metadata: authors: 0: author: Atkinson,Anthony B.
1: author: Lakner,Christoph
count: United States
docna: 0: docna: Capital and labor : the factor income
            composition of top incomes in the United States, 1962-2006
docty: Policy Research Working Paper
subtopic: Employment and Unemployment,Poverty Reduction Strategies,Wages, Compensation & Benefits,Public Sector Economics,Macro-Fiscal Policy,Economic Adjustment and Lending,Taxation & Subsidies,Public Finance Decentralization and
            Poverty Reduction,Economic Conditions and Volatility
teratopic: Macroeconomics and Economic Growth,Social Protections and Labor,Public Sector Development,Poverty Reduction
abstracts: cdata!: This paper finds that capital and labor
            incomes in the United States have become more closely
            associated since the 1980s. This contributed to the
            well-known increase in the top 1 percent's share of
            total income, exacerbating rising inequality in capital
            incomes and earnings. The paper shows that the trend in the
            association is U-shaped, as the recent increase contrasts
            with a tendency toward a weakening association until the
            1980s. The paper uses data derived from tax records, studies
            the asymmetries in the association, tests for robustness to
            alternative income definitions, and discusses the potential
            role of declining top marginal tax rates.
geo_reg_key: 119228,517191
repnme: repnme: Capital and labor : the factor income
            composition of top incomes in the United States, 1962-2006
display_title: Capital and labor : the factor
            income composition of top incomes in the United States, 1962-2006
listing_relative_url: /research/2017/12/29263583/capital-labor-factor-income-composition-top-incomes-united-states-1962-2006
topicv3: Macroeconomic Vulnerability and Debt
            m1326057 1104,Taxation m1327615 1121,Income Inequality m1503891 2668,Self-Employment m1504000 2698,Self-Employment Income m1504044 2741
publishtoextweb_dt: 2017-12-11T00:00:00Z
historic_topic: Macroeconomics and Economic Growth,Social Protections and Labor,Public Sector Development,Poverty Reduction
no_of_pages: 43
author_keys: 0: author_key: 675865
1: author_key: 1206398
model_extractions: WPS8268 Policy Research Working Paper 8268

# Capital and Labor

The Factor Income Composition of Top Incomes in the United States, 1962–2006 _Anthony B. Atkinson Christoph Lakner_ Development Economics Development Research Group December 2017
Policy Research Working Paper 8268

# **Abstract**

This paper finds that capital and labor incomes in the United States have become more closely associated since the 1980s. This contributed to the well-known increase in the top 1 percent’s share of total income, exacerbating rising inequality in capital incomes and earnings. The paper shows that the trend in the association is U-shaped, as the recent increase contrasts with a tendency toward a weakening association until the 1980s. The paper uses data derived from tax records, studies the asymmetries in the association, tests for robustness to alternative income definitions, and discusses the potential role of declining top marginal tax rates.

This paper is a product of the Development Research Group, Development Economics. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The authors may be contacted at clakner@worldbank.org.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
# Capital and Labor:

# The Factor Income Composition of Top Incomes in the United States, 1962-2006

Anthony B. Atkinson and Christoph Lakner<sup>_∗_</sup> JEL classication: D31, D33, E25. Keywords: Inequality, factor incomes, top incomes, United States.

> _∗_ Corresponding author: clakner@worldbank.org. A. Atkinson: Nueld College, Oxford, London School of Economics and Institute for New Economic Thinking at the Oxford Martin School. C. Lakner: World Bank and Centre for the Study of African Economies, University of Oxford. A. Atkinson sadly passed away in January 2017, before the new version of the paper was nalized. This paper would not exist without his contributions. Subsequent revisions were carried out by C. Lakner who is responsible for any errors. This paper has been developed from one of the chapters of C. Lakner's DPhil thesis. Emmanuel Saez participated in the rst stages of this project, and we should like to thank him for his assistance with the US data. We are most grateful to seminar participants at the Oxford INET, the World Bank and the ECINEQ conference, and to Rolf Aaberge, Francisco Ferreira, La-Bhus Fah Jirasavetakul, Stefan Klasen, Sebastian Königs, Branko Milanovic, Bob Rijkers, Francis Teal and Adrian Wood for discussions and comments on earlier versions. Lakner gratefully acknowledges nancial support from the Economic and Social Research Council [grant number ES/G011974/1], the Studienstiftung des deutschen Volkes, the Institute for New Economic Thinking (INET) at the Oxford Martin School and the Center for Equitable Growth at UC Berkeley. The ndings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its aliated organizations, or those of the Executive Directors of the World Bank or the governments they represent.
# Introduction

