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DEWEY J
working paper
department
of economics
MATCHING, HETEROGENEITY AND
THE EVOLUTION OF INCOME DISTRIBUTION
Daron Acemoglu
95-25
Oct. 1995
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
MATCHING, HETEROGENEITY AND
THE EVOLUTION OF INCOME DISTRIBUTION
Daron Acemoglu
95-25
Oct. 1995
MASSACHUSETTS INSTITUTE
MOV
1 4 1995
95-25
First Version: April 1995
This Version: October 1995
MATCHING, HETEROGENEITY AND
THE EVOLUTION OF INCOME DISTRIBUTION
Daron Acemoglu
Department of Economics
Massachusetts Institute of Technology
Keywords: Search, Matching, Mismatch, Human
Wage Inequality, Income Inequality.
JEL Classification: D31, J41.
Capital,
Abstract
This paper presents a model in which firms and workers have to engage in costly search
to find a production partner. In this setting, the
evolutions are
endogenized.
The presence
skill,
job and wage distributions and their
of search costs
implies that there
are
two
workers sometimes produce
with jobs intended for unskilled workers and vice versa, the gap between skilled and unskilled
workers gets compressed. Second, because skilled workers always have a better outside option,
unskilled wages are pushed down. We show that these forces can lead to a non-ergodic
equilibrium process whereby at high levels of initial inequality, wage inequality can keep rising
whereas it would be decreasing starting from lower levels of inequality. The model predicts
that increasing wage inequality is more likely to arise in economies with less frictional labor
markets, less redistributive taxation and less public schooling. These predictions are in line
with the diverse cross-country patterns that we observe.
redistributive forces in the labor market. First, because skilled
have benefited from comments of and discussions with Abhijit Banerjee, Roland Benabou,
Alberto Bisin, Olivier Blanchard, Peter Diamond, Kevin Lang, Thomas Piketty, Chris
I
Pissarides, Steve Pischke,
Andrew
participants at various seminars
Scott, Robert Shimer, Enis Uysal, Jaume Ventura, and
and conferences.
Introduction
I.
Income inequality
The main component
decades.
of this change
identified as increased earnings
is
1993). Similar trends are observed in other
OECD countries but
than in the U.S. For instance Katz,
al
et
that
from
The same
moderate increase
et al (1995)
it
wage
who
in the
has changed
The
.95 to 1.04 in Japan.
(between-group wage
has not increased at
all.
in
Japan
The same
find the highest increase in
Canada and not much action
in
statistic
and to some extent in the U.K., while
only changed by a small amount, and in France,
emerges from the study of Card
and from
over non-college graduates
earn
graduates
college
1.19 to 1.22 in.France
inequality) has also soared in the U.S.
in the U.S., a
pronounced
workers on the 90th and 10th percentiles of the wage distribution
0.88 to 1.10 in the U.K.,
premium
less
Pierce,
(1992) report that the logarithm of the
U.S. has increased from 1.23 to 1.38 between 1979 and 1987.
from
much
appear
a half
and wage
Murphy and
inequality (see for instance Levy, 1989, Levy and Murnane, 1992, Juhn,
differential for the
two and
in the U.S. has risen considerably over the past
it
pattern
wage inequality
in France. Davis' (1993)
analyzes wage changes in West Germany, Netherlands, Austria and
has
Sweden
who
as well as
the
above countries, also confirms the conclusion that wage inequality has risen mostly in the
Anglo-Saxon economies.
These trends towards higher inequality in
a
number of
countries pose
many academic
and policy questions. Although empirical labor research has accomplished reasonable success
in accounting for these changes in terms of relative supply
Murphy,
1992, Katz et
supplies have
al,
1992),
we
moved differently in
still
and demand of
lack an understanding of
different countries.
For
instance,
skills (e.g.
Katz and
why
relative
Katz
et al report that
demands and
from
1979 to 89, the growth rate of the relative supply of college educated workers has been 0.023
in the U.S. as
compared to 0.037
and 0.029 in Japan. But in a
in the U.K., 0.050 in France
general equilibrium model, these changes are endogenous, and they can be a consequence as
much
as a
cause of the wage inequality. Similarly, the relative
endogenously determined and
it is
institutional set-up of the labor
Card
for skilled labor
is
influenced by the composition of the labor force and the
market
1
.
Further,
(1990) finds that after the Mariel Boat
in unskilled labor force, there
demand
it is
lift
in
widely believed that the U.S. has a
Miami
was no change in unskilled
less
despite the unprecedented increase
relative
wage nor unemployment, thus
market than other
frictional labor
U.K. during the
is
decade or
last
one of the main
OECD
so]. It is
economies
[this
is
is
factors underlying the increase in
to provide such a theoretical
be true for the
then natural to ask whether this institutional difference
wage inequality
need a well-formulated general equilibrium model to answer
paper
also suggested to
in the U.S. Again,
The
this question.
framework to evaluate these
we
objective of this
while also matching
facts
the salient patterns that arise from a cross-country comparison.
Our model
builds
on the work of Diamond
(1982),
Jovanovic (1979), Mortensen (1982)
and Pissarides (1985) where workers and firms have to search for the
market.
We
extend this framework in three respects:
first,
right partner in the labor
both workers and firms are
potentially heterogenous, thus workers of different skill levels and jobs of different qualities
may
be simultaneously searching. Second, the composition of jobs
knowing the
distribution of skills in the
economy, and the wages they
decide what kind of jobs to open. Third, as
Heckman,
in other
1992), the distribution of
it is
income
is
it;
similarly, a
finds the
worker
highest. Therefore,
will not
skills
the distribution of
skills, will
conclusions
the income and thus
call
that the Walrasian auctioneer
frictionless market,
fact that the level of
mismatch.
would have
demand
Two
.
firm will
allocated to
is
The main
inequality,
results of this
through
its
impact on
influence the degree of mismatch in the labor market, and
are: first,
skill level
A
2
our economy will have a degree of
income
more heterogeneity
of workers will lead to
in the
form of
income
larger gaps
more mismatch which means
workers will be either under- or over-educated for their
2
we
skills;
for their offsprings
will introduce forces towards redistribution across agents of different
Our main
relative
(skills)
of workers and the requirements of jobs.
paper will be derived from the
Cameron and
look until he finds the job where his marginal product
compared to the
mismatch between the
mismatch
worker
pay in equilibrium,
allowed to influence the distribution of
In our model frictional labor markets lead to a feature
it
will
the case in the data (for instance
words, rich parents will be able to buy more education
stop looking before
endogenized; firms,
is
jobs. Additionally, since
levels.
between
that
more
mismatch
appears to have adjusted to accommodate the shift of relative supply.
other recent papers are related to our work. Davis (1995) analyzes the composition of
jobs and the impact of bargaining
technological shock
on
unskilled
distribution of skills and
on wages, and Saint-Paul (1993) discusses the impact of a skill biased
wages and unemployment. Neither of these models endogenize the
income and the underlying mechanisms
are different.
reduces profitability of jobs, aggregate investment
second, in
itself,
mismatch
a stabilizing force
is
and the economy grows
falls
on income
less.
And
yet,
mismatch implies that
distribution;
highly skilled workers sometimes end-up working in unskilled jobs and receive lower wages
while
less skilled
workers
get
matched to higher paying
two groups becomes narrower over
call
the outside option
effect:
jobs; therefore, the gap
time. Thirdly, however, there
wages in
a
non-competitive
economy
between the
another aspect which
we
are in general influenced
by
is
the outside options of workers (and firms) and these outside options counteract the equalizing
effect of
mismatch; because the outside option of the skilled workers
is
always better than that
of the unskilled, unskilled wages decrease relative to the earnings of the skilled workers.
shown
It is
that this outside option effect increases with the degree of inequality, and while an
economy with
limited inequality will exhibit decreasing inequality over time and have a steady
when
growth, the same economy,
it
starts
with higher
wage and income inequality and lower growth. Put
frictions
initial inequality, will
differently, the
on wage determination makes the dynamics
have increasing
impact of labor market
of inequality non-ergodic, and leads to
multiple limiting distributions of income.
The presence
of
two counteracting
redistributive forces implies that
our model can
account for both increasing and decreasing wage inequality depending on the institutional
features
form of
and the
relative
economy. [This contrasts with the simplest
starting level of inequality of the
supply-demand story which needs repeated positive shocks to the
demand
for skilled workers].
markets,
less redistributive
The model
also predicts that
economies with
less frictional
taxation and less public schooling should experience
inequality than the rest (or should have increasing
wage inequality while the
change). These findings accord with the cross-country patterns that
the U.K., Canada versus France, Japan,
we
observe
Germany and Sweden 3 ]. The model
labor
more wage
may have no
rest
[i.e.
the U.S.,
also suggests a
negative correlation between the growth rate and the level of inequality, as found
country studies of Persson and Tabellini
relative
by the
cross-
(1993), Perotti (1994,95).
Finally, labor economists have observed that in
most periods within group and between
An alternative approach is to explain the cross-country patterns by different technology shocks
for different countries. This approach
is
however not very
be affected by the same shocks. Indeed Card
et al
technological change to be identical for the U.S.,
satisfactory since
OECD
countries should
(1995) find that the measures of skilled biased
Canada and France.
group inequality move
workers of different
in the
same
skill levels,
direction; that
so do
wage
when wage
is,
differences
among
differences increase
between
equally skilled workers (see for
instance Goldin and Margo, 1992, for the 50s and 60s and Juhn,
Murphy and
Pierce, 1993).
Yet, despite the importance of the changes in this aspect of inequality, in Levy and Murnane's
words
(1992, p. 1372) "the most important unresolved puzzle concerns the reasons for the almost
20 years trend towards increased within group inequality
we
pattern
predict
is
when
that
".
skill level
moving
that can account for within
in opposite directions as in the U.S.
Before proceeding with the analysis
that
we
on
build
and costly
economy
it is
Ruhm
allocated to the jobs
(1991),
of the 1970s.
whether the microfoundations
Topel and Ward (1992) show
most suited to them. Barron
(1995).
The
how
is
employers spend
confirmed by the case studies in Levy
findings of high correlation between profitability of firms and wages
both in union and non-union sectors
[e.g.
Abowd and Lemieux,
1993, Dickens and Katz, 1987,
Estevao and Tevlin, 1995, Katz and Summers, 1989] support the view that there
in the labor market, thus bargaining
over- and under-education
(e.g.
is
workers
a careful paper using the
is
very
PSID by Sicherman
common. The most
(1991).
worker with the same education
level
and
He
finds that
required for the job (over-
educated workers) earn more than other workers doing the same job but
less
on
Rumberger, 1981) which suggests that the mismatch even with
who have more education than the reported amount that is
under-educated earn
rent-sharing
a large empirical literature
respect to easily observed characteristics such as years of schooling
is
is
powers and outside options should indeed be expected to
play an important role in wage determination. Finally, there
interesting piece for us
an important
workers are only slowly
et al (1989) find that
is
we will
and between group wage inequalities
useful to ask
considerable resources in recruiting activities, and this
and Murnane
group inequality. However,
are plausible. Considerable evidence suggests that job search
activity.
more diverse
sometimes end-up working with jobs of
different qualities; this therefore explains the rise in within
mechanism
theory can be useful here. The
inequality of skills increases, firms start creating a
job distribution, and workers of the same
also suggest a
Our
characteristics. Conversely,
less
than a typical
workers
who
appear
than others doing the same job but more than a typical worker with
the same characteristics. Although the evidence could also be interpreted as unobserved
heterogeneity,
i.e.
over-educated workers having
less
"unobserved
capital", this is
not consistent
with the
rest
of Sicherman's results. In particular, he also finds that over-educated workers
have a significantly higher probability of moving to
a better job,
which means that on average
these workers are truly over-qualified for the job they are performing and
this reallocation
To
is
quite slow implying that "mismatch"
mismatch would
give a concrete example,
equipment accepts workers
equipment with
The plan
less
who
is
as follows.
and characterizes the equilibrium of
benchmark
this
case. Section III analyzes the
The next
economy
new
firm which has installed a
some other workers who can use
somewhere
training are also available
of the paper
not a very transitory phenomenon.
is
arise if a
require training while
more importantly,
this
in the labor market.
dynamic model
section presents a simple
in the presence of frictionless
markets
dynamics of inequality with random matching
as a
in the
context of a two-class economy. This section also conducts the comparative static exercises
regarding labor market efficiency, redistributive taxation and public education. Section IV
analyzes the dynamics of inequality starting from general income distributions. Section
to different matching technologies.
