International Migration, Internal Migration, and Substitute and Complement Industries
Akira Shimada
Faculty of Economics, Nagasaki University

Abstract
By assuming an economy that comprises substitute and complement industries with efficiency
wages, we investigate the effects of immigrants on native workers and firms. Each industry
accepts immigrants and employs them in addition to native workers.

In a substitute(complement) industry, the marginal product of one input is negatively (positively) related to
employment of another input.

We show that under no internal migration, native worker

employment and utility decrease (increase), whereas the profits of the firms increase (decrease) in
the substitute (complement) industry as more immigrants are accepted. Accordingly, in bothindustries, the interests of native workers and firms are always in conflict. However, if native
workers internally migrate between industries, there are cases where both native worker utility andthe profits of the firms in the same industry become larger by accepting more immigrants or fewer
immigrants. Accordingly, under internal migration, the interests of native workers and firms inthe same type of industry are not always in conflict. Our results suggest that conflicts also exist
within each type of industry and that internal migration can relieve such conflicts.

JEL Classification: F22; J23; J41; J42; J61

Keywords: International migration of labor; Internal migration of labor; Efficiency wages;
Substitute industry; Complement industry; Native workers; Immigrants01. Introduction
In this paper, we investigate how changes in the number of immigrants affect native workers
and firms in an economy where there are substitute and complement industries and wages are
determined according to the efficiency wage hypothesis. We assume two cases, one where there
is no internal migration and one where native workers migrate between the two types of industry
to attain higher utility. We show how internal migration changes the effects of immigrants on
native worker utility and the profits of the firms in each industry and infer from these results
whether or not internal migration helps to relieve conflicts of interests between native workers andfirms in each industry.
One of the most contentious issues related to international migration is the effects of immigrants
on native workers. Whether or not we accept immigrants or how many immigrants are accepted
is usually determined by taking into account their effects on native workers. Accordingly, a large
number of theoretical and empirical studies on the effects of immigrants on native workers of their
host countries have been conducted. Greenwood and McDowell (1986), Borjas (1999), andStark (1991) are early contributors.
The effects on native workers differ, depending on whether native workers and immigrants aresubstitutes or complements as factors of production. When both native workers and immigrants
are unskilled, it is likely that they are substitutes. In this case, we intuitively consider that
increased immigration will have negative impacts on native worker employment. On the other
hand, when native workers are skilled and immigrants are unskilled, it is likely that they arecomplements. In this case, we intuitively consider that increased immigration will have positive
impacts on native worker employment (Straubharr and Zimmermann 1993, Zimmermann 1995,
1996).

1

However, empirical results on the effects of immigrants on native workers are not as conclusive
as intuition suggests. Borjas (1997) showed that the entry of large numbers of less skilled
immigrants has an adverse impact on the labor market opportunities of less skilled U.S. native
workers. On the other hand, Grossman (1982) concluded that although she verified that native
workers and immigrants are substitutes, moderately large inflows of immigrants do not poseserious economic threats to native workers. Friedberg and Hunt (1995) argued that even if
immigrants and native workers are close substitutes, it was not found that native workers suffer
significantly from increased immigration. Moreover, Zorlu and Hartog (2005) did not even finddominant robust patterns of substitution and complementarity between native workers andimmigrants.
By utilizing the monopoly union model, Schmidt, Stilz, and Zimmermann (1994) andZimmermann (1996) analyzed this problem theoretically. They showed that, even if native
workers and immigrants are substitutes, the job replacement effects of immigration might lead towage restraint and thereby a higher employment of native workers.
The results of these previous studies are suggesting that the intuitive conclusions mentioned
above cannot always be endorsed, either empirically or theoretically.
Moreover, as pointed out by Greenwood and Hunt (1995), immigrants influence native workers,
not only through the production channel, but also through other channels, which may offset or
reinforce the production channel effects.
However, even if the conclusions on the effects of immigrants on native workers are notuniform across various studies, as pointed out by Bodvarsson et al. (2007), it is certain that
immigrants have different effects on different individuals or different groups in the same economy,
with some losing and others gaining. In other words, their interests conflict when immigrants are2received. If not, accepting immigrants would not be so controversial. This suggests that wehave to make clear how we can reconcile the conflicting interests among different agents.
Many researchers have dealt with this problem by considering the political process of
immigration quota determination. Shughart et al. (1986) employed the interest group theory of
government.

Benhabib (1996) investigated the determinants of an immigration policy thatspecifies minimum or maximum skill or wealth requirements for immigrants under the majority
voting.

Flores (1997) constructed a single sector general equilibrium model to argue thatwhether a country admits further immigration depends on the country’s factor ownership
distribution, the level of transfers, and the prejudices held by individuals against immigrants.
Amegashie (2004) built a model in which the number of immigrants is determined through thelobbying.

Bodvarsson et al. (2007) determined the immigration quota endogenously by

assuming a political market for it.
However, many of the pervious studies on quota determination did not pay sufficient attention
to the possibility that the conflicting interests among different agents may change when accepting
immigrants. They determined the quota implicitly assuming that how the interests of different
agents are related would not be affected by immigration.
These analyses also did not include the possibility of native worker internal migration, whichmight be induced by immigration. In other words, native worker behavior is not unaffected byimmigrants. Many native workers actually move to the other domestic labor markets when
immigrants enter their labor markets. In his theoretical analysis, Rivera-Batiz (1981) assumed
internal migration in an economy that accepts immigrants. There are many empirical studies on
this issue. For example, Hatton and Tani (2005) investigated whether internal migration isbeneficial to the British labor market for adjusting to immigration, and concluded that it is one of
the mechanisms through which regional labor markets adjust to immigration shocks. Borjas
3

(2005) showed that the native workers’ response to migration reduces the impacts of immigration
on wages in a local labor market. Accordingly, internal migration seems to be beneficial tonative workers even if its effects are limited.1
This suggests that internal migration may alleviate the conflicts of interests among different
agents arising from immigration. Therefore, in order to relieve such conflicts when accepting
immigrants, we have to make clear how the conflicts of the interests will be affected by internalmigration.
For this purpose, following the model by Bodvarsson et al. (2007), we assume an economy
consisting of substitute and complement industries, each of which produces goods by employing
both native workers and immigrants.
However, this paper is markedly different from Bodvarsson et al. (2007) in that there exists
involuntary unemployment, since wages are determined according to the efficiency wage
hypothesis. In Bodvarsson et al. (2007), wages are determined to make the demand for andsupply of labor equal in each industry.

