Inequality and City Size∗ Nathaniel Baum-Snow, Brown University & NBER Ronni Pavan, University of Rochester October, 2010

Abstract Between 1969 and 2007 a strong monotonic relationship between wage inequality and city size has developed. In this paper, we investigate the causes of the city size inequality premium and its relationship with the growth in overall wage inequality. We find that one-quarter to one-third of the overall increase in hourly wage inequality in the United States from 1979 to 2007 is explained by city size independent of observable skill. While this influence has occurred throughout the wage distribution, the fraction of the increase in the lower half of the wage distribution explained by city size is at least 50 percent larger than that in the upper half of the wage distribution. More rapid growth in within skill group inequality in larger cities has been by far the most important force driving these city size specific patterns in the data. Differences in the industrial composition of cities of different sizes explain 19 to 32 percent of this city size effect. Our evidence on the evolution of wage inequality in cross- sections of cities improves understanding of the role of urban agglomerations in contributing to the expansion of inequality in marginal products of labor over time.

∗ We gratefully acknowledge financial support for this research from National Science Foundation Award SES - 0720763. Cemal Arbatli and Ee Cheng Ong provided excellent research assistance. The paper has greatly benefited from discussions at the University of Chicago Booth School micro lunch, CREI and the North American Regional Science Council meetings. In addition, we thank Satyajit Chatterjee for his helpful comments.

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1

Introduction

Juhn, Murphy & Pierce (1993), Card & DiNardo (2002), Lemieux (2006), and Autor, Katz & Kearney (2008) among others have documented a sharp rise in U.S. wage inequality since 1980, especially at the top end of the wage distribution. These studies discuss skill-biased technical change, capital-skill complementarity, the shifting composition of the working population, the decline in the real value of the minimum wage and the decline of unionization as potential causes of this rise in wage inequality. It is less widely recognized that over this same time period, a strong positive relationship between wage inequality and city size has also developed. In the 2004 to 2007 period, the variance of log hourly wages was 0.28 in rural areas and roughly monotonically increasing to 0.53 in the largest three metropolitan areas. By contrast, in 1979 the variances of log hourly wages for rural areas and the three largest metropolitan areas were 0.19 and 0.24 respectively. Similar patterns are also seen in other commonly used measures of wage inequality. In this paper, we investigate the causes of the emergence of the city size inequality premium from 1979 to 2007 and its relationship with the growth in overall wage inequality. Our analysis reveals mechanisms by which overall wage inequality has increased over time and also improves our understanding of the reasons behind emerging higher wages and wage inequality in larger cities. We find that 25 to 35 percent of the overall increase in inequality in the United States from 1979 to 2007 is explained by city size independent of observed skill potentially correlated with city size. While this influence has occurred throughout the wage distribution, the portion of the increase in inequality in the lower half of the wage distribution that can be attributed to city size-specific factors is more than 50 percent larger than that in the upper half of the wage distribution. Commensurate with Autor, Katz & Kearney’s (2008) evidence using national data, we demonstrate that growth in within group inequality has been the most important force driving these city size specific patterns in the data. That is, most of the impact city size has had on the increase in inequality nationwide has come because wages have become more unequal within skill groups in larger cities than in smaller cities. While this could reflect increased ability dispersion within observable groups in larger cities, we think it is more likely to reflect more rapid increases in the return to unobserved skill in these locations. We show that greater increases in observed skill premia in larger cities played a relatively small role in generating the strengthening relationship between inequality and city size. We also find that changes in the sorting of population sub-groups with higher 2

wage inequality toward larger cities has had little effect on overall inequality. For this reason, we suspect that increased sorting across locations on unobserved skill is also not an important explanation. After accounting for differences in one-digit industry composition across locations, our estimated relationship between the variance of wages and city size is reduced by about six percentage points. Most of this industry effect comes from faster growth in the variance of wages within industry/skill groups that have always been disproportionately located in larger cities. This evidence is consistent with Autor et al.’s (1998) evidence that skill upgrading, particularly in computer-intensive industries, has been an important mechanism behind the rise in wage inequality. It is also consistent with Bacold et al.’s (2009) evidence that while various measures of cognitive and noncognitive skills are similar across cities of different sizes, the returns to certain soft and technical skills are higher in larger agglomerations. Figure 1 documents recent trends in wage inequality for our sample of white men. As has been documented elsewhere, the 1980s saw a dramatic increase in wage dispersion at all points in the distribution and a decline in real wages for most workers. During the 1990s, those in the top quarter of the wage distribution saw wage increases and increasing inequality, with relative stability throughout most of the rest of the distribution. One lesson from the 1990s data is that it is impossible to fully understand trends in inequality without examining the upper and lower portions of the wage distribution separately. Between 1999 and 2007, the wage distribution evolved similarly as during the 1980s, with increased inequality emerging at all points in the distribution. The only difference is that wages in this later period kept up slightly better with inflation. Data on factor prices and quantities in cross-sections of cities at recent points in time exhibit many of the same features as factor price and quantity data in the time series since 1979. In particular, even though larger cities have greater log wage gaps between skill groups, they also have larger quantities of skilled workers relative to unskilled workers and this factor ratio has increased over time. The fraction of the prime-age male working population with a college degree grew by two percentage points in rural areas between 1980 and 2007, with this growth roughly monotonically increasing to 13 percentage points in the largest cities over this time period.1 Katz & Murphy (1992) and Bound & Johnson (1992) interpret analogous patterns in the time series as being driven by skill-biased technical 1

Figure A1 presents data on the fraction of workers who are college graduates over time by city size.

