CHAPTER 7
7-1. Debbie is about to decide which career path to pursue. She has narrowed her options to two
alternatives. She can either become a marine biologist or a concert pianist. Debbie lives two periods.
In the first, she gets an education. In the second, she works in the labor market. If Debbie becomes
a marine biologist, she will spend $15,000 on education in the first period and earn $472,000 in the
second period. If she becomes a concert pianist, she will spend $40,000 on education in the first
period and then earn $500,000 in the second period.
(a) Suppose Debbie can lend and borrow money at a 5 percent annual rate. Which career will she
pursue? What if she can lend and borrow money at a 15 percent rate of interest? Will she choose a
different option? Why?
Debbie will compare the present value of income for each career choice and choose the career with the
largest present value. If the discount rate is 5 percent,
and
PVBiologist = – $15,000 + $472,000/(1.05) = $434,523.81
PVPianist = – $40,000 + $500,000/(1.05) = $436,190.48.
Therefore, she will become a pianist. If the rate of interest is 15 percent, however, the present value
calculations become
and
PVBiologist = – $15,000 + $472,000/(1.15) = $395,434.78
PVPianist = – $40,000 + $500,000/(1.15) = $394,782.61.
In this case, Debbie becomes a biologist. As the interest rate increases, the worker discounts future
earnings more, lowering the returns from investing in education.
(b) Suppose musical conservatories raise their tuition so that it now costs Debbie $60,000 to become
a concert pianist. What career will Debbie pursue if the discount rate is 5 percent?
Debbie will compare the present value of being a biologist from part (a) with the present value of
becoming a pianist. The relevant present values are:
and
PVBiologist = – $15,000 + $472,000/(1.05) = $434,523.81
PVPianist = – $60,000 + $500,000/(1.05) = $416,190.48.
Debbie will, therefore, become a biologist.
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7-2. Peter lives for three periods. He is currently considering three alternative education-work
options. He can start working immediately, earning $100,000 in period 1, $110,000 in period 2 (as
his work experience leads to higher productivity), and $90,000 in period 3 (as his skills become
obsolete and physical abilities deteriorate). Alternatively, he can spend $50,000 to attend college in
period 1 and then earn $180,000 in periods 2 and 3. Finally, he can receive a doctorate degree in
period 2 after completing his college education in period 1. This last option will cost him nothing
when he is attending graduate school in the second period as his expenses on tuition and books will
be covered by a research assistantship. After receiving his doctorate, he will become a professor in a
business school and earn $400,000 in period 3. Peter’s discount rate is 20 percent per period. What
education path maximizes Peter’s net present value of his lifetime earnings?
The present discounted values of Peter’s earnings associated with each of the alternatives are
PV HS = 100,000 +
110,000 90,000
+
= $254,167 ,
1.2
1.2 2
PVCOL = −50,000 +
180,000 180,000
+
= $225,000 ,
1.2
1.2 2
PV PhD = −50,000 +
0 400,000
+
= $227,778 .
1.2
1.2 2
and
Thus, the best option for Peter is to start working upon completely high school.
7-3. Jane has three years of college, Pam has two, and Mary has one. Jane earns $21 per hour, Pam
earns $19, and Mary earns $16. The difference in educational attainment is due completely to
different discount rates. How much can the available information reveal about each woman’s
discount rate?
The returns to increasing one’s education from one to two years of college and then from two to three
years of college are
r1to 2 =
$19 − $16
$21 − $19
= 18.75% and r2to 3 =
= 10.53% .
$16
$19
Having observed their educational choices, we know that Mary’s discount rate is greater than 18.75
percent, Pam’s is between 10.53 percent and 18.75 percent, and Jane’s is less than 10.53 percent.
7-4. Suppose the skills acquired in school depreciate over time, perhaps because technological
change makes the things learned in school obsolete. What happens to a worker’s optimal amount of
schooling if the rate of depreciation increases?
If the rate of depreciation is very high, the payoff to educational investments declines. As a result, a
worker’s optimal amount of schooling will also fall as the benefits of education erode rapidly.
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7-5. Suppose workers differ in their ability, but have the same discount rate. Is it possible for the
more able workers to choose less schooling?
This result is possible as long as more able workers have lower marginal-rate-of-discount curves. For
example, if an 18-year-old basketball player can earn $3 million per year by entering the NBA after high
school whereas he would earn $3.25 million per year by entering the NBA after college, the opportunity
cost of college ($3 million per year) may be so great that the player opts to skip college. (A similar story
might explain why Bill Gates dropped out of Harvard.)
7-6. Suppose Carl’s wage-schooling locus is given by
Years of Schooling
6
7
8
9
10
11
12
13
14
Earnings
$10,000
$12,800
$16,000
$18,500
$20,350
$22,000
$23,100
$23,900
$24,000
(a) Derive the marginal rate of return schedule. When will Carl quit school if his discount rate is 4
percent? What if the discount rate is 12 percent?
The marginal rate of return is given by the percentage increase in earnings if the worker goes to school
one additional year.
Schooling
6
7
8
9
10
11
12
13
14
Earnings
$10,000
$12,800
$16,000
$18,500
$20,350
$22,000
$23,100
$23,900
$24,000
MRR
28.0
25.0
15.6
10.0
8.1
5.0
3.5
0.4
Carl will quit school when the marginal rate of return to schooling falls below his discount rate. If his
discount rate is 4 percent, therefore, he will quit after 12 years of schooling; if his discount rate is 12
percent, he will quit after 9 years of schooling.
