The Impact of Immigration on Wages, Internal Migration and Welfare

Transcription

1 The Impact of Immigration on Wages, Internal Migration and Welfare Suphanit Piyapromdee January, Abstract Over the past few decades, the number of immigrants entering the U.S. has increased substantially. The local impacts of immigration may differ from national impacts since some cities attract more immigrants. Even within a city, workers may be affected differently depending on the substitutability of their labor with that of the new arrivals as well as their abilities to move. This paper studies the impact of immigration on wages, internal migration and welfare. I develop and estimate an equilibrium model where labor differs by skill level, gender, experience and nativity. Workers are also heterogeneous in city preferences and place attachments. Cities vary in productivity levels, housing prices and local amenities. The results indicate that a 30 percent increase in the stock of immigrants has a small impact on the wages and welfare of natives. If workers are constrained to remain in their original locations, the initial wage impacts on previous immigrants are negative and much more severe in the popular destinations for new immigrants. When workers migrate in response to the immigration, the negative wage and welfare impacts in most locations are diffused. However, the negative impacts on the wages of low skill workers in some locations intensify. This is because low skill workers have stronger attachments to places, and hence are less mobile relative to high skill workers. The extent to which the migration responses reduce the adverse wage impacts depends on a city s labor composition. The model is also used to assess changes in the skill mix of immigrants and a location-specific immigration policy. The most recent version is available at Department of Economics, UW-Madison. piyapromdee@wisc.edu. I am very grateful to John Kennan, Rasmus Lentz and Chris Taber for their guidance. I thank Andrew Anderson, Mike Anderson, Mikael Andersen, Jesse Gregory, Erik Hembre, Kevin Hutchinson, Lindsay Jacobs and Richard Rogerson as well as seminar participants at UW-Madison for helpful comments. 1

2 1. Introduction Policy makers and economists have long tried to quantify the impact of immigration on natives. The large inflow of immigrants into the U.S. in the past few decades has raised concerns partly because of its composition. At least a third of the new immigrants are undocumented with little education and limited English skills Passel, 2005). Amongst the U.S. working-age population, the share of immigrants increased from 10 percent in 1990 to about 18 percent in At the city level, the immigrant share of the working-age population varies from more than 40 percent, for example in Los Angeles and, to roughly two percent in cities such as Flint, MI. The local impacts of immigration may differ from national impacts since some cities attract more immigrants. An inflow of immigrants may reduce wages for competing workers, but raise wages for complementary labor. Further, an inflow of new arrivals may change additional economic conditions such as housing prices as well as induce internal migration. 1 Thus even within a city, workers can be affected differently due to heterogeneity in labor types and degrees of mobility. In this paper, I study the wage and welfare implications of immigration as well as the migration responses in an equilibrium model. Accounting for the geographic sorting of workers is important because internal migration can change the distribution of the impacts. The estimated model is used to assess changes in the stock and skill mix of immigrants as well as a location-specific immigration policy. I quantify the effects of immigration on different groups of workers both when workers are constrained to remain in their original locations and when all workers are free to migrate. Overall, I find that the arrival of new immigrants has little impact on natives, but has substantial negative impacts on the wages and welfare of previous immigrants. Further, when workers migrate in response to the immigration, the negative wage impacts may be diffused or intensified. An outmigration of workers of a given type raises the local wages for workers of that type, while reducing the local wages of workers with complementary characteristics. However, the migration responses reduce the negative impacts in most locations. To illustrate these mechanisms, I specify and estimate a spatial equilibrium model with a nested CES technology. The basic setup is built on Roback 1982), Diamond 2013) and Ottaviano and Peri 2012). Since the impacts of immigration are likely to depend on the substitutability of labor, I modify the model to incorporate heterogeneity across workers in skill level, gender, age and birthplaces. I allow for the possibility that workers of different types are imperfect substitutes. Cities differ in productivity levels, housing prices and amenities. Workers maximize utility by choosing the most desirable city in a static discrete choice setting. Preferences for city characteristics are heterogeneous across types of workers. The value of a city among natives of the same type varies depending on the distance from the city to the individual s birthplace. For immigrants, the value of a city depends on their networks in that city. Since new immigrants tend to locate in the same cities as their fellow expatriates, I include a role for network effects by using the number of 1 Saiz 2007) estimates that a one percent increase in a city s population due to immigration is associated with approximately one percent increases in average rents and housing values. 2