It
V turns
demonstrates the robustness of our results and also links
within group inequality to changes in between group inequality. Section VI concludes.
Appendix
A
contains the proofs. Appendix
general transmission technologies.
Appendix
B extends our
C
which
is
results to
available
upon
game form used for wage determination and some further robustness
II.
II. 1.
The
Each agent
capital. In
lives for three periods.
The
first is
youth
in
the second, middle age, he works and decides
offspring's education
of his wage income.
and also
We
each generation, there
(1)
j
results.
Environment and the Walrasian Equilibrium
which he
is
of generation
how much to
refer to
a
an agent
consume. In
how much
to spend
save for the final, retirement, period of his
as of generation
continuum of agents with
size
t, if
he
is
middle aged
normalized to
t is;
(1
(he or she) acquires
how much to
period, each agent also conceives a unique offspring, and decides
of agent
request includes the
Timing of Events, Technology
Preferences,
human
Basic
economies with more
-8- y)logc. ,+Sloge.^j+Ylogc,^
1.
The
at
this
on
his
life
out
time
t.
In
utility function
where
c denotes
consumption, the superscript
O
used for old
is
and
age,
education expenditure. The utility function exhibits impure altruism
as
denotes the
e
the agent does not
obtain utility from the welfare of his child but from the education he gives him.
All that the agent decides to save this period
return
Rt+
i-
Thus the budget
constraint of the agent
invested
is
the gross market rate of
at
is
o
/F^ ^
C
(2)
where W:
t
is
1=
the wage income of the individual
j,
which
only income
his
is
when he
is
middle
aged (only the retired are rentiers).
With
we
this specification,
c°+1 =Rt+1 yw
c =(l-8-y)wjt
u
(3)
where
S:
t
obtain the following simple decision rules for agent
,
e
,
jf+1
jtt
denotes the savings. The
human
=Sw
Sj
,
jf
f
'
;>1
h
(4)
In other words, there
e.g.
what they
learn
is
.
e;>1 " /*
a certain level of
in the
human
is
determined by the
precisely,
min
human
capital,
h mln below which agents do not
,
from compulsory schooling or from
capital of the offspring
More
is
=f
their friends, etc. This
simple specification for the intergenerational transmission of
human
=—- (^i)-YV
capital of the next generation
education level (expenditure) given to them by their parents.
j;
linear in the
human
is
fall
a very
capital; in particular,
wage of the parent, which
capital of the parent in the competitive equilibrium.
-
the
will in turn be linear
Appendix B shows the
robustness of our results to a general transmission rule of the form h t+i = i/'(w ,h ).
t
We
assume that there
is
a
continuum of firms equal to
one worker, thus we can think of these firms
as jobs.
equal to that of workers so as to avoid issues of
4
1,
t
and that each firm employs
The measure of firms
unemployment 4
.
A
is
firm (job)
taken to be
i,
if
matched
Although unemployment is closely related to the issues of wage inequality [because
unemployed workers are poor and also because unskilled workers are more likely to be unemployed],
the issues of unemployment are distinct from the basic redistributive forces that frictions in the labor
market introduce. Therefore, in order to focus on our primary interest, we have chosen an economy
without unemployment. The issues of unemployment are analyzed in similar but simpler setting in
Acemoglu (1995b).
with worker
produces
j,
a
yUf*4rhIP
(5)
where k it
is
the capital associated to this job, and L,
The production function
to the relation.
between physical and human
benchmark
function has non-increasing returns to
scale,
exist,
and
also, since there is
capital that
worker
a result, there are diminishing returns to
capital. In this frictionless (competitive)
would not
human
the
brings
j
has constant returns to scale and complementarity
As
capital.
is
case,
it is
human
important that the production
because otherwise a competitive equilibrium
nothing that pins
down
jobs in this
economy,
it
should not be possible to produce more output by combining two jobs.
The
capital that the firms will use at time t
comes from the savings of generation
workers. All workers invest their savings in a mutual fund which
by the
profits
threat of potential entry
allocated to the firms.
(a
is
forced to
perfectly contestable market).
The intermediation by the mutual fund removes
The
make
t-1
zero
savings are then
uncertainty from
all
the rate of return on savings.
We also assume that the economy is small and open, thus it faces
world
R at which the mutual fund can borrow or invest, therefore, the
a constant
interest rate
cost of capital to a firm and the rate of return
We
decision
human
is
assume that firms hire their
completely
irreversible.
on
Thus
at that
do not
point, while they
necessarily
employ. In practice k does not only correspond to
t
equipment, the type of job, the location,
The
5
it is
we
denote this distribution by P
know
and
t
linear technology
and thus
investment opportunities,
a8<y
fixes the cost
all
course natural; growth in this
this
the distribution of
they will
but also to the quality of the
natural that
it
has a high degree of
t
(k).
An alternative is to have a linear household technology with a rate of return R. Then
condition, namely [(l-a)/4/i?]
our
result
economy
,
ensures that the
economy always
invests
a simple
some funds
in this
of capital at R. In the absence of a linear technology and other
would hold except those regarding the growth rates. This is of
is driven by the total amount of investment and in the absence
of an alternative investment opportunity,
all
that
is
saved will be invested and moreover, with
logarithmic preferences savings are independent of the rate of return
would
.
know which worker
capital stock
and so
5
of the firms' choices of capital stock will in general give a
result
distribution of jobs and
etc,
R
beginning of period
capital stock at the
capital in the labor market, the
irreversibility.
savings are fixed at
achieve the same growth rate irrespective of the rate of return
-
on
i.e.
(4)
-,
capital.
hence the economy
The Walrasian Equilibrium
II.2.
In this section
a la Bertrand for
all
we
the same
is
as
t
(k).
Then
capital
the one that a Walrasian auctioneer (or a Social Planner) would choose.
F
capitals,
t
firms' investment choices determine
(h),
human and
given the complementarity of
worker
Sattinger, 1993
(e.g.
is
k
allocated to a job with physical capital
at
time
Then
since workers are paid their marginal product, the
has a
human
capital
h
w (h)=aAk
l
-a
h
a- 1
also introduce the variable 6 such that
time
capital at
use the notation, (h,k)€:J>t
worker of generation
a
h-=h 6jt where h
t
is
the median level of
skills relative
the median. This distribution can be simply obtained from the distribution of h and
it
by
G(0). Let us also denote the starting distribution of
corresponding distribution of inequality by
paper that even the poorest agent
Proposition
1:
that
t
and thus the distribution of 6 measures inequality of
t,
t
.
where (h,k)£Tr
t
human
wage of
capital
given by
is
(6)
We
we
then
t,
see section
-
worker with human
V.l. for a formal definition). If according to this allocation rule, a
h
human
physical capital, the highest
allocated to the highest physical capital firm
is
firms compete
all
and the allocation of the heterogeneous workers to the heterogenous
Given the distribution of human
P
Thus,
as frictionless.
the workers in the labor market. This implies that workers will be paid
their marginal product,
firms
model the labor market
will
G
(.).
with
j* starts
hj,
Further,
human
capital
we assume
> >h min Then we
.
by F
as in
to
we denote
(.)
and the
the rest of the
can state
6
;
Let GJQ) be the distribution of initial inequality of human capital as defined above.
Then, with frictionless labor markets;
The physical
(i)
to
human
capital ratio for all jobs
is
given by
—=
h
(ii)
(Hi)
V
Gq,
G =G
t
for
all
thus inequality in this
The growth rate of the economy
The proof
6
If
is
always g
of this proposition, like
the poorest agent has
inequality and inequality
is
t,
would
less
than
all
h,,^,
c
\.
—
R
=8aA{(l-a)AR
then in the
a
.
>
self-replicates.
~1
}
a -1.
the others in this paper,
self-replicate forever
to avoid having the max{.,.} notation.
economy
-
first
is
in
Appendix A. Here
period there will be a contraction in
from then on. The reason we
start
away from h,^
main
suffices to give the
it
to steady
growth
as in
intuition.
The economy
the model of Rebelo (1991).
the feature that leads
More importantly
for the focus of this
linear
worker with twice
capital
with
j
as
much human
and thus earns twice
as
CRRA preferences), the
who
has twice as
much
to
make our
j'
we obviously have
results in the rest of the
paper
cannot pay their parents to get more
credit constraints).
as
constant; that
as
much
a special
Thus the
much
model
physical
also be true
offspring of
worker
capital as the offspring of
here, but
results to non-linearities
human
is,
it is
only chosen
transparent as possible, and this can be seen in
economy, there
finally that in this
linear.
will also have twice as
Appendix B where the robustness of our
Note
is
much. With logarithmic preferences (which would
capital as
This result emphasizes that
capital
another works with twice
capital as
accumulation rules are also
j'.
this
on human
paper, given constant returns to scale, the rate of return
a
and
is
is
are
many
is
demonstrated.
missing markets: young generations
youth
(a
form of intergenerational
But despite these missing markets, inequality
is
not harmful to aggregate
capital in their
performance. The economy has the same growth rate irrespective of the level of inequality.
III.
III.l.
Inequality
With Random Matching:
A Two
Class
Economy
Preliminaries
We now
market.
analyze the
The key assumption
beginning of the period,
all
economy
that
is
outlined in the
section but with a frictional labor
last
costly for agents to engage in search activities. In the
it is
firms and workers are matched 1-to-l, thus each job will have a
worker, and each worker will have a job. At this point, either party can terminate the relation
and look for
is
a
new
partner. Costly matching refers to the fact that engaging in further search
costly to both firms and workers.
working
lives are divided into
We
model
this
two segments of length
decides to look for another match, they
by assuming
l-?j
do not produce
and
rj.
If
that workers'
the worker (or the firm)
in the first
segment and only meet
another partner from the pool of unmatched agents for the second segment of their
this
they cannot break the match again 7 Thus
.
C
77
and firms'
lives; after
captures the degree of frictions in the labor
the discussion in the text.
unchanged if parties can change as many partners
one round of change is to simplify the expressions and
The advantage of having infinite sampling as in Appendix C is that the
economy would converge
to the perfectly competitive case analyzed in the last section as
Appendix
as
they like
-
shows
the reason
we
that
all
our
results are
limit ourselves to
t/-»1
(e.g.
market.
When
r]
= 0, mobility
we
In this section
implies that any
is
start
very costly; after separation no output
with the extreme assumption that matching
two workers, even
same probability of meeting
produced.
is
if
human
they have unequal
a given firm in the
economy and
random. This
is
capital levels, face exactly the
conversely, the same also applies
to firms. This assumption will be relaxed in section V.
There
are a
partners, each
We
assumed.
justification
is
number
of other issues to deal with. First, since there are costs of changing
match has some surplus to be shared, thus
The second
workers will want to separate from their
to avoid this issue as
deal
As
it
[this is trivially true
a result of this
issue
is
that
when
17
is
[a
•
near
1,
game
theoretic
some
firms and
match and look for another partner.
first
when
assumption in the
77
all
firms and workers produce with the
a
worker
this
(or a firm)
= 0, Appendix
C
minor problem;
who becomes
segment of
his or its
derives the exact expression for
capital level of the agent.
unmatched
will be of
We
and
this event
will analyze this
With
F
t
for the workers and
t
is
independent of the
economy
as v-^0,
F
t
(h)
and P
t
(k), is
n;[^,^X^)^Xfc )]=^%"
now
partner.
arises
exactly the
when
the distribution of
given by [see Appendix
a
a
-iS(l-/3)T^A:/-
from the labor market
To
human
is
same
avoid
matched
or physical
thus the set of agents
worker with human
who
are
independent
as their initial
frictions.
