The Role of Location in Evaluating Racial Wage Disparity

A standard object of empirical analysis in labor economics is a modified Mincer wage function in which an individual's log wage is specified to be a function of education, experience, and an indicator variable identifying race. We analyze this approach in a context in which individuals live and work in different locations (and thus face different housing prices and wages). Our model provides a justification for the traditional approach, but with the important caveat that the regression should include location-specified effects. Empirical analyses of men in U.S. labor markets demonstrate that failure to condition on location causes us to (i) overstate the decline in black-white wage disparity over the past 60 years, and (ii) understate racial and ethnic wage gaps that remain after taking into account measured cognitive skill differences that emerge when workers are young.


Introduction
In hundreds of studies social scientists have examined the role of minority status in wage determination by estimating variants of the Mincer earnings function, (1) ln(w i ) = β 0 + β 1 R i + β 2 E i + γ(X i ) + i ; the expected log wage of individual i is specified to be a function of an indicator variable for minority demographic status R i (e.g., race, immigrant status, ethnicity, or gender), education E i (or an alternative measure of human capital), and also, typically, some function of such covariates as age or experience, given by γ(X i ). An estimated negative value of β 1 is taken as an indication of wage disparity that adversely affects minority workers.
For example, a very large literature uses regression analyses along the lines of equation (1) to document racial disparities among blacks and whites in the U.S. The disadvantaged position of black workers is understood to be the consequence of discrimination in labor markets and racial differences in the pre-market development of human capital. 1 Here we study the properties of the estimates of regression (1) when individuals live in locations that have differing prices, e.g., different local wages and prices of other goods and services. 2 We are interested in both of the possibilities mentioned above-that wage gaps are due to discrimination or due to unmeasured differences in human capital. Either way, we find that in a standard model of local labor markets, β 1 is a meaningful parameter (under somewhat restrictive conditions) but it can be estimated consistently only if one includes location fixed effects into equation (1).
With this observation in mind, we turn to two empirical exercises. First, we conduct an analysis of black-white wage gaps for men in the United States from 1940 through 2000-work that roughly parallels (and extends) the seminal work of Smith and Welch (1989). We find that an analysis that omits location substantially overstates black-white wage convergence over this period.
Second, we estimate the black-white wage gap and the wage gap between Hispanics and non-Hispanic whites, using regressions in which we control for cognitive test scores (AFQT scores) taken when individuals were young, as in Neal and Johnson's (1996) important work.
Omission of fixed effects for location causes us to substantially understate the magnitude of both the race wage gap and ethnicity wage gap.
Our paper proceeds in four additional sections: The first section sets out a standard model of urban differences in prices, and in that context demonstrates that the race wage 1 Charles and Guryan (2008) document patterns of black-white wage disparity consistent with Becker's model of taste-based discrimination. Many important papers speak to the emergence of black-white differences in human capital development. For example, Cunha, Heckman, Lochner, and Masterov (2006) provide an insightful analysis of human capital development, including the emergence of racial and ethnic disparities, and provide an extensive reference to the literature; Card and Krueger (1992) analyze the consequences of school quality for black-white earnings differences; and Neal and Johnson (1996) and Neal (2006) demonstrate the importance of cognitive ability (measured when individuals are young) for subsequent labor market success. 2 Through out the paper, we will refer to these locally priced goods and services as "housing," which surely has the largest budget share among locally priced goods, but of course many goods and services are locally priced (e.g., haircuts are more expensive in Chicago than in Peoria).
gap-whether generated by discrimination or by human capital differences-is a constant across different local labor markets if and only if preferences are homothetic. Under that assumption, estimation of a Mincer wage regression typically requires inclusion of location fixed effects. The second section presents evidence of the importance of this idea for standard empirical exercises that examine racial or ethnic wage gaps. Section three revisits our theory, asking about the implications of our model if the key assumption of homotheticity is violated. Section four provides a discussion.

Race Wage Gaps in a Multiple-Location Model
Workers supply labor in local labor markets, and across those markets there are often substantial differences in wages and housing prices. Theoretical reasoning in the urban/regional economic literature, in the pioneering work of Haurin (1980) and Roback (1982) and in many papers that followed, suggests that observed location-specific price differences can generally be understood to be the consequences of differences in locations' attractiveness and location-specific differences in productivity. Our goal is to determine what these models have to say about racial wage disparities in local labor markets.
