㻌
㻌㻌Tsukuba Economics Working Papers
No. 2014-003

Pareto-improving Immigration in the Presence of Social Security
byHisahiro Naito
July 2014

UNIVERSITY OF TSUKUBA
Department of Economics
1-1-1 Tennodai
Tsukuba, Ibaraki 305-8571
JAPAN

Pareto-improving Immigration in the Presence ofSocial Security
Hisahiro Naito ∗†Department of Economics
Graduate School of Humanities and Social Sciences
University of Tsukuba, Japan
July 17, 2014

Abstract
The eﬀect of accepting more immigrants on welfare in the presence of a payas-you-go social security system is analyzed theoretically and quantitatively.
First, it is shown that if initially there exist intergenerational government transfers from the young to the old, the government can lead an economy to the(modified) golden rule level within a finite time in a Pareto-improving way byincreasing the percentage of immigrants to natives (PITN). Second, using thecomputational overlapping generation model, I calculate both the welfare gainof increasing the PITN from 15.5 percent to 25.5 percent and years needed toreach the (modified) golden rule level in a Pareto-improving way in a model
economy. The simulation shows that the present value of the Pareto-improving
welfare gain of increasing the PITN comprises 23 percent of the initial GDP.
∗

Email: naito@dpipe.tsukuba.ac.jp
I appreciate comments from the participants of the Canadian Public Economics Meeting(Kingston), the North American Summer Econometric Society Meeting (St. Louis), the European
Econometrics Society Meeting (Oslo), the Macroeconomics workshop at the University of Tokyo and15th Macroeconomics Conference (Japan), the economics workshop at the University of Victoria.
I appreciate comments from Kosuke Aoki, Masayoshi Hayashi, Ryo Kato and Shinichi Nishiyama,
Pierre Pestieau and Herbert Schuetze.
†1

It takes 112 years for the model economy to reach the golden rule level in a
Pareto-improving way.

1

Introduction

Transforming a pay-as-you-go (PYGO) social security system into a funded system
is not easy. When the PYGO social security system is changed to a funded system,some generations must bear the so called “the double burden”, such that a young
generation needs to pay the social security tax twice. Thus, although the transition
from a PYGO social security to a funded system is desirable since a PYGO social
security system causes under-accumulation of capital, it is diﬃcult to transit in a
Pareto-improving way.
On the other hand, in the face of the fiscal problems caused by negative demographic shocks in the presence of PYGO social security, policy makers have recently
shown increased interests in accepting more immigrants. For example, the IMF report on the Japanese economy (2012), which is suﬀering as a result of an extremely
low fertility rate and practical restrictions on the inflow of immigrants, states thatfacilitating a modest level of immigration could have a large payoﬀ for the Japanese
economy. However, although using immigrants to solve the fiscal problem of PYGO
social security in the short run is attractive, its desirability in the long run is unclear. Increasing the percentage of immigrants to natives implies a higher population
growth rate in a host country. With a neoclassical production function that exhibits
the diminishing marginal product of capital, the standard growth model (Ramsey(1928) and Solow (1964)) predicts that such a higher population growth rate leads toboth a lower level of capital stock per capita and income per capita, starting from a
dynamically eﬃcient initial steady state. Thus, it is not clear how such an increase of
immigrants can help to solve the problem of PYGO social security in the long run. The
literature also presents an unclear picture. In the literature on public finance, there
is increasing interest in the eﬀect of accepting more immigrants on the social welfare.
By using the computational overlapping generation model (Auerbach and Kotlikoﬀ
model(Auerbach and Kotlikoﬀ (1998)), Storesletten (2000) argues that accepting aparticular type of immigrants, (skilled who are of an age such that they will not beable to claim the social security benefit because they will not be able to satisfy theminimum requirement of the duration of social security tax payments) will increase
the social welfare in the presence of the retirement of the baby boom generation. In
contrast, Fehr, Jokisch and Kotlikoﬀ (2004) argue that there will be no such a welfare
gain. Feldstein (2006) analyzes the eﬀect of immigration in Spain and concludes that
immigration does not bring a welfare gain while Collado, Iturbe-Ormaetxeand andValera (2004) argue that accepting more immigrants brings a positive welfare gain to
Spain.
Given the mixed results in the literature regarding the eﬀect of accepting more
immigrants on the social welfare and the theoretical prediction of the standard growthmodel, a natural question arises whether or not, from a theoretical standpoint, accepting more immigrants Pareto-improves welfare in the presence of PYGO social
security. With a neoclassical production function that exhibits diminishing marginal
product of capital, a decrease of income per capita at the steady state seems inevitable
when the economy experiences a higher population growth rate due to an increased
inflow of immigrants. Yet, if this is so, one may wonder why the previous studies
have arrived at such diﬀerent results regarding the eﬀect on welfare of accepting more
immigrants.
Motivated by those questions, I analyze theoretically and qualitatively the eﬀect
on welfare of accepting more immigrants and increasing the population growth rate.
I conclude that accepting more immigrants and increasing the population growth rate
Pareto-improves welfare and, to a large extent, solves in a Pareto-improving way the
problem of under-accumulation of capital that is caused by implementing a PYGO
social security system.1 More specifically, firstly, using the overlapping generation
model developed by Diamond (1965), I show that in an economy with or without
distorting taxes it is Pareto-improving to increase the percentage of immigrants tonatives (PITN) if there exit upward intergenerational transfers, in the sense that themarginal product of labor of a young individual times labor supply is greater than thesum of resources that a young individual consumes when he or she is young and the1

The Matlab code which is used for this simulation is available from the Journal’ website andfrom the author.

2amount of resources that are transferred to future periods(the MPL condition). Notethat the former is the pre-tax income of a young individual and the latter is equal
to the sum of the after tax income, the amount of publicly provided private goods
for a young individual and the government saving per each young individual. In thepresence of a PYGO social security system, some of the pre-tax income of a young
individual is used for the consumption of the old. Thus, the above MPL conditionis likely to satisfied unless the government have a large amount of the government
savings. Thus, in the presence of PYGO social security, this MPL condition is likely
to be satisfied. Secondly, I show analytically that when this MPL condition is satisfied,
the government can lead the economy to the (modified) golden rule level in a Paretoimproving way within a finite time by putting in savings the government budget
surplus, which is obtained by increasing the PITN. Note that when the economy
reaches the golden rule level, the problem of under-accumulation of capital causedby the PYGO social security is solved for all practical purposes. Third, I quantify
this Pareto-improving welfare gain that is yielded by increasing the PITN in thepresence of a PYGO social security system and calculate the year needed to reach the(modified) golden rule level in a Pareto-improving way by using the computational
overlapping generation model developed by Auerbach and Kotlikoﬀ (1987). I consider
a moderate increase of the PITN, such that the PITN starts to increase from 15.5
percent, reaches 25.5 percent at the 80th year and remains constant at 25.5 percent
in later years.2 With this speed of increase of the PITN and in the model thatmimics important dimensions of the US economy, my simulation shows it takes aminimum of 112 years for the model economy to reach the golden rule level in a
Pareto-improving way. On the new balanced growth path, the capital stock per
eﬃcient unit of labor increases by 102 percent and the publicly provided private
goods per capita increases by 36 percent. When the target capital stock is set atthe modified golden rule level with 3 percent of intergenerational discount rate, ittakes 65 years to reach the modified golden rule level in a Pareto-improving way
2

15.5 percent initial PITN is obtained by using census 2000 data from the author’s calculation.
See section 4.2 for more detailed discussion.

3and the capital stock per eﬃcient unit of labor increases by 18 percent. The present
discounted value (PDV) of the Pareto-improved utility, measured by the expenditure
function, of natives and their descendants, which does not include the increased utilityof immigrants and their descendants, comprises 23 percent of the initial GDP. Whenthe time to reach the target PITN is shortened to 42 years, the economy reaches
the modified golden rule at the 59th year and the PDV of the Pareto improvement
comprises 28 percent of the initial GDP. Finally, I conduct robustness checks bychanging a number of parameter values, for example, the share of the surplus for thegovernment savings, the replacement rate, the time preference rate, the risk aversion,
the initial government debt(asset) level, the level of immigrants’ earnings and theconsumption of public services by immigrants. Those robustness checks show thatthe results of the simulation do not change substantially in magnitude for diﬀerent
parameter values. Both theoretical results and the computational results suggest therobustness of the welfare gain of increasing the PITN in the presence of PYGO social
security.
The remainder of the paper is organized as follows. Section 2 reviews the related
literature. Section 3 presents a theoretical analysis. Section 4 presents a simulationbased analysis using the computational overlapping generation model. Section 5 concludes.

