Columbia University

Macroeconomics Analysis I G6215

Fall 2009

Professor Xavier Sala-i-Martin

(1) Cobb Douglas Technology

Imagine that the production function is .

(a) Show that it satisfies the three key neoclassical assumptions.

Suppose that the depreciation rate is . Consider the case in which investment goods and output are the same good, and normalize the price of output to one. Suppose that the real interest rate is constant and equal to r. Firms take this interest rate as given and the rental rate of labor (the wage rate) is also constant and chooses capital, investment and labor so as to maximize the present value of all future profits. Aggregate labor supply is constant.

(b) Set up the Hamiltonian using the "present value shadow price". Find the first order conditions. Solve the model and find K as a function of r, L, A, and .

(c) Set up the Hamiltonian using the "current value shadow price". Find the first order conditions. Solve the model and find K as a function of r, L, A, and . Compare this result with the one you find in (b).

(d) Consider an unexpected and permanent increase in A. How do capital and investment behave over time?

(e) Consider an unexpected and permanent increase in r. How do capital and investment behave over time?

(f) Consider an unexpected and permanent increase in . How do capital and investment behave over time?

(g) Imagine that, at time 0, we learn that A will increase at some future time T. How do capital and investment behave over time?

(2) Accelerator Theory of Investment

In the 1950s, some economists postulated that net investment, , was proportional to changes in production, . Our neoclassical says that net investment is not a function of . Or does it?

(a) Consider again a Cobb-Douglas production function and the problem we set up in question (1). Use the first order conditions to express the level of capital as a proportional function of Y.

(b) Take derivatives of both sides and say whether our model is consistent with the accelerator theory of investment.

(c) Would this result go through if the production function was not Cobb Douglas?

(3) A Good Theory?

Discuss why the Neoclassical Theory of Investment is or is not a good theory of investment. Discuss its pros and cons, theoretical and empirical.