CHAPTER 8
GENERAL HUMAN CAPITAL AND
LONG-RUN EQUILIBRIUM IN LABOUR MARKETS
WORKERS ARE HOMOGENEOUS
Here we take the basics of an individual decision, which we just developed for an individual isolated from any external forces – Robinson Crusoe – and generalize it to all individuals, and put those individuals into an economy where markets exist. This will allow us to derive the conditions of Long-run equilibrium in those markets, the “skills markets”. Initially we treat all individuals as identical, or homogenoeous , which yields Long-run equilibrium conditions in skills markets which mimic precisely the Long-run equilibrium conditions of product markets. In the next chapter, we allow individual differences – we have heterogeneous individuals – which changes the Long-run equilibrium conditions somewhat.
I.
Fundamentals of the Flows Approach:
We saw in our analysis of Crusoe’s decision that a variable that was a factor in his decision was n , or how long he expected to use the increased productivity of his created Capital. That same n plays an important role in Human Capital analysis. So to apply the idea of human capital to observed labour market phenomena – that is, to give meaning to n – we must begin with the “flows approach” to labour markets. The flows approach is based on the obvious fact that workers age. They enter the labour market when young, work until old, and then retire (if they are lucky enough not to die before).
So labour markets are inherently dynamic. A fraction of the existing labour supply leaves at the end of each period. This is exit , which equals: quits + retirements + deaths. A number of new workers enter labour markets at the beginning of each period. This is entry . For simplicity we will assume a uniform
(rectangular) age distribution of workers. All workers expect to work n+1 periods. [Note: it is n +1 rather than n because we begin all working lives at the beginning of t = 0, and end them at the end of t = n , which makes n +1 periods of a working life.] It is a property of the simple model – the model we are developing here – that exit only occurs at the end of a working life. For now exit is just retirements.
All entry occurs at a given age. Thus the model requires that “Work” include the acquisition of human capital – that is the acquiring of skills. (This means that all of you, as college students, are “working”, even if you are not working for pay in the labour market.) We assume a constant number of people working at any period. This stock of workers is N . This assumption, together with the definitions of exit and entry, mean that each period: 1) exit equals entry and 2) both are equal to N / n+1 for the labour stock .
Time is critical. Time is human capital. Everything happens in real time. Time is actually continuous, but here we break time up into discrete time periods, which are denoted t , because the mathematics of discrete time periods is easier to deal with. We can make the analysis even simpler by defining the length of a period, t , as the length of time it takes to acquire the human capital, the skill . That is discrete, and the use of discrete time periods is mathematically allowed, and sensible.
Time is used in two different ways. First, there is real time . Real time moves inexorably forward, as we all know. 2005 follows 2004; t+1 follows t , etc.
Second, time is also used to measure individuals’ lives. To do this, first we divide the labour force into age cohorts . An age cohort is the group of the labour force that is all the same age. (Look around you labour economics class; all students who will graduate from college at the same time as you are part of your cohort .) For any cohort, t = 0 is the initial period of its working life, and it will retire at the end of period t = n .

Allen: labour economics, Chapter 8 page 2
Figure 8-1 illustrates the double use of time. Cohort

enters the labour market at the beginning of the year 2004. 2004 is t = 0 for cohort

. If n is 39, cohort

retires at the end of 2043. Between 2004 and
2005 real time moves to the right, along the horizontal axis, one period. Cohort

enters the labour market at the beginning of the year 2005. 2005 is t = 0 for cohort

. With n = 39, cohort

retires at the end of 2044. 2005 is also t = 1 for cohort

. So, visually, real time moves, but each cohort is still. As real time moves, each cohort ages until its t = n ; at that time the cohort retires. number of workers per cohort
Figure 8-1
100,000

