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Firm Heterogeneity
and the Impact of Immigration:
Evidence from German Establishments

WP 21-16

Agostina Brinatti
University of Michigan
Nicolas Morales
Federal Reserve Bank of Richmond

Firm Heterogeneity and the Impact of Immigration:
Evidence from German Establishments∗
Agostina Brinatti†
Nicolas Morales‡
December 2021
Most recent version here

Abstract
We use a detailed establishment-level dataset from Germany to document a new dimension
of firm heterogeneity: large firms spend a higher share of their wage bill on immigrants
than small firms. We show analytically that ignoring this heterogeneity in the immigrant
share leads to biased estimates of the welfare gains from immigration. To do so, we set
up and estimate a model where heterogeneous firms choose their immigrant share and
then use it to quantify the welfare effects of an increase in the number of immigrants
in Germany. Two new adjustment mechanisms arise under firm heterogeneity. First,
native workers reallocate across firms, which mitigates the competition effect between
immigrants and natives in the labor market. Second, the gains are largely concentrated
among the largest and most productive employers, which induces an additional aggregate
productivity gain. If we ignore the heterogeneity in the immigrant share across firms, we
would underestimate the welfare gains of native workers by 11%.
JEL: F16, F22, J24, J61
Keywords: Heterogeneous Firms, Migration, International Trade

∗

We would like to thank John Bound, Javier Cravino, Andrei Levchenko, Jagadeesh Sivadasan, Sebastian
Sotelo, and seminar participants at the University of Michigan, Inter-Fed Brown Bag, Richmond Fed, Central
Bank of Uruguay, Duke-Richmond-UVA Macro Jamboree, SOLE, ITO (FREIT), SITE (Stanford), Midwest
Trade (MSU), and SEA (Houston) for helpful comments and suggestions. We are grateful to the Research Data
Centre (FDZ) of the German Federal Employment Agency at the Institute for Employment Research (IAB) in
Germany for facilitating access to the data. The views expressed are those of the authors and do not necessarily
reflect those of the Federal Reserve Bank of Richmond or the Board of Governors.
†
University of Michigan, brinatti@umich.edu
‡
Federal Reserve Bank of Richmond, nicolas.morales@rich.frb.org, sites.google.com/view/nicolasmorales

Introduction

During the past two decades, the number of immigrants living in developed countries increased
by more than 80%, which has fueled the academic and public debate regarding the impact of
immigration on native workers. To study this question, most of the literature has assumed,
implicitly or explicitly, that a representative firm exists. However, firms are heterogeneous
along many dimensions such as size, productivity, export behavior, and demand for labor. In
this paper, we ask whether such heterogeneity across firms matters as we aim to understand
the effect of immigration on the welfare of native workers.
We start by using a detailed establishment-level dataset from Germany to document a new
dimension of heterogeneity: large employers are more immigrant-intensive than small employers. We then show analytically and quantitatively that ignoring this heterogeneity leads to
biased welfare gains from immigration. First, when firms are homogeneous, the elasticity of
substitution between immigrants and natives in the labor market coincides with the within-firm
elasticity. However, when firms are heterogeneous, the aggregate immigrant-native substitution
elasticity depends on the within-firm elasticity and the elasticity of substitution across firms
or goods. Thus, having different immigrant-intensities across firms allows for natives and immigrants to specialize in working for different employers, which makes them less substitutable
in the aggregate labor market. Second, when firms are heterogeneous, the gains are largely
concentrated among the largest and most productive employers, which induces an additional
aggregate productivity gain. We find that if we ignore this heterogeneity, the welfare gains
from an increase in immigration would be underestimated by 11%.
To characterize the relationship between employer size and immigrant intensity, we use a comprehensive employer-employee matched dataset of social security records in Germany between
2003 and 2011. We show that the median establishment in the top wage bill decile spends
5.6% of their wage bill on immigrants, while the median establishment in the fifth decile spends
almost half of that (2.9%), and the median establishment in the bottom decile spends even
less (0.4%). This relationship is stronger in the tradable sector, where the immigrant share of
the top decile is 8%, while the immigrant share at the bottom decile is zero. We explore the
mechanisms behind this relationship and provide evidence suggesting that firms may incur fixed
hiring costs to start recruiting immigrants. We also rule out confounders such as differences in
worker skills, production technologies, and local labor markets.
Next, we set up a model with heterogeneous firms to quantify the general equilibrium adjustment and welfare implications of an influx of immigrants. The model incorporates a tradable
and non-tradable sector, the decision to export (Melitz, 2003), and crucially, the decision to
hire immigrant labor. Consumers have preferences over a set of goods in each sector, which are
aggregated in a CES fashion. Each good is produced by a single firm that can use immigrant
and native labor as inputs, which we consider imperfect substitutes in production (Peri and
1

Sparber, 2009, 2011).
We model the immigrant hiring decision following the input-sourcing literature (Antràs et al.,
2017; Blaum, 2019; Blaum et al., 2018; Halpern et al., 2015). Firms can choose to hire immigrant
labor, but to do so they must incur two types of fixed costs: an initial fixed cost to start hiring
immigrants, and an additional fixed cost for any new country they source immigrants from.
Such fixed cost structure has two implications supported by the data. First, larger and more
productive firms will be more likely than small firms to hire immigrants in equilibrium. Second,
larger firms will also find it profitable to recruit immigrants from more countries and spend a
larger share of their wage bill on immigrants. To fully capture the rich relationships between
size and immigrant intensities across firms observed in the data, the model allows for two
sources of firm heterogeneity: innate productivity and the cost of hiring immigrants, which are
both drawn from a joint distribution.
We use a simplified version of this model to analytically show that the welfare predictions of
a model that ignores the relationship between firm size and immigrant share are biased. To
this end, we compare the welfare gains between our model with full heterogeneity and a model
without heterogeneity in immigrant intensities. The sign of the bias depends on whether the
elasticity of substitution between immigrants and natives is larger or smaller than the elasticity
of demand, which regulates the change in the scale of production. When the substitution effect
is stronger than the scale effect, immigrants crowd-out natives at immigrant-intensive firms who
are reallocated toward native-intensive firms. By specializing in producing different goods than
immigrants, natives become less substitutable in the labor market, and the downward pressure
on wages induced by competition with immigrants is weaker than when natives do not reallocate
across firms. Such reallocation across firms implies that the aggregate elasticity of substitution
in the model with full heterogeneity is lower than in the model without heterogeneity, which
makes the welfare gains from immigration larger.
The magnitude of the bias depends on the elasticity of demand, the elasticity of substitution
between immigrants and natives, and the joint distribution between firm-level productivity
and firm-level immigrant-hiring costs. Following Oberfield and Raval (2014), we estimate the
elasticity of demand from the average firms’ markups (i.e., the ratio of revenue to total costs).
The substitution between immigrants and natives is structurally estimated using the firm’s firstorder condition with respect to immigrant and native labor. We regress the firm-level relative
wage between immigrants and natives on relative employment, following an IV approach as in
Ottaviano and Peri (2012). Since the quantities in our model are in effective units of labor,
we provide a model-based method to back out the effective units from data on labor quantities
and wages.
Given the estimates of these two elasticities, we estimate the joint distribution of productivities
and costs to match the observed dispersion and correlation between firm-level revenues and

immigrant-intensities in the data. These parameters are jointly estimated with the remaining
parameters of the model through a Simulated Method of Moments (SMM) approach to match
key targeted micro- and macro-level moments in Germany between 2003 and 2011. We show
that the estimated model is capable of replicating the cross-sectional distribution of immigrant
intensities across firms, even for important untargeted moments in the distribution.
We validate the model by comparing our model-predicted treatment effects of an increase in
immigration across firm sizes with the observed treatment effects estimated independently from
the model. Specifically, we regress firm revenues and the relative wage bill between immigrants
and natives on the share of immigrants in the local labor market and its interaction with firm
size. To identify the causal effect, we follow Ottaviano and Peri (2012) and instrument the share
of immigrants in a labor market with a shift-share instrument that exploits country-of-origin
variation in the initial network of immigrants across local labor markets. For establishments
in the tradable sector, we find that a 1% increase in the share of immigrants in the local labor
market increases revenues for firms in the top decile by 2.16%, while it decreases revenues in the
bottom decile by 0.42%. We also show that large establishments in the tradable sector become
more immigrant-intensive than small establishments. For establishments in the non-tradable
sector, we find weak heterogeneous effects in their response to immigration. The model does
a good job in replicating the observed relative responses to immigration across firms in both
sectors.
We use the estimated model to measure the welfare effects of a 20% increase in the total
number of immigrants, which is what happened in Germany between 2011 to 2017 after the
country unified its labor market with other EU countries. We find that native workers in both
sectors benefit from immigration since wages are higher due to larger domestic and international
demand, and prices are lower due to lower production costs. Revenues and profits increase for
both sectors, but more so in the tradable sector, where firms are more intensive in immigrant
labor. Natives reallocate within sector toward less immigrant-intensive firms and across sectors
toward the non-tradable sector. In monetary terms, welfare gains from immigration amount to
$4 billion for native workers and $15 billion for firm owners.
Finally, for our welfare results, we quantify the significance of accounting for the heterogeneity
in the immigrant share. To do so, we keep the same estimates of the elasticity of substitution
and the elasticity of demand, and re-estimate the remaining parameters of our model for the
case where all firms spend the same share of their wage bills on immigrants. Such model is
equivalent to a quantitative model estimated without firm-level data on immigrant labor, a
data limitation commonly faced by the literature. Overall, the model without heterogeneity
understates the change in welfare of natives by 11%, which is driven by an underestimation of
both the drop in the price level and the increase in wages caused by immigration. The bias can
be explained by two main components. First, the aggregate elasticity of substitution between
immigrants and natives in the heterogeneous model is lower than when ignoring heterogeneity
3

in the immigrant share. Second, even when using the same aggregate elasticity in both models,
there is a complementarity induced by heterogeneity that increases aggregate welfare, as the
largest and most productive firms benefit the most from the endogenous productivity gains
generated by immigrants.
Our paper contributes to the literature in three main ways. First, while some notable papers
use general equilibrium models to study the impact of immigration (Burstein et al., 2020;
Caliendo et al., 2021; Desmet et al., 2018; di Giovanni et al., 2015; Khanna and Morales, 2018;
Morales, 2019), they tend to follow a neoclassical approach, where firms are assumed to be
homogeneous in their immigrant hiring decisions. Relative to the existing quantitative models,
we add the novel feature of firms endogenously choosing their immigrant intensities by following
the literature on intermediate input sourcing (Antràs et al., 2017; Blaum, 2019; Blaum et al.,
2018; Halpern et al., 2015). This approach allows us to consider the firm as a fundamental
channel where aggregate production and labor adjust to immigration. We document a large
heterogeneity in the immigrant share across firms and, in light of this heterogeneity, we find
that it matters for quantifying the aggregate impact of immigration.
Second, we also speak to an emerging literature that uses firm-level data to provide reducedform evidence on the effect of immigration on firms (Arellano-Bover and San, 2020; Card et al.,
2020; Dustmann and Glitz, 2015; Kerr et al., 2015; Mahajan, 2020; Mitaritonna et al., 2017;
Orefice and Peri, 2020). We contribute to this literature by documenting new facts regarding the
relationship between firm size and immigration and by assessing the aggregate consequences of
immigration with a general equilibrium model. In Section 8, we further discuss how our results
compare to the findings of this literature and how the institutional context of Germany matters
for our conclusions.
Third, we contribute to the literature that studies the importance of firm heterogeneity for
aggregate outcomes. In the context of international trade, Arkolakis et al. (2012) show that,
conditional on having the same trade elasticity, the welfare gains from trade are the same for a
class of heterogeneous and homogeneous firm models. As opposed to that class of heterogeneous
firm models, we allow firms to be heterogeneous in their input shares and, building on Oberfield
and Raval (2021), we show how this heterogeneity affects the aggregate elasticity of substitution
between immigrants and natives.1 Our new insight is that if firms are heterogeneous in their
immigrant share, immigration induces a reallocation of natives across firms. Such reallocation
affects the aggregate substitution between natives and immigrants and, in turn, the welfare
gains from immigration.
1

Oberfield and Raval (2021) show that the aggregate elasticity between two inputs of production, labor and
capital, depends on the elasticity of substitution within a firm and the reallocation of market shares across firms
that employ capital and labor differently.

Data

We use a detailed, employer-employee matched dataset from Germany provided by the Research Data Center (FDZ) of the Federal Employment Agency in the Institute for Employment
Research (IAB). The main data source is the Longitudinal Establishment Panel (LIAB), which
includes records for a large sample of establishments over the period 2003-2011.2 The dataset
contains full employment trajectories for each employee who worked at least one day for one
of the establishments in the sample during the period. It also includes employee information
on citizenship, occupation, education, and daily wage. Regarding citizenship, countries are
grouped into ten regions: 1) Germany, 2) France, United Kingdom, Netherlands, Belgium,
Austria, Switzerland, Finland, and Sweden, 3) Italy, Spain, Greece, and Portugal, 4) countries
that joined the EU after 2004, 5) countries of former Yugoslavia not in the EU, 6) Turkey, 7) all
other European countries including Russia, 8) Asia-Pacific, 9) Africa and Middle East, and 10)
the Americas. On the establishment side, the dataset contains information on industry, location, and establishment-level financials such as revenues, investment, and material use, among
others. More information on LIAB can be found in Heining et al. (2016).
A key variable needed for our analysis is workers’ immigration status at a given establishment,
but the German social security data records citizenship as opposed to country of birth. Since
we are interested in country of birth, we redefine this key variable to make sure we count
immigrants properly. The most common recoding is when observing individuals with a foreign
citizenship become Germans the next period. If a worker is recorded as a foreigner for at least
two periods, we classify them as an immigrant from the initial citizenship country.3
It is important to note that the German administrative data is at the establishment level, and
it is not possible to link multiple establishments to a single firm. Throughout the paper, we will
use establishment and firm interchangeably. Also, while LIAB is not directly a representative
sample of the population, we apply survey weights to get representative aggregates whenever
necessary. For establishment location within Germany, our data includes an administrative subdivision of German states into districts called “Kreis.” For part of our analysis, we also group
districts into local labor market areas following the analysis of Kropp and Schwengler (2011),
who use commuting flows to delineate functional labor markets. We complement the German
administrative data with publicly available datasets from the World Bank to deflate wages and
compute exchange rates, the World Input-Output tables for data on trade and international
GDP, and the OECD for aggregate migration data.
2

The data basis of this paper is the Longitudinal Model (version 1993–2014) of the Linked Employer-Employee
Data from the IAB. The data were accessed on-site at the Research Data Centre of the Federal Employment
Agency at the Institute for Employment Research (FDZ) and/or via remote data access at the FDZ.
3
A second challenge is that some workers might join the labor market with a foreign citizenship, but they
may have grown up in Germany to foreign parents. Our results are robust to recoding workers as natives if
they have foreign citizenship and either join the labor force at age 20 or younger without a college degree, or
join the labor force at age 25 or younger with a college degree.

