General equilibrium analysis of immigration
with cultural good and asset effects
Toshihiro Atsumi
Faculty of Economics, Meiji Gakuin University
Discussion Paper No.20-02
2020 年 12 月

General equilibrium analysis of immigration with cultural good and asset effects

Abstract
This paper examines the general equilibrium economic impact on a host. country when
immigrants differ from natives in terms of consumption behavior and property ownership. The
immigrants arrive with a culture different from that of the natives and consume their own cultural
goods. The natives own property such as land but the immigrants do not. How immigration
affects the natives under such circumstances is analyzed using the product variety approach. If
the cultural difference is significant, immigration harms the natives’ economic welfare by
reducing the range of goods and services offered for the natives. On the other hand, asset effects
due to increased demand for property raises the earnings of the natives and works as a pull factor
of immigration. In addition, among the various other impacts, immigration can affect the natives
in a non-monotonic way. That is, once the immigrants’ share within the population exceeds athreshold, immigration impact on the natives switches from negative to positive. In contrast, theimmigrants always gain from arrival of new immigrants with their culture. Therefore,
immigration can enlarge or reduce inequality between the natives and the immigrants.

Keywords: immigration; variety; monopolistic competition; cultural goods; property; land
JEL classification: F22, J61, Z19

1. Introduction
The home market effect explained by Krugman (1980) suggests that living in a larger economy isusually more beneficial in terms of the greater varieties of goods and services available. In thiscontext there is no economic reason for opposing immigration, if it is considered simply as apopulation increase that makes the country larger. However, immigration has always been under
debate in various parts of the world. This suggests that there are significant differences between a
population increase and immigration that need to be taken into account in order to study
immigration impacts.
As is well documented by Borjas (2014), the large body of research on immigration focuseson the labor market or wage impacts of immigration. In contrast, this study focuses on generalequilibrium, economy-wide impacts of immigration. In this regard, this paper is related to studies
by Epstein (1974), McCulloch and Yellen (1977), and Ottaviano and Peri (2006). However,while the pioneering studies by Epstein (1974) and McCulloch and Yellen (1977) are based oncompetitive models, I use the product variety approach under monopolistic competition in order
to incorporate cultural goods. Also, while Ottaviano and Peri (2006) shed light on the diversitythat the immigrants bring, which is found to raise wages and rents in U.S. cities, I consider the
possibility of immigrants’ consumption of cultural goods eroding the product variety for thenatives to consume. Cultural goods have been studied, for example, in the context of international
trade by Francois and van Ypersele (2002). This paper is concerned with the differences in thepreferences for cultural goods that the natives and the immigrants may well have. Another
difference I incorporate in the analysis is property. It is assumed that host country property isowned by the natives and, therefore, limited resources such as land need to be shared among thenatives and the immigrants.
One might expect an inverted U-shaped relation regarding immigration and its impact on thenatives. That is, a small level of immigration may help (or does no harm to) the natives, but they
may object when immigration becomes too large relative to the native population. However, if
this were the case, then we would observe a convergence in the levels of immigration acrosscountries. In reality, while levels of immigration remain low in some countries, there arecountries that keep attracting new immigrants. This suggests that there is a strong pull factor of
immigrants at work in some countries, among various factors that influence immigration.
Previewing the results, there are two possible economic outcomes of immigration on the1natives. One is that immigration is always welfare improving, and the other is a U-shaped
relation between the immigrants’ share within the population and the natives’ economic welfare.
That is, initially immigration reduces the natives’ economic well-being but once immigration
exceeds a threshold, the impact turns positive. These are results of two opposing forces that
immigration brings about under the present setting. One is the crowding out of the variety ofcultural goods for the natives due to immigrants demanding and consuming different varieties of
their cultural goods. The other force is an increase in land rent (relative to wages), which only thenatives receive, due to increased demand for land to satisfy the immigrants’ demands. Whether
the latter outweighs the former determines the economic impacts of immigration on the native
population. It is also shown that immigrants, in contrast, always gain from new arrival of
immigrants with their culture, due to the increase in the variety of their cultural goods. By
comparing the situations of the natives and the immigrants, inequality between them can also be
studied. It is found that immigration can work in either direction; that is, immigration can
increase or reduce inequality, but in a monotonic way. Therefore, in total, four types of outcomes
will be shown, when inequality is considered in addition to impact on the natives.
The rest of the paper is organized as follows. After setting the assumptions in Section 2,
behaviors of consumers and firms are derived in Section 3. Section 4 solves and explains theequilibrium of the model .The equilibrium is analyzed and welfare implications of immigration is
studied and summarized in Section 5, followed by a discussion of the results in Section 6.