: t = 0

: t = n
2004
2005 2043 2044

: t = 0

: t = 1 real time

: t = n
We use a closed labour market, one with no immigration or emigration. It has 4,000,000 workers: N =
4,000,000. The age of the workers is distributed uniformly from 21 to 60; each age-cohort has the same number of workers (100,000). The length of a period, t
, is one year. Everyone “works” from age 21 through age 60, that is for 40 years. So n = 39; t = 0 is age 21; t = 1 is age 22, and t = n is age 60 for each cohort. Each year, then, a cohort of 100,000 workers exit, and each year another cohort of 100,000 workers enter: 100,000 = 4,000,000/40 = N / n+1 .
On Figure 8-1 , there are two large rectangles: one with left and right borders that are bold solid lines, one with left and right borders that are dashed lines. The area of each of these rectangles is the total number of workers, that is the size of the labour force, N . The vertical lines are the number of workers in each cohort. Given the numbers in the illustration above, the vertical distance of these lines is 100,000.
The are of the two large rectangles is (100,000 workers/cohort) • (40 cohorts) = 4,000,000 workers.
II.
Human Capital and Flows – Workers make choices:
A. All Workers Alike – Two Skill Markets:
We begin with the simplest possible “case”: all workers are alike, and there are two skill markets. To integrate Human Capital with the flows approach, we need several fundamental assumptions and definitions.
Assumptions :
A1 : There exists at least one distinct skill, labelled the B -skill. Another skill, the A -skill, is the “default”.
All workers begin the initial period possessing the A -skill; the A -“skill” is basic unskilled labour. The markets for the two “skills” are referred to as the
A -market and the B -market, or, for simplicity of notation below, just A and B . B is distinct in two respects:

Allen: labour economics, Chapter 8 page 3
1) The demand for the B -skill is different from demand for the A -skill. A -skilled workers cannot do B -skill work, although B -skilled workers can do A -skill work.
2) This strictly implies that some way exists of identifying workers who possess the B -skill.
A2 : It requires real resources to produce (or identify) workers with the B -skill. In fact, this is really more than an assumption. If acquisition of the B -skill requires no resources, there is no economic problem. It is the fact that it requires resources to produce (or identify) workers with B -skill that makes the decision an economic decision – a decision that involves the allocation of scarce resources. These resources are expended in the initial period, t = 0. This also strictly implies that the wage paid to people with the B -skill must be higher than that paid to workers with the A -skill.
A3 : Workers make decisions which maximize the value of lifetime utility expected at the time of decision and suitably discounted .
Definitions :
N = N A + N B , so labour supply in A excludes B -skilled workers who could work in A .
EN is total entry in any period; EN A is entry to A ; EN B is entry to B .
EX is total exit in any period; EX A is exit from A ; EX B is exit from B .
Stasis : A skill-market is in Long-run equilibrium when its supply is constant over time. This does not mean the supply doesn’t vary. With the flows approach, it means that that exit equals entry, so the size of the supply is constant. Long-run equilibrium must exist in both skill-markets; if one is out of
Long-run equilibrium, the other must be out of Long-run equilibrium as well. Exits are determined by decisions well back in the past, and are treated as constant. Market A is in Long-run equilibrium when N A is constant, which requires that EN A = EX A . Market B is in Long-run equilibrium when N B is constant, which requires that EN B = EX B .
To illustrate, consider the closed labour market with 4,000,000 workers. Let Long-run equilibrium be three-quarters of workers in A , and one-quarter of workers in B , so N A / N = .75, and N B / N = .25. Exit from
A each period is 75,000: EX A = ( N A / N )•100,000. Exit from B each period is 25,000: EX B =
( N B / N
)•100,000. Long-run equilibrium requires that entry to
A also be 75,000: EN A = 0.75•100,000 =
75,000, and that entry to B be 25,000: EN B = 0.25•100,000. This will keep a constant 3,000,000 workers in A , and a constant 1,000,000 workers in B .
If fewer than EX A enter A , and more than EX B enter B , EN A < EX A and EN B > EX B . This will cause N A to shrink and N B to grow. For example, say 55,000 enter A one period, and 45,000 enter B that period.
After that period N A will equal 2,980,000 and N B will equal 1,020,000. If this continues for a second period, after that period N A will equal 2,960,000 and N B will equal 1,040,000. This is not stasis; neither
N A or N B is constant. This is also not a long-run equilibrium. (And the initial condition could not have been long-run equilibrium for this to have happened, as we will see below.)
And, obviously, vice-versa : If more than EX A enter A , and fewer than EX B enter B , EN A > EX A and EN B <
EX B . This will cause N A to grow and N B to shrink. This, too, is neither stasis nor long-run equilibrium.
Symbols:
1.
w is per-period real wage , subscripted (as with Crusoe) with periods, t , and superscripted with skillmarket, A or B .
2.
h is per-period hours of work , subscripted with periods, t and superscripted with skill-market, A or
B , as: h A t and h B t
.
3.
y is per-period earnings , so: y A t
= w A t
• h A t
, and y B t
= w B t
• h B t
.

