£0 156 803
OB 018 456
Snith, Janes P. ; Welch, Finis
Inequality: Race Differences in the Distribution of
Earnings. Band Paper Series P5481-1.
Rand Corp., Santa Honica, Calif.
The Band .Corporation, 1700 Hain Street, Santa Monica,
California 90^06 ($1.50)
HF-$0.83 HC-S1.67 Plus Eostage.
♦Black Employment; Census figures; ♦Data analysis;
♦Employment ; ♦Equal Opportunities (Jets); *Incoae;
Low Income Groups; ♦Bacial Differences; Bacial
Middle Income Groups'
Characteristics and determinants cf earnings
distributions for black and vbite males are repealed in samples from
the 1960 and 1970 censuses. Using this data, this paper describes and
contrasts the properties^ of black and shite male earnings
distributions. It also uses earnings functions est » iated frcm the
census to identify and rank variables in terms cf their contribution
in explaining relative earnings dispersion* Extensive statistical
analysis is used to make predictions about black and vbite earnings.
A short bibliography is included. (Autbcr/PB) '
************** *********************** **********************************
* Reproductions supplied by EDRS are the best that can be made *
♦ from the original document. *
************************************* *£***************** 4 ** 4** *********
•ML i V 1978
» — I
INEQUALITY: RACE DIFFERENCES IN THE DISTRIBUTION OF EARNINGS
James P. Smith and Finis Welch
MATERIAL HAS BEEN GRANTfc
THE "EPSON OR ^OA^fflX 60 F * 0 *
ATINC IT POINTS OF uViS * T »0N ORIGIN.
STATED 00 NOT MPre« OPINIONS
SENT Orp1?,i°L A ^"" A ""-V REPRE.
*OUCAT,ON C SZSXZ P^VcV 076 ° P
The Rand Paper Series
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The Rand Corporation
Santa Monica, California 90406
INEQUALITY: RACE DIFFERENCES IN THE
•k-k J A
DISTRIBUTION OF EARNINGS
James P. Smith
The Rand Corporation, 1700 Main Street, Santa Monica, CA 90406
The University of California, Los Angeles, and The Rand Corporation
Forthcoming in the International Economic Review .
Inequality: Race Differences in the Distribution of Earnings
» " * 1
* James P. Smith and Finis 'Welch
Too often incoroe inequality in the United States is characterized
by stressing differences in average earnings between various demographic
groups. In fact, repeated emphasis on race and sex differentials might
lead one to suspect that mean wage differentials represents large part
of total inequality. Yet, according to the most recent Census, the
story of income inequality in America can be told with little mention of
black-white differences in mean earnings: for males, the black-white
wage differential accounts for Uess than three percent of total (log)
earnings variances. 2 Moreover, within race, blacks' earnings are often
less equally distributed than earnings of whites. The economic^pie may
be smaller for blacks, but it is alsc sliced less evenly.
In this paper, we examine characteristics and determinants of
earnings distributions for black and white males as they are revealed in
"^This research was supported by a contract from the Department of
Labor and a grant from ASPER, the Department of Health, Education and Welfare
to The Rand Corporation and a grant from the Ford Foundation to the National
Bureau of Economic Research* We would like to thank William Gould for his as
2 If blacks were to represent half of the population and if both
average differences and variances within race were preserved, the mean dif-
ference would still represent only six percent of total variance*
the 1/100 Public Use Samples of the 1960 and 1970 Censuses. This
paper is divided into two sections. The first describes and contrasts
the salient properties of black and white male earnings distributions.
