Chapter 06 - Human Capital
CHAPTER 6
6-1. Debbie is about to choose a career path. 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 rate of interest between the
two periods. 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 greater present value. If the interest rate is 5 percent,
PVBiologist = –$15,000 + $472,000/(1.05) = $434,523.81
and
PVPianist = –$40,000 + $500,000/(1.05) = $436,190.48.
Therefore, she will become a concert pianist. If the rate of interest is 15 percent, however, the
present value calculations become
PVBiologist = –$15,000 + $472,000/(1.15) = $395,434.78
and
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 interest 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:
PVBiologist = –$15,000 + $472,000/(1.05) = $434,523.81
and
PVPianist = –$60,000 + $500,000/(1.05) = $416,190.48.
Debbie will, therefore, become a biologist, showing that as the cost of an invest increases, the
chance of pursuing that investment falls.
6-1
Chapter 06 - Human Capital
6-2. Peter lives for three periods. He is currently considering three alternative educationwork 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.
6-3. 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 (otherwise she would have invested in a second year of education and earned
18.75% on the investment), Pam’s is between 10.53 percent and 18.75 percent, and Jane’s is less
than 10.53 percent.
6-2
Chapter 06 - Human Capital
6-4. 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.
6-5.
(a) Describe the basic self-selection issue involved whenever discussing the returns to
education.
People choose their level of education knowing their own abilities, preferences, and financial
situation. Most important here is knowing one’s abilities. Highly capable people would likely
earn a large salary even if they didn’t attend college, but they choose to attend because they earn
even more (net of the cost of college) by doing so. Likewise, less capable people know they are
less capable and that they will not get very high paying jobs even with a college degree.
Consequently, highly capably people tend to go to college while less capable people are less
likely to go to college, and the average wage of college graduates is higher than the average wage
of non-college graduates largely because of self-selected education levels due to innate skills or
abilities.
To put numbers with the problem, suppose highly capable person would earn $50,000 without a
college education and $65,000 with a college education. Similarly, a less capably person would
earn $20,000 without a college education and $35,000 with a college education. All high ability
people go to college, while none of the low ability people do. Clearly in this example, if one
knows the numbers, one would say that the return to college is $15,000. If one just saw the raw
data of who went to college (and who did not) and each person’s income, one would falsely
conclude that the return to college is $45,000.
(b) Does the fact that some high school or college dropouts go on to earn vast amounts of
money (e.g., Bill Gates dropped out of Harvard without ever graduating) contradict the
self-selection story?
No. One, there are always exceptions. And two, if the cost of education gets large enough (or the
returns to education get small enough), even high ability people will forego college.
(c) Most government-provided job training programs are optional to the worker. Describe
how the self-selection issue might be used to call into question empirical results suggesting
there are large economic benefits to be gained by requiring all workers to receive
government-provided job training.
As job training programs are optional, and willingness to work or try to get a new job or to get
retrained is probably the most important factor in a person’s success, there is certainly a selfselection story to be told. In particular, the successful people coming out of job training programs
would likely have been successful even if left on their own because of their innate
ability/motivation. Similarly, the people who did not choose job training and failed to get a job
would likely have failed to get a job even if the government required them to pursue job training.
6-3
Chapter 06 - Human Capital
6-6. Suppose Carl’s wage-schooling locus is given by
Years of Schooling
9
10
11
12
13
14
Earnings
$18,500
$20,350
$22,000
$23,100
$23,900
$24,000
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
9
10
11
12
13
14
Earnings
$18,500
$20,350
$22,000
$23,100
$23,900
$24,000
MRR
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.
6-7. Table 217 of the 2006 U.S. Statistical Abstract shows that, among all 25-34 year-olds,
the average annual earnings of a high school graduate with no further education was
$26,073 while the average annual earnings of a college graduate with no further education
was $43,794 is 2003.
(a) Assuming college requires five years, show that the annual return to each year of college
education averages 10.9%.
The annual rate of return, x, solves (1+x)5 • $26,073 = $43,794, which yields x = 0.10928.
(b) It is typically thought that this type of calculation of the returns to schooling is biased,
because it doesn’t take into account innate ability (i.e., ability in the workplace not due to
college) or innate motivation. If this criticism is true, is the actual return to each year of a
college education more than or less than 10.9%?
It is typically argued that people who are innately skilled or motivated pursue more education
than those who are less innately skilled or motivated, because the cost (psychic and in terms of
the time spent in college) are less for the innately skilled or motivated. If true, then the returns to
education are over-estimated by this type of simple calculation. Of course, the typical story might
be wrong. The innately skilled or motivated might have to give up a lot in terms of foregone
earnings in order to attend college, which they might not need in the first place (e.g., Bill Gates,
NBA players). If so, then the returns to education could be under-estimated.
