Abstract
From 1992 to 2014, Brazil experienced a decline in income inequality along with a significant increase in schooling level, though the latter was more pronounced among women. Brazil also experienced a decline in returns to education, whereas an opposite trend was observed in several developed countries and China. In this paper, we evaluate the effects of educational, marital, and labor market factors on the income inequality of married couples. We also analyze how changes in educational assortative mating affect their income. Our findings suggest that changes in educational marital sorting parameters had a small but statistically significant effect on household income inequality. We show that growth in female labor force participation and a decrease in the gender wage gap explain part of the decline of the Gini coefficient. Educational factors also explain a part of that decline. Nevertheless, the main driver of the reduction in income inequality among couples appears to be the overall decrease in the returns to education.
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Notes
In this work, household income inequality considers income of (married or cohabiting) couples.
There was an increase in the proportion of educated men and women in some developed countries as well; however, in Brazil, this phenomenon was more pronounced.
Article “Gini back in the bottle,” published in 10/13/2012 by The Economist.
A married couple with similar traits, including educational level, religion, ethnicity, and socioeconomic status, can be said to be in a homogamous marriage.
The covered period varies by country and extends until 2013.
Data for China are discussed in Feng and Tang (2019).
Browning et al. (2014).
Barbosa (2019) presents data showing that the hours spent per week by women in household work decreased (31 to 24 hours) between 2001 and 2015 in Brazil, whereas among men it rose from 5 to 6 hours a week.
The levels of schooling are described in the Data section.
Household income inequality here refers to couples’ income inequality, and our sample is composed only of couples (married or cohabiting). The decompositions developed in this study disregard the effects of the general equilibrium.
We follow the notation used in DiNardo et al. (1996).
From now on, we omit the integration domain.
Female labor force participation, female wage gap, educational composition and returns to education.
Returns to education are evaluated in terms of couples’ education level unlike what we usually find in the literature, that is, by analyzing returns to education at the individual level.
DiNardo et al. (1996) began its sequential decomposition with institutional and labor markets factors, which are the main components they wanted to analyze.
The microdata and questionnaires are available at www.ibge.gov.br.
Between 1992 and 1995, only individual sampling weights were available. “Appendix D” provides a more complete description of the dataset.
We regard individuals with less than 5 years of schooling as uneducated and those with more than 12 years of schooling as educated.
The purpose of this regression is just to illustrate that the returns to education decreased from 1992 to 2014, but we do not use these coefficients in this work. We calculate the return to education according to equation 9. Additionally, this is what we hold fixed when we estimate the counterfactual.
See Fig. 3.
See “Appendix C”.
In 2014, the actual Gini coefficient was 0.499 and the counterfactual was 0.449.
We run 500 iterations of a resampling bootstrap to determine whether the differences between the actual Gini coefficient and the counterfactual one are significantly different from zero.
The methodology is presented in Sect. 4.2.1.
The respective results are labeled with FLFP*RM (female labor force participation*random matching, where the former refers to FLFP from base year 1992 (or 2014)).
We run 500 iterations of a resampling bootstrap to determine whether the differences between the actual Gini coefficient and the counterfactual one are significantly different from zero.
