Journal
of Econometrics
43 (1990) 121-133.
North-Holland
GENERALIZED
ENTROPY MEASURES
OF MOBILITY
DIFFERENT
SEXES AND INCOME LEVELS
Esfandiar
Southern Methodist
FOR
MAASOUMI
University, Dallm, TX 75275, USA
Sourushe
ZANDVAKILI
University of Cincinnati,
Cincinnati, OH 45221, USA
A family of mobility measures is investigated using the Michigan panel data on income dynamics,
These measures are decomposed in order to learn about components
that are due to differences in
gender and income groups, on the one hand, and within-group
components which are free of such
group characteristics.
These measures compute
the degree of equalization
over time of the
distribution
of ‘permanent
incomes’. Permanent or average income of an individual is computed
over an increasing
number of points within a time interval of interest. This degree of equality is
compared
(as a ratio) to the equality of incomes at any reference point of interest in order to give
us a measure of stability in the size distribution
of incomes. We take an average of inequality
values over the time interval as our reference value for ‘short-run’ inequality. Mobility is greater
the greater this degree of relative equalization.
Several aggregator
functions have been used to compute the ‘permanent
income’ variable. Their
justification
and role in robustifying
our inferences is briefly discussed. The mobility profiles here
suggest that there has not been much equalization between different sexes, but there is not a lot of
inequality
between these groups in our sample. Most of the inequality is within groups where the
annual distributional
changes belie the relative stability of the permanent
income distribution.
1. Introduction
Mobility may be defined as a measure of movements within the distribution
of a suitable welfare attribute such as income. The measurement
of such an
inevitably
latent concept requires a suitable point of reference, or a reference
distribution,
and entails many implicit value judgements
that may be ideally
explicated
with reference to implied welfare functions.
Such measurement
should also correct for at least some transitory movements in order to provide
meaningful
assessments of permanent
mobility.
Several approaches
are available for the study of mobility. The alternatives
not discussed in this paper generally require a generation
mechanism
or an
estimate
for the transition
probabilities
governing
individual
movements.
Measures of mobility are functions defined on such probability
matrices. For
modern
welfare-theoretic
discussions
of this approach
see, for example,
Shorrocks (1976) and Geweke et al. (1986) in the continuous
time domain.
0304-4076/90/$3.5001990,
Elsevier Science Publishers
B.V. (North-Holland)
122
E. Maasoumi, and S. Zandvakili,
Generalized entropy measures of mobility
There are other ‘structural’ approaches to earnings mobility that utilize
econometric analyses of a priori specified models relating mobility to its
hypothesized causes. Examples of this approach are Creedy (1985) and Lillard
and Willis (1978). The approach that is implemented here has been developed
and applied in Maasoumi and Zandvakili (1986,1988) and Zandvakili (1987).
Shorrocks (1978) proposed a special member of the family of mobility measures developed by us, but contains an excellent discussion of the basic ideas.
Our measures of mobility are anonymous in the sense that they are based on
relative and symmetric measures of ‘distance’ between an income distribution
and a uniform (rectangular) one. This is of course a relative inequality measure
as exemplified by Theil’s information measures. We also use ‘longer-run’ or
‘permanent’ real incomes by aggregating individual’s income over time intervals of interest. This is designed to control for some of the transitory
fluctuations that may be caused by (e.g.) windfalls and to account for other
temporary or cyclical movements. But interest usually centers on how the
long-run distributions compare with reference distributions at specific points
in time. We take the weighted averages of annual inequalities within an
interval of interest to be a representation of the ‘short-run’ income inequality
and a reasonable reference value. The ratio of these two inequality indices
serves as a bounded measure of ‘stability’ in the income distribution over the
corresponding time interval. A negative function of this ratio is then a measure
of anonymous mobility. Mobility is thus dejned as a relative degree of equalization as the time interval for measuring incomes is expanded.
In our investigation we compute permanent incomes using an ‘optimal’
family of aggregator functions proposed in Maasoumi (1986). We also look at
diverse members of an ‘optimal’ decomposable family of relative inequality
measures known as the Generalized Entropy (GE). The experimentation over
members of these families, as well as with different sets of weights given to
incomes at different points within the intervals of interest, represent an
attempt to robustify our summary findings. We are also able to investigate, in
some cases quite unambiguously, decompositions of inequality and mobility
profiles by gender and by income level. This level of detail is necessary for a
better understanding of where policy may be required.