3 previous immigrants born in the same country to represent the strength of the network. The model is estimated using U.S. Census data from 1980 through 2000 and the combined American Community Survey. Local labor demand is identified using the predicted inflow rate of immigrants based on historical settlement patterns Card, 2009). Given the enclave tendencies of immigrants, the number of new arrivals is fairly predictable. Additionally, I use the age structure of cities as an alternative instrumental variable to identify the substitutability between high and low skill labor Moretti, 2004). Labor supply is estimated using the discrete choice methods developed by McFadden 1973); Berry et al. 1995, 2004) which have been applied to estimate workers preferences for locations by Bayer et al. 2007) and Diamond 2013). 2 I adapt Diamond 2013) s approach and proceed in two steps. In the first step, I use maximum likelihood to estimate the mean utility levels from living in each city for each type of worker in each decade, controlling for network values. In the second step, I decompose the mean utility estimates into values for each city characteristic. Workers preferences are identified using local labor demand shocks, and the interactions of these shocks with housing supply elasticity determinants. I measure these shocks by interacting cross-sectional differences in industrial employment composition with national changes in industrial employment hours separately for workers of different skill levels, genders and ages. The housing supply elasticity is identified using the same instruments as for worker s preferences. Labor demand shocks induce workers to migrate to arbitrage increased wages, driving up rents. Cities with inelastic housing supplies would exhibit larger rent increases leading to relatively less in-migration. Hence, the interaction of these shocks with housing supply elasticity determinants provide variation in housing demand. The estimates indicate high, but imperfect, substitutability between natives and immigrants within the same skill-age-gender group. The substitutability between natives and immigrants is higher amongst low skill than high skill workers; this could be because factors such as the quality of education and English skills are less important for low skill labor. All workers prefer cities with higher wages. High skill workers are more sensitive to changes in prices, relative to low skill workers. Using the estimated model, I consider four counterfactual experiments. The first involves a 30 percent increase in immigrants, holding the skill mix fixed. The second examines the wage and welfare impacts of an increase in immigrants if the U.S. were to adopt a skill-selective immigration policy. The third experiment analyzes an influx of low skill immigrants. The final experiment is motivated by the Support Our Law Enforcement and Safe Neighborhoods Act or SB1070 in Arizona which includes a range of provisions intended to control unauthorized immigration. In this case, I consider a counterfactual location-specific policy that removes fifty percent of low skill immigrants in five states that have passed anti-illegal immigration laws similar to SB1070. In the first experiment, I find that a 30 percent increase in the stock of immigrants has little impact on natives wages and welfare. If workers are constrained to remain in their original loca- 2 Similar methods have been used by Mangum 2010) to study the dynamics of migration. 3

4 tions, the initial impacts on previous immigrants are negative and much more severe in the popular destinations for new immigrants. When all workers are free to migrate, the negative impacts in some cities, such as and Los Angeles, attenuate. However, in some locations, such as Abilene, TX the negative impacts on the wages of low skill workers intensify. The differential impacts are due to heterogeneity in workers attachments to places. More workers, in cities such as, already left their birthplaces, and hence are more mobile to migrate in response to the immigration. The migration responses, however, reduce the negative impacts on the welfare of workers in most cities. In the second experiment, I consider an influx of high skill immigrants. In this case the wages of low skill workers rise due to the complementarity of high and low skill labor. However, the wages of high skill workers fall and the negative impact is more severe for the incumbent high skill immigrants. The differential wage impacts between immigrants and natives are due to their imperfect substitutability. When all workers are free to migrate, the negative impacts on the wages and welfare of high skill workers attenuate. Overall, a skill selective immigration policy leads to welfare gains for natives. In contrast, the influx of low skill immigrants in the third experiment reduces in the wages and welfare of incumbent low skill workers, but raises the wages and welfare of high skill workers. The average net loss, however, is small. The last experiment shows that location-specific immigration control policy has a local and short-term positive impact on the wages and welfare of low skill workers but negative impacts on high skill labor. As workers reallocate, the impacts of removing low skill immigrants become negligible. Early studies on the effects of immigration provide mixed conclusions. Borjas et al. 1996) and Borjas 2003) claim that immigration has a pronounced negative effect on natives wages while Card 1990, 2001, 2005) finds little impact. More recently, Card 2009) argues that the discrepancy between these findings is reconciled by recognizing the high-degree of substitutability between high school graduates and dropouts as well as the imperfect substitutability between natives and immigrants. 3 Ottaviano and Peri 2008, 2012) estimate a nested-ces model using data at the national level; their models do not differentiate between male and female labor in the production function. They find that natives and immigrants exhibit particularly low substitutability among less educated workers. Furthermore, immigration has a small impact on natives wages, but a substantial negative impact on the wages of previous immigrants. 4 My findings are similar to Ottaviano and Peri 2012, 2008) in that the impacts of immigration are more concentrated on previous immigrants. However, in contrast to Ottaviano and Peri 2012), I find that native and 3 The Card 2009) argument is that the wage impact of low skill immigration is diffused across the 60 percent of the population who have no college education rather than concentrated on the 14 percent of the population who are high school dropouts. Further, the imperfect substitutability between immigrants and natives implies that the effects of immigration are more concentrated on immigrants themselves. 4 Using a similar approach to Ottaviano and Peri 2012), Manacorda et al. 2012) study the impacts of immigration in the UK. They find that natives and immigrants are imperfect substitutes and the effect of increased immigration is concentrated on the wages of incumbent immigrants. 4

