CHAPTER 10
DIFFERENCES IN ABILITY
I. Two Skill Markets:
This is another form of heterogeneity. Individuals clearly differ in ability. These differences are
widely cited – but rarely modeled – as being the reason for the large lifetime income differentials
we see in the world. Here we model these formally. We will leave the judgment of whether they
are adequate to explain the large real-world income differentials to the reader. There are two ways
to put differences in ability into the model.
A. Differences in opportunity costs: variable-time processes
The first, often used by intermediate microeconomics textbooks, is to note that the opportunity
costs – and in some circumstances, the direct costs as well – of acquiring the B-skill are smaller for
individuals with greater ability. Individuals with higher ability will acquire the B-skill more
quickly. This leads both to lower opportunity costs and to greater time receiving the wage
differential from having the B-skill.
This can only happen when the time it takes to acquire the B-skill is variable. If more able
individuals can get through faster, the skill-acquisition process must be able to accommodate
different individuals moving at different speeds; these processes we call variable-time processes.
Some processes of skill-acquisition we observe are obviously variable-time: a baseball player may
make the major leagues after one year in the minors, two years, three years, ... . He may have
played baseball in college for four years or have gone straight to the minors as an eighteen-year-old.
Some students finish university in four years, most finish in five, some finish in six, ... . So an
analysis of variable-time skill-acquisition is useful.
The analysis is similar to the analysis for individual differences in discount rates, or in
preferences for leisure, with one difference. The difference is that now, if we continue to assume
that all individuals die or retire at the same age, we have changes in two variables, one on each side
of the decision (in)equation.
One variable to change, on the RHS of equation (8-4), is opportunity costs of acquiring the Bskill, OCB. We have been assuming that all training takes place in one time period: t = 0. With a
variable time process, we have to make the time periods somewhat smaller, so that there can be
variation in expected time to complete skill-acquisition. For simplicity, we’ll hold the opportunity
hours, ohB, constant for all workers in each period. But now opportunity costs of acquiring B can
take more than the single period. The number of periods it takes to acquire the B-skill is m. Now
ohB = (h – hBt), with hBt constant for all t = 0,m-1. For simplicity, although we know this isn’t
quite right, I’ll ignore discounting during the period of skill acquisition, so discounting will begin
only after the skill is acquired. The opportunity cost of B , OCB, is now: OCB = m•(ohB•wA).
Since the more able will acquire the B-skill more quickly, they will have more periods over
which to receive the B-wage. This means that the variable on the LHS that we’ve been using for
those periods, Dn, must change. Since m is the number of the period over which individuals receive
the B-wage, we must now rewrite this variable as Dn-m.
The variable which is common to both OCB and Dn-m, that now varies between individuals is m,
the number of periods it takes an individual, i, to acquire the B-skill. To denote that this now varies
between individuals, it is written as m(i), for individual i. We now have OCBi = m(i)•(ohB•wA) on
Allen, labour economics, Chapter 10
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the RHS and Dn-m(i) on the LHS. Since we’re still holding the number of periods in a
working/training lifetime constant, at n+1, these two vary in opposite directions for as individuals
vary from the mean value of m. Nicely, both variations cause the equilibrium wage differential (or,
really, the wage relation, wBE = XwAE + Z), to be greater for individuals with greater ability.
Once again, consider equation (8-4), which has the return to choosing B on the LHS and the
costs of B on the RHS. Again, we rewrite it as an inequation, adding subscripts for individuals:
(10-1) Dn-m(i)•h•(wB – wA) >< OCBi + DCB .
Individuals choose B when the LHS is greater than the RHS. Given a long-run equilibrium wage
differential, (wB – wA), individuals with the lowest values of m(i) – the individuals with the highest
ability – are the first to choose B. For them, the value of the LHS on inequation (10-1) are the
highest; their values of [n – m(i)] are the highest because their values of m(i) are the lowest. And
individuals with the highest ability are the first to choose B; their values of OCBi are the lowest
because their values of m(i) are the lowest. For the highest ability individuals, both the LHS of
inequation (10-1) are the highest and their values of the RHS of inequation (10-1) are the lowest.
Equilibrium is still stasis: the number of new entrants to A is equal to exit from A, and the
number of new entrants to B equals exit from B. So there is an equilibrium proportion of the new
entrants choosing A, ENA/EN, and an equilibrium number choosing B, ENB/EN. m(E) is the value
of m(i) which splits the population of new entrants, EN, into their equilibrium proportions: ENA/EN
and ENB/EN. Now m(E)  Dn-m(E) , and OCBE , which in turn set the LHS equal to the RHS of
equation (10-1): Long-run equilibrium is now:
(10-2)
Dn-m(E)•h•(wBE – wAE) = OCBE + DCB , where OCBE = m(E)•(ohB•wA)
Long-run equilibrium is now a double equilibrium. Long-run equilibrium simultaneously
determines both the equilibrium wage differential (or, really, the wage relation, wBE = XwAE + Z),
and the equilibrium value of the “cutting” ability level: m(E)  Dn-m(E) and OCBE .
