Immigration, Worker-Firm Matching, and. Inequality

Transcription

1 Immigration, Worker-Firm Matching, and Inequality Jaerim Choi* University of Hawaii at Manoa Jihyun Park** KISDI August 2, 2018 Abstract This paper develops a novel framework of worker-firm matching to study the distributional impacts of low-skilled immigration on native workers. Adopting an assignment model from the trade literature, we build a theoretical model where the inflow of immigrants affects natives through the change in worker-firm matching. Theory predicts that the inflow of low-skilled immigrants pushes natives up to match with more productive firms. With complementarity assumption, the benefits are biased toward skilled workers, and inequality rises among native workers. Using Census and IPUMS American Community Survey over the period , we provide empirical evidence that immigration raises inequality through the matching channel. Keywords: Immigration; Matching; Inequality. JEL Code: F22, F66, J31. *Department of Economics, University of Hawaii at Manoa, choijm@hawaii.edu **Korea Information Society Development Institute, jhnpark@kisdi.re.kr

2 1 Introduction What are the distributional effects of immigration in the U.S.? This paper proposes a new framework, a matching between workers and firms, to think about the impacts of immigration on inequality. Immigrants in the U.S. labor market are characterized by less educated immigrants - there are relatively more immigrants at the lower end of the skill distribution. Thus, the influx of immigrants may differently affect natives with heterogeneous skills, which leads to a change in inequality in the U.S. Therefore, it is essential to incorporate heterogeneous skills into the theoretical immigration model, not based on representative agent models, to analyze the impact of immigration on inequality in the U.S. In addition to heterogeneity in the inflows of immigrants, the U.S. labor market is characterized by positive assortative matching (PAM) between workers and firms: High-skilled workers are matched with high-productive firms. Moreover, log wage schedules are increasing and convex in education levels (Lemieux, 2006, 2008). Based on these observations in the U.S. labor market, we build a theoretical immigration model of matching between heterogeneous workers and heterogeneous firms with complementary production technology in the framework of monopolistic competition (Sampson, 2014). Heterogenous firms are paired up with heterogeneous workers to produce differentiated goods. In equilibrium, our model replicates the positive assortative matching and the convex log wage schedule which are observed in the U.S. labor market. When immigrants arrive in the destination country, it changes the skill distribution of workers, and the influx changes the matching mechanism between workers and firms. Because the wages are determined by both the skill level of workers and the productivity level of firms, there would be distributional consequences after immigration. We analyze two hypothetical cases of immigration. First, we assume that the skill distribution of immigrants is identical to that of natives. In this case, the matching mechanism and the wage function do not change from immigration. Second, we consider a 1

3 case where immigrants are relatively less skilled than natives. The case of low-skilled immigration is of great importance because the low-skilled immigration characterizes the U.S. immigration. The distributional implication, in the second case, presents a starking contrast to the first case. The model predicts that immigrant inflow changes the matching mechanism between workers and firms. Native workers can now be paired up with firms with higher productivities, which increases inequality. Due to the complementary production technology between worker s skill and firm s productivity, more skilled workers benefit more from the re-matching process. We empirically test whether the theory applies to the U.S. local labor markets during the period from 1980 to We use U.S. Census 1980, 1990, 2000 and IPUMS American Community Survey and exploit the variation in the share of immigrants across commuting zone-year. We assume that immigrants tend to be less-skilled than natives since the foreign-born percentage of the U.S. population in a less educated group is overrepresented than in other groups (Peri, 2016) and the absolute number of immigrants in this group outweighs the number of other groups. We use a Baltik-type shift-share instrument to deal with potential endogeneity issues. Using the two-stage least squares estimation technique, we demonstrate that the influx of low-skilled immigrants increases inequality within natives. We further explore whether the increase in inequality comes from the worker-firm matching channel. Because we cannot directly observe worker-firm matching in our dataset, we define each commuting zone - industry pair as a proxy unit for a firm. Using a variant of Mincerian wage regression, we decompose worker s wage into components related to observable worker characteristics, unobservable firm heterogeneity (commuting zone - industry heterogeneity), other unobservable fixed effects, and residual variation. We use the estimated firm heterogeneity as a proxy for time-invariant productivities of native workers matching counterparts. Empirical results show that low-skilled immigration induces native workers to match with more productive firms. 2

4 2 Related Literature There has been extensive literature on the effect of immigrants on the wages of natives (Borjas, 2003; Card, 2009; Ottaviano and Peri, 2012; Basso and Peri, 2015). However, studies in this research arena have not come to a single conclusion as to the direction of the impact of immigration on the wages of native workers. Borjas (2003) argues that immigration adversely affects the wages of competing native workers. The author assumes that workers with the same education and different work experience in a national market are not perfect substitutes and uses the variations in schooling-experience groups to measure the wage impact of immigration. On the contrary, Ottaviano and Peri (2012) find a small positive effect on native wages. Using the national approach as in Borjas (2003), this study estimates the elasticity of substitution across a different group of workers and uses the estimated elasticities to calculate the total wage effect of immigration. They find a small but significant degree of imperfect substitutability between natives and immigrants under the same education-experience cell, which leads to the overall positive wage effect of immigration. Similar to Ottaviano and Peri (2012), Basso and Peri (2015) find a zero to a positive impact of immigration on the wages of native workers using the 2SLS method with a shift-share instrument. Card (2009) explores the impact of immigration on wage inequality using cross-city variations in the U.S. The author finds that immigration had a small effect on wage inequality among natives. However, considering immigrants are counted in total population, the effect of immigration on total wage inequality becomes positive as immigrants tend to locate in the upper and lower tails of the skill distribution. We contribute to this literature by building a new immigration framework to analyze the impact of immigration on the wages and inequality of native workers. The modeling framework is based on the heterogeneous workers and heterogeneous firms which allow us to analyze diverse aspects of immigration on natives. The prediction from the model and empirical results shed some light on the impact of immigration on the wages of native workers. 3

