Social Policy and Income Inequality

National Poverty Center Working Paper Series #08-02 January 2008 Social Policy and Income Inequality Christopher Bollinger, Department of Economics, University of Kentucky James P. Ziliak, Department of Economics, Center for Poverty Research, University of Kentucky This paper is available online at the National Poverty Center Working Paper Series index at: http://www.npc.umich.edu/publications/working_papers/ Any opinions, findings, conclusions, or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the view of the National Poverty Center or any sponsoring agency.

Social Policy and Income Inequality Christopher Bollinger Department of Economics University of Kentucky James P. Ziliak Department of Economics Center for Poverty Research University of Kentucky May 2007 Revised January 2008 * Address correspondence to James P. Ziliak, Department of Economics, University of Kentucky, Lexington, KY 40506-0034; Email: jziliak@uky.edu; Phone: 859 257 2776. We thank Luis Gonzalez for excellent research assistance, and seminar participants at the 2007 Institute for Poverty Research Summer Workshop for comments on an earlier version. All errors are our own.

Social Policy and Income Inequality Abstract: We document the economic, demographic, and social policy forces underlying changes in income inequality among single mother families over the past twenty-seven years in the United States. Using data from the 1980 2006 waves of the March Current Population Survey, we construct additively decomposable measures of after-tax income-to-needs inequality into the major income factors of earnings, transfers, other income, and taxes, and also into within- and between-group inequality based on employment status, education attainment, age, past marital status, and race. Our results indicate that income-to-needs inequality rose nearly 50 percent between 1980 and 2006. Employing Quandt-Andrews tests of unknown change point we identify a trend break toward higher inequality centered in 1994 just as major tax and welfare reform policies, and a business-cycle expansion were taking hold nationally. The post 1994 rise in inequality is driven by a 75 percent increase in the cross-sectional variance of income accruing in large part to strong income growth in the upper half of the distribution. A decomposition focusing on the components of total income indicates that the rise in income inequality among single mothers is driven by higher earnings inequality, and that most of the change in inequality is occurring within demographic groups in part because of large, offsetting between-group changes in population shares and relative incomes.

1 Documenting the sources of widening inequality in the United States continues to be a focal research priority in economics. This vast literature has linked the growth in inequality to expanding college-high school premiums (Bound and Johnson 1992; Katz and Murphy 1992), rising returns to unobserved skills (Juhn, Murphy, and Pierce 1993), falling rates of unionization and the real value of the minimum wage (DiNardo, Fortin, and Lemieux 1996; Lee 1999), social norms (Piketty and Saez 2003), and the composition of the workforce (Lemieux 2006), among others. While the debates on the relative contribution of each factor are ongoing, this research has deepened our understanding of the fundamental mechanisms behind rising inequality (Lemieux 2008). At the same time, much of the literature, and the complementary work on volatility (Gottschalk and Moffitt 1994; Dynarski and Gruber 1997; Haider 2001; Blundell, Pistaferri, and Preston 2006), has focused on men, and when women are considered, distinctions are typically not made based on important demographics such as employment status, marital status, and race. 1 This omission inhibits a more complete portrait of inequality in America, and especially a greater understanding of the possible links between social policy reforms affecting single mother families and inequality. In this paper we present new evidence on the economic, demographic, and social policy forces underlying changes in income inequality among single mother families over the past 27 years. Perhaps no other demographic group was singled out by policy in the 1980s and 1990s as prominently as single mothers with dependent children. President Reagan set in motion the retrenchment of the cash welfare program Aid to Families with Dependent Children by increasing the implicit tax rate on earnings and reducing the liquid asset level necessary to 1 Notable exceptions are the articles by Karoly (1993), Gottschalk and Danziger (1993), and Cancian, Danziger, and Gottschalk (1993). The data in these papers focus on the 1970s and 1980s, and thus do not span most of the major reforms to social policy affecting single mother families. Gottschalk and Danziger (2005) do make some allowance for nonemployment in some of their estimates of family income inequality. Keys (2007) is among the first to examine women in the volatility literature.

2 qualify for benefits as part of OBRA 1981. This retrenchment was completed by President Clinton with passage of the 1996 Personal Responsibility and Work Opportunity Reconciliation Act, which abolished AFDC and replaced it with the new time-limited, block-grant program Temporary Assistance to Needy Families (TANF). Concurrent to restrictions to cash welfare were enhanced incentives for single mothers to work via expansions in the Earned Income Tax Credit (EITC) as part of the Tax Reform Act of 1986 and OBRA 1990 and 1993, as well as expansions in Medicaid program eligibility and later the introduction of the Supplemental Children s Health Insurance Program as part of OBRA 1997. In 1991 Congress was required to modify rules for child eligibility in the Supplemental Security Income (SSI) program in light of the Supreme Court s 1990 Zebley decision that ruled unconstitutional previous guidelines. The revised rules resulted in a large increase in children participating in SSI, including many from single mother families on the AFDC program (Kubik 1999; Schmidt and Sevak 2004). A burgeoning literature, much of which is summarized in the survey chapters in Moffitt (2003), emerged in the wake of these reforms addressing outcomes such as employment, income, poverty, welfare participation, childbearing, marriage, and health, among others. Virtually all research on the effects of social policies on the well being of single mothers has focused on average impacts, and while this frequently is of first-order importance to program evaluation, it limits our understanding of program effects at other moments of the distribution including inequality. There are some recent exceptions to the focus on the first moment. Mills, et al. (2001) use non-parametric density re-weighting techniques to compare the single mother family income distribution in 1993 to 1999, and their counterfactual experiments suggests that most of the income gains in that six year period were from strong economic conditions and higher education attainment and not welfare reform. Schoeni and Blank (2000) compare the effects of

