NOTES COMPARING INCOME MOBILITY IN GERMANY AND THE UNITED STATES USING GENERALIZED ENTROPY MOBILITY MEASURES Esfandiar Maasoumi and Mark Trede* Abstract—Based on a derivation of the asymptotic sampling distributio n of the generalize d entropy mobility measures, this paper provides a statistically rigorous analysis of income mobility in Germany and the United States using the panel data set PSID-SOEP equivalen t data le. Several alternativ e measures of income aggregation , inequality measures, and grouping s are considere d to establish robustness . We nd that, to a high degree of statistical con dence, post-governmen t income mobility is much higher in Germany. Possible reasons for these ndings are revealed through disaggregatio n of the samples by population subgroups.
I.
Introduction
I
NTEREST IN the dynamics of income distributions has grown with the increasing availability of panel data for many countries. It has been recognized that static snapshots of income or earnings distributions alone are not suf cient for meaningful evaluation of welfare. A society with a rigid income distribution where everyone remains in the same position year after year is commonly regarded as less well-off than a “mobile” society. Therefore, mobility indices have developed and been interpreted as indicators of “opportunity.” There are many empirical studies of earnings and income mobility; see, for example, Bjo¨rklund (1993), Burkhauser and Holtz-Eakin (1993), Burkhauser and Poupore (1997), Hungerford (1993), Gustafsson (1994), OECD (1996) and Schluter (1996). For an earlier survey of empirical studies of earnings mobility, see Atkinson, Bourguignon , and Morrison (1992). Traditionally, empirical studies of earnings and income mobility have not addressed the issue of statistical signi cance and rigorous inference. Point “estimates” are reported but no con dence intervals are. However, the commonly used panel data sets are rarely larger than, say, 15,000 observations, over a short time frame, and therefore sampling errors should not be neglected a priori. We provide detailed formulae and the programming code necessary for statistical inference on a large class of mobility measures at a level that has come to dominate in other empirical areas. There is a loosely related body of literature in econometrics that studies earnings/income “dispersion” employing standard panel data models. These studies are focused on explaining the sources of variation in earnings and should be interpreted with caution if “mobility” is to be inferred from them in a welfare-theoretic sense. This paper reviews a large family of mobility measures, namely the generalized entropy mobility measures (GEMM) introduced by Maasoumi and Zandvakili (1986). The GEMM family is prominent in the tradition of “inequality reducing” class of mobility indices introduced by Shorrocks (1978). Shorrocks has demonstrated that members of GEMM that utilize the income aggregator functions in Maasoumi (1986) satisfy the greatest number of desirable properties for mobility measures. Also, recent deeper welfare-theoretic interpretations of the other tradition to mobility measurement, the transition matrix/Markov Received for publication October 27, 1997. Revision accepted for publication August 9, 2000. *Southern Methodist University and University of Cologne, respectively. We thank the editor, Robert Mof tt, four referees, and many seminar participants . The rst version of this paper was written in 1996 with the same title.
models, point to the same GEMM as an ideal family of indices in that context too. See Maasoumi (1998) for a synthesis. We establish the asymptotic sampling distribution of these measures using the “delta method” and the existing theory of method of moments estimators. Because different individuals face different panel inclusion probabilities, we also allow for weighted observations in our formulae. Similar results are available only for inequality measures. See Cowell (1989). Our empirical application concerns the United States and Germany before reuni cation. It may be viewed as a statistical investigation and broad generalization of the recent study of similar data by Burkhauser and Poupore (1997). These generalizations are (i) They utilize two of the members in the class of measures studied in the present article (based on Theil entropies); we offer a robusti cation of ndings over a broad class with different aversion parameters. (ii) To obtain “long-run” incomes, they aggregate by adding up incomes over time, as was done by Shorrocks (1978). This assumes in nite substitutability of incomes at different points of the life cycle. This is a serious issue in the conception of “permanent income” and mobility. We follow Maasoumi and Zandvakili (1986) by considering a range of aggregator functions that allow for different degrees of substitution, including the simple sum. Again we study the sensitivity of our inferences to the aggregation method. (iii) We report standard errors that allow construction of con dence bands and tests of hypotheses regarding the levels and differences of mobility estimates, over time as well as between countries. Previous studies merely permit a numerical comparison with quantities that may be casually judged as “too small” or “too large”. We nd that some differences in the third decimal places are statistically signi cant. Burkhauser and Poupore (1997) give an excellent review of the data, the reasons for interest in comparing the United States and (West) Germany with their different labor markets and welfare schemes, and the type of mobility measures studied here. In particular, they note that the existing econometric panel studies would be misread if the larger wage dispersion in the United States is extrapolated to a nding of greater mobility in the United States than Germany. Our ndings offer general and strong statistical support for their conclusions that German income distributions exhibit greater degrees of mobility for the whole sample, as well as for almost all socioeconomic subgroups, de ned by age, gender, and educational attainment. This is a statistical extension over a broader class of mobility measures, as well as more exible “permanent income” functions. The paper is structured as follows. Section II describes the generalized entropy mobility measures and derives their asymptotic distribution. Section III compares income mobility in Germany and the United States. Section IV concludes. II.
