CHAPTER 7
HUMAN CAPITAL
the basics
We refer to “Human Capital” as the model of “Long-run” supply in skills markets. It’s the precise
extension of the idea of Long-run equilibrium in the product markets, well known to all students
who have taken introductory microeconomics. In the product markets, Long-fun equilibrium is
detrmined by a single factor: that firms will enter an industry where existing firms are experiencing
excess profits (or rents), and firms will exit from an industry where existing firms are experiencing
losses. So the Long-run equilbrium result is determined by the behavior of firms: that they seek
profit. This behavior, combined with the classic textbook assumption that all firms in any industry
– that is, any product market – have identical costs, creates the two important Long-run equilibrium
conditions: that Long-run equilibrium is zero-profit, and that price in any product market is equal to
the average cost of producing the product.
The extension to skills markets works the same way. In the skills markets, individuals enter skills
markets by purchasing the “human capital” that gives them the skill. Income-maximizing
individuals will enter skills markets where the existing workers have higher lifetime incomes – rents
– and will choose not to enter skills markets where the existing workers have lower lifetime
incomes. This behavior, combined with the assumption that all individuals are alike, creates the
two important – initial – Long-run equilbrium conditions: that Long-run equilibrium wages will
equalize lifetime incomes among all individuals, and that wage differentials between skills markets
are equal to the cost of acquiring the higher skill.
So the analysis begins with the individual’s decision to acquire a skill. Acquiring a skill is
acquiring “human capital”. At its most fundamental level, Capital is time. So the basic intuition of
acquiring “human” Capital can be explored by first looking at a single individual operating alone,
with no markets. So we begin with Robinson Crusoe, marooned alone on a small island. In his
initial season on the island he has two choices involving the future, which I call A and B. A here,
and in the chapters on Human Capital that follow, is the do-nothing choice. B is the choice to
acquire capital.
A: Crusoe can spend initial season Working: gathering seeds and eating them. Then he can
spend the next season also Working: gathering seeds and eating them, and the next, and the
next....
Or
B: Crusoe can spend some of his initial season Working: gathering seeds and clearing some
land, planting some seeds in the cleared land. Then he can spend future seasons Working:
harvesting seeds from his clearing and planting some each year.
If Crusoe chooses the first option, A, he spends his time Working (gathering) and in Leisure. If he
chooses the second option, B, he spends time in his initial season Working (gathering, clearing, and
planting) and in Leisure. If he chooses B, in the initial periods he must Work more – gathering the
extra seeds to plant, clearing and planting. So if he chooses B Crusoe has less Leisure in his initial
season. But if he clears and plants in the initial season, he can harvest more in a given amount of
time in all future seasons, so in future seasons he has more Leisure.
Since with B he can harvest more in a given amount of time in all future seasons, this is an increase
in the productivity of his labour. The second option, B, creates "Capital"; the Capital increases
Crusoe's productivity in all seasons after the initial. The cost of the Capital is exclusively time: the
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opportunity cost of the Leisure time Crusoe must sacrifice in the initial season to clear, to plant, and
to gather the extra seeds he will plant.
For simplicity, we will measure all of Crusoe’s costs and benefits in hours of Leisure. We know
that Crusoe, as an economic agent, will maximize utility. By using only Leisure as a measure of all
costs and benefits, we implicitly assume that the utility value of pure subsistence – eating enough
seeds to survive – is zero. This is wrong, but starting at a positive number gives the same result,
and it is easier to start at zero. Also, by using hours of Leisure as a measure of all costs and
benefits, we implicitly assume that utility is a linear function of the hours of Leisure. That is, we
assume that utility is some constant times the number of hours of Leisure. The easiest constant to
use is 1. So in the analysis below, Crusoe gets 1 unit of utility for 1 hour of Leisure, and we can
use hours of leisure as utility here.
We know that this is wrong. Leisure, like all other goods, is subject to decreasing marginal utility,
as we saw when we did Work/Leisure choice. But it’s a useful assumption here and doesn’t bias the
results in any significant way.
Crusoe’s decision involves utility – Leisure – in both the present and the future. When considering
decisions about the future, we need to take into account that the future always involves some
uncertainty. The way we deal with that uncertainty in economics is to recognize that individuals
discount the future. As a simple example, consider $100 today and the promise of $100 a year from
now. Even if the “promise” is made by some institution that is fairly certain to be able to pay the
$100 a year from now – for example, the Canadian government – you may not be around to collect.
So you must discount the promise of $100 a year from now.
