CHAPTER 9
WORKERS ARE HETEROGENEOUS
I. Two Skill Markets:
One of outcomes of the model to this point is that lifetime incomes are the same for all workers.
Different skill-markets have different levels of income when observed at any specific time, but
summed over all periods from t=0,n , those incomes are equal in Long-Run equilibrium. In fact it’s
even stronger: the actual definition of Long-Run equilibrium, which has no net entry into, or exit
from, any skill-market, is that lifetime incomes are equal. If they are not, both net entry and net exit
will occur. This is the result of free entry. As in the model of pure competition for firms, where
free entry causes zero excess profits, here free entry causes zero “rents” to choosing to acquire a
skill.
Yet this is not what we observe in the economy. There we observe lifetime income differentials,
very substantial (and, currently, growing fast) lifetime income differentials. To develop a model in
which lifetime income differentials exist, we must assume that workers are different from one
another, or heterogenous. And ultimately, in Chapters 10 and 11, we must assume that one of these
differences – differences in ability – lead to restriction in entry to some skills. This isn’t hard to
believe: how many of your fellow high school students would you want operating on your brain, for
example? Or teaching you economics. Or, perhaps, wiring your house for electricity?
To keep things simple, I develop most of the rest of the analysis for only two skill-markets: A
and B. We have already seen that the extension to three and more skill-markets just to repeat the
analysis for A vs B, but one time period later, with B vs C substituted for A vs B. So the analysis of
two skill-markets generalizes, with no new insights, to three or more skill-markets in a hierarchy.
All of the exogenous variables currently in the model are candidates for introducing differences
between workers. Let's consider, first, that workers obviously differ in their values of n, the time
remaining in their working life. The way this model is currently constructed, variations in n don't
matter mcuh because, in Long-Run equilibrium, the wage differential is small enough that only
workers in the initial period will have any incentive to choose B. Differences in n will matter once
heterogeneity is introduced via other exogenous variables, however. To do that requires an analysis
with at least two types of heterogeneity: differences in n and differences in one other variable. That
comes later.
A. Differences in Discount Rates:
A second variable that obviously varies between individuals is , the rate of discounting the
future. To denote that this now varies between individuals, I subscript the deltas with i, for
individual: i . This means that the discount operator also varies between individuals, with each i
mapping to a unique discount operator: Dni . Varying discount rates are shown on Figure 9-1. For
simplicity this is represented as a uniform, or rectangular, distribution of the i's. The smallest
value of a delta in that distribution is i = 0, and the largest value is i = “max”. Since the Dnis are
inversely related to the is , the largest value of Dni is determined by i = 0, and the smallest value
of Dni is determined by i = “max”. Each individual entering the initial period chooses A or B
according to how he or she discounts the future, that is, based on his or her value, i . Since the
wage differential between B and A occurs in the future, the value of any given wage differential is
higher to individuals who discount the future the least – that is, for individuals with the lowest is
and, therefore, the highest Dnis.
Allen: labour economics, Chapter 9
page 2
Once again, consider equation (8-4), which has the return to choosing B on the LHS and the
costs of B on the RHS. We now slightly rewrite it as an inequation, adding subscripts for
individuals:
(9-1) Dni•h•(wB– wA) >< OCB + DCB .
Individuals choose B when the LHS is greater than the RHS. Note that with the “all other things
equal” condition, we assume all costs to be the same for all individuals. Given a long-run
equilibrium wage differential, (wB– wA), individuals with the lowest values of i (the highest values
of Dni) are the first to choose B. For them the value of the LHS of inequation (9-1) are highest.
And individuals with the highest values of i (the lowest values of Dni) are the first to choose A.
For them the value of the LHS of inequation (9-1) are lowest.
[Note: look back at the numbers of Exercise 8-A, Problem 3, a) and b). From these and inequation
(9-1) you can see that facing a wage differential (wB – wA) = $1.10, individuals with i =0  D5i =
5, will choose B; individuals with i  D5i = 4 will choose A.]
Figure 9-1
Number
Of
Workers
i(1) i(2)
i(E)
Cohort-B
i = 0
i increasing
i(99)
i(100)
Cohort-A
E
i = max
Equilibrium is still stasis: the number of new entrants to A is equal to exits from A, and the
number of new entrants to B equals exits from B. So there is an equilibrium proportion of the new
entrants choosing A, and an equilibrium number choosing B. Denote the total number of new
entrants in any period, the cohort, as EN; ENA is the cohort entering the A-market and ENB is the
cohort entering the B-market: EN = ENA + ENB. E is the value of i which splits the cohort, EN,
into its equilibrium proportions: ENA/EN and ENB/EN. And E  DnE , which in turn sets the LHS
equal to the RHS of equation (9-1): Long-Run equilibrium is now:
(9-2)
DnE•h•(wBE – wAE) = OCB + DCB .