Generalized Entropy Mobility Measures
The framework follows Maasoumi and Zandvakili (1986, 1990). Let Y it denote income of person i, i 5 1, . . . , n, in period t 5 1, . . . , T, and Y t 5 (Y 1t , Y 2t, . . . , Y nt ) 9 . Let S i 5 S i (Y i1 , . . . , Y iT ) by the ‘permanent’ or aggregate income of individual i over T periods.
The Review of Economics and Statistics, August 2001, 83(3): 551–559 2001 by the President and Fellows of Harvard College and the Massachusetts Institute of Technology
©
552
THE REVIEW OF ECONOMICS AND STATISTICS
This is just for convenience, of course, because one can de ne aggregate income over periods 1 to T 9 # T, say, and develop a mobility pro le as T 9 approaches T. Then S 5 (S 1 , . . . , S n ) 9 is the vector of aggregate incomes for a chosen time frame. Following Maasoumi (1986), the following type of aggregation functions are justi ed on the basis that they minimize the generalized entropy distance between S and all of the T income distributions:
SO D T
Si 5
a tY
2 1/b
for b Þ
2b it
1
Il~X ! 5
l~l 1
5
0, 21 1 n
5
I0~X ! 5
t5 1
O
A.
where a t are the weights attached to income in period t, ¥ t a t 5 1. The elasticity of substitution of income across time is constant at s 5 1/(1 1 b). The case b 5 2 1 corresponds to perfect intertemporal substitution that subsumes Shorrocks’ analysis for certain weights. This case is also the most common formulation of the “permanent income” concept in economics and used by Burkhauser and Poupore (1997). Mobility is measured as the ratio of long-run income inequality occurring when the accounting period is extended, and a measure of short-run inequality. The latter may be represented by any one period of interest, or a weighted average of the single-period inequalities. Other than this notion of “equality enhancing” mobility, no other welfare-theoretic bases have been put forward or implemented in favor of any other “mobility” or “dispersion” measures until recently.1 The extension of the time interval is meant to re ect the dynamics and smooth out the transitory or life cycle effects. Shorrocks (1978) proposed the following mobility measures using the weighted shortrun inequalities, as follows: 12
I ~S! T ¥ t5 1 a t I ~ Y t !
where I[ is the inequality measure. For convex inequality measures I[ , 0 # M # 1 is easily veri ed when S is the linear permanent income function. For other aggregator functions, see Maasoumi and Zandvakili (1990). A priori, there would be no reason for an analyst to give unequal weights to different years under study. Nevertheless, Shorrocks (1978) suggests the ratio of year t income to total income over the T periods as suitable values for a ts. We consider both weighting schemes here.2 The family of inequality measures used here is the generalized entropy (GE). For a weighted random vector X 5 (X i, . . . , X n ) 9 with weights w 5 (w 1, . . . , w n) 9 , the generalized entropy inequality measure is de ned as This is particularl y so in the case of econometri c studies of “wage dispersion” in which statistical causes of dispersio n are usefully identi ed, but welfare-theoreti c motivation is lacking in regards to “dispersion ” as a measure of “mobility”, or inequality. 2 In other work with PSID data, Maasoumi and Zandvakili (1986, 1990) have studied different weights, including principal component weights and unequal subjectiv e weights. They nd these weights are inconsequen tial for the qualitativ e inference s and rankings. 1
wi w #
i51
Xi X#
l11
1 2
G
for l Þ
0, 21
Xi X#
ln
Xi X#
O S D S D n
i51
21,
t5 1
M5
n
1 2 n
I 21~ X ! 5 a tY it for b 5
i51
wi w #
O S DS D S D
0
T
n
1
T
Y ait t for b 5
O S DFS D
where X# 5 ¥ ~wi/w # ! Xi and w # 5 ¥ w i/n. Usually (as in our n application), the weights are the reciprocal inclusion probabilities. For l 5 0, 21 this index converges to the rst and second Theil measures of information, respectively:
t5 1
P
1 1! n
wi Xi ln . w# X#
The Asymptotic Distribution of GEMM
Cowell (1989) noted that the GE measures of inequality are functions of the moments of the income variable. Similarly, we note that the GE mobility measures can be written as functions of moments of (transformed) income variables. Let m (X) 5 * `0 xdF( x), for any variable X. Then, for l Þ 0, 2 1 M l~m~ Y 1! , . . . , m~ Y T ! , m~ S ! ,
Ml 5
1 m~ Y l1 ! , . . . , m~ Y l11 ! , m~ S l11 !! 1 T
12 5
T
¥ t51
m~ S l11! 2 1 m l11~ S ! m~ Y l11 ! t a t l11 2 m ~Y t!
S
1
D
and M 0 ~m~ Y 1 ! , . . . , m~ Y T ! , m~ S ! ,
M0 5
m~ Y 1 ln ~Y1!!, . . . , m~YT ln ~YT !!, m~S ln ~S!!! m~ S ln ~S!! 2 ln ~m~S!! m~S! m~Yt ln ~Yt !! T ¥t51 at 2 ln ~m~Yt!! m~Yt!
S
12 5
D
M 2 1~m~ Y 1 ! , . . . , m~ Y T ! , m~ S ! ,
M 21 5
m~ ln ~Y1 !!, . . . , m~ln ~YT!!, m~ln ~S!!! 5
12
¥
T t51
ln ~m~S!! 2 m~ln ~S!! . at ~ln ~m~Yt!! 2 m~ln ~Yt !!!