Figure 10-1
Number
Of
Workers
i(1) i(2)
i(E)
Group B1
m(i) = min
m(i) increasing
i(99)
i(100)
Group A1
m(E)
m(i) = max
To illustrate, assume that 100 individuals enter the work force each period. Order these
individuals on the basis of their values of m(i), and denote them as i(1), i(2), i(3), … , i(98), i(99),
i(100). Individual i(1), the highest ability individual, has m(1) = min; individual i(2)has m(2) =
(min plus a small number), etc; individual i(100), the lowest ability individual, has m(100) = “max”;
individual i(99) has m(99) = (“max” minus a small number), etc. If skills A and B exist, then some
must choose A and some must choose B. The first to choose B will be i(1), since for her the return
to choosing B – that is, the value of the LHS of inequation (10-1) – is the highest, and the cost of
choosing B – that is, the value of the RHS of inequation (10-1) – is the lowest. The next to choose
B is i(2), the next i(3), … . Meanwhile, the first to choose A will be i(100), since for her the return
to choosing B – that is the value of the LHS of inequation (10-1) – is the lowest, and the cost of
Allen, labour economics, Chapter 10
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choosing B – that is, the value of the RHS of inequation (10-1) – is the highest. (Remember that A
exists, so some workers must choose A) The next to choose A will be i(99), the next i(98), … .
This self-selection process continues until, somewhere in the middle of the diagram on Figure
10-1, one individual is indifferent. This individual’s value of m(i) is such that the LHS of
inequation (10-1) equals the RHS. (Since she must make a decision, we will have her choose B.)
Her value of m(i) is the equilibrium m(E)  Dn-m(E) , and OCBE , which in turn set the LHS equal to
the RHS of inequation (10-1). Given long-run wage differential (wBE – wAE), this is individual i(E),
with a value m(E) , on Figure 10-1. Individual i(E) is the “cut” individual – the individual who
divides the group of all new entrants into those who choose A, Group A1 on Figure 10-1, and those
who choose B, Group B1 on Figure 10-1.
Reaching equilibrium is now a two-stage process analytically. To describe this, we will
consider an exogenous change that increases demand for the B-skill, so it increases the wage at B in
the short run. This is the process described at the end of Part A in Chapter 8. Its effects on
equilibrium ability levels are shown on Figure 10-2. We begin with long-run equilibrium, E1,
shown on Figure 10-2. The increase in the wage at B causes the wage differential to increase in the
short-run. More new entrants choose B. Some of these new entrants will have lesser ability, that is,
larger m(i)s , than the first cut, m(E1). So the “cut” increases in the initial short-run, to m(SR). This
will cause an increase in group B, from B1 to BSR, and simultaneously a decrease in group A from
A1 to ASR.
Figure 10-2
Number
Of
Workers
i(E1)
i(E2)
Group A1
Group B1
BSR
Group A2
Group B2
m(i) = min
ASR
m(E1)
m(i) increasing
m(E2)
m(SR)
m(i) = max
SR to LRE2
This increase in new entrants to B causes NB to increase and wages at B to decrease.
Simultaneously the decrease in new entrants to A causes NA to decrease and wages at A to increase.
This in turn reduces the wage differential between B and A, and causes the m(i)of the “cut”
individual to rise. The process continues until the new long-run equilibrium is reached. This will
be characterized by long-run wage differential, (wBE2 – wAE2) and a new “cut” individual, i(E2),
whose m(i) is m(E2). This process is illustrated on Figure 10-2.
Three things must be true of the new long-run equilibrium. Look back at the long-run
equilbrium condition, equation (10-2). The first change is that the wage at B rises from the
productivity increase in sector B, wB. The equation no longer holds. Now, to get back to equality,
both the LHS falls and the RHS rises. These occur simultaneously, both caused by the fact that the
fall in the LHS must be caused by a greater NB, which means a smaller NA. These in turn reduce the
B-wage, increase the A-wage, and reduce the wage differential. But as the size of the B-sector
increases, m(E) must rise. The approach from the short-run to the new long-run equilibirum is a
simultaneous reduction of m(E), which reduces Dn-m(E) while increasing OCBE , and a reduction of
Allen, labour economics, Chapter 10
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the wage differential, as a greater proportion of workers choose B. So the first thing that is true of
the new long-run equilibrium is that it must be characterized by a greater NB and a smaller NA than
the initial long-run equilibrium. This is shown at the end of Part A in Chapter 8, and is shown on
Figure 10-2, where Group B2 is larger than Group B1, and Group A2 is smaller than Group A1.