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change. Their empirical observations have precipitated an extensive theoretical literature, including Acemoglu (1998) and Galor & Moav (2000), that attempts to better rationalize the sources of skill-biased labor demand shifts. Krusell et al. (2001) alternatively propose that this pattern is attributable to a combination of capital-skill complementarity and declining capital rental rates relative to input costs of other factors, a hypothesis echoed in part by Autor et al. (2008). The fact that the skilled/unskilled labor factor ratio and wage inequality have both increased more rapidly over time in larger cities than in smaller cities is evidence that agglomeration economies represent a crucial ingredient to understanding changes in relative skill prices nationwide. Urban agglomerations may have directly facilitated the development of new more skill-intensive technologies that are then most efficiently implemented in the same larger cities. This "Jacobs" externality story of skill and agglomeration biased technical change is consistent with empirical evidence for young industries (Henderson et al., 1995 and Glaeser et al., 1992). Such technical change would have resulted in labor demand schedules for more skilled workers shifting out more rapidly over time in larger cities than in smaller cities. Given the right environment, capital-skill complementarity may also have been a key ingredient generating the emergent city size inequality premium. As the city size wage gap has increased since 1980 (Baum-Snow & Pavan, 2009), the relative cost of capital has declined more rapidly in larger cities. Agglomeration-biased, but skill-neutral, technical change may have generated this increase in the relative cost (and productivity) of labor in larger cities, thereby indirectly leading to greater increases in wage inequality in large locations through capital-skill complementarity. Factor-neutral technical change in the presence of agglomeration economies would also lead to larger increases in wage inequality in larger cities given capital-skill complementarity. Each of these two mechanisms generate larger movements along skilled workers’ labor demand schedules than those of unskilled workers in larger cities than smaller cities. Whether cities’ role in generating increased wage inequality has come from skill-biased technical change, agglomeration-biased technical change or skill-neutral technical change in the presence of agglomeration economies and capital-skill complementarities, it is clear that productivity differences across cities of different sizes have been instrumental in generating the emerging relationship between city size and wage inequality. Our empirical results therefore indicate that the nationwide rise in wage inequality would have been significantly less rapid if such agglomeration economies did not exist. 4

This paper proceeds as follows. Section 2 discusses in more detail the evolution of the city size inequality premium since 1979 and shows how we construct the data. Section 3 describes our empirical methodology. Section 4 investigates the independent role of city size in generating growth in several measures of wage inequality. Section 5 characterizes the role of shifting industry compositions. Finally, Section 6 concludes.

2

Wage Dispersion and City Size

2.1

Patterns in the Data

Table 1 presents a set of facts about the evolution of various measures of hourly wage inequality over time. It shows that in each decade since 1979 the variance of wages has increased, as has the 90-50 percentile gap. In particular, the variance of wages increased by 0.39 or 85 percent and the 90-50 gap increased by 0.28 or 54 percent between 1979 and 2007. The 50-10 percentile gap also increased in every decade except the 1990s, with a total increase over the study period of 0.17 or 11 percent. While the 1970 census does not contain the requisite data to calculate hourly wages, the variance of weekly wages increased slightly during the 1970s. Columns 4 and 5 of Table 1 present a decomposition of the total variance in wages into observed and residual components. The "Between" component is based on means of 330 age, education and city size cells. The "Residual" component is based on within-cell residuals from these means. We see that while the between variance increased at a faster rate between 1979 and 2007, the residual component of the variance increased by more in numerical terms. Columns 6 and 7 show the 90-50 and 50-10 percentile gaps in residuals. Both of these components increased during the study period, with the 90-50 gap increasing much more quickly than the 50-10 gap.2 Figure 2 demonstrates that a positive relationship between wage inequality and city size emerged over the full distributions of wages and city size concurrently with the growth in overall wage inequality. For the purpose of this paper, we index metropolitan area size to be 0 in rural areas and 1 to 10 to represent deciles of the urban population distribution in year 2000. That is, in 2000 approximately 10 percent of the metropolitan area population nationwide resides in each of our city size categories. For other years, we maintain the 2

While the total variance can be decomposed into "Between" and "Residual" components, there is no natural decomposition possible for other measures of distribution spread.

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same assignment of metropolitan areas to categories based on populations in 2000. We experimented with other similar indexes of metropolitan area size, including using contemporaneous deciles and/or fixed cutoff populations over time, and they all generate very similar results. We prefer our measure because it eliminates the possibility that changes in the relationship between city size and inequality could have been generated by a few metropolitan areas that changed locations in the city size distribution. In addition, our measure provides a clear way to assign metropolitan areas to city size categories in 20052007 for which there is no reliable MSA level population data. Incidentally, our measure also generates the steepest relationship between city size and inequality in 1979 of the four measures that we examined.3 Figure 2 Panel A shows that while the variance of hourly wages was almost flat as a function of city size in 1979, its slope increased in each subsequent decade. The variance of hourly wages differed by 0.05 between rural areas and the largest metropolitan areas in 1979 whereas by 2004-2007 this gap had increased to 0.25. The variance of weekly wages in 1970 was even flatter in city size than the variance of hourly wages in 1980. Figure 2 Panels B and C show the evolution of hourly wage distribution percentile gaps by city size over time. Panel C shows that the 50-10 percentile gap increased in all types of locations and changed very little in slope with respect to city size during the 1980s. It saw its greatest increase in slope with respect to city size during the 1990s, even though the average level actually changed very little. Both slope and level increased after 1999. Comparison with Figure 1 shows an interesting evolution of the relationship between inequality at the bottom end of the wage distribution nationwide and that as a function of city size. Over the course of the 1980s, the increase in inequality in the bottom part of the wage distribution occurred within cities of all sizes simultaneously. During the 1990s, while inequality changed little in the bottom part of the wage distribution nationwide, the 50th-10th percentile gap became strongly increasing in city size. A decline in wage inequality in rural areas and small cities and an increase in larger cities generated this pattern. Finally, after 1999 both the level and slope of the 50-10 gap with respect to city size increased. Comparison of Figure 2 Panel B and Figure 1 reveals a different story about the upper part of the wage distribution. It shows that the city size inequality premium in 3

Others including Ciccone & Hall (1996) use density rather than metropolitan area population as a way of capturing the extent of agglomeration forces. Depending on the importance of local transportation and communication costs, each measure can be justified by standard urban theory. We find population to be a more natural empirical measure as it does not require data on developed area. The correlation between year 2000 MSA population and population density in our data set is 0.44.

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the top part of the wage distribution increased in every period studied after 1979. This is consistent with the evolution of the overall 90-50 gaps seen in Figure 1.