The respective results are labeled with WG*RM (wage gap*random matching, where the former refers to the wage gap from base year 1992).
In our sample, almost 70% of men and 71% of women had less than 5 years of schooling in 1992.
The purpose of this observation is to illustrate this trend rather than to construct the counterfactual.
In the US case, it appears that marital sorting has no effect on income inequality.
To draw means to remove the couple from the sample.
This algorithm was used by Greenwood et al. (2014).
1) The codes presented in this section originate from the 2014 Brazilian National Household Sample Survey (PNAD). In the other years, we match the variables used in 2014 with their respective codes or similar variables. 2) The household sampling weights’ code is v4732. 3) The microdata and questionnaires are available at https://www.ibge.gov.br/en/npstatistics/social/population/20620summaryofindicatorspnad2.html?=t=microdados. For data before 1999, the microdata are available at https://loja.ibge.gov.br/pesquisanacionalporamostradedomicilios1987a1999microdados.html. Additional information and questionnaires are available at http://www.econ.pucrio.br/datazoom/english/pnad.html.
The methodology for \(s_{ij}\) is described in Sect. 4.1.
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This study is based on a chapter of Lorena Hakak’s doctoral dissertation at Sao Paulo School of Economics. The author received financial support from the Conselho Nacional de Desenvolvimento Cientifico e Tecnologico (CNPqBrazil) to pursue her doctoral studies. The authors declare that they have no conflicts of interest.
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We are indebted to Luis Araujo, Emerson Marcal, Daniel Monte, David Rubin, Adhemar Villani, and seminar participants at EESPFGV, SBE2016, UFABC, FEA/USP, the 2017 Annual Congress of the European Economic Association, and the 2017 Latin American Meeting of the Econometric Society. Special thanks go to Lasse Eika, who generously shared his Stata Codes with us. We thank NEREUS (The University of Sao Paulo Regional and Urban Economics Lab  http://www.usp.br/nereus/) for computational support. Financial support of CNPqBrazil is also acknowledged. Previous title: Household Income Inequality and Education in the Marriage Market in Brazil: an empirical study.
Appendices
A Appendix
A.1 The steps to obtain the reweighting function \(P_{x}^{\prime }(t_{e_{ij}}=b,t_{s}=t)\):
1) We draw one man from the husbands’ marginal distribution of education and one woman from the wives’ marginal distribution of education in \(t=1992\). With probability
we obtain a man and a woman with educational levels i, j.
2) Considering the product of marginal distributions and the marital sorting parameter \(s_{ij}(t)\), we decide whether the couple gets married. Hence, to construct the counterfactual we need to draw a man and woman from the marginal educational distributions of men and women and estimate \(s_{ij}(t)\) . The probability of them getting married is
3) If they get married, we draw^{Footnote 36} them from the marginal distributions of education and measure the probabilities in equation (19) again without that couple. We need to calculate the marginal distributions in every iteration. Then, we repeat the process until all couples have been formed.
B Appendix
B.1 standardized contingency table
To analyze whether positive assortative mating in Brazil increased from 1992 to 2014, we need to compare the joint distributions from these years. To this end, we standardize both joint distribution tables, considering the same marginal distributions. We use the Sinkhorn–Knopp algorithm that allows us to iterate over columns and rows, preserving the dependent relationship between the joint distribution and the marginal distributions.^{Footnote 37}
B.2 Sinkhorn–Knopp Algorithm
We perform the following steps to execute the algorithm:

1
Divide the husbands’ marginal educational distribution as of 1992 (or 2014) by the marginal distribution as of the year used to standardize the table. We obtain a weight for each education level.

2
Divide the joint distributions in each row by these weights.

3
Divide the wives’ marginal educational distribution as of 1992 (or 2014) by the marginal distribution as of the year used to standardize the table. We obtain a weight for each education level.

4
Divide the joint distributions in each column by these weights.

5
Repeat steps 14 until the desired marginal distributions are obtained.
Tables 9 and 10 show the standardized table for 1992, obtained using the marginal distributions as of 2014 and the actual joint distribution as of 2014. In Table 9, we estimate the joint distribution as of 1992, using the marginal educational distributions as of 2014, and hold the dependence structure pattern of the joint distribution constant. We assess this analysis by calculating the odds ratio of the joint distribution as of 1992 (using the marginal educational distributions as of 2014) and the joint distribution as of 1992 in Table 15, and observe that it remains unchanged in both cases (Sinkhorn and Knopp (1967), Tan et al. (2004)).
C Appendix
In this section, we describe in more detail the variables used in the empirical exercises. We use household sampling weights to construct the variables.^{Footnote 38}
C.1 family variables
C.1.1 Family ID
In Brazil, it is possible to have more than one family in the same household. To avoid counting two families as one, we create an identifier for each family. In the sample, we keep families constituted by couples of one man and one woman. Other family types (families with only one head of household or samesex couples) are excluded from the sample. To construct the family ID, we use the following variables from the Brazilian National Household Sample Survey (PNAD): control number (v0102), serial number (v0103), the number associated with the household member (v0301), the number associated with the family (v0403), and status within the family (v0402 equal to 1 and 2).
C.1.2 Number of children
We add up the number of children in every family using the family ID, gender (v0302), age (v8005) between 0 and 17, and status within the family (v0402 equal to 3).
C.1.3 Age
We analyze spouses aged between 26 and 60. Single individuals, when included, are also between 26 and 60 years of age.
C.1.4 Levels of education
We construct a “years of schooling” variable to represent 0 to 17 years of schooling and aggregate values of this variable into five groups. In other words, individuals are grouped by the number of years of schooling into five mutually exclusive groups. The first group consists of illiterate (less than one year of education) individuals; the second contains those with elementary school education (4–5 years of schooling); the third comprises those with middle school (8–9 years of schooling) education; and the fourth group contains those who dropped out of and those who graduated from high school (10–12 years of schooling). The last group consists of individuals who had at least some postsecondary education regardless of whether they earned an undergraduate degree or M.A. or Ph.D. degrees (more than 12 years of schooling).
We use the following codes to form this variable: the code for school type and educational stage (v6003), grade attended (v0605), elementary school duration (v6030), highest educational stage attended (v6007), last grade completed of the educational stage attended previously (v0610), elementary school duration (v6070), and being able to read and write (v0601).
C.1.5 Couple’s income
This variable is the sum of individual monthly incomes of the husband and wife. We use the variable monthly income from all sources for individuals aged 10 or above (v4720).
C.2 Labor counterfactual variables
C.2.1 Female labor force participation
We develop the female labor force participation counterfactual using the reweighting function shown in equations (7) and (8) and described in Sect. 4.2.1. The reweighting function is the ratio of the proportion of working women in the base year and the actual year of interest for each of 25 combinations of male and female educational levels. Dummy variable w is 0 if variable v4704 is two; alternatively, w is 1 if v4704 is one.
C.2.2 Female wage gap
We develop the female wage gap counterfactual reweighting functions shown in equations (9) and (10) and following equations (7) and (8). They are described in Sects. 4.2.2 and 4.2.1, respectively.
In the first step, we calculate \(\gamma \), which is the mean for each of 25 combinations of male and female educational levels, of men’s and women’s incomes (v4720). Next, for every woman, we calculate \(\vartheta \), which is the product of \(\gamma \) and the income of the woman’s husband.
We then create a dummy variable \(\omega \), which takes the value of 1 if the wife’s income is greater than \(\vartheta \) and is zero otherwise.
The reweighting function is the ratio of the proportion of women with \(\omega \) equal to one (or equal to zero) and the total number of women, in the base year and the actual year of interest for each of 25 combinations of male and female educational levels.
C.3 Marriage counterfactual variables
C.3.1 Marital sorting parameter
To construct the reweighting function, we follow equation (15) and steps (1) to (3) described in Sect. 4.2.3, setting \(t_{ij}=t\) and \(t_{s}=b\). The methodology for \(s_{ij}\) is described in Sect. 4.1.
C.3.2 Random matching
To construct the reweighting function, we follow equation (15), setting \(t_{ij}\)=t, and steps (1) to (3) described in Sect. 4.2.3. In this case, we set \(s_{ij}=1\).^{Footnote 39}
C.4 Educational counterfactual variables
C.4.1 Educational composition
To construct the reweighting function, we follow equation (15) and steps (1) to (3) described in Sect. 4.2.3.
C.4.2 Returns to education
We use the income distribution in year b. The reweighting function being calculated is the ratio of the joint distribution of the couples’ education in year t and year b for all education levels. We then calculate the income distribution using this reweighting function to evaluate the impact of returns to education on the distribution.
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Firpo, S., Hakak, L. Changes in the women’s labor market and education and their impacts on marriage and inequality: evidence from Brazil. Empir Econ 62, 1909–1950 (2022). https://doi.org/10.1007/s00181021020766
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DOI: https://doi.org/10.1007/s00181021020766
Keywords
 Household income inequality
 Assortative mating
 Marriage market
 Returns to schooling
 Labor market
 Family economics
JEL codes
 J12
 J16
 D31
 I24
 I21