Allen: labour economics, Chapter 8 page 4
4.
t is time periods. An operational definition of great convenience is to set the length of the periods equal to the length of time required to produce or identify skill B .
5.
DC B is the direct costs of acquiring skill B .
6.
 is the discount rate, the rate at which individuals discount the future.
Working Assumptions :
We need some simplifying assumptions to make the initial presentation easier to understand, and to make working the problems easier. Some of these will be systematically changed later as their effects on labour market outcomes is explored.
1) As noted above, workers are motivated to maximize lifetime utility. We assume a constant marginal utility of income and risk-neutrality. These two assumptions mean that we can treat the decision to maximize utility as being identical to the decision to maximize lifetime real income, suitably discounted.
This allows us to ignore the utility function and deal only with real income. Because we are interested in labour market decisions, we go farther and assume that real income is determined exclusively by earnings.
2) We assume initially that everyone has the same discount rate. As in the previous chapter, we simplify the mathematics of discounting by using the discount operator, D n
.
3) Expectations are treated very simply. Wages are not expected to change, so we essentially ignore expectations (for now). This means that w is both the actual, observed , real wage and the expected real wage. It also means we can set all w A t
, t = 0, n , equal and denote these just as w A . And we can set all w B t
, t = 0, n , equal, and denote these just as w B .
4) We assume initially that everyone works the same number of hours per period unless they are acquiring
Human Capital. So h A
0
= h A t
= h B t
, t = 1, n and we can denote this just h , dropping the subscript The only h that has to retain sub- and superscripts is h B
0
. This is because the opportunity costs of acquiring the
Bskill are incurred by working fewer hours while acquiring the B -skill, Human Capital. So during the initial period, individuals choosing B work h B
0
hours at wage w A , where 0
 h B
0
< h . It is necessary that h B
0
< h because it takes time to acquire the B -skill. Note that h B
0
can equal zero.
This “opportunity time” is “spent” during the initial period. In fact we defined the length of the time period to be the amount of time it takes to acquire the B -skill, so this will always be true. The critical variable is opportunity costs. Thus, what matters is not how many hours are worked while acquiring the
B -skill but how many hours of work are foregone while acquiring the B -skill. Opportunity hours are oh B
= ( h – h B
0
). The opportunity costs of acquiring the B -skill are opportunity hours times the wage the worker would have been paid for those hours: oh B • w A . For simplicity, this will be denoted as OC B , and
OC B = oh B • w A .
A reminder that this is the simplest model: with only two skill-markets and no differences between workers. General Human Capital means that the skill acquired, the B -skill required for entry into the B market, can be used in a large enough number of firms or situations (such as self-employment) that no employer has any monopsony power over the skill. In this case, the individual will have to pay all costs of acquiring the skill. So in this case, the decision is the worker’s decision.
Decision Rule :
The model of the worker’s decision is very similar to Crusoe’s decision. The worker chooses to work in the skill-market which gives her the higher expected lifetime income. This is the value of lifetime income expected at the time of decision.

Allen: labour economics, Chapter 8 page 5
As with Crusoe’s decision, there are two ways of modeling this decision. In the first the worker compares the values of the lifetime incomes of the A and B choices, at t = 0, and chooses to work in the market which gives the higher lifetime income. These lifetime incomes are:
(8-1-a) Y( A ) = y A + D n • y A = ( h • w A ) + D n •( h • w A ), and
(8-1-b) Y( B ) = ( h B
0
• w A – DC B ) + D n • y B = ( h B
0
• w A – DC B ) + D n •( h
• w B ).
Economic principles require that w A < w B , so that D n • y B > D n • y A (see below). Depending on the relative sizes of the variables, either market can yield the higher lifetime income. Workers choose to work in the market which generates the higher lifetime income.
Following is the second way of modeling the worker’s decision.
This is the way that will be used in most of the analysis that follows!
This way of modeling the worker’s decision is that the worker chooses to acquire the B -skill if her gross expected return from working in the B -market is greater than (or equal to) her gross cost of acquiring the B -skill. Saying the same thing, that the worker chooses to acquire the
B -skill if her net expected return from working in the B -market is positive (or zero). The gross return to choosing B is the difference in income from periods t = 1, n , properly discounted. The gross cost of choosing B is all “paid” during the initial period. This is because of our simplifying action in setting the length of t equal to the time required to acquire the B -skill, so the skill is completely acquired during t = 0.
The gross cost is the sum of the two costs: the opportunity cost of working fewer hours, and the direct cost. The net return to choosing B is positive if the gross return is greater than the gross costs. Formally, the worker will choose to work in the B -market if:
(8-2

) ( D n • y B > D n • y A ) = [ D n •( h • w B ) – D n •( h • w A )]

( OC B + DC B ), or if:
(8-2) D n
( y B – y A ) = [ D n • h
•( w B – w A )]