5 Researchers have also studied more micro-founded mechanisms for the impact of immigration on the wages of native-born workers. Peri and Sparber (2009) argue that native-born workers and foreign-born workers specialize in different production tasks so that large influx of less-educated immigrants may not necessarily substitute less-educated native workers. In the model, immigration will re-allocate task supply in which foreignborn workers specialize in manual tasks while native workers specialize in communication tasks. The task specialization mechanism attenuates the negative wage impact of immigration on native workers. Hunt and Gauthier-Loiselle (2010) find evidence of the positive spillovers of skilled migration on boosting innovation in the U.S. As immigrants are more concentrated in science and engineering occupations than natives in those occupations, immigrants increase innovation as measured by US patents per capital. Since scientific and engineering knowledge transfers occur easily, this positive spillover makes native workers better off. Ortega and Peri (2014) show that openness to immigration leads to higher income per capita. This positive effect operates through an increased total factor productivity from increased diversity and ideas in the host country. We introduce a new mechanism, worker-firm matching, to explain the distributional effects of immigrants on the natives. As immigration can be seen as matching between immigrant workers and native firms, we incorporate the standard matching framework into the model and analyze how different types of immigration affect the matching function, which maps workers skills to firms productivities, in the host country. Through the change in the skill distribution of workers from immigration, native workers are paired up with new firms with higher productivities. This matching mechanism affects the wages of native workers, thereby changing inequality among native workers. Our insight on immigration, defined as matching between immigrants and native firms, is based upon the concept of globalization in Kremer and Maskin (1996, 2006). Kremer and Maskin (1996) propose a model of production by workers of different skills with complementarity between two tasks. In a closed economy setting, they argue that 4

6 segregation of high-skilled and low-skilled workers into separate firms will lead to wage inequality. Kremer and Maskin (2006) extend the analysis to a two-country framework and analyze the impact of globalization, which is defined as workers from different countries being able to form a team in the same firm. Globalization enables high-skilled workers in poor countries to be matched with more skilled workers in rich countries. Low skilled workers in the poor countries are marginalized due to globalization because their skill levels are so low that rich country workers do not want to match with them. Our theoretical framework is also closely related to the burgeoning literature in international trade that uses an assignment model with heterogeneous workers (Antràs, Garicano and Rossi-Hansberg, 2006; Ohnsorge and Trefler, 2007; Costinot and Vogel, 2010; Sampson, 2014; Grossman, Helpman and Kircher, 2017; Grossman and Helpman, 2018). Antràs, Garicano and Rossi-Hansberg (2006) propose a knowledge-based hierarchy model to analyze the impact of cross-country team formation on the structure of wages. Ohnsorge and Trefler (2007) study the implication of two-dimensional worker heterogeneity and worker sorting on international trade. Costinot and Vogel (2010) develop tools and techniques to analyze the factor allocation and factor prices in a Roy-like assignment model with a continuum of workers and a continuum of tasks. Grossman, Helpman and Kircher (2017) develop two industries and two heterogeneous factors of production framework in which they study the distributional effects of international trade. In this paper, we base our model upon Sampson (2014) s monopolistic competition assignment model. Then, we apply the monotone comparative statics technique in Costinot and Vogel (2010) to derive analytical results about the distributional impacts of immigration. Immigration changes the skill distribution of workers, thus changes the matching mechanism in the model. The theoretical model developed in this paper, to the best of our knowledge, is the first application of the assignment model with heterogeneous workers in the immigration literature. 5

7 3 Some Observations in the U.S. Labor Market Fact I. Natives and immigrants are heterogeneous in skills. There are relatively more immigrants at the lower end of the education distribution, i.e., immigrant workers have lower education levels compared to native workers (See Figure 1). Native workers are concentrated in high school graduates, but a significant share of immigrant workers report their education as none. Moreover, immigrants are less likely to have excellent communication skills, such as language, than comparablyeducated, native-born workers (Peri and Sparber, 2009). Thus, the influx of immigrants into the U.S. labor market can be characterized as an increase in the supply of workers with low skills. Fact II. Worker-Firm matching shows positive assortative matching. We use U.S. census data to approximate firms by commuting zone-industry pairs. Because firm productivities are unobservable to researchers, we set up a variant of Mincerian type regression and compute time-invariant commuting zone-industry fixed effects to proxy for firm productivities (See Section for more details). Within each firm, we calculate average schooling years and average imputed log hourly wages of workers. In the top panel of Figure 2, more educated workers are paired up with more productive firms. In the bottom panel of Figure 2, higher paid workers are matched with more productive firms. There is a positive association between worker skills and firm productivities. In this literature, we use positive assortative matching (PAM) to denote this relationship. This relationship also holds for other years (1980, 1990, and 2010). Fact III. Log wage function is strictly increasing and convex in skills. Lemieux (2006, 2008) argues that log wages have become an increasingly convex function of years of education. In , the log wage appeared to be a linear function in years of education. However, in , this log wage function had become a convex 6