6-4
Chapter 06 - Human Capital
6-8. 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,
otherwise pursuing the diploma would be better than not pursuing the diploma. Thus, it must be
that K ≤ $45,000 to make sure low-ability people don’t pursue the diploma.
Similarly, in order for a high-ability worker to pursue education, it must be that
K – $8,000 ≥ $25,000,
otherwise not pursuing the diploma would be better than pursuing the diploma. Thus, it must be
that K ≥ $33,000 to make sure high-ability people pursue the diploma.
Thus, in order to use education as a signaling device in this example, it must be that educated
workers are paid between $33,000 and $45,000.
6-9. Some economists maintain that the returns to additional years of education are actually
quite small but that there is a substantial “sheepskin” effect whereby one receives a higher
salary with the successful completion of degrees or the earning of diplomas (i.e.,
sheepskins).
(a) Explain how the sheepskin effect is analogous to a signaling model.
The sheepskin effect is analogous (in fact it is identical) to the signaling model in that purchasing
the signal doesn’t actually change the person’s skills or productivity. Rather, purchasing the
signal in effect documents or reveals that the person is a high ability person. This is exactly the
same as the sheepskin effect. That is, paying the money and sitting through classes and doing the
work doesn’t change the person. Rather, no one without high skills would choose to do this, so
acquiring a sheepskin is a tool by which to “signal” one’s productivity even though achieving the
sheepskin had not direct effect on the individual.
6-5
Chapter 06 - Human Capital
(b) Typically in the United States, a high school diploma is earned after 12 years of
schooling while a college degree is earned after 16 years of school. Graduate degrees are
earned with between 2 and 6 years of post-college schooling. Redraw Figure 6-2 under the
assumption that there are no returns to years of schooling but there are significant returns
to receiving diplomas.
The Wage-Schooling Locus with Sheepskin Effects
Dollars
$68,000
$42,000
$30,000
$18,000
12
16
20
Years of Schooling
The bold line in the above graph gives the wage-schooling locus with sheepskin effects. In
particular, anyone without a high school diploma earns $18,000; anyone with a high school
diploma (and no college diploma) earns $30,000; someone with a college diploma (but not a
graduate school diploma) earns $42,000; and people with a graduate degree earn $68,000.
(c) Devise a difference-in-differences estimator (i.e., what data would you need and what
would you do with the data) that would allow one to get at whether completing each year of
school or completing degrees matters more when determining wages.
One way to do this without a difference-in-differences estimator (in fact, it is just a differences
estimator) would be to collect data on yearly salary, years of education, and diplomas on a
randomly chosen group of individuals. From these data, a wage-schooling locus can be
produced. One could then see if it looks like the graph above (suggesting a sheepskin effect) or
like the graph in Figure 6-2 (suggesting continuous returns to each year of education). The
problem here is that self-selection remains a problem.
In order to use a difference-in-differences estimator, one would want to grab a random group of a
cohort of individuals (say born in 1980) and observe each person’s years of schooling, diplomas,
and salary in two separate years. Differences in income can then be computed for all, as well as
differences in years of education and differences in diplomas if any exist. The difference-indifferences estimator is then straightforward. The difficult part is making sure some people don’t
change years of education or diplomas across the two time periods while others change just years
of education (and return to work) without ever receiving an additional diploma, while others
change their years of education and receive an additional diploma (and have returned to work).
This estimator still suffers from selection bias, but it is less than a simple differences estimator.
6-6
Chapter 06 - Human Capital
6-10. 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 Jill will invest less at each age as the return
to making the investment has fallen.
(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 as she no longer values the benefits from
making the investment as much as she had before her discount rate fell.
(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.
6-7
Chapter 06 - Human Capital
6-11. Suppose 3 million high school graduates start college each year. Those who earn a
college degree will do so in four years. However, some students will drop out along the way.
The first-year attrition rate is 20%, while the second and third-year attrition rates are 10%
and 2.5% respectively.
(a) What is the distribution of college students by year in college? How many students
graduate from college each year?
Consider an in-coming class. There are 3 million freshmen. The following year there will be 2.4
million sophomores. The following year there will be 2.16 million juniors. The following year
there will be 2.106 million seniors. Thus, at any time, there will be 9,666,000 college students –
31.0% will be freshmen, 24.8% will be sophomores, 22.3% will be juniors, and 21.8% will be
seniors. Moreover, each year will see 2,106,000 new college graduates.
