DP2012-29

Im migration C onflicts *J unko D OI
Laixun ZHAO

Revised December 6, 2012

* The Discussion Papers are a series of research papers in their draft form, circulated to encourage
discussion and comment. Citation and use of such a paper should take account of its provisional character.
In some cases, a written consent of the author may be required.

Immigration Con‡icts
Junko Doi and Laixun Zhaoy
November 12, 2012

Abstract
Almost all existing literature assumes immigrants immediately assimilate in
the receiving country. In contrast, the present paper considers the case of nonimmediate assimilation, and analyzes various immigration con‡icts in an overlapping generations dynamic system. We examine three types of con‡icts thatarise when immigrants come in: skill con‡icts that a¤ect the capital rental andalso cause the wage gap to change between skilled and unskilled workers; intergenerational con‡icts that lead to di¤erent impacts on the young and oldgenerations; and distributional con‡icts that a¤ect each generation’s life-time
utility unequally. The degree of substitution between natives and immigrants in
production plays a key role. We also analyze the welfare composition in detail
generation by generation, and provide policy recommendation for each case.

Keyword : immigration, overlapping generations, inequality, welfare
JEL classi…cation : F22

1

Introduction

The current wave of globalization is characterized by freer mobility of not only goods
and capital across countries, but also freer mobility of human resources. With thelaunch of the European Union, NAFTA, APEC, etc., international labor migration has
increased substantially. The world total stock of international migrants reached 214
million in 2010, representing about 3% of world population. Richer or bigger countries
are the main hosts of these migrants, in the order of the U.S., Russia, Germany, Saudi
Arabia, Canada, France and England, etc. Immigration is a routine issue in presidential campaigns in the US, Europe and Australia. Even in Japan, a country that has
traditionally been closed to foreign migrants, labor shortage especially in agriculture
and heavy manual work is forcing the government to reconsider immigration among
other alternatives (Already small numbers of seasonal foreign workers and nurses are
Faculty of Economics, Kansai University, 3-3-35 Yamatecho, Suita, Osaka, 564-8680, Japan; Email: jkdoi@ipcku.kansai-u.ac.jpyCorresponding author; Research Institute for Economics & Business, Kobe University, Kobe,
Japan; E-mail:zhao@rieb.kobe-u.ac.jp

1being introduced). Facing population aging, sooner or later more lenient immigration
policies may have to be adopted.
Nevertheless, there is evidence that some unskilled workers, local communities andlaw enforcement o¢ cers are more negative on this issue. They are afraid that immigrants may compete away jobs and lower wage rates, free ride on public services suchas schooling, parks, medicare, public universities and others, cause cultural con‡icts,
increase crime, etc., and eventually lower the net welfare of natives.1 Especially during
the current …nancial and economic crisis, some US law makers demand a tightening of
immigration laws. As a consequence of all these, despite being an integral part of globalization, immigration is perceived negatively in public opinion. Often when related
issues appear in the media, they are about illegal immigration, or some other negative
images such as taking jobs away and depressing wages, etc.
A particularly relevant issue is assimilation. While most theoretical analysis in
the literature has treated immigrants and natives identically, recent empirical studies
have found that immigrants do not exert enough e¤ort to absorb the mainstream
culture, customs and language, even though the potential economic returns (increased
productivity and enhanced earnings) to assimilation are high. It is also observed thatduring a recession, …rms tend to lay o¤ foreigner workers …rst; foreign guest workers
are limited to a certain sector, or in occupations with a speci…c skill only.2
The present paper attempts to examine the above immigration issues in an overlapping generations (OLG) model, especially incorporating non-assimilation. To keep
the model tractable, we consider an economy without trade but allows optimal immigration every period. Workers are divided into skilled and unskilled (also referred
to as non-skilled or low skilled) groups, who are combined with capital to produce asingle good, with skilled workers being more e¢ cient. Households consume this good
and an impure public good, the latter of which is …nanced through income taxes. To
incorporate non-assimilation, we assume that non-skilled and skilled immigrants havedi¤erent degrees of substitutability with natives, such that the possession of skill makes
a worker less substitutable. We examine the di¤erent impacts of allowing skilled andunskilled immigration on the receiving economy.
1

Hanson (2005) vividly documents that opinions on immigration are sharply divided in the U.S.,
even within each interest group, such as within the Democrat and Replican parties, labor groups,
environmental groups, etc. He argues that there are two main sources of opposition to immigration:
one concern is labor-market pressures and a second is about public …nances. Hatton and Williamson(2005) and Hanson, Scheve and Slaughter (2007) …nd evidence that while OECD countries have loweredbarriers to foreign trade and capital in recent decades, they have not commensurately reduced barriers
to immigration.
2
For instance, Both Lazear (1999) and Bauer et al. (2000) …nd assimilation incentives to be low
if it is costly for immigrants to learn the receiving country’s culture and language, or if immigrants’culture is strongly represented in the host country (such as by a large immigrant community); Djajic(2003) argues that immigrants assimilate at di¤erent rates which may also di¤er from those of theirchildren; Stark and Fan (2006) and Fan and Stark (2007) explain that non-assimilation arises because
immigrants want to keep close ties with families, friends and cultural roots at home. They even …ndthat the richer the natives, the weaker the e¤ort to assimilate, ceteris paribus; More recently, Harrie
et al. (2008) argue that immigrants’utility will be lowered if they assimilite because they have to lose
important parts of their own culture.

2

In particular, we …nd that immigration brings three types of con‡icts, even underfull employment:3 i). “Skill con‡icts” that cause the wage gap to change between
skilled and unskilled workers; ii). “Intergenerational con‡icts” that lead to di¤erent
impacts on the young and old generations, stemming from changes in the capital stock,
the interest rate and the impure public good; iii). “Distributional con‡icts”that a¤ect
each generation’s lifetime utility unequally, and in particular, the receiving generation
seems to be hurt the most. The details of each con‡ict depends crucially on the degreeof substitution between natives and immigrants.
Speci…cally, these con‡icts stem from the following e¤ects: A productivity e¤ect–
immigration changes the productivity of native skilled and unskilled workers, dependingon the degree of skill complementarity, and a capital allocation e¤ect–immigration
increases the labor supply, lowering per-capita capital for production. The intercourseof these e¤ects cause the high and low skilled wages to change, and thus the capital
return, public-goods contribution and steady states will all change.
For instance, the arrival of immigrants increases the total scale of production, raising
the productivity and wages of native skilled workers. On the other hand, the capital
allocation e¤ect lowers the wages for both skilled and unskilled workers. While theskilled wage may increase or decrease, the unskilled wage increases less or falls more,leading to a “skill con‡ict”.
Intergenerational con‡icts occur when di¤erent generations may face di¤erent levelsof capital stock, interest rates and levels of public goods. Immigration seems to bene…tthe old generation more, because they earn income from savings while the young generation earns wage income, which could lead to con‡icts between workers and ‘capitalowners’; Thus, while old cohorts may enjoy cheaper products made by immigrants,
young natives tend to worry about their jobs and wages being ‘competed away’.
Also we …nd that distributional con‡icts arise among cohorts (both young and old)
living in periods t 1, t and t + 1. To be speci…c, the generations living in t 1 andt enjoy the same level of capital stock because the economy stays in the same steadystate when immigrants arrive in period t. However, their utilities di¤er due to thewage di¤erence caused by an increase in the number of workers in period t. Further,
the capital stock changes after t, leading to more con‡icts.
Our strategy is to divide the impacts of immigration into three cases. In Case I,
both the high and low skilled wages increase and the capital stock also rises. In Case II,
the exact opposite occurs such that both wages fall. In Case III, the high skilled wage
rises but the low skilled wage falls. Subsequently in each case, we calculate the welfarechanges for the skilled and unskilled natives respectively. Using these comparisons, we
determine whether a particular immigration con‡ict is likely to arise or not. And …nally,
depending on the source and type of con‡ict, we provide some policy recommendations.
By conventional wisdom, natives worry that immigration may lower wages. We
amplify the problem by assuming that the wage e¤ect dominates other e¤ects (e.g.,
the capital-stock e¤ect, the interest-rate e¤ect, the public-goods e¤ect). Nevertheless,3

One might naturally think that immigration could reduce native employment in some sectors,
which certainly causes con‡icts. In the present paper, our concern is that even with full employmentof both natives and immigrants, various con‡icts still arise.

