Inequality Interactions: The Dynamics of Multidimensional Inequalities

This article offers a multidimensional, interdisciplinary and dynamic framework for understanding socio-economic inequalities. It uses the tools of economic inequality measurement to demonstrate the link between interpersonal and categorical inequalities and to show the effect of progressive redistribution on both. It then presents two new concepts for analysing interactions between varieties of inequality: inequality re-ordering is deﬁned as a reduction in categorical or group inequalities that leaves interpersonal inequality unchanged, and inequality diversion is deﬁned as a reduction in one form of inequality that increases another form of inequality. The argument that inequality diversion arises across economic and social dimensions is illuminated through two case studies: the transition to increasing meritocracy, and the relationship between professional female labour market participation and domestic service. Exploring these relationships is essential to a joined-up approach to inequality reduction, enabling us to understand trade-offs and complementarities in tackling inequalities, and to identify policies required to reduce them.


INTRODUCTION
Inequality has many dimensions, affects our lives in multiple domains, and has long been understood as fundamental to questions of economic development.Yet the outcomes across different dimensions can be highly variable.Most countries have experienced rising income inequality since the 1980s, meaning that majority populations have benefited less than proportionately from economic growth. 1 Over the same period a variety of inequalities The author would like to thank Ingrid Bleynat, Alice Evans, Aaron Reeves and Mike Savage for helpful comments and discussion, and the anonymous referees for their feedback.He gratefully acknowledges support from the Leverhulme Foundation through a Leverhulme Research Fellowship (grant number RF-2016-437).

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between categories such as broad social classes, genders and sexualities have declined in many countries. 2While inequality has been a core topic of development studies for both economists and sociologists, as McCall (2017: 874) notes, the expansion of research in specific areas 'has made it difficult to conduct conversations across these areas despite the reality -no less true today than in the past -of deeply intertwined dynamics'.The purpose of this article is to provide a framework that is interdisciplinary, multidimensional and dynamic to help clarify the relationships between different varieties of inequality.To this end I explore what I call inequality interactions, where I define an inequality interaction as a causal or analytical relationship between a change in one type of inequality, and the level of or change in another type of inequality.
Understandings of the multidimensional nature of well-being and thus inequality are deeply embedded in the development literature, drawing above all on Amartya Sen's capabilities framework (for example, Sen, 1992).Yet the underlying intuition regarding the more complex interactions between different types of inequality can be found in Crenshaw's (1989) foundational discussion of intersectionality.Crenshaw described the 1965 Moynihan report on the disadvantages suffered by African Americans, which claimed that the Black community's ills were exacerbated by the predominance of female-headed households.This, argued the report, overturned what it saw as the natural patriarchal order in which Black men should be at the head of their families.Moynihan 'depicted a deteriorating Black family, foretold the destruction of the Black male householder and lamented the creation of the Black matriarch' (Crenshaw, 1989: 163).Crenshaw pointed out the patriarchal nature of Moynihan's proposals: he hoped to reduce race inequality by strengthening the position of men in Black families, thereby worsening gender inequality. 3In our terms, this is a case of inequality interactions: Moynihan was proposing a change in one type of inequality which would imply a knock-on effect on another type of inequality.The intersectionality approach has since been applied widely in the literature on low-and middle-income countries.For example, the discrimination and human rights violations suffered by Indigenous women have been examined by Banda and Chinkin (2004), who consider the role of human rights law across countries, by Boesten (2010) in the case of Peru, and by Sylvain (2011) in the case of San women in Namibia.Numerous papers consider the intersections of caste and gender in India (for example, Anandhi, 2013;Rao, 2015).
2. On the international rise in social fluidity (intergenerational inequality between large social classes) see Breen (2005); on gender inequality globally see Stotsky et al. (2016); Sundström et al. (2017); Weichselbaumer and Winter-Ebmer (2005); on the changing categorical diversity of the US elite see Zweigenhaft (2001).For a wider discussion see Savage (2021).3.More recently and on the same basis, Crenshaw (2014) criticized President Obama's fiveyear, US$ 200 million programme, 'My Brother's Keeper', for targeting boys and young men from ethnic minorities while excluding girls and young women, despite the fact that on some indicators these girls do worse than their male counterparts.
In this article, I propose two new concepts representing different types of inequality interaction.Inequality diversion is defined as a reduction in one type of inequality that implies a rise in another type of inequality.Moynihan's proposal above would be one such case.Inequality re-ordering is defined as a reduction in inequality between groups or categories, holding constant the interpersonal distribution of income.My goal is to show that such inequality interactions are widespread in countries at all income levels, and that these concepts can help to clarify debates on inequality.
After defining terms and presenting the new concepts in the following section, I then consider two categories of inequality interactions.I first analyse inequality interactions across populations within a single quantifiable dimension of well-being, such as income or wealth.I use the tools of economic inequality measurement, including decomposable indices, to show how progressive redistribution and inequality re-ordering link interpersonal inequality with categorical inequality.The relationships explored in this section are not causal in an empirical sense, but analytical, meaning they follow from the logic of the definitions.I then consider a different set of relationships that are empirical and causal, as opposed to analytical.These are inequality interactions across single dimensions, including the relationship between economic inequality and social inequalities that are not quantifiable.I discuss two such cases: the rise of meritocracy, and the employment of domestic workers.Because these are based on contingent sociological mechanisms, different empirical conditions may help to reduce the degree of inequality diversion.