10
C
economy
who
capital h: t
human and
is
physical
for details];
/^ *dFjh)+(l-P)fajAk
Gale, 1987) and this shows that the only difference between the
are analyzing
is
actual
for the firms.
level of a
a firm of physical capital k; „
capital are respectively
(7)
wage
P
77*].
in equilibrium,
measure zero. Also since the event of being unmatched
this formulation, the
matched with
new
no
face a probability v of not getting
of characteristics, the distribution of these unmatched agents
distributions, thus
no separations
since there are
separated will not be able to find a
life,
will
economies that we analyze, outside options will
class of
problem we assume that each worker and firm
in the first
want
partner
first
influence wages through the threat of separations, but in equilibrium, there will be
separations. This introduces a
We
unemployment. Therefore, throughout the paper we
will lead to
with the case where r\<r{ such that
they meet
Nash bargaining
will follow the usual practice of using
given in appendix C].
a bargaining rule needs to be
of section
l
'%dP (k).
t
II
and the one we
The
first
term
the share of total output that the worker gets.
is
point of the firm and
is
thus deducted from the wages, and the third
point and adds to his share of the surplus [since
which there
are looking at an equilibrium in
separates,
it
human
of
will
The second
we have
no
are
by F
produce with the worker
obtains a proportion
/?
it
t
(h).
Since there
meets and
as
separations. Thus,
there
no
is
it is
also multiplied
by
|3.
The
This term
rest.
third term
is
is
subtracted
threat point.
from the
total surplus
The sum
which the firm
will
and then
maximize by choosing k
Thus, given the assumption that
rj<r)*
.
t
worker
subtracted from
/3
of the total
firms,
P
t
(k)
and
added to the total wage of the worker
also
of these three terms give
has to
it
explained similarly as the deviation of the
worker to make one more round of search among the pool of unmatched
is
a distribution
possibility of separation, the
of the surplus and the firm gets the
(7).
It is
and
a firm deviates
if
only one more round of matching,
is
we
transferable utility]. Recall that
the total surplus that the parties share and since the worker obtains a proportion
surplus,
the threat
the worker's threat
is
meet one of the unmatched workers, and these workers have
capital given
is
The
firm's profit function
easily seen that (8)
which ensures that there
is
strictly
will be
no
this
as his
is
concave in k
separations,
.
t
all
firms choose the same level of physical capital investment equal to;
(l-P)(l +Pr,)(l-a)fh
(9)
k
We
=
a
dF (h)
a
t
R
can note a number of important features in equation
unable to capture the
full
Acemoglu, 1995a] and for
marginal product of
this reason, the
its
growth
investment,
rate of this
(9).
First because the firm
will always underinvest [see
it
economy
will always different
than the growth rate of the competitive economy, g c More interestingly, however, firm
.
investments also depend on the distribution of
from
(9)
that a mean-preserving spread of
F
t
human
capital,
F
t (.).
will reduce investment.
mean-preserving spread which maintains the total amount of
11
is
We
can see immediately
The
human
level
intuition
capital the
is
that, a
same but
more unequally
makes
it
in this
economy
is
distributed, increases
reflected
mismatch and thus reduces
by the distribution of human to physical
profits.
all
firms choose to have the same level of physical capital,
human
mismatch.
A
workers than
gains
it
from
economy,
since
also gives the distribution of the
t
simplify the analysis
;
we then
We
From
(10)
v
equation
V
human
We
let
t
capital of the rich
= h /h
lt
2t ,
the ratio of
starts
with two
by h 2t and
human
wages of each group,
j
=
1,2,
less
poor
relative to the rich.
can be written as
characterize the evolution of this system, as long as
t
= h /h 2t
_h u+1 jw
(11)
Therefore,
we have
]t
as follows;
lt
[i + (i-p)(i-x) V
h lt
is
away from h min we
,
]tf-(i-m-^
a simple non-linear first-order difference equation describing the
evolution of inequality. In other words, equation (11) gives the law of motion of the
capital ratio of the
poor to that of the
rich
any given period. This equation,
0=1. The questions to ask
(locally
and
capital
a
can write the law of motion of
is
that of
/^/c/- /z/-/3(l-/3)A^^^
Now to
are in
human
the proportion of poor agents be A. This proportion
poor agents may become gradually
(7),
to note that
Classes
with the case where the economy
denote the
human
worker.
capital
study the dynamics of
income between the two groups.
will never change, but
start
is
firm loses more from a low
capital, a
human
high
a
we
groups of workers; rich and poor.
the poor by h lt
human
of seeing the intuition
Growth Dynamics with Two
Analysis of Inequality and
To
way
slightly different
because there are decreasing returns to
III.2.
recall
to physical capital ratios and an increase in inequality in the distribution of
capital increases
capital
F
and
capital ratios
that in the Walrasian case these ratios are always constant. In contrast, in this
Mismatch
and globally)
are
stable.
(i)
and then from
(11),
(9)
and
(10)
we
can see what wages
obviously has a stationary point
whether there
Using subscript
12
are others;
(ii)
whether
human
at full equality,
this stationary point
to denote initial values,
we
have;
g
Proposition 2:
If
(i)
rj
V^
= 0,
monotonically converges to
(/>,
</>„
~
monotonically converges (from below) to g^=f38A{(l-a)(l-P)AR
lft]>0 and a[l + rj(l-(3)]<
(ii)
to$ x = l an d the growth
V
<
4>
(Hi)
If
4>(k), 4> t
<
The
represents
equality.
from
and
left
now
E(4>\1),
(f>
4>
£
G
then 3 <f
1,
<f> o:,
<
we have
= mm >0, and g
(t>
V
(0,1] such that
<
</>„
1
77
= 0, and
However,
all
as
possible inequality levels
soon
as
rj
> 0,
be able to approach
it
line. If it
only points to the right of
A
it
<
<f>\
g„
<
<f> t
—
(12)
<j>
00
= 0.
g„,\
from Figure
1;
the broken line
approaches
it
Whether
at
this line
<f> t
=1
equality will be
full
needs to cut the 45° line
by looking
-1.
}
= mm and
-*<t> ao
some point A,
are in the basin of attraction of full equality.
45° can be determined
a
1
are in the basin of attraction of full
of A, wage and income inequality increase over time.
from above or below
<t>
approaches from below,
from above,
t
monotonically converges
the curve starts below the 45° line and
below or above the 45°
To
(all 4>'s)
g
converges (from above) to g oo = 0.
t
<
and
rate
a -1.
}
(4>(k),l], 4> t
technical intuition of the result can be obtained
either
unstable.
V
V
growth
~
monotonically converges to
then
1,
£.(0,1):
(j>(k)
the
rate g t converges (from below) to g^=fi8A{(l-a)(l-ll)(l+{iri)AR
lfr\>0 and u[l + i)(l-&)]>
4>*
then 3
1,
1
= 1 and
To
approaches
4> x
the
=
1
at
=a+a(i-pyn.
09,
For
(12) to
be
less
than
1 - i.e.
2 needs to hold [note that
converges to
local stability of
when
worker where y
implies that high
is
is
effect
capital
(i)
is
of Proposition
the case where the frictions are so
on wages. The wage
the output produced. Since
get higher wages.
all
is
is
simply
/3y for
each
workers (h^ produce more than low-human capital workers
However, given decreasing returns to human
worker who has twice
not twice
rate
firms have the same level of capital, this
as
much human
capital,
output. Since the accumulation rules (h t+1 as a function
his offspring
(ii)
,
the main result of the paper,
human
returns to scale), a
much
the condition in case
the poor hit h min the curve no longer applies and the system
high that there are no outside option
and
-,
^n - see proof of Proposition 2 in Appendix A].
Proposition 2
(hj)
=1
as rich,
and therefore the gap
process takes us to full equality. Expressed differently,
13
is
wj
capital (constant
does not produce twice
are linear, this
as
means that
getting narrower. Overtime, this
when = 0,
17
the only redistributive force
from our
that emerges
economy where
high
human
human
the
frictional labor
human to
This
is
is
mismatch.
Compared
physical capital ratio was constant in
capital agents are
capital agents.
market
all
knowing who they
economy,
firms, in this
working with lower physical to human
because firms not
to the frictionless
capital ratio
will
than low
match with choose
a
certain level of capital stock that they cannot change.
Next consider the
due to costly matching:
leaves the
17
> 0. There
the outside option effect
workers. Intuitively, a firm,
if it
which
case in
which
is
now
an additional redistributive force
increases the relative
when bargaining with a skilled worker,
worker the next worker
will be
wages of the skilled
has a worse outside option
worse on average. In contrast,
it
has a strong
bargaining position against the unskilled worker because the next worker will be better on
average. This outside option effect,
the larger
is
A
redistributes
from the poor to the
at
high
low
at
levels of inequality
is
stronger
have been that the mismatch
is
why
while the outside option effect
levels.
possible conjecture after realizing the presence of
conjecture
rich,
the gap between the skilled and the unskilled. This gives the intuition for
the mismatch effect dominates
dominates
which
wrong.
effect
Why? The
two counteracting
would always dominate the outside option
intuition
lies
effects
effect.
could
But
this
in realizing that the distribution of jobs
is
endogenous in this economy, and outside options are determined by the composition of jobs.
In the frictionless economy, there are firms
employ low
physical capital firms targeting to
in the
rj
< rf,
economy
of this section
who
prefer to
skill
employ the unskilled workers (low
workers
-
see also Section V.l). In contrast,
firms dislike employing unskilled workers (but given that
all
they are happy not to segregate into two groups, one targeting the unskilled) and this
depresses the relative wages of the unskilled below their level in the frictionless
Note
also that as
we
converge to
increasing until
it
finally reaches g^*.
the growth rate
is
decreasing.
The
In this economy, accumulation
the
less is
This
is
is
Also
full equality,
as
the
the growth rate of the
economy converges
more
is
economy
is
to maximal inequality,
intuition of this result can be obtained
the engine of growth, and the
economy.
from expression
the inequality of
(9).
skills,
investment because firms anticipate the mismatch that will reduce their profitability.
the feature that leads to a negative link between inequality and growth in our model
[see the cross-country evidence
on
this respect discussed in the introduction].
14
V.3.
Remarks and Discussion
First, increasing the
number
human
as there are decreasing returns to
worker
is
in place.
decreasing once
The
by opening
imperfections to
some
i.e.
the return to the
whether
is
a large
For
if
employ
regarding the qualifications of
firms do not have a large
one
mitigate the matching
a
It
When two
continuum of workers, there would be no uncertainty
new employees. However,
number
skilled
can also be thought that in
in practice there are limits to
become, especially when they have diverse jobs and workforces.
large firms can
in
may
job. But, this will clearly not solve all problems.
a firm could
capital of
employ two workers can open one
instance, a firm that will
or two unskilled workers arrive, there will again be mismatch.
the limit,
human
results as long
firm could avoid the mismatch problems:
a series of jobs simultaneously, a firm
extent.
and one unskilled
capital,
would not change our
the other workers and the irreversible attributes of the job are
interesting question
for instance,
skilled
all
of workers per firm
how
And even large
of job openings at the same time; for instance, a large firm
need of an engineer in a given period would face the same problems.
Secondly, decreasing returns to
model has
this respect the
human
a close link to the
Fernandez and Rogerson (1994) where
Tamura
inequality and to
capital plays
(1991)
where
work
it
Benabou
of
a concavity in
an important role in this
result. In
(1993,1995), Durlauf (1995),
one the key functions
creates costs of
leads to convergence to full equality [see
Benabou
1995 for general conditions for convergence in economies with local externalities].
difference
factor,
is
that in our case,
human
capital,
economy analyzed
and in
in section
all
we need
fact,
II
is
without
the usual assumption of decreasing returns to a
this
would not be
assumption, the Walrasian analogue of this
well-defined.
Another
in contrast to the technological externalities in these models, in our
effects are
among
our
results relate to
is
economy
it
is
Newman,
that
the external
A second
1993, Piketty, 1995]. In this
emphasized that the dynamics of income distribution can be non-ergodic,
however, these papers have no frictions in the labor market, and there
wage
all
is
the one of credit market imperfections and inequality [see
others Galor and Zeira, 1992, Banerjee and
literature,
related difference
pecuniary and are derived from the frictional nature of the labor market.
literature that
The
inequality.
