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Worker-firm matching in a global economy Tito, Maria Domenica 2015

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WORKER-FIRM MATCHING IN A GLOBAL ECONOMYbyMaria Domenica TitoB.A., Universita` del Salento, 2007M.Sc., Universita` di Bologna, 2009A THESIS SUBMITTED IN PARTIAL FULFIlLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Economics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)July 2015c© Maria Domenica Tito, 2015AbstractThis thesis investigates the impact of international trade on the sorting patterns of workers acrossfirms and analyzes the implications for welfare.The first essay builds a model of matching between heterogeneous workers and firms in presenceof search frictions. Variation in the worker type at the firm level exists in equilibrium only becauseof search costs. When firms gain access to foreign markets their revenue potential increases. Whenstakes are high, matching with the right worker becomes particularly important because deviationsfrom the ideal match quickly reduce the value of the relationship. Hence exporting firms select setsof workers that are less dispersed relative to the average.The second essay documents the difference in the sorting patterns of workers between exportersand non-exporters in a French matched employer-employee dataset. We proxy the type of eachworker using her average wage over her job spells and construct measures of the average type andtype dispersion at the firm level. We find that exporting firms tolerate a lower dispersion in thepool of workers they hire. The matching between exporting firms and workers is even tighter insectors characterized by better exporting opportunities as measured by foreign demand or tariffs.We also confirm the conjecture in the literature that exporters pay higher wages because, amongother factors, they employ better workers.The final chapter explores the implications for wage inequality using the French Employer-Employee Data. We find that the differences in sorting in large part account for the existingdifferences in the wage structure between exporters and non-exporters. Exporting firms tend tohave higher wages but tolerate a lower dispersion. Using an alternative theory-based measure ofresidual wage inequality, we also find that the unexplained component tends to be smaller in export-ing and more productive firms, even when controlling for some differences in workforce composition.This finding suggests that exporters are better able to overcome frictions in the labour market inorder to move closer to their ideal worker.iiPrefaceParts of Chapter 2 - sections Welfare implications, Revenue loss in partial equilibrium andWelfare analysis: calibration and simulation - and Chapter 3 is based on work joint with Pro-fessor Matilde Bombardini and Gianluca Orefice. I performed the model calibration and simulationfor Chapter 2 and conducted most of the empirical analysis for Chapter 3. The sections Welfareimplications, Revenue loss in partial equilibrium in Chapter 2, Empirical specification 1and 2 in Chapter 3 were originally drafted by Matilde Bombardini; the Data section was writtenby Gianluca Orefice. Gianluca Orefice prepared the empirical results in Chapter 4 (Tables 25-46).iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Trade and Worker Sorting under Constant Costs of Search . . . . . . . . . . . . . 42.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 A model of assortative matching: closed economy . . . . . . . . . . . . . . . . . . . . . 62.3 A model of assortative matching: open economy . . . . . . . . . . . . . . . . . . . . . 182.4 Trade liberalizations and matching sets . . . . . . . . . . . . . . . . . . . . . . . . . . 232.5 Open economy and heterogeneous trade costs . . . . . . . . . . . . . . . . . . . . . . . 252.6 Welfare implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.7 Welfare analysis: calibration and simulation . . . . . . . . . . . . . . . . . . . . . . . . 282.8 Testable implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303. Does Exporting improve Matching? Evidence from French Employer-EmployeeData . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2 Empirical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.4 Empirical trends of wage inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.5 Constructing worker types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.6 Firm types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.7 Empirical specification 1: export status and matching set . . . . . . . . . . . . . . . . 433.8 Empirical specification 2: market access and tariff shocks . . . . . . . . . . . . . . . . 473.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494. Search, Matching, Trade and Wage Inequality . . . . . . . . . . . . . . . . . . . . . 69iv4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.3 Wage inequality in a search and matching model . . . . . . . . . . . . . . . . . . . . . 714.4 Empirical specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.5 Policy implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85A. Mathematical appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85B. Numerical simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95C. Additional figures and tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97vList of TablesTable 1 - Model Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Table 2 - Model Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Table 3 - Changes in Real Expenditure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Table 4 - Real Relative Revenue Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Table 5 - Residual Wage Inequality: Worker Decomposition . . . . . . . . . . . . . . . . . 52Table 6 - Residual Wage Inequality within Sectors: Worker Decomposition. . . . . 52Table 7 - Unconditional Wage Components: Firm Decomposition . . . . . . . . . . . . . 53Table 8 - Residual Wage Inequality within Sectors: Firm Decomposition . . . . . . 53Table 9 - Residual Wage Inequality within Sectors: Complete Decomposition . . 53Table 10 - Rank Correlation Matrix, proxies for firms’ types . . . . . . . . . . . . . . . . . 54Table 11 - Measuring Sorting Patterns, Manufacturing Sectors . . . . . . . . . . . . . . . 55Table 12 - Pooled Cross-Sectional Regressions: Average Lifetime Wage, morethan 5 workers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56Table 13 - Pooled Cross-Sectional Regressions: Standard Deviation of LifetimeWage, more than 5 workers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Table 14 - Average Lifetime Wage, 25th percentile, more than 5 . . . . . . . . . . . . . 58Table 15 - Average Lifetime Wage, 75th percentile, more than 5 . . . . . . . . . . . . . 59Table 16 - Pooled Cross-sectional Regressions: Average of Workers’ Fixed Ef-fects, more than 5 workers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60viTable 17 - Pooled Cross-sectional Regressions: Standard Deviation of Workers’Fixed Effects, more than 5 workers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Table 18 - IV Regressions: Standard Deviation of Lifetime Wage, more than 5workers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Table 19 - Group-Weighted Regressions: Standard Deviation, more than 5workers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Table 20 - Pooled Cross-sectional Regressions: Inter-quartile Range, more than5 workers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Table 21 - Market Access Regressions: Average, more than 5 workers . . . . . . . . . 65Table 22 - Market Access Regressions: Standard Deviation, more than 5 workers 66Table 23 - Tariff Regressions: Average, more than 5 workers . . . . . . . . . . . . . . . . . 67Table 24 - Tariff Regressions: Standard Deviation, more than 5 workers . . . . . . . 68Table 25 - Average Wages, more than 5 workers . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Table 26 - Standard Deviation of Wages, more than 5 workers . . . . . . . . . . . . . . . 78Table 27 - Residual Wage Changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79Table 28 - Wage Changes when Moving to a New Job. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100Table 29 - Classification of CS Occupation into ’white’ and ’blue’ collar workers. 100Table 30 - Summary Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101Table 31 - Summary Statistics: Market Access Shocks . . . . . . . . . . . . . . . . . . . . . . 101Table 32 - Pooled Cross-sectional Regressions: Standard Deviation of newlyhired workers, more than 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102viiTable 33 - Pooled Cross-sectional Regressions: Standard Deviation of currentworkers, more than 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103Table 34 - Cross-Sectional Regressions: Average Lifetime Wage, more than 5 . . 104Table 35 - Cross-Sectional Regressions: Standard Deviation of Lifetime Wage,more than 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105Table 36 - Pooled GLS Regressions: Average Lifetime Wage . . . . . . . . . . . . . . . . . 106Table 37 - Pooled GLS Regressions: Average of Workers Fixed Effects . . . . . . . . 107Table 38 - Pooled GLS Regressions: Standard Deviation of Lifetime Wage . . . . . 108Table 39 - Pooled GLS Regressions: Standard Deviation of Worker Fixed Effects109Table 40 - Pooled Cross-sectional Regressions: Standard Deviation by group,more than 5 workers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110Table 41 - Sectoral Rank Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110Table 42 - GLS Regressions: Sectoral Rank Correlations . . . . . . . . . . . . . . . . . . . . 111Table 43 - IV Regressions: Standard Deviation of Lifetime Wage, more than 5workers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111Table 44 - Standard Deviation of Wages, White Collars, more than 5 workers . . 112Table 45 - Standard Deviation of Wages, Blue Collars, more than 5 workers . . . 113Table 46 - Firm-level Residual Wage (Empirical Measure) . . . . . . . . . . . . . . . . . . . 114viiiList of FiguresFigure 1 - Matching Bounds for Worker θ when σ η−1η = 1. . . . . . . . . . . . . . . . . . . . 14Figure 2 - Firm Type Space by Export Status . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Figure 3 - Wage Decomposition: Time Trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Figure 4 - Variability in Wages: Comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Figure 5 - Matching Set for the Simulated Economy . . . . . . . . . . . . . . . . . . . . . . . 95Figure 6 - Standard Deviation of the Matching Set by firm type, normalized bythe average worker type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Figure 7 - Standard Deviation of the Matching Set by firm type, normalized bythe average worker type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Figure 8 - Export Cut-off . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97Figure 9 - Distribution of Sampled Workers - trimming the 95th percentile . . . . 97Figure 10 - Distribution of Value Added per Worker by Export Status . . . . . . . . . 98Figure 11 - Distribution of Individual Effects, largest connected group . . . . . . . . . 98Figure 12 - Distribution of Firm Effects, largest connected group . . . . . . . . . . . . . 98Figure 13 - Wage Changes by Wage Quartile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99ixAcknowledgementsI would like to express my sincerest gratitude to my supervisors, Dr. Matilde Bombardini and Dr.Keith Head, for their invaluable guidance and support throughout the completion of my doctoraldegree. Thank you for teaching me the fundamentals of research and enhancing both my personaland professional developments.I would like to thank the members of my committee for devoting their time, effort, support, andexpertise towards my thesis research. I would like to thank Dr. Hiroyuki Kasahara and Dr. FlorianHoffman for their guidance throughout my project. I am also grateful to Dr. Tomasz Swiecki forhis time and insights.I offer my gratitude to all other faculty, staff and my fellow students at the Vancouver School ofEconomics for their support. Special thanks are owed to the Faculty of Graduate Studies and theUniversity of British Columbia for giving me the opportunity and financial assistance to pursue myPhD.All of my success would not have been possible without the love and support of my family. I wouldlike to thank my parents, Rosa and Michele Tito, who have supported me throughout my years ofeducation, both morally and financially and always encouraged me to follow my dreams.The most special thanks to my husband, Nereus A. Joubert. There are not enough words to expressall my gratitude to you; your infinite support makes everything possible.xTo Nereus A. Joubert and my parentsxi1. IntroductionThe pattern of sorting of workers across firms has fundamental implications for the efficiency of theeconomy as well as for the inequality of wages in the labor force. The first implication has been aconcern of the literature on assignment starting from Shapley and Shubik [1971] and Becker [1973].From those contributions we know that when firms and workers are complementary in productionthen the allocation of the right worker to the right job maximizes output. The second implicationhas received attention more recently for example by Card et al. [2013], who show that sorting ofgood workers to good firms can explain as much as 35% of the recent increase in wage inequality inWest Germany. The logic by which highly skilled workers are paid more not only because of theirinnate higher productivity, but also because they work with highly productive firms and co-workers,is common to the contribution by Kremer and Maskin [1996] as well.In this paper we start from the premise that the optimal allocation of workers cannot be reachedbecause of search costs, and therefore firms accept some degree of mismatch in equilibrium becausethe cost of search exceeds the benefit from a more suited partner. We then explore whether thematching of firms and workers is affected by access of the former to the export market. But how canmarket integration affect how firms and workers are matched? When firms gain access to foreignmarkets their revenue potential increases. When stakes are high, matching with the right workerbecomes particularly important because deviations from the ideal match quickly reduce the value ofthe relationship.Using matched employer-employee data from France, we show that exporters select pools ofworkers characterized by higher average type and lower type dispersion than non-exporting firms.While the first effect is predicted by other models (Helpman et al. [2010] and Sampson [2014])we believe we offer a novel way of testing this prediction, which disentangles pure exporter wagepremia (deriving from profit-sharing with workers as in Amiti and Davis [2012]) from the selection ofbetter workers by exporting firms. The second effect, i.e. the influence of exporting on worker typedispersion, is unexplored in the literature and is quantitatively as strong as the effect of exportingon average worker type. We explore further the effect of exporting by building measures of theexporting opportunities in different sectors using tariffs and aggregate imports from all the countriesthat France exports to. We find that when exporters face lower tariffs or larger demand for importsin a foreign market, the dispersion of types in their pool of workers declines further. We believe thisresult is harder to reconcile with a view that the exporting and tightening of the matching are bothdriven by a common excluded factor.In the final chapter, we analyze the link between export status and variation in wages acrossfirms. We find that the differences in sorting in large part account for the existing differencesin the wage structure between exporters and non-exporters. Exporting firms tend to pay higherwages, even if accounting for differences in unobserved employment composition. Exporters alsotolerate a lower dispersion in wages; the difference in wage dispersion, however, is fully accountedfor by differences in employment composition. We look deeper into the within-component of wagedispersion by constructing a theory-consistent measure of residual wage inequality. This measureexploits the changes in wages that occur when a worker moves across jobs. We find that workers1moving to exporters experience on average an increase in wages: this is consistent with a reductionin residual wage inequality. This finding suggests that exporters are better able to overcome frictionsin the labour market in order to move closer to their ideal worker.To study the impact of exporting on matching we employ the model proposed by Atakan [2006]and Eeckhout and Kircher [2011], where we show that exporting is identical to an increase in thefirm type. Heterogeneous workers and firms face a dynamic problem where they meet at randomand decide whether to accept to match or not. If they do not accept to match they pay a searchcost and keep searching for an additional period. The presence of search costs creates an acceptanceset, rather than a unique assignment outcome that prevails in the frictionless model. As shown byEeckhout and Kircher [2011], the boundaries of such acceptance set are increasing in the firm type,confirming the pattern of positive assortative matching in a model with frictions.We focus on a different dimension: we take the width of the acceptance set as a measure of thevariability in the worker type tolerated by the firm. On the one hand, because of complementarity, aworker with type below the firm’s ideal contributes relatively less to output, with a lower contributionthe higher the productivity of the firm. On the other hand, a worker type that is above the averagetype requires an increasing compensation due to her outside option. Such compensation rises muchfaster at firms that are more productive because they employ on average more productive types.The result is that firms that are more productive tolerate less relative dispersion from their idealworker type.From a welfare perspective, we show that a more productive firm (or an exporting firm) featuresa lower deviation from the optimal level of revenues created under frictionless matching. This isonly a partial equilibrium result and cannot inform us as to whether there are overall gains relatedto this matching channel. In particular there are two counteracting forces. On the one hand import-competing firms receive a negative shock to their revenues and therefore their matching range tendsto widen. On the other hand, exporting firms receive a positive shock and choose smaller deviationsfrom the optimal. We therefore proceed to simulate the model with two symmetric countries andcalibrate it to French moments in order to recover the parameters for the search costs, the transportcosts and the elasticity of demand. We numerically show that the gains from trade are larger aswe increase the cost of search. We interpret this result as providing support to the idea that whenan economy is characterized by high frictions, trade opening can be more beneficial than when theeconomy is essentially very close to the optimal worker-firm allocation. This explicit result on welfareis novel in the literature and we believe it could be further explored in a richer model.This paper contributes to the growing literature on international trade with heterogeneous laborand firms, which is surveyed in a recent chapter by Davidson and Sly [2012]. More specifically itbelongs to a strand of research that investigates the effect of openness on the process of matchingbetween firms and workers, which is at the core of the contribution by Sampson [2014], who studiesits consequences for wage inequality.1The most closely related work is a recent paper by Davidson et al. [2012], which shows, using1Our paper is also related to the large literature on the impact of trade on inequality, which includes, amongmany others, Feenstra and Hanson [1999], Costinot and Vogel [2010], Bustos [2012], Amiti and Davis [2012], Ver-hoogen [2008] and Fr´ıas et al. [2012].2Swedish data, that export-oriented sectors display a higher correlation between firm and workertypes, estimated as firms’ and workers’ fixed effects in a wage regression as in Abowd et al. [1999](henceforth AKM).Our approach shifts the focus on the firm-level decision rather than looking at the aggregatestrength of matching and therefore relies on a different type of variation to detect different matchingbehavior by firms that are differentially exposed to international trade. In particular, it exploitswithin-sector variation between exporting and non-exporting firms, therefore isolating and control-ling for other sector-level characteristics of the labor market that may affect the sorting of workersacross firms.Moreover, because Eeckhout and Kircher [2011] prove that firms fixed effect deriving from a wageregression a` la AKM might be negatively or not correlated with the true firm type, we are careful toavoid using those fixed effects as a proxy for firm type. We use instead variables constructed fromfirm-level data, such as market shares, value added and total employment.From a theoretical standpoint our approach differs from Davidson et al. [2008] in that we havea different focus. We are interested in deriving predictions at the firm level, rather than at theaggregate level and therefore we allow for a rich heterogeneity on both the worker and the firmside. Davidson et al. [2008] simplify those dimensions in order to obtain clean aggregate results. Inparticular they have high and low types of workers and high and low technology firms. Globalizationcan take the economy from an equilibrium in which high-tech firms employ high type workers andlow-tech firms employ both high and low type workers to an equilibrium where there is perfectassortative matching. The firm-level predictions in their set-up between exporters and non-exportersare stylized in that there is no predicted variation in the type of workers hired by different types offirms under trade.The relationship of this paper to the theoretical framework in Helpman et al. [2010] and Helpmanet al. [2013] deserves a more detailed analysis, since both models describe the matching of hetero-geneous firms to heterogeneous workers in the presence of search frictions. The main conceptualdifference between the two theoretical approaches is the nature of worker heterogeneity. In Helpmanet al. [2010] workers are not ex-ante different, but they have a productivity draw that is firm-specific.Therefore there is no sense in which an ex-ante high-type worker is more likely to match with a high-type firm, since a firm simply selects the workers that have better productivity draws relative tothat firm only. In general our estimation procedure, which presumes the existence of fixed workertypes is incompatible with their view of ex-ante identical workers. Let us for a moment set asidethis difference and investigate the predictions of their model in terms of the dispersion of workertypes within firms. Under the assumption of a Pareto distribution, exporters (and more productivefirms in general) choose a higher cut-off for hiring workers. This results in a distribution of workerswithin firm that has higher standard deviation, higher mean and a constant coefficient of variation(the ratio of standard deviation to mean). Therefore we need an alternative theoretical frameworkto investigate the impact of exporting on matching of permanently heterogeneous workers and firms:this is the objective of Chapter 2. Chapter 3 documents the empirical strategy and the differencesin average worker type and dispersion between exporters and non-exporters. Chapter 4 analyzes theimplication for wage inequality.32. Trade and Worker Sorting under Constant Costs of Search2.1 IntroductionEmployment relations, like mate selection, involve the search for an optimal match between theparticipants in the relationship. The success of the resulting partnership depends to a large extenton the interaction between worker and firm characteristics: both sides agree to match when theyperceive a net gain, even if they did not meet the ideal companion. The alternative, screening for apartner with optimal characteristics, might either be time consuming or require investing excessiveresources.The opportunity cost of a mismatch, however, increases when the value achievable in an idealmatch is higher, e.g. in a larger market. Trade represents an analogous mechanism: gaining accessto foreign revenues affects the incentives of firms and workers to match, inducing partnerships closerto the ideal worker-to-firm assignment.Becker [1973] develops a theoretical model addressing the problem of sorting of workers acrossfirms.2 His solution depends on the interaction of worker and firm types in production. In partic-ular, if the production function is supermodular in worker and firm types, positive assortativematching occurs, i.e. the optimal assignment requires matches between agents of similar types; if,instead, the production function is submodular, the optimal matches are between divergent types(negative assortative matching).Becker’s conditions crucially rely on the costless observability of worker and firm characteris-tics. The realized worker-to-firm allocation may diverge from the optimal assignment if acquiringinformation on partners is costly. Under this framework, firms and workers face a trade off betweenmatching with suboptimal partners and investing additional resources to find the ideal companion.Either agent agrees to match if the costs of additional search outweighs the gain from a betterpartner.Mechanisms affecting the incentives of workers and firms to match may act to reduce deviationsfrom the optimal assignment. Trade is one of such mechanisms. This chapter develops a model char-acterizing the effect of trade on the set of acceptable matches between firms and workers - hereafter,the matching set - under constant costs of search and derives the related empirical implications. Icalibrate the model to the features of the French economy and quantify the gains from an episodeof trade liberalization.The theoretical framework is based on Atakan [2006] and Eeckhout and Kircher [2011]. The closedeconomy version of the model combines double-side heterogeneity and labour market frictions. Inpresence of production complementarities between worker skills and firm productivity,3 the optimalmatches are between firms and workers of similar types. However, if meeting a potential partnerrequires paying a strictly positive search cost and agents’ types are not observable before a meetingoccurs, firms and workers agree to match even if they do not meet their optimal partner. In fact,2In his original contribution, Gary Becker approaches the problem of sorting in the marriage market. Subse-quent contributions have applied the results derived by Becker [1973] to other assignment problems, involving differ-ent groups of heterogeneous agents.3Although the model is developed under the assumption of a supermodular production function, its qualitativeproperties extend to the case of a submodular production function.4agents accept to match as long as the search cost exceeds the benefit from a more suited partner.Therefore, the matching set of each agent is a subset of the space of potential partner types.We first introduce export decisions a` la Melitz [2003] in this framework to analyze the dynamicimpact of trade liberalization on sorting. In each period firms face a trade off between the opportunityto achieve larger revenues by selling abroad and the higher fixed costs to access foreign markets; moreproductive firms self-select into exporting. Additional revenues induce firms and workers to becomemore selective when choosing the partners to match with. The benefits of additional revenues dueto exporting are fully achieved only when matching with the ideal partner; since deviations from theideal match quickly reduce the value of the relationship, matching with the right worker becomesparticularly important. As a result, in an open economy each agent that has access to the exporttechnology enjoys a smaller matching set compared to the corresponding closed economy version.However, a model with the export choice a` la Melitz [2003] implies that we should never observetwo firms of the same productivity, but different export status. Motivated by the empirical evidenceon the distribution of firm productivity by export status, in a second extension of the model we followHelpman et al. [2013] and allow heterogeneous fixed costs of exporting across firms, disentangling theeffect of exporting from that of firm productivity. In this framework, becoming an exporter remainsidentical to an increase in firm type. We find that exporting firms tolerate a smaller matching setcompared to non-exporters of similar productivity.From a welfare perspective, we also show that a more productive firm (or an exporting firm) fea-tures a lower deviation from the optimal level of revenues created under perfect assortative matching.This is only a partial equilibrium result and cannot inform us as to whether there are overall gainsrelated to this matching channel. In particular, there are two counteracting forces. On the onehand, import-competing firms receive a negative shock to their revenues and therefore their match-ing range tends to widen. On the other hand, exporting firms receive a positive shock and choosesmaller deviations from the optimal assignment. We therefore proceed to simulate a version of themodel with two symmetric countries and calibrate it to French moments in order to recover theparameters for the search costs, the transport costs and the elasticity of demand. We numericallyshow that the gains from trade are larger as we increase the cost of search. We interpret this result asproviding support to the idea that when an economy is characterized by high frictions, trade openingcan be more beneficial than when the economy is essentially very close to the optimal worker-firmallocation. This explicit result on welfare is novel in the literature and we believe could be furtherexplored in a richer model.The present