(b) Suppose the government imposes an income tax of 20 percent on both labor earnings and
interest income. What is the effect of this income tax on Carl’s educational attainment?
This is a tricky problem. If Carl is making his educational decision by comparing the marginal rate of
return to schooling to some rate of discount that does not depend on the government’s tax policies, then it
turns out that Carl’s optimal schooling level is unchanged. It is easy to verify that if the government
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imposes a 20 percent tax rate on labor earnings in the second column of the table above, the marginal rate
of return to schooling (column 3) remains unchanged. If, however, Carl’s rate of discount is affected by
the government’s tax policies (for example, Carl’s rate of discount might be affected by the rate of
interest banks pay), then Carl’s educational decision will be affected. For example, if Carl’s rate of
discount falls by 20 percent then the amount of schooling acquired goes up because the marginal rate of
return schedule in column 3 of the table has changed.
7-7. In the typical signaling model, it is assumed that the costs of acquiring an education are higher
for low-ability than for high-ability workers. Suppose the government subsidizes low-ability
workers for the higher costs they incur in obtaining an education. What happens to the signaling
value of education? Can there be a perfectly separating equilibrium in this labor market?
If the government subsidizes schooling so that the cost of schooling is the same for all workers, then the
signaling value of schooling is lost. There cannot be a perfectly separating equilibrium because all
workers would have the same incentive to obtain the same amount of schooling.
7-8. Suppose there are two types of persons: high-ability and low-ability. A particular diploma costs
a high-ability person $8,000 and costs a low-ability person $20,000. Firms wish to use education as a
screening device where they intend to pay $25,000 to workers without a diploma and $K to those
with a diploma. In what range must K be to make this an effective screening device?
In order for a low-ability worker to not pursue education, it must be that $25,000 ≥ K – $20,000 which
requires K ≤ $45,000. Similarly, in order for a high-ability worker to pursue education, it must be that K –
$8,000 ≥ $25,000 which requires K ≥ $33,000. Thus, in order to use education as a signaling device, it
must be that educated workers are paid between $33,000 and $45,000.
7-9. It has been argued that the minimum wage prevents workers from investing in on-the-job
training and discourages employers from providing specific training to low-income workers. Why
would the minimum wage have an adverse effect on human capital accumulation for low-income
workers?
First, when the minimum wage is high, the marginal return to on-the-job investment falls (assuming one
can always find a job), and therefore the lowest skill workers may no longer find it useful to engage in onthe-job training.
Second, a firm that offers general or specific training in the first period pays the worker a wage below his
or her marginal product while the investment is taking place and above his or her marginal product in the
post-investment period. If the minimum wage prevents the investment-period wage from falling
sufficiently, however, firms may not be able to offer the training.
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7-10. Jill is planning the timing of her on-the-job training investments over the life cycle. What
happens to Jill’s OJT investments at every age if
(a) the market-determined rental rate to an efficiency unit falls?
The marginal revenue of investing in OJT declines so that Jill will invest less at each age.
(b) Jill’s discount rate increases?
If Jill’s discount rate increases she becomes more “present oriented”, reducing the future benefits
associated with OJT. Thus her OJT investments fall.
(c) the government passes legislation delaying the retirement age until age 70.
The marginal revenue of investing in OJT increases because the payoff period to the investment is longer.
Thus, she undertakes more OJT in this case.
(d) technological progress is such that much of the OJT acquired at any given age becomes obsolete
within the next 10 years.
The marginal revenue to investing in OJT declines and the amount of OJT acquired falls.
7-11. In 2000, there were about 9 million students in four-year college institutions in the United
States. Believing that education is the key to the future, a presidential candidate proposes that the
federal government pay the first $3,000 of college expenses each year for everyone attending a fouryear college. It is expected that this proposal will encourage 3 million more people to enroll in a
four-year college each year, but that the graduation rate will fall from 80 percent to 75 percent.
What is the yearly projected cost of the program? What is the average cost of the plan per new
student attending a four-year college? What is the average cost of each new four-year college
graduate?
Under the old plan, 9 million students attended a four-year college each year, with 80 percent (7.2
million) eventually graduating. Under the new plan, 12 million students will attend a four-year college
each year, with 75 percent (9 million) eventually graduating. The annual cost of the program, therefore, is
12 million × $3,000 = $36 billion. The average cost of the plan per new college student is $36b / 3m =
$12,000. The average cost of each new graduate is $36b / 1.8m = $20,000.
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7-12. In 1970, men aged 18 to 25 were subject to the military draft to serve in the Vietnam War. A
man could qualify for a student deferment, however, if he was enrolled in college and made
satisfactory progress on obtaining a degree. By 1975, the draft was no longer in existence. The draft
did not pertain to women. Using the data in Table 255 of the 2002 edition of the U.S. Statistical
Abstract, use women as the control group to estimate (using the difference-in-differences
methodology) the effect abolishing the draft had on male college enrollment.
The difference-in-differences table is
Men
Women
1970
55.2
48.5
College Enrollment (percentage)
1975
Diff
Diff-in-diff
52.6
-2.6
-3.1
49.0
0.5
Thus, abolishing the draft is estimated to lower the college enrollment rate of men by 3.1 percentage
points.
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