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” discount rate: E  DnE .
To illustrate, assume the cohort is 100 individuals entering the work force each period. Order
these individuals on the basis of their values of i , and denote them as i(1), i(2), i(3), … , i(98),
i(99), i(100). Individual i(1)’s i = zero, individual i(2)’s 1 = (zero plus a small number), etc;
individual i(100)’s i = “max”, individual i(99)’s i = (“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 (9-1) – is the
highest. The next to choose B is i(2), the next i(3), … . Meanwhile, the first to choose A will be
Allen: labour economics, Chapter 9
page 3
i(100), since for her the return to choosing B – that is the value of the LHS of inequation (9-1) – is
the lowest. (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 rectangle on Figure
9-1, one individual is indifferent. This individual’s value of i is such that the LHS of inequation
(9-1) equals the RHS. (Since she must make a decision, we will have her choose B.) Her value of
i is the equilibrium E  DnE, where DnE sets the LHS of inequation (12) equal to the RHS. Given
long-run wage differential (wBE– wAE), this is individual i(E), with a value E , on Figure 9-1.
Individual i(E) is the “cut” individual – the individual who divides the group of all new entrants into
those who choose A, Cohort-A on Figure 9-1, and those who choose B, Cohort-B on Figure 9-1.
Reaching equilibrium is now a two-stage process analytically. To describe this, we will
consider an exogenous change: an increase in productivity in the B-sector that increases demand for
the B-skill and increases the wage at B in the Short Run. (This is the process described on pages 7
and 8 of Chapter 8.) Its effects on equilibrium discount rates are shown on Figure 9-2. We begin
with Long-Run equilibrium, E1, shown on Figure 9-2. 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 higher discount rates, i , than the first cut, E1. So the “cut” increases in the
initial Short-Run, to SR . This will cause an increase in Cohort-B, from B1 to BSR, and
simultaneously a decrease in Cohort-A from A1 to ASR.
Figure 9-2
Number
Of
Workers
i(E1)
Cohort-A1
Cohort-B1
BSR
i = 0
i(E2)
Cohort-A2
Cohort-B2
i increasing
ASR
E1
2
SR
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.
(See the graph and description on pages 7 and 8 of Chapter 8.) These dual changes reduce the wage
differential between B and A, which causes the i of the “cut” individual to fall. 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 i is E2. This process is
illustrated on Figure 9-2.
Three things must be true of the new Long-Run equilibrium. Look back at the long-run
equilbrium condition, equation (9-2). The initial change is that the wage at B, wB, rises from the
productivity increase in sector B. In the Short-Run, the equation no longer holds. The RHS
remains constant, so to get back to equality, to Long-Run equilibrium, the LHS must fall.
That fall, the movement to Long-Run equilibrium, must be caused by a greater supply of labour
to the B-market, NB, which means a lesser supply of labour to the A-market, NA. These in turn
reduce the B-wage, increase the A-wage, and reduce the wage differential. And as the size of the
Allen: labour economics, Chapter 9
page 4
B-sector increases, DnE must fall. The approach from the Short-Run to the new Long-Run
equilibirum is a simultaneous reduction of DnE 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 on pages 7 and 8 of Chapter 8, and is shown on Figure 9-2, where Cohort-B2 is larger
than Cohort-B1, and Cohort-A2 is smaller than Cohort-A1.
Figure 9-2 also makes it clear that this first effect strictly implies the second – that the value of
the discount rate of the “cut” individual must have risen. This too is shown on Figure 9-2. There
individual i(E2) is further along the continuum of individuals ordered by their discount rates than is
individual i(E1). So 2 is greater than E1, and, since 2 > E1 , DnE2 < DnE1. The third effect, then,
is that the equilibrium wage differential, (wBE – wAE), also must have risen. From equation (9-2),
with DnE2 < DnE1 and all other things equal, (wBE2 – wAE2) must be greater than (wBE1 – wAE1).