3

we …nd that immigration can bene…t the receiving country in many cases. But con‡icts
can still arise.
The various cases and con‡icts we examine are inspired by a long list of empiricalstudies, whose …ndings sharply contrast with each other. The results in the present
model can help to explain these contrasting …ndings. For instance, Smith and Edmonston (1997) and Borjas (1999) …nd weak positive complementarity e¤ects of immigration and estimate the net bene…t from the present stock of immigrants to be approximately $10 billion in the U.S. Ottaviano and Peri (2005) …nd strong complementarities
between comparably skilled immigrants and natives such that overall immigration generates a large positive e¤ect on the average wages of US-born workers. On the other
hand, Bauer and Zimmermann (1999) calculate that an annual in‡ow of 200,000 nationals from central and east European countries would potentially decrease the wagesof the European Union workers by about 0.8 percent; Davis and Weinstein (2002) show
negative terms of trade e¤ects of immigration based on technology superiority. Borjas,
Freeman, and Katz (1997) and Borjas (2003) …nd that immigration depresses wages fornative workers who are likely to substitute for immigrant labor. Borjas, Grogger andHanson (2010) …nd that a 10-percent immigrant-induced increase in the supply of aparticular skill group reduced the black wage by 2.5 percent, lowered the employment
rate of black men by 5.9 percentage points, and increased the incarceration rate ofblacks by 1.3 percentage points.
There is voluminous theoretical literature on how immigration a¤ects host-country
welfare. With regard to models that analyze dynamics e¤ects of immigration, Galor andStark (1991) study the pattern of migration in an OLG model with two countries thatare di¤erent in technology. Dolado, Goria, and Ichino (1994) show that immigration
can increase the stock of human capital in the host country, and so facilitate growth.
Lundborg and Segerstrom (2000, 2002) argue that since immigrants lower wages, …rms
can spend more on R&D, raising growth. In Stark and Wang (2002), migration toa richer foreign country, by raising both the level of human capital and the averagelevel of human capital of non-migrants in the home country, can enhance welfare andnudge the economy toward the social optimum. Ben-Gad (2008) shows that an in‡ux
of high-skilled immigrants lowers the wages of skilled workers, raises the wages of
unskilled workers, and substantially raises the return to native-owned capital. With
regard to static models, Ethier (1985) studies the relationship between international
trade and labor mobility in the host country, focusing on how immigrants help preserve
import-competing industries with their cheap labor. Felebermayr and Kohler (2007)
decompose the native welfare e¤ect into a standard complementarity e¤ect, augmented
by a Stolper-Samuelson e¤ect, and a terms-of-trade e¤ect. They calibrate this model toa generic OECD economy and provide simulation results. However, in all these models,
immigrants and natives are "undi¤erentiated", i.e., immigrants assimilate immediately
when arriving in the receiving country. A most recent paper by Stark and Jakubek(2013) studies how migrants help each other out by forming migration networks andthere exists an optimal size of such networks and hence multiple networks coexist.
The rest of the paper is organized as follows. Section 2 constructs the basic model,
section 3 examines various immigration con‡icts, section 4 analyzes the impacts on

4

welfare, and section 5 provides some policy recommendations and concluding remarks.
Analysis of the dynamic system and the steady state is relegated to the Appendix.

2

The Basic Model

Consider an overlapping-generations economy where people work while young and retire
when old. Workers can be either high skilled or low skilled. Skills are born with andcannot be changed. The initial native population size is at 2Nt = Nty + Nto , with Nty
and Nto being the numbers of young and old at t.
Before time t, only natives live in the host country. From time t, the host country
receives It number of immigrants (young generation only) every period, Ith of whichbeing high skilled and Itl = It Ith being low skilled. Immigration makes the total
number of workers in the host country 2Nt + It at t: We assume that one immigrantbears one child in the next period, and this child is treated as a native since it isborn in the host country. Then the total population at time t + 1 becomes T Pt+1 =
y
y
y
o
o
Nt+1
+ Nt+1+ It+1+ It+1, where Nt+1
is the number of young native people, includingyorepresents immigrants
the children of natives and immigrants, i.e., Nt+1
= Nt + It ; It+1yarriving in t and becoming old at t + 1; and It+1 stands for immigrants arriving in t + 1:

2.1

Consumers

We consider a one-good economy. Let j denote the four types of cohorts: respectively
for high skilled natives and immigrants, and low skilled natives and immigrants. Each
type of cohorts maximizes the per capita utility as follows:
max U = log cy;jt + log gt +s:t:cy;j
t
o;j
ct+1

=
=

(1
)wtj sjt ;Rt+1 sjt ; wherelog co;jt+1 + log gt+1 ;
Rt+1

1+

(1)

t+1 ;

where cy;jis cohort j’s consumption when young at time t and co;jt
t+1 is the counterpart
when he becomes old; gt is an impure public good per capita at time t, supported by
tax at t 1: Parks, roads, bridges, hospitals, clean environment, schools and libraries
are some examples of the impure public good; is the time preference; is the income
tax rate; sjt is savings; wtj is the wage rate, to be determined in (11) soon; is the rateof return from savings; and R is the gross rate of return. All tax revenue, T; is used to
produce the impure public good for the next period,4Gt = Tt 1 = (wth 1 Nth 1 + wtl 1 Ntl 1 );Gt+1 = Tt =wth (Nth + Ith ) + wtl (Ntl + Itl ) ;

(2)
(3)

where Gt is the aggregate public goods, and N h and N l are the number of high and low
skilled natives respectively. The per-capita public goods decreases with immigration,
4

To simplify the model, only labor income is taxed. Interest income is not taxed because the sourceof which is labor income.

5

e.g., hospitals, roads, schools and parks could become crowded and congested.5 Utility
maximization results in consumption for the young and old and the savings as:
cy;j
=
t
co;j
t+1 =
sjt =

2.2

1
(1
1+
1+
1+

)wtj ;(4)(1)wtj Rt+1 ;

(5)

(1

)wtj :(6)

Production

To model the di¤erence between natives and immigrants, we assume that they are
combined slightly di¤erently with capital to produce the …nal good, by the following
CES (constant elasticity of substitution) type production function,
Yt = Kthf h Nth

h

+

h

b Ith

h

1

g h + Ktl

f l Ntl

l

+

l

b Itl

l

g

1
l

;

(7)

where Ktx (x = h; l) denotes the physical capital used by high (h) and low (l) skilledworkers at t; and x is a parameter that regulates the shares of natives and immigrants
in their respective contributions to output. Production requires both capital and labor.
But for labor, either natives or immigrants or both can be used, and both of them can
11
be either high or low skilled. We assume 1 > h h > l l ; that is, high skilledcohorts are relatively more productive than low skilled ones. The parameter b captures
possible impacts of the distinct cultures and customs of immigrants that may stem fromdi¤erent institutions, education, region, etc., which may improve production or cause
frictions that delay production. We assume b < 1; roughly implying that immigrants
are less productive than natives for each type of workers.6
Finally, we describe the parameter x (x = h; l), which represents the degree ofsubstitutability between natives and immigrants, for high and low skilled agents respectively. Following a long research tradition on immigration,7 we assume that skilledworkers possess speci…c skills that enable them to be less substitutable. A worker becomes more substitutable if he is less skilled, and in the extreme, unskilled natives andunskilled immigrants are perfect substitutes. It is then natural to assume that h < l ;i.e., the substitutability of high skilled workers must be lower. More generally, immigrants and natives can be either substitutes or complements in production, dependingon the value of x , as will be examined in detail later.
5