We begin with a population in which individuals belong to one of two racial groups: a minority group, indicated by R = 1, and a majority group, indicated by R = 0. These people live in one of n cities, and consume two goods: a non-housing good that has a price 1 in every location, and housing, which has a price that varies across cities. We designate the rental price of housing p j per unit (j = 1, . . . n). Wages also differ across location, and we want to allow for the possibility of race-based differences in wages: individuals from racial group R = 0 earn wage w 0 j in city j, while those from group 1 earn w 1 j . For simplicity, we assume that all workers supply one unit of labor, regardless of where they work. We assume also that all individuals have the same preferences. Finally, we assume that there is costless migration between locations. Let the expenditure function for workers of each group (R = 0 or 1) living in city j be e R j = e(p j , u R j ). The key equilibrium condition is that workers of both groups must be indifferent over their city of residence; for individuals in each racial group, utility u R j is the same in each city. Therefore we can drop the subscript j on utility, and note that equilibrium entails (2) e(p j , u 0 ) = w 0 j and e(p j , u 1 ) = w 1 j for j = 1, . . . , n.
While u 0 and u 1 must each be invariant across cities, utility might differ between demographic groups if their earnings differ. As we have noted, this latter outcome is possible if the two groups differ in terms of productivity or if there is racial discrimination that results in wage disparity. With this in mind, we consider an "equality index" in location j, which we define to be the ratio of the wage for the minority group 1 relative to the wage of the majority group 0, (3) This equality index will be less than one if the minority group is disadvantaged or greater than one if the minority group is advantaged. Importantly, in general this ratio is seen to depend on the housing price p j .
When is the equality index independent of location-specific price variation? First, note that if preferences are such that individuals' expenditure functions takes a "separable" form e(p, u) = ψ(p)f (u), the equality index in location j is , which does not depend on local prices. Second, and more importantly, note that the converse is true.
The proof is simple: Let I j = g(u 0 , u 1 ), so that the index in location j does not depend on that location's prices. Without loss of generality we can take u 0 = 1, u 1 = u. Then I j = e(p j , u) e(p j , 1) = g(u, 1), so we can write e(p j , u) = e(p j , 1) · g(u, 1). Setting ψ(p) ≡ e(p, 1) and f (u) ≡ g(u, 1) we find that the expenditure function has the form e(p, u) = ψ(p)f (u).
A familiar result from price theory is that the expenditure function takes the form in city j we have where R i indicates the individual i's race (R i = 0 or 1). Since ψ(p j ) is independent of utility, local wage levels vary with local prices, but log racial wage disparity is a constant that is invariant with regard to location. Thus we can let β j 0 ≡ ln(ψ(p j )) + ln(f (u R 0 )) and β 1 ≡ ln(f (u R 1 )) − ln(f (u R 0 )), and we have the structural relationship where β 1 the penalty (or premium) to minority status.
Mincer famously provided theoretical reasoning to expect that log wage increases linearly in years of education (E i ), and in addition is increasing in experience. If so, and if the returns to education and experience are the same for minority and majority workers, f (u) should be proportional to e β 2 E i +γ(X i ) , where g(X i ) is an increasing function of experience (X i ). 3 Then, assuming additional variation in the observed log age can be represented by an independent additive error term ij , we have This is the same as the familiar equation (1) with the important proviso that local labor market fixed effects (j = 1, · · · , n) must be included. 4 More generally, it might be advisable to treat the return to education as non-linear (see, e.g., Heckman, Lochner, and Todd, 2006), which is the approach we take in our empirical application below.
In the U.S. there is large variability in housing prices across cities. In general, utility losses individuals experience by locating in a particularly expensive city will differ among individuals. Furthermore, these utility losses might systematically be correlated with race, if only because blacks are on average poorer than whites. This is where the homotheticity assumption comes into play. When an individual experiences a price increase, real income 3 There is an important caveat to the Mincer theory: Black, Kolesnikova and Taylor (2009) show that β 2 , the "return to education," will differ by location if preferences are not homothetic. Here, though, we are assuming homotheticity, and in that case returns are the same for each location. 4 Location fixed effects can be safely omitted only if the vector of location indicator variables is orthogonal to other included variables, in which case those effects are absorbed into the error term. As we show below, though, black and white workers differ substantially in location patterns.
of course decreases. But if preferences are homothetic, the proportional decrease in real income is the same for all individuals. 5 Put another way, the proportional increase in the wage needed to induce people to live in a particularly expensive city will be the same across all individuals, and therefore the proportional wage gap between whites and blacks will be the same in each location. All that is required to estimate this gap is that the researchers estimate the wage regression using the log of wage as the dependent variable (i.e., use the Mincer specification) and include fixed effects to capture location-specific price differentials.
On the other hand, if there are serious violations of homotheticity, equilibrium blackwhite wage gaps will differ across cities. The same is true if markets are substantially out of equilibrium. We return to these issues in Section 3 below. First, though, we turn to empirical implementations of our key regression (6), asking if the inclusion of location fixed effects matters for inferences about racial wage gaps in the U.S.