2.2

Data Construction

Our primary data source for demographic information and wages is the Census Public Use Microdata 5 Percent Samples from 1980, 1990 and 2000 plus the 2005-2007 American Community Surveys (ACS).4 We choose these data sets so as to achieve large enough samples within metropolitan areas in order to precisely estimate and decompose wage distributions by metropolitan area size categories. We limit our analysis to white men ages 25-54 who report working at least 40 weeks, 35 usual hours per week and who earn at least 75 percent of the federal minimum wage in each year. The full-time full-year limitation allows us to measure marginal products of labor for individuals who are less likely to be constrained in their residential locations by family or education considerations. We use white men only to limit the possibility that changes in discrimination and patterns of labor market attachment for women and non-whites influence our estimates. Our earnings measure is the hourly wage calculated by dividing annual income by weeks times usual hours worked.5 Annual income from the census is for the previous calendar year while that from the ACS is for the year ending in the (unobserved) survey month. Therefore, we sometimes report ACS wages as being for the period 2004-7. Many other studies that examine trends in inequality in the United States use the Current Population Surveys instead. We found that the CPS does not provide sufficient sample sizes and geographic detail to be the optimal data set for our purposes. We consistently use year 1999 definition county based metropolitan area (MSA) geography throughout the analysis. Unfortunately, the most disaggregated census micro data geography of County Groups in 1980 and Public Use Microdata Areas in 1990 and 2000 in many cases does not match up to MSA geography. As such, our spatial allocation of individuals reported as living in regions that straddle MSA boundaries is imperfect. We allocate those living in straddling county groups or PUMAs to the subregion with the largest population. We assign each metropolitan area to one of ten population size categories and all other areas to the remaining non-metropolitan size category. 4

We do not use the 2008 ACS data because it only reports intervalled weeks worked. Measurement error is an additional justification for using full-time full-year workers. Baum-Snow and Neal (2009) demonstrate that there exists significant measurement error in hourly wages for part-time and part-year workers in the census. 5

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An examination of the population distribution across city size categories reveals some shifts over time. The distribution of total U.S. population across size categories in 1980, 1990 and 2000 built using aggregates of county populations reported in the census shows that while the population of rural areas declined from 22 percent in 1980 to 20 percent in 2000, the fraction of the population living in one of the top three metropolitan area size categories increased from 24 percent to 26 percent during this period. While this shift of population toward larger metropolitan areas is interesting, we find no evidence that it can explain any part of the increase in the city size inequality premium.

3

Measuring the Role of City Size

In this section, we develop a methodology for evaluating the effects of city size independent of observed skill on changes in various measures of wage inequality. We begin with the nonparametric statistical decomposition of quantity and price components of changes in the wage distribution proposed by DiNardo et al. (1996) and adopted by Autor et al. (2008) for analysis of U.S. data and Dustmann et al. (2009) for analysis of German data. We use similar types of decompositions to build counterfactual nonparametric wage distributions holding the elements of prices and quantities influenced by city size constant at their 1979/1980 profiles. We use these counterfactual distributions to calculate counterfactual inequality measures absent city size effects. In the construction of counterfactual wage distributions absent city size effects, we impose that skill price distributions in rural areas are unaffected by city size effects in each year. As such, we construct counterfactual skill price distributions to maintain their 1979 shapes relative to rural distributions within demographic cells but impose changes in spread over time that are unconditional on location. This benchmarking to rural locations is a natural choice. As seen in Figure 2, rural wage inequality consistently increased the least of all location types since 1979. Additionally, the industrial structure of the economy has changed the least in rural areas. Therefore, our counterfactual distributions capture how inequality would have evolved had larger cities’ technological gaps with rural areas not expanded beyond their 1980 levels. In other words, we measure inequality absent the agglomeration-biased technical change that has taken place since 1980. Inspection of the following modified Herfindahl index by location confirms that rural areas have experienced the most stable industrial composition since 1979.  denotes the

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share of employment in industry  at time . 

 =

1X ( −  )2 2

(1)

=1

This index can be thought of as a monotonic transformation of the distance in  dimensional space between industry compositions at times  and . It is straightforward to show that the index is bounded between 0 and 1. Using this index, we find that rural areas experienced the smallest change in industry composition between 1979 and 2007 and during each of the intervening study periods of any location size category. This pattern holds for all education groups as well except for "dropouts" in the 1990s and "more than college" in the 1980s. In addition, this pattern looks very similar when the index is instead calculated using three-digit industries. These results are reported in Table A1. We now specify how we construct counterfactual residual and wage distributions absent city size effects. We denote the observed distribution of wages in each year as  (). The wage distribution can be decomposed into "price" components, or the distribution of wages conditional on skill group  and city size category , and "quantity" components, represented by the distribution of skill groups across locations. Z (2)  () = [ ( −  ( ) | )] (|) () In this expression,  ( ) is the mean wage of skill group  residing in location type  at time , while  (·| ) is the residual wage distribution for this skill group-location combination. Conditional on observables, the distribution of wages coincides with the distribution of residuals plus a mean shift. Once we integrate this distribution over quantities in each skill group-location combination, we obtain the full wage distribution  (). To facilitate working with city size-specific components of quantities, we decompose the joint quantity distribution  ( ) into shares of each skill group  in each city size category , denoted  (|), and overall shares of the population in each skill group, denoted  (). To build counterfactual wage distributions, we replace each component with an analog that removes all city size influences that developed after 1980. To hold the quantity component of wages or residuals at their 1980 city size profiles, we replace  (|) with 1980 (|) in Equation (2) above. We also present some counterfactual results in which we

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return the full quantity component of wages or residuals to their 1980 state by additionally replacing  () with 1980 (). Generating counterfactual price components of these distributions that take out changes in city size effects since 1979 is a more complicated process. Our goal is to maintain the 1979 distributions of prices across city sizes within skill groups while allowing the distribution of prices to change between skill groups. To achieve this goal, we hold the relative price distributions fixed across city size categories within skill over time while allowing price distributions by skill group unconditional on location to change. We center these counterfactual distributions around the percentiles at which residuals equal zero in the actual  (| ) distributions. As discussed above, we treat rural areas as the reference location. The full counterfactual price distributions have residual distributions  (| ) and mean components  ( ) which we explain in turn. Construction of the counterfactual residual price distributions  (| ) is carried out as follows. Because the key adjustment is done by percentile, it is convenient to initially −1 express residuals in terms of their inverse cumulative distribution functions  (| ). We expand (or contract) observed inverse cumulative distribution functions by adding on the growth in the difference between residuals at the given percentile  and the reference percentile for the location  and location 0 distributions in 1980. We additionally adjust for the possibility that the percentile of the mean of each residual distribution is not the same in 1980 as it was in subsequent study years. In mathematical terms, this amounts to nonparametrically calculating e −1 (| ) = −1 (| 0) + [−1 (| ) − −1 (| 0)]   1980 1980 −1 −[−1 1980 ( | ) − 1980 ( | 0)]