Figure 10-2 also makes it clear that this first effect strictly implies the second – that the m(i) of
the “cut” individual must have risen. This now means two things: 1) the value of the ability level of
the “cut” individual must have fallen; and 2) the opportunity costs of acquiring the B-skill of the
“cut” individual must have risen. This too is shown on Figure 10-2. There i(E2) is further along the
continuum of individuals ordered by decreasing ability levels, reflected in increasing m(i)s, than is
individual, i(E1). So m(E2) is greater than m(E1), and m(E2) > m(E1)  Dn-m(E2) < Dn-m(E1).
The third effect then is that the equilibrium wage differential, (wBE – wAE), also must have risen,
but now for two reasons. The first is that, since OCBE2 > OCBE1 , the RHS of equation (10-2) has
risen. So the LHS must have risen as well. There are two changes in the LHS. One of these is to
Dn-m(E); this has fallen, since Dn-m(E2) < Dn-m(E1). Even had the RHS remained constant, the
equilibrium wage differential would have to have also risen, but since the RHS has risen, the
equilibrium wage differential has risen by even more. With all other things – all things not
functions of m(i) – held equal, (wBE2 – wAE2) must be greater than (wBE1 – wAE1).
Again, the Long-run equilibrium here has several nice characteristics. The first is that
heterogeneity again eliminates the small analytical problem faced by models where everyone is
alike. Here it is individual differences in ability, reflected in the time it takes different individuals to
acquire the B-skill, that cause individuals to self-select into the different skill markets and to keep
the system in stasis and long-run equilibrium.
The second nice characteristic is that, again, an increase in demand for the B-skill causes an
increase in the long-run equilibrium wage differential. It does so because the new long-run
equilibrium requires that more workers choose B. These additional workers have somewhat lower
ability than in the initial long-run equilibrium, so the wage differential must increase in order to
attract them in the self-selection process. Moving from homogeneous workers to heterogeneous
workers gives upward-sloping supply curves for all skill markets.
Note also that, as observed by an outsider, the long-run equilibrium is again characterized by
positive lifetime income differentials. Lifetime incomes are still equalized at the “cut”. And
lifetime incomes are the same here for all individuals in the A-market. But the mean lifetime
income for individuals in the B-market will be higher than the lifetime income for the “cut”
individual, because the mean ability level of all those in the B-market must be higher than the
ability level of the “cut” individual. And the higher the level of ability in the B-market, the higher
the lifetime income, because those with high ability receive the equilibrium wage differential for a
longer time, for more time periods. Again, since the equilibrium wage differential is larger the
greater is the elasticity of demand for the B-skill, as shown in part D of the exercise below, lifetime
income differentials are larger the greater is the elasticity of demand for the B-skill.
------------------------------------------------------Exercise 10-I:
The equilibrium here has a single equation (10-2) with two unknowns: m(E) and the wage
differential: (wBE – wAE). To solve the system we need a specification of the distribution of the Dnis,
which means adding another unknown, NBE and Demand functions for both the A and the B skill
Allen, labour economics, Chapter 10
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markets. This is very messy algebraically; it is inherently non-linear and it’s not useful to use a
linear approximation; and it involves four equations and four unknowns.
Luckily, however, we can again closely approximate a solution using EXCEL Below is an
EXCEL exercise using the short-cut we used before: use wAE as a “numeraire”.
This would be very difficult to do for non-zero values of  because we would have to calculate
the value of Dn-m(i) for all i. Since we must now have a large number of time periods, those
calculations would take 31 columns and 61 rows in the spreadsheet. So, to make it easy, set  = 0,
so Dn-m(i) is just equal to [n-m(i)]. I’m breaking time up into, sort-of, quarter-years, and the m(i)
vary uniformly from 1 to 31 of these quarter-years. Because the time periods are so short, n = 121,
so the Dn-m(i) vary uniformly from 120 for the most able to 90 for the least. Other numbers used in
this exercise are: h = 500 and ohB = 250 (these are now for quarters, not years, so are ¼ the size
used before); wAE = $10/hour, and DCB = 10,000. So opportunity costs vary uniformly from $2,500
to $77,500. The reduced form of equation (10-2) is: wBE = wAE + (OCBE + DCB)/Dn-m(E)•h. With
these numbers, the reduced form equation becomes:
wBE = $10 +{[$2500•m(E) + $20,000]/(500•D121-m(E))} = $10 + [$5•m(E)+40]/D121-m(E) .
ROUND ALL COLUMNS TO TWO DECIMALS!
A The exercise assumes 31 workers with a uniform distribution of the m(i)s. Column A is for the
individuals; this is also the number of workers in the B-market, NB. In A1 type “N(B)”. In A2 type
“1”; in A3 type “=A2+1”; copy A3 to A4:32. Column B is for the m(i)s, so in B1 type “m(i),1”. In
B2 type “1”. In B3 type “=B2+1”; copy B3 to B4:32. There are two variables determined by the
m(i)s; opportunity costs and number of periods the individual will get the B-wage. So use Column
C for the opportunity costs. In C1 type “OC,1”; in C2 type “=2500*B2”; copy C2 to C3:32.