The value an individual uses to discount the future is called that individual’s discount rate. We use
the symbol  for this discount rate. Using the example above, an individual values the promise of
$100 a year from now, which is called the present value as: PV = $100/(1+). If it were a promise
to pay $100 two years from now: PV = $100/(1+)2; if three years from now: PV = $100/(1+)3. In
general, a promise to pay $100 n years from now has present value: PV = $100/(1+)n.
Thus Crusoe must discount the values of Leisure in the future.
To make the algebra easier I use a discount operator. The symbol for the discount operator, which
you will use solving problems and doing proofs, is Dn, where:
n
Dn =

t 1
1
(1   ) t
.
If any value X is constant over time, that is, if X = X1 = X2 = X3 = … = Xn, then:
n
 X (1   )
t
= Dn•X
t 1
Note that if  = 0, Dn = n, and Dn•X = nX.
There are two ways of looking at Crusoe’s choice. Both give the same result, but it is useful to
consider each separately. The first way of looking at Crusoe’s choice is that he will choose the
option with the greater value to him at the time of choice – that is, at the beginning of the initial
season.
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The second way of looking at Crusoe’s choice is that he will choose to create Capital – option B – if
the expected future value of the increased productivity is greater than its initial cost. Both ways of
looking at Crusoe’s choice give the same result. The first is probably easier for you to use solving
problems in Exercise I below; but I introduce the second because it is easier to use when we shift to
the labour market and you need to get familiar with it.
We now need to define some other symbols:
Time is in discrete “seasons”, and is denoted t. Time will normally be a subscript on another
variable, and it takes values from 0 to n.
The initial season is t = 0.
n is the number of seasons, after the initial season, that Crusoe expects to spend on the island
before he is rescued.
hWt is the number of hours Crusoe spends Working in season, t.
hLt is the number of hours of Leisure Crusoe takes in season, t.
I use the convention that superscripts refer to the different options (A or B), and subscripts
will be time periods.
As with Work/Leisure choice, if Crusoe is not working, he is in Leisure. So hours of Work
determine hours of Leisure, and we can use only one of the two variables in the analysis. I
will use hours of Leisure.
Crusoe must Work more hours in the initial season if he chooses to create Capital (choice B), so he
has less Leisure in period 0 with option B: hLB0 < hLA0 . But he must Work more hours all other
seasons if he chooses not to create capital (choice A), so he has less Leisure in periods 1, n with
option A: hLAt < hLBt , t = 1,n .
I make a final simplifying assumption – once Crusoe chooses an option, he sticks to it and nothing
changes. So his hours of Leisure are the same for all time periods after the initial, whether he
chooses A or B: hLA1 = hLA2 = … = hLAn, and hLB1 = hLB2 = … = hLBn .
So we have four things that can vary and which will affect Crusoe’s decision: Crusoe’s cost of
acquiring the Capital, (hLA0 – hLB0); the increase in Crusoe’s productivity from having the Capital,
(hLBt – hLAt), t = 1,n ; the rate at which Crusoe discounts the future, ; and how long Crusoe expects
to be on the island, n. [NOTE: In the story, Crusoe did not know for sure when he will be rescued;
that is, he did not know the value of n. This introduces a big analytical problem, known as risk or
uncertainty. Analysis of risk can be done, it just adds complexity; analysis of uncertainty is tricky.
We develop the theory assuming Crusoe does know n. After all, we’re not really trying to describe
Crusoe; we’re developing the basics of Human Capital using a made-up story to illustrate it.]
Looking at Crusoe’s decision in the first way has him compare the total values of the two choices
The values of the A and B choices, in the intial period, t = 0, are:
(7-1-a) V(A) = hLA0 + hLA1/(1+ + hLA2/(1+ + … + hLAn/(1+n, and
(7-1-b) V(B) = hLB0 + hLB1/(1+ + hLB2/(1+ + … + hLBn/(1+n.
Since hLA1 = hLA2 = … = hLAn , and hLB1 = hLB2 = … = hLBn , we can write these equations more
concisely by: 1) setting hLA1 , hLA2 , …, and hLAn equal to just hLA ; setting hLB1 , hLB2 , …, and
hLBn equal to just hLB ; and 2) using the discount operator, so:
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(7-2-a) V(A) = hLA0 + Dn•hLA , and
(7-2-b) V(B) = hLB0 + Dn•hLB .
We know that Dn •hLB > Dn•hLA, because, with Capital, Crusoe is more productive and, therefore,
has more leisure. And hLA0 > hLB0 because Crusoe must spend more time Working – clearing,
planting and gathering the extra seed to plant – in the initial season as he creates Capital. Therefore
he has less Leisure in the initial season. Depending on relative sizes of the variables, then, either
option may have the higher value. Crusoe, as a utility (Leisure) maximizer, chooses the option with
the higher value, ie with the most hours of Leisure with the future values appropriately discounted.