(b) 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 four-year college. It is expected that this proposal will encourage 1 million more
high school graduates to enroll in a four-year college each year. Of these 1 million new
college students, the first, second, and third-year attrition rates are 40%, 20%, and 5%.
Why is it likely that attrition rates will be higher among these groups of students?
The policy will encourage 1 million new students to attend college. Some of these students will
be “college material” so that the $3,000 subsidy simply allows them to now afford the cost of
college. Others, however, will not be “college material” but since the cost of college has fallen
because of the subsidy, they give college a shot (and eventually drop out).
(c) What is the yearly projected cost of the program in part (b)? What is the average cost of
each new four-year college graduate?
To calculate the yearly cost of the program, we first need to track the 1 million new students.
There are 1 million additional freshmen. The following year there will be 600,000 additional
sophomores. The following year there will be 480,000 new juniors. The following year there will
be an additional 456,000 seniors. Thus, in the long-run, there will be an additional 2,536,000
college students in the system, for a total of 12,202,000 college students in the system in any
given year.
The yearly cost of the program is $3,000 • 12,202,000 = $36,606,000,000 = $36.6 billion.
Notice that the subsidy is given to every student whether or not they would attend college without
the subsidy. Under the subsidy program, which costs $36.6 billion each year, only 456,000 more
college graduates are created. Thus, the cost of the program for each of these new college
graduates is about $80,000.
6-8
Chapter 06 - Human Capital
6-12. 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 269 of the 2008
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 have lowered the college enrollment rate of men by 3.1
percentage points.
6-13.
(a) Draw the wage-schooling locus for someone for whom the returns to schooling decrease
through college but increase after college. (Assume college is completed after 16 years of
schooling and that one can receive at most 6 years of post-college schooling.)
The wage-schooling locus for someone who experiences a negative return to
each year of college but a positive return for each year of graduate study.
Dollars
This is a strange case, so the slope of
the locus during the college years (12
to 16) can be almost anything as long
as it is negative.
12
16
6-9
22
Years of Schooling
Chapter 06 - Human Capital
(b) On a new graph, plot the marginal rate of return to schooling implied by the wageschooling locus described in part (a).
Yearly Marginal
Return Schooling
0
12
16
22
Years of Schooling
The marginal return to each year of schooling is plotted above in the bold line to match the wageschooling locus of part (a).
(c) What can be said about a college graduate who faces the wage-schooling locus described
in part (a)?
If the returns to each year of college are negative, the immediate consequence is that no one will
go to college without then going on to graduate school. One either ends schooling after high
school or with some graduate school.
6-10
Chapter 06 - Human Capital
6-14. A high school graduate has to decide between working and going to college. If he
works, he will work for the next 50 years of his life. If he goes to college, he will be in
college for 5 years, and then work for 45 years. In this model, the rate of discount that
equates the lifetime present value of not going to college and going to college is 8.24% when
the cost of each year of college is $15,000, each year of non-college work pays $35,000, and
each year of post-college work pays $60,000. For each of the parts below, discuss how the
rate of discount that equalizes the two options would change and who would make a
different schooling decision based on the change. (Extra credit: Use Excel to show that the
rate of return to schooling is 8.24% in the above case, and solve for the rates of discount
associated with each of the parts below.)
Calculating the rate of return for each case is straightforward in Excel by using the IRR function.
(a) Each year of college still costs $15,000 and each year of post-college work still pays
$60,000, but each year of non-college work now pays $40,000.
As the dollar benefit from not attending college has increased (from $35,000 to $40,000
annually), the return to college must fall. In fact, it falls to 5.98%.
(b) Each year of college still costs $15,000 and each year of non-college work still pays
$35,000, but each year of post-college work now pays $80,000.
As the dollar benefit from college has increased (from $60,000 to $80,000 annually), the return to
college must also increase. In fact, it increases to 13.66%.
(c) Each year of non-college work and post-college work still pays $35,000 and $60,000
respectively, but now each year of college costs $35,000.
As the dollar cost to college has increased (from $15,000 for four years to $35,000 for four
years), the return to college must fall. In fact, it falls to 5.86%.
(d) Each year of college still costs $15,000. The first year of non-college work pays $35,000
but then increases by 3 percent each year thereafter. The first year of post-college work
pays $60,000 but then increases by 5 percent each year thereafter.
This problem boils down to the rates of change in salaries. As the non-college salary is
increasing at a lower rate than the college salary is increasing, the benefits from attending college
are increasing relative to the benefits from not attending college. Thus, the return to college must
increase. In fact, it increases to 12.73%.
6-11