Column D is for the wage from the supply side; it is essentially the reduced form equation above.
In D1 type “w(B),S1”; in D2 type “=10+(5*B2+40)/(121-B2)”; copy D2 to D3:D32. Column E is
for the demand in the B-market. In E1 type “w(B),D1”. In E2 type “=11.45-0.03*A2”; copy E2 to
E3:32. Now run down Column E remembering the rule: a worker chooses B if the wB found from
her w(i) is less than the market wage. The market wage is in Column E and declines as more
workers enter B. Run down Column E until the number in Column D is lower. The equilibrium,
wBE, is the last value of Column E which is greater than or equal to the value of Column D in the
same row. From this equilibrium value of wBE , move left and find NBE , w(E) and OCBE .
Now find the median value of m(i), and the median value of opportunity costs, for the B-group and
the median value of m(i) and the median value of opportunity costs for the A-group. The median
here is the middle value. If there are two in the middle, which will happen if there is an even
number in the group, take the midpoint of the two. Here the median will be the same as the mean
because the distribution is uniform. Use the two median values of m(i) to find the median values of
D121-m(i). Use the four median values you’ve found, the values above, and the equilbrium value of
wBE that you found to compute lifetime income differences an observer would find between A and
B. Remember the equilibrium wage comes from the Demand column, and remember that an
observer will “see” only n, not Dn.
B An increase in demand in the B-market. Use column F; in F1 type “w(B),D2”. In F2 type “=120.03*A2”; copy F2 to F3:32. Now run down Column F comparing with Column D, and repeat what
you did in A above with Columns E and D, until you find a new wBE . From this value, move left
and find new values of NBE and hE . What changes in each of these variables was caused by the
increase in Demand in the B-market?
Allen, labour economics, Chapter 10
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Now find the median value of m(i), and the median value of opportunity costs, for the B-group and
the median value of m(i) and the median value of opportunity costs for the A-group. The median
here is the middle value. If there are two in the middle, which will happen if there is an even
number in the group, take the midpoint of the two. Here the median will be the same as the mean
because the distribution is uniform. Use the two median values of m(i) to find the median values of
D121-m(i). Use the four median values you’ve found, the values above, and the equilbrium value of
wBE that you found to compute lifetime income differences an observer would find between A and
B. Remember the equilibrium wage comes from the Demand column, and remember that an
observer will “see” only n, not Dn.
C Second a new supply curve. But now we’ll have an increase in the difficulty of acquiring the Bskill, which is modeled by increasing the time required to acquire the B-skill ability by everyone.
Use Column G for new m(i)s: in G1 type “m(i),2”; in G2 type “10”; copy G2 to G3:G32. There are
two variables determined by the m(i)s; opportunity costs and number of periods the individual will
get the B-wage. Use Column H for the new opportunity costs. In H1 type “OC,1”; in H2 type
“=2500*G2”; copy H2 to H3:32. Column I is for the new wage from the supply side; it is
essentially the reduced form equation above. In I1 type “w(B),S2”; in I2 type
“=10+(5*G2+40)/(121-G2)”; copy I2 to I3:I32. Now run down Column E comparing with Column
I, and repeat what you did in A above with Columns E and D, until you find a new wBE . From this
value, move left and find new values of NBE and hE . What changes in each of these variables was
caused by the change in preferences?
Now find the median value of m(i), and the median value of opportunity costs, for the B-group and
the median value of m(i) and the median value of opportunity costs for the A-group. The median
here is the middle value. If there are two in the middle, which will happen if there is an even
number in the group, take the midpoint of the two. Here the median will be the same as the mean
because the distribution is uniform. Use the two median values of m(i) to find the median values of
D121-m(i). Use the four median values you’ve found, the values above, and the equilbrium value of
wBE that you found to compute lifetime income differences an observer would find between A and
B. Remember the equilibrium wage comes from the Demand column, and remember that an
observer will “see” only n, not Dn.
D Finally, an increase in the elasticity of demand in the B-market. Use column J: in J1 type
“w(B),D3”. In J2 type “=11.45-0.015*A2”; copy J2 to J3:32. Now run down Column J comparing
with Column D, and repeat what you did in A above with Columns E and D, until you find a new
wBE . From this value, move left and find new values of NBE and hE . What changes in each of
these variables was caused by the increase in Demand in the B-market?