( OC B + DC B ).
The Left-Hand Side (LHS) of inequation (8-2) is the return to working in the B -market; the Right-
Hand Side (RHS) is the cost of acquiring the B -skill. The worker chooses to acquire the B -skill and work in the B -market if the LHS

RHS. (We have to have the worker 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 shows that there are six things that can vary and affect the choice of skill market: oh B and DC B Costs), ( w B – w A ) (wage increase from choosing B ), h ,

(which determines D ), and n
, The relation between these variables and workers’ decisions is clear from either equations (8-1), or from inequation (8-2), but it is a little easier to see from inequation (8-2). These relations generate the six
“all-other-things-equal” comparative static results.
The comparative statics of the model are that workers are more likely to choose to acquire the B -skill – that is, the LHS of inequation (8-2) is more likely to be greater than the RHS – when:
1.
the opportunity cost of acquiring the B -skill is smaller, which is when oh B is smaller;
2.
the direct cost of acquiring the B -skill is smaller, ie when DC B is smaller;
3.
the wage differential, ( w B – w A ), is larger;
4.
the income differential associated with a given wage differential is greater, ie when h is larger;
5.
the rate at which workers discount the future,

is smaller, so, for any n , D n is larger, and
6.
the amount of time workers expect to receive the income differential is greater, ie when n is larger.

Allen: labour economics, Chapter 8 page 6
Long-run Labour Market Equilibrium :
As with the micro-economics of product markets, the difference between the Short-run and the Longrun is that in the Long-run there is time for entry and exit. (Or, also analogously, the Long-run is long enough for new Capital to be created. Here the Long-run is long enough that Human Capital is created.)
In the micro-economics of product markets, Long-run equilibrium equalizes profits. If one market has a higher rate of profit, Capital enters that market, Short-run supply increases, prices fall and profits fall.
This continues until profits are equalized. And if one market has a lower rate of profit, Capital exits that market, Short-run supply falls, prices rise and profits rise. This continues until profits are equalized.
The analysis of labour markets is precisely equivalent. Long-run equilibrium equalizes lifetime incomes. If the B -market has a higher lifetime income it will attract Human Capital to enter that market.
The number of new entrants to the B -market rises: EN B > EX B . This causes Short-run labour supply in B to rise, that is to shift to the right. The real wage in B , w B , and therefore lifetime income in B , Y( B ), fall.
Simultaneously the number of new entrants to the A -market falls for one or more periods: EN A < EX A .
This causes Short-run labour supply to fall in A to fall, that is to shift to the left. The real wage in A , w A , and therefore lifetime income in A , Y( A ), rise. These effects continue until lifetime incomes are equalized. (This is described in more detail, and graphed, below, in the demonstration of what happens if there is a productivity increase in the B -market. That productivity increase creates a Short-run situation in which lifetime income is temporarily higher in the B -market.)
The Long-run equilibrium condition of equal lifetime incomes in both markets is a necessary condition of all workers being alike, and of free entry into B
. “Free” entry here does not mean that there is no cost, but that there are no barriers other than cost .
We can write the Long-run equilibrium in two ways, following the two ways we modeled workers’ choices. In the first way, lifetime incomes are equalized:
(8-3) Y( A ) = Y( B )

D n • y A = D n • y B

( h • w A ) + D n •( h • w A ) = ( h B
0
• w A – DC B ) + D n •( h • w B ).
The second way of writing the Long-run equilibrium sets gross returns to working in the B -market equal to gross costs, which is the same as setting net returns to working in the B -market equal to zero.
Note that this can be derived by rewriting equation (8-3), so it is the same condition as setting lifetime incomes equal.
(8-4

) ( D n • y A – D n • y B ) = ( OC B + DC B )

[ D n •( h
• w B )
–
D n •( h
• w A )] = ( OC B + DC B ), or if:
(8-4) [ D n • h •( w B – w A )] = ( OC B + DC B ).
Long-run equilibrium is now simultaneously:1) equilibrium in both of the two separate skill markets, the A -market and the B -market, and 2) an equilibrium wage differential between the markets. That is, if we know the values of “all other things being held equal”:
D n
, h , oh B , and DC B , we have a single equation and two unknowns, w A and w B . Thus the Long-run equilibrium condition, equation (8-4), cannot determine unique values of w A and w B without either another equation (the demand equations would work) or by making one of the unknowns a “known”. In the exercises below, you do the second most of the time by treating the wage in the A -market as the base wage . In that treatment, the wage in the A market is the wage to which all other wages are pegged.

