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\NA GE INEQUAtmr AND
SEGREGA TION BY SKILL
Michael Kremer
Eric
96-23
Maskin
Aug. 1996
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
,
WAGE INEQUALHYAND SEGREGATION BY SKILL
Michael Kremer
Eric
96-23
Maskin
Aug. 1996
^lASSACVlilsn^nnbUiU^
vn
23
Wage
Inequality and Segregation by Skill'
Michael Kremer and Eric Maskin
August
2,
1996
Abstract
Evidence from the United
States, Britain, and France suggests that recent growth in wage
accompanied by greater segregation of high- and low-skill workers into
model in which workers of different skill-levels are imperfect substitutes
inequality has been
separate firms.
A
can simultaneously account for these increases in segregation and inequality either through
technological change, or, more parsimoniously, through observed changes in the skilldistribution.
'We thank Daron Acemoglu, V.V. Chari, Benjamin Friedman, Lawrence Katz, Thomas Piketty, Sherwin
Rosen, Kenneth Troske, and Finis Welch for suggestions; Andrew Bernard, Eli Berman, Steve Davis, Brad
Jensen, Francis Kramarz, Steve Machin, and Ken Troske for suggestions and data; and Sergei Severinov and
Charles
Morcom
Studies, U.S.
for exceptionally helpful research assistance.
Bureau of the Census, for providing
We
are grateful to the Center for
Economic
We
acknowledge research support from the National
Science Foundation. All opinions, as well as mistakes, are our own, and not those of the U.S. Bureau of the
data.
Census.
'M.I.T., National
Bureau of Economic Research, and Harvard
"Harvard University.
Institute for International
Development.
Introduction
One of
the
most disturbing economic trends of the
last fifteen to
been a growing wage gap between high- and low-skill workers
twenty years has
in several industrialized
economies, including that of the United States. This trend has been well documented.
and Murphy [1992], for example, find
high school education and
1
that,
Katz
between 1979 and 1987, wages of men with a
-5 years of experience fell
by twenty percent, while those of male
college graduates rose by ten percent.
This paper provides evidence that this increase in inequality has been accompanied by
growing segregation of workers by
high- and low-skill workers to
skill.
work
That
in the
is,
same
over time,
it
has
Economic
firm.
become
less
common
activity has shifted
for
from
firms such as General Motors, which use both high- and low-skill workers, to firms such as
Microsoft and McDonald's, whose workforces are
much more homogeneous. For example,
France, between 1986 and 1992, the correlation of log wages
among workers
in the
in
same
establishment rose from 0.36 to 0.44, the correlation of experience rose from 0.11 to 0.16,
and the correlation of seniority grew from 0.24
to 0.31.
between the wages of manufacturing production workers
1975 to 0.80
in 1986,
and the correlation of a
dummy
In the United States, the correlation
in the
same
plant rose
from 0.76
in
variable for being a production worker
rose from 0.195 in 1976 to 0.228 in 1987.
We
propose a simple model that can account for the simultaneous increases
inequality and in segregation, as well as for the absolute decline in
workers.
in
wages of low-skill
In the model, workers of different skill-levels are imperfect substitutes, and output
2
is
more
sensitive to skill in
some
and mean of the
as the evidence bears out (see below), the dispersion
We
increase?
show
that a rise in skill-dispersion causes firms that
one
low-skill workers to specialize in
Although,
(Proposition 3).
wages of the poor
we
~
demonstrate that
skill-distribution
States
—
What happens
tasks than in others.
in
such a model when,
skill-distribution
once hired both high- and
level or the other, thereby fostering segregation
model, an increase in mean skill-level can enhance the
in this
thus formalizing the so-called "trickle-down" principle (Proposition 1)
this principle relies crucially
becomes
sufficiently dispersed
a further increase in
mean
~
on a
as
"tight" distribution
it is
skill-level raises the
now
likely to
of
be
skills.
in the
Once
~
the
United
wages of high-skilled workers but
causes those of poorly-skilled workers to decline, thereby aggravating inequality (Proposition
2)-
Note
articulated
that this last implication contrasts with a
by Schmitt [1992]:
widespread view of the labor market
"Rising returns to skills in the face of large increases in the
supply of skilled labor suggest a substantial shift in labor
workers."
Whereas labor economists such
demand
in favor
as Schmitt see rising mean-skill levels as
deepening the mystery of increasing inequality, our analysis suggests
may have
this rise in mean-skill
helped cause the increase in inequality.
The model's implication
workers by
evidence
of skilled
skill is
that,
promote segregation of
consistent not only with the time-series data cited above, but also with
among U.S.
states,
more segregated by education
countries, with
that increases in skill-dispersion
in
those with wider dispersion in educational attainment are
employment. Casual empiricism suggests
wide dispersion of
skill,
are particularly prone to dualism.
that
developing
3
We
noted above that there
skill-distribution
(see
have increased
Kahn and Lim
evidence that both the mean and the dispersion of the
is
in the
economies studied
[1994]) that skill-biased technological change
effective skill-level of high-skill vi'orkers,
distribution of skills.
there
is
Of
course, such technological change
Murphy, and Pierce
[1993]).
It
has been argued
may have
which would have had the posited
also documentation of increases in the
(see Juhn,
in this paper.
These
shows how they could have been magnified
mean and
shifts
is difficult
raised the
effects
to measure.
on the
However,
dispersion of observable skills levels
have been
into significant
fairly
modest, but our model
income
changes
in the
Section
I offers
distribution
and degree of segregation.
The remainder of
model and the
the paper
theoretical results.
is
organized as follows.
Section
and France have become more segregated
propositions in detail.
dispersion are
n
a sketch of our
presents evidence that the United States, Britain,
in recent years.
Section IE develops our theoretical
Section IV provides evidence that U.S. states with greater
more segregated by
skill,
as the
model
predicts.
Section
V
skill
interprets recent
increases in inequality and segregation in light of the model.
The conclusion
implications and speculates on the possibility that segregation
may
discusses policy
spur inequality by
reducing opportunities for low-skill workers to learn from higher-skill co-workers, giving rise
to a vicious cycle of increasing inequality
I.
A
and segregation.
Sketch of the Theory
A
model consistent with the
stylized facts about segregation
and inequality discussed
above must incorporate three ingredients.
it
should be the case that
imperfect substitutes for one another;
(i)
workers of different
(ii)
different tasks within a firm are
skills are
Specifically,
complementary;
and
(iii)
different tasks within a firm are differentially sensitive to skill.
To
why
see
(i),
imperfect substitutability,
perfect substitutability obtains.
2q'.
Then,
other operational
change.
skill
makes no prediction about what
skill
q
two of
as
is
us examine a model in which
two
skills in the
skill-levels
skill q' .
we
4
skill-levels
q can be replaced by two of
skill q',
and q -
with no
much
as q'-
economy. Moreover, such a model
will see in a firm: there
To make any headway
segregation, therefore, imperfect substitutability of skill
In fact, there
let
Clearly, ^-workers will be paid exactly twice as
workers regardless of the distribution of
worker of
needed,
Specifically, consider workers of
every firm, a worker of
in
is
.
is
could as easily be one
in explaining either inequality or
required.
a body of work within the labor economics literature that assumes
such imperfect substitutability (see Sattinger [1993] for a review).
evidence to justify the assumption; see Katz and
Murphy
There
[1992] and
is
also empirical
Murphy and Welch
[1992].
To
there are
understand the role of assumption
two
tasks
-
(ii),
a "g" task and an "h" task
complementarity, consider a firm in which
~
but no complementarity between them.
Specifically, take a production function in which, if a (^-worker
is
hired for the g-task and a
^'-worker for the
/i-task,
output
is
(1)
nq,q') =giq)^Kq').
Different skills are imperfect substitutes in this formulation: only one worker can be hired for
each
task,
means
and so the
that relative
sort of two-for-one substitution
wages may depend on the
number of ^-workers
contemplated above
distribution of skills;
Thus, unlike the
something about inequality. However,
it still
g-task
This
for example, the
of ^'-workers, the former workers' wages are
rises relative to that
likely to experience a relative decline.
firms.
if,
is ruled out.
first
model,
this
formulation
may
say
implies nothing about the skill-composition of
In particular, because tasks are not complementary, the optimal choice of skill for the
is
independent of that for the
Finally, to understand the
/j-task.
need for
(iii),
differential skill-sensitivity, consider a
production function in which different tasks are complementary but identical in their
sensitivity to skill:
<2)
This
Aq,q') = qq'is
the production function used
and also by Kremer [1993]
implication that there
is
by Becker [1981]
in his study of labor
in his analysis of assortative marriage,
markets and development.
always complete segregation of
matched only with other ^-workers,
etc.
To
see
this,
skill
It
has the
within firms: ^-workers are
imagine that there are two ^-workers
and two (^'-workers. Then either the q-workers can be matched with each other and the same
for the ^'-workers
( self-matching 't
or else there can be two matches consisting of one q- and
one ^'-workers each ( cross-matching V But self-matching leads
to total output of
^+^'^
6
which
is
greater than the output of 2qq' resulting
from cross-matching. Hence, regardless of
the distribution of skills, production function (2) implies that
we should
see only self-
matching, and so the evolution from General Motors to Microsoft/McDonald's discussed
above could not have taken place.
We
shall therefore introduce
an asymmetry between tasks
by working with a production function of the form,
where
Q<c<d.
By
redefining the unit of
skill,
we
fiq,q') =qq",
(3)
can rewrite
this as
e>l.
For ease of computation (although none of the qualitative results depend on
we
shall, in
take e=2, so that
fact,
<^)
We
this),
Mq')
can think of the q '-task (which
and the ^-task (which
is
is
= qq'^-
relatively sensitive to skill) as the "managerial" task
relatively skill-insensitive) as the "assistant's" role.
Hence,
let
us
rewrite the production function as
^^^
fiqa^qj = qaql
Production function (6) has the implication that
(L
<
//), it is
more
production function
Lucas [1978]:
efficient for the
is
H-worker
if
a firm employs workers of skills
to be the
closely related to one used
manager
(since
Uf
>
L and
HI}).