(3)

for each percentile . We denote the percentile at the mean of the residual distribution for demographic group  in location  at time  as  , such that  solves −1  ( | ) = 0. In order to maintain mean 0 residual distributions within demographic group and city size category, we demean the resulting probability distribution functions e (| ) =  e  6   (| ) to generate counterfactual residual price distributions  (| ). We calculate the counterfactual mean component of wages  ( ) in one of three ways. Our version "a" of counterfactual means maintains the same gap between demographic 6 The choice of the reference percentile  makes little difference as long as it is near the center of the relevant price distribution.

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means and city size cells as in 1980 as follows. Z Z   ( ) =  ( ) (|) + 1980 ( ) − 1980 ( )1980 (|)

(4)

When also maintaining the 1980 distributions of workers across locations within demographic cells, this calculation becomes the following, which we call ””. Z Z  (5)  ( ) =  ( )1980 (|) + 1980 ( ) − 1980 ( )1980 (|) The assumption underlying the formulations in Equations (4) and (5) is that absent changes in technology or market conditions affecting skill prices biased towards agglomerations, mean wages would have evolved in exactly the same way except that the slope with respect to city size within demographic group would not have increased after 1980. A reasonable alternative assumption is that had agglomeration-biased changes not occurred, the profile of mean wages in each demographic group would have the 1980 slope but instead be anchored at the location 0 mean. In mathematical terms, this counterfactual "b" can be expressed as follows:  ( ) =  ( 0) + 1980 ( ) − 1980 ( 0)

(6)

Note that this formulation contains no embedded quantity components. As we show below, the counterfactual version a wage distributions are only slightly less spread out than actual wage distributions while distributions built using counterfactual version b are significantly tighter than those observed in equilibrium. Table A2 mathematically specifies all of the counterfactual distributions that we use to generate the results reported in Tables 2 and 3.

4

Main Results

Table 2 presents the fraction reduction in the total growth of measures of residual inequality reported in Table 1 Columns 5-7 under four counterfactual environments. Column 1 presents reductions after holding the distribution of individuals across skill group and location size constant at 1980 levels. This is similar to Lemieux’s analysis (2006), in which he finds that shifts in quantities across skill groups played a central role in explaining the rise of residual wage inequality during the 1990s. Panel A reports that shifts across skill

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groups and locations accounted for 19 percent of residual wage growth between 1980 and 2000, dropping to 13 percent between 1980 and 2007. Inspection of Column 1 in Panels B and C reveals that this effect is seen throughout the wage distribution, but is more pronounced below the 50th percentile. Columns 2 and 3 show the independent effects of city size on changes in residual wage inequality. They both show that larger increases in the dispersion of skill prices in larger cities than smaller cities represents a more important force behind expanding residual wage inequality than the fact that the fraction of the population in skill groups featuring high residual dispersion has increased. In particular, city size-specific factors independent of observed skill explain 38 percent of residual variance growth, 28 percent of growth in the residual 90-50 gap and 61 percent of growth in the residual 50-10 gap from 1979 to 2007. Most of these effects are being driven by slower growth in the counterfactual residual distribution during the 1990s than in the other two periods studied. In fact, the counterfactual 50-10 gap actually fell during the 1990s, indicating that increasing price dispersion in larger cities counteracted compression of residual skill prices at the bottom ends of these distributions. Comparisons of results in columns 2 and 3 reveal that changes in the distribution of the population across locations within skill group has had a negligible impact on residual wage inequality. Table 2 Column 4 indicates that 48 percent of residual variance growth from 1979 to 2007 can be explained with demographic quantities and city size. The remainder is explained by changes in skill prices independent of location. This importance of skill prices is somewhat more pronounced for the top half of the wage distribution than the bottom half and during the 1980s and after 1999 than during the 1990s. Table 3 presents results analogous to those in Table 2 but for total hourly wages rather than just their residual component. Columns 1 and 2 show that as with residuals, changes in skill group quantities over time account for about 12 percent of the growth in the variance of wages between 1979 and 2007. Additionally accounting for changes in sorting across locations hardly changes this conclusion. In Table 3 Columns 3 and 4, we present counterfactual reductions in wage inequality due to city size, not taking into account differences across skill groups. Results in these columns demonstrate that more than half of the increase in inequality between 1979 and 2007 is related to city size, though very little has to do with changes in the distribution of the population across cities of different sizes. City size appears slightly more important for understanding increased inequality at the bottom end of the wage distribution than at 12

the top end. These results form a baseline against which we compare results in Columns 5-9, which account for the distribution of skill groups across locations. Column 5 reports the effects of adding on counterfactual residuals used for Table 2, Column 2 onto observed skill/location group means. It shows that adjusting only residuals and not means for changes in city size effects over time generates a 23 percent reduction in the growth of the variance, a 16 percent reduction in the growth of the 90-50 gap and a 33 percent reduction in the growth of the 50-10 gap from 1979 to 2007. As seen in Column 6, adjusting means using counterfactual type a, constructed using Equation (4), hardly changes these results. This indicates that either city size had a much greater effect on changes in the distributions of residual inequality or the "a" mean adjustment does not fully capture how city size changed cell means. Column 7 demonstrates that as with the residual results, changes in the distribution of workers across locations within demographic group had virtually no effect on wage inequality. Columns 8 and 9 report reductions in inequality using the alternative "version b" counterfactual means. These consistently show larger reductions in inequality due to city size of 35 percent of the growth in the variance, 28 percent of the growth of the 90-50 gap and 51 percent of the growth of the 50-10 gap. Columns 10 and 11 show that changes in the quantity of skill groups over our sample period accounts for additional 9 to 14 percentage point reductions in the growth of our three inequality measures of interest. The remaining 56 to 66 percent must be generated by changes in the means and residual price distributions of skills over time. Overall, evidence in Table 3 indicates that factors specific to city size and not skill group generated 25 to 35 percent of the reduction in the variance of log hourly wages between 1979 and 2007 and one-third to one-half of the increase in the 50-10 percentile gap during this period. Increases in residual inequality due to city size and independent of skill group account for the bulk of the importance of city size. This means that residual inequality grew much faster in larger cities than in smaller cities, and this relative growth is not explained by the fact that larger cities are more skilled and the higher skill groups had more rapid growth in residual inequality in all locations. One reasonable interpretation of our results is that the return to unobserved skill increased more quickly in larger cities. Our "version b" results show that the return to observed skill also increased more in larger cities but this increase did not pass through to account for as much of the expansion in observed wage dispersion. Results in Tables 2 and 3 indicate that the effects of city size on inequality in the 1990s 13

