# microeconomic hard grader

1 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 Î´ 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. ï¬nd 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, Î´, and s.

3. Solve for the steady state level of capital per worker as a function of Î´ and s.

4. Solve for the steady state level of output per worker as a function of Î´ 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 Î´ and s.

2. Solve for the steady state level of output per worker as a function of Î´ 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 Î´ = 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 eï¬€ective worker, output per eï¬€ective worker, the growth rate of output per eï¬€ective 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 oï¬€ in situation 1) or 3)? Explain.