---------------------------------------------------------------------The Long-Run equilibrium here – in fact all Long-Run equilibria with heterogeneity – have
several nice characteristics. The first is that heterogeneity eliminates the small analytical problem
faced by models where everyone is alike. That problem is that, if all individuals are identical, all
must make the same decision at the same period. In the equilibria with homogeneity, all equilibriua
in Chapter 8, since everyone is the same, everyone must all choose either A or B at t=0. So
"equilibrium" with homogeneity is a state where everyone chooses, say B until the LHS of equation
(8-4) is less than the RHS, then everyone chooses A until the LHS of equation (8-4) is greater than
the RHS, then everyone chooses B ... . That "equilibrium" is cyclical rather than steady-state,
characterized by “bouncing” from having everyone choose B to having everyone choose A to
having everyone choose B to having everyone choose A to … . With heterogeneity, the equilibrium
is steady-state: ENB/EN proportion of the new cohorts choose B at their t=0, and ENA/EN proportion
of the new cohorts choose A. Individuals’ discount rates, i , which are here the way individuals
vary, cause them to self-select who goes to which market.
The second nice characteristic is that now 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 a greater proportion of the cohorts choose B. These additional workers are
workers at the margin of discount rates, so they have somewhat higher discount rates than does
Cohort-B in the initial Long-Run equilibrium. The wage differential must increase in order to
attract these marginal workers in the self-selection process. This means that moving from
homogeneous workers to heterogeneous workers gives upward-sloping Long-Run supply curves for
all skill markets.
Note also that, as observed by an outsider, the Long-Run equilibrium is now characterized by
ambiguity about lifetime income differentials. An outsider observing lifetime incomes would have
to use a single discount rate to evaluate them. If that observer used the “cut” discount rate, E, no
lifetime income differential would be found. But if she used a lower discount rate than E, lifetime
income at B would be observed to be greater than lifetime income at A; if she used a higher
discount rate than E, lifetime income at A would be observed to be greater than lifetime income at
B. Since the “cut” discount rate is unobservable, an outside observer could find anything.
Keep in mind that while lifetime income differentials are ambiguous, the situation of each
individual is not. Those choosing B get a lifetime income at B that is higher to them because their
discount rates are lower than the “cut” discount rate, E; those choosing A get a lifetime income at
A that is higher to them because their discount rates are higher than E.
Allen: labour economics, Chapter 9
page 5
Finally, note that this analysis has had a substantial effect on public policy. In the 1960s, a
sociologist named Daniel Patrick Moynihan – who was to gain greater fame as a United States
Senator of some prominence – studied, empirically, the situation of children raised in poor families
in the US. He found that they had much higher discount rates than middle class chldren. This
makes sense: kids raised in urban ghettos had good reason to discount the future at very high rates.
But our analysis shows that individuals with low discount rates are less likely to choose Human
Capital.
At the time Moynihan did his study, there also existed in North America a strong ethic that
government should do its best to equalize opportunity. So, to encourage kids from poorer families,
who had relatively high discount rates, the student loan program was approved, with very low
interest rates on those loans.
Exercise 9-I:
The equilibrium here has a single equation (9-2) with two unknowns: DnE 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.
[Even simplified hugely by assuming that the A-market is very large relative to the B-market so wAE
is a given, the equations that must be solved simultaneously are: 1) wBE = (DnE•h•wAE + OCB +
DCB)/DnE•h; 2) distribution of the Dnis: DnE = Dnmax – d•NBE ; and 3) demand in the B-market: wBE
= $x – y•NBE , where d, x, and y, are parameters. This is three equations and three unknowns, and
the first equation combined with the third makes it non-linear, so soultions are difficult.]
Luckily, however, we can closely approximate a solution using EXCEL Below is an EXCEL
exercise using the short-cut we used before: wAE is a “numeraire”, a “given”.
The numbers used in this exercise are: wAE = $10/hour, h = 2000, ohB = 1000, DCB = $10,000,
and n = 10. The reduced form of equation (9-2) is: wBE = wAE + (OCB + DCB)/DnE•h . With these
numbers, the reduced form equation becomes: wBE = $10 +[$20,000/(2000•D10E)] = $10 + $10/D10E.
ROUND ALL COLUMNS TO TWO DECIMALS!
A The exercise assumes 31 workers with a uniform distribution of the Dnis. 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 Dnis, so in B1 type “D(10)”. In
B2 type “10”. In B3 type “=B2-.2”; copy B3 to B4:32. B32 should be 4. Column C is the “supply”
curve; it is the wBEs determined from the D10is, so in C1 type “w(B),S1”. In C2 type “=10+10/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 “=11.99-0.03*A2”; copy D2 to D3:32. Now run down Column D remembering the rule: a
worker chooses B if the wB found from her D10 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 wage in B, 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 D10E .