exceed those in the other two study periods. Examination of analogous results using 1990 and 2000 instead of 1980 as base years confirms this observation. During the 1990s, city size explains 61 to 72 percent of the growth in the variance of log wages and 39 to 47 percent of the 90-50 percentile gap. Counterfactual 50-10 gaps absent city size effects decline by 49-74 percent more than the actual decline of 0.04 reported in Table 1. In contrast, between 1999 and 2007 city size had a negligible effect on the variance and small effects on the other two measures of distribution studied when using 1999 as a base year, despite the fact that wage inequality grew robustly during this period. Our evidence thus indicates that some sort of agglomeration-biased technical change was important in the 1980s, and especially the 1990s, but less important since then in driving increases in wage inequality.

5

The Role of Industry

To help understand why wage inequality increased more in larger cities than in smaller cities independent of skill group, we take a closer look at changes in the composition of industries as a function of city size. To do so, we add one-digit industries to the set of skill variables accounted for in the previous section.7 Because some two and three-digit industries do not hire workers of all education levels in cities of some sizes, it would be impossible to disaggregate the set of industries examined much more than we do here. We find that 19 to 32 percent of the city size effect independent of skill documented in the previous section is attributable to industry, with most coming through residual wages. This means that industries with increases in within-skill worker inequality are disproportionately located in larger cities. Given the fact that input costs are higher in larger cities, the location choices of such industries is evidence that their production technologies exhibit strong agglomeration economies or capital-skill complementarities. With the strong assumption that the production technology is the same across locations within industry, we would therefore conclude that the role of city size explained by industry comes from movements along labor demand schedules because of differing relative input costs rather than heterogeneity in technologies. In order to operationalize this additional cut of the data, we must make some mild 7

In fact, we use a set of industry categories that additionally disaggregates non-durable and durable manufacturing, transportation from communications and public utilities, and professional services from other types of services.

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parametric assumptions and limit ourselves to using variance as a measure of inequality. Even the 5 percent census samples do not have sufficient sample sizes to allow for nonparametric analysis of wage distributions by age, education, industry and city size. We decompose the variance of hourly wages into "Between" and "Residual" components based on the following regression equation. ln  =  +   +   + 

(7)

where  indexes individual,  indexes age/education (skill) group,  indexes one-digit industry,  indexes location size and  indexes time.8 Note that this specification nests the fully nonparametric specification used for the analysis of skill groups and city size in the previous section. Based on this empirical formulation, we calculate counterfactual variances absent city size effects analogously to those performed for the results in the previous section. As a basis for these counterfactuals, we can write the variance of log wages in year  as  (ln  ) =

X

  ( +   +   ) +



X

  ( )

(8)



where  denotes the share of the total population in year  that is in skill/industry/location cell . It is possible to write the variance in this way because membership in any cell  is exclusive from membership in any other cell. We calculate  ( ) by regressing the squared errors estimated using Equation (7) back on the same set of indicator variables as in Equation (7).9 As in the exercises performed above, we replace any of the three elements in Equation (8) either with values from 1980 or values that adjust for city size effects that emerged after 1980. To calculate counterfactual variances using 1980 quantities, we replace  with 1980 . Our adjustment for city size effects proceeds analogously to that used for the previous section. For means, or components of  , we apply Equation (5) or (6) exactly as is done above. We calculate counterfactual residual variances as follows.   ( ) =  (0 ) +  (1980 ) −  (01980 )

(9)

8 To attain sufficient variation within education and industry, we use five year age ranges for this analysis rather than the two year age groups used for the analysis in the previous section. 9 Instead of using observed shares  , one could estimate the shares using a regression equation analogous to (7). We find that both methods produce very similar results.

15

This method is analogous to the nonparametric method used for the previous section. Because it is more constrained, it generates counterfactual variances that respond slightly less to city size effects than the results reported in Table 2. Table 4 presents the fraction reduction in the growth of between residual and total variances of log wages under various counterfactual environments. Because it includes a more saturated specification, this empirical model generates residual variances that are 0.01 smaller in all years than those reported in Table 1. In each of the three panels of Table 4, we adjust for a different set of observable characteristics when constructing counterfactual variances. Results in Panel A are constructed using the underlying regression specification in (7). For the counterfactual exercises in Panels B and C, we break the between component of the variance into observed and an additional residual component. We only adjust elements that affect this observed component, listed in the panel headers, when constructing the counterfactual variances used to construct numbers in Panels B and C. Results in Table 4 Panel C confirm evidence from Tables 2 and 3 that the fact that skill groups are not randomly allocated across locations reduces the growth in the raw city size inequality premium by 40 to 50 percent. Table 4 Panel B reports results using the same set of predictors as used for Tables 2 and 3 with very similar results. In particular, as reported in Columns 1-3, we find that shifts in the distributions of workers across skill groups and locations account for about 10 percent of the growth in both Between and Residual variance between 1979 and 2007. Column 4 shows that using counterfactual version a, we find that city size had only a 9 percent impact on growth of the between component of the variance whereas column 7 indicates that city size had a 32 percent impact on the between component using counterfactual version b. We find that city size accounts for 31 percent of the growth of residual variance, adding up to a total of 22 percent using counterfactual version a and 31 percent using version b. These numbers are slightly smaller than those reported in Table 3 because the construction of counterfactual residual variances are more constrained and the age cells are larger. In Table 4 Panel A we investigate how much of our estimated effects of city size absent effects of skill can be explained by one-digit industry composition. Comparisons of Panels A and B reveal that one-fifth to one-third of the growth in total variance accounted for by city size independent of skill can be attributed to industry, coming primarily from the residual component. This means that industries with greater wage dispersion within skill groups were more concentrated in larger cities. In addition, Columns 1-3 show that the shifting worker composition across industries generated a similar fraction of the increase in 16