However,
as in
our paper, non-ergodicity
different wealth levels receive different returns
15
on
is
their assets.
is
no
due to the
direct relation to
fact that agents of
economy, human
Finally, note that in this
looking; parents,
when
human
consider the rate of return on
qualitative results
would hold
investments;
long
i.e.
as
how much
deciding
as
capital.
long
capital investments are
education to give to their offspring, do not
For our purposes
as there
The Importance
Labor Market
human
"efficient" labor
US
parameter
greater
is
r?
rj,
(e.g.
Labor market
Katz
as a
as
it is
less
frictionless), the
Looking
rj.
than rf
more
the reason for the limited
as
measure of the degree of inefficiency of the labor market. The
-
thus
likely
is
at
unionized
is less
equations (11) and
we
(12),
is
naturally thought to correspond
we
can see that the higher
77.
A
reduction in
77
shifts the solid
consider
two
search costs], then
identical
it is
the sense of a lower
F = F a =F
b
t
economies a and b except that
immediate from Figure
than economy
b.
1
Moreover, there
economy a converges
to
77
is
rich (or
slow
maximum
down
harmful to equalization of incomes, thus
Implication
III.l: Inequality
Implication
III.2:
More
is
more
(as
it
<
more
t]
z
<
up and
inequality).
rf
[i.e.
more
Thus
a has lower
inequality in
77
b will converge to
inequality and
in this
full
equality and
no growth.
economy
is
to redistribute
creates
more
inequality and
with more
efficient
more mismatch.
labor markets.
markets can lead to poor long-run performance.
16
1
likely
the reverse redistribution), a higher level of
likely to increase
efficient labor
rj
exists a set of initial inequality levels
levels,
In other words, since the main role of higher
income from the poor to the
h
wage
that a will always have
such that starting from these inequality
steady growth, while
t]
is
curve in Figure
enlarges the basin of attraction of full equality (the region of decreasing
we
is
not letting the economy become frictionless or near-
are
inequality to increase. In particular, the system
to have a stable region for lower
if
less
1994 and Bertola and Ichino, 1995). In our model, the
et al,
labor market that has easier turnover and
long
institutions differ greatly across countries. Large
mobility costs and the closer are wages to marginal product. Thus a
less are
to one with lower
capital of their
and European labor markets are often emphasized and the
can be treated
the
in these
capital investments].
markets of European countries are suggested
increase in inequality
human
our
of Institutions and Implications for Cross-country Trends
Efficiency:
between
differences
suffices to say that
it
some backward looking element
is
offspring [see Acemoglu, 1995a, for forward-looking
III.4.
here,
the income level of the parents influences the
as
completely backward
These implications are
US
inequality in the
are
with the view that the increasing wage and income
in line
due to more
efficient labor
market institutions (easy turnover, low
and weak unions), however, they question the conventional wisdom that
firing costs
this
is
necessarily a desirable outcome.
We now
Redistributive Taxation:
turn to a simple characterization of the impact of
different levels of redistributive taxation. Let us
on labor income
that
is
More
effects,
is
a flat tax rate equal to t
then redistributed in a lamp-sum fashion 8 This means that there will
.
human
be redistribution from the high
wealth
assume that there
the tax redistribution
is
capital (rich) agents to
poor
agents. Since there are
directly reflected in the transmission of
human
no
capital.
specifically,
(13)
0, + r
wp
h i,M Sw \t
(1-t)w 1 ,+Atw u +(1-A)tw 2/
h 2,t +
(HK+H^^K
P
8W'It
l
,
w u +T(l-X)(w -w u )
2t
W2t- T K™2TW ld
w denotes the actual wages and the
subscripts are again used to distinguish the two groups. Now substituting from (10), we obtain
where
denotes the post tax income and
(14)
0, + f
^(l-^ATj^l-^^^-Artl^l-^KlO
This difference equation, like
properties,
we
again look
as before,
(11),
has a steady state at
0=1. To
investigate the stability
at;
^LR
d
(15)
=a[ i +(
i^ )v . T( i +( i-p )7l)]
d(p
t
Thus, the economy with redistributive taxation will tend to be more stable since the slope of
the curve
(t>
t+l (4>^)
6
(16)
is
near the steady state
increasing in
t.
Thus an
(6
0=1
is
always decreasing in
-QV -(1-/6)(1-A)77
t.
Also,
+ T(l-A)ri + (1-/3)T7]
increase in t shifts the solid curve in Figure
1
upward (and
also
There is also the issue of capital income taxation. We can assume that the capital income is
taxed in the same proportion and redistributed among the retired generation or is not taxed at all. This
does not change our results. The only short-cut we are taking here is that we are ignoring the possible
effects
of labor income taxation
on the Nash Bargaining between
17
firms and workers.
flattens
t,
it).
Moreover,
the system
c/>
t+] (</> t
may become
= 0)
is
no longer always
negative,
and for high enough values of
globally stable, hence converge to full equality
from
all
starting
distributions. Thus;
Implication
III. 3:
Redistributive taxation reduces the forces that lead to inequality in the pre-tax
income and wage distribution, thus
stabilizes the system. It also tends to increase
growth in the long-
run.
Therefore, this comparative static suggests that societies with
systems, again the U.S. and the U.K., should have had a
pre-tax
wage and income
more pronounced
and perhaps more adverse
inequality,
less redistributive
increase in their
on
effects
their overall
performance. Also, interestingly, despite the obvious and often emphasized disincentive
of redistributive activities
on growth,
tax
effects
Perotti (1995) finds that measures of such activities are
positively correlated with subsequent growth.
A
Public Education:
education.
The
effects are quite intuitive.
taxation, reduces the link
more
reliance
similar analysis can be conducted for the impact of public
between
is
public education, similar to redistributive
wage and the
on public education reduces the
and thus increases growth. This
we
a parent's
More
economy with
IV. Inequality Dynamics
predictions.
path].
still
r{*
t
The question
As long
as
in
wage
inequality.
is
two
classes.
Our
(7)
analysis will also give us
[that
is,
17
is
smaller than
such that again there are no separations along the equilibrium
t
G (0).
of public education
most dramatic increase
assume that wages are given by
We start with an arbitrary distribution F (h)
wealth
amount
the least
will demonstrate that the results of the previous section are robust
We
an appropriately defined
wage inequality
With General Income Distributions
to richer starting distributions of inequality than
some new
capital, as a result
forces that lead to increasing
(especially at the college level), has experienced the
we
human
again in line with the pattern of cross-country trends that
observe whereby the U.S., the
In this section,
offspring's
when
the poorest agent
G t+1 (0)
is
and
will be
a
corresponding distribution of relative
more
disperse than
away from h min the dynamics of
,
determined from the following equation;
18
G
t
(0).
inequality can
now
be
_hUtl j[u(i-pynyiZ-P(i-pynfh°dFSh)
;>1
Ki P[Wi-p>nW-P(i-pynJh°dF,(h)
_
)6[i + (i
-gjgjg jjjj^gxg)
flUii-pynWii-pynlPdGXd)
where
when
recall that
/f,
the median of the
is
human
capital distribution at
the poorest agent hits h min then for that agent
,
dynamics of inequality are determined by
possible to solve these kinds of equations,
a
Proposition 3: A) Suppose
i)
= 0,
then VGq,
it is still
G
we have 0.=6min = —
—
t.
In contrast,
Therefore, the
.
dynamic functional equation. Although
dynamics of inequality. The main
features about the
time
t
possible to ascertain a
results are
summarized
number
of useful
in Proposition
converges monotonically to full equality
growth rate of the economy converges monotonically
not
it is
and
3.
the
to g^*.
B) Suppose 7]>0. Then;
(i)
Full equality
is
always a stationary distribution.
There always exist a stationary distribution with two or three groups.
(ii)
positive measure at h min; one group
more group may
also
No
(iv)
If a[l+(l-^)rj]<l
The
first
with
with positive measure at some h above the median; and one
stationary distributions other than the ones in
Maximal
is
have positive measure at the median.
(Hi)
(v)
is
One group
,
then full equality
inequality with two groups
part of the result
is
is
is
and
(i)
locally stable
always locally
immediate. With
rj
and
(ii) exist.
otherwise,
With
not.
stable.
= 0, only the mismatch
as before, this leads to global stability of full equality.
it is
77
> 0,
full
effect
equality
is
is
present
always a
stationary distribution as inspection of (17) shows immediately. Also, the exact same condition
that
To
we had in the
see
what other
plots (17) with
economy
ensures the local stability of this stationary distribution.
stationary distributions are possible,
0j t+1
inequality at time
and
two-class
on the
t [for a
strictly concave. It
vertical
and
ft.
we make
on the horizontal
given value of the integral \d
always intersects the 45° line
19
a
use of Figure
2.
This Figure
axis for a given distribution of
dG (0)~\. The curve is upward sloping
at
t
0=1 and never
for
8< 1
and
it
may
intersect 45°
that 6 h i+\
at a level
at
= 6ht
one more time
,
for
all
j,
at
some 8>
1.
A
we cannot have
thus
below the median unless for
stationary distribution must have the property
equation does not apply,
this group, this
h min Also, by the same argument, the
weight
a stationary distribution that has positive
i.e.
a
group
stationary distribution can have positive weight
.
on
one group above the median.
Which
IV.2.
What
Distributions are Likely to Lead to Increasing
types of distributions of income and
terms of equation
(17) this
Wage
Inequality
skill create forces that lead
to inequality? In
corresponds to investigating the role that the integral term plays
in determining convergence.
To answer
this question
a "poor" agent getting richer are affected
by the
we
how
investigate
the conditions for
integral term.
i
Lemma
mfimum{8 }<a
,
then
is
increasing in
t
The economic
option of firms
is
such that6t <6
get poorer
l
and
0€E(0,,1) get
a
J6
dG (6).
t
intuition of this result
sufficiently high
bottom of the
the
,
t
the outside option of firms [observe that
at
36
t
6
richer.
Suppose
1:
la
distribution
-
is
straightforward.
when
r?
= 0,
The
term
is
related to
these terms disappear]. If the outside
a large value of the integral
who
integral
-,
then there
exists a section
have to accept a very low wage to get employed and
therefore, this group will get poorer while the rest of the population converges to a higher
level of skills
When
and income.
is
the integral term (the outside option) high? Intuitively, the outside option of
the firms will be high
Therefore,
when
if
they can get a highly skilled worker with a high likelihood.
the distribution of
skills is
it is
wage
inequality, convergence to full equality
to the right [that
is,
when
there are a large
is
most
much
difficult
number of
who are sufficiently poor relative to them].
[thus not
is
not high.
the differential outside options of firms that introduces the forces towards increasing
Since
agents
unequal, the outside option of firms
when the
rich agents
still
exist
In this case, the integral term
some workers who
20
are
is
skewed
with a smaller group of
uncertainty about the workers that the firm can get
matching] but there will
distribution
at
would be
large
the second round of
poor enough that they have to
accept lower wages and thus get poorer over time.
Implication IV. 1: Increasing wage inequality
distribution of income
and
skills
we
real
world economies,
If
think of
income distributions
are
is
likely to be
are skewed to the right.
skewed to the
this
is
a blessing
However,
inequality are not as closely related as in our model.
suggest that the skill distribution
increase the risk of increasing
wage
is
and
in the real
The
increasingly
world
a curse. Firstly, real
result, increasing
wage
world income and
skill
and thus according to our
left
inequality will be a less serious problem.
may
a more serious problem when the
recent increase in college attendance
skewed to the
and
right
this
would
inequality.
IV.3. Inequality Cycles
Proposition
3
only dealt with stationary distributions. However, a dynamic system
also settle into a cycle.
interest as
around
it
illustrates
Lemma
1
Whether our economy can generate endogenous
the interaction of the counteracting forces in the model.
suggests that inequality cycles
more
poor agents
are
get poorer,
income inequality
again get richer.
likely to
may
increases,
no
cycles. In fact,
economy
In
odd numbered
to
#!
consisting of three groups:
to
is
H
2
<
limited, but
when they
(17) falls,
we
will find that
two groups
is
they
may
a special case
are a generic possibility.
H
> M= 1 >
periods, the share of the
$i
H
-
3
L
with proportions X H X M and
,
draws the possibility of a two-cycle.
periods, inequality increases and the share of high group goes
Similarly, the share of the
numbered
discussion
prove the possibility of two-cycles by constructing an example. Consider the
.