2. Assumptions
The population of the economy consists of natives and immigrants. The native population isdenoted as L N and the immigrant population is denoted as LI . The share of immigrants istherefore LI (L N + LI ) ≡ λ . Both the natives and the immigrants work and consume. The twogroups are homogeneous except for a difference in their preferences. Specifically, they consume
different cultural goods in addition to the goods that the natives and the immigrants consume incommon. 1 Francois and van Ypersele (2002, p.359) define cultural goods as “goods which arevalued differently by consumers at home than by individuals abroad, and which are produced2

under scale economies.” This paper follows their definition of cultural goods and defines them as
goods which are valued differently by the natives than by the immigrants, and which areproduced under scale economies. The consumers’ preferences are described by the two-tierstructure detailed below, which is based on the consumers’ love of variety originally developed
by Dixit and Stiglitz (1977) with constant elasticity of substitution (CES) preferences.
For the natives, the utility function isU N ≡ X µCN
nXρwhere X =  ∫ x(i ) di 
 0


1ρ

1− µ

,

(1)

1γ

nCN
γ
and C N =  ∫ c N (i ) di  . 2 U N is a Cobb-Douglas function of the 0

consumption of an aggregate of the common goods ( X ) and the cultural goods for the natives( C N ). The second tier defines X to be a CES function such that x(i ) is the consumption ofeach variety of the common goods. It is assumed that 0 < ρ < 1 to ensure that the varieties areimperfect substitutes. X is therefore a CES composite of the total mass of the varieties of thecommon goods, n X . The elasticity of substitution between any two varieties of the common
goods is 1 (1 − ρ ) ≡ σ ( σ > 1 ). Similarly, C N is a CES composite of the total mass of thevarieties of the cultural goods for the natives ( nCN ), and the elasticity of substitution between any
two varieties of the cultural goods is 1 (1 − γ ) ≡ δ ( δ > 1 ).
Turning to the immigrants, their utility function isU I ≡ X µ CInXρwhere X =  ∫ x(i ) di 
 0


1ρ

nCI
γ
and C I =  ∫ c I (i ) di 
 0


1− µ

,

(2)

1γ

. The only difference between the natives

and the immigrants is in the consumption of the cultural goods. C I is the immigrants’ CES
composite of the total mass of the varieties of the cultural goods for the immigrants ( nCI ). The
elasticity parameters are assumed to be the same between the natives and the immigrants.

1

The common goods can include services such as housing, but these will be labelled ‘goods’ throughout. And, ofcourse, the cultural goods can include cultural services.
2
andare two extreme cases.
implies there are no common goods and all goods consumed are
implies there are no cultural goods and the natives and the immigrants consume only thecultural goods.
common goods. This can be interpreted as a case of complete cultural assimilation.

3

On the production side, both the common goods and the cultural goods are produced bymonopolistically competitive firms using land and labor as inputs. The economy is endowed witha fixed area of land ( T ) and, importantly, land is entirely owned by the natives. A firm producing
a variety of the common goods needs a unit of land and m X units of labor per unit output. The
firm thus faces increasing returns to scale. Its total cost for producing a given amount q X is thenc(q X ) = t + wm X q X ,(3)

where t is land rent and w is wage. A firm producing a variety of the cultural goods for thenatives needs a unit of land and mCN units of labor per unit output. Hence its total cost for
producing a given amount qCN isc(qCN ) = t + wmCN qCN .