Firms Are Heterogeneous in Their Immigrant Share

We present a series of facts that provide insight on how employers have different intensities
on immigrants and use these facts to ground our model.4 As a first step, we document that
larger employers are more intensive in immigrant labor. We rank the establishments in our
sample into wage bill deciles, where decile 1 includes the smallest establishments, and decile
10 includes the largest.5 For each decile, we plot the median share of immigrant labor in the
establishment wage bill to capture the firm-level intensity on immigrants. As shown in the solid
blue line in Figure 1, there is a monotonic and increasing relationship between employer size
and immigrant intensity. The median establishment in decile 10 spends 5.6% of their wage bill
on immigrants, while the median establishment in decile 5 spends only 2.9%, and the median
establishment in the lowest decile spends even less, 0.4%.
Figure 1: Immigrant share of the wage bill across establishments

The relationship between employer size and immigrant intensity is not driven by specific confounders such as industry or labor markets. Large employers could be concentrated in industries
that are more intensive in skills provided by immigrants. At the same time, immigrants might
also concentrate in large cities where immigrant networks are larger, which also happens to
be where large employers are located. However, none of these channels seem to explain the
observed heterogeneity in immigrant intensities. As shown in the dashed lines in Figure 2a,
the pattern remains strong after controlling for three-digit industry fixed effects and local labor market fixed effects, indicating that differences in production technologies or geographic
4

In Appendix A, we present summary statistics on the sample of establishments, and the distribution of
immigrants across sectors and origin regions.
5
We use wage bill as our main measure to rank establishments, but results are robust to using employment
or revenues. We focus on establishments with more than 10 employees, but the relationship between size and
immigrant intensity is still positive and strong when including smaller establishments.

Figure 2: Immigrant share across industries, labor markets, and skill groups
(a) All establishments

(b) By education group

Note. We divide all establishments with more than 10 employees into total wage bill deciles, with 1 being the
smallest establishments and 10 the largest. For each decile, we plot the median immigrant share of the total
establishment wage bill. Decile 1 is normalized to 0. Left panel: we plot the observed median immigrant
share, the residual median share after removing industry-time fixed effects, and the residual median share
after we remove industry-time and location-time fixed effects. Right panel: we divide all establishments
with more than 10 college and non-college employee, respectively, into total wage bill deciles. For firms in
each decile, we plot the median immigrant share of total wage bill spent in each education group.

destinations of immigrants alone cannot explain the observed relationship between size and
immigrant-intensity.
Our relationship of interest is also not driven by immigrant skills. Large firms tend to be more
intensive in high-skill labor (Burstein and Vogel, 2017), and if immigration policy in Germany
would be skewed toward workers with a specific education, this could drive the relationship
between size and immigrant intensity. As shown in Figure 2b, the relationship between size
and immigration holds for workers with and without a college education. Additionally, we
corroborate that the observed patterns are not driven by the establishment being foreign-owned,
or being part of a multi-unit firm.
The evidence presented thus far is consistent with the existence of fixed costs to hire immigrants, which act as a barrier for small firms to hire their optimal immigrant labor shares. The
immigration literature has well documented that immigrants and natives are imperfect substitutes as they perform different tasks in production (Peri and Sparber, 2009, 2011). Hence,
all firms would optimally choose to hire immigrants in the absence of hiring costs. In the real
world, however, many firms do not hire immigrants, and immigrant intensities across firms are
very different even when controlling for industry, labor market, and skill differences.
For the most part, costs to recruit immigrants take the form of fixed costs since they do not
depend on the number of immigrants to hire. These costs include training legal and human
resources staff to comply with immigration law and learning how to screen foreign workers.
7

For example, employers may not be familiar with foreign institutions where the immigrant
accumulated work experience or the foreign universities that granted their educational degrees.
Firms may need to pay a one-time cost to learn about individual countries and their educational
and business institutions. In Germany, particularly before the EU labor market integration in
2011, most immigrants needed a guaranteed employment offer in order to migrate there. Given
this context and our data window between 2003 and 2011, it makes sense to focus on the
decision of firms to explicitly decide to pay these costs and recruit immigrants.6
Based on this anecdotal evidence, in Appendix B, we show evidence consistent with the presence
of fixed costs to hire immigrants. Importantly, we find that there is a significant mass of small
firms that do not hire immigrants, and there is lumpiness in the hiring process. We also find
that large firms recruit immigrants from more countries, which is consistent with the learning
costs to understand immigrant backgrounds.7
Finally, we argue it is important to explicitly separate establishments in the tradable and nontradable sectors throughout the analysis. As shown by Burstein et al. (2020), the tradability of
the output produced by immigrants is a key feature to account for, as immigrants are absorbed
differently in the labor market when working in tradable versus non-tradable occupations.
Tradable sectors face a more elastic demand and can expand output more than non-tradable
sectors in response to an influx of immigrants. Figure 3 shows that establishments in the
tradable sector are more intensive in immigrants than similar sized establishments in the nontradable sector. The tradable sector presents a stronger positive relationship between size and
immigrant intensity than the non-tradable sector.8
The differences in the immigrant share across firm sizes documented in this section imply that
firms can benefit differently from immigration: immigrant-intensive firms are likely to experience a larger drop in cost of production than native firms. Given that firms make production
decisions that determine employment and wages of natives, the documented heterogeneity can
potentially affect our current understanding of the effects of immigration on natives’ welfare.
To quantify the welfare gains of immigration under the observed differences in firm immigrant
shares and characterize the bias introduced by ignoring this heterogeneity, we set up a quantitative model presented in the following section.
6

Our framework is well suited to study cases where firms have an active role in finding and sponsoring
immigrants. The US H-1B program where firms sponsor workers’ visas, and Canada’s point system that gives
high weights to guaranteed employment offers, are good examples of cases similar to Germany prior to 2011.
7
The relationship between immigrant share and firm size could also be explained by recent theories on the
internal organization of firms, as in Caliendo et al. (2015). If larger firms with more layers of management
can supervise and hire more immigrants than smaller firms, it could also rationalize the patterns in Figure
1. Alternatively, large firms could have a technology that is biased toward immigrants, which would also
rationalize these patterns. However, these theories would not rationalize that larger firms also hire workers
from more countries, as shown in Appendix B.
8
Our definition for the tradable sector considers manufacturing, professional services, and wholesale trade.
While immigrants do concentrate in some small establishments in the non-tradable sector (e.g., restaurants),
the representative establishment captured by the median tends to have a low immigrant intensity.

Figure 3: Tradable and non-tradable sector.

Note. We divide all establishments with more than 10 employees into total wage bill deciles, with 1 being
the smallest establishments and 10 the largest. For each decile, we plot the median immigrant share of the
total establishment wage bill. We separate establishments in each decile on whether they belong to the
tradable or non-tradable sectors. We calculate the 95% confidence interval using 200 bootstrap repetitions.

The Model

Our quantitative model has two main components: the labor demand and the labor supply. On
the labor demand side, heterogeneous firms choose their optimal immigrant share, following
the setup proposed by the literature on importing intermediate inputs (Antràs et al., 2017;
Blaum, 2019; Blaum et al., 2018; Halpern et al., 2015). Firms also choose whether to export
their goods by paying a fixed cost as in Melitz (2003). The labor supply side of the model is
based on the combination of Eaton and Kortum (2002) model of comparative advantage with
Roy (1951), commonly referred to as EK-Roy models.9 We focus on the main components of
the model and relegate some derivations to Appendix C.
Consumption:
Domestic workers (indexed by i), supply Ld effective units of labor inelastically and have CobbDouglas preferences for goods from two sectors indexed by k as shown in equation 1:

Ui = (YiT )α (YiN T )1−α

(1)

where Y T stands for a tradable sector and Y N T for the non-tradable sector. Each sector k is
composed by a CES aggregate of varieties indexed by z as in equation 2:

Yik =

(y(z)ki )

σ−1
σ

σ
σ−1

(2)

Jz
9
The so-called EK-Roy models have been used to model individual choices across sectors (Lagakos and
Waugh, 2013; Lee, 2020) and across countries to migrate (Morales, 2019), among many other applications.

where Jz represents the set of varieties available in the country, and σ > 1 is the elasticity of
demand.

Production:
In each industry k, there is a mass of N firms indexed by j that produce a specific variety.
Firms employ only labor inputs, which can be native “domestic” workers or immigrants. There
is a long tradition in immigration literature to think about immigrants and natives as imperfect substitutes in production, as they have different comparative advantages across tasks and
specialize in different occupations (Peri and Sparber, 2009, 2011). We assume that firms combine domestic and foreign effective units of labor (dj and xj , respectively) in a CES manner as
shown in equation 3. For simplicity, we omit the subscript k from the equations below, but all
parameters except for the elasticities are industry-specific:

−1
−1 −1
yj = ψj βdj + (1 − β)xj

(3)

where β is a sector-specific distributional parameter that captures the average intensity in
immigrant labor, is common across sectors and captures the degree of substitution between
native and immigrant workers within the firm, and ψj is an firm-specific productivity draw.
Using CES properties, the unit cost for firm j can be written as in equation 4:

1−
uj = β wd1− + (1 − β) Wx,j

1
1−

(4)

where wd and and Wx,j are the wage per effective unit of native and immigrant labor, respectively. Following CES properties for the expenditure share in a given input, we can write the
domestic share as in equation 5:

sd,j =

β wd1−
β wd1−
=
1−
β wd1− + (1 − β) Wx,j
u1−
j

(5)

If the wage per effective unit of immigrant labor, Wx,j , was the same across firms, the unit cost
of production would also be the same. In that case, all firms, regardless of their productivity
or size, would have the same immigrant and domestic shares. However, as shown in Section 3,
the data suggests that the immigrant share is not constant across firms, and large firms have
a larger intensity in immigrants than small firms. To incorporate this into the model, we need
a theory on why firms hire different shares of immigrants and face different immigrant costs
Wx,j .
As discussed in Appendix B, we find multiple features in the data that suggest that firms face
fixed costs of hiring immigrants, and part of it seems to be dependent on the origin region
10

of the immigrants hired. Larger firms are not only more intensive on immigrants than small
firms, but also hire immigrants from more countries. Additionally, there is lumpiness in the
observed hiring patterns when firms start hiring immigrants from a given region. Finally, the
immigrant share of the firm has a strong correlation with the number of regions that the firm
recruits from, even after controlling for the total number of immigrants hired. These features
of the data are consistent with the idea that firms must invest resources into learning how to
recruit immigrants from additional origin regions.
Environment to Recruit Immigrants:
To theorize on the firm choice of its immigrant share that accommodates those facts and remains
tractable in a general equilibrium framework, we follow Blaum et al. (2018) and Blaum (2019),
who develop a theory of how firms choose their intermediate input share. We assume that the
immigrant input of labor, xj , is a composite of labor from different origin countries (indexed
by o) as in equation 6:

κ−1
κ

δo xj,o do

xj =

κ
! κ−1

(6)

Σj

κ is the elasticity of substitution between origin countries, such that every additional origin
country the firm hires from will have a positive impact on productivity and lower the effective
immigrant unit cost Wx,j faced by firm j. The hiring strategy of the firm, denoted by Σj ,
represents those countries where the firm hires immigrants from, out of a total of O origins.
Firms must pay a fixed cost fimm to begin hiring immigrants from abroad and a firm-specific
fixed cost fj for each additional origin country it wants to hire from. For example, if the
firm hires immigrants from two origins, it spends wd × (fimm + 2 × fj ) in hiring costs. One
interpretation is that the fixed cost fimm captures the costs of setting up a legal department
or training HR staff on the immigration hiring process in order to start hiring immigrants.
The cost fj captures the learning cost that is country-specific, such as understanding foreign
education credentials and labor experience necessary to screen workers.
We assume that hiring costs fj are jointly drawn with the firm-specific productivities ψj , from
a multivariate sector-specific log normal distribution with mean [µψ , µf ], dispersion [σψ , σf ],
and covariance between firm productivity draws and hiring costs of σψ,f .
Choosing Σj becomes computationally challenging because it requires computing profits for 2O
possible combinations of countries. To overcome this difficulty, we make a series of simplifications. First, we assume that foreign countries are perfectly ranked in terms of productivity δo ,
such that firms will first source from the foreign country with the largest δo and move down the
ladder as they source from more countries. This assumption simplifies the sourcing problem
as it now boils down to choosing the mass of countries, n ∈ [0, 1), to hire from. Second, we
11

assume δo is a random variable distributed Pareto with shape parameter ξ and scale parameter
δ̄. This assumption allows us to get a closed form expression for the wage index of immigrants
as in equation 7:10
1 ξ−κ
−
κ−1 ξ
1

1−κ
{z
}
|
1
ξ
ι
Wx,j = wx κ
(7)
nj
δ̄ κ−1 ξ − κ
{z
}
|
z̄

where ι > 0 can be interpreted as the elasticity of the immigrant unit cost to expanding
the mass of countries the firm hires from. Intuitively, imperfect substitution of immigrants
generates productivity gains from hiring immigrants from additional origins. This reduces the
wage index of immigrants and the unit cost of production.
Pricing Decision:
For a given domestic share (and unit cost of production), firms choose the price that maximizes
variable profits. Given that consumers have CES preferences, the optimal price is a constant
markup over the marginal cost:
σ
pj =
uj
(8)
σ−1
where pj is the price charged in the domestic market.
Optimal Domestic Share:
An advantage of this setup is that we can write the unit cost uj , price pj , and the optimal mass
of countries nj as a function of the key object sd,j , as in equations 9 and 10:
pj =

σ
β wd1− s−1
σ − 1 | {z d,j}

(9)

sd,j =

β wd1−

ι(−1)

β wd1− + (1 − β) wx1− (z̄)1− nj

−→ n(sd,j ) = χ̄

1
ι(−1)
1
−1
sd,j

(10)

where χ̄ is a combination of parameters and wages wd , wx . Equation 9 follows from equation
4 and the consumer’s optimization problem. Equation 10 follows from equations 4, 5, and
7.
Firms maximize their profits by choosing the optimal native share sd,j , as shown in equation
11:
10

The specific implementation of these assumptions can be found in Appendix C.

max Πj = (pj (sd,j ) − uj (sd,j )) yj − nj (sd,j )fj wd − wd fimm I(nj (sd,j ) > 0)
sdj
{z
} |
{z
}
|

(11)

Sourcing cost

profits

The main takeaways of the model are as follows: firms benefit from an immigration inflow
because the wage of immigrants drops and so does the unit cost of production. The size of the
drop in the unit cost of production is firm-specific, and it depends on the firm’s domestic share.
In other words, the domestic share acts as a firm-exposure to a common immigration shock
and becomes the key empirical object to learn about how much each firm (and the economy
as a whole) benefits from immigration. The native share sd,j can be directly observed in our
firm-level data and is the fundamental link between the model and the data.
How do firms choose their optimal domestic share? They face a trade-off between the drop in the
marginal cost of production induced by complementarity of hiring from an additional country
and the fixed cost to source from that additional country. Given their scale of production,
larger firms earn higher profits and can afford paying fj more times than small firms. Thus,
larger firms hire immigrants from more countries than small firms, and they become more
immigrant-intensive.
Export Decision and the Rest of the World (RoW):
Consumers in the RoW are assumed to have identical preferences over local and German varieties as in equation 2 with elasticity of demand σx .
German firms in the tradable sector can decide to export their goods by paying a fixed cost fx ,
as in Melitz (2003). Therefore, a firm will choose to export if the variable profits from export
sales are larger than fx . The exporters choose the price to charge abroad to maximize export
profits. The optimal price in that market is again a constant markup over total marginal cost,
which now includes an iceberg cost τ > 1 that represents a fraction of the good that gets “lost”
in transit as in equation 12:

pxj =

σx
uj τ
σx − 1

(12)

Finally, conditional on its export decision, the firm chooses sd,j by solving a problem analogous
to 11.11
Since our focus is the German economy, we make several simplifications to the modeling of the
RoW. We assume it has a single tradable sector, foreign firms are equally productive, and use
only domestic labor to produce with a constant return to scale production function yjx = ψ̄ x dxj .
11

The model predicts that firms that hire immigrants are more likely to export, which provides a micro
foundation for the empirical literature looking at the relationship between exports and immigration (Bonadio,
2020; Cardoso and Ramanarayanan, 2019; Gould, 1994; Hiller, 2013).