Hanson (2005, p1) states that "taxpayers in high-immigration U.S. states shoulder most of immigration’s …scal costs, which they bear in the form of higher taxes that go to pay for public services
used by immigrant households."
6
If b > 1, then in equilibrium the number of new immigrants must be bigger than natives eachperiod. We exclude such an abnormal case in the present model by assuming b < 1.
7
For instance, in Borjas (1995) immigrants and natives with equal skills are perfect substitues.
Other studies assume a certain degree of imperfect substitutability, see for example, Ethier (1985),
Angrist and Kugler (2003) and Ottaviano and Peri (2005).

6

Let the …nal good be the numeraire. Then the pro…t of the representative …rm canbe written as
Kth + Ktl Rt

t = YtwtI;h Ithwth Nthwtl Ntl

wtI;l Itl :Pro…t maximization yields the return to capital, which must be equalized working witheither high or low skilled labor, and this condition then determines capital allocation
between the two types of labor:(Ktx )Rt =1

f x (Ntx )

x

+

x

x

1

b (Itx ) g x :(8)Firms producing the …nal good choose the quantity of each type of labor input fora given wage type.
wtx = (Ktx ) (1wtI;x = (Ktx ) (1) f x (Ntx )) f x (Ntx )

x
x

+
+

x
x

x

1

x

1

b (Itx ) g xb (Itx ) g x

1 x
1 x

(Ntx )

x

1

;

(9)

b (Itx )

x

1

:

(10)

We assume that the immigration policy set by the receiving-country governmentis, it allows only those immigrants employed by domestic …rms and that immigrants
receive the same wage rates as natives (e.g., H visa holders in the U.S.), given eachworker’s type. This can be justi…ed on the grounds that in the absence of illegal
immigration, it is unlawful to pay legal immigrants a discriminatory wage. Then …rms
will hire immigrants up to the point where their marginal productivity is equal to thatof native workers, which is again equal to their wage, resulting in
wtI;x = wtx :(11)

However in equilibrium the number of immigrants must be smaller than that of natives
due to the assumption of b < 1. There is full employment in this economy. Conditions(9) and (10) give the ratio of the number of immigrants and natives as,
Ith =Nth = b 1

1h; Itl =Ntl = b 1

1
l

(12)

:

That is, more immigrants are hired as their productivity rises and as the elasticity of
substitution becomes higher.
Using the above, the production function can be rewritten as
Yt = Kth1h 1(E h )1

+ Ktl

1

l 1

El

1

;

where E x f1 + b 1 x g x > 1 (x = h; l as before), which can be called the immigratione¤ects, since they stem from immigration of respectively high and low skilled workers,
1
and x ( x ) x (Ntx ).

7

2.3

Market Equilibrium

By Walras’law, there are two market equilibrium conditions we need to take care of.
One is in the goods market, where …nal output is consumed by the young and oldgenerations, re-invested as capital and collected as government tax revenue:
Yt =

N
+I
X

Cty;x +xN
X

Cto;x + Kt+1 + Tt :

(13)

x

where C is the aggregate consumption. The other is in the capital market, where total
capital in each period is combined respectively with high and low skilled workers to
produce the …nal output:
Kt = Kth + Ktl :(14)
The dynamic system and the steady state are analyzed in the Appendix.

3

Basic E¤ects of Immigration

Since agents born at t are a¤ected by economic conditions in periods t 1; t and t + 1
only, we assume that they do not care about other periods and consider only thosecon‡icts that arise among the three generations. For analytical tractability, we assume
that half of the initial natives are high skilled and the other half low skilled, N h = N l .
Immigration increases labor supply, which a¤ects the productivity of natives andthe wages of both skills. These lead to changes in their consumption and saving choices,which further a¤ect the tax revenue and capital stock economywide. Hence, in thissection, we …rst consider changes in the marginal productivity and compare the wages
with and without immigration. Then we analyze how immigration a¤ects other endogenous variables. In the next section we shall examine the welfare e¤ects under di¤erent
types of immigration con‡icts.

3.1

Marginal Productivity and Wages

First consider how the marginal productivities and wages of high and low-skilled native
workers change with immigration.8 The production function before immigration is
obtained by setting I h = I l = 0 in (7); and in the steady state, we also have the wages
as (for all x = h; l):wtx 1 = (11

) ( x) x

1

K
h+

l

;

(15)

1
the skill di¤erence e¤ect between high and low skilled workers; h +where ( x ) xlmeasures the e¤ectiveness of the labor force (meaning it is adjusted by the share parameter x . As their names show, they denote the e¤ect coming from the di¤erence
8Note that they are unde…ned ath= 0 andl.

8

parameter x between skilled and unskilled workers when x changes. Hence, hK+ l is
the steady-state e¤ective capital-labor ratio, or the capital per unit of e¤ective labor.
After immigration takes place, we obtain,
wtx = (11Kx

) ( x ) x (E x )1hE h +(16)lEl

With immigration, the capital allocation per worker decreases, since the existing capitalstock must be shared among more workers, as is shown by h E hK+ l E l in (16). On theother hand, due to technology complementarity, each worker’s productivity increasesxby (E x )1 :9
Comparing the marginal productivity conditions before and after immigration yields:
Lemma 1 Given h < l and b < 1 by assumption, we have E h E l > 0, i.e., theimmigration e¤ect is stronger for high skilled workers than for low skilled workers.
Proof. Note that1h

> 1l : From the de…nition of E h and E l , we see that the signof E h E l depends on their component di¤erence b 1
11
b 1 h b 1 l > 0; leading to E h E l > 0:

3.2

1

1

b1

h

l

: Since b < 1; thenWage Changes under Immigration

Next we examine how wages change after immigration. Taking the ratio of the highskilled wage between periods t 1 and t gives
wth
=
wth 1
h

h

+
hE h +

l
lEl

Eh

1

h

(17)

;
1

l

h

+
where
can be called the capital allocation e¤ect and E h
the prohEh+ lEl
ductivity e¤ect, both for high skilled workers. The former stems from changes in thecapital allocation per worker and the latter from changes in the productivity post immigration. The capital allocation e¤ect decreases the wage, since immigration increases
the supply of workers in the host country. On the other hand, the productivity e¤ect
increases the wage. The intersection of these two opposing e¤ects determines whether
the high-skilled wage increases or not. Analogously, a similar condition can be obtained
for the ratio of the low-skilled wage between periods t 1 and t.
As shown above, the capital allocation e¤ect and the productivity e¤ect causechanges in the marginal productivity and wages of native skilled workers. Recall thatthe capital allocation e¤ect is negative, but the productivity e¤ect is positive. Combining both e¤ects, we can divide the wage impact of immigration into three cases: CaseI, both native wages increase; Case II, both wages decrease; Case III, the high skilled
wage increases, but the low skilled wage decreases.
9

Without immigration, I h = I l = 0 and E h = 1 = E l ; then (16) boils down to (15).