(4)

Similarly, a firm producing a variety of the cultural goods for the immigrants needs a unit of land
and mCI units of labor per unit output. Its total cost for producing a given amount qCI is thenc(qCI ) = t + wmCI qCI .

(5)

3. Consumer and firm behavior
3.1 Consumers
For a given income ( y ) and the prices of the common and the cultural goods, which are p X andpCN , respectively, a native consumer’s problem is to maximize her utility subject to the budget
constraint∫ p (i )x(i )di + ∫
nX

0

X

nCN

0

pCN (i )c(i )di = y .

(6)

Since the preferences of the common and the cultural goods are separable and their second tiers
are homothetic in x(i ) and c(i ) , the problem can be solved in two steps. The first step
comprises the decisions to choose x(i ) and c(i ) . The second step is to allocate expenditure
between X and C N . First, the consumer should choose x(i ) to minimize the cost of
consuming X . This implies minimizing expenditure ∫nX0p X (i )x(i )di subject to1

 n X x(i )ρ di  ρ = X .
 ∫0


4

(7)

The first-order condition of the problem is that the marginal rate of substitution between any twovarieties i and j is equal to its price ratio, that is,

x(i )

ρ −1

x( j )

ρ −1

=

p(i )
.
p( j )(8)

Substituting (8) into (7), the following compensated demand function for variety j of X is
obtained:p( j ) ρ −1
1x( j ) = ∫0 p X (i )
nX

ρ
ρ −1

1

X.

(9)

ρ
di 


Using (9), the minimum expenditure to consume X is
ρ −1

∫

nX

0

ρ
 nX
 ρ
p X ( j )x( j )dj =  ∫ p X (i ) ρ −1 di  X .
0


(10)

ρ −1

ρ
 nX
 ρ
The term  ∫ p X (i ) ρ −1 di 0in (10) is called the price index ( G X ), and
ρ −1

1

ρ
nX
1−σ
1−σ
 nX
 ρ
G X ≡  ∫ p X (i ) ρ −1 di =  ∫ p X (i ) di  .

 0
0


(11)

Using G X , (9) can be simplified as p ( j )x( j ) =  X 
 GX 

−σ

X.

(12)

Similarly for the cultural goods, we obtain p ( j )cCN ( j ) =  CN  C N , GCN −δ

(13)

where GCN is the price index of the natives’ cultural goods, and
γ −1

1

γ
nCN
1−γ
1−γ
 nCN
 γ
GCN ≡  ∫ pCN (i )γ −1 di  =  ∫ pCN (i ) di  .


0
0





(14)

The second step of the solution for the consumer’s problem is to allocate income between X
and C N , that is, to maximize the upper-tier utility function (1) subject to the budget constraint

5

G X X + GCN C N = y . The first-order conditions give us that µy should be spent on thecomposite of the common goods, and (1 − µ ) y on the composite of the cultural goods.
Therefore, the demand functions for each variety of the common goods and the cultural goods,
respectively, arex( j ) = p X ( j ) G X
−σ

σ −1

µYN ,

(15)

(1 − µ )YN ,

(16)

and

cCN ( j ) = pCN ( j ) GCN
−δ

δ −1

where YN is total income of the natives. Also from (15), the price elasticity of demand[dx( j ) dp X ( j )] x( j ) p X ( j ) can be derived as σ + [p X ( j )1−σ (1 − σ )] ∫0 p X ( j )1−σ dj , butnX

n → ∞ with a continuum of varieties of the common goods, so the second term approaches zero.
Therefore, the price elasticity of demand is simply σ . Similarly, the price elasticity of demand
for the natives’ cultural goods is δ .
The immigrant consumers’ behavior is derived similarly, and the demand functions for eachvariety of the common goods and the cultural goods for the immigrants, respectively, arex( j ) = p X ( j ) G X
−σ