Foreign firms also pay the iceberg trade costs to export their goods but do not have to pay a
fixed cost to export.
Labor Supply:
Consumers are either firm owners, whose income are firms’ profits, or workers who earn wages.
We treat workers as heterogeneous in their sectorial skills by combining tools from the Eaton
and Kortum (2002) model of trade and the Roy (1951) model of occupational selection. Specifically, we assume that each country o = {g, x} has an exogenous number of workers born in o
o
) from a Frechet
(No ). Each worker i from o draws a sector k, location ` specific ability (ηi,`,k
distribution with shape parameter ν > 1, and scale parameter Ao,k as in equation 13:
!
F (η) = exp −

Ao,k (η)−ν

(13)

where Ao,k can be interpreted as the comparative advantage of workers from o in industry
k. Workers within a country are ex-ante identical but ex-post heterogeneous due to different
ability draws across sectors, while workers from different countries also differ in that they draw
their abilities from different distributions. Workers choose the industry and country that yield
the highest utility as shown in equation 14:

o
Ui,`,k

o
w`,k ηi,`,k
φ−1
=
o,`,k
P`

(14)

w`,k η o

i,`,k
is the real wage, and φk,o,` are iceberg frictions for workers from country o to
where
P`
work in industry k and country `. The iceberg cost captures both the cost of working in a given
sector and the migration cost of moving. For example, if Germany is very restrictive in letting
migrants into the country, φk,o=x,`=g will be very high. For simplicity, we will assume the cost of
migration out of Germany is infinity, such that German workers are immobile across countries.
Following the properties of the Frechet distribution, the fraction of workers from country o who
choose to work in industry k in destination location ` can be expressed as in equation 15:

πo,k,`

w`,k
P`

φ−ν
o,`,k

ν
=P
w`,k
φ−ν
o,`,k
`,k Ao,k
P`
Ao,k

(15)

This expression shows that reducing migration costs from any o to Germany increases the
supply of immigrants into the country.
Equilibrium and Market Clearing:

The equilibrium in this model can be defined as a set of prices, wages, and labor allocations
such that: workers optimally choose the industry and destination country `, k to work for,
consumers in each location choose how much of each variety to purchase to maximize utility,
firms choose the sourcing strategy and export status to maximize profits, labor markets clear,
and trade is balanced. Appendix C includes the main equilibrium conditions.

4.1

Firm Heterogeneity and Welfare Gains

In this section, we show that ignoring heterogeneity in the immigrant share across firms may lead
to biased estimates of the welfare gains of immigration. To that end, we compare the analytical
welfare gains of a simplified version of our fully heterogeneous model with that of a model that
ignores heterogeneity in immigrant share (but allows for heterogeneity in innate productivity).12
We will refer to these models as the “heterogeneous model” and the “homogeneous model,”
respectively. The homogeneous model can be a special case of the heterogeneous model with
fimm = fj = 0, or any model in the class of heterogeneous and homogeneous models following
the Arkolakis et al. (2012) framework. Alternatively, it could be a model with CES preferences
over goods coupled with the canonical production framework of immigration, with constant
elasticity of substitution between immigrants and natives (Card, 2009; Dustmann and Glitz,
2015; Ottaviano and Peri, 2012; Peri and Sparber, 2009).
To simplify the model, we focus on a closed economy with one sector. We assume that native
workers are homogeneous and set fimm = 0, but leave the firm-specific fixed cost fj unrestricted.
In this model, the welfare gains of immigration are given by the increase in real wages wPd as
shown in equation 16:

dlog

w
d

P
=−

ωj dlog(sdj )
dlog(S agg )
=−
−1
−1 }
| {z

Prediction without
heterogeneity in sdj

1
1 + (σ − ) Γ({sdj , ωj })
|
{z
}

(16)

≥0

p y

where ωj is the market share of firm j ( ωj ≡ R pjj yjj dj ) and measures firm j’s weight in the
consumption basket, S agg stands for the immigrant share in the total wage bill in the economy,
while Γ is a function that depends on the joint distribution of firm-level market shares (ωj ) and
native shares (sdj ).
The first component of expression 16 coincides with the welfare prediction of models that ignore
heterogeneity in sdj . In these models, immigration reduces the unit cost of production for all
firms and, as firms become more competitive, they increase their scale of production, demand
for native labor, and wages. The size of these gains depends on the size of the inflow and on
as it regulates how substitutable immigrants and natives are in the labor market. The more
substitutable immigrants and natives are, the lower the productivity gains for firms, and the
12

All derivations are included in Appendix D.

lower the welfare gains for natives.
The welfare predictions of the homogeneous model may be biased if there is heterogeneity in
the presence of immigrants across firms. Under heterogeneity, a new adjustment mechanism
arises, because native workers reallocate across firms with different immigrant intensities. Such
reallocation has two main implications. First, when firms are heterogeneous, the aggregate
elasticity of substitution between immigrants and natives depends on the within-firm elasticity
() and the elasticity of substitution across firms or goods (σ). Thus, having different immigrantintensities across firms allows natives to specialize in working for specific employers, which can
make them more or less substitutable with immigrants in the aggregate labor market. Second,
there is a complementarity between firm efficiency and the firm-specific endogenous productivity
gains from immigration. As these gains are largely concentrated among the largest and most
productive employers, there is an additional aggregate productivity gain that is not present in
the homogeneous model. Hence, even if we estimate the homogeneous model with the same
aggregate elasticity than the one predicted by the heterogeneous model, there can still be
first-order differences between their welfare predictions.
When firms are heterogeneous in their immigrant share, the aggregate elasticity of substitution
between immigrants and natives (agg ) is a weighted average between the elasticity of substitution within the firm () and the elasticity of demand or elasticity across firms (σ):
agg = (1 − π) + π σ

(17)

where π, and hence agg , depend on the distribution of sdj . The weight π is proportional to
the cost-weighted variance of immigrant shares and lies between zero and one (see Oberfield
and Raval (2021) for a derivation), taking the value of zero if firms employ the same immigrant
share. The first term, (1−π) , measures the substitution effect within firms; whereas the second
term, π σ, measures a reallocation effect across firms with different immigrant-intensities.
In the edge case of = σ, the substitution and scale effects cancel out, immigrants do not crowdin or crowd-out native workers, and native employment at the firm level does not change.13
Given that the reallocation of natives across firms is muted, the demand response for native
labor and welfare gains are the same as those predicted by the homogeneous model.
When the elasticity of substitution within the firm is stronger than the elasticity of demand ( >
σ), immigrants crowd-out natives from immigrant-intensive firms, and natives are reallocated
toward native-intensive firms. Such increase in specialization between natives and immigrants
in producing different varieties makes them less substitutable in the labor market than when
13

The relative change in employment
of natives
across firms is proportional to the change in immigrant share.
0
Let x̃ ≡ dlog(x), then d˜j − d˜j 0 = −σ
s̃
−
s̃
and, to a first order approximation, s̃dj ≈ (−1)(1−sdj )(w̃imm −
dj
dj
−1
w̃d ). Thus, the drop in relative wage of immigrants induced by an immigration inflow reallocates natives toward
native-intensive firms if > σ and toward immigrant-intensive firms if < σ.

natives do not reallocate across firms. Given that this reallocation adjustment is absent if firms
employ the same immigrant share, the increase in both, the aggregate demand for natives and
welfare are larger in the heterogeneous world.
When the elasticity of substitution is weaker than the elasticity of demand ( < σ), the opposite
happens. Immigrants crowd-in natives toward immigrant-intensive firms, and this reallocation
pattern increases the concentration of immigrants and natives in producing a similar set of
varieties. As a result, immigrants and natives become more substitutable in the labor market
when compared to the homogeneous world, and the increase in real wages and welfare are
lower.
Overall, equation 16 shows that the sign of the bias depends on the race between and σ.
In Section 5, we estimate these elasticities and find that ˆ > σ̂, suggesting that welfare gains
predicted by the homogeneous model are downward biased. Equation 16 also shows that the
size of the bias depends not only on these two elasticities, but also on the joint distribution of
firm size and immigrant share through Γ({sdj , ωj }). We estimate our model to match moments
on the joint distribution of sdj and ωj and find that the homogeneous model underestimates
welfare by 11%.14
As noted by Arkolakis et al. (2012), there is a class of heterogeneous and homogeneous models
where, if calibrated to the same aggregate elasticity and change in aggregate share, would yield
the same welfare gains. In our case, however, we would still expect a bias even if we assign
the same aggregate elasticity to both models. The reason is that the endogenous productivity
gains generated by firms choosing their sdj are stronger for larger and more productive firms,
an adjustment channel that is absent in the homogeneous model. Intuitively, conditioning on
{sdj }, agg is independent from ωj , meaning that agg is not informed by which firm benefits
by how much (e.g., the joint distribution of {sdj , ωj }). Consequently, agg will not capture the
first-order heterogeneous response and resulting reallocation of natives across firms that arises
when firms are heterogeneous.
The discussion on whether the fully heterogeneous firm model provides new welfare implications
of immigration has similarities and differences with the discussion offered by Melitz and Redding
(2015) about the welfare implications of trade. Similar to their paper, our heterogeneous model
differs from the homogeneous model in that the elasticity (of substitution) is endogenous,
and the homogeneous model does not capture the extra adjustment mechanism that arises
when we allow for heterogeneity. However, opposite of their paper, the differences in welfare
predictions in our setup are of first-order importance and do not vanish for small immigration
14

Additionally, equation 16 shows that the size of immigration shock does not affect the size of the bias, which
we also corroborate quantitatively in Appendix G.3.

inflows.15

Estimation

As discussed in Section 4.1, the key parameters of the model are , σ, and parameters that
determine the joint distribution of firm productivities and fixed costs to hire immigrants. In
this section, we explain how we use the German administrative data to estimate these key
parameters of the model.
Elasticity of Demand
We use micro-data to identify the elasticity of demand that firms face. Following Oberfield and
Raval (2014), we infer the demand elasticity from firms’ markups, i.e., the ratio of revenue to
total costs. According to the model, the following condition holds for every firm j:
Revenuej
σ
=
Costj
σ−1
where Revenuej stands for the revenues of firm j, and Costj denotes production costs. Although
the model assumes that the only production costs are labor costs, we compute total cost as the
sum of wage bill and material bill. The average markup is 1.4, which implies that the elasticity
of demand is 3.08. This estimate is consistent with the values used in the literature, where this
parameter takes values between 3 and 4.
We use data on markups for exporters relative to non-exporters in the tradable sector to back
out the implied demand elasticity from the RoW. The observed markup for exporters can be
expressed as a weighted average between the domestic markup (depending on σ) and the export
markup (depending on σx ). Using the exports as a share of revenues as weights, we calibrate
σx = 3.62.16
Elasticity of Substitution Between Native Workers and Immigrants
In the model, firm j’s demand of immigrant labor relative to native labor is given by (18):
βk 1 d
wd
j
j
− ln
ln x = ln
wj
1 − βk

xj
15

(18)

In Section 7.2, we show quantitatively that the welfare prediction of the homogeneous model with the
aggregate elasticity generated by the heterogeneous model reduces, but does not eliminate the bias. Such bias
remains large even for inflows of immigrants as small as 0.1%.
16
x
More specifically, we use the following equation: markup exporters = share exports × σxσ−1
+ (1 −
σ
share exports) × σ−1 . As we observe the markup for exporters and export share in the data, we can back
out σx using our estimated value of σ.