Let m be a vector of moments and M 5 M( m ) a differentiable function. Then, for general processes, the following asymptotic disˆ , a (weighted) estimator of m, (Amemiya, tribution is well known for m 1985):
Î
n 11
2 w
ˆ2 ~m
m! ®
d
N~0, S!
NOTES with S 5 Cov(mˆ ) and 2w the coef cient of variation of the weights w. By an application of the Delta method, it follows that
Î
n
M~mˆ ! 2 M~m! ® 2 1/ 2 ~1 1 sM w!
d
N~0, 1!
where
S D S D S D
2
sM 5
] M 9 ]M S ]m ]m
]M denoting the derivative vector. ]m Of course, S is usually unknown and has to be substituted by a ˆ ; also, the gradient will have to be evaluated at m ˆ. suitable estimator S However, this does not alter the asymptotic distribution. See, for example, Amemiya (1985). The partial derivatives and the asymptotic covariance matrices mentioned above are derived in appendix A to this paper. A GAUSS program implements the computations. with
III.
Empirical Application
This section is concerned with income mobility in Germany and the United States in the second half of the 1980s. The reuni cation of Germany after 1989 is apt to produce income distributions that are transitory. Mobility measures employed here are designed to avoid temporal effects. The study of the post-reuni cation data, as a new regime, deserves a separate study. Our data are taken from the PSID and the GSOEP equivalent data les. These les provide data about many characteristics of individuals between 1984 and 1989. In particular, there are comparable pre- and post-government income variables for both countries. (See Butrica and Jurkat (1998) for detailed descriptions of the data sets.) Our data are essentially the same as the ones described and analyzed by Burkhauser and Poupore (1997). All of our results are obtained for the individual level gures, using equivalence scales when necessary. Burkhauser and Poupore report point estimates of Gini and two members of our mobility family, and they use only a simple sum of incomes over time for their long-run income measure. It is known that Gini, as well as some GEMM indices (with aversion parameter l close to that of Gini) are relatively insensitive to the tail areas of distributions. This makes them unrevealing when policy programs are often directed toward these tails (especially the lower). Our application covers the cases considered by Burkhauser and Poupore, but includes a sensitivity analysis by considering other values for the “inequality aversion” parameter, as well as the income aggregation functions. We also solidify the empirical ndings by statistical tests of signi cance, as well as provide a new dynamic mobility graph. A.
Data
Pre-government income is the sum of total family income from labor earnings, asset ows, private transfers, and the imputed rental value of owner-occupied housing. Labor earnings include wages and salary from all employment including training, self-employment income, and bonuses, overtime, professional practice or trade, and pro t sharing. Asset ows include income from interest, dividends, and rent. In Germany, private transfers include payments from individuals outside of the household. These include alimony and child support payments. In the United States, private pensions, Veterans Adminis-
553
tration pensions, and annuities are also included. Estimation of the imputed value of housing differs between both countries: in Germany, imputed rental value of owner-occupied housing is a respondentsupplied estimate of the monthly rental value of their dwelling. This estimate is annualized. In the United States, imputed rental value of owner-occupied housing is 6% of the difference between the house value and the remaining mortgage principle. Post-government income is the sum of total family income from labor earnings, asset ows, private transfers, public transfers, and the imputed rental value of owner-occupied housing, minus total family taxes. Earnings, asset ows, private transfers, and imputed rental value of housing are de ned as in pre-government income. In Germany, public transfers include housing allowances, child bene ts, subsistence allowance from the Social Welfare Authority, special circumstances bene ts from the Social Welfare Authority, Social Security pensions for old age, disability, or widowhood, government student assistance, maternity bene ts, unemployment bene ts, unemployment assistance, and unemployment subsistence allowance. In the United States, public transfers include AFDC payments, supplemental security income (SSI), social security payments, unemployment compensation, worker’s compensation, and the face value of food stamps. For Germany, total family taxes are derived by micro-simulation routines developed by Schwarze (1995). They include income taxes, church taxes, and social security payments (health, unemployment, and retirement insurances). These routines assume that all married couples le jointly, all ling units take the standard deductions and not other deductions, and all employees are subject to the average national contribution rates for health, unemployment, and retirement insurance. Under these assumptions, rather accurate estimates of the tax burden are obtained for Germany. Post-government (post.gov) income data are obtained using these estimates. For the United States, total family taxes includes income taxes of the head, partner, and other family members, as well as payroll taxes of the head and partner. Income tax values are provided by the PSID. Payroll taxes are calculated by bracketing labor income and applying the average payroll tax rate for that bracket as reported by the Social Security Bulletin, Annual Statistical Supplement (1990). Other variables that we used are the number of persons in the household, age, years of education, participation level in the labor market, and sex of the individual. The data set offers two equivalence scales: the OECD scale and the ELES scale. Inspecting the data closely, one nds that some observations are implausible. For instance, it is not likely to survive on an annual post-government income of merely DM 1,000. Some GE inequality and mobility measures are known to be especially nonrobust to measurement errors in the tails of income distributions; see, for example, Cowell and Victoria-Feser (1996) and Cowell and Schluter (1998). To avoid this kind of sensitivity, we excluded individuals with annual post-government equivalent incomes of less than US$450 and DM 1,000. The proportion of deletions is always less than 1.1% of the number of observation. B.