σ −1

µYI ,

(17)

(1 − µ )YI ,

(18)

and

cCI ( j ) = pCI ( j ) GCI
−δ

δ −1

where YI is total income of the immigrants and GCI is the price index of the immigrants’cultural goods, that is,
γ −1

1

γ
nCI
1−γ
1−γ
 nCI
 γ
GCI ≡  ∫ pCI (i )γ −1 di  =  ∫ pCI (i ) di  .
 0

0


(19)

Summarizing the consumption behaviors of the natives and the immigrants, the demand forvariety j of the common goods ( X ) is µp X ( j ) G X
−σ

σ −1

(YN + YI ) , the demand for variety ( j )of the cultural goods for the natives ( C N ) is (1 − µ ) pCN ( j ) GCN
−σ

σ −1

YN , and the demand forvariety j of the cultural goods for the immigrants ( C I ) is (1 − µ ) pCI ( j ) GCI
−σ

6

σ −1

YI .

3.2 Firms
The firms engaging in the Dixit-Stiglitz monopolistic competition are identical, and take no
strategic actions against other firms because the mass of firms is large. This simplifies the supply
side, and it is sufficient to consider a typical firm’s behavior in the three sectors. Hereafter,
therefore, subscripts i and j will be omitted. A typical profit-maximizing monopolisticallycompetitive firm producing a variety of the common goods will set its price so that marginal
revenue equals marginal cost, that is,p X (1 − 1 σ ) = m X .

(20)

Similarly, a variety of the cultural goods for the natives and the immigrants will be priced as
pCN (1 − 1 δ ) = mCN ,(21)

pCI (1 − 1 δ ) = mCI ,

(22)

and

respectively. These pricing behaviors are known as mark-up pricing, where firms always set theirprices above their marginal costs. Since rival firms are producing more or less substitutablevarieties, the mark-up depends on σ or δ : if the varieties are close substitutes (or the
consumers’ love of variety is weak), i.e., if σ or δ is high, then the consumers are sensitive to
price and prices are closer to the marginal costs. By substituting (20) into (11), the mark-up
pricing by each firm leads to the price index of the common goods being
G X = (n X )1−σ
1

σm X
,
σ −1

(23)

and those for the natives’ and the immigrants’ cultural goods being
GCN = (nCN )1−δ

δmCN
,
δ −1

(24)

GCI = (nCI )1−δ

σmCI
,
δ −1

(25)

1

and
1

respectively.

7

4. Equilibrium
We can now consider the equilibrium. An equilibrium is defined as a situation in which, allowing
free entry, the goods and factor markets clear. The profit of a typical firm producing a variety of
the common goods ( π X ) isπ X = p X q X − t − wm X q X .

(26)Under free entry, the firms bid up the land rent ( t ) until π X is zero. Then in equilibrium,
operating profit p X q X − wm X q X equals t . Substituting (20) into (26) and setting it equal tozero, we haveqX =t (σ − 1)
.
mX

(27)

This indicates a positive linear relation between the land rent ( t ) and q X , which is per firm
output (or size). It also indicates that a higher σ , that is, weaker love of variety, gives a larger

q X . Similarly, free entry in the cultural goods leads toqCN =t (δ − 1)
,
mCN

(28)

qCI =

t (δ − 1)
.
mCI

(29)

and

Market clearing in the common goods market requires−σq X = µp X G X

σ −1(YN + YI ) ,(30)

where YN = wL N + tT and YI = wLI . Hereafter, the wage ( w ) will be set equal to 1 so that
YN = L N + tT and YI = LI . Market clearing in the natives’ and the immigrants’ cultural goods

markets requires

qCN = (1 − µ ) pCN GCN
−δ

δ −1

YN ,

(31)