Recent literature has documented the increase in income inequality at the very top of the distribution (e.g. Atkinson and Piketty, 2007, 2010; Alvaredo et al., 2017). Between the mid1980s and the mid-2000s, the income share of the top 1% in the US approximately doubled, while around the same time their share of income from capital declined, and salaries and selfemployment incomes became more important (Figure 1, also see Piketty and Saez, 2007a). These changes in the income composition have been even more pronounced over the long-run, with the share of income from capital among the top 1% dropping from close to 50% in the 1920s to less than 20% in the 2000s (Piketty and Saez, 2003, 2007a). Such an increase in the labor share at the top could arise from a change in the association between labor and capital, or because the share of tax units in the top who receive only capital income declines while there remain separate classes of laborers and capitalists. As part of the hypermeritocratic society (Piketty, 2014), the inequality in the wage distribution has increased substantially (Piketty and Saez, 2007a), providing support for the second channel. At the same time, tax units increasingly have income from both capital and labor (Wol and Zacharias, 2009), and pure rentiers have virtually disappeared (Atkinson, 2009).

While the literature (e.g. Piketty and Saez, 2007a) has focused on the distribution of total income, as well as the distributions of capital and labor income separately, the association between the two income sources has received little attention. To extend the existing literature, this paper examines the association between capital and labor incomes at the top of the US income distribution. Using data based on US tax returns, we can directly measure the association. In a classical society, capitalists are at the top of the capital distribution and bottom in the wage distribution, i.e. the correlation between capital and labor incomes is negative. On the other hand, this correlation is generally positive (but less than one) in modern economies (Piketty, 2014). Our paper also links the study of the functional (capital v. labor) and the personal (rich v. poor) income distributions. For 2
<!-- Start of picture text --> 100 16<br>= 80 14<br>e 3<br>5 °<br>g &<br>= 60 12 9<br>fe)+S) £Cc<br>g ic<br>8 2<br>2 40 10 0<br>a (0)<br>ie)o£o<br>g oO<br>H 20 8<br>0 6<br>1965 1970 1975 1980 1985 1990 1995 2000 2005<br>Year<br>ly Salaries ly Self-employment<br>lh S-corporations ha Capital income<br>—*—-_ Share of total income<br>Excluding capital gains and negative observations. Income categories are defined below in Section 1.<br><!-- End of picture text -->
while capital incomes and rentiers have not disappeared from the top. This asymmetry is not found at the top of the distribution (top 5% and upwards). Our results are robust to treating negative incomes and capital gains dierently, and to how we allocate income from self-employment and closely-held corporations to capital and labor.

We examine the association in two ways. We begin by decomposing the top 1% income share by factor incomes, which is frequently done with standard inequality measures, but has not been applied to top income shares. The inequality in total income is decomposed into the labor share in total income, the top 1% share within the distributions of capital and labor incomes, and the alignment coecient, which captures the association. The alignment coecient, like the Pearson correlation coecient, is aected by monotone transformations in the marginal distributions. In the second part of the paper, we thus use a rank-based measure of association that is more general and invariant to such transformations. Specifically, we analyze the association matrices between labor and capital, which are a discrete approximation to the copula density and equivalent to transition matrices in the study of economic mobility. The literature on mobility also provides us a test of increasing association based on cumulative association matrices.

This remainder of the paper is structured as follows. Section 1 describes the data. In Section 2, we develop and estimate a decomposition of top income shares by factor incomes. Section 3 reports the results from the association matrices. Section 4 concludes and discusses how the changes in the association may be related to top marginal tax rates. The Appendix provides additional results and several robustness checks.<sup>1</sup>

# 1 The data

We use the Statistics of Income public use les (PUF) by the US Internal Revenue Service over the period 1962 to 2006, which are (nearly) annual public-use samples based on federal

> 1A discussion of our US results after the mid-1990s in comparison with Norway can be found in an unpublished note by Aaberge et al. (2017).

4
income tax returns.<sup>2</sup> The data are based on a random sample of tax records led during a particular calendar year. Importantly for our analysis, high-income returns are over-sampled, and we use the appropriate sampling weights to adjust for this. Following Piketty and Saez (2007a) and the literature using tax records more generally, the unit of analysis is a tax unit as dened under US tax law. We thus include singles and married couples without adjusting for dierences in tax unit size.<sup>3</sup> Not every tax unit les a tax return; historically there were high exemption levels and income taxes only applied to the most auent taxpayers. We ignore the non-ling issue in this paper because we would have to make an arbitrary assumption about the income composition of the non-lers. Furthermore, this is unlikely to aect our results, since the fraction of lers is quite high and stable over the period of analysis (94% on average, compared with 9% before World War II). The PUFs are subject to some adjustments, especially at the top, that try to minimize the risk that individual taxpayers can be identied (Winglee et al., 2002). As a result, an observation in the PUF never contains all the information on a tax return, and may include information from other returns. Further information is provided in Appendix A.3, where we also show some robustness checks.

Income is dened as (taxable) gross market income, as reported on federal income tax returns.<sup>4</sup> We follow Piketty and Saez (2007b), Piketty et al. (2018) and Saez and Zucman

> 2Like other papers using these data (e.g. Saez and Zucman, 2016 or Piketty et al., 2018), we exclude the micro data for 1960, since it contains fewer tax return variables. There exist no PUFs for 1963 and 1965. More information on these data is available at http://users.nber.org/~taxsim/gdb/.