On the basis of the approximately 2300 households which remained in the
Michigan panel over the period 1969-1981, we conclude: (i) There is not a
great deal of inequality between the men- and women-headed households. (ii)
The dominant within-group component of inequality is either increasing over
this period or, when incomes are smoothed by time aggregation, relatively
stable. (iii) This larger within-group component of inequality is due to high
levels of inequality within lower income groups (such as women-headed
households). (iv) Grouping by real income brackets leads predictably to very
large between-group inequality values. (v) Some equalization of real incomes
has occurred over time within most income groups, but this is very hard to
E. Maasoumi, and S. Zandvakili, Generalized entropy measures of mobility
123
judge by a comparison of annual inequality measures and most clearly
revealed by using our ‘permanent income’ distributions. (vi) Modest levels of
mobility are recorded as the aggregation interval is expanded, but the corresponding profiles flatten- out after about eight or nine years.
The plan of this paper is as follows. Section 2 briefly outlines the theoretical
framework and the notation. Section 3 contains a discussion of our empirical
findings with decompositions based on gender. Section 4 deals with similar
decompositions based on income groups, and section 5 concludes.
2. The framework
Let q, denote the income of individual i, i = 1,. . . , N, in period t, t =
1, . . . , T, and q = cf”_ ,Y,, the total income over any subperiod M I T. A class
of aggregate or ‘permanent’ income functions which can treat incomes in
different periods as distinct and substitutable attributes may be denoted by
M=l,...,T.
si=sj(~r,...J,+f)Y
(1)
Following the arguments in Maasoumi (1986), we choose S,(e) according to
generalized information criteria which insure that the distribution of the
resulting aggregate income shares has the minimum ‘distance’ from the M
distributions of its M annual income components. Examples of such functional families are
1
-l/8
S,a
[
EC&~
,
f
P+0, -1,
(2)
P=O,
(2’)
p=
-1,
(2”)
with (Y, denoting the weight given to income at time t, c,a, = 1, and the
constant elasticity of substitution of income across time is u = l/(1 + /3).
The family of measures used here to compute inequality (the degree of
equalization compared with the rectangular distribution) is the Generalized
Entropy (GE). For an axiomatic analysis of the many useful properties of
these measures see, for instance, Shorrocks (1984). This family includes many
of the well-known measures, such as both of Theil’s information indices [see
Theil (1967)] and monotonic transformations of a class of measures proposed
by Atkinson (1970). The GE family is defined as follows (e.g.) for the
124
E. Maasoumi, and S. Zanduakili, Generalized entropy measures of mobility
aggregate income shares, S,* = Si/cy_ Sj:
I,(S)
= ?[(NS,*)‘+‘-
=
l]/Ny(l+y),
fs;*
= &Vi
log(Nsi*),
log(l/Ns;*),
Y#O,-I.
Y=
0,
y= -1,
(3)
(3’)
(3”)
where (3’) and (3”) may be recognized as, respectively, Theil’s first and second
inequality measures. The well-known utilitarian, equality-prefering, Schurconcave welfare functions corresponding to this family of indices reveal some
of the subjective but traditionally implicit value judgements attached to
measuring mobility. Roughly speaking, these welfare functions incorporate a
subjective preference for mobility toward an equal distribution of income.
Useful additive decomposition properties of these measures have been given
elsewhere, for example in Maasoumi and Zandvakili (1988). Accordingly, we
are able to distinguish the contribution to total inequality-mobility
coming
from (averaged) within-groups from that which exists between groups. The
least ambiguous of these decompositions is afforded by Theil’s second measure
(y= -1).
To measure mobility, let Y = (Yi,, ...,Y,,)be the income vector at point t.