Allen: labour economics, Chapter 8 page 7
---------------------------------------------------------------------------------
[NOTE: What equation (8-4) does determine is the Long-run equilibrium wage at B as a function of the wage at A . That is, with known values of D n
, h , oh B , and DC B , equation (8-4) determines the values of X and Z in the equation: w B = Xw A + Z .
The algebra of this is as follows. Equation (8-4) becomes: ( D n • h )( w B –
w A ) = ( oh B • w A ) + DC B

( D n • h )( w B ) – ( D n • h )( w A ) = ( oh B • w A ) + DC B 
( D n • h )( w B ) = ( D n • h )( w A ) + ( oh B • w A ) + DC B
 w B = [(( D n • h )+ oh B )( w A )]/( D n • h ) + DC B /( D n • h ). So equation (8-4) determines the function: w B = Xw A + Z , where X = (( D n • h ) + oh B )/( D n • h ), and Z = DC B /( D n • h ).
If Z = 0, the wage ratio , w B / w A , is a constant, X .]
-----------------------------------------------
Process of moving from one Long-run equilibrium to another
Here I show the process of moving from one Long-run equilibrium to another. The change is caused by a productivity increase in the B -market. The example, then, also shows how a productivity increase in one sector of the economy is transmitted to the other sectors. The productivity increase in the B -market will cause wages in both market to increase. But because we assume that all workers are alike, and because we assume free entry and exit to each skill-market, the relation between wages in the two markets doesn’t change at all.
[The relation, derived above, is: w B = Xw A + Z , where X = (( D n • h ) + oh B )/( D n • h ), and Z = DC B /( D n • h ). It doesn’t change because no elements that make up X or Z change. X and Z are entirely determined by
( D n • h ), oh B , and DC B . None of these changes when productivity increases at B.
Since all the elements of the relation remain constant, the relation remains constant. The productivity increase in B must increase wages in B and therefore, from the relation, must also increase wages in A !]
Observe Figure 8-2 . We start in a Long-run equilibrium, with wages w A
1
and w B
1
, and with N A
1 workers in A and N B
1
workers in B . Demand in the two markets is D A and D B
1
, which reflects productivity in the two markets before the productivity increase. Then there is a productivity increase in
B , which shifts demand in that skill-market to D B
2
.
Figure 8-2 w A w A
2
1
A -market
N A
2
N A
1
D A w w w
B
B
B
SR
2
1
B -market
N B
1
N B
2
D
D
B
B
L-r: S
2
1
B

Allen: labour economics, Chapter 8 page 8
The first effect of this shift is Short-run, and raises the wage at B from w B
1
to w B
SR
. It is labelled w B
SR because, as we see, this is a Short-run situation and not permanent. The wage differential has now risen from ( w B
1
– w A
1
) to ( w B
SR
– w A
1
). This increase in the wage differential will cause the gross return to B – the LHS of equation (8-4) – to now be larger than the RHS. So this is Short-run and not a Long-run equilibrium.
All new entrants will now choose to enter the B -market. As they do, they shift N B to the right, ultimately from N B
1
to N B
2
. As N B shifts to the right, the wage at B falls, ultimately to w B
2
. w B
2
is still higher than w B
1
, but lower than w B
SR
. Meanwhile, since all new entrants are choosing B , N A shifts left, ultimately from N A
1
to N A
2
. As N A shifts to the left, the wage at A rises, ultimately to w A
2
. The shifts continue as long as the LHS of equation (8-4) is greater than the RHS. The shifts stop, and the system is back in Long-run equilibrium, when the LHS of equation (8-4) once again equals the RHS.
[Once back in Long-run equilibrium, wages in both markets will have risen. It will again be true that w B
= Xw A + Z , where X = (( D n • h ) + oh B )/( D n • h ), and Z = DC B /( D n • h ).]
Above I have described this process – the process of transmitting the productivity increase in B to a wage increase in A – as having only new entrants to the labour markets choose B . I could also have described it by also having some workers already at A shift to B . That is I could have described this process as one of mobility between sectors. If the productivity increase at B is large enough, it will cause some, usually the younger, workers already at A to want to choose to pay the costs of acquiring the B skill. These workers will shift from the A -market to the B -market. This shift will speed up the movement to a new Long-run equilibrium.
Finally, note that the Long-run Supply curve in the B -market is unequivocally positively sloped.
-------------------------------------------------------------------
For an Exercise, write out and draw the graphs of what happens if the A -market experiences a productivity increase.