The Importance of Location for Evaluating the Black-White Wage Gap
We consider here two important applications-inferences about black-white wage convergence over the past few decades, and inferences based on regressions that include a cognitive skills measure. Like Smith and Welch (1989), and many other authors, we estimate black-white wage gap using public use samples from the U.S. Decennial Census. There are substantial advantages to these data for this purpose. First, they provide us with an opportunity to examine the 5 The is is a central point in the literature on price indices. See, for example, Samuelson and Swamy (1974). In our context homotheticity means that the income elasticity of housing is 1. This might not be too far off the mark. In Epple and Seig's (1999) general equilibrium model, the permanent income elasticity of housing is estimated to be 0.94. Drawing on evidence from partial equilibrium empirical analysis, Harmon (1988) places it at 1, while Haurin and Lee (1989) give an estimate of 1.1. economic progress of African Americans relative to whites over a long period using data from instruments that are similar both in terms of content and mode of administration.
Second, the data provide extremely large samples, and therefore allow for precise estimates.
While there are some serious limitations with the Census data in regard to the variables available, we do have data on key economic outcome variables like earned income and labor supply, along with race, age, and education, and we have some information on location of residence. Even though the variables are quite limited, we are able to establish quite convincingly our central point-that treatment of location is very important in the estimation and interpretation of the decline in black-white disparity in U.S. labor markets over the past six decades.
For our analysis, we restrict attention to men. 6 We begin by dividing respondents on the basis of race-black and non-Hispanic white-and exclude other racial/ethnic groups.
We are interested in wages earned by "prime aged" full-time working men, so we restrict attention to men aged 25-55 who worked at least 27 weeks in the previous year. 7 In our analysis we use age, which we have in 31 discrete categories (individual years, 25 though 55 inclusive), education, which we have in 10 categories ("no schooling or kindergarten only" through "more than a bachelor's degree"), and location, which we have for several hundred unique localities. 8 Our primary focus is on the measurement of black-white wage disparity, conditional on observable characteristics. To give the simplest possible example, suppose we are interested in conditioning only on age. We can proceed as follows. Let b index black individuals and w index white individuals, and let x i be the exact year age of individual i. Let y i be the log wage of individual i, and let E(y b,i |x) be the expected value of the log wage of that (black) 6 Female labor markets are equally interesting, and we intend to evaluate black-white wage gaps among women in future work. 7 We exclude unpaid family workers, military personnel, the self-employed, and those employed in agriculture. See the data appendix for more detail. In general we pattern our exclusion rules after Smith and Welch (1989), although there are some substantial differences. The data appendix also outlines how we construct the key wage variable. 8 The data appendix discusses our location variables. These vary somewhat over the 60 years of analysis.

BLACK, KOLESNIKOVA, SANDERS, AND TAYLOR
individual given that his age is x. Our interest then is in . of age among black workers. The idea of looking at the object is of course that E(y w,i |x) provides a missing counterfactual to the question: What would be the expected log wage of a black worker age x if he were treated in the labor market as a similarly aged white worker? 9 Then by averaging difference over the age distribution of black workers we are looking at the "average treatment effect on the treated." Our theoretical reasoning suggests that we need to evaluate ∆ within locations. Since we are interested in the "average treatment effect" over all locations, we can follow an approach comparable to that given in (7) but now let x index a location-age cell (e.g., one cell will be men aged 31 residing in Houston). Notice that in the Census data there will be thousands of such cells, which again we index with x. Now we have where N is the number of age-location cells.
Finally, there is a tradition in race wage regressions of controlling also for schooling. Given that education in our data is categorized in discrete cells (as discussed in the appendix), we continue to adopt a non-parametric approach. In this instance we simply let x index a location-age-education cell (e.g., high-school educated men aged 31 in Houston), and now let f b (x) represent the distribution of the black population over these cells.
We could directly estimate equation (8) by calculating the conditional means at each point in the distribution of covariates and then taking averages. As a practical matter, we implement an estimation procedure that returns us to the traditional regression framework.
Let∆ be the non-parametric matching estimator based on the direct approach of (8), and 9 Notice that the "treatment" here is not merely the absence of potential racial discrimination in the labor market. Being treated as a white person includes other facets, including improved pre-market conditions that affect human capital.
letβ 1 be the weighted OLS estimator of the regression With a bit of algebra it is possible to establish that∆ ≡β 1 if the weights are constructed as follows: The first step in constructing the weights is to realize that the Census data themselves come with weights that allow one to mimic the U.S. population. In the appendix we describe our treatment of missing data. Our approach is to assume that data are, conditional on the age-race-education-race cell, missing at random. We thereby construct new weights; for an individual in a particular cell x 0 the weight, adjusted for missing data, is w 1 (x 0 ). Now consider the conditional "probability of being black" for that particular cell: Having calculated this probability for each cell, we proceed by defining a new final set of weights, say w 2 (x 0 ), as follows: if the worker is black, and if the worker is white.
