Chapter 10  Labor Unions
CHAPTER 10
101. Suppose the firm’s labor demand curve is given by:
w
= 20  0.01
E
,
where
w
is the hourly wage and
E
is the level of employment. Suppose also that the union’s
utility function is given by
U =
w
×
E
.
It is easy to show that the marginal utility of the wage for the union is
E
and the marginal
utility of employment is
w
. What wage would a monopoly union demand? How many
workers will be employed under the union contract?
Utility maximization requires the absolute value of the slope of the indifference curve equal the
absolute value of the slope of the labor demand curve. In this case, the absolute value of the slope
of the indifference curve is
E
w
MU
MU
w
E
=
.
The absolute value of the slope of the labor demand function is 0.01. Thus, utility maximization
requires that
01
.
=
E
w
.
Substituting for
w
with the labor demand function, the employment level that maximizes utility
solves
01
.
0
01
.
0
20
=

E
E
,
20 – 0.01
E
= 0.01
E
20 = 0.02
E
E
= 1,000 workers.
The highest wage at which the firm is willing to hire 1,000 workers is 20 – 0.01(1000) = $10.
Thus, the monopoly union requires the firm to employ 1,000 workers, each at $10 per hour.
101
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102. Suppose the union in problem 1 has a different utility function. In particular, its utility
function is given by:
U
= (
w

w
*
)
×
E
where
w
*
is the competitive wage. The marginal utility of a wage increase is still
E
, but the
marginal utility of employment is now
w
–
w
*
. Suppose the competitive wage is $8 per hour.
What wage would a monopoly union demand? How many workers will be employed under
the union contract? Contrast your answers to those in problem 1. Can you explain why they
are different?
Again equate the absolute value of the slope of the indifference curve to the absolute value of the
slope of the labor demand curve:
01
.
0
*
=

=
E
w
w
MU
MU
w
E
.
Setting
w*
= $8 and using the labor demand equation yields:
01
.
0
8
01
.
0
20
=


E
E
,
12 – 0.01
E
= 0.01
E
12 = 0.02
E
E
= 600 workers.
The highest wage at which the firm is willing to hire 600 workers is 20 – 0.01(600) = $14.
Thus,
the monopoly union requires the firm to employ 600 workers, each at $14 per hour.
In problem 1, the union maximized the total wage bill. In problem 2 the utility function depends
on the difference between the union wage and the competitive wage. That is, the union
maximizes its rent. Since the alternative employment pays $8, the union is willing to suffer a cut
in employment in order to obtain a greater rent of $6 per hour ($8 up to $14).
103. Figure 103 demonstrates some of the tradeoffs involved when deciding to join a
union.
(a) Provide a graph that shows how the presence of union dues affects the decision to join a
union.
(Assume all workers pay a flat rate for dues.)
Show on your graph how the presence
of union dues may lead the worker to be less inclined to join the union.
102
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 Fall '10
 SCHWENKENBERG
 Economics, Supply And Demand, Utility

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