model contributes to the literature on international trade with heterogeneous laborand firms, which is surveyed in a recent chapter by Davidson and Sly [2012]. More specifically itbelongs to a strand of research that investigates the effect of openness on the process of matchingbetween firms and workers, which is at the core of the contribution by Sampson [2014], who studiesits consequences for wage inequality. Sampson [2014], however, abstracts from search costs and it isuninformative on the welfare losses caused by deviations from the optimal allocation.The most closely related work is a recent paper by Davidson et al. [2008], which shows thatglobalization can take the economy from an equilibrium in which high-tech firms employ high typeworkers and low-tech firms employ both high and low type workers to an equilibrium with perfect5assortative matching. Our approach differs from Davidson et al. [2008] in that we have a differentfocus. We are interested in deriving predictions at the firm level, rather than at the aggregate leveland therefore we allow for a rich heterogeneity on both the worker and the firm side. Davidsonet al. [2008] simplify those dimensions in order to obtain clean aggregate results. The firm-levelpredictions in their set-up between exporters and non-exporters are stylized in that there is nopredicted variation in the type of workers hired by different types of firms under trade.The relationship of this paper to the theoretical framework in Helpman et al. [2010] and Helpmanet al. [2013] deserves a more detailed analysis, since both models describe the matching of hetero-geneous firms to heterogeneous workers in the presence of search frictions. The main conceptualdifference between the two theoretical approaches is the nature of worker heterogeneity. In Helpmanet al. [2010] workers are not ex-ante different, but they have a productivity draw that is firm-specific.Therefore there is no sense in which an ex-ante high-type worker is more likely to match with a high-type firm, since a firm simply selects the workers that have better productivity draws relative tothat firm only. In general our estimation procedure, which presumes the existence of fixed workertypes is incompatible with their view of ex-ante identical workers. Let us for a moment set asidethis difference and investigate the predictions of their model in terms of the dispersion of workertypes within firms. Under the assumption of a Pareto distribution, exporters (and more productivefirms in general) choose a higher cut-off for hiring workers. This results in a distribution of workerswithin firm that has higher standard deviation, higher mean and a constant coefficient of variation(the ratio of standard deviation to mean). Therefore we need an alternative theoretical frameworkto investigate the impact of exporting on matching of permanently heterogeneous workers and firms:this is the objective of the present paper.The rest of the paper is organized as follows. Section 2.2 introduces the closed economy model; theopen economy version is characterized in section 2.3; the impact of trade liberalization is analyzed inSection 2.4. Section 2.5 analyzes an extension with heterogeneous trade costs; the associated welfareanalysis is in Sections 2.6 and 2.7. Section 2.8 reviews the theoretical predictions and section 2.9concludes.2.2 A model of assortative matching: closed economyThe economy is composed by two groups of infinitely-lived heterogenous agents, workers and firms.Workers differ in their ability level, θ, distributed according to a smooth density, g (θ), on the interval[0, 1]; we follow the standard convention that higher θ denotes a worker of higher ability. Firms areheterogeneous in productivity levels, ψ, distributed according to a smooth density, h (ψ), on [0, 1];4also here, a higher ψ indexes a more productive firm. Production occurs if matches are formedbetween firms and workers; we analyze the matching problem between one firm and one worker.54Here, we interpret the types of workers and firms as their percentiles in the productivity distributions. Thisinterpretation naturally implies that agent types are distributed over the interval [0, 1].5In this setting we can think of a firm with n workers as solving the same problem n times where the matchingwith one worker does not affect matching with the others. Nevertheless it is possible to introduce more than oneworker in the production function and allow complementarities among workers as well as between firm and workers.We believe that the qualitative results of the 1 worker-1 firm model would not be altered in an extended frame-work with many workers, as additional complementarities across workers induce all agents to search for partnerseven more intensively. A final extension would require endogenizing the number of workers selected by the firms.6Individual agents do not create output when unmatched. Time is continuous and at each point intime, agents are either matched or unmatched. If a firm ψ agrees to match with a worker θ, itreceives the blueprint for a new variety ω; they produce output according to the production functionf (θ, ψ) = (θψ)σ , σ > 0We embed the matching problem in a monopolistic competition model a` la Krugman [1979]. Eachfirm produces a differentiated variety of a given product. Demand for an individual variety isisoelastic with elasticity η > 1. Therefore firms selling their output in the domestic market obtaintotal revenues:R (θ, ψ) = (θψ)σ(η−1)η E1ηwhere E is the domestic total real expenditure. Firm revenues are increasing in firm and workertype and feature complementarity between the two types, i.e. fθψ > 0. Complementarity is key forwhether there is positive assortative matching in equilibrium between firms and workers.Under these assumptions, in the absence of frictions, we would observe perfect positive assortativematching. Under that scenario every type of firm would be matched with a unique type of worker.In particular, a more productive firm would be matched with a more productive worker, but therewould be no variation within the set of workers matched with firms of a given type ψ, as in Sampson[2014].Here we are interested in analyzing the variation between workers employed by the same type offirm. We therefore introduce frictions as in Chade [2001] and Atakan [2006]. Agent types are notobservable before a meeting occurs and searching for a partner is costly. At each point in time, ifengaging in search, agents incur a strictly positive cost, c > 0,6 in order to meet a potential partner.Meetings occur at random;7 after meeting, workers and firms perfectly observe one another’s typeand decide whether to produce. If both agents agree to match, they leave the market and splitthe surplus they generated according to Nash Bargaining, with a fraction γ accruing to the workerand a complementary fraction (1− γ) captured by the firm. If unmatched, each agent continuessearching. Let w (θ) and pi (ψ) be the steady-state option value of remaining unmatched for workersand firms. Regardless of how the surplus is split, the worker and the firm will accept to match if thesurplus from the relationship is non-negative, i.e. profitable matches satisfy a non-negative surpluscondition,s (θ, ψ) = R (θ, ψ)− w (θ)− pi (ψ) ≥ 0where s (θ, ψ) represents the surplus from a match. LetM (ψ) be the steady-state set of acceptableIntroducing fixed costs for each additional worker hired by the firm, the model would predict that more productivefirms are bigger in terms of workers. Such an extension would still accommodate our prediction on the within-firmdispersion of worker ability, as additional complementarities induce more productive firms to search more for theirpartners. We showed this result in a simplified two-period model, as in Bombardini et al..6We assume that search costs do not vary with the agent type. In particular, we abstract from search costs a`la Shimer and Smith [2000], as the empirical implications of a model where search costs take the form of foregoneoutput are counterfactual. In fact, in Shimer and Smith [2000] export shocks would not change the matching sets ofagents, since the cost of search increases proportionally to potential output.7Models of directed search with agent heterogeneity introduce wage dispersion among heterogeneous workersand among identical workers depending on their employer, but are unable to generate variation between workersemployed by the same type of firm. See Shi [2002] and Shimer [2003].7matches for firm ψ and letM (θ) be the steady-state set of acceptable matches for worker θ. If workerθ agrees to match with firm ψ, ψ ∈ M (θ); if firm ψ agrees to match with worker θ, θ ∈ M (ψ).Formally,M (ψ) = {θ ∈ [0, 1] : s (θ, ψ) ≥ 0}M (θ) = {ψ ∈ [0, 1] : s (θ, ψ) ≥ 0}Then, a match occurs iff θ ∈ M (ψ) ⇔ ψ ∈ M (θ). In what follows, we will assume that thedistribution functions are identical, h (ψ) = g (θ) if ψ = θ; this guarantees that the individualmatching strategies are mutually consistent.Finally, the agent pay-off from a match consists of her continuation value and the share of thesurplus she captures, as implied by Nash bargaining,Firm’s Payoff: pi (ψ) + s(θ,ψ)1−γWorker’s Payoff: w (θ) + s(θ,ψ)γOur characterization of the stationary equilibrium departs from Atakan [2006] in determininghow the distribution of the unmatched agents evolves over time. Atakan [2006] adopts the clonesassumption, i.e. the assumption that, after matching, the agents leaving the market are replacedby agents of identical types. Although this assumption is convenient for tractability, we will adopt amore realistic specification that endogenizes the distribution of unmatched agents.8 Let ν (θ) ≤ g (θ)and v (ψ) ≤ h (ψ) be the measure of unmatched workers and firms. In each period, only unmatchedagents engage in search for a potential partner. The unmatched agents meet random partners at theflow rate ρ > 0. After meeting, they observe each other’s type; if both agree, a match is formed andthey leave the market. In case they do not agree to match, each of them incurs the fixed cost c tokeep searching. In order to maintain a population of unmatched agents in equilibrium, we assumethat nature randomly destroys matches according to a constant flow probability λ > 0. After amatch is destroyed, both agents reenter the pool of searchers.Now we have all the elements to define a stationary search equilibrium (SSE).Definition A stationary search equilibrium (SSE) consists of a pair of functions w : [0, 1] → R,pi : [0, 1] → R, a pair of strategies M (θ), θ ∈ [0, 1], M (ψ), ψ ∈ [0, 1] and a pair of distributions,ν (θ) ≤ g (θ) and v (ψ) ≤ h (ψ) such that• given M (θ), M (ψ), ν (θ), and v (ψ), w (·) and pi (·) solvew (θ) =∫[0,1]max{−c+ w (θ) +s (θ, ψ)γ,−c+ w (θ)}v (ψ) dψ (1)pi (ψ) =∫[0,1]max{−c+ pi (ψ) +s (θ, ψ)1− γ,−c+ pi (ψ)}ν (θ) dθ (2)8See Appendix A.3 for a comparison of the two assumptions.8• given w (θ) and pi (ψ),ψ ∈M (θ) iff s (θ, ψ) = R (θ, ψ)− w (θ)− pi (ψ) ≥ 0 (3)θ ∈M (ψ) iff s (θ, ψ) = R (θ, ψ)− w (θ)− pi (ψ) ≥ 0 (4)• given M (θ), M (ψ), ν (·) and v (·)λ (g (θ)− ν (θ)) = ρ · ν (θ)∫M(θ)v (ψ) dψ (5)λ (g (ψ)− v (ψ)) = ρ · v (ψ)∫M(ψ)ν (θ) dθ (6)Three sets of conditions are at the core of the definition of the equilibrium. Conditions (1) and(2) define the equilibrium option value of remaining unmatched for workers and firms. Each agentsolves an optimal stopping problem, choosing the optimal time to stop searching for a partner inorder to maximize the sequence of flow pay-offs from her search. A worker receives−c if she rejects the match−c+ w (θ) +s (θ, ψ)γif she accepts the matchIn a recursive formulation, as in conditions (1) and (2),9 each period the agent has the option ofmatching with a -random- partner or searching further. Hence, the value of being unmatched isattained by maximizing over the two options of matching or waiting.Conditions (3) and (4) characterize the matching strategies of workers and firms. Agents acceptto match if the surplus from the relationship is non-negative.Finally, conditions (5) and (6) describe the stationary distribution of unmatched agents. Toguarantee stationariness, in each unit of time the flow of matches that are dissolved (left-hand sideof the equations) must equate the flow of newly formed matches (right-hand side).The mutual consistency of option values, of matching sets and of the distributions of the un-matched defines an implicit continuous mapping, which admits at least one fixed point. This guar-antees the existence of the equilibrium, as proven in Shimer and Smith [2000] and Atakan [2006].Theorem 1. Under constant costs of search, a SSE exists; moreover, it entails positive assorta-tive matching, i.e. the matching sets are convex and the matching bounds are non-decreasing inthe agent type.Proof. See Atakan [2006].The main contribution of the present chapter lies in the characterization of the properties of theequilibrium and of the matching sets; we’ll face this task next.9Conditions (1) and (2) are the Bellman equations associated to the optimal stopping problem.92.2.1 Properties of the equilibriumFirst, we take a closer look at the functions characterizing the equilibrium. Those functions arewell-behaved and have some useful properties.Proposition 1. The equilibrium surplus function s (θ, ψ) is continuous, symmetric, and strictlysupermodular.Proof. See Atakan [2006].Proposition 2. In a SSE, the option value functions w (θ) and pi (ψ) are differentiable and mono-tonically increasing in worker’s ability θ, ∀ θ ∈ (0, 1), and firm’s productivity ψ, ∀ψ ∈ (0, 1).Proof. Let θ1, θ2 ∈ (0, 1). From condition (1),w (θ1)∫M(θ1)v (y) dy + γ · c =∫M(θ1)[R (θ1, y)− pi (y)] v (y) dyandw (θ2)∫M(θ1)v (y) dy + γ · c ≤∫M(θ1)[R (θ2, y)− pi (y)] v (y) dyThen,∫M(θ2)[R (θ1, y)−R (θ2, y)] v (y) dy∫M(θ2)v (y) dy≤ w (θ1)− w (θ2) ≤∫M(θ1)[R (θ1, y)−R (θ2, y)] v (y) dy∫M(θ1)v (y) dySince the revenue function is differentiable over its entire open domain, w (·) is differentiable at allθ ∈ (0, 1), and∂w (θ)∂θ=∫M(θ)∂R(θ,y)∂θ v (y) dy∫M(θ) v (y) dyThe monotonicity of the worker option value follows from the monotonicity of the revenue function.A similar argument applies to prove the differentiability and the monotonicity of pi (ψ).Intuitively, the monotonicity of the option value of being unmatched depends on the proper-ties of the revenue function and on the matching patterns: workers of higher ability make largercontributions to revenues and tend to match with more productive firms. This implies that theiroption value must be larger. The monotonicity of option values is consistent with empirical findingsdocumenting that high ability workers receive higher wages,10 and more productive firms tend to bebigger both in terms of revenues and employment.1110In absence of direct observables for ability, numerous empirical contributions study the correlation between ed-ucation (or other correlates of ability) and wages; see Card [2001] for a recent survey. Abowd et al. [1999] proposea strategy to disentangle the effect of education from workers’ time invariant unobservable characteristics; they findthat the latter correlate positively with wages.11Bernard et al. [1995] were the first to document that exporting firms are bigger both in terms of revenues andemployment. They also document a positive correlation between exporting and productivity.10Proposition 3. For all θ such that s (θ, 0) ≤ 0 and s (θ, 1) ≤ 0,12 the workers’ total pay-off from amatchW (θ, ψ) =s (θ, ψ)γ+ w (θ)is non-monotonic in ψ. In particular, a worker pay-off is maximized under the Beckerian assignment,ψ = µθ (θ) = θ. Similarly, the firms’ total pay-off from a matchΠ (θ, ψ) =s (θ, ψ)1− γ+ pi (ψ)is non-monotonic in θ and maximal at θ = µψ (ψ) = ψ, for all ψ such that s (0, ψ) ≤ 0 ands (1, ψ) ≤ 0.Proof. The workers’ total pay-off from a match is differentiable13 and∂W (θ, ψ)∂ψ=1γ∂R (θ, ψ)∂ϕ−∫M(ψ)∂R(x,ψ)∂ψ ν (x) dx∫M(ψ) ν (x) dx=σγ(η − 1η)E1ηψση−1η −1θση−1η −∫M(ψ) xσ η−1η ν (x) dx∫M(ψ) ν (x) dxLet l (θ) = min {ψ ∈ [0, 1] : s (θ, ψ) ≥ 0} and u (θ) = max {ψ ∈ [0, 1] : s (θ, ψ) ≥ 0}.14 If s (θ, 0) ≤ 0and s (θ, 1) ≤ 0, by continuity, s (θ, l (θ)) = s (θ, u (θ)) = 0. Therefore, ∃ y ∈ [l (θ) , u (θ)] such that∂s (θ, ψ)∂ψ∣∣∣∣ψ=y= 0This implies that, for ψ = yθση−1η =∫M(y) xσ η−1η ν (x) dx∫M(y) ν (x) dxThe function ∫M(ψ) xσ η−1η ν (x) dx∫M(ψ) ν (x) dxis strictly increasing in ψ. It follows that∂s(θ,ψ)∂ψ < 0 ψ > y∂s(θ,ψ)∂ψ > 0 ψ < yDue to the symmetry of the surplus function, the optimal assignment coincides with the Beckerian12Our result does not apply when the boundaries are binding, i.e. s (θ, 0) > 0 or s (θ, 1) > 0 over the entire typespace. The boundaries are not binding for some θ ∈ (0, 1) if the cost of search c is small enough.13The differentiability of w (·) and pi (·) implies that the surplus function is differentiable over the matching setsof either agent.14Minima and maxima are well-defined since s (θ, ψ) is continuous.11allocation, y = θ.15 An analogous argument can be used to prove the non-monotonicity of Π (θ, ψ)with the respect to θ and that the function has a maximum when θ = ψ.The agents’ incentives to match shape the non-monotonicity of the pay-off functions. A workergets her highest pay-off when matched with a firm of similar productivity. However, deviating fromthe beckerian assignment induces the worker to accept a lower outcome. On the one hand, if shematches with a low productivity firm, such a match would not generate much output. On the otherhand if she agrees to work for a more productive firm - relative to her type -, she needs to compensatethe firm for not matching with a more appropriate worker.The non-monotonicity of the total pay-off in the partner’s type is not a new in the literatureon labour market sorting. In Eeckhout and Kircher [2011] the non-monotonicity is the result of asimilar mechanism. In Bagger and Lentz [2008], instead, the non-monotonicity stems from wagegrowth expectations. In their model, the higher the productivity of the firm where the worker isemployed, the higher her ability to extract surplus from the next high productivity firm she meets.Therefore, a worker, even if employed at a relatively high productive firm, is willing to accept a lowwage with the expectation of high future wage growth.16Propositions 2 and 3 suggest an empirical strategy to identify agent types. In particular, propo-sition 3 clarifies that the information contained in wages is not sufficient to characterize both firmand worker types. Firms’ fixed effects from a wage decomposition a` la Abowd-Kramarz-Margolis(AKM) might not be correlated with the actual firm’s productivity, as shown in Eeckhout andKircher [2011].17 Agent pay-offs, such as wages and profits, are instead appropriate to characterizeworker and firm types.We conclude this section by describing the behaviour of the matching bounds.Proposition 4. For all workers θ such that s (θ, 0) ≤ 0 and s (θ, 1) ≤ 0, the matching bounds, l (θ)and u (θ), are strictly increasing in θ. In particular,∂l (θ)∂θ=θση−1η −1l (θ)ση−1η −1∂u (θ)∂θ=θση−1η −1u (θ)ση−1η −1For all firms ψ such that s (0, ψ) ≤ 0 and s (1, ψ) ≤ 0, the matching bounds l (ψ) and u (ψ) are15Suppose not and assume that the surplus function has a maximum at y < µ (θ) = θ. Then,s (θ, θ) < s (θ, y)s (y, y) < s (θ, y)Such conditions contradict the supermodularity of the surplus function. Allowing for different maximum pointsacross workers and firms violates the symmetry of the surplus function.16A similar effect is also observed in Lopes de Melo [2011].17See the appendix for a proof of this result in this model.12strictly increasing in ψ,∂l (ψ)∂ψ=ψση−1η −1l (ψ)ση−1η −1∂u (ψ)∂ψ=ψση−1η −1u (ψ)ση−1η −1Proof. s (θ, ψ) is differentiable over the type space. By the Implicit Function Theorem,∂l (θ)∂θ= −∂R(θ,ψ)∂θ −∂w(θ)∂θ∂R(θ,ψ)∂ψ −∂pi(ψ)∂ψ= −θση−1η −1l (θ)ση−1η −1[l (θ)ση−1η −∫M(θ) yσ η−1η v(y)dy∫M(θ) v(y)dy][θση−1η −∫M(l(θ)) xσ η−1η ν(x)dx∫M(l(θ)) ν(x)dx]=θση−1η −1l (θ)ση−1η −1[θση−1η − l (θ)ση−1η][θση−1η − l (θ)ση−1η]=θση−1η −1l (θ)ση−1η −1Since θ ≥ l (θ), the behaviour of the matching bound depends on the value of α ≡ σ η−1η . Inparticular,∂l (θ)∂θ=> 1 if α > 1= 1 if α = 1< 1 if α < 1As also shown by Atakan [2006], proposition 4 confirms that the matching bounds are increasingin agent type, implying a pattern of positive assortative matching in presence of frictions. However,proposition 4 also focuses on the variation in the length of the matching set across the type space. Anatural candidate to measure the length is the matching range d (ψ), defined as the difference betweenu (ψ) and l (ψ). By Proposition 4, the behaviour of the matching range reflects the incentives ofeach agent to deviate from the optimal assignment: the decision to deviate trades off the marginalloss of revenues from accepting a mismatch with the benefits of further search. For a firm ψ,the marginal loss due to a deviation of length k from the optimal assignment is given by L =σ η−1η E1ηψση−1η −1 (|ψ − k|)ση−1η −1. If σ η−1η = 1, the marginal loss is constant.18 High and lowproductivity firms have equal incentives to deviate from their optimal worker. If σ η−1η > 1, the18If σ η−1η = 1,∂d (ψ)∂ψ=∂u (ψ)∂ψ−∂l (ψ)∂ψ= 0, if ση − 1η= 113 𝜃 𝑙(𝜃) 𝑢(𝜃) 𝜓 Figure 1: Matching Bounds for Worker θ when σ η−1η = 1marginal losses are larger for agents of higher productivity; more productive firms, therefore, tendto display matching sets of smaller measures.However, d (ψ) might not be an appropriate measure to compare dispersion of worker typeswithin firms since firms exhibit differences also in the types of worker hired. Let us consider twofirms, ψH > ψL. Firm ψH hires on average very high worker types and firm ψL tends to hire very lowworker types. If σ η−1η = 1, we should observe the same d (ψ) for both firms, but we would probablynot conclude that the two firms tolerate the same degree of worker variation. This is because firmψH tolerates less variation relative to the workers hired than firm ψL. Hence we argue that thecorrect way to analyze the matching range is to adopt a scale-free dispersion measure, a normalizedmatching range d1 (ψ) where we divide the matching range by the optimal worker type hired by firmψ, i.e. θ = ψ. Define d1 (ψ) = u1 (ψ)− l1 (ψ), where u1 (ψ) =u(ψ)ψ and l1 (ψ) =l(ψ)ψ .The variation of the normalized matching range is independent from the parameters of the model,as the following proposition establishes.Proposition 5. Dispersion of worker types working at firm ψ, as measured by the normalizedmatching range d1 (ψ) is decreasing in firm type for all firms ψ such that s (0, ψ) ≤ 0 and s (1, ψ) ≤ 014Proof. Let α ≡ σ η−1η ;∂d1 (ψ)∂ψ=∂u(ψ)∂ψ ψ − u (ψ)−∂l(ψ)∂ψ ψ + l (ψ)ψ2=ψα−1u(ψ)α−1ψ − u (ψ)−(ψα−1l(ψ)α−1ψ − l (ψ))ψ2=u (ψ)[ψαu(ψ)α − 1]− l (ψ)[ψαl(ψ)α − 1]ψ2< 0Intuitively the result in proposition 5 follows from the concavity of matching sets. Attractingpartners of higher productivity becomes increasingly costly; this effect is stronger for more productiveagents, as their partners demand increasingly larger option values.Before introducing the exporting decision in our framework, we will work through some compar-ative statics exercises that capture some qualitative features of the open economy model.2.2.2 Equilibrium matching sets under changes in revenuesThis subsection explores the effect of an increase in revenues on matching sets. We are going to buildon this result when characterizing the impact of trade liberalizations. Let us consider a scenariowhere the revenues from a match increase across the entire type space,Assumption 1. (A.1) Let χ > 1 and RX (θ, ψ) = χ · RD (θ, ψ), where RD (θ, ψ) = E1η θαψα,α ≡ σ η−1η .Clearly, a SSE exists under the revenue function RX (θ, ψ). However, changes in the revenuefunction affect the properties of the equilibrium functions, and, in particular, of matching sets.Intuitively, additional revenues increase the opportunity cost of a mismatch. Both firms and workerswould achieve the higher potential revenues only if they choose to match with a partner closer totheir ideal type. If the search costs are constant, they both have an incentive to keep searching forbetter partners. The following proposition summarizes the main comparative static result of thissubsection.Proposition 6. Under constant costs of search the measure of the matching sets is inversely relatedto the (mass of) revenues generated from a match, i.e. the matching sets under the revenue functionRX are a subset of the matching sets under RD, as specified in (A.1).The proof of proposition 6 is by contradiction, following the logical steps described below.Changes in revenues affect the total surplus from a match, which in equilibrium would impacteither the option values, the matching sets or the probability to form a match. However, the condi-tion such that the increase in revenues is fully absorbed by the option values violates the constant15surplus condition (CSC),19For worker θ: γ · c =∫M(θ)s (θ, ψ) v (ψ) dψ (7)For firm ψ: (1− γ) · c =∫M(ψ)s (θ, ψ) ν (θ) dθ (8)i.e. the condition such that an agent is willing to search for a potential partner as long as the searchcost equates the share of the surplus she expects to extract over her set of acceptable matches.If, by contradiction, option values would entirely absorb changes in revenues, agents would expectto extract a larger surplus over their existing matching set and therefore, would have additionalincentives to search for better partners. Then, the conjecture that matching sets do not respond toan increase in revenues is inconsistent with the CSC.Proof. Let wD (θ), piD (ψ), αDθ (θ, ψ), αDψ (θ, ψ), νD and vD be the option values, the matchingfunctions20 and the distributions of the unmatched under the revenue function RD (θ, ψ); let wX (θ),piX (ψ), αXθ (θ, ψ), αXψ (θ, ψ), νX and vX be the option values, the matching functions and thedistributions of the unmatched under RX (θ, ψ).Step 1. Matching set are not invariant to changes in revenues. By contradiction, suppose thatmatching sets do not respond to a change in revenues,αDθ (θ, ψ) = αXθ (θ, ψ)αDψ (θ, ψ) = αXψ (θ, ψ)By lemma 2, νD = νX and vD = vX . Then, for worker θ, let(lD (θ) , uD (θ)):{lD (θ) ≡ min{ψ ∈ [0, 1] : sD (θ, ψ) = 0}uD (θ) ≡ max{ψ ∈ [0, 1] : sD (θ, ψ) = 0}(lX (θ) , uX (θ)):{lX (θ) ≡ min{ψ ∈ [0, 1] : sX (θ, ψ) = 0}uX (θ) ≡ max{ψ ∈ [0, 1] : sX (θ, ψ) = 0}Then,lX (θ) = lD (θ) = l (θ) ⇔{wX (θ) = wD (θ) + κθpiX (l (θ)) = piD (l (θ)) + κl(θ)where κθ + κl(θ) = (χ− 1)RD (θ, l (θ)). The expected share of surplus over all admissible matches19This condition follows immediately from the equilibrium characterization of the agents’ option values. SeeAtakan [2006] for a proof.20Define αθ (θ, ψ) = 1 if ψ ∈M (θ) and αψ (θ, ψ) = 1 if θ ∈M (ψ).16for θ must equate the search costs, as implied by the CSC,γ · c =∫MX(θ)[RX (θ, ψ)− wX (θ)− piX (ψ)]v (ψ) dψ=∫MD(θ)[RD (θ, ψ)− wD (θ)− piD (ψ) + (χ− 1)RD (θ, ψ)− (κθ + κψ)]v (ψ) dψTherefore, ∫MD(θ)[κψ + κθ] v (ψ) dψ =∫MD(θ)(χ− 1)RD (θ, ψ) v (ψ) dψ (9)Let α ≡ σ η−1η ; using the conditions wi (θ) + pii (l (θ)) = Ri (θ, l (θ)) and wi (l (ψ)) + pii (ψ) =Ri (l (ψ) , ψ), i = D,X, and the definition for κθ, κψ,κψ + κθ = (χ− 1)E1η [l (θ)α θα + ψαl (ψ)α]−(κl(ψ) + κl(θ))(10)whereκl(ψ) + κl(θ) ≥ (χ− 1)E1η l (ψ)α l (θ)αHowever, equation (10) does not satisfy the CSC because the function (χ− 1)E1ηψαθα is supermod-ular.21 By a similar reasoning, it is possible to prove that uX (θ) 6= uD (θ).Step 2. Matching sets are inversely related to changes in revenues. By contradiction, assume thatαDθ (θ, ψ) < αXθ (θ, ψ)αDψ (θ, ψ) = αXψ (θ, ψ)Then,lD (θ) ≥ lX (θ)uX (θ) ≥ uD (θ)and22piF(lX (θ))≤ piD(lX (θ))+ κlX(θ)where κlX(θ) is as defined above. By lemma 1, the conditions on the matching sets also imply that21The proof applies to any supermodular revenue function.22If lD (θ) ≥ lX (θ),RD(θ, lX (θ))− wD (θ)− piD(lX (θ))≤ RX(θ, lX (θ))− wX (θ)− piX(lX (θ))= 0This implies the conditions on the option value functions.17∫MX(θ) vαXψ dψ ≥∫MD(θ) vαDψ dψ. Then, from the CSC,γ · c =∫MX(θ)[RX (θ, ψ)− wX (θ)− piX (ψ)]vαXψ (ψ) dψ≥∫MD(θ)[RX (θ, ψ)− wX (θ)− piX (ψ)]vαDψ (ψ) dψ≥∫MD(θ)[χ ·RD (θ, ψ)− wD (θ)− piD (ψ)− κθ − κψ]vαDψ (ψ) dψ=∫MD(θ)[RD (θ, ψ)− wD (θ)− piD (ψ)]vαDψ (ψ) dψ︸ ︷︷ ︸γ·c++∫MD(θ)[(χ− 1)RD (θ, ψ)− (κθ + κψ)]vαDψ (ψ) dψThe implied inequality is violated over the type space due to supermodularity. Assuming thatαDψ (θ, ψ) ≶ αXψ (θ, ψ) would produce similar contradictions. The only case compatible with theCSC requires thatαDθ (θ, ψ) > αXθ (θ, ψ)αDψ (θ, ψ) > αXψ (θ, ψ)The comparative statics exercise in the present subsection suggests that mechanisms increasing thevalue achievable in a match induce agents to search more intensively for their ideal partner. Inpresence of a trade liberalization, however, the increase in revenues affects only a subset of all firmsand induces interesting additional effects for non-exporters, as we will show in the following sections.2.3 A model of assortative matching: open economyWe consider a world made up with two symmetric countries, a domestic and a foreign economy; weindex foreign economy variables with stars. Within each country, firms face the additional decisionto trade. In this section we introduce exporting a` la Melitz [2003], with fixed cost of exportingcommon to all firms. The decision to export and to match are simultaneous. After meeting, if afirm of type ψ and a worker of type θ agree to match and it is profitable for the firm to export, theoutput from the match has to be allocated between the domestic and foreign markets,θ · ψ = qd + qxwhere qd is the quantity demanded on the domestic market and qx is the quantity demanded on theforeign market. If the firm decides to produce exclusively for the domestic market, the output from18the match has to meet only the domestic demandθ · ψ = qdWe maintain the monopolistic competition structure a` la Krugman [1979]. Demand for an individualvariety is isoelastic with elasticity η > 1, common to both countries. Transport costs τ , in thestandard form of iceberg melting costs, are paid by consumers when purchasing a foreign variety.An exporting firm maximizes its total revenues by equalizing marginal revenues across markets; witha CES utility, this requires that the producer prices are equal across the two markets. Therefore,the revenues from a match between a worker and an exporting firm areRX (θ, ψ) =[E + τ1−ηE∗] 1η ψση−1η θση−1ηIf the firm