Chapter 11. Quantitative Answers
1. Suppose the supply curve for labor in a competitive industry is given by Ls = −50 + 15ω and the
demand curve for labor is Ld = 130 − 15ω .
(a) What is the equilibrium wage and employment? Unemployment rate?
Answers:
The equilibrium wage is $6/hr. The equilibrium employment is 40.
Ls = Ld = −50 + 15ω = 130 − 15ω
30ω = 180
ω = $6
Ld = 130 − 15ω = 130 − (15)(6) = 40
The unemployment rate is 0%. The quantity of labor supplied equals the quantity of labor
demanded.
(b) Suppose now that all firms pay an efficiency wage of $7/hour. How many workers lose
their jobs? What is the increase in the size of the labor force? What is the increase in the
labor force participation rate?
Answers:
With an efficiency wage of $7/hour, the quantity of labor supplied is 55 and the quantity of
labor demanded is 25. Due to the efficiency wage, 15 workers lose their jobs because the
quantity of labor demanded decreased from 40 to 25.
Ls = −50 + 15ω = −50 + (15)(7) = 55
Ld = 130 − 15ω = 130 − (15)(7) = 25
Labor force participation rate is total labor force divided by population. Although we know
that 15 more people are willing to supply labor, without an estimate of population we cannot
calculate the labor force participation rate.
(c) What is the new unemployment rate as a result of the efficiency wages?
Answers:
Ls − Ld = 55 − 25 = 30
Unemployment rate = unemployed / labor force = 30 / 55 = 54.5%
2. First, for the individual to accept the job, the discounted present value of the contract must
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exceed that of the alternative pay stream. Consequently,
So, the firm must pay at least $14,400 in the second period or else the worker will reject the
job offer. However, for the firm to willing to enter into the contract, it must be profitable. As
a result, the present discounted value of the worker’s MRP must exceed that of the worker’s
cost:
Thus, any second period wage offer between $14,400 and $17,000 will be mutually acceptable
to both the firm and individual.
3. (a)
The lines cross when T = 16.67. To see this set MRPL = w and solve for T :
MRPL = w
12 + 0.2T = 7+0.5T
5 = 0.3T
=⇒ T ∗ = 5/0.3 ≈ 16.67
(b) For the proposal to be profitable, the discounted PV of a typical worker’s MRPL must
exceed the discounted PV of a typical worker’s salary. This implies
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(c) If T = 25, then 16.67(25) = 416.75 and
profitable.
= 325. Thus, the proposal would be
4. If a = 0.1, then Jack expects to sell S = 10+90(0.1) = 19 cars per year. At Lemons R Us he
would be paid $500 per car, for an annual income of 500(19) = $9, 500/yr. Thus, he would
clearly be better off accepting the job paying a fixed salary of $20,000. If a = 0.6, then Jack
expects to sell S = 10+90(0.6) = 64 cars per year. At Lemons R Us he would be paid $500
per car, for an annual income of 500(64) = $32, 000/yr. Thus, he would clearly be better off
accepting this job that the job at Used Lemons. To find the value of a such that Jack would
be indifferent between the two job offers, solve:
500 ∗ (10 + 90a) = 20, 000
=⇒ a∗ = 0.33
For all values of a > a∗, Jack is better off being paid per car sold at Lemons R Us; for all
values of a < a∗, Jack is better off accepting the fixed salary job at Used Lemons.
5. (a) The worker’s payment under scheme A is 100; under scheme B, his payment is 5(c/2).
So,
5(c/2) > 100
=⇒ c∗ > 2(100)/5
=⇒ c∗ > 40
(b) Scheme B is preferable if c > 40. Since c = 20+5e, this implies that e > 4.
(c) If the worker opts for scheme A, his utility is U = 100. Note, e = 0 if the worker opts for
this scheme since effort provides disutility and the worker is not rewarded for expending
any effort. If the worker opts for scheme B and expends e = 4 (see part (b)), then
U = 5(40/2) − √4 = 100 − 2 = 98. Thus, scheme A is now preferable. If the worker sets
e = 10, then U = 5((20 + 5 ∗ 10)/2) − √10 = 175 − 3.16 = 171.84. So, now scheme B
provides greater utility.
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