INEQUALITY INTERACTIONS AMONG INDIVIDUALS, GROUPS AND CATEGORIES
Any study of inequality must define, implicitly or explicitly, two parameters: the dimension of well-being, and the population among which it is distributed.To lay the groundwork I start by distinguishing two types of inequality based on the population of interest: interpersonal inequality on the one hand, and inequality between groups, or simply group inequality, on the other hand.This allows us to draw connections between inequalities as studied by economists and as studied by sociologists, including what are often called 'categorical' inequalities (Tilly, 1998).
Interpersonal inequality is inequality in the distribution of some variable(s) of interest across a population of individuals.Standard approaches to economic inequality fall under this heading, where the variable of interest might be income, consumption, or wealth.Inequality in the distribution of these variables is typically measured using an inequality index such as the Gini, Theil, or Mean Log Deviation (MLD).However, we might also be interested in interpersonal inequality of power, of education levels or health status, or of access to some public good or service, and the variable may or

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may not be quantifiable.Whatever the variable, the unit of analysis is the individual and normative judgements about income inequality -such as the statement that less inequality is better than more inequality -are explicitly based on the normative assumption that all individuals have equal weight.
In contrast, group inequality refers to inequalities between groups of individuals, including inequality between genders, races and castes.Class analysis is a form of group inequality but it has the crucial difference that the groups are defined by the inequalities between them, according to their role in the process of economic production (for instance as a worker or a capitalist), or their position in socio-cultural hierarchies (Devine and Savage, 2005).Just like interpersonal inequality, group inequalities may be quantifiable, such as differences in average incomes or health outcomes, or not, such as status differences and stigma associated with a group.
Categorical or group inequalities can refer to different underlying ontologies: they can refer to systematic differences in the characteristics of the individuals who comprise those groups, such as the fact that women earn less than men.Or they can refer to social relations between categories, such as relations of domination between men and women.Within the latter approach, inequalities between groups are sometimes described as a 'matrix of domination' or 'vectors of oppression and privilege' (Collins, 2000;Ritzer and Stepnisky, 2008).Crucially for our purposes, the two can interact.For instance, following Tilly's (1998) theoretical framework, Mosse's account of poverty in western India highlights 'the effect of social categorisation and identity in reproducing inequality and making exploitation socially viable' (Mosse, 2010(Mosse, : 1157) ) -meaning that the nature of social relationships enables one group to remain richer than another group.
With these clarifications we can now define our key concepts.An inequality interaction is a change in one inequality that has either a causal or an analytical relationship with another inequality.The two inequalities could be of the same dimension across different populations, such as when a change in interpersonal income inequality leads to a change in categorical income inequality.Or the two inequalities could be in different dimensions, such as when a change in income inequality leads to a change in status inequality.
I analyse two broad types of inequality interaction.Inequality diversion is defined as a reduction in one form of inequality that is dependent on sustaining, or worsening, another form of inequality, whether analytically or through causal mechanisms.Inequality re-ordering is defined as a reduction in inequality between groups or categories, holding constant the interpersonal distribution of income.This means that the categorical identities of those holding different positions in the distribution change in such a way as to reduce inequality between categories, but the positions themselves do not change.

INEQUALITY INTERACTIONS WITHIN A SINGLE DIMENSION
In this section, I present two cases to demonstrate how interpersonal inequality interacts with categorical or between-group inequality within a single dimension.This allows us to use the analytical tools of economic inequality measurement.I first explore progressive redistribution, and then show how decomposable inequality measures can illuminate cases of inequality re-ordering.

