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http://www.archive.org/details/howmuchdoessortiOOkrem
IHB31
.M415
working paper
department
of economics
HOW MUCH DOES SORTING INCREASE INEQUALITY?
Michael Kremer
96-18
March 1996
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
HOW MUCH DOES SORTING INCREASE INEQUALITY?
Michael Kremer
96-18
March 1996
MASSAC^' *"™
I
lJULlS
—
'"""WE
1996
HOW MUCH
DOES SORTING INCREASE INEQUALITY'
Michael Kremer"
March
am
29, 1996
comments from Ricardo Caballero, Steve Durlauf, Nada Eissa, Glenn
Loury, Larry Katz, Casey Mulligan, Paul Romer, Danny Quah, and two anonymous referees.
I thank Raghav Kapoor, Amanda Merryman, and especially Matt Barmack and Sergei
Severinov for outstanding research assistance. Research support was provided by the National
Science Foundation grant number SBR9414126, and by the MacArthur Foundation.
*I
grateful for
"Massachusetts
Harvard
Institute
Institute
of Technology, National Bureau of Economic Research, and
of International Development.
96-18
Abstract
Social commentators from William Julius Wilson to Charles
increased sorting of people into internally
marriages
is
spurring long-run inequality.
fears are misplaced.
Murray have argued
homogeneous neighborhoods,
Calibration of a formal
that
schools, and
model suggests
that these
In order to increase the steady-state standard deviation of education
by
one percent, the correlation between neighbors' education would have to double, or the
correlation
between spouses' education would have
correlations have declined slightly over the past
significant effects
to increase
few decades.
by one-third.
Sorting has
on intergenerational mobility than on inequality.
In fact, both
somewhat more
Michael Kremer
E52-251C MIT
MA 02139
Cambridge,
1
Several influential social commentators have argued that Americans are increasingly
sorting into internally
homogeneous neighborhoods,
schools, workplaces, and marriages, and
William Julius Wilson [1987] contends
that this will increase future inequality.
that black
middle-class flight from urban centers leaves inner-city blacks without role models; Robert
Reich [1991] argues that a "fortunate
and Herrnstein and Murray [1994] assert that people are sorting more by intelligence,
society;
with a "cognitive
elite" increasingly living in the
same
schools, working in the
children
may be
Corcoran
I
and marrying each other.
in a vicious cycle
A
started this project,
I
in sorting will
same
series of papers indicate that
neighborhood [Borjas, 1994; Crane, 1991;
shared these concerns.
influenced by their parents and neighbors.
the
This suggests that
of increasing sorting and inequality.
on misleading intuition from models
changes
same neighborhoods, attending
1989; Cutler and Glaeser, 1995; Case and Katz, 1991].
et al.,
When
firms,
substantially affected by their
America may be caught
largely
has seceded from close contact with the rest of
fifth"
in
which
I
now
childrens'
believe that they are based
outcomes are very strongly
Calibration of a simple model suggests that
have only a small impact on steady-state inequality of characteristics
such as education and income.
I
assume
characteristics
childrens' inherited characteristics equal a linear combination of their parents'
and
their neighbor's characteristics, plus an error term.
greater correlation between spouses or
among neighbors
in
This implies that
any heritable characteristic will
lead to greater inequality of that characteristic in the next generation.
It is
possible to solve
for the steady-state standard deviation of a characteristic as a function of the correlations
between spouses' and among neighbors'
characteristics.
2
Data from the Panel Survey of Income Dynamics (PSID) and the 1970 and 1980
Matching Census Extract Data Sets indicate
that,
on average, a child
will obtain an extra 0.38
A
years of education for each additional year of his or her parents' education.
child will gain
an extra 0.13 years of education for each additional year of education in the neighborhood
(census
tract).
Given these parameters, the steady-state standard deviation of educational
attainment would increase by approximately
1%—or
less than
between neighbors' education doubled from 0.2 to 0.4 or
if
one week— if the correlation
the correlation
between spouses'
education increased from 0.6 to 0.8.
In fact, these correlations have been stable or declining.
among people
in the
same census
tract fell
correlation between spouses' education fell
The finding
seems
from 0.18
from 0.649
that increased sorting has only a
to be an artifact of the
to
The
0.178 from 1960 to 1990.
to
0.620 between 1940 and 1990.
minor impact on inequality does not
Estimation of a general Markovian model
for intergenerational transmission of education yields similar results.
moderate observed
heritability of education
Even under strong assumptions about
between spouses' education
deviation of education.
will
still
There
masks a highly inherited
somewhat
evidence
latent variable.
have only minor effects on the steady-state standard
Although the paper focuses on education, changes
these variables have roughly the
is little
the heritability of such a latent variable, the correlation
unlikely to have significant impacts on inequality of
Sorting has a
The
assumption that children's characteristics are a first-order linear
function of parents' and neighbors' characteristics.
that the
correlation of education
same
in sorting are also
income or occupational
status, since
heritability as education.
greater effect on the persistence of educational status across
3
generations than on inequality of education.
4
A
doubling of the correlation
among
neighbors'
education would increase the correlation between parents' and childrens' education by 8
percent,
from 0.329
0.6 to 0.8
percent,
to 0.354.
An
increase in the correlation between spouses' education from
would increase the correlation between
from 0.329
to 0.367.
in the steady-state
Persistence of economic status
strongly influenced by educational sorting
within their
own
and childrens' education by 12
Either of these changes in sorting
approximately a 3 percent increase
discounted education.
parents'
if
would produce
standard deviation of dynastic
among
ethnic groups will not be
people draw spouses and neighbors largely from
ethnic group.