5 immigrant labor is more substitutable amongst low skill workers. My interpretation of this result is that differences in the quality of education and English skills are less important for low skill labor. A similar result is obtained by Card 2009) using city-level estimation. Concerning natives migration responses, Borjas 2006) investigates the link between native internal migration and the wage impact of immigration in local labor markets. 5 In his model, previous immigrants are assumed to remain in their original locations and the native migration flows only depend on wages. He shows that the native migration responses can substantially reduce the negative wage impacts of immigration. In contrast to Borjas 2006), I demonstrate that the migration responses do not always reduce the wage impacts. The extent to which the migration responses reduce the adverse local impacts of immigration depends on the substitutability between different types of labor as well as the local labor composition. Kennan 2012, 2013) analyzes the wage impacts of immigration in a multi-sector economy. Under the assumptions that product markets are competitive and firms produce a sufficient number of common products as well as share the same technology, then wages in efficiency units are equalized. 6 With a constant returns technology, immigration has no impact on wages in the long run. My paper assumes a single tradeable good sector with a constant returns production function. 7 When different types of workers are imperfect substitutes, immigration impacts wages, even in the long run, by changing the composition of the labor force. However, migration responses partially re-balance these changes which reduces the overall impacts even without arbitrage in product markets. My model is also related to work studying location choices. Moretti 2013); Notowidigdo 2011); Diamond 2013), among others, generalize the Roback 1982) model to analyze geographic locations of different skill groups. Diamond 2013) extends Roback 1982) to include heterogeneity in workers preferences for cities. She focuses on welfare inequality between college and noncollege workers. In her model, labor is only differentiated by skill levels: high and low; immigrants and natives are perfect substitutes. The main contribution of this paper is to quantify the wage and welfare impacts of immigration in an equilibrium model, taking into account the migration responses of workers. The framework sheds light on immigration policies such as skill-selective entry and location-specific immigration control, the focus of current political debate. The rest of the paper is organized as follows: Section 2. presents an overview of the data. Sec- 5 Studies on native migration responses to immigration provide mixed conclusions. Frey 1995) finds a strong correlation between out-migration rates of natives and inflow rates of immigrants, while Card 2001); Card and DiNardo 2000) argue that migration flows of natives and older immigrants are not sensitive to inflows of new immigrants. See Card and DiNardo 2000) for a review of these studies. 6 For example, if there are three factors of production and at least three goods are commonly produced in all locations, then the factor prices would be the same. 7 Basu and Fernald 1997) find that production functions in most industries exhibit roughly constant returns to scale. Further, I assume a single tradeable good in order to derive simple expressions for local labor demand. Other papers assuming single sector models include Card 2009); Diamond 2013); Ottaviano and Peri 2012). 5

6 tion 3. specifies the model and Section 4. describes the estimation procedures. Section 5. presents the baseline results. Section 6. shows counterfactual experiments and Section 7. concludes. 2. Data Overview 2.1 Sample Description The analysis is based on data from the five percent samples of the 1980, 1990 and 2000 U.S. Census as well as the combined American Community Surveys ACS) from the Integrated Public Use Microdata Series IPUMS) Ruggles et al., 2010). Throughout the analysis, I refer to the combined ACS as the 2007 sample period. The key characteristics of workers are skill level, potential experience, gender and nativity. I define cities as the metropolitan statistical areas MSA s) from the 2000 Census. I use information on definitions of MSAs at the detailed level to match the 2000 MSAs to 1980 and The ACS uses the same geographic coding as the 2000 Census. The Census includes 218 MSAs consistently across the three rounds. I focus on the 148 MSAs which have at least 100 full-time and non self-employed of each type of workers based on the key characteristics described above. All other areas are combined together and treated as the outside option. Workers in the sample are restricted to individuals over the age of 18 with 1 to 45 years of potential experience who report positive earnings and worked at least one week in the previous year. Labor supply is a count of employed people multiplied by the individual person weight. Immigrants are defined as individuals born abroad. Years of potential experience are calculated using the difference between current age and the age at which the individual entered the labor force. I assume that high school dropouts enter the labor force at age 17, high school graduates enter at 19, people with some college education enter at 21, and people with at least a college degree enter at 23. Young workers are defined as workers with 1-20 years of potential experience, and old defined as workers with years of experience. 8 High skill workers are defined as as those with 1-3 years of college or more. Low skill workers include high school graduates and dropouts. This classification of two skill groups is supported by Ottaviano and Peri 2008) who estimate the inverse elasticity of substitution between dropouts and high school graduates to be between zero and 0.04, and the inverse elasticity of substitution between workers with some college, and those with a college degree or more to be small. Similarly, Card 2009) shows that the inverse elasticity of substitution between high school and dropouts is approximately zero. The wage sample is a subset of the employment sample where workers who are self-employed and workers who work less than 35 hours a week and 40 weeks per year are eliminated. Cities wages are measured by the hour-weighted average wage for each type of workers in each city. The price of national goods is set at the CPI-U index of all goods measured in 2011 dollars. City rents 8 In section 5.5), I consider measuring experience of immigrants using experience in the U.S. only. 6

7 are measured as the average gross monthly rent which includes both the housing rent and the cost of utilities) per household member. For households owning houses, I calculate imputed rents from housing values using a discount rate from Peiser and Smith 1985) where monthly expenditures for utilities are added to obtain gross imputed rent. Additional data on land unavailability are taken from Saiz 2010). The main estimation of local labor demand, labor supply and housing supply use prices and employment information from the 1990, 2000 U.S. Census and the combined ACS. The 1980 U.S. Census is only used for constructing instrumental variables and network effects. See the Online Appendix Section A. for further details on variable construction and summary statistics. 2.2 Characteristics of Immigrants Table 1) reports the numbers and characteristics of immigrants from The share of immigrants in the U.S. working-age population increased from 10 percent in 1990 to 17.7 percent in More than half of immigrants have only high school diplomas or less, are male and aged 40 years and older. 9 The ratio of immigrants to natives among high school dropouts in the U.S. i.e. the number of high school dropouts who are immigrants divided by the number of high school dropout natives) rose from 0.30 to 0.81 from 1990 to The rapid rise in this ratio is due to a decline in the number of high school dropout natives; although the rate of immigrants who were high school dropouts also fell over the same period, the proportion remained high nearly 25 percent in Among those with college degrees, this ratio rose from 0.12 to 0.21, roughly corresponding with the overall increase in the share of the working age population who are immigrants. The ratio of immigrants to natives among all male workers rose from 0.12 to A similar increase is found among workers aged 40 years or older. Tables 2) and 3) report the numbers of immigrants and educational attainment by country of origin, respectively. Nearly 40 percent of immigrants are from Mexico and Central America, with 70 to 80 percent having at most a high school education. Large fractions of Immigrants from Europe, India, Japan and China, on the other hand, have at least college degrees. Finally, the lower panel of Table 3) reports the aggregate native immigrant wage differential. The estimates are the wage of natives, relative to the base estimates for immigrants from a regression of log hourly wages. Controls include skill level, rural, race dummies, age, age squared, and years spent in the U.S. interacted with skill level. As documented in Card 2005), a wage gap emerged between natives and immigrants over the 1970s and has persisted. The wage differential between natives and immigrants rises from 17 log points in 1990 to 26 log points in For high skill workers, which include individuals with some college education or more, this differential is smaller and has fallen across time. The wage gap could reflect differences in the quality of native 9 The descriptive statistics presented in this paper come from the U.S. Census which may over-represent legal immigrants. Passel 2005) estimates that at least a third of the new arrivals are undocumented from Mexico and Central America with little education. 7