H
This
by Rosen [1981, 1982], Miller [1983], and
(7)
/(9i v..,9„9
J
=
qjC£ «.).
/'>0 ./"<0.
i=l
where ^„
is
the skill of the manager, ^,
subordinates,
of
is
skill at the
a choice variable.
is
the skill of subordinate
Under production function
/,
and
r,
(7), there is
the
no
number of
substitutability
managerial level, but several low-skill subordinates can be substituted for one
Hence,
high-skill subordinate.
high-skill workers
in equilibrium, low-skill
workers become subordinates, and
become managers. The higher a manager's
skill,
the
more employees he
supervises.
By
contrast, our production function (6)
imposes imperfect substitutability on
subordinates as well as on managers. (In this sense,
and
(7).)
This has the implication
same or lower
that, in
it
combines the approaches inherent
equilibrium, a
in (2)
manager of one firm may be of the
than a subordinate in another firm (even though, as already noted,
skill
managers are more highly skilled than subordinates within the same
M.B.A. might choose
firm).
For example, an
either to accept an entry-level position with high-skill colleagues at an
investment bank, or a more senior position at a less prestigious firm.
We
consider a one-good
workers of different
too
is
endogenous,
interest to us.
We
skill.
it is
We
in
which there
is
an exogenous distribution of
take this distribution as given because, although ultimately
likely to
(6).
it
change more slowly than the wage and matching patterns of
suppose that there
have the production function
to scale.
economy
is
an indefinite supply of potential firms,
Hence,
this is a
competitive
all
of which
economy with constant
returns
8
Our
theoretical results (Propositions 1-3 in Section HI)
equilibrium
wage
distribution
in the distribution of skills.
and the pattern of
examine how the competitive
by
skill-levels within firms are affected
Here we give a simplified account of these
shifts
results.
Proposition 3 shows that, as the dispersion of skills increases in the economy, the
relative dispersion of
two
of
L and
skill-levels,
skill
H,
it is
within firms
;skill
more
H, where
L<
falls.
To
get a feel for this, suppose that there are just
H. Now, given two workers of
efficient for the
L and two
skill
workers
workers to cross-match (L-workers match with H-
workers) rather than self-match (workers of the same skill-level match)
if
and only
if
L'*H'< 2LH\
(8)
Thus because competitive equilibrium
and only
is efficient,
we
will
have equilibrium cross-matching
if
if (8) is satisfied, i.e.,
H< ^"^L.
(9)
2
Imagine
close in value.
that the
From
economy begins with a
(9),
we
will
have cross-matching
—
will
employ both low- and
following the G.M. pattern
literally true if there are different
there are
more ITs than
skill-distribution in
in equilibrium,
i.e.,
L and
in the
H are fairly
each firm
high-skill workers.
numbers of L- and H-workers
L's, all the L's will
which
~
(This cannot be
economy.
If,
say,
be matched with ITs, and the remaining ffs will
be self-matched.)
Suppose now
that
H increases
economy
and
that the dispersion
L
decreases.
of
skills increases.
This can be modeled by supposing
Eventually (9) will no longer hold,
will re-align so that there is only self-matching,
that
is,
at
which point the
each firm will have only
9
H- workers (Microsoft)
To
or only L-workers (McDonald's).
understand this result less formally, note that there are two forces
at
work
in
determining the equilibrium matching pattern.
On
in the production function militates in favor of
cross-matching between workers of different
skills, in
which the more highly
skilled
worker
is
the one hand, the
assigned the managerial task.
hand, the complementarity between tasks promotes self-
when
(i.e.,
tasks are perfectly symmetric, only the second effect
assortative matching.
matched up to the point
effect
overwhelms the
is
in
which the difference
At
first.
switches to self-matching.
is,
On
present,
and
we have
tasks
the other
Now,
assortative) matching.
For any given degree of asymmetry between
be some deviation from self-matching. That
asymmetry of the
perfect
however, there will
tasks,
unequally skilled workers will be cross-
in their skills is
that point, as in the
so great that the second
above two-point example, the economy
Proposition 3 generalizes this logic to the case of
many
skill-
levels.
Let us turn to the issue of wages.
of
skills
workers decline.
is sufficiently
demonstrates that
To
see
skill-levels,
wages of
mean
highly dispersed.
why
L<
That
wage
is,
that there are equal
let
the distribution
skill-level reduces
is
true if the distribution of skills
with a diffuse skill-distribution, an increase in the
inequality.
these propositions hold in a simple case, consider an
M < H, and
if
low-skill workers rise, whereas those of high-skill
Proposition 2 establishes that the opposite
skill-level aggravates
assume
1
has sufficiently low dispersion, then an increase in the
inequality in the sense that the
mean
Proposition
us suppose that
numbers
(say,
H and L satisfy
(9).
economy with
three
For easy computation,
x each) of L- and H-workers and
that the
number
10
of M-workers
is at
least 2x.
Because
H and L satisfy
and //-workers to be self-matched. In
fact,
we
(9),
we know
that
it is
inefficient for L-
claim that both L- and //-workers should be
matched with M-workers. The alternative would be for the L- and //-workers
with each other.
But notice
that, since
I//2 +
(10)
Condition (10) implies that
who
are
total
matched
<M and M <H,
M^
< LAf2 + MH^.
output would be raised by taking an L- and an H-worker
matched with each other and rematching both with M-workers (who were previously
self-matched).
make
L
to be
Now, because
there
zero profit in equilibrium.
more than Ix M-workers,
at least
is
perfect competition and constant returns to scale, firms
Hence workers must absorb
all
one pair must be self-matched
revenue.
Since there are
in equilibrium, in
which case
.
11
their firm's
revenue
is
M^
M-wage
Therefore, the equilibrium
w(M)
(11)
=
is
^
2
Thus, the zero profit requirement implies that the equilibrium L-wage
LM^-
w(I) = LM"^ - w{M) =
(12)
is
—
Similarly,
wiH) =
(13)
MH^
-
^.
2
To model
increase in
(14)
the idea of an increase in the
M. The
derivatives of
\^{1-)
mean
and wifJ) with respect
2IM
-
Im^
and
(15)
skill-level, let us
/f2 - 'iu''.
to
M
examine a small
are, respectively.
12
dispersion of skill
The assumption of "low"
-
~
the assumption treated
H ~ L.
corresponds in this three-skill model to the condition that
positive,
whereas (15)
which
increases,
By
is
contrast,
is
But
negative.
what Proposition
it is
readily
this
means
w(L)
rises
In this case, (14)
and wiH)
falls as
1
is
M
asserts.
1
shown
that if
L<^M< ^H,
(16)
4
which corresponds
To
4
to the higher dispersion of Proposition 2, then (14) is negative
Hence w(L)
positive.
falls
while w(H) rises
understand Proposition
1
at
slowly than does the
total
a more intuitive level, notice that
L-M
Hence an
product LM^.
L: the net marginal revenue left over after the
and (15)
is
M increases, as required by Proposition
when
share of the revenue going to the A/-worker in an
match
increase in
M-worker
is
(i.e.,
AfV2
when L = Af
)
rises with
2.
the
M more
M confers a positive benefit on
paid for his higher product
is
This can be viewed as a manifestation of the "trickle-down" principle, the theory
positive.
that
that
by Proposition
propounds that some of the
fruits
generated by improving the productivity of middle- and
upper-class workers will accrue to the poor.
In our model, however, the trickle-down principle pertains only to skill-distributions
with low dispersion.
benefits
If
L and
M are
far
from cross-matching between
matching.
Therefore since w{Af)
rises,
enough
L and
apart, a further increase in
M to deteriorate relative to those from
w(L) must
enhances the quality of a match between
M causes the
fall.
At the same time,
self-
this increase in
M
M and H (provided M and H were far enough apart
13
Hence, an increase
to begin with), raising w{H).
between H- and L- workers
in
mean
skill-level causes the
to increase, as Proposition 2 establishes.
Notice that our model has the feature that workers of median
above example) can serve
they are matched with.
(see Section EI).
II.
We
wage gap
in
This
skill
(Af-workers in the
equilibrium as either managers or assistants, depending on
is
a property that generalizes to the case of
will speculate
on
its
political
many
and social implications
whom
skill-levels
in the conclusion.
Rising Segregation by Skill
We
offer evidence that segregation of workers
States, Britain,
has increased in the United
skill
and France. Subsection n.A defines the index we use to measure segregation,
and Subsection II.B shows
that the index has increased in these three countries
measured by any of three proxies: wages, worker
II.A.
by
classification,
when
skill is
and experience.
The Segregation Index
Sociologists have used an index of correlation to measure segregation
variables, such as race [Bell, 1954; Robinson, 1950].
In this section,
segregation that generalizes this measure of correlation so that
taking on
many
values.
We
also adjust
it
it
by dichotomous
we propose
can be applied to variables
to take account of the concentration of high-skill
workers that would have been expected purely from chance.
(Ellison and Glaeser [1994]
develop an index of geographic concentration designed to address similar issues.)
discusses
how
an index of
Appendix
I
confidence intervals can be constructed around the estimated segregation index.
14
Suppose
be the
set of
skill, q,
that there are
workers
J firms
firm j and
in
z,
(or plants) in the
the
sample indexed by 7 =
number of workers
in this firm.
Let Z,
J.
1
Given a measure of
such as wages, education, or worker classification, denote the mean skill-level
sample by
q
{i.e.,
q
=^y^ q fN,
j-i
where
A^
is
the total
number of workers
in the
in the sample).
i€Z,
Define p as our index of correlation or segregation:
p =
(17)
where workers are indexed by
of zero indicates that
all
/
and k (so
firms have the
indicates complete segregation, in
keZj
J=l ieZj
that q^ is the skill-level of
same
which
all
worker
/).
A
correlation
skill-mix of workers, and a correlation of one
workers within a firm have the same
q.