Then, the measure of mobility described in section 1 is any negative function
of the following measure of relative stability, R,, defined over M periods:
where the sum in the denominator is a measure of ‘short-run’ inequality and S,
are defined over the same M periods in a consistent manner. It is clear that for
any convex measure 1, we have 0 I R, I 1 if the permanent income function,
S, is a weighted average of incomes with the same weights (Y,[as in (2”)]. This
special case was shown by Shorrocks (1978). The same bounds on R, may be
established using the decompositions of multidimensional measures of inequality given in Maasoumi (1986, propositions 1 and 2). We may use 1 - R, as a
measure of mobility. P, = (R,,
...,RT) is a stability profile which can also
reveal the effect of increasing smoothing of the income variable starting from
R, = 1. Other than this smoothing out of the short-run effects, R, is capable
of revealing durable ‘mobility’ toward equalization in a way that may be
obscured by looking at each I,( Y,), t = 1,. . . , T.This is most clearly seen by
considering a situation in which only a permutation of the income vector has
occurred between two periods. As is well known, this leaves our anonymous
E. Muusoumi,
and S, Zanduukili,
Generulized entropy meusures
ofmobility
125
(symmetric),
relative inequality
measures unchanged,
but the distribution
of
the aggregated
incomes over the two periods will be changed,
possibly
dramatically
(unless there is perfect equality to begin with!). Consequently,
I,(S) and R M will measure any mobility over the two periods.
3. Mobility and gender
‘Household’
income data for the period 1969-1981
are taken from the
Michigan Panel Study. Household’s income (head and spouse, if any) consists
of the following:
income from wages, salaries, rents, dividends,
interest,
business, bonuses, commissions,
professional
practice, aid to dependent
children, social security, retirement
pay, pension or annuities,
unemployment
compensation,
child support, and other transfer payments. Real total income
is obtained
using the current consumer price index. Income is adjusted for
family size (in 1975) to provide a better measure of family income since family
members
effectively
pool their incomes [see Kakwani
(1984) and Rosen
(1984)]. We refer to this adjusted income as the ‘Per Capita Family Income’
(PCFI).
In computing
the permanent
incomes three different schemes were used in
order to weight income at different times. These (Y, weights are (i) equal
weights for all years, (ii) the ratio of mean income at time t to the mean
income over the entire M periods (MIW in tables), and (iii) the normalized
elements of the eigenvector corresponding
to the first principal component
of
the x’x matrix [see Maasoumi (1989) and Ram (1982)]. We did not find any
qualitative
differences in our results between these three cases, and thus report
only the computations
based on ratio of means weights. The other two cases
are reported in Zandvakili
(1987).
In our computations
the substitution
parameter
/? is restricted
by the
relations
- y = 1 + p. We computed four different aggregator functions corresponding
to four inequality
measures with - y = V= (2,1,0.5,0.0).
V= 0.0
and 1.0 correspond
to Theil’s first and second inequality
measures, respectively, combined with the linear and the Cobb-Douglas
forms of the aggregator
function.
Tables l-3 provide, respectively, the annual short-run inequalities,
the inequalities
in the aggregated (long-run) incomes, and the income stability
of each based on gender is also given. Note that
measures R M.Decomposition
as one moves toward 1981 the number of periods over which S,, Z,(S), and
R, are calculated is increasing from 1 to 13. The results for every other year
are reported to save space.
Short-run inequality in table 1 has generally increased. As expected, inequality is greater with larger degrees of relative inequality aversion (V). There are
1776 male- and 529 female-headed
households
in the sample. The ‘withingroup’ component
of short-run inequalities is dominant. The absolute value of
the ‘between-group’
component, however, has increased over the 13 years. For
both sexes annual inequalities have a rising trend (less uniformly so for V = 2).
126
E. Muasoumi, and S. Zandvakili, Generalized entropy measures of mobility
Table 1
1969-81 per capita family income: Short-run inequalities.