where wjd is the effective wage paid by firm j to native workers, and dj is native employment
in effective units, wjx is the effective wage paid for the immigrant labor bundle, and xj is the
composite immigrant labor defined by 6.
Estimating equation (18) presents a number of challenges. First, effective wages and quantities
are not observed directly in the data. Second, estimating equation (18) by OLS would yield
biased estimates of , since unobserved demand shocks at the firm level can affect the relative
quantities of immigrants and natives and the wages firms pay to each labor type.
To address these challenges, we proceed sequentially. First, as we explain in Appendix E.2, we
use the structure of the model to estimate the immigrant composite xj based on observed data
on labor quantities and wages across origin countries and industries. Second, we propose an
instrument to structurally estimate from equation (18).
To summarize our empirical strategy, we construct a shift-share instrument that exploits immigrant networks to create a supply push at the local labor market level that is plausibly
independent from demand shocks at the firm level. We bootstrap the standard errors to account for using generated regressors. The first stage is strong with an F-stat above 20, and
our preferred estimate for is 4.28, which is close to the estimates of Burstein et al. (2020),
who find an elasticity of substitution between immigrants and natives within occupations of 5.
Appendix E.2 describes the dataset construction, instruments, and results in detail.
Additional Parameters
Given the estimates for the elasticity of demand and the elasticity of substitution between immigrants and native workers, we calibrate the parameters of the model by simulated method of
moments to match micro- and macro-level moments. This approach serves as a bridge between
aggregate data on trade and immigration and what we have learned about firm heterogeneity
from the firm-level data.
As a first step, we proceed to do some normalizations, since not all parameters can be separately identified. The mean fixed costs of hiring immigrants (µf,k ), the mean productivity
of immigrants (Ao,k ), and the migration cost (φo,`,k ) cannot be separately identified from the
immigrant share in the production function (βk ), so we normalize the first one to 0 and the
remaining two to 1. We assume the mean productivities in each sector are equal to 1 (µψ,k = 1)
and set the elasticity of labor supply ν = 6.17 following Morales (2019). Finally, we calibrate
the Cobb Douglas parameter α = 0.68 to match the domestic expenditures in the tradable and
non-tradable sectors using World Input-Output Tables (WIOT).
As a second step, we are left with fourteen parameters, which we jointly estimate using a SMM
approach by minimizing the distance between fourteen moments simulated by the model and
fourteen empirical moments computed from the data. While all parameters are estimated together, there is strong intuition regarding which parameters identify which moments. The vari19

ance of log revenues conditional on the immigrant share and exporter status is used to identify
the dispersion parameter on productivities σψ,k . The observed variance of the immigrant-share
relative to the domestic share identifies the variability of fixed costs σf,k , while the difference
in the mean of sd,j between firms in percentile 90 relative to percentile 50 are used to identify
the correlation between productivities and hiring costs σψ,f,k . These three parameters for each
sector estimate the joint distribution between size and immigrant intensity, a key ingredient for
the quantitative model.
For the remaining parameters, we use the aggregate immigrant share by sector to identify βk ,
the distributional share parameter in the production function. The fraction of firms that hire
immigrants helps identify the base fixed hiring costs fimm,k . The average immigrant share across
all firms and sectors is used to identify ι, the elasticity on how the immigrant cost changes with
the mass of countries the firm hires from. For trade moments, we match the mean ratio of
export to domestic revenues for exporters to identify the iceberg cost and the fraction of firms
that export in the tradable sector to match the fixed cost of exporting fx . Finally, we use
aggregate data to compute the relative GDP per capita between Germany and the RoW, which
helps identify the mean productivity of the RoW ψ̄ x .
Table 1 shows the fourteen moments that are targeted in the estimation, their observed values in
the data and the ones generated by the model. For all fourteen moments, the model does a good
job in approximating their observed values. Table 2 contains the final calibration of the fourteen
parameters that minimize the distance between simulated and empirical moments.
While the model matches the targeted moments, we want to make sure it also matches nontargeted moments that are relevant to our main mechanisms. As shown in Appendix E.2, the
model does a good job in matching the cross-sectional means and medians of the immigrant
share by size decile.
Table 1: Simulated vs data moments
Moment description
Aggregate sd,T
Aggregate sd,N T
Var(log(revj )|sd,j , exporterj ), T
Var(log(revj )|sd,j ), NT
Var((1 − sd,T )/sd,T )
Var((1 − sd,N T )/sd,N T )
E(sd,T,p90 ) − E(sd,T,p50 )

Simulated

Data

Moment description

0.91
0.93
1.38
1.23
1.36
1.48
0.015

0.91
0.93
1.38
1.29
1.39
1.58
0.021

E(sd,N T,p90 ) − E(sd,N T,p50 )
Share of firms hiring immigrants, T
Share of firms hiring immigrants, N T
GDP per capita RoW to Germany
Share of firms exporting, T
E(Export to Domestic Revj ), T
E(sd )

Simulated

Data

0.009
0.57
0.63
0.32
0.34
0.80
0.93

0.008
0.62
0.61
0.32
0.37
0.79
0.93

Table 2: Parameter estimates using Simulated Method of Moments
Parameter description
Share of natives, T
Share of natives, N T
Dispersion in ψj , T
Dispersion in ψj , NT
Dispersion in fj , T
Dispersion in fj , NT
Covariance of ψ and fj , T

Parameter

Estimate

βT
βN T
σψ,T
σψ,N T
σf,T
σf,N T
σψ,f,T

0.84
0.86
1.02
0.35
1048
1710
-2.65

Parameter description
Covariance of ψ and fj , NT
Fixed cost of immigrants, T
Fixed cost of immigrants, NT
Productivity in RoW
Fixed cost of exporting
Iceberg trade cost
Elasticity sd to n

Parameter

Estimate

σψ,f,N T
fimm,T
fimm,N T
ψx
fg
τ
ι

8.17
3.41E-04
9.66E-04
1.52
0.011
1.49
0.013

Model Validation: Heterogeneous Response

Before quantifying the aggregate implications of a change in the number of immigrants in
Germany, we evaluate whether the data validates the main mechanisms proposed by the model.
First, the model predicts that large firms, who are more immigrant-intensive than small firms,
will experience a larger increase in terms of revenues. Second, given that ˆ > σ̂, larger firms
will increase their immigrant share relative to smaller firms. Such heterogeneity in the response
to immigration is expected to be larger in the tradable sector, where the relationship between
size and immigrant intensity is stronger.
We begin by estimating a regression as shown in equation 19:

agg
agg
ln(yj,m,k,t ) = θ1 Sm,t
+ θ2 Sm,t
log(empj,t−1 ) + θ3 Xj,t + δj + δk,t + δm t + j,m,k,t

(19)

where yj,m,k,t is an establishment-level outcome such as sales, for establishment j located in
agg
labor market m, industry k, in year t. The regressor Sm,t
is the share of immigrants in the total
wage bill of labor market m in year t, empj,t−1 is establishment size measured by employment,
and Xj,t are establishment-level control variables. This model allows for labor markets to be
in different linear trends as captured by δm t. It also includes industry-time fixed effects to
control for factors affecting all establishments in an industry over time and an establishment
fixed effect to control for unobservable characteristics that are time-invariant.
agg
We define the immigrant shock Sm,t
at the local labor market level as we aim to understand
how different establishments adjust within a labor market whenever there is an immigration
influx. The key parameter of interest is θ2 : if positive, it implies that a rise in the share of
immigrants in a labor market promotes faster growth for larger establishments compared to
smaller ones in the same market. Thus, θ2 > 0 will suggest that larger establishments respond
more to immigration than small establishments.

Even though the fixed effects and controls included in the empirical specification aim to capture
unobservable shocks and establishment heterogeneity, ordinary least squares (OLS) estimates
will be upward biased if, for example, productivity shocks at the local labor market level
improve establishment outcomes and attract migration inflows into the region. To address
these endogeneity concerns, we follow an IV approach inspired by Card (2001) and Ottaviano
et al. (2018), and define a shift-share instrument as shown in equation 20:

Zm,t =

GER
X Wage Billo,m,2003 1 + γo,t
Wage Billm,2003 1 + γtGER
o

(20)

where Wage Billo,m,2003 is the wage bill earned by immigrants from origin country o in labor
market m in our initial year 2003. Wage Billm,2003 is the total wage bill spent across all foreign
P
origin countries in 2003 ( o Wage Billo,m,2003 ). The initial share is interacted with a time-shifter
that captures the national growth rate, from 2003 to year t, of immigrants from origin o relative
to the working-age population growth in Germany. Thus, this shift-share instrument interacts
country-specific flows of migration with their initial differential presence in local labor markets
in Germany. The validity of this instrument relies on the assumption that the geographic
distribution of immigrants by origin in 2003 is not correlated with local economic conditions
in any year t once we control for fixed effects that capture unobservable differences across
establishments, industries, and local labor markets. The interaction term is instrumented by
Zmt log(empj,2003 ).
For the sake of the economic interpretation of the effect of an immigration shock, we compute
agg
the elasticity or semi-elasticity of yj,m,k,t to Sm,t
, denoted as yj,m,k,t , as follows:
yj,m,k,t

agg
≡ θ1 + θ2 log(empj,t−1 ) Sm,t

(21)

when the outcome variable of the regression is log(y), yj,m,k,t equals the elasticity of y, and when
the regression outcome variable is y, it equals the semi-elasticity.17 The elasticity of firm j’s
outcome yj,m,k,t to an immigration shock depends on both its size and the share of immigrants
in the labor market where it operates.

6.1

Results

We present the estimates of equation 19 using total revenues and the ratio of immigrant to
native wage bill as the outcome variable to show that larger firms expand more and become
more immigrant-intensive in response to an immigration shock.
Table 3 presents estimates for total revenues for the full sample in columns 1 to 3 and separately
for the tradable and non-tradable sectors in columns 4 and 5. Columns 6 to 8 present results
agg

Specifically, equals

∂yj,m,k,t Sm,t
agg
yj,m,k,t
∂Sm,t

and

∂yj,m,k,t agg
agg Sm,t ,
∂Sm,t

respectively.

using the immigrant to native wage bill ratio as the outcome. The OLS estimate in column 1
shows that, on average, establishments in local labor markets with larger increases in the share
of immigrants register larger revenue growth. Column 2 shows that the 2SLS estimate is lower
than the OLS estimate consistent with the hypothesis that OLS estimates are upward biased.18
The 2SLS estimate suggests that immigration into a local labor market has no statistically
significant impact on establishments’ revenues. However, the average effect masks significant
heterogeneity, uncovered in column 3. After accounting for the heterogeneous effect across
establishment sizes, the average effect is negative and strong. That is, an increase in the share
of immigrants in the labor market shrinks firms’ revenues on average, and increases the revenue
of large establishments relative to small establishments. The implied threshold size of the
establishment, above which the elasticity is positive, is 71 employees.
Columns 4 and 5 show that the heterogeneity in size is driven primarily by establishments in the
tradable sector, where large establishments grow their revenues significantly more than small
establishments. Establishments in the non-tradable sector do not seem to differentially respond
to the immigration shock, consistent with the patterns in Figure 3, where establishments in the
non-tradable sector presented a low correlation between immigrant share and size.
Columns 6 to 8 show the the 2SLS estimates for the firm-level ratio between immigrant and
native wage bill. Column 6 suggests that immigration into a local labor market has no impact
on the immigrant intensity of establishments, but once again, this result masks significant
heterogeneity across sectors. Column 7 shows that large firms in the tradable sector increase
their immigrant-intensity relative to small firms: firms with more than 33 employees increase
their immigrant-intensity, while smaller firms become more native-intensive. However, Column
8 shows that this heterogeneous effect across firm size is absent in the non-tradable sector, as
expected based on the relatively flat relationship between firm size and the immigrant-share
shown in Figure 3.
Table 4 presents the results in terms of elasticities by firm size and sector, which will be used
to compare the elasticities implied by our quantitative model. In the tradable sector, a 1%
increase in the immigrant share decreases establishments’ revenues in the lowest size decile by
0.42% while increasing establishments’ revenues in the highest decile by 2.16%. The elasticity of
revenues in the non-tradable sector, on the other hand, seems to be similar across establishments
of different size.
We find a similar pattern in each sector when looking at the response of the relative wage bill
between immigrants and natives across size deciles. In the tradeable sector, a 1% increase in
the share of immigrants in the labor market would increase the ratio of an establishment in
the lowest decile by 0.01 while increasing the ratio for an establishment in the highest decile
by 0.21. The elasticities across deciles in the non-tradable sector seem to be decreasing with
18

First stages can be found in Appendix Table 16.

size but are not statistically significant.
Table 3: Heterogeneous benefits of immigration
Log of Revenues
Sector

θ1

Immigrant-Native Wage Bill

All

Tradable

(1)

(2)

(3)

(4)

NonTradable
(5)

5.83***
(1.98)

2.99
(3.29)

-31.86***
(11.47)
7.49***
(2.46)

-57.56***
(16.95)
13.28***
(3.66)

0.28

θ2
Average y

All

Tradable

(6)

(7)

NonTradable
(8)

6.81
(17.78)
-0.44
(3.48)

0.2
(1.7)
0.18
(0.36)

-3.13*
(1.72)
0.9**
(0.43)

6.14
(4.12)
-1.07
(0.78)

0.54

0.26

0.06

0.08

0.05

N observations
N establishments

3507
949

1974
532

1533
417

3507
949

1974
532

1533
417

Estimation
1st stage F-stat

OLS

2SLS
372.23

2SLS
35.85

2SLS
29.47

2SLS
15.53

2SLS
35.85

2SLS
29.47

2SLS
15.53

Table 4: Response to immigration by firm size
Size deciles
Tradeable

-0.28
0.02

-0.06
0.03

0.03
0.05

0.2
0.07

0.41
0.08

0.57
0.1

0.25
0.14

0.23
0.11

0.24
0.1

Revenues
-0.42
Relative Immigrant WB 0.01
Non-Tradeable
Revenues
Relative Immigrant WB

0.81 1.41
0.11 0.16
8

0.13 0.22 0.22 0.21 0.21
0.08 0.07 0.06 0.04 0.03

10
2.16
0.21

0.2
0

0.27
-0.06

Note. We rank establishments in terms of employment and for each decile, compute the mean elasticity of
revenues and semi-elasticity of spending in immigrants relative to natives in response to a 1 percent change in
the local labor market immigrant share. We compute the average of 21 for each decile using the same sample
as in Table 3.

In Appendix F, we also show that export revenues are more elastic than domestic revenues, as
predicted by the model. These estimates imply that for every 1% increase in the immigrant
share of the labor market, domestic revenues increase by 0.44%, whereas export revenues increase by 1.15%. Since the response of export revenues is stronger than domestic revenues, this
channel can explain part of the heterogeneous effects found in Table 3. Large establishments,
which are more likely to be exporters, may adjust more to the immigration shock because they
are able to expand their export revenues, whereas for small firms, expansion is constrained by
the size of the domestic market.

Appendix F also shows alternative specifications of equation 19, where we remove the industrytime fixed effects, the local labor market time trends, and the firm controls. Overall, the
qualitative implications of our results hold under the alternative specifications. We also run a
set of specification tests to verify the validity of our instrument following the recent literature on
shift-share instruments as suggested by Goldsmith-Pinkham et al. (2020) and Borusyak et al.
(2021), among others. We find no evidence of pre-trends, and other labor market characteristics
drive little variation in the initial shares used to construct the shift-share instrument.

6.2

Predicted Treatment Effects: Data vs. Model

As a final step, we assess whether our model can generate counterfactual predictions that match
the observed heterogeneous treatment effects across employer sizes estimated in Table 4. This
is a key validation of the model as the reduced form estimates in this section have not been
targeted at all for the estimation of the model. First, we use our estimated model to compute,
for each firm, the revenue and relative wage bill elasticities in response to a 1% change in the
immigrant share in each sector. Then, we divide the firms in the model into size deciles and
calculate the mean elasticity for each decile.19 Second, we take the estimated elasticities by
decile from Table 4 and compare them to the estimated elasticities in the model.
As shown in Figure 4, the model does a good job in replicating the relative treatment effects from
our empirical exercise. The changes in the tradable sector predicted by the model replicate the
revenue responses in the data almost exactly until decile seven and predict a more conservative
response to immigration for firms in the highest three deciles. For the non-tradable sector,
the model does a good job in replicating the treatment effects in the data across deciles, where
establishments of different sizes do not respond differently to the immigration shock. The model
also captures that large firms become more immigrant-intensive than small firms, particularly
in the tradable sector.20

Aggregate implications

We proceed to quantify the economic and welfare consequences of an inflow of immigrants
into Germany. Section 7.1 evaluates the main forces shaping the adjustment of the economy
to the immigration shock. Section 7.2 quantifies the bias in the estimated welfare gains for
native workers when using a model that does not capture the observed heterogeneity in the
19

Similar to the counterfactual discussed in Section 7, we lower migration costs to each sector such that the
total number of immigrants in Germany increases by 1%.
20
The model-generated elasticities include general equilibrium changes in prices and quantities due to immigration, while in the data, we control for aggregate changes through industry-time fixed effects and local labor
market trends. Given this discrepancy, we should not expect the levels of the elasticities to necessarily match
between model and data. Instead, the key object to compare when judging whether the model can replicate
the heterogeneous responses observed in the data is the relative elasticity across size deciles.