Benabou [1993a, 1993b], Durlauf [1992, 1994] and Fernandez and Rogerson [1992]
examine the transmission of inequality under neighborhood
similar topics using a modified version of a biological
effects.
This paper examines
model created by Galton [1889] and
Pearson [1896], and generalized by Cavalli-Sforza and Feldman [1981].
Economists such as
Becker and Tomes [1979], Becker [1981], Blinder [1973; 1976], and Atkinson [1975], have
used related models to examine inheritance.
Blinder shows that changes in inheritance rules,
or in marital sorting, can have only a slight effect on income distribution through inheritance
of physical capital, since inherited wealth accounts for only a small part of inequality.
This
paper differs from earlier work by allowing the correlation between spouses' characteristics to
depend on the variance of characteristics
effects,
in the
population; by incorporating neighborhood
by calibrating the model to empirical data on the transmission of education rather than
simply the inheritance of financial assets; and by using the model to quantitatively examine
4
I
am
grateful to
Glen Loury for pointing
this out to
me.
4
contemporary concerns about increased sorting
The remainder of
the paper
is
American
in
organized as follows.
society.
Section
sorting on inequality using a modified version of the Galton-Pearson
Section
II
calibrates the model, arguing that sorting into
neighborhoods
Section
III
is
likely to
examines the
significantly
models the
model of
inheritance.
homogeneous households and
on the persistence of educational
Section IV argues that the conclusions are unlikely to change
changed under more general models.
rather than appealing to incentive responses
I
among
status
This paper evaluates claims that sorting will increase inequality on their
conclusion,
effect of
have only minor effects on steady-state inequality of education.
effect of sorting
families and ethnic groups.
I
which
are difficult to quantify.
own
However,
terms,
in the
argue that endogenous behavioral responses are likely to further counteract
tendencies for sorting to increase inequality.
The conclusion
also discusses policy
implications: arguing that to the extent social policy seeks to reduce inequality,
better to invest directly in the
poor than to
try to
change sorting patterns.
it
is
probably
The Model
I.
Suppose
t+1
that the educational attainment of a
member
of the
+
0<a + P<l,
ith
dynasty
in
generation
is
(1)
+
^. =*H
where
a
«
h, +Zi>,,
P^
—
£,m
represents the additional years of education a child will attain given an increase of
one year of average parental education,
i'
is
the spouse of agent
i,
P represents the additional
years of education a child will attain given an increase in average education of one year in
the neighborhood, n
and
£, t+
,
is
an
i.i.d.
is
the
number of neighbors,
random shock
is
the education of neighbor
to educational attainment.
dynasties, but could vary with time
linear, first-order, genderless
z){
if
there
is
human
model should be seen
capital, as
implications of sorting,
I
that fertility
I
is
bounded.
down
is
time
t,
constant across
in education.
This
In
will generally refer to zJt as
In order to focus
and the transmission of human
neighbors and parents are completely inelastic to incentives.
'This approximation clearly breaks
k t+1
at
as an illustrative approximation.'
measured by years of education.
assume
that
an exogenous time trend
general, zJt could represent any asset, but for specificity,
representing
Note
j
on the
capital
All households are
by
assumed
to
for the tails of the distribution, since education
have two children, one boy and one
2
girl.
Squaring both sides of equation (2) and taking expectations shows that the variance of
education
at
time t+1 will be an increasing function of cr„ the variance of education
at
time
p m the correlation between spouses' education; and p n the correlation between neighbors'
,
,
education.
Assuming
that n
is
large, so that a child's parents
can be treated as an
insignificant part of the neighborhood,
tf^JPJ
(2)
+p2pn
r+l
Solving for the steady
a2
-
a
1
+Pm )
more general case with both
effects.
2
+ (P 2 +
2ap>
n
parental and neighborhood effects.
If
Although tendencies for poor households to have more children
any case
it is
is
also likely to be
P=0
or p n =0,
the
will clearly affect the
minor [Mare, xxx]. In
not clear that this will interact with changes in the assortativeness of marriage.
therefore abstract
patterns
.
E
This case illustrates principles which carry over to
steady-state distribution of education, this effect
I
+a
useful to first consider the effects of sorting in marriage in the special case in
which there are no neighborhood
2
,
=
1
the
2
state,
(3)
It is
+2a p pn a
on steady
from
differential fertility in calibrating the effect of
state inequality.
changes
in
marriage
t;
—
steady-state variance simplifies to
<
oK-.
"
(4)
1-a 2 (
—pJ
1+
\
2
This paper will generally take p m as exogenous.
examining the possibility
may
education
is
Lam
and Schoeni [1993] find
is
worth briefly
between
Figure
shows
I
have a higher correlation between
that the correlation
between spouses's
To
0.77 in Brazil compared to 0.57 in the relatively more equal United States.
understand the possible reasons for
the
it
lead to interesting feedback effects.
that U.S. states with a higher variance of education also
3
However,
that greater inequality will lead to a greater correlation
spouses' characteristics, since this
spouses' education.
J
same observed
quality, q,
this relationship,
suppose that people choose spouses of
where q lt equals z it plus an
i.i.d.
variable, denoted
be taken to represent a non-heritable characteristic which people value
in
spouses.
Alternatively, %u could be thought of as noise term that obscures the true quality
either interpretation, p m will equal
intersection of the functions p m (a)
(T/Co
2
^
+ a2 ,).
The
and c(p m ), as shown
cj
it
.
cj
jt
can
4
z,
.
t
Under
steady-state will be given by the
in
Figure
I.
A
change
in
matching
technology or preferences which reduces cr^ will create a multiplier effect on steady-state
3
As
will
become
marital sorting
4
if
If there is
there
clear below, this effect
on inequality, even
transferable
utility,
if
is
far too big to
there were
be explained by the effect of
no mobility between
states.
people of the same observed quality would match together
were complementarity between quality of marriage partners. If spouses cannot
same quality might match together even if there
transfer utility to each other, people of the
were no complementarity between spouses' quality [Becker, 1981].
inequality.