> 3Our results are robust to including only tax units that have two adults. Between the 1960s and 2000s, the average tax unit size declined from 2.6 to almost 2 persons. The proportion of married tax units declined by 10pp in the whole population but slower at the top of the distribution (Saez, 2004). Lakner (2014) shows that the trend in top shares is robust to accounting for tax unit size. Piketty et al. (2018) also nd very similar trends for tax units and adults. Tax units tend to be smaller than households; Hungerford (2010) estimates 75% of households to have one tax unit, and another 17% to have two tax units.

> 4We exclude any income that is not taxable (e.g. non-taxable fringe benets such as health insurance), since it is not reported on the tax return. Imputing such non-taxable incomes from the US Current Population Survey does not have a large eect on top income shares (Bivens and Mishel, 2013; CBO, 2012). We also exclude non-market or transfer income such as Social Security and unemployment insurance benets. Since the PUFs do not capture all tax liabilities (e.g. exclusion of state and local taxes), we focus on gross incomes, like Piketty and Saez (2007a) and most of the literature on top incomes. It is also unclear how to split total federal income taxes between capital and labor without additional imputations. Finally, our analysis uses income that is observed from tax records (like Piketty and Saez, 2007a), which does not match national income (like Piketty et al., 2018) or macro totals in the nancial accounts (like Saez and Zucman, 2016) (also see Figure A.5).

5
(2016) to construct income components from the raw data that are comparable over time. We dene labor income as the sum of wages and (taxable) pensions.<sup>5</sup> Self-employment income is the sum of sole-proprietorship (Schedule-C) and partnership income. In the baseline, capital income is dened as the sum of dividends, (taxable) interest, rents, estate income, royalties, and prots from S-corporations.<sup>6</sup> Because negative incomes can result in top shares for the distribution of capital or labor income that are greater than one, we drop observations which are negative in labor, self-employment or capital income.<sup>7</sup> Self-employment income reects returns to both human and physical capital, so it needs to be split between labor and capital. In the baseline, we allocate two-thirds of self-employment income to labor and one-third to capital. While these weights are arbitrary, they are similar to earlier literature, close to factor shares found in national accounts (Gollin, 2002; Feldstein, 2008; Elsby et al., 2013; Karabarbounis and Neiman, 2014), and intermediate compared with more extreme weights considered in the robustness checks (Appendix A.5).<sup>8</sup> Table 1 presents summary statistics for the baseline income denition. The number of observations is 87,000 per year on average.

In Appendix A.5, we show that our results are robust to several alternative income denitions. First, instead of dropping negative observations, we set them to zero (similar to Saez and Stantcheva, 2017). Second, we include capital gains, which are an important income source at the top. The tax data only report realized capital gains, which are lumpy because realizations respond to changes in the tax code or asset prices. An accruals-based approach changes the timing of capital gains in the short run, but the long-run trend remains

> 5 Stock options are taxed as wage income when they are exercised.

> 6 S-corporations are businesses with few shareholders that are taxed at the personal instead of the corporate level. Including S-corporation prots with capital is similar to CBO (2012), and tries to address the shift of corporate income to the personal sector following the Tax Reform Act of 1986 (TRA86) (see below). Since we only include income that is taxable at the personal level, our denition of capital income excludes undistributed corporate prots.

> 7This aects on average 7.5% of the (weighted) sample, due to negatives in self-employment (4.9% of self-employment incomes are negative) and/or capital income (3.0% of capital incomes are negative). The median total income of the excluded observations is similar to the 70th percentile. Using the same dataset, Auerbach and Hassett (2002) also drop observations with negative adjusted gross income.

> 8Johnson (1954) attributes 65% of non-farm entrepreneurial income to labor. Kaymak and Poschke (2016) include 64% of entrepreneurial income with labor income. Piketty et al. (2018) allocate 70% of non-corporate business income to labor.

6
Table 1: Summary statistics

||1966|1970|1980|1985|1990|2000|2006|
|---|---|---|---|---|---|---|---|
|Overall mean|37,095|39,871|38,455|38,536|40,682|49,330|49,582|
|Top 5%: mean|147,774|153,513|151,149|156,166|200,505|297,669|310,457|
|Top 5%: threshold|88,910|96,650|98,188|100,965|108,510|135,043|136,985|
|Top 1%: mean|288,018|286,433|278,975|293,308|453,733|755,730|799,571|
|Top 1%: threshold|170,001|174,502|166,212|164,912|205,199|300,179|314,160|
|Top 0.5%: mean|379,552|373,430|368,796|402,933|666,268|1,152,718|1,217,897|
|Top 0.5%: threshold|234,381|233,496|218,958|211,241|294,606|444,401|479,634|
|Number observations|73,606|73,949|122,446|68,743|67,727|114,160|114,662|