Accordingly, native workers’ preferences toward

immigration policies and their conflicting interests between industries mainly come from theeffects of immigration on native worker wages. On the other hand, in this paper, immigration
affects both native worker wages and employment so that their interests and preferences toward
immigration policies are determined by both of them. In addition, we focus on the conflicts ofinterests within each industry whereas Bodvarsson et al. (2007) dealt with the problem of theconflicts across industries.

1

However, results by Kritz and Gurak (2001) do not support the claim that native workers makemigratory response to immigration.

They suggested that internal migration dynamics aregoverned by factors other than recent immigration.
4All native workers are assumed to be identical, which suggests that they may migrate internallyto a different industry from the one initially assigned and can be employed there. All immigrants
are also assumed to be identical. These two kinds of workers are imperfect substitutes in thesubstitute industry, whereas they are complements in the complement industry. Native workers
and immigrants do the same job in the substitute industry, but their productivities are different.
Accordingly, even in the substitute industry, their wages are determined in such a manner as tosatisfy their respective non-shirk conditions. Native worker wages and immigrant wages in thecomplement industry are determined to satisfy their respective non-shirk conditions. In order toinvestigate the effects of changes in the number of immigrants on native workers and firms, we
assume two cases. In one case, there is no internal migration. Native workers are always in theindustry to which they were initially assigned, and immigrants are always in the industry into
which they were accepted. In another case, native workers internally migrate to achieve a higher
steady state expected lifetime utility, whereas immigrants do not internally migrate. Accordingly,
the actual number of native workers in each industry can be different from its initial number ineach industry.
We demonstrate that in the case where native workers do not internally migrate, native worker
employment and their steady state expected lifetime utility are smaller, whereas the profits of thefirms are larger in the substitute industry when it accepts more immigrants, suggesting that in thesubstitute industry we cannot improve native worker utility and the profits of the firms at the sametime by manipulating the number of their own immigrants.

Likewise, native worker

employment and their steady state expected lifetime utility are larger, whereas the profits of thefirms are smaller in the complement industry when it accepts more immigrants, suggesting that in
the complement industry we likewise cannot improve native worker utility and the profits of thefirms at the same time by manipulating the number of their own immigrants. Therefore, in both5

industries, when there is no internal migration, the interests of native workers and firms are always
in conflict.
On the other hand, under internal migration of native workers, native worker employment in thesubstitute industry does not necessarily decrease even if it accepts more immigrants. Similarly,
native worker employment in the complement industry does not necessarily increase even if it
accepts more immigrants. This is because native worker employment in each industry is affected,
not just by the number of immigrants in that industry, but also by the number of immigrants
accepted into the other industry. Native worker steady state expected lifetime utility is higher if
the sum of their employment in both industries increases, and immigrants into the substituteindustry and those into the complement industry have different impacts on the sum of native
worker employment. Accordingly, we can simultaneously improve native worker utility and the
profits of the firms in the substitute industry by accepting more immigrants into the substituteindustry, if the complement industry significantly increases the number of its own immigrants.
This is because large increases in the number of immigrants in the complement industry can makenative worker employment, and thereby their steady state expected lifetime utility, larger, andincreases in the number of immigrants in the substitute industry make the profits of the firms inthat industry larger although this partly offsets increases in the native worker utility. Likewise,
we can simultaneously improve native worker utility and the profits of the firms in thecomplement industry by accepting fewer immigrants into the complement industry, if thesubstitute industry significantly decreases the number of its own immigrants. This can be
explained in a similar manner. Large decreases in the number of immigrants in the substituteindustry can make native worker employment, and thereby their steady state expected lifetime
utility, larger, and decreases in the number of immigrants in the complement industry make theprofits of the firms in that industry larger although this partly offsets increase in the native worker
6utility. Therefore, when native workers internally migrate, the interests of native workers andfirms in the same industry are not always in conflict.
Our results suggest that internal migration can relieve the conflicts between native workers andfirms in each industry.
The remainder of this paper is organized as follows:

Section 2 presents an economy

comprised of the substitute industry and the complement industry, into which immigrants are
accepted. Section 3 examines the effects of changes in the number of immigrants on native
workers and firms when there is no internal migration. We try to make clear that the interests ofnative workers and firms in each industry are in opposition when accepting immigrants. Section
4 examines the effects of changes in the number of immigrants on native workers and firms when
native workers internally migrate between the industries. We try to find the possibilities thatinternal migration alleviates conflicts in each industry. Concluding comments are presented in
Section 5.

2. The ModelIn this section, we model an economy consisting of the substitute and complement industries.
There are no transactions of goods between the two industries. They are become interdependentwhen native workers migrate between them. Firms in each industry are perfectly competitive
and they use two inputs in production – native labor and immigrant labor. As mentioned, allnative workers are identical, whichever industry they belong to.

All immigrants are also

identical, whichever industry they are accepted into. We do not make a specific assumption on
the skill of native and immigrant workers. Physical capital is fixed and we do not assume it
explicitly as an input in the production function. In both industries, wages are determinedaccording to the efficiency wage hypothesis.
7When the two inputs are substitutes as factors of production, the marginal product of one input
is negatively related to employment of the other input. We assume that native labor andimmigrant labor are imperfect substitutes in the substitute industry. To capture these ideas most
simply, we assume that firms in the substitute industry are faced with the following production
function:
YS = FS ( N S + θM S ), 0 < θ < 1, FS′ > 0, FS′′ < 0,where N S is the number of employed native workers in the substitute industry, M S is thenumber of employed immigrants in the substitute industry, and θ is a constant. Per capita
effective labor of immigrants is assumed to be smaller than that of native workers. This paper
assumes that immigrants do not affect the product demand. Their effects are only through the
production.
Firms demand native workers and immigrants to maximize their profits. Native workers andimmigrants are demanded to satisfy the following conditions:FS′ = wSN ,(1sn)

θFS′ = wSM ,(1sm)

where wSN is native worker wages in the substitute industry, and wSM is immigrant wages inthe substitute industry.2
Firms in both industries set native worker wages and immigrant wages in such a manner as toprevent shirking by employed native workers and immigrants, respectively (Shapiro and Stiglitz
1984).