Overall
Between
Within
Men
Women
0.793
0.729
0.636
0.815
0.981
1.211
1.520
1.676
1.604
1.600
3.015
2.195
2.580
3.292
Degree of inequality aversion = 2.0
1969
1971
1973
1975
1977
1979
1981
1.109
1.049
1.002
1.636
1.569
1.895
2.441
0.014
0.016
0.019
0.023
0.038
0.041
0.047
1.095
1.033
0.982
1.613
1.530
1.854
2.394
-
Degree of inequality aversion = 1.0
1969
1971
1973
1975
1977
1979
1981
0.430
0.466
0.464
0.531
0.578
0.613
0.706
0.013
0.014
0.017
0.021
0.033
0.035
0.039
0.417
0.452
0.446
0.510
0.545
0.579
0.667
0.340
0.375
0.362
0.422
0.467
0.498
0.578
-
0.676
0.709
0.727
0.808
0.808
0.849
0.964
Degree of inequality aversion = 0.5
1969
1971
1973
1975
1977
1979
1981
0.375
0.407
0.404
0.456
0.494
0.505
0.571
0.012
0.014
0.017
0.019
0.030
0.032
0.036
0.363
0.394
0.387
0.436
0.463
0.473
0.535
0.306
0.341
0.331
0.380
0.419
0.427
0.490
-
0.601
0.618
0.635
0.693
0.688
0.707
0.782
Degree of inequality aversion = 0.0
1969
1971
1973
1975
1977
1979
1981
0.367
0.402
0.395
0.448
0.492
0.483
0.549
0.012
0.013
0.016
0.018
0.029
0.030
0.034
0.355
0.388
0.380
0.430
0.463
0.453
0.515
0.304
0.344
0.332
0.383
0.429
0.417
0.481
0.606
0.615
0.631
0.693
0.682
0.688
0.753
But short-run inequality amongst female-headed
households is always greater
than amongst men. These annual values, however, contain many transitory
components
which are partially removed from the aggregated values in table 2.
Table 2 values exhibit much less volatility. After a decline in the initial
years, Z,(S) has increased back to about its original value. Also, long-run
inequality
is always smaller than the corresponding
short-run inequality. Once
again, inequality
among women is greater than among men, and within-group
inequality is several times the between-group
component.
These relative values
are somewhat sensitive to the family size adjustment of incomes. For instance,
E. Maasoumi, and S. Zandvakili, Generalized entropy measures of mobility
127
Table 2
1969-1981 per capita family income (MIW): Long-run inequality.
Overall
Between
Within
Men
Women
0.793
0.665
0.599
0.588
0.602
0.653
0.690
1.616
1.430
1.365
1.533
1.507
1.509
1.563
0.340
0.325
0.316
0.319
0.327
0.336
0.352
0.676
0.651
0.648
0.654
0.662
0.654
0.674
0.306
0.296
0.291
0.294
0.302
0.310
0.323
0.601
0.579
0.577
0.579
0.585
0.579
0.591
0.304
0.295
0.292
0.295
0.306
0.312
0.326
0.606
0.578
0.572
0.573
0.578
0.566
0.573
Degree of inequality aversion = 2.0
1969
1969-71
1969-73
1969-75
1969-77
1969-79
1969-81
1.109
0.949
0.889
0.951
0.974
1.035
1.100
1969
1969-71
1969-73
1969-75
1969-71
1969-79
1969-81
0.430
0.414
0.408
0.414
0.426
0.434
0.455
1969
1969-71
1969-73
1969-75
1969-77
1969-79
1969-81
0.375
0.364
0.360
0.363
0.373
0.381
0.396
1969
1969-71
1969-13
1969-75
1969-77
1969-79
1969-81
0.367
0.355
0.351
0.354
0.366
0.371
0.386
0.014
0.016
0.018
0.020
0.025
0.030
0.034
1.095
0.933
0.871
0.931
0.950
1.005
1.067
Degree of inequality aversion = 1.0
0.013
0.014
0.016
0.018
0.021
0.025
0.029
0.417
0.400
0.392
0.396
0.404
0.409
0.425
Degree of inequality aversion = 0.5
0.012
0.014
0.015
0.017
0.020
0.024
0.027
0.363
0.350
0.345
0.347
0.353
0.357
0.369
Degree of inequality aversion = 0.0
0.012
0.013
0.014
0.016
0.019
0.023
0.026
0.355
0.342
0.337
0.339
0.341
0.349
0.361
the between-group
component increases to 15-25% of overall inequality for all
unadjusted
incomes [see Zandvakili
(1987)]. This is partly due to a larger
proportion
of two income earners being among the male-headed
families.
The corresponding
stability measures are presented in table 3. Again, seven
of the thirteen
possible values are reported without any qualitative
loss.
0 < R,,, < 1 in all cases. The following may be concluded from table 3:
(i)
There is a tendency for the profiles to fall and then level off as the number
of aggregated years increases from one to thirteen.