2

Literature

In the theoretical literature on the eﬀect of accepting immigrants, Razin (1999) is,to the best of my knowledge, the first paper that shows that accepting immigrants
improves welfare. He shows that in a small open economy model in which factor
prices are fixed, accepting more immigrants can improve welfare in the presence ofPYGO social security. However, in a subsequent paper, Razin and Sadka (2000) showthat in an closed economy model in which capital accumulation and factor prices areendogenous, the result obtained under the assumption of a small open economy is not
likely to hold.
4For the empirical side, there is a large volume of literature that analyzes theeﬀect of accepting immigrants. (Huddle(1993), Borjas(1994), Passel (1994), Simon(1984) and Akbari (1989), Lee and Miller (1997), Auerbach and Preopoulous (1999),
Storesletten (2003), and, Collado and Valera (2004)). More recently, this literature analyzes the eﬀect of accepting immigrants in a dynamic general equilibrium
model (Storesletten(1995, 2000), Canova and Ravn (2000), and Fehr, Joisch and Kotlikoﬀ (2004)). Storesletten (1995) is the first paper to analyze the eﬀect of accepting
immigrants in a dynamic general equilibrium mode with aggregate uncertainty andStoresletten (2000) analyzes without aggregate uncertainty. He argues, in the US context, that by selecting a particular type of immigrants, the acceptance of immigrants
can have a positive eﬀect on the welfare of the native. Fehr, Joisch and Kotlikoﬀ(2004) argues that there is no such welfare gain, or, even if there were to be a gain,
it would be very small.
This paper contributes to the existing literature in several ways. First, I identify the conditions under which it is Pareto-improving to accept more immigrants.
(MPL condition). Second, I show that the government can lead the economy to the(modified) golden rule level by accepting more immigrants within a finite time in a
Pareto-improving way. This is in sharp contrast to the literature on the social security
reform, in which it is argued that it is diﬃcult to increase the capital stock throughsocial security reform in a Pareto-improving way (Geanakoplos, Mitchell and Zeldes(1998)). Third, I develop the method that allows me to show that accepting more
immigrants Pareto-improves all generations in the presence of distorting taxes andchanging factor prices in a dynamic economy. This technique was originally developed in the international trade literature to show the superiority of free trade overrestricted trade in the absence of lump-sum taxes and transfers (Dixit and Norman(1980)). To the best of my knowledge, this is the first paper that applies this technique to the analysis of a dynamic economy. Fourth, I quantify the welfare gain of
this Pareto-improvement, which is predicted by the theoretical model. Consistent
with the results of Storesletten(2000), I show that there is a non-trivial welfare gainof increasing immigrants in the US and demonstrate that my results are robust for5

diﬀerent values of the parameters. The theoretical results and the robustness of thesimulation results show that a policy of increasing immigrants in the US can be an
important policy option to be considered.

3

The model

The model uses the standard overlapping generation model with a neoclassical production function developed by Diamond (1965). Each individual lives for two periods.
When individuals are in the first period, they work and are called “young”. Whenthey are in the second period, they are retired and are called “old”. I assume that
immigrants come to the host country only when they are young and that the government of the host country prohibits the immigrants from immigrating when they
are already old. I define individuals who are born at the beginning of period t in
the host country as natives of cohort t, regardless of the nationality of the parents.
Immigrants who move to the host country at the beginning of period t are consideredas immigrants of cohort t. Let j be the index indicating nationality. If an individual
is a native, j = n and if she or he is an immigrant, j = m. Let Ntj be the numberof the young of type j in period t. Let (cy,jco,jtt+1 ) be the consumption in the young
period and the old period of a type j (j = n, m) individual of cohort t. Let gty,j be
the amount of publicly provided private goods, such as education and government
provided health care service for the young, for each young individual of type j whichis consumed at period t. Let gto,j be the amount of publicly provided private goods,
such as medicaid and publicly provided nursing home, for each old individual of typej at period t. Let gtind,j be the amount of age-independent publicly provided private
goods that are consumed by a young individual and an old individual of type j.3 I
3In this paper, I ignore non-rivalry public goods. Note that the presence of non-rivalry public
goods will favor immigration because accepting immigrants means that the cost of non-rivalry public
goods will be shared by more individuals without decreasing their consumption.

6

assume that the utility function of cohort of type j is
j
yj
oj
ind,j
ind,j
ind ojyj y,j jyj yj
U j (cyj)
t , lt , gt , gt , ct+1 , gt+1 , gt+1 ) = u (ct , lt ) + v (gt , gt

+

1
ind,j
oj o,j[uoj (co,jt+1 ) + v (gt+1 , gt+1 )] .
1+ρ

(1)

j
ind,j
i ij
I assume that uij (ci,j) (i = y, o; j = n, m) are strictly int , lt ) and v (gt , gt

creasing and concave functions. I assume additive separability of publicly provided
private goods so that the provision of publicly provided private goods does not aﬀect
the consumption and saving decisions of individuals. This assumption simplifies theanalysis because I also assume that the government redistributes the welfare gain of
accepting more immigrants in the form of increased publicly provided private goods
to individuals.4
For the production side, let F (Lt , Kt ) be a production function where Lt and Kt
are the total amount of labor and the total capital stock used at period t. Let δ be
the depreciation rate of the capital. I assume that F (Lt , Kt ) exhibits constant returns
to scale and that both the marginal product of labor and capital are diminishing. I
assume that the standard Inada condition is satisfied.
I assume that the economy is at the steady state initially and that the initial
economy is dynamically eﬃcient.5 Furthermore, for the welfare analysis of accepting
more immigrants, I make the following additional assumptions.
AS1: The amount of publicly provided private goods per person, (gtyj , gtoj , gtind,j ) isconstant at the initial steady state and (gty,j , gto,j , gtind,j ) = (g y,j , g o,j , g ind,j ).
AS2: The government uses a PYGO social security system at the initial steady state.
AS3: For one unit supply of labor by a native, φn eﬃcient units of labor is supplied.
For one unit supply of labor by an immigrant, φm eﬃcient units of labor is supplied
where φm ≤ φn . For normalization, I assume that φn = 1.
4

There are several ways to redistribute the welfare gain to individuals. I use this method tosimplify the analysis. The main conclusion does not change when other way of redistributing thewelfare gain are used.
5
In the literature, it is well-known that if the market interest rate is lower than the population
growth rate, it is possible to Pareto-improve welfare (dynamic ineﬃciency). Since this paper’s interest
is not such a dynamic ineﬃciency problem, I postulate that at the initial steady state, the market
interest rate is higher than the population growth rate (Cass(1972)).

7

AS4: The descendants of immigrants integrate with the native population and earn
the same income as natives.
AS5: Immigrants and their children stay permanently in the host country.
AS6: The fertility rate of immigrants is equal to or higher than the fertility of thenative.
AS7: If immigrants and natives have the same productivities, then the government
treats immigrants and native in the same way in the tax and social security system.
I use AS1 to focus on the issue of immigration, rather than on issues of public
expenditure. I need AS2 to see the eﬀect of increasing the number of immigrants inthe presence of a pay-as-you go social security. AS7 needs more discussion. Clearly,
if the government can treat the immigrants in a discriminating way in the tax andthe public pension systems, there is a way to increase the utility of both natives andimmigrants. Normally, the wage rate of the immigrants in their country of origin islower than the wage rate in their host country. Thus, it is possible to Pareto-improve
the welfare of both natives and immigrants if (a) the government in the host country
sets a high tax rate on incoming immigrants in such a way that the net wage rateof immigrants in their host country is still higher than their net wage rate of theircountry of origin, (b) the government redistributes to natives the tax revenue collectedfrom immigrants. AS4 precludes such an obvious case from occurring.
Let FK and FL be the partial derivative of the production function with respect
to capital and labor. Let wtj and rt be the wage rate of an individual type j and theinterest rage at period t. Let sjt and at be the amount of savings made by cohort tof type j and the total amount of the government savings divided by the number ofcohort t. Then, wtj and rt are determined as follows:
wtn = FL (Lt , Kt ), wtm = φm FL (Lt , Kt ), rt = FK (Lt , Kt ) − δ
� j j
� j
� j
j
where Lt =φj lt Nt and Kt =st−1 Nt−1+ at−1
Nt−1
j

8

(2)

The resource constraint at the period t is as follows:=�

F (Lt , Kt ) + (1 − δ)Kt
� j
j
j
y.j
ind.j
o.j
{cy,j
+
s
+
g
+
g
+
a
}N
+
Nt−1 {co,j+ g ind.j }
t
t
t
t
t +g

(3)

Let the fertility rate of natives and immigrants be πn and πm , respectively. Given
πn and πm , Ntn can be written as follows:mn
Ntn = (1 + πm ) × Nt−1
+ (1 + πn ) × Nt−1

(4)

Let αt be the immigrant-to-native ratio (INR) of cohort t. The immigration policy
is expressed in terms of αt . For example, a one-time increase of INR means thatα0 = α ∗ , α1 = α
� and αt = α∗ for t ≥ 2 where α
� > α∗ . Permanently increasing INR

means that α0 = α∗ and αt = α
� for all t ≥ 1 where α
� > α∗ . The sum of natives andimmigrants of cohort t is
�

j=n,m

Ntj = Ntn × (1 + αt )mn
= {(1 + πm ) × Nt−1
+ (1 + πn ) × Nt−1
} × (1 + αt )n= {(1 + πm ) × αt−1 + (1 + πn )} × Nt−1
(1 + αt )(5)nThe total number of cohort t − 1 is Nt−1
(1 + αt−1 ). Thus, at the steady state immi-

gration policy α∗ , the growth rate of cohorts is (1 + πm ) × α∗ + (1 + πn ) − 1. DefineR(α) as follows:
R(α) ≡ (1 + πm ) × α + (1 + πn )We can interpret R(α) as one plus the cohort population growth rate when the immigration policy α is implemented. To avoid a situation in which the total populationof period t becomes zero or negative, I assume
R(αt ) ≡ (1 + πm ) × αt−1 + (1 + πn ) > 0 for αt−1 ≥ 0 .

