Question

Consider a production function for an economy: Y = 20(L^0.5K^0.4N^0.1)

where L is labor, K is capital, and N is land. In this economy the factors of production are in fixed supply with L = 100, K = 100, and N = 100.

a. What is the level of output in this country?

b. Does this production function exhibit constant returns to scale. Demonstrate by example.

c. If the economy is competitive so that factors of production are paid the value of their marginal products, what share of total income will go to land?

Answer 1

Y = 20L^0.5K^0.4N^0.1

(a) Plugging in given values,

Y = 20 x (100)^0.5 x (100)^0.4 x (100)^0.1 = 20 x (100)^{(0.5 + 0.4 + 0.1)} = 20 x 100 = 2,000

(b) Let us double all inputs such that new production function becomes

Y1 = 20 x (2L)^0.5(2K)^0.4(2N)^0.1 = 20 x 2^0.5 x 2^0.4 x 2^0.1 x L^0.5K^0.4N^0.1 = 20 x (2)^{(0.5 + 0.4 + 0.1)} x L^0.5K^0.4N^0.1 = 2 x Y

Y1 / Y = 2

Since doubling all inputs exactly doubles the output, there is constant returns to scale.

(c) Since coefficients of L, K and N add up to 1, this is a Cobb-Douglas production function, for which

Share of income going to a factor = Its coefficient in production function. Therefore

Share of income going to land = Coefficient of N in production function = 0.1 (= 10%).