.
The
some
of
However, we know that with two-groups, Proposition 2 gave the complete
X L such that X L eL +X M+X HeH =l and X L +X M+X H =l Figure
H
is
and because the integral term in
and with richer starting distributions of income, cycles
following
is
be possible in this model. Intuitively,
become poor when income inequality
characterization and there were
We
cycles
may
Low
Low
L
group goes from
group goes up to
L
2
an d of course that of the median/mean group
2
to
>
0j
is
L
0j
L.
up from
In contrast, in even
and that of the High
always
at 1.
in even
numbered
Figure 3 applies, and this curve
is
a tilted
falls
Since inequality
higher in odd numbered periods, the curve in Figure 3 that describes transitions
broken one. Whereas
H
2
is
the
periods, inequality decreases, thus the solid curve in
image of the broken one around 0=1. The intuition
21
of this figure follows from
and hence the
get richer
It
tilt
1.
If
the integral increases, the poor get poorer and the rich
around 0=1 to the
solid curve.
when we have more than
can also be verified that
are possible
cycles of
Lemma
and thus from Sarkovski's Theorem
[see
any periodicity and chaotic behavior can
Proposition 4:
(i)
three groups, three-period cycles
Grandmont,
arise in this
In a three group economy, there always
exists
1985, Li and York, 1975],
economy.
a vector (k
L
,
\
H
,
6/-,
H
Bf, 6 2L,
2
J
thus a corresponding set of starting values, which lead a two-cycle.
(ii)
any
In a four group economy, there
periodicity
and
a
exists
set
of starting distributions that lead to
cycles
of
chaotic behavior.
V. More General Matching Technologies, Mismatch and Implications for
Within versus Between Group Inequality
The premise
that an increase in heterogeneity will lead to
an important role in our analysis so
matching. However,
it
is
more mismatch has played
and was derived for the
far
quite clear that for
many
situations
random
special case of
random matching
is
not an
appropriate assumption. For instance, skilled workers do not look for unskilled jobs nor do
firms post vacancies that are open to
all skill
categories of workers.
Of equal importance
is
that
so far a change in the skill composition affected the investment of firms but not the diversity
of jobs that were offered, and this
issues.
We will find that more
is
again unrealistic. In this section
general matching technologies
the previous sections. Also, an interesting
to
more types of
matching.
worker
most
is
start
If
way
that
is
results of
follow; as higher inequality leads
by defining the polar extreme to the random matching technology;
the matching technology of the
is
do not change the key
observed in the data.
economy
allocated to the firm with the highest
skilled
to analyze these
jobs becoming available, the between and within group measures of
inequality will be linked in a
We
new prediction will
we want
allocated to the second highest
is
amount of physical
and so on.
rule applies within the set of separated workers. This
If
is
there
is
II.
We
then the highest skilled
capital,
then the second
a separation,
clearly the
Walrasian auctioneer chose in the equilibrium of section
22
efficient,
efficient
then the same
same allocation that the
can give the definition of
efficient
matching
a little
more
formally,
Definition: Let us define for every worker
(!„,(/)=
j,
and
ds
I
similarly for each
firm
J seSv& >hj
s
i,
fip(/)=
/
Js€.S F :k :.>k
ds where SF and S w are the
and
match. Then £lF (i)
and
offirms and workers who are looking for a
sets
i
are respectively the ranks offirm
ft^X/)
i
and worker j in
in the set of workers looking for a match. Let us also again use the notation (ijjET* if i
j will be
matched
either
The matching technology
together.
=>Viv
efficient
iff
The matching technology we
(1-q)
will use
and
a hybrid
is
between the
that at the beginning of the period,
all
the firms before deciding which one to apply to.
only a probability q that he will correctly assess the firm's type and
is
the costs of search, and the
he will be in
(1-q),
efficiency of the
economy we
analyze
effect
applying to a random firm.
matching technology
still
is
77
= 0. This removes the outside option
It is clear
For the
effect
2:
also
it is
all
random and
we use the
q>0, V
This lemma
no second
from the model and
notation
efficient
(k',h')
(k',h'J
from
(77
is
a
stage
simplifies
> 0) would
would not change the
important that, since there
they were both chosen to match
For
is
to see that the presence of the outside option effects
analysis
distributions. Further,
Lemma
a separate issue
take the costs of mobility to be sufficiently high so that there
the subsamples selected for
t, if
is
has costly search. In fact, to simplify the
again add an additional force towards increasing inequality, but
time
random
a motivation consider
However, the degree of
at
a
As
otherwise, with probability
results.
and random
efficient
among themselves, and
However, there
the exposition.
{i).
of firms and of workers are chosen to match randomly
the case where a worker looks at
matching, that
^F {i')<^F
(i\f)E<P, then
the remaining q of the firms and workers are matched efficiently.
we
<=>
F (i*)<nF (i) and (i'/jET, then Cljj ')<(!„(]).
we assume
technologies. In particular,
analysis,
(ijjE'P
(l
=>v/: £ljj*)<£ljj).
or (in) if Cl„(j)<nF (i)
proportion
is
and
Ctjj)^^)
(i)
or (n) if ClJj)<(lF (i)
our
the set offirms,
rest
of
continuum of workers,
matching are identical to the overall
E 3*,
to denote h' and k' matching together
efficiently.
ETV F/h')=P/k'J.
states that if a
firm and a worker will match together
23
when
selected for
matching, then they must have exactly the same rank in their respective distributions
efficient
in
(i.e.
terms of the definition in the above paragraph,
we
will always be in case
human and
surprising that given efficient matching and complementarities between
capital, a
stronger than this:
is
instance, the possibility of
matching with
To
high
a
workers of different
it
skill
two
it
human
e
level
and since q > 0,
this
the better worker. For
result of
no
clustering.
all
would
when they
human
small enough that this
e
know that
Moreover, we
must have the same form
transformation of P
Now
both selected for
know
and thus we
as
is
worker
capital of the
investment
its
matching with
profitable,
hence the
the subsamples of firms and workers selected
each other [that
F and P„
is
therefore, these
t
F must be
t
a one-to-one
a lot about the equilibrium distributions.
the profits of a firm with physical capital k are given by
(18)
(k)={\-q){l-li)Ak
Tr
t
It is
t ],
are
give the firm a strictly higher probability of
q > 0, we can find
Why?
ruled out.
is
worker could increase
for efficient matching are identical to the initial distributions
distributions
firms; for
k expect to match with two
matching. But the profits of a firm are increasing in the
less skilled
one
less skilled
capital in the case
employs, and the firm that matches with the
by
physical
firms choosing the same level of physical capital and one
worker, and the other with a
levels of
by
rules out the possibility of clustering
suppose that two firms of the same capital
see this
efficient
not
highly skilled worker and low quality worker will not match together, but the result
lemma
of the
It is
(i)).
clear that
all
firms have to
l
-a
jv adF {vyq{l-^)Ak
l
-a
h a -Rk
s.t.
t
make
(kJi)eP
t
the same level of profits. Thus,
7rt (k)
= ir
t
for
all
levels of
investment k that are chosen in equilibrium.
Next we can
(19)
also write the first-order condition for a typical firm:
(l-aXl-qXl-^k^fv^F^+il-^qil-^k^h^R+aqil-P^k
J
Lemma
3: For
constant.
For
This
q=l,
all
is
the physical to
q<l,
human
capital ratio,
fi(k),
1
^
-
for workers matching
1
—
dk
|
y
=0.
'
efficiently is
this ratio, jx(k), is decreasing in k.
a very strong result.
It
demonstrates that for
24
q<l,
as well as
the workers
matching randomly,
those
allocated
equalization of incomes. Let us
capital ratio
with
full efficient
to
matching will
efficient
human
understand the result of the constant
first
matching (q=
we
In this case,
1).
forces
create
towards
to physical
are very close to the Walrasian
allocation with the only difference that because of the ex post bargaining, physical capital does
not receive
its full
order condition
marginal product and thus firms underinvest. In
(19)
with q=l,
to physical capital ratio and
we
can see that
make zero
profits.
constant; but each firm can also be selected for
of physical capital will lose
when matching
4:
For
Proposition
q<l, a mean-preserving
Suppose
5:
t]
allocated
(8).
Lemma
work
at a
-
strict
3, it is
initial
human
/x(k)
is
concavity of the profit
Thus to be compensated, such
human
first
and, a firm with a high level
a firm,
to physical capital ratio.
straightforward that:
spread o/F/.J reduces investment
= 0. Consider an
constant
q<l, and suppose
consider
random matching
should work with a higher
efficiently,
all
Now
case as in equation
Next, given mismatch established in
Lemma
firms have to
more from being randomly
random matching
function in the
all
by inspecting the
fact,
human
capital
and
output.
F and
distribution
a
corresponding distribution of relative wealth Gq.
(i)
Suppose q=l, then VG^,
gt=g~* for
(ii)
V
all
is
In this
hold
as
long
unrealistic as
for all
t,
thus inequality self-replicates
and
the
growth rate
t.
0<q< I
economy, gp
G,=G
and V G&
always
less
G
->
t
Gx
where
Gx
exhibits full equality.
than g^* and monotonically converges to
economy with hybrid matching technology,
q<l
as
it
relies
all
The growth rate of the
it.
the key results of the section
[the limiting case of fully efficient technology,
on an
invisible
hand arranging the
efficiency of the matching technology
is
q=l,
is
III
obviously
right matches]. Also, the degree of
another measure of labor market efficiency and
we
can see that a higher q (more efficient matching technology) makes inequality more longlasting.
Therefore this result complements the one obtained in section
labor markets are
more
likely to lead to higher
25
wage
inequality.
III.3
that
more
efficient
V.2.
Between and Within Group Inequality
from the
Finally,
we
analysis of this section
can draw some implications about the
interaction of between group and within group inequality. In this section both of these types
of wage inequality are present. Let us define the expected wage of a worker with
W
by
This wage depends on the equilibrium choices of firms. In
capital hj at
time
particular let
P be the equilibrium
t
(hj).
t
distribution of firms and kj be such that
t
W {h^{l-q#A(hfjk
(20)
l
clearly for h,
difference
< h 2 W^hj)
what we
is
W^hj) >0.
-
,
measure
will
as
(h',k')
£
"Pv
Then
-a
dP (kyqjlA{hy{ky- a
.
t
t
And
human
these
If
two
skill
premium or
the "skill"
groups are observable, this
as
the between group wage
inequality.
There
is
also within
group wage inequality in
wage distribution for workers who have human
and the
rest
suppose
we
and in
fact,
of
its
mass
distributed as
is
take a mean-preserving spread of
the gap between the
inequality. But
its
a
k
~
l a
F This
t
.
and the more
mean
falls
skilled
too),
more
1-q)
two
see this, note that the
mass of q
at
fiAQi^fQc^)
where k has distribution P
will
mean
is
that there
larger,
is
more
t
(k).
Now,
inequality
hence more between group
then from
Lemma
2
P
t,
also
and the within group wage inequality
between group inequality
sometimes (with probability
to these jobs, and hence,
capital hj has a
a mean-preserving spread,
t
increases. Intuitively, higher
two
less
when F undergoes
becomes more spread (though
jobs. But,
(ZA(h^)
economy. To
this
leads to a
identical
more
diverse distribution of
workers end-up randomly allocated
diversity in these jobs implies a larger
wage gap between these
identical workers.
Finally, although
always
move
together.
it
For
is
often the case, between and within group inequality do not
instance, in the 1970s the U.S. experienced an increase in within
group inequality but no increase
explained
if
q
labor markets].
is
changing
From
(20)
in
[e.g. as a
we
between group wage inequality. Such
a pattern can be
consequence of the changing organization of firms and
can see that an increase in q will increase the wages of the h 2 [the
high skilled] group more than that for h, [the low skilled] group, leading to an increase in
between group inequality but
it
will also increase the proportion that are
thus will generally lead to a compression of within group inequality.