Substituting the formula for p m into (4) yields
o.» -
o}
o* 1
-
2„
2
+4o/of(l
r
- cr)
=
a<l;
2
2(1 - a )
2
a
^
-
2
2.- 2
1
0=1,
o >2o,
2'
;
(6)
otherwise.
oo,
As
is
apparent from the denominator of
state-inequality depends critically
on a.
If
a
(4), the effect
equals
1,
of changes in sorting on steady-
the correlation
between spouses' assets
determines whether inequality increases without bound, or society eventually reaches some
steady-state level of inequality.
still
If
a
is
close to
1,
the correlation between spouses' assets will
have a major impact on steady-state inequality.
correlation between spouses' assets has only a
Because they may be responsible for
inequality,
it is
useful to
first
minor
However,
effect
if
a
is
small or moderate, the
on steady-state inequality.
intuition about the effects of sorting
on
consider the effect of sorting on steady-state inequality under
savings models in which shocks to dynastic wealth are fully transmitted to children, so
equals
1.
Let z represent physical, rather than
and returns are independent of wealth, and
as exogenous.
that
If there is perfect correlation
human
capital,
and suppose
that savings rates
shocks cannot be perfectly insured.
between spouses' wealth, z it+1
a
Take "»
will equal z it
+
9
e 11+1
In this case, assets in each dynasty will follow a
.
among
assets
However,
if
of
will
be
finite for
If
between
5
spouses' assets
is
Rich people will tend to marry
bounded away from one,
less
wealthy spouses because
This "Maria Maples effect" implies that rich people will on
who
are poorer than themselves, so the steady-state variance of assets
p m less than one.
marriages take place only within countries, there will be a steady-state variance of
assets within a country, but the variance of assets
The
the variance of
be poorer than them, and some of these people will have very large values
average have children
will
6
the idiosyncratic variable.
£,
indefinitely.
the correlation
inequality will be bounded.
most people
grow
dynasties will
random walk and
data are consistent with this prediction.
the United States, but the variance of log
among
countries will
Inequality has not
grow
indefinitely.
grown monotonically within
income (and presumably wealth) among countries
has grown over time.
Changes
to
1,
in sorting will
but will have a tiny effect
larger
if
a
is
close to
1
.
Table
have a dramatic effect on steady-state inequality
if
I
a
is
moderate.
shows
The
if
a
multiplier effect will also be
is
close
much
the direct and indirect effects of a reduction in the
noise in the marriage process which causes the correlation between spouses' assets to increase
from 0.6
5
to 0.8.
If there are
decreasing returns to accumulable factors, inequality will eventually be
bounded, but only because
it
will
be impossible for the very rich to continue accumulating
because of economy-wide diminishing returns.
6
If
a^
approach
1
is
small enough relative to
sufficiently quickly that
aE2
between spouses' assets will
the variance of assets will grow without bound.
the correlation
10
To
see the intuition for
simply note that
indirect effect of
if
a
II,
the direct effect
The
a
falls.
in sorting
much
declines
faster than the direct effect as
or
I,
As
empirical estimates suggest that in the contemporary United States,
a
approximately 0.4 for education and income, which suggests that the direct effect of
changes
in sorting
on inequality
small and the indirect effect
is
unimportance of the multiplier effect for
straightforward to calculate
It is
p m to cause specified increase in c„.
were
how
7
big
l1
,
N
Solving
this
The
equation implies that
a would
if
will abstract
from
it
qualitatively similar if P
is
(
1+ g ) „2
a
2
need to be for a specified increase
an increase in p m from 0.6 to 0.8
in the
is
l-H^a
N
have to equal 0.84.
and the steady-state standard deviation of human
variance
is
am
is
affected by
is
Increases in sorting will always increase steady-state
inequality, but this effect will only be strong
so that the denominator of (3)
2
greater than zero, so childrens' education
neighborhood.
(4),
-
-
a would
in the
e
= 1.1
relationship between sorting
average education
I
Given the
neglible.
effect.
For example,
-^
(7)
7
I
is
cause a ten percent increase in the standard deviation of education, then by
to
capital
of a,
realistic values
remainder of the paper and will focus on the direct
in
so sensitive to a, examine Figure
is
high, the denominator of equation (4) will be close to zero.
changes
discussed in Section
is
is
why
if
close to zero.
a+
P
Even
is
close to
if
a+
P
is
1,
and p m and p n are
close to
1
,
the steady-state
only moderately sensitive to the correlation between spouses' assets
grateful to a referee for the derivation below.
large,
if
p m and p n
11
are small.
Equation
(3) indicates that the effects of
simultaneous increases
marriage
in sorting in
and neighborhoods are greater than the sum of the effects of separate increases.
However,
discussed below, this complementarity between sorting in marriage and neighborhoods
relatively
minor
for realistic values of
a
and
p\
This section calibrates the model to the distribution of
human
capital in the United
Using data from the PSED, and from the Matching Census Extract Data
that a, the parental effect,
neighborhood
likely to
I
is
effect, is 0.13.
is
Calibration
II.
States.
as
approximately 0.38, and that an upper bound on
Given these parameters, increases
in sorting
sets, I find
p\ the
by education are
have a minor impact on the steady-state standard deviation of education.
focus on education, rather than income, because spouses' income
marriage, and sorting
neighborhoods.
is
is
endogenous
to
only well-defined for variables which are exogenous to marriages and
Moreover, as Solon [1992], and
measurement error or fluctuations
in
income
neighborhood income on childrens' income.