Note: All mean and threshold incomes refer to sum of income components, and are expressed in 2006 USD. Excluding capital gains and negative observations. CPI data are taken from Saez (2013).

similar (Larrimore et al., 2016). Third, we report results for two alternative ways of splitting income from self-employment and closely-held businesses: (a) We show that our results are robust to including 54% of S-corporation prots with labor income instead of allocating them entirely to capital income.<sup>9</sup> As Figure 1 shows, this pass-through income has become increasingly important at the top (also see Cooper et al., 2016).<sup>10</sup> Using evidence from rm owner deaths, Smith et al. (2017) estimate that 54% of S-corporation prots at the top represent labor income. (b) Our conclusions are also robust to moving in the opposite direction and allocating a greater share of self-employment income to capital. Following Saez and Zucman (2016), we include all of self-employment income (as well as S-corporation prots) with capital, instead of only one-third in the baseline.

# 2 Decomposition by factor incomes

We begin the analysis with a decomposition of top income shares by factor incomes. This is a formal derivation and the rst empirical application of the decomposition by Atkinson (2007), who builds on Meade (1964). It is closely related to factor income decompositions of

> 9Wages that the owner-manager of the S-corporation pays herself would have already been included in wage income.

> 10S-corporation ling status became more attractive following TRA86, which reduced the top personal tax rate below the corporate tax rate (Slemrod, 1996; Auerbach and Slemrod, 1997). Since then, the incentives to le as an S-corporation or a C-corporation have remained similar (Smith et al., 2017).

7
other inequality measures (Shorrocks, 1982; Lerman and Yitzhaki, 1985; Milanovic, 2017). Like these decompositions, inequality in total income is decomposed into three elements: The share of each factor in total income, the inequality in the distribution of income from each of the factors, and a term capturing the association between the incomes from dierent factors and total income.

The income share of top quantile _i_ can be written as _Si_ =<sup>_<u>Y</u>_</sup> _Y_<sup>_<u>i</u>_,where</sup><sup>_Y_istotalincome</sup> in the data and _Yi_ denotes total income of tax units with income greater or equal to _yi_ , the threshold income (e.g. the 99th percentile in the case of the top 1% income share). For any individual _j_ , total income _yj_ is derived from _M_ components, such that _yj_ =<sup>�</sup><sup>_M_</sup> _m_ =1<sup>_xj,m_.</sup> Dening _X_<sup>�</sup> _i,m_ =<sup>�</sup><sup>_N_</sup> _j_ =1<sup>_xj,m ×_1</sup><sup>_{yj≥yi}_,thetopincomesharecanbewrittenas</sup> , where _Xm_ =<sup>�</sup><sup>_N_</sup> _j_ =1<sup>_xj,m_isthetotalincomefromfactor</sup><sup>_m_,and</sup><sup>_Xi,m_=�</sup><sup>_N_</sup> _j_ =1<sup>_xj,m×_</sup> 1 _{xj,m ≥ xi,m}_ . The rst term in the nal expression is the share of total income derived from income source _m_ . _S_ =<sup>_Xi,m_theshareoftotalincomefromfactor</sup><sup>_m_that</sup> _i,m Xm_<sup>denotes</sup> accrues to the top quantile _i_ of recipients of income from factor _m_ , e.g. the share of capital income going to the top 1% of capitalists. It thus captures inequality in the marginal distribution. Atkinson (2007) refers to the nal term as the alignment coecient, which captures the extent to which the rankings under income from factor _m_ and total income coincide. It is dened as _A_ = _S_ � _i,m_<sup>where</sup> _S_<sup>�</sup> = _X_ � _i,m_<sup>theshareoftotalincomefrom</sup> _i,m Si,m_<sup>,</sup> _i,m Xm_<sup>is</sup> factor _m_ received by the top quantile _i_ of total income recipients.<sup>11</sup> The alignment coecient lies between 0 and 1 since shares are non-negative and _Si,m ≥ S_<sup>�</sup> _i,m_ .<sup>12</sup> If top income recipients

> 11Following Shorrocks (1982), _S_ � _i,m_ may be called the pseudo share. It is dierent from _Si,m_ because observations are ranked according to total, not factor, income.

> 12After canceling out the incomes of tax units who are in the top quantile _i_ of both income from factor _m_ and total income, we can write

8
(according to total income) receive no labor income, _X_<sup>�</sup> _i,l_ = 0 and _Ai,l_ = 0 . On the other hand, if everybody in the top quantile _i_ of the total income distribution is also found in the top quantile _i_ of the distribution of labor income, then _S_<sup>�</sup> _i,l_ = _Si,l_ and _Ai,l_ = 1 .