YI ,

(32)

and

qCI = (1 − µ ) pCI GCI
−δ

δ −1

respectively. The factor market clearing conditions aren X m X q X + nCN cCN qCN + nCI mCI qCI = L N + LI

8

(33)

for the labor market, andn X + nCN + nCI = T(34)

for the land market. (33) and (34) imply full employment in the labor market and the land market,
respectively.
Using (27) and (30), we obtain an expression for the mass of the firms producing thecommon goods asnX =µ (YN + YI )
,
σt

(35)

which is equivalent to the mass of varieties of the common goods available for the natives andthe immigrants. Similarly, using (28) and (31), the mass of the firms producing the natives’common goods is

nCN =

µYN
,
δt

(36)

and, using (29) and (32), the mass of the firms producing the immigrants’ common goods is

nCI =

µYI
.
δt

(37)

The expressions (35), (36), and (37) state that rises in incomes support new firm (and variety)creation, but rises in the land rent hinder it. The former is the income effect and the latter is thecost effect that determines the masses of firms.
Substituting (30), (31), and (32) into (33), we obtain the equilibrium land rent (relative towage) as
 L + LI 
t = A N
,
 T


(38)

where A is a composite of the parameters defined asA≡µ (δ − σ ) + σ,µ (σ − δ ) + σ (δ − 1)

(39)

and A > 0 . 3 Importantly, (38) indicates that the land rent ( t ) linearly increases with population
including immigrants ( L N + LI ). This implies that immigration always increases the natives’nominalincomethrough

a

rise

in

the

9

land

rent.

Further,

it

is

found

that

∂A ∂σ = − δ 2 µ [δ (µ − σ ) + σ − µσ ] < 0 and ∂A ∂δ = − (1 − µ )σ 2 [δ (µ − σ ) + σ − µσ ] < 0 .
2

2These imply that if σ and/or δ increases then A decreases, so the effect of immigration on
driving up the land rent is weakened. This is because larger σ and/or δ implies weaker
demand for variety, meaning that the mass of the firms is less important, and that the prices are
more important than variety. Lower demand for firms then leads to lower demand for land. It isalso found that ∂A ∂µ = δ (δ − σ )σ [δ (µ − σ ) + σ − µσ ]2 . Therefore, if δ > σ ( δ < σ ) then∂A ∂µ > 0 ( ∂A ∂µ < 0 ), so a higher µ implies a stronger (weaker) impact of immigration on
driving up the land rent. The intuition is as follows. If δ > σ ( δ < σ ) then the consumers’ loveof variety is stronger in the common goods (cultural goods). Recalling that µ is the expenditure
share on the common goods, an increase in µ will shift demand towards the common goods(cultural goods), which leads to increased (decreased) demand for the firms and, correspondingly,
for land.
Substituting (38) into (35), (36), and (37) gives the equilibrium masses of the firms in thethree sectors,
nX =

nCN =

µδT
,
µδ + σ (1 − µ )(40)(1 − µ )T (1 − λ ) + (1 − µ )T ,(41)

(1 − µ )T λ ,

(42)

δA

δ

and

nCI =

δA

which can be confirmed to satisfy (34). (40) indicates that the equilibrium mass of the firms
producing the common goods ( n X ) is neutral to population. The reason for this can beunderstood by inspecting (35), recalling the equilibrium land rent in (38). Population increase(including immigrants) drives up the land rent, which raises the natives’ income ( YN ). This itself
supports creation of new firms, but at the same time the increase in the land rent raises the cost ofsetting up new firms. These two effects cancel out, and any change in population does not affectthe equilibrium mass of the firms in the common goods sector. In other words, irrespective of
3

See appendix for proof.

