# ECON 3100 MSU Calculate the Competitive Market Wage Economics Questions

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CHAPTER 10
10-1. Suppose the firms labor demand curve is given by:
w = 20 – 0.01E,
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
MU E
w
=
.
MU w
E
The absolute value of the slope of the labor demand function is 0.01. Thus, utility maximization
requires that
w
= 0.01 .
E
Substituting for w with the labor demand function, the employment level that maximizes utility
solves
20 ? 0.01E
= 0.01 ,
E
20  0.01E = 0.01E
20 = 0.02E
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.
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10-2. 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:
MU E
w ? w*
=
= 0.01 .
MU w
E
Setting w* = \$8 and using the labor demand equation yields:
20 ? 0.01E ? 8
= 0.01 ,
E
12  0.01E = 0.01E
12 = 0.02E
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 option 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).
10-3. Figure 10-2 demonstrates some of the tradeoffs involved when deciding to join a
union. Suppose in addition to higher wages the union negotiates a 10 percent employer
contribution to a defined contribution pension plan. Provide a graph similar to Figure 10-2
that incorporates this retirement benefit into the decision of whether to join a union. Show
on your graph how additional fringe benefits such as a retirement plan may cause the
worker to be more inclined to join the union.
Negotiating a 10% employer contribution to a defined contribution pension plan is almost the
same as negotiating an additional 10% increase in the wage. Thus, the budget line (BT) in Figure
10-2 will rotate out. As long as the firm does not respond by cutting hours too much (such as to
h0 in Figure 10-2), workers will have more incentive to join the union as they will receive higher
hourly compensation (though possibly asked to work fewer hours).
C
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B
P1
U1
h1 h2
T
The graph above is a simplified version of Figure 10-2 from the text. The negotiated contribution
rotates the budget line from BT to CT. As long as the firm does not reduce hours from h1 to less
than h2, the worker is better off.
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10-4. Consider a two-sector economy with homogeneous labor and jobs in both sectors. Two
million workers supply their labor perfectly inelastically. Labor demand in both sectors
can be written as:
E1 = 1,800,000  100,000w1 and E2 = 1,800,000  100,000w2.
(a) If both sectors are competitive, what is the market-clearly wage and how many workers
are employed in both sectors?
As labor demand is identical in both sectors and labor is homogeneous, 1 million workers will
work in both sectors. Using this for E in the labor demand equations, we find that
1,000,000 = 1,800,000  100,000w
100,000w = 800,000
w* = \$8 per hour.
(b) Suppose a labor union forms in sector 1. The union negotiates a wage of \$12 per hour,
and firms choose how much labor to employ. Anyone not employed in sector 1 is relegated
to sector 2. How many workers will be employed in sector 1 (unionized)? How many
workers will be employed in sector 2, and what wage will they receive?
At a wage of \$12 per hour, the unionized sector (sector 1) will employ:
E1 = 1,800,000  100,000w1
E1 = 1,800,000  100,000 × 12
E1 = 1,800,000  1,200,000
E1* = 600,000 workers.
This forces 2 million  0.6 million = 1.4 million workers into the non-unionized sector (sector 2).
With this many workers relegated to sector 2, wages are:
1,400,000 = 1,800,000  100,000w
100,000w = 400,000
w* = \$4 per hour.
Therefore, 600,000 workers are employed at \$12 per hour in the unionized sector while 1,400,000
workers are employed at \$4 per hour in the non-unionized sector.
(c) What is the union-wage gap in part (b)? What would the union-wage effect be if one
controlled for the spillover effect?
Using the answers to part (b), the union-wage effect is (\$12  \$4) / \$4 = 200% (or it could be
expressed as \$8 = \$12  \$4 per hour as well).
The spillover effect refers to the infusion of workers into sector 2 because the union formed and
the unionized firms restricted labor. If we compare the union wage of \$12 to the competitive
wage of \$8 per hour that would have come about without a union (part a), the union-wage effect
is (\$12  \$8) / \$8 = 50% (or it could be expressed as \$4 = \$12  \$8 per hour as well).