128
E. Maasoumi,
and S. Zandvukili,
Generalized entropy measures of mobility
Table 3
1969-1981
per capita family income (MIW):
Overall
Between
Degree of inequality
1969
1969-71
1969-73
1969-75
1969-71
1969-79
1969-81
1.000
0.916
0.885
0.828
0.786
0.744
0.692
1.000
0.928
0.903
0.877
0.855
0.832
0.813
1969
1969-71
1969-73
1969-75
1969-77
1969-79
1969-81
-
1.000
0.932
0.911
0.885
0.867
0.855
0.840
-
1.000
0.927
0.904
0.879
0.864
0.852
0.839
aversion
0.033
0.034
0.037
0.039
0.045
0.052
0.056
1.000
0.907
0.873
0.822
0.772
0.723
0.670
1.000
0.918
0.887
0.823
0.786
0.747
0.700
-
1.000
0.912
0.881
0.851
0.824
0.801
0.779
-
1.000
0.949
0.930
0.907
0.887
0.860
0.841
= 0.5
0.967
0.896
0.873
0.844
0.820
0.801
0.782
aversion
Women
= 1.0
0.970
0.896
0.869
0.839
0.812
0.783
0.761
aversion
Men
= 2.0
0.987
0.900
0.868
0.811
0.167
0.723
0.671
0.033
0.035
0.038
0.041
0.047
0.054
0.058
Degree of inequality
1969
1969-71
1969-73
1969-75
1969-77
1969-79
1969-81
aversion
0.030
0.032
0.035
0.038
0.043
0.049
0.052
Degree of inequality
1969
1969-71
1969-73
1969-75
1969-77
1969-79
1969-81
Within
0.013
0.016
0.017
0.018
0.020
0.021
0.021
Degree of inequality
Income stability.
1.000
0.917
0.891
0.862
0.841
0.830
0.813
-
1.000
0.955
0.942
0.920
0.904
0.885
0.870
= 0.0
0.967
0.892
0.868
0.840
0.819
0.800
0.783
1.000
0.915
0.889
0.861
0.846
0.835
0.822
1.000
0.950
0.933
0.911
0.895
0.871
0.855
(ii)
The profiles for households headed by men fall faster and further than
those of women-headed
households.
(iii) These patterns
are robust with respect to the choice of aggregation
function, family size adjustment,
and inequality measure.
The fact that the profiles are becoming flatter is an indication that, although
there have been some transitory movements in the size distribution
of income,
there is a lack of any permanent
equalization.
Further, while some equaliza-
E. Muusoumi, und S. Zmdvakili, Generalized entropy measures of mobility
tion has taken place within each group of households, inequality between
and women-headed
households has increased in absolute value.
129
men-
4. Mobility and income level
Maasoumi
and Zandvakili
(1988) give inequality
and mobility decompositions by age, education, and race. Similar decompositions
by income level can
reveal the aggregate impact of all such nonincome
characteristics
(including
gender). It is anticipated
that if the major causes of variation in incomes are
transitory in nature, the length of time spent in any income class will be short.
‘Permanent’
income inequality changes will be very revealing in this context.
The total sample is divided into seven income groups (GI-G7).
The
assignment
to groups is on a one-time basis and according
to the simple
arithmetic
mean income of the individual
household over the thirteen-year
period. These real income levels begin with mean incomes of less than $4,999,
and increasing in increments of $5,000. The last group contains mean incomes
of $35,000 or more.
Short-run
inequalities
and their decompositions
based on income level are
given in table 4. There are several recognizable patterns. The ‘between-group’
inequality
has increased steadily over this period. The ‘within-group’
component of inequality
fluctuates around a relatively constant
mean value. The
observed patterns suggest that the nonincome
differences do contribute
to the
increase in between-group
inequality. Over 70% of women-headed
households
earn less than $15,000. Of course, this is confounded
by the differential impact
of inflation on different income groups (we use real incomes).
Overall long-run
inequality
levels have risen after an initial decline. Its
decomposition
by income level shows that, as Z,(S) has changed, its betweengroup component
has increased uniformly. At the same time the within group
inequality
has decreased steadily. This change has been dramatic so that in the
later years the between-group
component is larger than the within component.
These changes include the well-known life cycle and human capital effects and
are not inconsistent
with the cummulative
effects predicted by discrimination
theories.