26
matching
efficiently,
VI. Concluding
This paper has presented
distribution.
equilibrium model with endogenous job and
a general
The model provides
Comments
a simple
skill
framework with which to analyze the interaction
between labor markets and the evolution of income and wage inequality. Our investigation
has revealed that labor market frictions introduce a
one,
which we dubbed mismatch,
number of
from
redistributes
rich to
redistributive forces.
effect
was increasing with the degree of inequality and
but can increase starting from high
initial levels
is
wage inequality
is
more
less redistributive taxation,
likely to increase in
and
with the diverse experiences of
also
less
the
decreasing at low
of inequality.
This framework has also enabled us to carry out simple comparative
that
we found that
this led to the conclusion that
the dynamics of income and wage inequality are non-ergodic: inequality
levels
first
poor workers while the second,
the outside option effect, redistributes from the poor to the rich. Moreover,
second
The
economies with more
statics.
We found
efficient labor markets,
public schooling. Interestingly, these patterns are in line
OECD
countries over the past twenty-five years. Finally,
we
found that our model implies that within and between group inequality should move
together which again appears to be the case in the data.
This paper
is
a first attempt at a
inequality and labor market organization.
framework for the
As
such,
it
leaves
theoretical analysis of
many
wage
issues unresolved. First,
we
have restricted our attention to parameter values for which there are no separations or
disagreements along the equilibrium path. Thus our model
of
unemployment and
inequality
which
are interesting
makes the model technically much more complicated
is
not well-suited to analyze issues
and important. Introducing separations
[see for instance
the
full
dynamic models
of search with ex ante heterogeneity of Sattinger (1995), Shimer and Smith (1995), Burdett and
Coles (1995)], but this
is
an important extension to consider [Acemoglu, 1995b]. Second,
natural to question whether segmentation in the labor market
limiting
may
also
mismatch and which of our conclusions
open the way for
a theory of
is
likely to arise as a
Is
mismatch higher
way
of
will be affected. Future research in this area
endogenous segmentation in the labor market. Finally,
our model poses a number of new empirical questions; can we find
mismatch?
it is
in labor markets
with more inequality?
markets with higher inequality worse?
27
Is
a
good measure of
performance in labor
~
Appendix A: Proofs of Lemmas and Propositions
Proof of Proposition
in Proposition
with capital
1
1:
Take the equilibrium with the physical to human
(i)
and suppose
level
k makes
workers are paid their marginal product
all
capital ratio as
Then
as in (6).
a firm
profits equal to
Tr(k)=Ak
1
(Al)
l
~a
h a -Rk-w(h)h
-R-Rk-—Rk
\-a
.
\-a
=0
Thus
irrespective of the value of k.
the allocation where
at
firms are allocated according to
all
the Walrasian rule (see the definition in section V.l) and k/h
Given the value of
profit.
h,
is
constant,
make
firms
all
zero
each firm would also exactly choose this level of capital ratio
since
l
(A2)
/
Thus no firm wants
equilibrium.
to deviate and change
At no other
capital stock, a firm
equilibrium in which firms are
Given
(ii)
human
(l-a)A) a
k=
7r (yt)=0 at
R
its
h.
and the allocation
capital stock,
would be maximizing
profit, thus there
is
an
is
no
at a different capital ratio.
a constant physical to
human
we
capital ratio,
can write the wage of worker with
capital h: t as
(A3)
w(h)=aA
Thus from
(3)
({l-a)A\ a
hjf
R
in the text;
\-a
V rH»)"V
(A4)
for
(iii)
all
j.
h..
6 .,=—j-
Next,
From
the
and since
which firm
transitions are linear,
above equations, the
( (\-a\A\~
<— a -1 and given
g =8a ±
at
all
2:
(i)
grow
<£ t+ i(0t)
1
constant
all
-a
a
h,
and hence,
'
'
grows
the time, this
growth
the
at
is
rate
the rate
rate.
in the text; stationary points (distributions)
with the 45°
28
R
agents
all
too, hence also the output
Consider Figure
correspond to the intersection of
is
h t+l =8aA
of
capital
that the capital ratio
level capital stocks
Proof of Proposition
human
l
(l-a)A
line
and
</>
t+
i($ t = 1)
=
1,
thus
full
equality
is
always a stationary point.
Now
consider
0M(^o)
(A6)
This
negative for
is
all
>0
17
60 M
_
and equal to zero for
[a0 gl + (l-^)(l-A)77
^
(A7)
-a-/3)q-^
=
.
= 0.
17
aC ]x[l-(l-i8)Ai7(/)>(l-j3)^]
1
[l-(l^)A^r + (l-)8)Ar,] 2
aC ]x[^ + (l-^)(l-A)7 0r-(l-iS)(l-A)7
1
+
[(l-/3)AT7
7]
7
.
[HI-^At^I-^Atj] 2
This derivative in (A7) always
and never
limit
(ii)
Next for
t
and
=l,
all
r\
>0, the curve
0=1 from
we
45° line at
above,
it
and
(9)
in Figure
must cut the 45°
= l from above
t
available
77
+ (l-@)rj]<
/30 3
1,
1).
must cut the 45°
horizontal coordinate of A, 0(X). Then,
Euclidian distance between
two
V
t
G
points. Thus,
line
attraction of full equality; or in other words,
growth
(9)
rate at the limit
is
given
shows that investment
Similarly,
thus for
all
are defined
between
O
V
to the
t
G
left
1,
is less,
of A,
(9)
and
we
where
diverge
min
E
and thus
cuts the
either
[full details
cannot turn from convex to
1
as in
at
1
shows that
Differentiation
Figure
1.
some point A. Let us
(0(X),1),
<
d[0„l] where
t
A
-» 0oo
this limit
is
call
d[.,.] is
the
the
are in the basin of
=
l.
And
again the
reached, inspection of
lower.
full equality.
functions, and {0 t } forms a
thus over a closed and bounded
29
going to
if it is
will be defined below, d[0 t+1 ,l]
from
origin
rate at the
the function in Figure
d[0 t+ i,l]
and before
is
The growth
points to the right of
O
(4);
thus growth
((^n, 0(X))
by continuous
and
by
once
(0(X),1),
all
from the
unambiguously positive
is
t
starts
it
once before 0=1. Evaluating (A7)
request]. Therefore, the function in Figure
a result, the curve
=
negative at
concave and hence, the shape of the function can only be
As
= 0,
= 0.
line at least
t+1
t
upon
for
(with a slope of less than
3
30 t+1 2/d0 2 <O or d0 t+1 2/d0 2 >O but then d
T7
globally stable.
is
(4)
1 starts
get (12). This implies that for a[l
t
and when
positive,
is
Thus the system
45° line.
obtained straightforwardly from
is
approach
at
below the
falls
exists
set.
The dynamics
>
d[0
t
,l]
and
for both groups
monotonic sequence and
is
defined
This implies that {0,} must have a
1
1
1
we must converge
convergent subsequence, therefore,
below by h min
,
this
sequence {0
t
}
1
to a certain point. Since
can only tend to zero,
if
h 2t goes to
h lt
is
bounded
Thus we need
infinity.
to check whether growth in h 2t can be sustained with a proportion (1-X) of the agents
accumulating and the
rest at
bound h min In
the lower
.
this case the
law of motion of h 2t would
be given by;
(A8)
We
/z^^S^^^/^-^Cl-ie^/^^lHl-A^^^l-iS^^/c/-^^
will
show
contradiction:
h 2 can grow only
k
if
t
This would imply from (A8) that
linear in
is
h2
x
level of
function of h 2
,
h 2 such
.
(at
a constant rate)
So
let
is
to be a concave function of h 2t and therefore, there
their lower
bound, h min Then,
.
(iii)
more than
Finally,
equality
is
we
a
#,
<
that represented
4>(\).
by
is
linear in
set
is
set. If
maximum
is
From
a
k
(9),
unique
t
is
immediately seen
level of
h 2 such that
growing when the poor
By
curve intersects the 45°
stable ergodic sets;
line, it
Thus, the economy that
starts
with a
level of
no long-run growth. This proves part
= 1,
full
the
which corresponds to 4>*=1 and the economy has
the one described above in
must do so twice, thus
maximum
(a)
is
hit
definition there can be
the function never intersects the 45° line before
unique stable limiting distribution that
.
t
a stationary point.
0(X) exhibits
inequality
h2
are suitably defined
turn to the alternative configuration with a[l + (l-0)r)]>l. In this case,
not a stable ergodic
unique ergodic
= h min /h 2
is
k were an everywhere concave
capital of the rich class stops
<£ min
no other stationary point when
inequality
if
the same conclusion would apply a fortiori.
h 2t =h 2t+1 = h 2 and thus the human
C
t
not possible for h 2t and there exists a
Similarly,
.
not possible by
is
us suppose that k
=Bh 2 +Ch 2f where B and
h 2t+1 = h 2t = h 2
that
h2
growth
constants. But this implies that constant
unique
bound
that h 2t growing without a
4>*
<1
(ii).
Alternatively,
if
the
and then there are two locally
inequality without any growth and (b) another locally stable
ergodic set with a certain degree of inequality but also positive growth but at a rate less than
g^. This proves case
(iii).
Proof of Proposition
<
A9 >
3:
A)
?j
= 0, then
W£ V >w
f°r
30
Thus
B)
jt
(i)
(ii) 0;
0; >t
t
>1
=l> Vj implies that
=lj implies that
stays there.
is
implies that 6 )l+l
Take
a
0j t+1
<6
0j I+1
=l
and for
ht
=
l
Vj.
jt
<
Thus
1, 0j, t+
Jit
equality
full
at
is
h 2 This
.
at
to full equality.
a stationary point,
irrespective of the distribution. Thus, the
proportion of the population X L to be
above the median
group
h min and suppose
a
at
the median
proportion X H
will be a stationary distribution iff
h^iUil-^vjh^il-^iX^il-X^X^XXin]
(A10)
Thus any vector
given these
(X L ,
XH
,
hj) that satisfies
two equations such
(A 10) will constitute
a stationary distribution
and
vectors always exist.
This follows from the inspection of equation
(iii)
i>0 hence convergence
intersection with the 45° above 0=1, hence at
(17)
and Figure
2. (17)
can have
at
most one
most one group above the median. Also
it
cannot have any intersections below 0=1, thus no group below the median other than one
at
h min
(iv)
.
Finally,
Take
t
we
can have a third group
< 1. For
convergence
at
we need
the median.
t+1
>0
.
t
This implies
ffi + (i-pyn]-P(i-pynJPdG,(8)
or, equivalently,
@-= 8[l + (l- 8)T
(A12)
J
Similarly, for
(A13)
For
t
j
7
](0r-0f )-i8(l-i3)77 (l-0,)/0^G r (0)>O.
>l, convergence requires
®
t+1
<0
t,
thus
a
+
=p[l+(l-(3)ri](O-0 )+P(l-P)r1 (e-l)f6
local stability, the relevant conditions can be written as
(A14)
@-(0;=i
@
+
(0S
=1
v;*7,0^r)>o
Vz*;,0J^r)<O
31
a
dG (8)>O.
t
.
a@
d®(e'=i)
<0 and
These in turn are equivalent to
+
(6»;=i)
>0. Thus,
local stability requires
36
36
3®(6i=l)
za-.cn
(A15)
^^= 8(a-l)
+ iS(l- S)r7 (a-l) +iS(l-j8)77<0.
^-^= 8(l-a)
+i
)
)
and
(A16)
i
8(l- 8)r? (l-a)^(l-iS)77 >0.
i
(A 15) and (A 16) are equivalent to each other and in turn to our condition for the
full
(v)
equality in the text a+ai(l-/3)r/<l
The diagrammatic
representation of the solution to (A 10) corresponds to Figure 2b
intersection above the
Proof of
Lemma
1:
median
is
obtain
80-
^-^[l + (l-i8)rj](aC
1
-l)^(l-)8)77/^GX0)
36t
This expression
sufficiently large,
level of
is
a
unambiguously negative
is
it is
income (and
most
skills).
decreasing in the
amount of
.
That
is
if
relative inequality is
the gap between the poorest agents and the
difficult to close
This expression
6t <a
if
Xa
is
(A18)
^
4:
(i)
From
inequality.
we
the text,
can see the condition for a cycle to be
^[i + (i^h](^r-^(i-^)^[^(^r + (i-^-^) + ^(^)
8[l + (l-^)r7 ]- 8(l- 8)77 [A
j
J
L
)
iS[l + (l-i8)77]-^(l-/3) T? [X
(A20)
mean
also increasing in the integral term, thus its zero
t
Proof of Proposition
(a
necessary), hence the local stability.