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10-5. Consider a firm that faces a constant per unit price of \$1,200 for its output. The firm
hires workers, E, from a union at a daily wage of w, to produce output, q, where
q = 2E½.
Given the production function, the marginal product of labor is 1/E½. There are 225
workers in the union. Any union worker who does not work for the firm can find a nonunion job paying \$96 per day.
(a) What is the firms labor demand function?
The problem stipulates that the price of output is constant at \$1,200. This means that the firm
also faces constant marginal revenue at \$1,200. That is, p = MR = \$1,200. The labor demand
function, or the value of marginal product of labor, is
VMPE = MR × MPE = 1200 / E ½.
(b) If the firm is allowed to specify w and the union is then allowed to provide as many
workers as it wants (up to 225) at the daily wage of w, what wage will the firm set? How
many workers will the union provide? How much output will be produced? How much
profit will the firm earn? What is the total income of the 225 union workers?
If the firm offers w < \$96, no workers will be provided. This would leave the firm with no output and no profit. The workers would all receive \$96 per day, making their total daily income \$21,600. If the firm offers a wage of w > \$96, all 225 workers will be provided. These 225 workers would
produce q = 2 × sqrt(E) = 2 × sqrt(225) = 30 units of output. The firm would then earn a profit of
30(\$1,200)  225w. Profit, therefore, is maximized when w is minimized subject to the constraint.
If the union would supply all 225 workers at a wage of \$96, for example, the firm would offer w
= \$96 and earn a daily profit of \$14,400. The total daily income of the 225 workers would remain
at \$21,600. (If the firm needs to offer strictly more than \$96 per day to attract workers, it would
offer a daily wage of \$96.01. All 225 workers would work for the firm, making 30 units of
output. The firms daily profit would be \$14,397.75. And the total daily income of the 225
workers would be \$21,602.25.)
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10-6. Consider the same set-up as in problem 10-5, but now the union is allowed to specify
any wage, w, and the firm is then allowed to hire as many workers as it wants (up to 225) at
the daily wage of w. What wage will the union set in order to maximize the total income of
all 225 workers? How many workers will the firm hire? How much output will be
produced? How much profit will the firm earn? What is the total income of the 225 union
workers?
To solve this with Excel, the spreadsheet looks like the following, where the union specifies the
wage, labor demand comes from part (a), and everything else follows naturally:
Union
Labor
Labor
Daily
wage Demand
Costs Output
Price
Revenue
Profit
Income
\$96
156.25 \$15,000.00 25.00 \$1,200 \$30,000.00 \$15,000.00 \$21,600.00
\$97
153.04 \$14,845.36 24.74 \$1,200 \$29,690.72 \$14,845.36 \$21,753.04
\$98
149.94 \$14,693.88 24.49 \$1,200 \$29,387.76 \$14,693.88 \$21,899.88
\$99
146.92 \$14,545.45 24.24 \$1,200 \$29,090.91 \$14,545.45 \$22,040.77
\$100
144.00 \$14,400.00 24.00 \$1,200 \$28,800.00 \$14,400.00 \$22,176.00

\$190
39.89 \$7,578.95 12.63 \$1,200 \$15,157.89 \$7,578.95 \$25,349.58
\$191
39.47 \$7,539.27 12.57 \$1,200 \$15,078.53 \$7,539.27 \$25,349.90
\$192
39.06 \$7,500.00 12.50 \$1,200 \$15,000.00 \$7,500.00 \$25,350.00
\$193
38.66 \$7,461.14 12.44 \$1,200 \$14,922.28 \$7,461.14 \$25,349.90
\$194
38.26 \$7,422.68 12.37 \$1,200 \$14,845.36 \$7,422.68 \$25,349.60
\$195
37.87 \$7,384.62 12.31 \$1,200 \$14,769.23 \$7,384.62 \$25,349.11
Thus, the union sets a daily wage of \$192. The firm responds by hiring 39.06 workers, who
produce 12.5 units of output. The firm earns a daily profit of \$7,500, while the 225 workers,
39.06 of whom are in the union and 185.94 of whom are not in the union, earn a total of \$25,350
each day.