Figure 4: Predicted treatment effects: Model vs data
(a) Revenues - Tradable sector

(b) Revenues - Non-Tradable sector

(d) Relative Wage Bill - Non-Tradable sector

Note. For the model, we rank establishments in terms of revenues into 10 deciles, with decile 1 being
the establishments with lowest revenues. In the top two panels, we compute the elasticity of revenues to
a 1% increase in the immigrant share and calculate the mean elasticity for firms in each decile. For the
data, we use the sector-specific elasticities by size decile presented in Table 4. In the bottom two panels,
we calculate, for each establishment, the change in the ratio between the wage bill of immigrants and the
wage bill of natives in response to a 1% change in the immigrant share. We then compute the average for
each size decile in both the data and the model.

immigrant share across firms. Finally, Section 7.3 discusses the role of trade for our quantitative
results.

7.1

Quantitative Exercise

The economic adjustment to the immigration shock takes the form of equilibrium changes in
prices, wages, welfare, and the reallocation of workers across sectors and firms. The size of the
shock mimics the magnitude of the immigration wave that occurred in Germany between 2011
and 2017. According to the OECD, the total number of immigrants in Germany went from
10.55 million in 2011 to 12.74 million in 2017, a 20.7% increase. While our data ends in 2011,
we can use the model to calculate the new equilibrium when the total number of immigrants
in Germany increases exogenously by 20%. To do so, we change the migration cost from the
RoW to Germany, φk,x,g , such that it increases the total stock of immigrants by 20%.21 For our
quantitative results, we set the numeraire to be the wage in the RoW, wx .
We define welfare of natives, denoted by Wg , as their real labor income:
P
Wg =

k (Lg,k wg,k )/Ng

(22)

As shown in Table 5, the welfare of native workers would increase by 0.24%, which represents
$113 per native worker every year or $4 billion for the aggregate economy. Such welfare gains
are mainly explained by the drop in the cost of the consumption basket: 70% of the gains
can be explained by the drop in the price index, while only 30% is explained by the increase
in per capita labor income. The decrease in the price index is mainly driven by the tradable
sector because its price index drops more strongly than the non-tradable sector, and because it
accounts for a larger share of the consumption basket of Germans (almost 70%). Welfare also
increases because wages are higher due to immigration, as the increase in the scale of production
and associated demand for native labor offsets the substitution effect between natives and
immigrants.
The welfare gains of firm owners is significantly larger than for native workers because they
experience the same price decreases but do not compete with immigrants in the labor market.
Their real income from firm profits increase by 1.22% due to the drop in production costs and
increase in profits induced by immigration, amounting to a gain of $15 billion.
Table 6 narrows the analysis to the sector level and shows the sectoral effects on employment
and wages in terms of labor units (i.e., number of workers) and effective units. The influx
of immigrants decreases the relative wage between immigrants and natives, and both sectors
become more immigrant-intensive. As they become more competitive, both sectors expand
21

In Appendix G, we show our results for different changes in the stock of immigrants.

Table 5: Effect of immigration on welfare

Native Workers
Firm Owners

Real Income

Price Index

Nominal Income

Monetary Gains

0.24%
1.22%

-0.17%
-0.17%

0.07%
1.04%

$4B
$15B

Note. We compute the changes on the key endogenous variables of going from the observed
equilibrium to an equilibrium where the number of immigrants is 20% higher. Income refers
to wages for workers and profits for firm owners. Monetary gains are computed using average
wages PPP adjusted at 2019 dollars and total workforce numbers from the OECD. We use data
from LIAB to separate the share of the wage bill by sector.

their production and total employment in terms of effective units. Employment of native
workers decreases in the tradable sector as the least productive native workers are substituted
by immigrants, and they reallocate to the non-tradable sector. This result differs from the wellknown Rybczynski (1955) theorem, which predicts that production of the immigrant-intensive
sector increases and production of the native-intensive sector decreases, so natives reallocate
from the native-intensive sector to the immigrant-intensive sector. This theorem builds on the
assumption that the domestic share of labor does not respond to an immigration shock, which
does not hold in our setting. In our model, the domestic share decreases in both sectors but
decreases more in the immigrant-intensive sector. Thus, even though output increases more
in the immigrant-intensive sector than in the native-intensive sector, the immigrant-intensive
sector does it by hiring more immigrants. Some of these immigrants replace less productive
native workers, who are now reallocated to the native-intensive sector.
Wages per native worker increase in both sectors. In the tradable sector, this is due to selection
as lower ability natives reallocate to the non-tradable sector, and those natives who stay in
the tradable sector are, on average, of higher ability. In the non-tradable sector, there are two
counteracting effects. On one hand, lower ability natives get in the sector decreasing average
wages. On the other hand, the additional domestic demand created by the new immigrants
increases demand for the sector pushing effective wages up. Overall, the latter effect dominates,
and workers in both sector earn higher wages due to immigration.
The benefit of immigration for firms is large in the aggregate, but it masks significant heterogeneity for firms of different sizes in the tradable sector. From the top panel of Figure 5, three
facts stand out. First, there is a large dispersion in the within-sector price responses and the
initial exposure to the immigration shock, which can be a quantitatively important determinant
of the aggregate results described before. Second, the cross-sectional differences in the initial
exposure (1 − sdj ) go a long way in explaining differences in price responses (Figure 5a). Third,
the exposure to the shock is significantly higher for larger firms (Figure 5b). Thus, the positive
relationship between firm size and immigrant intensity, as observed in the data, drives the positive relationship between firm size and price decrease in the model. Larger firms, by virtue of
being immigrant-intensive, are more exposed to the decrease in immigrant wage than smaller
28

Table 6: Effect of immigration on employment and wages
Labor units
Employment Tradable
Total
Native
Immigrant

Effective units

Non-Tradable

Tradable

Non-Tradable

2.49%
-0.11%
20.01%

2.09%
0.23%
20.01%

4.49%
-0.09%
16.51%

3.78%
0.20%
16.51%

0.07%
-6.32%

0.07%
-6.26%

0.05%
-3.51%

0.11%
-3.45%

Wages
Natives
Immigrants

Note. We compute the changes on the key endogenous variables of going from
the observed equilibrium to an equilibrium where the number of immigrants is
20% higher.

firms, and their unit cost of production and price decrease more than the cost of small firms.
As a result of immigration, larger firms increase their market share. Even though larger firms
gain market share to small firms (Figure 5c), they reduce their share in the labor market of
natives (Figure 5d) because immigrants crowd-out natives at immigrant-intensive firms (large
firms), and these natives are reallocated to native-intensive firms (small firms).

7.2

Role of Heterogeneity in Immigrant Share

In this section, we asses the importance of the documented heterogeneity in quantifying the
adjustment of the German economy to an immigration inflow. To that end, we compare the
model predictions to the same immigration shock across two models: the heterogeneous model
and the homogeneous model. The heterogeneous model is the general model presented in Section
4, whereas the homogeneous model is a particular case where the parameters generating the
heterogeneity in immigrant share are turned off. Importantly, both models are recalibrated
to match the same aggregate moments and are subject to the same immigration shock (20%
increase in the stock of immigrants).22 The homogeneous model, however, does not match the
observed cross-sectional heterogeneity in the immigrant share; that is, Var(sdj ), Cov(sdj , revj ),
and the share of firms hiring immigrants. To estimate the homogeneous model, we impose the
following restrictions: σf,T = σf,N T = σψ,f,T = σψ,f,N T = fimm,T = fimm,N T = 0.
As shown in the last row of Table 7, the homogeneous model underestimates the welfare gains
by 11% because it predicts a weaker increase in workers’ income and a weaker drop in the price
index. As explained in section 4.1, the increase in real wages is stronger in the heterogeneous
model because firms choose different immigrant shares. Hence, immigration increases the specialization of immigrants and natives in producing different varieties, which makes them less
22

In terms of equation 16, it means that in both economies, dlog(S agg ) is the same. The estimates of
parameters that are not estimated by SMM (e.g. and σ) are the same in both models. Appendix G.3 presents
the recalibrated parameters under homogeneity.

Figure 5: Responses to immigration across sectors and firms.
(a) Change in domestic price

(b) Immigrant intensity

(d) Change in native employment share

Note: The x-axis of figure 5a groups firms into deciles in terms of their immigrant intensity (1 − sdj ), and the
ex-axis of figure 5b, 5c, and 5d does it in terms of their total revenues. The y-axis in all figures measures the
average change in the variable in the counterfactual equilibrium where immigrant stock increases by 20%
relative to the initial equilibrium.

substitutable in the aggregate. Since the competition faced by natives in the labor market
due to immigration is weaker, there is a lower downward pressure on the wages of natives.
The stronger drop in prices in the heterogeneous model, especially in the tradable sector,
is explained by larger firms being more immigrant-intensive: large firms, by virtue of being
immigrant-intensive, experience a relatively strong drop in the price of the good they produce
and, given that they account for a larger share of the consumption basket, their price drops
affect the aggregate price index of the economy.

The results of this section highlight the importance of firm-level hiring decisions in understand-

Table 7: Welfare effects with and without firm heterogeneity on the immigrant share
Welfare Nominal
Workers
Wage
Heterogeneous
Homogeneous
Homog/Heterog

0.24%
0.22%

0.07%
0.06%

Price
Index

Price Index
Tradable

Price Index
Non tradable

-0.17%
-0.16%

-0.18%
-0.16%

-0.15%
-0.15%

89%

Note. For both models, we compute the changes on the key endogenous variables of going
from the observed equilibrium to an equilibrium where the number of immigrants is 20%
higher. The heterogeneous model is our baseline model. The homogeneous model is an
alternative model where all firms are equally intensive on immigrants.

ing the consequences of immigration. Immigration leads to within-industry reallocations of
native workers across firms. One reason why this reallocation matters in the aggregate is that
it affects the (endogenous) immigrant-native elasticity of substitution. However, even with the
same aggregate elasticity, the homogeneous model would underestimate the welfare gains of
immigration. In Appendix G, we quantify the welfare gains of the homogeneous model with
the same aggregate elasticity that the one implied by the heterogeneous model, and show that
the bias is not eliminated and remains large (8% approx.). Thus, even after conditioning on
the same change in domestic labor share and aggregate native-immigrant elasticity of substitution, the micro structure of the economy affects the measurement of the welfare gains from
immigration.

7.3

The Quantitative Role of Trade

Exports and trade have a key role in the quantitative results of increasing immigration and the
size of the bias. We compare our baseline model with an alternative model where Germany
and the RoW are in autarky, such that trade is not allowed between countries. This model
is analogous to a model where the fixed cost of selecting into trade goes to infinity (e.g.,
fx → ∞).
As shown in Table 8, if countries cannot engage in international trade, the price decrease
induced by immigration is too strong. The model with no trade overstates the decrease in the
price index by more than double the decrease predicted by the baseline model. Both trade and
migration lower the marginal cost of production and, in turn, the price index. When trade is
not allowed, migration becomes more important as a source of reducing the cost for consumers
as they cannot adjust their consumption through trade.
However, the relationship between trade and welfare goes in the opposite direction when considering the wage component. In the baseline model with trade, demand is more elastic, and total
production expands more than in the no-trade model in response to immigration. The more
elastic product demand increases labor demand for both immigrants and natives and partially

compensates the competition effect in the local labor market. As shown in Table 8, the model
with no-trade predicts a negative impact on wages, as demand does not respond as much, and
the competition effect between natives and immigrants dominates. As explained by Burstein
et al. (2020), if immigrants work for a sector where goods are traded, immigration imposes less
of a downward pressure on wages because the demand is more inelastic. While both effects are
at play, the change in price index dominates the quantitative difference in terms of real wages
between the baseline and the no-trade model. The model with no trade overstates the welfare
gains of immigration by 41%.
Finally, we compare the no-trade model with a model with no trade and homogeneous immigrant intensities. The homogeneous model underestimates the gains from immigration by 9%,
which is lower than the bias in the model with trade (11%). Trade amplifies the inequality in
sizes across firms in the tradable sector, which in turn, amplifies the differences in immigrant
intensities across firms.
Table 8: Comparing the baseline model with a model no-trade model
Welfare

Nominal Wage

Price Index

Revenues

Baseline
No Trade

0.24%
0.34%

0.07%
-0.04%

-0.17%
-0.37%

1.05%
0.98%

No Trade and homogeneous

0.31%

-0.02%

-0.33%

0.98%

Note. The values represent the percent change of key variables after a 20% increase in the
stock of migrants.

Comparing our results with the literature

To put our results into context, it is important to understand the institutional framework in
Germany during our study period. We focus on the years between 2003 and 2011, before
Germany unified its labor market with other EU countries. Hence, this is a period where a
majority of immigrants needed a guaranteed employment offer in order to migrate. Such policy
context is important because firms had a fundamental role in determining what immigrants
came into the country. Similar setup can be found in the United States, the largest destination
country of immigrants, through the H-1B, H-2B, and L-1 visa programs, among others. In
these programs, firms need to sponsor workers’ visas for them to be able to migrate to the
country. The Canadian immigration system is similar with its point-based system, where
immigrants with a guaranteed employment offer get substantially more points to qualify for
immigration.
Differences in immigration policy across countries can reconcile why firm-level studies find,
what at first may seem contradictory. Mitaritonna et al. (2017) find that larger French firms
32

are more immigrant-intensive, but small and low-productivity firms experience the most gains
from immigration. Arellano-Bover and San (2020) find that immigrants in Israel initially select
into small firms, while Mahajan (2020) finds that high-productivity firms in the United States
benefit the most from immigration. In the context studied by Mitaritonna et al. (2017) and
Arellano-Bover and San (2020), immigrants were easily available to firms, while in our setup
and Mahajan (2020), migration policy required firms to invest resources for recruiting and
sponsoring immigrants. Therefore, our framework is well suited to study immigration whenever
migrants are not easily available in the labor market, and firms have an active role in deciding
which immigrants come into the country.
In terms of the magnitude of our findings, our quantitative estimates are somewhat larger than
those estimated by Caliendo et al. (2021), who predict immigration after the EU labor market
integration increases welfare for the original EU members by just 0.04%. Our larger gains
can be explained due to allowing immigrants and natives to be imperfect substitutes, while in
Caliendo et al. (2021) they are considered perfect substitutes within skill group. Their estimates
also are mainly driven by the UK, which opened their goods and labor market simultaneously.
They conclude that a phased policy like Germany, where the labor market was opened in a
later period, would likely have created higher welfare gains.

