Suppose the production function is given by Q = L 0.75 K 0.25 , where L is labor

Question

Suppose the production function is given by Q = L0.75K0.25, where L is labor units and K is capital units employed in the production.  Derive the equations for the average product of labor and marginal product of labor (holding capital constant). (See 5.2, p. 129-135).

Average product of labor (APL) =

Marginal product of labor (MPL) =

Marginal product of capital (MPK) =

The marginal product of labor is (Choose one: diminishing/increasing/constant).

I know that MPLis decreasing because __________________________________________.

Even though land is approximately in fixed supply, agricultural output has increased greatly in the past hundred years because of technological innovations including ________________, ___________________, and _________________.

These innovations caused the agricultural production function Q(L)to shift _____.

Explanation / Answer

APL = Q/L

        = (L^0.75 × K^0.25) / L

        = L^(0.75 – 1) × K^0.25

        = K^0.25 / L^0.25 (Answer)

MPL = Derivative of Q with respect to L

        = K^0.25 × 0.75L^(0.75 – 1)

        = 0.75K^0.25 / L^0.25 (Answer)

MPK = Derivative of Q with respect to K

         = L^0.75 × 0.25K^(0.25 – 1)

         = 0.25L^0.75 / K^0.75 (Answer)

Increasing / Decreasing / Constant:

If the derivative of MPL is greater than 0, then MPL is increasing.

If the derivative of MPL is smaller than 0, then MPL is decreasing.

If the derivative of MPL is equal to 0, then MPL is constant.

MPL = K^0.25 / 0.75L^0.25

Derivative of MPL with respect to L = 0.75K^0.25 × (-0.25)L^(-0.25 – 1)

                                                            = -0.25 × 0.75K^0.25 / L^1.25

The function becomes negative; therefore, it is smaller than 0 and decreasing.

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