operates only on the domestic market, the revenues from the match amount toRD (h, ϕ) = E1η · ψση−1η θση−1ηIn this framework, exporting firms enjoy higher revenues. Since accessing the foreign market requiresthe payment of a fixed cost, fx > 0, per instant of time, only firms able to generate revenues largeenough to cover the fixed costs self-select into exporting. As in other trade models with firmheterogeneity, the exporting decision is associated to a cut-off rule. Here, however, the relationbetween firm productivity and export status is not 1-to-1. In fact, since the revenues depend onboth agents’ types, the decision to export is conditional on firm and worker types.We now move to analyze the interaction between matching and exporting. We assume that aftermeeting a worker and observing his type, each firm decides whether to match and, conditional on thematch taking place, which market to serve. The simultaneity between matching and the exportingdecision implies that trade affects the surplus from a matchψ ∈MX (θ) ⇔ max{sD (θ, ψ) , sX (θ, ψ)}≥ 0⇔ max{RD (θ, ψ)− wX (θ)− piX (ψ) , RX (θ, ψ)− wX (θ)− piX (ψ)}≥ 0as well as the structure of pay-offsFirm’s Payoff: piX (ψ) +max{sD(θ,ψ), sX(θ,ψ)}1−γWorker’s Payoff: wX (θ) +max{sD(θ,ψ), sX(θ,ψ)}γwhere piX (ψ) and wX (θ) are the option value of remaining unmatched in an open economy. Changesin option values ultimately affect the measure of unmatched agents.At this point, we are ready to introduce the definition of a Trade SSE equilibrium.Definition A Trade Stationary Search Equilibrium (SSE) consists of a pair of functions wX :[0, 1] → R, piX : [0, 1] → R, a pair of strategies MX (θ), θ ∈ [0, 1], MX (ψ), ψ ∈ [0, 1], a pair of19distributions, νX (θ) ≤ g (θ), vX (ψ) ≤ h (ψ) and a cut-off rule ψˆ (θ) such that• given MX (θ), MX (ψ), νX (θ), and vX (ψ), wX (·) and piX (·) solvewX (θ) =∫ 10max{−c+ wX (θ) +max{sD (θ, ψ) , sX (θ, ψ)}γ,−c+ wX (θ)}vX (ψ) dψ (11)piX (ψ) =∫ 10max{−c+ piX (ψ) +max{sD (θ, ψ) , sX (θ, ψ)}1− γ,−c+ piX (ψ)}νX (θ) dθ (12)• given wX (θ) and piX (ψ),ψ ∈MX (θ) iff max{sD (θ, ψ) , sX (θ, ψ)}≥ 0 (13)θ ∈MX (ψ) iff max{sD (θ, ψ) , sX (θ, ψ)}≥ 0 (14)• given MX (θ), MX (ψ),λ(g (θ)− νX (θ))= ρ · νX (θ)∫MX(θ)vX (ψ) dψ (15)λ(h (ψ)− vX (ψ))= ρ · vX (ψ)∫MX(ψ)νX (θ) dθ (16)• conditional on θ ∈MX (ψ), firms with ψ ≥ ψˆ (θ) exports; firms with ψ < ψˆ (θ) serve only thedomestic market.The definition of a Trade SSE clarifies how exporting influences equilibrium outcomes. Additionalrevenues modify the incentives of workers and firms to match, through their option value of beingunmatched. The option values, in turn, influence the matching sets and the distributions of theunmatched.Before any further analysis, we must first establish that a Trade SSE exists and characterize itsbasic properties.Theorem 2. Under constant costs of search, a Trade SSE exists; moreover, it entails positive as-sortative matching, i.e. the matching sets are convex and the matching bounds are non-decreasingin the agents’ type.Proof. Step 1: A Trade SSE exists.. The equilibrium conditions for a Trade SSE are equivalent tothe conditions in the definition of a SSE with revenue functionR (θ, ψ) = max{RD (θ, ψ) , RX (θ, ψ)} (17)The revenue function (17) is continuous and supermodular. By theorem 1, a SSE exists and displayspositive assortative matching.Step 2: Agents’ continuation value do not affect the decision to export; a SSE is a Trade SSE. LetwX (·) , piX (·) be the option value in the SSE with revenue function (17). Then, conditional on20θ ∈ M (ψ), a firm exports if the total pay-off from serving all markets is higher than the pay-offobtained when serving exclusively the domestic marketpiX (ψ) +sX (θ, ψ)1− γ≥ piX (ψ) +sD (θ, ψ)1− γ⇔ RX (θ, ψ)− fX ≥ RD (θ, ψ)Let ψˆ (θ) be such that, conditional on θ ∈M (ψ),RX(θ, ψˆ (θ))− fX = RD(θ, ψˆ (θ))Therefore, conditional on θ ∈M (ψ), firms with ψ ≥ ψˆ (θ) will serve both the domestic and the for-eign market. The SSE with revenue function max{RD (θ, ψ) , RX (θ, ψ)} satisfies also the exportingcut-off rule; it is, therefore, a Trade SSE.Using the explicit form of the revenue function, we can characterize more precisely the cut-offrule,23Corollary 1. Exporting Cut-off. Under costly trade, the marginal productivity of an exportersatisfiesψˆ (θ) =[fx[E + τ1−ηE∗]1η − E1η] 1α1θ(18)Conditional on θ, firms with productivity ψ > ψˆ (θ) export.Proof. A marginal exporter is indifferent between serving the domestic and serving both domesticand foreign markets if[E + τ1−ηE∗] 1η ψˆαθα − fx = E1η · ψˆαθαCondition (18) immediately follows.The export cut-off described in corollary 1 differs from the condition in Melitz [2003] and Chaney[2008]. Here both firm and worker type determine the productivity and the revenues achievable ina match. Firm type space can be partitioned into three sections. A first interval contains firms thatare productive enough to always generate sufficient revenues to cover the fixed costs of exporting.A second set includes firms of intermediate productivity whose ability to export is conditional onmeeting a worker with an ability level high enough. Finally, a third interval encompasses firms oflow productivity that, independently from the type of match they agree to, they will not be ableto profitably access the foreign market. Figure 2 shows the cut-off rule over the type space. Ourframework is equivalent to Melitz [2003] in terms of the productivity of a match.As in Melitz [2003], the export cut-off is also affected by the foreign market size and the costs ofexporting. A larger foreign market size increases the revenue potential from exporting and allowsmore firms to start exporting; higher export costs, instead, raise the barriers to access the foreignmarket and increase the minimal productivity threshold to profitably export.23Figure 8 shows the export cut-off function.21 𝜓(?̅?) 𝜓�𝜃� Non Exporters Conditional  Exporters Exporters Figure 2: Firm Type Space by Export Status2.3.1 Properties of the trade equilibriumIn what follows, we analyze the properties of a Trade SSE and characterize the effects of tradeliberalizations (next section). First, we focus on the behaviour of option values.Proposition 7. The agents’ option values under trade are no smaller than the option values underautarky. The pay-off functions are monotonic in the agent’s type and non-monotonic in the partner’stype.Proof. Let wX , wD be the option values for a worker under trade and under autarky. From the flowequation characterizing the option value for a worker of ability θ,wD (θ) =∫[0,1]max{−c+ wD (θ) +sD (θ, ψ)γ, −c+ wD (θ)}vD (ψ) dψ≤∫[0,1]max{−c+ wD (θ) +RX (θ, ψ)− wD (θ)− piD (ψ)γ, −c+ wD (θ)}vD (ψ) dψThen, either wD (θ) ≤ wX (θ) or vX (ψ) ≥ vD (ψ) or both. The rest of the proof is as in propositions2 and 3.Matching bounds preserve some useful properties in open economy.Proposition 8. For all workers θ such that s (θ, 0) ≤ 0 and s (θ, 1) ≤ 0, the matching bounds, l (θ)and u (θ) are continuous functions of θ and a.e. differentiable. In particular, at points at which theyare differentiable∂l (θ)∂θ=θα−1l (θ)α−1∂u (θ)∂θ=θα−1u (θ)α−1Proof. The surplus function is continuous and monotone along the border of the matching sets.Therefore, the matching bounds l (θ) and u (θ) are continuous.24 The differentiability fails onlyalong the export cut-off.By proposition 8, the matching bounds must be well-behaved since the surplus function inheritsthe continuity and monotonicity of the revenue function. If, instead, the surplus function was not24See Jittorntrum [1978] for a proof.22continuous, then the minimal productivity level of a potential partner would be subject to discretejumps (the matching bound would lose points); if the surplus function was not monotone aroundthe border of the matching set, then, the productivity levels associated to zero surplus would beconstant over some region of the type space (the matching bounds would explode).2.4 Trade liberalizations and matching setsThe present section analyzes episodes of trade liberalization. Trade liberalizations are specified as areduction in the variable trade costs from the autarkic level τ →∞.25Proposition 9. Suppose that τ <∞, decreasing from the autarkic level τ →∞. Then, the matchingsets of agents whose matches generate sufficient revenues to export are smaller after opening to trade.The matching sets of agents whose matches never generate sufficient revenues to export, instead, arelarger after opening to trade.Proof. Let ψˆm be such that ∀ψ ≥ ψˆm, ∃ θ ∈MX (ψ) such thatRX (θ, ψ) = RD (θ, ψ)and let ψˆM be such that ∀ψ ≥ ψˆM and ∀ θ ∈MX (ψ)RX (θ, ψ) ≥ RD (θ, ψ)Then, it is possible to identify three separate regions of the firm type space: Lψ =[0, ψˆm), Mψ =[ψˆm, ψˆM)and Uψ =[ψˆM , 1]. Analogously for workers, Lθ =[0, θˆm), Mθ =[θˆm, θˆM)andUθ =[θˆM , 1]Step 1: Matching Sets are not invariant to trade opening. By contradiction, suppose that ∀ θ ∈[0, 1] and ∀ψ ∈ [0, 1], MX (θ) = MD (θ) and MX (ψ) = MD (ψ). By lemma 2, this implies thatνD (θ) = νX (θ) and vD (ψ) = vX (ψ). On Lθ,RD (θ, l (θ))− wX (θ)− piX (l (θ)) = 0RD (θ, u (θ))− wX (θ)− piX (u (θ)) = 0Since under autarky,RD (θ, l (θ))− wD (θ)− piD (l (θ)) = 0RD (θ, u (θ))− wD (θ)− piD (u (θ)) = 025A reduction in fixed costs of trade would deliver similar predictions.23then,wX (θ) + piX (l (θ)) = wD (θ) + piD (l (θ))wX (θ) + piX (u (θ)) = wD (θ) + piD (u (θ))On Mθ,wX (θ) + piX (l (θ)) = wD (θ) + piD (l (θ))wX (θ) + piX (u (θ)) = wD (θ) + piD (u (θ)) +RX (θ, u (θ))−RD (θ, u (θ))and on Uθ,wX (θ) + piX (l (θ)) = wD (θ) + piD (l (θ)) +RX (θ, l (θ))−RD (θ, l (θ))wX (θ) + piX (u (θ)) = wD (θ) + piD (u (θ)) +RX (θ, u (θ))−RD (θ, u (θ))The characterization of option values, however, violates the CSC, as proven in proposition 6.26Step 2: Matching sets do not change monotonically over the type space. By contradiction, sup-pose that ∀ θ ∈ [0, 1] and ∀ψ ∈ [0, 1] either MX (θ) ⊂ MD (θ) and MX (ψ) ⊂ MD (ψ) orMD (θ) ⊂ MX (θ) and MD (ψ) ⊂ MX (ψ). Consider, first, the case such that MX (θ) ⊂ MD (θ)and MX (ψ) ⊂MD (ψ). This implies that νX (θ) > νD (θ) and vX (ψ) > vD (ψ). On LθwX (θ) + piX(lX (θ))= wD (θ) + piD(lD (θ))+RD(θ, lX (θ))−RD(θ, lD (θ))wX (θ) + piX(uX (θ))= wD (θ) + piD(uD (θ))+RD(θ, uX (θ))−RD(θ, uD (θ))On Mθ,wX (θ) + piX(lX (θ))= wD (θ) + piD(lD (θ))+RD(θ, lX (θ))−RD(θ, lD (θ))wX (θ) + piX(uX (θ))= wD (θ) + piD(uD (θ))+RX(θ, uX (θ))−RD(θ, uD (θ))and on Uθ,wX (θ) + piX(lX (θ))= wD (θ) + piD(lD (θ))+RX(θ, lX (θ))−RD(θ, lD (θ))wX (θ) + piX(uX (θ))= wD (θ) + piD(uD (θ))+RX(θ, uX (θ))−RD(θ, uD (θ))On Lθ, Mθ and Uθ,wX (θ) + piX(lX (θ))≥ wD (θ) + piD(lD (θ))(19)26To follow similar steps to proposition 6, set κθ + κl(θ) ≡ RX (θ, l (θ))−RD (θ, l (θ)).24The CSC is violated over Lθ. In fact,γ · c =∫MD(θ)[RD (θ, ψ)− wD (θ)− piD (ψ)]vD (ψ) dψ<∫MX(θ)[RD (θ, ψ)− wD (θ)− piD (ψ)]vX (ψ) dψ≤∫MX(θ)[RD (θ, ψ)− wX (θ)− piX (ψ)]vX (ψ) dψ= γ · cOn the other hand, ifMD (θ) ⊂MX (θ) andMD (ψ) ⊂MX (ψ), the CSC, then, would be violatedover Uθ and Uψ.Step 3: After opening to trade, matching sets expands for non-exporting matches and contract forexporting matches. Given the conclusion in Step 2, it must be that MD (θ) ⊂MX (θ) over Lθ andMX (θ) ⊂ MD (θ) over Uθ. The variation of the matching sets, however, cannot be characterizedex-ante over Mθ. In fact, for θ ∈ Mθ lD (θ) ≤ lX (θ) but uX (θ) ≤ uD (θ): matching sets over Mθmight expand or contract after opening to trade.In presence of a trade liberalization, the trade-off between higher option values and larger measure ofunmatched agents produces different outcomes over the type space. In fact, the expansion in revenuescreates incentives to search more intensively over the space where exporting matches occur, leadingto tighter matching sets over that part of the space. However, the reduction in the measure of thematching sets over the upper portion of the space induces the opposite effect for the least productiveagents. Firms and workers whose match is unlikely to produce enough revenues to cover the fixedcosts of exporting tend to match with agents further away from their ideal companion as thoseagents lose potential partners they were able to match with before opening to trade.The effect of additional revenues due to exporting on the measure of the matching set cannot besigned ex-ante for firms and workers of intermediate productivity. It depends upon the features ofthe initial distributions of agents and the parameters of the revenue function. We expect, however,that for agents with low probability of achieving an exporting match, the matching set is likely toexpand - those agents need to balance the loss of their most profitable matches with an increasein the probability of finding a match; for the agents with a higher probability of meeting a partnersufficiently productive to export the matching set is likely to contract - those agents are sufficientlycompensated in terms of higher revenues and would not need to adjust their matching probabilities,through their matching sets.2.5 Open economy and heterogeneous trade costsThe original contribution by Melitz [2003] that we followed in previous sections implies that weshould never observe two firms of the same productivity, but different export status. The starkprediction that all exporters should be more productive than non-exporters is clearly not supported25by the data, as argued for example by Bernard et al. [2003] and Helpman et al. [2013]. In both USand Brazilian data the distribution of productivity of exporters has a higher mean, but also displaysa substantial overlap with the productivity distribution of non-exporters, a feature that is clearlyshared by the French sample used for the empirical analysis in Chapter 3, as shown in Figure 10.Similarly to Helpman et al. [2013], we allow different firms to have different costs of exporting inorder to disentangle the effect of exporting from that of firm productivity. This may reflect variousidiosyncratic factors such as better knowledge of the export market that makes setting up an exportoperation less costly. Because our interest in this section is exclusively in comparing exporters andnon-exporters and not in the endogenous sorting into exporting or the estimation of the fixed costof exporting, we make one further simplifying assumption. We assume that some firms draw aprohibitively high fixed cost of exporting, while the rest of the firms draw a negligible fixed cost. Tobe more precise, the fixed costs are drawn from the following distributionP (f = fL) = λP (f = fH) = 1− λwhere fL = 0 and fH → ∞. By the law of large numbers, a fraction λ [h (ψ)− v (ψ)] of firmsexports; the complementary fraction operates only on the domestic market. All exporting firms alsoface an iceberg transport cost τ > 1.It is straightforward to verify that, for given θ and ψ, revenues of an exporting firm are largerthan those of a non-exporting firm. It is useful to rewrite revenues of an exporting firm and anon-exporting firm with given productivity ψ as follows:Rd (θ, ψ) = (Adθψ)σ(η−1)η , (20)Rx (θ, ψ) = (Axθψ)σ(η−1)η (21)where Ad = E1σ(η−1) and Ax =(E + E∗τ1−η) 1σ(η−1) and Ax > Ad. We therefore establish thefollowing property.Remark 1 Exporting is isomorphic to an increase in productivity for a firm of initial productivityψ.Based on Remark 1 it is possible to analyze the effect of export status on matching by char-acterizing the matching behavior of more productive versus less productive firms. In fact, startingfrom a specific distribution of firm types h (ψ) and introducing export opportunities, we can derivea distribution based on adjusted firm types ϕ ≡ Aiψ, i = d, x, h′ (ϕ).The rest of the economy displays the same features as described in the previous sections. Theproductivity rescaling implies that the equilibrium characterization is equivalent to that of a closedeconomy under h′ (ϕ). Theorem 1 and propositions 2-5 all apply in this context. Exporting (ormore productive) firms match on average with workers of higher productivity and tolerate lowervariation in the set of workers they match with, relative to their optimal partner. In section 2.8 wewill lay-out the theoretical predictions in greater details.262.6 Welfare implicationsWe have so far not discussed the consequences of trade liberalizations in terms of welfare. Searchcosts induce firms and workers to deviate from the efficient worker-to-firm assignment, generatingaggregate output losses. Opening to trade affects the incentives of workers and firms to match. Aftera trade liberalization occurs, the model features two counteracting effects. While newly exportingfirms have stronger incentives to tighten their matching range, non-exporting firms see their revenuesdecline because of import competition and therefore will see an increase in their normalized matchingrange. If the tightening of the matching sets for exporters prevails, the deviation of the realizedoutput from the optimal level contracts. Our model, thus, identifies worker selection as a newpossible channel for the realization of the gains from trade. It is important to note, however, thatsuch a channel would be absent in a model with worker-firm heterogeneity and no search frictionsas in Sampson [2014]. The worker selection effect starts operating only as search frictions increasefrom a zero level.We present our analysis in two steps. First, we report partial equilibrium results that examinethe revenue loss as a function of firm productivity and confirm that the revenue loss is lower formore productive firms. Second, we turn to a general equilibrium model where we introduce a secondsymmetric country and compute overall welfare changes, taking into account variety effects and pricechanges.2.6.1 Revenue loss in partial equilibriumWe start by analyzing the revenue loss as a function of firm productivity. We choose to present ameasure of revenue loss relative to the optimal allocation as in Eeckhout and Kircher [2011]. Weassume that the initial distribution of firms and workers, g (θ) and h (ϕ), are uniform, implying thatthe optimal assignment is between worker and firms of similar type.27 Then, for each firm ϕ andworker θ we can define a revenue loss relative to the optimumL (ϕ, θ) =12(θα − ϕα)2The assumption in creating such a measure is that in the optimal allocation a worker or type θwould generate a revenue of θ2α and is allocated half of that revenue. Holding the type of the firmconstant at ϕ we sum the revenue loss from the optimal level for each possible worker type in theacceptance set. We then divide by the optimal revenue summed across the same range. We obtainthe share of revenues lost relative to the optimum for a firm of type ϕ (and the workers in that firm’sacceptance range),RL (ϕ) =∫ u(ϕ)l(ϕ)12 (θα − ϕα)2 dθ∫ u(ϕ)l(ϕ)[12θ2α + 12ϕ2α]dθ(22)The following proposition shows that such deviations from the optimal revenues are smaller for moreproductive firms.27Different distributions of types imply a different equilibrium assignment and, therefore, different loss functions27Proposition 10. The share of output lost relative to the optimal revenues, RL (ϕ), is decreasing inthe type of the firm ϕ for any c if α ≤ 1Proof. See Appendix A.2.Although we are not able to prove proposition 10 for all values of α, our simulations show thatthe share of output lost relative to the optimal revenues is always decreasing in the type of the firmϕ. This proposition is a comparison across firm types observed in a given equilibrium, so that forexample we can use this proposition to compare revenue losses of an import-competing firm versusan exporting firm of the same original underlying productivity. The exporting firm will feature alower share of revenues lost because of mismatch across all the possible matches in her matching set.This partial equilibrium result cannot be employed, instead, to evaluate the overall welfare impactof trade opening because the equilibrium under autarky will feature different type distributions anddifferent shapes of the matching set compared to the equilibrium under trade. For the the generalequilibrium analysis, we resort to simulation and calibration that we describe in the next section.2.7 Welfare analysis: calibration and simulationWe calibrate the open-economy version of the heterogeneous fixed cost model to match momentsof the French Economy. The parameters to be calibrated are as follows: the elasticity of demand,η; the curvature of the production function, σ; the variable trade cost τ ; the fraction of firmsλ drawing the low fixed cost of export; the rate at which matches are dissolved, δ; the rate atwhich matches are created, ρ; the search cost c; the worker ability distribution, g (θ), and the firmproductivity distribution, h (ψ). We need to calibrate those parameters for both a domestic anda foreign economy; in what follows we will assume that the home country and the foreign countryshare the same characteristics.In a model with monopolistic competition and CES preferences, the elasticity of demand η mapsinto the trade elasticity, ε = η − 1. Following the literature on trade elasticity estimates,28 we setη = 4, the median value over the range of those estimates.We calibrate the curvature of the production function to the elasticity of a CES-aggregate ofworker types with respect to firm revenues. In the estimation, σ = 0.653.We use the implied relationship between foreign and domestic shipments at an exporting firm toidentify a plausible parametrization of the variable trade cost. From the model, the ratio betweenthe output sold in the domestic market and the exported output depends on the relative marketsize,qddqdx=(EE∗)1/ητη−1 (23)The domestic market size E and the foreign market size E∗ depend on the set of initial parameters.However, if the primitives are identical, the domestic and the foreign country share the same size,E = E∗. Therefore, equation (23) implies τ = 1.513.28See Head and Mayer [2013].28We maintain the normalization fL = 0 and fH → ∞; therefore, the fraction of firms λ drawingthe low fixed cost of export corresponds to the fraction of exporters. In our data, λ = 0.8.29We set the search cost c to match the average within-firm wage dispersion in the data. Thisimplies c = 0.025. We use the estimates from Hairault et al. [2012] to calibrate the job findingprobability ρ and the separation probability δ.30Finally, we parametrically estimate the worker ability distribution g (θ) and the firm productivitydistribution h (ψ) using our proxies for agent types, average worker wage and (the rank of the)domestic market share. We assume that both distributions are Beta with parameters (αθ, βθ) forworkers and (αψ, βψ) for firms. Table 1 summarizes the parameters from the calibration.We simulate the model with n = 1000 firms for T = 1000 periods. The model does a goodjob at matching the average ratio of foreign to total shipments and the average within-firm wagedispersion. The results from Table 2 are not surprising since those moments were targeted by ourcalibration.Next, we move to the welfare analysis. Our analysis focuses on two welfare proxies: changesin real expenditure and changes in relative deviations from the optimal worker-to-firm assignment,expression (22). We characterize the steady state equilibrium in the model with two symmetriccountries under the parametrization from Table 1; we, then, compare the steady state autarkyequilibrium that we derive when setting the share of exporters λ = 0 to the initial equilibrium. Inour formulation this is equivalent to a very large increase in the fixed export costs, fL = fH →∞.We find that moving to autarky reduces real expenditure by 22%. The change in real expenditure,however, captures 3 effects: the change in the number of available varieties, the change in the workerselection patterns and a price effect. While the first two effects are positively related to welfare, thethird effect, instead, acts to reduce welfare after a trade liberalization. This is due to the capacityconstraint in our model: the output in a match is fixed and proportional to the agents’ types. Thedemand shocks due to a trade liberalization have no impact on the firm’s production decision; aftertrade opening, a firm only reallocates part of its output from the domestic to the foreign market.The second measure we chose, instead, changes in relative deviations from the optimal worker-to-firm assignment, captures exclusively selection effects. We construct deviations from the optimalworker-to-firm assignment comparing the realized real revenues in presence of search costs to the realrevenues in the frictionless equilibrium; we normalize by the revenues in the frictionless equilibrium.We find that, in open economy, the losses are −86.08% of the real revenues under optimal allocation;moving back to autarky implies an increase in the losses to −87.44%. Therefore, opening to tradeis associated with a reduction in the relative deviation from the efficient allocation by −1.36%.Further, we look at how our two measures are affected by different levels of search cost c andα.31 The results are shown in Tables 3 and 4. The numerical results suggest that gains from tradeand frictions are substitutes in our framework. Both changes in real revenues and changes in relative29In our dataset, firm-level information is available only for firms with more than 20 employees. This thresholdexcludes a large portion of non-exporting firms.30Using administrative data on the labour market, Hairault et al. [2012] estimate the job finding probability andthe separation probability from 1994 onwards. They find that the average separation probability is 1.7%, while theaverage job finding probability is 13.5%31α collects demand and supply parameters, α = σ η−1η , where η is the elasticity of demand and σ is the curva-ture of the production function.29real revenue losses seems to be non-decreasing in the level of search cost, for a given α. While wewould still expect gains from trade due to variety effects, changes in real revenue losses would bezero in absence of any frictions and positive if the search costs are strictly positive. There existsa complementarity, instead, between gains from trade and α, as expected. Fixing the search costand comparing across values of α, we find that real expenditure is increasing in α. Revenue losses,instead, are inversely related to α.2.8 Testable implicationsThe model delivers novel cross-sectional and dynamic testable predictions. The ability to test thosepredictions is based upon the identification of worker and firm types. In what follows, we will assumethat measures capturing worker and firm types are available; we will relegate the discussion on theidentification strategy of agent types to next chapter.A first prediction of this chapter is that exporters match with workers that are characterized bylower relative dispersion of ability. The theory shows that the only robust prediction regarding thelink between worker type dispersion and export status (or productivity) requires expressing suchdispersion relative to the optimal worker type. Similar variation can be identified across sectors andover time, if firms across sectors and over time differ in the degree of openness: we should expectthat exporters in sectors with lower tariffs or higher aggregate demand would select an even lessdispersed pool of workers if compared to non-exporters.A second prediction focuses on the dynamic impact of trade liberalizations across firms. Areduction in tariffs or a foreign demand shock induces firms that gain access to foreign markets tobecome more selective. Low productivity firms, however, are adversely affected in an episode oftrade liberalization: in fact, firms of low productivity tend to match with agents further away fromtheir ideal companion as those agents lose potential partners they were able to match with beforeopening to trade. The differential impact of trade opening magnifies the matching set differencesbetween more productive and less productive firms.Finally, our model also predict that exporting firms hire workers of higher average type, as inSampson [2014] and, under the interpretation of permanent worker heterogeneity, Helpman et al.[2010]. The next chapter develops an empirical strategy to test these predictions.2.9 ConclusionThe present chapter develops a model to analyze the impact of trade on the sorting patterns ofworkers across firms. If agents’ types are not observable before a meeting occurs and searching fora partner is costly, both workers and firms agree to match if the cost of search exceeds the benefitfrom a more suited partner. Mechanisms that expand the revenues achievable in an ideal matchinduce both agents to engage in more intensive search. We analyze trade, one of such mechanisms.Gaining access to foreign markets expands the revenues achievable in the ideal match and inducesworkers and firms that have a positive probability to export to become more selective. We will testthis prediction in the next chapter.30Table 1: Model CalibrationParameter Model Data Momentη = 4 Demand Elasticity Average trade elasticityσ = 0.653 Production Curvature Worker type elasticityτ = 1.513 Variable trade cost Average foreign to domestic shipmentsλ = 0.8 Share of exporters Average share of exportersδ = 1.7% Destruction rateAverage separation probabilityHairault et al. [2012]ρ = 13.5% Meeting rateAverage number of new hiresHairault et al. [2012]c = 0.025 Search cost Within-firm wage dispersionB (αθ = 39.92, βθ = 28.96) Worker distribution Empirical distributionB (αψ = 0.89, βψ = 1.09) Firm distribution Empirical distributionTable 2: Model FitMoment Data ModelAverage foreign to total shipments 0.28 0.25Average within-firm wage dispersion 0.91 0.92Table 3: Changes in Real Expenditureα = 0.75 α = 1 α = 1.25cH = 0.025 22.0% 27.2% 34.1%cM = 0.005 15.6% 16% 16.1%cL = 0.001 15.4% 15.2% 15.9%Notes: Changes in real expenditure relative toautarky, by search cost, ci, i = {H,M,L} and α.