Zimmerman
[1992], have shown,
will bias estimates of the effect of parental
Of
course, there
may be measurement
and
error in
education as well, to the extent that years of education do not acurately represent the correct
theoretical concept of
human
capital, but
some purposes, education may
income.
in fact
it is
likely to be smaller than that for income.
be a better indicator of socio-economic status than
For
12
Table
II
21 waves of the
reports
PSID
summary
statistics
on childrens' and parents' education from the
(up through the 1988 interview year).
Only children over twenty-five
years old in 1988 were included in the sample, to avoid including children
completed
their education.
17 years of schooling.
first
who had
not yet
People with any post-graduate education were classified as having
In order to use self-reported data
used split-offs from the original
from both parents and children,
PSID sample. This means
sample of the population because
it
that the
sample
misses the replenishments to the
PSID
is
not a
I
random
to replace people
lost to attrition.
Data on the average education
in the
taken from taken from the 1970 census.
census
Census
people, so this represents a fairly small area.
average education of males over 25
in the
tract in
which the child grew up were
tracts typically
comprise five thousand
The neighbors' education was calculated
census tract in which the child resided between
1968 and 1975, weighted by the number or years the child spent
People were excluded from the sample
tract in
which they grew up, or
Children
who grew up
whom
Column
if
in the
census
tract.
information was not available on the census
self-reported data on both parents
in single parent
available on both parents.
those for
if
as the
was not
available.
households were included, as long as data was
Similar results were obtained for a larger sample, including
all
child-reported measure of parental education were available.
(1) of
Table HI shows the baseline specification. The parental effect
estimated to be 0.379, the neighborhood effect
deviation of the shock to education
neighborhood effect
is
large:
is
is
is
estimated to be 0.127, and the standard
estimated to be 1.74 years.
living in an educated
The estimated
neighborhood increases the expected
13
education for one's children two- thirds as
much
marrying an educated spouse.
as
discussed below, the coefficient on parents' characteristics
and the coefficient on neighborhood characteristics
The remaining columns of Table in
approximates
childrens' education.
and
Column
reality.
An
general, (but
much
(3) distinguishes
likely to be biased
is
how
likely to be biased
downwards
upwards.
well this linear, genderless model
between mothers' and
fathers' effects
on
their
F-test cannot reject the null hypothesis of equal effects of mothers'
A
fathers' education.
test
is
As
previous version of this paper obtained similar results using a more
messier),
model
that distinguished
between fathers and mothers, and
between sons and daughters.
Column
(4) of
Table
III
checks
how
well the linear model approximates the process of
intergenerational transmission by adding quadratic and interaction terms to the regression.
Although the difference
Section
III
estimates a
in
adjusted
R
2
between the models
more general Markovian model of
is
small, an F-test rejects linearity.
intergenerational transmission that
does not impose linearity.
Given the estimated parameters, education
is
remarkably insensitive to the correlation
between spouses' or among neighbors' education. The standard deviation of steady-state
education would increase from 1.87 to 1.89 years
spouses' education increased from 0.6 to 0.8.
It
between neighbors' education doubled from 0.2
-
0.8 percent
would increase
to 0.4.
would increase
by about a week.
in sorting in
these magnitudes
would increase
the correlation between
1.3 percent if the correlation
Assuming
these hypothetical increases in segregation
Simultaneous increases
- if
a 180
day school year,
the standard deviation of education
marriage and in neighborhoods of
the steady-state standard deviation of years of education to
14
2.2%, slightly more than the sum of the individual
Although these changes are small,
effects.
they happen relatively quickly: the standard deviation of education will increase 85 percent of
the
way
to
its
Note
steady-state value after in single generation following the increase in sorting.
that sorting
by education has been quite stable
hypothesized above are extreme.
the
same census
In 1960, the correlation
was 0.180. 8
tract
In 1990,
it
had
historically, so that the
between the education of people
fallen to 0.178.
from the public use microsample of the 1940 census
in
changes
sample of couples
In a
which the husband was between the
ages of 30 and 45, the correlation of years of education between spouses was 0.649.
1980,
it
had
fallen to 0.633,
and by 1990,
it
had
in
fallen to 0.620.
9
By
(Note, however, that
Mare
[1991] and Kalmijn [1991] find slight increases in assortativeness of marriage using a
measure of sorting based on the likelihood of crossing educational
school graduate/some college barrier.
correlation as a
measure of
stability in sorting patterns
Measurement
sorting.
barriers,
Kremer and Maskin [1995] discuss
Whichever measure
is
such as the high
the justification for
used, the overall impression
is
of
over long periods.)
error in parents' education will bias the estimated
a
downwards.
Ashenfelter and Krueger [1994] estimate that 8 to 12 percent of measured variance in
schooling
8
This
is
due to
error,
may be due
and they
cite
previous estimates by Siegel and
in part to equalization of
Hodge [1968] and
incomes among regions of the country during
the period.
9
men and women have different educational distributions, and that these
distributions have mass points, the maximum feasible correlation between spouses' education
is less than one.
For example, in 1990, the maximum feasible correlation between spouses'
Given
that
education was 0.96.
correlation to the
There was a
maximum
slight decline
feasible correlation
over the period
in the ratio of the actual
between spouses' education.
15
Bielby [1977] that between 13 and 20 percent of the variance of schooling
measurement
Measurement
error.
extent that parents' education
overestimated
if
who
parents
is
error in parents' education will bias P
correlated with neighbors' education,
fj
is
due
upwards
to
to the
be
will also
Under
value education tend to live in educated neighborhoods.
the extreme assumptions that twenty percent of the variance in parents' characteristics were
measurement
neighborhood,
in
and
error,
a would
that
none of the
be 0.38 * 5/4
=
effect of parents' education
0.47 and
(3
would be
was picked up by
the
In this case, an increase
0.13.
p m from 0.6 to 0.8 or in p n from 0.2 to 0.4 would increase the steady-state standard
deviation of education by approximately 1.5 percent.
increase in p n from 0.6 to 0.8
by only 2.8 percent.