The long-run
within-group inequalities
reveal a falling trend for each of the
seven income groups. This is anticipated
since transitory
components
are
smoothed out and individual incomes have approached
group mean incomes
in the long run. These long-run grouping observations
are somewhat sensitive
to the family size. Within-group
aggregate income inequalities
are noticably
smaller when income is not adjusted for family size, and there is generally less
inequality
within the higher income groups.
The stability profiles in table 6 reveal much higher degrees of permanent
equalization
within income groups than was observed for the gender groups of
the last section. Note that as the stability profiles of the whole sample flatten,
--_.
0.367
0.402
0.395
0.448
0.492
0.483
0.549
1969
1971
1973
1975
1977
1979
1981
^.
0.375
0.407
0.404
0.456
0.494
0.505
0.571
1969
1971
1973
1975
1977
1979
1981
.,I_.
0.430
0.466
0.464
0.531
0.578
0.613
0.706
1969
1971
1973
1975
1977
1979
1981
._ -
1.109
1.049
1.002
1.636
1.569
1.895
2.441
1969
1971
1973
1975
1977
1979
1981
Overall
-
_
,_,
0.272
0.264
0.224
0.231
0.219
0.212
0.249
0.103
0.144
0.180
0.225
0.275
0.293
0.322
_
0.270
0.268
0.230
0.242
0.238
0.218
0.256
0.316
0.306
0.259
0.272
0.262
0.267
0.322
0.114
0.161
0.205
0.259
0.316
0.346
0.384
0.0%
0.134
0.165
0.207
0.254
0.266
0.293
0.958
0.824
0.692
1.225
1.045
1.271
1.716
Within
0.151
0.224
0.310
0.412
0.524
0.624
0.725
Between
1969-81
Table 4
-
G2
1.429
0.560
0.415
0.539
0.487
0.683
0.985
0.413
0.334
0.284
0.324
0.310
0.374
0.490
0.356
0.299
0.263
0.292
0.282
0.331
0.432
0.402
0.398
0.348
0.333
0.343
0.345
0.382
.
-
0.339
0.290
0.261
0.284
0.274
0.320
0.424
.----
0.391
0.323
0.272
0.290
0.306
0.311
0.362
0.389
0.325
0.279
0.291
0.303
0.310
0.361
Degree of inequality aversion = 0.0
0.396
0.388
0.336
0.326
0.337
0.332
0.364
Degree of inequality aversion = 0.5
0.429
0.413
0.352
0.354
0.364
0.353
0.393
0.421
0.356
0.302
0.312
0.322
0.335
0.394
0.668
0.626
0.430
0.459
0.479
0.544
0.679
Degree of inequality aversion = 1.0
0.879
0.672
0.524
0.950
0.633
0.639
0.722
G3
Short-run inequalities.
Degree of inequality aversion = 2.0
Gl
per capita family income:
--_-,/_
0.349
0.367
0.313
0.293
0.263
0.271
0.311
0.353
0.373
0.314
0.294
0.260
0.269
0.315
0.385
0.411
0.336
0.314
0.273
0.286
0.342
0.608
0.725
0.479
0.449
0.365
0.460
0.525
G4
0.243
0.235
0.192
0.237
0.214
0.196
0.236
0.233
0.228
0.188
0.224
0.205
0.191
0.234
0.236
0.231
0.190
0.224
0.208
0.194
0.252
0.282
0.281
0.217
0.274
0.290
0.227
0.570
G5
0.263
0.246
0.214
0.215
0.176
0.200
0.222
_
0.243
0.230
0.202
0.205
0.167
0.192
0.218
0.240
0.227
0.199
0.204
0.165
0.193
0.227
0.293
0.265
0.220
0.236
0.178
0.232
0.310
G6
0.187
0.228
0.202
0.223
0.245
0.191
0.233
0.179
0.219
0.192
0.212
0.224
0.183
0.221
0.179
0.224
0.190
0.211
0.218
0.183
0.225
0.209
0.450
0.212
0.242
0.242
0.211
0.299
G7
1.109
0.949
0.889
0.951
0.974
1.035
1.100
0.430
0.414
0.408
0.414
0.426
0.434
0.455
0.375
0.364
0.360
0.363
0.373
0.381
0.396
0.367
0.355
0.351
0.354
0.366
0.371
0.386
1969
1969-71
1969-73
1969-75
1969-77
1969-79
1969-81
1969
1969-71
1969-73
1969-75
1969-77
1969-79