9

3.3

E¤ects on Capital Goods and Interest Income

The wage changes just considered may a¤ect the household’s behavior in consumption
and savings patterns, eventually leading to changes in the capital stock, public goods,
and interest rate. We examine them in detail now.
Comparing (44) and (45) in the Appendix yields
1

1

h h
l l
Nth + Ith
Ntl + Itl
Kt+1
wtl
wth
+ l
:
= h
K
wt 1 ( h ) 1h N h + ( l ) 1l N l
wt 1 ( h ) 1h N h + ( l ) 1l N l
t
t
t
t

(18)

K is from natives only, but Kt+1 is generated by the disposable income of the fourtypes of agents. The ratio of capital stock before and after immigration depends on not
only the ratio of high and low skilled wages, but also their respective contribution to
the capital stock: Each ratio is less than 1, and they sum up to 1 without immigration
but more than 1 with immigration10 .
To examine in detail the e¤ects of immigration on capital goods, we rewrite (18) as
Kt+1
=
K
h

h

l

h

Eh +
h+

l

El

l

1

(19)

l

Since Eh ++ l E > 1, the capital stock increases with immigration in all three cases.
Further, the ratio of interest rates can be derived as
1

Kt+1
Rt+1
=
:
(20)
Rt
K
Thus, changes in the interest rate depend on changes of the capital stock, in the
opposite direction though.
Note that the impure public goods in each period is provided with the tax revenueof the previous period as shown in (2) and (3). Then the di¤erence between pre andpost immigration is,
1h h
E + lEl
Gt+1
=
:
(21)
h+
l
Gt
Similar to the capital stock, the provision of public goods increases after immigrants
come in.

4

Immigration Con‡icts

Having investigated the various e¤ects of immigration, now we put them together andconsider the possible con‡icts immigration may cause in the receiving country.
10

The ratio of capital stock pre and post immigration depends on the population structure. Recall
the assumption N h = N l . Since high skilled immigrants possess more complimentary technology
than low skilled ones, the economy accepts more of the former immigrants. As time passes, thecontribution ratio of high skilled immigrants will be larger, and the wage e¤ect becomes stronger forthem. However, we do not need to care too much about this issue since we focus on only periods t
and t + 1.

10

4.1

Wage Related Con‡icts

First, let us look into the skill con‡ict on wages between high and low skilled workers.
Table 1 combines both the capital allocation e¤ect and the productivity e¤ect, andsummarizes the main comparative statics results on wages. In Case I, both h and l
are less than 1and both types of immigrants tend to be complementary to natives,
hence both wages increase with immigration. But Case II is the exact opposite, where
both h and l are bigger than 1, and both wages decrease. In these two cases,
since both wages move in the same direction, immigration is less likely to cause con‡icts
between high and low skilled workers. In contrast, in Case III, the high skilled wage
increases, but the low skilled wage decreases. Hence a con‡ict arises, and we name it
the skill con‡ict.
Table 1 Skill con‡icts
Case I Case II Case III
h
w
+
+
wl
+
con‡ictnono
yes
These results are summarized in the following proposition.
Proposition 2 A skill con‡ict arises in Case III when the high- and low-skilled wages
move in opposite directions since the productivity e¤ect dominates the capital allocatione¤ect for the high skilled, but exactly the opposite arises for the low skilled; In contrast,
skill con‡icts are less likely to occur in Cases I and II, where both types of immigrants
tend to be either complementary or substitutable to natives simultaneously, leading both
wages to move in the same direction.
The next question is, how does the skill-premium (i.e., the wage gap between highand low skilled workers) change after immigration takes place? Straightforward calh
1
1
wth 1
1 h
( h ) h (E h )wth>=1] > 0; which arises since E hculations yield wl
1 [
l
l
1
w
t
t 1
( l ) l (E l )1lEl
and by using Lemma 1. That is, the wage gap widens with immigration.
Speci…cally, in Cases I and II, while both high and low skilled wages move in the same
direction, the high-skilled wage changes more in Case I and less in Case II than the
low-skilled wage. And in Case III, immigration raises (reduces) the high (low) skilled
wage as shown. Therefore,
Proposition 3 The wage gap between high and low skilled natives will be enlarged withimmigration, regardless if both wages rise or fall.
Indeed, Scheve and Slaughter (2001) …nd that opposition to immigration is higher
among the less educated, in a U.S. survey of public opinion on immigration policy, as
the above Proposition would predict.

11

4.2

Welfare Related Con‡icts

Now we examine how immigration impacts the host country’s per capita welfare ofgeneration t. From previous sections, we know that immigration a¤ects not only thecurrent generation but also future generations through changes in the amount of capital
goods. Moreover, it causes income redistribution which may lead to con‡icts. Here wefocus on the case when immigration increases capital goods; that is, con‡icts may stillarise even though the host country as a whole bene…ts from immigration.
Using (1)~(5), substitution and rearrangement of equations lead to
Utwith;N l

Utwithout;N lwtwith;l= (1 + ) log without;l + logw| t {z }wage e¤ect

+ log

Gwith
t+1
without
G
| t+1
{z }

with
Rt+1
Rwithout
| t+1
{z }

interest rate e¤ect+ (1 + ) log

2N|2N{z+ I}(22)

scale e¤ectpublic-goods e¤ect

The immigration-induced change in the per-capita welfare can be decomposed into
four e¤ects: the wage e¤ect, the interest rate e¤ect, the public-goods e¤ect, and the scalee¤ect, which can be explained as follows. By (1) utility is obtained from consumption
at young and old ages and from the impure public goods. Consumption at young age
comes from wage income, and that at old age depends on savings and interest income.
Moreover, the public goods at t + 1 is produced by tax revenue at t, which also comesfrom wage income. Therefore, the welfare di¤erence with and without immigration
depends on the di¤erence of wages, interest rates, public goods and the scale e¤ect ofimmigration–the ratio of immigrants to total population (We assume the scale e¤ect
to be dominated by other e¤ects; Otherwise if the scale e¤ect dominates, there are
too many immigrants and welfare always falls). The ratio of interest rates depends
negatively on the ratio of capital stock from (20); and the ratio of public goods is (21),which works in the opposite direction as the interest rate e¤ect.
Consider a small open economy. Capital-stock increases via immigration do nota¤ect the interest rate in the host country, which is given by the world market, andextra capital is lent to other countries. In this setting, we have:
Utwith;nx

Utwithout;nx = (1 + ) log

wtwith;x
wtwithout;x

|

{z

}

wage ef f ect+ (1 + ) log

2N
:
2N
+
I
| {z }(23)

scale ef f ect

That is, the welfare only depends on the wage e¤ects, and we obtain Table 1 again.
Then, we can calculate the welfare di¤erence according to skills, as in the following
subsections.

124.2.1

Low-skilled natives

First, we analyze the immigration-induced change in the welfare of low-skilled natives.
Substituting (20) and (21) into (22) yields the welfare di¤erence with and without
immigration. Using (19) and ignoring the scale e¤ect, we get
Utwith;nl Utwithout;nl
h h
wl
E +
= log( l t )1+ (
h+
wt 1

l

E l (1
)
l

2

)

(24)

The welfare di¤erence stems from the wage di¤erence, and includes not only the wagee¤ect but also the interest and public goods e¤ects.
In Case I, the welfare is higher with immigration than without. For Cases II and
III, we rewrite the above as,
Utwith;nlh+hE h +

Utwithout;nl = log(

l
lEl

+

)

2+

El

(1+ )(1

l

)

(25)

:

2
If +
+
< ; since both the productivity and capital allocation e¤ects are
positive, welfare increases with immigration; If + 2 +> , the capital allocatione¤ect becomes negative but the productivity e¤ect is still positive, resulting in an
ambiguous net e¤ect. Thus we can summarize the above discussion in Table 2L.