the variance of wages as did shifts across skill groups. However, unreported calculations indicate that as with shifts in the composition of skill groups, these industry composition shifts were not accompanied by shifts in the distributions of workers across location within industry. Table 5 summarizes our results for each of the time periods studied. The first row of each panel presents the fraction of the growth in the variance of each component of wages due to all factors related to city size. The subsequent two rows break down this effect of city size into components due to skill and industry compositions of the workforce. The final row in each panel, marked "Remainder", indicates the portion of the growth in the variance that is due to city size and that we cannot attribute to another factor correlated with city size. These results clearly indicate that understanding changes in wage dispersion within skill groups and industries disproportionately located in larger cities is key to explaining the growth in U.S. wage inequality since 1979.

6

Conclusions

In this paper, we demonstrate that cities have played an important role in the rise of inequality over time. In 1979, there was only a weak positive relationship between inequality and city size while by 2007 a much stronger relationship between these two variables had developed. We demonstrate that city size specific factors can explain one-quarter to onethird of the overall increase of the variance in wages between 1979 to 2007 independent of observed skill. City size has been about 50 percent more important for generating increased inequality in the bottom half of the wage distribution than in the top half, although it accounts for a greater amount of the increase in inequality in the top half of the wage distribution. The most important factor generating the city size specific component of inequality growth is that demographic groups and industries disproportionately located in larger cities experienced larger increases in their wage dispersion in larger cities than in smaller cities. That is, city size has become more complementary with wage dispersion within observed skill groups. It is also true that the skill premium has grown more in larger cities than in smaller cities and rural areas. However, the emergence of the city size inequality premium is not due to systematic migration of demographic groups with greater wage dispersion toward larger cities. We hope that our analysis sparks further research examining reasons for changes in 17

the structure of labor demand using metropolitan area level data. In particular, while we provide evidence that agglomeration has interacted with technical change of some sort, whether skill-biased or skill-neutral, there remains much to be learned about the extent to which the increase in wage inequality that has been driven by this technical change has been augmented by capital-skill complementarity and potentially declining capital costs. As such, a ripe area for future research is to understand how increases in inequality attributable to movements along labor demand schedules have augmented those caused by technical change particularly oriented toward larger cities.

18

References Acemoglu, Daron. 1998. "Why Do New Technologies Complement Skills? Directed Technical Change and Wage Inequality." Quarterly Journal of Economics, 113:3, 10551089. Autor, David H., Lawrence F. Katz and Melissa S. Kearney. 2008. "Trends in U.S. Wage Inequality: Revising the Revisionists." Review of Economics and Statistics, 90:2, 300-323. Autor, David H., Lawrence F. Katz and Alan B. Krueger. 1998. "Have Computers Changed the Labor Market?" Quarterly Journal of Economics, 113:3, 1169-1213. Baum-Snow, Nathaniel and Derek Neal. 2009. "Mismeasurement of Usual Hours Worked in the Census and ACS" Economics Letters, 102:1, 39-41. Baum-Snow, Nathaniel and Ronni Pavan. 2009. "Understanding the City Size Wage Gap" manuscript. Bacold, Marigee, Bernardo S. Blum and William S. Strange. 2009. "Skill and the City" Journal of Urban Economics, 65:2. Bound, John and George Johnson. 1992. "Changes in the Structure of Wages in the 1980s: An Evaluation of Alternative Explanations" American Economic Review, 82:3, 371-392. Card, David. 2009. "Immigration and Inequality" American Economic Review, 99:2, 1-21. Card, David and John E. DiNardo. 2002. "Skill-Biased Technological Change and Rising Wage Inequality: Some Problems and Puzzles" Journal of Labor Economics, 20:4 733-783. Chay, Kenneth and David S. Lee. 2000. "Changes in Relative Wages in the 1980s: Returns to Observed and Unobserved Skills and Black-White Wage Differentials" Journal of Econometrics, 99:1, 1-38. Ciccone, Antonio & Robert Hall. 1996. "Productivity and the Density of Economic Activity" American Economic Review, 86:1, 54-70. Ciccone, Antonio & Giovanni Peri. 2005. "Identifying Human-Capital Externalities: Theory With Applications" Review of Economic Studies, 73, 381-412. DiNardo, John, Nicole Fortin and Thomas Lemieux. 1996. "Labor Market Institutions, and the Distribution of Wages 1973-1992: A Semiparametric Approach," Econometrica, 64, 1001-1044. 19

Dustmann, Christian, Johannes Ludsteck & Uta Schoenberg. 2009. "Revisiting the German Wage Structure," Quarterly Journal of Economics, 124:2, 843-881. Galor, Oded and Omer Moav. 2000. "Ability-Biased Technological Transition, Ability Bias and Economic Growth," Quarterly Journal of Economics, 115:2, 469-497. Glaeser, Edward L., Matt Resseger and Kristina Tobio. 2008. "Urban Inequality", NBER Working Paper #14419. Glaeser, Edward L., Heidi Kallal, José A. Scheinkman, and Andrei Schleifer. 1992. "Growth in cities" Journal of Political Economy 100:6,1126—1152. Henderson, J. Vernon, Ari Kuncoro and Matthew Turner. 1995. "Industrial Development in Cities" Journal of Political Economy 103:5,1067—1090. Juhn, Chinhui, Kevin Murphy & Brooks Pierce. 1993. "Wage Inequality and the Rise in Returns to Skill" Journal of Political Economy, 101:3, 410-442. Katz, Lawrence and Kevin Murphy. 1992. "Changes in Relative Wages 1963-1987: Supply and Demand Factors" Quarterly Journal of Economics, 107:1 35-78. Krusell, Per, Lee E. Ohanian, Jose-Victor Rios-Rull, and Giovanni L. Violante. 2000. "Capital-Skill Complementarity and Inequality: A Macroeconomic Analysis." Econometrica, 68:5 1029-1053. Lemieux, Thomas. 2006. "Increasing Residual Wage Inequality: Composition Effects, Noisy Data, or Rising Demand for Skill?" American Economic Review, 96:3 461-498. Machado, Jose and Jose Mata. 2005. "Counterfactual Decomposition of Changes in Wage Distributions Using Quantile Regression," Journal of Applied Econometrics, 20:4, 445-465.