8 and immigrant labor. Bratsberg and Terrell 2002) note that the lower quality of education in immigrant sending countries and limited English skills imply lower initial human capital. Over time, immigrants catch up to natives through training and acquisition of language skills. Using data from the U.S. Census, I find that immigrants with at least college degrees who have been living in the U.S. for at least ten years earn more than the average native. This paper abstracts from earnings assimilation of immigrants. To estimate a model with wage assimilation, panel data is required. 10. Most panel immigration data, such as the New Immigrant Survey is not large enough to precisely estimate my model. Nonetheless, given that the average wage gap between immigrants and natives remains sizable, the current framework provides a reasonable representation of average workers. 2.3 Immigrant Settlement Patterns The motivation for the city-level analysis in this paper is illustrated in Table 4), which presents immigrant densities in the 15 most popular destinations of immigrants. The immigrant shares of the working-age population in these cities range from about 30 to 60 percent. Further, cities that attract more immigrants in 1990 continue to attract more immigrants over time. A well-known immigrant settlement pattern is that they tend to locate in country-specific enclaves. Prominent examples include the concentration of Arab immigrants in Detroit Abraham, 2000), Polish immigrants in Chicago Pacyga, 1991), and Mexican immigrants in Los Angeles and Chicago. Figure 1) illustrates the enclave pattern for Filipino and Polish immigrants in the 186 MSAs. The x-axis of the top bottom) panel represents the ratio between the share of immigrants from Philippines Poland) who lived in a given city in 1980 and the share of all immigrants who lived in that city in The y-axis represents the same relative share of the Filipino Polish) immigrants observed in the 2000 census who arrived after These figures imply that cities where Filipino immigrants were concentrated in 1980 continued to attract more Filipino immigrants in the following decades. Card 2009) notes that similar correlation holds for immigrants from most other source countries. 3. Model The present framework is built on Roback 1982), Diamond 2013) and Ottaviano and Peri 2012). To analyze the effects of immigration on different types of workers, I extend Roback 1982) by introducing additional heterogeneity across workers via a nested-ces technology similar to Ottaviano and Peri 2012). The model incorporates heterogeneity across workers in skill level, gender, age and nativity. I allow for the possibility that workers of different types are imperfect substitutes. Cities differ in productivity levels, housing prices and amenities. Cities productivity levels vary across types of workers. Preferences for city characteristics may vary across worker types. 10 See Lessem and Sanders 2012) for example 8

9 The value of a city among natives of the same type depends on the distance between that location and their birthplaces. To account for immigrants tendency to locate in the same regions as their fellow expatriates, I allow them to derive utility from cities networks. I use the number of previous immigrants born in the same country group in the past to represent the strength of the network. As in Diamond 2013), housing supplies differ across cities due to differences in geographic constraints. The amount of developable land within a city influences housing markets by restricting new construction Saiz, 2010). Cities with less developable land thus have relatively inelastic housing supplies. I begin this section by specifying labor demand, then go on to discuss housing supply, city amenity and network, workers location decisions, and finally present the equilibrium conditions. 3.1 Labor Demand To derive simple expressions for city-specific labor demand, I assume a one-sector economy. 11 While I do not explicitly incorporate multiple sectors into the model, I allow cities production functions to differ in productivity to reflect differences in cities sectoral compositions. Firms are competitive and produce identical tradeable goods using capital and labor with a constant returns technology. 12 Each city c has many homogeneous firms in year t. In what follows, I drop the firm s subscript for ease of exposition. The firm s production function takes the following form Y ct = A ct L α ctk 1 α ct 1) where A ct is city-specific productivity, K ct is capital, L ct is a CES aggregate of different types of labor, and α 0,1) is the income share of labor. 13 An immigrant is defined as a person born outside the U.S. Since workers are heterogeneous, the effects of immigration depends on the substitutability between labor types and the magnitude of the inflow. Intuitively, immigrants may put downward pressure on the wages of substitute labor and upward pressure on the wages of complements. The model incorporates imperfect substitution among labor inputs via a nested CES production function similar to Ottaviano and Peri 2012); the main difference between my setup and Ottaviano and Peri 2012) is that I allow male and female labor to be imperfect substitutes. As reported in Johnson and Keane 2013), unconditional on occupations, the substitutability between men and women is low. The CES nests are ordered by skill, gender, age and nativity see Figure 2) for a graphical representation). I place skill levels in the upper nests since education is the primary determinant of labor substitutability. Gender and 11 A single tradeable good sector allows simple local labor demand functions which are convenient for estimation purposes. Other papers using single sector models include Card 2009); Diamond 2013); Ottaviano and Peri 2012). 12 As estimated by Basu and Fernald 1997), production functions in most industries exhibit roughly constant returns to scale. Using plant-level data, Baily et al. 1992) find that firms produce with approximately constant returns technology. 13 The functional function in 1) is supported by the empirical observation that the share of labor income is constant in the long run Kaldor, 1961; Gollin, 2002); see Ottaviano and Peri 2012) for more discussion. 9

