) Derive the Euler equation.

(c) Discuss the issues with steady-state optimal growth paths for consumption, capital and output.

(d) Derive the per capita production function, y_t = F(k_t). [Note: y_t = Y_t/N_t and k_t = K_t/N_t denote per capita output and capital in period t, respectively]