9

(6)

3.1

An Economy with Lump-sum tax, Same Productivities
and Preferences: Deriving MPL condition

The purpose of the section is to derive the MPL condition which plays an importantrole in the analysis of an economy with distorting taxes and in a quantitative analysis. To obtain MPL condition in a clear way, first I consider an economy where the
lump-sum tax is available for the government and immigrants and natives have same
productivities and preferences. This implies that there are no distorting taxes. Once
we get the MPL condition in this economy, I will show that this MPL condition plays
the critical role for evaluating the eﬀect of increasing the number of immigrants on
welfare and capital accumulation in an economy with distorting taxes and with diﬀerent productivities and preferences between immigrants and natives. Readers who areinterested in the analysis of the economy where immigrants and native are diﬀerent interms of productivities, consumption of public services and preferences can skip thissection and go to sub-section 3.4 directly.
Let bjt+1 be the social security benefit for the cohort t at the period t + 1. Let τtj
be the amount of the lump-sum tax at the period t for type j. The assumption of
the same productivities and the same preference between natives and immigrants andoj j j jj
AS7 imply that φj , wtj , cyj
t , ct+1 lt , st bt+1 τt do not change for diﬀerent value of j. Thus,

we eliminate superscript j for those variables and from the utility functions. Cohort t
maximizes the life-time utility function subject to the budget constraint. The budget
constraint of the cohort t of type j is
wt lt − τt = cyt + st and bt+1 + (1 + rt+1 )st = cot+1 .

(7)

The government budget constraint at period t is(τt − g y − g ind − at )

�

j=n,m

Ntj − (bt + g o + g ind − (1 + rt )at−1 )

�

j
Nt−1
=0

j=n,m

Using the individual budget constraint, the homogeneity of the production function and equation (2), it is straightforward to show that the government budget10

constraint is equivalent to the following resource constraint:6
F (Lt , Kt ) + (1 − δ)Kt ≥� j� j{cyt + st + g y + g ind + at }Nt + {cot + g o + g ind }Nt−1
where Lt =

�

j=n,m

lt Ntj and Kt = (st−1 + at−1 )

j=n,m

j=n,m

�

j
Nt−1

(8)

j=n,m

Using (5), the above resource constraint can be rewritten as
F (Lt , Kt ) + (1 − δ)Kt ≥nn{cyt + st + g y + g ind + at }Nt−1
R(αt−1 )(1 + αt ) + {cot + g o + g ind }Nt−1(1 + αt−1 )Before analyzing the eﬀect of accepting more immigrants, we characterize the initialsteady state. Let w∗ and r∗ be the wage rate and the interest rate at the initialsteady state respectively, where the immigration policy at the initial steady state isα∗ for all t. Let s∗ be the amount of savings of each individual and the number ofold native at the initial steady state. Let a∗ and b∗ be the government savings (ordebt if it is negative) divided by the number of young at the initial steady state andthe social security benefit at the initial steady state. The government will choose
the steady state lump-sum tax policy τ ∗ so that it satisfies the government budget
constraint. This implies that at such b∗ and τ ∗ , the resource constraint must be
satisfied. Conversely, when the resource constraint is satisfied, then the government
budget constraint is also satisfied.
The initial steady-state economy with the steady state immigration policy α∗ is6Using the homogeneity of the production function, we have F (Lt , Kt ) + (1 − δ)Kt = Fk Kt +FL Lt + (1 − δ)Kt . Then, using (2), F (Lt , Kt ) = (rt + δ)Kt + wt Lt + (1 − δ)Kt . Using the definitionof Kt and Lt and using the individual budget constraint, we have (8).

11characterized as follows:(s∗ , l∗ ) = arg max uy (w∗ l − τ ∗ − s, l) + v y (g y , g ind ) +s,l1[uo ((1 + r∗ )s + b∗ ) + v o (g o , g ind )]1+ρ
(9)

where w∗ =

∗

∗

∗

∗

∂F (L , K ) ∗∂F (L , K ), r +δ =∂L∂K

L∗ = l∗ R(α∗ )N0n∗ (1 + α∗ ) and K ∗ = (s∗ + a∗ ) × N0n∗ (1 + α∗ )F (K ∗ , L∗ ) + (1 − δ)K ∗ = {cy∗ + s∗ + g y + g ind + a∗ }R(α∗ )N0n∗ (1 + α∗ )

(10)
(11)
(12)

+{co∗ + g o + g ind }N0n∗ (1 + α∗ )(13)cy∗ = w∗ l∗ − τ ∗ − s∗ and co∗ = (1 + r∗ )s∗ + b∗(14)

N0n∗ is some positive number(15)

The the utility level at the initial steady state is defined as follows:u∗ ≡ uy (cy∗ ) + v y (g y , g ind ) +3.1.1

1[uo (co∗ ) + v o (g o , g ind )] .
1+ρ

(16)

Welfare Eﬀect of Increasing the INR

In this sub-subsection, I examine, starting from period 1, whether or not increasing
the INR to a higher level permanently will Pareto-improve welfare. Increasing the
INR permanently is defined as α0 = α∗ and αt = α where α > α∗ for t ≥ 1. For
the analysis, consider the following constrained maximization problem, which is afunction of α:12V (α) =max{cyt ,cot ,st , at |t=1,2,..}s.t. uy (cyt , lt ) + v y (g y , g ind ) +1[uo (co1 ) + v o (go , g ind )]1+ρ

1[uo (cot+1 ) + v o (g o , g ind )] ≥ u∗ for t = 1, 2, ... (17)1+ρF (Lt , Kt ) + (1 − δ)Kt ≥nn{cyt + st + g y + g ind + at }Nt−1
R(αt−1 )(1 + αt ) + {cot + g o + g ind } Nt−1
(1 + αt−1 ) for t = 1, 2, ...

(18)n
Kt = (st−1 + at−1 )Nt−1(1 + αt−1 ) for t = 2, ... and s0 = s∗ and a0 = a∗n
Lt = lt Nt−1
R(αt−1 )(1 + αt )(19)

α and α∗ are given
The above programming problem deserves several comments. First, V (α) is theutility of cohort 0 at period 1 when the government is accepting immigrants with aconstant ratio α∗ at the initial steady state and starts to accept immigrants withratio α from period 1. Because at the period 1, the consumption of the cohort 0 at
the young period is already determined, I do not include the consumption of cohort
0 at the young period. Second, the first constraint is related to Pareto improvement
and requires that all cohorts except cohort 0 need to have at least as the same utility
as they would have at the initial steady state. Note that in the first constraint, there
are α and α∗ . From the point of period 1, the immigration policy at the period 0 is
pre-determined. Such an variable is denoted as α∗ . The policy that is determined at
the period 1 or later periods is denoted as α.
nnnNote that Nt−1
is determined by α and Nt−2
, but Nt−1
is also aﬀected by α andnNt−3. This implies that a change of immigration policy α aﬀects all Ntn for t = 1, 2, ...
n
To make the calculation easy, it is useful to divide the resource constraint by Nt−1

13nwhen t = 1 and by Nt−1
(1 + α) when t = 2, 3, 4....7 Then, (18) becomes as follows:

F (R(α∗ )(1 + α)l1 , (s∗ + a∗ )(1 + α∗ )) + (1 − δ)(s∗ + a∗ )(1 + α∗ ) ≥{cy1 + s1 + g y + g ind + a1 }R(α∗ )(1 + α) + {co1 + g o + g ind } (1 + α∗ ) for t = 1(20)

F (R(α)lt , st−1 + at−1 ) + (1 − δ)(st−1 + at−1 ) ≥{cyt + st + g y + g ind + at } × R(α) + {cot + g o + g ind } for t = 2, 3, 4, ...