Now find the median value of m(i), and the median value of opportunity costs, for the B-group and
the median value of m(i) and the median value of opportunity costs for the A-group. The median
here is the middle value. If there are two in the middle, which will happen if there is an even
number in the group, take the midpoint of the two. Here the median will be the same as the mean
because the distribution is uniform. Use the two median values of m(i) to find the median values of
D121-m(i). Use the four median values you’ve found, the values above, and the equilbrium value of
wBE that you found to compute lifetime income differences an observer would find between A and
B. Remember the equilibrium wage comes from the Demand column, and remember that an
observer will “see” only n, not Dn.
Write out your answers and turn them in along with the completed spread-sheet.
--------------------------------------------------------------------------------------------------------
Allen, labour economics, Chapter 10
page 7
Note that the exercise shows the three effects of an increase in demand in the B-market: 1) NBE
is higher after the demand increase; 2) m(E) is higher after the demand increase, which means both
that OCBE is higher and Dn-m(E) is lower; and 3) with wAE held constant, wBE is higher after the
demand increase. Finally, note the effects 1) and 3) taken together give us a long-run supply curve
for the B-skill which is upward sloping. (And, if we weren’t holding wAE constant, we would also
have a long-run supply curve for the A-skill which is upward sloping.)
B. Differences in probability of succeeding: fixed-time processes:
The process of acquiring B, on the other hand, may be relative fixed in time. If this is true, or
close to true, the opportunity costs of acquiring B can be treated as the same for all. But, an
individual can opt for B in the initial period and fail to acquire it. This approach resembles many of
the processes we see: a football player (hockey player) usually makes a major league team right out
of university (junior hockey) or doesn't make it at all; a university semester is fixed in length and
most students take the same number of courses per semester. But a large proportion of entrants
don't finish with a degree.
In this case a third path is required in addition to the two we now have. Those two are: choose A
in the initial period and stay in A for t=1,n; or choose B in the initial period then get the return to B
during t=1,n. The third path added now is: choose B in the initial period, but fail to acquire the Bskill, so the individual ends up spending t=1,n in A. Individual differences in ability are now
modeled as individual differences in the probability of succeeding to acquire the B-skill, given that
the individual opts to try in the initial period. The variable representing this probability – which is
how we now model the ability of individuals – is Pi(B); higher ability  higher Pi(B). We assume
that individuals accurately understand their Pi(B). One interesting result of this approach is that,
unlike all the other ways we’ve modeled heterogeneity, it can generate quite large equilibrium
lifetime income differentials, even at the margin.
i. Risk-neutral agents
Once uncertainty enters the model, we must formally deal with behavior toward risk. We begin
with the simplest assumption, risk-neutral behavior. Risk-neutral agents maximize expected
income. Individual ability, Pi(B), is the individual's probability of successfully acquiring the Bskill, given that she chooses B in the initial period – that is, given that she pays the costs of
attempting to acquire the B-skill. (Note, for simplicity I will now write Pi(B) as simply Pi .) Each
individual now faces three possibilities, although she has only two choices. The choices are still A
or B in t=0.
The three possibilities are:
(10-3a) Y(A) = (h•wA) + Dn•(h•wA)
(10-3b) Y(B-f) = (h•wA – OCB) – DCB + Dn•(h•wA), and
(10-3c) Y(B-s) = (h•wA – OCB) – DCB + Dn•(h•wB).
Y(B-f) is lifetime income if B is chosen in t=0 and the individual fails to acquire the B-skill; Y(B-s)
is lifetime income if B is chosen in t=0 and the individual succeeds in acquiring the B-skill.
The choice to attempt to acquire B is now a prospect:
(10-4) [1-Pi],[Pi ] : [ (h•wA – OCB) – DCB + Dn•(h•wA)],[ (h•wA – OCB) – DCB +
Dn•(h•wB)].
The expected income from choosing to attempt to acquire the B-skill is the expected value of the
prospect. Note that the initial period is the same for both possibilities, so we can simplify by
bringing it outside the prospect, then rewriting the prospect as its excpected value:
Allen, labour economics, Chapter 10
page 8
(10-5) Ye(B) = (h•wA – OCB) – DCB + [1-Pi][ Dn•(h•wA)] + [Pi][ Dn•(h•wB)].
Because individuals are risk-neutral, the expected utility of the prospect is equal to it's expected
value, so individuals choose B in the initial period when Ye(B) is greater than Y(A). Or, saying the
same thing a little differently, individuals choose B when the expected return from choosing B is
greater than the (known) cost of choosing B. That is, individuals choose B when the LHS of
inequation (10-6) is greater than the RHS:
(10-6) {[1-Pi][ Dn•(h•(wA – wA))] + [Pi][Dn•(h•(wB – wA))]}
= [Pi][Dn•(h•(wB – wA))]}
>
<
OCB + DCB .