Progressive Redistribution
I start by analytically demonstrating a positive inequality interaction: how progressive redistribution in the interpersonal distribution (of income or any other redistributable variable) leads to a decline in categorical or group inequality.This is motivated by empirical findings such as Gonzales et al. (2015), who find that gender inequality tends to be lower in countries and in years with lower income inequality.Similarly, in Brazil, Ikemura Amaral and Loureiro (2022) find that declining inequalities between men and women and between racial groups were driven by the more general decline in income inequality between households, due to a rising minimum wage and other policies that targeted low-income households.The reverse can, of course, also occur.According to Manduca (2018), African Americans saw an improvement in their relative position in the income distribution in rank terms, reducing racial inequality, but rising interpersonal inequality in the USA (equivalent to a regressive redistribution) has offset this development to the point that racial inequality has not declined overall.That is, rising interpersonal inequality increased inequality between racial groups.I now suggest a simple way to understand these findings analytically.
First, we partition the population into different groups.For any standard (Lorenz-consistent) inequality measure I, between-group inequality I B is defined as the level of inequality if all individuals are assigned the mean of their group: we imagine that all women have the average women's income, all men have the average men's income, etc.We now model a standard progressive redistribution that is a reasonable approximation to many tax and benefit systems: a proportional tax combined with a flat universal benefit payment that is funded by the tax (Ferreira and Leite, 2003;Lakner et al., 2020). 4The policy transforms an individual's income y i to y i as follows: where t is the proportional tax rate and b is the benefit, which is equal to the income raised by the tax divided among the population: b = t ȳ where ȳ is Paul Segal mean income.This policy raises all incomes below the mean and reduces those above the mean, by a share that is increasing with distance from the mean.The new distribution Lorenz-dominates the old distribution so interpersonal inequality necessarily declines.I now show that this reduction in interpersonal inequality will automatically reduce between-group inequality.This occurs because group means experience the same transformation as individual incomes: for any group G with population N G and mean income ȳG before the policy, mean income after the policy is (2) The effect on group means is therefore the same as the effect on individual incomes, resulting in a reduction in between-group inequality.This simple analytical framework explains our first inequality interaction: the reduction of interpersonal inequality through a canonical progressive redistribution also reduces between-group or categorical inequalities, however those groups may be defined.This occurs despite the fact that the definition of the policy makes no reference to groups or categories. 5t is worth noting that this analysis applies to intersectional inequalities, which refer to inequalities involving groups that are defined by a combination of characteristics.If Black people have lower average earnings than white people, and Black women have lower average earnings than Black men, then the above findings imply that progressive redistribution will reduce the inequality between Black women and any other group.For this reason, progressive redistribution supports Crenshaw's (1989: 167) insistence on 'addressing the needs and problems of those who are most disadvantaged'.Crucially, we do not need to know which groups are most disadvantaged in order for progressive redistribution to help them.

Inequality Re-ordering
I now turn to inequality re-ordering, whereby inequality between groups or categories declines but interpersonal inequality does not.This is one way to look at Fraser's (2000) concern that the politics of identity, or 'recognition', has displaced the politics of distribution, or economic justice.Fraser argues that the rise in energy spent on valorizing disadvantaged groups is displacing energy or political capital away from attempts to reduce economic inequality more broadly.While Fraser's concern is with political capital, I consider the shift identified by Fraser from the perspective of inequality measurement: I will show that a decline in categorical or between-group inequality that is not accompanied by a decline in interpersonal inequality necessarily implies a rise in inequality within categories or groups -a variety of inequality diversion.This suggests an additional problem to complement Fraser's argument.
We now need to consider a group-decomposable inequality index such as the Theil or the Mean Log Deviation.For decomposable measures, total interpersonal inequality is the sum of between-group inequality (defined above) and within-group inequality, which is defined as a weighted average of inequality within the groups. 6Thus for decomposable inequality measure L, where L B is between-group inequality and L w is a weighted average of inequality L j within each group j.
The effect of inequality re-ordering now follows directly: if betweengroup inequality L B declines while overall interpersonal inequality L stays constant, then it implies a rise in within-group inequality L W . (And if overall inequality L rises while L B declines, then the rise in L W must be even greater.)This is therefore a case of inequality diversion: a decline in one type of inequality comes at the cost of a rise in another type of inequality.
Returning to Fraser's concerns, this implies that a reduction in inequality between, for instance, men and women, that is not accompanied by a reduction in interpersonal inequality, will necessarily imply a rise in inequality among men and/or among women.Indeed, unless the rise in within-group inequality is absorbed entirely by men, a decline in inequality between men and women will necessarily increase inequality among women -a point we will return to below in the context of professional women and domestic service.
In the USA, where interpersonal inequality has risen dramatically over the last several decades, we can also see this at the level of the power elite: Zweigenhaft (2001) reports an increase in the participation of women and minorities in the US elite since the 1950s, while the inequality between those women and minorities who have reached the elite and the rest has become wider.We observe a similar pattern at the level of the global elite: Anand and Segal (2017) find that the geographic composition of the global top 1 per cent became more diverse over the period 1988 to 2012, with poorer regions seeing their participation in the global top 1 per cent rise at the expense of the advanced economies.In this sense, inequality between countries declined at the level of the global elite.But over the same period, total global 6.How this average is weighted depends on the specific measure.See Anand and Segal (2008: 85) for a discussion of decomposable inequality indices, with application to decomposing global inequality into within-and between-country inequality.