(21)

Let L be the Lagrangian function. Let γt and λt be the Lagrangian multiplier ofthe minimum utility constraint (17) and the resource constraints (20) and (21). Letγt∗ and λ∗t be the Lagrangian multipliers when α = α∗ . Then, we have the following
observation.
Observation 1
When α = α∗ the solution of MPP is
cyt = cy∗ , co0 = co∗ , st = s∗ , at = a∗ lt = l∗ for t = 1, 2, ...
∗1R(α ) ∗
uoc (co1 ) and λ∗t+1 =λ1+ρ
1 + r∗ t1γt∗ = y y∗ ∗ λ∗t R(α∗ ) and for t = 1, 2, ...
uc (c , l )

λ∗1 =

(22)
(23)
(24)

For the proof of observation 1, see appendix B1.
Observation 1 implies that when the immigrant-native ratio α is fixed at α∗ , theinitial steady state allocation is Pareto-eﬃcient and it is not possible to have Pareto
improvement from the initial steady state holding α = α∗ . Now suppose that the
government increases the INR from α∗ . Whether or not such an increase of the INR
Pareto-improves welfare can be analyzed by calculating dV /dα and evaluating it atα = α∗ . From the envelope theorem, dV /dα|α=α∗ is equal to7nn
The reason that we did divide the resource constraint at the period 1 by Nt−1
, not Nt−1
(1 + α)n∗nis that the population of the old at the period 1 is Nt−1 (1 + α ), not Nt−1 (1 + α).

14�R(α∗ )λ∗1 +

∞
�
t=2

λ∗t R� (α∗ )���× FL (R(α∗ ), s∗ + a∗ )l∗ − (cy∗ + s∗ + g y + g ind + a∗ ) .

(25)

where R(α) = 1 + πn + α(1 + πm )(26)

The first bracket is positive because the Lagrangian multiplier of the resource
constraint is positive and the marginal eﬀect of increasing α on one plus the population
growth rate is positive. In the second bracket, the first term is the marginal product oflabor times labor supply, which is what an individual contributes to the economy when
he or she is young at the initial steady state. When an young individual contribute
FL l∗ to the economy, there are three choices to distribute this contribution from thepoint of the government. The first choice is to let the young individual consume thiscontribution. The second choice is to transfer this contribution to the future periods
and let this young individual or future cohort to consume it. The third choice is totransfer this contribution to the old individuals. cy∗ + g y + g ind is the amount of theresource consumed by the current young individual at the initial steady state. s∗ + a∗is the amount of the resource that is transferred to future periods. Note that cy∗ +��g y +g ind +s∗ +a∗ does not include co∗ and g o . Thus, FL l∗ − cy∗ − g y − g ind − s∗ − a∗is the amount of the resource transferred to the old individuals at the initial steadystate. We call this amount as the upward intergenerational transfer.
Definition: When FL l∗ − cy∗ − g y − g ind − s∗ − a∗ is positive, we say that the MPL

condition is satisfied and call the amount FL l∗ − cy∗ − g y − g ind − s∗ − a∗ as upward
intergenerational transfers.

Proposition 1 (MPL condition version) If there exit upward intergenerationaltransfers, in the sense that the marginal product of labor of a young individual times
labor supply is greater than the sum of resources that a young individual consumes
when he or she is young and the amount of resources that are transferred to future
periods, then accepting more immigrants Pareto-improves the welfare of all genera-

15

tions.8 Because the marginal product of labor of the young times labor supply is the
pre-tax earning of a young individual and because cy∗ + s∗ is the after-tax income of
the young by the definition, FL l∗ − cy∗ − s∗ is the amount of tax paid by a young

individual. g y + g ind + a∗ is the amount of resources provided by the government to a
current young individual and the amount of resources to the future cohorts per eachyoung. Thus, we have the following Corollary:
Corollary (Tax-expenditure version)Alternatively,if the amount of tax paidby the young is greater than the sum of the amount of resources provided by thegovernment to a current young individual and the amount of resources to the futurecohorts per each young at the initial steady state, accepting more immigrants Paretoimproves welfare.
This MPL condition plays a critical role for the analysis not only of the casewhere the government has an access to the lump-sum tax but also of the case where
the government does not have an access to the lump-sum tax. In addition, in thesimulation analysis, this MPL condition plays an important role.
The Tax-expenditure condition has also an important implication for the costbenefit analysis of accepting immigrants. Note that for the government expenditure
part, the social security benefit and publicly provided private goods for the old are notincluded in the Tax-expenditure condition. Only the tax that young immigrant pays
and the resource that is used for the young or the future cohorts should be included.
910

Graphically, Proposition 1 is explained in Appendix A1.
8Note that this condition does not change even in the presence of public goods because an increaseof the number of immigrants does not aﬀect the consumption of public goods by the nature of public
goods.
9
When immigrants and natives are diﬀerent in preferences and productivities, then the sum of
the redistribution from immigrants to natives and MPL condition become important. See section
3.3.
10
In a typical study of assessing the fiscal eﬀect of accepting more immigrants, it calculates thepresent value of the government expenditure such as the consumption of publicly provided private
goods and the social security benefit that immigrants receive and the tax revenue (including income
tax and the social security tax) that immigrants pay. However, if the social security benefit thatthe retired immigrants receive is included in the cost-benefit calculation, then the social security tax
that the children of immigrants pay should also be included because the social security benefits arefinanced by the social security tax that the children of natives and immigrants pay. Of course, if thesocial security tax that the children of immigrants pay is included, then the social security benefit
that the children of the immigrants receive should also be included. Again, if the social security

16

3.2

Presence of Distorting Taxes and Implementation of Pareto
Improvement

In the preceding subsection, it was shown that it is Pareto-improving to accept more
immigrants when the lump-sum transfers are possible and when there are upward
intergenerational transfers from the young to the old in the sense that the marginal
product of labor of the young times labor supply is greater than the resources thatare allocated to the young or to the future cohort. With a neoclassical production
function that exhibits diminishing marginal product, the pre-tax wage decreases andthe pre-tax interest rate increases when more immigrants are accepted. In the absenceof the lump-sum transfer, it is not clear whether it is possible to Pareto-improve all
generations.
In this subsection, I will show that, it is possible to Pareto-improve welfare, withoutchanging incentives of individuals, by increasing INR in the absence of the lumpsum transfer as long as the MPL condition is satisfied11 In addition, I show that a
relatively simple adjustment of taxes(wage tax and interest tax) and social security
benefit achieves this Pareto improvement while the government is increasing the INR.
For the analysis, let α� be a time invariant new immigration policy from period 1

where α� > α∗ . As in the previous section, I assume that the economy is dynamicallyeﬃcient at the initial steady state. This implies that

FK (R(α∗ ), s∗ + a∗ ) > δ + R(α∗ ) − 1(27)

In this sub-section, we analyze whether government can make the economy reach thegolden rule by accepting more immmigrants in a Pareto-improving way. To avoid abenefit of the children of the immigrants receives is included, then the social security tax that the
grand-children of the immigrants should be included. Note that in the PYGO social security system,
the social security benefit that the immigrants receives is roughly balanced by the social security
tax paid by the children of the immigrants. This implies that in the cost-benefit calculation, thesocial security benefit that the immigrant receive is roughly canceled out by the social security tax
that the children of the immigrants pay. Thus, in the cost benefit calculation, only the tax thatyoung immigrants pay and the publicly provided private goods for the young immigrants should be
included.
11
In the presence of non-optimal taxation, it is obvious that it is possible to Pareto-improve welfare
if the government can change incentives of individuals.

17

situation that the golden rule level capital stock does not exist, I assume that the sumof the population growth rate and the depreciation rate is striclty greater than zero:R(αt−1 ) − 1 + δ > 0 for αt−1 ≥ 0 .

(28)

As for taxes, I assume that the government uses a capital income tax and a wage
tax at the initial steady state. I assume that those taxes do not need to be the second
best optimal. Let τwt and τrt be the wage tax rate and capital income tax rate at theperiod t. Let the individual budget constraint (7) is modified as follows:
wt lt (1 − τwt ) = cyt + st and bt+1 + (1 + (1 − τrt+1 )rt+1 )st = cot+1
Let τw∗ and τr∗ be the wage tax rate and capital income tax rate at the initial steadystate. At the initial steady state, the above budget constraints become as follows:∗w∗ l∗ (1 − τw∗ ) = cy + s∗ and b∗ + (1 + (1 − τr∗ )r∗ )s∗ = co∗
I assume that at the initial steady state the social security benefit is proportional topre-tax earnings:b∗ = Ω × w ∗ l ∗(29)

When the government increases the INR, the wage rate falls and the interest rateincreases initially. To achieve Pareto improvement by increasing the INR, first I
assume that the government sets the tax rates such that after-tax wage and interest
rate after an increase of the INR are equal to the after-tax wage rate and interest
rate at the initial steady state. Then, the wage tax rate and the interest tax rate for
period t are set as follows:
wt (1 − τwt ) = w∗ (1 − τw∗ ) and rt (1 − τrt ) = r∗ (1 − τr∗ )(30)

Second, I assume that the government re-scales Ω so that the social security benefit
becomes proportional to after-tax earnings, not pre-tax earnings, and that an indi18

vidual receives the same benefit when wage rate, wage tax rate and labor supply areat the same level as at the initial steady state. This implies thatbt+1 =Ω
wt (1 − τwt )lt
1 − τw∗(31)