Inequality Results
As a rst step, it is necessary and informative to analyze the GE indices of inequality for both countries. We have exempli ed these results in table 1. We report aggregator functions and inequality measures that are different from the ones reported by Burkhauser and Poupore (1997). This is merely to economize on space. Our results con rm their ndings, and more extensive results are available from
1984
1985
1986
1984
1985
1986
1985
0.0630 (0.0030)
1985
0.0679 (0.0029)
from
1984 1985 1986 1987 1988
from
1984 1985 1986 1987 1988
0.0933 (0.0029) 0.0583 (0.0019)
1986
0.0865 (0.0034) 0.0563 (0.0024)
1986
1987
0.1147 (0.0031) 0.0918 (0.0025) 0.0619 (0.0021)
until 1987
0.1067 (0.0040) 0.0870 (0.0030) 0.0607 (0.0024)
until 1987
PSID
0.2014 (0.0044) 0.2114 (0.0046) 0.2226 (0.0046) 0.2360 (0.0049) 0.2651 (0.0067)
1988
0.2106 (0.0080) 0.2225 (0.0085) 0.2327 (0.0083) 0.2502 (0.0092) 0.2910 (0.0135)
1988
0.1386 (0.0038) 0.1196 (0.0034) 0.0969 (0.0029) 0.0668 (0.0028)
1988
0.1223 (0.0041) 0.1071 (0.0035) 0.0871 (0.0030) 0.0575 (0.0022)
1988
TABLE 2.—GE MOBILITY
0.2030 (0.0043) 0.2138 (0.0045) 0.2259 (0.0044) 0.2455 (0.0047)
until
1984 0.2198 (0.0041) 0.2086 (0.0042) 0.2062 (0.0043) 1985 0.2326 (0.0053) 0.2211 (0.0049) 1986 0.2415 (0.0048) 1987 1988 1989
from
1987 0.2040 (0.0074) 0.2158 (0.0080) 0.2236 (0.0068) 0.2417 (0.0068)
until
1984 0.2086 (0.0054) 0.2079 (0.0075) 0.2061 (0.0076) 1985 0.2380 (0.0101) 0.2230 (0.0088) 1986 0.2371 (0.0073) 1987 1988 1989
from
PSID
1989
1984
b 5 21 and l 5 0 1985
POST-G OVERNMENT INCOME, OECD EQUIVALIZED
1986
1984
1985
1986
0.1137 (0.0045)
1985
0.1495 (0.0040) 0.1313 (0.0035) 0.1108 (0.0029) 0.0877 (0.0025) 0.0510 (0.0023)
1989
0.1231 (0.0053)
1985
b 5 0 and l 5 21
0.1235 (0.0048) 0.1106 (0.0042) 0.0935 (0.0039) 0.0731 (0.0034) 0.0439 (0.0027)
1989
b 5 21 and l 5 0
0.1737 (0.0075) 0.1088 (0.0086)
1986
0.1629 (0.0057) 0.0975 (0.0054)
1986
POST-GOVERNMENT INCOME, OECD EQUIVALIZED
1987
0.1984 (0.0078) 0.1536 (0.0092) 0.1142 (0.0121)
until 1987
0.1925 (0.0069) 0.1467 (0.0076) 0.1027 (0.0079)
until 1987
SOEP
1988
0.2158 (0.0076) 0.1825 (0.0085) 0.1525 (0.0101) 0.0816 (0.0036)
1988
1989
0.0861 (0.0017) 0.0907 (0.0018) 0.0946 (0.0019) 0.1004 (0.0021) 0.1078 (0.0023) 0.1188 (0.0029)
1989
0.0881 (0.0020) 0.0921 (0.0022) 0.0970 (0.0024) 0.1000 (0.0024) 0.1082 (0.0029) 0.1189 (0.0036)
1989
0.2463 (0.0077) 0.2154 (0.0082) 0.1889 (0.0092) 0.1355 (0.0050) 0.0909 (0.0049)
1989
0.2443 (0.0082) 0.2127 (0.0092) 0.1854 (0.0102) 0.1278 (0.0046) 0.0878 (0.0045)
0.0915 (0.0019) 0.0974 (0.0021) 0.1023 (0.0022) 0.1105 (0.0024) 0.1242 (0.0029)
1988
0.0941 (0.0026) 0.0998 (0.0030) 0.1051 (0.0031) 0.1084 (0.0029) 0.1231 (0.0037)
1988
0.2142 (0.0079) 0.1809 (0.0087) 0.1499 (0.0096) 0.0801 (0.0035)
0.0935 (0.0019) 0.1003 (0.0021) 0.1056 (0.0022) 0.1166 (0.0025)
until 0.2069 (0.0047) 0.1259 (0.0028) 0.1067 (0.0022) 0.0986 (0.0019) 0.2164 (0.0050) 0.1230 (0.0031) 0.1086 (0.0024) 0.2272 (0.0051) 0.1232 (0.0044) 0.2385 (0.0055) 0.2574 (0.0067) 0.2760 (0.0069)
1989
b 5 0 and l 5 21
1987 0.0983 (0.0028) 0.1052 (0.0034) 0.1118 (0.0041) 0.1130 (0.0028)
until
SOEP
0.2223 (0.0093) 0.1308 (0.0040) 0.1126 (0.0033) 0.1052 (0.0033) 0.2335 (0.0099) 0.1279 (0.0047) 0.1162 (0.0046) 0.2443 (0.0101) 0.1333 (0.0089) 0.2595 (0.0112) 0.2860 (0.0138) 0.3056 (0.0137)
OF
OF
TABLE 1.—GE INEQUALITY
554 THE REVIEW OF ECONOMICS AND STATISTICS
NOTES FIGURE 1.—GE MOBILITY INDEX
the authors that demonstrate a qualitative robustness to the choice of inequality measure and the aggregation functions. For the PSID, the left-hand panels of table 1 report GE inequality of nominal post-government income equivalized by the OECD scale. Two representative parameter combinations are considered to re ect different degrees of inequality aversion/intertemporal substitution. Inequality is reported for each single year and for aggregated income over increasing number of years. The row “from” indicates the starting period, and the column “until” reveals the last year of aggregation. As an example, with l 5 0 and b 5 21 (Theil’s measure
555 FOR THE
PSID