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inequality increased and, with it, inequality within countries.Thus the rise in country diversity in the global elite occurred as the members of this elite increased their distance from their own national populations.Segal (2021a) shows that Chinese elites in particular converged with their international counterparts from rich countries while diverging from their compatriots in both economic and social terms.Finally, the decline in gender inequality globally since 1980, documented by Stotsky et al. (2016), coincided with rising income inequality within countries over the same period, implying that inequality among men and/or among women has risen.
For those who believe that inequalities in general should be reduced, inequality re-ordering presents a dilemma.The reduction in categorical or between-group inequality is desirable, while the increase in inequality within groups is undesirable, and risks harming the most disadvantaged highlighted by Crenshaw -namely, those in the disadvantaged group who remain at the bottom of that group after the inequality re-ordering.But equation (3) also indicates the solution to the dilemma: the only way to avoid the undesirable increase in within-group inequality is to ensure that interpersonal inequality also declines -that is, to produce the decline in betweengroup inequality not through inequality re-ordering, but through some form of progressive redistribution.This does not imply that one should not directly target categorical inequalities, but rather that one needs to consider the effect on the interpersonal distribution.For instance, consider a gender equality policy that has the effect of reducing between-sex inequality by reducing all men's pay and increasing all women's pay by a fixed proportion.This would keep within-group inequality constant while reducing between-group inequality, implying a reduction in interpersonal inequality in equation (3).The challenge, then, is to construct policies that reduce categorical inequality through improvements in the interpersonal distribution, as opposed to through inequality re-ordering.

INEQUALITY INTERACTIONS ACROSS INCOME AND SOCIAL DIMENSIONS
Progressive redistribution and inequality re-ordering can be transparently modelled using the analytical tools of inequality measurement.But inequality interactions can also occur through more subtle social mechanisms.I now present two cases that involve interactions across dimensions, involving status hierarchies and relations of domination as well as income.The first is meritocracy, where social arrangements designed to reduce inequalities due to birth can lead to inequality diversion that increases both income and status inequalities.The second is the interaction between rising female labour market participation and the employment of domestic labour, where a reduction in one type of gender inequality may produce inequality diversion by increasing status inequalities and relations of domination.Since merit and status are not readily quantifiable the analysis is necessarily qualitative, focusing on the variety of sociological mechanisms that underlie the interactions.

Meritocracy
Meritocracy is a form of social organization under which those who demonstrate greater ability and effort are given positions of greater power, wealth and social standing.It is typically contrasted with forms of social organization in which these positions are assigned on the basis of family or social connections.Its purported benefits are to both increase efficiency, and to reduce inequalities due to social position at birth.I argue that it implies inequality diversion because in practice meritocracy tends to exacerbate two other forms of inequality: income inequality and status inequality.
First, meritocracy can increase income inequality through its role in legitimating highly unequal incomes.This legitimation function has been found across a range of countries in the recent literature on elites.Luci and Gessaghi's (2016) study of Argentina shows that in the 1990s, foreign direct investment by multinationals led to new channels of access to elite positions through the development of a new cadre of meritocratically selected managers.But this was also a period of rapidly rising inequality, and members of this new elite justified their own success, in contrast with the (far more numerous) losers from the disruptive economic changes, on an explicitly meritocratic basis.Even traditional Argentine elite families justify their privileged position in society on the basis of the merit of their forefathers who constructed the nation, and at the same time attempt to renovate the family's merit each generation through their own new achievementsif necessary by incorporating middle class 'men of talent' into the family through marriage.Hecht (2017) and Krozer (2018) interviewed members of the elites in London and in Mexico respectively and found that the very rich explicitly justified their outsized incomes based on the claim that the market rewards merit.The ideology of meritocracy can also be used to actively oppose egalitarian policies: Mosse (2018: 431) finds that in India opposition to affirmative action for disadvantaged castes is based on the argument that it will 'displace competitive merit so as to undermine the efficiency of public institutions'.More generally, Mijs (2021) finds that the level of inequality correlates positively across countries and over time with belief in meritocracy. 7here is a long recognition of this interaction.The term 'meritocracy' was famously coined by British sociologist and revisionist socialist Michael