α collects demand and supply parameters, α =σ η−1η , where η is the elasticity of demand andσ is the curvature of the production function.Simulated results with variable trade cost τ =1.513, share of exporters (after trade opening)λ = 0.8 and empirical distributions for workerand firm types.31Table 4: Real Relative Revenue Lossesα = 0.75 α = 1 α = 1.25cH = 0.025 -1.36% -1.11% -1.01%cM = 0.005 -1.12% -0.95% -0.85%cL = 0.001 -1.05% -0.95% -0.82%Notes: changes in relative revenue losses com-pared to autarky, by search cost, ci, i ={H,M,L} and α. α collects demand and supplyparameters, α = σ η−1η , where η is the elasticity ofdemand and σ is the curvature of the productionfunction. The revenue losses are relative devia-tion of the realized from the revenues under theoptimal assignment; we normalized by the rev-enues under the optimal assignment. Simulatedresults with variable trade cost τ = 1.513, shareof exporters (after trade opening) λ = 0.8 andempirical distributions for worker and firm types.323. Does Exporting improve Matching? Evidence from FrenchEmployer-Employee Data3.1 IntroductionDoes exporting influence the sorting patterns of workers across firms? In presence of frictions inthe labour market, the equilibrium worker-to-firm assignment does not depend excusively on theinteraction between worker and firm characteristics in production, as in Becker [1973]. Firms acceptsome mismatch in equilibrium if the search cost exceeds the benefit from a more suited partner. Thedecision to export, however, influences the expected pay-offs of agents and contributes to shape thematching decisions of workers and firms. In the previous chapter, we show that when firms gainaccess to the foreign market their revenue potential increases; since deviations from the ideal matchquickly reduce the value of the relationship, matching with the right worker becomes particularlyimportant.In this chapter we test the theoretical predictions on the differences in the ability distributionbetween exporters and non-exporters using matched employer-employee data from France. We showthat exporters select pools of workers characterized by a higher average type and a lower typedispersion than non-exporting firms. While the first effect is predicted by other models (Helpmanet al. [2010] and Sampson [2014]) we disentangle pure exporter wage premia (deriving from profit-sharing with workers as in Amiti and Davis [2012]) from the selection of better workers by exportingfirms. The second effect, i.e. the influence of exporting on worker type dispersion, is unexploredin the literature and is quantitatively as strong as the effect of exporting on average worker type.We explore further the effect of exporting by building measures of the exporting opportunities indifferent sectors using tariffs and aggregate imports from the rest of the world of the various countriesthat France exports to. We find that when exporters face lower tariffs or larger demand for importsin a foreign market, the dispersion of types in their pool of workers declines further.This paper contributes to the growing literature on international trade with heterogeneous laborand firms, which is surveyed in a recent chapter by Davidson and Sly [2012]. The most closelyrelated work is a recent paper by Davidson et al. [2012], which shows, using Swedish data, thatexport-oriented sectors display a higher correlation between firm and worker types, estimated asfirms’ and workers’ fixed effects in a wage regression as in Abowd et al. [1999] (henceforth AKM).Our approach shifts the focus on the firm-level decision rather than looking at the aggregatestrength of matching and therefore relies on a different type of variation to detect different matchingbehavior by firms that are differentially exposed to international trade. In particular, we exploitwithin-sector variation between exporting and non-exporting firms, therefore isolating and control-ling for other sector-level characteristics of the labor market that may affect the sorting of workersacross firms.Moreover, following Eeckhout and Kircher [2011], we are careful to avoid using firm fixed effectsfrom a wage regressions as a proxy for the firm types. We use instead variables constructed fromfirm-level data, such as market shares, value added and total employment.33The remainder of the paper is divided into two parts. Sections 3.2-3.6 develop the empiricalstrategy, while Sections 3.7-3.8 focus on the empirical specifications and present the results. Section3.9 concludes.3.2 Empirical analysisOur empirical analysis proceeds in two steps. First, following the theory, we construct worker typesusing the average wage of the worker over her job spells. As a robustness check, we also estimatethe worker types employing a methodology pioneered by Abowd et al. [1999] (AKM) and recentlyenriched by Card et al. [2013]. We are careful to separately construct measures of the firm typefollowing a recent analysis of the AKM methodology by Eeckhout and Kircher [2011]. In a secondstep we propose various measures that approximate the matching range of individual firms and showthat those measures are systematically different between exporters and non exporters, both in thecross section and when export markets are subject to shocks that affect the profitability of exporting.Before describing our empirical strategy in details, we offer a brief overview of the features ofthe wage-setting institutions in France and of the data employed in this paper.3.2.1 Institutional BackgroundOur model assumes that wages are the outcome of a bargaining game between firms and workers.This condition is key to the empirical anaylisis in order for wage outcomes to reflect worker andfirm characteristics. Here we analyze the institutional features of French labour market and whetherthey will be a good approximation to the characteristics of the model.Since 1950, wage-setting institutions in France are organized according to hierarchical principles.Wages are bargained at three different levels: (i) at the national level, a binding minimum wage(called Salaire Minimum Interprofessionnel de Croissance, or SMIC hereafter) is set by the govern-ment;32 (ii) at the industry level, employers’ organizations and unions negotiate pay scales; wagesare, then, negotiated occupation by occupation; (iii) at the firm level, employers and unions usuallynegotiate wage increases.Typically, in 1970s and 1980s collective agreements were negotiated within different sectors be-tween unions and employer associations, then extended by the Ministry of Labour to the entireindustry to become binding also for workers and firms not part of the original negotiation. At theend of the 1980s, more than 95% of the workforce was covered by those collective agreements. How-ever, different laws have strengthened the decentralization of the wage bargaining process in Franceover the last thirty years. Three channels have been used to promote firm-level agreements: (i) theobligation for firms to negotiate on wages each year, (ii) more possibilities offered to firms to deviatefrom industry-level agreements (escape clauses), and (iii) fiscal incentives.33 In 1982, the Auroux32Until 2010, the SMIC was raised each year in July according to a legal formula based on partial indexation topast inflation and to past wage growth.33In 2008, reduction of social security contributions paid by the employers became conditional upon wage negoti-ations occurring within the firm.34Law introduced the duty for firms with at least 50 employees and an elected union representative tonegotiate wages with unions every year, although not the obligation to reach an agreement. Subse-quent legislations concerning the working time reduction (Robien’ s laws in 1996, the first Aubry’slaw in 1998, the second Aubry’s law in 2000) allowed the application of escape clauses to workinghour arrangements, reinforcing the trend towards decentralization.Since the 1980s, firm-level negotiations acquired progressively more importance. By 2005, 41%of the workers employed in private firms with more than 10 employees were covered by a wageagreement signed that very same year (Carlier and Naboulet [2011]).34 The number of such agree-ment grew also significantly, from about 3.000 in 1993 to more than 7.500 in 2006. Although suchevidence supports trends towards decentralization, bargaining at the firm-level does not guaranteethat workers employed at a given firm within the same occupation earn different wages. To provideevidence of worker-firm bargaining, we will analyze the variability in wages across workers withinfirm-occupation in Section 3.4.3.3 DataThe data for our project come from three main sources, the De´claration Annuelle des Donne´esSociales (DADS), the Enquete Annuelle d’Entreprises (EAE) and the French Customs Data.35DADS is an administrative database of matched employer-employee information collected by theINSEE (Institut Nationale de la Statistique et des Etudes Economique). The data are based onthe mandatory reports, filed by employers, of the gross earnings of each employee in compliancewith French payroll taxes. All paying-wages individuals and legal entities established in France arerequired to file payroll declarations; only individuals employing civil servants are excluded from filingsuch declarations. The INSEE prepares extracts of the original database for research purposes. Werely on the panel version of DADS, which covers all individuals employed in French enterprises bornin the month of October of even-numbered years until 2001 and every year after that.36 This choiceis motivated by the need to follow workers across years and job positions in order to recover theirtypes (see Section 3.4).Our extract stretches from 1995 to 2007. The initial data set contains around 24 million obser-vations (corresponding to the triplet worker-firm-year) which are identified by worker and firm ID(respectively, nninouv and siren).For each observation we have information on the individual’s gender, year and place of birth,occupation (both 2-digit CS and 4-digit PCS-ESE classification), job spell,37 full-time/part-timestatus, annualized real earnings, total number of hours worked as well as the industry of the employ-ing firm (NAF700, 4-digit industry classification). We restrict our sample to full-time employees inmanufacturing (NAF 10-33), reducing the total number of observations to 2, 662, 411. Most full-time34In 1992, 40% of the workforce was covered by some firm-level agreement. Source: Abowd et al. [2005]; authors’calculation based on data from wage structure survey in 1992.35These data are subject to statistical secrecy and have been acceded at CEPII.36In 2002, the sampling methodology has been extended to include all individuals born in the month of Octoberof every year. Currently, the DADS panel represents 1/12th of the total French workforce.37DADS records both the job start date and the number of days the individual worked in a given firm during thecalendar year.35workers are employed at a single firm during the year. Only 6% has more than one employer in agiven year; for those, we selected the enterprise at which the individual worked the largest numberof days during the year. Finally, to control for possible outliers, we remove those observations whoselog annualized real earnings are more than 5 standard deviation away from the predicted wage, basedon a linear model including gender, an ile-de-France dummy and in-firm experience. We obtain afinal sample of 2, 579, 414.Following EK, in order to construct appropriate measures of firm types, we enrich the available setof firm-level variables by merging DADS with EAE, a survey-based dataset containing balance-sheetinformation on French firms in manufacturing over the period 1995-2007. The unit of observation inEAE is a firm-year combination; the firm identifier is the same as the firm ID in DADS (siren). EAEsamples only medium-large enterprises with at least 20 employees. From EAE we collect informationon sales (domestic and exports), total employment, value added and also on the main sector of thefirm (NAF700 4-digit classification).38 The merge with EAE further reduces the sample availability.We restrict our sample to individuals working for firms whose characteristics are available from EAE.Furthermore, we remove those firms whose number of sampled employees from DADS is larger thanthe effective employment reported in EAE. This provides us a final sample of 1, 673, 992 observationson which we implement our empirical strategy.Export-related information on French firms come from the French Customs. The custom dataincludes export records at firm-, product- and destination-level for the universe of exporters locatedin France.Finally, aggregated trade flows and applied tariff levels come from standard sources, respectivelyCOMTRADE and WITS. Aggregated trade flows are used to compute aggregated market shocksas (weighted) import demand by all potential French trade partners, while applied tariff levels areused as a second proxy for foreign market openness - average tariff reduction (across all French tradepartners) representing a measure of higher market access for French firms.3.4 Empirical trends of wage inequalityIn this section we present a sequence of variance decompositions to document the components ofwage inequality and its pattern over time. We start with a Mincerian specification of log wagescontrolling for workers’ observable characteristics,logwit = αtxit + ηit (24)where logwit denotes the wage of worker i at time t; xit represents the set of observable charac-teristics. We control for in-firm experience, gender and occupation. We estimate regression (24)separately for each year. This exercise serves to understand the contribution of between- vs within-group components to the overall wage variance,V ar (logwit) = V ar (αˆxit) + V ar (ηˆit) (25)38We compare the firm’s industry classification between EAE and DADS and keep only those observations whoseindustry information coincides between the two sources.36where the hat denotes estimates from regression (24). V ar (ηˆit) represents the portion of wagevariability unexplained by workers characteristics and it is known as residual wage inequality.Table 5 reports the results for variance decomposition (25) for the first year (1995) and the lastyear (2007) in our sample. We find that the wage inequality has been trending upwards between1995 and 2007; the residual component accounts for most of the level and the change in wagevariance between 1995 and 2007. The pattern is similar if estimating regression (24) separately ineach sector-occupation-year cell; the results are reported in Table 6.39The relatively small contribution of worker observables to wage variation might be due to workerunobserved heterogeneity; the residual wage component might be overestimated if it captures allunmeasured sources of wage variability uncorrelated to other observable characteristics available inour sample. This might be of concern since we are unable to include education among the controlsof regression (24). If, instead, the unobservable characteristics were correlated with the controls ofregression (24), the contribution of observables to wage variation would be overestimated. We findthat controlling for firm fixed effects tends to further reduce the contribution of worker observables,as we will show next.We now present the results from a variance decomposition that controls for firm fixed effects. Fol-lowing Abowd et al. [1999], we regress the log wage of the worker on worker observable characteristicsand firm fixed effects,logwijt = βtxit + ψjt + it (26)where j denotes the firm where worker i is employed at time t. As before, we run first specification(26) separately for each year. In this specification, ψjt represent a firm time-varying component ofwages. The associated variance decomposition is composed of 4 terms,V ar (logwijt) = V ar(βˆxit)+ V ar(ψˆjt)+ 2 · Cov(βˆxit, ψˆjt)+ V ar (ˆit) (27)where V ar(βˆxit)represents the contribution of worker observables to the wage variance; V ar(ψˆjt)is the between-firm wage component; Cov(βˆxit, ψˆjt)is the covariance between worker observablesand the firm fixed effect; V ar (ˆit) constitutes the residual.To identify the exclusive contribution of the firm component, we first run a wage decompositionthat excludes worker observables; Table 7 reports the result from such decomposition. We findthat the part unexplained by firm fixed effects accounts for the largest share of wage variabilityand its contribution has been increasing over-time. Table 8 confirms a similar pattern when thedecomposition is at the sector-occupation-year level. The role of the within-firm component is evenmore relevant in the more disaggregated decomposition.Table 9 and Figure 3 report the wage decomposition conditional on worker observables. Con-trolling for worker characteristics does not alter the dominant role of the within-firm component.Figure 3 shows that the within variability of wages has been also slowly increasing over the last13 years. Figure 4 looks further into the within-component: it compares the overall (demeaned)39We weigh the wage decomposition within each sector-occupation-year cell by the number of workers in the cell.The weighting implies that the sum of the individual components of the variance decomposition does not need to be100%.37wage distribution with a occupation-firm demeaned wage distribution. Although firm and occupa-tion characteristics account for a big part of the overall wage variation, a substantial part of thevariability in wages can be still observed across workers employed in the same occupation at thesame firm. This suggests that bargaining at the firm-level does not imply that workers employed ata given firm with the same occupation earn the same wages.Let us consider the other components from the wage decomposition in Table 9. Worker observ-ables contribute to around 1/3 of the total wage variation, with its contribution decreasing overtime. The share of wage variation explained by the between-firm component is much smaller than inTable 7, indicating that differences in employment composition explain a large part of the differencesin wages across firms. Finally, the correlation between worker observables and firm fixed effects ispositive and increasing over-time. However, following Eeckhout and Kircher [2011], we are carefulto interpret the correlation between worker characteristics and firm fixed effects as a measure ofsorting, since the fixed effects are not necessarily correlated with the true firm type.Although the correlation between worker characteristics and firm fixed effects does not necessarilycapture the effect of worker sorting across firms, we believe that sorting patterns influence theindividual components in the wage decomposition and determine their evolution over time.3.5 Constructing worker typesThis section describes our methodology to construct worker types. There is no obvious strategy tocreate a proxy for worker types; we propose two methods:1. average lifetime wage2. AKM fixed effectThe first method is based on our model and uses the average wage of the worker over her jobspells - average lifetime wage- to proxy for the worker type. However, the average lifetime wage alsocaptures the average productivity of the firms the worker is willing to match with. To remove the ef-fect of the average firm productivity, we adopt an alternative strategy, the AKM methodology. Thismethodology aims at decomposing individual workers’ wages into a firm component and a workercomponent.40 Such decomposition, however, is not consistent with the theory: a log-linearizedversion of the wage equation from equation (1) only features a worker component. Therefore, weconsider the average lifetime wage as our preferred proxy for worker type.41First Method: Average Lifetime Wage as proxy for Worker Type40The AKM methodology has seen a very large number of applications, e.g. Abowd and Kramarz [2003], Abowdet al. [2006] Abowd et al. [2007], Abowd et al. [2008], Abowd et al. [2009a], Abowd et al. [2009b], Carneiro et al.[2012], Torres et al. [2012].41We report also results using the worker fixed effects from the AKM regression as a robustness.38The theory suggests that the average lifetime wage is a good proxy for worker types since it ismonotonically related to θ, the ability of a worker. In fact, a more productive worker obtains, onaverage, a higher wage since she makes a larger contribution to revenues and tends to match withbetter firms. From the model, the average lifetime wage of a worker of type θ takes the followingexpressionw¯ (θ) =∫ u(θ)l(θ)[w (θ) + s(θ,ψ)γ]v (ψ) dθu (θ)− l (θ)= w (θ) +cu (θ)− l (θ)(28)In the appendix we formally show that the average lifetime wage is increasing in the worker type θ.Second Method: AKM Fixed EffectsThe AKM decomposition is based on a specification relating a measure of log compensation forworker i employed in firm j at time t to workers and firms’ effects:lnwit = x′itβ + θi + ψJ(i,t) + εit (29)where θi is worker i’s component and ψJ(i,t) is the firm component. The function J (i, t) = j identifiesthe firm employing worker i at time t. The vector xit includes time-varying worker characteristics,therefore the component θi captures persistent differences in compensation explained by ability andother time-invariant worker characteristics. Although persistent differences in compensation couldarise also for reasons other than ability differences (e.g. negotiating skills), our theoretical frameworksuggests that θi is an appropriate proxy for the individual’s unobservable true type. In our model,the person effect from a wage decomposition captures the variation in wages across the firms in hermatching set.42 Workers of higher ability make a larger contribution to revenues and tend to matchwith more productive firms, obtaining on average higher wages. This behaviour creates a mappingof higher ability into higher person effects.We assume that the error term εit is iid across time and workers with mean zero. This assumptionrequires that employment mobility is exogenous, depending only on observable characteristics, personand firm effects. More precisely, the fixed effects estimator conditions on the whole sequence ofestablishments at which each worker is observed; this implies that the exogenous mobility assumptionis not violated in presence of systematic mobility patterns driven by the person effect θi and/orthe sequence of firm effects(ψJ(i,t), ψJ(i,t+1), . . . , ψJ(i,T )). The random mobility assumption is,instead, violated if mobility depends, for example, on match-specific components of wages. 4342The person effect captures also changes in the matching bounds, but this effect is negligible if the tail of firm’sdistribution is not decaying too fast.43The results estimated under the assumption that the error term εit includes a match effect ηiJ(i,t) and anidiosyncratic term as in Card et al. [2013] and Woodcock [2008] are qualitatively similar to those in Tables 12 and13.39Following Card et al. [2013], we perform a diagnostic tests on the interaction between wage changesand mobility patterns. Figure 28 reports wage changes associated with job transitions classifiedbased on the quartile of the firm type - proxied by the domestic market share - for the origin anddestination workplace. We should expect little or no variation in wage before the job change ifselection or transitory wage components do not affect mobility patterns. Workers moving fromthe fourth quartile do not experience a reduction in wages prior to their transition, while workersleaving firms in the lowest quartile of the firm productivity distribution seem to be influenced intheir decision by transitory wage components. We believe that for those workers, selection mightnot exercise a strong influence as their wages after the transition appear to be of a similar level asbefore. As an additional indication against systematic mobility patterns, Table 28 documents thesign of wage changes for workers moving across jobs. In presence of systematic mobility patternswe should expect all wage changes to be positive; in our dataset, instead, only half of the movers(around 52%) experience an increase in wages.We follow AKM for the explicit specification of (29). Our dependent variable is the log of annu-alized real wages.44 We include as time-varying controls a quartic in employer-specific experience,45time-dummies, a dummy for workers residing in ile-de-France and time-varying gender effects (ex-actly, the interactions of sex with all the other variables).The panel version of DADS does not contain information on education. AKM obtain informationon the highest degree attained from the permanent demographic sample (Echantillon De´mographiquePermanent, EDP). However, this information would be available, in our case, only for about 20%of the workers in our sample. Thus, we decided not to include a control for schooling in ourdecomposition.46As described in Abowd et al. [2002], fixed effects for workers and firms can be separately identifiedonly for sets of firms and workers that are ‘connected’ by moving workers. In fact, the person effect iscommon to the individual’s job spell; its identification requires observing the individual at differentemployers. Similarly, a firm effect is common to all employees of the firm; identifying the firm effectrequires observations on multiple employees of the firm. Identifying both effects requires mobility ofworkers across firms.47 The movement of workers across firms characterizes a connected group. Aconnected group is defined by all workers who ever worked for any firm in the group and all firmswhose workforce is included in the group. A second group is unconnected to the first if no firm inthe first group has ever employed any worker from the second group and no firm in the second hasever employed workers from the first. Within each group, we normalize the mean of the fixed effectsto zero, therefore it is not possible to identify 1 individual and 1 firm effects per group.44Working hours are often not reported. The restriction to full-time workers absorbs possible differences in hoursworked across individuals.45DADS contains information on the job starting date at a certain firm - we compute the employer-specific expe-rience as a difference between the current year and the first year of employment at the firm.46In addition, most of the effect of schooling would be absorbed by the person effect. AKM mention that school-ing does not vary over time in their sample.47Let us consider a simple example of how to implement the AKM methodology. Consider a connected groupwith 2 firms and N workers and suppose that at least one worker, individual 1, is employed in both firms over thesample period. The observed wage differential for individual 1 is entirely attributed to the difference between firmsfixed effects. Normalizing the mean firm effect to zero, it is possible to identify one of the fixed effects. A similarargument applies to the identification of the person effect.40Due to the normalization, comparing fixed effects between groups has no real meaning. Therefore,when comparing workers and firms, we only employ estimated fixed effects from the largest connectedgroup, which represents 88% of the workers in our final sample.The estimation of the fixed effects is performed using the Gauss-Seidel algorithm, proposed byGuimaraes and Portugal [2010]. Such algorithm consists in solving the partitioned set of normalequations, associated to (29), given an initial guess on the coefficients. Workers’ and firms’ fixedeffects are recovered as coefficients on the dummy variables identifying the worker and the firm atwhich he’s employed. According to Smyth [1996], the Gauss-Seidel algorithm achieves a stable butslow convergence, depending on the correlation between the parameter estimators. This implemen-tation has the advantage of not requiring an explicit calculation of inverse matrices to determinethe vector of coefficients nor forces us to drop small firms, due to the large number of firm effects toestimate.48We recover fixed effect estimates for 406404 individuals and 31649 firms. In the appendix toChapter 1, we include the distribution of the worker fixed effects (Figure 11) and firm fixed effects(Figure 12) for the largest connected group.With estimates of worker types at hand, we now proceed to construct measures of the averageworker type and dispersion of the worker type at the firm level. Specifically, we construct thevariables AvWorkerTypejt, SdWorkerTypejt and IQR WorkerTypejt as:AvWorkerTypejt =1njt∑i∈Ijtw¯iSdWorkerTypejt =1njt√∑i∈Ijt(w¯i −AvWorkerTypejt)2IQRWorkerTypejt = w¯j,75th − w¯j,25thwhere Ijt is the set of workers employed by firm j at time t, w¯j,75th and w¯j,25th are the types ofworker at the 75th and at the 25th percentile of firm j’s employee type distribution.We build these measures only for firms with more than 5 sampled workers.49 The choice of thethreshold is a compromise between retaining a sample of satisfactory size and constructing samplemeasures that approximate the true underlying measures. On the one hand, a larger thresholdforces us to cut a larger percentage of the sample. On the other hand, a larger number of sampledworkers reduces the noise in the estimation of a firm’s matching set. We consider each employmentrelationship as a realization of a match along the set of acceptable matches within a firm’s matchingset. In the limit, increasing the number of match realizations, the constructed statistics of workertypes converges to the true measure.Choosing a higher threshold of sampled workers does not affect the results. If including firmswith less than 5 sampled workers, instead, the coefficients on our variables of interest are of the48The number of firms’ fixed effect is too large for e.g. the felsdv estimator. In such case, Andrews et al. [2006]suggest pooling small plants into a single super-plant. However, we prefer not to implement a similar strategy, as,in our case, firms - not plants - are the units of observation.49Figure 9 reports the distribution of sampled workers, cut at the 95th percentile. The full distribution is simi-lar, but with a longer tail.41correct sign but in some specifications are not significant.503.6 Firm typesFor the purpose of comparing matching choices of exporting and non-exporting firms, we need tocontrol for the type of the firm. Eeckhout and Kircher [2011] show that the relationship betweenthe true firm type and firm fixed effects estimated from a AKM-style wage regression is theoreticallyambiguous, i.e. it can be positive, negative or zero. Eeckhout and Kircher [2011] also argue thatthe ideal firm component is a measure of firm type that