Question

Suppose a firm has a labor demand curve given by w = 20 - 0.01E. Furthermore, suppose that the union representing workers in the firm derives utility from the wage rate and the level of employment according to the utility function U = w · E where the marginal utility of an increase in the wage is MUw = E and the marginal utility of an increase in employment is MUE =w.

(a) Graphically depict the union's utility-maximization problem, as well as its optimal wage-employment choice.

(b) Write down the tangency condition for the union's utility maximization problem.

(c) Write down the constraint for the union's utility maximization problem.

(d) Solve algebraically for the optimal wage-employment choice.

(e) How does your answer to (d) compare to the perfectly-competitive equilibrium wage-employment combination?

Answer 1

We are given that labor demand curve is given by

w = 20 - 0.01E , which can be written as

w + 0.01E = 20

where w is wage and E is employment.

The utility function of union is

U = w * E.

The union's utility is thus maximized with subject to labor demand constraint, i.e.,

maximize U = w * E

such that w + 0.01E = 20

a). Graphically, this can be represented as :

We are taking employment of X axis and Wage on Y axis. We have U = w*E, which is represented by indifference curve on the employment-wage space. The indifference curve IC1 have the lowest utility among indifference curve and IC3 has highest. IC2 has medium utility. We have to maximize the union's utility with respect to labor demand curve. Labor demand curve is represented by downward sloping line AB, where an increase in employment will reduce the wages. The union's utility will be maximized when its indifferent curve is tangent on labor demand curve, that is the combination is wage and employment will maximize utility which is on labor demand curve. This optimal wage-employment choice is represented by point P, where employment is E* and wage is w*, and utility is maximized as indifference curve IC2 is tangent on AB.

d). Algebraically, this utility-maximization problem can be solved through Lagrangian Function, such as

To get optimal values, we will partially differentiate it with respect to w, E, and , and setting them equal to zero we get

................eqn1

..........eqn2

w + 0.01E = 20 .........eqn 3

Equating eqn 1 and 2 we get

.................eqn4

Putting this in eqn 3, we get

0.01E + 0.01E =20

0.02E = 20

E* = 1000..................eqn5

putting this in eqn4, we get

w* = (0.01)E* = 0.01(1000) = 10 ............eqn6

thus union's optimal employemt wage choice is 1000 units of labor and 10 units of wage .