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Young in his 1958 satire The Rise of the Meritocracy.He hoped to draw attention to the inequalities which would be generated in a society in which birth no longer played a role in socio-economic status, but in which 'merit', defined as ability plus effort, would determine everyone's position.To varying degrees this is the transition made by many societies since the late 19th century, driven (argued Young) by the universalization of education.Yet in Young's portrayal meritocracy does not lead to a less stratified or unequal society.Instead, it becomes still more unequal: he predicted that classification by ability would lead to wider gaps between classes (Young, 1958: 106).Consistent with this prediction, OECD (2018) finds that there was a general rise in upward mobility for people with lower educated parents born between 1955 and 1975, and that this was followed by the well-documented rise in income inequality after 1980, during the working lives of the beneficiaries of this mobility.Young's argument would be that the ideology of meritocracy helped justify some of the policies that contributed to this rise in inequality, such as declining taxes on the rich and stagnating real minimum wages.
In addition to widening the income gaps between classes through changing social norms that legitimize those income gaps, meritocracy may also intensify status inequality.Young feared that stratification by 'merit' would create a new morality of desert: the poor would know that their poverty was their own fault, rather than a reflection of an unfair society.They would be 'bound to recognise that they have an inferior status -not as in the past because they were denied opportunity; but because they are inferior' (Young, 1958: 107-08).Meanwhile at the top of the distribution, 'the eminent know that success is a just reward for their own capacity, for their own efforts, and for their own undeniable achievement.They deserve to belong to a superior class'.This accurately predicted the findings of the new sociology of the elite in Argentina, Mexico and the UK noted above (as well as Markovits, 2019 on the USA), and is consistent with Ridgeway's (2014) argument that material conditions become embedded in social status hierarchies.
This theme has been elaborated by educational sociologists, who have shown that while educational attainment has often been a major factor permitting upward social mobility, this does not imply that inequalities in social mobility have declined.Rather than contributing to equalizing outcomes, education becomes another arena in which to express and reproduce existing inequalities.Parrado and Salvador (2011) find that Brazil is the most meritocratic country in Latin America, scoring four out of five on their scale.Yet Almedia (2001) shows that, in São Paulo, expensive private schools attended by the rich have a near-monopoly on entry to prestigious (and mostly public) universities, which then dominate access to privileged occupations.Those schools effectively transmit the knowledge and skills required to pass competitive university entry examinations, maintaining the meritocratic character of the system: the children of the rich do indeed acquire merit as a result of this training.Byun and Kim (2010) find that in South Korea in 2008, nine out of ten students with monthly family incomes above US$ 6,300 received private tutoring, compared with four out of ten students whose monthly family income was below about US$ 900.Markovits (2019) describes the enormous investments that rich parents in the US make in order to get their children into the 'right' schools and colleges that will help them procure elite positions in adulthood.This is why educational expansion need not reduce inequality: inequalities get reproduced at every level of education, because richer families use their greater resources to cultivate educational merit in their children.Put another way, education and merit become used by elites as mechanisms of opportunity hoarding (Tilly, 1998).
The rise of meritocracy can therefore be as a case of inequality diversion because the decline in inequality due to social position at birth comes at the cost of more inequality in two other senses.First, it contributes to the widely documented rise in individual economic inequality, by acting as a legitimizing discourse.Even where it succeeds in reducing inequality between groups defined by parental class, it does so at the expense of increasing inequality within those groups, and in the interpersonal distribution more broadly, by widening the gap between those 'with merit' and those without.Second, as elaborated by Young, it worsens the inequality of social status between the rich and poor: the rich feel deserving and superior, while 'the poor and the disadvantaged … can easily become demoralised by being looked down on so woundingly by people who have done well for themselves.It is hard indeed in a society that makes so much of merit to be judged as having none.No underclass has ever been left as morally naked as that' (Young, 2001).
Yet the inequality interactions do not stop there.The irony is that where meritocracy has contributed to rising income inequality, the rise in inequality has led in turn to a decline in social mobility.Atkinson (1998: vi) notes that 'inequality of outcome today is a cause of inequality of opportunity in the next generation'.Jackson and Segal (2004) argue that countries with lower levels of economic inequality also have higher levels of intergenerational mobility, a phenomenon that has more recently been denoted the Great Gatsby Curve (Corak, 2013).Empirically, OECD (2018: 15-16) finds that since the 1990s, after widespread rising inequality, income positions at the top and bottom of the distribution have become more persistent, implying lower income mobility.They also confirm that earnings mobility across generations tends to be lower in countries with higher inequality. 8he dynamics of inequality interactions thus suggest that attempts to increase social mobility through meritocracy are likely to be self-defeating.Put another way, meritocracy should not be confused with equality of opportunity.Indeed, recognition of this problem also has a long pedigree: R.H. Tawney anticipated it in his 1931 work Equality, arguing that as long as