3 welfare waivers and TANF across education groups at the 20th and 50th percentiles of family income, and find that waivers did not alter the distribution of income but that TANF raised incomes at the 50th percentile while leaving the 20th percentile unchanged. Meyer and Sullivan (2006) compare trends in the distribution of income and consumption of single mother families in the Consumer Expenditure Survey, arguing that consumption is a superior measure of well being especially at the low end of the distribution. Bitler, et al. (2006) use data from a random assignment experiment in Connecticut to examine heterogeneous treatment effects of welfare waivers on employment. Bollinger, et al. (2007) estimate quantile treatment effects of social policies and the business cycle on income and earnings, finding that TANF raised disposable incomes an average of eight percent among higher skilled mothers, but that it also resulted in a significant equal-size loss of after-tax total income among the low-skilled, and income fell upwards of 20 percent in response to TANF in the lower-tail of the distribution. We extend the literatures on inequality and on social policy by quantifying the contributions of economic and demographic factors on changes in the level and trend of after-tax income inequality among single mother families. Using data from the 1980-2006 waves of the March Current Population Survey, our focal measure of inequality is the squared coefficient of variation in age-adjusted disposable income-to-needs ratios. This index is in the family of generalized entropy measures and has several desirable properties, the most important of which for our purpose being that it is additively decomposable into the components of income as well as into within-group and between-group inequality (Shorrocks 1980, 1982; Mookherjee and Shorrocks 1982; Jenkins 1985). The additivity property is not guaranteed with more common measures of inequality such as the Gini coefficient, the variance of log income, or 90-10 ratios. We focus on income inequality rather than hourly wage inequality because of the large number

4 of non-working single mothers prior to the mid 1990s and the subsequent changes in the composition of their income from transfers to earnings (Bollinger, et al. 2007). 2 This leads to another key advantage of the squared coefficient of variation over other measures like the variance of logs or 90-10 ratios because the squared CV is still defined even if some of the income factors are zero, such as earnings for non-workers or transfers for non-welfare recipients. Most of the inequality literature examines changes in trend inequality across decades or by 5-year intervals, and we do so as well. However, there is no formal statistical basis for decades or 5-year intervals, and so an additional contribution of our paper is to treat the break point as an unknown parameter to estimate by employing Quandt-Andrews type tests of structural change (Quandt 1960; Andrews 1993), coupled with confidence intervals around the breakdate (Bai 1997). With the estimated breakdates in trend inequality we then use methods proposed in Mookherjee and Shorrocks (1982) and Shorrocks (1982) to additively decompose changes in inequality into changes in the key income factors of earnings, transfers, other nonlabor income, and tax payments and EITC credits, as well as changes in within-group and between-group inequality based on employment status, education, age, marital status, and race. We identify a rise in disposable income inequality of about 50 percent among all single mothers between 1979 and 2005, and this rise in inequality appears to have affected all major subgroups of single mothers. The Quandt-Andrews tests of structural change, which control for the business cycle, indicate that the trend break in inequality is centered around periods of policy reforms the breakdate for each of disposable income, earnings, and taxes is 1994 with a 95 percent confidence interval of one year, which coincides with the implementation of welfare reform at the state level, the expansion of the EITC, and the robust macroeconomy. The post 2 As noted by Lemieux (2008) most of the inequality literature focuses on wages because trends in within-group inequality of men were not robust to the use of weekly wages as in Juhn, et al. (1993). In future work we plan to explore whether this non-robustness result extends to the single mother population as well.

5 1994 rise in the squared coefficient of variation is driven by a 75 percent increase in the crosssectional variance of income accruing in large part to strong income growth in the upper half of the distribution, the latter of which corroborates recent work on wage inequality (Autor, et al. 2005). The results of our decompositions indicate that nearly all of the increase in income inequality among single mothers is attributable to rising earnings inequality, though the rise in inequality was tempered by the progressive income tax system including the EITC. Moreover, most of the change in inequality is occurring within groups in part because of large, yet offsetting, between-group changes in population shares and relative incomes. II. Data The data derive from the 1980 2006 waves (1979 2005 calendar years) of the March Annual Social and Economic Study of the Current Population Survey (CPS). The unit of observation is single female family heads between the ages of 16 and 54 with dependent children present under the age of 18. Single heads include never married women as well as those divorced, separated, or widowed. In a bid to minimize measurement error in some of our subgroup analyses we allocate each mother to one of forty-five five-year birth by education cohorts, where three separate education groups of less than high school, high school graduate, and more than high school are assembled, and drop cohort-education cells with fewer than 50 observations (Blundell et al. 1998). The key variable of interest is disposable family income-to-needs, defined as gross income less net tax payments relative to the family-size and year-specific poverty threshold. For our purpose gross income is the sum of family income and the imputed value of public food assistance programs. Family income is the same as that used in official Census Bureau calculations of poverty and inequality and includes earnings, Social Security (retirement,

