Question

Suppose a competitive firm’s production function is Y= 20
L^{1/2} K^{1/3}. L is Labor , K is capital and Y is
output.

a) (4) Find the marginal product of labor and capital.

b) (4) What is Marginal Rate of technical Substitution of Labor for Capital?

c) (2) Does this production function exhibit increasing, decreasing or constant returns to scale? Show your work.

Answer #1

Ans. The production function, Y = 20 L ^{1/2}
K^{1/3}

a) MP_{L} = dY/dL = 10 L^{- 1/2}
K^{1/3}

MP_{K} = dY/dL = 20/3 L^{1/2} K^{-
2/3}

b) the marginal rate of technical substitution (
MRTS_{LK}) = MP_{L}/MP_{K}

= 10 L^{-1/2} K^{1/3}/(20/3)L^{1/2} K
^{-2/3}

= 3K/2L

c) let F ( L, K ) = 20 L^{1/2} K ^{1/3}

For a > 1, we have

F ( aL, aK) = 20 (aL)^{1/2} ( aK)^{1/3}

= 20 a^{1/2} L^{1/2} a^{1/3}
K^{1/3}

= a^{5/6} ( 20 L^{1/2}
K^{1/3})

= a^{5/6} F ( L, K )

Since a < 1, therefore, this production function exhibits decreasing returns to scale.

^{}

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