10population, equilibrium n X is constant. The equilibrium mass of the firms producing thecultural goods for the natives ( nCN ) in (41) has two terms. The first term indicates that nCN
linearly decreases with the immigrants’ share ( λ ) within the population, meaning that
immigration crowds out the production of the cultural goods for the natives. However, there is asecond term which is constant. These facts imply that immigration lowers nCN towards its
minimum, (1 − µ )T δ . On the other hand, as (42) indicates, the equilibrium mass of the firms
producing the cultural goods for the immigrants ( nCI ) linearly increases with λ .
Particularly important in evaluating the economic impact of immigration on the natives are
the land rent ( t ) and the mass of firms producing the cultural goods for the natives ( nCN ). The
natives’ economic welfare depends on whether the increase in t is large enough to compensate
for the decrease in nCN .

5. Patterns of immigration impacts
5.1 Possibility of U-shaped non-monotonicity
In this section, using the equilibrium results obtained in Section 4, we analyze the natives’economic welfare based on indirect utility which is their real income. The natives’ per capita real
income ( ω N ) isωN =YN L NµG X GCN

1− µ

.

(43)

Substituting the results obtained in equilibrium into (43), we obtainωN =(1 − λ + A)µ

(1 − λ ) σc X  


µδT

σ − 1  µδ + σ (1 − µ ) µ
1−σ

σcCN σ − 11− µ (1 − µ )T  1 − λ + A 


 δ
A




1− µ
1−δ

(44)

and
1− µ

 (1 − µ )(1 − λ + A)T  1−δ
 {(δ − 1)A + (λ − 1)(1 − µ )}
δA
∂ω N


=
.
µ
∂λ
1− µ
µ
σ
−
1
 σcCN µδT σc X  2σ − 1  µδ + σ (1 − µ )   σ − 1  (δ − 1)(λ − 1)

 

 

11

(45)

Inspecting (45), the sign of ∂ω N ∂λ is ambiguous, because {(δ − 1)A + (λ − 1)(1 − µ )} can takeeither sign. Importantly, therefore, immigration can affect ω N

in both directions. If{(δ − 1)A + (λ − 1)(1 − µ )} is positive and correspondingly ∂ω N ∂λ > 0 , then an increase in theshare of the immigrants within the population ( λ ) will always increase the natives’ per capita
real income ( ω N ). That is, ω N monotonically increases with immigration. However, if{(δ − 1)A + (λ − 1)(1 − µ )} is negative, then ∂ω N ∂λ < 0 .4 Specifically, if condition
λ = 1−(δ − 1)A > 0
1− µ

(46)

holds, then a U-shaped non-monotonic relation between λ and ω N exists as illustrated infigure 1. There is a turning point at λTP = 1 − (δ − 1)A (1 − µ ) . If λ < λTP ( λ > λTP ), then∂ω N ∂λ > 0 ( ∂ω N ∂λ > 0 ), meaning that an increase (decrease) in the immigrants share
reduces (raises) the natives’ welfare. Hence, if λ < λTP ( λ > λTP ) then the natives are likely totake anti- (pro-) immigration positions. The result suggests that once a country’s share of λexceeds a certain level ( λTP ), immigration impact on the natives turns positive and the natives’
attitudes may well change from anti- to pro-immigration. Otherwise, the country stays
anti-immigration and low immigration level persists.
What is the economic meaning of the non-monotonicity condition in (46)? Satisfaction ofthis condition requires (δ − 1)A (1 − µ ) < 1 . First, therefore, A must be small. Small A means,as presented in (38), the positive impact of immigration on the land rent is weak. This implies
that satisfaction of condition (46) is likely to reduce the natives’ income gain from immigration.
Second, (1 − µ ) (δ − 1) must be large, which requires low µ and/or low δ . Low µ means
higher expenditure share on the cultural goods, whereas low δ means that the love of variety of
the cultural goods is strong. Therefore, the inevitable loss of variety of the cultural goods for thenatives caused by immigration, as presented in (41), is stronger.
In summary, condition (46) is likely to be satisfied when 1) the two cultures areheterogeneous in consumption and/or 2) the variety of cultural goods is important.
4∂ω N ∂L N > 0 . Therefore, a native population increase always raises their per capita real income.