20

Figure 1: Log Hourly Wage Growth by Percentile, 1979-2007

0.2

0.1 1989-1999 0 1999-2007 ‐0.1 1979-1989 ‐0.2 0

25

50

75

100

P Percentile til Notes: The sample includes all full-time white male workers ages 25-54 working at least 40 weeks in the listed years. Data is from the census 5% PUMS in 1980, 1990 and 2000 and 1% American Community Surveys (ACS) in 2005, 2006 and 2007. Hourly wages are deflated by the CPI-U and calculated as the logarithm of wage and salary income divided by the product of weeks worked and usual hours worked per week. Observations with imputed demographics, labor supply or wages, the self-employed and those who earned less than 75% of the federal minimum wage in the earnings year are excluded from the sample. Calculations are weighted by sampling weights except for those using the 1980 census which is an unweighted sample. Data listed as being for 2007 actually represents average wages from full years ending in 2005, 2006 or 2007.

Figure 2: Wage Inequality by City Size Panel A: Variance of Hourly Wages 0.6 2004-7 1999

0.4

1989 1979

0.2

0 0

2

4

City Size

6

8

10

Panel B: 90-50 Percentile Gap 1 2004-7 0.8

1999 1989

0.6 1979 0.4 0

2

4

City Size

6

8

10

Panel C: 50-10 Percentile Gap 0.95

2004-7

0.8

1999 1989

0.65

1979

0.5 0

2

4

City Size

6

8

10

Notes: See the notes to Figure 1 for a description of the sample. City size categories are based on 2000 metro area populations. Size 0 corresponds to non-MSA locations. Sizes 1-10 correspond to ten-percentile bins from the year 2000 MSA population size distribution.

Figure A1: Fraction College or More by City Size

0.45

2004-7 1999 1989

0.35 1979 0.25

0.15 0

2

4

City Size

6

8

10

Notes: Because of a high rates of allocated income in the American Community Surveys for those with a college degree, we do not drop observations with allocated income or labor supply information in any year for the purpose of generating this graph.

Table 1: Trends in Hourly Wage Inequality 1 Year 1979 1989 1999 2004-7

Variance 0.21 0.29 0.34 0.39

2 Total 90-50 Gap 0.53 0.62 0.72 0.81

3 50-10 Gap 0.63 0.70 0.67 0.74

4 Between Variance 0.05 0.08 0.10 0.12

5 Variance 0.16 0.21 0.25 0.27

6 Residual 90-50 Gap 0.45 0.50 0.56 0.61

7 50-10 Gap 0.55 0.60 0.60 0.64

Notes: See the notes to Figure 1 for a description of the sample. Residuals are calculated using fully interacted age, education and city size cell means of hourly wages. We use 5 education bins and 15 age bins. Residual variance from a more saturated semiparametric specification that also includes 1-digit industry but only 6 age categories is 0.01 smaller in all years.

Table 2: Contributions to Residual Hourly Wage Inequality

Year

1 1980 Quantities Demographics & City Size

2 3 4 Both 1980 City Size Profile Full Demographics Prices Only Prices & Quant. Prices & Quant. Panel A: Variance

1979 to 1989 1979 to 1999 1979 to 2004-7

0.00 0.19 0.13

0.28 0.53 0.38

0.28 0.54 0.38

0.28 0.67 0.48

Panel B: 90 - 50 Percentile Gap 1979 to 1989 1979 to 1999 1979 to 2004-7

-0.03 0.11 0.08

0.22 0.37 0.28

0.22 0.37 0.28

0.23 0.53 0.40

Panel C: 50 - 10 Percentile Gap 1979 to 1989 1979 to 1999 1979 to 2004-7

-0.14 0.19 0.13

0.05 1.05 0.59

0.07 1.08 0.61

-0.08 1.09 0.61

Notes: Entries indicate the fraction reduction in growth of residual wage inequality measures shown in Table 1, Columns 5-7 due to each of the factors listed in column headers. These fractions are calculated by comparing growth in various counterfactual residuals to growth in actual residuals. In Column 1, we maintain the 1980 demographic and city size joint distribution. In Column 2, we maintain the 1979/1980 residual profile with respect to city size within demographic group as calculated using Equation (3) in the text. In Column 3, we additionally maintain the 1980 distribution of individuals across locations within demographic groups but not between demographic groups. In Column 4, we additionally maintain the 1980 distribution across demographic groups. Table A2 mathematically specifies how we calculate each counterfactual residual distribution.

Table 3: Contributions to Total Wage Inequality

Counterfactual X Set Variety Year

1 2 1980 Quantities Dem Dem. & City Size

3

4

5

No Demographics Prices Prices & Only Quantities

6 7 8 1980 City Size Profile Full Demographics Residual Prices a Prices a & Prices b Prices Only Quantities Only

9

Prices b & Quantities

10

11 Both Full Demographics Prices a & Prices b & Quantities Quantities

Panel A: Variance 1979 to 1989 1979 to 1999 1979 to 2004-7

0.01 0.15 0.12

0.00 0.14 0.12

0.36 0.57 0.51

0.36 0.56 0.51

0.18 0.35 0.23

0.24 0.38 0.24

0.22 0.38 0.25

0.24 0.44 0.34

0.23 0.44 0.35

0.23 0.48 0.34

0.25 0.54 0.44

-0.07 0.19 0.20

-0.01 0.25 0.28

-0.02 0.26 0.28

-0.08 0.27 0.28

-0.05 0.36 0.40

0.17 0.79 0.33

0.18 1.04 0.51

0.16 1.00 0.51

0.24 1.10 0.49

0.25 1.36 0.65

Panel B: 90 - 50 Percentile Gap 1979 to 1989 1979 to 1999 1979 to 2004-7

-0.23 0.02 0.09

-0.21 0.07 0.12

0.21 0.52 0.53

0.21 0.52 0.54

-0.10 0.16 0.16

-0.05 0.17 0.18

Panel C: 50 - 10 Percentile Gap 1979 to 1989 1979 to 1999 1979 to 2004-7

-0.12 0.51 0.25

-0.14 0.25 0.19

0.27 0.94 0.63

0.24 0.94 0.60

0.13 0.71 0.33

0.21 0.87 0.35

Notes: Entries indicate the fraction reduction in growth of total wage inequality measures shown in Table 1, Columns 1-3 due to each of the factors listed in column headers. These fractions are calculated by comparing growth in various counterfactual wages as compared to growth in actual wages. See Table A2 and the text for complete explanations and mathematical expressions showing how we construct each counterfactual.

