Question

An electronics plant’s production function is Q = L 2K, where Q is its output rate, L is the amount of labour it uses per period, and K is the amount of capital it uses per period.

(a) Calculate the marginal product of labour (MPL) and the marginal product of capital (MPK) for this production function. Hint: MPK = dQ/dK. When taking the derivative with respect to K, treat L as constant. For example when Q = L 3K2 , we have: MPK = 2L 3K.

(b) Does this production function exhibit diminishing marginal returns to labour? Explain.

(c) Is this plant subject to decreasing or increasing returns to scale? If the use of both inputs is increased by 100%, will the output rise by less or more than 100%? Please explain and show mathematically.

(d) The firm’s vice president for manufacturing hires you to determine which combination of inputs the plant should use to produce 1000 units of output. The price of labor is $4 per unit, and the price of capital is $2 per unit. What advice would you give her?

i. In what ratio should inputs be used to maximize profits?

ii. How many units of labour and capital should be employed to produce 1000 units of output?

Answer #1

Q = L^{2}K

(a)

MPL = Q / L = 2LK

MPK =
Q /
K = L^{2}

(b)

When L increases, MPL also increases, therefore production increases at an increasing rate with addition of more labor. Therefore production function does not exhibit diminishing returns to labor.

(c)

If both inputs are increased by 100%, it means that both L and K are doubled. New production function becomes

Q* = (2L)^{2} x 2K = 4 x L^{2} x 2K = 4 x
L^{2}K = 4 x Q

Q* / Q = 4

Therefore, output increases by more than double (more than 100%), so there is increasing returns to scale.

(d)

Total cost (C) = wL + rK = 4L + 2K and Q = 1,000

(i) Profit is maximized when MPL / MPK = w/r = 4/2 = 2

MPL / MPK = 2LK / L^{2} = 2K / L = 2

2K = 2L

K = L (Optimal input ratio)

(ii) Substituting K = L in production function,

Q = 1,000 = L^{2} x L = L^{3}

Taking cube root on each side,

L = 10

K = L = 10

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