6 disability, and survivors benefits), Supplemental Security Income, Unemployment Insurance, workers compensation, AFDC/TANF and other forms of public cash welfare, veterans payments, pension income, rent/interest/dividend income, royalties, income from estates, trusts, educational assistance, alimony, child support, assistance from outside the household, and other income sources. We define earnings as total family earnings from wage and salary income, nonfarm self employment, and farm self employment. Because the Census Bureau defines a family as two or more persons related by birth, marriage, or adoption, family earnings contains earnings of the mother as well as dependent children and other related adults such as a resident grandparent. It does not contain earnings of cohabiting partners or other non-family members in the household. We append to family income the (Census Bureau s) imputed dollar value of public food assistance programs, which includes the Food Stamp Program and the National School Lunch and Breakfast Programs. To construct after-tax total income we subtract tax payments from gross income and add back refundable EITC income. Tax payments are the sum of Federal, state, and payroll taxes that are estimated for each family in each year using the NBER TAXSIM program. The TAXSIM module calculates Federal, state, and payroll marginal tax rates and tax payments using basic information on labor income, taxable nonlabor income, dependents, and certain deductions such as property tax payments and child care expenses. 3 The Federal and state taxes include the respective EITC code for each tax year and state, thus allowing for the possibility of negative tax payments. We assume that the family only bears the employee share of the payroll tax rate. To control for changes in average family size over the twenty-seven years of our sample, we deflate after-tax income by the family-size specific poverty threshold. Because the poverty thresholds 3 The CPS does not have information on certain inputs to the TAXSIM program such as annual rental payments, child care expenses, or other itemized deductions. We set these values to zero when calculating the tax liability.

7 have been critiqued over the years on many dimensions (Citro and Michael 1995), including the quality of the adult equivalent scale used, we also present our main inequality estimates unadjusted for family size. If the respondent refuses to supply earnings or transfer information, then the Census Bureau uses a hotdeck imputation method to allocate income to those with missing data. Bollinger and Hirsch (2006) argue that including allocated data generally leads to an attenuation bias on estimated regression coefficients based on imputed data. Bollinger and Hirsch (2007) also show that, for women, there appears to be no selection bias for dropping employed women who fail to report earnings. Hence, we follow their recommendation and drop those mothers with allocated earnings or transfer income. In addition, 0.7 percent of the remaining sample has negative or zero values for total income, and we drop these observations. All income sources are deflated by the personal consumption expenditure deflator with 2005 base year. The total number of observations is 99,769 single female-headed families. Basic summary statistics are provided in Appendix Table 1. III. Trends in Income Inequality of Single Mothers, 1979 2005 Our focal measure of inequality is a member of the generalized entropy class of inequality indices, which is given as (1) I κ κ n 1 1 yi = 1, κ 0,1 κκ ( 1) n i= 1 y where y i is disposable income-to-needs for family i, y is average disposable income-to-needs, and κ reflects an aversion to inequality with lower values implying greater aversion to inequality (Shorrocks 1980; Cowell 2000). Although a wide variety of inequality measures are available, the generalized entropy class has several desirable properties including that it satisfies

8 the Pigou-Dalton principle of transfers so that it records a rank-preserving increase in inequality with transfers from a poor person to a less poor person, it is scale invariant which is useful for making inequality comparisons across groups and time, it has useful stochastic dominance properties for ranking income distributions (Fomby, et al. 1999), and perhaps most important for our purpose here, the generalized entropy class provides consistent and additively decomposable measures of inequality. These exact decompositions can be made by subpopulations such as employment status, education, age, and race into within and between group inequality (Mookherjee and Shorrocks 1982), and into income factors such as earnings and transfers both at a point in time and over time (Shorrocks 1982). More common measures such as the Gini coefficient, the Atkinson index, the variance of log income, or percentile ratios such as 90-10 are not additively decomposable and thus do not provide consistent decompositions across groups. For example, Cowell (1988) notes that it is possible with the log variance to have a change in the income distribution that leaves between group inequality constant, raises inequality within each group, and yet results in lower total inequality, which is clearly an undesirable property. For most of our analyses we set κ = 2, which yields I2 0.5* 2 = CV or one-half of the squared coefficient of variation. Aside from the decomposability properties mentioned above a key advantage of I 2 is that this summary measure is still defined even when some of the income components used in factor-share decompositions are zero. This is important in our sample because many single mothers do not work, do not receive transfers, or do not have other forms of nonlabor income (e.g. transfer income for the non-poor). To net out any life-cycle age effects on income to needs we use an age-adjusted measure of inequality. Specifically, we estimate via OLS the linear model of disposable income-to-needs on a quartic in age. That is, for person i, i = 1,, N, at time period t, t = 1,, T, we estimate