Note
15
that this index is invariant to affine transformations of the units in
The index of
correlation, p,
is
which
skill is
measured.^
equivalent to one minus the variance of q within firms
divided by the overall variance of q, and thus to the R^ obtained by regressing ^ on a series
of firm dummies.
To
see this, add and subtract the
mean
qj
within each firm; to obtain:
keZ,
(18)
P
^In the case
of a
"J
=
dummy
variable, the correlation specializes to the index used
by Bell
[1954] and Robinson [1950],
T^P'-D)'
p=M.
NDil-D)
where
D
denotes the
dummy
Note
variable.
plants of the proportion of workers for
the variance of the
dummy
that this index normalizes the variance across
whom
the
dummy
variable equals one by dividing by
variable in the total population.
Without normalization,
segregation by production-worker status, for example, would appear higher in an
economy
with five firms consisting only of non-production workers and ninety-five consisting only of
economy with one firm consisting of only non-production
workers and ninety-nine firms consisting of production workers. After normalizing, however,
production workers than in an
we
find that both economies have the
maximum
segregation index of one.
16
we
After deleting the terms which equal zero,
can simplify to
J
E^/^'-^)'
p =
(19)
H
J
J
^l
2,22
+s^
^-
St
Sb
where sj and
s,,^
represent the variance of q within and between firms, respectively, and
represents the total variance of
economy. For example,
in the
q
(20)
2
Sj-'
s,^=J:Zj{^-qflN.
One
virtue of this index
since p depends only on the variance of q in the
is that,
population and on the variance of
mean q between
data sources for workers and firms.
This
is
useful
firms,
if
it
can be calculated using separate
data linking employees and their firms
are unavailable, as in the United States.
Note
that if firms are of finite size, there will
levels of workers in the
some
same
firm,
even
if
be some correlation between the
workers are assigned to firms randomly.
firms will happen to receive workers of above-average
workers of below-average
skill.
assignment, the variance of
If all firms consisted
mean
skill
of
skill,
skill-
That
and others will receive
K workers,
then, with
random
between firms would be \/K times the over-all
is,
17
variance of
skill
between workers, and the expected correlation would thus be \IK.
To compare
sizes, or to
the extent of segregation across economies with different average firm
determine whether an economy exhibits more segregation than would be expected
purely from chance,
it
may be
helpful to use the adjusted R^ as a corrected index of
segregation:
K^-J)
Srl{N-\)
To
see
how
this
index corrects for finite firm size, suppose that a proportion y of the
population matches with other workers of exactly the
matches randomly. Let
generating workers'
randomly
will
o/
converge to
the variance of
skill,
and that a proportion
-
1
y
denote the variance of the underlying stochastic process
As noted above,
skills.
same
OjVK
as A^—>«>,
s^^ for
where
the set of firms
K is
the
whose workers
number of workers per
are
drawn
Since
firm.
wages within firms plus the variance of mean wages between firms must
equal the total variance of wages, the variance of wages within the set of firms with randomly
drawn workers converges
consisting of perfectly-matched workers
i\-y)il-\/K)Gj^.
Note
Since the variance of wages within the firms
to {\-\/K)c^^.
that in a
is
sample of
Substituting these expressions for
sj and
zero,
sj
for the set of all firms converges to
A^ workers, the expectation of
Sj^ into (21)
and using the
5/
is
fact that
{N-\)aj/N.
K = N/J,
that Pj converges to y, which, as desired, is the proportion of the population that
with others of the same
N
is
small, and even
skill.
when
Simulations suggest that the expectation of
firms vary in size.
p.. is
we
see
matches
y even when
18
Two
properties of
correspond to increases in
of
Given
skill affect p.
will
are worth noting.
p^.
p^.
that
In Section HI,
we
First,
if
N and J constant,
we demonstrate how changes
increases in p
in the distribution
hold the number of workers per firm constant, changes
always be of the same sign as changes in
negative
holding
p.
Second, the adjusted /?-squared
the correlation between the skills of workers in the
same firm
in p^
may be
than that
is less
expected from random assignment of workers to firms.
II.B.
Trends
in
Segregation
Wages, experience, and worker
classification are all imperfect indicators of skill, but
together they paint a consistent picture of
wages among workers
compensating
in the
same firm may be due
differentials, efficiency
However, there
is little
increasing segregation.
to segregation
thus
is
skill
or to
reason to think these latter factors increased in importance over the
and
Britain.
Worker
example, into production and non-production worker categories)
somewhat problematic
as of the worker.
by
wages, or pressures for equity within the firm.
period, especially given the decline of unions in the United States
classification (for
Positive correlation of
as a
measure of
skill,
since
it
reflects characteristics of the job, as well
Nonetheless, non-production worker status
presumably correlated with
skill.
also
is
is
correlated with wages, and
Herman, Bound, and Machin [1994] show
that in the
United States, non-production workers are more highly educated than production workers and
much more
likely to
Where
be managers or professionals.
possible,
we measure
Plant data are likely to be
indicators of skill at plant level rather than firm level.
more economically meaningful than firm
data, since
it is
not clear
19
that anything of
economic importance changes
if
and McDonald's. However, even the plant data
company
a holding
may
acquires both Microsoft
be misleading since they include only
workers employed by the firm owning the plant and therefore indicate increased segregation
when
The American and
firms contract out work.'
British data are at the plant level, whereas
the French data are at the level of the business unit,
American firm than
The
is
conceptually closer to the
to the plant.
best available data
performed by
which
INSEE
in
come from
France.
The Wage
Structure Surveys (ESS),
1986 and 1992, examined a sample of 10,719 business units and
318,332 workers
in 1986,
was drawn from
all
and 3,803 business units and 42,783 workers
in 1992.^*
The sample
business units with at least ten employees, excluding agriculture,
transportation, telecommunications,
and services to individuals. Table
I,
drawn from
Kramarz, Lollivier, and Pele [1996], shows that between 1986 and 1992, the correlation of
log
wages among workers
in the
same business
unit rose
from 0.36
to 0.44, the correlation of
experience rose from 0.11 to 0.16 and the correlation of seniority rose from 0.24 to 0.31.
Segregation according to each of the six categories of worker classification rose as well.
example, unskilled blue-collar workers were more likely to be grouped together
were engineers, professionals, and managers. Worker
'
classification is likely to
In the U.S., increases in the importance of pensions,
requiring
all
workers to be treated equally,
may have
in firms, as
be a good
combined with pension laws
provided a small impetus toward such
contracting out over the period studied, but any increase in correlation from this source
likely to
due
For
have been more than counter-balanced by declines
weakening of unions.
in
wage
is
correlation within plants
to the
^
The
ratio of
sampled per firm.
workers to business units
is
smaller in 1992 because fewer workers were
20
measure of
mention
skill in
that
France, because
it
is
governed by
fairly precise rules.
We
should also
Kramarz, Lollivier, and Pele found that besides increasing segregation, the
period from 1986 to 1992 was also a period of rising inequality.
Table
I:
Segregation Indexes for Various Indicators of Skill in France
(95% confidence
interval in brackets.)
1992
1986
Log Wages
Experience (Age)
Seniority
Unskilled Blue-Collar Worker
Skilled Blue-Collar
dummy
Worker dummy
Foreman dummy
Clerk
dummy
Technician
dummy
Engineer, Professional, or
Source:
Manager dummy
0.36
0.44
[0.3554; 0.3663]
[0.4316; 0.4518]
0.11
0.16
[0.1081; 0.1127]
[0.1554; 0.1665]
0.24
0.31
[0.2364; 0.2450]
[0.3027; 0.3203]
0.31
0.34
[0.3058; 0.3159]
[0.3324; 0.3508]
0.23
0.26
[0.2265; 0.2349]
[0.2535; 0.2693]
0.08
0.10
[0.0785; 0.0820]
[0.0970; 0.1044]
0.08
0.15
[0.0785; 0.0820]
[0.1457; 0.1568]
0.14
0.17
[0.1376; 0.1433]
[0.1652; 0.1768]
0.14
0.21
[0.1376; 0.1433]
[0.2044; 0.2180]
Kramarz, Lollivier, and Pele [1996]
In the United States, data linking workers and their plants are hard to
come
by.
However, segregation of workers by wage can be calculated using the Davis and Haltiwanger
[1991] estimates of the variance of wages
among
all
workers from the
CPS and
variance of average wages between plants from the Census of Manufactures.
taking the ratio of estimates from two different data sources
makes
of the
Unfortunately,
the measure of segregation
21
Moreover, data are available only for manufacturing workers and reliable data only
noisier.
for production workers within manufacturing.
the
economy, and services such
However, since services are a growing share of
as finance, restaurants,
their exclusion is likely to lead to underestimation of
Table
EI
reports between-
and law are highly segregated by
any increase
correlation
in sorting.
and within-plant variances of production worker wages
manufacturing,^ taken from Davis and Haltiwanger [1991].
The
Both within-plant
plant.*
and between-plant variances of wages have increased, reflecting the widely documented
skills
However,
if
there
had been no realignment of the way
were grouped within firms, both variances would have increased
fact, the
in
final line reports the
between wages of production workers within the same
in inequality in recent years.
skill,
in the
same
rise
relative
ratio.
In
between-plant variance increased proportionately more than the within-plant variance.
Thus, although the data are imperfect, and estimates for 1984 seem particularly suspect, the
correlation
between wages of production workers within the same firm increased from 1975
shown
to 1986, as
^It is
in the final row.'
not possible to obtain reliable data on non-production workers because of
differences in the sampling frames between the
two surveys used by Davis and Haltiwanger.
However, the available data show no trend in segregation among non-production workers.
The segregation index was .447 in 1975 and .443 in 1986.
*
Note
that these correlations are higher than those reported in France,
include only production workers, whereas the French data include
of production workers' wages within plants
'Given the large sample
estimates
is
is
much
all
because these data
workers.
smaller than that of
all
The variance
workers.
sizes, sampling error is likely to be small, and noise in these
probably due to other factors, such as business-cycle effects.