Allen: labour economics, Chapter 8 page 9
Exercise Set 8 A :
Find the below, using the values of the “givens” given here. That is, use the values here unless they are specifically changed in the questions below. For Problem 1, find w B as a function of w A : w B = Xw A + Z .
Find X and Z and write out the value of w B as a function of w A .
1. Given: i) all individuals work 2000 hours per year ( h =2000); ii) all individuals work 20 periods after the initial period ( n =20); iii) training for B takes 1000 hours ( oh B =1000): a) If

=0 and DC B =0, find w B as a function of w A . b) If

=0 and DC B =$10,000, find w B as a function of w A . c) If

is such that D 20 =15 and DC B =$10,000, find w B as a function of w A . d) Given

=0 and DC B =$10,000, productivity in B increases, increasing the wage in B in the
Short-Run. Find w B as a function of w A .
[Hint: this is very easy. What’s interesting is why, so Why?]
For the rest of this exercise use the numeraire . Set w A =$10/hour. You start in Problem 2 with i) h =2000; ii) n =20; iii) oh B =1000; iv) DC B =0. For #s 3 through 6, make the changes described. Continue with those changes unless told otherwise.
2. Given the values above: a) If

=0, find the Long-Run equilibrium wage at B : w B . b) If

is such that D 20 =15, find the Long-Run equilibrium wage at B : w B .
3. Change to n =5; all other variables remain the same. a) If

=0, find the Long-Run equilibrium wage at B : w B . b) If
 is such that D 5 =4, find the Long-Run equilibrium wage at B : w B .
4. Change opportunity costs of the training for B to 1500 hours; oh B a) If

=0, find the Long-Run equilibrium wage at B : w B .
=1500. b) If
 is such that D 5 =4, find the Long-Run equilibrium wage at B : w B .
5. oh B is back to 1000 hours . Change direct costs of the training for B to $10,000: DC B =$10,000 (and ignore the fact that anyone without financing will starve before he can enter B).
1.
If

=0, find the Long-Run equilibrium wage at B : w B .
2.
If
 is such that D 5 =4, find the Long-Run equilibrium wage at B : w B .
6. Change h to 2500 hours.
1.
If

=0, find the Long-Run equilibrium wage at B : w B .
2.
If
 is such that D 5 =4, find the Long-Run equilibrium wage at B : w B .
Very briefly, write what these exercises showed you about the comparative statics of how the Long-
Run equilibrium wage in the B -market changes as the “other variables”: DC B ; oh B ; n ;

; and h , change.

Allen: labour economics, Chapter 8 page 10
B. All Workers Alike – Three Skill Markets:
Now we assume that a third skill, C , exists. The demand for skill C is different from demand for skills
A or B . B -skilled workers cannot do C -skill work, although C -skilled workers can do B -skill work.
Some way exists of identifying workers who possess C -skill.
It requires real resources to produce (or identify) workers with the C -skill. So far C looks like B , but there us one difference. A worker must already possess the B -skill in order to get training for the C -skill.
So there is a hierarchy of skills: C is above B , and B is above A . (Think about the necessity to have an undergraduate degree, or at least some years of undergraduate education, before one can get training in medicine or law.)
The existence of the hierarchy has the following effect on the model: To get the Bskill a worker has to have training in the initial period, t = 0. So to get the C -skill a worker must have training in period one, t
= 1. Conceptually this means that workers make a decision in the initial period about whether to stay A skilled, or whether to get training for the B -skill. Then, in period one, workers must make a second decision: whether to stay B -skilled or whether to get training for the C -skill.
MORE DEFINITIONS:
N = N A + N B + N C , so labour supply in A and B excludes C -skilled workers who could work in A or B .
Entry to the C -market is EN C ; Exit from the C -market is EX C
Long-run equilibrium for the C -market requires that EN C = EX C = N C / N .
Additional Symbols:
1.
w C t
is per-period real wages in market C . Set all w C t
, t = 1, n , equal, and denote these just w C .
2.
h C t
is per-period hours in market C . Since these are all equal after training, h C t
= h for t = 2, n .
3.
The opportunity costs of acquiring the Cskill are incurred by working fewer hours while acquiring the C -skill. So during period one , individuals choosing C work h C
1
hours at wage w B t
, where h C
1
< h .
The opportunity “hours”, hours of work foregone to acquire the C -skill, are: oh C = ( h – h C
1
). The opportunity cost of acquiring the C -skill is OC C = oh C • w B .
4.
DC C denotes the direct costs of acquiring skill C .
Because the old initial period, t =0, is in the past when the decision between B and C is made, the new decision period, period one or t =1, should not be discounted when analyzing the decision between B and
C . The new first period after the decision between B and C is period two , or t =2. This should now be discounted by 1/(1+