Section II relies on earnings functions estimated from the Census to
identify and rank variables in terms of their contribution in explain-
ing relative earnings dispersion; These earnings equations are used
to predict the full distributions of earhings for blacks and whites
^-/-Separately so that predicted and observed distributions can be compared
throughout the complete range of the distributions. We think that the
predictions capture many important features of the observed distribu-
tions. In particular, predicted earnings variances in 1960 and 1970 are
highdr for blacks than whites and, in 1970, this difference is also re- '
fleeted in our predictions. A concluding part of this section, briefly,
presents results for a more generalized random-coefficients model that
aims at identifying sources of residual variation.
j. An Overview
Because comparisons of full distributions may reveal differences not
conveyed by summary measures, we begin with a convenient method of con-
trasting distributions illustrated in Figure 1. In the curves labelled
3 The sample is restricted to non-self-employed males with positive
earnings. Including the self-employed would increase inequality and im-
part a more positive skew to the distribution. Ignoring non-earnings
income probably leads to an underestimation of total income at bpth the
lower and upper tails of the income distribution - the lower tail because
of government transfers, the upper because of non-human wealth income.
"actual", earnings of black males at selected percentiles of black
earnings distributions are presented relative to the earnings of white
males at the same percentiles of white distributions. (The curves
labelled "predicted" are discussed below.) Since the curve is positively
sloped, there exists more relative dispersion for blacks up to the 70
to 80 percentile. In the top tail oi the distribution, the relative
dispersion of white earnings is greater, indicating greater positive
skewness in the white distribution. The reversal at the 80th percentile
shows that a unique ranking of inequality by race is not possible-. * Those
summary measures which weight the bottom tail of the distribution more
heavily (e.g., log variances) will tend to rank blacks over whites in
inequality. Other measures (e.g., coefficients of variation) could
produce the opposite result. Although we will initially rely on only
one measure of inequality, logarithmic variance, we will also deal
with some distinctions between this summary measure and the full dis-
Variances of log male earnings and weekly wages are listed in
Table 1 separately by .year (1960 and 1970), and race (blacks and whites).
These variances exhibit their familiar U-shaped age profile. For both
races, total variance is dominated by the within-age cell variances with
over 70 percent of the aggregate variance consisting of within-cell variance.
Between age cell variance is lower for blacks both absolutely and
as a proportion of total variance reflecting a less steeply-graduated
age earnings profile for blacks.
MEASURES OF DISPERSION
IN LOG MALE
JD *t V/
0.6087 * 0.7732
IN LOG MALE
TOTAL EARNINGS INEQUALITY USING VARIANCE IN LOG
Assumes equal number of Che blacks and whites in population.
Using log variance as the criterion, we find in the Census data that
earnings of blacks are less evenly-distributed than those of whites.
For those aged 18-65, variances in log weekly or yearly earnings -rose
for white males but remained relatively constant for black men between
1960 and 1970. The lower between-race variance in 1970 was hot suffi-
cient to prevent aggregate inequality from rising.
2. Sources of Difference in Earnings Dispersion
2.1 Regression Accounting
Earnings distributions are determined by the functional relationship
between earnings and personal characteristics and the underlying joint
distribution of these characteristics in/ a population. The regression
technique has become the standard format for accounting for group dif-
ferences in mean earnings but is less often used for examination of full
distributions. Yet, the" interest of policymakers often resides in the
tails of distributions rather than in comparisons of representative or
"average 11 persons belonging to the respective population under study.
Because the number of explanatory variables is large, the distributions
of these variables will.be initially characterized x simply by their
variances and covariances. Th(e regressions are based on the 1/100
Public Use Samples of* the 1960 and 1970 Censuses. Individuals are
"*This results from aggregate variance being heavily weighted by
the white sample.
partitioned according to estimated years of work experience^ and within
each experience class (1-5, 6-10, 11-15, 16-20, 21-30, and 31-40), the
regression estimated is of, the form:
(1) 7 - x'(b Q + + d 2 6 2 + d^S^) + u
where y = In (earnings last year/weeks worked last year), 7 x 1 is a vector
of characteristics of the individual and
d i =
1 if black ,
1 if 1960
In accounting for explained variance, let represent the estimated
parameter vector for the i-th group. Explained variance is:
(2) a 2 = b'.V. (x)b.