A
would increase
Even
if
a were
0.6 and
(3
were 0.13, an
the steady-state standard deviation of education
doubling of p m from 0.2 to 0.4 would have a slightly smaller
effect.
Substituting the estimated parameters into (4) implies that the steady-state standard
deviation of education will be 1.87 years, which
of education
among
inequality
above
is
Note
less than the 2.1
children in the parents' generation.
its
steady-state value
is
10
The model's implication
that this is the standard deviation of years of education for a
is
some evidence
that
a may
have been greater
an artifact of reduced differences between
among
among
sons
is
fathers in the
sample
is
men and women;
a
and Duncan [1967], find
sample of people
thus
much
smaller than
regressions of fathers'
of 0.56.
This
is
not merely
the standard deviation of
3.49 years, while the standard deviation of education
2.15; similarly, the standard deviation of education
while the standard deviation of education
is
in the past:
education on fathers' reports of grandfathers' education yield an
education
that
The variance of education may have been
young enough to have parents in the PSID. The standard deviation
it would be in a sample containing both recent and older cohorts.
"There
year standard deviation
1
consistent with the fact that the standard deviation
has been dropping over the past generation."
10
is
among daughters
is
among mothers
2.06.
that the correlation of occupational status
On
is
2.62,
the other hand, Blau
on fathers and sons
in
16
greater in previous generations due to greater regional and racial variation in education
spending, and more limited public support of higher education.
Results are likely to be similar for characteristics other than education.
heritability of
income
is
approximately 0.4, according to Solon [1992] and
Zimmerman
Blau and Duncan [1967] find similar correlations for occupational
[1992].
and Murray [1994] claim
that intelligence is
The
status.
Herrnstein
about 60 percent heritable and issue dire
warnings about the effects of alleged recent increases
in assortative
mating by intelligence.
Herrnstein and Murray's estimates of the heritability of intelligence are accepted
then increased correlation of spouses' intelligence
is
face value,
have a minor impact on the
likely to
(However,
steady-state variance of intelligence under the model.
at
If
model may not be
this
Murray believe
appropriate for examining the genetic transmission that Herrnstein and
drives
intergenerational correlation of intelligence.)
The main exception
[1995],
parental
who
to findings of heritability in this
finds the intergenerational correlation of
income
is
income influences
upward biases
correlation in
consumption
used as an instrument for parental consumption.
children's
in estimates
consumption
of heritability.
It is
In any case,
play a somewhat important role:
a=
an increase
0.4, indicating that there has
status.
if
not clear
been
0.8 and
in
little
is
[3
in
=
how
is
Mulligan
approximately 0.8 when
However,
IV techniques may
directly,
consumption with the lower correlations
and Zimmerman [1992].
1964 was
economic
range or below
if
parental
lead to significant
to reconcile
such a high
income documented by Solon [1992]
0, assortative
marriage begins to
p m from 0.6 to 0.8 will increase the standard
change
in father-son correlation of
17
deviation of consumption by 7.3 percent.
A
fuller analysis
would examine
the case of
multiple, correlated, heritable characteristics, as proposed by Goldberger [1979].
III.
Although changes
somewhat more
in sorting
significant impact
steady-state correlation
have only a small impact on inequality, they have a
on the intergenerational correlation of education. The
between the educational attainment of a parent and
a
,„,
corrSz^^)
(8)
A
Sorting and Persistence of Inequality
=
his or her child
is
(1 +
pj
—^-
* P Pn
.
doubling of p n from 0.2 to 0.4 will increase the intergenerational correlation of education
by 8 percent, from 0.329 to 0.354.
An
increase in p m from 0.6 to 0.8 will increase the
intergenerational correlation by 12 percent,
hypothesized changes in p m and p n are
from 0.329
many
to 0.367.
(Recall, however, that these
times greater than the changes which have been
experienced historically.)
Since sorting increases the intergenerational correlation of education, inequality
dynasties will be
more
sensitive to sorting than inequality
"discounted dynastic education" as
(9)
- M
A,
-
*£ s'v
»=1
among
individuals.
Define
among
18
where 6
is
the discount rate.
(10)
12
Assuming
"
A,
£
that the population
in steady-state.
V
+ P»
1
»'
is
Pp„ iv
-*,-;
r=l
where p denotes the mean
level of education.
discounted dynastic education
This implies that the steady-state variance of
is
2
_
(11)
1-5
Assuming an
M2
1+ P»
(
intergenerational discount rate of 0.5, an increase in p m
from 0.6
to 0.8
would
increase the standard deviation of discounted dynastic education by 3.4 percent, and an
increase in p n from 0.2 to 0.4
Under the model,
if
would increase
it
by
3.1 percent.
spouses and neighbors are chosen with regard to education but
not ethnicity, ethnic groups will converge to the
mean
level of education at a rate
corresponding to the intergenerational correlation of education.
convergence among ethnic groups.
Sorting will therefore slow
This effect will be exacerbated
if
influenced by the "ethnic capital" of their group, as in Borjas [1994].
substantial sorting
on purely ethnic grounds, sorting by education
childrens' education
However,
will only
if
there
is
is
have a minor
impact on the rate of convergence among ethnic groups.
To
12
see this, consider a
Obviously,
this
concept
exponential in education, and
model with both neighborhood
is
a bit ad hoc.
utility is log,
it
On the other
may not be a
effects
hand,
if
and ethnic group
wages
effects.
are approximately
terrible welfare indicator.