9 (2) y = α + β age + β age + β age + β age +ε, 2 3 4 it 1 it 2 it 3 it 4 it it where ε it is a mean zero random error. Because the mean of the fitted OLS residual, ˆit ε, used in the denominator of equation (1) is zero, the residual-based version of I 2 is undefined. Thus, in order to ameliorate this shortcoming, we add to each observation in a given year the year-specific mean of the predicted dependent variable, ˆ ε + yˆ, which results in a non-zero mean but has no impact on the year-specific estimated variance. A. Aggregate Trends in Inequality i Figure 1 depicts the trends in inequality for disposable income-to-needs and for t comparison purposes we also depict inequality trends in disposable income levels. The trends in both series are identical, but in most years the level of inequality is higher once adjustments are made for family need standards. We also present the 95 percent confidence interval for the income-to-needs measure of I 2 constructed from 100 replications of the nonparametric bootstrap. Because the unadjusted income inequality series is generally contained within the income-toneeds confidence interval we focus our ensuing discussion on income-to-needs. Figure 1 shows a trend increase in inequality from 1979 to 1985, which coincides with the inequality literature on men, but little change for the next decade except for 1987. After 1994, however, inequality begins a strong upward drift rising by over 60 percent between 1994 and 2004 before settling back down in 2005 at a level that is over 20 percent higher than 1994. What is striking about the trend in inequality is how volatile it becomes after 1994. The time series was relatively stable for the 15 years prior, but it becomes unstable after 1994 with large spikes in 1996 and 2004. Even though the I 2 is estimated precisely in each year (the bootstrap t- statistic rejects the null that it is zero with test statistics ranging from 7 to 22), the confidence

10 intervals widen markedly after 1994 such that the lower bound indicates little change over time in inequality whereas the upper bound suggests that inequality has doubled. Although the inequality series appears to change in the mid 1990s around the time of major social policy reforms, it is also important to determine whether the series differs in a statistical sense. One approach is to parametrically fit a series of trend break points in the mid 1990s and test for a structural change point. In lieu of arbitrarily choosing break points in the inequality series, we instead turn to the recent time series literature on testing for structural change with an unknown change point (Andrews 1993; Bai 1997; Hansen 2001). Because these tests have not to our knowledge been applied in the inequality or social policy literatures a brief summary of the procedure is warranted. The tests build on an idea due to Quandt (1960), who proposed splitting the sample at every possible breakdate, estimating the model parameters separately on the two sub samples, and constructing the associated Chow test statistic for all possible sample splits (any subsample must have more observations than parameters estimated). The estimated breakdate is the sample split with the largest value of the Chow test statistic. If the breakdate is known a priori then one can appeal to the usual chi-square tables for critical values. However, in many cases the breakdate is not known and the chi-squared critical values are not valid. Andrews (1993) developed the asymptotic theory for the case of unknown change point and provided tables of critical values, and consequently the new tests are generally known as the Quandt-Andrews sup- Wald statistic. As noted by Hansen (2001), this method of least squares testing for structural change is valid for the linear regression model with homoskedastic variances, and Bai (1997) proposed a straightforward method of constructing confidence intervals around the breakdate.

11 To implement the tests we take our estimated univariate time series of residual inequality displayed in Figure 1 and run the following regression: (3) 2 I ( t) = σ + θt+ ζ, t = 1,..., τ 2 1 1 1 2 I ( t) = σ + θ t+ ζ, t = τ + 1,..., T 2 2 2 2 where I () 2 t is the estimated residual inequality, σ 2, j = 1, 2 is a constant term (reflecting the constant inequality), θ, j = 1,2 is the coefficient on the linear trend t, ζ is an iid random error j term, τ is the unknown breakdate, and T = 27 for the years calendar years 1979 2005. For each possible breakdate, τ, we conduct the joint test of the null hypothesis of constant inequality 2 2 ( σ1 = σ 2 and θ1 = θ2) by constructing the following Wald test statistic (4) W = ( SSE ( SSE + SSE ))/(( SSE + SSE )/( T 2* l) ) pooled 1 2 1 2 where SSE is the sum of squared errors for the pooled regression with no break, SSE1 and pooled j SSE 2 are the sum of squared errors for the pre- and post-break periods, respectively, and l is the number of parameters in each subsample. The estimated breakdate is the ˆ τ with the maximum test statistic ˆ W, i.e. the supwald statistic. The associated Bai (1997) confidence interval for c with trending regressors is ˆ τ ± ( + 1) Lˆ. The term c is the critical value for a test of size α (c=7 when α = 0.1 and c=11 when α = 0.05) and ˆL is the outer product of the fitted valued of the ˆ τ regression standardized by the estimated error variance, ˆ ˆ γ ZZ ' γˆ L =, with ˆ γ = [ ˆ σ ˆ 2 j, θ ] ˆ j and σ Z = [1, t] and where t is set at the estimated breakdate ˆ τ (See Bai (p. 555) for additional details). In Table 1 we report the Quandt-Andrews supwald test statistics for age-adjusted residual inequality, along with the estimated breakdate, and the associated 95 percent confidence interval. Because the determinants of income, and thus income inequality, are affected by the

