On the geography of inequality: labour sorting in general equilibrium

Abstract

ABSTRACTWe study how cities’ amenities and limited housing supply contribute to aggregate wage inequality and affect housing prices through the sorting of heterogeneous skilled workers. We develop a general equilibrium model where workers differ along a continuum of skills and compete for limited housing. Our analysis suggests that spatial sorting accounts for 7.5% of the aggregate wage dispersion, increases average housing prices by 20–40% in constrained cities, and makes the economy 1.9% more productive. In addition, we evaluate a place-based policy that aims to expand the supply of houses in 1% of constrained cities and find that it improves aggregate productivity between 0.2% and 0.4%. However, the place-based policy has the unintended consequence of aggravating aggregate wage inequality by the same magnitude.KEYWORDS: labour sortinginequalityhousingplace-based policiesJEL: D44D58F16J24R13 ACKNOWLEDGEMENTSWe are extremely grateful to Ernesto Dal Bó, William Fuchs and John Morgan for their support. We also thank Scott Baker, Victor Couture, Cecile Gaubert, Rui de Figueiredo, William Grieser, William Hardin, Enrico Moretti, Gonzalo Maturana, Steve Tadelis, Joachim Voth, Reed Walker, Zhonghua Wu and Noam Yuchtman, as well as numerous seminar and conference participants, for their helpful discussions and comments. We would also like to thank Diogo Duarte who contributed to this project on an earlier version. This paper was originally part of Santiago Truffa’s PhD dissertation titled ‘Essays in urban economics’.DISCLOSURE STATEMENTNo potential conflict of interest was reported by the authors.Notes1 In this study we will focus on wage and housing price inequality. In particular, since we are able to compute wages at the individual level, we can analyse both between- and within-city inequality. When we refer to aggregate inequality, we mean the total variance of all individual wages.2 Shapiro (Citation200Citation6), Glaeser and Gottlieb (Citation2008), Couture (Citation2015), Albouy et al. (Citation2016) and Albouy (Citation2016) have empirically shown the importance of amenities in accounting for sorting patterns. We build on this literature, and we quantify the trade-off between amenities versus restrictions on the housing supply. Related literature has explored the sorting of heterogeneous firms (Behrens et al., Citation2014; Gaubert, Citation2018; Serrato & Zidar, Citation2016) to study the welfare implications of taxes and firm incentives. We complement this literature by focusing on the worker side. Further work is required to join these two threads in the literature.3 Frameworks that divide the workforce into discrete categories are empirically sensitive since the results depend on dichotomous definitions of what type of worker qualifies for each type of category. Indeed, Baum-Snow et al. (Citation2018) show that if we change the definition of high-skilled worker to a worker with some college education, some of the results shown by Diamond (Citation2016) no longer hold.4 To do so, we follow recent literature that models the housing market with bidding wars. For a review, see Han and Strange (Citation2015).5 A notable exception is Kline and Moretti (Citation2014), who develop a methodology to estimate their aggregate effects.6 Similar to Costinot and Vogel (Citation2010), we assume that Ai(s,σ)>0 is twice differentiable and strictly log-supermodular to capture the idea that high-skill workers have a comparative advantage in more complex tasks. This feature partially compensates for the fact that we have a common agglomeration elasticity and do not differentiate between low- and high-skilled workers’ elasticities.7 Notice that Li(s,σ)=vi(s)1{Mi(s)=σ}, where 1 is an indicator function. Also, notice that Bi=Vi(s¯).8 Following Albrecht et al. (Citation2016), we assume that Bi and Si are large enough so that the arrival rate of buyers visiting a particular seller follows a continuous Poisson process with parameter θi.10 The preference in (9) is a monotonic transformation of the homothetic preference U=Tailog⁡(xi) commonly adopted in the literature. Thus, all the properties of this utility representation are preserved under the monotonic transformation in (9).11 In fact, this is consistent with empirical distributions of talent, as shown by Bacolod et al. (Citation2009). Although we see differences between cities in the fraction of high- to low-skilled workers, we still observe a positive mass of workers at every level of talent. Moreover, if we restrict attention to two cities, we can prove that for any pair of non-overlapping skill distributions, this configuration is never in equilibrium, since the lowest-skilled worker in the high-skilled city will always have an incentive to move to the low-skilled city, where she is the most skilled worker. We present the two cities case in the Appendix in the supplemental data online.12 The author develops a methodology that derives hedonic measures of local productivity and local amenities from data, such as local wages, housing prices and taxes. The quality-of-life measure positively correlates to measures of natural amenities relating to climate and geography.13 This measure stems from satellite-generated data on terrain elevation and the presence of water bodies to estimate the amount of developable land in each MSA. We focus on the part of the elasticity that is determined by geographical restrictions.