10 age seem to be the second most important determinants. I put gender at the second level and then age as in Johnson and Keane 2013). Finally, I place workers immigration status in the last level. The first-level nest of labor aggregate is a combination of high and low skill labor according to L ct = θ et L ρ E ect e ) 1 ρe, 2) where the skill levels are high and low e {H,L}, and σ E = 1 ρ 1 E is the elasticity of substitution between skill levels. 14 The parameters θ Ht,θ Lt represent the relative productivity levels of high and low skill labor, respectively. These may vary over time due to skill-biased technical change. Further, θ Ht + θ Lt = 1 and similarly I standardize the relative productivity levels in the lower CES levels to sum to one; any common multiplying factor is absorbed in A ct. At the next level, the skill-specific labor L cet is a CES aggregate of male and female labor ) 1 L ect = φ egt L ρ ρ G G egct, 3) g where g {F,M} denotes female and male respectively, φ Fet + φ Met = 1, and σ G = 1 ρ 1 G is the elasticity of substitution between genders. 15 The parameters φ Fet,φ Met vary by skill level and over time. Under this setup with a single national product, the imperfect substitutability between men and women pertains to occupational differences across genders. Johnson and Keane 2013) estimate that conditional on education and occupation, men and women are close substitutes. However, the unconditional substitutability between genders is relatively low. Next, the gender-skill-specific labor L egct is a combination of a set of age-specific labor inputs according to the following L egct = δ aet L ρ A egact a ) 1 ρa, 4) where age a {Y,O} denotes young and old respectively, δ Yet + δ Oet = 1, and σ A = 1 ρ 1 A is the elasticity of substitution between young and old workers. 16 Eq 4) allows the relative productivity δ aet to vary across skill levels and time This specification is commonly used in labor economics see for example Katz and Murphy 1992); Card and Lemieux 2001)). The alternative is to have four skill groups: college, some college, high school and dropouts Borjas, 2003 and Borjas and Katz, 2007). However, as noted in Card 2009), the inverse elasticities of substitution between college and some college, as well as between high school graduates and dropouts are approximately zero. 15 In Section 5.5), I consider a CES specification where the relative productivity levels between men and women are fixed over time. 16 I group workers into two broad experience categories since Ottaviano and Peri 2012) find that the elasticity across the broad experience groups is not very different from the elasticity across narrow experience groups. 17 An alternative specification where the relative productivity levels between young and old workers are fixed over time is considered in Section 5.5). 10

11 Finally, L egact is a combination of labor supplied by natives, N egact and immigrants, M egact. I allow the elasticity of substitution between natives and immigrants to vary across skill levels as follows L egact = βegacts s ρ M,E egact s ) 1 ρ M,E, 5) where s {M,N} denotes immigrant and native, respectively, βegact N + βegact M = 1, and σ M,E = 1 1 ρ M,E denotes the elasticity of substitution between natives and immigrants in each skill level. 18 The relative productivity levels between natives and immigrants, βegact,β N egact M are allowed to vary by skill, age, gender, city and time. This allows wages of natives relative to immigrants in a specific group to vary over time due to changes in the productivity level of immigrant labor. Further, the wages of natives relative to immigrants of the same characteristics can vary across cities. This is to reflect variation in cities productivities based on differences in industrial compositions. Capital is perfectly elastically supplied at a common price κ t. 19 Let P t denote the output price. Firms operate in a perfectly competitive output market so real wages equal the marginal product of labor. The city s demands for workers of characteristics: e,a,g,s) in city c in year t is given by lnw s egact/p t = lna ct + η t + lnθ et + 1 σ E lnl ct lnl ect ) + lnφ get + 1 σ G lnl ect lnl egct ) 6) +lnδ aet + 1 σ A lnl egct lnl egact ) + lnβ s egact + 1 σ M,E lnl egact lns egact ), and 1 α)act η t = ln α κ t /P t ) 1 α ) α. 3.2 Housing Supply Housing supply serves as a congestion force. Since I focus on the labor market structure, I model the housing market using a simple setting. Each city is endowed with a fixed amount of land suitable for construction. Developers are price-takers and sell identical houses. Let P h,ct denote local hosing prices which are set through equilibrium in the competitive market. Following Davis and 18 I also estimate the elasticity of substitution between natives and immigrants under various specifications and definitions of immigrations in Section 5.5). This includes removing immigrants who have been living in the US for less than 5 years as well as excluding immigrants who arrived before aged 20. Additionally, I consider a specification where the relative productivity levels between immigrants and natives do not vary across cities. 19 Ottaviano and Peri 2008) estimate the speed of capital adjustment to be 10 to 20 percent per year. Thus decadal changes are well approximated by the long run equilibrium where capital is fully adjusted. 11