12

ωN

∂ω N

0

∂λ

∂ω N

<0

1−

∂λ

>0

λ

(δ − 1)A
1− µFigure 1. Case of a U-shaped impact of immigration on the natives

5.2 Inequality between the natives and the immigrants
Up to this point, the welfare of the immigrants has not been presented. The immigrants’ percapita real income ( ω I ) is

ωI =YI L IµG X GCI

1− µ

=

1
µ

G X GCI

1− µ

.

(47)

Given result (42) that an increase in the share of immigrants ( λ ) within the population always
increases the variety of the immigrants’ cultural goods ( nCI ), immigration always reduces theprice index of their cultural goods ( GCI ). Therefore, the immigrants always gain from further
inflow of new immigrants. In other words, ∂ω I ∂λ > 0 always holds, hence the existing
immigrants always gain from the arrival of new immigrants.
The present model also speaks to inequality between the natives and the immigrants. It canbe measured by the relative real income, ω N ω I . Using (43) and (47), it is13σ + µ −2ω N  1 − λ + A  σ −1,=ωI  1 − λ (48)and ω N ω I > 1 . Therefore, the natives, on average, always enjoy higher real incomes than theimmigrants. Further,
µ −1

σ −1
( A + 1) 1 − λ + A  (2 − µ − σ )∂ (ω N ω I )
λ


=
.
∂λ
(σ − 1)λ2

Inspecting (49), its sign depends onand σ . Specifically, if

µ

(49)

µ +σ < 2then∂ (ω N ω I ) ∂λ > 0 , that is, immigration monotonically enlarges inequality. On the other hand, ifµ + σ > 2 then ∂ (ω N ω I ) ∂λ < 0 , that is, immigration monotonically reduces inequality
between the natives and the immigrants. Why does immigration enlarge the inequality when µ
and/or σ is small? This can be explained by recalling the analysis of the equilibrium land rentin (38). Smaller σ means that the variety of the common goods is important for the consumers.
This leads to larger demand for firms and, correspondingly, for land. This raises the land rent,
which works to the advantage of the natives through their income increase. In addition, a smallerµ with smaller σ means that the consumers’ expenditures shift to the common goods in which
the demand for variety is large. This also increases the land rent and equivalently the natives’
income, to enlarge the inequality between the natives and the immigrants.
In summary, immigration impact on inequality between the natives and the existing
immigrants is monotonic, but can act in either direction.

5.3 The four patterns of immigration impacts
Summarizing the results obtained, we find that there are four patterns in terms of the impacts ofimmigration on the natives and on the inequality between the natives and the immigrants (Table
1).
The existence of U-shaped non-monotonicity suggests that economies can be separated into
two groups. One group comprises the countries that keep accepting new immigrants, since
immigration is self-reinforcing. The other group comprises those that sustain only a low level ofimmigration, because additional immigration harms the natives economically.

14

Table 1. Four possible patterns of immigration impacts on welfare and inequality
Impact on the natives

Inequality between the natives
and the immigrants

1) Non-monotonic (U-shaped)
impactandincreasing
inequality
2) Non-monotonic (U-shaped)
impactanddecreasing
inequality

