Consider a firm using labor and capital as its only inputs. The price of capital

Question

Consider a firm using labor and capital as its only inputs. The price of capital is $40 where the price of labor (wage) is $60. Using 500 units of labor and 500 units of capital the firm is producing 1200 units of output. At this mix of input the firm's MPL is 10 while its MPK is 5.

a. Write the firm's isocost equation. What is the slope of the isocost?
b. Determine if the firm's mix of inputs is optimal. Explain.
c. If your answer to "b" is no, what should the firm do to improve its performance? Explain.
d. Now suppose as a result of a mandated increase in the minimum wage the wage increases to $80. What would be the implication of this change for this firm?

Explanation / Answer

(a) The firm’s Isocost equation is as follows –

TC = $60L + $40K

Where,

TC = Total Cost of production of firm

$60 = Price of per unit of labor

$40 = Price of per unit of capital

L = Units of labor

K = Units of capital

Calculate slope of the Isocost –

Price of per unit of labor (w) = $60

Price of per unit of capital (r) = $40

Slope of the Isocost = -w/r = -$60/$40 = -1.5

The slope of the isocost is -1.5.

(b)A firm’s mix of inputs is said to be optimal, if the following condition is fulfilled –

MPL/w = MPK/r

Putting respective values in above arrangement with respect to given firm –

10/60 = 5/40

1/6 1/8

As condition for optimal mix of inputs is not fulfilled at the current level of inputs used by the firm, the firm’s mix of inputs is not optimal.

(c) In part (b), we can see that MPL/w > MPK/r.

In such case, if firm wants to improve its performance then it must increase the quantity of labor. This is because increase in quantity of labor will decrease the marginal product of labor or MPL and ratio of MPL and wage rate will decline and it will become equal to MPK/r leading to attainment of optimal mix of inputs.

(d) It has been provided that wage rate has increased to $80.

However, quantity of labor units used (500 units) had remained same.

Plugging the new wage rate in condition for optimal mix of inputs –

MPL/w = MPK/r

10/80 = 5/40

1/8 = 1/8

As we can see that condition for optimal mix of inputs is fulfilled at new wage rate.

Thus, the implication of this change in wage rate for firm is that at this new wage rate the firm’s current mix of inputs has become optimal.

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