12 Palumbo 2008); Diamond 2013), the inputs to housing production include construction materials and land. The production function takes the following form Q ct = a ct l ϕ ctm 1 ϕ ct, where Q ct is the quantity of houses in city c in year t, a ct is city-specific productivity, l ct is the amount of developable land and m ct is the quantity of construction materials. 20 The parameter ϕ represents the share of land in the housing production. 21 The developer s profit function is π = P h,ct Q ct P l,ct l ct P mt m ct, where the price of construction materials P m,t is exogenous and the price of land P l,ct is a function of city population. The latter assumption is based on the Alonso-Muth-Mills model. The basic idea is that as a city s population increases, the land available for development decreases and hence land prices rise. 22 Hence, I assume that the land price takes the following form ) γc P l,ct = Q ict i where i Q ict is the quantity of local houses and γ c measures the elasticity of land price with respect to houses. Since developers are price takers, the housing price is set at the average cost of production, P h,ct = AC p mt, p l,ct ). 7) In the steady state equilibrium, housing prices equal the discounted values of rents, R ct = i t P h,ct, where i t is the interest rate. For simplicity, I assume that absentee landlords initially own and sell land to developers. Housing is owned by natives collectively and the rental income is redistributed equally to all native workers. 23 The log housing rent is given by lnr ct ) = lncc ct ) + γ c ln s a g Ŝ egact ). 8) e 20 Thorsnes 1997) estimates the elasticity of substitution between land and non-land inputs in the housing production to be around one. He concludes that the housing production technology is well approximated by the Cobb-Douglas production function. 21 Davis and Palumbo 2008) document that the land s share in home values varies by region. I incorporate these regional differences by allowing the land price elasticity with respect to local population to vary by city. 22 See Diamond 2013) for more discussion. 23 The assumption of rent redistribution can be easily modified 12

13 where s a g e Ŝ egact is the effective population [ adjusted for relative housing consumption, and ) 1 ϕ ) ] ϕ the construction cost: lncc ct = lni t + ln a 1 ϕ ct 1 ϕ + ϕ 1 ϕ Pm,t 1 ϕ. The elasticity of rent with respect to city population γ c varies by the geographic characteristics of each city. Scarcity of land suitable for development limits new construction and induces a less elastic housing supply small γ c ). I approximate γ c using a log-linear function γ c = γ + γ geo lnx geo c ). 9) In Eq 8), γ measures the base housing supply elasticity. The second term, γ geo measures the contribution of geographic constraints on the inverse elasticity of housing supply where xc geo measures the share of land within 50 km of each city s center that is unavailable for development due to wetlands, water bodies or steep slopes. Saiz 2010) provides measures of unavailable land at the MSA level. 3.3 City Amenity and Network In reality, cities differ in many dimensions. Exogenous factors such as climate may influence the attractiveness of a city. To better understand how individuals make their location decisions, I allow cities to differ in amenities, defined as non-pecuniary characteristics that influence a city s desirability. This includes climate, proximity to natural features as well as the quality of goods and services. All residents within a city have access to these amenities, but different groups of workers do not need to value these amenities equally. The amenities in city c in year t is denoted by the vector xct. A A well-known settlement pattern of new immigrants is that they tend to locate in countryspecific enclaves Card, 2009). 24 This could be because it is easier to move or adjust to a city where an individual has a larger network. For instance, there are more ethnic grocery stores in areas where immigrants are concentrated Largent et al., 2013). 25 Therefore, I consider the utility value an immigrant gains from a city-specific network size. I use the city s number of previous immigrants born in the same country group as a proxy for network size. Define x rb c,t τ as a 22 element vector where each component contains city c s number of immigrants in year t τ born in each of the 22 country groups see Tables 2) and 3) for the descriptive statistics of immigrants from the 22 country groups). For natives, I allow workers to derive the network values from living in their birthplaces. Since I only observe birthplaces at the state level, I allow natives to gain utility from living in or near their 24 Prominent examples include the concentration of Arab immigrants in Detroit see Abraham, 2000) and Mexican immigrants in Los Angeles and Chicago. 25 Curran and Rivero-Fuentes 2003); Massey and Espinosa 1997) find that networks affect immigration decisions. Additionally, Munshi 2003) documents that a Mexican immigrant is more likely to find a job in the U.S. if his network is larger. 13