Table 2LE¤ect name Case I
wage+interest+
public goods
+
scale
total
+

4.2.2

Case II

Case III

+
+

+
+
2

+ if +
ambiguous if

+
2

+

< ;
+
>

+ if +
ambiguous if

2

+

+

High-skilled natives

Analogously, the high skilled natives’welfare changes as followsh+hE h +l2Nt:
2Nt + It
(26)
Ignoring the scale e¤ect and using the same method above, the welfare di¤erence canbe summarized as in Table 2H.
Utwith;nh Utwithout;nh = log(

lE

+

)
l

Table 2H

13

2+

Eh

(1+ )(1h)+(1+ ) log

2

< ;
+
>

E¤ect name Case I
Case II
wage+
interest++
public goods
+
+
scale
total
+
+ if + 2 +
<
2
total
ambiguous if ++4.2.3

Case III++++>

Summary on welfare

From Tables 2H and 2L, the welfare of both high and low skilled workers move in thesame direction, both increasing in Case I but both decreasing in Case II. In these twocases, skill con‡icts are less likely to occur. In other words, if high- and low-skilled
immigrants bring complementary technology with high- and low-skilled natives, welfare
increases for both types of natives, as in Case I. On the other hand, if the host country
accepts immigrants with high h and l (i.e., more substitutable technology), then thewelfare of natives (irrespective of skill) will fall, as in Case II. However, in Case III,
since welfare for the low-skilled falls but that for the high skilled rises, skill con‡icts
are more likely to occur.

4.3

Intergenerational Con‡icts

In this subsection, we examine how immigration a¤ects di¤erent generations living attime t, which could possibly lead to intergenerational con‡icts.
The t-period welfare of the old and young generations11 are respectively
Uty = log Cty + log Gt ;Uto = log Cto + log Gt ;where Cty (Cto ) stands for the total consumption of the young (old) cohorts. Since thepublic goods at t is provided by tax collected at t 1; both generations enjoy the sameamount of Gt (since immigrants arrive at t). Then the welfare di¤erence is determinedby the di¤erence in consumption (or income),
Uty
Uto

wl N l +wh N hlog wl Nt l t +wth tN h
Uty 1
t 1 t 1t 1 t 1=
Uto 1
log RRt t 1
l

= 1+

log
log

1

(E l )lh1+ h (E h )hEh+ lEl

h+
hEh+

l
lEl

1

(27)

This expression states that the welfare di¤erence depends on the total-income ratio
and the interest-rate ratio. In the second term on the last line, since E h > E l > 1, theantilogarithm in the numerator is less than 1, leading the numerator to be negative.
Similarly, the antilogarithm in the denominator is also less than 1 but with a negative11

We consider all young and old agents irrespective of their skills.

14power, so the denominator is positive. Thus, the second term is negative. Then the
RHS of (27) becomes less than 1. That is, the welfare increase is smaller for the young
generation than for the old generation, which gives:
Proposition 4 The old generation enjoys more bene…ts from immigration than theyoung generation, leading to an intergenerational con‡ict.
The intuition can be explained as follows. Post immigration, the wage is a¤ected by
the capital allocation e¤ect and the productivity e¤ect, while the interest-rate increaseis caused by the fall in the amount of capital per worker and the immigration e¤ect
shown by (8). Old cohorts bene…t more from immigration because their income comesfrom savings.

4.4

Distributional Con‡icts

Finally, we look into how immigration a¤ects the life-time utility of each generation.
It may seem that immigration at t lowers the welfare of the natives born at t 1
irrespective of their skill levels, because the public goods they contributed must be
divided by immigrants also, i.e., by the number 2N +I at t. In other words, immigrants
can enjoy the bene…ts at t, although they did not pay tax for the public goods at t 1:
However, since the steady state and all variables change, natives born at t 1 are
not always harmed by immigration, compared with t or t + 1 generations. Here we
investigate this issue in detail.
4.4.1

Welfare of low-skilled natives

Low-skilled natives born at t
t 1 and t is,

1 First, the welfare di¤erence between generations

Utnl=

Utnl 1
Kt+1
)
log(
K

1

+ log(

Kt+1
wl
2Nt 1
) + (1 + )[log l t + log]:
K
2Nt + It
wt 1

(28)

Note that the di¤erence between (24) and (28) lies in the interest rates. In (24), theproduction functions at t + 1 with and without immigration are di¤erent, in addition
to the capital-stock di¤erence. On the other hand, in (28), the shape of the production
function is the same at t + 1 and t; then the interest di¤erence between the two periods
stems only from the capital-stock di¤erence.
As previously shown, the capital stock e¤ect works in the same direction as the
public-goods e¤ect, but opposite to the interest-rate e¤ect. Using (19) and ignoring
the scale e¤ect, we can rewrite (28) as,h

Utnl

Utnl 1 = log+
hE h +

l
lEl

(1+

)

El

1

l

:

(29)

In Case I, since the capital allocation e¤ect is dominated by the productivity e¤ect,
the welfare di¤erence is positive. On the other hand, in Case II, the former e¤ect15

ldominates the latter e¤ect; moreover, (29) lies above the curve EII
due to the bigger
power of the capital allocation e¤ect, giving rise to a negative sign for (29). And thisholds true in Case III too. Then, we obtain Table 3L1.

Table 3L1 Welfare di¤erence between t and t 1
E¤ect nameCase I Case II Case III
wage e¤ect+interest e¤ectpublic goods e¤ect+++scale e¤ect
total+
Low skilled natives born at t + 1 Similarly, the welfare di¤erence between generations t and t + 1 can be obtained as,
nl
Ut+1

Utnl

Kt+1
Kt+2
) + log (= (1 + ) log () 1 + log
KK
| {z }{z }| t+1
wage ef f ect

1+(E h )twhere TTPPt+1=

interest rate e¤ecth+ 1+(E l )

l

Kt+1
)
(
|K
{z }

+ log

T Pt
T Pt+1

public goods ef f ect

At t + 1; young immigrants who
l .
l
h
h
(E h ) 1+(E h ) +(E l ) 1+(E l )came to the host country at t become old. By (12), It+1 =Nt+1 is a constant, and thus
the immigration e¤ect at t + 1 is the same as at t: Then, both the per-capita capital
allocation ratio, 1=( h E h + l E l ) and the productivity e¤ect remain the same in thesetwo periods, and the wage ratio is a¤ected by the capital goods e¤ect only. Thus,
wages are di¤erent only because the capital stock has changed. Agents who save att receive interest income at t + 1. Further, the di¤erence in interest income betweent + 1 and t + 2 depends on Kt+2 =Kt+1 > 1:12 Therefore, the interest rate e¤ect lowers
the welfare di¤erence between t + 1 and t: We thus obtain Table 3L2.
Table 3L2 Welfare di¤erence between t and t + 1
E¤ect nameCase I Case II Case III
wage e¤ect+++interest e¤ectpublic goods e¤ect+++scale e¤ect
total
+
+
+
Using Tables 3L1 and 3L2, if the scale e¤ect is dominated by the other three e¤ects,
we further obtain Table 3L3 below, which provides the details on the patterns of gains
from immigration.
12

See the Appendix.

