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Causal analyses typically focus on average treatment effects. Yet for substantive research on topics like inequality, interest extends to treatments' distributional consequences. When individuals differ in their responses to treatment, three types of inequality may result. Treatment may shape inequalities between subgroups defined by pretreatment covariates, it may induce more inequality in one subgroup than another, or it may polarize people across multiple dimensions of well-being. We introduce a model, called a covariance regression, that captures all three types of inequality via the means, variances, and correlations between multiple outcomes. The model can test for heterogeneous treatment effects, quantify the heterogeneity, and explain its structure using covariates. Finding that a treatment creates inequalities could drive theoretical refinement and inform policy decisions (targeting groups where payoffs will be most predictable). We illustrate the utility of covariance regressions by analyzing the effects of sharing information about income inequality on redistributive preferences.

In the early twentieth century, the cotton-growing regions of the U.S. South were dominated by families of tenant farmers. Tenant farming created opportunities and incentives for prospective tenants to marry at young ages. These opportunities and incentives especially affected African Americans, who had few alternatives to working as tenants. Using complete-count Census of Population data from 1900-1930 and Census of Agriculture data from 1889-1929, we find that increases in tenancy over time increased the prevalence of marriage among young African Americans. We then study how marriage was affected by one of the most notorious disruptions to southern agriculture at the turn of the century: the boll weevil infestation of 1892-1922. Using historical Department of Agriculture maps, we show that the boll weevil's arrival reduced the share of farms worked by tenants as well as the share of African Americans who married at young ages. When the boll weevil infestation altered African Americans' opportunities and incentives to marry, the share of African Americans who married young fell accordingly. Our results provide new evidence about the effect of economic and political institutions on demographic transformations.

Policy reforms and rising income inequality transformed educational and economic opportunities for Americans approaching midlife in the 1990s. Rising income inequality may have reduced mobility, as income gaps increased between rich and poor children. Against the effects of rising inequality, Civil Rights reforms may have increased mobility, as opportunities expanded across cohorts of black students and workers. We compare educational and income mobility for two cohorts of black and white men, the older born in the late 1940s and the younger born in the early 1960s. We find that educational mobility increased for black men, but income mobility declined for both races. Economic mobility declined despite unchanged or improved educational mobility because of increased returns to schooling and increased intergenerational income correlations, independent of schooling.