Question

Assume that the production function of an economy is given by Y = 20K0.5L0.5, where Y is GDP, K is capital stock, and L is labor. In this economy, the factors of production are in fixed supply with K = 100 and L = 100.

(a) Does this production function exhibit constant returns to scale? Demonstrate by example. (4 points)

(b) If the economy is competitive so that factors of production are paid the value of their marginal products, what is the share of total income that will go to capital? (3 points)

(c) How would a one-time, 50-percent decrease in the quantity of capital (perhaps the result of war damage) change capital’s share of total income? Explain. (3 points)

Answer 1

a) Y = 20K^0.5L^0.5

^{Yea this production function exhibits constant returns to scale.}

tY= 20(tK)^0.5Lt)^0.5

tY = _t^0.5+0.5( 20K^0.5L^0.5) = Y

Thus, this function exhibits constant returns to scale.

b) MP_{​​​​​L}​= 20*0.5*L^{​​​​​​-0.5}K^0.5

MP_{​​​​​L} = 10(K/L)^0.5 = 10

MP_{​​​​​K} = 20*0.5*K^-0.5L^{0.5​​​​​​} = 10

Share of capital = K*MP_{​​​​​k} = 100*10 = 1000

c) when capital is reduced to 50. The share of capital will reduce too.

MPk = 20*0.5*K^-0.5L^0.5 = 2*2^0.5

Share of capital = 50*2*2^0.5 = 100*2^0.5