22
Table
Sorting of Manufacturing Production Workers by Wages:
II:
U.S. Plants
1975
1977
1979
1982
1984
1986
s^
9.00
9.77
10.78
11.81
11.96
13.84
.2
2.13
2.37
2.19
1.82
1.00
2.76
s^
6.86
7.40
8.58
9.99
11.0
11.19
p
0.76
0.76
0.80
0.85
0.92
0.80
Source: Davis and Haltiwanger [1991], p calculated as s^Vs/.
Sorting by production worker/non-production worker status also increased.
shows
that the correlation of a
dummy
American manufacturing workers
variable for non-production worker status
in the
same
plant increased
from 0.195
to
Table HI
among
0.228 between
1976 and 1987.^
Table
III:
Sorting by Production Worker/Non-Production
Worker
Status: U.S. Plants
Year
1975
1977
1980
1981
1983
1984
1985
1986
1987
hfPW/Emp
0.261
0.260
0.282
0.287
0.302
0.291
0.304
0.314
0.310
P
0.195
0.192
0.196
0.199
0.215
0.218
0.225
0.231
0.228
Source: Census of
Manu factures
British evidence also
shows increased segregation by worker
classification.'
and 1990 Workplace Industrial Relations Surveys (WIRS) divide employment
British establishments into
thank
Andrew Bernard, Brad
that
n
is
non-manual employment
then
is
Jensen, and the U.S. Bureau of the Census for
Data for 1976 and 1977 may not be strictly
from other years, since a different sampling methodology was used by
supplying the information in Tables
comparable with
each of 402
manual and non-manual components. Manual employment
further subdivided into unskilled, semi-skilled, and skilled; and
*We
in
The 1984
and
III.
the census in those years.
^e thank Stephen Machin for providing the information
in
Table IV.
23
subdivided into
(iii)
(i)
clerical, secretarial,
staff; (ii)
supervisors and foremen;
junior technical/professional; (iv) senior technical/professional, and (v) middle/senior
management. Table IV shows
example, the correlation of a
the
and administrative
that sorting increased in
dummy
same firm increased from 0.058
most worker
classifications.
variable for middle/senior managers between workers in
in
1984
to 0.077 in 1990.
Sorting of senior
professional/technical staff also increased, contributing to an increase in sorting
upper-grade, non-manual workers.
and among
all
States
among
skilled
unskilled manual workers.
and France, indicates
than just in the
There was a
rise in segregation
among
among
all
clerical workers,
non-manual workers taken as a whole. Within the category of manual
workers, sorting declined
among
For
economy
Note
and semi-skilled manual workers, but increased
that the British evidence, unlike that
that segregation increased
among
from the United
a fixed sample of plants rather
as a whole.
Section HI explores
how
segregation and wages are jointly determined given the
technology and the distribution of
skill.
24
Table IV: Sorting
in
402 British Establishments, 1984-1990
Mean
1984
Mean
P
1990
1984
P
1990
Managers
0.074
0.080
0.058
0.077
Senior. Prof/Tech
0.072
0.083
0.230
0.276
Junior Prof/Tech
0.095
0.090
0.294.
0.301
Ttl.Upper Gr. Non-Manual
0.241
0.253
0.299
0.323
Clerical
0.233
0.222
0.273
0.322
0.511
0.525
0.446
0.469
0.162
0.163
0.341
0.323
0.143
0.132
0.391
0.367
0.185
0.180
0.415
0.444
0.489
0.475
0.446
0.469
Total
Non-Manual
Skilled
Manual
Semi-skilled
Manual
Unskilled Manual
Total
Manual
25
III.
Theory
Let us extend the analysis of Section
point skill-distributions
~
to
more general
values 6
respectively.
assume
e {0,1
=
and 6 =
Let
|i(e)
}.
1
it
maintain the hypotheses of
6, let p{n;Q)
I that, if
mode
and
let
be the number of workers of
n
,
skill-level n.
efficient for
^(9) = I,np{n;Q)/ Zp(n;e)
•
We
two workers'
skill-levels
was
greater than
A
each of them to match with another of his or her
match with one another (cross-matching). Let
less than An'(l); let Mj
n^ be the smallest integer
0, 1, p(n;Q) is "sufficiently
is,
of the distribution.
the ratio of
skill-level (self-matching), rather than
(3/2)"^n'(l);
and
n
correspond to the "pre-shift" and "post-shift" distributions,
was more
be the smallest integer no
6 =
For given
that |i(6) also equals n(Q), the
2/(1+5"^), then
own
We
three-
(6).
be the mean of p(;e). That
Recall from Section
=
which was conducted for two- and
consider a pair of distributions of types on the integers between
parametrized by
The
~
distributions.
competitive markets and production function
We
I
no
peaked" near the
be the smallest integer no
less than
mode
n*(6).
^n
*(1)
More
•
n,
less than
We
assume
that, for
precisely, "sufficiently
26
peaked" means
n'(0)-l
p(n\Q);e) >
(22)
n'iQVA
pin'iQ) +
J)
p(n';e),
j^
n'=n'(e)+l
n'=An'iQ)
An'(B)
(23)
J]
< pin;Q), for aU n e [n„n'(e)]
;,(7,';e)
,
and
ti/A
(24)
;?(/i;e)
>
p(n';e), for aU ne[n*(e),n3].
j^
n'=V^'{e)
As
will
become
clear below, condition (22),
computations,'" ensures that at least
skill in
condition, this will serve to tie
down
level
imposed mainly
skill will
competitive equilibrium.
their
wage (which
is
why
to simplify the
have to self-match,
between
n,
and n'(6) and between n(Q) and n^
level
and the dispersion of
consider a shift in the distribution of
skill in
'"Condition (22) can be relaxed considerably.
correspondingly more complicated.
will also
skill
(22)
is
so convenient to
skill
match with modal workers.
have increased
in recent decades.
which, as 9 increases from
to
1
,
However, the argument becomes
s'
i.e.,
Given the zero-profit
Conditions (23) and (24) imply that in equilibrium, some workers of every
Both the mean
We
is
some workers of modal
match with others of the same
impose).
which
(i)
the
27
mean
rises, i.e.,
„•(!) = n*(0)+l;
(25)
the lower and upper tails get fatter symmetrically but retain their "relative shapes,"
and
(ii)
i.e.,
there exists
ri
>
1
such that"
^^"'
(26)
^
= a, for
all
n^n,' and n^n,.
^
Pin;0)
III.A.
Wages
We
first
show
sufficiently close to
the lower
tail
Proposition
"We
1
that if the distributions /?(•;•) are "tight"
n
—
-- i.e., if
n
is
then an increase in 9 promotes equality in the sense that wages in
of the distribution rise and wages in the upper
(The Trickle-Down Principle):
adopt (26) not because
presentation of our results.
enough
it is
of the distribution
we
worker
wish to argue
in the
fall.
that /?(;•) satisfies (22)-(25).
necessarily realistic, but because
In particular,
sufficiently dispersed, then a typical
Suppose
tail
it
Given
facilitates the
that, if the skill-distribution is
lower end of the distribution will see his
wage fall and one in the upper end of the distribution will experience a wage increase when
the mean skill-level rises. We demonstrate this (Proposition 2) by showing that the average
wages
in the
lower and upper
tails,
respectively, fall
and increase. But a change
average wage would not be a good measure of what happens to a given worker
distribution of skills also changed.
qualitative results
do not depend on
Hence, (26) makes comparisons
it.
in the
if
the relative
easier, although
our
28
n
,
if
n
is
near enough
so that
n
« > (2/3)"^ n
•
then
w(/i;0) < w(n;l), for aU 7i<n*(0),
(27)
and
(28)
w(n;0) > w(n;l),
We
Lemma
1:
establish Proposition
For any
n', if
n
is
1
aU /i>
for
n*<l)-
through a series of Lemmas.
such that
n<An\
(29)
then workers of skills n and n' cannot be matched in equilibrium.
Proof
of
:
As demonstrated
skill n'
matched
Lemma
Proof:
when they
(i.e.,
2:
From
in Section
I,
total
are cross-matched
output for two workers of
(i.e.,
2nn'^
) is
be
skill
less than that
n and two workers
when
they are self-
n^+n'^) if n/n'<A.
Q.E.D.
w(n'(6);e) = {Ti*{Q)fl2.
(22),
even
if all
matched with workers of
workers of
skill n*(e), there
all skills
would
/iVn'(e) between An*(e) and n*(0)/A were
still
be some n*(e)-workers
left over.
29
Moreover, from
n{Q)
Lemma
But
,
no
skill
outside the interval [An'(Q), n(Q)/A] can be matched with
Therefore, at least
in equilibrium.
equilibrium.
1
this
means
some n*(6)-workers must be self-matched
that they earn 1/2 the output of a firm with
in
two workers of
skill
Q.E.D.
n\Q).
Lemma
If n
3:
<
n*(0) and n'
>
n*(6), then
workers of type n and n' cannot be matched
in
equilibrium.
Proof
:
Total output with one («,«') firm and one {n,n) firm
'^
Total output with the assignments {n,n) and (n',n')
/i*=n'(9).
prove the lemma,
we need
(30)
Defining z = n/n', and
z'
A
^
(31)
Since n'
Lemma
Proof
:
> n,z' >
4:
If n, n'
1,
that
T
>
Y.
Note
7=
n'^n
isY* = n
+
*^n
n^, where
+
n'^n.
To
that
Y'-Y
= n*^-n + n' ^n' - n' ^n -
n'ln,
we
n*\
obtain
^^ =z*z'^
-
z'h-l
n<n,z<l.
Thus the
=
(2'
^-1)(1
-z).
Q.E.D.
result follows.
e (An'(e), n{Q)), then n and n' cannot be matched in equilibrium.
'^We thank Charles
we
show
and since
Assume without
than the ones
=
to
is
loss of generality that n'
Morcom
>
for the proofs of
originally developed.
n.