Conclusion

In this paper, we document a large degree of heterogeneity across employers regarding their
immigrant share, and revisit the old question of the impact of immigration on the welfare of
native workers. When immigration increases by 20%, our model predicts that both the tradable
and non-tradable sectors expand in terms of revenues and profits due to the drop in unit cost
induced by the inflow of immigrants. This expansion is more pronounced in the tradable sector,
where firms are more intensive in immigrant labor. The immigration inflow also induces the
tradable sector to become more immigrant-intensive, which triggers a reallocation of the least
productive natives from the tradable sector toward the non-tradable sector. We find that native
workers and firm owners in both sectors experience higher wages and profits, respectively, and
lower prices due to immigration. The welfare gains amount to $4 billion for native workers and
$15 billion for firm owners.
Most of the literature has assumed that firms are homogeneous in terms of hiring decisions of
immigrants, which is at odds with the data and leads to biased welfare gains from immigration.
First, when firms are homogeneous, the elasticity of substitution between immigrants and
natives in the labor market coincides with the within-firm elasticity. However, when firms
are heterogeneous, the aggregate immigrant-native elasticity of substitution depends on the
within-firm elasticity and the elasticity of substitution across firms or goods. Thus, having
different immigrant-intensities across firms allows for natives and immigrants to specialize in
33

working for different employers, which makes them less substitutable in the aggregate labor
market. Second, when firms are heterogeneous, the gains are largely concentrated among the
largest and most productive employers, which induces an additional aggregate productivity
gain. These two forces lead to potentially large biased estimates of the welfare gains from
immigration. We find that if we ignore this heterogeneity, the welfare gains from an increase
in immigration would be underestimated by 11%.

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Summary statistics

In Table 9, we present the average employment, college employment, and immigrant distribution
by origin region for our sample. We split the establishments in the sample into the tradable
and non-tradable sectors and calculate summary statistics for years 2003 and 2011.
Table 9: Descriptive Statistics
Tradable

N establishments (unweighted)
Mean Employment
Mean Employment - College

Non-Tradable

2003

2011

2003

2011

1,530
45.0
4.5

1,426
45.9
5.8

2,148
39.2
3.0

2,379
36.5
2.9

Share of employment by origin region
Germany
EU (FR, GB, NL, BE, AT, CH, FI, SE)
EU (ES, IT, GR, PT)
EU, joined after 2004
Europe, other
Turkey
Former Yugoslavia
Asia - Pacific
Africa and Middle East
Americas

90.97% 91.15%
1.03% 0.97%
1.94% 1.69%
0.63% 0.74%
0.80% 1.10%
2.73% 2.55%
0.79% 0.61%
0.41% 0.52%
0.52% 0.46%
0.16% 0.21%

92.66% 91.13%
0.74% 0.70%
1.22% 1.40%
0.68% 1.22%
0.73% 1.02%
1.71% 2.06%
0.73% 0.70%
0.76% 0.64%
0.63% 0.75%
0.14% 0.36%

Note: The sample is restricted to establishments with more than 10 employees.

Empirical Evidence for Fixed Cost Assumptions

This section presents stylized facts that motivate the modeling assumption that firms face fixed
costs to hire immigrants and that these costs have to be paid whenever the firm expands the
set of countries where it hires immigrants from. In the data, countries of origin are grouped in
nine blocks as explained in Section 2.
First, as shown in Table 10, there is a significant mass of small firms that do not hire any
immigrants. If immigrants and natives are imperfect substitutes, as documented extensively in
the literature (Peri and Sparber, 2009, 2011), firms would optimally choose to hire a strictly
positive level of both native and immigrant workers, which contradicts the results in Table 10.
This pattern could be rationalized if firms have to pay a fixed cost to hire immigrants. Even
if immigrants and natives are complementary inputs, if profits earned by small firms are not
enough to afford the fixed cost of hiring immigrants, their choice set is restricted to native
38

workers. Fixed costs thus imply that small firms are more likely to hire only native workers, as
shown in Figure 1.
Table 10: Share of firms that hire immigrants by firm size
Size deciles
Share of firms

0.39

0.36

0.43

0.50

0.53

0.63

0.66

0.80

0.87

0.97

Second, larger firms source immigrants from more countries. Figure 6 shows a positive relationship between firm size, measured by the wage-bill decile, and the median and mean of the
number of origins the firm sources immigrants from. Table 11 shows the OLS estimate of a regression of the number of source countries on firm size in the previous year after controlling for
sector-year fixed effects. The estimate is positive and statistically significant at a 1% confidence
level.
Figure 6: Number of origin regions by establishment size

Note. We divide all establishments with more than 10 employees into total wage bill
deciles, with decile 1 including the smallest establishments and 10 the largest.

Third, firms that increase the number of sourcing countries tend to do it by adding a single
additional origin, as opposed to multiple origins at the same time. Each row in Table 12
shows the number of countries that an establishment sourced immigrants from in period t − 1
(N ct−1 ), each column shows that number for period t (N ct ), and each cell contains the number of
establishments that keep or increase the number of countries between t−1 and t. Establishments
that increase the number of origins where they hire immigrants from are more likely to go from
N ct−1 to N ct−1 + 1 than to any other number of countries. This fact would not arise if firms
were supposed to pay a fixed cost to source immigrants from any origin as firms would optimally
start hiring from all countries after paying that cost. However, if firms were supposed to pay a
cost for every additional origin they source immigrants from, they would start hiring from one
country at a time.

Table 11: Number of sourcing countries and firm size: OLS estimates
Number of
countries
Employment (in logs)

Number of
countries

1.40***
(0.03)

Wage Bill (in logs)

1.12***
(0.03)

N observations
N establishments

15,095
2,478

Note. ∗ ∗ ∗ = p < 0.01, ∗∗ = p < 0.05, ∗ = p < 0.1. We
control for 2-digit industry-time fixed effects and local
labor market time trends. Standard errors are clustered
at the establishment level. Sample is restricted to establishments with more than 10 employees.

Table 12: Number of immigrant origin countries
N ct−1
0
1
2
3
4
5
6
7
8
9

0
5,108

1
368
2,014

2
41
319
1,160

N ct
3
4
5
*
*
*
64
*
*
259 47
*
766 179 40
512 144
125

6
7
*
*
*
*
*
*
*
*
33
*
372 106
332 107
310

8
*
*
*
*
*
26
26
88
436

9
*
*
*
*
*
*
*
*
70
406

Note. Sample is restricted to establishments with more than 10 employees. N ct
stands for the number of regions the establishment hires immigrants from at time
t. Number of regions can go from 1 to 9. Cells with an “*” have less than 20
observations and cannot be disclosed.

Fourth, the year that the firm adds an additional country, it starts hiring a large number of
employees from that country. This jump in the number of employees hired from the additional
country is consistent with firms paying a fixed cost for any additional sourcing country. If this
were not the case and the cost were variable, firms would tend to start hiring small quantities
of those immigrants. Table 13 shows the distribution of the number of new hires with respect
to the size of the workforce of the firm for two sample of firms. The first sample (“All”) is
the sample of firms that started hiring from a new source country, and the second sample
(“Top 5 deciles”) is the subsample of them that are in the top 5 deciles of the employment size
distribution. The first row of the following table shows that the average number of employees
from the new source is 3.8% of the total employment of the firm, and there is a significant
mass of firms (10%) that hire approximately 10% or more of their employment in new-country
40

immigrants. These results do not seem to be driven by firms hiring only few workers that still
represent a large share of their small workforce because results remain in the subsample of the
Top 5 deciles.
Table 13: Immigrants from new source as a share of firm total employment
Sample
All
Top 5 deciles

Mean
3.80
3.90

1% 5% 10%
0.00 0.06 0.10
0.00 0.05 0.9

Percentiles
25% 50% 75%
0.27 0.87 2.98
0.24 0.75 2.93

N
90% 95% 99%
9.02 16.85 44.63
10.00 18.55 46.57

3617
3224

Note: An observation is an establishment-year. We rank establishments who start hiring from a new origin
region in terms of the employment from the new region relative to the establishment’s total employment.
The sample “All” includes those observations that increase the number and the sample “Top 5 deciles”
contains the subsample of firms that belong to the top 5 deciles in terms of employment.

Fifth, firms hiring immigrants from more countries tend to be more immigrant-intensive. This
is exactly what the model predicts in equation 10 and is corroborated by Figure 7, where we
group firms by the percentage of their payroll spent on immigrants. Figure 7 shows that firms
that are more intensive on immigrants also source immigrants from more countries.
Figure 7: Number of origin regions by immigrant share

Note. We group establishments by the share of the wage bill spent on immigrants
into 20 bins (those who spend 0-1%, 1-2%, etc.). For firms in each bin, we plot
the mean and median number of origin countries. In our sample, we have 9
immigrant origin regions, which are listed in section 2.

There may be a mechanical correlation between the number of sourcing countries and the
number of immigrants, as the total number of immigrants that the firm hires can drive the
observed relationship between number of countries and immigrant share. To suggest that the
changes in immigrant share are mainly associated to the number of sources countries, Table
14 shows that, even after controlling for the total number of immigrants hired, the correlation
between immigrant share and the number of countries is significant and strong. Moreover,
41

a variance decomposition based on these estimates suggests that 10% of the variance in the
immigrant share is explained by differences in the extensive margin (number of countries), and
only 3% is explained by the intensive margin (number of immigrants).
Table 14: Immigrant share: Intensive vs Extensive Margin: OLS estimate
Immigrant Immigrant
share
share
N countries

0.016***
(0.0008)

0.012***
(0.0009)
5.23e-03
(1.07e-06)

17,501
2,485

N immigrants
N observations
N establishments

Note.∗ ∗ ∗ = p < 0.01, ∗∗ = p < 0.05, ∗ = p < 0.1.
We control for 2-digit industry-time fixed effects
and local labor market time trends. Standard errors are clustered at the establishment level. Sample is restricted to establishments with more than
10 employees.

To conclude, we interpret these stylized facts as evidence in favor of an environment where
large firms are more immigrant-intensive than small firms because they can afford to pay more
fixed costs to hire immigrants from different origins.

C
C.1

Model Derivations
Sourcing Decision Details

In this section, we describe step by step how we get to the immigrant wage index expression in
equation 7. Following equation 6, we know the price index for foreign labor is as in equation
23:

Z
Wx,j =

1
! 1−κ

δoκ wx1−κ do

(23)

Σj

where δo is a source-country specific productivity assumed to be a Pareto random variable with
the following cumulative distribution and density function:
ξ
δ̄
F (δ) = 1 −
δ

and g(δ) = ξ δ̄ ξ δ −ξ−1

(24)

where δ̄ and ξ are the scale and shape parameters, respectively. Since the firm needs to pay a
fixed cost fj for each additional country they hire from, they will just hire from countries with
42

a δ > δj∗ , for a given δj∗ . The mass of countries that the firm hires from is then nj = F (δ >
δj∗ ) = δ̄ ξ (δj∗ )−ξ . With this result, we can calculate the price index of foreign labor as in equation
25:

Wx,j =

wx1−κ

1
! 1−κ

∞

δj∗

δoκ ξ δ̄ ξ δ −ξ−1 dδ

ξ δ̄ ξ κ−ξ
δ
κ−ξ

1
∞ ! 1−κ

δj∗

ξ δ̄ ξ ∗ −(ξ−κ)
(δ )
ξ−κ j

1
1−κ

if ξ − κ > 0
(25)

Since the mass of countries the firm sources from is nj = δ̄ ξ (δj∗ )−ξ , we can now compute the
foreign price index as in equation 26:

−

Wx,j = wx

C.2

1
κ

δ̄ κ−1
|

ξ
ξ−κ
{z
Z̄

1
1−κ

1 ξ−κ
κ−1 ξ
|
{z
}
ι

(26)

}

Equilibrium Equations

The equilibrium in this model is defined as a set of prices, wages, and labor allocations such
that: workers optimally choose the industry and destination country d, k to work for, consumers
in each location choose how much of each variety to purchase to maximize utility, firms choose
the sourcing strategy and export status to maximize profits, labor markets clear, and trade is
balanced. We set the wage in the RoW (wx ) to be the numeraire. Formally, the equilibirum
conditions are the following:
1) Consumer budget constraint. In a given country, natives and immigrants have identical
preferences. The total expenditure in Germany (Yg ) and RoW (Yx ) are shown in equation
27:

Yg =

(wg,k Lg,k + wg,x,k Lg,x,k + Πg,k )

Yx = w x L x + Π x

(27)

where Lg,k is the total number of German effective units of labor in sector k, Lg,x,k is the
number of effective immigrant units in Germany working in sector k, and wg,k , wg,x,k are the
respective effective wages. Πg,k are the total profits in sector k in Germany. wx , Lx , and Πx
are the effective wages, effective labor, and total profits in RoW.
2) Trade balance. Total income from exports in Germany is equal to the total import expenditure as in equation 28:

T
1 (exporterg,j = 1) pTj,x,g yj,x,g
=

1 (exporterx,j = 1) pj,g,x yj,g,x

(28)

3) Total labor market clearing. In each industry, the expenditure of labor by industry k equals
the number of effective units supplied by the labor market times the effective wage paid by
that industry. The market clearing conditions 29-31 require that demand for effective units of
native and immigrant labor equals supply in each industry and country:
X

1
ν
dj,k = Ag,k
(πg,k )

ν−1
ν

H̄Ng

(29)

1
ν

xj,o,k = Ax,k (πx,g,k )

ν−1
ν

H̄ Nx

(30)

ν−1
ν
dj,x = Ax,k
(πx,x,k ) ν H̄ Nx

(31)

Equation 29 stands for the market clearing condition for natives in Germany, equation 30 for the
market clearing condition for immigrants in Germany, and equation 31 for the market clearing
of workers that stay in RoW. The parameter H̄ stands for the Gamma function evaluated at
1 − κ1

Welfare Response to Immigration

We focus on a closed economy with one sector, we choose the wage of natives as the numeraire,
and assume that the fixed cost fimm is zero (but the firm-specific fixed cost fj is unrestricted).
We present the expression for the change in the welfare of natives workers in four steps.
Step 1: Express dlog(sdj ) as proportional to dlog(sd1 ).
The profit function and the corresponding first order condition with respect to sdj are:
θ+1
Πj = Aψjσ−1 sχdj − Bfj (s−1
dj − 1)

ψjσ−2 s−χ+1+θ
= fj C(1 − sdj )θ
dj
where A, B, and C are general equilibrium variables that are common to all firms, χ =

−1
−1>0 .
and θ = ι( − 1)

σ−1
−1

The first order condition for firm j and firm 1 implies that:
(χ + 1 + θ +
or
dlog(sdj ) =