12 business cycle (Solon, et al. 1994; Ziliak, et al. 1999), we conduct our Quandt-Andrews tests controlling for the macroeconomy, including the unemployment rate, constructed by the Bureau of Labor Statistics, and the growth rate in real chain-weighted GDP, constructed by the U.S. Department of Commerce. Because our test contains a constant term, trend, and the two business-cycle controls, l = 4, which reflects the fact that by construction the tests allow for structural change in the coefficients of all included regressors. The corresponding critical values for the l = 4 case are 13.82, 15.84, and 20.24 for the 10 percent, 5 percent, and 1 percent levels of significance, respectively (Andrews 1993). In the first row of Table 1 the supwald statistic of 23.7 clearly rejects at the 1 percent level the null hypothesis of no structural change for disposable income-to-needs inequality. Moreover, what appears as obvious in Figure 1 is indeed statistically validated in that the estimated breakdate occurs in 1994 and with a tightly estimated one-year confidence interval between 1993 and 1995. Because the tests of structural change in Table 1 are not based on fullyspecified multivariate models we are cautious about making causal statements attributing increasing inequality to welfare reform and the EITC, but we do note that the result is robust to exclusion of business cycle factors in results not recorded (though with wider confidence intervals). To better understand why inequality is more volatile over the past decade in Figure 2 we depict trends in the mean and variance of disposable income-to-needs. The figure makes transparent that the instability in I 2 in Figure 1 is driven by a massive increase in the variance of income among single mother families. Mean income is stable through the early 1990s and then begins a steady secular climb before plateauing in 2001. The variance likewise is stable into the early 1990s, but then rises by 75 percent between 1994 and 2005, with an even larger spike in

13 2004. The rise in cross-sectional income variance in this population has not been previously established and suggests that the social policies and macroeconomy of the mid 1990s affected much more than the first moment of the income distribution. Before proceeding it is important to address two measurement issues, one related to top coding at the high end of the distribution and one related to income underreporting at the low end of the distribution. Burkhauser, et al. (2007) recently showed that the Gini coefficient and 90-10 ratio are sensitive to income top coding in the CPS. Because the coefficient of variation, and thus I 2, is top sensitive, it is possible that our measure also is affected by top coding in the CPS. We examined the extent of top coding across all the subcomponents of income in our sample of single mother families and in most years there were no top-coded observations, but top coding became more prevalent in beginning in 1998, which is consistent with Burkhauser, et al.; however, unlike the general population, top coding never affected more than 0.2 percent of the sample of single mother families. Thus, top coding appears to be an inconsequential issue for this population. Meyer and Sullivan (2006), on the other hand, argue that income among single mother families in the CPS at the lower tail of the distribution is mismeasured because of underreporting of transfer income, especially beginning in the mid 1990s, and thus consumption is a superior measure of well being. 4 Although a comparison of income to consumption is beyond the scope of the current project, we first examine the possible influence of underreporting of transfers, as well as the sensitivity of I 2 to outliers at the high end of the distribution, by depicting various percentiles of the income distribution from the 2 nd percentile (p2) to the 99 th percentile (p99) in Figure 3. We present the percentiles in terms of real 2005 dollars, and not income to needs, in order to 4 Though Bollinger (1998), using data in the CPS matched to Social Security records, shows that if anything the poor overstate earnings in the CPS relative to administrative Social Security data.

14 highlight the comparatively low incomes of this population relative to percentiles from the whole population as presented in Piketty and Saez (2003). For example, among single mothers, the 99 th percentile in 1998 is $75,250 compared to over $230,000 reported for the entire tax-paying population. In Figure 3 we observe no discernable long-term downward trend in income at the low end of the distribution. For example, the average level at the 2 nd percentile is $533 from 1979 1994 and $545 from 1995 2005 (recall 1994 is the estimated breakdate in inequality). Like Meyer and Sullivan (2006) we do find a 30 percent reduction in after-tax income at the 2 nd percentile across the 1993 1995 and 1997 2000 periods, which spans the enactment of PRWORA. But we also find a 42 percent reduction when comparing the 1980 1982 and 1983 1986 periods which spans the major reforms to AFDC enacted by President Reagan, though there is no evidence of an increase in transfer underreporting in the mid 1980s. We also note that there is a 15 percent increase in after-tax income at the 2 nd percentile when comparing the 1997 2000 and 2001 2004 periods. Although this does not speak directly to the dominance of consumption relative to income as a metric of well being, it does suggest that such a case be based on strong theoretical arguments in addition to measurement because there is no consistent long-term pattern of income declines among very poor single mothers (at least in the CPS). At the same time Figure 3 does reveal evidence of a strong trend in income beginning in the mid 1990s in the upper half of the distribution. We next attempt to gauge the impact of the lower and upper tails of the distribution on our estimated inequality series in Appendix Figure 1. In the figure we depict the original untrimmed series along with the estimated I 2 where (a) income at the second percentile and below is trimmed, and (b) income at the ninety-eighth percentile and above is trimmed. The appendix figure shows that inequality is lower as expected