y i i i
.where V^(x) is the observed variance-covariance matrix of characteris-
tics for the i-th group. Characteristics are partitioned according to:
^ = ^2£ 2> ' —3 * ^4 ' ""5^ ^*
^Estimates by Hanoch by ages of beginning work level by schooling:
Schooling: 0-7 8 9-11 12 ' 13-15 16 17+,
Age at begin-
ning work: . 14 16 18 20 23 25 28
This is equivalent to an earnings equation with log annual earnings
as the dependent variable and weeks worked included as an independent
variable with unit coefficient.
x£ = (years of grade school, years of college) "= schooling;
x' 2 = (North Central; South, West, Metropolitan, Central City
years in current residence) = location;
x_2 = (federal employee, employee of regulated industry,
federal share of industry, state and local governments
share of industry) » government employment;
x| = (years of experience, years of experience squared) =
/ experience; . N
X5 = (log of weeks worked) = weeks worked •
The earnings variation attributed to the j-th set of explanatory variables
is: , - .
where V(x ) is the diagonal block in V(x) describing the variance of x. . .
Similarly, that part of the explained -variance attributable, to covariance
between x. and x, is:
(4) a2 C (x.,x k ) = 2b : ;c(x j ,x k )b k
Where b^ and b fc refer to x^ and x fc , and C(x^, x fc ) is the covariar\ce of
x j^ x fc* A summary of the more important accounting effects is contained
in Table 2.
With respect to black-white inequality, the partial effect of weeks
worked accounts for roughly half of explained variance. Depending 'upon
one's view of underlying causes of hours variation, it fray be preferable
12 ' ,
CONTRIBUTION OF SELECTED FACTORS TO VARIANCES PREDICTED
FROM ESTIMATED EARNINGS EQUATION
fotal* Variance iiofnnual Earnings
Contributions of '
» V %A. Log Weeks Worked
) B. Years of Schooling
; C. Regional Variables
D. Government* Variables
E. Schooling-Weejygtoked Interaction
F. Location-Scho^jS^Cn tor action
Total Variance in Annual Earnings
A. Log Weeks Worked
B. Years of Schooling
C. Regional Variables
D. Government Variables
E. Schooling-Weeks Worked Interaction
F. Location-Schooling Interaction
Total Variance in Annual Earnings
A. Log Weeks Worked
B. Years of Schooling
C. Regional Variables
D. Government Variables
E. Schooling-Weeks Worked Interaction
F. Location-Schooling Interaction
Total Variance in Annual Earnings
Contributions of .*
}A. Log Weeks Worked
Years of Schooling
Schooling-Weeks Worked Interaction
especially for welfare statements, to partition out that part of total
earnings dispersion due to hours worked. Clearly, if leisure time has
value and if the hours decision is voluntary, earnings f luctuatipns re-
sulting from fluctuations in time worked should not be viewed as equi-
valent to variance associated with wage differentials. Even though em-
ployment instability is an obvious cause of inequality, it is important
to note that the interracial differences are not solely due to employment
factors. Using variance in log weekly wage to measure inequality, the
dispersion among blacks still exceeds that among whites.
The distribution of schooling is a second factor underlying black-
white differences in inequality, for all but the most recent cohorts,
schooling is more unequally distributed among black males. There exists
a clear secular trend for both rac£s towards less dispersion in schooling
and a narrowing of the differentials in -variance between races. Given
similar average returns to education within experience classes, this
larger variance in black schooling would imply more black earnings in-
equality. However, proportionate variation in human capital, as mea-
sured only by years »of schooling completed, accounts for little of the
difference in inequality. The variance attributed to schooling declines
as work experience increases. The lower schooling variances in 1970
also lead to a reduction in inequality for both races. Since both
schooling and weeks^worked increase annual earnings, the positive cor-y
relation between them adds to earnings dispersion. Evidently labor sup-
ply behavior "builds positive correlation between wages and time worked
and spreads the distribution of earnings. Because schooling coefficients
8 ~ *
This is a result of lower estimated returns to schooling in the
more experienced groups.
