For a firm in the widget industry, X = 100L - L^2; where X is the quantity of wi

Question

For a firm in the widget industry, X = 100L - L^2; where X is the quantity of widgets produced, and L is the number of labor hours hired. The demand for this firm's output is perfectly elastic at a price of $1 per widget. The firm can hire equally productive labor from two identifiable groups, 1 and 2. The supplies of group-1 and group-2 labor to the firm are w1/4 - 1 and 3w2/4 - 3 hours, where wL and w2 denote the wage rates (in dollars per hour) offered by the firm to labor from these groups, respectively. First suppose that the firm is required to set w1 = w2, and calculate the value of these wage rates in equilibrium. Next suppose that the firm is allowed to set w1 w2, and calculate the values of these wage rates in equilibrium.

Explanation / Answer

A. when w1 = w2 =w

L = w1/4-1 + 3w2/4-3 = w1/4 + 3w2/4 - 4   = w-4

At perfect elasticity P = MR = 1

R = X*P = 100L - L^2

MR = 100-2L = 1 so L=50

w = 50+4 = 54 = w1 = w2

B. when w1 and w2 are different

L = w1/4-1 + 3w2/4-3 = w1/4 + 3w2/4 - 4

At perfect elasticity P = MR = 1

R = X*P = 100L - L^2

MR = 100-2L = 1 so L=50

54 = w1/4+3w2/4

216 = w1 + 3w2 this is eq 1

Now if the labour are equally productive then suppy of labor of both the groups will be same.

w1/4-1=3w2/4-3

w1 = 3w2-16 i.e. w1-3w2 = -16 this is eq2

solving eq1 and eq2 we get

w1 = 100 and w2 = 116/3 = 38.67

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