15 with the bottom 2 percent trimmed, but the series lies within the 95 percent confidence interval of the untrimmed series. However, with the top 2 percent trimmed inequality is constant and lies below the confidence region in every year. This suggests that among single mother families rising inequality in the 1990s is an phenomenon heavily concentrated in the upper tail of the income distribution similar to the trend in the general population (Piketty and Saez 2003; Autor, et al. 2005). Indeed, in results not tabulated, the Quandt-Andrews test places the structural break at 1998 for p75 thru p98 with a one-year confidence interval, and at 1993 with a three-year confidence interval for p99. B. Factor Decomposition of Inequality We now explore whether the rising inequality depicted in the previous section can be attributed to changes in one or more of the income sources derived by families. As highlighted in Grogger (2003) and Bollinger, et al. (2007), there have been substantial changes in the level and composition of income among single mothers, with a massive shift away from transfers and toward labor market earnings. We examine whether the shift in income composition contributed to the rise in inequality. Specifically we decompose disposable income-to-needs into four factors (5) yit earningsit + transfersit + otherit taxesit where earnings refers to total labor market earnings in the family, transfers refers to income from the AFDC/TANF program, the SSI program, the Social Security and Disability Insurance programs, and the Food Stamp Program, other refers to other nonlabor income from both public and private sources, and taxes refers to the sum of federal, state, and payroll tax payments inclusive of the refundable portion of the federal and state EITC. These four major factors are isolated to highlight the potential contribution of rising employment rates to earnings, the contribution of declining welfare and food stamp income and rising disability payments in the

16 1990s, the contribution of income from other household members, and finally the inequality reducing impact of progressive income taxes including the EITC. 5 Shorrocks (1982) shows that the generalized entropy class of inequality measures is additively decomposable into the contributions of income factors such as that described in equation (5), which implies that we can write I 2 as (6) I 2 4 = S f = 1 f σ f where f = earnings, transfers, other, taxes and Sf ρ f I2. ρ f is the correlation coefficient σ between factor f and total disposable income to needs y, σ f is the standard deviation of income factor f, and σ y is the standard deviation of disposable income-to-needs. Note that the product of the correlation coefficient and the ratio of the standard deviations is simply the coefficient from a least squares regression of disposable income-to-needs on income factor f. The advantage of I 2 is clear in this decomposition because the values of factor f may be zero for many households, e.g. zero earnings for nonworkers or zero transfers for non welfare recipients, and yet I 2 is still defined in these cases (note that the variance of log income fails here). Figure 4 depicts trends in the cross sectional decomposition of disposable income-to- y needs inequality into its factor shares. 6 Both earnings and other nonlabor income lie above the x-axis because they are disequalizing and contribute to inequality and both transfers and taxes lie below the x-axis as they are equalizing factors in the distribution of income. The four factors of income inequality shown in the figure are quite stable from 1979 to 1994, but after 1994 earnings inequality accelerates, and to a lesser extent so too inequality of other income but 5 See the series of papers in Slemrod (1994) on the role of tax policy on overall income inequality. 6 As with total income, we conduct our decomposition of factor shares based on residual earnings, transfers, other nonlabor income, and taxes from a regression of each factor on a quartic in age.

17 only until 1998 when it falls back to the level in 1994 and remains constant thereafter. The Quandt-Andrews tests reported in rows 2 5 of Table 1 generally corroborate the picture in Figure 4 in that the null of no structural break in inequality of the income factors is rejected and with the exception of transfers the estimated breakdate is 1994. Furthermore, Figure 4 suggests that the bulk of the cross-section instability after 1994 depicted in Figure 1 emanates from rising inequality in the labor market. The major equalizer is the tax system, both through the refundable EITC and progressive marginal tax rates. In a typical year prior to 1994 the tax system reduced inequality by about 33 percent, but after 1994 this share rose 8 percentage points to 41 percent. Perhaps surprising, income transfers have never played a significant role in reducing inequality among single mothers, averaging about 8 percent prior to 1994 and 3 percent after 1994. So while there was a massive rundown in welfare caseloads after 1994, the transfer income losses do not appear to be a major contributor to the significant rise in inequality. C. Sub-Group Decomposition of Inequality The factor share decompositions indicated that the inequality of earnings is the major factor determining the inequality of income among single mothers in any given year. The share attributable to earnings rose further in the mid 1990s, which coincides with rising employment of this demographic group (Meyer and Rosenbaum 2001; Grogger 2003; Bollinger, et al. 2007) and suggests a potentially important role of employment for the trend in inequality. Figure 5 shows the trend in I 2,k where k represents workers and non-workers respectively. The figure shows that income inequality actually rose much more dramatically among non-workers than workers between 1994 and 2005 a 90 percent increase for non-workers and 23 percent increase for workers. Table 2 presents Quandt-Andrews tests of structural change for the group-specific inequality series and the tests indicate structural breaks in 1994 for workers and 1998 for non-