2

The profits of the firms in the substitute industry are defined as,

π S ≡ FS (NS + θMS ) − wSN NS − wSM MS ,where the product price is assumed to be one throughout the analysis.
8

The instantaneous utility of a representative employed native worker in the substitute industry
who does not shirk is equal to wSN − eN , where eN , that is a constant, is effort exerted by him.
We assume that effort exerted by a representative employed native worker who does not shirk is
identical in the substitute and complement industries. If a representative employed native worker
in the substitute industry shirks, his instantaneous utility is equal to wSN .

However, in such acase, he will be detected and fired by firms at the probability, i.e., the detection rate, ρ N , which
is a constant and is assumed to be identical for all employed native workers in the substitute andcomplement industries who shirk. Moreover, even if employed native workers in the substituteindustry are not fired on the grounds of shirking, some of them separate from their jobs at theprobability, i.e., the separation rate, β N , which is a constant and is defined as a ratio ofseparations of employed native workers in the substitute industry due to reasons other thanshirking to the number of employed native workers in this industry. We assume that employednative workers in the complement industry also separate at the same probability.
Native workers who separate from their jobs and become unemployed in the substitute industry
constitute flow into unemployment of native workers in this industry. 3

Flow out of

unemployment of native workers in the substitute industry arises from reemployment of
unemployed native workers in this industry. They are reemployed at the probability, i.e., theaccession rate, α SN , which is defined as the ratio of new hires of unemployed native workers in

3

Even in a case where native workers migrate internally between the two industries, once native

workers become unemployed in either industry, we assume that they do not move to the otherindustry seeking employment opportunities. However, they can be reemployed in the same
industry where they become unemployed.

See Section 4 for the assumptions on internalmigration by native workers.
9the substitute industry to the number of the unemployed native workers in this industry. In
steady state, the number of native workers in the substitute industry who flow into unemployment
is equal to the number of native workers in the substitute industry who flow out of unemployment.
Under these assumptions, the expected lifetime utility of a representative employed native
worker who shirks in the substitute industry VESSN is,rVESSN = wSN + ( β N + ρ N )(VU SN − VESSN ),where r is the discount rate.

VU SN is the expected lifetime utility of a representative

unemployed native worker in the substitute industry, which is given by,
rVU SN = w + α SN (VESSN − VU SN ),where w is the unemployment benefit, which is a constant and is assumed to be identical for all
unemployed native workers and unemployed immigrants in the substitute and complement
industries. The expected lifetime utility of a representative employed native worker who does
not shirk in the substitute industry VENSN is,rVENSN = wSN − eN + β N (VU SN − VENSN ).

Imposing the non-shirk condition VENSN = VESSN (≡ VE SN ) and noticing that in steady state thenumbers for the flow into and flow out of unemployment of native workers in the substituteindustry are equal, i.e., β N N S = α SN ( N S − N S ), we derive steady state native worker wages inthe substitute industry,wSN = w + eN +{N S ( N S − N S )}β N + r

ρN

eN .

(2sn)

where N S is the number of native workers existent in the substitute industry.4 Native worker
wages increase with their employment and decrease with their number existent in the industry.

4

If there is no internal migration, the number of native workers existent in the substitute industry

10

In addition, the steady state expected lifetime utilities of representative employed andunemployed native workers in the substitute industry are derived as follows:
VE SN =w
1 ⎡ {N S ( N S − N S )}β N ⎤1++⎥ eN .
r ρ N ⎢⎣
r
⎦

(3sn)

VU SN =w
1 {N S ( N S − N S )}β N
+
eN .
r ρN
r

(4sn)

No matter whether employed or unemployed, native workers attain higher steady state expected
lifetime utilities with increases in their employment.
By a similar argument, we derive steady state immigrant wages in the substitute industry wSM
as follows:
wSM = w + eM +{M S ( M S − M S )}β M + r

ρM

eM ,

(2sm)

where eM , that is a constant, is effort exerted by a representative employed immigrant in thesubstitute and complement industries, M S is the number of immigrants accepted into thesubstitute industry, which is exogenous and manipulated by the industry, M S is the number of
employed immigrants in the substitute industry, β M is the immigrant separation rate in thesubstitute and complement industries, and ρ M is the immigrant detection rate in the substituteand complement industries.

N S is equal to the number of native workers initially assigned to this industry, which does notchange throughout the analysis. On the other hand, when native workers internally migrate
between the substitute and complement industries, the number of native workers existent in thesubstitute industry does not remain unchanged.
migration of native workers.
11

See Section 4 for the case with internalIn the complement industry, native workers and immigrants perform different jobs thatcomplement each other in production.

Accordingly, the marginal product of one input ispositively related to employment of the other input. We assume that in the complement industry,
goods are produced according to the following production function:
YC = FC ( N C , M C ), FC1 , FC 2 > 0, FC12 , FC 21 > 0, FC11 , FC 22 < 0, FC11FC 22 − FC12 FC 21 > 0,
where N C is the number of employed native workers in the complement industry, and M C is
the number of employed immigrants in the complement industry.5
Firms demand native workers and immigrants to maximize their profits, and they are employed
to satisfy the following conditions:
FC1 = wCN ,(1cn)

FC 2 = wCM ,(1cm)

where wCN is native worker wages in the complement industry, and wCM is immigrant wages
in the complement industry.
We can derive steady state wages and expected lifetime utilities of native workers in thecomplement industry in a similar manner as in the substitute industry:wCN = w + eN +5

{N C ( N C − N C )}β N + r

eN ,

(2cn)

VECN =w
1 ⎡ {N C ( N C − N C )}β N ⎤1++⎥ eN ,r ρ N ⎢⎣
r
⎦

(3cn)

VU CN =w
1 {N C ( N C − N C )}β N
+
eN ,
r ρN
r

(4cn)

ρN

FC1 ≡ ∂FC ∂N C , FC 2 ≡ ∂FC ∂M C , FC12 ≡ ∂ (∂FC ∂N C ) ∂M C , FC 21 ≡ ∂ (∂FC ∂M C ) ∂N C ,FC11 ≡ ∂ (∂FC ∂N C ) ∂N C , FC 22 ≡ ∂ (∂FC ∂M C ) ∂M C .
12where N C is the number of native workers existent in the complement industry.6
Steady state immigrant wages in the complement industry are also determined as follows:
wCM = w + eM +{M C ( M C − M C )}β M + r

ρM

eM ,

(2cm)

where M C is the number of immigrants accepted into the complement industry, which isexogenous and manipulated by the industry.