8 function in that the return to post-secondary education is much higher than the return to elementary and secondary education. We revisit this finding using US Census dataset. First, we show that log hourly wage is strictly increasing in years of education in each year and returns to an additional year of education is much higher for those with high school education or more compared to those with secondary education or less. Moreover, the log wage gap within the same education level between the year 1980 and the year 2010 increases in years of education (See Figure 3). 4 The Model We build a theoretical immigration framework that can replicate these three observations in the U.S. labor market: 1. Heterogeneity in skills between immigrants and natives, 2. Positive assortative matching between workers and firms, and 3. Log wage function is increasing and convex in education levels. More specifically, our model is based upon Sampson (2014) s matching model in which heterogeneous firms hire heterogeneous workers to produce differentiated goods in a monopolistic competition framework. Using this framework, we extend the model to allow for immigration and study the distributional effects of different cases of immigration. 4.1 Environment Consider an economy populated by a mass N of workers indexed by skill level z, and a mass M of firms indexed by productivity level ϕ. The cumulative distribution of skills is given by H(z), which is twice differentiable and has a positive density H (z) > 0 on the bounded support [z min, z max ] with z min > 0. Similarly, the cumulative distribution of firm productivities is given by G(ϕ), which is twice differentiable and has a positive density G (ϕ) > 0 on the bounded support [ϕ min, ϕ max ] with ϕ min > 0. Firms with different productivities hire different types of workers to produce differentiated goods, indexed 7

9 by ω, using worker as the sole input to production. The preference of each individual is given by a CES utility over a continuum of differentiated goods: [ U = ω Ω ] σ x(ω) σ 1 σ 1 σ dω where Ω is the set of available goods and σ > 1 is the elasticity of substitution between goods. It follows that the demand for any variety ω is given by: x(ω) = XP σ p(ω) σ where X is the aggregate output of composite good, P := [ ω Ω p(ω)1 σ dω ] 1 1 σ is the aggregate price, and p(ω) is the price of differentiated good ω. Without loss of generality, we normalize the aggregate price P equal to one. Then, the demand for variety ω can be expressed as: x(ω) = Xp(ω) σ. Consider a firm that produces variety ω using productivity ϕ and that hires a set L ω of workers with densities l w (z). Then, the output is given by: x(ω) = ψ(ϕ, z)l w (z)dz. z L ω Assumption 1. The productivity function ψ(ϕ, z) is twice continuously differentiable, strictly increasing, and strictly log supermodular. Assumption 2. The log productivity function ln ψ(ϕ, z) is twice continuously differentiable, strictly increasing, and second partial derivatives with respect to z are strictly positive. 8

10 4.2 Profit Maximization Each firm chooses the employment size and the worker skill level to maximize the following profit: π(l, z; ϕ) = X 1 σ 1 σ [ψ(ϕ, z)l] σ w(z)l, where w(z) is the wage paid to workers of skill level z. The first order conditions of the profit maximization problem can be written as follows: π z = σ 1 σ X 1 1 σ [ψ(ϕ, z)l] σ ψz (ϕ, z) w (z) = 0, π l = σ 1 σ X 1 1 σ [ψ(ϕ, z)l] σ ψ(ϕ, z) w(z) = 0. Rearranging the two conditions, we can obtain the differential equation that characterizes the trade off relation between the productivity and the wage as follows: ψ z (ϕ, z) ψ(ϕ, z) = w (z) w(z). (1) Assumption 1 and equation (1) dictate the equilibrium matching pattern, which is the positive assortative matching (PAM) between firm types and worker types. Denote m(z) as a matching function that maps worker skill to firm productivity, and w(z) as an equilibrium wage function. Assumption 2 dictates the equilibrium log wage schedule. From equations (1), we know that the partial derivatives of log productivities with respect to skill z are identical to partial deriviatives of log wage function with respect to skill z. Because the log productivities are strictly increasing and convex in skill z from Assumption 2, log wage schedule ln w(z) is strictly increasing and convex in skill z. The equilibrium matching function and wage function replicate the patterns in the U.S. labor market: the positive assortative matching between workers and firms (Fact II) and the log wage schedule is 9

11 strictly increasing and convex in years of eduaction (Fact III). Using the profit maximization condition, the optimal number of workers, the price, and the profit that a firm with productivity ϕ, are given by, ( ) σ σ 1 l(z; ϕ) = Xψ (ϕ, z) σ 1 w(z) σ, σ p(ϕ) = σ w(z) σ 1 ψ(ϕ, z), π(ϕ) = σ σ (σ 1) σ 1 X ( ) 1 σ w(z). ψ(ϕ, z) 4.3 Labor Market Clearing Consider a set of workers [z min, z] and the set of firms [ϕ min, m(z)] that matched with these workers in the equilibrium. The labor market clearing condition can be expressed as: M m(z) ϕ min ( ) σ σ 1 Xψ ( ϕ, m 1 (ϕ) ) z σ 1 w(m 1 (ϕ)) σ dg(ϕ) = N σ z min dh(z) where the left-hand side is the demand for workers by firms with productivity level between ϕ min and m(z) and the right-hand side is the supply of workers matched with those firms. Differentiating this equation with respect to z yields, m (z) = N ( ) σ σ w(z) σ H (z) MX σ 1 ψ(m(z), z) σ 1 G (m(z)), for all z [z min, z max ] (2) with the two boundary conditions ϕ max = m(z max ) and ϕ min = m(z min ). This differential equation with two boundary conditions, together with equation (1), uniquely determine the matching function m(z) and the wage function w(z). 10