over simple sum of incomes), the inequality of incomes aggregated over 1986–1988 is 0.2327. Standard errors are in parentheses. For the SOEP, the right-hand panels of table 1 depict the GE inequality measures for the same income variable as above. Consistent with the PSID entries, the changes of the inequality values are generally statistically signi cant. Given the nature of approximations in the Delta method, the high degree of signi cance in most of these entries is comforting. In particular, we note that income inequality in the United States rst declines, then increases signi cantly in the latter years of the 1980s (so much so, that
FIGURE 2.—GE MOBILITY INDEX
FOR THE
SOEP
THE REVIEW OF ECONOMICS AND STATISTICS 0.2957 (0.0150) 0.2765 (0.0164) 0.2411 (0.0169) 0.1543 (0.0136) 0.1148 (0.0117) 0.1376 (0.0105) 0.1500 (0.0081) 0.1316 (0.0074) 0.1114 (0.0063) 0.0934 (0.0055) 0.0674 (0.0044)
Age 26–35
0.2194 (0.0136) 0.1416 (0.0107)
0.2575 (0.0150) 0.2019 (0.0137) 0.1373 (0.0109)
0.2862 (0.0158) 0.2502 (0.0161) 0.2028 (0.0153) 0.0983 (0.0101)
0.3864 (0.0200) 0.3431 (0.0254) 0.2972 (0.0280) 0.2144 (0.0164) 0.1530 (0.0146) 0.3396 (0.0168) 0.2878 (0.0212) 0.2260 (0.0236) 0.1260 (0.0114) 0.2871 (0.0141) 0.2134 (0.0204) 0.1376 (0.0210)
0.2061 (0.0151) 0.1768 (0.0184) 0.1628 (0.0225) 0.0760 (0.0072)
0.2279 (0.0128) 0.1452 (0.0142) 0.1480 (0.0145) 0.2128 (0.0189) 0.1924 (0.0187) 0.1528 (0.0172) 0.0958 (0.0147) 0.0483 (0.0088)
1989
Age 0–15
Age 16–25
until 1987 1986 1985
BY
POST-GOVERNMENT INCOME
0.1406 (0.0074) 0.1169 (0.0066) 0.0922 (0.0057) 0.0596 (0.0042) 0.1298 (0.0069) 0.0986 (0.0060) 0.0656 (0.0043)
0.1079 (0.0062)
0.0852 (0.0060)
1984 1985 1986 1987 1988
1984 1985 1986 1987 1988
0.1118 (0.0070) 0.0647 (0.0056)
0.1926 (0.0095) 0.1530 (0.0079) 0.0949 (0.0063)
0.0544 (0.0033) 1984 1985 1986 1987 1988
0.1603 (0.0079) 0.1079 (0.0060)
1988 until 1987 1986 1985 from
PSID
0.2098 (0.0120) 0.1843 (0.0111) 0.1394 (0.0099) 0.0872 (0.0089)
OF
Looking at overall post-government mobility in both countries, we nd that income mobility is clearly higher numerically in Germany than in the United States. These differences are statistically signi cant at all reasonable levels. We report only a few typical tables here, but this nding is robust with respect to the weighting scheme, the consideration of in ation, the equivalence scale, the aggregation method (b), and the inequality measure (l). These provide signi cant and extensive generalizations of Burkhauser and Poupore’s numerical results. Both short-run and long-run mobility are higher in Germany. It seems that aggregating income over six years or so gives a reliable picture of lifetime inequality in the United States. In Germany, the aggregation period would have to be longer, but how much longer cannot be inferred from the available sample period. Table 2 shows the generalized entropy mobility measures for the two countries. For instance, taking l 5 0, b 5 21 as in Burkhauser and Poupore (1997), over the period 1985–1988, mobility estimates for Germany and the United States are 0.1809 and 0.1071, respectively. Standard errors based on the formulae derived in the previous section are reported in parentheses. The asymptotic distribution theory suggests that these gures are statistically signi cant. A relatively plausible assumption of independence between these two samples allows for quick computations and the statistical rejection of the null hypothesis of equality between these two estimates. In this example, in ation was not taken into account, and the equivalence scale is the OECD scale. Figures 1 and 2 depict the same indices (with l 5 0 and b 5 21). The lowest lines show the two-period mobility, the next lines show the three-period mobility, and so forth. Note that the traditional mobility pro le (that is, extending the accounting period while keeping the starting period xed) is obtained by connecting the rst points of each line. We agree with Burkhauser and Poupore that these ndings demonstrate that the inferences made about earnings/wage dispersion in the panel data studies fail to re ect “mobility” that is produced in Germany by government programs and taxes. The explanation is that our measures of mobility are conceived in terms of the usual increasing and Schur-concave (equality preferring) social welfare functions that acknowledge a subjective direction to income movements. Variance and dispersion are questionable measures of mobility; they are merely indicators that suggest the potential for higher mobility in the less rigid and less organized labor markets of the United States. This potential is not realized.