# 2.1 Results

Figure 2 shows the results of the factor income decomposition for the top 1%. Panel A shows _S_ 1 , the top 1% share of total income, which roughly doubled over this period (as was already shown in Figure 1). While we estimate the top share at a somewhat lower level than Piketty and Saez (2007a), the two series track each other very closely, as discussed in Appendix A.4. The remaining three panels of Figure 2 refer to the dierent components of the decomposition: The labor share (B); the share in total labor (capital) income of the top 1% of labor (capital) income recipients (C); and the alignment coecients (D). The results for the top 5% and top 0.5% are shown in the Appendix (Figures A.1 and A.2). In almost all our results, the 1980s mark a turning point, which coincides with the sweeping changes to the US federal income tax introduced by the Tax Reform Act of 1986 (TRA86).<sup>13</sup> Tax reforms may lead to re-timing or accounting responses (Slemrod, 1992), which can explain short-run uctuations but they are unlikely to account for the long-term trends that we analyze in this paper (Alvaredo et al., 2013). Furthermore, it is reassuring that our results are robust to alternative income denitions (see Appendix A.5), such as the inclusion of capital gains, an important channel of tax avoidance (Piketty et al., 2014).

The labor share in total income (panel B) uctuated between 87% and 90%, with no clear trend. At around 75%, Piketty et al. (2018) nd a much lower labor share over this period, which is also approximately constant. Piketty et al. estimate a higher capital share because they include capital incomes that are not reported on personal tax returns, such

> This is weakly positive because in the rst term all _xj,m_ are at least as big as the cut-o level _xi,m_ , while in the second term they are strictly less than _xi,m_ .

> 13As we already discussed, TRA86 brought the personal top marginal income tax rate below the corporate rate, thus providing incentives to move income from the corporate to the personal income tax base (Auerbach and Slemrod, 1997). TRA86 also raised tax rates on capital gains by including all realized capital gains in taxable income (Slemrod, 1996).

9
<!-- Start of picture text --> 16 A. Share of total income (%) 91 B. Labor share of entire population (%)<br>14 90<br>12<br>89<br>10<br>88<br>8<br>87<br>6 L<br>1965 1970 1975 1980 1985 1990 1995 2000 2005 1965 1970 1975 1980 1985 1990 1995 2000 2005<br>14 C. Share of factor income (%) 60 96 D. Alignment coefficient (%) 85<br>a ge<br>12 Li 50 94g A. me f [80<br>oat” oS ° @ gee? Sei<br>10 P 40 92 Sag " 6<br>8 o-e-aee CFB, di :x;<br>See gy 30 90 wy $ 70<br>Beige q Q<br>6 | Bad<br>K-20 ssql Q K-65<br>1965 1970 1975 1980 1985 1990 1995 2000 2005 1965 1970 1975 1980 1985 1990 1995 2000 2005<br>——®—- Labor —©— Capital<br>Separate axes: labor (L) on left, capital (K) on right. Excluding capital gains and negative observations.<br>Capital includes S-corporation profits. Self-employment income split between labor (2/3) and capital (1/3).<br><!-- End of picture text -->
follow a very similar trend to the taxable capital income shares reported by Saez and Zucman (2016), who allocate all self-employment income to capital (also see Appendix A.5).<sup>15</sup> The alignment coecient for labor income declined slightly from 91% to 88% in the late-1970s, before rising to 95% in the 2000s (panel D, left axis).<sup>16</sup> For capital income, the alignment coecient is lower and follows a U-shaped pattern; it declined from almost 80% in the 1960s to 65% in the 1980s, before rising to 83% by the end of the period.<sup>17</sup> A value of 83% for the capital alignment coecient means that 83% of total capital income of the top 1% of capitalists goes to tax units who are also in the top 1% of total income. Given that the top 1% capitalists receive 55% of capital income (panel C), this implies that 46% of all capital income goes to tax units who are in the richest 1% (this is _S_<sup>�</sup> 1 _,c_ above). The same statistic was 20% of all capital income in the mid-1980s. These estimates suggest that over the last 20 years capitalists are increasingly also at the top of the income distribution. Labor income has an even stronger association with total income: Around 95% of labor income of the top 1% of earners is received by tax units that are also in the richest 1%, compared with 83% for capital.<sup>18</sup> We will examine this asymmetry in the association in more detail below.

# 3 Rank-based measure of association

The association measure that we have used so far, the alignment coecient, is not independent of monotone transformations in the marginal distributions. For example, a doubling of all labor incomes would tend to change the ranking in the distribution of total income, and thus aect the alignment coecient through _S_<sup>�</sup> _i,m_ . Only a rank-based measure of associa-

> 15In their main results for wealth, Saez and Zucman include other assets that do not generate taxable incomes, e.g. primary housing. The top 1% share of wealth has increased less than the top 1% share of taxable capital income.

> 16When all of self-employment income is included with capital, the labor alignment coecient is much lower and increasing during most of the period (Figure A.6).

> 17The U-shape is also visible for alternative income denitions (Figure A.6). It is less pronounced when capital gains are included, although this is largely due to a spike in capital gains realizations before TRA86.