Notice that if there is a white worker who is not matched to a black worker at all in the data (i.e., p(x 0 ) = 0), that individual is dropped from the analysis, and if his characteristics are quite dissimilar from typical black workers in the sample he will be given low weight.
Conversely, white individuals who have characteristics that are more typical of the black individuals in the sample are weighted more highly. Intuitively, our re-weighting scheme forces the distribution of covariates in the sample of whites to be identical to the distribution of covariates in the sample of blacks. In the matching context, this is often referred to as "inverse probability weighting." 10 It is important to keep in mind that the estimate of the average treatment effect contains the impact of "unobservables." Thus, for example, if we implement our estimator by matching on all available observables (age, location, and education), we are still leaving out important ways in which black and white workers differ in the labor market. For example, Black, Haviland, Sanders, and Taylor (2006)  Finally, we are analyzing wages of men who work 27 weeks a year or more. While the wages of working individuals are indeed important, so are the issues concerning racial differences in labor force nonparticipation. Three well-known facts are germane: First, nonparticipation rates of African American men are higher than the corresponding nonparticipation rates of whites. Second, nonparticipation rates are inversely correlated with education, and presumably nonparticipation also varies with unobservable skills as well.
Third, nonparticipation rates have been growing over time. 11 Chandra (2000) gives an excellent review of the issues involved. Here we ignore these concerns, and focus instead on the role of location for understanding the racial wage gap among those who are working. Table 1 gives results. We estimate regression (9) using weighted OLS with weights given in (11). In column (1) we report the outcome in which we match on age only. There is, of course, a compelling reason to match on age, since productivity is related to age, and since age is, from the perspective of labor market participant, exogenous. Having done so, we estimate the black-white log wage gap to be an astonishing −0.74 in 1940. This gap declines to a still-substantial −0.31 in 1980 and to −0.26 in 2000. 12 An important feature 11 Furthermore, much evidence (e.g., Black, Daniel, andSanders, 2002, andAutor andDuggan, 2003) suggests this nonparticipation due to disability is quite sensitive to prevailing economic opportunities, particularly for the low skilled. In addition, the increased incarceration of black males, noted by Western (2006) and others, also makes the use of observed wages problematic. Given the assumptions of our model-that equilibrium always holds and that preferences are homothetic-wage gaps can be thought of as money-metric measures of the welfare disadvantage to being black in the American labor market. These measures are correctly estimated only after matching black and white individuals within labor markets. Column (2) of Table 1 reports the resulting estimates of this latter sort (but not adjusting for educational differences among blacks and whites). There are substantial differences in the inferences we draw using estimates in column (2), which makes location adjustments, and column (1) As we have noted, it is common in the literature to condition on both age and education when evaluating wage gaps. The idea is to try to sort out how much of the race "treatment effect" is due to differences in years of formal schooling acquired by workers. We thus conduct our exercise matching on age and education in column (3)  What are the shifting patterns of residence that have such an important impact when we estimate black-white wage gaps? Table 2 provides the basic answer. In that table we report the results of the following exercise: We begin by calculating the extent to which black men disproportionately reside in the South. We do this by constructing an index equal to the ratio of "the fraction of black men aged 25-55 living in the South" to "the Although Southern residence and urban residence are important elements of the story, it is not sufficient to simply include a Southern indicator variable and an urban indicator variable in our regressions. When we try that exercise we find that we still substantially overestimate black-white wage convergence. 14 Importantly, we show that failure to condition on detailed location causes us to overestimate the absolute degree of the racial wage disparity in the early years of our analysis (prior to 1970) but to underestimate disparity in each year after 1970.
In a highly influential paper, Neal and Johnson (1996) provide an important critique of the standard modified Mincer wage regression as applied to the analysis of race wage disparities. They note that (i) blacks and whites typically have different levels of human capital, even conditional on observed years of schooling, and, in any event (ii) completed years of schooling is an endogenous choice variable that will depend on any number of factors, including the quality of schooling to which a young person has been exposed. They argue, therefore, in favor of an alternative approach in which the wage regression includes a measure of cognitive ability, measured while an individual is still quite young (i.e., while 14 For instance, when we add an indicator for residing in the South and indicator for residing in an MSA to our nonparametric regression, we find that convergence is overstated 29 percent relative to conditioning fully on location as we do in column (4) of Table 1. he or she is a teenager), rather than the more traditional "years of schooling" variable. 15 In such an empirical exercise, estimated black-white gaps are found to be quite small in absolute value. Indeed, Bollinger (2003), in his reanalysis of Neal and Johnson's data (which looks at the role of measurement error inherent in any test score) summarizes by suggesting that human capital "attainment at age 18 may explain all of the gross differences in wages between blacks and whites" (p. 583).