18 workers, though the latter has a 4-year confidence interval. This highlights growing instability among single mothers disconnected from the workforce (Blank 2007), but it also poses a prima facie puzzle in light of the results indicating the primacy of earnings inequality in explaining income inequality. To examine in more detail the relative roles of between-group (e.g. workers versus nonworkers) inequality and within-group inequality, in this section we consider sub-group inequality decompositions based on employment status. We also consider other demographic splits based on education attainment, age, past marital status, and race because the groups historically at highest risk of welfare use and thus likely affected by social policy reforms of the 1990s are the less educated, the young, the never married, and African Americans (Moffitt 1992; Blank 1997). The generalized entropy measure of inequality is once again useful here because it is additively decomposable into within-group and between-group contributions to inequality, and only depends on a few, easily obtained factors. Mookherjee and Shorrocks (1982) show that the I 2 index can be decomposed as (7) I 1 [ 1] K K 2 2 2 = ωμ k ki2, k + ωk μk k= 1 2 k= 1 where the first term is within-group inequality and the second term is between-group inequality, and the three determinants are the group-specific population share ω ( k = 1,..., K ), the square of k the relative mean of group k, 2 2 yk μk y, and the group-specific inequality I 2,k. Figures 6 through 10 depict the within- and between-group decomposition of equation (7) for our selected demographic groups (employment status, educational attainment, age, marital status, and race). For each figure we depict overall within- and between-group inequality, and also show the contribution of each sub-group to within inequality. In Figure 6 this means that we

19 have a total of four lines based on employment status overall within-employment and betweenemployment inequality, as well as the contributions to within-employment inequality by workers and non-workers (the sum of these latter two figures yields the overall within-employment inequality). Figure 6 reveals that most of the inequality in any given year is accounted for by inequality within employment status (88 percent on average), and because approximately 90 percent of within-group inequality is attributable to workers, the figure also confirms the importance of earnings to within-group inequality highlighted earlier. Figure 6, coupled with Table 3, helps reconcile the puzzle raised earlier in Figure 5 which showed rising inequality among non-working families. Note that the inequality trends of workers and non-workers in Figure 5 do not account either for changes in the population shares of each group or the average income accruing to each group. However, Table 3 shows that the population share and relative mean income of non-working single mothers plummeted by over one-third between 1980 and 2005, and while inequality within the group of non-workers increased in Figure 5, the declining population shares and relative incomes reduced their contribution to overall inequality as depicted in Figure 6. Figures 7 10 suggest that a similar story to Figure 6 in that nearly all of the inequality in a given year is due to within-group inequality and very little is attributed to between group differences. Within-group inequality among single mother families is most prominently affected by those with more than a high school education, those age 31 and older, those widowed or divorced, and those who are white. Table 3 provides five-year snapshots of population shares, relative mean incomes of each group, and group-specific inequality. The table indicates that the trend towards greater inequality (a) within higher educated mothers is explained in part by their rising share of the population, (b) within older mothers is explained by an upward shift in the age

20 distribution of mothers, (c) within widowed and divorced mothers is explained both by the rising inequality within the group and their rising share of relative income even though the population share of never married single mothers more than doubled over the past 25 years, and (d) within white mothers is explained by rising inequality within the group coupled with their sizable shares of the population and mean income. D. Changes in Inequality over Time The decompositions in Sections B and C are a time series of cross sectional relationships and thus provide a snapshot at various points of time of the income sources and/or demographics determining inequality. However, the trends do not speak directly to the underlying sources of change in inequality over time. A long standing approach in the inequality literature to examine changes in inequality is to adopt the so-called shift-share method that takes a given factor, say education, and asks questions such as what would inequality in 2000 be if education attainment remained fixed at levels in 1980? (DiNardo, et al. 1996; Mills, et al. 2001; Autor, et al. 2005; Lemieuz 2006). This approach is attractive because it offers transparent counter-factual decompositions of income distributions. However, as argued by Mookherjee and Shorrocks (1982) it is less useful when there are multiple changes occurring simultaneously (as affected single mothers over the past two decades) because it is difficult to determine the relative importance of each factor to trend inequality, and the combined effects of the changes do not necessarily sum up to total inequality, i.e. it is possible to over- or under-explain trend inequality with shift-share analyses. A preferred alternative is to adopt the decompositions described in equation (6) and (7) as they aggregate changes in sources exactly into the changes in total inequality; that is, changes in income factors or changes in within and between group inequality add up to changes in total

