Question

Y = Kºn1-a Assume a = 1/3 and answer the following questions; A) Calculate marginal productivity of capital. B) Show whether there are decreasing returns to capital. C) Calculate marginal productivity of labor. D) Show whether there are decreasing return to capital. r share of outnut Anmol E) Define labor share of output (income) as Wage paid to labor WEN = "*N, where W is total Output wage paid to all workers. Calculate labor share of income (Hint: Wage=Marginal productivity of labor). F) Define capital share of output (income) as ** , where W is total Output wage paid to all workers. Calculate labor capital of income. G) Transform the production function into a relation between output per worker and capital per worker. H) For a given saving rate, s, and depreciation rate, 8, give an expression for capital per worker in the steady state. I) Solve for the steady-state level of output per worker when s=0.32 and 8=0.08 J) Suppose that the depreciation rate remains constant at 8=0.08, while the saving rate reduced by half, to s=0.16. What is the new steady-state output per worker?

Answer 1

Y = K$\alpha$N1 - $\alpha$   

$\alpha$ = 1/3  

Y = K1/3N1 - 1/3   

Y = K1/3N2/3  

A)

$\partial$Y/$\partial$K =   MPK = (1/3)(K1/3 - 1N2/3 )

   MPK = (1/3)(K- 2/3N2/3 )

B)

$\partial$MPK/$\partial$K = (1/3)(-2/3)K-2/3 - 1N2/3  

= - (2/9)(K- 5/3 N2/3 ) < 0  

Therefore there are decreasing returns to capital

C)

$\partial$Y/$\partial$N = MPN = (2/3)( K1/3N2/3 - 1 )

MPN = (2/3)( K1/3N- 1/3)

D)

$\partial$MPN/$\partial$N =   (2/3)(- 1/3)(K1/3N- 1/3 - 1)  

= - (2/9)(K1/3N- 4/3) < 0

Therefore there are decreasing returns to labor