The calculus solution is: given any wage, the firm will employ (1200/w)2 workers. This is
derived by setting the value of marginal product equal to the wage and solving for employment:
.
As the unions objective is to maximize total income, it chooses w to maximize the income of the
workers employed by the union plus the income of the workers not employed by the union.
Therefore, we have:
2
2
?
138,240,000
? 1,200 ?
? 1,200 ? ?? 1,440 ,000
?
+ 21,600 ?
Max w?
.
? + 96 225 ? ?
? =
?
?
w
? w ?
? w ? ?
w2
?
The first order condition, therefore is
? 1,440 ,000
w
2
+
276 ,480 ,000
w3
= 0 which solves as w = \$192.
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10-7. Suppose the unions resistance curve is summarized by the following data. The unions
initial wage demand is \$10 per hour. If a strike occurs, the wage demands change as
follows:
Length of Strike:
1 month
2 months
3 months
4 months
5 or more months
Hourly Wage Demanded
9
8
7
6
5
Consider the following changes to the union resistance curve and state whether the
proposed change makes a strike more likely to occur, and whether, if a strike occurs, it is a
longer strike.
(a) The drop in the wage demand from \$10 to \$5 per hour occurs within the span of 2
months, as opposed to 5 months.
If the union is willing to drop its demands very fast, the firm will find it profitable to delay
agreement until the wage demand drops to \$5. A strike, therefore, is more likely to occur. If \$5 is
the lowest wage the union is willing to accept, the strike is much more likely to last 2 months now
than the probability it would have lasted 5 months under the original resistance curve.
(b) The union is willing to moderate its wage demands further after the strike has lasted for
6 months. In particular, the wage demand keeps dropping to \$4 in the 6th month, \$3 in the
7th month, etc.
If the union is willing to accept even lower wages in the future, some firms will find it optimal to
wait the union out. Thus, strikes will be more likely and last longer.
(c) The unions initial wage demand is \$20 per hour, which then drops to \$9 after the strike
lasts one month, \$8 after 2 months, and so on.
Conditioning on a strike occurring, the length of strike will be unchanged as the resistance curve
after the initial demand stays the same. The probability of a strike ever occurring increases,
however, when the initial demand increases but everything else remains the same.
10-8. At the competitive wage of \$20 per hour, firms A and B both hire 5,000 workers (each
working 2,000 hours per year). The elasticity of demand is -2.5 and -0.75 at firms A and B
respectively. Workers at both firms then unionize and negotiate a 12 percent wage increase.
(a) What is the employment effect at firm A? How has total worker income changed?
At firm A, ?A = %?EA ÷ %?wA = -2.5. When wages increase 12%, therefore, employment falls
by 30%. The firm will start to employ 70% of 5000 × 2000 = 7 million work-hours per year,
possibly by hiring 3,500 workers for 2,000 hours each. Total income was 10 million work-hours
times the wage of \$20 per hour for a total of \$200 million. Total income will now be 7 million
work-hours times the new wage of \$22.40 (a 12 percent increase above \$20), for a total income of
\$156.8 million plus any income earned by the workers who no longer work at firm A because of
the reduction in labor used.
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(b) What is the employment effect at firm B? How has total worker income changed?
At firm B, ?B = %?EB ÷ %?wB = -0.75. When wages increase 12%, therefore, employment falls
by 9%. The firm will start to employ 91% of 5000 × 2000 = 9.1 million work-hours per year,
possibly by hiring 4,550 workers for 2,000 hours each. Total income was 10m work-hours times
the wage of \$20 per hour, for a total of \$200 million. Total income will now be 9.1 million workhours times the new wage of \$22.40 (a 12 percent increase above \$20), for a total income of
\$203.84 million plus any income earned by the workers who no longer work at firm B because of
the reduction in labor used.
(c) How much would the workers at each firm be willing to pay in annual union dues to
achieve the 12 percent gain in wages?