12 4.4 Equilibrium Definition 1. The equilibrium is characterized by a set of functions, m(z) (the matching function) and w(z) (the wage function), such that (i) Optimality: Firms maximize profits that satisfy equation (1), (ii) Labor Market Clearing: The labor markets clear as in equation (2) 4.5 The Distributional Effects of Immigration Suppose that this economy, entirely composed of natives, starts to open doors to immigrants. Because immigrants have lower education levels and communication skills compared to natives (See Fact I), we consider a case where low-skilled immigrant workers migrate to the destination country. We assume that they are integrated into the pool of workers and that they are relatively less-skilled than native workers. We can model this case as follows. First, the number of workers increases such that N > N where N denotes the number of workers after immigration, and N represents the number of workers before immigration. Second, the skill distribution of workers after immigration changes such that h (z ) h (z) h(z ) h(z) for all z z where h (z) is the skill distribution function after immigration and h(z) denotes the skill distribution function before immigration. We can decompose the effect of immigration into two channels: 1) the change in the number of workers and 2) the change in the worker skill distribution. Proposition 1. Suppose that the number of workers increases such that N > N. Then, (i) The matching function, m(z), does not change; (ii) The wage function, w(z), does not change. Proof. See Appendix In this case, the increase in the size of the workers does not affect matching patterns and wages. The impact of immigration on native workers is neutral. However, based on the Fact I, we analyze the impact of the change in the worker skill distribution as follows. 11

13 Proposition 2. Suppose that there are relatively more low-skilled workers after immigration such that h (z ) h (z) h(z ) h(z) for all z z. Then, (i) The matching function, m(z), shifts upward; (ii) The wage inequality rises such that w (z ) w (z) w(z ) w(z) for all z z. Proof. See Appendix Since immigrant workers are less-skilled than native workers, the lower part of the worker s skill distribution thickens, which induces native workers to match with more productive firms: the upward shift of the matching function m(z). Due to the log-supermodularity of the production function, more skilled workers benefit more from the increase in match quality than less skilled workers. Therefore, the wage inequality rises. 4.6 Extension We extend the impacts of immigration to a case in which the skill distribution of immigrants is more diverse than that of natives. Proposition 3. Suppose that there are more diverse workers after immigration such that h (z ) h (z) h(z ) h(z) for all z z ẑ and h (z ) h (z) h(z ) h(z) for all ẑ > z z. Then, (i) There exists a skill level z such that m (z) m(z) for all z [z min, z ], and m (z) m(z) for all z [z, z max ]; (ii) The wage inequality increases among low-skilled workers such that w (z ) w (z) w(z ) w(z) for all z > z z > z min and the wage inequality reduces among high-skilled workers such that w (z ) w (z) w(z ) w(z) for all z max > z z > z. Proof. See Appendix Because immigrant workers are concentrated in the extreme tails of the skill distribution, the impacts of immigration differ across groups. Among high-skilled natives, 12

14 immigration causes native workers to match with less productive firms. However, immigration induces native workers to be paired up with more productive firms among low-skilled native workers. The combined effects lead to the increase in wage inequality among the low-skilled group and the decrease in wage inequality among the high-skilled group. Hence, the mid-skilled native workers relatively benefit the most from the case of immigration with diverse immigrants. 5 Empirical Analysis In this section, we analyze whether our theoretical model can explain the effect of the influx of immigrants in the U.S. from 1980 through First, we give an overview of our dataset and construct key variables. Then we empirically show that the immigrant inflow increases the inequality in the U.S., consistent with our theory, and show that the worker-firm matching channel operates. 5.1 Data and Variables Data overview Our primary datasets are 5% 1980, 1990, 2000 U.S. Census and year IPUMS American Community Survey for We restrict samples to all individuals in age 18 to 64. Immigrants refer to individuals who were born in a foreign country, and natives refer to those who were born in the U.S. 1 We use commuting zones as geographical units following Autor, Dorn and Hanson (2013) and Basso and Peri (2015). An instrumental variable for the share of immigrants in each year and commuting zone is estimated by a shift-share method based on the foreign-born population in the 1% 1950 U.S. Census. Hourly wages are calculated by dividing the wage and salary income by weeks worked 1 U.S. territories are excluded. 13

15 and usual hours worked per week. Hourly wages in the lowest and the highest 1% of the distribution and those of less than one dollar are dropped. Table 1 presents descriptive statistics for natives and immigrants. The share of immigrants at the national level in our sample gradually rises from 6.45% in 1980, 9.37% in 1990, 13.83% in 2000, to 17.11% in It also shows that immigrants tend to have a lower share of female and lower share of managers. The share of female and the share of manager in 2010 are 43.44% and 6.62% among immigrants, respectively, while they are 49.31% and 9.04% among natives. The level of education shows that immigrants are less-skilled than natives. In 2010, 6.30% of natives had less than high school while 26.08% of immigrants have less than high school. In the same year, 39.38% of natives have more than a high school degree while 35.45% of immigrants have more than a high school degree Commuting zone We use commuting zones as primary geographical units following Autor, Dorn and Hanson (2013) and Basso and Peri (2015). States are too large to represent local labor markets, and counties are too small and often do not overlap with labor market regions. Metropolitan areas include regions around urban areas and have a limitation in representing rural areas. Commuting Zones are developed by Tolbert and Sizer (1996) to approximate local labor markets, and they can be consistently constructed over the full period of our analysis. The U.S. Census and American Community Survey files provide geographical units that are smaller than states State Economic Areas (1950), Country Groups (1980), and Public Use Micro Area (PUMA, 1990, 2000, ). These geographical units are mapped to commuting zones using the mapping file ( While many units fall into a single CZ, some geographical units are matched to several CZs with each probability. For example, 20% of workers in a PUMA commute to the first CZ and 80% to the second CZ. Each observation in that PUMA is now split into two CZs with probability 14