From (A 12), we
(A17)
6
stability of
L
a
]
w
(^r + (l-A -^) + A (^r]
L
(4r + (l-A
L
-A //) + ^(^r]
^ ^[iHi-^i^r-^i-^M^rHi-^-^^gfn
L
w
L
^[l + (l-^)77 ]-/3(l-)8)77 [A
32
(^r + (l-A -^) + A (<r]
^_
(A21)
j8[i + (i-jS)i ? ](^r-iB(i-/s)T ?
[A^r + (i-^-A w )^ w (gy)
(A22)
X L 6 l+ X H6 l=X L + X H
(A23)
X L 9^ + X H6^=X
L
Thus the problem of finding
For
six equations, (A18)-(A23).
from the bounded,
closed,
such a vector will always
(ii)
a cycle
all
convex
exist.
is
set [0,1]
Thus
6
]
,
L+
XH
L
to find a vector (X
values of a,
tt
j3, rj
into
,
.
.
XH
,
6^-,
6^,
L
,
2
6 2 H) to satisfy these
these six equations are continuous and
itself.
Thus by Brouwer's
fixed-point
map
theorem
a two-cycle always exists.
This part of the proposition follows part
(i)
with nine equations in nine unknowns. The
details are omitted.
Proof of
Lemma 2:
First
cannot have any atoms
human
highest
definition,
suppose
either.
capital will
F(.)
then
<x(k)
3 a set
of firms
efficient
match with the highest physical
cr(k')
has
show
that in this case P(k)
matching technology, the
capital
and so on and thus, by
no atoms by contradiction. Suppose P(k) has an atom
such that
all
iEa(k') have the same capital
positive probability q, firms in a(k') will
and h 2 where h 2 >h]. But,
could increase
efficient
first
F(h)=P(k) v(h,k)GP.
no atoms, with
levels hj
We will
Then, by definition of the
We will now show that P(k)
at k',
has no atoms.
its
capital
matching. For
all
by
e
this
level.
Since F(.) has
match with workers of human
capital
cannot be an equilibrium because one of the firms in
and make sure that
q>0, we can
it
find a small
meets h 2 whenever
enough
e
it is
selected for
such that this strategy
is
profitable.
Now
show
suppose that
F(.)
has an
that F(h) = P(k) V(h,k)e0>,
then k' = k"; and
(ii) if
it is
atom
at h'
and denote the measure of
sufficient to
prove that
(i)
if
this
by
size at k'
which
33
is
To
(h',k')eP and (h',k")6P,
(h',k')e:Pand (h",k')GlP, then h' = h". (PS: in other words,
have an atom of exactly the same
f(h').
P(.)
must
the level of physical capital that matches
with
when matching
h'
Suppose
(i)
efficiently).
k", such that (h',k")£!P. But given the
3
concave, thus either k' or k"
Suppose
(ii)
£
i
i
i
£
would
tf(k')
then
3
would
increase
Lemma
Proof of
3:
capital at h', the profits 7r(k)
its
investment by
investment by
its
and match with
e
h>h'
h'. If
then some of
and match with h with probability
e
Take two workers with
(i)
different
Hence
h'.
human
will
make
human
the same profit level,
a contradiction.
<
capital levels h,
by
jii,
and
\i 2 .
From the
fact that
h 2 and
< k and
2
both firms
can write
k^l-^A^-RYk^l-^A^-R}.
(A24)
> k
Since k 2
human
as
1;
long
capital ratio
have constant
Now
as
both firms are making positive
decreasing.
is
human
capital
definition of the
And
if
profits,
fi 2 ,
the profits are equal to zero,
by noting
that dk/dh</z(k),
thus the physical to
p. x
=
a
(l-a)(l-p)An(ky +a(l-P)Ati(ky-
this implies that the firm
is
making negative
profits
and
human
capital, the rate of return
constant. Since
fx(k) is
profits.
firms
from
(19)).
to physical capital ratio [this
from
to physical capital ratio being increasing
(A25)
all
i.e.
/x 2 ,
a unit increase in capital will
i.e.
by more than the current human
human
ju(k) is
<
fj, t
to physical capital ratios (this can also be checked directly
re-write (19)
human
increase
But
we
capital ratios
h<h',
q. If
denote the capital stocks that these two workers will work respectively with by kj
the corresponding physical to
is
not profit maximizing. Hence a contradiction.
such that (h,k')£CP for h different from
o(k')
increase
is
human
a
Thus
_1
iJL(k)
all
(A24)].
is
by
Hence
<i?
firms must be
making
zero-
constant and independent of the distribution of
on physical
capital
and
total
output are independent of
heterogeneity.
(ii)
Take
k
and k 2 >k
t
a range of values of
t
and
T(h 1 ,k1) = T(h.2,k2) = 0.
is
decreasing in
hj<h 2
T(.,.)
k and
evaluate (19) in the text and
such that
(hjjk^EJ1
and
has three terms (not counting
k and does not depend on
let
us
)€.7>
(h. 2 ,k 2
.
R which
is
call this T(k,h).
Then, by definition
constant): the first
h, this therefore implies that the
terms have to be higher for k2 than for k^ This implies that either
/x(k),
capital ratio, has to be decreasing in
k or dh/dk have to be increasing
sufficiently close to each other /x(k)
is
decreasing
34
if
and only
if
dh/dk
Take
sum
of the other
human
the physical to
in k.
is
term
But for k and k2
t
increasing (since
by
Lemma
dh/dk
2
Lemma
Proof of
Proposition
human
in
(ii)
is
5:
(i)
continuous) and therefore
4:
This follows from equation
q= 1:
human
since
3,
(19).
from
Proposition
as in
wages are a concave function of human
Proposition 3 for the case of
Appendix
far
has to be decreasing.
capital ratios are constant
thus inequality self-replicates
capital,
From Lemma
So
fi(k)
we have
B:
77
=
Lemma 2,
wages are linear
1.
capital,
thus the argument of
applies.
Dynamics With Non-linear Transitions
analyzed a simple
economy
in order to clearly illustrate the innovation
of this paper. In particular, the decision rules and the accumulation equations were linear and
this led to the result that inequality self-replicates in the competitive
is
well-known that the competitive economy
Tamura,
often lead towards equality
(e.g.
we proposed
interact
here
would
whether our main conclusions
transitions that transform the
level for the offspring.
That
is;
(i)
may
be wondered
it
appendix
and the wage of
a
how the new
important to determine
we
take general non-linear
worker into
forces towards equality, thus the
(ii)
forces
is
a
human
framework, we demonstrate the robustness of our
this
full equality;
also
we had
e t+1 =Sw
capital
results.
economy with mismatch
outside options introduce forces towards inequality and
,
thus the education expenditure was linear in wage income.
h t+1 =min{h min ,et+] } thus the human
,
expenditure and in particular,
We know both
it
capital of the offspring
did not depend
on the human
B1 )
where the range of
we assume
Vi = *KA<)'
\f-(.,.)
is
[h min ,+ 00).
35
was
linear in education
capital of the parent directly.
of these assumptions to be untrue (see Borjas, 1992,
1992 for empirical evidence). Instead here,
(
it
lead to a multiplicity of limiting distributions.
In the text
And
Thus
are robust. In this
mismatch introduces
converges faster to
may
Using
1991).
capital
it
also includes certain redistributive forces that
with these. In particular,
human
economy. However,
Cameron and Heckman,
Further
we assume
that
a
\p is
continuous, strictly concave and strictly increasing in
a=A ( -—
(\-a\A\-'°
both arguments. Also for notational convenience define
IV with
Proposition B.l: Consider the frictionless economy of section
inequality Gq. Then, for all
G^
limiting distribution
Proof:
The
\J/f.,.J
such that ^(aah mtn,hmiT)> h min)
G
t
and since
2,
w
= ah Concavity of
.
t
t
xf/(aah. min ,h. mil)
> h min
,
all
\f/
economy
exhibits full equality.
Proof:
The
labor market
and convergence to
To
look
at
see that
full
it is
of inequality
w
l-a
VfP-W
a
faster
given respectively
G& G
Then, for all
converges to
t
G„
that
hadF (h
t
[I
)}
\
K
h
»\
•
immediate.
than in the competitive economy, take two agents h lt
by
\l/(.,.)
t
a
is
= 0.
= /3y Thus
l-a
equality
rj
.
.
t
/
a
a point such
always faster than the frictionless economy.
the distance between their offsprings in the competitive
economy
h 2M
h 2t+l
and
economy and
Thus we
are
l-a
l-a
kSL
pi
< h
2t
and
the mismatch
comparing
'"
(B3)
ty(aah ltJi lt)
since
is
unchanged, in particular
is
(B2)
to full equality
is
unchanged,
dynasties will reach this level as t-»oo
J> hmin> and starting distribution
Convergence
are
ensures that there
Proposition B.2: Consider the economy of section IV, with frictions and
such that \}/(aah min,h mi
distribution of
converges to a unique stationary
labor market and wage determination in the competitive
\l/(cxah* ,h*)
a starting
that exhibits full equality.
thus from our analysis of section
that h* =
Th en;
R
x//
is
always increasing in
of the values of
h
lt
its first
t
a
y{B(l-p) * a[fh dFt(h)]
)'
h?
t
argument, the second term in (B3)
and h 2t and thus the
,
a
,
is
h lt
smaller irrespective
faster convergence.
This proposition establishes the generality of our result that mismatch introduces
additional forces towards equalization of income, or alternatively that
relatively rich to the poor.
36
it
redistributes
from the
We now turn to the economy with outside options. Investment in this economy is still
given by
(9)
since the production side
is
exactly the same as in the
main
text.
Thus;
l-a
(B4)
a
a
w(h )=c{fh dF (h)\
t
where
and
c=j6+£(1-/3)t7
t
h"-d^fh
a
dF (h)
l
d=P(l-p)T).
Proposition B.3: Consider the economy of section IV, with frictions
an open
of continuous functions
set
inequality
G&
such that
G
\p(.,.J
\l/(aah min) h mi
with
and
J > h mtn and
,
converges to a stationary distribution
t
t]>0. Then, there exists
starting distributions of
Gx
that exhibits limiting
inequality.
Proof:
It suffices
show
to
that a limiting stationary distribution with inequality
Consider the following where a proportion X of agents are
For
this
we
L^(^C
(B5)
h
0(o:ah min ,h min)
satisfied.
We
h min and
possible.
(1-X) are at h*.
require,
\-a
Choose
at
is
*=W
C
hZ-d
\-a
+
(l"^
= h min + e then
can choose the
a
+ (i-^*'
)
*"
°
C
hL-d
^
+
(l"^ 1" AmJ
m+ (l-X)h
for h* large enough, the
rest of ^(.,.) in
any form
(in
\h mm )
equation in (B5) can be
first
an open
*"
set)
to satisfy the second
equation.
Therefore, this proposition establishes that outside options in general introduce forces
that redistribute
from the poor to the
rich,
hence even
when the competitive economy exhibits
decreasing wage and income inequality everywhere, the frictional
options can have increasing wage and income inequality.
It
outside
thus follows that our conclusions
are independent of the linear transitions utilized in the paper.
37
economy with
References
Human
Acemoglu, D. (1994);"A Microfoundation For Increasing Returns in
-
Accumulation"
MIT Working
Paper.
-
Acemoglu, D. (1995);"Unemployment and Inequality:
-
Banerjee, A. and A.
Newman
Capital
A
Dynamic Framework"
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41
Young Men"
Quarterly
LU
O
CO
LU
cr
O
LL
LU
DC
O
LL
Appendix C: Wage Determination, Investment Decision By Firms and
Stability Conditions for Infinite
Sampling
Wage Determination
C.l) Equilibrium
Consider the game
respective probabilities
(8
illustrated in Figure
and
1-/3
CI. Nature moves
whether the worker or the firm
and decides with
first
will
make
the
first offer.
This introduces the feature that the bargaining powers are potentially unequal. After the offer
(worker) can say Yes in which case there
of the worker (firm), the firm
offered wage.