21 inequality. Specifically in the case of determining how changes in income factors affect changes in inequality between any two periods t and t+1 Jenkins (1985) shows that we can rewrite equation (6) as (8) ΔI2 I2( t+ 1) I2( t) = ΔSf 4 f = 1 which means that the change in inequality across any two years is the simple sum of changes in the factor components. Likewise we can decompose the changes in within-group and betweengroup inequality by taking the difference in equation (7) between periods t and t+1 and rearranging to yield (9) K K K 2 2 2 2 ωk μk 2, k ωkμk 2, k ωk μk 2, k k= 1 k= 1 k= 1 ] Δ I = ( t) ( t) Δ I + Δ ( t+ 1)[ I ( t+ 1) + 0.5] + ( t) Δ [ I ( t+ 1) + 0.5 which says that the change in inequality is due to (a) a change in within-group inequality, (b) a change in population shares, and (c) a change in relative mean incomes. 7 The next step is to choose the relevant periods t and t+1 to conduct the decompositions of inequality change. The bulk of the literature on both inequality and volatility proceeds by choosing decades such as the 1970s, 1980s, and 1990s or 5-year intervals within decades to conduct counter-factual experiments. This choice is based largely on convention. However, Figure 1 and Table 1 are striking in that there is compelling evidence of a structural break in trend inequality around 1994, and that this break point overlaps with major reforms to social policies including welfare reform and EITC expansions. Consequently, we consider three time decompositions the change in inequality across the entire sample period of 1979 to 2005, the change in inequality prior to the structural break (1979 1994) and the change after the structural 7 Mookherjee and Shorrocks (1982) note that there is an index number problem here in that the decomposition in equation (9) uses current period values of population shares and (t+1) values of inequality and income shares, but it is possible to reverse the order. For transparency we just present the results from equation (9).

22 break (1994 2005), and the change in inequality across five-year intervals (we include the extra period in the first change of 1979 to 1985). To facilitate comparisons across the multiple changes, and to guarantee adding up, we divide both sides of equations (8) and (9) by the base year inequality I 8 2 (1979) and report the results as percentage changes. Table 4 presents the results of the income factor change decompositions of equation (8). In the first row of Table 4 we see that by 2005 inequality as measured by I 2 rose by 49 percent above the baseline value in 1979, and that in the absence of the progressive U.S. tax system inequality would have been 18 percentage points higher. Prior to the 1994 breakdate, inequality rose by 23 percent, and then it rose an additional 26 percent thereafter (relative to 1979). Quantitatively there is little difference across periods in the relative contribution of each factor to inequality, though transfers actually became disequalizing after 1994. The five-year inequality change decompositions show considerable within-period variation in the relative roles of earnings, transfers, other income, and taxes on inequality. In most cases inequality would be significantly higher in the absence of income taxes, though there are some exceptions. Taxes were disequalizing between 1985 and 1990 and again between 2000 and 2005, perhaps reflecting the tax cuts associated with the 1986, 2001, and 2003 tax reforms. This merits future research on both the population of single mothers as well as the broader population of taxpayers. In Table 5 we record the results of the demographic change decompositions of equation (9). Across the entire sample the bulk of the 49 percent rise in inequality is attributed to a rise in within-group inequality regardless of the group selected to conduct the decomposition. However, this is often because of large, offsetting changes in the two terms affecting betweengroup inequality. For example, there was a large and disequalizing increase in the share of single 8 In some cases rounding error after the calculations may result in some rows not adding up.

23 moms with more than high school between 1979 and 2005, but the relative mean incomes of this population fell over the period suggesting a shift in the placement of higher educated single mothers toward lower income earners relative to two decades ago (see Table 3). A similar scenario unfolded based on employment status of single mothers as well as past marital status. After 1994 there was a substantial increase in the contribution of employment to inequality, but these new workers were placed in the lower end of the distribution and thus reducing the relative mean income of workers, which equalized incomes in the overall population of single mothers. In the case of marital status, there was an equalizing shift in the population toward never-married mothers, but their relative mean incomes fell compared to widowed, separated, and divorced mothers. Examining the inequality changes across sub-periods reveals a similar trend in favor of within-group changes being the prominent factor in changes in inequality, but again often because of substantial offsetting changes in relative population shares and relative mean incomes. This suggests that the time-series of cross-sectional decompositions of within- and between-group inequality in Figures 6 10 understate the role of important between group changes in inequality over the past two decades arising from large shifts in employment status, education attainment, age, and previous marital status. IV. Conclusion We identify an increase in disposable income-to-needs inequality of about 50 percent among single mothers in the United States over the past 27 years. Our statistical tests identified a break in trend inequality toward higher inequality between the years 1993 and 1995 a period that was characterized by major changes in the U.S. tax and transfer system, and was also in the early stages of the longest post-war business-cycle expansion. Further analysis suggests that the rise in income inequality was driven largely by higher earnings inequality, and most of the