Question

Consider the following production function: Y = K^0.5(AN)^0.5, where both the population and the pool of labor are growing at a rate n= .07, the capital stock is depreciating at a rate d= .03, and A is normalized to 1 (A=1). [N=L]

a. What are capital’s and labor’s shares of income?
b. What is the form of this production function?
c. Find the steady-state values of k and y when s =.20.
d. At what rate is per capita output growing at the steady state? At what rate is total output growing? What if total factor productivity is increasing at a rate of 2 percent per year ( g= .02)?

Answer 1

a.  capital’s shares of income = 0.5 (given by the index of capital)

and labor’s shares of income = 0.5 (given by the index of labor)

b. the form of this production function = Cobb- Douglas

c. d. at the steady state, per capita output is growing at the rate of n = 0.07

and, at the steady state, total output is growing at the rate of n + d = 0.07 + 0.03 = 0.10

Now, total factor productivity is increasing at a rate of 2 percent per year ( g= .02), then the new steady state equilibrium and growth rates are as follows:

Now, at the steady state, per capita output is growing at the rate of n + g = 0.07 + 0.02 = 0.09

and, at the steady state, total output is growing at the rate of n + g+ d = 0.07 + 0.02 + 0.03 = 0.12,

Thanks!