To answer this question, assume that reductions in employment come from reducing the number
of workers hired, and not by reducing the number of hours worked by each worker. So, for firm
A, assume the number of workers falls to 3,500 but hours remain at 2,000. Similarly, for firm B,
assume the number of workers falls to 4,550 but hours remain at 2,000. In this case, income has
increased from 2,000 x \$20 = \$40,000 per year per worker to
2,000 x \$22.40 = \$44,800 per year for each worker continuing to have a job. So, workers at
either firm, as long as they retain their job, are willing to pay up to \$4,800 annually in union dues.
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10-9. Several states recently passed laws restricting bargaining rights for public employees.
Most notably the changes tended to restrict the unions right to negotiate over fringe
benefits such as health care and retirement benefits. What problems were these legislative
changes trying to address? Even assuming such a law survives a constitutional challenge
(which some did not), why might restricting bargaining rights not fully address the
problems lawmakers were aiming to solve?
Following the Great Recession, many private employee labor contracts and conditions were
changed. Fewer fringe benefits were being paidemployees were asked to pay more of their
health insurance costs, and contributions to and benefits paid from retirement plans fell
(especially to and from defined benefit plans). Salary increases were marginal from 2007  2010.
And so on. Its harder to have such shifts in the public sector in part because public organizations
are not arranged nor operate the same way as private firms. Public sector unions had also
negotiated terrific healthcare and retirement benefits in the decades leading up to the Great
Recession. The defined benefit retirement plans for public sector workers in many states became
vastly insolvent as benefits steadily increased at the same time state budgets became strapped due
to lower state tax revenues. Thus, the states that passed these laws did so because, in their mind,
the compensation package previously offered to public sector unions was out of line with the
private sector marketplace, and so out of line that the current state of things threatened state
solvency.
There are at least two reasons to think that these laws may not be as successful in lowering the
state burden in terms of public sector employees as advertised. First, the unions are still allowed
to bargain over salaries. As money (and compensation) is fungible, the union may demand that
cost-savings on healthcare or retirement be offset with higher direct salaries. Second, the unions
still have the right to strike and have significant political power. Thus, even though their
bargaining rights may be hypothetically restricted, it remains unclear if the state would actually
take hard stands during negotiations.
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10-10. Suppose the economy consists of a union and a non-union sector. The labor demand
curve in each sector is given by L = 1,000,000  20w. The total (economy-wide) supply of
labor is 1,000,000, and it does not depend upon the wage. All workers are equally skilled
and equally suited for work in either sector. A monopoly union sets the wage at \$30,000 in
the union sector. What is the union wage gap? What is the effect of the union on the wage in
the non-union sector?
In a competitive economy, both sectors would hire half of the workers as labor is supplied
inelastically. Therefore, to solve for the competitive wage, solve
500,000 = 1,000,000  20w ? wComp = \$25,000.
If the union wage is set at \$30,000, the union sector employs
L = 1,000,000  20(30,000) = 400,000 union workers.
The remaining 600,000 must be employed in the non-union sector, which will happen if the wage
in the non-union sector is (1,000,000  600,000)/20 = \$20,000.
Hence, the wage gap between the union and the non-union sectors equals:
Union Wage Gap: \$30,000  \$20,000 = \$10,000.
Thus, the union wage gap represents 50% of the non-union wage as \$10,000 ÷ \$20,000 = 50%.
The effect of the union wage gap is that more people now work in the non-union sector, which
depresses wages there. In particular, although the union only negotiated a pay raise of \$5,000
above the competitive wage, the wage gap is \$10,000 as the workers who no longer work in the
union sector compete wages down in the non-union sector.
10-11. In Figure 10-6, the contract curve is PZ.
(a) Does point P represent the firm or the workers having all of the bargaining power?
Does point Z represent the firm or the workers having all of the bargaining power?
Explain.
The firms isoprofit curves improve as it hires the same number of workers at a lower wage,
which means improvements are achieved by moving down (to the south). So, from the firms
perspective, ?* > ?M > ?Z.. From the unions point of view, indifference curves increase to the
northeast (as more people are hired at the same wage or when the same number of people are