As
Lemmas
in the
3, 4,
proof of
and
6,
Lemma
which
3,
we
are simpler
30
show
by the firms
that total production
firms (n
,n')
and
(n',n')(i.e.,
Y
(«,n')
Here,
').
Y=
and (n,n)
n'^n
+ n^
(i.e.,
JO
is
less than that
by the
and Y' = n'^n + n^n'. This
.
implies that
Y-Y
(32)
= n*^n + n'h' - n'^n -
Dividing the right-hand side of (32) by «*^
(33)
z ^
Now,
z'
-
z'h
point
= zil -z'2) + z'-l = (1 -z')(z(l +z')-l).
increasing in both z and
it is
=
non-negative at the bottom of the range: z
Lemma
5:
If
and so
w(/i;0)
Proof
If
n e (An'(e),
/i'(9)),
=
{n\<d)fll.
n(n*(e))^
n e (An'(9),
-
n'(9)), then
then
some n-workers
from (23) and
exclusively with skills no greater than An(Q).
with
skills greater
than
An(Q) and
than n*(9) and, from
less than n'(9).
matched with /i*(9)-workers
nin'mf
-
z'
=
A.
z',
But
so
we
at this
Q.E.D.
vanishes.
it
:
it is
1
obtain
n'<n. As for z(l+z')-\,
l-z' is positive, since
need only check that
-
we
n'\
(«'(9))V 2.
Lemma
Lemma
But from
4, they
Thus by process of
in equilibrium.
matched with «'(0)
are
1
,
the n-workers cannot be
Lemma
3,
elimination,
2,
we
matched
they cannot be matched
cannot be matched with
From Lemma
in equilibrium
skills greater
some n-workers must be
conclude that w(/i;9)
Q.E.D.
=
31
Lemma
6:
If
n,
J2n
n'e («'(9),
*(9)),
then n and n' cannot be matched in equilibrium.
Proof Assume without loss of generality that n' >
:
assignments («',«) and
(/i*,n'),
where
n*=/j*(6).
n.
Then
=
ri^n
+
production with the
Let
n'\*.
Y
denote
Y'
-
Yhy
obtain
^2^ ^.2_
(34)
Now,
Y'
total
Then Y = n'^n+n^. Dividing
production with the assignments {n,n') and {n,n).
n^ we
Let Y' denote
z-1 is positive.
minimum when
attains a
Lemma
As
7: If
n e {n*
^.2
for z+1
z
=
- z'^, it is
and
1
^ ^,2 (i_^)^ 22_i ^ (2-l)(2 + l_2'2),
^_i
z'=
(9), \/2"« '(9)).
increasing in z and decreasing in
yjl,
^t
which point
some ^-workers
are
it
Hence
z'.
Q.E.D.
vanishes.
matched with n'(9)
it
in
equilibrium
and so
M;(n;e) =/i-(0)n2 - Oi:(?))!.
(35)
Proof
:
If
n e(n*(B), yjln
matched exclusively with
matched with
skills less
*(9)),
^^en from (24) and
skills greater
than n{<d).
Lemma
than yf2n '(9).
And, from
Lemma
1,
the n-workers cannot be
From Lemma
6, they
3,
they cannot be
cannot be matched with
skills
32
greater than «'(9) and less than or equal to
matched with «'(e)-workers
yj2n
Thus some n-workers must be
'(9).
From Lemma
in equilibrium.
2,
we deduce
(35).
Q.E.D.
We
all
can
now
establish Proposition
workers will have
equilibrium.
better
at least
Hence, an increase
managers and therefore
that their
.
Given a
than An'(O).
skill greater
below the mean,
skill level
1
Lemma
some workers
in the
mean
sufficiently tight distribution of skills,
skill
that their firms will
will
5 implies, therefore, that, for each
have managers of mean
implies that low-skill workers will have
produce more output. Because the wage
managers can obtain from self-matching increases by
less than this increase in
output, the assistants will receive part of the increased value of their matches.
wages of the low-skilled
greater than
mean,
mean
at least
skill
^n
*(0).
will rise.
Lemma
some workers
will
produces an increase
matching for those of mean
A
tight distribution also
have assistants of mean
1
:
skill will
Suppose
skill in
output for such matches.
increase by
more than
high-skill workers will fall in response to this shift in
Proof of Proposition
means
that
7 implies, therefore, that for each
in
fi
is
skill in
near enough
mean
n
That
is,
no worker has
skill level
equilibrium.
skill
above the
An
increase in
However, the wage from
output,
skill.
so that
the
self-
and hence wages for
More
formally,
we
n > (2/3)"^«.
have:
Because
33
(2/3)"^
Lemma
>
3/4
> A, we have
5 implies that w{n;Q)
>
Since n
3«*(e)/2).
„
Because
and
<
v/T,
> «
(3/2)"^
n >
n*(e).
Lemma
7 implies that w(n;e)
(36), this derivative is negative,
n<
and so w{n;Q)
n'(e).
-
increasing in 6.
is
|/i'(0)<V2n-(e),
=
rfw(n;e)/dn'(e) = n^ -
(37)
if
Thus dw(n;e)/dn'(e) = n\Q){2n
positive,
n < n ^
so, if
From
(/2'(e))V2.
-
is
(36)
And
= n{n\e)f
3n*(e)/4, this derivative
'(0)
Hence,
n > {2l2,y^h > 3«'(e)/4 > An\Q).
and so w(«;G)
«'(0)«^
-
-(n\eyf
is
Hence
(n'(e))V2.
.
decreasing in 9.
Q.E.D.
We now
turn to the case in
case, an increase 9
greater than
An '(9)
greater than ni(l).
works a gainst
Lemma
8 below will
matched
Lemmas
/ii(0)-
and
equality.
(recall that n,=/i,(l)),
n,(0)- or n,(l)- worker, is
3, 4,
which the distributions p(;) are more
8,
any
in
To
and
see
let S,
show
that
this, let Mi(9)
consist of
all
any worker
diffuse.
In that
be the smallest integer
workers with
in 5,,
skills
no
except possibly for an
equilibrium with another worker in
5,.
Moreover, from
or n,(l)- worker not matched with another worker in 5,
matched with an n*(9)-worker. Hence, the average wage of workers
in S, equals the
is
sum
of
34
(i)
and
the total output of matches between workers in 5,
workers matched with «'(6)-workers,
all
in equilibrium there are
of calculating
(ii),
from an n,(l)-n*(9) match. Moreover,
no matches between
we can assume
(ii)
must
in S,
Next consider the
integer
no
workers
in 52-
nj
self-matching increases
is
(i).
workers of
skill
in S2,
no
(26), the
Formally,
big enough so that, for n >
more slowly than
the output
Proposition 2:
average wage in S2 also
we can
Suppose
1
implies that
purpose
Furthermore, from
Hence, the
conclude that the average
less than n^ (the smallest
and
(ii)
1X2,
the
from an
in S2 equals the
the
rise.
We
sum
wages of workers
of
in S2
wage of an «'(9)-worker from
n'{<d)-n
9.
match.
Hence, the wage
Moreover, the argument of
Hence, the wage of an n-
conclude that
(ii)
must
rise
rises.
state:
that the distributions
/?(•;•)
(i)
divided by the number of
implies that, for n'e (n'(9),n,), vv(n';9) falls with 9.
1
from
We
skills in the interval [«'(9),/Z2-l], all
worker (n > n^ matched with an n'-worker must
and, so
increases.
The average wage of workers
of an «-worker matched with an /j'(0)-worker increases with
Proposition
But
S,.
so, for the
does not change as 9 increases.
from matches between workers
Now,
Lemma
and «'(l)-workers, and
that h'('«,(0);9) also falls as
set S2 consisting of
matched with workers with
and n,(l)-
fall.
less than (3/2)"^n'(l)).
the total output
ni(0)-
cannot be made up for by an increase in
wage of workers
/i,(0)-
wage of an n'(6)-worker from self-matching
(26), the relative distribution of skills in 5,
decrease in
wages of
divided by the number of workers in
wf«,(l);6) falls as 6 increases because the
increases faster than the output
the
(ii)
satisfy (22)-(26)
and are
sufficiently
with 9
35
diffuse so that
increases
from
and n
n < n
to
1
.
More
<
Then
n.
the average
<
E
n=a
in the tail
above ^2 increases with
£
w(«;l);7(n;l)
"'"^
>
(39)
E
8:
=
is
/7(/i;0)
n^tij
is
given in Appendix
matched with a worker of
2.
An(Q)
skill
n'
(so that n,(l)
> An(Q)
=
w,).
If
n <
in equilibrium, then
n,(e).
The
of
.
Let «,(6) be the smallest integer no less than
An'(6) and an «-worker
n'
w(n;0);7(n;0)
E
/7(«;1)
The formal proof of Proposition 2
0:
"'"^
i»=ii2
Lemma
n, falls as 9
-^li5-
B=fl
wage
below
Y. >v(«;0)p(n;0)
^
Similarly, the average
in the tail
we have
precisely,
X; win;l)pin;l)
(38)
wage
skill
proof,
which
n < An{Q).
one of greater
skill.
It
is in
shows
Appendix n,
calculates the profit for a firm hiring an assistant
that profit will
be higher with a manager of
skill /z,(9)
than with
36
Segregation
We now
dispersion accompanied by a rise in the
mean
segregation, provided that the shift in the
To make
integers alone.
=
/
1, 2,
this last
proviso formal,
we
is
will
Instead fix (rational) numbers
will
number of workers of
the
is
small
when 9
increases
skill nir.
from
to
no longer work with
b and b with
that
< ^ < ^
•
For each integer
h = n'lr and ^ = n ^ For each
all
integers
on the
distributions
n e
\n',
r,
h%
we
p
(n;6)
From
(25), notice that, if r is big, the shift in the
a>
and P <
mean
1
1
such
that, for all r, {pji\0), /?X;1))
and
(40)
pjji-^ 1)
Condition (40) says that the
shift
< p;7>;0),
from 6 =
uniformly in the middle of the distribution.
this
an increase in the index of
1.