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
“=12.48-0.03*A2”; copy E2 to E3:32. Now run down Column E comparing with Column C, and
Allen: labour economics, Chapter 9
page 6
repeat what you did in A above with Columns C and D, until you find a new wBE . From this value,
move left and find new values of NBE and D10E . Comparing with the initial equilibrium of A, what
changes in each of these variables was caused by the increase in Demand in the B-market?
C Last, a new value of DCB, which shifts the supply curve. The new DCB = 6,000. The new
reduced form equation for wBE is wBE = 10 + 16,000/2000•D10E = 10 + 8/D10E . Use Column F: in
F1 type “w(B),S2”. In F2 type “=10+8/B2”; copy F2 to F3:32. Now run down Column D
comparing with Column F, and repeat what you did in A above with Columns C and D, until you
find a new wBE . From this value, move left and find new values of NBE and D10E . Comparing with
the initial equilibrium of A, what changes in each of these variables was caused by the increase in
Demand in the B-market?
Write out your answers and turn them in along with the completed spread-sheet.
-------------------------------------------------------------------------------------------------------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) D10E 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.)
EXERCISE: Show that if an individual can borrow at a low interest rate, r, to pay for the Bskill, the individual’s decision must be rewritten with r used where i was used before. Individuals’
discount rates will no longer cause a selection process for the B-skill. If that B-skill is a university
education, university education should no longer select against kids with high discount rates.
Allen: labour economics, Chapter 9
page 7
B. Differences in Preference for Leisure (Hours Worked):
A third variable that obviously varies between individuals is the preference for Leisure. Here
that preference will be viewed as the preference for Hours Worked, since it is Hours Worked, h, that
is in the decision equations and in the long-run equilibrium equation. To note that this now varies
between individuals, I subscript the hs, Hours Worked, with i: hi. Varying Hours Worked are shown
on Figure 9-3. For simplicity this is shown as a uniform, or rectangular, distribution of the his. On
that Figure, the Hours Worked vary between a minimum of 1000 Hours per year, and a maximum
of 3000 Hours per year. Each individual entering the initial period chooses A or B according to
how many Hours per year she plans to work, that is, based on her value of hi . Since the wage
differential between B and A is worth more the more Hours the individual works, the value of any
given differential is higher to individuals who intend to work the most.
Once again, consider equation (8-4), which has the return to choosing B on the LHS and the cost
of B on the RHS. We now slightly rewrite it as an inequation, adding subscripts for individuals:
(9-3) Dn•hi•(wB– wA) >< OCB + DCB .
Individuals choose B when the LHS is greater than the RHS. Note that with the “all other things
equal” condition, we assume all costs to be the same for all individuals. Given a long-run
equilibrium wage differential, (wB– wA), individuals with the highest values of hi are the first to
choose B. For them the value of the LHS of inequation (9-3) are highest. And individuals with the
lowest values of hi are the first to choose A. For them the value of the LHS of inequation (9-3) are
the lowest.
[Note: look back at the numbers of Exercise 8-A. Compare parts a) between Problems 3 and 6.
From these and inequation (9-3) you can see that facing a wage differential (wB– wA) = $1.10,
individuals with hi = 2500, will choose B; individuals with hi = 2000 will choose A.]
Figure 9-3
Number
Of
Workers
i(E)
i(1) i(2)
Cohort-A
hi = 1000
hi increasing
i(99)
i(100)
Cohort-B
h
hi = 3000
Equilibrium is still stasis: the number of new entrants to A is equal to exits from A, and the
number of new entrants to B equals exits from B. So there is an equilibrium proportion of the new
entrants choosing A, and an equilibrium number choosing B. ENA is the cohort entering the Amarket; ENB is the cohort entering the B-market. hE is the value of hi which splits the cohort, EN,
into its equilibrium proportions: ENA/EN and ENB/EN. And hE sets the LHS equal to the RHS of
inequation (9-3): Long-Run equilibrium is now:
(9-4)
Dn•hE•(wBE – wAE) = OCB + DCB .
Long-Run equilibrium is again 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” Hours Worked: hE.