Looking at Crusoe’s decision in the second way, Crusoe chooses to create Capital if his gross
expected value of the increased productivity from that Capital is greater than his gross costs – the
foregone Leisure. Saying the same thing, Crusoe chooses to create Capital if his net expected value
of the increased productivity from that Capital is positive. Formally, this way of viewing Crusoe’s
decision is that he will choose B if:
(7-3) (Dn•hLB – Dn•hLA)  (hLA0 – hLB0), or if:
(7-3) [Dn•(hLB – hLA)]  (hLA0 – hLB0).
The Left-Hand Side (LHS) of inequation (7-3) is the return to creating Capital; the Right-Hand Side
(RHS) is the cost of creating Capital. Crusoe chooses to create Capital if the LHS  RHS.
Otherwise he does not create Capital, the A-choice. (We have to have Crusoe do something when
there is a “tie”, when the LHS = RHS. The convention in economics is to have the agent make the
postive decision when the net benefit is zero.)
Looking at the decision this way again shows the four things – variables – that can vary and affect
Crusoe’s decision: (hLA0 – hLB0), (hLB – hLA),  (which determines D), and n. The relation between
the four variables and Crusoe's decision is clear from either equations (7-2), or from inequation (73), but it is a little easier to see from inequation (7-3). These relations generate the four
comparative static results.
The comparative statics of the model of this decision are that Crusoe is more likely to choose B –
that is, the LHS of inequation (7-3) is more likely to be greater than the RHS – when, all other
things held equal:
i) the cost of B is lower, ie when (hLA0 – hLB0) is smaller;
ii) the productivity increase with B is larger, ie when (hLB – hLA) is greater;
iii) Crusoe discounts the future by less – ie his  is smaller – so, for any n, Dn is larger; or
iv) he expects to spend a longer time on the island – ie his n is larger – so, for any , Dn is
larger.
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Exercise Set 7:
We begin with Crusoe having available, after he eats, sleeps and takes care of other necessities of
life, 5000 hours per season. Crusoe can either Work or take Leisure in those 5000 hours. Crusoe
values only Leisure, and we measure his “utility” simply in hours of Leisure.
Crusoe needs 300kg of edible seeds to live reasonably, and Crusoe gets no utility from overeating.
It takes him 10 hours to gather 1kg of seeds on uncleared land. It would take him 1600 hours to
clear the land for planting, to gather the seeds for planting, and to plant seeds which will let him
harvest 300kg on the cleared land. It takes Crusoe 8 hours to gather 1kg of seeds on cleared land.
Crusoe wants to maximize his Leisure (his “utility”) subject to the constraints descried above and
his discounting of the future.
1. Work out, with numbers, the values of equations (7-2), and inequation (7-3) with n = 3 and
D3 = 2.5. If Crusoe expects to be rescued after the initial plus three more seasons, will he
choose A or B?
2. Now use equation (7-2-b) with  = 0 and n = 3 to show that if Crusoe is to clear-and-plant,
he will do so in the initial season (t = 0), not the second season: t = 1.
3. Crusoe's discount rate is zero ( = 0).
a. Crusoe expects to be rescued after the initial plus two more seasons (n = 2). Will
Crusoe choose A or B?
b. Crusoe expects to be rescued after the initial plus three more seasons (n = 3). Will
he choose A or B?
4. It now takes Crusoe 8.5 hours to gather 1kg of seeds on cleared land. Crusoe's discount rate
is zero ( = 0). Crusoe expects to be rescued after the initial plus three more seasons (n = 3).
Will he choose A or B?
5. OPTIONAL: Here is a preliminary use of risk in theory. In this case Crusoe is what is
known as risk-neutral. This means he bases his decision only on expected values. Expected
Value is equal to: (the probability of event number 1 times the value of event 1) + (the
probability of event number 2 times the value of event 2).
It takes Crusoe 8 hours to gather 1kg of seeds on cleared land. Crusoe's discount rate is zero
( = 0).
a. Crusoe expects: a fifty percent chance, or a probability of .5, of being rescued at the end
of the second season (n = 2); a fifty percent chance, or a probability of .5, of being
rescued at the end of the third season (n = 3). Will he choose A or B?
b. Crusoe expects: a twenty percent chance, or a probability of .2, of being rescued at the
end of the second season (n = 2); and an eighty percent chance, or a probability of .8, of
being rescued at the end of the third season (n = 3). Will Crusoe choose A or B?