AGE GROUPS, PART I
Mobility Results
TABLE 4.—GE MOBILITY
C.
SOEP
aggregated income inequality increases when these latter years are included). We also nd that inequality is statistically signi cantly lower in Germany than in the United States, at any level of aggregation. These ndings are predictive of the mobility pro les.
0.1690 (0.0131) 0.1387 (0.0152) 0.1183 (0.0167)
0.154 0.131 0.111 0.169 0.112 0.056
0.1311 (0.0099) 0.0823 (0.0092)
0.232 0.122 0.160 0.116 0.074 0.050
0.0980 (0.0074)
0–15 16–25 26–35 36–50 51–65 661
0.1175 (0.0077) 0.1033 (0.0070) 0.0851 (0.0065) 0.0736 (0.0035) 0.0449 (0.0027)
SOEP
0.1197 (0.0056) 0.1049 (0.0053) 0.0835 (0.0050) 0.0508 (0.0026)
PSID
1988
Age class
0.2330 (0.0180) 0.2078 (0.0216) 0.1891 (0.0267) 0.1167 (0.0115) 0.0854 (0.0122)
PROPORTIONS
0.1044 (0.0044) 0.0882 (0.0038) 0.0639 (0.0035)
AND
0.0804 (0.0045) 0.0548 (0.0036)
TABLE 3.—AGE CLASSES
1989
556
0.1911 (0.0232) 0.1291 (0.0167) 0.1071 (0.0127) 0.0940 (0.0113) 0.0807 (0.0127) 0.1745 (0.0210) 0.1180 (0.0147) 0.0814 (0.0140) 0.0567 (0.0107) 0.1474 (0.0179) 0.0948 (0.0099) 0.0733 (0.0103) 0.0664 (0.0078) 0.0637 (0.0057) 0.0418 (0.0064) 0.0927 (0.0067) 0.0691 (0.0070) 0.0616 (0.0050) 0.0581 (0.0079) 0.0718 (0.0114) 1984 1985 1986 1987 1988
0.0740 (0.0062) 0.0335 (0.0073)
0.0896 (0.0079) 0.0657 (0.0050) 0.0488 (0.0061)
Age 661
0.1671 (0.0202) 0.0584 (0.0093)
0.1778 (0.0255) 0.1133 (0.0210) 0.0794 (0.0273)
0.2189 (0.0247) 0.2128 (0.0270) 0.1968 (0.0294) 0.1342 (0.0095) 0.0844 (0.0088) 0.1854 (0.0247) 0.1741 (0.0275) 0.1600 (0.0294) 0.0807 (0.0080) 0.1645 (0.0199) 0.1433 (0.0226) 0.1098 (0.0220) 0.0832 (0.0068) 0.1002 (0.0157) 0.0972 (0.0147) 0.0849 (0.0120) 0.0708 (0.0109) 0.0420 (0.0084) 0.0993 (0.0145) 0.0932 (0.0130) 0.0814 (0.0102) 0.0620 (0.0083) 0.0447 (0.0084) 1984 1985 1986 1987 1988
0.0604 (0.0098) 0.0393 (0.0059)
0.0754 (0.0117) 0.0639 (0.0092) 0.0463 (0.0068)
Age 51–65
0.1412 (0.0152) 0.1024 (0.0160)
0.1693 (0.0111) 0.1455 (0.0106) 0.1194 (0.0093) 0.0779 (0.0067) 0.1657 (0.0115) 0.1227 (0.0112) 0.0700 (0.0065) 0.1041 (0.0089) 0.1180 (0.0122) 0.1078 (0.0116) 0.0994 (0.0104) 0.0695 (0.0080) 0.0417 (0.0064) 0.1266 (0.0088) 0.1154 (0.0085) 0.0980 (0.0079) 0.0538 (0.0046) 0.1143 (0.0067) 0.0999 (0.0062) 0.0736 (0.0061) 0.0918 (0.0063) 0.0705 (0.0056) 0.0628 (0.0048) 1984 1985 1986 1987 1988
until
PSID
TABLE 5.—GE MOBILITY
OF
Age 36–50
POST-GOVERNMENT INCOME
BY
AGE GROUPS, PART II
0.1380 (0.0101) 0.0790 (0.0088)
until
SOEP
0.1956 (0.0112) 0.1677 (0.0103) 0.1405 (0.0088) 0.1052 (0.0070) 0.0606 (0.0065)
NOTES
557
Our comparative ranking inferences are not particular to group characteristics. To explore this question in our approach, one decomposes mobility measures. The parallel to this is the inclusion of the corresponding characteristics as covariates in regression studies. Thus, we split the samples according to various criteria and investigate mobility within each group. (We did not take into account income mobility between groups.) We note that the ordering is robust relative to adjustment for in ation by the consumer price indices. It turns out that many group characteristics do not have a signi cant impact on comparative income mobility pro les. For instance, looking at males and females separately, the mobility pictures look very similar to the ones in gures 1 and 2. The same may be said of which equivalence scale is used, OECD or ELES. Using household income instead of equivalized individual income, we nd that the mobility estimates are somewhat lower for Germany but unchanged for the United States. The same phenomenon was observed using per capita income. Age. To investigate the impact of age, we split the sample into six age groups. Table 3 clari es the age classes and their sample shares averaged over the 1984–1989 period. The age structure of the two countries differ. Tables 4 and 5 show the GEMM for each of the age groups for both countries. The parameter combinations correspond to Theil’s rst measure of inequality and the arithmetic income aggregators. This facilitates