3. Effects of Immigrants on Native Workers and Firms without Internal Migration

In this section, we examine how changes in the number of immigrants affect native workers andfirms when there is no internal migration.
To see their effects in the substitute industry, we combine Equations (1sn) and (2sn) andEquations (1sm) and (2sm) and totally differentiate them:( FS′′ − ESN 1 )dN S + θFS′′dM S = ESN 2 dN S ,(5sn)

θFS′′dN S + (θ 2 FS′′ − ESM 1 )dM S = ESM 2 dM S ,(5sm)

where ESN 1 ≡

βN
∂w
βN
∂wSN
NS
NS
=
eN > 0, ESN 2 ≡ SN = −eN < 0,
2
2∂N S ( N S − N S ) ρ N∂N S(NS − NS ) ρN

6

Similar to the definition of N S , if there is no internal migration, the number of native workers

existent in the complement industry N C is equal to the number of native workers initiallyassigned to this industry, which does not change throughout the analysis. On the other hand,when native workers internally migrate between industries (see Section 4), the number of native
workers existent in the complement industry will be different from the number of initiallyassigned native workers. However, even under internal migration, the sum of native workers
existent in the two industries does not change.

13

ESM 1 ≡

∂w
βM
MS
βM
∂wSM
MS
eM < 0,=eM > 0, ESM 2 ≡ SM = −22∂M S(M S − M S ) ρ M∂M S ( M S − M S ) ρ M

and dN S = 0 from the assumption.
Solving Equations (5sn) and (5sm) for dN S , we see that as the substitute industry accepts
more immigrants, native worker employment in this industry becomes smaller:

θFS′′ESM 2
dN S
=−
< 0.
dM S
ESN 1ESM 1 − FS′′( ESM 1 + θ 2 ESN 1 )(6sn)

Since native worker employment in the substitute industry is independent of the number of
immigrants accepted into the complement industry, it always becomes smaller when the substituteindustry accepts more immigrants.7
Since firms set higher wages to prevent shirking as more native workers are employed, i.e.,
ESN 1 > 0, their wages also decrease as more immigrants are accepted:
dwSN
dN S
= ESN 1
< 0.
dM S
dM S

(7sn)

These results are easily confirmed by drawing the curves for the total differentiation of thenative worker non-shirk condition (Equation 2sn) and the total differentiation of the demand fornative workers. The former curve is upward sloping on the coordinates, where dwSN is takenon the vertical axis and dN S is taken on the horizontal axis. By substituting the solution of
Equations (5sn) and (5sm) for dM S into the total differentiation of Equation (1sn), we derive thetotal differentiation of the demand for native workers,

7

This result does not hold if native workers migrate internally. See Section 4 for effects ofimmigrants on native worker employment in the substitute industry under internal migration.
14

FS′′dN S +θFS′′( FS′′ − ESN 1 ) ESM 2
dM S = dwSN .
ESN 1ESM 1 − FS′′( ESM 1 + θ 2 ESN 1 )This equation shows that the demand curve is downward sloping and shifts downward to the left
with increases in dM S , which makes both dN S and dwSN lower in equilibrium.
As shown by Equations (3sn) and (4sn), the steady state expected lifetime utilities ofrepresentative employed and unemployed native workers in the substitute industry are increasing
with respect to native worker employment in this industry. Accordingly, their utilities decrease
with increases in the number of immigrants accepted into the substitute industry:∂VE SN∂M S< 0,∂VU SN
∂M S

< 0.

(8sn)

Equations (6sn), (7sn), and (8sn) suggest that in the substitute industry, native worker welfare
worsens as more immigrants are accepted. Since their employment, wages, and utility do notdepend on the number of immigrants accepted into the complement industry, increases in thenumber of immigrants accepted into the substitute industry always make their welfare worse.
We can infer from this result that native workers in the substitute industry may prefer strict
immigration restrictions.
On the other hand, immigrants in the substitute industry have positive impacts on firms in thisindustry.

In other words, their profits increase as they accept more immigrants.

This isbecause,dπ S
dw
dw= − SN N S − SM M S ,
dM S
dM S
dM S
and immigrant wages as well as native worker wages in the substitute industry are decreasing withrespect to the number of immigrants, i.e., dwSM dM S = θdwSN dM S < 0.
firms in the substitute industry may prefer loose immigration restrictions.
15

This suggests that

We can infer from the above results that the interests of native workers and firms in thesubstitute industry do not coincide when accepting immigrants. This suggests that if firms havemore influence over the immigration policy than native workers, the substitute industry will
accept more immigrants. On the other hand, if native workers have more influence than firms, itwill accept fewer immigrants.
We derive the effects of immigrants on native workers and firms in the complement industry
utilizing Equations (1cn) and (2cn) and Equations (1cm) and (2cm):( FC11 − ECN 1 )dN C + FC12 dM C = ECN 2 dN C ,(5cn)

FC 21dN C + ( FC 22 − ECM 1 )dM C = ECM 2 dM C ,(5cm)

where ECN 1 ≡ECM 1 ≡

∂w
βN
NC
βN
∂wCN
NC
eN < 0,=eN > 0, ECN 2 ≡ CN = −2
2∂N C( NC − NC ) ρ N∂N C ( N C − N C ) ρ N

