Question

Suppose output is given by

Y = K ^1/2 (AN) ^1/2

As in the basic model, the workforce grows at rate n, capital depreciates at rate d and the savings rate is s. In addition, suppose that TFP grows at a constant rate g. That is:

∆A/A = g

We will refer to the product AN as the “effective workforce”. It follows that the effective workforce grows at rate n + g.

a. Express the production in per “effective worker” terms by letting y_e = Y/AN and k_e = K/AN .

b. The growth in capital per effective worker can be written as

∆k_e / k_e = ∆K/K − ∆A/A − ∆N/N

Using the same steps as described in class, derive the dynamics of capital per effective worker.

c. Illustrate the steady-state for this economy on a diagram and derive an expression for the steady-state capital stock per effective worker, k^∗_e . How does it depend on g?

d. Derive an expression for the steady-state output per effective worker, y ^∗_e . In this steady-state, at what rate must output per worker, y = Y/N , be growing?

Answer 1