Table 4: Counterfactual Reductions in Variance Growth 1

Year

2 1980 Quantities Between Residual

3 Total

4 5 6 1980 City Size Profile, Version a Between Residual Total

7 8 9 1980 City Size Profile, Version b Between Residual Total

Panel A: Demographics, Industry and City Size 1979 to 1989 1979 to 1999 1979 to 2004-7

0.06 0.13 0.14

0.10 0.27 0.23

0.08 0.22 0.19

0.08 0.07 0.04

0.16 0.36 0.22

0.13 0.26 0.15

0.10 0.25 0.31

0.16 0.36 0.22

0.13 0.32 0.25

0.15 0.33 0.22

0.10 0.26 0.32

0.22 0.45 0.31

0.16 0.39 0.31

0.33 0.57 0.49

0.36 0.55 0.62

0.30 0.58 0.41

0.33 0.57 0.49

Panel B: Demographics and City Size 1979 to 1989 1979 to 1999 1979 to 2004-7

0.02 0.09 0.10

-0.01 0.17 0.12

0.00 0.14 0.11

0.07 0.10 0.09

0.22 0.45 0.31

Panel C: City Size Only 1979 to 1989 1979 to 1999 1979 to 2004-7

NA

0.36 0.55 0.62

0.30 0.58 0.41

Notes: Entries are calculated analogously to those in Tables 2 and 3, except using a baseline regression model that also includes onedigit industry indicators interacted with age and education and separately interacted with city size categories. Because this model is more richly specified, the residual variance is 0.01 smaller than that reported in Table 1. While results for Panels B and C are based on the same decomposition of total variance into between and residual components as in Panel A, they use only demographics and/or city size cells in the construction of counterfactuals. Columns 4-6 use the same method for building counterfactual means as in Table 3 Column 7. Columns 7-9 use the same method for calculating counterfactual means as in Table 3 Column 9.

Table 5: Fraction of Variance Growth Due to Various Factors Between

Residual

Total

0.36

0.30

0.33

0.73 0.00 0.27

0.28 0.18 0.54

0.50 0.09 0.41

0.55

0.58

0.57

0.52 0.02 0.46

0.22 0.17 0.62

0.32 0.12 0.56

0.62

0.41

0.49

0.49 0.02 0.49

0.24 0.23 0.53

0.37 0.12 0.51

Panel A: 1979-1989 Total City Size-Specific Skill Sorting Across Locations Industry Sorting Across Locations Remainder

Panel B: 1979 to 1999 Total City Size-Specific Skill Sorting Across Locations Industry Sorting Across Locations Remainder

Panel C: 1979 to 2007 Total City Size-Specific Skill Sorting Across Locations Industry Sorting Across Locations Remainder

Notes: Each entry labeled "Total City Size-Specific" gives the fraction of the total growth in the variance of hourly wages during the time period indicated in panel headers due to factors correlated with city size. Remaining entries give the fraction of the growth in variance related to city size due to the factors listed at left. Entries are calculated using numbers in Table 4 Columns 7-9.

Table A1: Shifts in Industry Composition by City Size City Size Index Rural 1 2 3 4 5 6 7 8 9 10

Sum of Squared Industry Share Changes 1980-1990 1990-2000 2000-2004/7 1980-2004/7 0.000 0.002 0.002 0.003 0.003 0.002 0.003 0.003 0.002 0.003 0.004

0.001 0.001 0.002 0.002 0.002 0.003 0.004 0.003 0.003 0.003 0.004

0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.002 0.001 0.001

0.006 0.011 0.015 0.015 0.017 0.015 0.017 0.018 0.017 0.016 0.020

Notes: Each entry is the sum of the squared difference in 1-digit industry shares between the years in the column headers for the city size category indicated in each row.

Table A2: Calculation of Counterfactual Distributions Table 2

Column Reported Counterfactual 1

 g ( | x, s)h (x, s)dsdx  g ( | x , s ) h ( x , s ) dsdx  g ( | x, s)h ( s | x)h ( x)dsdx  g (  | x , s ) h ( x , s ) dsdx t

2

2

1980

c

t

2

3

t

c

t

2

4

a1980

c

t

3

1

3

2

3

3

3

4

1980

 g ( w  m x , s  | x , s ) h ( s | x ) h ( x ) dsdx  g ( w  m ( x , s ) | x , s ) h ( x , s ) dsdx c a  g t ( w  m t ( s ) | s ) h t ( s ) ds  g ( w  m ( s ) | s ) h ( s ) ds  g ( w  m ( x , s ) | x , s ) h ( x , s ) dsdx c a  g t ( w  m t ( x , s ) | x , s ) h t ( x , s ) dsdx  g ( w  m ( x , s ) | x , s ) h ( s | x ) h ( x ) d sd x  g ( w  m ( x , s ) | x , s ) h ( x , s ) dsdx  g ( w  m ( x , s ) | x , s ) h ( s | x ) h ( x ) d sd x  g ( w  m ( x , s ) | x , s ) h ( x , s ) dsdx  g ( w  m ( x , s ) | x , s ) h ( x , s ) dsdx t

t

t

at

t

5

c

aq

6

3

7

3

8

3

9

3

10

3

11

t

1980

c

t

3

b1980

1980

t

3

bt

t

c

t

aq

t

a 1980

c

t

c

t

t

b

t

c

t

bt

b

t

a 1980

aq

t

c

t

t

1980

b

t

1980

bt