16 .2 and.8, and personal weights are also split by.2 and.8 (Autor, Dorn and Hanson, 2013) Share of immigrants The number of immigrants in each commuting zone is endogenously determined by the pull factors of each region, such as labor market conditions. To remove the endogeneity, we construct an instrumental variable for the share of immigrants in each commuting zone and year (ImmigShare g,t ) by a shift-share method based on the foreign-born population in 1950 US Census as follows: P op g,o,t = P op g,o,1950 P op US,o,t P op US,o,1950 ImmigShare IV g,t = (o = U S and other origin countries) o US P op g,o,t P (3) op g,o,t o where g denotes a commuting zone, o represents a origin country, and t is year. We group origin countries into 16 origin regions. 2 For each origin region o, immigrant population in 1950 in commuting zone g (P op g,o,1950 ) is increased by the national increase rate between 1950 and year t ( P op US,o,t P op US,o,1950 ). The stock of the native population in each commuting zone and year is estimated similarly and noted as o = US. We calculate the stock of all immigrants in each commuting zone and year by aggregating the number of immigrants over all origin regions, excluding the U.S. Lastly, the share of immigrants in commuting zone g and year t, ImmigShare IV g,t, is calculated by the stock of all immigrants ( o US immigrants and natives ( o P op g,o,t ). P op g,o,t ) divided by the sum of both We then examine the weak instrumental variables problem. The first stage regression result shows that the F statistics is which is higher than % critical value 2 Sixteen origin regions are United States, Other North America, Central America and Caribbean, South America, Northern Europe, United Kingdom and Ireland, Western Europe, Southern Europe, Central/Eastern Europe, Russian Empire, East Asia, Southeast Asia, India/Southwest Asia, Middle East/Asia Minor, Africa, and Oceania. 15

17 for one endogenous variable and one excluded instrument. Hence, we reject the null hypothesis that our shift-share instrument is weak Inequality To measure inequality within group k (k T otal, Nat, Imm), we first compute the uth percentile of log wages within commuting zone g and year t, which we denote as ln(wage) k,g,t (u). Then, we define four different measures of inequality as the difference between the higher percentile of log wage minus the lower percentile of log wages as follows: Inequality p(90) p(10) k,g,t ln(wage) k,g,t (90) ln(wage) k,g,t (10), Inequality p(75) p(25) k,g,t ln(wage) k,g,t (75) ln(wage) k,g,t (25), Inequality p(90) p(50) k,g,t ln(wage) k,g,t (90) ln(wage) k,g,t (50), Inequality p(50) p(10) k,g,t ln(wage) k,g,t (50) ln(wage) k,g,t (10). The first two measures of inequality represent overall inequality. The third measure indicates the upper-tail inequality and the last measure denotes the lower-tail inequality. Figure 4 shows the geographical variation of the share of immigrants and the inequality in the year The inequality measure is based on the difference between the 90th percentile and the 10th percentile, and we only include natives. There are sufficient variations across commuting zones for both measures. The share of immigrants and inequality measure appear to be higher in coastal areas. During the period between 1980 and 2010, we note that the national level inequality, which is based on the difference between the 90th percentile and the 10th percentile, rose from 1.54 to 1.85 for the total population. The inequality also rose from 1.53 to 1.84 among natives and from 1.57 to 1.92 among immigrants. 16

18 5.1.5 Firm productivity In the theory part, we show that immigration changes the matching mechanism such that native workers with the same skill level can pair up with more productive firms after immigration. To test this worker-firm matching channel, we need to define the unit of a firm and the measure of firm productivity. However, there is no direct way to measure the unit of firm and productivities from our dataset. We investigate this question by focusing on the proxy firms and their productivities. First, we define each commuting zone - industry pair as a proxy for a firm. Next, we set up the following variant of Mincerian type regression to compute time-invariant firm productivities. We compute commuting zone - industry fixed effects, F irm g,j, and use it to proxy for firm productivity: ln(wage) i,g,j,t = α 0 + α 1 Sex i + α 2 Age i + α 3 Age 2 i + α 4 Edu i + α 5 Immigrant i + α 6 Manager i + α 7 F irm g,j + φ t + ɛ i,g,j,t (4) In this specification, we control for individual i s characteristics such as sex, age, age squared, level of education dummies (less than high school, high school, some college, more than college), immigrant dummy, manager dummy, and year fixed effects. We can interpret commuting zone - industry fixed effects, F irm g,j, as unobserved time-invariant firm characteristics. The higher value of firm fixed effect, F irm g,j, represents that the firm has higher productivity and thus it can pay higher wages. Table 2 reports the coefficients for the Mincerian type regression. The estimation results confirm the previous findings in the standard wage regression. Figure 5 shows the distribution of estimated firm productivity, F irm g,j. There is total 24,324 number of firms in the sample. The distribution shows that firm productivity lies between and 3.48, giving us enough variation to work with. Given the time-invariant firm productivity, we are interested in estimating whether immigration re-allocates native workers to match with firms with higher productivities. 17