16

Table 3L3 Welfare Pattern 1E¤ect nameCase I Case II Case III
wage e¤ect
P1
P3
P3
interest e¤ect
P2
P2
P2
public goods e¤ect
P1
P1
P1
scale e¤ect
total
P1
P3
P3
where Vi is value at period i, and P 1 fVt 1 < Vt < Vt+1 g; P 2 fVt+1 < Vt <Vt 1 g; P 3
fVt < Vt+1 ; Vt 1 g and P 4
fVt > Vt+1 ; Vt 1g stand for the patterns inwhich the wage, interest-rate and public goods e¤ects are compared. For example, P 1
in the wage row in Case I means wt 1 < wt < wt+1:
Table 3L3 implies that distributional con‡icts may occur among di¤erent generations. While the welfare di¤erence between t and t 1 comes from the capital allocationand productivity e¤ects after immigration, that between t and t + 1 stems from capitalstock changes. Note that the per capita welfare of generation t is given by
U = (1 + ) log wtl + log Rt+1 + log wtl 1 + log wtl ;
{z
} |
|
{z
}
total consumption(30)

total public goods

which shows that total consumption depends on the wage income and the interest
income, because households save a part of wages for consumption in old age.
In Case I, both the high and low skilled wages increase after immigration, so thatthe capital stock also increases at t + 1: Though the interest rate is lowered by theincrease in the capital stock, aggregating all e¤ects we …nd that welfare is the highest
in t + 1.
In Cases II and III, welfare at t is the lowest, because the wage is lower at t thanat t 1 due to the capital allocation e¤ect, and it is also lower than at t + 1 due to thecapital stock e¤ect:4.5

Welfare of high-skilled natives

High-skilled natives born at t 1 Using the same procedure as in the previous
subsection yields the welfare di¤erence between high skilled workers of generations t
and t 1 as follows,
Utbh1
Kt+1
)
log(
K

Utnh
=

1

+ log(

Kt+1
wh2N) + (1 + )[log ht + log];
K
2N + I
wt 1

which is the same as equation (26) without the interest rate e¤ect. We thus have
Table 3H1 Welfare di¤erence between t and t

171

E¤ect nameCase I Case II Case III
wage e¤ect++interest e¤ectpublic goods e¤ect+++scale e¤ect
total
+
+
High skilled natives born at t + 1 Given the scale e¤ect is dominated, the welfare
di¤erence between high skilled workers at generations t and t + 1 is,
nh
Ut+1

Utnh

= (1 + ) log(

Kt+1
Kt+2
)
) + log(K
Kt+1

1+ log(

Kt+1):
KThis di¤erence depends on the capital stock for each period, since the immigratione¤ect does not change between t + 1 and t. We thus obtain Table 3L2 as Table 3H2
again. Combining Tables 3L2 and 3H1 also gives Table 3H3.
Table 3H3 Welfare Pattern 2E¤ect nameCase I Case II Case III
wage e¤ect
P1
P3
P1
interest e¤ect
P2
P2
P2
public goods e¤ect
P1
P1
P1
scale e¤ect
total
P1
P3
P1
The welfare di¤erence between t and t 1 stems from changes in the wage andcapital goods e¤ect. In contrast, the di¤erence between t and t + 1 comes from capitalstock changes. Table 3H3 shows that in Cases I and III, the welfare at t + 1 is thehighest among the three periods. The wage at t is higher than at t 1; because theproductivity e¤ect dominates the capital allocation e¤ect. Moreover, since the capitalstock increases at t + 1, the wage at t + 1 is higher than at t: Hence, Pattern P 1
emerges in Cases I and III. In Case II, the welfare at t is the lowest; i.e., welfare falls
after immigration, following the wage decrease.
Summarizing the above yields:
Proposition 5 Regardless of skills, those born in generation t cannot obtain the highest welfare among the three generations, t 1; t; and t + 1. As such, generation tnatives may oppose immigration, even though their wages increase.

4.6

Generation t

1

In this subsection, we consider how the welfare of generation t 1 is a¤ected byimmigration, since they become old at t when immigrants come in. Because the e¤ects

18

are similar for both high and low skilled workers, we only consider the former ones.
The welfare di¤erence between generations t 1 and t 2 can be written as
Uth 1

Uth 2 =

log

Rt
2N
+ log
Rt 1
2N + I:(31)

With immigration at t; the capital stock increases in t + 1 and after. However, theinterest rate at t changes due to the immigration e¤ect. It follows that the welfare ofgeneration t 1 (who becomes old at t) changes via the interest rate and scale e¤ects.
Substituting the interest rate using (8) into (31) gives
Uth 1
We …nd that the welfare of t

Uth 2 =

log

2N
Rt
Rt 1 2N + I:

1 generation increases if the following condition is met,
Rt
Rt 1

1 2N > I;

which requires the scale of immigration to be small. This arises because the increasein the interest rate exceeds the decrease in public goods per capita.

5

Policy Discussion and Concluding Remarks

We now use the above results to derive some policy implications. In Case I, since
immigration increases the wages of both high and low skilled workers and the natives’
welfare, skill con‡icts are less likely to arise and immigration can be more actively
adopted. But distribution problems occur since the skill premium is widened. The
government must take into account intergenerational con‡icts between the young andold, and the skill premium between the high and low skilled workers. In contrast, inCase II, both native wages and welfare fall, and the welfare of those born at t is thelowest among the three generations, leading to con‡icts between generations t and t 1
or t + 1. It is no wonder that natives at t may oppose immigration in this case. Finallyin Case III, high skilled natives will bene…t from immigration, but the low skilled will
be hurt, leading to a higher possibility of con‡ict. In fact, such skill con‡icts are notdi¢ cult to …nd in developed countries. In this case, …ne tuning, such as transfers andother redistribution measures are needed. In addition, since the root is a wage shock
stemming from immigration, allowing free trade might mitigate these con‡icts.
To smoothe out the impacts of these immigration con‡icts, a rights-based approach
proposed by Hanson (2005) seems practical. Such a policy aims to phase in slowly overtime immigrant access to government bene…ts, by creating a graduated set of rights todraw on public bene…ts to which immigrants would gain access after having worked in
the host country for a speci…c time period.
Our results have been based on non-immediate assimilation. If immigrants can
assimilate soon after coming into the host country, then b = 1 in our model. The
detailed discussion of this case is contained in the Appendix.

19We have also abstracted from population growth in the formal model. In the
Appendix, we discuss this issue in more detail. Since high skilled immigrants are more
complimentary than low skilled ones, the host country may accept more immigrants
with higher skills by Lemma 1, which will a¤ect the accumulation of capital goods.

The Appendix
Figures
If immigration did not a¤ect both high and low skilled wages, (17) would be equal to
1. Then we obtain the relationship between E h and E l as,
h

El =

+

l

Eh

1

h

h

l

l

E h:

(32)

In Figures 1–3 to follow, we name the curve from (32) Eih (i = I; II; III) for case i.
Similarly we can also obtain
h

Eh =

+

l

El
h

1

l

l
h

El;

(33)

which is represented by the curve Eil (i = I; II; III) for case i:The slopes of these curves
on E h E l space can be calculated as in general,
h
h
h
1
@E l
+ l1
h
=
E
l
@E h
h
l
@E
=
l
1
1
@E h
l
( h + l) 1
(E l )

h
l

(34)

;
:

(35)

l

To determine their signs for each case, we make the following assumption:hh
ASSUMPTION 1: h = l > (1)=(1) and h = l > (1
l
l
)=(1
):
Assumption 1 simply implies that the slopes are positive at the origin E h = E l = 1.
hNext, setting (34) and (35) equal to 0, we …nd the slopes change at E x = ( h + l 1 x ) 1
where x = l; h: When x < 1, E x does not lie in the region 1 < E l < E h ; because
the values in brackets in E x are less than 1; when x > 1; E x lies in the region
E l < E h < 1: Thus, we obtain
LEMMA 2: The slopes of (34) and (35) do not change signs in the region 1 < E l <E h , and their values at E h = E l = 1 are positive when h < 1
and l < 1
;
h
l
h
When
> 1
and
> 1
, the corresponding slopes change signs at E =
h
l
l
h
1
( h+ l 1 h )and E = ( h + l 1 l ) 1 l :

