Question

A firm's production function is Q = 70L0.6 K0:3. Its marginal product of labor is thus MP2 = 42L-0.4 0.3 and its marginal product of capital is MPK = 21L0.6 K-0.7. a. What returns to scale does this production function exhibit: constant, increasing, or decreasing? Show mathematically. b. Suppose the wage rate is $12 and the rental rate for capital is $48. Show that the firm is not minimizing cost when it employs 40 workers (L) per day and 15 machines (K) per day. c. How should this firm adjust its use of capital and labor to minimize the cost? You do not need to solve for an actual amount of each. Just indicate the direction of change of each assuming it stays on the same isoquant. Briefly explain your answer.

Answer 1

Given, $\large Q = 70L^{0.6}K^{0.3}$

MP = 42L-0.460.3

and $\large MP_K = 21L^{0.6}K^{-0.7}$

a. Assume the inputs increases by a factor of lambda.

Q' = 70(AL)06(AK)0.3

Q' = 70106206,0.350.35

Q' = 10.6+0.370L0.660.31

$\large \implies Q' = \lambda^{0.9}Q$

From the above equation we can see the change in output is less than the increase in input. Thus, the given production function exhibits decreasing returns to scale.

The production function is Cobb-Douglas function.

$\large Q = AL^{\alpha}K^{\beta}$

In case of Cobb-Douglas production function, when

$\large \alpha + \beta > 1, \ Increasing \ returns \ to \ scale$

$\large \alpha + \beta = 1, \ Constant \ returns \ to \ scale$

$\large \alpha + \beta < 1, \ Decreasing \ returns \ to \ scale$

b. Wage rate = $ 12, Rent = $ 48

Number of workers = 40, capital unit = 15

At optimal input level the firm will be minimizing cost when the following condition is satisfied

$\large \frac{MP_L}{MP_K} = \frac{w}{r}$

$\large \implies \frac{42L^{-0.4}K^{0.3}}{21L^{0.6}K^{-0.7}} = \frac{12}{48}$

$\large \implies \frac{2K}{L} = \frac{1}{4}$

$\large \implies \frac{K}{L} = \frac{1}{8}$

Number of labour = 40, Number of Capital = 15

$\large LHS = \frac{K}{L} = \frac{15}{40} = \frac{3}{8}$

But the optimal capital and labour ratio is not equal.

$\large \implies \frac{3}{8} \ne \frac{1}{8}$

Thus, the firm is not minimizing cost.

c. We are required to reduce the number of capital unit.

From the above ratio we can see the ratio is greater thus in order to reduce the ratio on LHS we are required to reduce the input.

Reduce capital and increase labour.