19 5.2 Empirical Results In the theoretical model, we predict that the inflow of immigrants increases the inequality within natives. A significant share of immigrants are low-skilled immigrants, and lowskilled immigrants tend to be even lower skilled than low-skilled natives. Thus, under the positive assortative matching, we predict that natives would be pushed up to be matched to more productive firms when immigrants arrive at the destination country. Due to the log supermodularity, better workers benefit more from the increase in the quality of matched firms and thus the inequality within natives increases. In this section, we empirically analyze the impact of immigrants in the U.S. from 1980 to The effect of immigrant inflow on inequality First, we test whether the inflow of immigrants increases the inequality in the U.S. We fit models of the following form: Inequality g,t = β 0 + β 1 ImmigShare g,t + ρ g + φ t + ɛ g,t ImmigShare g,t = γ 0 + γ 1 ImmigShare IV g,t + ρ g + φ t + ε g,t where Inequality g,t represents the inequality in commuting zone g and year t. The share of immigrants ImmigShare g,t is instrumented by the variable ImmigShare IV g,t. Unobserved commuting zone and year effects are controlled with fixed effects, ρ g and φ t. The unit of observation is commuting zone-year and each observation is weighted by sum of personal weights in each unit. Standard errors are clustered at the commuting zone level to account for error correlations over year. The first two columns of Table 3 estimate the model using all individuals, including both natives and immigrants. The coefficient of in column 1 indicates that one percentage point increase in the share of immigration is predicted to increase the relative wage gap between the 90th percentile and the 10th percentile by 0.75 percent. Column 2 18

20 shows that one percentage point increase in the share of immigration is associated with an increase in the relative wage gap between the 75th percentile and the 25th percentile by 0.79 percent. In columns 3 and 4 of Table 3, we confine our analysis to natives. The coefficients of and are higher than the case of all individuals, which provides substantial evidence that immigration affects native workers. Columns 5 and 6 of Table 3 provide the results of the case of immigrants. The coefficients of and are lower than the case of natives, which indicates that the distributional impact of immigration is stronger for natives. In Table 4, we repeat the two-stage least squares regression analysis using different measures of inequality. We define the upper-tail inequality, p(90) - p(50), as the log wage difference between the 90th percentile and the 50th percentile. We also define the lowertail inequality, p(50) - p(10), as the log wage difference between the 50th percentile and the 10th percentile. In columns 3 and 4 of Table 4 focusing on natives, both the uppertail inequality and the lower-tail inequality increase from immigration, and the impact is stronger for the lower-tail inequality. This is different from total (column 1 and 2) or immigrant (column 5 and 6) population where the increase in inequality is mostly in the upper-tail, suggesting that inflow of low-skilled immigrants affects low-skilled natives more than the high-skilled natives The effect of immigrant inflow on matching Using the estimated firm productivities, we can assign values to each native worker as follows: Matching i,g,j,t F irm g,j where Matching i,g,j,t refers to a firm productivity level that native worker i who works at the firm (g, j) in year t. In each commuting zone - year, we further compute the uth quantile value of firm productivities among native workers and denote this as Matching g,t (u). We expect that low-skilled immigration shifts the matching function upward, which 19

21 pushes natives to work in more productive firms after immigration because natives are pushed up by immigrants. To test this hypothesis, we specify the following regression form: Matching g,t (u) = δ 0 + δ 1 ImmigShare g,t + ρ g + φ t + ɛ g,t ImmigShare g,t = η 0 + η 1 ImmigShare IV g,t + ρ g + φ t + ε g,t where Matching g,t (u) represents the uth quantile value of firm productivities among native workers in commuting zone g and year t. The share of immigrants ImmigShare g,t is instrumented by the variable ImmigShare IV g,t. Unobserved commuting zone and year effects are controlled with fixed effects, ρ g and φ t. The unit of observation is commuting zone-year and each observation is weighted by sum of personal weights in each unit. Standard errors are clustered at the commuting zone level to account for error correlations over year. Table 5 confirms that the increase in the share of immigrants re-allocates native workers to match with higher productive firms because the signs of coefficients are positive and significant in most cases. This result validates our central prediction such that the matching function shifts upward after low-skilled immigration. Quantitatively, one percentage point increase in the share of immigrants is predicted to increase the productivity of median native worker s matching firm by percent. During the period between 1980 and 2010, the share of immigrants at the national level increased by percentage points. Through the matching channel, low-skilled immigration bid up wages of median native workers by 1.78 percent The effect of immigrant inflow on wage We now examine whether the increase in match quality of natives leads to differential effects on wages of heterogeneous native workers. We expect that natives in the upper 20

22 tail of wage distribution gain more from immigrant inflow compared to natives in the lower tail of wage distribution since better workers benefit more from increased match quality due to the complementarity. To test this hypothesis, we specify the following regression form: W age g,t (u) = δ 0 + δ 1 ImmigShare g,t + ρ g + φ t + ɛ g,t ImmigShare g,t = η 0 + η 1 ImmigShare IV g,t + ρ g + φ t + ε g,t where W age g,t (u) represents the uth quantile value of wages among native workers in commuting zone g and year t. The share of immigrants ImmigShare g,t is instrumented by the variable ImmigShare IV g,t. Unobserved commuting zone and year effects are controlled with fixed effects, ρ g and φ t. The unit of observation is commuting zone-year and each observation is weighted by sum of personal weights in each unit. Standard errors are clustered at the commuting zone level to account for error correlations over year. Figure 6 provides clear evidence that low-skilled immigration generates inequality. The effects of immigration on wages of native workers are positive and significant for the upper part of wage distribution while it is harmful and insignificant to the lower part of the wage distribution. Figure 6 shows that one percentage point increase in the share of immigrants raises the log wage of the median native worker by 0.36 percent. Because the share of immigrants at the national level increased by percentage points during the period 1980 and 2010, low-skilled immigration increases wages of median native workers by 3.84 percent. Because low-skilled immigration through the matching channel increases wages of median native workers by 1.78 percent, we quantify that the matching channel consists of 46.4 percent of the increase in median native worker s wage from immigration. 21