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privileged families were able to provide education, cultural and financial resources to benefit their children, formal equality of opportunity would remain a sham, reduced to 'the impertinent courtesy of an invitation offered to unwelcome guests, in the certainty that circumstances will prevent them from accepting it' (Tawney, 1931: 143, quoted in Jackson andSegal, 2004: 25) Unlike the cases of progressive redistribution and inequality re-ordering described above, these inequality diversions are not analytical relations but instead are contingent on empirical social processes.This suggests they might be avoidable, or could at least be mitigated.What factors might condition the impact of rising meritocracy on income and status inequalities, and reduced social mobility?First, since the disequalizing effect works via inequality in parents' investments in their children's education, we would expect the mechanism to be blunted where the inequality in parental means to buy such investments is lower -including when the starting level of income inequality is lower.The lower the level of income inequality, the less excess income the rich have to make the investments in private schooling, tutoring and cultural activities that extend their children's 'merit' beyond the levels provided by public education.
Second, the greater the provision of public education for the non-rich, the greater the investment in 'merit' will be for the majority of the population, reducing the gap with the rich.This is consistent with findings in educational research.Schlicht et al. (2010) find that education expenditures correlate negatively with inequality in educational outcomes across European countries, while Neidhöfer et al. (2018) find that a rise in intergenerational educational mobility across Latin America was driven mainly by investments in education that benefited children from families with low education.Unsurprisingly, it matters what kind of public education is financed, as indicated by Brazil's regressive investment in public universities discussed above; moreover, the literature on educational inequality has found a range of other determining factors, such as the extent of pre-school education and tracking within school.The key point for present purposes is that once undesirable inequality interactions have been identified, there may be policies available to mitigate them.

Professional Women and Domestic Service
Our analysis of inequality re-ordering above demonstrated that a reduction in gender inequality that is not accompanied by a reduction in interpersonal inequality necessarily implies a rise in within-gender inequality.The relationship between gender inequality and domestic service provides an illustration of this principle, and -like meritocracy -involves inequalities in both income and social dimensions.While gender inequality has declined globally, and in particular many highly paid professions are now open to women that were closed to them some decades ago, there has been an accompanying growth in the low-paid service sector in many countries, both rich and poor (see Autor and Dorn, 2013 on the USA; Goos et al., 2009 on Europe;Hairong, 2008 on China).This has often involved new professional classes of working women employing domestic workers to perform social reproduction work that is assumed to be women's responsibility.In Beijing, Sun (2009: 10-16) reports that maids are 'indispensable to the smooth running of the household' for the 'emerging urban middle class'.Referring to the US case, Flanagan (2004) puts it starkly, stating, 'many of the gains of professional-class working women have been leveraged on the backs of poor women'.Bowman and Cole (2009: 165) argue that hiring a housecleaner 'displace[s] inequality from the male-female axis to the householder-housekeeper axis'.Like the analysis of inequality reordering above, the argument is that a reduction in inequality between men and women has worsened inequality among women, specifically between domestic workers and their employers.Moreover, the women employed to do this domestic labour are often from ethnic minorities, and/or immigrants from poorer countries, meaning they suffer from intersecting, and thereby intensified, disadvantage. 9I now expand on these arguments in two ways by distinguishing between the impacts on income inequality and social inequality, and by considering different mechanisms of inequality diversion.First consider income.Women occupying highly paid professional positions formerly occupied by men represents a form of inequality re-ordering as analysed above, implying a reduction in inequality between men and women while increasing inequality among women: the higher rungs of the income distribution ladder are now open to women, allowing a small share of them to move up, further above other women.However, in this case there is a countervailing economic mechanism which may reduce inequality among women: a rise in demand for domestic workers would be expected to increase pay for these workers, which may reduce inequality between them and their employers. 10This is consistent with cross-country studies that find rising inequality also leads to rising demand for low-wage service occupations (Mazzolari and Ragusa, 2013), which would be a feedback mechanism ameliorating the rising income inequality.The net effect on inequality among women is therefore ambiguous: inequality re-ordering increases it, while rising demand for domestic workers reduces it.But the effect on overall interpersonal 9. See Cano and Soffia (2009) on the case of Chile, where female immigrants from neighbouring countries typically enter domestic service; Sassen (2002) discusses the case of 'global cities' where both highly paid professionals and their domestic servants typically come from other countries.10.Another way to look at this mechanism is to note that the entry of women into the professions reduces the supply of labour performing social reproduction work, which would be expected to raise the wage for that labour.