Our theory, presented in Section 1, suggests that one must include location fixed effects in the wage regression. With this in mind, we conduct here a reexamination of the central results of Neal and Johnson (1996) and Bollinger (2003), using, as did these authors, data from the National Longitudinal Study of Youth, 1979, and also updating the results using the 1997 cohort. We examine black-white wage gaps, and also wage gaps between Hispanics and non-Hispanic whites. 16 We begin by considering the simplest wage equation of Neal and Johnson (1996), and Bollinger (2003): where ln(w i ) is the natural logarithm of the respondent's wage on the last job, A i is the respondent i's age in months, R i in a race indicator variable equal to 1 if the respondent is black, and H i is an indicator variable of Hispanic ethnicity. We use data from the 1979 Cohort of the National Longitudinal Survey of Youth (NLSY) for the year 1990. We report the estimates of β B and β H in column (1) of Table 3.
Our estimates and sample differ somewhat from those of Neal and Johnson (1996) and Bollinger (2003)  In column (2), we repeat the analysis but, motivated by our theoretical reasoning above, we now allow for the error term to have a location specific intercept, that is, for each location j we have ij = η j + e i . The inclusion of the location fixed effects increases the absolute value of the estimated disparity coefficients for blacks, and especially for Hispanics.
Apparently, blacks and Hispanics disproportionately live in locations in which the wages of non-Hispanic whites are relatively high.
We turn next to the regressions in which we include a standardized measure of cognitive skills. In this case we use the results of the Armed Forces Qualification Test (AFQT), as do Neal and Johnson (1996). Our specification now, like Bollinger's (2003), is the AFQT score T i is entered linearly. We report the estimates of this equation, without the location fixed effects, in column (3). Results are again quite similar to those of Neal and Johnson (1996) and Bollinger (2003). We cannot reject the hypothesis that non-Hispanic men earn the same as non-Hispanic white men. Indeed, the point estimate for β H is positive.
For black men, the coefficient is remains negative and statistically significant, but is reduced substantially in absolute value, from −0.25 to −0.06.
Finally, in column (4) we estimate model (13)  Comparison of columns (3) and (4) indicate, in summary, that location plays a large role in determining the level of the earnings gaps. Indeed, implications are striking. We obviously cannot conclude, on the basis of available evidence that all of the wage gap is due to cognitive skill differences between the races that develop at young ages. Even conditioning on the AFQT skill measure, black men are found to earn approximately 13 percent less than their white counterparts in their same labor markets.
In Lang and Manove (forthcoming) outline a model of statistical discrimination and educational sorting which predicts that black workers will acquire more education than whites with similar levels of cognitive ability. 18 Their theoretical reasoning leads them to estimate a wage regression like Neal and Johnson's, but in which both the education and the AFQT skill measure are included as conditioning variables. They find a substantial negative coefficient on the "black" indicator variable when they do so. Of course, given the concerns we raise above, we would also want to include location in such a specification. Thus we estimate regressions (3) and (4)  In any event, we view the results in Tables 3 and 4 as presenting compelling evidence of the importance of conditioning on location when comparing the earnings of groups with differing locations.
17 Interestingly, in comparison to the estimates for the 1979 cohort, there is a sharp decline in the magnitude of the coefficient on the AFQT test. There are many possible explanation for this decline. We mention five. First, the cohorts are much different in age at this analysis: 25 to 32 for the 1979 and 22 to 26 for the 1997. Participation patterns may differ dramatically between those ages, as well as by the type of jobs held by respondents. Second, for the 1979 cohort, the Department of Defense provided the test scores using item response theory. For the 1997 cohort, however, the Department of Defense has been unwilling to provide the norming, so the test was normed by staff at the Center for Human Resources Research at Ohio State without benefit of the individual test items. It is possible that the inability to use individual test items may have substantially reduced the accuracy of the test norming. Third, the Department of Defence has moved to computer assisted testing for the AFQT; it is possible that the new AFQT is less predictive of civilian labor market success. Fourth, it is possible, though in our view unlikely, that the economic rewards to cognitive skills of young workers have declined. Finally, the respondents in 1979 cohort were much older (ages 15 to 23) when they took the test than the 1997 cohort (ages 12 to 17). It is possible that AFQT test is a less reliable predictor of labor market performance when given at young ages. 18 See also Lang and Lehmann (forthcoming) for an enlightening discussion of discrimination in labor markets, including an extensive set of references to the extant literature.

What if Preferences are Non-Homothetic?
The structural models we estimate above rely on homotheticity of individual preferences. As it turns out, matters become substantially more complicated if preferences are not homothetic. In particular, the steps we took in deriving equation (6) no longer pertain; equilibrium racial wage disparity varies by location. We examine these issues by looking at two cases-one in which there are location-specific differences in productivity and one in which there is variation in local amenities.
