Questions

1. Cobb-Douglas production function

This exercise discusses properties of the Cobb-Douglas production function and its use in the Solow model.

Suppose we have the following aggregate production function Y = F(K, L) = AK^αL^1-α, where L is the number of workers, K is the capital stock, A is technology and α ∈ (0, 1) is the capital share of income.

a) Calculate the marginal product of capital and of labor from the aggregate production function.

b) Show that the share of total income that is paid to workers is equal to 1 - α, assuming that markets are competitive.

c) Derive output per worker from the aggregate production function.

2. Solow model

Suppose output per worker can be described by the per worker production function y_t = Ak^α_t. We assume that there is no population growth (L is constant). On average each worker saves a fraction γ ∈ (0, 1) of her output. In a closed economy, savings equal investments, thus, the amount of investment that is added to the per capita capital stock is i_t = γyt. Each year, a share δ of the capital stock is lost due to depreciation, i.e. d_t = δk_t.

a) Write down the capital accumulation equation per worker, i.e. how the capital stock per worker evolves from t to t + 1.

b) Plot y, i, and d as functions of k together in one graph. Label the steady state capital stock with k* and the corresponding output y*.

c) Find the growth rate of capital per worker in the steady state.

d) Find the expression for the steady state capital stock per worker (k*) as a function of the parameters γ, A, δ, and α. (Hint: use the condition for the steady state.) Insert this expression in the production function to get an expression for output per worker in the steady state (y*).

e) Assume the economy currently has a capital stock per capita below its steady state level: k < k*. Show in the graph consumption per capita, depreciation and investment. What is the sign of k_t+1 - k_t?

f) Make a new graph with time on the horizontal axis and k on the vertical. At time t₀ the capital stock is as before k0 < k* . Sketch how you expect the capital stock to evolve over time.

g) Assume that two countries A and B have the same production function, saving rate and depreciation rate. Country A is in the steady state, but country B is below the steady state. What can you say about the growth rates of these two countries over time?

3. Numerical example for the Solow model.

Assume the same model as in the previous exercises with the following parameter values: α = 1/3, A = 1/2, γ = 0.4 and δ = 0.05.

a) Calculate the steady state value of the capital stock (per worker) and income (per worker).

b) Suppose now that the depreciation rate of capital increases from δ = 0.05 to δ = 0.075. Calculate the new steady state capital stock (per worker) and output (per worker).