14 states of birth I also include U.S. outlying areas as natives birth places: American Samoa, Guam, Puerto Rico and U.S. Virgin Islands). Define xc st as a 54 element vector where each component k is equal to one if part of city c is contained in state k. For natives who live outside their birth states, the network value depends on the distance from one s birth state to the destination city. Define xc d as a 54 element vector where each component k contains the distance from the population centroid in state k to the population centroid in city c. The vector of network value and amenities to worker i in city c in period t is ) ) N ct = 1 {S=N} xc st,xc d,1 {S=M} xc,t τ,x rb ct A 10) where 1 {S=N} is an indicator function equals one if a worker is native. 3.4 Labor Supply Each worker i chooses the most desirable location taking all cities characteristics as given. 26 For simplicity, the original immigration decision is taken to be exogenous; upon arrival in the U.S., an immigrant must choose a city of residence. Natives are born and initially live in their birth locations. Upon entering the labor market, they choose a city of residence. The worker maximizes utility by choosing a city c, the quantity of a housing good Q t which has a local price of R ct, and a national good G t which has a common price of P t. Let z denote a vector of the worker s characteristics which includes skill level e {H,L}, gender g {F,M}, age a {Y,O}, and nativity s {M,N}. A worker of type z inelastically supplies one unit of labor and earns a wage of Wct. z 27 The utility of worker i living in city c, U ict is defined as ) ) Q ζ i + ln + u i N ct ) 11) subject to U ict = max ln Q,G it G 1 ζ i it P t G it + R ct Q it W z ct where u i N ct ) is the utility from city amenities and networks, and 0 ζ i 1 is a parameter which can be trivially identified as the share of income on housing. 28 Most empirical studies find that 26 I focus on individual location decisions for computational reasons. In my model, workers differ based on skill level, age, gender and nativity. Modeling individual decisions involves estimating parameters for sixteen types of workers. While the role of joint location decisions of couples is important, this requires estimating parameters for twenty-four possible types of households in each city and year see Section 4.) for more discussion). 27 I abstract from labor supply decisions to keep the model simple; however in reality immigration may impact the employment incentives as well as occupational choices of natives. For example, natives may respond to immigration by leaving the labor force or switching to language-intensive occupations. These labor supply adjustments can influence the wage and welfare impacts. Thus, analyzing the effect of immigration on the labor supply of natives is an important area for future work. 28 The distribution of workers across cities can impact the level of rental income. However, it is assumed that workers do not take their own effect on housing rents into account when making location decisions. 14

15 housing is a normal good, with an income elasticity of Polinsky and Ellwood, 1979). This suggests that housing expenditure shares are lower for higher income workers. Since income inequality is most pronounced between college and noncollege workers Katz and Autor, 1999), I restrict ζ i to only vary across skill-nativity groups. 29 The relative value of the national to housing goods can be represented as ζ i = λ r z. Maximizing 11), the worker s indirect utility from living in city c in year t is given by V ict = w z ct λ r z r ct + u i N ct ) 12) where w z ct = lnw z ct + I t )/P t ) and r ct = lnr ct /P t ). The value of amenities and networks to worker i in city j in period t is defined as u i N ct ) = β A z x A ct + β st zt st i x st c + β d ztst i x d c + β rb zt rb i x rb c,t τ + λ σ z ε ict 13) where ε ict is drawn from a type I extreme value distribution; rb i is a 22 element binary vector with each component equal to one if the worker was born in the country group; and st i is 54 element binary vector where each component equals one if the worker was born in the state. Each worker has an individual idiosyncratic taste for cities, ε ict drawn from a Type I Extreme Value distribution. To keep the number of parameters manageable, I assume that the variance of workers idiosyncratic tastes for each city only differ across immigration status and skill levels. However, worker i s marginal utility of the amenities βz A, and the value of networks βzt st,βzt,β d zt rb can vary across all types of workers. Furthermore, I allow the value of networks to vary across time for two reasons. First, this greatly simplifies the computation see Section 4.)). Second, this allows the model to capture the cohort effects and account for the rapid growth of immigrants into nontraditional cities. 30 For identification purposes, I normalize the standard deviation of workers idiosyncratic taste for cities to one by dividing 12) by λz σ, and redefine the parameters of the normalized optimized utility function as V ict = λz w w z ct λz r ) r ct +λz A xct A + λzt st st i xc st + λztst d i xc d + λzt rb rb i xc,t τ rb + ε ict. 14) where w z i ct λz r ) r ct is the worker s income net of housing expenditure or local real wage. The preferences amongst workers of type z for a given city differ due to their birthplaces and idiosyncratic taste for cities. Let Γ z ct denote the common utility value of city c for all workers of type z, Γ z ct = λz w w z i ct λz r ) r ct + λ A z xct. A 29 Additionally, Baum-Snow and Pavan 2010) find that expenditures shares are similar across cities of different size. 30 Kritz and Gurak 2006) document a rapid growth of immigrants into cities with historically small immigrant populations from ; examples includes Atlanta, Dallas, Orlando, and Sacramento. 15

16 The term Γ z ct represents the mean utility of workers of type z from living in city c net of the home or network values. Eq 14) can be rewritten as V ict = Γ z ct + λ A z x A ct + λ st zt st i x st c + λ d ztst i x d c + λ rb zt rb i x rb c,t τ + ε ict. Since the preference shocks are drawn from an extreme value distribution, the probability of a person choosing to live in city c is Pr ict = exp Γ z ct + λzt st st i xc st + λztst d i xc d + λzt rb rb i xc,t τ rb )) Σ k C exp Γ z kt + λzt st st i xc st + λztst d i xc d + λzt rb rb i xc,t τ rb )). 15) Therefore the labor supplies for each worker type in city c in year t are Z ct = Σ i Zt Pr ict where Z ct is the number of workers of type z in city c in year t, and Z t is the set of workers of type z in the economy McFadden, 1973). 3.5 Equilibrium Equilibrium is defined by a set of prices w z ct,r ct) and populations of each type Z ct) such that 1. Every worker i maximizes his or her utility by choosing the optimal city c : c = argmaxv i jt 16) j C 2. Every firm j chooses an optimal production plan y jt to maximize its profit: P t y jt P t y jt, y jt Y jt 17) 3. The labor demand and labor supply of each worker type are equal: Zct = Σ i Zt Pr ict 18) w z ct = lna ct + η t + lnθ et + 1 lnl ct lnl ect ) + lnφ get + 1 lnl ect lnl egct ) σ E σ G 19) +lnδ aet + 1 σ A lnl egct lnl egact ) + lnβ s egact + 1 σ M,E lnl egact lnz ct) 4. Total housing demand satisfies the housing rent equation: rct = lncc ct ) + γ + γ geo lnxc geo )ln 16 s a g Ŝ egact ). 20) e