∂wCM
MC
βM
∂w
βM
MC
=
eM > 0, ECM 2 ≡ CM = −eM < 0,2
2∂N C(M C − M C ) ρ M∂M C(M C − M C ) ρ M

and dN C = 0 from the assumption.
Solving Equations (5cn) and (5cm) for dN C , we obtain the results on the effects of accepting
immigrants into the complement industry on native worker employment and wages of the same
industry:dN C
ECM 2 FC12
=−
> 0.
dM C
FC11FC 22 − FC12 FC 21 − ( FC11ECM 1 + FC 22 ECN 1 ) + ECN 1ECM 1

(6cn)

dwCN
dN C
= ECN 1
> 0.
dM C
dM C

(7cn)

According to Equations (6cn) and (7cn), as the complement industry accepts more immigrants,
native worker employment and wages in this industry become larger. Since native worker
employment and wages are independent of the number of immigrants accepted into the substitute16

industry, they always become larger when the complement industry accepts more immigrants.8
These results can be confirmed in a similar manner as in the substitute industry. Increases in
the number of immigrants accepted into the complement industry shift the demand curve fornative workers in the complement industry upward to the right. On the other hand, the curve forthe non-shirk condition (Equation 2cn) does not shift.
From Equations (6cn), (3cn), and (4cn), the expected lifetime utilities of representative
employed and unemployed native workers in the complement industry become higher as more
immigrants are received into this industry:
∂VECN
∂M C

> 0.

∂VU CN
∂M C

> 0.

(8cn)

Equations (6cn), (7cn), and (8cn) suggest that in the complement industry, native worker
welfare improves as more immigrants are accepted. Since their employment, wages, and utility
do not depend on the number of immigrants accepted into the substitute industry, increases in thenumber of immigrants accepted into the complement industry always improve their welfare. We
can infer from this result that in the complement industry, native workers may prefer loose
immigration restrictions.
In contrast to the substitute industry case, the effects of immigrants on the profits of the firms in
the complement industry are ambiguous, since,

8

In the complement industry also, the resulting effects of accepting immigrants on native worker

employment and wages do not hold if native workers migrate internally. See Section 4 foreffects of immigrants on native worker employment and wages in the complement industry underinternal migration.
17

dπ C
dw
dw
= − CN N C − CM M C ,
dM C
dM C
dM C
and dwCN dM C > 0 and dwCM dM C < 0 .

However, if the number of employed immigrants

are sufficiently small compared with the number of employed native workers, it is likely that in
the complement industry, the profits of the firms decrease as they accept more immigrants. Thissuggests that firms in the complement industry may prefer strict immigration restrictions.
We can infer from the above results that in the complement industry also, the interests of native
workers and firms do not coincide when accepting immigrants. This suggests that if firms havemore influence over the immigration policy than native workers, the complement industry will
accept fewer immigrants. On the other hand, if native workers have more influence than firms, itwill accept more immigrants.
To summarize the results derived in this section, in the substitute industry native workers arenegatively impacted by the inflow of immigrants and in the complement industry they are
positively impacted by the inflow of immigrants. On the other hand, the profits of the firms
increase in the substitute industry and they are likely to decrease in the complement industry as
each industry accepts more immigrants. Therefore, in each industry, the interests of native
workers and firms are in opposition when accepting immigrants, which implies that it is
impossible for each industry to simultaneously improve native worker utility and the profits of thefirms by controlling the number of its own immigrants.

4. Effects of Immigrants on Native Workers and Firms under Internal Migration

In this section, we examine how changes in the number of immigrants affect native workers andfirms when native workers migrate between the substitute and complement industries.
In actual economies, not only do native workers migrate internally between these industries, but
18

so do immigrants. However, immigrants are not as mobile as native workers since, comparedwith native workers, it is not easy for immigrants to find jobs at a different industry from the one
into which they were initially accepted. Therefore, we consider the simplest situation where only
native workers internally migrate.
We assume that they migrate to the other industry without any costs if they can attain higher
expected lifetime utility there that takes the possibilities into account that they are employed at theprobability N i N i and unemployed at the probability 1 − N i N i ,i = S , C , where underinternal migration N S and N C are different from the numbers of native workers initiallyassigned in substitute and complement industries, respectively. Once internally migrated, native
workers are assumed not to leave that industry even if they are not employed there.
Specifically, if the native worker steady state expected lifetime utility in the substitute industry
that takes the possibilities of employment and unemployment there into account VSN ,⎛ N ⎞NSVE SN + ⎜⎜1 − S ⎟⎟VU SN ,NS⎝ NS ⎠is larger (smaller) than the native worker steady state expected lifetime utility in the complement
industry that takes the possibilities of employment and unemployment there into account VCN ,⎛ N ⎞NC
VECN + ⎜⎜1 − C ⎟⎟VU CN ,NC
⎝ NC ⎠

then native workers in the complement industry (the substitute industry) migrate to the substituteindustry (the complement industry).
VSN decreases (increases) as more native workers immigrate to (emigrate from) the substituteindustry, and VCN increases (decreases) as more native workers emigrate from (immigrate to) the19

complement industry.9 Accordingly, in equilibrium, the steady state expected lifetime utilities ofnative workers in the substitute and complement industries are equalized, i.e., VSN = VCN .
Utilizing Equations (3sn) and (4sn) and Equations (3cn) and (4cn), this equilibrium condition
can be rewritten as,
N S NC β N ⎛ N S N S
NC NC ⎞
⎟ = 0.
⎜⎜−+−N S NCr ⎝ 1 − N S N S 1 − N C N C ⎟⎠To establish this condition for any β N and r ,N S NC=,N S NC

has to hold. Therefore, if native workers migrate to the other industry to achieve higher steadystate expected lifetime utility, then native workers are employed at the same probability in bothindustries, which is equal to N N , where N ≡ N S + N C and N ≡ N S + N C .
Accordingly, under internal migration, steady state native worker wages in the substitute andcomplement industries wSN im , wCN im are expressed as follows:
wSN im = wCN im = w + eN +{N ( N − N )}β N + r

ρN

(9)

eN .

We derive the employed native worker steady state expected life utility in the substitute andWe can derive dNS dNS < NS N S from Equations (5sn) and (5sm) and dNC dNC < NC N C

9from Equations (5cn) and (5cm). This suggests d ( Ni Ni ) dNi < 0, i = S ,C.