Finally, the
offers.
Or
answer can be
However,
is
a probability
if
this
is
for any
the answer
s
as in
may
No
in
be
Out
in
which
Binmore, Rubinstein and Wolinsky
let s
come
is
a
No decision,
Ww and W
new partner. We
will analyze this
VF
And
.
t
and that to the worker
also
Vw
.
denote the value to the parties
game
These
the
if
(obviously these values are the same irrespective of whether
F
voluntary end to the relation or a separation due to Nature's choice).
assumptions in the
there
tend to zero.
determine in the distribution of total output y
by
(1986), after the
to an end before the next round of offers and
Let the value of the game to the firm be
relation ends
the
at
which Nature again randomly picks another party to make
that the relationship will
and then
agreement
both parties look for a new partner.
case
the case, again both parties have to look for a
s
is
And
it
given our
text, these "disagreement" values are as follows:
Ww^MK^k dP {k)
WF=(l-P)riAkl-afh adF (hj
l
-a
t
t
Also to
start with,
Let us also suppose that
has offered
we
are
w to the worker.
s)V w +sW w and
(CI)
suppose that the relation
if
he says
If
No
is
jointly beneficial, so that
on the higher branch so
that the firm
the worker says Yes he gets w.
he would get
Ww
w F =max{Ww
C-l
.
If
is
making the
WW + W
offer
F
.
and
he says No, he obtains,
Thus, the firm would offer
;(l-s)Vw+sWw }
y>
(1-
Q
I
O
-<
o
CD
O'
a
7T
c
CD
C/3
CD
A» y
J^
CD
-^
J.
o
c
T]
Q
m
o
D
CD
O'
a
V
Similarly,
on the lower branch the worker would
w w=y-min{WF
(C2)
And
offer,
;(l-s)VF+sWF }.
obviously by construction both of these would be accepted. Since Nature always chooses
between the firms and the worker making
and
V
/3,
offers (the
two branches) with
(1-/5)
then
Vw =Pw w+ (l-P)w F
(C3)
Now
probabilities
suppose that
V W >W W
V F >W F
and
,
then,
by
.
rearranging,
s
cancels and
we
obtain
Vrto+{\-P)Ww-fiWp
(C4)
This
is
obviously greater than
V F >WF
Ww
(if it
were
less
y>
Ww + W
F
would be violated) and
similarly
.
WW >VW Then we can write
(C5)
Ww=(l-P)Ww+P\y-(l-S)VF-sWF
which on rearranging terms yields W w + W = y, thus a contradiction and therefore
the case that W W <VW By a similar argument, W <V can be established.
Next suppose
that
.
].
F
F
.
Next,
make
of
let s -» 0,
from (CI) we
exactly the same offer
who makes the
offer,
which
we have
is
get
C.2) Investment
Now
we
(7)
the one given in the text (equation
F.
Thus both
(7));
parties
thus irrespective
the same equilibrium: the worker obtains a proportion
W
F ; this is
of course exactly the
jS
of
Nash Bargaining
in the text.
By Firms:
will establish that there exists a cut-off level
model of section EI,
all
77*
such that for
77
firms will choose the same investment level and accept
Suppose a firm adopts the strategy used in the
firm will choose
F
w F = w w = w and y-w F =y-ww =
the "net surplus" plus his "reservation utility"
Solution, given in
must be
it
its level
of capital,
kn by
,
text,
solving
C-2
which
is
to accept
all
<
all
17*,
in the
workers.
workers, then this
max, {X[(l-(3)AhX~
a+
V-W'n^h?k!;-
a
n
+ (l-/B)P V (l-X)Ah^-
(C6)
+ (l-X)[(l-P)AhX~
+ (l-P)P V (l-X)AhX~
where k
a
is
mapping
interact.
the capital stock that
level of
Then
the equilibrium
is
a
-(l-li)fi7p4hX
a+
<l-WnAh
a
2
a
a
k'- )-Rk n }.
other firms have chosen. This maximization will lead to
since there
no term
is
1
~a
-Rk}.
Suppose
now
a firm deviates
down low
skill
workers and wait for one more round of sampling. Obviously, such
would
also
from the
choose a different
profits of this deviant are given
strategy of accepting
level of capital
C8 )
+(i-X)[(i-p)AhX~
+(i-P)pv (i-X)AhX~
(C8) has exactly the same
expressions
is
that the
last
first
all
two
a
lines as (C6)
a
which we denote by kd Then the
.
a+
1
^ -0)77(1 -A)/!/^
0--P)P'n^hX~
high skilled workers more often.
a
-(i-W'nAh?P~ ]-Rk d }.
and thus the only difference between the two
make more
Now
look
at
profits
the
if
kd >k
since
(l-a){X[(l-fi)AhX
a+
^-^V^AKkn
+ (l-(3)PV (l-X)Ah
+ (l-X)[(l-H)AhX
1
a+
a
2
kn
a
]
a
(l-X)Ah 2 kn ]}-R=0.
for the deviant:
C-3
C
^-mv^AhXa
a
first
it is
two terms
in
producing with
order conditions for k and k d from
first
(C6) and (C8).
+ (l-P)P7
""]
a
four terms in (C6) have been replaced by the
(C8). Obviously, the deviant can only
(C6')
workers and decides to
by
a+
ETT4 (kd)=maxk {X[{l-p)T)XAh"P
d
And
which these two
in
by kn =k Therefore,
given
Ev(k)=k(l-P)Ah?P^+(l-X)(l-p)AhZk
deviant firm
(
]
(l-Wv^Ah?k>-
k n which does not depend on k
(C7)
turn
all
a
a
(C8')
+ {\-X)[(\-$)AhZk~
d
a
+(l-p)PT)XAh"k d
a
a
+(l-P)PV (l-\)Ah?kd ]}-R=0.
Comparing
for
and
(C6')
kd >k, we need
(C8'),
we
now
can see that
only the
first
three terms are different and
these three terms in (C8') to be larger. Thus, a sufficient condition for a
deviation not to be profitable
is
\(\-P)T)Ah°+{\-X)(\-P)T)Ah".
<
and recalling that there will be a maximum value, $ min that
(i-/3)A + (i /3)(i-;i)<r
r
1
t=
is sufficient for the
can take in any limiting distribution, r}<7]
(l-i8)A + (l-)3)(l-A)<^
Or,
17
,
.
<f>
n
equilibrium in which
nmin
~* 0>
^mm
°°
""*
all
and thus
investment, they will be
Remark:
It
firms to choose the same level of capital stock. Also note that
all
17*
-» 0. Finally, since all firms
happy to accept
all
workers in the
choose the same
first
when
level of
round.
should also be noted that because the dynamics of the system are backward
looking, even
if
<
17
17*
does not hold (for instance because h min = 0), in the region where the two
groups are close to each other, the results in the text exactly apply. Only after inequality
reaches a certain level that firms start choosing different capital levels.
C.3) Conditions For Local Stability of Full Equality
In the text
we
meet a new partner
Infinite
Sampling
analyzed the simplified case where agents had only one more chance to
after
of exposition and the
With
they separated from their
more
realistic case
decide to wait an arbitrarily large
first
match. This was done for simplicity
of course corresponds to the one where a firm can
number of
Again under a suitably chosen condition on
17,
periods for a high skill worker
this will
chooses
to.
not happen along the equilibrium path
but this possibility will influence the wages. Especially,
C4
if it
as this
channel from the overall
distribution of skills to wages
it is
important to check that
we
appendix,
As
7]
it is
the crucial one that leads to non-convergence in section
III,
not driven by the two-period assumption. In this part of the
investigate this question.
we investigate wage determination under the hypothesis that
before,
the same level of capital, k
that
is
should be
With the
t,
and
this will lead to a similar condition to the
than a cut-off level
less
infinite
all
firms choose
one in C.2 above
rf.
human
sampling assumption, the wage of a worker with
capital
h
t
is
given as
w(hypAhX~ a -fa{j{Ah
(CIO)
The explanation
total surplus
point.
The
randomly
for this equation
minus
/3
a
-w{h)]dFt {h)}+(l-(S)T)w{h).
straightforward.
is
The worker
times the threat point of the firm and plus
threat point of the firm
selected
X
is
the average of the output
worker minus the wage
it
would pay to
this
gets a
(1-/3)
it
proportion
times his
(S
of the
own
threat
would produce with
a
by
17
worker,
all
multiplied
since the firm loses potential output in the process of waiting for the next partner.
this averaging uses
agree
and
as in
a firm deviates
this
F
t
as
the relevant distribution, because in equilibrium
the text, there
is
a set of agents of measure v
and breaks the match,
measure zero
set
is
it
assumed to be exactly the same
same and the problem faced by the worker
7j,
in the
Markov
is
is
as
straightforward; since
(Cll)
two
types,
all
we
F
t
.
firms are the
worker
will agree to the
same wage
Osborne and Rubinstein, 1990, or Fudenberg and
[see
if
distribution of
the unconditional distribution,
1991].
In the case of
unmatched and v*0;
an infinite horizon one with constant discount
Perfect equilibrium, the
he would in the second period
are
that
firms and workers
would meet with one of those and the
Finally, the threat point of the worker, the last term,
factor,
who
all
Note
can write the wages as
m^;*:/-^^^
C-5
now
as
Tirole,
—
MIT LIBRARIES
3 9080 00940 1222
for
j
= 1 and
2.
Combining
this equation for
(C12)
=
j
l
and = 2, we can write
j
[l-(l-P) v ][w»-WlA=l3AhZk?-
Then, the wages of the two types of workers
a
-pAh
will be given as
a. \-a
(C13)
wu
1
—±
ih^-faXAhftl'"-^
i-v
'
-jSrKl-X)
^'
1 -(1-/3)77
i
and
a 7 \-a
(C14)
=-
iv
lt
I-77
yMh^k,
[
"
'
-p
Lr
r\
k
-firj(l-X)Ah
%
2t t
"
'
'
'
"
'
--fax
Proceeding in a similar fashion to the analysis in the
text,
"l/"-/
l-(l-/8)77
1-(1-J8)r7
we
can write
w u Ch?r CV X.h?rPr,(l-X)h?rV (l-k)(C-P)hZ
w
Ch£-(C-P)r)kh? -P'nMiZ-'n(l-X)ChZ
2.
(C15)
t
a
C<f) t
a
a
-CT)X4>t -pT)(\-X)4>t -T)(\-X)(C-P)
C-(C-jS)77A0"-j377A-77(l-A)C
where
Thus,
C=1-(1-/3)t;. Again,
we
it
can be checked readily that
find that as in the text, for
below the 45° and thus
as in
77
> 0,
<£ t+1 (0 t
=l) =
l
and that
the difference equation that related
the proof of Proposition
2,
we
<f>
1
t+l
to
<f> t
starts
can see that there must be a set
of inequality level from which full equality, <j>=l, will not be reached.
Next, to investigate the local
stability of full equality,
1
we look
at
To see that the term in (C16) is negative note that the numerator is always negative and the
denominator is positive. The positivity of the denominator, D, follows from the fact that when 77= 1,
D=0 and 3D/3t/<0 and as 7/ is always less than 1, D>0.
C-6
X
8# M
(C17)
l
1
x
aC4>"~ -aCT)X4>"~ -apit)(\-'k)4>"~
a
d^t
C-(C-P)r)k<j>t -pr)),-r)(l-X)C
a.T)k(C-P)4>"~
<l> t
^
C-(C-pyn*.tf-pnk-nQ.-k)C
thus
^m(^=1 ) _
/
C-(C-/3)r7 A-j377A-»7(l-A)C
30,
C18 \
aC-aC-nA-a/^l-A)
.
aa-q-jBh^) .
C(l-rj)
ar;A(C-j3)
+
C-(C-/3)r? A-i877A-'r7(l-A)C
g
l-(l-j3)ij
Therefore, the stability condition becomes a<l-(l-/3)i7 rather than
a<
.
These
1+(1-)8)17
conditions are very similar to each other and have the same comparative static properties, in
particular, a high
infinite
sampling
a
all
or a high
rj
make
it
harder for the system to be stable. Therefore, with
the conclusions of Proposition 2 hold.
C-7
2554
3
I
Date Due
Lib-26-67