), not by 1/(1+

)
2
. And the new second period after the decision between B and C is period three , or t =3, and it should now be discounted by 1/(1+

)
2
, not by 1/(1+

)
3.
And so on for periods four through n . So the final period should now be discounted by 1/(1+

) n -1 , not by 1/(1+

) n . Thus the discount operator for the B vs C decision is D n -1
.
Once again, we can drop the time subscripts for all but period one, since w A t
, w B t
, and w C t
are constant for t =2, n .
Decision Rule :
The model of the worker’s decision is almost identical to the decision of
A vs B . The worker chooses to work in the skill-market which gives her the higher expected lifetime income. This is the value of lifetime income expected at the time of decision.

Allen: labour economics, Chapter 8 page 11
Again, there are two ways of modeling this decision. In the first way the worker compares the values of the lifetime incomes of the B and C choices, at t = 1, and chooses to work in the market which gives the higher lifetime income. Because the decision is being made in period one, these lifetime incomes are:
(8-5-a) Y( B ) = D n-1 • y B = ( h
• w B ) + D n-1 •( h
• w B )
(8-5-b) Y( C ) = D n-1 • y C = ( h C
0
• w B – DC C ) + D n-1 •( h
• w C ).
D n -1 • y C > D n -1 • y B , because the C -skill commands a higher wage. But h C
0
< h because it takes time to acquire the C -skill, time that is spent during the initial periods. Depending on relative sizes of the variables, either market yields the higher lifetime income. Workers choose to work in the market which generates the higher lifetime income.
The second way of modelling the workers decision is that the worker chooses to acquire the C -skill if her expected return from working in the C -market is greater than the cost of acquiring the C -skill.
Formally, the worker will choose C if:
(8-6

) ( D n-1 • y B – D n-1 • y C )

( OC C + DC C )

[ D n-1 •( h
• w C )
–
D n-1 •( h
• w B )]

( OC C + DC C ), or if:
(8-6) [ D n-1 • h
•( w C – w B )]

( OC C + DC C ).
The LHS of inequation (8-6) is the return to working in the C -market; the RHS is the cost of acquiring the
C -skill. The worker chooses the acquire the C -skill and work in the C -market if LHS

RHS of inequation (8-6).
Looking at the decision this way shows that there are six things that can vary and affect the choice of skill market: oh C and DC C Costs), ( w C – w B ) (wage increase from choosing C ), h ,

(which determines D ), and n-1
, The relation between these variables and workers’ decisions is clear from either equations (8-5), or from inequation (8-6), but it is a little easier to see from inequation (8-6). These relations generate the six “all-other-things-equal” comparative static results.
The comparative statics of the model are that workers are more likely to choose to acquire the C -skill – that is, the LHS of inequation (8-6) is more likely to be greater than the RHS – when:
1.
the opportunity cost of acquiring the C -skill is smaller, which is when oh C is smaller;
2.
the direct cost of acquiring the C -skill is smaller, ie when DC C is smaller;
3.
the wage differential, ( w C – w B ), is larger;
4.
the income differential associated with a given wage differential is greater, ie when h is larger;
5.
the rate at which workers discount the future,

is smaller, so, for any n-1 , D n-1 is larger, and
6.
the amount of time workers expect to receive the income differential is greater, ie when n-1 is larger.
Long-run Labour Market Equilibrium :
The analysis here is simply a repeat of the one for A vs B , but with one time period fewer for individuals to receive the return to choosing C . In the labour markets, entry to the C -market occurs only after workers have acquired the C -skill – ie they have acquired more Human Capital. Entry to the B market is a default at this stage, choosing not to acquire additional Human Capital.
Long-run equilibrium in labour markets equalizes lifetime incomes. If the C -market has a higher lifetime income, workers enter that market. The number of new entrants to the C -market rises: EN C >

Allen: labour economics, Chapter 8 page 12
EX C . (In fact, in the model we’re using now, where all workers are alike, for one or more periods all new workers with the B -skill enter C .) This causes Short-run labour supply to rise, and wages, and therefore lifetime incomes, to fall. Meanwhile the number of new entrants to the B -market falls for one or more periods: EN B < EX B . (In fact, in the model we’re using now, where all workers are alike, for one or more periods no new workers with the B -skill enter B .) This causes Short-run labour supply to fall, and wages, and therefore lifetime incomes to rise. These effects continue until lifetime incomes are equalized.
(For an exercise, write out and draw the graphs of what happens if the C -market has a higher lifetime income in the Short-run, because, for example, of a productivity increase in the C -market.)
We can write the Long-run equilibrium conditions two ways, following the two ways we modeled workers’ choices. In the first, lifetime incomes are equalized:
(8-7) Y( B ) = Y( C )