Let us suppose that there exist
satisfies (22)-(26)
that an increase in skill
not too large relative to that of dispersion.
consider a pair of distributions (pX;0), pji;\)), where for
is
show
skill-level leads to
mean
choose integers n' and n ^ such
...,
We
turn to the issue of segregation.
for
to 9
=
aU nE{n^,n^).
1
reduces probability more-or-less
Henceforth,
we
will drop the subscript r
when
does not create ambiguity.
For
n, ri
« -workers. Let
and
9, let zin, n'; 9)
|i(n, n')
=
(«
+
n')l2.
be the equilibrium number of matches between n- and
Then, from (19), the index of segregation can be
37
expressed as
(41)
^^
p(e) =
"-=^^
^-^
Y:^i\i(n,n')-\imfzin,n';Q)
+££(«n=n
n=B n'=n
Suppose
Proposition 3:
and
(40).
To
and
(40).
that, for
each r
=
1,2,...,
.
\iin,n'))h(n,n';Q)
»'=fl
the pair (Pri;0), pX;l))satisfies (23)-(26)
Then, for r sufficiently big, p(l)>p(0).
understand Proposition
The
matters, let us
3,
effect of such a shift
assume
for the
consider a shift in the skill-distribution satisfying (26)
is
moment
to
move
probability weight to the
that this shift
tails.
does not affect the mean.
To
simplify
Now, because
of the complementarity between tasks, a firm will not hire workers that are too different in
skill-level.
Thus,
if
the shift pushes
push more firms' means
~
the numerator
more workers
|i(n,«') into the tails.
and also the
first
into the tails of the distribution,
mean p(9) does
~
rises
+
on whether the
middle of the distribution
skill-levels are in the tails or the
much by
the shift
—
since
it is
with the
not change too much.)
the dispersion of skill within a firm, (n-ii(n,n')f
affected very
will
This means that the dispersion of firms' means
term of the denominator of (41)
(This argument assumes that the overall
it
(n'-|i(n,n'))^does not
shift.
In contrast,
depend very much
—
and thus
is
not
determined mainly by the trade-off between task-
complementarity (promoting matching of similar skill-levels) and differential sensitivity to
38
skill
(enhancing matching of different
skill-levels).
Hence
the second term in the
denominator of (41) remains about the same. The over-all effect of the
The formal proof can be found
increase in p.
in
A
skill, as in
Roy
Second, the model assumes that
[1951].
good, so that there
no
is
First,
we
model would allow
fuller
an
Appendix n.
Several limitations of our model are worth noting.
can be reduced to a single dimension.
shift, therefore, is
all
treat all skills as if they
for multi-dimensional
workers are producing a single
between workers through the product market. To the
interaction
extent that high- and low-skill workers specialize in production of different goods, increases
in mean-skill will tend to
reduce the price of skill-intensive goods, and thus will be more
likely to increase relative
wages of low-skill workers. This would
described in Proposition
partially offset the effect
2.
Second, the production function allows no substitutability between quantity and quality
of workers. The number of workers
variable.
However, many of the
in a firm is taken as fixed, rather than as a choice
results of this
model would presumably hold
as partial
equilibrium results in models that allowed for choice of quantity of workers, as long as
quality and quantity
workers specialize
were not perfect
producing different goods,
in
mean-skill level of the
There
is
economy
some gap between
linear
wa ges
.
and endogenous
Under
it
the extent that high
becomes more
will increase relative
to increased segregation of workers
of workers by
To
substitutes.
wages of
and low-skill
likely that an increase in the
low-skill workers.
the theoretical finding that increased skill-dispersion leads
by
skill
and the empirical finding of increased segregation
the model, the relationship between
to the skill-distribution.
wages and
For a high dispersion of
skill is
skill,
non-
there will be a
39
strong correlation between the wages of workers in the
of
skill,
there will be a
wages of workers
weak
in the
correlation.
same firm
same
may
low dispersion
Thus, on average, the correlation between the
However, the
will increase with skill-dispersion.
correlation need not increase monotonically, because over
dispersion
firm, and, for a
not change the pattern of matching by
schedule so as to reduce the correlation of wages.
some
skill,
but
ranges, small increases in
may change
Thus there may be
the
wage
segregation by wages could increase without a concomitant rise in segregation by
contrasts with conventional efficiency unit
wages within firms
will
models
in
which an increase
always correspond to an increase
which
special cases in
skill.
in correlation
in the correlation
of
This
of
within
skills
firms.
IV. Skill Dispersion and Segregation across U.S. States
We
next argue
that, as
proposition 3 implies, U.S. states with greater variance of
education are more segregated by education.
Casual empiricism suggests the same
is
true for
countries.
Kremer and Troske [1995]
create segregation indices
by
state
based on several
indicators of skill using the Worker-Establishment Characteristics Database
WECD
(WECD). The
matches 199,557 manufacturing workers to 16,144 manufacturing establishments
[Troske, 1994].
It
was created by linking individual data from the 1990 census with
establishment data from the Longitudinal Research Database, using detailed location and
industry information.
It
should not be taken as representative of American firms, since the
40
match
are
rate varies
computed
dummies
by industry, portion of the country, and plant
as the adjusted R^
in a state.
from regressing worker
The sample was
characteristics
in
Table V.
A
set of firm
states.
States with greater variance of education'^ tend to be
shown
on a
indices
which data on more than two
restricted to firms for
workers were available, which led to the exclusion of four
education, as
The segregation
size.
more highly segregated by
regression of the correlation index on the variance of
education, with observations heteroscedasticity weighted by the square root of the sample
yielded a
size,
t-statistic
Although the model may be a good description of how
of 5.47.
workers sort into firms by education,
it is
not a good description of
how
regression of segregation by age within plants on the variance of age
state yielded a t-statistic
of -1.69.
relationship between segregation
the log
There was a positive, but not
and the variance of
wages or the predicted wage given
" The variance of worker
skill
among workers
characteristics in each state
it
is
in the
the skill indicator
was
either
race.'''
was calculated using the Sample
based on a large number of observations
and should be representative of manufacturing workers
in the state as a
whole.
predicted wage is calculated from a country-wide regression of log wages on age,
and dummies for female, married, black, female* married, female*black, and various
'''The
education levels.
A
statistically significant,
age, education, sex, marital status, and
Detail File from the l-in-6 census sample, so
age^,
when
they sort by age.
41
Table V: Effect of Variance of
Regression of p^ for
skill indicator
Heteroscedasticity-weighted using the
Education
Dependent
Skill
on Segregation by
Skill: U.S. States
on the variance of the indicator
number of firms per
Age
in the state.
(t-statistics in
state,
parentheses)
Log Wage
Predicted
Wage
Variable
0.0014
0.3037
0.117
0.218
(0.056)
(2.51)
(1.36)
(1.802)
Variance of
0.0257
-0.0017
0.444
0.431
Variable
(5.471)
(-1.687)
(0.591)
(0.948)
N
46
46
46
46
R^
0.3995
0.0595
0.0077
0.0196
Constant
Kremer and Troske [1995]
Source:
Although comparable data are not available for rigorous comparisons across countries,
the
model
is
consistent with the widespread view that developing countries, with wide skill-
dispersion, are prone to "dualism."
The model suggests
that if skill-differentials are
comparatively small, as in highly traditional societies'^ or
and low-skill
range of
will
skills,
mix together within
in the
paid workers in the traditional sector.
sectors of developing countries'
and highly paid
flight attendants
seems
developing economies with a wide
in
In fact, even the less skill-intensive tasks in the formal
economies are often performed by workers who are highly
comparison to workers
on Thai Air are sought
their estates,
after
in the informal sector.
by college graduates
by the educated
in
For example, jobs as
China, and jobs as
in Thailand.
pre-modem agricultural societies, such as medieval Europe,
many of the better-educated workers helped landlords
and thus worked in the same "firms" with serfs or peasants.
likely that in
Russia under serfdom, or colonial India,
manage
workers of high-
advanced sector will earn more than relatively high-
clerks in international hotels are highly prized
It
in
societies,
such as India or Brazil, low- and high-skill people will work in separate firms,
and relatively low-paid workers
skilled
However,
firms.
modem
42
V. Interpreting Recent Changes in
Under our model,
Wages and Segregation
the recent simultaneous increases in inequality and segregation
could be due either to skill-biased technological change or a shift in the observable
distribution.
Both hypotheses are consistent with the
skill-
data, although the latter provides a
more
parsimonious explanation.
On
the technological front, the
model
predicts that both segregation and inequality
should increase following change in which the effective
There
is
some evidence
skill
that recent technological progress has
Kahn and Lim [1994] argue
that total factor productivity
of high-skill workers improves.
been of
this type.
growth across American industries
has been proportional to the fraction of skilled workers in the industry.
Bound, and Griliches [1994] find evidence
placed a premium on high-skill workers.
attractive to leave the firms
the relative
wage of
complementary
that adoption of
new
Moreover, Berman,
technologies
may have
This means that although the old technologies in
which high- and low-skill workers mix are
more
For example,
still
high-skill workers
available,
now
find
it
and form firms with other high-skill workers. This reduces
low-skill workers because firms hiring
them must pay more
to attract
high-skill workers.
Our model, however, does not
require appeal to largely unobserved skill-biased
technological change to account for changes in wages and segregation;
explained through observed changes in the distribution of
skill in
the data can be
the United States and the
United Kingdom.
There has been a substantial increase
in mean-skill in the U.S.
over recent decades.
43
From
1960-1 to 1987-88, the flow of bachelor's degrees as a percentage of the labor force
increased 1.58 times.
The flow of
professional degrees increased 1.65 times, masters' degrees
2.14 times, and doctorates 1.87 times. [U.S. Department of Education, 1990, Table 220, cited
Bureau of the Census, 1992]. Most of the increase
in Ehrenberg, 1992;
in the
flow took place
during the earlier part of the period, but the stock increased throughout the period, as less
educated people
the labor force.
left
Evidence that the increase
accompanied by a moderate
Pierce [1993].