17 The proof of existence is based on Bayer and Timmins 2005). Eq 16)-20) implicitly define the vector of population Z t = Σ i Zt Pr it that maps [0,Z t ] C into itself where Z t is the total population of type-z workers and C is the number of cities in the choice set. An equilibrium is a fixed point, Z t = g Z t,ω ) Σ i Zt Pr it 21) where Ω is the vector of parameters. The following proposition provides sufficient conditions under which a fixed point of 21) exists. Proposition 1. If i) ε ict is drawn from a continuous well-defined distribution function, ii) each consumer s utility u i is continuous in Z t and iii) each firm s production possibility set y j is closed, bounded, convex and 0 y j R n, then an equilibrium exists. Proof. Assumption iii) and the continuity of the firm s objective function ensure that the solution to 17) exists. Assumptions i) and ii) imply that the mapping g in 21) is a continuous mapping of a closed and bounded interval into itself. By the Brouwer fixed-point theorem, there exists a fixed point of this mapping g. Given that ε ict is drawn from a Type I Extreme Value distribution, it is continuous; u i is continuous; and the firms production possibility set satisfies iii), thus Proposition 1 implies that an equilibrium exists. As discussed in Bayer and Timmins 2005), the uniqueness of an equilibrium depends on the following features of the problem: i) the magnitude of the agglomeration and congestion forces; ii) the total number of cities; iii) the importance of individual tastes in the utility function; and iv) the variation and importance of fixed attributes across cities such as home premiums and network values. Bayer and Timmins 2005); Bayer et al. 2004) show that a congestion effect gives rise to a unique equilibrium. The present model incorporates a congestion force through housing supply. To develop the intuition on how a unique equilibrium arises, let us consider a special case where all labor types are perfect substitutes and the elasticities of housing supply across cities are identical, γ c = γ, c. Let Ψ denote the following vector Ψ = Z t g Z t,ω ) = 0. The following proposition is similar to the uniqueness result developed by Bayer and Timmins 2005). Proposition 2. If i) ε ict is drawn from a continuous well-defined distribution function, ii) each consumer s utility u i is continuous in Z t, iii) all labor types are perfect substitutes, and iv) the elasticities of housing supply across cities are identical, the equilibrium is unique. Proof. With constant returns to scale technology and full capital adjustment, if all labor types are perfect substitutes then changes in cities population have no impact on wages and hence no 17

18 effect on g. Let Ψ 1 denote the matrix of partial derivatives of Ψ with respect to Z t. Given the housing congestion force, Ψ 1 contains diagonal elements: 1 dg j /dz j ) > 1 since dg j /dz j < 0. Further, since the housing supply elasticities are identical, increasing each city s population by a fixed amount has no effect on the choice probabilities, k dg j /dz k = 0. This implies that k j dg j /dz k = dg j /dz j. Hence the diagonal elements of Ψ 1 are greater than the sum of the off-diagonal elements, and Ψ 1 is a matrix with a positive dominant diagonal. Consequently, the solution to Ψ = 0 is unique. A congestion effect causes workers to disperse which preserves the preference rank-ordering of locations. However, a sufficiently strong agglomeration effect can alter the rank-order of locations leading to multiple equilibria. In the present framework, network values are measured by the numbers of previous immigrants and independent of the current number of immigrants in a given city; hence there is no agglomeration incentive due to current networks for immigrants. 31 However, heterogeneity in labor types may induce complementary workers to concentrate in the same locations. Nonetheless, provided that the housing supply congestion effect is sufficiently strong, a unique equilibrium can be obtained. Given the number of parameters, the restrictions on the model primitives for which a unique equilibrium arises cannot be easily characterized. However, as noted in Bayer and Timmins 2005), it is possible to verify whether an equilibrium is unique. Consider the two sequences defined by { Zt,g Z t ),g g Zt )),... } starting at the endpoints of Zt. If an agglomeration effect induces multiple equilibria, then these sequences converge to at least two points. So one may verify uniqueness by applying g.) iteratively starting near the endpoints of Z t and determining whether the sequences converge to distinct fixed points see Bayer and Timmins, 2005 for further details). 4. Estimation 4.1 Labor Demand In general, the labor demand functions can be estimated in one step using nonlinear techniques. However, since the firm s production function takes a four-level nested CES form, estimating the parameters using a nonlinear system of equations generates numerical difficulties. Thus, I follow Card and Lemieux 2001) by proceeding iteratively from the lowest nest to the top I impose that the network effects are independent of the current number of immigrants to reduce the multiplicity of equilibria and for computational convenience. Intuitively, we can think of the number of previous immigrants as a proxy for the availability of place-specific information as well as ethnic goods and services e.g. restaurants and grocery stores). A small change in the immigrant population will not change these city attributes in the medium run. Additionally, given the enclave patterns of immigrants in the data, holding the network strength fixed enables us to pin down an equilibrium that is likely to realize. 32 Manacorda et al. 2012); Ottaviano and Peri 2012) also use this iterative estimation method. 18