Allen: labour economics, Chapter 9
page 8
To illustrate, assume the cohort is 100 individuals enter the work force each period. Order these
individuals on the basis of their values of hi , and denote them as i(1), i(2), i(3), …, i(98), i(99),
i(100). Individual i(1)’s hi = 1000, individual i(2)’s hi = 1020, etc; individual i(100)’s hi =3000,
individual i(99)’s hi = 2980, 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(100), since for her the return to choosing B – that is, the
value of the LHS of inequation (9-3) – is the highest. The next to choose B is i(99), the next i(98),
… . Meanwhile, the first to choose A will be i(1), since for her the return to choosing B – that is the
value of the LHS of inequation (9-3) – is the lowest. (Remember that A exists, so some workers
must choose A) The next to choose A will be i(2), the next i(3), … .
This self-selection process continues until, somewhere in the middle of the rectangle on Figure
9-3 one individual is indifferent. This individual’s value of hi is such that the LHS of inequation (93) equals the RHS. (Since she must make a decision, we will have her choose B.) Her value of hi is
the equilibrium hE , where hE sets the LHS of inequation (9-3) equal to the RHS. Given long-run
wage differential (wBE – wAE), this is individual i(E), with a value hE , on Figure 9-3. Individual i(E)
is the “cut” individual – the individual who divides the group of all new entrants into those who
choose A, Cohort-A on Figure 9-3, and those who choose B, Cohort-B on Figure 9-3.
Reaching equilibrium is now a two-stage process analytically. To describe this, we will
consider an exogenous change: an increase in productivity in the B-sector that increases the demand
for the B-skill and increases the wage at B in the Short Run. (This is the process described on pages
7 and 8 of Chapter 8.) Its effects on equilibrium Hours Worked are shown on Figure 9-4. We begin
with long-run equilibirum, E1, shown on Figure 9-4. The increase in the wage at B causes the wage
differential to increase in the short-run. More new entrants choose B, and some of these will have
lower preferences for Hours Worked, hi – that is, higher preferences for Leisure – than the first cut,
hE1 on Figure 9-4. So the “cut” decreases in the initial Short-Run to hSR . This will cause an
increase in Cohort-B, from B1 to BSR, and simultaneously a decrease in Cohort-A from A1 to ASR.
Figure 9-4
Number
Of
Workers
i(E2)
i(E1)
Cohort-A2
Cohort-A1
ASR
hi = 1000
Cohort-B2
hSR
Cohort-B1
BSR
hE2
h1
hi increasing
hi = 3000
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.
(See the graph and description on pages 7 and 8 of Chapter 8.) These dual changes reduce the wage
differential between B and A, which causes the hi 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 hi is hE2. This process is
illustrated on Figure 9-4.
Three things must be true of the new Long-Run equilibrium. Look back at the Long-Run
equilbrium condition, equation (9-4). The initial change is that the wage at B, wB, rises from the
Allen: labour economics, Chapter 9
page 9
productivity increase in sector B. In the Short-Run, the equation no longer holds. The RHS
remains constant, so to get back to equality, to Long-Run equilibrium, the LHS must fall.
That fall, t he movement to Long-Run equilibrium, must be caused by a greater supply of labour
to the B-market, NB, which means a lesser supply of lbaoaur to the A-market, NA. These in turn
reduce the B-wage, increase the A-wage, and reduce the wage differential. And as the size of the
B-sector increases, hE must fall. The approach from the Short-Run to the new Long-Run
equilibirum is a simultaneous reduction of hE 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 on pages 7 and 8 of Chapter 8, and is shown on Figure 9-4, where Cohort-B2 is larger
than Cohort-B1, and Cohort-A2 is smaller than Cohort-A1.
Figure 9-4 also makes it clear that this first effect strictly implies the second – that the
preference for Hours Worked of the “cut” individual must have fallen – preference for Leisure of
the “cut” individual must have risen. This too is shown on Figure 9-4. There individual i(E2) is not
as far along the continuum of individuals ordered by their preference for Hours Worked than is
individual i(E1). So h2 is less than hE1 . The third effect, then, is that the equilibrium wage
differential, (wBE – wAE), also must have risen. From equation (9-4), with h2 < hE1 and all other
thins equal, (wBE2 – wAE2) must be greater than (wBE1 – wAE1).
-----------------------------------------------------------------------Again, the Long-Run equilibrium here has those same several nice characteristics. The first is
that heterogeneity here also eliminates the small analytical problem faced by models where
everyone is alike. The problem is described above, on page 4. Here it is individual preferences for
Leisure, reflected in preferences for Hours Worked, that cause individuals to self-select into the
different skill markets. This keeps the system in stasis, and it makes the Long-Run equilibrium a
steady-state.
