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Worker-firm matching in a global economy
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This thesis investigates the impact of international trade on the sorting patterns of workers across firms and analyzes the implications for welfare.
The first essay builds a model of matching between heterogeneous workers and firms in presence of search frictions. Variation in the worker type at the firm level exists in equilibrium only because of search costs. When firms gain access to foreign markets their revenue potential increases. When stakes are high, matching with the right worker becomes particularly important because deviations from the ideal match quickly reduce the value of the relationship. Hence exporting firms select sets of workers that are less dispersed relative to the average.
The second essay documents the difference in the sorting patterns of workers between exporters and non-exporters in a French matched employer-employee dataset. We proxy the type of each worker using her average wage over her job spells and construct measures of the average type and type dispersion at the firm level. We find that exporting firms tolerate a lower dispersion in the pool of workers they hire. The matching between exporting firms and workers is even tighter in sectors 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, among other 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 existing differences in the wage structure between exporters and non-exporters. Exporting firms tend to have higher wages but tolerate a lower dispersion. Using an alternative theory-based measure of residual wage inequality, we also find that the unexplained component tends to be smaller in exporting 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 in order to move closer to their ideal worker.
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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(C) 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 B 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 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 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??as 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 age 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 ac 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 wi 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
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 heter 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 formedbetwe 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.Meet7 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’ 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 thato 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.8o 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)o 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 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
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) see Card [2001] for a recent survey. Abowd et al. [1999] proposea strategy to disentangle the effect of education from workers’ time invariant unobserv 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 = uθ (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 = uψ (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)??-∫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 & u (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 ? l(?) u(?) ? Figure 1: Matching Bounds for Worker θ when σ η-1η = 1marginal losses are larger for agents of 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 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 compa