Suppose that minority workers have (unobserved) lower levels of human capital than majority-group workers. Suppose also that there is variation across cities in productivity. 19 The city with higher productivity will have higher wages and in consequence will typically have higher housing prices. In this setting, we are interested in learning how race wage gaps vary across locations.
Continue to let u 1 and u 0 be utility levels, respectively, of minority and majority workers.
Given that minority workers have a lower level of human capital, and thus within each city lower wages, their utility will also be lower; u 1 < u 0 . The equality index in a given city with a housing price p is I = e(p, u 1 ) e(p, u 0 ) .
We want to know how this index in a low-price, low-productivity city compares to the index in a higher-price city. We conduct this thought experiment by evaluating the derivative of the equality index with respect to the housing price: With a bit of algebraic manipulation we can rewrite (14): Shephard's lemma indicates that the derivative of the expenditure function with respect to p is the demand for housing. So (15) can in turn can be written in terms of the budget 19 While there has been much work on possible causes of such productivity differences (e.g., see Acemoglu, 1996, Glaeser andMare, 2001, and other work on agglomeration), we are agnostic here about the source of this variation.
shares of housing for minority workers and majority workers, respectively s 1 H and s 0 H : This latter expression is positive if minority workers allocate a higher share of their income to housing than do their majority counterparts. Given that minority workers have relatively lower income, this amounts to the assumption that the income elasticity of housing is less than one. As we mention above, there are some estimates that place the permanent income elasticity of housing demand near one, but others do suggest that it is less than one. 20 If so, Given that cities with relatively high productivity are also cities with higher housing prices in this example, we expect that the wage equality index will be higher in high-productivity cities than in low-productivity cities. This means that for a disadvantaged minority, the equality index will be closer to 1; the proportional nominal wage gap will be smaller (in absolute value) in the high-productivity city.
It is quite easy to explain the logic of this proposition. Suppose individuals live in one of two locations-to take a concrete example, say Memphis and Chicago in 1940-and suppose that in each location black workers (the minority in this example) earn less than their white counterparts because of differences in human capital. Suppose further that all workers are more productive in Chicago, owing perhaps to Chicago's industrial agglomerates. In equilibrium we expect Chicago to have higher wages than Memphis and also to have a relatively higher housing price. What about the black-white wage gap in the two cities?
Given that the elasticity of demand for housing is less than one, the relatively high housing price in Chicago places a greater burden on the (poorer) black workers than the (richer) white workers. Thus if both black and white workers are indifferent between living in Memphis and Chicago, as they must be in equilibrium, black workers will require a larger "Chicago wage premium" than will white workers; the proportional gap between black and white wages will be smaller (in absolute value) in Chicago than in Memphis. Thus the equality index is higher in the more expensive city (Chicago). This is what (17) shows.
The same conclusion follows if the minority wage gap is instead generated by labor market discrimination rather than human capital differences. Notice, first of all, that under our assumption of costless mobility, a discriminated-against black worker will be willing to live in either city, Memphis and Chicago, only if utility is the same in the two locations. Thus the equilibrium condition (2) continues to hold, as do our subsequent derivations, leading to (17). Again, the resulting wage disparity must be smaller in Chicago than in Memphis.
Intuitively, the utility cost must be the same in the two places, and this can happen only if the proportional wage gap is smaller in the location with higher housing prices.
Another mechanism for generating price differentials across locations is differentials in location-specific amenities. In general, the value of a location-specific amenity will vary according to individuals' incomes. For example, good public transportation might be more valuable to individuals who cannot afford a car and good public education is more important to people who don't view private education as a viable option. Similarly, variety in gourmet restaurants is typically more valuable to wealthy individuals. In turn, the value of amenities will be correlated by race if there are race-related differences in income.
As in the example above (with location-specific differences in productivity), the equilibrium black-white wage gap varies across locations. But in this case we can be less certain about the relationship between the black-white wage gap and housing prices.
Our theoretical analysis implies, in short, that there will likely be differences across cities in equilibrium black-white wage gaps if our assumption of homothecitiy in preferences is violated. With this in mind, we provide in Table 5 (4) of Table 1). 21 The most striking feature of these statistics is the wide variation across cities in estimated log wage gaps. Consider, for example, the estimates from 1940. Southern cities generally 21 As discussed in the appendix, we lack the necessary data to conduct this exercise for 1960, and so omit that year from our analysis.
had the largest gaps-in the neighborhood of −0.70 to −0.80-and these gaps are in some cases twice as large (in absolute values) as the gaps observed in cities with the smallest wage gaps. If this variation is an equilibrium phenomenon, it could reflect that productivity in general is higher in cities outside the South.