Note that bt+1 = Ωw∗ lt when wt (1 − τwt ) = w∗ (1 − τw∗ ) .
When the government sets taxes and social security benefit in this way, saving
behavior and labor supply behavior do not change because the budget constraint of
a consumer at any period t is the same as at the initial steady state. This implies
that the equilibrium social security benefit at any period t is the same as at the initialsteady state. If the government provides at least as the same level of publicly provided
private goods, the levels of the utility of all cohorts are at least as the same as at theinitial steady state.
As for the extent of the increased immigrants, motivated by Proposition 1, I assume
that the MPL condition is satisfied at the initial steady state:l∗ FL (l∗ R(α∗ ), s∗ + a∗ ) > cy∗ + g y + g ind + s∗ + a∗

(32)

The LHS of the above equation is a marginal increase of the output due to a
one unit increase of the population. The RHS is the amount of resources that anindividual receives when he or she is young.
The result that was generated in the preceding subsection shows that a marginalincrease in the number of immigrants Pareto-improves welfare if the MPL conditionis satisfied. However, that result does not imply that an unlimited acceptance ofimmigrants always Pareto-improves welfare. I impose two conditions on the extent
that immigrants are accepted. The first condition is regarding the MPL condition
when the new immigration policy α� is implemented. I assume that when the newimmigration policy α� is operative and when the government savings per each young

is held constant, the marginal increase of the output due to a one unit increase of the
population is greater or equal to the amount of resources that an individual receives
when he or she is young at the initial steady state. This implies that
19

n
m
n
n
l∗ FL (l∗ (1+�α)((1+πn )Nt−1+(1+πm )Nt−1), (s∗ +a∗ )(Nt−1+Nt−1)) ≥ cy∗ +g y +g ind +s∗ +a∗ .

(33)
m
n
Note that (1 + πm ) × Nt−1
+ (1 + πn )Nt−1
is the population of young native at periodmnt. Using the homogeneity of the production function and Nt−1=α
�Nt−1
, (33) can be

written asl∗ FL (l∗ (R(�α), s∗ + a∗ ) ≥ cy∗ + s∗ + g y + g ind + a∗ .

(34)

The second condition is regarding the golden rule. When the government accepts
more immigrants according to (34), and it adjusts the wage tax rate and capitalincome tax rate so that the after-tax wage rate and after-tax interest rate are thesame as at the initials steady state as defined in (30), it is possible that the golden
rule binds even the government savings per each young is held constant. This is
more likely when the capital stock per capita at the initial steady state is lower than
the golden rule level very close to the golden rule level. This implies that at α�, thefollowing golden rule is satisfied.

FK (R(�α), s∗ + a∗ ) = δ + R(�α) − 1(35)

On the other hand, when the capital stock per capita at the initial steady state is
suﬃciently lower than the golden rule level, the MPL condition (34) binds first insteadof the golden rule (35) when the government accepts more immigrants. This implies
that the following condition holds at α
� when the government savings per each young
is held constant:

FK (R(�α), s∗ + a∗ ) > δ + R(�α) − 1(36)

If (35) holds, then the government does not need to increase the government savings
to reach the golden rule and the analysis becomes trivial. Thus, in the following
analysis I assume that (36) holds. In other words, I assume that the capital stock per
capita at the initial steady state is suﬃciently lower than the golden rule level.
When the government accepts more immigrants, the government can increases the20

government savings balance as I show below. This implies that it is possible that,at some point, the marginal product of capital (MPK) becomes equal to the golden
level of capital stock per capita. But when the MPK becomes equal to the golden
rule level, it is clearly better to use the entire government surplus to increase thesupply of publicly provided private goods rather than to increase the government
savings balance. Thus, I assume that as long as the MPK is higher than the golden
rule level, the government uses some of the government budget surplus to increase
the government saving balance and the remainder to increase the supply of publicly
provided private goods. When the MPK reaches the golden rule level, the government
uses all of the surplus to increase the publicly provided private goods. Thus, we havethe following MPK condition:
FK (R(�α), s∗ + at ) ≥ δ + R(�α) − 1(37)

where at > a∗(38)

Now we examine whether social security benefit and taxes determined by (31) and(30) are feasible from the point of the government budget constraint. To check thefeasibility of such taxes, consider the net government budget surplus for period 1,SP1 .
SP1 = (w1 τw1 l∗ −g y −g ind −a∗ )

�

j=n,m

N1j +(r1 τr1 s∗ −b∗ −g o −g ind )�

N0j +(1+r1 )a∗

j=n,m

�

j=n,m

(39)

Note that τw1 and τr1 are defined in (30). Also I assume that the government will
have at least the same amount of the government saving per the number of the young
as at the initial steady state. By substituting τw1 and τr1 into SP1 and using thehomogeneity of the production function, we have (see appendix B2)
SP1 = N1n

� 1+�α1+α∗FL (N1n l∗ z, (s∗ + a∗ )N0n (1 + α∗ ))l∗ − cy∗ − s∗ − g y − g ind − a∗ ]dz (40)where N1n = N0n (1 + πn ) + N0n (1 + α∗ )(1 + πm ). Thus, from the MPL conditions

21

N0j

(32) and (33), the inside of the integration is positive for z ∈ [1 + α∗ , 1 + α
�]. Thismeans that this tax plan is feasible in period 1. The government can use some of thesurplus of the budget to increase the supply of publicly provided private goods andput remainder in savings. Let a1 be the balance of the government savings per eachyoung individual at the end of period 1where a1 > a∗ . What will be the net budget
surplus in period 2, SP2 ?
Note that the natives and immigrants of cohort 1 will save the same as the natives
and immigrants at the initial steady state, because they face the same after-tax wage
rate and interest rate under the proposed tax policy as at the initial steady state.
This implies that s1 = s∗ for both groups of cohort 1. Assume that at the period2,
the government will save at least a∗ per the number of the young. Thus, SP2 becomes
SP2 =

�

j=n,m

N2j ×(w2 τw2 −g y −g ind −a∗ )+

�

j=n,m

N1j ×(r2 τr2 s∗ −b∗ −g o −g ind )+(1+r2 )a1

�

N1j .

j=n,m

(41)

Note that the pre-tax wage for period 2, w2 , and the pre-tax interest rate for period
2, r2 , are equal tow2 ≡ FL (l∗ (N2n +N2m ), (N1n +N1m )(s∗ +a1 ) and r2 ≡ FK (l∗ (N2n +N2m ), (N1n +N1m )(s∗ +a1 ))−δ(42)
Again the government sets τw2 and τr2 such that after-tax wage rate and after-tax
interest rate become the same as at the initial steady state. Thus, SP2 becomes (seeappendix B3)
SP2 = N1n (1 + α�)+ (1 + α�)N1n� s∗ +a1s∗ +a∗� α�

[FK (l∗ N2n (1 + α�), zN1n (1 + α�)) + 1 − δ]dz

R� (α) [FL (l∗ R(z), (s∗ + a∗ ))l∗α∗�−{cy∗ + s∗ + g y + g ind + a∗ } dz

(43)

The first term of (43) measures the welfare gain that arises from the additional savings
that the government accumulates at the end of period 1. The second term measures

22the welfare gain that arises from the increased population growth rate in the presenceof the PYGO social security. From the MPK condition (??) and the conation on the
population growth rate the inside of the first integration is positive. From (34), theinside of the second integration is positive. Thus, SP2 is positive and the government
can implement the proposed tax policy. Again, at the end of period 2, the government
can use some of the above surplus to increase the supply of publicly provided private
goods and put the rest to increase the balance of government savings. Similarly, thegovernment surplus for period t becomesn
SPt = Nt−1
(1 + α�)n+ (1 + α�)Nt−1� s∗ +at−1s∗ +a∗� α�

[FK (l∗ R(�α), z) + 1 − δ]dz

R� (α)[FL (l∗ R(z), (s∗ + a∗ ))l∗α∗

− {cy∗ + s∗ + g y + g ind + a∗ }]dz

(44)where at is the balance of government saving per each young individual at the end of
period t − 1. Again the government uses some of the surplus for increasing publicly
supplied private goods and puts the rest in savings. This implies that SPt > 0 for allt = 1, 2, .... Thus, we have the following proposition:
Proposition 2. Consider an economy in which the wage and interest taxes areused at the initial steady state. If the MPL condition is satisfied at the initial steadystate, accepting more immigrants with tax rule (30) and social security benefit rule(31) Pareto-improves the welfare of all generations.
3.2.1

Government Savings and the Golden RuleIn this subsection, I examine the capital stock path and government saving pathwhen the government increases the INR. To examine the government saving path, I
need to specify how much of the government surplus, SPt , is put into the additional
government savings. For the analysis, I assume that the surplus that arises from theincreased government savings at period t − 1, which is the first integration of SPt ,23

is put into the additional government savings that is added to a∗ at period t. Notethat the government could use some part of the second integration in SPt , the surplus
that is generated directly from the increased immigration. Thus, my assumption
is a conservative value for the government savings. I will show that, even with thisconservative level, the economy reaches the golden rule level of capital stock per capita
within a finite time in a Pareto-improving way. Note that the total number of youngn
individuals at the period t is Nt−1
R(�α)(1 + α
�). Thus, the government savings pereach young individual at the end of period t for t ≥ 2, at , becomes Q
N n (1 + α�)at − a = n t−1
Nt−1 R(�α)(1 + α�)∗� s∗ +at−1s∗ +a∗[FK (l∗ R(�α), z) + 1 − δ]dz

