Constant Population

Consider the following production function:

Yt = F(Kt,Nt) = K

1 2 t N

1 2 t

Assume that capital depreciates at rate δ and that savings is a constant proportion s of output:

St = sYt

Assume that investment is equal to savings:

It = St

Finally, assume that the population is constant:

Nt = Nt+1 = L

1. The production function above expresses output as a function of capital and labor (workers). Derive a function that expresses output per worker as a function of capital per worker (i.e. find yt = f(kt)).
2. Write down the capital accumulation equation in terms of capital per worker (i.e. an equation with only kt+1, kt, δ, and s.
3. Solve for the steady state level of capital per worker as a function of δ and s.
4. Solve for the steady state level of output per worker as a function of δ and s.
5. What is the steady state growth rate of output per worker?
6. What is the steady state growth rate of output?

2 Growing Population

Consider the following production function:

Yt = F(Kt,Nt) = (K

1 2 t + N

1 2 t )2

Assume that capital depreciates 5% each year and that households save 5% of their income. Assume that investment is equal to savings. Finally, assume that the population is growing 15% each year.

1

Econ 320 - Growth Homework

1. Solve for the steady state level of capital per worker as a function of δ and s.
2. Solve for the steady state level of output per worker as a function of δ and s.
3. What is the steady state growth rate of output per worker?
4. What is the steady state growth rate of output?

3 Technological Growth

Suppose that production is given by

Y = K

1 2 (AN)

1 2

The savings rate is s = 0.16 and the rate of depreciation is δ = 0.1. Suppose further that the number of workers grows at 2% per year and that the rate of technological progress is 4% per year.

1. Find the steady-state values of the the following variables: capital per effective worker, output per effective worker, the growth rate of output per effective worker, the growth rate of output per worker, and the growth rate of output.
2. Suppose that the rate of technological progress doubles to 8% per year. Recompute your answers to part 1). Explain.
3. Now suppose that rate of technological progress is still equal to 4% per year, but the number of workers now grows at 6% per year. Recompute your answers to part 1). Are people better off in situation 1) or 3)? Explain.