19
Suppose
<
12 )
z
,
k+ a
=
\t+*l'J
where y measures the strength of ethnic transmission, and z g
group g
at
time
t.
(I
assume
that a child of parents
equal probability of becoming a
of each ethnic group
wm
member
of either.)
,
Suppose
that a
Given the
the society as a whole.
(1
t
+
1
lives in an ethnic
linearity of the
as if a fraction
w m+
(1
-
w m )(l
z
w+1
u,
p m ) marries a spouse with the average education
= * +e
+ [a +
Zftf
denotes the average education of
persistence of differentials
p n )(3.
-
Assuming
that
among
a+
f3
all
members of
groups, equals
+ y<
1, all
p+Y-e]ii,
a +
(3
the population,
+ y
-
(1
-
w m )(l
and
-
the group
0, the
pm)
oc/2
-
(1
-
w
n)
groups will converge to the same steady-state
independently of sorting, but they will approach the steady-state more quickly
members of
in
Thus
(13)
(1 -
period
in
wn
of the ethnic group marries a spouse with the average education of people in the
ethnic group, and a fraction
where
capital of ethnic
randomly drawn fraction
marries within the ethnic group, that a fraction
model, one can calculate average education of a group
w m )p m
human
from two different ethnic groups has
neighborhood, and that the remainder sort by education alone.
-
the
is
if
more
match with spouses and neighbors from outside the group, and
of these people have spouses and neighbors with the average education
in the
if
more
population as a
whole.
Changes
in sorting
by education
will
have only have a minor impact on
6, persistence
20
among
of inequality
ethnic groups,
if
groups sort strongly by ethnicity
circumstances under which disparities
For example,
significance.
if
wm =
among
0.7,
and
-
precisely the
ethnic groups are likely to have the most social
w
n
=
0.5, then the increases in sorting in
marriage and neighborhoods described above would increase 6 from 0.432 to 0.444 and 0.445
respectively.
13
There has been a debate about whether social policy should focus on reducing sorting
by race and ethnicity, or sorting by socio-economic
status.
The
analysis in previous sections
on inequality among individuals. The
implies sorting by education will have
little
analysis in this section suggests that
groups sort highly by ethnicity, reducing sorting by
ethnicity
sorting
is
likely to
if
effect
have a larger effect on disparities between ethnic groups than reducing
by education.
IV.
The previous
process.
sections
Extensions
assumed education was transmitted by
a first-order linear
Theoretically, sorting could have a bigger impact on inequality
transmitted through a non-linear process, or
if
if
education were
there were a highly heritable latent variable
which influenced education. This section argues
that an initial
examination of the data does
not suggest that relaxing the assumptions of the model will significantly change the
conclusion that sorting has minor effects on steady-state inequality.
13
In a regression of childrens' education
on parents' education, neighbors' education, and
the average education of their ethnic group, ethnic education
I
assume y =
0.
is
insignificant (and negative), so
21
examine the
In order to
Markovian context,
I
effect of marital sorting
on inequality
in a
divided the population into six education categories:
some
more general
elementary
1)
school and below, 2)
some high
more than
then estimated the probability that the child will be in each category
college.
I
school, 3) high school, 4)
conditional on the category of each parent.
matrix,
Given the estimated Markovian
a proportion *F of the population chose spouses with the
randomly.
*P chose spouses
15
As
*F increases
from 0.6
deviation of education increases by approximately
A
transition
would obtain
possible to solve for the steady-state distribution of education that
is
it
14
college, 5) college, and 6)
similar transition matrix
neighborhood.
was constructed
.
1
-
to 0.8, the steady-state standard
percent.
1
to
a proportion
examine the
effect of sorting
by
divided the population into four quartiles, corresponding to the quartiles of
I
the distribution of education by neighborhood,
and then constructed a transition matrix.
analyzing the effect of sorting by neighborhood,
was
spouses' education
0.6,
education of the parents.
an increase
1
same education, and
if
and
The
in the steady-state
I
assumed
that the correlation
that parents' education could be
between
summarized by
analysis indicates that an increase in p n
from 0.2
In
the average
to 0.4 leads to
standard deviation of education of approximately 3/10ths of
one percent.
14
Scarcity of data
made
it
impossible to distinguish the effects of mothers' and fathers'
education separately.
"One weakness of
this
approach
not match with others of the
education.
Thus,
this
is
that
same education
it
does not take into account that those
are likely to
procedure gives equal weight to
all
the off-diagonal cells in the
transition matrix, rather than weighting the cells near the diagonal
from the diagonal.
who do
match with people of similar
more heavily than those
far
22
Scarcity of data
made
for non-linearities in parental
couples
in
it
impossible to use finer categories or to simultaneously allow
For example, there are too few
and neighborhood transmission.
which the mother has a college education, the father has an elementary school
education, and the typical neighbor had a high school education to accurately estimate the
distribution of education for children with this background.
Under Markovian models,
parents'
in
which childrens' education
a non-linear function of
and neighbors' education, average steady-state education will typically depend on the
pattern of matching.
Under
the the estimated
state education increases slightly
Markovian
16
if utility is
people will consider
within the household
this cost
is
joint,
falls
Average steady-state education would
1.4 percent in response to an increase in the correlation
Recall that
transition matrices, average steady-
with the correlation between neighbors' education, but
with the correlation between spouses' education.
to 0.8.
is
fall
between spouses' education from 0.6
transferable, an efficient marriage pattern will be chosen, so
of sorting in choosing spouses.
it
may be
educated people for marrying them.
difficult for less
However,
if
consumption
educated people to compensate more
In any case, as discussed in the conclusion, the
implication that the average level of education will be reduced by increased correlation
between spouses' education
is
likely to be overturned
by endogenizing a, the
heritability of
education.