Moreover, we

can also derive dViN d ( Ni Ni ) > 0, i = S , C from Equations (3sn) and (4sn) and Equations(3cn) and (4cn). Therefore, dViN dNi < 0, i = S ,C.

For example, if VSN > VCN , then native

workers in the complement industry migrate to the substitute industry.
increases and N C decreases. This makes VSN smaller and VCN larger.
20

As a result, N S

complement industries under internal migration VE SNim , VECNim and the unemployed native worker
steady state expected life utility in the substitute and complement industries under internalmigration VU SNim , VU CNim :VE SNim = VECNim =w
1 ⎡ {N ( N − N )}β N ⎤+1+⎥ eN .
r ρ N ⎢⎣r⎦

(10)

VU SN = VU CN =w
1 {N ( N − N )}β N
eN .
+
r ρN
r

(11)

im

im

Noticing that dN = dN S + dN C , we infer from Equations (9), (10), and (11) that under internalmigration native worker steady state wages and expected lifetime utilities in each industry
increase, not only with native worker employment in its own industry, but also with that in theother industry.
The substitute industry under internal migration can be modeled by Equations (1sn) and (9) andEquations (1sm) and (2sm). We totally differentiate them to derive the following equations:′ im )dN S + θFS′′dM S = ESCN
′ im dN C ,( FS′′ − ESCN

(5sn’)

θFS′′dN S + (θ 2 FS′′ − ESM 1 )dM S = ESM 2 dM S ,(5sm)′ im ≡
where ESCN

∂wSN im∂N S=∂wSN im∂N C=∂wCN im∂N C=∂wCN im
∂N S

=

N
βN
eN > 0 and dN = 0 from2(N − N ) ρN

the assumption.
Equations (5sn’) and (5sm) are taken as describing how changes in native worker employment
and immigrant employment in the substitute industry are related to each other for given changes in
the number of immigrants accepted into the substitute industry and native worker employment in
the complement industry.
By deleting dM S from Equations (5sn’) and (5sm), we derive the relationship betweenchanges in native worker employment in the substitute and complement industries for given
21

changes in the number of immigrants accepted into the substitute industry:′ im ) − ESCN
′ im ESM 1}− {FS′′( ESM 1 + θ 2 ESCN
θFS′′ESM 2
dN C =dN S +dM S
2′ im (θ FS′′ − ESM 1 )′ im (θ 2 FS′′ − ESM 1 )
ESCN
ESCN

(12s)

(≡ φS 1dN S + φS 2 dM S ).
Since the coefficients of the first and second terms on the right-hand side of Equation (12s) φ S1 ,φ S 2 are negative, the curve for this equation is downward sloping and shifts downward to the left
with increases in dM S on the coordinates where dN C is taken on the vertical axis and dN S is
taken on the horizontal axis.
Similarly, we can model the complement industry under internal migration by using Equations(1cn) and (9) and Equations (1cm) and (2cm). The total differentiation of these equations gives
us the following equations that describe how changes in native worker employment andimmigrant employment in the complement industry are related to each other for given changes in
the number of immigrants accepted into the complement industry and native worker employment
in the substitute industry:′ im )dN C + FC12 dM C = ESCN
′ im dN S ,( FC11 − ESCN

(5cn’)FC 21dN C + ( FC 22 − ECM 1 )dM C = ECM 2 dM C .

(5cm)

We delete dM C from Equations (5cn’) and (5cm) to derive the relationship between changes
in native worker employment in the complement and substitute industries for given changes in thenumber of immigrants accepted into the complement industry:dN C =′ im ( FC 22 − ECM 1 )
ESCN
dN S
′ im − FC11 ) − FC 22 ESCN
′ im
FC11FC12 − FC12 FC 21 + ECM 1 ( ESCN+− FC12 ECM 2
dM C′ im − FC11 ) − FC 22 ESCN
′ im
FC11FC12 − FC12 FC 21 + ECM 1 ( ESCN

(12c)(≡ φC1dN S + φC 2 dM C ).
Since the coefficient of the first term on the right-hand side of Equation (12c) φC1 is negative andthe coefficient of the second term φC 2 is positive, the curve for this equation is downward22sloping and shifts upward to the right with increases in dM C .
We are now in position to show how changes in the numbers of immigrants accepted into thesubstitute and complement industries affect native worker employment in each industry. For thispurpose, we solve Equations (12s) and (12c) for dN S and dN C .
dN S =− φS 2
φC 2
dM S +
dM C .
− φC 1 + φ S 1− φC 1 + φ S 1(13s)

dN C =φ S1φC 2− φ S 2φC1
dM C +
dM S .
− φC 1 + φ S 1− φC 1 + φ S 1(13c)

Equations (13s) and (13c) suggest that native worker employment in each industry depends not
only on the number of immigrants accepted into that industry but also on the number of
immigrants accepted into the other industry. This result is contrasted with the one derived underno internal migration.
Specifically, since − φC1 + φS 1 < 0, which implies that the curve for Equation (12s) is steeper
than the curve for Equation (12c), not only do increases in the number of immigrants accepted into
the substitute industry make native worker employment in the substitute industry smaller, but so
do increases in the number of immigrants accepted into the complement industry, i.e.,∂N S ∂N S,< 0.
∂M S ∂M C(14s)

Similarly, not only do increases in the number of immigrants accepted into the complement
industry make native worker employment in the complement industry larger, but so do increases
in the number of immigrants accepted into the substitute industry, i.e.,∂N C ∂N C,> 0.
∂M C ∂M S(14c)

These results are easily confirmed by drawing the curves for Equations (12s) and (12c). As
23

mentioned, the curve for Equation (12s) is steeper than the curve for Equation (12c). With
increases in the number of immigrants accepted into the substitute industry, the curve for Equation(12s) shifts downward to the left. This makes the intersection of the two curves move upward to
the left, resulting in decreases in native worker employment in the substitute industry(∂N S ∂M S < 0) and increases in native worker employment in the complement industry(∂N C ∂M S > 0).