Individuals with the largest values of Pi – individuals with the highest ability – are the first to
choose B because for them the values of the LHS of inequation (10-6) are highest. [Note: again
look back at your answers to Exercise A in Chapter 8, Problem 3, a) and b). From these and
inequation (10-6) you can see that facing a wage differential (wB – wA) = $1.20, individuals with Pi
= 0.9, will choose B; individuals with Pi =0.8 will choose A.] Given an equilibrium wage
differential, (wB – wA), the new entrants with the largest Pi's choose B: group 1 on Figure 10-3,
which has the highest expected return from choosing B.. The values of Pi for the next group of new
entrants – group 2, found as we move to the left along the horizontal axis of Figure 10-3 – is
smaller, so the expected return from choosing B to this group is smaller. But it is still greater than
the (known) value of choosing A.. This selection process continues until the equilibrium PE , is
reached. See Figure 10-3.
The long run equilibrium is now
(10-7) [PE][Dn•(h•(wBE – wAE))] = OCB + DCB .
Figure 10-3
Number
Of
Workers
i(1) i(2)
i(E)
Group B1
Pi = 1.0
Pi decreasing
i(100)
i(101)
Group A1
PE
Pi = 0
To illustrate, assume that 101 individuals enter the work force each period. Order these
individuals on the basis of their values of Pi , and denote them as i(1), i(2), i(3), … , i(99), i(100),
i(101). Individual i(1), the highest ability individual, has P1 = 1.0; individual i(2)has P2 = 0.99, etc;
individual i(101), the lowest ability individual, has P101 = 0; individual i(100) has P100 = 0.01, etc.
If skills A and B exist, then some must choose A and some must choose B. The first to choose B
will be i(1), since for her the return to choosing B – that is, the value of the LHS of inequation (106) – is the highest. The next to choose B is i(2), the next i(3), … . Meanwhile, the first to choose A
will be i(101), since for her the return to choosing B – that is the value of the LHS of inequation
(10-6) – is the lowest. (Remember that A exists, so some workers must choose A) The next to
choose A will be i(100), the next i(99), … .
This self-selection process continues until, somewhere in the middle of the diagram on Figure
10-3, one individual is indifferent. This individual’s value of Pi is such that the LHS of inequation
(10-6) equals the RHS. (Since she must make a decision, we will have her choose B.) Her value of
Allen, labour economics, Chapter 10
page 9
Pi is the equilibrium PE which sets the LHS equal to the RHS of inequation (10-6). Given long-run
wage differential (wBE – wAE), this is individual i(E), with a value PE , on Figure 10-3. Individual
i(E) is the “cut” individual – the individual who divides the group of all new entrants into those who
choose A, Group A1 on Figure 10-3, and those who choose B, Group B1 on Figure 10-3.
Reaching equilibrium is now a two-stage process analytically. To describe this, we will
consider an exogenous change that increases demand for the B-skill, so it increases the wage at B in
the short run. This is the process described at the end of Part A in Chapter 8. Its effects on
equilibrium ability levels are shown on Figure 10-4. We begin with long-run equilibrium, E1,
shown on Figure 10-4. The increase in the wage at B causes the wage differential to increase in the
short-run. More new entrants choose B. Some of these new entrants will have lesser ability, that is,
smaller Pis , than the first cut, PE1. So the “cut” increases in the initial short-run, to PSR. This will
cause an increase in group B, from B1 to BSR, and simultaneously a decrease in group A from A1 to
ASR.
Figure 10-4
Number
Of
Workers
i(E1)
i(E2)
Group A1
Group A2
Group B1
BSR
Pi = 1.0
Group B2
Pi decreasing
ASR
PE1
PE2
PSR
SR to LRE2
Pi = 0
This increase in new entrants to B causes NB to increase and wages at B to decrease.
Simultaneously the decrease in new entrants to A causes NA to decrease and wages at A to increase.
This in turn reduces the wage differential between B and A, and causes the Pi of the “cut”
individual to rise. The process continues until the new long-run equilibrium is reached. This will
be characterized by long-run wage differential, (wBE2 – wAE2) and a new “cut” individual, i(E2),
whose Pi is PE2. This process is illustrated on Figure 10-2.
Three things must be true of the new long-run equilibrium. Look back at the long-run
equilbrium condition, equation (10-7). The first change is that the wage at B rises from the
productivity increase in sector B, wB. The equation no longer holds. Now, to get back to equality,
the LHS must fall. As the size of the B-sector increases, PE must rise. The approach from the
short-run to the new long-run equilibirum is an increase of PE, and a reduction of the wage
differential, as a greater proportion of workers choose B. So the first thing that is true of the new
long-run equilibrium is that it must be characterized by a greater NB and a smaller NA than the initial
long-run equilibrium. This is shown at the end of Part A in Chapter 8, and is shown on Figure 10-4,
where Group B2 is larger than Group B1, and Group A2 is smaller than Group A1.
