CHAPTER 7
HUMAN CAPITAL
the basics
At its most fundamental level, Capital is time. So the basic intuition of “human” Capital can be
explored by considering 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: He 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: He 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 he chooses the first option, A, Crusoe 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 he 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
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.
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.
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.
For simplicity, we will measure all 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 are implicitly assuming 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 are
implicitly assuming that utility is a linear function of the hours of Leisure. That is, utility is just
some constant – the easiest constant being 1 – times the number of hours of Leisure. So below,
Crusoe gets 1 unit of utility for 1 hour of Leisure, and we can just use hours of leisure as utility
here.
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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 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:
n

Dn =
t 1
1
(1   ) n
. This also implies, in the mathematical sense of “implies”:
n
 X (1   )
n
= Dn•X if X = X1 = X2 = X3 = … = Xn.
t 1
Note also that if  = 0, Dn = n, and Dn•X = nX.
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
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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. 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 six things that can vary and which will affect Crusoe’s decision: , n, hLA0 , hLB0 , and
hLAt and hLBt , t = 1,n .
[NOTE: In the story, Crusoe did not know for sure when he will be rescued; that is, he did not know
n. This introduces a big analytical problem, known as risk or uncertainty. Analysis of risk is tricky;
analysis of uncertainty is even trickier. I develop the theory assuming Crusoe does know n. After
all, I’m not really trying to describe Crusoe; I’m 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:
(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. But 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 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 Sides (LHSs) of inequations (7-3) are the return to choosing to create Capital; the
Right-Hand Sides (RHSs) are 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
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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 reduces to four the number of things that can vary and affect
Crusoe’s decision:  (which determines D), n, (hLA0 – hLB0), and (hLB – hLA). 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).
The comparative statics of this 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 productivity increase with B is larger, ie when (hLB – hLA) is greater;
ii) the cost of B is lower, ie when (hLA0 – hLB0) is smaller;
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 can 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?