20

x

,

Note that the threshold of the slope of (35) is not based on the horizontal axishbut the vertical axis. When E l = ( h + l 1 l ) 1 l ; the expression for E h beh

l

l

1

h

l

h

1

h

+
) 1 l ; at which point the slope ofcomes[( h + l 1 l ) 1 l ]hh ( h+ l
1 l(35) changes.
Finally, the second derivatives from (34) and (35) can be found as

2

@ El
=
@2E h
2
@ El
=
@2E h

h

+

l

h

1

l

El

1

l

[E h 1l[(E h + E l
l

+ El

h
l
h

h

1

Eh

1
l

) 1
1

l

(36)

;
+

l
h

]3

Eh

El ]

(37)

but negative otherwise, and similarly, (37) is
Thus, (36) is positive when h < 1lnegative when < 1
but positive otherwise. The denominator in (37) is always
positive because the …rst term always dominates the second term. Since l < 1 andE h > E l , the sign of (37) is determined by the sign of the …rst term in the numerator.
Summing up the above derivations, we can determine the shapes of (32) and (33).
If h < 1; the locus of (32) is depicted by the convex shaped curve named EIh on
E h ~E l space in Figure 1. Analogously, if l < 1; the locus of (33) is depicted bylthe concave shaped curve named EII .
From Figure 1, (17)< 1 on the left hand side (the curve shown by (32)) and (17)> 1
on the right hand side. It means that when E l < E h ; the high skilled wage increases
after immigration. Similarly we can do the analysis for the low skilled wage. The
shaded area in Figure 1 represents the region where both wages increase. In otherwords, this region indicates the combination of h and l that gives rise to higher
wages with immigration for both types of workers. It occurs when both immigratione¤ects are big under small h and l ; which leads to high productivity e¤ects that
dominate the capital allocation e¤ects. Under EIl ; Case III arises, where E l is too
small compared with E h , and the capital allocation e¤ect dominates the productivitye¤ect, resulting in a fall in the low skilled wage.
On the other hand, if h > 1; the locus of (32) is depicted by the inverseh
U shaped curve named EII in Figure 2. Analogously, if l > 1; the locus ofl(33) is depicted by the inverse C-shaped concave named EII
on E l ~E h space. The
shaded region indicates the combination of h and l that leads to lower wages withimmigration for both types of workers; That is, Case II arises in this region, becausehlthe high (low) skilled wage decreases above the curve EII(EII):Similar to the above,hunder EII ; Case III arises again.
hlFinally to cases h < 1
and l > 1
, where we obtain EIII
and EIII. The
former is convex and the latter is inverse C-shaped in Figure 3. Then, under the curve
E h = E l ; the high skilled wage increases but the low skilled wage decreases.
Summing up, in Case I under the conditions h < 1
and l < 1
, Case II underhlh>1
and > 1; and Case III when E is su¢ ciently small, if immigrants
bring skills that are complementary with natives, both the high- and low-skilled wages
increase. But if immigrant skills are substitutable with natives, both wages decrease.

21

The Dynamic System
This subsection considers the dynamic impacts of immigration on the host country.
After entering in period t, immigrants start to contribute to capital accumulationand government tax. Rewriting (13) using the budget constraints gives the capital
movement equation,
Kt+1 = St :(38)
where St is the aggregate savings. Substituting (6) into (38) gives,
Kt+1 =

1+)(wtn;h Nth + wtn;l Ntl + wtI;h Ith + wtI;l Itl ):

(1

(39)

From (8) we have the capital allocation equation,
l

Ktl =
where

x

El h
K ;
hE h t(40)1( x ) x (Ntx ) : Substituting (40) into (14) further gives,
x

Ex
K:
(41)
hE h +lEl t
That is, capital allocation between high-skilled and low-skilled workers depends onlabor productivity. An increase in labor productivity (of either type of workers) attracts
more capital to work with it, leaving less capital for the other type of workers. Thismechanism works through demand linkages in both the capital and labor markets,
causing reallocation of capital between workers of di¤erent skills.
And substituting (9) and (41) into (39) gives the equation of motion for capital asKtx =Kt+1 =

1+

(1

) (1

)

h

Eh +

l

El

1

Kt

(42)

Similarly, we can derive the equation of motion for public goods as
Gt+1 = (1

)

h

Eh +

l

El

1

Kt

(43)

Equations (42) and (43) together describe the dynamic system of this economy.
Steady state
In order to isolate the impacts of immigration, we assume that the economy is initiallyin a steady state.
De…nition 6 The original steady state is de…ned as:
The amount of aggregate capital in period t is equal to that in period t
Kt = K t 1 = K .

22

1, i.e.,

The production function in t

1 period can be rewritten ashf h Nth 1

Yt 1 = Kth 1

1l

f l Ntl 1

g h + Ktl 1

g

1
l

:

Equalization of capital earnings giveslKtl 1 =

hKth 1 :

Capital in period t is determined by savings in period t

1,

Kt = S t 1 N t 1
=

1+

(1

) (1

)

h

+

h

+

l 1

Kt 1 :The value of capital at the steady state is then
K =[

1+

(1

) (1

)

l 1

]1

1

(44)

:

Since the capital stock is determined by the disposable income, it becomes higher
under a lower tax rate and with a higher ratio of skilled workers, which in turn leads
to higher output. Substituting (44) into (42), we can further obtain Kt+1 ;Kt+1 =

(1

1+

) (1

h

)

Eh +

l

1

El

(45)

Kt

Note that (44) and (45) may be di¤erent from each other, since population grows
with immigration, increasing labor input. Population grows at
T Pt+n+1
T Pt+n

t+n
h

(1

b1

(1
= ~b

b1

1 =

t+n
h

h

)(E h ) [(E h )

+1

t+n 1
h

b1

h

t+n
h

+ 1] + (1)(E h )

h

b1

l

t+n 1
l

+ (1

b1

+1

See the subsection "Population structure" below and (51).
If the economy moves to a new steady state where ( KKt+n+1
t+ni.e., Kt+n+1 =Kt+n = 1 + ~b; after some periods we must have,
Kt+n+1 =
~ = f
K

1+

(1

1+
) (1

(1

) (1
h

)

Eh +

1 + ~b

h

)
l

El

Eh +

l

)(E l ) [(E l ) + 1]
t+n
h

b1

)(E l )

1)=( TTPPt+n+1
t+n

l

El

1

l

1

1) = 1;

Kt+n

1

g1

1

(46)

~ stands for the capital labor ratio at the new steady state.
where KThe di¤erence between (44) and (45) (or (46) ) is one of the sources that cause
various con‡icts between natives and immigrants in the present model.

23

t+2
Capital Stock Ratio K
Kt+1

In calculating the welfare di¤erence between t and t + 1; we need to know Kt+2 =Kt+1 :Kt+2 is from the aggregate capital accumulation at t + 1:
Kt+2 =
=

1+
1+

(1

n;h
n;l
I;h h
I;l l
h
l
)(wt+1
Nt+1+ wt+1
Nt+1+ wt+1
It+1 + wt+1
It+1 )

(1

) (1

h
h
t+1 E +

)

l
l 1
t+1 E

Kt+1 ;

1x: The population structure is di¤erent at t and t + 1;because
where xt+1 = ( x ) x Nt+1
the numbers of high and low skilled immigrants di¤er. Using Kt+1 and Kt ; and rearranging giveKt+1 =
Kt+2
=f
Kt+1

(1

1+
h

1
h

) (1
(E h )1+

1
h+

h

)
h

l

+

Eh +

1
l

l

El

1

Kt

1+ l

El

1
l

1
1
( )
( )
] [( h ) h E h + ( l ) l E l ][11( h ) h E h +( l ) l E l
h

l

g1

> 1:

WelfareUsing (1), (2), (3), (4) and (5) , we obtain the welfare di¤erence with and without
immigration as;
Utwith;nx

Utwithout;nx = (1 + )[log(wtwith;nx ) log(wtwithout;nx )]withwithout+ log Rt+1
log Rt+1+ log (wtwith;h N h + wtwith;l N l + wtwith;h Ith + wtwith;l Itl )log (wtwithout;h N h + wtwithout;l N l )+(1 + ) (log 2N log(2N + I)) :Rearranging this leads to (22).