Figure 10-4 also makes it clear that this first effect strictly implies the second – that the Pi of
the “cut” individual must have fallen. This means that the value of the ability level of the “cut”
individual must have fallen. This too is shown on Figure 10-2. There i(E2) is further along the
continuum of individuals ordered by their decreasing ability, reflected in decreasing Pis, than is
individual, i(E1). So PE2 is less than PE1.
Allen, labour economics, Chapter 10
page 10
The third effect then is that the equilibrium wage differential, (wBE – wAE), also must have risen.
The RHS of equation (10-7) has not changed. So the new equilibrium LHS must be the same as in
the initial equilibrium. Since PE2 < PE1 , and all other things are constant, (wBE2 – wAE2) must be
greater than (wBE1 – wAE1).
Again, the Long-run equilibrium here has several nice characteristics. The first is that
heterogeneity again eliminates the small analytical problem faced by models where everyone is
alike. Here it is individual differences in ability, reflected in different probabilities of success in
acquiring the B-skill by different individuals, that cause individuals to self-select into the different
skill markets and to keep the system in stasis and long-run equilibrium.
The second nice characteristic is that, again, an increase in demand for the B-skill causes an
increase in the long-run equilibrium wage differential. It does so because the new long-run
equilibrium requires that more workers choose B. These additional workers have somewhat lower
ability – a somewhat lower probability of success if they choose B in the initial period – than in the
initial long-run equilibrium, so the wage differential must increase in order to attract them in the
self-selection process. Moving from homogeneous workers to heterogeneous workers gives
upward-sloping supply curves for all skill markets.
Note also that, for the first time, the equilbrium of the model is not a zero lifetime income
differential for anyone. The equilibrium is now that the lifetime income in B-market is always
higher. Here a worker either makes into the B-market, or she doesn’t. Because there is a non-zero
probability of failing, (1 – Pi) > 0, the long-run equilibrium is characterized by positive, and not
insubstantial (15% or so) lifetime income differentials for all who succeed into B. Lifetime
incomes are the same here for all individuals in the B-market and all who choose the A-market, but
they are lower for those who attempt the B-market in the initial period and fail. The substantial
lifetime income differentials generated by this version of the model are demonstrated in Exercise
10-II below.
This version of the model generate substantial lifetime income differences between A and B.
So, with this version, if we start pyramiding the markets – adding a C-market to follow success in
the acquisition of the B-skill, then a D-market to follow success in acquiring the C-skill, … – this
version of the Human Capital model will generate extremely large lifetime income differences.
------------------------------------------------------Exercise 10-II:
The equilibrium here has a single equation (10-7) with two unknowns: PE and the wage
differential: (wBE – wAE). To solve the system we need a specification of the distribution of the Dnis,
which means adding another unknown, NBE and Demand functions for both the A and the B skill
markets. This is very messy algebraically; it is inherently non-linear and it’s not useful to use a
linear approximation; and it involves four equations and four unknowns.
Luckily, however, we can again closely approximate a solution using EXCEL Below is an
EXCEL exercise using the short-cut we used before: use wAE as a “numeraire”.
The numbers used in this exercise are: D10 = 8, ohB = 1000, h = 2000, wAE = $10/hour, DCB =
10,000, and n = 10. The reduced form equation for wBE , from equation (10-7)is: wBE = wAE + (OCB
+ DCB)/PE•Dn•h . Solving with the numbers given here, that becomes wBE = 10 + (20,000)/16000PE
= 10 + 1.25/PE .
ROUND ALL COLUMNS TO TWO DECIMALS!
Allen, labour economics, Chapter 10
page 11
A The exercise assumes 31 workers with a uniform distribution of the Pi s. Column A is for the
individuals; this is also the number of workers in the B-market, NB. In A1 type “N(B)”. In A2 type
“1”; in A3 type “=A2+1”; copy A3 to A4:32. Column B is for the Pi s, so in B1 type “P(succ)-1”.
In B2 type “1.0”. In B3 type “=B2-.033”; copy B3 to B4:32. Column C is for the wBEs determined
from the Pi s, so in C1 type “w(B),S1”. In C2 type “=10+1.25/B2”; copy C2 to C3:32. And
Column D is for the demand in the B-market. In D1 type “w(B),D1”. In D2 type “=14.9-0.15*A2”;
copy D2 to D3:32. Now run down Column D remembering the rule: a worker chooses B if the wB
found from her hi is less than the market wage. The market wage is in Column D and declines as
more workers enter B. Run down Column D until the number in Column C is lower. The
equilibrium, wBE, is the last value of Column D which is greater than the value of Column C in the
same row. From this equilibrium value of wBE , move left and find NBE and hE .
Now use the equilbrium value of wBE that you found, and the values above, to compute observed
lifetime incomes for the three groups: 1) those who choose B initially and fail to acquire the B-skill;
2) those who choose A initially; and 3) those who choose B initially and succed in acquiring the Bskill median value of hours for the B-group and the median value of hours for the A-group.