Of course there are other plausible explanations for this variation. For example, Charles and Guryan (2008) emphasize that there is substantial variation across U.S. locations in prejudice, and that this variation is responsible for some of the variation in black-white gaps in wages. Also, Card and Krueger (1992) document that in the early part of the twentieth century the quality of education afforded African American children was particularly poor in much of the South. Both of these factors are surely at work in shaping the observed patterns.
More generally, there is good reason to think that observed outcomes in 1940 are not an equilibrium outcome. After all, this year was near the beginning of the epochal "second great migration," during which millions of African Americans migrated out of the South, in part, no doubt, to escape poor economic conditions in the South. While we are convinced about the value of thinking about racial wage gaps in an economically interpretable equilibrium setting, we are mindful also that the equilibrium assumption is often unrealistic in location models, and that this concern might be especially germane for the application at hand.
Our general point, in any event, is that if we want to look at black-white wage convergence in the U.S. since 1940 we miss a great deal if we simply look at national averages. Returning to Table 4, which considers cities separately, we notice interesting differences in trends black-white convergence that occurred uniformly across the country.

Concluding Remarks
We have described a conventional model in which prices vary across location. Our model rationalizes a simple approach to estimating racial wage gaps. We show, in particular, that the traditional approach of including a race indicator variable in a Mincer wage regression provides an economically interpretable estimate if one includes location fixed effects in the regression. Of course, most empirical analyses of the black-white wage gap do not include such fixed effects. Thus we revisit two important applications: Our first empirical exercise entails an update of Smith and Welsh's classic work on the evolution of black-white wage disparity among men. We find that for the 1940 to 2000 period, we overestimate black-white wage convergence by 45 percent when we fail to condition on location. 22 Our second application is based on the important work of Neal and Johnson (1996).
Following their approach, when we do not condition on location, we find that a very large proportion (approximately three quarters) of the black-white wage gap among men is accounted for by differences in measured cognitive ability that emerge when these workers were young. This same approach shows that measured cognitive ability differences account for all of the wage gap between Hispanic and non-Hispanic white men. Inferences are quite different when we include location fixed effects. In particular, the absolute value of the estimated racial and ethnic log wage gaps are considerably larger when location fixed effects are included in regressions.
The results of this latter exercise reinforces Lang and Manove's (forthcoming) arguments that the black-white wage gap is not due primarily to cognitive skill gaps that emerge in childhood. Furthermore, our approach of controlling for location shows that Lang and Manove may be actually understating the empirical evidence in support of their contention.
More generally, we present concerns about the assumptions that lead to our structural model. In particular, if preferences are not homothetic, it is no longer the case that the racial log wage gap is a constant across locations, even when the racial utility gap is the 22 As we note above, pessimistic as our results are, they may still understate the current levels of labor market disparity, as we do not account for racial differences in participation or unemployment. See, e.g., Black (1995), Chandra (2000), Neal (2006), and Ritter and Taylor (forthcoming).
same across locations (as it must be in equilibrium). Moreover, for many applications, the equilibrium assumption is probably not tenable. A great deal of work remains to resolve these issues as economists seek to better understand the nature of racial inequality in labor markets.      1940 1950 1970 1980 1990

Data Appendix for Census Analysis
All of the Census data for this paper are taken from integrated data sets of the Public Use Micro Samples (IPUMS) that were released in each of these Censuses, 1940through 2000. See Ruggles, et al. (2008 for details.
Respondents were asked about their earnings in the previous year, the number of weeks worked that year, and, at least for the 1980-2000 Censuses, the usual hours worked that year. Baum-Snow and Neal (2009), however, document systematic biases that differ by race and sex in responses to hours worked. We thus investigate results for both weekly earnings and hourly earnings, finding similar results. We report results for weekly earnings.
Our goal is to provide an analysis similar to that of Smith and Welch (1989). Toward that end, we make many data-use decisions that parallel theirs, though there are differences that we outline here. Like Smith and Welch we restrict our analysis to workers who work at least 27 weeks. As for age restrictions, Smith and Welch consider men aged 16 to 64. We are concerned about the growth of enrollment in high school and college, and we do not want to worry about decisions of "early retirement," so we limit our analysis to men 25 to 55 inclusive. To deal with the issue of schooling, Smith and Welch drop men from their sample who are enrolled in school if they work less than 50 weeks a year.