≡ Q(at−1 )

(45)
(46)

Note that from (37), we haveN n (1 + α�)at − a ≥ n t−1
Nt−1 R(�α)(1 + α�)∗� s∗ +at−1

R(�
α)dz

s∗ +a∗= at−1 − a∗Thus, at is increasing over time as long as at is determined according to equation (45)and (37). Now, consider the graph of at = Q(at−1 ) + a∗ where at is measured on thevertical axis and at−1 is measured on the horizontal axis. Q(at−1 ) + a∗ is equal toa∗ at at−1 = a∗ and Q(at−1 ) + a∗ is increasing. It is also concave due to the diminishingmarginal product of capital. The slope of Q(at−1 ) + a∗ at at−1 = a∗ isQ� (a∗ ) =1(FK (l∗ R(�α), s∗ + a∗ ) + 1 − δ)
R(�
α)

(47)

Because of assumption (36), Q� (at−1 ) at at−1 = a∗ is strictly greater than 1. On theother hand, due to the diminishing marginal product of capital, Q� (at−1 ) becomesclose to zero as at−1 becomes bigger. Thus, the at = Q(at−1 ) + a∗ and 45 degree line
intersect at at−1 = a∗ and at a∗∗∗ where a∗∗∗ > a∗ . Let a∗∗ be the point where Q� (a∗∗ )24

= 1. This implies that at a∗∗∂F (l∗ R(�α), s∗ + a∗∗ )+ 1 − δ = R(�α)∂K
In other words, at a∗∗ the golden rule is satisfied. Note that the government canchoose a1 so that a1 > a∗ because the surplus at period 1 is strictly positive. From thegraph of at = Q(at−1 ) + a∗ , at keeps increasing starting from a small a1 > a∗ . Before
it reaches a∗∗∗ , it reaches a∗∗ within a finite time. This implies that the economy
reaches the golden rule level of capital stock per capita within a finite time.
Note first that in this analysis, I assume that only the first integration of SPt ,i.e. the surplus that arises from the increased government savings at period t − 1, is
put into savings at period t. However, the second integration of SPt , i.e. the surplus
which arises directly from increased immigrants, also can be put into savings. Thus,
the government can shorten the time to reach the golden rule level by putting thesurplus that arises directly from increased immigrants. Secondly, we can note that the
government can induce the economy to reach the modified golden rule level within afinite time in a Pareto-improving way, because the modified golden rule level is lower
than the golden rule level.
Proposition 3. (Reaching (modified) Golden Rule Level) Suppose that a
PYGO social security system is used initially and that the MPL condition is satisfied.
Also assume that the capital stock per capita at the initial steady state is suﬃciently
lower than the golden rule level. Then, by accepting more immigrants and using
the proposed tax rule, social security benefit rule and government saving policy, thegovernment can induce the economy to reach the (modified) golden rule level withinin a finite time in a Pareto-improving way.

3.3

Intra-redistributional Channel and Diﬀerence of Productivities and Preferences

When immigrants earn less than natives or immigrants consume more publicly provided private goods than natives, accepting more immigrants could decrease the wel25

fare of natives, because it means that more resources are taken from natives and used
for immigrants. This is the intra-redistributional channel of accepting more immigrants. A similar redistribution could also happen when the preferences of immigrants
and natives are diﬀerent and the labor supply or savings of immigrants diﬀers fromthose of natives. To analyze this redistributional channel of accepting more immigrants, consider again a permanent change of immigration policy such that α
� > α∗ .

Let (cy,j∗, ,co,j∗ , sj∗ , lj∗ ), be the consumption at the young period, the consumption
at the old period, savings and labor supply of type j(j = n, m), respectively at theinitial steady state.Since I assume that the preferences and the productivities of thenative and immigrant are diﬀerent in this section, I put superscript j even for thevariables at the steady-state situation. Also, let g i,j be the publicly provided private
goods for goods type i and type j nationality where i = y, o, ind and j = n, m. For
conditions that guarantee that accepting more immigrants Pareto-improves welfare,I assume that the following condition is satisfied:
Modified MPL condition=R� (α)[+J � (α)[+(∂F n∗l − (cy,n∗ + g y,n + g ind,n + sn∗ + a∗ )]∂L∂F m m∗φ l − (cy,m∗ + g y,m + g ind,m + sm∗ + a∗ )]∂L∂F+ (1 − δ))(sm∗ + a∗ ) − (co,m∗ + g o,m + g ind,m ) > 0
∂K

where

for all α ∈ [α∗ , α�] :n∂F (R(α)ln + φm J(α)lm , sn∗ Nt−1
+ αsm∗ )
∂F
=
,
∂L
∂L
∂F
∂F (R(α)ln + φm J(α)lm , sn∗ + αsm∗ )=∂K∂K

and J(α) = (1 + πn + α(1 + πm ))α(48)

In the first line and the second line, the inside of the bracket is the amount of the
upward intergenerational transfer made by a native and an immigrant, respectively.
Thus, the sum of the first line and the second line is essentially the same as the MPL
condition in the preceding section. The third line captures the intra-redistribution
from the natives to immigrants. The third line measures the amount that an old

26

immigrant contribute to the economy though savings minus the resource used for an
old immigrant. When the productivities and preferences of immigrants and natives
are the same, the third line becomes equal to the inside of the first line, the amountof the upward intergenerational transfer. Thus, the above modified MPL condition
states that the total upward intergenerational transfer is large enough to oﬀset the
intra-redistribution from an old immigrant to an old native. From (48), we have thefollowing propositions.12
Proposition 4 Assume that natives and immigrants do not have same productivities and preferences. If the Modified MPL condition is satisfied, it is possible toPareto-improve the welfare of all generations by accepting more immigrants. For the
proof, see Appendix B4.
Proposition 5 If the Modified MPL condition is satisfied, the government can
induce the economy to reach the (modified) golden rule level within a finite time in a
Pareto-improving way by accepting more immigrants.

4

Quantifying the Welfare Gain of Accepting More
Immigrants in the Presence of a PYGO Social
Security System

Propositions 1 and Proposition 2 above show that accepting more immigrants canPareto-improve the welfare of all generations if there are intergenerational transfersin the sense that the marginal product of labor of a young individual is greater thanwhat he or she receives including publicly provided private goods while he or she isyoung (the MPL condition). Furthermore, Proposition 3 shows that if the government
can put some of the welfare gain into savings, it is possible to induce the economy toreach the golden rule level of capital in a Pareto-improving way within a finite time.
This result is in sharp contrast to findings in the literature on social security reformwhich shows that some generation must bear a double burden to increase the capital
12

In Appendix B5, I show that the modified MPL condition becomes the MPL condition when
immigrants and natives have the same producitvies and preferences.

27

stock of the economy in the presence of a PYGO social security system (Geanakoplos,
Mitchell and Zeldes 1998). In addition, the proposition 4 and 5 show that even if
an immigrant earns less than a native, accepting more immigrants Pareto-improves
welfare when the intra-redistribution is not as large as the intergenerational transferfrom the young to the old.
A number of issues arise as a result of accepting proposition 1-5. First, those
propositions are based on a two-period overlapping generation model. In a realistic
multi-period overlapping generation model, it might not be possible for an economy
to reach the (modified) golden rule level of capital stock per capita within a finite
time in a Pareto-improving way by accepting more immigrants. Second, althoughproposition 3 shows that the economy reaches the (modified) golden rule level ofcapital stock per capita within a finite time, in practice it might take a long time, aslong as 1000 years, to reach the golden rule level. Third, propositions 1-5 are silent
on the quantitative eﬀect on welfare. Given that it might take a quite long time toreach the golden rule level, the welfare gain of accepting more immigrants can be verysmall. In addition, propositions 4 and 5 say nothing about the degree to which thediﬀerence in productivity between immigrants and natives is allowed for with respect
to Pareto improvement when more immigrants are accepted.
This section addresses these issues. To address these issues, I use the computational overlapping generation model developed by Auerbach and Kotlikoﬀ (1987). I
assume that the model economy consists of overlapping generations in which eachgeneration lives for 80 periods and that the probability of death increases with eachpassing period. I assume that the model economy is similar to the US economy inseveral dimensions.13 In the analysis, I assume that initially the model economy ison the balanced growth path and the percentage of immigrants to natives (PITN),
which is defined as 100 times the number of immigrants divided by the number ofnatives, is similar to that which the US 2000 census data indicates. Then, I will ex13

However, the model economy is diﬀerent in several important dimensions as well. For example,
the model economy does not include a certain aspect of open economy such as international tradeand capital mobility. The model economy does not incorporate the accumulation of human capitalof the natives and immigrants.