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inequality should be to reduce it: inequality re-ordering on its own has no impact, whereas rising demand for domestic workers reduces it.The impact on social inequality is also ambiguous.Tronto (2002) argues that domestic work is different from other forms of employment both because it takes place in the home, and because it involves affective labour.This implies a more intimate relationship with the employer, and expectations on the part of the employer that the domestic worker should treat their labours differently, and be motivated differently, from other types of work.This means that two related types of social inequality may be worsened: status hierarchies, whereby domestic workers are seen as low status, and relations of domination, in which domestic workers may be subject to the arbitrary command of their employers (Segal, 2021b).Ethnographic work confirms that both of these are commonplace in domestic work.Rollins (1985: 8-9) describes the mistress-servant relationship in the USA as 'an extreme and "pure" example of a relationship of domination in close quarters' and reports that in her interviews with domestic servants, 'all domestics concurred that employers appreciated some forms of deference and outward signs of subservience ' (ibid.: 147).With this in mind, Ehrenreich (2001: 53) reports that she finds the idea of employing a cleaner 'repugnant' because 'this is not the kind of relationship that I want to have with another human being'.Hairong (2008) describes the social hierarchy that developed in China between domestic workers and their employers in the three decades after the 1978 reforms, which Segal (2021a) links with the rise in economic inequality.Thus, if rising gender equality within the professional class leads to rising employment of domestic workers, it may also exacerbate relationships of domination and subordination.
However, Meagher (2002) points out that there are conditions under which domestic work does not involve domination and subordination, and in her own interviews finds that domestic workers do not perceive their work as degrading.Similarly, Rollins's interviewees, despite the deference their employers demanded, mostly found the work unobjectionable and 'an acceptable occupation' (Rollins, 1985: 133).Meagher (2002) argues that when domestic workers have legal rights, decent levels of pay, or work for multiple employers, then the degree of both domination and personalization in the relationship need not be any greater than in many other jobs.A cleaner may clean many homes in a week, spending only a few hours in each; the same applies to a babysitter or part-time nanny.Even a full-time nanny, while expected to develop an intimate relationship with the children under their care, need not develop one with the parents.Indeed, where a nanny is employed in order to allow both parents to work, the nanny would be expected to see little of the parents precisely because they are working.
As in the inequality diversion created by meritocracy, we would expect both the level of inequality and the role of public services to influence these processes.The degree of economic inequality between employers and employees in domestic service is likely to influence the degree of both status inequality and domination in the relationship.Segal (2021b) defines entitlements over labour as the ratio between the disposable income of rich households and the cost of employing a typical worker, arguing that the lower this ratio -and thus the higher the wages of domestic workers relative to the incomes of their employers -the lower we might expect the hierarchical distance between employer and employee to be; rather, the relationship is more likely to be as described by Meagher.Looked at from another angle, if competition between employers has bid up wages for domestic workers, then it is likely that it will also bid up the respect with which those workers need to be treated.
Second, if social reproduction services are provided by the state, then increasing employment of professional women need not imply so much employment of domestic service.Social reproduction can be performed by the family, the market, or the state (Orloff, 1993), so a reduction in family provision could be compensated by an increase in state provision, rather than private provision.This is clear for care work including childcare, though perhaps not for house cleaning.Faur (2011) describes the case of Argentina where a substantial rise in the share of two-earner couples since 1980 was accompanied by efforts to increase public provision of childcare (albeit with limited success, owing to limited financing); Meyers et al. (1999) show that supportive childcare policies in some social democratic European countries helped to reduce the extent to which women with young children left employment.The design and focus of such policies vary: Staab and Gerhard (2010) argue that early childhood education and care services in Mexico are aimed more at supporting poor women to enter the labour market, whereas in the case of Chile this is secondary to the goal of increasing educational achievements among children from low-income families.As in the general case of inequality re-ordering, the dilemmas presented by this case cannot be resolved by reversing the decline in gender inequality, but instead should be addressed through more generalized reductions in interpersonal inequality, and institutional arrangements that sever the link from a decline in one form of inequality to rises in others.