With increases in the number of immigrants accepted into the complement

industry, the curve for Equation (12c) shifts upward to the right. This also makes the intersectionof the two curves move upward to the left, resulting in decreases in native worker employment in
the substitute industry (∂N S ∂M C < 0) and increases in native worker employment in thecomplement industry (∂N C ∂M C > 0).
According to Equations (14s) and (14c), accepting more immigrants into the substitute industry
negatively impacts its own native worker employment, and accepting more immigrants into thecomplement industry positively impacts its own native worker employment. These results are
the same as those derived without assuming internal migration (see Equations 6sn and 6cn).
However, being different from the case where there is no internal migration, native worker
employment in the substitute industry is not always smaller when more immigrants are accepted
into the substitute industry. This is because even if the substitute industry increases the numberof its own immigrants, native worker employment in that industry may be the same or may
possibly be larger when the complement industry significantly decreases the number of its
immigrants. Likewise, native worker employment in the complement industry is not always
larger when more immigrants are accepted into the complement industry. This is because even if
the complement industry increases the number of its own immigrants, native worker employment
in that industry may be the same or may possibly be smaller when the substitute industry
significantly decreases the number of its immigrants. These results are contrasted with those24

obtained under no internal migration.
Moreover, native worker steady state expected lifetime utilities increase with the sum of native
worker employment in both industries (see Equations 10 and 11). Being different from the casewhere there is no internal migration, increases in native worker employment in an industry do notalways lead to higher native worker steady state expected lifetime utility in the same industry.
For example, even if native worker employment in the substitute industry increases, the sum ofnative worker employment and thereby the native worker steady state expected lifetime utility inthe substitute industry (and in the complement industry also) may be the same or may possibly belower when native worker employment in the complement industry significantly decreases.10
The results on native worker steady state expected lifetime utility are also in contrast with thoseobtained under no internal migration.
Combining Equations (13s) and (13c), we calculate changes in the sum of native worker
employment due to changes in the numbers of immigrants:dN =φ (1 + φ S 1 )− φ S 2 (1 + φC1 )dM S + C 2
dM C ,− φC 1 + φ S 1− φC 1 + φ S 1(15)

where {−φ S 2 (1 + φC1 )} (−φC1 + φ S1 ) < 0 and {−φC 2 (1 + φS 1 )} (−φC1 + φS1 ) > 0.
According to Equation (15), when the complement industry increases the number of its own
immigrants sufficiently, the sum of native worker employment will become larger even if the10

If both industries reduce the number of immigrants, native worker employment increases in thesubstitute industry (see Equation 14s). However, as will be shown by Equation (15), the sum ofnative worker employment in both industries and hence the native worker steady state expected
lifetime utility in both industries will become smaller if the decrease in the number of immigrants
accepted into the complement industry is sufficiently large.
25

substitute industry increases the number of its own immigrants in order to make the profits of thefirms in the substitute industry larger.11 In this case, not only is there an increase in the profits ofthe firms in the substitute industry but also native worker steady state expected lifetime utility inboth industries becomes higher.12 Therefore, accepting more immigrants into the substituteindustry can be beneficial to both native workers and firms in this industry, and there is no conflictof interests between them.
Similarly, when the substitute industry decreases the number of its own immigrants sufficiently,
the sum of native worker employment will become larger even if the complement industry
decreases the number of its own immigrants in order to make the profits of the firms in thecomplement industry larger.13 In this case, not only is there an increase in the profits of the firms

11

As in the case where there is no internal migration, the profits of the firms in the substituteindustry increase with the number of its own immigrants, i.e., dπ S dM S = −(dwSN im dM S ) N S− ( dwSM dM S ) M S > 0.

This is because under internal migration also, native worker wages andimmigrant wages in this industry are decreasing with respect to the number of its own immigrants.
In other words,

dwSN im dM S = (1 θ )(dwSM dM S ),and they are equal to (dwSN im dN )(dN dM S ), where dwSN im dN > 0, dN dM S < 0.
12

Native worker employment in the substitute industry is smaller because both industries are

accepting more immigrants as suggested by Equation (14s). On the other hand, native worker
employment in the complement industry is larger as suggested by Equation (14c). However, as
we have already derived, native workers in the substitute industry are employed at the sameprobability as in the complement industry. This is, of course, due to internal migration.
13

As in the case where there is no internal migration, the profits of the firms in the complement26

in the complement industry but also native worker steady state expected lifetime utility in bothindustries becomes higher. 14

Therefore, accepting fewer immigrants into the complement

industry can be beneficial to both native workers and the firms in this industry, and there is noconflict of interests between them.
These results suggest that under internal migration, the interests of native workers and firms inthe same industry do not always conflict.
If native workers in the complement industry have more influence over the immigration policy
than firms in this industry, the complement industry will accept more immigrants. This improves
the native worker steady state expected lifetime utility in both industries if increases in the number

industry are likely to decrease with the number of its own immigrants, i.e.,dπ C dM C = − (dwCNim dM C ) NC − (dwCM dM C )M C < 0.

We can explain this as follows: As

under no internal migration, native worker wages in the complement industry increase with thenumber of immigrants accepted into this industry:dwCN im dM C = (dwCN im dN )(dN dM C ) > 0.
On the other hand, immigrant wages in the complement industry decrease with their number:
dwCM dM C = FC 21 ( dN C dM C ) + FC 22 ( dM C dM C ) < 0.

However, if the number of employed immigrants in the complement industry is sufficiently small
compared with that of employed native workers in the complement industry, the profits of thefirms in this industry decrease as more immigrants are accepted.
14