and the covariance between education and weeks worked decline, over the
life cycle, this interaction has its primary influence in earlier ex~
perience intervals. Moreover, it Orally has a slightly larger effect
for whites than blacks and thus does not help explain race differences. ,
The regional distribution of blacks combined with t|e large variance
between regions in black earnings is important in explaining higher
black inequality. Among our region variables, Southern residence was
the most important. Holding constant schooling, experience and weeks
worked, black-wiiice "ratios of earnings range from 12 to 29 percent
lower (depending on the amount of job experience) for Southern residents
than they do for Northeastern residents. Regional disparities in
earnings are far more important for older (more experienced) workers ^
and are more important in 1960 than in 1970. Also, part of schooling f s
contribution to explained -variance is captured via covariance between
schooling and 'geographic location \- reflecting_the fact that average
school completion levels are higher where wages are high. 1 ~~ . - ~~
We attempted to measure the direct and indirect influence of
government on aggregate wage dispersion. The direct influence is cap-
tured simply with a dummy variable for employment in the government sec-
tor. Since government has the potential for influencing wages in other
sectors, as well, we also included variables indicating employment in
those sectors that seem most susceptible to governments power - indus-
tries regulated by the government and those that sell a significant frac-
tion of their product to government. Compared to the other factors in-
cluded in our regressions, government, employment, both direct and indi-
rect, proved j to be relatively unimportant, accounting for around 10 per-
cent of explained weekly wage variance.
2.2 Full Distributions
Although larger black relative 'dispersion was on average an ac-
curate characterization, we have noted that a comparison of the top
quarter of black and white earners reveal^ more relative dispersion
in white earnings. We examine next complete distributions of earnings
to see how closely our predictions compare to observations throughout
the entire distribution.. Using our estimated wage equations, earnings .
for all males in the Census samples with one to- forty years of market
experience are predicted, In Figure 1, black-white earnings ratios at
deciles of the predicted black and white distributions are compared to
the wage ratios based on the actual black-^white distributions* Because
a positiyely sldped curye indicates larger black relative variance, the
rising predicted earnings ratios until the 80 decile confirm our ability
to capture "some factors leading to larger black inequality. Although
the decline is not as rapid as that in actual earnings, our predicted
earnings ratios in 1970 decrease after the 80th decile. In 1960, our
predicted distributions fail to track the decline in black-white earnings
ratios in the upper deciles, but the rate of increase in the predicted
ratios is clearly attenuated. The factors used earlier to explain lar-
ger black inequality apparently also cause the reversal in relative
variance by race in the upper section of the earnings distribution.
We will illustrate why the reversals in relative variance occur
using the marginal distributions of the two variables identified by our
earnings equation as the dominant causal factors in explaining wage dis-
persion - education and geographical location.
In Table 3, years of schooling completed at deciles of the black and
vThite schooling distribution for 1960 are listed. The larger dispersion
A similar pattern (not shown) exists in 1970.
EDUCATION DISTRIBUTIONS 1960
A. Deciles of the Schooling Distribution
B. Marginal Returns to Schooling
Years of Schooling
0-8 9-12 13+
Black .0490 .1118 .1135
White .0601 .0972 .1048
C. Residential Distribution by Deciles of Predicted Earnings
in black schooling is not present throughout the education range.