θ
θ
) dlog(sdj ) = (χ + 1 + θ +
) dlog(sd1 )
1 − sdj
1 − sd1

αj
1
dlog(sd1 ) with αj =
>0
α1
χ + 1 + θ + θ(1 − sdj )−1

(32)

Step 2: Express dlog(sdj ) as proportional to dlog(Sdagg ).
By definition, the aggregate domestic share is the total wage bill spent on natives divided by
the total wage bill:
Sdagg

P
X W Bj
X
j W Bdj
P
=
= P
sdj =
ωj sdj
W
B
j
j W Bj
j
j |
j
{z }
ωjW B =ωj

where ωjW B is the share of firm j in the wage bill of natives and happens to also be the share
in revenues, ωj . In what follows, we use this fact and keep the notation as ωj .
The change in the aggregate domestic share is then given by:
dlog(Sdagg )

X
j

ωj sdj
P
dlog(ωj ) + dlog(sdj )
j ωj sdj
| {z }

(33)

ωjS

where ωjS is the share of firm j in the aggregate domestic share.
Next, we find an expression for dlog(ωj ) as a function of dlog(sdj ). To that end, we use firm
j’s optimal demand for natives and the definition of ωj :
σ−1
D
rj = s−χ
→ dlog(W Bj ) = dlog(D) − χdlog(sdj )
σ
ψj dj
X
W Bj
ωj = P
→ dlog(ωj ) = dlog(W Bj ) −
ωl dlog(W Bl )
l W Bl
l

W Bj =

where D is a general equilibrium variable common to all firms.
The expression of dlog(ωj ) as a function of dlog(sdj ) follows from combining these last two
expressions:

X
dlog(ωj ) = −χ dlog(sdj ) −
ωl dlog(sdl )
(34)
l

This expression, together with 32 and 33, implies that the change in aggregate share can be

expressed as a function of the change in sd1 :
dlog(Sdagg ) =

dlog(Sdagg )

ωjS (−χ dlog(sdj ) −

ωl dlog(Sdl ) + dlog(sdj )

ωjS

− χ(αj −

(35)

ωl αl ) + αj dlog(sd1 )

In a more compact way, it reads as:
dlog(Sdagg ) =

ωjS

− χ(αj − ᾱ) + αj dlog(sd1 )
{z
}
|

(36)

βj

with ᾱ ≡

ωl αl .23

Expressions 37 and 32 let us express individual changes in domestic share as a function of the
aggregate change:
X
αj
βl
(37)
dlog(sdj ) = dlog(Sdagg ) with β =
β
l
Step 3: Express welfare change into a component observable with aggregate data and a component that requires micro-level data.
The welfare gains from immigration in this simplified model are given by the drop in the price
index induced by immigration. The change in the price index (relative to the numeraire good)
is a weighted average of the changes of individual prices which, in turn, are proportional to the
change in the domestic share:
dlog(P ) =

ωjrev dlog(pj )

ωjrev dlog(uj )

dlog(sdj )
=
dlog(wd ) +
−1
j
P
j ωj dlog(sdj )
= dlog(wd ) +
−1
X

where we used the fact that ωj =

p1−σ
,
P 1−σ

ωjrev

(38)

ωjrev = 1, and equations 5 and 8.

We can express the change in the price index as a function of the change of the aggregate share
and an additional factor by plugging equation 37 into equation 38.
23

If all firms choose the same immigrant-share, dlog(Sdagg ) = dlog(sdj ).

The last two expressions and the optimal pricing implies:
dlog

P
wd

dlog(Sdagg )
−1

αj
β
j
|
{z
}

ωj

Γ̃ {sdj ,ωj };σ,

This expression shows that the change in the price index can be computed only if firm-level
data on the market share and immigrant intensity are available.

Step 4: Determine if the bias is larger or smaller than one.
For the sake of the mathematical exposition, we work with the inverse of Γ̃, which takes the
following shape:

P S

−1 P ω S βj
j ωJ − χ(αj − ᾱ) + αj
j j
=
Γ̃ {sdj , ωj }; σ,
= P
ᾱ
j ωj αj
and can be rewritten as in 39 by adding and subtracting

ωjS ᾱ:

P
P

−1
− σ j ωjs αj − j ωj αj
P
Γ̃ {sdj , ωj }; σ,
=1+
−1
j ωj αj

(39)

The bias will be higher or lower than one, depending on whether is larger than σ, as the sign
of the second term on the right side is always negative. To see this, notice that there is a tight
relationship between ωj and ωjS :
sdj
ωjS = ωj P
j ωj sdj
which implies that the weighting system ω s assigns lower weight to immigrant-intensive firms
than the weighting system ω. Given that αj is strictly increasing in the immigrant-share of
the firm, the average of αj under the weighting system ω s must be lower than that under ωj
and
P s
P
ω
α
−
j
j
j
j ωj αj
P
<0
j ωj αj

−1
Thus, if > σ, equation 39 shows that Γ̃ {sdj , ωj }; σ,
is lower than one and vice versa.

It also follows that Γ {sdj , ωj } in Section 4.1 is always positive:
P s
P

1
j ωj αj −
j ωj αj
P
Γ {sdj , ωj } ≡ −
>0
−1
j ωj αj

E
E.1

Estimation of
Dataset Description

To estimate the elasticity of substitution between native and immigrant effective units, , we
use an alternative administrative dataset called SIAB, which is also provided by the German
Social Security Administration.24 SIAB contains the full labor biographies for 2% of the German workforce between 1975 to 2014 and includes information on employer size, citizenship,
workplace, industry, occupation, and other covariates similar to the labor market component
of our main dataset LIAB. A few advantages of SIAB include a representative sample of the
German workforce, a longer time span, and a significantly larger sample size. As will be explained in section E.2, the estimation procedure requires constructing generated regressors at
the firm-time-origin level and control for a rich set of time-varying fixed effects. Given these
constraints, this alternative dataset allows us to exploit the larger sample size and longer time
panel.
One limitation of the SIAB dataset is that it does not contain information on every employee
at the establishments in the sample. Since we need the migrant and native employment at the
establishment level, we group establishments in SIAB into bins by time, geographic district,
three-digit industry, and size quartile. We then construct our firm level dataset by considering
all employees in the sample working for establishments in a given bin as if they would work for
the same “synthetic” firm.

E.2

Estimation Details

To get an expression for the immigrant composite, we start from the supply side of the model.
Using the Frechet properties, we can write the number of effective units supplied to firm j in
industry k by workers from origin country o as in equation 40.:
1

ν
(πo,k,` )− ν H̄ Njo
xj,o = Ao,k
|
{z
}

(40)

γo,k

where Njo is the number of workers employed at firm j, and the expression γo,k is the average
ability per worker from o at firm j.
Using the first order condition of profits from firm j with respect to each xj,o relative to the
first order condition with respect to a base origin country o0 , xj,o0 , and using equation 40, we
can get an expression as in equation 41:
24

The data basis of this section of the paper is the weakly anonymous Sample of Integrated Labour Market
Biographies (SIAB) 1975 - 2014. The data were accessed on-site at the Research Data Centre (FDZ) of the
Federal Employment Agency (BA) at the Institute for Employment Research (IAB) and/or via remote data
access at the FDZ. For more information on SIAB please check Antoni et al. (2016).

wo xj,o
wo0 xj,o0

= Ln

δo,k
δo0 ,k

γo,k Njo
0
γo0 ,k Njo

κ−1
+
Ln
κ

!
(41)

Using equation 41 and assigning a value for κ, we can get to the first estimating equation, 42,
which gives us an estimate for the average effective units provided by each migrant worker at
firm j:25

κ̂ − 1
κ−1
0
Ln Wage billo,j −
Ln(Njo ) = Ln(δo,k ) +
Ln(γo,k ) + Ln(δo0 ,k ) − Ln(γo0 ,k Njo ) (42)
κ̂
{z
}
|
{zκ
} |
Firm FE

ζo,k Origin-Industry FE

To estimate equation 42, we pool all years between 1995 until 2014 and run a regression at the
firm-origin-time level. We include origin-industry-time and firm-time fixed effects, such that
we only exploit the cross-sectional variation to estimate the fixed effects. From equation 42,
we obtain the fixed effects ζo,k , which will allow us to compute the immigrant composite at
the firm level using data on the number of immigrants by country, the ζo,k estimates, and the
assigned value of κ as shown in equation 43:

x̂j =

κ̂−1
κ̂

δo xj,o

κ̂
κ̂−1

κ̂−1
δo (γo,k Njo ) κ̂

κ̂
κ̂−1

ζ̂o,k

Njo

κ̂
κ̂−1
κ̂−1
κ̂

(43)

Once we calculate x̂j , we can proceed to estimate our key elasticity . We can use the firm
first order condition with respect to the number of native effective units dj and the immigrant
composite xj to get to estimating equation 44:

Ln
|

d
wj,t
dj,t
βk
−1
= Ln
+
Ln
x
k
wj,t xj,t
1−β

{z
}

γd,k Njd
x̂j,t

!
(44)

Relative wage bill

With some additional structure, we reach estimating equation 45, as shown in Section 5. We
proceed to take logs and reorganize equation (18) into estimating equation 45:

Wage bill Nativesj,t
Wage Bill Immigj,t

−1
=
Ln

Njd
x̂j,t

+ Ln
|

βtk
1 − βtk

+ Ln(γd,k,t ) +ζj + ξj,t
{z
}

(45)

Industry-time FE

κ stands for the degree of substitution across immigrant origin countries for production. We assume κ = 20,
close to the upper bound of the elasticity of substitution between immigrants and natives estimated by Ottaviano
and Peri (2012). We show results are very robust to other values of κ between 10 and 30.

We assume the error term can be written as a firm fixed effect ζj and an unobserved component
ξj,t . We also use the labor supply property that the number of effective units of native workers
can be expressed as an interaction between an industry-time constant γd,k,t and the observed
number of German workers at firm j, Njd as in equation 40. While the model is static, once
again we add time subscripts as we pool several years of data to maximize our sample size.
The OLS estimates will not provide a consistent estimate of the elasticity of substitution under
the presence of unobservable shocks affecting both the relative labor demand and relative wage.
If, for example, firms face productivity shocks that are biased to immigrants, the OLS estimate
will be upward biased. To address endogeneity concerns, we instrument the firm’s relative
demand of workers with the following shift-share instrument:

f
Zj,m,t
=

X Wage Billo,m,1995 EmploymentImm
o,m,t
Ger
Wage
Bill
m,1995 Employmentm,t
o

(46)

The initial share component of the instrument is the wage bill of immigrants from origin o in
market m in year 1995 relative to the total wage bill in market m in 1995.26 We use “kreis”
as the market concept (m) of this instrument, which is the finest geographical area in our
dataset. The shift component of the instrument captures the employment level of immigrants
from country o relative to Germans in market m in year t. This instrument exploits countryof-origin-driven variation in the relative supply of immigrant across markets and “assigns” the
increase of immigrants from each origin in that market to firms according to their market-share
in 1995.
The validity of the instrument depends on this market share not being correlated with shocks
determining the relative wage that firms pay in period t. Larger firms tend to have a larger
market share and may also tend to pay systematically different average wages to immigrants
relative to natives. Even though we control for time-invariant firm heterogeneity, there may be
serially correlated time-varying productivity shocks that affect the relative size of firms in 1995
and their hiring decisions in the future. This would bias the 2SLS estimate upward. The timeindustry fixed effect will help control for unobserved time-varying shocks. Finally, we cluster
standard errors at the firm level to account for the correlation within firm over time.
Table 15 presents the OLS and the 2SLS estimates of 45. The OLS estimate of −1
is larger

than 1 and implies an unreasonable elasticity of substitution between immigrants and natives
of -35.1. The 2SLS estimate in column 2 is lower than one and statistically significant. This
estimate implies that the elasticity of substitution between immigrants and native workers
within the firm is 4.28. As expected, the OLS estimate is upward biased, since the error term
includes demand-side shocks that positively affect the wages and employment of immigrants
26

While the data is available since 1975, we use 1995 as our base year since administrative data for East
Germany only becomes available after 1993.

relative to natives. The instrument is strong, as shown by the F-stat in Table 15.
Table 15: Estimates for
OLS

2SLS

Estimate for ( − 1)/

1.029***
(0.003)

0.81***
(0.355)

Number of observations
Implied

458,308
-35.1

458,308
4.28

First stage
Instrument

1st stage F-stat

-0.00025***
(0.00005)
458,308
21.29

Note.∗∗∗ = p < 0.01, ∗∗ = p < 0.05, ∗ = p < 0.1. OLS and 2SLS estimates for equation 45.
We include industry-time and firm fixed effects. Industry-time FEs are defined according to
our tradable and non-tradable industries used in the model. Standard errors are clustered
at the firm level and bootstrapped with 200 repetitions. Time period used is 1995 to 2014.

Model Fit
While the model matches the targeted moments, we want to make sure it also matches nontargeted moments that are relevant to our main mechanisms. As shown in Figure 8, the model
does a good job in matching the cross-sectional means and medians of the immigrant share by
size decile. The medians are completely untargeted by the estimation routine, and the model
does a good job in replicating the positive slope in the tradable sector and somewhat misses the
slight increasing slope in the non-tradable sector. However, the observed correlation between
size and immigrant share in the non-tradable sector is weak and the model captures the levels
reasonably well. The means are also informative of the distribution within decile. These are
not completely untargeted since we are matching the mean immigrant share across all establishments in our estimation routine as well as the difference in the means of P90 and P50 for
each sector. However, we are not targeting the mean by sector nor the relationship between
any deciles other than 5 and 9. As shown in Figure 8, the model does a good job matching
both means but underestimates the mean for the first deciles in the tradable sector.

Figure 8: Immigrant share across establishments: model vs data
(a) Median Tradable

(b) Mean Tradable

(d) Mean Non-Tradable

Note: We divide establishments in the model and the data into size deciles, where 1 groups the smallest
establishments. We plot the mean and median for each decile and each sector as shown by the data as in
Figure 1. For the model, we plot the size distribution predicted by our estimated model.

F
F.1

Empirical Results Details
Heterogeneous Response to Immigration: Additional Results
Table 16: First stage regressions
Full sample
S agg
m,t

Zm,t
Zm,t × log(size)

1.49***
(0.256)
-0.02
(0.05)

S agg
m,t

Tradable sector
S agg
m,t

× log(size)
0.59
(1.420)
1.15***
(0.298)

1.35***
(0.374)
0.02
(0.069)

S agg
m,t

× log(size)
-0.86
(2.013)
1.45***
(0.413)

Non-Tradable sector
S agg
m,t

1.50***
(0.377)
-0.07
(0.074)

S agg
m,t

× log(size)
2.79
(1.918)
0.61
(0.241)

3507

1974

1533

Kleinberg-Paap F-stat

35.86

29.48

15.53

Note. ∗ ∗ ∗ = p < 0.01, ∗∗ = p < 0.05, ∗ = p < 0.1. We restrict the sample to years between 2008 and 2011. We
control for establishment fixed effects, 2-digit industry-time fixed effects, local labor market-time trends, and
lagged firm level controls such as log employment and investment. Sample is restricted to establishments with
more than 30 employees. Standard errors are clustered at the establishment level.The Kleibergen-Paap F-stat
tests for the joint significance of both instruments. The first two columns are the first stages for the full
sample, columns 3 and 4 restrict the sample to establishments in the tradable sector, and columns 5 and 6 to
the non-tradable sector.