They
average
in the
rise in dispersion
partition increased
wage
of
skill in the
skill is
United States has been
provided by Juhn, Murphy, and
inequality
from 1964
to
1988 into increases
in
the dispersion of observable indicators of skill (education and experience), increases in returns
to those observable skills,
and increases
Somewhat more than 20% of
in inequality
due
to unobservable factors.'*
the growth in inequality due to observable factors can be
attributed directly to increased dispersion of skill, holding the
We
do not have data on the
wage
."
skill-distribution in France, but British data also indicate
an increased mean and dispersion of
'*They measure
wage schedule constant
skill.
Schmitt [1992] reports the distribution of the
inequality by the gap between log
wages of workers
at the
10th and
90th percentiles of the wage distribution.
'^The single biggest source of increased inequality
within education-experience groups.
is
the increase in dispersion of
This could be due to increased variance in
unobservable to the econometrician or to increased returns to these unobservable
wages
skills that are
skills.
If
one assumes that the decomposition into quantity and price changes is the same for
unobserved as for observed variables, then slightly more than 20 percent of the total increase
in inequality is accounted for by the direct impact of widening of the skill-distribution.
Under the assumption that the entire change in the residual is due to changes in prices, rather
than quantities, of unobserved skill, widening of the skill-distribution explains just under 10
percent of the increased inequality.
44
British workforce
the General
by educational qualifications
1978-80 and 1986-88, based on data from
in
Household Survey. In 1974-76 the median worker had no qualifications beyond
primary education.
By
vocational education.
1986-88, the median worker had
We
calculated the variance of expected log
education in both periods, holding constant the
of the observable
skill distribution.'*
wage
in
wages conditional on
The variance of
skill,
as
measured by the variance of
20% from
0.0410
in
1978-80 to
1986-88.
Thus,
at least in the
and dispersion of the
U.S and
proposition 2 and
more
3).
Britain, there
skill-distribution.
in the wage-distribution, but
require a
or lower-middle
schedule, as a measure of the variance
expected log wages conditional on education, increased
0.0488
some 0-levels
Of
have been significant changes
in the
These changes have not been as profound
such a discrepancy
is
as the shift
from our theory (see
to be expected
course, to assess the predictions of our
mean
model
quantitatively
would
detailed and richer formulation of the production function than that of section
m.
One
issue that arises in interpreting the empirical results
relationship between changes in the skill-distribution
and changes
there are search costs, changes in the skill-distribution
entrants to the labor force
'*
in the skill-distribution
The wage schedule
may
the timing of the
in
worker-matching.
new
who were
Matching patterns among workers may thus respond
only with a lag.
If
primarily affect matching of
and of the unemployed, rather than causing workers
already matched to switch jobs.
changes
is
to
Juhn, Murphy, and Pierce [1993] find the
is based on Schmitt's estimates of differences
group for male workers with twenty years of experience in 1986-88.
in
wages by education
45
from 1964 to 1979. There was a
greatest increase in dispersion of skill in the U.S. occurred
smaller increase in
increase in the
skill
wage
dispersion from 1979 to 1988, with about ten percent of the total
differential
between workers
at the 10th
and 90th percentiles of the wage
distribution accounted for
by increasing dispersion of observable
percent of the increase in
wage
to
measured increases
skills.
Overall, only three
dispersion between 1979 and 1988 can be attributed directly
in skill dispersion.
The data on segregation of workers by
the period 1975-86, with the increase in segregation
skill
coming between 1977 and 1984.
cover
It is
possible that changes in sorting during the later period reflect either the small
contemporaneous widening of the
skill-distribution, or the larger earlier
widening of
this
distribution.
In sum, the data are consistent either with the hypothesis that technological change
raised the effective skill of high-skill workers, or with the
more parsimonious
were simply observed increases the mean and dispersion of
can explain the simultaneous
Our model
is
rise in segregation
skill.
story that there
In either case, the
model
and inequality.
also consistent with the absolute decline in
wages
for low-skill workers
found by Katz and Murphy [1992]. The simplest models of skill-biased technological change
do not generate such a decline. Technological change
high-skill workers does not necessarily
technologies remain available.
skill
make
that
opens up new opportunities for
low-skill workers
worse
off, since the
older
Technological change that increases the productivity of low-
workers could reduce their wages by reducing the prices of products they produce.
There
is,
however, mixed evidence about whether there has been a
produced by low-skill workers and,
if so,
fall in
prices of
sufficient to explain the reduction in
goods
wages
for low-
46
skill
workers [Lawrence and Slaughter, 1993;
Shatz, 1994;
Krugman and Lawrence, 1993; Sachs and
Krueger, 1996].
VI. Policy Implications and Directions for Future Research
To
conclude,
us consider
let
different implications in our
dynamic
interaction
In
many
within a firm.
intra-firm
firm
wage
may have
two prominent
model than
social policy issues that
have rather
more standard frameworks, and speculate on
in
the
between inequality and segregation.
countries, there are strong social
Freeman [1980],
differentials.
norms promoting equality across workers
for example, provides evidence that unions typically reduce
Our model implies
norms
that social
perverse effects on the distribution of income in the
for equality within the
economy
as a whole.''
If
low- and high-skill workers were perfect substitutes, pressures for equality within the firm
would lead
in
to increased segregation
by
skill,
but would not affect wages or output.
our model, pressures for equality within the firm will reduce output.
However,
Moreover, they
may
perversely reduce wages for low-skill workers by causing high-skill workers to sort into
separate firms, thus depriving low-skill workers of the benefit of cross-matching.^"
'^e
are grateful to
Daron Acemoglu
^°For example, suppose that
skill 1.5,
for suggestions
some workers had
skill 1,
on
this point.
and
that slightly
more workers had
so that rents from cooperation would accrue to the scarce low-skill workers.
firms would match together the two types of worker, and produce output 2.25.
Some
Firms with
only high-skill workers would produce output 3.375, so high-skill workers would earn half of
this,
or 1.6875.
Low-skill workers would earn 2.25
intra-firm equity
would self-match
made
it
-
1.6875, or .5625.
If pressures for
impossible to have a wage gap this great, then .11 high-skill workers
together, and earn a
wage of 1 .6875 ~
exactly what they earned otherwise.
47
The model
also bears on tax policies, such as those in the Clinton health care
proposal, that encourage increased segregation of high-wage workers [Cutler, 1994;
Elmendorf and Hamilton, 1994; Sheiner, 1994]. Under standard efficiency unit models,
workplace composition would be perfectly
calculations based
revenue.
such incentives, and so revenue
elastic to
on the current workplace composition would seriously overestimate
However, such policies would have no harmful
effect
on efficiency. Under our
model, by contrast, workforce composition would be only imperfectly elastic to incentives for
segregation (and so naive revenue estimates might not perform badly).
Still,
changes
in
workforce composition would create deadweight losses.
We
have taken the distribution of
endogenize the distribution of
patterns on skill acquisition.
skill,
more
old.^'
is less
effect of
to
matching
which people learn from high-skill colleagues, a
to a high-skill
manager when young, and
For example, young lawyers and doctors
from the wisdom of
may wish
as the
start
Later,
their elders.
when human
manager of
out assisting
skill.
In the
capital accumulation
important, most take positions of higher rank and supervise less-skilled subordinates.
relatively
cheap to provide training
in this
way
if
Low-skill workers would also self-match and earn a
have earned
^'
examine the
senior colleagues, performing tasks that are only moderately sensitive to
process, they profit
It is
when
exogenous. Future researchers
in particular to
In industries in
worker often serves as assistant
low-skill assistants
and
skill as
This
in the
is
there are tasks that are suited to
wage of
.5
-
less than
what they would
absence of pressures for equity within the firm.
not to deny that there are also learning environments in which people maximize
learning by performing a high-skill job, even
if their
co-workers are of lower
skill.
48
workers of
medium
skill,
as in
law or medicine."
These examples suggest
skill
that increases in segregation will spur future inequality of
by reducing opportunities for low-skill workers to learn from high-skill co-workers.
This raises the possibility that exogenous increases in either segregation or
may
skill inequality
spur a vicious cycle of ever-rising inequality and segregation.
Besides
its
implications for income distribution and the evolution of the skill
distribution, the pattern of
matching
may have
political
which high- and low-skill workers match together
homogenous
culture than one in
and social implications.
in firms is likely to
which people match with others of
Language, customs, and fashions are likely to be more similar
between groups.
States, Brazil,
Societies with
wide distributions of
and India, are more
likely to
skill,
if
assistant)
may develop
rigid hierarchy.
is
own
more
skill.
interaction
such as the contemporary United
develop separate subcultures and perhaps to
experience political and social conflict between these subcultures.
pattern of matching in
society in
have a more
their
there
A
On
which groups of people always play the same
into a caste-like system, in
the other hand, a
role
(manager or
which participants work together, but
There may be advantages to matching patterns
in
in a
which over wide ranges
people are indifferent between being the assistant to a higher level manager or the manager of
a lower level assistant, so that they play different roles in different situations.
" Of
For example, young baseball players
would presumably learn a great deal from playing in the major leagues, but the production
function makes it important to match the best players together, and so young players train in
the minor leagues, probably less effectively.
course, this condition does not always obtain.
49
Appendix
I
It is
possible to construct a confidence interval for the segregation index under the
assumption that the sampling errors
firms are independent.
in the estimates of the variances
This will be the case
if
independent of the average q within a firm or
the variance of
is
of q within and between
q within the firm
is
a linear function of the average q within the
firm.
Given
that s^J^ld^^
and s\J/c!\ are distributed as independent x^ variables with N-J
and 7-1 degrees of freedom respectively,
1
1
s^ w
(A.l)
^
1
where
bf is
+
1
+
b/
J
—a\-
an F-distributed random variable with (N-J,J-1) degrees of freedom.
50
Thus a 95-percent-confidence
interval for the index of segregation is
F{N-J,J-1)^^
F(N-J,J-ll0575
i p £
Sb
where F(N-J, J-1 )o.gi5
.