28

amine whether it is possible to Pareto-improve the welfare of all the generations in themodel economy by increasing the PITN by a reasonable size. In addition, I examine
how long it takes for the model economy to reach the (modified) golden-rule level ina Pareto-improving way and quantify the Pareto-improving welfare gain. To checkthe robustness, I recalculate the model by changing the value of the following parameters: the replacement, initial government debt (asset) level, the level of earnings ofimmigrants, immigrants’ consumption of publicly provided private goods, probabilityof immigrants returning to the country of origin, the CRRA and the time preference
rate.

4.1

The Model Economy: Auerbach and Kotlikoﬀ Modelwith Immigration

Agents will show up in the model from age 1. I assume that age 1 corresponds to age
20 in real life. They work from age 1 until age 45. From the beginning of age 46, they
retire. At each age, they die with some probability and they can live until age 80. Let
i be the index of age. For i ≥ 2, let pi be the probability that an agent is alive at theage i, given that he or she is alive until age i − 1. Due to the lack of data, I assume
that pi is the same for natives and immigrants. To simplify the notation, I assume
that p1 = 1.14 An agent who enters the model at period t maximizes the following
utility function:

max

45
�
i=1

βi

i
�
q=1

pq

�

� 80
�
�
i
i,j
i,j
ζ
1−ζ 1−γ
ζ(1−γ)
��[c][(ci,j
)
(1
−
l
)
]
ij
ij
t−1+i
t−1+i
t−1+i
+ gt−1+i+
β i pq+ gt+1−i
1−γ
1
−
γ
q=1
i=46
(49)

i,j
where ci,jt−1+i and lt+1−i are the amounts of private consumption and labor supply ofitype j agent of age i at period t − 1 + i. gt+1−i
is the amount of publicly provided
ij
private goods to age i agent in period t + 1 − i. I assume that the amount of gt+1−iischosen by the government. In this formulation, as in Storesletten (2000), I postulate
14

Infant and child mortality is defined in (55).

29that the immigrants assume they will stay in the host country until the end of theirlives.15 Let si,jt be the savings of type j of age i at time t. The budget constraint ofan agent at age i is
i,j
i,j
i,j
si−1,j× lt−1+i= ci,jt−2+i (1 + rt−1+i (1 − τr,t−1+i )) + (1 − τw,t−1+i )wt−1+i H
t−1+i + st−1+i for 1 ≤ i ≤ 45

(50)
i,j
i,j
i,j
si−1,jt−2+i, (1 + rt−1+i (1 − τr,t−1+i )) + bt−1+i = ct−1+i + st−1+i for 46 ≤ i ≤ 80i,jsi,jt−1+i = 0 for i = 0 and st−1+i ≥ 0 for 1 ≤ i ≤ 80

(51)
(52)

where H i,j is the eﬃcient unit of human capital of type j at age i and H i,j > 0 for
1 ≤ i ≤ 45 and H i,j = 0 for i ≥ 46. wt is the wage rate for one eﬃcient unit of labor atperiod t. I assume that an individual cannot have a negative savings balance. Oncean individual dies, the government imposes a 100 percent inheritance tax. bi,jt−1+i isthe social security benefit for type j of age i that is given at period t − 1 + i. Fori,ji ≤ 45, bi,jt−1+i = 0 and for i ≥ 46, bt−1+i is determined as follows:jjbi,jt−1+i = 12 × RR × AIM E (t) and AIM E (t) =

�45

i,j
wt−1+i lt−1+i
H i,j
45 × 12i=1 (1 + µ)45−iwhere RR is the replacement and AIM E j (t) is the average income monthly index of
the cohort who become age 1 at time t.
For the production function, I assume that the economy’s aggregate production
can be described by the Cobb-Douglas production function:
Yt = Ktθ (Et Lt )1−θ and µ = (Et+1 − Et )/Et

(53)

where θ is the capital share and Et represents the level of technology. µ is income percapita growth rate. Lt is the eﬃcient unit of labor supply at period t.

15

Let 1 − p�i be the probability that an immigrant returns to her/his home country

Alternatively, we can assume that immigrants will enjoy the same wage level in their homecountry as in the host country even if they return to their home countries. The two assumptions
generate the same results.

30

at the beginning of age i, given that she or he stays in the host country at the agei − 1 for i ≥ 2. To simplify the notation, I assume that p�1 = 1. Let Ntij be the numberof agent of age i of type j at time t. Then, Ntim = pi p�i Nti−1,m and Ntin = pi Nti−1,n . Lt
is defined as follows:

Lt =

45 �
�

H i,j Nti,j lti,j(54)

i=1 j=n,m

I assume that all immigrants will arrive in the host country at age 1. Let σn be
the growth rate of the number of natives of age 1 at the steady state and let σm be
the growth rate of the number of immigrants of age 1 at the steady state. The total
�1,n �80in−(i−1)
number of natives is 80
× Πiq=1 pq . The total number
i=1 Nt = Nt
i=1 (1 + σn )�−(i−1)of immigrants is Nt1,m 80
× Πiq=1 pq p�q . This implies that the ratio of
i=1 (1 + σm )the total number of immigrants to the total number of natives is constant if and only

if the growth rates of Nt1,n and Nt1,m are the same. Thus, for calculating the steadystate, I assume that σn = σm = σ. I assume that the children of immigrants become
native if their parents stay in the host country until the children attain adulthood.
Then, Nt1,n is determined by the fertility rates of both immigrants and natives andthe return rate of immigrants. Let ηji,j be the age-specific fertility of type j at age i.
The number of children born of type j parents of age i is Nti,j η i,,j . Let d be the infant1,nchild mortality. Then the number of natives of age 1 at t + 20, Nt+20, is determined

as follows:
1,n
Nt+20
= (1 − d) ×

� 80
�
i=1

η i,m Nti,m Π20
�i+x +
x=1 p

80
�
i=1

η i,n Nti,n

�

(55)

The percentage of immigrants to natives (PITN) at the steady state (see Appendix
A6) becomes
�
�80
1
1
i,n
i,m
− 80
× Πiq=1 pq
N
i=1 (1+σ)i+19 η
1−d
t
i=1
�80
i,n × 100 = �80
1i,m × Πi�q Π20�i+x
q=1 pq px=1 pi=1 Nti=1 (1+σ)i+19 η�80−(i−1)
× Πiq=1 pq p�qi=1 (1 + σ)
× �80
× 100
−(i−1) × Πi
q=1 pqi=1 (1 + σ)
31

(56)

(56) says that the steady-state PITN is determined once the age 1 population growthrate, σ, the return rate of immigrants and the fertility rates of natives and immigrants
are set. Conversely, we can choose σ so that the resulting PITN is consistent withthe data16 .
The capital stock at period t is the sum of the balance of individual savings andgovernment savings. Let at−1 be the balance of the government asset (or debt if it
is negative ) per capita at the end of period t − 1. Then, the total capital stock at
period t is
Kt =

80 �
�

i,j
Nt−1,isi,jt−1 + at−1

i=1 j=n,m

80 �
�

i,j
Nt−1

(57)

i=1 j=n,m

The (eﬃcient unit) wage rate at time t, wt and the pre-tax interest rate at time t,
rt , are determined as
wt = (1 − θ)Ktθ Et1−θ L−θtandrt = θKtθ−1 Et1−θ L1−θt

(58)

Now, consider the initial balanced growth path where the capital to labor ratio(in eﬃcient units) stays constant. Let w∗ (1 + µ)t and r∗ be the wage rate and interest
rate at period t on the initial balanced growth path. Let s∗i,j (1 + µ)t and b∗i,j (1 + µ)t
be the savings and the social security benefit of type j of age i at period t on theinitial balanced growth path. Let a∗ (1 + µ)t and g ∗i,j (1 + µ)t be the government
asset(debt) per capita and the publicly provided private goods for type j of age i atperiod t on the initial balanced growth path. It is possible that immigrants use more
public services, for example, children’s education. Thus, I put superscript j. Let w∗
and r∗ be the eﬃciency unit wage rate and the interest rate on the initial balanced
growth path determined from (57) and (58). Then the government budget constraint
16

The RHS is an increasing function of σ at the parameter values that are consistent with the US.
data.