direct comparison with the Burkhauser-Poupore ndings. Other values have been tried and are available from us. As can be gleaned from these results, age is a rather important source of heterogeneity in mobility indicators. For both countries, mobility is generally highest for the second age group (young adults aged 16–25) and decreasing thereafter. This is consistent with prior expectations regarding life cycle earning patterns, and one of the main reasons one prefers to look at mobility rather than static single-year estimates of inequality. For all age groups, mobility is statistically signi cantly higher in Germany than in the United States, sometimes dramatically so. Trede (1998) obtains similar ndings for the “earnings” pro les in West Germany. This phenomenon was noted by Burkhauser and Poupore (1997), and is contrary to somewhat misplaced prior expectations that freer United States labor markets will lead to greater mobility compared to more rigid markets in Germany, emphasizing credentials and union management-negotiated wage levels. Our inferences put these contrary ndings on a statistically sound and comparable basis with those gleaned from the existing econometric studies that often nd statistically larger dispersions for the U.S. earnings. IV.
Conclusion
We have derived the asymptotic distribution of the GE measures of mobility proposed by Shorrocks-Maasoumi-Zandvakili. Our results cover the case of independent but not necessarily identical processes. Our results also allow for the common phenomenon of weighted observations, for instance, by inclusion probabilities. Our primary focus was the post-government income mobility in Germany and the United States. Our main empirical nding is that mobility estimates are statistically signi cantly higher in Germany than in the United States. We nd that this ordering exists for all of our six age groups. Our preliminary look at pre-government income distributions has so far indicated the same orderings. We have also looked at the traditional mobility pro les and a “moving” pro le. The traditional graphical analysis is supplemented
558
THE REVIEW OF ECONOMICS AND STATISTICS
and supported by our statistical techniques and ndings. We nd that, for the United States, the estimated pro le becomes almost at after six years. This may suggest that a six-year aggregation period is adequate for representing long-run incomes in the United States. In contrast, the German mobility pro le is steeper and has a positive slope even at the end of the six years. Longer sample periods are required to study this issue further, but our analysis stops at about the German reuni cation. The consequent structural changes deserve a separate study after suf cient time has elapsed. One welfare implication of these results is that the combination of higher annual inequality and lower mobility in the United States indicates a higher ranking for Germany in these respects. But given that our inequality measures are relative indices, these inferences neglect the higher average income in the United States and should not be used to indicate welfare levels in the two countries. This is best done using tests of stochastic dominance and generalized Lorenz relations. See Maasoumi and Heshmati (2000).
Schluter, C., “Income Mobility in Germany: Evidence from Panel Data,” Distributiona l Analysis Research Programme, London School of Economics discussio n paper no. 17 (1996). Schwarze, J., “Simulating German Income and Social Security Tax Payments Using the GSOEP,” Cross-Nationa l Studies in Aging, Syracuse University program project paper no. 19 (1995). Shorrocks, A. F., “Income Inequalit y and Income Mobility,” Journal of Economic Theory 19(2) (1978), 376–393. Social Security Bulletin, Annual Statistical Supplement, U.S. Department of Health and Human Services (1990). Trede, M., “The Age-Pro le of Earnings Mobility: Statistical Inference of Conditional Kernel Density Estimates,” Journal of Applied Econometrics 13(4) (1998), 397–409.
APPENDIX
]M l 5 ]m~ Y t !