is specific to every job within the firm,but measurable variables such as output and profits are obviously only observed at the aggregatefirm level, not for each relationship within the firm. We therefore adopt three proxies for firm typeto investigate the behaviour of firms fixed effects ψ: value added per worker of firm j, V Apwj ,the logarithm of total employment in firm j, logEmpj and domestic market share DomSharej ,defined as the ratio of firm j’s domestic sales to total domestic sales in the firm’s sector (each firmis classified as belonging to only one sector in each year).51 While the first two proxies are standardmeasures of the productivity or demand intensity for a firm product, the third is motivated by Eatonet al. [2011]. In particular, while the first two proxies contain a measure of success over all markets,including the foreign ones, the third variable better captures the success of the firm with respect tothe domestic market, before the choice of exporting. We average each proxy over the years the firmappears in the sample to smooth out the effect of changes in the workforce.52We first explore the hypothesis put forward by EK regarding the ability of the AKM firm fixedeffects to capture the firm type. Table 10 shows the pairwise correlation between the AKM firmfixed effect, the three proxies for firm type and the average worker type at firm j as measured bythe average AKM worker fixed effect, AvWorkerTypej over the sample period at firm j. The firststriking fact is the negative and large correlation (−0.80) between average worker type and the AKMfirm fixed effects ψ, confirming previous findings by Abowd et al. [2004]. If instead we employ thethree proxies for firm type, we observe for each of them a positive and significant correlation withthe average worker type, proxied either with the average lifetime wage or the average of the workerfixed effects. The three firm type proxies are in turn all positively correlated with one another, butdisplay small and opposite correlations with the AKM fixed effect ψ. In particular DomSharej andV Apwj have a positive correlation of 0.01 and of 0.001 with ψ, respectively, while and logEmpjdisplays a negative correlation of −0.01.Table 11 shows that this correlation pattern is not unique to a few sectors. In column 4 we reportthe correlation between AvWorkerTypej and ψj by two-digit sector, while column 6 displays the50In the appendix, we report the results from GLS regressions, weighting by the number of observations. SeeTables 36-39.51We consider sectors at the 4-digit level for the constructions of market shares.52Our model confirms the positive correlation between productivity, value added per worker and domestic mar-ket share. According to our theory, more productive firms tend to match with better workers, realizing on averagelarger revenues. Therefore, firms of higher productivity should display larger value added per worker and a largershare in the domestic market. The model is silent about employment differences due to variations in productivitysince we focus on the matching problem between one firm and one worker. If introducing homogeneous labour inthe production function, the model will also address the implication that more productive firms hire a larger work-force.42analogous correlation between DomSharej and AvWorkerTypej . While the first set of correlationsis always negative and significant, the second set of correlations is positive and significant, except inone case where the correlation is positive but not significant. The evidence presented in tables 10and 11 is consistent with the hypothesis put forward by EK, that the AKM firm fixed effect may notbe correlated with the true firm type. Although it is still possible that, as Abowd et al. [2004] pointout, there is truly negative assortative matching between workers and firms, Andrews et al. [2008]remark that the negative correlation between fixed effects might reflect the limited mobility bias inthe sample. The fraction of movers in our sample (around 18%), however, is much larger than thatin the sample used by Andrews et al. [2008] (around 1.3%); moreover, since the estimated factor tocorrect for the mobility bias in Andrews et al. [2008] is unable to completely absorb the negative signof the correlation, we believe that the correlation between fixed effects would also remain negativein our case.3.7 Empirical specification 1: export status and matching setWe now proceed to illustrate the specifications employed to describe the different matching behaviorof exporting and non-exporting firms. The first implication of our model is that exporting firmshire workers of higher average type. This is a similar prediction to the models of Sampson [2014]and, under the interpretation of permanent worker heterogeneity, Helpman et al. [2010]. We believethis is a novel method of corroborating such prediction since it shows directly that an exporter payshigher wages because it employs better workers, not because it shares higher revenues with the sametype of workers. The former is the mechanism involved in explaining the exporter wage premium inHelpman et al. [2010], but we believe it has not been tested before.In a pooled cross-section of firms over the sample period, the basic specification we employ isthe following:AvWorkerTypejt = β0 + β1Exportjt + β2 Firm Typejt +Dst + ujt (30)where Exportjt = 1 if firm j exports at time t and Firm Typejt is one or all of the three proxies forfirm productivity, V Apwj , logEmpj and DomSharej .Differences in average worker type between exporters and non-exporters also reflect differencesin the occupational structure. If, for example, exporters employ workers in occupations with higheraverage wage, they might also have higher average type, since the person effect contains all time-invariant characteristics, like occupation that rarely changes over time for a given worker.53 Weadd the number of occupation, N.occjt and the share of white collar workers,54 whiteshare, tospecification (30) to control for differences in occupational structure across firms. Similarly, thenumber of exported products, log Products, which we include in the specification with all controls,is intended to capture structural differences in occupational complexity that might cause a spuriouscorrelation of the exporting status with the average firm type.53Around 80% of the workers in our sample does not switch occupation during the time period analyzed.54The blue vs white collar classification is based on occupational code. We report the classification we adopt inTable 29.43In addition, all specifications but the first include a quadratic in the number of sampled workers tocontrol for the precision of our left-hand side estimates. Finally, all specifications include sector-yeardummies, Dst.The novel contribution of this paper is the prediction that exporters match with workers thatare characterized by lower relative dispersion of ability. The specification that we employ is thefollowing:SdWorkerTypejt = β′0 + β′1Exportjt + β′2Firm Typejt +Dst + u′jt (31)The theoretical section shows that the only robust prediction regarding the link between workertype dispersion and export status (and productivity) requires expressing such dispersion either inpercentage terms or relative to the average worker type. In this regard, it is essential to rememberthat the fixed effects are estimated from a log-linearized equation, where types are therefore alreadyexpressed in percentage differences from one another. Nevertheless we will add the average workertype in the specification with all controls.Similarly to specification (30), we include the number of occupation, N.occjt, the share of whitecollar workers, white share and the number of exported products, log Products, to control for dif-ferences in the occupational structure across firms with different export status. All specificationsinclude sector-year dummies, Dst.Both specifications (30) and (31) are estimated by OLS and standard errors are clustered at thelevel of the firm.We also develop an alternative strategy to test the prediction that exporters select a set of workerscharacterized by a lower dispersion. We compare the rank correlation between the average workertype by firm and the firm type among exporters to the rank correlation between the average workertype and firm types among non-exporters. A lower dispersion of the workforce among exportersimplies better sorting and should be associated with a larger rank correlation. We test the existenceof systematic differences in correlation between exporters and non-exporters employing the followingspecificationCorr(AvWorkerTypejt, DomSharejt)st= β′′0 + β′′1 Exportst +Ds +Dt + u′′st (32)where Exportst = 1 if the correlation is constructed for the set of exporting firms in sector s at timet. In addition to sector and time dummies, we also include some controls at the sector-level, theaverage (log) employment and the average domestic market share, that might differentially affectthe matching patterns between exporters vs non-exporters.3.7.1 ResultsThe estimation results relative to specifications (30) and (31) are presented in Tables 12 and 13.Column 1 of Table 12 reports a positive and statistically significant relationship between exportstatus and the average type of the worker employed by the firm. The positive relationship is ofsimilar strength when we introduce in turn the three controls for firm type (domestic share, valueadded per worker and employment).44As predicted by theory, the coefficient on all three proxies for firm type is positive and significant,like the one on export status. In the specification reported in column 5 we include the three controlsfor firm type in the same regression, and the coefficient on export (the one of our interest) remainspositive and significant, like the ones on value added per worker and employment. The coefficient onexport status loses significance only in column (7) when we introduce firm fixed effects. We believethat the non-significant coefficient is to be attributed to the low variability in export status. In fact,around 80% of the firms are exporters and such percentage does not fluctuate much over time.Table 13 reports the results of the estimation of specification (31) and has a similar structureto Table 12. Starting from column 1 where no controls are added, we document the expectednegative and significant relationship between export status and variability of worker type. Theeffect persists with a similar magnitude when we control for the above mentioned firm type controls(domestic share, value added per worker and employment). The inclusion of all the control variablesin column (6) does not alter the significance of the coefficient on the export dummy. As in Table12, our coefficient of interest loses significance only in column (7), when we add firm fixed effects.Interestingly, while, as predicted by theory, the coefficient on two proxies for firm type is negative(columns 2 and 3), the table documents a positive and significant correlation between value addedper worker and the dispersion of worker type (column 4 - 6) a pattern that is not in line with thepredictions of the model.It is important to quantify the effect at the core of this paper. Based on our preferred specificationin Table 13, column 6 where we include all controls, the expected difference on the dispersion ofworker type between exporter and non-exporter firms is about 0.020 points (holding the othervariables constant). Considered that the dependent variable has a standard deviation of 0.41, anexporter features worker variability that is lower by 4.9% standard deviations. The effect on themean worker type can be calculated using the results from Table 12 and is of the same order ofmagnitude, but a little smaller: an exporting firm displays an average worker type that is 3%standard deviations higher.55In the appendix, Tables 32 and 33 report the results separately for the sample of newly hiredworkers and for the stayers. As expected, the export dummy is negative and significant only inTable 32, suggesting that exporters hire workers whose ability is closer to the average ability of theirworkforce.We investigate further the variation of the matching sets by firms’ characteristics. Tables 14and 15 disentangle the effect of the 25th and of 75th percentile in the within-firm worker abilitydistribution on the differences in dispersion between exporters vs non-exporters. We find that thelower ability dispersion at exporters is mainly driven by the stricter requirements imposed on workersof lower ability (Table 14). Only if including all controls (column 6 in Table 15), the export dummybecomes negatively associated to the 75th percentile of the ability distribution, suggesting a tendencyfor exporters to deviate less from the average ability.The negative association between the first and second moment of the ability distribution andthe export status is robust to the inclusion of Occ Pred Avg Wagejt in Table 34 and Occ Pred Std55This magnitude has been computed by using export coefficient of Table 12 column 6. The standard deviationof the average worker type is 0.81.45Wagejt in Table 35.56 Those variables serves the purpose of controlling for the average wage andthe standard deviation of wages as implied by the firm occupational structure.Tables 16 and 17 report estimates for the same specifications as in tables 12 and 13, but employa different proxy for the worker type, i.e. worker fixed effect from the AKM regression. This isa strategy for approximating the ability of the worker that is supported theoretically by Eeckhoutand Kircher [2011] and that we employ to check for robustness. Table 16 reports again a positiverelationship between export status and average worker type; the coefficient on export status remainspositive but loses its significance when adding controls for firm type and the occupational structure.Table 17 confirms a negative relationship between the dispersion of worker type and export status.Controlling for the type of the firm (by using employment, domestic market share and value addedper worker) the coefficient on export is negative and once again we find that firms with higheremployment and higher domestic share have tighter worker type dispersion - coherently with themodel. But, again, firms with high value added per worker have a wider variation of worker type(which contrasts with theoretical predictions).To address problems of endogeneity and reverse causality, we develop an IV strategy. We instru-ment export status using a firm level measure of tariff,Firm Tariffjt = ln1 +1∑sr τsrtExportsjsr,t−1Salesj,t−1 (33)where τsrt is the tariff faced by firms exporting good s to country r at time t; we aggregate acrosscountries using as weights the share of exports of good s to country r of firm j at time t − 1 overthe total sales of firm j at time t− 1,Exportsjr,t−1Salesj,t−1. Table 18 reports the second stage results57. Dueto the use of previous year export share, the number of observations in our sample drops to 15,826.The coefficient on export status remains negative and significant in all specifications. In particular,in our preferred specification, the coefficient is smaller than the coefficient from the OLS regressionin Table 13; this is consistent with the idea that more productive firms possess a better technologyto search for their workers.Table 19 presents the results for specification 31 with an alternative measure of dispersion, aweighted average of the standard deviation of ability for different groups of workers. In particular,we divide occupations into ‘managers’, ‘executives’ (white collar occupations) and ‘blue’ collar58 (asreported in Table (29)) and we construct average employment shares of those occupational groupswithin firm over time. We then weight the standard deviation of lifetime wages for each group by its56The variable Occ Pred Avg Wagejt is defined asOcc Pred Avg Wagejt =∑mwmt × smjtwhere wst is the average wage paid to workers with occupation m in year t and smjt is the share of workers em-ployed by j with occupation m in year t over the total number of workers employed; Occ Pred Std Wagejt is de-fined asOcc Pred Std Wagejt =∑mstdwmt × smjtwhere stdwmt is the dispersion in wages for workers with occupation m in year t.57The first stage results are reported in Table 4358Table 40 reports the result for blue collar workers, managers and executives.46average employment share to construct our new dependent variable. The coefficient on the exportdummy remains negative and significant in all specifications; in most columns, the magnitude of thecoefficient is not significantly different from what is reported in Table 13. This suggests that ourresult is not due to compositional differences between exporters and non-exporters.Looking at the coefficients on firm’s type controls (columns 2, 3 and 4 in Table 19) we discoverthat employment and domestic market share have the expected sign, i.e. they are negatively relatedto the standard deviation of workers types. The coefficient on value added per worker is positiveand significant only in column 6, suggesting that the ’puzzling’ effect discussed above for tables 13and 17 might be related to changes in employment composition over time.Table 20 presents a further robustness check. In particular, we employ the interquartile range ofworker type at firm j, as described earlier. It is easy to verify that all previously described patternsappear again in this table. Exporting firms choose a narrower range of worker types.Finally, Tables 41 and 42 in the appendix confirm that differences in dispersions translate intohigher rank correlations between average worker type and firm type for exporters compared tonon-exporters.3.8 Empirical specification 2: market access and tariff shocksOur first empirical strategy has relied on cross-sectional differences between exporting and non-exporting firms. Plausibly, the export dummy may be capturing the effect of other firms charac-teristics that are not included in our firm type proxies and that affect the matching behavior offirms.Our second strategy to detect the impact of exporting on matching between firms and work-ers aims at addressing this concern. We exploit differences in the opportunities offered by foreignmarkets, approximated by demand shocks and tariffs across sectors and countries over time. Thesedifferent shocks, which we indicate as ‘market access’ shocks should affect exporting firms differen-tially from non-exporting firms. A positive demand shock in a foreign market or a lower tariff facedby French exporters should induce the exporting firm to select an even less dispersed labor force.The specification that we estimate is the following:AvWorkerTypejt = γ0 + γ1Mkt Accessst × Exportjt + γ2Mkt Accessst+γ3Exportjt +Dst + vjt, (34)SdWorkerTypejt = γ′0 + γ′1Mkt Accessst × Exportjt + γ′2Mkt Accessst+γ′3Exportjt +Dst + v′jt (35)47whereMktAccessst =∑rMktAccesssrt ×French exportssr,t−1French exportss,t−1, (36)MktAccesssrt =Tariffssrt orImportssrt orImportssrtTariffssrt,Importssrt is the total value of imports by country r from the rest of the world59, Tariffssrt is thetariff faced by a French firm exporting to country r in sector s60 at time t, and French exportssr,t−1is the value of exports from France to country r in sector s at time t− 1 (with total exports in thesector in that year indicated as French exportss,t−1). The variable MktAccessst measures the costof access or demand size in foreign markets for firms in a given sector s, weighted by the importanceof French firms in that sector the previous year. The model predicts that a positive export shockshould result in an increase in the average worker type and further tightening of the acceptance setfor an exporting firm, so we expect γ1 < 0 and γ′1 > 0 for the case of MktAccesssrt =Tariffssrt andthe opposite when market access is measured as Importssrt orImportssrtTariffssrt.3.8.1 ResultsTable 21 and 22 report estimates of the coefficients in specifications (34) and (35) when marketaccess for a firm in sector s is measured by total import demand faced by an exporter in sector s.We do not present results for the case when total import demand is deflated by the tariff faced byFrench exporters because they are very similar. Table 21 reports results on the average worker type;our coefficient of interest is positive and significant on all specifications. However, if evaluated atthe mean of the market access measured by Importssrt - 13 (in log-s) - an exporter does not featurea higher average worker ability; only exporters in sectors with a degree of openness larger than theaverage will enjoy an effect on the average worker type.In Table 22 we find for all specifications that the estimated coefficient γ′1 is negative and signifi-cant, so that exporters seem to choose a less dispersed workforce in particular when having betteraccess to foreign markets. The inclusion of firm type controls does not affect the magnitude andsignificance of this result. The coefficient on export status is negative and significant in all specifica-tions. If evaluated at the mean of the market access measured by Importssrt, an exporter featuresworker variability that is lower by 3.4% standard deviations than a non-exporter firm.Tables 23 and 24 report estimates of the coefficients in specifications (34) and (35) when marketaccess for a firm in sector s is measured by the average tariff faced by an exporter in sector s. Onlycolumns 4-6 of Table 23 report a negative coefficient γ1 - which is line with the prediction - but notstatistically significant.Table 24 reports very similar results to Table 22: better export market conditions as measuredby a lower tariff faced on the export market result in a tighter matching set for exporting firms. So,59The inclusion of French exports to country r does not affect the results.60Sectors are defined as 4-digit HS codes.48contrary to Table 23, the effect of export opportunities on standard deviation of worker type seemsmore robust to the definition of market access. In particular, firms exporting in country-sector with’mean’ market access (mean value equals to 5.58% in our sample) have a lower worker variabilitythan non-exporters (13.7% standard deviation units), with such gap increasing with the marketaccess of the firm.3.9 ConclusionsUsing linked employer-employee data from France, we show that exporters and non-exporters matchwith sets of workers that are different. Exporters employ workers of higher average type and lowertype dispersion. We also find that when exporters face lower tariffs or larger demand for imports ina foreign market, the dispersion of types in their pool of workers declines further. These findings areconsistent with the search and matching model we developed in the first chapter. The same findingsbear some implications on the wage variation within and across firms. This is the topic that we willanalyze in the following chapter.49Figure 3: Wage Decomposition: Time Trends50Figure 4: Variability in Wages: Comparison51Table 5: Residual Wage Inequality: Worker Decomposi-tionConditional Wage Components1995 2007Worker Observables 36.0% 29.2%Residual 64.0% 70.3%Wage Inequality0.49 0.59(Level)Note: Wage Variance decomposition from a mincerian equa-tion that controls for worker observables (in-firm experience,gender and occupation). The regressions are run separatelyby year cells.Table 6: Residual Wage Inequality within Sectors:Worker DecompositionConditional Wage Components1995 2007Worker Observables 34.6% 32.0%Residual 57.5% 74.9%Wage Inequality0.47 0.52(Level)Note: Wage Variance decomposition from a mincerian equa-tion that controls for worker observables (in-firm experienceand gender). The regressions are run separately by sector-occupation-year cells.52Table 7: Unconditional Wage Com-ponents: Firm Decomposition1995 2007Between-Firm 35.9% 32.8%Within-Firm 64.1% 67.2%Wage Inequality0.27 0.36(Level)Note: Wage Variance decompositionfrom a wage equation that controlsfirm fixed effects. The regressions arerun separately by year cells.Table 8: Residual Wage Inequality within Sectors:Firm DecompositionUnconditional Wage Components1995 2007Between Firm 14.9% 19.6%Within Firm 84.9% 89.1%Wage Inequality0.44 0.51(Level)Note: Wage Variance decomposition from a mincerianequation that controls for firm fixed effects. The regres-sions are run separately by sector-occupation-year cells.Table 9: Residual Wage Inequality within Sectors: Com-plete DecompositionConditional Wage Components1995 2007Between-Firm 9.2% 13.5%Within-Firm 55.5% 63.3%Worker Observables 32.5% 26.6%Cov. observables-firm 1.4% 1.5%Wage Inequality0.44 0.51(Level)Note: Wage Variance decomposition from a mincerian equa-tion that controls for worker observables (in-firm experienceand gender) and firm fixed effects. The regressions are runseparately by sector-occupation-year cells.53Table 10: Rank Correlation Matrix, proxies for firms’ typesψAvg. Avg. Avg.Dom. Avg.VA Avg.Type Wage Share per w. Empl.ψ 1Avg. Worker Type by Firm -0.80 1Avg. Wage by Firm 0.13 0.35 1Avg. Dom. Share 0.01 0.08 0.20 1Avg. VA per worker 0.001 0.05 0.13 0.64 1Avg. Empl. -0.01 0.06 0.12 0.78 0.72 1ψ: Firms’ fixed effects, from the AKM decomposition.Avg. Wage by Firm: average of the workers’ wages over job spells.Avg. Worker Type by Firm: Average of workers’ fixed effects by firm, from the AKM decomposition.Avg. VA per worker: Average value added per worker, normalized by 4-digit industries.Avg. Dom. Share: Average domestic market share at a 4-digit level.Avg. Empl.: Average employment, normalized by 4-digit industries.Notes: Rank correlation between proxies of firms types. We do not report the p-values but all rankcorrelations are significantly different from zero.54Table 11: Measuring Sorting Patterns, Manufacturing Sectors(4) (5) (6) (7)ψ, Avg. Avg.Share,T ype Avg.WageNAF Industry Label No Firms ρS1 p-val2 ρS1 p-val210 Food 9 -0.96 0.00 - -11 Beverage 8 -1 - - -12 Tobacco prods - - - - -13 Textiles - - - - -14 Clothing 270 -0.84 0.00 0.18 0.0015 Leather/shoes - - - - -17 Paper 1317 -0.85 0.00 0.14 0.0018 Printing 1286 -0.86 0.00 0.14 0.0019 Refining 402 -0.88 0.00 0.42 0.0020 Chemical 666 -0.86 0.00 0.17 0.0021 Pharma 780 -0.79 0.00 0.30 0.0122 Plastics 2070 -0.76 0.00 0.13 0.0023 Non-metallic prods 59 -0.64 0.00 0.13 0.3324 Metalworking 1565 -0.72 0.00 0.33 0.0025 Metal prods 1987 -0.83 0.00 0.25 0.0026 Info/elec/opt 947 -0.82 0.00 0.27 0.0027 Elec equip 595 -0.84 0.00 0.14 0.0028 Machinery 5433 -0.81 0.00 0.21 0.0029 Automotive 2898 -0.82 0.00 0.28 0.0030 Other trans equip 126 -0.74 0.00 0.16 0.0731 Furniture 969 -0.81 0.00 0.25 0.0032 Other mfg 878 -0.71 0.00 0.13 0.0033 Repairs 1197 -0.79 0.00 0.23 0.00Manufacturing 23388 -0.80 0.00 0.20 0.001 Spearman correlation coefficient.2 p-value from testing independence between the variables.Notes: Columns (4)-(5): Rank correlation and significance level between the aver-age worker type, (Avg.Worker), and the firm fixed effect (ψ) from an AKM decom-position including a quartic polynomial in experience, a dummy for workers residingin Ile-de-France, time dummies and all the interactions with the gender dummy.Columns (6)-(7): Rank correlation and significance level between the average life-time wage of workers, (Avg.Wage), and the firm type, proxied by the average do-mestic market share in 4-digit sectors Avg.Share.55Table 12: Pooled Cross-Sectional Regressions: Average Lifetime Wage, more than 5 work-ers(1) (2) (3) (4) (5) (6) (7)Variables Average Lifetime Wage, more than 5Export 0.142a 0.0583a 0.0598a 0.0751a 0.0356a 0.0246c 0.00658(0.0122) (0.0118) (0.0118) (0.0114) (0.0113) (0.0130) (0.00905)N.Occ 0.0173a 0.0329a 0.0345a 0.0127a -0.00248 -0.0149a(0.00208) (0.00187) (0.00176) (0.00197) (0.00196) (0.00158)log empl 0.114a 0.113a 0.118a 0.0128(0.00604) (0.00618) (0.00596) (0.00969)log dom.share 0.0249a 0.00412b 0.00328c -0.000103(0.00212) (0.00202) (0.00197) (0.00224)log VA per worker 0.166a 0.165a 0.105a -0.00490(0.00843) (0.00831) (0.00805) (0.00567)white share 0.526a 0.621a(0.0190) (0.0235)log N. Products 0.00627c 0.00348(0.00334) (0.00316)Sector-Year y y y y y y yObs. 57,469 57,469 57,469 57,469 57,469 57,469 57,469R2 0.136 0.201 0.188 0.213 0.236 0.301 0.809N.Occ.: number of occupations, based on 2 digit occupational codes for France.log empl. log-employment.log VA per worker: log-value added per worker.log dom.share: log-domestic market share, at the 4 digit sector level.white share: share of non-production worker.log N. Products: log-number of exported products (HS6 codes). This variable is zero for non-exporters.Legend : a significant at 1%, b significant at 5%, c significant at 10%.Notes: Cross-Sectional Regressions for firms with more than 5 workers, years 1995-2007. Different specifi-cations in the columns. Standard errors, clustered at the level of the firm, are reported in parenthesis. Allspecifications but the first include a quadratic in the number of sampled workers, to control for the preci-sion of the left-hand side variable.56Table 13: Pooled Cross-Sectional Regressions: Standard Deviation of Lifetime Wage, morethan 5 workers(1) (2) (3) (4) (5) (6) (7)Variables Standard Deviation of Lifetime Wage, more than 5Export -0.0353a -0.0149 -0.0369a -0.0525a -0.0204b -0.0198b 0.00892(0.0102) (0.0102) (0.0104) (0.0102) (0.0103) (0.00856) (0.00777)N.Occ 0.0313a 0.0131a 0.0102a 0.0301a 0.0249a 0.0172a(0.00170) (0.00151) (0.00147) (0.00169) (0.00127) (0.00145)log empl -0.107a -0.108a -0.0221a -0.0381a(0.00526) (0.00543) (0.00409) (0.00761)log dom.share -0.00797a 0.000615 0.00220 0.00239(0.00168) (0.00170) (0.00138) (0.00187)log VA per worker 0.0451a 0.0430a 0.108a 0.0107b(0.00647) (0.00643) (0.00528) (0.00496)white share 0.482a 0.415a(0.0127) (0.0195)log N. Products 0.0132a 0.00308(0.00237) (0.00269)Avg Lifetime Wage -0.730a -0.998a(0.00950) (0.0213)Sector-Year y y y y y y yObs. 57,469 57,469 57,469 57,469 57,469 57,469 57,469R2 0.062 0.089 0.067 0.068 0.092 0.542 0.834N.Occ.: number of occupations, based on 2 digit occupational codes for France.log empl. log-employment.log VA per worker: log-value added per worker.log dom.share: log-domestic market share, at the 4 digit sector level.white share: share of non-production worker.log N. Products: log-number of