23 6 Conclusion This paper introduces a novel channel - a matching between workers and firms - to the literature on the effect of immigrants on wages and inequality of natives in the destination country. Based on the matching framework adopted from the assignment framework in trade literature and observations in the U.S. labor market, we construct a theoretical model where the influx of immigrants affects heterogeneous native workers through the worker-firm re-matching channel. Low-skilled immigrants push natives up to be matched with better firms, and due to the complementarity between worker skill and firm productivity, the re-matching benefits better workers more and leads to an increase in inequality. Data show that immigrants in the U.S. tend to have lower skills than native workers. Using the U.S. data from 1980 to 2010 and exploiting the variation in immigrant share across commuting zones and years, we investigate whether our theoretical model can explain the change in the inequality of natives from the inflow of low-skilled immigration. First, we have shown that immigration increases inequality in the U.S. Next, we test whether the inflow of immigrants changes the matching and wages of native workers. Consistent with our theory, natives are matched to more productive firms due to the inflow of immigrants, and better native workers benefit more than other natives, which explains why immigration leads to widening inequality in the U.S. 22

24 7 Appendix 7.1 Proofs Proof of Proposition 1 Proof. Prove first that the matching function, m(z), and do not change. Rearranging equation (2) yields, ( ) σ ln w(z) = ln + σ 1 ln ψ(m(z), z) + 1 σ 1 σ σ ln X 1 σ ln NH (z) MG (m(z))m (z), (5) By differentiating both equations with respect to z, we obtain the following second-order differential equation for the matching function m(z): m (z) m (z) = (σ 1)ψ ϕ(m(z), z) ψ(m(z), z) σ ψ z(m(z), z) ψ(m(z), z) + G (m(z))m (z) H (z) G (m(z)) H (z), (6) In equation (6), the matching function does not depend on the number of workers N. This implies that the matching function, m(z), does not change. In equation (5), one percent increase in N is associated with 1 σ percent decrease in wage function w(z) z. Also, one percent increase in X is associated with 1 σ percent increase in wage function w(z) z. One percent increase in N is associated with one percent increase in X. Hence, the wage function, w(z), does not change Proof of Proposition 2 Proof. (i) Suppose that there exists z [z min, z max ] such that m h (z ) (z) < m(z). h (z) h(z ) and the positive assortative matching property of the matching function imply that h(z) m(z min ) = ϕ min m (z min ) and m (z max ) = ϕ max m(z max ). So there must exist z min 23

25 z 1 z 2 z max and ϕ min ϕ 1 ϕ 2 ϕ max such that i) m(z 1 ) = m (z 1 ) = ϕ 1 and m(z 2 ) = m (z 2 ) = ϕ 2, ii) m z (z 1 ) m z(z 1 ) and m z(z 2 ) m z (z 2 ), iii) m(z) > m (z) for all z (z 1, z 2 ). m z (z 1 ) m z(z 1 ) and m z(z 2 ) m z (z 2 ) implies that: m z (z 1 ) m z (z 2 ) m z(z 1 ) m z(z 2 ). Using equation (2), we can derive the following inequality: [ ] w(z 1 σ ) h(z 1 ) w(z 2 ) h(z 2 ) h (z 2 ) h (z 1 ) h(z2 ) h(z 1 ) requires that: w (z 2 ) w (z 1 ) w(z2 ) w(z 1 ). [ ] w (z 1 σ ) h (z 1 ) w (z 2 ) h (z 2 ). However, this is a contradiction. Since m (z) < m(z), it must be that w (z 2 ) w (z 1 ) < w(z2 ) w(z 1 ). Consequently, if h (z ) h (z) h(z ) h(z), then m (z) m(z) for all z [z min, z max ]. (ii) From equation (1), z z ψ z (ϕ, z) z ψ(ϕ, z) dz = w (z) z w(z) dz. The right-hand side is the measure of wage inequality, ln w(z ) ln w(z). Since the productivity function ψ(ϕ, z) is strictly log supermodular, m (z) m(z) for all z [z min, z max ] 24

26 implies that the left-hand side is weakly increasing for all z [z min, z max ]. Consequently, the wage inequality rises Proof of Proposition 3 Proof. (i) Suppose that there does not exist z [z min, z max ] such that m (z) m(z) for all z [z min, z ] and m (z) m(z) for all z [z, z max ]. Since h (z ) h (z) h(z ) for all h(z) z z ẑ and h (z ) h (z) h(z ) h(z) for all ẑ > z z, the positive assortative matching property of the matching function implies that m(z min ) = ϕ min m (z min ) and m(z max ) = ϕ max m (z max ). Hence there must exist z min z 0 < z 1 < z 2 z max and ϕ min ϕ 0 < ϕ 1 < ϕ 2 ϕ max such that i) m (z 0 ) = m(z 0 ) = ϕ 0, m (z 1 ) = m(z 1 ) = ϕ 1 and m N (z 2 L) = m S (z 2 L) = ϕ 2, ii) m z(z 0 ) m z (z 0 ), m z(z 1 ) m z (z 1 ) and m N (z 2 L) m S (z 2 L), iii) m (z) < m(z) for all z (z 0, z 1 ) and m (z) > m(z) for all z (z 1, z 2 ). There are two possible cases: z 1 < ẑ and z 1 ẑ. Suppose that z 1 < ẑ. m z(z 0 ) m z (z 0 ) and m z(z 1 ) m z (z 1 ) implies that: m z(z 1 ) m z(z 0 ) m z(z 1 ) m z (z 0 ). Using equation (2), we can derive the following inequality: [ ] w(z 0 σ ) h(z 0 ) w(z 1 ) h(z 1 ) h (z 1 ) h (z 0 ) h(z1 ) h(z 0 ) requires that: w (z 1 ) w (z 0 ) w(z1 ) w(z 0 ). [ ] w (z 0 σ ) h (z 0 ) w (z 1 ) h (z 1 ). 25

27 However, this is a contradiction. Since m (z) < m(z) for all z (z 0, z 1 ), it must be that w (z 1 ) w (z 0 ) < w(z1 ) w(z 0 ). Suppose that z 1 ẑ. m z(z 2 ) m z (z 2 ) and m z(z 1 ) m z (z 1 ) implies that: m z(z 1 ) m z(z 2 ) m z(z 1 ) m z (z 2 ). Using equation (2), we can derive the following inequality: [ ] w(z 2 σ ) h(z 2 ) w(z 1 ) h(z 1 ) h (z 1 ) h (z 2 ) h(z1 ) h(z 2 ) requires that: w (z 1 ) w (z 2 ) w(z1 ) w(z 2 ). [ ] w (z 2 σ ) h (z 2 ) w (z 1 ) h (z 1 ). However, this is a contradiction. Since m (z) > m(z) for all z (z 1, z 2 ), it must be that w (z 1 ) w (z 2 ) < w(z1 ) w(z 2 ). Consequently, there exists a skill level z such that m (z) m(z) for all z [z min, z ], and m (z) m(z) for all z [z, z max ]. (ii) The proof is identical to that of Proposition 2 - (ii). 26

28 7.2 Figures and Tables Figure 1: Education distributions of natives and immigrants Notes: The figure plots cumulative distribution functions for natives and immigrants by years of education. The blue line denotes natives who were born in the U.S. in age 18 to 64 while the red short-dashed line represents immigrants who were born in a foreign country in age 18 to 64. The source of data is the US Census

29 Figure 2: Positive assortative matching between workers and firms Notes: The figure shows the relation between firm s productivity and average education level of workers (top) and the relation between firm s productivity and average imputed log hourly wages of workers (bottom) in each commuting zone-industry pair. Firm s productivity is estimated as in section and wage is imputed with observable characteristics of each worker. Each circle indicates the commuting zone-industry pair, and the size of the circle and fitted line is weighted by the total number of workers. The source of data is the US Census

30 Figure 3: Log wage function Notes: The figure shows log wage functions for workers by schooling group and year. We regress log wage on education and square of education in each year as follows: W i = β 0 + β 1 S i + β 2 S 2 i + ɛ i. Then, we plot the fitted values from the regression. The sources of data are 1980, 1990, 2000 U.S. Census and year IPUMS American Community Survey for

31 Figure 4: The share of immigrants and inequality across commuting zones ( , ] ( , ] ( , ] [ , ] No data ( , ] ( , ] ( , ] [ , ] Notes: The data are taken from 2010 US Census. The upper panel shows geographical variation of the instrumental variable for the share of immigrants in year 2010, ImmigShare IV g,2010. The lower panel depicts the geographical variation of the inequality measure defined as the difference between the log hourly wages of the 90th percentile and the log hourly wages of the 10th percentile, Inequality p(90) p(10) g,2010. The inequality measure only includes natives. 30

32 Figure 5: The distribution of estimated firm productivity Notes: Data are from 1980, 1990, 2000 U.S. Census and year IPUMS American Community Survey for N = 24, 324. The figure shows estimated firm fixed effects. We regress the following form: ln(w age) i,g,j,t = α 0 + α 1 Sex i + α 2 Age i + α 3 Age 2 i + α 4Edu i + α 5 Immigrant i + α 6 Manager i + α 7 F irm g,j + φ t + ɛ i,g,j,t. Then, we plot the distribution of the estimated values of F irm g,j. 31

33 Figure 6: Effects of immigrants on natives wages Notes: Data are taken from 1980, 1990, 2000 US Census and IPUMS ACS Each circle reports estimated coefficients from a separate regression. We report the impact of immigration on uth quantile value of wages among native workers. The unit of observation is commuting zone by year. Each observation is weighted by person. Clustered robust standard errors are reported. 32

34 Table 1: Summary Statistics Year National Immigrant Share (%) Panel A: Native Characteristics Female Share (%) Manager Share (%) Age Log Hourly Wage Education Level None Primary Secondary High school Some college College Master/Professional Ph.D Panel B: Immigrant Characteristics Female Share (%) Manager Share (%) Age Log Hourly Wage Education Level None Primary Secondary High school Some college College Master/Professional Ph.D Notes: Data are from 1980, 1990, 2000 US Census and IPUMS ACS Immigrants refer to stock of foreign born between age 18 and 64, and natives refer to stock of population between age 18 and 64 who were born in the U.S. Unit of observation is individual and summary statistics are weighted by personal weights. 33