REFERENCES Amemiya, T., Advanced Econometrics , (Cambridge: Harvard University Press, 1985). Atkinson, A. B., F. Bourguignon , and C. Morrisson, Empirical Studies of Earnings Mobility, (Switzerland: Harwood, 1992). Bjo¨rklund, A., “A Comparison between Actual Distribution s of Annual and Lifetime Income: Sweden 1951–89,” Review of Income and Wealth 39(4) (1993), 377–386. Burkhauser, R. V., B. A. Butrica, and M. C. Daly, The Syracuse University PSID-GSOEP Equivalent Data File: A Product of Cross-Nationa l Research, Cross-National Studies in Aging, program project paper no. 25 (1995). Burkhauser, R. V., and D. Holtz-Eakin, “Changes in the Distribution of Wage Earnings in the United States and Germany During the 1980s” (pp. 27–35). in R. V. Burkhauser, and G. G. Wagner, (Eds.), Proceeding s of the 1993 Internationa l Conference of German Socio-Economi c Panel Study Users, Vierteljahreshef t zur Wirtschaftsforschung, Heft 1/2 (1993). Burkhauser, R. V., and J. G. Poupore, “A Cross-Nationa l Comparison of Permanent Inequality in the United States and Germany,” this REVIEW 79(1) (1997), 10–17. Butrica, B., and D. Jurkat, Codebook for PSID-GSOEP Equivalent File 1980–1996, www-cpr.maxwell.syr.edu /gsoep/equiv l.htm (1998). Cowell, F., “Sampling Variance and Decomposabl e Inequalit y Measures,” Journal of Econometrics 42(1) (1989), 27–42. Cowell, F., and M.-P. Victoria-Feser, “Robustness Properties of Inequalit y Measures,” Econometrica 64(1) (1996), 77–101. Cowell, F., and C. Schluter, “Measuring Income Mobility with Dirty Data,” CASEpaper 16, CASE, STICERD, LSE (1998). Gustafsson, B., “The Degree and Pattern of Income Immobility in Sweden,” Review of Income and Wealth 40(1) (1994), 67–86. Hungerford , T. L., “U.S. Income Mobility in the Seventies and Eighties,” Review of Income and Wealth 39(4) (1993), 403–417. Maasoumi, E., “The Measurement and Decomposition of Multi-Dimensional Inequalit y,” Econometrica 54(4) (1986), 991–997. “On Mobility,” chapter 5, in D. Giles and A. Ullah (Eds.), Handbook of Applied Economic Statistics (Marcel Dekker, 1986). Maasoumi, E., and A. Heshmati, “Stochastic Dominance amongst Swedish Income Distributions, ” Econometric Reviews 19(3) (2000), 287–320. Maasoumi, E., and S. Zandvakili, “A Class of Generalized Measures of Mobility with Applications, ” Economics Letters 22(1) (1986), 97–102. “Generalized Entropy Measures of Mobility for Different Sexes and Income Levels,” Journal of Econometrics 43(1–2) (1990), 121–133. OECD, “Earnings Inequalit y, Low-Paid Employment and Earnings Mobility” (pp. 59–109), in Employment Outlook (Paris: Organizatio n for Economic Cooperation and Development (OECD) 1996.)
The partial derivative s for M l , l Þ 0, 2 1 are
Lemma 1.
2
] Ml 5 ]m~ S !
S
S
S
T
¥t51
1!
DD
S
1 !m~ S l11 ! m~ Y l11 ! t a t l11 2 m ~Y t!
~l 1 T
m l12 ~ S ! ¥ t51
S
]M l l11 5 ]m~ Y t !
T
¥ t51
D
m~Sl11 ! l11 2 1 m~Yt ! l11 m ~S ! , 2 l11 m~Yt ! l12 1 Y at l1 1 m ~ ! t 2 m ~Y t !
at ~l 1
S
m~ S l11 ! 2 1 m l11 ~ S ! l1 1 m~ Y t ! a t l11 2 1 m ~Y t!
S
at
DD
1
D
D
,
,
2
m
l11
~ Yt !
and ]Ml 5 ]m~ S l11 !
S
1
2
l11
m~Yt ! 1 2 ml1 ~Yt !
T
ml11 ~S! ¥t51 at
1
D
For M 0 , the derivative s are
]M 0 5 ]m~ Y t !
] M0 5 ]m~ S !
]M 0 5 ]m~ Y t ln ~Yt !!
at 2
S
m~S ln ~S!! 2 m~S!
S
T
m~Yt ! ¥t 51 at
T
m~S! ¥t51
S
F
ln ~m~S!!
DS
m~Yt ln ~Yt !! 1 m~Yt !
GD
m~Yt ln ~Yt !! 2 m~Yt !
ln ~m~Yt !!
m~ S ln ~S!! 1 1 m~S! , m~Yt ln ~Yt !! ln Y at ~m~ t !! 2 m~Yt !
F
D
G
m~S ln ~S!! 2 ln ~m~S!! m~S! m~Yt ln ~Yt !! T m~Yt ! ¥t 51 at 2 ln ~m~Yt !! m~Yt ! at
S
F
GD
and ] M0 5 ]m~ S ln ~S!!
2 T
m~S! ¥t51
F
1
m~Yt ln ~Yt !! at 2 m~Yt !
ln ~m~Yt !!
G
,
2
,
1 2
D
,
NOTES and, for M 21 , ] M 21 5 ]m~ Y t !
559
and a t ~ ln ~m~S!! 2 m~ln ~S!!! T 2 , Y m~ t !~¥t 51 at @ln ~m~Yt !! 2 m~ln ~Yt !!#!
] M 21 5 ]m~ ln ~S!! Lemma 2.
] M 21 5 ]m~ S ! ] M 21 5 ]m~ ln ~Yt !!
2
1 T
m~S! ¥t51 at @ln ~m~Yt !! 2
m~ln ~Yt !!#
at ~ln ~m~S!! 2 m~ln ~S!!! 2 T 2 , ~¥t51 at @ln ~m~Yt !! 2 m~ln ~Yt !!#!
,
1 T
¥t51 at @ln ~m~Yt !! 2
m~ln ~Yt !!#
.
The covarianc e matrix S can be found using the formula
Cov ~mˆ ~X1 !, mˆ ~X2 !! 5
~1 1
2 w
!
Cov~X1 , X2 ! , n
where X 1, X 2 are to be replaced by the variables Y t , Y lt 11, Y t ln (Y t ), or ln (Y t) and S, S l11 , S ln (S), or ln (S), as appropriate . These formulae have been implemented in GAUSS.