∂ω N
∂ω N
> 0,
<0
∂λ
∂λ

∂ (ω N ω I )
>0
∂λ

∂ω N
∂ω N
<0
> 0,
∂λ
∂λ

∂ (ω N ω I )<0∂λ3) Monotonic (positive) impact
and increasing inequality

∂ω N
>0
∂λ

∂ (ω N ω I )>0∂λ4) Monotonic (positive) impact
and decreasing inequality

∂ω N
>0
∂λ

∂ (ω N ω I )<0∂λ

6. Discussion
Figure 2 shows the share of foreign born within the population of various OECD member
countries. There are, of course, various reasons for the variation observed in these data, and theycannot be explained solely by the theory presented in this paper. However, at first glance, Korea
and Japan are outliers given their low shares of immigrants within their populations. Korea andJapan are both Asian OECD countries, and likely to be culturally different from larger culturalgroups in the rest of the world. Japan, in particular, was analyzed by Huntington (1996) as an‘isolated culture’, meaning that the nation does not share its culture with any foreign country, ordoes not belong to any other cultural groups of nations. If such is the case, the present theory
suggests that Japan (and probably Korea) is more likely to be classified in the non-monotonic
impact group in Table 1. This will then explain why in Japan the share of immigrants is relatively
low, and more importantly, suggests that the trend may reverse and the native population will
enjoy the economic gains from having more immigrants. This forecast becomes more likely when
one takes into account that the native population is now decreasing in Japan. In the present model,
a decrease in native population implies an automatic rise in the share of immigrants, λ .
Recalling Figure 1, then, a native population decrease in Japan raises λ , leading to a rise in thereal wages of the natives ( ω N ) so that the natives’ attitudes towards immigration change fromnegative to positive.

15

There is growing concern in Japan that the nation’s population will keep decreasing,accompanied by a rapid aging of the society. The present theory, however, suggests that there isan opposite force that may put a brake on the population decrease: if the Japanese situation can
be classified as the non-monotonic or the U-shaped type, then the natives’ attitudes towards
immigration are likely to turn positive (at least from economic grounds) as the share of
immigrants rises in Japan. After crossing the turning point as illustrated in figure 1, theself-reinforcing pull factor of immigration sets in to bring in new immigrants, potentially leading
to a population increase in Japan.
Some limitations of the analysis should be noted. First, external trade was ignored. This
rules out potential effects of people’s movement between countries on international flow ofgoods and services. Second, the analysis assumed only two different cultures, those of the natives
and the immigrants. Therefore, it may not capture the real-world complexity that arises when
immigrants arrive from multiple cultures. Third, the model assumed homogeneous workers. Thatis, the analysis abstracted from possible differences in skills between the natives and theimmigrants. Finally, in the very long run, immigrants may assimilate to the native culture, inwhich case, the crowding-out problem of the cultural goods that the analysis incorporates may
become less important.

30

28.2 27.6

25
20
15

20.0
16.7 16.4 16.0

15.5

13.9 13.4

13.1 12.8

12.3 11.6
8.5

10

5.6
5
0

1.9 1.7
NZL AUS CAN AUT IRL SWE BEL NOR ESP USA DEU GBR NLD DNK FIN KOR JPN

Source: OECD (2015) and Japanese Ministry of Health, Labor and Welfare

Figure 2. Share of foreign born within the population, %16

Appendix
Proof of A ≡µ (δ − σ ) + σ> 0:µ (σ − δ ) + σ (δ − 1)

Recall assumptions σ > 1 , δ > 1 , and 0 < µ < 1 .
a. If σ > δBoth the denominator and the numerator of A are positive, that is, µ (σ − δ ) + σ (δ − 1) > 0 andµ (δ − σ ) + σ > 0 . Therefore, A > 0 .
b. If σ < δThe denominator of A can be rearranged as δ (σ − µ ) − σ (1 − µ ) . By the assumptions,

σ − µ > 1 − µ > 0 . In addition, σ < δ . Hence, δ (σ − µ ) − σ (1 − µ ) > 0 . The numerator of A
can be rearranged as σ (1 − µ ) + δµ . Given the assumptions, it is positive. Therefore, A > 0 .
c. If σ = δThen A =

1
>0
δ −1

From a, b, and c, therefore, A > 0 .

Acknowledgement
I appreciate the financial support of a Grant-in-Aid for Scientific Research (16H06548) from theJapan Society for the Promotion of Science.

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