Table 17 evaluates how the controls added to the regression affect our estimates. Column 2
removes the firm-level controls, column 3 removes the industry-time FEs, and column 4 removes
the local labor market trends.
Table 18 presents the heterogeneous effects of the immigration shock on profits, total employment, and labor productivity. Profits are measured as revenues net of wage bill and material
bill, and labor productivity is measured as the ratio between revenues and employment. The
2SLS estimates in Table 18 reassures the previous findings on the heterogeneous effect of immigration. Relative to small establishments, larger establishments hire more workers and show
a larger labor productivity (columns 2 and 3). Estimates for profits are imprecisely estimated,
so we cannot reject a null effect of changes in response to the immigrant share.

F.2

Export Revenues vs Domestic Revenues

A second prediction is that the drop in unit costs generated by immigration would expand
export revenues more than domestic revenues because an exporter faces a demand curve from
the RoW that is more elastic than its domestic demand.
Table 19 presents the estimated results of regression 19 for domestic revenues and export
revenues for the sample of exporters. The average response of export revenues is stronger
53

Table 17: Robustness exercises for main specification
Baseline
θ1

No firm-level No industry-time
controls
FEs

No local labor
time trends

-31.86***
(11.47)
7.49***
(2.46)

-37.39**
(15.41)
8.56***
(3.28)

-52.91***
(12.79)
12.38***
(2.74)

-25.32**
(10.99)
5.93**
(2.4)

N observations
N establishments

3507
949

1st stage F-stat

35.85

8.76

33.67

18.18

θ2

Note.∗ ∗ ∗ = p < 0.01, ∗∗ = p < 0.05, ∗ = p < 0.1. Dependent variable in all cases is log
revenues. We restrict the sample to years between 2008 and 2011. We control for establishment
fixed effects, 2-digit industry-time fixed effects, local labor market time trends, and lagged firm
level controls such as log employment and investment. Standard errors are clustered at the
establishment level. Sample is restricted to establishments with more than 10 employees.
Column 1 shows the baseline specification with full controls. Column 2 removes the firm-level
controls. Column 3 removes the industry-time fixed effects and controls only for time fixed
effects. Column 4 removes the local labor time-trends.

Table 18: The impact of immigration on other outcomes
Log Profits

Log employment

Log Revenue per
employee

-136.7
(101.31)
29.6
(17.35)

-4.82
(6.43)
1.64
(1.4)

-26.99**
(11.4)
5.83**
(2.51)

Average y
Threshold size

0.47
101

0.18
19

0.09
102

N observations
N establishments
Estimation
1st stage F-stat

2901
853
2SLS
30

3507
949
2SLS
35.86

3507
949
2SLS
35.85

θ1
θ2

than domestic revenues, and in both cases, the heterogeneous effect significantly favors large
establishments relative to small establishments. These estimates imply that by each 1% increase
of the labor market immigration share, domestic revenues increase by 0.44%, whereas export
revenues increases by 1.15%. Since the response of export revenues is stronger than domestic
revenues, this channel can explain part of the heterogeneous effects found in Table 3. Large

establishments, which are more likely to be exporters, may adjust more to the immigration
shock because they are able to expand their export revenues whereas for small firms, expansion
is constrained by the size of the domestic market.
Table 19: Revenue regressions by sector and exporter status
Log Export
Revenues

Log Domestic
Revenues

-87.99**
(39.31)
20.64**
(8.07)

-78.45***
(29.77)
16.6***
(5.92)

Average y
Threshold size

1.15
71

0.44
113

N observations
N establishments
Estimation

1654
466
2SLS

1st stage F-stat

20.72

θ1
θ2

Note. ∗ ∗ ∗ = p < 0.01, ∗∗ = p < 0.05, ∗ = p < 0.1. We
restrict the sample to years between 2008 and 2011. We
control for establishment fixed effects, 2-digit industrytime fixed effects, local labor market-time trends, and
lagged firm level controls such as log employment and investment. Standard errors are clustered at the establishment level. Sample is restricted to establishments with
more than 30 employess and that report positive export
revenues.

To summarize our findings, the reduced-form evidence presented in this section shows that larger
employers benefit more from an increase in the immigrant share of the local labor market than
small establishments. Establishments’ export revenues are more responsive than its domestic
revenues. This evidence is consistent with the mechanisms put forward in the model: given
that large firms are more immigrant-intensive than small firms (Figure 2a), large firms face a
larger drop in the labor cost of production than small firms when the economy receives a new
wave of immigrants. This drop in the cost of production drives large firms to expand their
production at the expense of putting downward pressure on the market price of the good they
sell. This downward pressure is weaker the more elastic the demand. Given that large firms are
likely to export and foreign demand is more elastic, they find it optimal to increase production
to all markets and especially to export markets. As a result, an influx of immigrants is mostly
absorbed by large firms that find it profitable to expand production.

F.3

Shift-share Instrument Diagnostics

Our instrument falls into the category of shift-share instruments, and as such, we run a series
of diagnostics suggested by the literature on the validity of shift-share instruments (Borusyak
et al., 2021; Goldsmith-Pinkham et al., 2020). Our setup is not exactly the standard shiftshare case because in addition to the shift-share instrument, we have an interaction between
the instrument and the log size of the establishment. However, we can still use the guidance of
these methodological papers to understand the variation driving our instruments.
As a first step, we follow the suggestions in Goldsmith-Pinkham et al. (2020) and Borusyak
et al. (2021) and test for pre-trends. The shift-share design implies that the common shock is
the main driver of the observed changes, so we need to make sure there were no preexisting
differences explaining such observed changes. As shown in Table 20, we lag the outcome 5
years and 1 year and use them as outcomes in our baseline regression. The instrument is still
strong, but the second stage coefficients are not significant. This corroborates that the observed
changes are not driven by preexisting differences across establishments. Borusyak et al. (2021)
also suggest that if the sum of the initial shares does not add up to one within local labor
market, we should control for the sum of the exposure shares in our regression. We do so in
a non-parametric fashion by including an establishment fixed effect in our regressions which
would absorb the sum of initial shares at the local labor market level.
Table 20: Pre-trends tests
Log Total Revenues t − 5

Log Total Revenues t − 1

2.51
(9.28)
-1.29
(1.93)

-7.48
(9.61)
2.09
(1.99)

N observations
N establishments

3329
907

3434
937

1st stage F-stat

41.16

40.85

θ1
θ2

Note.∗ ∗ ∗ = p < 0.01, ∗∗ = p < 0.05, ∗ = p < 0.1. We restrict the sample to years
between 2008 and 2011. We control for establishment fixed effects, 2-digit industrytime fixed effects, local labor market time trends and lagged firm level controls such as
log employment and investment. Standard errors are clustered at the establishment
level. Sample is restricted to establishments with more than 30 employess. The first
column includes the outcome variable lagged by 5 periods, the second column includes
the outcome variable lagged by one period.

As a second step, we focus on the case of testing for exogenous shares, and run a set of
diagnostics proposed by Goldsmith-Pinkham et al. (2020). We perform the tests for a simplified
version of equation 19, where we do not include the size interaction term nor the industrytime fixed effects and labor market trends. While the regression is different than our main
specification, the analysis is still useful to understand what is driving the main shift-share
56

instrument.
In our case, we can write the first stage coefficient on the shift-share instrument as a combination of the estimates of nine separate first stage regressions. Each of these “just identified”
regressions uses an instrument that is constructed with the initial share and shock of only one of
our nine origin regions. The weights in which each of these nine instruments affects the overall
IV are called Rottemberg weights. We proceed to use the code provided by Goldsmith-Pinkham
et al. (2020) to calculate such weights and denote them α. Each origin region is affected each
year by a national level shock we denote by G. The just identified coefficients are denoted by
β.
As shown in panel A of Table 21, 89% of the Rottemberg weights are positive, meaning that our
regression is likely not subject to misspecification. In panel B, we show the correlation between
the weights, the shocks, and the just-identified coefficients. Panel C shows the top five origin
regions in terms of the Rottemberg weights. For the time period between 2003-2011, countries
of former Yugoslavia have the largest weight with 0.28. These are followed by Asia-Pacific
(0.24), other non-EU countries which include predominantly Russian immigrants (0.17), Africa
and Middle East (0.15), and Turkey (0.07). These regions are expected to drive most of the
variation in our instrument. It is reassuring however, that no single region accounts for a large
majority of the variation in our instrument.
Table 21: Shift-share diagnostics
Panel A

Sum

Mean

αs ≤ 0
αs > 0

-0.014
1.014

-0.014
0.127

0.111
0.889

Panel B

αs

βs

αs
G
βs

1
0.149
0.013

1
-0.402

Panel C

Countries of former Yugoslavia
Asia-Pacific
Europe other
Africa and Middle East
Turkey

0.28
0.24
0.17
0.15
0.07

0.98
1.11
1.23
1.13
0.83

β
1.54
4.46
3.89
3.97
1.47

Note.We run the shift-share diagnostics suggested by GoldsmithPinkham et al. (2020). Panel A shows the share of Rottemberg
weights that are positive and negative. Panel B shows the correlation between the Rottemberg weights, the time-shifter shock G,
and the just-identified coefficients β. Panel C summarizes α, G,
and β for the top 5 origin regions in terms of weights.

Finally, we look into the correlation between the initial shares used in the instrument and other
57

covariates at the local labor market in the initial period. The intuition behind this exercise is
that the variation in the initial shares should not be explained by other covariates that can also
affect the change in outcomes at the regional level. As shown in Table 22, key characteristics
at the regional level only explain 4.4% of the total variation in the shares, indicating that the
shares are not significantly driven by other observables.
Table 22: Correlation between initial shares and observables
Initial share 03
Avg Age
Share Female
Share College
Share Manual Occupation
Share Services Occupation
Share Manufacturing
Average Wage
N
R-sq

-0.0008
(0.0003)
-0.0086
(0.007)
0.0207
(0.014)
0.0096
(0.009)
0.0129
(0.007)
-0.004
(0.002)
4.60E-07
(1.08E-07)
936
0.0436

Note.We pool 104 local labor market and 9 origin
regions. Regressions include an origin region FE,
but results are consistent to not controlling for origin FEs or running a separate regression for each
origin. As covariates, we include average age, share
of women, share of college graduates, share in manual and services occupations, share in manufacturing
industry, and average wage. Key statistic for analysis is the R-squared.

G
G.1

Additional Quantitative Results
Size of the Inflow of Immigrants
Table 23: Change in real wages for alternative counterfactuals
Percent change in immigrant stock
0.1%

10%

20%

30%

50%

0.001%

0.01%

0.06%

0.12%

0.24%

0.36%

0.58%

Homogeneous/Heterogeneous
Homogeneous (agg)/Heterogeneous

0.82
0.63

0.88
0.88

0.90
0.91

0.89
0.91

0.89
0.92

0.89
0.91

0.89
0.92

Aggregate Elasticity

4.154

4.216

4.224

4.206

4.203

4.202

4.185

Real wages

Note.We compute real wage changes for different aggregate changes in the number of immigrants. The row
“Homogeneous/Heterogeneous” presents the relative real wage changes between the homogeneous model and
our baseline heterogeneous model. The row Homogeneous (agg)/Heterogeneous, computes the relative real
wage changes between a homogeneous model and our baseline model, where the homogeneous model has the
same aggregate elasticity than the one predicted by the heterogeneous model. The aggregate elasticity is the
endogenous elasticity of substitution between immigrants and natives in the baseline heterogeneous model.

G.2

Homogeneous Model

This section presents the estimates of the parameters estimated by simulated method of moments for the homogeneous model, conditioning on ˆ = 4.28, σ̂ = 3.08, and σˆx = 3.62.
Table 24: Simulated vs data moments
Moment description

Simulated

Data

Moment description

0.91
0.93
1.38
1.29

GDP per capita RoW to Germany
Share of firms exporting, T
E(Export to Domestic Revj ), T
E(sd )

Aggregate sd,T
Aggregate sd,N T
Var(log(revj )|sd,j , exporterj ), T
Var(log(revj )|sd,j ), NT

Simulated

Data

0.32
0.37
0.79
0.93

Table 25: Parameter estimates using simulated method of moments
Parameter description

Parameter

Estimate

Parameter description

Share of natives, T
Share of natives, N T
Dispersion in ψj , T
Dispersion in ψj , NT

βT
βN T
σψ,T
σψ,N T

0.82
0.84
1.03
0.38

Productivity in RoW
Fixed cost of exporting
Iceberg trade cost
Elasticity sd to n

Parameter Estimate
ψx
fg
τ
ι

1.64
0.014
1.55
0.014

G.3

Homogeneous Model with aggregate elasticity

This section presents the estimates of the parameters estimated by simulated method of moments for the homogeneous model, conditioning on the aggregate elasticity of substitution
implied by the heterogeneous model (ˆ = 4.20) and, as before, σ̂ = 3.08, and σˆx = 3.62. We
compute the aggregate elasticity of substitution implied by the heterogeneous model as the
weighted average of the elasticity in the labor market for tradable and for non-tradable sector.
The weights are given by the number of firms in each sector and equal to 0.5. The elasticity in
each labor market is computed as follows:

d ln Lg /Lg,x
d ln wg,x /wg

Table 26: Simulated vs data moments
Moment description

Simulated

Data

Moment description

0.91
0.93
1.38
1.29

GDP per capita RoW to Germany
Share of firms exporting, T
E(Export to Domestic Revj ), T
E(sd )

Aggregate sd,T
Aggregate sd,N T
Var(log(revj )|sd,j , exporterj ), T
Var(log(revj )|sd,j ), NT

Simulated

Data

0.32
0.37
0.79
0.92

0.32
0.37
0.79
0.93

Table 27: Parameter estimates using simulated method of moments
Parameter description

Parameter

Estimate

Parameter description

Share of natives, T
Share of natives, N T
Dispersion in ψj , T
Dispersion in ψj , NT

βT
βN T
σψ,T
σψ,N T

0.82
0.84
1.03
0.38

Productivity in RoW
Fixed cost of exporting
Iceberg trade cost
Elasticity sd to n

Parameter Estimate
ψx
fg
τ
ι

1.64
0.008
1.56
0.014

Full text of Working Papers (Federal Reserve Bank of Richmond) : Firm Heterogeneity and the Impact of Immigration : Evidence from German Establishments, Working Paper 21-16

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