Besides
education
16
is
its
assumption of
linearity,
another restrictive assumption of the model
transmitted by a first-order process
- i.e.,
that childrens' educations
is
that
influenced
Sorting has similar effects on average predicted income given education, which
a non-linear function of education.
is
is itself
23
only by parents' education.
mask
Theoretically, moderate heritability of education could potentially
stronger heritability of an underlying latent variable.
persistent latent variable influencing education,
where 6
is i.i.d.
with
2
mean zero and variance ae
spouses' latent variables
education
17
,
and
Under
is
variables models.
same
the
that there are
this
and
model, the estimate of
a
see this, suppose that z
that education,
Assume
as the correlation
no neighborhood
The estimated
.
To
y,
equals z
that the correlation
between
effects.
denoted
their
is
a
+
6,
between
observed years of
18
will be subject to attenuation bias, as in errors-in-
heritability of education will
be
2
(14)
plim
d =
a.
o e + °z
Note
that
17
To
az : + a e 2 = g
2
.
The
probability limit of the estimated standard error will be
the extent that people care about their spouse's education, they will
basis of education, rather than the latent variable.
latent variable will
more educated
match on the
However, people with high values of the
be considered more desirable partners because they will typically have
children.
Moreover, people may have some information about the latent
variable by observing relatives.
18
Neighborhood
effects are statistically insignificant in the estimated latent variable model.
I
24
/1C\
(15)
-2
,.
plm
Under
- N2
,
oe =
2
(o-aro^.x
—^
(l +
al 2 -
the one hand, greater values of
will be
more
greater
a
2
+
°e
2
°e
+
- g D^
2^1
-2
a °e
+
'
a
oe2
,
^
2
cc
(l
* o\
.
+ p)
imply that steady-state inequality of the
sensitive to the correlation
given d, a greater
The
+
the latent variable model, the steady-state variance of education will be
<">
On
p"\
between spouses' education.
implies a greater variance of 0, the
i.i.d.
On
latent variable
the other hand, for a
variable influencing education.
the less sensitive inequality will be to the correlation
between spouses'
education.
Instrumental variables estimation using grandparents' education as an instrument for
parents' education suggests that
a
is
approximately 0.44.
the regression, the
IV estimate of a increases
the regression; the
change
of the data
in the
0.44 for estimate
in the coefficient is
sample for which data
may
to 0.58.
is
When
neighbors' education
Neighbors' education
is
due to the loss of approximately
available on neighbors.
is
put in
insignificant in
30%
percent
This suggests that the
be preferable, but both estimates should be treated with caution.
If
grandparents directly contribute to their grandchildren' education, the IV estimate will be
biased upwards.
Equations (13) through (15) imply that
0.6 to 0.8 will increase
change
in sorting
c by 0.9%, compared
would increase
since this implies a large
These
results with
c e2
G„,
by
1%
if
to
if
a
0.8% under
a was
0.6, but
is
0.44, an increase in p m
the baseline model.
by only 0.7%
if
from
This
a was
0.7,
.
more complicated models should be considered
provisional.
It is
25
possible that sorting has stronger effects at the extreme tails of the distribution.
may
analysis
in
not have been fine enough to determine the effects of segregation on segregation
inner city ghettos, or
estimation of
inequality.
The Markov
among Native Americans
more complex
However,
latent variable
initial efforts in
in
South Dakota.
Theoretically, the
models could yield larger
effects of sorting
on
these directions suggest that the results of previous
sections are fairly robust.
VI.
Authors from a variety of
caught
in a tide
Conclusion
political perspectives
have argued that America
of increasing inequality and segregation.
The data reviewed
is
in this
becoming
paper cast
doubt on the view that Americans are sorting more than before, and the calibration suggests
that increased correlation
effect
between spouses or among neighbors'
on inequality of moderately heritable characteristics.
dramatic increases
in sorting in either
deviation of education by only
likely to
likely to
As shown
in
have a minor
Table IV, even
marriages or neighborhoods would increase the standard
percent, or approximately one week.
Increased sorting
fit
the data but yield different long-run predictions, there
presume these more complicated models would suggest a
is
While
have a moderate impact on the intergenerational correlation of education.
other models might also
to
1
is
is
no reason
larger, rather than smaller, effect
of segregation on long-run inequality.
In fact, the analysis in this paper
since
it
may
overestimate the effect of sorting on inequality,
takes a, the effect of parents' education on their children's education, as exogenous to
26
sorting paterns.
Reductions
in the cost
of sorting are likely to raise the private return on
educational investments closer to the social return.
parents to adjust their consumption to the
Becker [1981] argues
education
if this
that parents
mean
to
They may
also reduce incentives for
smooth consumption across generations.
have greater incentive to invest
in childrens'
increases the child's chance of marrying a desirable spouse.
assortative marriage can be seen as a tax
proceeds going to their children-in-law.
on parents' investments
If
pm
=
0.6,
Imperfectly
in their children,
with the
average education of a child and his or
her spouse will increase by 0.8 years for every extra year of education the child's parents
provide.
If
p m rises by
1/3, to 0.8,
average education of a child and his or her spouse will
increase by 0.9 years for every extra year of education the child's parents provide.
likely that this twelve
and a half percent increase
would increase investment
in
in the return to educational
seems
It
investment
education by more than enough to offset the approximately one
percent decline in education from increased marital sorting in the estimated Markovian
model.
19
If
improvements
in the
matching technology lead parents
to increase investment in
education, and thus increase a, both the variance and the average level of education will
increase [Becker 1981].
In situations in
which the average
coefficient of variation, a/u, provides a better
deviation of education.
