Answer to Question #206454 in Microeconomics for Humphrey

Question

A firm has the following production function:

Q = 9L^2/3K^1/3

Where Q is output, L is labor and K is Capital.

Where Q is output, L is labor and K is Capital.

(a) What sort of returns to scale characterize this production function?

(b) Derive the marginal product of labor and capital.

(c) What do we mean when we say the production function is linearly homogenous?

Answer 1

(a)

$Q=(f(L,k)=9L^{\frac{2}{3}}K^{\frac{1}{3}}\\P(3L,3K)=9(3L)^{\frac{2}{3}}( 3K)^{\frac{1}{3}}\\=9×3^{\frac{2}{3}}×3^{\frac{1}{3}}×L^{\frac{2}{3}}×K^{\frac{1}{3}}\\=3×9L^{\frac{2}{3}}K^{\frac{1}{3}}$

When multiplied to 3, the total output also increases 3 times. Therefore this production function represents constant returns to scale.

(b)

$Q=9L^{\frac{2}{3}}K^{\frac{1}{3}}$

For marginal product of labor

$\frac{∆Q}{∆L}=9×\frac{2}{3}×L^{\frac{2}{3}-1}×K^{\frac{1}{3}}\\MPL=6L^{-\frac{1}{3}}K^{\frac{1}{3}}\\MPL=6(\frac{K}{L})^{\frac{1}{3}}$

For marginal product of capital

$\frac{∆Q}{∆K}=9L^{\frac{2}{3}}×\frac{1}{3}×K^{\frac{1}{3}-1}\\MPK=3(\frac{L}{K})^{\frac{2}{3}}$

(c)

When a production function is linearly homogeneous,it means that,when factors of production are changed proportionally,the level of output also changes in same proportion. This also known as constant returns to scale.

Justification

$F(L,K)=9K^{\frac{1}{3}}L^{\frac{2}{3}}\\F(2L,2K)=9(2K)^{\frac{1}{3}}(2L)^{\frac{2}{3}}\\=9×2×L^{\frac{2}{3}}K^{\frac{1}{3}}\\=18K^{\frac{1}{3}}L^{\frac{2}{3}}$

This justifies , when labor and capital are doubled, output also doubles.