> 18The labor alignment coecient tends to exceed the capital coecient also for other income denitions (Figure A.6). The exceptions are a few years when capital gains are included, and early in the period when all self-employment income is included in capital.

11
tion is invariant to all monotone transformations in the marginals (Dardanoni and Lambert, 2001). In the remainder of the paper, we use an analytical framework based on the copula function, which oers a clean separation of the joint distribution of labor and capital into the marginal distributions and a rank-based measure of the association.<sup>19</sup> Our rank-based association measure is also more general because it considers the entire distribution, while the alignment coecient for say the top 1% is determined only by whether observations cross the 99th percentile.

Total income is a two-dimensional vector _X_ = ( _L, K_ ) , where _L_ denotes labor income and _K_ refers to capital income. By Sklar's Theorem (Nelsen, 2006), there exists a copula function _CX_ such that _HX_ ( _l, k_ ) , the joint distribution function of _X_ , can be written as where _F_ ( _l_ ) and _G_ ( _k_ ) are the marginal distribution functions of labor and capital income. The density of the joint distribution is obtained by dierentiating with respect to _l_ and _k_ where _CFG{F_ ( _l_ ) _, G_ ( _k_ ) _}_ is the copula density. The joint density can thus be expressed as the product of the marginals, and the copula density, which is a rank-based measure of the association. The association matrix between labor and capital, shown in Table A.1 for 2006, is a discrete approximation to the copula density (Bonhomme and Robin, 2009).<sup>20</sup> The bins are dened in terms of ranks, splitting the distributions of labor and capital income into eight quantile groups: The bottom 50% ( _≤_ P50 ), the next 10% (P50-P60), the next 20%

> 19Copula functions have been widely used in actuarial science to describe multidimensional risks. In economics, they have been used to study the joint distribution of income and wealth (Kennickell, 2009; Jäntti et al., 2015), the horizontal equity of the tax system (Dardanoni and Lambert, 2001), income mobility by considering the dependence over time (Bonhomme and Robin, 2009), and multi-dimensional inequality and poverty (Atkinson, 2011; Ferreira and Lugo, 2013; Decancq, 2014).

> 20Parametric copulas tend to impose symmetry. Since the association between capital and labor is asymmetric, we will adopt a non-parametric approach and use association matrices directly.

12
(P60-P80), the next 10% (P80-P90), the next 5% (P90-P95), the next 4% (P95-P99), the next 0.5% (P99-P99.5) and the top 0.5% (>P99.5).<sup>21</sup> The association matrix is equivalent to transition matrices used to study economic mobility. Following Atkinson (1981), who examines transition matrices, we can test whether the degree of association between labor and capital has increased. Consider the following two association matrices _A_ and _A_<sup>_∗_</sup> where _i_ and _j_ are particular quantile groups (of labor and capital), _pi,j_ is the frequency in the association matrix, and _γ >_ 0 . _A_<sup>_∗_</sup> is obtained from _A_ by a correlation-increasing (or diagonalizing) switch, which adds _γ_ to the diagonal elements and subtracts it from the o-diagonal elements. This switch increases the weight on the diagonal, such that _A_<sup>_∗_</sup> exhibits a stronger association between labor and capital, but it leaves the marginal distributions unchanged. Let _α_ and _α_<sup>_∗_</sup> be the survival association matrices of _A_ and _A_<sup>_∗_</sup> , which are obtained by cumulating the association matrices from above. Table A.2 shows the survival association matrix observed for 2006. These are the survival copulas for a discrete distribution (Dardanoni and Lambert, 2001).<sup>22</sup> Taking the dierence between _α_<sup>_∗_</sup> and _α_ yields the following result (see Appendix A.1 for intermediate steps)

> 21Table A.1 shows that 0.14% of observations were both in the top 0.5% of the labor and the top 0.5% of the capital income distribution. This is greater than 0.0025%, which would be the frequency if the two variables were independent, but less than 0.5%, the frequency with perfect association.

> 22Given our interest in the top tail of the distribution, it makes sense to consider � the survival copula. Similar to the expression above, the joint survival function can be written as _HX_ ( _l, k_ ) = _C_<sup>�</sup> _X {F_<sup>�</sup> ( _l_ ) _, G_<sup>�</sup> ( _k_ ) _}_ , where _C_<sup>�</sup> _X_ is the survival copula, and _F_<sup>�</sup> ( _l_ ) = 1 _− F_ ( _l_ ) and _G_<sup>�</sup> ( _k_ ) = 1 _− G_ ( _k_ ) are the survival distributions (or complementary cumulative distribution functions) (Nelsen, 2006).

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Therefore, if the dierence between the survival association matrices in years _t_ +1 and _t_ is everywhere positive, labor and capital incomes have become more closely associated between those years, thus moving away from a class model, where one class is at the top of the labor distribution and the other at the top of the capital distribution.<sup>23</sup> This is a test of rst-order dominance, which will be sucient for this paper. To go beyond rst-order dominance, one would need to place additional restrictions on the social welfare functions, eectively giving a dierent weight to the association in dierent parts of the distribution (Atkinson, 1981; Aaberge, 2009; Aaberge et al., 2017).