Immediate Assimilation
Under immediate assimilation of immigrants, the production function is changed to
Yt = Kthf h Nth

h

+

h

Ith

h

1

g h + Ktl

f l Ntl

l

l

Itl

1 x

(Ntx )

x

1 x

x

+

l

g

1
l

:

In this case, wage rates are
wtx = (Ktx ) (1wtI;x = (Ktx ) (1) f x (Ntx )) f x (Ntx )

x
x

24

+
+

x
x

x

1

x

1

(Itx ) g x(Itx ) g x(Itx )

1
1

;

;

(47)
(48)

Immigrants and natives receive the same wage for their type. Setting (47) equal to(48) we obtain the number of immigrants the host country accepts,
Ntx = Itx :Moreover, the interest rate is given by
Rt+1 =

1Ktl

l

f l Ntl

Itl

l

+

l

g

1
l

:

And then the capital allocation between skilled and low skilled is determined by
Ktl
Kth
Given

h

<

l

=

h

f2 h Nth

1

0 =

1

g l

:

1

g h

;we have 1= h > 1= l : Combining the above and (14) yields,
Ktx =

where

l

f2 l Ntl

l

1

h

1
g h

f l (Ntl 1 ) g l
f h (Nth 1 )

=

l
h

2 x
2

1
h

x
1

h+2 l

l

K ;

: Comparing the wages of the assimilation case withthat of no immigration , we have
x
1
wtx
x
=
2
[ 1
wtx 1
2 h

h

+

h+2

l
1
l

l

] ;

The immigration e¤ect depends on ; h and l : While qualitatively it is the sameas when immigrants do not immediately assimilate, the quantitative level is di¤erent.
Moreover, under immediate assimilation, the host country accepts the same number of
immigrants as natives, since their productivity is the same. This means the population
structure will not change. In contrast, in the non-assimilation case, the population
structure gradually changes because the numbers of high-skilled and low-skilled immigrants the economy accepts are di¤erent.

Population Structure
Total population
Only natives live in the host country before time t: From time t; the host country
receives It number of immigrants every period. Then there are Nty 1 workers andNto 1 retirees at t 1; since the number of original natives does not grow. The total
population at t 1 is
T Pt 1 = Nth;y1 + Ntl;y1 + Nth;o1 + Ntl;o1 = 2Nt 1
At t; with immigrants It = Ith;y + Itl;y ; the total population becomes25

T Pt = 2Nt + It = Nth;y + Ntl;y + Nth;o + Ntl;o + Ith;y + Itl;yThe representative …rm’s optimization determines the number of immigrants accepted,
1x
so that we have (12). Then, Ntx;y + Itx;y = Ntx;y [1 + (b) 1 x ] = Ntx;y (E x ) : Using theseand Nth;o + Ntl;o = Nth;y1 + Ntl;y1 = Nth;y + Ntl;y ; we obtain another expression for total
populationhl

T Pt = Nth;y [1 + (E h ) ] + Ntl;y [1 + (E l ) ] + Nto(49)

Next, the total population at t + 1 is
h;y
l;y
h;o
l;o
h;y
l;y
h;o
l;o
T Pt+1 = Nt+1
+ Nt+1+ Nt+1+ Nt+1+ It+1+ It+1+ It+1+ It+1

Here, the number of original native olds at t+1 is equal to that of original native youngsh;ol;oat t; i.e., Nt+1
+ Nt+1= Nt : Similarly, since young immigrants coming at t become oldh;ol;oat t + 1; we have It+1 + It+1
= It :Optimization determines the number of immigrants,
x;y
x;y
x;y
so that we have (12) in each period. Then, since Nt+1
+ It+1= Nt+1
1 + (b) 1

1

x

=

x
x;y
Nt+1
(E x ) ; we obtain
h

l

h;y
l;y
h;y
l;y
Nt+1
+ Nt+1+ It+1+ It+1= Nth;y [(E h ) ]2 + Ntl;y [(E l ) ]2 :Moreover, since immigrants coming in at t become old at t + 1;
h

h;o
l;o
h;o
l;o
Nt+1
+ Nt+1+ It+1+ It+1= Nth;y (E h )l+ Ntl;y (E l ) :In addition, we have another expression from (49),
l

l

h

h

T Pt+1 = Nth;y (E h ) [1 + (E h ) ] + Ntl;y (E l ) [1 + (E l ) ]:(50)

From (49) and (50), the ratio of total population at t and t + 1 is
h

h

l

l

T Pt+1(E h ) [1 + (E h ) ] + (E l ) [1 + (E l ) ]
=
h
l
T Pt
[1 + (E h ) ] + [1 + (E l ) ](51)

Subtracting the numerator from the denominator gives
h

h

l

l

(E h ) [1 + (E h ) ] + (E l ) [1 + (E l ) ]h= [1 + (E h ) ][(E h )hhl1] + [1 + (E l ) ][(E l )

l

h

l

f[1 + (E h ) ] + [1 + (E l ) ]gl1] > 0;since (E h ) and (E l ) are larger than 1. Therefore, the host-country population growsby the number of immigrants. However, since we only focus on periods t and t + 1, thenumber of immigrants are too small compared to native population and we can thus
ignore the scale e¤ect in population when we consider welfare.

26

Population growth at the new steady state
After period t, the host country receives both high and low skilled immigration atconstant but di¤erent rates, and thus the population grows at di¤erent rates. Let uscompare between periods t + n and t + n + 1. The rate is expressed by=

T Pt+n+1
T Pt+n
h;y
l;y
h;o
l;o
h;y
l;y
h;o
l;o
Nt+n+1+ Nt+n+1+ Nt+n+1+ Nt+n+1+ It+n+1+ It+n+1+ It+n+1+ It+n+1
h;y
h;y
l;y
l;y
h;o
l;o
Nt+n
+ It+n+ Nt+n+ It+n+ Nt+n+ Nt+n

(1

=
(1

t+n
h

b1

t+n
h

b1
h

h

)fA(E h ) [(E h )l)f(E h ) A + [(E l ) ]Bg + (1
1

h

l

l

+ 1] + B(E l ) [(E l ) + 1]g
t+n 1
h

b1

h

l

;

)f[1 + (E h ) ]A + [1 + (E l ) ]Bg

1

where A 1 b 1 l and B 1 b 1 h : Generally the growth rate of the total population
depends on which period we choose. However, when n is su¢ ciently large, the growtht+nrate becomes constant. In particular, when n approaches in…nity, expressions b 1 h
t+n 1
and b 1 h become 0.
Acknowledgement: We wish to thank B. Chen, K Futagami, S. Ikeda, T. Kikuch, K.
Mino, A. Momota, K. Tabata, Y. Wang, Q. Xia, J. Yoshida, Y. Yoshida, A. Yanase andother participants of seminars at Kobe, Kyoto and Osaka Universities and at various
academic meetings for their helpful comments and suggestions on previous versions.
The usual disclaimer applies.

27

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