exported products (HS6 codes). This variable is zero for non-exporters.Avg. Lifetime Wage: workers’ lifetime wage, averaged by firm.Legend : a significant at 1%, b significant at 5%, c significant at 10%.Notes: Cross-Sectional Regressions for firms with more than 5 workers, years 1995-2007. Different specifi-cations in the columns. Standard errors, clustered at the level of the firm, are reported in parenthesis. Allspecifications but the first include a quadratic in the number of sampled workers, to control for the precisionof the left-hand side variable.57Table 14: Average Lifetime Wage, 25th percentile, more than 5(1) (2) (3) (4) (5) (6)Variables Average Lifetime Wage, 25th percentile, more than 5Export 0.116a 0.038 0.061b 0.074a 0.038 0.017c(0.026) (0.027) (0.027) (0.027) (0.027) (0.010)N.Occ 0.003 0.024a 0.026a 0.003 -0.003b(0.004) (0.004) (0.004) (0.004) (0.001)log empl 0.124a 0.127a 0.080a(0.013) (0.014) (0.004)log dom.share 0.012a -0.003 -0.002(0.005) (0.005) (0.001)log VA per worker 0.024 0.028c -0.017a(0.015) (0.015) (0.005)white share -0.182a(0.013)log N. Products -0.006b(0.002)Avg. Lifetime Wage 1.052a(0.004)Sector-Year y y y y y yObs. 58,335 58,335 58,335 58,335 58,335 58,335R2 0.023 0.035 0.030 0.029 0.036 0.873N.Occ.: number of occupations, based on 2 digit occupational codes for France.log empl. log-employment.log VA per worker: log-value added per worker.log dom.share: log-domestic market share, at the 4 digit sector level.white share: share of non-production worker.log N. Products: log-number of exported products (HS6 codes). This variable iszero for non-exporters.Avg. Lifetime Wage: workers’ lifetime wage, averaged by firm.Legend : a significant at 1%, b significant at 5%, c significant at 10%.Notes: Cross-sectional Regressions for firms with more than 5 workers, years 1995-2007. Different specifications in the columns. Standard errors, clustered at the levelof the firm, are reported in parenthesis. All specifications but the first include aquadratic in the number of sampled workers, to control for the precision of the left-hand side variable.58Table 15: Average Lifetime Wage, 75th percentile, more than 5(1) (2) (3) (4) (5) (6)Variables Average Lifetime Wage, 75th percentile, more than 5Export 0.040 0.026 0.013 0.005 0.016 -0.020b(0.026) (0.027) (0.027) (0.027) (0.027) (0.010)N.Occ 0.024a 0.017a 0.015a 0.022a 0.007a(0.004) (0.004) (0.003) (0.004) (0.001)log empl -0.036a -0.034b -0.073a(0.012) (0.013) (0.0041)log dom.share 0.002 -0.0003 -0.001(0.004) (0.005) (0.001)log VA per worker 0.096a 0.096a 0.018a(0.015) (0.015) (0.005)white share 0.155a(0.0125)log N. Products 0.003(0.003)Avg. Lifetime Wage 0.941a(0.004)Sector-Year y y y y y yObs. 58,335 58,335 58,335 58,335 58,335 58,335R2 0.029 0.033 0.032 0.035 0.036 0.876N.Occ.: number of occupations, based on 2 digit occupational codes for France.log empl. log-employment.log VA per worker: log-value added per worker.log dom.share: log-domestic market share, at the 4 digit sector level.white share: share of non-production worker.log N. Products: log-number of exported products (HS6 codes). This variable is zerofor non-exporters.Avg. Lifetime Wage: workers’ lifetime wage, averaged by firm.Legend : a significant at 1%, b significant at 5%, c significant at 10%.Notes: Cross-sectional Regressions for firms with more than 5 workers, years 1995-2007. Different specifications in the columns. Standard errors, clustered at the levelof the firm, are reported in parenthesis. All specifications but the first include aquadratic in the number of sampled workers, to control for the precision of the left-hand side variable.59Table 16: Pooled Cross-sectional Regressions: Average of Workers’Fixed Effects, more than 5 workers(1) (2) (3) (4) (5) (6)Variables Average of Workers’ Fixed Effects, more than 5Export 0.079a 0.030 0.036 0.039 0.025 0.013(0.027) (0.028) (0.028) (0.027) (0.028) (0.031)N.Occ. 0.014a 0.022a 0.022a 0.012a 0.001(0.004) (0.004) (0.004) (0.004) (0.004)log empl 0.053a 0.055a 0.059a(0.013) (0.014) (0.014)log dom.share 0.007c -0.002 -0.003(0.004) (0.004) (0.005)log VA per worker 0.060a 0.062a 0.014(0.014) (0.015) (0.015)white share 0.421a(0.036)log N. Products 0.007(0.009)Sector-Year y y y y y yObs. 54,633 54,633 54,633 54,633 54,633 54,633R2 0.020 0.027 0.026 0.027 0.028 0.040N.Occ.: number of occupations, based on 2 digit occupational codes for France.log empl. log-employment.log VA per worker: log-value added per worker.log dom.share: log-domestic market share, at the 4 digit sector level.white share: share of non-production worker.log N. Products: log-number of exported products (HS6 codes). This variable iszero for non-exporters.Legend : a significant at 1%, b significant at 5%, c significant at 10%.Notes: Cross-sectional Regressions for firms with more than 5 workers, years1995-2007. Different specifications in the columns. Standard errors, clustered atthe level of the firm, are reported in parenthesis. All specifications but the firstinclude a quadratic in the number of sampled workers, to control for the precisionof the left-hand side variable.60Table 17: Pooled Cross-sectional Regressions: Standard Deviation ofWorkers’ Fixed Effects, more than 5 workers(1) (2) (3) (4) (5) (6)Variables Standard Deviation of Workers’ Fixed Effects, more than 5Export -0.036a -0.020c -0.039a -0.052a -0.029b -0.042a(0.011) (0.011) (0.011) (0.011) (0.011) (0.013)N.Occ. 0.029a 0.013a 0.010a 0.028a 0.025a(0.002) (0.001) (0.001) (0.002) (0.002)log empl -0.096a -0.096a -0.092a(0.005) (0.006) (0.006)log dom.share -0.007a 0.0004 -0.001(0.002) (0.002) (0.002)log VA per worker 0.036a 0.035a 0.024a(0.007) (0.007) (0.007)white share 0.130a(0.016)log N. Products 0.010a(0.003)Avg Worker Type -0.086a(0.005)Sector-Year y y y y y yObs. 54,633 54,633 54,633 54,633 54,633 54,633R2 0.065 0.087 0.069 0.070 0.088 0.120N.Occ.: number of occupations, based on 2 digit occupational codes for France.log empl. log-employment.log VA per worker: log-value added per worker.log dom.share: log-domestic market share, at the 4 digit sector level.white share: share of non-production worker.log N. Products: log-number of exported products (HS6 codes). This variable iszero for non-exporters.Avg Worker Type: average worker fixed effect, estimated by the AKM decomposi-tion, by firm.Legend : a significant at 1%, b significant at 5%, c significant at 10%.Notes: Cross-sectional Regressions for firms with more than 5 workers, years 1995-2007. Different specifications in the columns. Standard errors, clustered at thelevel of the firm, are reported in parenthesis. All specifications but the first includea quadratic in the number of sampled workers, to control for the precision of theleft-hand side variable.61Table 18: IV Regressions: Standard Deviation of Lifetime Wage, more than 5workers(1) (2) (3) (4) (5) (6)Variables Standard Deviation of Lifetime Wage, more than 5Export -0.0399b -0.0547a -0.0202 -0.0520a -0.0587a -0.115a(0.0171) (0.0180) (0.0187) (0.0197) (0.0179) (0.0242)N.Occ 0.0123a 0.0276a 0.0126a 0.0116a 0.0211a(0.00237) (0.00241) (0.00232) (0.00233) (0.00160)log empl -0.0984a -0.0128b(0.00668) (0.00515)log dom.share -0.00156 0.00980a(0.00248) (0.00149)log VA per worker 0.0337a 0.101a(0.00743) (0.00567)white share 0.464a(0.0153)log N. Products 0.0361a(0.00596)Avg. Lifetime Wage -0.733a(0.0119)Sector-Year y y y y y yObs. 15,826 15,826 15,826 15,826 15,826 15,826R2 0.062 0.065 0.084 0.066 0.067 0.545N.Occ.: number of occupations, based on 2 digit occupational codes for France.log empl. log-employment.log VA per worker: log-value added per worker.log dom.share: log-domestic market share, at the 4 digit sector level.white share: share of non-production worker.log N. Products: log-number of exported products (HS6 codes). This variable is zero fornon-exporters.Avg. Lifetime Wage: workers’ lifetime wage, averaged by firm.Legend : a significant at 1%, b significant at 5%, c significant at 10%.Notes: Cross-sectional Regressions for firms with more than 5 workers, years 1995-2007.Different specifications in the columns. Standard errors, clustered at the level of thefirm, are reported in parenthesis. All specifications but the first include a quadratic inthe number of sampled workers, to control for the precision of the left-hand side vari-able.62Table 19: Group-Weighted Regressions: Standard Deviation, more than5 workers(1) (2) (3) (4) (5) (6)Variables Standard Deviation of Average Lifetime Wage, more than 5Export -0.012b -0.023a -0.026a -0.030a -0.022a -0.031a(0.005) (0.005) (0.005) (0.005) (0.005) (0.005)N.Occ. 0.016a 0.012a 0.011a 0.016a 0.017a(0.001) (0.000) (0.000) (0.001) (0.001)log empl -0.023a -0.022a 0.025a(0.003) (0.003) (0.003)log dom.share -0.003a -0.000 -0.001(0.000) (0.001) (0.001)log VA per worker -0.002 -0.001 0.117a(0.003) (0.003) (0.003)white share -0.150a(0.008)log N. Products 0.003c(0.002)Avg. Lifetime Wage 0.011a(0.001)Sector-Year y y y y y yObs. 54,436 54,436 54,436 54,436 54,436 54,436R2 0.037 0.072 0.069 0.068 0.072 0.108N.Occ.: number of occupations, based on 2 digit occupational codes for France.log empl. log-employment.log VA per worker: log-value added per worker.log dom.share: log-domestic market share, at the 4 digit sector level.white share: share of non-production worker.log N. Products: log-number of exported products (HS6 codes). This variable is zerofor non-exporters.Avg. Lifetime Wage: workers’ lifetime wage, averaged by firm.Legend : a significant at 1%, b significant at 5%, c significant at 10%.Notes: Group-weighted Regressions for firms with more than 5 workers, years 1995-2007. The dependent variable is a weighted average of the standard deviations formanagers, executives and blue collar workers, using as weights the average employ-ment compositions of those groups within firm over time. Different specifications inthe columns. Standard errors, clustered at the level of the firm, are reported in paren-thesis. All specifications but the first include a quadratic in the number of sampledworkers, to control for the precision of the left-hand side variable.63Table 20: Pooled Cross-sectional Regressions: Inter-quartile Range,more than 5 workers(1) (2) (3) (4) (5) (6)Variables Inter-quartile of Lifetime Wage, more than 5Export -0.083a -0.010 -0.049a -0.073a -0.021 -0.024b(0.015) (0.015) (0.015) (0.015) (0.015) (0.012)N.Occ. 0.019a -0.012a -0.016a 0.016a 0.005a(0.002) (0.002) (0.002) (0.002) (0.002)log empl -0.178a -0.179a -0.068a(0.007) (0.007) (0.006)log dom.share -0.011a 0.003 0.004a(0.002) (0.002) (0.002)log VA per worker 0.084a 0.079a 0.142a(0.010) (0.010) (0.007)white share 0.789a(0.019)log N. Products 0.019a(0.003)Avg. Lifetime Wage -0.930a(0.016)Sector-Year y y y y y yObs. 57,469 57,469 57,469 57,469 57,469 57,469R2 0.056 0.094 0.062 0.066 0.099 0.493N.Occ.: number of occupations, based on 2 digit occupational codes for France.log empl. log-employment.log VA per worker: log-value added per worker.log dom.share: log-domestic market share, at the 4 digit sector level.white share: share of non-production worker.log N. Products: log-number of exported products (HS6 codes). This variable is zerofor non-exporters.Avg. Lifetime Wage: workers’ lifetime wage, averaged by firm.Legend : a significant at 1%, b significant at 5%, c significant at 10%.Notes: Cross-sectional Regressions for firms with more than 5 workers, years 1995-2007. Different specifications in the columns. Standard errors, clustered at the levelof the firm, are reported in parenthesis. All specifications but the first include aquadratic in the number of sampled workers, to control for the precision of the left-hand side variable.64Table 21: Market Access Regressions: Average, more than 5 workers(1) (2) (3) (4) (5) (6)Variables Average Lifetime Wage, more than 5 workersMarket Access*Export 0.014a 0.012a 0.011a 0.012a 0.013a 0.013a(0.003) (0.003) (0.003) (0.003) (0.003) (0.003)Market Access -0.016a -0.014a -0.013a -0.012a -0.015a -0.018a(0.004) (0.004) (0.003) (0.004) (0.004) (0.003)Export -0.052 -0.078 -0.106b -0.114b -0.108b -0.168a(0.051) (0.050) (0.047) (0.050) (0.050) (0.050)N.Occ. 0.039a 0.015a 0.032a 0.035a -0.002(0.002) (0.002) (0.001) (0.001) (0.001)log empl 0.125a 0.125a(0.005) (0.005)log dom.share 0.031a 0.004b(0.002) (0.002)log VA per worker 0.158a 0.096a(0.008) (0.006)white share 0.500a(0.024)log N. Products 0.008b(0.003)Sector1-Year y y y y y yObservations 44,728 44,728 44,728 44,728 44,728 44,728R-squared 0.142 0.184 0.209 0.196 0.215 0.2991 2-digit sector dummiesN.Occ.: number of occupations, based on 2 digit occupational codes for France.log empl. log-employment.log VA per worker: log-value added per worker.log dom.share: log-domestic market share, at the 4 digit sector level.white share: share of non-production worker.log N. Products: log-number of exported products (HS6 codes). This variable is zerofor non-exporters.Market Access: weighted-average - across destinations - of the demand faced by agiven industry i at time t, where the weights are the share of world exports to thatparticular destination in that industry the previous year.Legend : a significant at 1%, b significant at 5%, c significant at 10%.Notes: Cross-sectional Regressions for firms with more than 5 workers, years 1995-2007. Different specifications in the columns. Standard errors, clustered at the level ofthe firm, are reported in parenthesis. All specifications but the first include a quadraticin the number of sampled workers, to control for the precision of the left-hand side vari-able.65Table 22: Market Access Regressions: Standard Deviation, more than 5workers(1) (2) (3) (4) (5) (6)Variables Standard Deviation of Lifetime Wage, more than 5Market Access*Export -0.009a -0.009a -0.009a -0.009a -0.009a -0.009a(0.003) (0.003) (0.003) (0.003) (0.003) (0.003)Market Access 0.011a 0.012a 0.011a 0.011a 0.012a 0.011a(0.003) (0.003) (0.003) (0.003) (0.003) (0.003)Export 0.096b 0.089c 0.113b 0.096b 0.080c 0.094b(0.048) (0.047) (0.046) (0.048) (0.046) (0.041)N.Occ. 0.011a 0.031a 0.012a 0.010a 0.026a(0.001) (0.001) (0.001) (0.001) (0.001)log empl -0.107a -0.105a(0.004) (0.004)log dom.share -0.006a 0.002(0.002) (0.002)log VA per worker 0.050a 0.033a(0.006) (0.005)white share 0.158a(0.018)Avg Lifetime Wage -0.104a(0.004)log N. Products 0.009a(0.003)Sector1-Year y y y y y yObs. 44,728 44,728 44,728 44,728 44,728 44,552R2 0.068 0.071 0.094 0.072 0.075 0.1431 2-digit sector dummiesN.Occ.: number of occupations, based on 2 digit occupational codes for France.log empl. log-employment.log VA per worker: log-value added per worker.log dom.share: log-domestic market share, at the 4 digit sector level.white share: share of non-production worker.log N. Products: log-number of exported products (HS6 codes). This variable is zerofor non-exporters.Avg. Lifetime Wage: workers’ lifetime wage, averaged by firm.Legend : a significant at 1%, b significant at 5%, c significant at 10%.Market Access: weighted-average - across destinations - of the demand faced by agiven industry i at time t, where the weights are the share of world exports to thatparticular destination in that industry the previous year.Notes: Cross-sectional Regressions for firms with more than 5 workers, years 1995-2007. Different specifications in the columns. Standard errors, clustered at the level ofthe firm, are reported in parenthesis. All specifications but the first include a quadraticin the number of sampled workers, to control for the precision of the left-hand side vari-able.66Table 23: Tariff Regressions: Average, more than 5 workers(1) (2) (3) (4) (5) (6)Variables Average Lifetime Wage, more than 5Weighted Tariff*Export 0.002 0.001 0.001 -0.001 -0.001 -0.004(0.004) (0.003) (0.003) (0.003) (0.003) (0.003)Weighted Tariff 0.001 0.002 0.002 0.00395 0.006c 0.012a(0.004) (0.003) (0.003) (0.003) (0.003) (0.003)Export 0.128a 0.082a 0.045b 0.050b 0.071a 0.039c(0.023) (0.021) (0.020) (0.020) (0.021) (0.021)N.Occ. 0.039a 0.016a 0.033a 0.035a -0.002(0.001) (0.002) (0.001) (0.001) (0.001)log empl 0.123a 0.124a(0.005) (0.005)log dom.share 0.031a 0.005a(0.002) (0.002)white share 0.512a(0.021)log VA per worker 0.161a 0.099a(0.008) (0.007)log N. Products 0.004(0.003)Sector1-Year y y y y y yObservations 48,280 48,280 48,280 48,280 48,280 48,280R-squared 0.143 0.185 0.210 0.197 0.217 0.3031 2-digit sector dummiesN.Occ.: number of occupations, based on 2 digit occupational codes for France.log empl. log-employment.log VA per worker: log-value added per worker.log dom.share: log-domestic market share, at the 4 digit sector level.white share: share of non-production worker.log N. Products: log-number of exported products (HS6 codes). This variable is zero fornon-exporters.Weighted Tariff : weighted average - across destination - of tariff levels in a given in-dustry i at time t, where weights are the share of world exports to that particular desti-nation in that industry and year.Legend : a significant at 1%, b significant at 5%, c significant at 10%.Notes: Cross-sectional Regressions for firms with more than 5 workers, years 1995-2007. Different specifications in the columns. Standard errors, clustered at the level ofthe firm, are reported in parenthesis. All specifications but the first include a quadraticin the number of sampled workers, to control for the precision of the left-hand side vari-able.67Table 24: Tariff Regressions: Standard Deviation, more than 5 workers(1) (2) (3) (4) (5) (6)Variables Standard Deviation of Lifetime Wages, more than 5Weighted Tariff*Export 0.007b 0.007b 0.007b 0.007b 0.006b 0.002(0.003) (0.003) (0.003) (0.003) (0.003) (0.002)Weighted Tariff -0.013a -0.012a -0.012a -0.013a -0.011a -0.001(0.003) (0.003) (0.003) (0.003) (0.003) (0.002)Export -0.059a -0.069a -0.037b -0.062a -0.073a -0.032b(0.017) (0.017) (0.017) (0.017) (0.017) (0.013)N.Occ. 0.011a 0.031a 0.013a 0.010a 0.025a(0.001) (0.001) (0.001) (0.001) (0.001)log empl -0.108a -0.022a(0.004) (0.003)log dom.share -0.007a 0.005a(0.002) (0.001)white share 0.480a(0.010)log VA per worker 0.047a 0.103a(0.006) (0.004)log N. Products 0.014a(0.002)Avg Lifetime Wage -0.727a(0.010)Sector1-Year y y y y y yObservations 48,280 48,280 48,280 48,280 48,280 48,280R-squared 0.068 0.071 0.094 0.072 0.074 0.5501 2-digit sector dummiesN.Occ.: number of occupations, based on 2 digit occupational codes for France.log empl. log-employment.log VA per worker: log-value added per worker.log dom.share: log-domestic market share, at the 4 digit sector level.white share: share of non-production worker.log N. Products: log-number of exported products (HS6 codes). This variable is zero fornon-exporters.Avg Lifetime Wage: workers’ lifetime wage, averaged by firm.Weighted Tariff : weighted average - across destination - of tariff levels in a given indus-try i at time t, where weights are the share of world exports to that particular destina-tion in that industry and year.Legend : a significant at 1%, b significant at 5%, c significant at 10%.Notes: Cross-sectional Regressions for firms with more than 5 workers, years 1995-2007.Different specifications in the columns. Standard errors, clustered at the level of thefirm, are reported in parenthesis. All specifications but the first include a quadratic inthe number of sampled workers, to control for the precision of the left-hand side vari-able.684. Search, Matching, Trade and Wage Inequality4.1 IntroductionWhat is the impact of international trade on within-firm wage inequality? So far, the trade literaturehas only focused on between-firm wage inequality, despite the fact that the within-firm componentis at least as important as the between-firm component of wage dispersion. Within-firm variationin wages accounts for at least half of the overall wage inequality in Brazil and in Sweden.61A framework that generates within-firm variation in wages requires both worker and firm hetero-geneity and frictions in the labour market. Sampson [2014] introduces worker and firm heterogeneityin a model of labour market sorting. In absence of frictions, however, every firm matches with aunique type of worker and there is no variation within the set of workers matched with firms of agiven type. Helpman et al. [2010] combine labour market frictions and heterogeneous agents. How-ever, in Helpman et al. [2010] workers are not ex-ante different, but they have a productivity drawthat is firm specific. Moreover, the only information known by firms and workers when bargainingon wages is the minimal worker ability threshold required by the firm. Therefore also in Helpmanet al. [2010] there is no variation in wages within firms.In the present chapter, we built on the theoretical framework from Chapter 2 to analyze thefirm-level contributions to the various components of wage inequality. The model provides onlyclear predictions on the cross-sectional differences between exporters and non-exporters in terms ofbetween-firm wage inequality. The within-firm wage dispersion is, instead, influenced by two factors,differences in employment composition - due to matching sets - and differences in wage schedules.Those factors might exercise diverging effects. On the one hand, exporters tend to match with lessdispersed pool of workers if the marginal loss from a mismatch increase rapidly with agent types;on the other hand, exporting firms generate larger revenues and tend to pay higher wages. Whicheffect prevails is an empirical question.We use the French employer-employee data to investigate the firm-level contributions to within-and between-firm wage inequality. We find that the differences in sorting in large part account forthe existing differences in the wage structure between exporters and non-exporters. Exporting firmstend to pay higher wages, even if accounting for differences in unobserved employment composition.Exporters also tolerate a lower dispersion in wages; the difference in wage dispersion, however, is fullyaccounted for by differences in employment composition. We look deeper into the within-componentof wage dispersion, by constructing a theory-consistent measure of residual wage inequality. Thismeasure exploits the changes in wages that occur when a worker moves across jobs. We find thatworkers moving to exporters experience an increase in wages on average: this is consistent witha reduction in residual wage inequality. This finding suggests that exporters are better able toovercome frictions in the labour market in order to move closer to their ideal worker.The present model belongs to a strand of research that investigates the interaction betweeninternational trade and wage inequality. The most closely related works are the contributions bySampson [2014] and Helpman et al. [2010]. To the best of our knowledge, our paper is the first61See Helpman et al. [2013] and Akerman et al. [2013].69to analyze the firm-level contributions to between- and within-firm wage inequality in presence oftrade.The rest of the paper is organized as follows. Section 4.2 describes the data. Section 4.3 developsthe empirical implication from the theoretical framework built in Chapter 2. Section 4.4 presentsthe empirical specifications. The results are described in subsection 4.4.1. Section 4.5 draws theimplications for policy.4.2 DataThe data for our project come from three main sources, the De´claration Annuelle des Donne´esSociales (DADS), the Enquete Annuelle d’Entreprises (EAE) and the French Customs Data.62DADS is an administrative database of matched employer-employee information collected by theINSEE (Institut Nationale de la Statistique et des Etudes Economique). The data are based onthe mandatory reports, filed by employers, of the gross earnings of each employee in compliancewith French payroll taxes. All paying-wages individuals and legal entities established in France arerequired to file payroll declarations; only individuals employing civil servants are excluded from filingsuch declarations. The INSEE prepares extracts of the original database for research purposes. Werely on the panel version of DADS, which covers all individuals employed in French enterprises bornin the month of October of even-numbered years until 2001 and every year after that.63 This choiceis motivated by the need to follow workers across years and job positions in order to identify workerand firm types.Our extract stretches from 1995 to 2007. The initial data set contains around 24 million obser-vations (corresponding to the triplet worker-firm-year) which are identified by worker and firm ID(respectively, nninouv and siren).For each observation we have information on the individual’s gender, year and place of birth,occupation (both 2-digit CS and 4-digit PCS-ESE classification), job spell,64 full-time/part-timestatus, annualized real earnings, total number of hours worked as well as the industry of the employ-ing firm (NAF700, 4-digit industry classification). We restrict our sample to full-time employees inmanufacturing (NAF 10-33), reducing the total number of observations to 2, 662, 411. Most full-timeworkers are employed at a single firm during the year. Only 6% has more than one employer in agiven year; for those, we selected the enterprise at which the individual worked the largest numberof days during the year. Finally, to control for possible outliers, we remove those observations whoselog annualized real earnings are more than 5 standard deviation away from the predicted wage, basedon a linear model including gender, an ile-de-France dummy and in-firm experience. We obtain afinal sample of 2, 579, 414.We enrich the available set of firm-level variables by merging DADS with EAE, a survey-baseddataset containing balance-sheet information on French firms in manufacturing over the period 1995-62These data are subject to statistical secrecy and have been acceded at CEPII.63In 2002, the sampling methodology has been extended to include all individuals born in the month of Octoberof every year. Currently, the DADS panel represents 1/12th of the total French workforce.64DADS records both the job start date and the number of days the individual worked in a given firm during thecalendar year.702007. The unit of observation in EAE is a firm-year combination; the firm identifier is the sameas the firm ID in DADS (siren). EAE samples only medium-large enterprises with at least 20employees. From EAE we collect information on sales (domestic and exports), total employment,value added and also on the main sector of the firm (NAF700 4-digit classification).65 The mergewith EAE further reduces the sample availability. We restrict our sample to individuals workingfor firms whose characteristics are available from EAE. Furthermore, we remove those firms whosenumber of sampled employees from DADS is larger than the effective employment reported in EAE.This provides us a final sample of 1, 673, 992 observations on which we implement our empiricalstrategy.Finally, export-related information on French firms come from the French Customs. The customdata includes export records at firm-, product- and destination-level for the universe of exporterslocated in France.4.3 Wage inequality in a search and matching modelWe build on the model from the first chapter to characterize the impact of trade through sortingon wage inequality. We refer to the framework with heterogeneous fixed costs of exporting whereexporting is isomorphic to higher productivity. Let’s start by defining the average wage at firm ϕ,Avg W (ϕ) =∫ u(ϕ)l(ϕ)[w (θ) + s(θ,ϕ)γ]ν (θ) dθu (ϕ)− l (ϕ)In our framework, differences in average wages are due to revenue sharing and sorting patterns.66Firms of higher productivity generate larger revenues and tend to match with workers of higherability which demand larger compensations. Schank et al. [2007] document the first channel. Theyfind that exporting firms pay wage premia compared to non-exporters, conditional on the ability ofthe workers. In the empirical section, we will disentangle the effect of workforce composition frompure differences in wage premia across firms.The theory also suggests that changes in between-firm wage inequality are driven by changes insearch frictions. For c→ 0, the average wage Avg W (ϕ)→ w∗ (µ (ϕ)) - w∗ (µ (ϕ)) is the wage underthe optimal assignment, θ = µ (ϕ) - inducing an increase in the between-firm wage inequality. In fact,the wage paid by firms under the optimal assignment is higher than the average wage in presence offrictions. The average deviation tends to be higher for firms on the upper bound of the productivitydistribution as their matching sets includes relatively more workers of lower productivity than theideal. Economy-wide changes in the market size would also act to increase between-firm variationin wages. Trade liberalization shocks that differentially affect firms might still increase the between-firm wage inequality if the increase of average wage for more productive firms dominates the increasein wage for less productive firms due to the expansion in their matching sets.65We compare the firm’s industry classification between EAE and DADS and keep only those observations whoseindustry information is consistent between the two sources.66It is easy to see that average wage is increasing in ϕ. See Appendix A.1 for a proof.71The novel implication of our framework is that matches between firms and workers generatewithin-firm variability in wages. The within component of wage inequality is related to within-firmvariation in wages. We define the within-firm variance as follows,Var W (ϕ) =∫ u(ϕ)l(ϕ)[w (θ) + s(θ,ϕ)γ −Avg W (ϕ)]2ν (θ) dθu (ϕ)− l (ϕ)In the appendix we show that the wage variance is driven by two factors, differences in employmentcomposition - due to matching sets - and differences in wage schedules. Those factors might exercisediverging effects on within-firm wage variance. On the one hand, more productive firms tend tomatch with less dispersed pool of workers if α > 1;67 on the other hand, more productive firmsgenerate larger revenues and tend to pay higher wages. Both effects are of the same sign if α ≤ 1.Therefore, if α ≤ 1, within-firm wage inequality would be larger at exporting firms. Whetherexporters or non-exporters are empirically responsible for most of the residual component of wagevariation is question that we will address in the next section using French data.If controlling for differences in wages across firms, we can define an alternative