VARIETIES OF INEQUALITY INTERACTIONS
The above discussions considered interactions between interpersonal inequality and categorical inequality, and between inequalities in the different dimensions of income, social status and domination.I also suggested that they can be divided into analytical relationships, and contingent or empirically determined relationships.In the case of progressive redistribution and inequality re-ordering, the (positive and negative) inequality interactions were derived analytically in the sense that they follow from the logic of the definitions.In the case of meritocracy and gendered dimensions of the labour market, on the other hand, the processes are contingent and depend on a range of intervening empirical factors.This means they can in principle be mitigated, and I suggested that in both cases the mitigating factors would include the initial level of income inequality, and the role of the state in providing relevant public services -education in the case of meritocracy, and social reproduction services in the case of women in the professional labour market.The four types of inequality interactions and their potential mitigating factors are summarized in Table 1.
The purpose of this article is not to give a normative analysis of how to respond to trade-offs between different types of inequality, but rather to help understand them.In some cases, this will reveal opportunities to avoid them, and in others it will simply make explicit what trade-offs are unavoidable.In that light, what lessons can we derive?First, the discussion here supports the argument that concerns about categorical inequalities cannot be divorced from concerns about interpersonal inequalities.Keeping an eye on the interpersonal distribution is required to avoid inequality diversion where a reduction in one type of categorical inequality leads to an increase in another type of categorical inequality -as in Crenshaw's discussionor to an increase in inequality within groups -as in the case of inequality re-ordering.
Second, the discussion suggests that standard proposals for reducing income inequality are likely also to have desirable effects on social inequality.In the case of meritocracy, progressive redistribution reduces the extent to which the rich can buy excess 'merit', stacking the deck in favour of their offspring and against social mobility.And by reducing entitlements over labour, or the affordability of domestic workers to richer households (Segal, 2021b), progressive redistribution reduces the extent of domination involved in the employment of domestic workers, allowing these workers to enjoy a more professional and less interpersonally hierarchical relationship with their employers.
Third, the analysis also indicates the role of public services in reducing social inequality.Studies on the distributional incidence of fiscal policy have long shown that when we include the value of public services in household income, they have a dramatic progressive impact on income inequality.They do this by defining 'final income' as disposable income plus the value of public services consumed, such as healthcare and schooling that are free or subsidized at the point of use (Lustig, 2018).We have now seen how public services can also reduce inequality in non-economic dimensions: public education reduces the educational advantage conferred by higher incomes, reducing the extent to which income inequality under meritocracy translates into educational inequality and then social immobility; and public provision of social reproduction services reduces the personalization of care work, reducing relations of domination and social hierarchy.

CONCLUSION
This article has presented an analytical framework and set of concepts to clarify the interactions between different forms of inequality and, it is hoped, to facilitate interchange between economic and sociological approaches to inequalities.The economic approach to inequality contributes a rigorous specification of the relationship between individual and group or categorical inequalities.The sociological approach broadens the scope of the variables that define inequalities, their causes, and the mechanisms of transmission of inequalities -including transmission across different dimensions of inequality.I have used this framework to show that inequality interactions are widespread, and may be negative or positive.
The concepts developed here have wide applicability beyond the examples that I have examined.The question of inequality diversion is at the heart of debates over immigration, for example.When someone from a poor country emigrates to a rich country, this tends to reduce the inequality between groups defined by place of birth: someone born in a poor country raises their income level closer to the average income of someone born in a rich country.It may also reduce global inequality, meaning inequality among all individuals regardless of which country they live in.But there are two ways in which scholars have argued it may imply inequality diversion by increasing inequality within the destination country.The first is the claim that immigration reduces low-skill wages, thereby increasing market inequality.The second is Alesina and Glaeser's (2004) argument that large welfare states require ethnic homogeneity, so that for instance 'if Europe becomes more heterogeneous due to immigration, ethnic divisions will be used to challenge the generous welfare state ' (ibid.: 11).Both effects are highly contested, however (Lecture and Card, 2009;Stichnoth and Van der Straeten, 2013).Another example of potential inequality diversion is quantitative easing, a form of monetary expansion that could have opposing effects on income and wealth inequality: by reducing unemployment it is expected Paul Segal (paul.segal@kcl.ac.uk) is Reader in Economics of Development at the Department of International Development, King's College London, UK.He is an affiliated scholar at the Stone Center on Socio-Economic Inequality, City University New York, USA, and the International Inequalities Institute, London School of Economics and Political Science, UK.His research focuses on global inequality and poverty, and on structural approaches to inequality within developing economies.He also works on interdisciplinary approaches to socio-economic inequality and is currently working on linkages between inequality and social reproduction.

Paul Segal Table 1 .
Varieties of Inequality Interactions