# 3.1 Results

We begin by examining the long-run evolution of some statistics from the association matrix before testing for rst-order dominance for selected years. Figure 3 shows several conditional probabilities that are obtained from the survival association matrix.<sup>24</sup> The gure exhibits a distinct U-shape over this 40-year period, with a decline during the initial 20 years and a rise in the 20 years leading up to 2006. For instance, in the 1960s tax units that were among the top 1% earners were among the top quintile of capitalists with an 80% probability. This number fell to less than 60% by the early-1980s, before again approaching 80% in the 2000s.

> 23Decancq (2014) derives dominance criteria for continuous copula functions that are equivalent to the discrete case considered here.

> 24For instance, Table A.2 shows that in 2006 0.8% of tax units were both in the top 20% of capital incomes and the top 1% of labor incomes. In other words, of those tax units that were among the top 1% earners, 80% were also among the top quintile of capitalists, which is shown in Figure 3.

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<!-- Start of picture text --> 807] \ av a<br>\ Nt. ne yt 80<br>70 \ a ao “<br>N rn " p<br>60 \\ =/ \ N-------—-) " 704—i)=<br>z<br>S :<br>£& _— x<br>50 ee \ \ 60 &<br>&° 40 _*SA HV en ee 50 28<br>. } FQ<br>30<br>a \ So eaiestentientia ne 40<br>/ a<br>20 \ /<br>\ 77 30<br>1965 1970 1975 1980 1985 1990 1995 2000 2005<br>Year<br>—+— P(top 5% labor | top 1% capital) ——+— P(top 20% labor | top 1% capital)<br>——+——_ P(top 5% capital | top 1% labor) —+——_ P(top 20% capital | top 1% labor)<br>—~— P(top 1% labor | top 1% capital) —-—-—-—- Top marginal tax rate<br>Excluding capital gains and negative observations. Capital includes S-corporation profits. Self-employment income<br>split between labor (2/3) and capital (1/3). Source of top marginal tax rates: Piketty et al. (2014).<br><!-- End of picture text -->
suggest a high degree of association: If the top 1% of earners had randomly been assigned capital incomes, 20% of them would be among the richest quintile of capitalists, compared to the observed 80%.

We test for rst-order dominance between 1966, 1985 and 2006, capturing the two 20year periods that we have just described.<sup>25</sup> The dierences between the survival association matrices is shown in Table 2, where we have highlighted the negative cells. We do not nd dominance between 1985 and 1966, since there are both positive and negative dierences between the two survival association matrices (panel a). At the top of the distribution, 1966 appears to dominate 1985, suggesting a fall in the association, consistent with what we observed above. The period between 1985 and 2006 presents a stark contrast of increasing association (panel b); the survival association matrix in 2006 lies everywhere above the 1985 matrix. In other words, over the same period as the top 1% income share doubled, tax units increasingly occupied similar positions in terms of earnings and capital income. Viewed over the entire period from 1966 to 2006 (panel c), the results also point towards increasing association, although the dierences are smaller and there are some small negative values (all less than 0.05 in absolute value). These dominance results are robust to using alternative income denitions (see Appendix A.5).

> 25We have chosen 1985 instead of 1986, because 1986 may be aected by the anticipation of TRA86, which was announced in 1986 and became eective in 1987. However, the results are robust to choosing 1986.

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Table 2: Dierence in survival association matrices (in percentage points)

|||(a) 1|985 compar|ed with 196<br>Capital|6|||
|---|---|---|---|---|---|---|---|
|Labor|Top 50%|Top 40%|Top 20%|Top 10%|Top 5%|Top 1%|Top 0.5%|
|Top 50%|0.68|-1.59|-1.05|-0.95|-1.08|-0.23|-0.09|
|Top 40%|1.11|-1.15|-0.96|-0.87|-1.06|-0.25|-0.09|
|Top 20%|0.67|-0.78|-0.94|-0.75|-0.77|-0.27|-0.10|
|Top 10%|0.33|-0.42|-0.82|-0.71|-0.65|-0.26|-0.10|
|Top 5%|0.10|-0.30|-0.66|-0.59|-0.53|-0.25|-0.10|
|Top 1%|-0.01|-0.06|-0.25|-0.26|-0.24|-0.16|-0.08|
|Top 0.5%|0.00|-0.02|-0.11|-0.13|-0.13|-0.08|-0.06|

