Problem:
Suppose that the production function is:
Y = AK.
Where Y is output, A is a positive constant reflects the level of productivity, and K is the capital stock. Think of K in a broad sense to include human capital.
Capital stock grows at:
.
K = s.Y - dK
Where s is a constant saving rate, d is a constant depreciation rate.
The dot over the variable is time derivative. Assume that d = 0 for simplicity.
Labour force grows at;
.
N = n.N
Where n is constant growth rate of labour.
Define:
k = K/N,
y = Y/N
Where k is per worker capital stock and y is per worker output.
Q1. Find first and second derivative of the production function with respect to K. Interpret your result.
Q2. Find y as a function of k.
Q3. Derive the equation of motion for k.
Q4. Do you think the economy will reach a steady state? Why?
.
Q5. Find the equation of the growth rate of y; (y/y); and answer the following questions:
(i) Suppose that “n” is reduced. What will happen to the growth rate of y in the long run? Explain graphically and mathematically.
(ii) Suppose that “s” is increased. What will happen to the growth rate of y in the long run? Explain graphically and mathematically.
(iii) According to this model, does there tend to be convergence across economies? Why?