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 a greater proportion of the cohorts choose B. These additional workersare
workers at the margin of preferences for Hours Worked, so they have somewhat lower preferences
for Hurs Worked than does Cohort-B in the initial Long Run equilibrium. The wage differential
must increase in order to attract these marginal workers to B in the self-selection process. This
means that moving from homogeneous workers to heterogeneous workers gives upward-sloping
Long-Run supply curves for all skill markets.
Note also that, as observed by an outsider, for the first time the Long-Run equilibrium is
characterized by unambiguously positive lifetime income differentials. To illustrate, let’s assume
that hE2 = 2000, and that the equilibrium wage differential – here (wBE2 – wAE2) – is wA = $10/hour
and wB = $12/hour. For the “cut” individual, whose preference for Hours Worked is 2000, this pair
of wages makes the lifetime income of the A-market equal to the lifetime income of the B market.
But the average Hours Worked for all workers in the B-market is 2500, and the average Hours
Worked for all workers in the A-market is 1500. So for the actually observed B-workers, the
lifetime income in the B-market is 20% higher than that of the “cut” individual. And for the
actually observed A-workers, the lifetime income in the A-market is 25% lower than that of the
“cut” individual. The lifetime income differential depends on the equilibrium wage differential,
but, for the first time in the model, we actually generate substantial lifetime income differentials.
And, since the equilibrium wage differential increases as the elasticity of demand for the B-skill
Allen: labour economics, Chapter 9
page 10
increases, now, as Part D of the Exercise below shows, lifetime income differentials will be higher
the more elastic the demand for the B-skill.
-------------------------------------------------------------------Exercise 9-II:
The equilibrium here has a single equation (9-4) with two unknowns: h and the wage
differential: (wBE – wAE). To solve the system we need a specification of the distribution of the his,
which means adding another unknown, NBE and Demand functions for both the A and the B skill
markets. This is, again, 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.
[Even simplified by assuming that the A-market is very large relative to the B-market so wAE is a
given, the simultaneous equations one must solve are: 1) wBE = (D10•hE•wAE + wAE• + OCB +
DCB)/D10•hE; 2) distribution of the hEs: hE = hmax – d•NBE; and 3) demand in the B-market: wBE = $x
– y•NBE , where d, x, and y, are parameters. This is three equations and three unknowns, and the
first equation combined with the third makes it non-linear, so soultions are difficult.]
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: wAE = $10/hour, n = 10; D10 = 8, ohB = 1000, and DCB =
$10,000. The reduced form equation for wBE is wBE = wAE + (OCB + DCB)/Dn•hE . Solving with the
numbers given here, that becomes wBE = 10 + (20,000)/8•hE = 10 + 2500/hE .
ROUND ALL COLUMNS TO TWO DECIMALS!
A The exercise assumes 31 workers with a uniform distribution of the hi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 his, so in B1 type “hours-1”. In
B2 type “2700”. In B3 type “=B2-50”; copy B3 to B4:32. Column C is for the wBEs determined
from the his, so in C1 type “w(B),S1”. In C2 type “=10+2500/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 “=11.470.03*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 or equal to the value of
Column C in the same row. From this equilibrium value of wBE , move left and find NBE and hE .
Now find the median value of hours for the B-group and the median value of hours 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 these values, 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.
Now three changes:
B First, an increase in demand in the B-market. Use column E; in E1 type “w(B),D2”. In E2 type
“=12.5-0.03*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 . Comparing with the initial equilibrium of A, what
changes in each of these variables was caused by the increase in Demand in the B-market?
Allen: labour economics, Chapter 9
page 11
Now find the median value of hours for the B-group and the median value of hours 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 these values, 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 a change in behavior to a lesser preference for
leisure, or a greater preference for work, by everyone. Use Column F for new hours: in F1 type
“hours-2”; in F2 type “3200”; in F3 type “=F2-50”; copy F3 to F4:F32. Column G is new “supply”
curve. In G1 type “w(B),S2; in G2 type “=10+2500/F2”; copy G2 to G3:G32. 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 .
Comparing with the initial equilibrium of A, what changes in each of these variables was caused by
the increase in Demand in the B-market?
Now find the median value of hours for the B-group and the median value of hours 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 these values, 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 H: in H1 type
“w(B),D3”. In H2 type “=11.47-0.015*A2”; copy H2 to H3:32. Now run down Column H
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 . Comparing with
the initial equilibrium of A, what changes in each of these variables was caused by the increase in
Demand in the B-market?
Now find the median value of hours for the B-group and the median value of hours 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 these values, 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.
-------------------------------------------------------------------------------------------------------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) hE 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.)