(b) 2006 compared with 1985

|Labor|Top 50%|Top 40%|Top 20%|Capital<br>Top 10%|Top 5%|Top 1%|Top 0.5%|
|---|---|---|---|---|---|---|---|
|Top 50%|1.78|2.05|2.53|2.08|1.43|0.35|0.18|
|Top 40%|1.87|2.11|2.55|2.08|1.41|0.37|0.18|
|Top 20%|1.35|1.63|2.03|1.57|1.07|0.36|0.18|
|Top 10%|0.58|0.92|1.41|1.17|0.80|0.31|0.16|
|Top 5%|0.22|0.50|0.96|0.81|0.59|0.26|0.15|
|Top 1%|0.03|0.07|0.27|0.26|0.21|0.12|0.09|
|Top 0.5%|0.01|0.03|0.12|0.13|0.11|0.05|0.05|

Description: Share of tax units who were in the top 0.5% of both labor and capital increased by 0.05pp between 1985 and 2006.

|||(c) 2<br>|006 compar<br>|ed with 196<br>Capital<br>|6<br>|||
|---|---|---|---|---|---|---|---|
|Labor|Top 50%|Top 40%|Top 20%|Top 10%|Top 5%|Top 1%|Top 0.5%|
|Top 50%|2.46|0.47|1.49|1.13|0.35|0.13|0.09|
|Top 40%|2.98|0.96|1.59|1.20|0.35|0.12|0.09|
|Top 20%|2.02|0.86|1.09|0.82|0.30|0.09|0.08|
|Top 10%|0.91|0.50|0.60|0.46|0.14|0.05|0.06|
|Top 5%|0.32|0.20|0.30|0.22|0.06|0.01|0.05|
|Top 1%|0.02|0.01|0.02|0.00|-0.03|-0.05|0.01|
|Top 0.5%|0.01|0.01|0.01|0.00|-0.02|-0.03|-0.01|

Note: Bold cells: Value of survival association matrix is lower in nal year (e.g. 2006) than initial year (e.g. 1985).

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# 4 Conclusion

This paper has studied the association between capital and labor incomes at the top of the distribution using tax return data. This helps to understand the driving forces behind the rise in the top 1% income share, that has been documented before (Piketty and Saez, 2003). We nd that capital and labor incomes have become more closely associated between 1985 and 2006, such that top capitalists and top earners are increasingly the same people. This rising association has contributed to the well-known increase in the top 1% income share, exacerbating the eects of rising inequality within capital incomes and earnings. In contrast, the 20 years leading up to 1985 saw a tendency towards a declining association, thus resulting in a U-shaped pattern. The association is asymmetric in some parts of the distribution, as a top earner is almost guaranteed to also be among the richest fth of capitalists, while a sizable share of top capitalists fall into the bottom four-fths of earnings. The association is more symmetric for richer quantiles, such as the top 5%. Our conclusions are robust to alternative treatments of negative incomes and capital gains, and how prots from self-employment and closely-held businesses are split between capital and labor.

The reversal from declining to increasing association coincided with a strong fall in the top marginal income tax rate in the US (dotted line in Figure 3). The top marginal rate declined from 91% in the early 1960s to 28% in 1986, and remained below 40% for the rest of the period. Lower taxes at the top raise the reward to bargaining more aggressively for higher pay, and therefore may explain the rapid rise in (gross) salaries at the top, which account for a large share of the increase in top income shares (Bakija et al., 2012; Alvaredo et al., 2013; Piketty et al., 2014).<sup>26</sup> Lower taxes may also account for the increasing association: When top marginal tax rates are low, tax units can save a greater share of their wages, thus accumulating more capital income over time.<sup>27</sup> This explanation assumes high saving

> 26The decline in top marginal tax rates is not the only possible explanation for these patterns (also see Alvaredo et al., 2013). Other explanations may include a superstar theory together with a globalized economy (Atkinson, 2008), the spread of performance-based pay (Lemieux et al., 2009) and the role of the nancial industry (Kaplan and Rauh, 2010; Philippon and Reshef, 2012).

> 27Saez and Zucman (2016) discuss the eect of increasing top incomes and high savings rates for wealth

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rates and only limited mobility at the top of the wage distribution, which is conrmed by the empirical evidence.<sup>28</sup> Our nding of an asymmetric association in some parts of the distribution also ts a model in which high earners accumulate capital incomes out of labor income.

Our paper shed light on the evolution of the association between capital and labor incomes during the last 40 years, when the top marginal tax rate declined strongly. It it unclear how the association will evolve in the future as there are two opposing forces. On the one hand, the high concentration of labor incomes coupled with low top marginal tax rates and high saving rates at the top, is unlikely to go away. On the other hand, we may see a reemergence of rentiers, as the high earners retire, which would reduce the association.<sup>29</sup>

> inequality, which is closely related to the distribution of capital incomes. Kaymak and Poschke (2016) present a formal model where a decline in income tax progressivity leads to an increase in wealth inequality.

> 28Saez and Zucman (2016) nd high and increasing saving rates for the top 1%. Kopczuk et al. (2010) show that around two-thirds of the top 1% of earners are still there after three years, with little change since the late 1970s.

> 29The average age at the top of the income distribution has increased since the 2000s, after having fallen for 20 years (Piketty et al., 2018).

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