Given our age restrictions, we find that adjustment to be unnecessary. Smith and Welch exclude unpaid family workers, military personnel, and the self-employed who are not in agricultural. We also exclude unpaid family workers, military personnel, the self-employed, and all agricultural workers. 23 We also follow Smith and Welch in limiting the sample to workers whose reported weekly earnings meet a minimum limit on weekly wages and an upper limit. The adopted limits are as follows: 23 Because of the increased mechanization of agricultural production in the U.S., there has been a dramatic reduction in farm labor and a corresponding increase in the size of farms; farming has become quite capital intensive. It is therefore difficult to separate the returns to capital from the returns to labor. We exclude wage-and-salary agricultural workers because payments to workers often involve payments in-kind, which makes the valuation of the wage paid difficult. Of course, the exclusion of agricultural workers has little effect in 2000, but represents a major exclusion for the early years. Excluding self-employed agricultural workers has the added advantage of rendering the 1940 Census compatible with subsequent Decennial Censuses, as the Census did not ask for farm earnings in 1940. An important concern with the Census data is item nonresponse. Respondents occasionally choose not to answer questions about their age, race, ethnicity, or education level.
More frequently, respondents omit answers to questions about hours worked or earnings.
Our approach is to drop respondents who do not answer questions about age, race, Hispanic status, education, or earnings. We do, however, increase the weights on other respondents with identical ages, race, and education levels to reflect the missing data by using inverse probability weighting. To be precise, we estimate the probably of a nonresponse, or where x indexes the age-race-education-location cell, and then we construct weights, w 1 , where w 0 are the initial Census weights. Thus, if half the people in the age-race-educationlocation cell do not respond to their earnings or hours worked questions, the responders within the cell have their weights doubled. 24 This procedure implicitly assumes that data are, conditional on the age-race-education-location cell, missing at random. Because we condition on age, race, education, and location, this procedure also replicates the Census joint distribution of the age-race-education-location variables. that the education questions exhibit significant measurement error and that the degree of measurement error is correlated with race. Moreover, there was a dramatic increase in the educational attainment of Americans over the period. For instance, in 1940 88 percent of blacks and 64 percent of whites between the ages of 25 and 60 did not have a high school education, and only 2 percent of blacks and 8 percent of whites had a bachelor's degree or better. By 2000, only 9 percent of blacks and 5 percent of whites did not have a high school degree while fully 33 percent of whites and over 19 percent of blacks had a bachelor's degree or better.
In our regression analysis we treat education in a non-parametric way, and given the available data, we use the following ten education categories: no formal education or kindergarten only, 1 to 4 years, 5 to 8 years, 9 years, 10 years, 11 years, 12 years, some college but no bachelor's degree, bachelor's degree, and more than a bachelor's degree.
Finally, there is the issue of the measurement of location. Because of the growth in cities and changes in disclosure policy, the identification of metropolitan statistical areas (MSAs) varies over time. In 1960 (the first public use micro sample that the Census Bureau released), the only geography identified was State of residence. As a result, we cannot conduct the same location analysis of interest to us with the 1960 data; we use only an urban indicator interacted with an indicator for state of residence. In 1940In , 1950In , and 1970In through 2000 we use MSA of residence for those respondents living in a MSA. For those respondents not living in an identified MSA, we use an indicator for state of residence. Hence, we exploit the geographical variation that is generally available to us. There are, however, a few additional noteworthy limitations: First, residents of some current MSA's are not separately identified in the early censuses, but are so identified subsequently. For example, in the 1940, Orlando residents are treated as individuals living in "rural" Florida, but in later years are broken out as part of an Orlando MSA. Similarly, Las Vegas is identified only starting in 1970. There are a host of smaller towns that are only identified in later years. Moreover, MSAs can be created from regions that were previously a part of different MSAs. This is a particular problem in the densely populated areas of the east and west coasts. Finally, for areas that are only identified as "rural" we may be mixing residents from very different areas of a given state.
For example, this designation mixes residents of the desert areas of Southern California with residents of rural Northern California, who may face very different labor markets and price levels.
For the 1940 through 1970 Censuses, we use a one-percent sample of respondents, and from 1980 to 2000, we use the five-percent sample, which, along with population growth, provides much larger sample sizes and much more precise estimates. 25 Finally, we note an important data limitation with the 1950 Census. In 1950, only the "sample line" respondents were asked about education and earnings by the Census Bureau. Hence, only about 3.3 percent of the population was given these questions. Thus, estimates from the 1950 Census are considerably less precise than estimates from even the 1940 Census.
In 1960 and 1970, the Census asked only for hours of work and weeks of work on intervals.
To impute the actual levels, we took information from the 1980 Census and calculated the average weeks (or average hours) conditional on the being in the relevant category. The imputations are as follows: Interval Imputed weeks Interval Imputed hours 1-13 weeks Prior to 1980, the Census did not ask the usual hours worked so we used hours last week as a proxy. In 1980, conditional on both reports being positive, the correlation is only 0.61.
While quite low, this correlation is not materially different than those found in validation studies; see Barron, Berger, and Black (1997) for a discussion.