Consider a production function for an economy: Y = 20(L.5K.4N.1) where L is labo

Question

Consider a production function for an economy:
Y = 20(L.5K.4N.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 an 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?

Explanation / Answer

A) Y= 20*(L^0.5)*(K^0.4)*(N^0.1) Y = 20*(100^0.5)*(100^0.4)*(100^0.1) Y = 2000 B) Yes, we can tell this because the exponents on labor, capital, and land add to 1. This means that doubling the inputs will result in double the production. Y= 20*(L^0.5)*(K^0.4)*(N^0.1) Y = 20*(200^0.5)*(200^0.4)*(200^0.1) Y = 4000 C) 10% will go to land, 50% will go to labor, 40% to capital. We get these from the exponents. But lets calculate the marginal products just for fun. Y = 20*(L^0.5)*(K^0.4)*(N^0.1) MPL = 10*(L^-0.5)*(K^0.4)*(N^0.1) MPK = 8*(L^0.5)*(K^-0.6)*(N^0.1) MPN = 2*(L^0.5)*(K^0.4)*(N^-0.9) Total income is: L*w + K*r + N*d = I We are told that the market is competitive so MPL = w, MPK = r, and MPN = d L*MPL + K*MPK + N*MPN = I So, the share of income that goes to land is: N*MPN / (L*MPL + K*MPK + N*MPN) Let's substitute in the fun formulas we derived just a second ago. Share = N*2*(L^0.5)*(K^0.4)*(N^-0.9) / ( L*10*(L^-0.5)*(K^0.4)*(N^0.1) + K*8*(L^0.5)*(K^-0.6)*(N^0.1) + N*2*(L^0.5)*(K^0.4)*(N^-0.9) ) Share = 2*(L^0.5)*(K^0.4)*(N^0.1) / ( 10*(L^0.5)*(K^0.4)*(N^0.1) + 8*(L^0.5)*(K^0.4)*(N^0.1) + 2*(L^0.5)*(K^0.4)*(N^0.1) ) Share = 2/(10 + 8 + 2) Share = 0.1

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