Between the 10 and 70 percentile, black schooling increased by 5.9
years while white schooling increases by 4.3 years. Given similar
returns to schooling by race, this is consistent with the rising
income by decile curve observed in Figure 1. But after the 70 per-
centile, we increment black schooling by only 1.8 years and white
schooling by 3.4 years. Therefore, in the top three deciles, the
schooling distributions predict more relative earnings dispersion
among whites. This pattern of differential -variances in schooling is
reinforced by rising income returns with schooling level. Although
schooling coefficients are similar within schooling class by race,
whites achieve higher schooling categories at earlier percentiles. Com-
paring individuals in the top third of the earnings distribution, an
additional year of schooling adds more to white earnings, spreading out
the distribution of white earnings compared to that among blacks.
The geographical distribution of blacks and whites is also listed
in Table 3 at selected percentiles of the 1970 predicted earnings distri-
bution. White males are fairly uniformly distributed over their deciles.
For blacks, however, the disparities between the deciles are large. For
example, over 70 percent of blacks with less than the median black in-
come live in the South, but less than 20 percent of the blacks in the
top three deciles are Southerners. Moving across the lower 75 percent
q£ the .earnings distributions, the region variables contribute to a
rising black-white earnings ratio. This is due to the larger black wage
differentials between regions and the extensive outmigration of blacks from
the South over this range of the distribution. With substantial black
North-ScHith wage differentials, black earnings will rise relative to
whites as bJLack representation in the South is decreased.
Within region\ dispersion is also significantly larger in the South
so that those sections of the distribution more heavily weighted by
Southerners will exhibit more ^dispersion. Among high earners, the
movement across regions is considerably smaller so that the power of
region variables in affecting these earnings ratios is largely eliminated.
2.3 Residual Variation
After adjusting for personal attributes, the residual variances ob-
tained from our regressions were substantial and were also larger among
blacks. Although these residuals are "unexplained" by our regressions,
they contain useful. information about the process determining individual
earnings. In this section, we employ a random coefficients framework to
determine whether the residual variation about our estimated earnings
equations is systematically related to individual differences in the
returns to certain characteristics,
10 • .
As ,an approximation, the expected value of an individual's
squared residual can be written as
k ■ kk *
T?he diagonal terms in the double summation are the variance in the coef-
ficient among Individuals; the off diagonal terms represent the covariances
in these coefficients. For example, if the earnings equation included
_.only schooling and a constant term, we would write \
' E < E i> - °u + 2W 5 ! ■' a Bi S i
The coefficient of schooling squared measures the variance in the return
to schooling among individuals; the schooling coefficient measures the
covariance between the individual intercept and the return to schooling.
^Instead of the standard linear model with randomness only in the
intercept, assume that individuals differ in all parameters.
,-1 ,~ . ...... ...s ,„.„v-l
V(&) - (X'X) (Z.x.x'Vx.x') (X f X)"
The intercept measure the underlying residual variation unrelated to
characteristics. We estimated equation (5) for blacks and whites in
six experience groups in 1970, An individual's residual was "computed
by subtracting from his actual earnings his earnings predicted by our
OLS earnings equation, ^ After squaring these residuals, we estimated
equation (5) •
The only variables that ployed significant were schooling and
(footnote 10 continued)
For the -th individual, we have the prediction,
Y = xj3 with the observation Y = X *3 + n 0
The expectation of the squared calculated resfdual (e = Y - Y ) is
X* Z %
E(ej|) = xJV(|) Xjl - 2x^(3-3)11^ + E<h*)
since E(3-3)n. £ = (X'X)" 1 x^Vx^ and
=X £ Vx )f " e haVe
B(«J) = x'VC3)x £ + (l-^CX'X)- 1 ^) a^.
In general, estimation of V, the individual covariance structure of
parameter differences from the population mean presents a formidable
problem. But, in this case, since the number of observations ranges
between 4,000 and 7,000, we can appeal to large numbers. Notice, in
particular, \ that both x^V(3)x^ and x^ CX f X) -1 x^ are of order, T~\
where T refers to the number of observations. It follows that
2 \ *
plim i = xlVx rt , = o 2 „ and as an approximation e£ - xJVx 0 + W 0 is used
T ^oo ^ a n * * * * *
along with the assumption that W has zero expectation and is independent of X.
