Consider a firm using labor and capital as its only inputs. The price of capital
ID: 1188176 • Letter: C
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
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?
Ans: a) The firms isocost equation can be written as:
C = wL + rK ----------- (1);
Where, C = Total cost of the firm, w = Wage of labor, L = Number of labor, r = Rent of capital and
K = Amount of capital.
Differentiating equation (1) totally, we get,
dC = w*dL + r*dK
From first order condition, dC = 0, we get,
w*dL + r*dK = 0
or, (-dK/dL) = (w/r)
or, (dK/dL) = -(w/r)
(dK/dL) is the slope of isocost line and it is (-w/r).
Now according to the given information, w = $60 and r = $40
Thus, Slope of isocost = (dK/dL) = -(w/r) = -(60/40) = -1.5
So, isocost line is downward slopping.
b) For optimality, we can formulate the problem as:
Max, Q = f(L.K); where Q = Amount of output produced.
Subject to, C = wL + rK
The relevant Lagranze multiplier is:
M = f(L.K) + (C - wL rK)
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