D n-1 • y B = D n-1 • y C

( h
• w B ) + D n-1 •( h
• w B ) = ( h C
0
• w B – DC C ) + D n-1 •( h
• w C )
The second way of writing the Long-run equilibrium sets returns to working in the C -market equal to costs of acquiring C . Note that this can be derived by rewriting equation (8), so it is the same condition as setting lifetime incomes equal.
(8-8') ( D n-1 • y B – D n-1 • y C ) = ( OC C + DC C )

[ D n-1 •( h
• w C )
–
D n-1 •( h
• w B )] = ( OC C + DC C ), or if:
(8-8) [ D n-1 • h
•( w C – w B )] = ( OC C + DC C ).
Long-run equilibrium is now simultaneously equilibrium in both the two separate labour markets, the
B -market and the C -market, and an equilibrium wage differential between the markets. Note from equation (8-8), if we know the values of D n-1
, h , o h C , and DC C , we have a single equation and two unknowns, w B and w C . Thus the Long-run equilibrium condition cannot determine w B and w C without another given.
[NOTE: What equation (8-8) does determine is the Long-Run equilibrium wage at C as a function of the wage at B . That is, with known values of D n-1 , h , oh C , and DC C , equation (8-8) determines the values of
X
’
and Z
’
in the equation: w C = X
’ w B + Z
’.
The algebra of this is as follows. Equation (8-8) becomes: ( D n-1 • h )( w C –
w B ) = ( oh C • w B ) + DC C

( D n-1 • h )( w C ) – ( D n-1 • h )( w B ) = ( oh C • w B ) + DC C

( D n-1 • h )( w C ) = ( D n-1 • h )( w B ) + ( oh C • w B ) + DC C
 w C = [(( D n-1 • h ) + oh C )( w B )]/ ( D n-1 • h ) + DC C /( D n-1 • h ). Equation (8-8) determines the function: w C = X
’ w B + Z
’
, where X
’
= (( D n-1 • h ) + oh C )/( D n-1 • h ), and Z
’
= DC C /( D n-1 • h ).
If Z
’
= 0, the wage ratio : w C / w B , is a constant X
’
.]
Comparing equation (8-4) to (8-8), we can see an important result. That is that if the costs of C are the same as the costs of B , the equilibrium wage differential ( w C –
w B ) must be greater than the equilibrium wage differential ( w B –
w A ). That’s because
D n -1 • h times the first differential must equal cost in equilibrium, while D n • h times the second differential must equal cost. If the costs are equal, ( w C –
w B ) must be greater than ( w B – w A ) because D n -1 must be less than D n . This generates the empirically observed result that age/wage profiles are steeper the greater the skill.

Allen: labour economics, Chapter 8 page 13
Exercise Set 8 B :
1.
Given: i) all individuals work 2000 hours per year ( h =2000); ii) all individuals work 20 years after the initial ( n =20); iii) training for C takes one year of working 1000 hours ( oh C =1000); iv) direct costs of training for C are zero ( DC C =0). a) If

=0, find w C as a function of w B . b) If

=0 and DC C =$10,000, find w C as a function of w B . c) If

is such that D 19 =14, and DC C =$10,000, find w C as a function of w B . d) Describe why your answers to a), b) and c) are different from your answers to Question 1, a), b) and c) in Exercise Set II -A. What does this tell us about the relation between time required for acquiring human capital and equilibrium wage differentials?
2.
Given

=0 and DC C =$10,000, productivity in C increases, increasing the wage in C in the Short-Run.
Find w C as a function of w B . a)
[Hint: this is very easy. What’s interesting is why, so Why?]
For the rest of this exercise use the numeraire . Set w B equal to your solution to Exercise 8 -A, Problem 3, part a).
3.
Given the values above, n changes to 5 and DC C =0. a) If

=0, find the Long-run equilibrium wage at C : w C . b) If

is such that D 4 =3, find the Long-run equilibrium wage at C : w C .
4.
Change direct costs of the training for C to $10,000, DC C =$10,000. a) If

=0, find the Long-run equilibrium wage at C : w C . b) If

is such that D 4 =3, find the Long-run equilibrium wage at C : w C .
5. Begin with the equilibrium of Question 3 a). above. Technology increases the long-run equilibrium wage in the C skill-market to w C = $18/hr. Find the new values of the equilibrium wages in the A and B skill-markets: w A and w B .