The random coefficient model suggests that a GLS approach may
have been more appropriate. The absence of any meaningful heteroscedascity
(noted below) indicates that this would not alter our results.
schooling squared. The coefficients of these two variables along
with the mean squared residual for each experience group is reported
in Table 4.
A. Coefficients from Regressions on Squared Residuals
1970 Whites 1-5 6-10 11-15 16-20 21-30 31-40
Schooling ,0069 .0062 ' .0050 .0030 .0047 .0027
Schooling -.2073 -.1861 -.1355 -.0964 -.0807 -.0572
Mean White .453 .284 .245 .283 .298 .294
Schooling .0066 .0045 .0025 .0080 .0001 .0063
Squared , ^
Schooling -.3176" -.1913 -.0461 -.0107 -.0186, -.0717
Mean Black .772 .457 , .388 .374. .-402 .453
In all twelve regressions, the schooling squared variable had the required
positive coefficient. The negative coefficient on schooling may indicate
that earnings from other pursuits and schooling are substitutes. Indivi-
duals who are able to obtain -high earnings in endeavors unrelated to
schooling may well behave so that they achieve low ex post returns to
schooling. Based on these regressions, variation in rates of return to
schooling are large and are an important source of the residual variances.
For racial comparisons > variation in schooling returns may be slightly
Uarger for whites so that they explain little of the black-white difference.
Our results suggest that at least additional research on this topiq
usinft (more) appropriate panel data may be fruitful.
We have examined the p6tential for using earnings equations esti-
mated from two large cross-sectional data bases Uri explaining
the complete distributions of black and white male earnings.
Although we have achieved some success iri documenting the contribution
of several variables, there are numerous factors omitted in our study.
Perhaps, the most serious omission involves the role of differences in
the underlying distribution of ability within population, Assortative
mating patterns alone could produce differential distributions: of
ability. The more positive the degree of assortative mating, the lar-
ger the dispersion iQ genetic traits in succeeding generations. Dis-
crimination against blacks'* may also operate in a manner that increases
dispersion in black incomes If discrimination takes^the form of quotas
or non-price rationing, it is the least able and qualified blacks who
will bear the major burden ♦ Unionism and minimum wage laws will tend to
produce similar results as the least skilled blacks are crowded into the
unprotected sectors ♦ Although the evidence we have suggests that on net
government is relatively unimportant and that its contribution probably
receives too much emphasis, the influence of a broad package of govern-
mental, welfare legislation should surely * be investigated.
Brimmer, Andrew, fl The Negro in the National Economy," T}ie American Negro
Reference Book, John P. Davis, editor (New Jersey, Prentice Hall, 1966).
Chiswick, Barry, "income Inequality," National Bureau of Economic Research,
, "Racial Differences in the Variation in Rates of Return from
Schooling," printed in Patterns of Racial Discrimination^ Volume II,
Employment and Income, Voxt Turstenberg et al. , (1974).
Jensen, Arthur,, "How Much Can We Boost I.Q. and Scholastic Achievement,"
Harpard~%ducational Review, 39, No. 1, (Winter 1960).
Lillgra, Lee and Robert Willis, "Dynamic Aspects of Earnings Mobility,"
unpublished manuscript, (June 1976).
Mincer, Jacob, Schooling > Experience and Earnings (New York: National*
Bureau of Economic Research, 1974) v f
Oster, Sharon M. , Are Black Incomes More Unequally Distributed," American
Economists? all 1974).
Smith, James P., The Distribution of Family Earnings, unpublished paper,
, and Finis Welch, Black-White Male Earnings and Employment: 1960-1970,
R-1666-DOL, (Santa Monica: The Rand Corporation, July 1975).
Wohlstetter, Albert and Sinclair Coleman, Race Differences in Income^ R-578-OEO
(Santa Monica: The ^and Corporation, 1970).
. • 23