measure of wagevariance relative to the average firm wage,Rel Var Wage (ϕ) =∫ u(ϕ)l(ϕ)[w (θ) + s(θ,ϕ)γ −Avg W (ϕ)]2ν (θ) dθAvg W (ϕ)Although we are unable to prove that the relative wage dispersion is decreasing in firm type, we simu-lated the model under different parametrizations and found that in our simulations Rel Var Wage (ϕ)is always decreasing in ϕ.68Let us now consider the forces driving changes in within-firm wage dispersions over time. Re-ductions in search costs tend to shrink the within-firm variation in wages; in the limit, if c → 0,both Var W (θ) ,Rel Var Wage (θ)→ 0. Then, the between-firm wage component would explain theentire wage variability.A similar outcome would emerge in presence of economy-wide shocks to market size. If, instead,shocks have differential impacts on matching sets of firms of different productivities, the changein within-firm wage inequality depends on the specific parameters of the model. In presence of atrade liberalization, for example, aggregate within-firm wage inequality might increase if the importcompetition effect across non-exporting firms dominates the tightening of the matching sets forexporting firms.Although the within-firm variation in wages is the closer measure to the empirical within-firmwage component, the theory suggests a different measure of residual wage variation. The wagerealization of a worker θ differs from the ideal wage outcome in presence of matching frictions;precisely, W (θ, µ (θ)) −W (θ, ϕ), where µ (θ) represents the frictionless assignment. We aggregate67If α > 1, the marginal losses of a mismatch increase rapidly with agent types. The dispersion relative to theoptimal assignment is smaller for firms of higher productivity. See chapter 1 for more details.68See Appendix A.2 for a longer discussion on the behaviour of Rel Var Wage (ϕ).72the theory-based measure of residual wage across all workers employed at firm ϕ,Res Wage Ineq (θ) =∫ u(θ)l(θ) [W (θ, µ (θ))−W (θ, ϕ)] ν (θ)u (ϕ)− l (ϕ)(37)The properties of the theory-based residual wage depend on the surplus function. In particular,if α ≥ 1, the surplus generated in matches between more productive agents tend to decay faster,inducing a smaller residual wage inequality.Empirically estimating the behavior of (37) requires identifying the types of workers and firms,the optimal assignment and the wage schedule of firms. Although the agents’ types can be inferredusing their pay-off over their lifetime and the optimal assignment requires estimating the empiricaldistribution of workers and firms, the wage schedule needs to be model-specific.However, changes in wage residuals do not require the estimation of the wage schedule. If a workermoves from a firm ϕ to a firm ϕ′, the change in her residual wage mainly reflect the difference infirm productivities relative to the worker abilityW (θ, µ (θ))−W (θ, ϕ)− [W (θ, µ (θ))−W (θ, ϕ′)] = W (θ, ϕ′)−W (θ, ϕ)The difference W (θ, ϕ′) −W (θ, ϕ) is positive if the worker moves to a firm whose productivity ismore similar to her own, |θ − ϕ′| ≤ |θ − ϕ|. Such movement implies a reduction in the residual wageinequality. We can build a firm-level measure of changes in residual wage inequality aggregatingacross all workers belonging to the intersection of matching sets of two firms,Res Wage Change (ϕ′) =∫I [W (θ, ϕ′)−W (θ, ϕ)] ν (θ) dθu (ϕ′)− l (ϕ′)for all ϕ such that I = M (ϕ′) ∩ M (ϕ) 6= ∅. The residual change in wage inequality would bepositive if firms are better at selecting their workers. It is also an empirical question whether theworkers moving across firms tend to end up with better matches at more productive firms ratherthan at less productive firms. Next we outline the empirical strategy to investigate our predictions.4.4 Empirical specificationsIn this section we investigate the relation between firm-level measures of wage inequality and aggre-gate wage variation. We start by analyzing how differences in wages across firms contribute to thethe between-firm wage inequality.A large literature in international trade documents differences in wages between exporters andnon-exporters. Those differences are attributed either to differences in the employment compositionor to rent-sharing policies. In order to disentangle those two possible mechanisms, we run thefollowing specification:Avg W jt = β0 + β1Exportjt + β2Firm Typejt + β3Compositionjt +Dst + ujt (38)73where Exportjt = 1 if firm j exports at time t; Firm Typejt is one or all of the three proxiesfor firm productivity, V Apwj , logEmpj and DomSharej ; Compositionjt captures the employmentcomposition. We proxy the employment structure by using the number of occupation, N.occjt, theshare of white collar workers,69 whiteshare, the number of exported products, log Products and anaverage measure of workers’ ability, AvWorkerTypejt.Our interest is in the relation between β1, β2 and β3. Conditional on the employment structure,β1 and β2 capture the effect of pure wage premia across firms; β3 extracts the effect of the workercomposition from wages.In addition, we include a quadratic in the number of sampled workers to control for the precisionof our left-hand side estimates, and sector-year dummies, Dst.Whether exporters are responsible for a larger contribution to the within-wage inequality dependson the parameters of the model. However, accounting for differences in average wages, we shouldexpect, according to our simulations, that the within component of wage inequality is, instead,smaller for exporters (and more productive firms). The specification that we employ is the following:Var W jt = β′0 + β′1Exportjt + β′2Firm Typejt + β′3Compositionjt +Dst + u′jt (39)Similarly to specification (38), we include the number of occupation, N.occjt, the share of white collarworkers, white share, the number of exported products, log Products, and the average unobservedability of workers, AvWorkerTypejt, to control for differences in the occupational structure acrossfirms with different export status. All specifications include sector-year dummies, Dst.Following the theory we develop an alternative strategy to study the variation across firms inthe residual component of wage inequality. We adopt the following specification,Res Wage Changejt = β′′0 + β′′1 Exportjt + β′′2 Firm Typejt +Dst + u′′jt (40)where Res Wage Changejt is the change in wage for movers that are employed at firm j at timet. We expect β′′1 > 0 if exporting firms are better at matching. In regression (40) we include onlythe number of occupation, N.occjt, the share of white collar workers, white share, the number ofexported products, log Products. In fact, according to our model, the coefficient on export statuswould become insignificant if adding AvWorkerTypejt as the differences in matching across firms areentirely accounted for by worker types.70In the appendix we report the results of a specification where the dependent variable is the firm-level empirical residual wage, estimated from a wage regression on workers characteristics and firmfixed effects,Res Wagejt = β˜0 + β˜1Exportjt + β˜2 Firm Typejt +Dst + u˜jt (41)to compare our theoretical measure of residual wage with the empirical counterpart.All specifications are estimated by OLS and standard errors are clustered at the level of the firm.69The blue vs white collar distinction is based on occupational code. We report the classification we adopt inTable 29.70In our model, the incentives to match are entirely driven by the type of worker and the type of firm that meet.744.4.1 ResultsThe estimation results relative to specifications (38) and (39) are presented in Tables 25 and 26.Column 1 of Table 25 reports a positive and statistically significant relationship between exportstatus and the average wage of the workers employed by the firm. The positive relationship remainssignificant when we introduce the three controls for firm type (domestic share, value added perworker and employment) or other controls for the firm-level employment composition. Exportingfirms tend to pay higher wages compared to similar non-exporters even when controlling for theunobserved average ability of their workforce (Column 7).As predicted by theory, the coefficient on all three proxies for firm type is positive and significant,like the one on export status. However, in accordance with the variance decompositions fromsection 3, the bulk of the variability in the average wage is captured by the contribution of theemployment composition. More than 35% of the variability in the average wage is due to theunobserved characteristics of workers, captured by Avg Worker Type, the average of a worker fixedeffects from an AKM regression.Table 26 reports the results for specification (39) and has a similar structure to Table 25. Startingfrom column 1 where no controls are added, we document a negative and significant relationshipbetween export status and the wage variability. The effect persists with a similar magnitude whenwe add all other controls, in columns (2)-(6). However, the wage dispersion is not significantlyrelated to export status if controlling for the average worker ability within the firm (column (7)),suggesting that differences in dispersions are mainly driven by unobserved worker characteristics.Similarly to the results in Table 13, while the coefficient on two proxies for firm type is negative(columns 2 and 3), Table 26 documents a positive and significant correlation between value addedper worker and wage dispersion.In the appendix, Tables 44 and 45 present robustness checks where specification (39) is estimatedfor different groups of workers. In particular, we divide occupations into ‘white’ and ‘blue’ collar(as reported in Table (29)). Table 44 reports results for white collar workers and Table 45 for bluecollar workers. The coefficient on the export dummy is negative both for blue and white collars, butsignificant only for blue collars. This is perhaps due to the higher share of blue collar workers in thesample. Even here, the coefficient of the export dummy loses significance if introducing a controlfor the unobserved worker characteristics.The results on the variation of residual wage inequality across firms are reported in Table 27. Theexport dummy is positively related to the change in residual wage of workers moving across firms.This seems to suggests that exporting firms tend to select workers closer to their ideal assignment;the coefficient on export is positive and significant in all specifications and his magnitude is almostunchanged across all columns. If adding a control for the average employment composition, thecoefficient on export loses significance, as expected. Consistent with our theory, all other proxy offirm types are also positively related to changes in residual wages of movers.In the appendix we report a specification where the firm-level residual wage inequality is con-structed as a residual from a wage regression on worker observables and firm fixed effects (Table 46).The results, however, are much less robust compared to the theory-based measure from Table 27.754.5 Policy implicationsThe bulk of the wage variation in France during 1995-2007 is not explained by observable workercharacteristics or firm fixed effects. Looking more closely, we find that the unexplained componenttends to be smaller in exporting and more productive firms, even when controlling for some differ-ences in the workforce composition. This finding seems to suggest that exporting firms are better atovercoming labour market frictions compared to non-exporting firms. In our framework, the differ-ences in matching across firms are due to differences in the opportunity cost from a mismatch. Wecannot exclude, however, that our empirical findings are driven by differences in matching technology,availability of resources for labour market search or the differential impact of credit frictions acrossfirms if these correlate with within-firm worker ability. Although the obvious policy implication toreduce within-firm wage inequality is to remove the frictions, the optimal policy might depend onthe particular nature of the friction. In our framework, export financing is a potential policy toolsto be added to the list of other supply-side programs aimed at overcoming search frictions in thelabour market. A rigorous analysis of the which policy instruments are optimal to use is left forfuture work.Reducing market imperfections would induce efficient sorting patterns, but not necessarily trans-late in a decrease of the overall wage inequality. In presence of complementarities, efficient sortingenhances between-group variation in wages. Anecdotal evidence suggests that sorting is also asso-ciated with a strong persistence in ability and assets between one generation and the next, leadingto an increases in the long-run inequality.However, KREMER [1997] finds that sorting has only small first-order effect on wage inequality.He calibrates a simple model using data from the Panel Survey of Income Dynamics and arguesthat changes in sorting have a small impact on steady-state inequality of characteristics that areonly moderately heritable, such as education and income. In addition, Heathcote et al. [2010] evenfind that, in a model with worker heterogeneity, skill-biased demand shocks which magnify sortingincentives produce welfare gains.A careful evaluation is therefore required for policies targeted to reduce wage inequality: whileinstruments aimed at removing labour market frictions increase efficiency in presence of worker-firm complementarities, policies that offset the rise in between-group wage inequality might notnecessarily be welfare improving.76Table 25: Average Wages, more than 5 workers(1) (2) (3) (4) (5) (6) (7)Variables Average Wages, more than 5Export 0.152a 0.063a 0.065a 0.082a 0.039a 0.028b 0.026c(0.013) (0.012) (0.012) (0.012) (0.012) (0.014) (0.014)N.Occ. 0.016a 0.034a 0.036a 0.012a -0.004c -0.004c(0.002) (0.002) (0.002) (0.002) (0.002) (0.002)log empl 0.126a 0.124a 0.130a 0.122a(0.006) (0.007) (0.006) (0.006)log dom.share 0.027a 0.004b 0.003c 0.003c(0.002) (0.002) (0.002) (0.002)log VA per worker 0.172a 0.171a 0.110a 0.108a(0.009) (0.009) (0.008) (0.008)white share 0.536a 0.484a(0.020) (0.020)log N. Products 0.006c 0.006(0.004) (0.004)Avg Worker Type 0.123a(0.006)Sector-Year y y y y y y yObs. 57,469 57,469 57,469 57,469 57,469 57,469 57,258R2 0.117 0.178 0.164 0.186 0.210 0.266 0.308N.Occ.: number of occupations, based on 2 digit occupational codes for France.log empl. log-employment.log VA per worker: log-value added per worker.log dom.share: log-domestic market share, at the 4 digit sector level.white share: share of non-production worker.log N. Products: log-number of exported products (HS6 codes). This variable is zero fornon-exporters.Legend : a significant at 1%, b significant at 5%, c significant at 10%.Notes: Cross-sectional Regressions for firms with at least 4 workers, years 1995-2007. Dif-ferent specifications in the columns. Standard errors, clustered at the level of the firm, arereported in parenthesis. All specifications but the first include a quadratic in the number ofsampled workers, to control for the precision of the left-hand side variable.77Table 26: Standard Deviation of Wages, more than 5 workers(1) (2) (3) (4) (5) (6) (7) (8)Variables Standard Deviation of Wages, more than 5Export -0.031a -0.012 -0.032a -0.050a -0.016 -0.034a -0.002 -0.002(0.011) (0.011) (0.011) (0.011) (0.011) (0.013) (0.007) (0.007)N.Occ 0.033a 0.015a 0.011a 0.032a 0.028a 0.006a 0.006a(0.002) (0.002) (0.002) (0.002) (0.001) (0.001) (0.001)log empl -0.110a -0.108a -0.110a -0.027a -0.027a(0.006) (0.006) (0.006) (0.003) (0.003)log dom.share -0.010a -0.001 -0.002 -0.002c -0.002c(0.002) (0.002) (0.002) (0.001) (0.001)log VA per worker 0.043a 0.042a 0.031a 0.010a 0.010a(0.007) (0.007) (0.007) (0.003) (0.003)white share 0.094a 0.007 0.011(0.017) (0.007) (0.007)log N. Products 0.009b 0.002 0.002(0.004) (0.002) (0.002)Dispersiom Worker Type 0.872a 0.869a(0.007) (0.007)Avg Worker Type -0.009a(0.002)Sector-Year y y y y y y yObs. 57,469 57,469 57,469 57,469 57,469 57,469 56,906 56,906R2 0.049 0.071 0.054 0.054 0.073 0.075 0.621 0.621N.Occ.: number of occupations, based on 2 digit occupational codes for France.log empl. log-employment.log VA per worker: log-value added per worker.log dom.share: log-domestic market share, at the 4 digit sector level.white share: share of non-production worker.log N. Products: log-number of exported products (HS6 codes). This variable is zero for non-exporters.Notes: Cross-sectional Regressions for firms with at least 4 workers, years 1995-2007. Different specificationsin the columns. Standard errors, clustered at the level of the firm, are reported in parenthesis. All specifica-tions but the first include a quadratic in the number of sampled workers, to control for the precision of theleft-hand side variable.78Table 27: Residual Wage Changes(1) (2) (3) (4) (5) (6) (7)Variables Residual Wage ChangesExport 0.028a 0.018c 0.024b 0.027a 0.025b 0.031b 0.018(0.010) (0.010) (0.011) (0.010) (0.011) (0.013) (0.013)N.Occ 0.004b 0.004b 0.001 0.003c 0.001a -0.006a(0.001) (0.002) (0.002) (0.001) (0.002) (0.002)log empl 0.033a 0.032a 0.033a 0.035a(0.005) (0.005) (0.005) (0.005)log dom.share 0.009a 0.008a 0.008a 0.0001(0.002) (0.002) (0.002) (0.002)log VA per worker 0.023a 0.017a 0.007a -0.033a(0.006) (0.006) (0.006) (0.007)white share 0.093a -0.144a(0.015) (0.016)log N. Products -0.003 -0.009a(0.003) (0.003)Avg Worker Type 0.454a(0.010)Sector-Year y y y y y y yObs. 64,586 64,586 64,586 64,586 64,586 64,586 64,586R2 0.035 0.036 0.037 0.036 0.037 0.038 0.104N.Occ.: number of occupations, based on 2 digit occupational codes for France.log empl. log-employment.log VA per worker: log-value added per worker.log dom.share: log-domestic market share, at the 4 digit sector level.white share: share of non-production worker.log N. Products: log-number of exported products (HS6 codes). This variable is zero fornon-exporters.Notes: Cross-sectional Regressions, years 1995-2007. Different specifications in the columns.Standard errors, clustered at the level of the firm, are reported in parenthesis. All specifica-tions but the first include a quadratic in the number of sampled workers, to control for theprecision of the left-hand side variable.795. ConclusionsUsing linked employer-employee data from France, we show that exporters and non-exporters matchwith sets of workers that are different. Exporters employ workers of higher type and lower typedispersion. We also find that when exporters face lower tariffs or larger demand for imports in aforeign market, the dispersion of types in their pool of workers declines further. We rationalize thisfinding using a model of matching with search frictions where more productive firms and exportingfirms match with better workers and tolerate a lower degree of dispersion among the workers em-ployed. We also show numerically that the welfare gains from trade are higher when search costs arehigher, which points to a substitutability between trade opening and the lowering of trade frictions.Trade liberalization seems more important when frictions are high and the worker-firm allocation isrelatively further away from the optimal.In the final chapter we find that the differences in sorting in large part account for the existingdifferences in the wage structure between exporters and non-exporters. Exporting firms tend to havehigher wages but tolerate a lower dispersion. Using an alternative theory-based measure of residualwage inequality, we also find that the unexplained component tends to be smaller in exportingand more productive firms, even when controlling for some differences in workforce composition.This finding suggests that exporters are better able to overcome frictions in the labour market inorder to move closer to their ideal worker assignment. 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Since therate at which matches dissolve remains constant, smaller matching sets lead to a higher fraction ofunmatched agents of similar type. On the other side of the market, e.g. from the perspective ofthe firms, a larger population of unmatched workers increases the probability of finding a match,everything else equal.Lemma 1. Given θ ∈ [0, 1], let αθ (θ, ψ) = 1 ⇔ ψ ∈ M (θ). If αθ (θ, ψ) ≤ α′θ (θ, ψ) ∀ψ ∈ M (θ),then ν (θ) ≥ ν′(θ), where ν (θ) , ν′(θ) satisfy (5). Moreover, if v (ψ) satisfies (6), ν (θ) ≥ ν′(θ) andv ≤ v′.Similarly, given ψ ∈ [0, 1], let αψ (θ, ψ) = 1 ⇔ θ ∈ M (ψ). If αψ (θ, ψ) ≤ α′ψ (θ, ψ) ∀ θ ∈ M (ψ),v (ψ) ≥ v′(ψ), where v (ψ) , v′(ψ) satisfy (6). Moreover, if ν (θ) satisfies (5), v (ψ) ≥ v′(ψ) andν ≤ ν′.Proof. Step 1. The map α→ να is well-defined and continuous.71Step 2. Given v, the operatorH (αθ, ν) = ν (θ)[λ+ ρ∫αθ (θ, y) v (y) dy]− λ · g (θ) (42)is monotonic in αθ, for any v.From the flow condition (5), H (αh, ναh) = 0 = H(α′h, να′h). Moreover,H(α′θ, ναθ)= ναθ (θ)[λ+ ρ∫α′θ (θ, y) v (y) dy]− λ · g (θ)≥ ναθ (θ)[λ+ ρ∫αθ (θ, y) v (y) dy]− λ · g (θ) = 0Thus, να′θ≤ ναθ , with strict inequality for functions differing over sets of positive measure.Step 3. The operatorT (αθ, ν) = ν (θ)[λ+ ρ∫αθ (θ, y)u (y) dy](43)is monotonic in αθ if v satisfies (6).71See Shimer and Smith [2000] for a proof.85From (6),v (ψ) =λ · h (ψ)λ+ ρ∫αψ (x, ψ) ν (x) dxwe can rewriteT (αθ, ν) = ν (θ)[λ+ ρ∫αθ (θ, y)λ · h (y)λ+ ρ∫αψ (x, y) ν (x) dxdy](44)Then, T (αθ, ναθ ) = λg (θ) = T(α′θ, να′θ), where ναθ , να′θsatisfy the flow condition (5). Moreover,for αθ ≤ α′θ,T (α′θ, ναθ ) = ναθ (θ)[λ+ ρ∫α′θ (θ, y)λh (y)λ+ ρ∫αψ (x, y) ναθ (x) dxdy]≥ ναθ (θ)[λ+ ρ∫αθ (θ, y)λh (y)λ+ ρ∫αψ (x, y) ναθ (x) dxdy]= να′θ (θ)[λ+ ρ∫α′θ (θ, y)λh (y)λ+ ρ∫αψ (x, y) να′θ (x) dxdy]Thus, it must be ναθ ≥ να′θ , with strict inequality if the matching set functions differ over sets ofpositive measure.72 Moreover, vαψ ≤ vα′ψ .Joint variations of the matching sets for both types of agents over the entire space, instead, implyequally-signed effects on the distribution of the unmatched, as the following lemma clarifies.Lemma 2. Given θ, ψ ∈ [0, 1], let αθ (θ, ψ) = 1 ⇔ ψ ∈ M (θ) and αψ (θ, ψ) = 1 ⇔ θ ∈ M (ψ).Assume that the g (θ) = h (ψ), ∀ θ = ψ. If α′θ ≥ αθ ∀ θ ∈ [0, 1], α′ϕ ≥ αϕ ∀ϕ ∈ [0, 1] and αθ (θ, ψ) =αψ (ψ, θ), then in a symmetric equilibrium νθ,α′θ,α′ψ ≤ νθ,αθ,αψ and vψ,α′θ,α′ψ ≤ vψ,αθ,αψ , where ν andv satisfy respectively (5) and (6). Moreover,∫α′θ (θ, ψ) vψ,α′θ,α′ψ (ψ) dψ ≥∫αθ (θ, ψ) vψ,αθ,αψ (ψ) dψand∫α′ψ (θ, ψ) νθ,α′θ,α′ψ (θ) dθ ≥∫αψ (θ, ψ) νθ,αθ,αψ (θ) dθProof. Using (6), the flow conditions can be rewritten asvψ,αθ,αψ (ψ)[λ+ ρ∫αψ (x, ψ) ν (x) dx]= λh (ψ)νθ,αθ,αψ (θ)[λ+ ρ∫αθ (θ, y) v (y) dy]= λg (θ)In a symmetric equilibrium, vψ,αθ,αψ (ψ) = νθ,αθ,αψ (θ) if ψ = θ. DefineT (αθ, αψ, ν) = ν (θ)[λ+ ρ∫αθ (θ, y) ν (y) dy](45)72Note that the operatorJ (νθ) =u (θ)λ+ ρ∫αψ (x, ψ) ν (y) dyis increasing in ν. The above condition on the distribution functions, then, easily follows.86If α′θ ≥ αθ, ∀ θ ∈ [0, 1]T(α′θ, α′ψ, ν)= ν (θ)[λ+ ρ∫α′θ (θ, y) ν (y) dy]≥ ν (θ)[λ+ ρ∫αθ (θ, y) ν (y) dy]= T (αθ, αψ, ν)where T(αθ, αψ, νθ,αθ,αψ)= T(α′θ, α′ψ, νθ,α′θ,α′ψ). Therefore, νθ,α′θ,α′ψ ≤ νθ,αθ,αψ . Similarly, it mustbe that vψ,α′θ,α′ψ ≤ vψ,αθ,αψ . Moreover, by the flow conditions,∫α′θ (θ, ψ) vψ,α′θ,α′ψ (ψ) dψ ≥∫αθ (θ, ψ) vψ,αθ,αψ (ψ) dψ∫α′ψ (θ, ψ) νθ,α′θ,α′ψ (θ) dθ ≥∫αψ (θ, ψ) νθ,αθ,αψ (θ) dθThe inverse relation between matching sets of an agent and unmatched distributions of hispotential partners does not hold in presence of contemporaneous variations of matching sets forboth workers and firms.73 The resulting effect is based upon the steady state flow equilibriumconditions requiring that the matches that dissolve equate the matches that are created. If thematching set expands, the probability of finding a partner increases, implying that the measure ofunmatched agents for both workers and firms decreases.The effects described in both lemmas, however, remain a partial equilibrium analysis. The fullcharacterization of the equilibrium requires that matching sets, unmatched distributions and optionvalue functions are mutually compatible. In fact, changes in the distributions of the unmatched, inturn, have an effect on the matching set themselves, through the firms’ and workers’ option values.The general equilibrium effect is characterized in proposition 6.A.2 Relative losses: variation by firm typeIn order to show that RL (ϕ) is decreasing in ϕ it is convenient to rewrite it as follows:RL (ϕ) = 1−∫ u(ϕ)l(ϕ) θαϕαdθ∫ u(ϕ)l(ϕ)12[θ2α + 12ϕ2α]dθIt is easy to verify that RL (ϕ) is decreasing if and only if∂∂ϕ∫ u(ϕ)l(ϕ) θαϕαdθ∫ u(ϕ)l(ϕ)[12θ2α + 12ϕ2α]dθ > 073Inverse inequalities for the distributions of the unmatched would be preserved if the matching sets are expand-ing for one type and contracting on the other side of the market.87Given,∂∂ϕ∫ u(ϕ)l(ϕ) θαϕαdθ∫ u(ϕ)l(ϕ)12[θ2α + ϕ2α]dθ =[∫ u(ϕ)l(ϕ) αθαϕα−1dθ + ϕαu (ϕ)α ∂u(ϕ)∂ϕ − ϕαl (ϕ)α ∂l(ϕ)∂ϕ] ∫ u(ϕ)l(ϕ)12[θ2α + ϕ2α]dθ[∫ u(ϕ)l(ϕ)12 [θ2α + ϕ2α] dθ]2−[∫ u(ϕ)l(ϕ) αϕ2α−1dθ + 12 [ϕα + u (ϕ)α] ∂u(ϕ)∂ϕ −12 [ϕα + l (ϕ)α] ∂l(ϕ)∂ϕ] ∫ u(ϕ)l(ϕ) θαϕαdθ[∫ u(ϕ)l(ϕ)12 [θ2α + ϕ2α] dθ]2it is sufficient to prove thatαϕα−1∫ u(ϕ)l(ϕ)[θα − ϕα] dθ +∂u (ϕ)∂ϕ[ϕαu (ϕ)α∫ u(ϕ)l(ϕ)[θα + ϕα] dθ −12[ϕα + u (ϕ)α]∫ u(ϕ)l(ϕ)θαϕαdθ]−∂l (ϕ)∂ϕ[ϕαl (ϕ)α∫ u(ϕ)l(ϕ)[θα + ϕα] dθ −12[ϕα + l (ϕ)α]∫ u(ϕ)l(ϕ)θαϕαdθ]is non-negative. For all α, αϕα−1∫ u(ϕ)l(ϕ) [θα − ϕα] dθ = 0.74 If α = 1, ∂u(ϕ)∂ϕ =∂l(ϕ)∂ϕ = 1. Then, it iseasy to see that the condition is verified,[u (ϕ)ϕ∫ u(ϕ)l(ϕ)[θ + ϕ] dθ −12u (ϕ)ϕ∫ u(ϕ)l(ϕ)θdθ]−[l (ϕ)ϕ∫ u(ϕ)l(ϕ)[θ + ϕ] dθ −12l (ϕ)ϕ∫ u(ϕ)l(ϕ)θdθ]≥ 0(46)∀ϕ ∈ [0, 1]. If α < 1, ∂u(ϕ)∂ϕ > 1 >∂l(ϕ)∂ϕ . Then, inequality (46) is also verified.A.3 Comparison across Models: Changes in the Meeting RateThe purpose of this section is to compare the equilibrium properties of matching sets under differentassumptions on the evolution of the unmatched. The comparison becomes quite easy as it relies onspecific values of the matching rate. In particular, a model with ρ = 0 is equivalent to assumingthat agents who match leave the market and are replaced by agents of identical type, e.g. clonesassumption. In this section, we will show that the qualitative properties of matching sets areindependent from the assumption of the evolution of the unmatched.Let us start our analysis developing the predictions of changes in the meeting rate on the distri-bution of the unmatched agents for given matching sets.Lemma 3. Let u0, uρ and M0,Mρ be the distributions of the unmatched and the matching setsassociated to the meeting rates ρ = 0 and ρ > 0, respectively. Then, ν0j > νρj , j = h, ϕ.Lemma 3 is based on the idea that if the probability to meet a potential partner tends to zero,the balancing of the flow equation implies that all matches that are formed must be destroyed bythe end of the period. Therefore, if ρ = 0, the distribution of the unmatched will coincide with theinitial distribution of agents. Exploiting lemma 3, we can easily characterize the full effect of themeeting rate on the equilibrium matching sets.74This is because, at the optimal assignment, ϕα =∫ u(ϕ)l(ϕ)θαu(ϕ)−l(ϕ)88Proposition 11. Let ν0, νρ and M0,Mρ be the distributions of the unmatched and the matchingsets associated to the meeting rates of ρ = 0 and ρ > 0, respectively. Then, M0 (j) ⊂ Mρ (j),j = θ, ψ.Proof. Let w0, pi0, wρ, piρ be the pay-off functions associated to the meeting rates of ρ = 0 and ρ > 0,respectively. By contradiction, suppose that Mρ (j) ⊆M0 (j). Then,{wρ (θ) ≥ w0 (θ)piρ (ψ) ≥ pi0 (ψ)From the CSC,γ · c =∫M0(θ)[RD (θ, ψ)− w0 (θ)− pi0 (ϕ)]v0 (ψ) dψ>∫Mρ(θ)[RD (θ, ψ)− w0 (θ)− pi0 (ψ)]vρ (ψ) dψ≥∫Mρ(θ)[RD (θ, ψ)− wρ (θ)− piρ (ψ)]vρ (ψ) dψ = γ · cTherefore, M0 (j) ⊂Mρ (j).A.4 Identification of worker type: average lifetime wageAgent types are positively correlated with their average pay-offs over their job spells. In particular,a more productive worker makes a larger contribution to revenues and tends to match with betterfirms, obtaining, on average, a higher pay-off. Following the model, we propose to identify the agenttypes using the average wage. In fact, the average wage of a worker θ is increasing in its type. Theexpected wage of a worker over her matching set is given byAvgW (θ) = w (θ) +cu (θ)− l (θ)The expected wage of a worker θ is increasing in her type∂AvgW (θ)∂θ=∂w (θ)∂θ− c∂u(θ)∂θ −∂l(θ)∂θ[u (θ)− l (θ)]2= α∫ u(θ)l(θ) E1η θα−1ϕαu (θ)− l (θ)− cθα−1u(θ)α−1− θα−1l(θ)α−1[u (θ)− l (θ)]2=1u (θ)− l (θ)α∫ u(θ)l(θ)E1η θα−1ϕα − cθα−1u(θ)α−1− θα−1l(θ)α−1[u (θ)− l (θ)]89If α ≥ 1, then ∂u(θ)∂θ −∂l(θ)∂θ < 0; therefore,∂AvgW (θ)∂θ > 0. Let’s consider the case when α < 1. Since∫ u(θ)l(θ) E1η θαϕα ≥ c, it is sufficient to show thatαθ≥θα−1u(θ)α−1− θα−1l(θ)α−1[u (θ)− l (θ)](47)For α → 0, LHS, RHS of (47) both tend to zero;75 moreover, LHS of (47) is increasing in α, whilethe RHS is decreasing. This proves that the inequality (47) is also verified for all α < 1.A.5 Firm fixed effects and firm typesLet W (h, ϕ) be the observed wage for a worker of ability h when matched with a firm of type ϕ.Then, a standard wage decomposition proposes to identifyW (h, ϕ) = θ (h) + ψ (ϕ) + εhϕ (48)where θ (h) correspond to the worker’s fixed effect, ψ (ϕ) represent the firm’s fixed effect and εhϕaccounts for the idiosyncratic elements of the match. Precisely,θ (h) =∫M(h)[W (h, ϕ)− ψ (ϕ)] v (ϕ) dϕψ (ϕ) =∫M(ϕ)[W (h, ϕ)− θ (h)] ν (h) dhCombining the two definitions,ψ (ϕ) =∫M(ϕ)W (h, ϕ) ν (h) dh−∫M(ϕ)∫M(h)[W (h, ϕ)− ψ (ϕ)] ν (h) v (ϕ) dhdϕ=∫M(ϕ)[γ − 1γw (h) +R (h, ϕ)− pi (ϕ)γ]ν (h) dh−∫M(ϕ)∫M(h)[W (h, ϕ)− ψ (ϕ)] ν (h) v (ϕ) dhdϕThe variation of firms’ fixed effect across firms’ types is captured byΓ (ϕ) =∫M(ϕ)[γ − 1γw (h) +R (h, ϕ)− pi (ϕ)γ]ν (h) dhNote that∂Γ (ϕ)∂ϕ=1γ∫M(ϕ)∂ [R (h, ϕ)− pi (ϕ)]∂ϕν (h) dh+ [w (u (ϕ)) ν (u (ϕ))− w (l (ϕ)) ν (u (ϕ))]Assume ν (h) = κ, ∀h ∈ [0, 1]; then,∂Γ (ϕ)∂ϕ=κγ∫M(ϕ)∂ [R (h, ϕ)− pi (ϕ)]∂ϕdh+ κ [w (u (ϕ))− w (l (ϕ))]75In fact, if α = 0, R (θ, ϕ) = 0 and s (θ, ϕ) = 0; this implies that matching sets are empty for all θ.90Then,∫M(ϕ)∂[R(h,ϕ)−pi(ϕ)]∂ϕ dh = 0 and w (u (ϕ))−w (l (ϕ)) ≥ 0. However, in presence of a distributionwith a long tail, fixed effects from a wage decomposition would not necessarily vary positively withthe firm types.A.6 Average wage by firmWe define the average wage at firm ϕ as the average over the set of profitable matchesAvg W (ϕ) =∫ u(ϕ)l(ϕ)[w (θ) + s(θ,ϕ)γ]ν (θ) dθ∫ u(ϕ)l(ϕ) ν (θ) dθSince w (θ) + s(θ,ϕ)γ is increasing in θ and firms of higher productivity match with workers of higherability, Avg W (ϕ) is increasing in θ.A.7 Wage variance by firmBefore defining the variation in wages at the firm level, let’s first consider the second moment of thewage distribution by firm,Sq Wage (ϕ) =∫ u(ϕ)l(ϕ)[w (θ) + s(θ,ϕ)γ]2ν (θ) dθ∫ u(ϕ)l(ϕ) ν (θ) dθThe second moment of the wage distribution also varies positively with firm type θ; in fact,∂Sq Wage (ϕ)∂ϕ=2γ∫ u(ϕ)l(ϕ)[w (θ) + s(θ,ϕ)γ]∂s(θ,ϕ)∂ϕ ν (θ) dθ∫ u(ϕ)l(ϕ) ν (θ) dθ++∂u(ϕ)∂ϕ u (ϕ) ν (u (ϕ))∫ u(ϕ)l(ϕ) ν (θ) dθ{[w (u (ϕ)) +s (u (ϕ) , ϕ)γ]2[∫ u(ϕ)l(ϕ)ν (θ) dθ]− Sq Wage (ϕ)}++∂l(ϕ)∂ϕ ν (l (ϕ)) l (ϕ)∫ u(ϕ)l(ϕ) ν (θ) dθ{Sq Wage (ϕ)−[w (l (ϕ)) +s (l (ϕ) , ϕ)γ]2[∫ u(ϕ)l(ϕ)ν (θ) dθ]}≥ 0since w (θ) + s(θ,ϕ)γ and∂s(θ,ϕ)∂ϕ are increasing in θ.The wage variance of firm ϕ is defined asVar W (ϕ) =∫ u(ϕ)l(ϕ)[w (θ) + s(θ,ϕ)γ −Avg W (ϕ)]2ν (θ) dθ∫ u(ϕ)l(ϕ) ν (θ) dθ= Sq Wage (ϕ)− [Avg W (ϕ)]291Taking the derivative with respect to ϕ,∂Var W (ϕ)∂ϕ=2γ∫ u(ϕ)l(ϕ)[w (θ) + s(θ,ϕ)γ −Avg W (ϕ)] [∂s(θ,ϕ)∂θ]ν (θ) dθ∫ u(ϕ)l(ϕ) ν (θ) dθ++∂u(ϕ)∂ϕ u (ϕ) ν (u (ϕ))∫ u(ϕ)l(ϕ) ν (θ) dθ{[w (u (ϕ)) +s (u (ϕ) , ϕ)γ−Avg W (ϕ)]2[∫ u(ϕ)l(ϕ)ν (θ) dθ]−Var W (ϕ)}++∂l(ϕ)∂ϕ l (ϕ) ν (l (ϕ))∫ u(ϕ)l(ϕ) ν (θ) dθ{Var W (ϕ)−[w (l (ϕ)) +s (l (ϕ) , ϕ)γ−Avg W (ϕ)]2[∫ u(ϕ)l(ϕ)ν (θ) dθ]}The variation in within-firm wages is driven by two components, changes in wage schedule andchanges in the matching set. The term2γ∫ u(ϕ)l(ϕ)[w (θ) + s(θ,ϕ)γ −Avg W (ϕ)] [∂s(θ,ϕ)∂θ]ν (θ) dθ∫ u(ϕ)l(ϕ) ν (θ) dθ(49)captures the differences in wage schedule between a firm ϕ and a firm ϕ+ δ, as δ → 0. If α ≤ 1, theexpression (49) is positive, since both w (θ) + s(θ,ϕ)γ −Avg W (ϕ) and∂s(θ,ϕ)∂θ are increasing in θ, ∀ϕand w (θ) + s(θ,ϕ)γ is symmetric around Avg W (ϕ). The term∂u(ϕ)∂ϕ u (ϕ) ν (u (ϕ))∫ u(ϕ)l(ϕ) ν (θ) dθ{[w (u (ϕ)) +s (u (ϕ) , ϕ)γ−Avg W (ϕ)]2[∫ u(ϕ)l(ϕ)ν (θ) dθ]−Var W (ϕ)}+∂l(ϕ)∂ϕ l (ϕ) ν (l (ϕ))∫ u(ϕ)l(ϕ) ν (θ) dθ{Var W (ϕ)−[w (l (ϕ)) +s (l (ϕ) , ϕ)γ−Avg W (ϕ)]2[∫ u(ϕ)l(ϕ)ν (θ) dθ]}captures the variation in matching sets between a firm ϕ and a firm ϕ+ δ, as δ → 0. Assuming thatthe tail of the distribution of the unmatched is not decaying too fast, the term above is positive ifα ≤ 1, since ∂u(θ)∂θ ≥ 1 ≥∂l(θ)∂θ and w (u (ϕ)) +s(u(ϕ),ϕ)γ > w (l (ϕ)) +s(l(ϕ),ϕ)γ . This implies that∂Var W(ϕ)∂θ > 0 for α = 1 and by continuity for 1 − ε < α < 1 + ε, with ε → 0. For α → ∞,u (ϕ) , l (ϕ)→ θ = µ (ϕ); therefore, ∂Var W(ϕ)∂θ → 0.Finally, let’s introduce the relative wage varianceRel Var (ϕ) =Var W (ϕ)Avg W (ϕ)The behaviour of the relative wage variance shares some similarity with the behaviour of the wagevariance∂Rel Var (ϕ)∂ϕ=2γ∫ u(ϕ)l(ϕ)[w (θ) + s(θ,ϕ)γ −Avg W (ϕ)] [∂s(θ,ϕ)∂ϕ]ν (θ) dθAvg W (ϕ)+∂u(ϕ)∂ϕ u (ϕ) ν (u (ϕ))[Avg W (ϕ)]2{[w (u (ϕ)) +s (u (ϕ) , ϕ)γ−Avg W (ϕ)]2Avg W (ϕ)− w (u (ϕ)) Var W (ϕ)}+∂l(ϕ)∂ϕ l (ϕ) ν (l (ϕ))[Avg W (ϕ)]2{w (l (ϕ)) Var W (ϕ)−[w (l (ϕ)) +s (l (ϕ) , ϕ)γ−Avg W (ϕ)]2Avg W (ϕ)}92The behaviour of the relative variance depends on the behaviour of the term[w(θ)+ s(θ,ϕ)γ −Avg W(ϕ)]2w(θ)+ s(θ,ϕ)γ,which is non-monotonic in θ∂[w(θ)+ s(θ,ϕ)γ −Avg W(ϕ)]2w(θ)+ s(θ,ϕ)γ∂θ= 2(∂w (θ)∂θ+1γ∂s (θ, ϕ)∂θ)[(w (θ) +s (θ, ϕ)γ)2−Avg W (ϕ)2]≥ 0 if w (θ) + s(θ,ϕ)γ ≥ Avg W (ϕ)≤ 0 if w (θ) + s(θ,ϕ)γ ≤ Avg W (ϕ)Let θ¯ such that w(θ¯)+s(θ¯,ϕ)γ = Avg W (ϕ); then,∂[w(θ)+ s(θ,ϕ)γ −Avg W(ϕ)]2w(θ)+ s(θ,ϕ)γ∂θ≥ 0 if θ ≥ θ¯This implies that[w (u (ϕ)) + s(u(ϕ),ϕ)γ −Avg W (ϕ)]2w (u (ϕ)) + s(u(ϕ),ϕ)γ≤Var W (ϕ)Avg W (ϕ)If α ≤ 1,76∂u(ϕ)∂ϕ[Avg W (ϕ)]2{[w (u (ϕ)) +s (u (ϕ) , ϕ)γ−Avg W (ϕ)]2Avg W (ϕ)− w (u (ϕ)) Var W (ϕ)}+∂l(ϕ)∂ϕ[Avg W (ϕ)]2{w (l (ϕ)) Var W (ϕ)−[w (l (ϕ)) +s (l (ϕ) , ϕ)γ−Avg W (ϕ)]2Avg W (ϕ)}≤ 0A.8 Residual wage inequalityLet W (θ, µ (θ))−W (θ, ϕ) be the deviation of the optimal wage from the realized wage for a workerθ when employed at a firm ϕ, where µ (θ) represents the frictionless assignment. Note thatW (θ, µ (θ))−W (θ, ϕ) = w (θ) +s (θ, µ (θ))γ− w (θ)−s (θ, ϕ)γ=1γ[s (θ, µ (θ))− s (θ, ϕ)]Aggregating across all workers employed at firm ϕ, we obtain a firm-level based measure of residualwage inequality,Res W In (ϕ) =∫ u(θ)l(θ) [W (θ, µ (θ))−W (θ, ϕ)] ν (θ) dθ∫ u(ϕ)l(ϕ) ν (θ) dθ76We maintain the assumption that the distribution of the unmatched does not have a fast decaying tail.93Then,∂Res W In (ϕ)∂ϕ=∂u(ϕ)∂ϕ u (ϕ) ν (u (ϕ))∫ u(ϕ)l(ϕ) ν (θ) dθ{1γ[s (u (ϕ) , µ (u (ϕ)))− s (u (ϕ) , ϕ)][∫ u(ϕ)l(ϕ)ν (θ) dθ]− Res W In (θ)}+−∂l(ϕ)∂ϕ l (ϕ) ν (l (ϕ))∫ u(ϕ)l(ϕ) ν (θ) dθ{1γ[s (l (ϕ) , µ (l (ϕ)))− s (l (ϕ) , ϕ)][∫ u(ϕ)l(ϕ)ν (θ) dθ]− Res W In (θ)}The variability of residual wage inequality across firms depends on the behavior of s (θ, µ (θ)) −s (θ, ϕ),∂∂θ[s (θ, µ (θ))− s (θ, ϕ)] =∂s (θ, µ (θ))∂θ+∂s (θ, µ (θ))∂ϕ∂µ (θ)∂θ−∂s (θ, ϕ)∂θ= −∂s (θ, ϕ)∂θ≤ 0 if θ ≤ ϕ≥ 0 if θ ≥ ϕThis implies that ∂Res W In(ϕ)∂ϕ ≤ 0 if α ≥ 1.94B. Numerical simulationWe simulate the model using the empirical distribution of worker and firm types to show that theproperties of matching bounds are verified under this specification (see Figure 11 for the distributionof worker types and 12 for the distribution of firm types). Figure 5 shows the matching set of theeconomy when normalizing the aggregate price index to unity and assuming the search cost c = 0.01,the meeting rate ρ = 1 and the exogenous separation rate δ = 1. Using the simulation results,we construct two measures of dispersions of worker types by firm, the length and the standarddeviation77 of the firm matching set. Figure 6 and 7 show that both measures are decreasing in firmtype, when normalized by the average worker type.Firm TypeWorker Type0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.10.20.30.40.50.60.70.80.91Figure 5: Matching Set for the Simulated Economy77In the empirical analysis, our preferred measure of dispersion is the standard deviation of the worker types byfirm, since it is less sensitive to outliers.950 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.31.41.51.61.71.81.92Firm TypeLength of Matching SetFigure 6: Standard Deviation of the MatchingSet by firm type, normalized by the averageworker type0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.511.522.53Firm TypeStandard Deviation of Worker Types relative to the AverageFigure 7: Standard Deviation of the MatchingSet by firm type, normalized by the averageworker type96C. Additional figures and tables0.125 0.25 0.375 0.5 0.625 0.75 0.875 10.1250.250.3750.50.6250.750.8751Figure 8: Export Cut-offFigure 9: Distribution of Sampled Workers - trimming the 95th percentile97Figure 10: Distribution of Value Added per Worker by Export StatusFigure 11: Distribution of Individual Effects, largest connected groupFigure 12: Distribution of Firm Effects, largest connected group98Figure 13: Wage Changes by Wage Quartile99Table 28: Wage Changes when Moving to a New JobWage Change PercentagePositive 54.82%Negative 45.18%Table 29: Classification of CS Occupation into ’white’ and ’blue’ collar workers.CS code White Collar Jobs3 Executives and Higher Intellectual Professions31 Health Professionals and Lawyers33 Senior Official in Public Administration34 Teachers, Scientific Professions35 Information, arts and entertainment37 Administrative and Commercial skilled workers38 Engineers and technical managers4 Intermediate Occupations42 Teachers and related43 Intermediate occupations, health and social work44 Religious45 Intermediate administrative professions in Public Administration46 Intermediate administrative and commercial occupation in Enterprises47 Technicians48 Foremen, supervisorsCS code Blue Collar Jobs5 Clericals52 Civilian Employees and officers in Public Service53 Protective Services54 Administrative Employees55 Commercial workers56 Personal services workers6 Labourers62 Qualified Industrial Labourers63 Qualified craft labourers64 Drivers65 Storage and Transport workers67 Non-Qualified Industrial Labourers68 Non-Qualified craft labourers69 Farm Workers100Table 30: Summary StatisticsMean Median Std DeviationAvg. Worker Type -0.04 -0.02 0.86Std Dev. Worker Fixed Effects 0.62 0.52 0.41Std Dev. Worker Fixed Effects, White Collars 0.55 0.47 0.36Std Dev. Worker Fixed Effects, Blue Collars 0.50 0.36 0.41Std Dev. Worker Fixed Effectsa 0.62 0.52 0.41Std Dev. Worker Fixed Effects, White Collarsa 0.55 0.47 0.36Std Dev. Worker Fixed Effects, Blue Collarsa 0.50 0.36 0.41Num. Occupation 4.90 4.00 2.44Domestic Market Share 0.03 0.01 0.08Employment 290.48 134.00 715.65Products 8.57 9.01 4.22Share of Non Production Worker 0.34 0.29 0.25Value Added per worker 70.76 45.71 161.35a Conditioning on a sample of firms with more than 5 sampled workers.Table 31: Summary Statistics: Market Access ShocksMean Median Std DeviationWeighted Tariff 5.58 5.03 3.49Market Access Shock1 12.93 14.32 6.15Market Access Shock2 12.89 14.27 6.12Weighted Tariff : Weighted average - across destination - of tarifflevels in a given industry i at time t, where weights are the shareof world exports to that particular destination in that industryand year.Market Access Shock1: Weighted average - across destinations,excluding France - of the demand faced by a given industry i attime t, where the weights are the share of world exports to thatparticular destination in that industry the previous year.Market Access Shock2: Weighted-average - across destinations -of the demand faced by a given industry i at time t, where theweights are the share of world exports to that particular destina-tion in that industry the previous year.101Table 32: Pooled Cross-sectional Regressions: Standard Deviation of newly hiredworkers, more than 5(1) (2) (3) (4) (5) (6)Variables Standard Deviation of lifetime wage, hired, more than 5Export -0.025 -0.048b -0.057a -0.063a -0.056a -0.057a(0.020) (0.020) (0.020) (0.020) (0.020) (0.017)N.Occ. 0.031a 0.024a 0.022a 0.030a 0.0198a(0.00289) (0.00234) (0.00223) (0.00289) (0.00209)log empl -0.028a -0.031a -0.023a(0.007) (0.008) (0.006)log dom.share -0.0002 0.002 0.004c(0.003) (0.003) (0.002)log VA per worker 0.038a 0.038a 0.048a(0.010) (0.010) (0.008)white share 0.332a(0.019)log N. Products 0.018a(0.004)Avg. Lifetime Wage -0.444a(0.011)Sector-Year y y y y y yObs. 14,971 14,971 14,971 14,971 14,971 14,971R2 0.154 0.168 0.166 0.168 0.169 0.483N.Occ.: number of occupations, based on 2 digit occupational codes for France.log empl. log-employment.log VA per worker: log-value added per worker.log dom.share: log-domestic market share, at the 4 digit sector level.white share: share of non-production worker.log N. Products: log-number of exported products (HS6 codes). This variable is zero for non-exporters.Avg. Lifetime Wage: workers’ lifetime wage, averaged by firm.Legend : a significant at 1%, b significant at 5%, c significant at 10%.Notes: Cross-sectional Regressions for firms with more than 5 workers, years 1995-2007. Dif-ferent specifications in the columns. Standard errors, clustered at the level of the firm, arereported in parenthesis. All specifications but the first include a quadratic in the number ofsampled workers, to control for the precision of the left-hand side variable.102Table 33: Pooled Cross-sectional Regressions: Standard Deviation of cur-rent workers, more than 5(1) (2) (3) (4) (5) (6)Variables Standard Deviation of lifetime wage, stayers, more than 5Export 0.020b 0.010 0.002 -0.003 0.006 -0.014(0.009) (0.009) (0.009) (0.009) (0.009) (0.010)N.Occ. 0.025a 0.020a 0.018a 0.024a 0.018a(0.001) (0.001) (0.001) (0.001) (0.001)log empl -0.032a -0.033a -0.017a(0.004) (0.004) (0.004)log dom.share -0.001 -8.06e−5 0.001(0.001) (0.001) (0.001)log VA per worker 0.040a 0.041a 0.080a(0.005) (0.005) (0.005)white share 0.381a(0.013)log N. Products 0.011a(0.002)Avg. Lifetime Wage -0.447a(0.012)Sector-Year y y y y y yObs. 40,579 40,579 40,579 40,579 40,579 40,579R2 0.043 0.071 0.066 0.072 0.077 0.253N.Occ.: number of occupations, based on 2 digit occupational codes for France.log empl. log-employment.log VA per worker: log-value added per worker.log dom.share: log-domestic market share, at the 4 digit sector level.white share: share of non-production worker.log N. Products: log-number of exported products (HS6 codes). This variable is zerofor non-exporters.Avg. Lifetime Wage: workers’ lifetime wage, averaged by firm.Legend : a significant at 1%, b significant at 5%, c significant at 10%.Notes: Cross-sectional Regressions for firms with more than 5 workers, years 1995-2007. Different specifications in the columns. Standard errors, clustered at the levelof the firm, are reported in parenthesis. All specifications but the first include aquadratic in the number of sampled workers, to control for the precision of the left-hand side variable.103Table 34: Cross-Sectional Regressions: Average Lifetime Wage, more than5(1) (2) (3) (4) (5) (6)Variables Average Lifetime Wage, more than 5Export 0.112a 0.044a 0.049a 0.066a 0.025b 0.022c(0.012) (0.011) (0.011) (0.011) (0.011) (0.013)Occ. Pred. Avg Wage 2e−5, a 2e−5, a 2e−5, a 2e−5, a 2e−5, a 1e−5, a(6e−7) (6e−7) (6e−7) (6e−7) (6e−7) (6e−7)N.Occ 0.0018 0.020a 0.023a -0.001 -0.009a(0.002) (0.002) (0.002) (0.002) (0.002)log empl 0.123a 0.120a 0.123a(0.006) (0.005) (0.006)log dom.share 0.024a 0.003c 0.003c(0.002) (0.002) (0.002)log VA per worker 0.151a 0.150a 0.103a(0.008) (0.008) (0.008)white share 0.458a(0.020)log N. Products 0.004(0.003)Sector-Year y y y y y yObs. 57,469 57,469 57,469 57,469 57,469 57,469R2 0.193 0.239 0.223 0.242 0.268 0.313N.Occ.: number of occupations, based on 2 digit occupational codes for France.log empl. log-employment.log VA per worker: log-value added per worker.log dom.share: log-domestic market share, at the 4 digit sector level.white share: share of non-production worker.log N. Products: log-number of exported products (HS6 codes). This variable is zero fornon-exporters.Occ. Pred. Avg Wage: average wage implied by the firm-level occupation composition.Legend : a significant at 1%, b significant at 5%, c significant at 10%.Notes: Cross-sectional Regressions for firms with more than 5 workers, years 1995-2007.Different specifications in the columns. Standard errors, clustered at the level of thefirm, are reported in parenthesis. All specifications but the first include a quadratic inthe number of sampled workers, to control for the precision of the left-hand side variable.104Table 35: Cross-Sectional Regressions: Standard Deviation of LifetimeWage, more than 5(1) (2) (3) (4) (5) (6)Variables Standard Deviation of Lifetime Wages, more than 5Export -0.049a -0.021b -0.042a -0.057a -0.025b -0.022a(0.010) (0.010) (0.010) (0.010) (0.010) (0.008)Occ. Pred. Std. Wage 1e−5, a 8e−6, a 9e−6, a 9e−6, a 8e−6, a 2e−5, a(6e−7) (6e−7) (6e−7) (6e−7) (6e−7) (5e−7)N.Occ 0.025a 0.006a 0.004a 0.024a 0.015a(0.002) (0.001) (0.001) (0.002) (0.001)log empl -0.104a -0.104a -0.013a(0.005) (0.005) (0.004)log dom.share -0.008a 0.0004 0.002c(0.002) (0.002) (0.001)log VA per worker 0.037a 0.036a 0.108a(0.006) (0.006) (0.005)white share 0.401a(0.013)log N. Products 0.010a(0.002)Avg. Lifetime Wage -0.754a(0.010)Sector-Year y y y y y yObs. 57,469 57,469 57,469 57,469 57,469 57,469R2 0.075 0.097 0.077 0.078 0.099 0.570N.Occ.: number of occupations, based on 2 digit occupational codes for France.log empl. log-employment.log VA per worker: log-value added per worker.log dom.share: log-domestic market share, at the 4 digit sector level.white share: share of non-production worker.log N. Products: log-number of exported products (HS6 codes). This variable is zero fornon-exporters.Avg. Lifetime Wage: workers’ lifetime wage, averaged by firm.Occ. Pred. Std Wage: standard deviation of wages implied by the firm-level occupationcomposition.Legend : a significant at 1%, b significant at 5%, c significant at 10%.Notes: Cross-sectional Regressions for firms with more than 5 workers, years 1995-2007.Different specifications in the columns. Standard errors, clustered at the level of the firm,are reported in parenthesis. All specifications but the first include a quadratic in thenumber of sampled workers, to control for the precision of the left-hand side variable.105Table 36: Pooled GLS Regressions: Average Lifetime Wage(1) (2) (3) (4) (5) (6)Variables Average Lifetime WageExport 0.174a 0.047a 0.050a 0.072a 0.021b -0.009(0.011) (0.010) (0.010) (0.010) (0.010) (0.012)N.Occ. -0.004 0.013a 0.019a -0.006b -0.009a(0.003) (0.002) (0.002) (0.003) (0.003)log empl 0.097a 0.085a 0.075a(0.007) (0.006) (0.006)log dom.share 0.032a 0.006a 0.004b(0.003) (0.002) (0.002)log VA per worker 0.177a 0.167a 0.103a(0.010) (0.009) (0.009)white share 0.559a(0.022)log N. Products 0.014a(0.004)Sector-Year y y y y y yObs. 148,784 148,784 148,784 148,784 148,784 148,784R2 0.181 0.253 0.244 0.268 0.289 0.360N.Occ.: number of occupations, based on 2 digit occupational codes for France.log empl. log-employment.log VA per worker: log-value added per worker.log dom.share: log-domestic market share, at the 4 digit sector level.white share: share of non-production worker.log N. Products: log-number of exported products (HS6 codes). This variable is zerofor non-exporters.Legend : a significant at 1%, b significant at 5%, c significant at 10%.Notes: GLS Regressions for firms with more than 5 workers, years 1995-2007. Dif-ferent specifications in the columns. Standard errors, clustered at the level of thefirm, are reported in parenthesis. All specifications but the first include a quadraticin the number of sampled workers, to control for the precision of the left-hand sidevariable.106Table 37: Pooled GLS Regressions: Average of Workers Fixed Effects(1) (2) (3) (4) (5) (6)Variables Average of Workers Fixed EffectsExport 0.107a 0.030 0.026 0.035 0.018 -0.004(0.025) (0.024) (0.024) (0.024) (0.025) (0.028)N.Occ. 0.007 0.011a 0.013a 0.007 0.0057(0.005) (0.003) (0.003) (0.005) (0.005)log empl 0.028b 0.021 0.010(0.014) (0.013) (0.014)log dom.share 0.012b 0.005 0.003(0.005) (0.005) (0.005)log VA per worker 0.065a 0.060a 0.013(0.017) (0.016) (0.016)white share 0.404a(0.037)log N. Products 0.011(0.008)Sector-Year y y y y y yObs. 79,689 79,689 79,689 79,689 79,689 79,689R2 0.052 0.073 0.073 0.075 0.076 0.089N.Occ.: number of occupations, based on 2 digit occupational codes for France.log empl. log-employment.log VA per worker: log-value added per worker.log dom.share: log-domestic market share, at the 4 digit sector level.white share: share of non-production worker.log N. Products: log-number of exported products (HS6 codes). This variable iszero for non-exporters.Legend : a significant at 1%, b significant at 5%, c significant at 10%.Notes: GLS Regressions for firms with more than 5 workers, years 1995-2007.Different specifications in the columns. Standard errors, clustered at the levelof the firm, are reported in parenthesis. All specifications but the first include aquadratic in the number of sampled workers, to control for the precision of theleft-hand side variable.107Table 38: Pooled GLS Regressions: Standard Deviation of LifetimeWage(1) (2) (3) (4) (5) (6)Variables Standard Deviation of Lifetime WageExport -0.017 -0.017 -0.038a -0.050a -0.025b -0.035a(0.011) (0.012) (0.012) (0.012) (0.012) (0.010)N.Occ. 0.024a 0.012a 0.010a 0.024a 0.017a(0.003) (0.002) (0.002) (0.003) (0.003)log empl -0.055a -0.060a -0.004(0.009) (0.008) (0.006)log dom.share -0.004 0.004c 0.006a(0.003) (0.002) (0.002)log VA per worker 0.037a 0.040a 0.095a(0.008) (0.009) (0.007)white share 0.508a(0.016)log N. Products 0.014a(0.003)Avg. Lifetime Wage -0.713a(0.012)Sector-Year y y y y y yObs. 88,790 88,790 88,790 88,790 88,790 88,790R2 0.099 0.119 0.108 0.111 0.123 0.553N.Occ.: number of occupations, based on 2 digit occupational codes for France.log empl. log-employment.log VA per worker: log-value added per worker.log dom.share: log-domestic market share, at the 4 digit sector level.white share: share of non-production worker.log N. Products: log-number of exported products (HS6 codes). This variable is zerofor non-exporters.Avg. Lifetime Wage: workers’ lifetime wage, averaged by firm.Legend : a significant at 1%, b significant at 5%, c significant at 10%.Notes: GLS Regressions for firms with more than 5 workers, years 1995-2007. Dif-ferent specifications in the columns. Standard errors, clustered at the level of thefirm, are reported in parenthesis. All specifications but the first include a quadraticin the number of sampled workers, to control for the precision of the left-hand sidevariable.108Table 39: Pooled GLS Regressions: Standard Deviation of WorkerFixed Effects(1) (2) (3) (4) (5) (6)Variables Standard Deviation of Worker Fixed EffectsExport -0.023c -0.026b -0.045a -0.054a -0.034a -0.043a(0.012) (0.013) (0.013) (0.013) (0.013) (0.014)N.Occ. 0.022a 0.011a 0.010a 0.021a 0.021a(0.003) (0.002) (0.002) (0.003) (0.003)log empl -0.048a -0.053a -0.056a(0.009) (0.008) (0.008)log dom.share -0.004 0.003 0.003(0.003) (0.002) (0.002)log VA per worker 0.032a 0.035a 0.023a(0.008) (0.008) (0.008)white share 0.152a(0.020)log N. Products 0.005(0.004)Avg Worker Type -0.092a(0.006)Sector-Year y y y y y yObs. 79,689 79,689 79,689 79,689 79,689 79,689R2 0.106 0.123 0.115 0.117 0.126 0.158N.Occ.: number of occupations, based on 2 digit occupational codes for France.log empl. log-employment.log VA per worker: log-value added per worker.log dom.share: log-domestic market share, at the 4 digit sector level.white share: share of non-production worker.log N. Products: log-number of exported products (HS6 codes). This variable iszero for non-exporters.Avg Worker Type: average worker fixed effect, estimated by the AKM decomposi-tion, by firm.Legend : a significant at 1%, b significant at 5%, c significant at 10%.Notes: GLS Regressions for firms with more than 5 workers, years 1995-2007.Different specifications in the columns. Standard errors, clustered at the levelof the firm, are reported in parenthesis. All specifications but the first include aquadratic in the number of sampled workers, to control for the precision of theleft-hand side variable.109Table 40: Pooled Cross-sectional Regressions: Standard Devia-tion by group, more than 5 workers(1) (2) (3) (4) (5) (6)Variables Blue CollarsExport -0.088a -0.069a -0.083a -0.099a -0.070a -0.024b(0.014) (0.014) (0.014) (0.014) (0.014) (0.011)Obs. 38,835 38,835 38,835 38,835 38,835 38,835R2 0.051 0.066 0.054 0.053 0.068 0.616ExecutivesExport -0.137a -0.105c -0.115b -0.123b -0.101c -0.016(0.052) (0.054) (0.054) (0.053) (0.053) (0.027)Obs. 11,732 11,732 11,732 11,732 11,732 11,732R2 0.153 0.164 0.158 0.157 0.165 0.586ManagersExport 0.237a 0.231a 0.235a 0.226a 0.226a 0.093b(0.056) (0.054) (0.054) (0.054) (0.053) (0.042)Obs. 7,440 7,440 7,440 7,440 7,440 7,440R2 0.273 0.289 0.280 0.279 0.289 0.578Notes: Cross-sectional Regressions for firms with more than 5 workers,years 1995-2007. Different specifications in the columns, as in Table 19.Standard errors, clustered at the level of the firm, are reported in paren-thesis. All specifications but the first include a quadratic in the number ofsampled workers, to control for the precision of the left-hand side variable.Table 41: Sectoral Rank Correlations(1) (2) (3) (4) (5) (6) (7) (8)Variables Rank CorrelationExport 0.042a 0.029a 0.023b 0.017c 0.042a 0.037a 0.026a 0.027c(0.008) (0.010) (0.010) (0.010) (0.008) (0.014) (0.010) (0.014)log empl 0.009c 0.004 0.003 -0.001(0.005) (0.005) (0.009) (0.009)log VA per worker 0.077a 0.075a 0.064a 0.064a(0.017) (0.017) (0.023) (0.024)Sector,Year y1 y1 y1 y1 y2 y2 y2 y2Obs. 3,836 3,836 3,836 3,836 3,836 3,836 3,836 3,836R2 0.041 0.042 0.049 0.049 0.195 0.195 0.198 0.1981 2 digit sector dummies.2 4 digit sector dummies.log empl. log-employment.log VA per worker: log-value added per worker.Legend : a significant at 1%, b significant at 5%, c significant at 10%.Notes: Industry regressions, years 1995-2007. Different specifications in the columns. Standard er-rors, clustered at the sector-level, are reported in parenthesis.110Table 42: GLS Regressions: Sectoral Rank Correlations(1) (2) (3) (4) (5) (6) (7)Variables Rank CorrelationExport 0.037a 0.033a 0.018b 0.024a 0.035a 0.011 0.022b(0.009) (0.009) (0.009) (0.009) (0.011) (0.007) (0.011)log empl 0.002 -0.004 -0.001 -0.008(0.004) (0.004) (0.007) (0.007)log VA per worker 0.074a 0.078a 0.096a 0.100a(0.014) (0.015) (0.020) (0.021)Sector,Year y1 y1 y1 y1 y2 y2 y2Obs. 3,812 3,812 3,812 3,812 3,812 3,812 3,812R2 0.082 0.082 0.094 0.094 0.333 0.343 0.3431 2 digit sector dummies.2 4 digit sector dummies.log empl. log-employment.log VA per worker: log-value added per worker.Legend : a significant at 1%, b significant at 5%, c significant at 10%.Notes: GLS industry regressions, years 1995-2007. Different specifications in the columns.Standard errors, clustered at the sector-level, are reported in parenthesis.Table 43: IV Regressions: Standard Deviation of Lifetime Wage, more than5 workers(1) (2) (3) (4) (5) (6)Variables Standard Deviation of Lifetime Wage, more than 5Export (Second Stage) -0.075a -0.095a -0.061a -0.099a -0.102a -0.153a(0.019) (0.020) (0.021) (0.022) (0.020) (0.035)Mkt Access (First Stage) 0.120a 0.112a 0.106a 0.099a 0.110a 0.043a(0.005) (0.004) (0.004) (0.004) (0.004) (0.001)F-stat (First Stage) 528 551 543 540 551 599Obs. 16,072 16,072 16,072 16,072 16,072 16,072Market Access: tariffs faced by firms in sector s exporting to country r at time t,weighted by the share of exports to country r of firm j at time t − 1 over the totalexports of firm j at time t− 1.Legend : a significant at 1%, b significant at 5%, c significant at 10%.Notes: IV Regressions for firms with more than 5 workers, years 1995-2007. Dif-ferent specifications in the columns. Standard errors, clustered at the level of thefirm, are reported in parenthesis.111Table 44: Standard Deviation of Wages, White Collars, more than 5 workers(1) (2) (3) (4) (5) (6) (7) (8)Variables Standard Deviation of Wages, White Collar, more than 5Export -0.035 -0.026 -0.031 -0.037 -0.030 -0.044c -0.005 -0.004(0.025) (0.025) (0.025) (0.025) (0.025) (0.026) (0.010) (0.010)N. Occ 0.009a 0.0002 -0.001 0.008a 0.012a 0.008a 0.007a(0.002) (0.002) (0.002) (0.002) (0.002) (0.001) (0.001)log VA per worker 0.015c 0.015c -0.005 0.011a 0.011a(0.008) (0.008) (0.008) (0.005) (0.005)log empl -0.032a -0.034a -0.029a -0.030a -0.032a(0.005) (0.006) (0.006) (0.002) (0.002)log dom.share -0.002 0.002 0.001 0.001 0.001(0.002) (0.002) (0.002) (0.001) (0.001)white share 0.211a 0.105a 0.098a(0.022) (0.012) (0.010)log N. Products 0.007b 0.005c 0.004(0.003) (0.003) (0.003)Dispersiom Worker Type 0.523a 0.511a(0.010) (0.010)Avg Worker Type -0.015a(0.004)Sector-Year y y y y y y y yObs. 22,756 22,756 22,756 22,756 22,756 22,756 22,756 22,756R2 0.177 0.180 0.177 0.177 0.180 0.192 0.434 0.577N.Occ.: number of occupations, based on 2 digit occupational codes for France.log empl. log-employment.log VA per worker: log-value added per worker.log dom.share: log-domestic market share, at the 4 digit sector level.white share: share of non-production worker.log N. Products: log-number of exported products (HS6 codes). This variable is zero for non-exporters.Notes: Cross-sectional Regressions for firms with more than 5 workers, years 1995-2007. Different specifi-cations in the columns. Standard errors, clustered at the level of the firm, are reported in parenthesis. Allspecifications but the first include a quadratic in the number of sampled workers, to control for the precisionof the left-hand side variable.112Table 45: Standard Deviation of Wages, Blue Collars, more than 5 workers(1) (2) (3) (4) (5) (6) (7) (8)Variables Standard Deviation of Wages, more than 5Export -0.085a -0.069a -0.081a -0.100a -0.067a -0.079a -0.010 -0.010(0.015) (0.015) (0.015) (0.015) (0.015) (0.017) (0.008) (0.008)N.Occ. 0.024a 0.008a 0.004c 0.023a 0.022a 0.006a 0.006a(0.002) (0.002) (0.002) (0.002) (0.003) (0.001) (0.001)log empl -0.096a -0.094a -0.096a -0.018a -0.018a(0.008) (0.008) (0.009) (0.004) (0.004)log dom.share -0.012a -0.005b -0.006b -0.004a -0.004a(0.002) (0.002) (0.002) (0.001) (0.001)log VA per worker 0.025a 0.038a 0.037a 0.011a 0.011a(0.008) (0.008) (0.008) (0.004) (0.004)white share -0.001 -0.003 -0.004(0.034) (0.014) (0.014)log N. Products 0.006 0.001 0.001(0.005) (0.002) (0.002)Dispersiom Worker Type 0.917a 0.913a(0.008) (0.008)Avg Worker Type -0.008b(0.003)Sector-Year y y y y y y yObs. 38,835 38,835 38,835 38,835 38,835 38,835 38,353 38,353R2 0.037 0.051 0.041 0.040 0.052 0.053 0.625 0.625N.Occ.: number of occupations, based on 2 digit occupational codes for France.log empl. log-employment.log VA per worker: log-value added per worker.log dom.share: log-domestic market share, at the 4 digit sector level.white share: share of non-production worker.log N. Products: log-number of exported products (HS6 codes). This variable is zero for non-exporters.Notes: Cross-sectional Regressions for firms with more than 5 workers, years 1995-2007. Different specifica-tions in the columns. Standard errors, clustered at the level of the firm, are reported in parenthesis. All speci-fications but the first include a quadratic in the number of sampled workers, to control for the precision of theleft-hand side variable.113Table 46: Firm-level Residual Wage (Empirical Measure)(1) (2) (3) (4) (5) (6)Variables Firm-level residualExport -0.002 -0.004b -0.002 -0.004b -0.003 -0.001(0.002) (0.002) (0.002) (0.002) (0.002) (0.002)N.Occ 0.001b -0.001c 0.0002 0.0002 0.00002 -0.00004(0.0003) (0.0003) (0.0003) (0.0003) (0.0003)log empl 0.006a 0.006a 0.006a(0.001) (0.001) (0.001)log dom.share 0.009a 0.008a 0.008a(0.002) (0.002) (0.002)log VA per worker 0.012a 0.011a 0.010a(0.001) (0.001) (0.001)white share 0.006b(0.003)log N. Products -0.001(0.0005)Sector-Year y y y y y yObs. 89,545 89,545 89,545 89,545 89,545 89,545R2 0.005 0.005 0.006 0.006 0.006 0.006N.Occ.: number of occupations, based on 2 digit occupational codes for France.log empl. log-employment.log VA per worker: log-value added per worker.log dom.share: log-domestic market share, at the 4 digit sector level.white share: share of non-production worker.log N. Products: log-number of exported products (HS6 codes). This variable is zerofor non-exporters.Notes: Cross-sectional Regressions, years 1995-2007. Different specifications in thecolumns. Standard errors, clustered at the level of the firm, are reported in parenthesis.All specifications but the first include a quadratic in the number of sampled workers, tocontrol for the precision of the left-hand side variable.114

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