Remember the equilibrium wage comes from the Demand column, and remember that an observer
will “see” only n, not Dn.
Now two changes:
B First, an increase in demand in the B-market. Use column E; in E1 type “w(B),D2”. In E2 type
“=15.9-0.15*A2”; copy E2 to E3:32. Now run down Column E comparing with Column C, and
repeat what you did in A above with Columns D and C, until you find a new wBE . From this value,
move left and find new values of NBE and hE . What changes in each of these variables was caused
by the increase in Demand in the B-market?
Now use the equilbrium value of wBE that you found, and the values above, to compute observed
lifetime incomes for the three groups: 1) those who choose B initially and fail to acquire the B-skill;
2) those who choose A initially; and 3) those who choose B initially and succed in acquiring the Bskill median value of hours for the B-group and the median value of hours for the A-group.
Remember the equilibrium wage comes from the Demand column, and remember that an observer
will “see” only n, not Dn.
C Second a new supply curve. We’ll have an increase in the difficulty of acquiring the B-skill,
which is modeled by decreasing the probability of succeeding to acquire the B-skill by everyone.
Use Column F for new hours: in F1 type “P(succ)-2”; in F2 type “0.5”; in F3 type “=F2-.033”; copy
F3 to F4:F17. Column G is new “supply” curve. In G1 type “w(B),S2; in G2 type “=10+1.25/F2”;
copy G2 to G3:G17. Now run down Column D comparing with Column G, and repeat what you
did in A above with Columns D and C, until you find a new wBE . From this value, move left and
find new values of NBE and hE . What changes in each of these variables was caused by the change
in difficulty of acquiring B?
Now use the equilbrium value of wBE that you found, and the values above, to compute observed
lifetime incomes for the three groups: 1) those who choose B initially and fail to acquire the B-skill;
2) those who choose A initially; and 3) those who choose B initially and succed in acquiring the Bskill median value of hours for the B-group and the median value of hours for the A-group.
Remember the equilibrium wage comes from the Demand column, and remember that an observer
will “see” only n, not Dn.
Write out your answers and turn them in along with the completed spread-sheet.
--------------------------------------------------------------------------------------------------------
Allen, labour economics, Chapter 10
page 12
Note that the exercise shows the three effects of an increase in demand in the B-market: 1) NBE
is higher after the demand increase; 2) PE is lower after the demand increase, and 3) with wAE held
constant, wBE is higher after the demand increase. Finally, note the effects 1) and 3) taken together
give us a long-run supply curve for the B-skill which is upward sloping. (And, if we weren’t
holding wAE constant, we would also have a long-run supply curve for the A-skill which is upward
sloping.)
Note also that lifetime incomes for any skill-level, or occupational group are a decreasing
function of the Pi s. That is, the more difficult it is to succeed in acquiring the B-skill, the greater
will be the lifetime income gain from succeeding to acquire the B-skill. To put it another way, the
greater the barriers to acquiring entering the B-market, the greater the lifetime income gain from
succeeding in doing so. This explains several things. There are a number of skills/occupations in
which the structure of the market itself puts up large barriers to success. There are two types of
these. First, consider rock stars, tv stars, movie stars, and professional athletes. The structure of
these markets causes the existence of only a few high-profile stars ("superstars"). (An economist,
Sherwin Rosen, has analyzed these markets in a paper, "The Economics of Superstars".) That only
a very few will make it to "super-stardom" is the same as having a very small P in equilibrium, very
small. In these situations, then, we should (and do), see very large lifetime income gains from
success in acquiring super-stardom.
The other type of barrier is artificial, deliberately designed to limit the number of people who
can succeed in entering the B-market. This second type of barrier implies a strategy for any
occupational group trying to increase its lifetime income: erect barriers that make it difficult to
succeed, either in acquiring the skill, or in entering the occupation. This stragegy has been adopted
been adopted by medical doctors, dentists, pharmacists, and lawyers – barriers to the acquisition of
the B-skill (very limited acceptance to medical and law schools, and to dentistry and pharmacy
programs – and by chartered accountants – exams required which limit the number who can pass.
Large lifetime income gains from success in acquiring the B-skill are associated with small
values of the Pi s. Yet the existence of barriers to success at B – whether "natural" (ie, a feature of
the structure of the market, as in "natural" monopoly), or artifical – by causing large lifetime income
gains, also cause a relatively large number of individuals to attempt to acquire these skills, and to
fail. This is a type of market failure: "too many" people trying to acquire these skills. This diverts
the productive time of "too many" people, and diverts resources to the attempt to train people who
will fail. This market failure is a special case of the more general market failure called “positional
externalties” and made famous by Robert Frank. If the market failure is caused "naturally",
remedies are difficult to suggest. If it is caused by artificial barriers, this misallocation must be
considered part of the cost of these barriers.
END OF ALL REQUIRED