19
To
Under
this
level of education changes, the
measure of inequality than the standard
model, with
its
additive error term, increases in
a
will
the extent that imperfectly assortative marriage arises not because of informational
problems, but because people also value other noninherited characteristics, reduced variance
in these non-inherited characteristics will not necessarily lead parents to invest
education.
more
in
27
reduce the coefficient of variation, but
in parental
investment
no reason
in general, there is
in children will affect the coefficient
Improved matching
to
presume
of variation one
that increases
way
or the other.
consumption smoothing across
will also reduce incentives for
generations, and this will
make
because improvements
matching technology will cause poor people to expect worse
in
the long-run distribution of assets
more
equal.
This
is
marriages for their children, and rich people to expect better marriages for their children.
consequent reduction
poor people
to save
sorting on inequality
in the desire for
more and
may be
consumption smoothing across generations
rich people to save less.
effects
the direct effect of increased
by
skill in
workplaces
may have
larger
on inequality than sorting by parental characteristics, since an individual's future
characteristics are
presumably more highly correlated with
than with his or her parents' characteristics.
which
will cause
by behavioral response.
partially counteracted
Sorting by academic ability in schools or
Thus
is
his or her current characteristics
Ironically, this suggests that the
seen as least objectionable from an egalitarian point of view
in increasing inequality.
increasing.
increased.
There
is
some evidence
that sorting
Kremer and Maskin [1994] present evidence
form of sorting
may be most
important
by these characteristics
that sorting in
is
workplaces has
Hernstein and Murray present [1994] anecdotal, but fairly convincing, evidence of
increased sorting by academic ability in higher education.
Public policy toward sorting should depend on
inequality. Increased sorting
conflict, or
some
The
widen
social
may make
inequalities
many
more
things other than
may choose
effect
on
persistent, exacerbate political
and cultural gaps within the population.
conditions, decentralized agents
its
On
the other hand, under
efficient sorting patterns, as
Becker has
28
argued.
People
case, sorting
is
may
sort according to their preferences for public goods, for
In
any
often difficult and expensive to change through policy, as exemplified by the
history of school busing.
would
example.
This paper suggests that
typically be better spent directly
if
on investing
the rich and the poor to be nearer each other.
the goal
is
to reduce poverty, resources
in the poor, rather
than on re-shuffling
29
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Table
a
I:
Direct and Indirect Effect of Increasing p m from 0.6 to 0.8
%
direct increase
p m after multiplier
c„
effect
in
%
indirect increase
in a.,
0.9
14.0
0.850
4.0
0.8
1.0
0.825
1.0
0.6
2.7
0.809
0.11
0.4
1.0
0.803
0.01
34
Table
II:
Years of Schooling
Years of Education
Father
Mother
Children
0.38%
1.63%
2.63%
2.42%
4.14%
4.51%
7.73%
6.73%
8.19%
5.06%
30.13%
4.39%
5.89%
1.55%
6.02%
8.61%
0%
0%
.46%
.21%
1.25%
2.01%
2.97%
8.86%
4.89%
7.61%
8.82%
41.79%
4.26%
4.85%
2.63%
5.73%
3.68%
0.04%
0.13%
0.33%
0.59%
.09%
2.84%
4.93%
40.45%
9.53%
10.95%
5.31%
14.08%
9.74%
M
11.12 yrs
11.48 yrs
13.33 yrs
o
3.49 yrs
2.62 yrs
2.11 yrs
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
n
0%
0%
1
2393 obs 2393 obs 2393 obs
35
Table
III:
Regression Estimates of Childrens' Education as a Function of Parents'
Education and Neighbors' Education
VARIABLE
PARED
(1)
(2)
0.379
0.413
(0.022)
NEIGHED
MOMED
-0.262
(0.018)
0.127
0.525
:
(0.044)
(0.450)
0.204
~
—
:
DADED
0.179
.
(0.019)
-
:
-
(0.007)
:
:
:
(0.016)
~
~
0.032
7.51
8.53
7.45
7.554
(0.417)
(0.225)
(0.425)
(3.448)
0.266
0.262
0.266
0.274
1413
1413
1413
1413
0.012
Z
-0.028
_
ADJ. R"
N
-
_
P*NEIGHED
CONS
(0.165)
(0.025)
PARED"
NEIGHED
(4)
_
0.127
(0.044)
(3)
(0.0194)
Note.- standard errors are in parentheses.
MIT LIBRARIES
3 9080
009781565
36
Table IV: Effect of Changes in Sorting on Steady-State Inequality, Intergenerational Correlation of
Education, Inequality of Discounted Education, and Persistence of Inequality between Ethnic Groups*
(Percentage increase over base case
shown
in
parentheses.
Note: Maintained assumptions are that a=0.379, p=0.127,
)
at =
1.74,
y=
0,
w m=
0.7,
and
w
n
=0.5.
a.
P~(ZiAt+ i)
n
0.2
1.874
0.329
2.242
0.432
(-)
(-)
(-)
(-)
(-)
(-)
0.8
0.2
1.889
0.367
2.313
0.444
(33%)
(-)
(0.8%)
(11.5%)
(3.2%)
(2.6%)
High Neighborhood
0.6
0.4
1.899
0.354
2.307
0.445
Sorting
(-)
(100%)
(1.3%)
(7.7%)
(2.9%)
(2.9%)
High General Sorting
0.8
0.4
1.915
0.392
2.382
0.457
(33%)
(100%)
(2.2%)
(19.3%)
(6.2%)
(5.6%)
Base case
High Marital Sorting
Pm
Pn
0.6
J
*
e
Figure
1.
Inequality as a function of a and p
Date Due
OCT 12 1397
Lib-26-6"