Native worker employment in the complement industry is smaller since both industries are

accepting fewer immigrants as suggested by Equation (14c). On the other hand, native worker
employment in the substitute industry is larger as suggested by Equation (14s). However,internal migration makes the native worker employment probability the same in both industries.
27

of immigrants into the complement industry are significant. At the same time, this will make itpossible for the substitute industry to accept more immigrants in order to increase the profits of thefirms in this industry. Accordingly, in such a case, native workers and firms in the substituteindustry will agree about increasing the number of immigrants.
Similarly, if native workers in the substitute industry have more influence over the immigration
policy than firms in this industry, the substitute industry will accept fewer immigrants. This
improves the native worker steady state expected lifetime utility in both industries if decreases in
the number of immigrants into the substitute industry are significant. At the same time, this will
make it possible for the complement industry to accept fewer immigrants in order to increase theprofits of the firms in this industry. Accordingly, in such a case, native workers and firms in thecomplement industry will agree about decreasing the number of immigrants.
This suggests that if native workers in one industry have more influence over the immigration
policy than firms in the same industry, in the other industry native workers and firms can realize
higher utility and higher profits simultaneously.
Therefore, we can conclude that internal migration is likely to alleviate internal conflicts in eachindustry and make it easier for each industry to determine the number of its immigrants.
Moreover, according to the above result, the economy will accept more immigrants if native
workers in the complement industry have more influence over the immigration policy than firms
in this industry. On the other hand, the economy will accept fewer immigrants if native workers
in the substitute industry have more influence over the immigration policy than firms in thisindustry.
To summarize the results derived in this section, as in the case where there is no internalmigration, immigrants negatively impact native workers in the substitute industry and positively
impact native workers in the complement industry. However, under internal migration, native28

worker employment and utility in the substitute industry are not necessarily smaller when more
immigrants are accepted into this industry, and native worker employment and utility in thecomplement industry are not necessarily larger when more immigrants are accepted into thisindustry. Consequently, under internal migration, the interests of native workers and firms in thesame industry are not always in conflict. In other words, there are cases where native worker
utility and the profits of the firms in the same industry become larger simultaneously by accepting
more immigrants or by accepting fewer immigrants. In such cases, native workers and firms inthe same industry will come to an agreement as to whether they increase or decrease the numberof its immigrants.

5. Conclusions

This paper investigated the effects of immigrants on native workers and firms in the substituteand complement industries, assuming a case where there is no internal migration and a case where
native workers migrate between these industries.
We demonstrated that when there is no internal migration, native workers and firms are always
in conflict in both industries. Accordingly, there is no room to simultaneously have positive
impacts on both native workers and firms in the same industry by manipulating the number of
immigrants. On the other hand, if native workers migrate internally seeking higher utility, there
are cases where native workers and firms in the same industry can simultaneously make theirutility and profits larger by manipulating the number of immigrants. In such cases, the interestsof native workers and firms in the same industry are not in conflict.
Previous studies on the effects of immigrants, or on the determination of immigration quotas,
have highlighted the conflicts among different industries or different sectors. However, conflicts
are also existent in each industry or sector and they can be affected by internal migration.
29

The results of this paper suggest that we should encourage internal migration to relieve suchconflicts, and that this will make the manipulation of the number of immigrants easier.

Acknowledgements

I am deeply indebted to George M. Agiomirgianakis. I am very thankful to him. This paper
was presented at the 7th annual meeting of the European Economics and Finance Society. I am
also grateful to the participants at the conference. Any shortcomings of this paper are, of course,
my responsibility.

30

References

Amegashie, J. A. (2004). “A Political Economy Model of Immigration Quotas.” Economics ofGovernance 5: 255-267.

Benhabib, J. (1996). “On the Political Economy of Immigration.” European Economic Review 40:
1737-1743.
Bodvarsson, Ö. B., W. H. Kaempfer, A. D. Lowenberg, and W. Mertens. (2007). “The PoliticalMarket for Immigration Restrictions: Model and Test.” Journal of International Trade andEconomic Development 16: 159-192.

Borjas, G. J. (1997). The Economic Impact of Mexican Immigration. In Coming Together?
Mexico-United States Relations, edited by B. Bosworth, S. M. Collins, and N. C. Lusting,155-171, Washington, D. C. : Brookings Institution Press.
Borjas, G. J. (1999). Heaven’s Door: Immigration Policy and the American Economy. Princeton:
Princeton University Press.
Borjas, G. J. (2005). “Native Internal Migration and the Labor Market Impact of Immigration.”
NBER Working Paper No. 11610.
Flores, O. (1997). “The Political Economy of Immigration Quotas.” Atlantic Economic Journal
25: 50-59.
Friedberg, R. M. and J. Hunt. (1995). “The Impact of Immigrants on Host Country Wages,
Employment and Growth.” Journal of Economic Perspectives 9: 23-44.
Greenwood, M. J. and G. L. Hunt. (1995). “Economic Effects of Immigrants on Native andForeign-Born Workers: Complementarity, Substitutability, and Other Channels of
Influence.” Southern Economic Journal 61: 1076-1097.
Greenwood, M. J. and J. M. McDowell. (1986). “The Factor Market Consequences of U.S.
Immigration.” Journal of Economic Literature 24: 1738-1772.
31

Grossman, J. B. (1982). “The Substitutability of Natives and Immigrants in Production.”
Review of Economics and Statistics 64: 596-603.

Hatton, T. J. and M. Tani. (2005). “Immigration and Inter-regional Mobility in The UK,
1982-2000.” Economic Journal 115: F342-F358.
Kritz, M. M. and D. T. Gurak. (2001). “The Impact of Immigration on the Internal Migration of
Natives and Immigrants” Demography 38: 133-145.
Rivera-Batiz, F. L. (1981). “The Effects of Immigration in a Distorted Two-Sector Economy.”
Economic Inquiry 19: 626-639.

Schmidt, C. M., A. Stilz, and K. F. Zimmermann. (1994). “Mass Migration, Unions andGovernment Intervention.” Journal of Public Economics 55: 185-201.
Shapiro, C. and J. E. Stiglitz. (1984). “Equilibrium Unemployment as a Worker Discipline
Device.” American Economic Review 74: 433-444.
Shughart, W. F., II, R. D. Tollison, and M. S. Kimenyi. (1986). “The Political Economy of
Immigration Restrictions.” Yale Journal on Regulation 4: 79-97.
Stark, O. (1991). The Migration of Labor. Cambridge: Blackwell Publishers.
Straubhaar, T. and K. F. Zimmermann. (1993). “Toward a European Migration Policy.”
Population Research and Policy Review 12: 225-241

Zimmermann, K. F. (1995). “Tackling the European Migration Problem.” Journal of Economic
Perspectives 9: 45-62.

Zimmermann, K. F. (1996). “European Migration: Push and Pull.” International Regional Science
Review 19: 95-128.

Zorlu, A. and J. Hartog. (2005). “The Effect of Immigration on Wages in Three European
Countries.” Journal of Population Economics 18: 113-151.

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