^b
and F(N-J, J-l)om5 are numbers
distribution function F(N-J,J-1) takes the values 0.975
at
which the cumulative probability
and 0.025 respectively.
This derivation assumed that the variance of q within a firm was independent
linear in, the firm's average q.
a firm, then
when
If the variance
a sample of firms
also tend to have a high variance of
precise than the calculations
convex
in
average
q, for
is
if
q within
is
concave
in
confidence interval.
firms,
indicate.
which
will
convex
in the
average q in
On
make
the estimate
q,
q within
the other hand,
if
the firm
it
will
more
The variance of q within firms
the standard deviation of
proportional to the average q within the firm.
firms
is
picked that has a high variance of average
below would
example,
of q within firms
or
of,
will
be
is
the variance of
q within
average q, then the formulas above will understate the width of the
Note
that this will
be the case for dichotomous variables, such as race or
production worker status, because the variance of the characteristic within the firm equals
- |i),
where
|i
is
the share of people with the characteristic.
However, when
\i(\
this share is
close to one or zero, the variance within firms will be approximately linear in average q, and
the estimated confidence intervals should not be that far off.
51
Appendix
Proof of
n'
II
Lemma
> An(Q). From
consider n' e
Consider n < An\Q).
8:
Lemma
[/J,(e), n*(e)).
1,
Suppose an n-worker
is
matched with some
an n worker cannot be matched with n' > n'(9).
Given
that,
from
Lemma
5,
w(n';e)
= n'(n\Q)f
-
skill
Hence
in\Q))V2,
we have
v(n,n';d) =
(A.3)
It
suffices to
show
n(n'f - w(/z';e) = n(n'f - n'(n*(eyf + i^I^I^L = Hin').
that,
on the interval
[n,(9), n\Q)],
establish this, note that the derivative of
2A\n{Q)f <
(n*(9))^,
H{)
and so the derivative
w,(9) (not necessarily an integer) such that
such number. Then dH(p)/dn' >
dH(n')/dn'
>
(A.4)
p.
Inn'
n
matched with
if
We
conclude that there
is
n' in
is
-
is
(n'(6))l
Hip) = H(n,(0)). In
Moreover, because 2nn'
=
Since H(p)
maximized
Now
negative at this point.
-
at n'
at n'
=
=
«,(9).
/z,(0),
Suppose there
fact, let
(n*(9))^
p be
is
To
2nn' <
exists
p>
the smallest
increasing in n',
i/(n,(9)),
Hin'iQ)) > Hin') for aU
But
=
>
0, for all n'
0.
is
is
H{n')
n'
e [n^id),n'ie))
.
equilibrium, then Hin') > Hin'iQ)), a contradiction of (A.4).
no p such
that
Hip) =
//(/z,(9)),
and so Hin')
is
maximized
at n'
n,(9) as claimed.
Q.E.D.
Proof of Proposition
2:
Let S,={nin < n,}.
Competitive equilibrium maximizes
total
52
product, and, for any match, the
Moreover, for any n <
worker
in 5,.
ni(0).
Finally from
sum
Lemma
Lemmas
of the equilibrium wages equals the match's product.
8 implies that an n-worker
1, 3,
and
another worker in 5, or with an n*(9)-worker.
53 win;Q)p(n;QX%{Q))
matched with another
an n,(0)- or n, -worker
Hence, the
sum
is
matched
either with
of wages of workers in 5,
equals
max^Y^
(A.5)
4,
is
z(n,n')nin'f ^
z(*iO n'=n n=a
J]
z(n,n'(0))v(n,n'(e);0)
n=iii(0)
subject to
(A.6)
J2 z(n,n')
^ p(n;Q), n = a,...,»i(0) -
1
n'=a
and
(A-^)
z(n,/i') +
X;
zin,n*m
^ pin;B),
n =
/ZjCO),
n,
,
n'=H
where, for
all n, n',
z{n,n') is the
worker, and v(«, n';Q)
may
or
w(/i';e).
may
is
the
number of matches between an «-worker and an
wage
for an o-worker if
n'-
matched with an n'-worker (which
not occur in equilibrium), assuming the latter
is
paid his equilibrium
wage
53
Let P,' be the variant of program (A.5)-(A.7)
(A.6)-(A.7).
Denote by
n*(e)(2«-3/i'(e)/2)
is
R/
the
negative
P/ are
when n = n,(0
We
which 6 =
for /=0,1.
/?i(0).
proportional to (A.6)-(A.7) with 6
=
1.
^
(A.8)
and so from (A.8) and the
From Lemma
must be matched
3,
fact that
=
R/ <
But from
worker
A <
(A.9)
the
sum
of wages in of workers in
max j^
a(-)
E
n'=n2n=n2
S2,
,
we
conclude that (38) holds.
in Sj or
i.e.,
z(n.«')n(n')' ^
%,
n,(i)(/z'(e))^-
skill
no
less than ^2
with a worker of type n'e [n(Q),n2-\].
n
Hence
in
(26), the constraints in
a worker in the set consisting of S2 of workers of
either with another
Because
=
Hence,
^
/?,(0)
in (A.5) but G
Hence, v(n,(i),n'(e);e)=
<
conclude that
1
the objective in P,'.
i?,'
V2(n*(0))' is decreasing in 6.
program
maximized value of
in
/?
(0) =
E E
"=n2 n'=n*(e)
X^
w(n';B)p(n';Q), equals
z(n',/i)v(/z./i';e)
,
54
subject to
(A.IO)
Y^
zOi',n) ^ p(n;B), n=n2,
n.
n'=n'(e)
From Lemma
2,
v(7z,«*(0);e) = n'iQ)n^ -
(A.ll)
From Lemma
7, (35),
and the
fact that
if
(A.ll)
^n
.
nj,
is
n ^
n^.
^^^
«'e (/i'(6),n2) and n ^
/ij
.
the fact that n^ >(3/2)"^«*(l) definition of nj implies that the right-hand side of
increasing in 9.
increasing in 6.
(A.IO), the
for
'(G),
v(«,n';e) = n'n^ - n'(0)(n')' +
(A 12)
For n >
<
n
^^*f^^
We
Similarly, because n'
conclude that
if
Pj'
is
the
<
1X2,
the right-hand side of (A.12)
program
in
which 9 =
maximized value R2' of the corresponding objective
1
is
in (A.9) but
satisfies /?2'>i?2(0)-
also
9
=
in
But from
55
(26), the constraints (A. 10)
when 9=0
are proportional to (A. 10)
when 9 =
1.
Hence
53p(/z;l)>v(n;l)
-^
(A.13)
=
!^=^^_
.
"="2
n=n2
This yields (39).
Q.E.D.
Proof of Proposition
dispersions of
3:
We
can decompose (41) as follows.
wages between and within
Let Bg and Wq be the
firms, respectively, for those firms that hire only
workers from the interval [n,+l,n2-l]. Then
(A.14)
E
^6=
iii(n,n')-n\Q)fzin,n';d)
n,ii'e[n,+l,n2-l]
and
(A.15)
W,=
Y,
in-\iin,n'))^zin,n';Q)
n,n'e[i:,+l,ii2-l]
Similarly, let Bq' and Wq' be the dispersions, respectively,
between and within
all
other
56
firms.
We
have:
^e'
=
(A.16)
E
E
iViin,n')-n'mhin,n';Q)^
n«[n,+l.n2-l]
n'=a
n'e[n, +l,nj-l] neLnj+l.iij-l]
and
^e'
=
(A.17)
E
E
(n-Ji(«,«'))'z(n,n';e) +
n'=n netni+l.Hj-l]
5;
j;
(/i-H(«,n'))2z(n,n';0).
n'cEni+l.iij-l] nelni+l.Hj-l]
We
can rewrite (41) as
(A.18)
p(0)=
'e
"e
B,^B,'^W,*W,'
Now,
the proofs of
Lemmas
5 and 6 imply that
if n, n'
e
(/z„n2),
then z(n,n';6)
>
if
and
57
only
if
either n
=
n'(d) or n'
=
(A.19)
n\Q).
But then
\vLin,n')-n\Q)\ =
\n -
\iin,n%
and so
Bg = Wq.
(A.20)
Moreover, from
Lemma
3, if z{n, n';Q)
(A-21)
with
strict
and n <
>
|/i-Ji(n,/j')|
inequality unless n'
=
n(Q).
or n >
/ij,
then
^|n(/i,n')-«'(e)|.
Hence,
Wq' <
(A.22)
From (A.20) and (A.22) we can
n,,
fig'
re-express (A. 18) as
(A.23)
p(e)=
'e
^e
25g+2Bg'-A
where
A>
0.
Now, below, we
will
show
Bj < pBjj
(A.24)
and (26) implies
that
(A.25)
Now,
the fact that
(A.26)
that, for r sufficiently big, (40)
5j' = uBq' and W^' «
A>0 and
aW^.
(A.23)-(A.25) imply
p(l)
>
f
2pB(,+2aBo'-aA
implies that
58
But because
a>
> P and A >
1
0, the
6=0, establishing that p(l) > p(0).
To
It
right-hand side of (A.26) exceeds that of (A.23) with
remains to establish (A.24) and (A.25).
derive (A.24), note that,
B,=
{Y^{n,n')-n'{V)fz{n,n';\)
53
ivi'eln,+l.n2-l]
=(A.27)
{n-n\\)fz{n,n\iy,V)
53
from Lemmas 3-7
2„e[„j+i^.i]
<1
2
(n-/i'(0))^z(/2,n*(0);0)
53
from (39) for r big enough
ne[n,-l,n2-l]
establishing (A.24).
To demonstrate
and so from (A.5)
(A.28)
(A.25), note that, for r sufficiently big, v(n, n'(0);0)
to (A.6)
z(«,/i';l)
=
v(n, n'(l);l)
and (A.8) to (A.8).
= az(n,n';0), for n ^ n' ^ n^ or
/ij
^ n ^ n',
which implies (A.25).
Q.E.D.
59
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