1969-81
1969
1969-71
1969-73
1969-75
1969-77
1969-79
1969-81
1969
1969-71
1969-73
1969-75
1969-11
1969-19
1969-81
Overall
0.096
0.117
0.137
0.159
0.186
0.210
0.235
0.103
0.128
0.150
0.175
0.204
0.231
0.257
0.114
0.144
0.170
0.200
0.233
0.264
0.294
0.151
0.208
0.255
0.311
0.365
0.425
0.481
Between
1969-81
0.270
0.238
0.214
0.195
0.180
0.161
0.152
0.272
0.236
0.210
0.188
0.169
0.150
0.139
0.316
0.270
0.238
0.213
0.192
0.171
0.161
0.958
0.741
0.634
0.640
0.609
0.610
0.619
Within
G2
(MIW):
1.429
0.708
0.504
0.404
0.341
0.313
0.308
0.413
0.311
0.261
0.225
0.200
0.181
0.179
0.356
0.274
0.236
0.205
0.182
0.164
0.162
0.402
0.339
0.298
0.263
0.246
0.230
0.220
0.339
0.261
0.226
0.197
0.174
0.153
0.153
Degree of inequality aversion
0.396
0.334
0.294
0.260
0.245
0.229
0.219
Degree of inequality aversion
0.429
0.354
0.309
0.275
0.259
0.241
0.232
=
=
0.0
0.5
Degree of inequality aversion = 1.0
0.879
0.610
0.486
0.490
0.444
0.422
0.401
inequality.
0.391
0.318
0.256
0.219
0.191
0.171
0.165
0.389
0.320
0.262
0.225
0.196
0.172
0.163
0.421
0.346
0.282
0.242
0.210
0.180
0.167
0.668
0.548
0.425
0.348
0.303
0.270
0.244
G3
Long-run
Degree of inequality aversion = 2.0
Cl
per capita family income
Table 5
0.349
0.335
0.296
0.262
0.214
0.182
0.173
0.353
0.342
0.304
0.269
0.219
0.185
0.175
0.385
0.374
0.328
0.290
0.233
0.192
0.180
0.608
0.601
0.506
0.426
0.322
0.274
0.247
G4
0.243
0.207
0.188
0.176
0.168
0.144
0.136
0.233
0.199
0.182
0.168
0.157
0.135
0.128
0.236
0.198
0.181
0.166
0.150
0.129
0.122
0.282
0.230
0.208
0.188
0.166
0.142
0.147
G5
0.263
0.223
0.212
0.190
0.169
0.147
0.135
0.243
0.208
0.198
0.179
0.161
0.139
0.127
0.240
0.204
0.193
0.176
0.158
0.135
0.122
0.293
0.240
0.222
0.208
0.188
0.160
0.146
G6
0.187
0.187
0.179
0.173
0.173
0.161
0.152
0.179
0.177
0.169
0.163
0.160
0.150
0.138
0.179
0.176
0.166
0.159
0.154
0.143
0.130
0.209
0.261
0.222
0.200
0.186
0.172
0.155
G7
132
E. Maasoumi, and S. Zanduakili. Generalized entropy treasures of mobiliiy
?
_
II
E. Muusoumi.
and S. Zunduukili,
Genercrlized entropy measures of mobilit,:
133
the corresponding
within-group
profiles continue
to fall. In our view some
equalization
has occurred, but this is mostly confined to within income groups.
5. Conclusion
A family of stability profiles was introduced and decomposed by gender and
income level. The sensitivity of such profiles with respect to the choice of
income aggregation
function, the weights given to annual incomes in such
aggregation,
measure of inequality,
and the length of time was investigated
within certain limits. It was learned that the stability profiles generally flatten
after a decline in the earlier years. The decomposition
of both incomes and the
stability
values by gender revealed greater degree of equalization
among
households
headed by men. Income level decompositions
show income movements within each group and not across them. This is partly because we have
studied a panel grouped by incomes and partly a reflection of nonincome
effects. Inference about the U.S. on the basis of the Michigan data must be
made with caution.
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