Question

• Answer the following questions about the producer’s production function: Q = 2K^1/2L²

• Does the production function display increasing, constant, or decreasing returns to scale? [Prove your answer by increasing all inputs by a factor of c in your analysis.]

• Find MP_L if capital is fixed at K₀=9 and determine whether the production process follows the law of diminishing returns (LDR) to labor.

• If input prices are r=5 and w=4 for capital and labor, respectively, and suppose MP_K=40 and the firm is currently using 3 units of labor (L=3). Is the firm in part b (above) minimizing its costs? If not, how could it improve its use of inputs in attempting to minimize costs? Explain.

Answer 1

Ans. Production function, Q = 2K0.5L2

a) To check for returns to scale, suppose both the inputs are increased by factor c, then new output,

Q’ = 2(cK)0.5(cL)2 = c2.5 Q

As Q’ > cQ, so, the production function exhibits increasing return to scale.

b) MPL = dQ/dL = 4K0.5L

At K = 9

MPL = 4*3L = 12L

Differentiating MPL by L we get,

dMPL/dL = 12 > 0

Therefore, the production function doesn’t exhibit diminishing returns to labour.

c) MPL at L = 3 is 12*3 = 36

MPK = 40

r = 5 and w = 4

At cost minimizing level,

MPL/MPK = w/r

But here, MPL/MPK = 9/10 > w/r =4/5

So, firm is not minimizing its cost. To minimise cost capital should be used less which will increase MPK making the required ratios equal.

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