32

on the initial balanced growth path is(1 + µ)t−1

�τwt w (1 + µ)Lt + τrt r−a∗ (1 + µ)
−

80
�

∗

∗

80 �
�

i−1,j ∗i−1,jpi Nt−1s+ (1 + r∗ )i=2 j=n,m

80 �
�

�

Nti,j + (1 + rt )a∗i=1 j=n,mi=46 j=n,mb∗i,j (1 + µ) × Nti,j − κ

i=2 j=n,m80 ��

i=1 j=n,m

80 �
45
�

i=46 x=11

80 �
�

i,j
Nt−1
−

80 �
�

i−1,j ∗i−1,j(1 − pi )Nt−1
st−1

i,j ∗i,jNt,ig (1 + µ)i=1 j=n,mb∗i,rm,x (1 + µ) × Nti,rm,x

�

=0

(59)

where τw∗ and τr∗ are the wage tax rate and capital income tax rate at the initialbalanced growth path. b∗i,rm,x and Nti,rm,x are, respectively, the social security benefit
and the number of immigrants of age i who returned to the home country at age x, but
who are still eligible to claim social security benefits. κ is the share of those returned
immigrants who actually claim social security benefits. The US social security does
not require the residence as long as the individual is eligible for benefits. I assume
that an immigrant is eligible to receive social security benefits as long as he or she
pays social security contributions for at least 10 years. The first term of (59) is therevenue from wage tax and the second term is the revenue from capital income tax.
The third term is the revenue from inheritance tax. In my simulation, I assume thatthe government imposes a 100 percent inheritance tax. The fourth term is the revenue
from issuing government bonds and the fifth term is the expenditure on the principal
and interest of the bonds. The sixth and seventh terms are the expenditure on the
publicly provided private goods and the expenditure on social security benefits ofnatives and immigrants who reside in the host country. The last term is the social
security benefits of immigrants who returned to the home country and who are eligible
to receive social security benefit.
When the government increases the PITN, the wage rate decreases and the interest
rate increases. To Pareto-improve welfare, I assume that the wage tax rate, interesttax rate and social security benefit are adjusted as in the theoretical analysis. Morespecifically, I assume that the government keeps the after-tax wage rate and interest
rate at the same level as on the initial balance growth path. This implies that the33

wage tax rate τwt and interest tax rate τkt are set as follows:
wt (1 − τwt ) = w∗ (1 + µ)t × (1 − τw∗ ) and rt (1 − τrt ) = r∗ (1 − τr∗ )(60)

The social security benefit is adjusted as follows:jjbi,jt−1+i = 12×RR×AIM E (t) and AIM E (t) =

�45

i,j
wt−1+i (1 − τwt )lt−1+i
H i,j(1 − τw∗ ) × 45 × 12(61)

i=1 (1 + µ)

45−iWhen the taxes and social security benefit formula are adjusted according to (60)and (61), the individual budget constraints is the same as on the initial balanced
growth path17 . Thus, consumption, labor supply and savings do not change. Then,
there would be a surplus for the government budget even if the government were tospend the same amount of publicly provided private goods per person as at the initialbalanced growth path, as analyzed in the previous section. The government can use
this surplus to increase savings or to increase the level of publicly provided private
goods. Let V be the distributional parameter that indicates what percentage of thebudget surplus is put into savings. The government can keep increasing the government savings balance until the economy reaches the golden rule level or a modified
golden rule level. As long as the MPK is greater or equal to the (modified) golden
rule level, the balance of the government savings at period t is determined from thefollowing equation:t(1 + µ) at

80 ��

i=1 j=n,mNti,j = V × SPt

where SPt is the government budget surplus at period t. If the MPK is equal to the17

Note that when wt (1 − τwt ) = wt∗ (1 − τw∗ ), AIM E j (t) =

34

�45

i=1 (1+µ)

45−i

i,j
w∗ (1+µ)t−1+i lt−1+i
H i,j
45×12

(modified) golden rule level, V becomes 0. SPt is defined as follows:

SPt = τwt wt Lt + (1 + µ)t−1

(1 + rt ) × at−1
−

80
�

�

i=46 j=n,m

�τrt rt80 ��i−1,j ∗i−1,jpi Nt−1s+ (1 + rt )i=1 j=n,m80 ��

i=1 j=n,mi=1 j=n,m

i,j
Nt−1
−

b∗i,j (1 + µ) × Nti,j − κ

80 �
�

80 �
�

i=1 j=n,m

80 �
45
�

i=46 x=11i−1,j ∗i−1,j(1 − pi )Nt−1s

Nti,j × g ∗,i,j (1 + µ)b∗i,rm,x (1 + µ) × Nti,rm,x

�

(62)

The rest of the surplus that is available for increasing publicly provided private
goods is (1 − V ) × SPt . This is distributed equally to the entire population (including
immigrants) at period t.18 Then, after increasing the PITN, the amount of publicly
provided private goods, gti , becomes(1 − V ) × SPt
gti = g ∗i,j (1 + µ)t + �80i,ni,mi=1 (Nt + Nt )(63)

For the (modified) golden rule, given the intergenerational discount rate for themodified golden rule, we say that the economy reaches the (modified) golden rule level
if the marginal product of capital becomes equal to the sum of the growth rate of theeﬃciency unit of labor, the depreciation rate, and the inter generational discount rate
for the modified golden rule. In the case of the golden rule, the intergenerational
discount rate for the modified golden rule is set to zero.

4.2

Policy Experiments and Parameters Values for Simulation

Policy Experiments
For the percentage of immigrants to natives (PITN), I calculate the data from the
US census at 2000. US Census 2000 shows that the PITN above the age of 20 is
15.5 percent. Thus, I assume that the PITN at the initial balanced growth path is18

The government could distribute this surplus more to the natives than to the immigrants. Thus,
this assumption underestimates the welfare gain to the natives.

35

15.5 percent. For the target PITN, I look at past US data. Historically, the PITN
was 5 percent in 1970 and increased to 18.3 percent in 2010.19 The PITN increased
more than 10 percent point within 40 years from 1970 to 2010, so I assume that a
10 percentage point increase in the PITN over 80 years is tolerable. Therefore, I set
the target PITN at 25.5 percent. In the slow benchmark case, the PITN starts toincrease from 15.5 percent and reaches the target PITN at the 80th year. After the80th year, it remains on the same level. To see how accelerating the speed at which
the PITN is increased aﬀects my results, I also consider intermediate and fast cases,in which the PITN hits the target PITN at the 62nd or 42nd year.20 I assume that
all immigrants arrive at age 1. The graph for the PITN over time is shown in Figure
1.21
Fertility, Mortality and Return Rate of Immigrants and Other Parameter
ValuesParameter values regarding fertility, mortality, return rate of immigrant, government
expenditure, government debt, taxes, preferences and production function are quite
standard. To save the space, I discuss those parameter values in Appendix A2.
19

Data in 1970 is from the US Census. Data in 2010 is from the CPS. After 2000, data on theforeign-born are available only through the American Community Survey and the CPS, not the
Census. For calculating the PITN, one concern is the comparability of the CPS and the Census
data. In 2000 when both the census and the CPS are available, the PITN in the Census is 12.42%
and 11.54% in the CPS. Thus, the diﬀerence between the Census and CPS would seem to be very
minor. This view is shared by the Census bureau. Schmidley and Robinson (2003) conclude thatthe diﬀerence of the estimates based on the Census data and CPS data are trivial.
20
In the fast and intermediate cases, to allow smooth transition of the PITN, the PITN keeps
increasing even after hitting the target PITN. See Figure 1.
21
More specifically, each of the three cases is calculated as follows. Let f15.5 be the steady-stateage1 immigrant-native ratio when the PITN is 15.5 percent. Define f25.5 in a similar fashion. Then,the age 1 immigrant-native ratio at period t, ft , in the slow case is ft = f25.5 for all t. For thetintermediate case, ft = f25.5 + 0.1 × f25.5 × ( 15
) for 1 ≤ t ≤ 15 , ft = 1.1 × f25.5 − 0.1 × f25.5 × (t − 15)tfor 16 ≤ t ≤ 30 and ft = f25.5 for 31 ≤ t. For the fast case, ft = f25.5 + 0.4 × f25.5 × ( 15) for1 ≤ t ≤ 15 , ft = 1.4 × f25.5 − 0.4 × f25.5 × (t − 15) for 16 ≤ t ≤ 30 and ft = f25.5 for 31 ≤ t.
Note that the steady-state age 1 immigrant-native ratio is defined in the first term of the RHS inequation (40). Thus, once the PITN is determined, the steady state age 1 immigrant-native ratio is
calculated from equation (40).

36

4.3

Results

Figures A3, A4 and A5 in the appendices show the balance of assets, consumption
and leisure for the life cycle of an individual at the initial balanced growth path in thebenchmark analysis. At the age of 46, the consumption of leisure becomes 1 due tomandatory retirement. At the initial balanced growth path, the capital to output ratiois 2.98, which is higher than the valued used in Storesletten (2.4) and Nishiyama andSmetter (2.7), but lower than the value used in the standard business cycle research,which is 3.2 (Cooly and Prescott(1995)). To check how my results are aﬀected by thecapital to output ratio, I change the time preference rate and examine how the resultschange for diﬀerent values of the time preference rate in the robustness checks.
Table 1 shows the parameter values and the welfare eﬀect of increasing the PITN
for diﬀerent values of V in equation (62), which is the share of the government surplus
that is put into savings. As mentioned above, the initial PITN is set at 15.5 percent
and the target INR at 25.5 percent. With respect to rate at which immigrants are
accepted, I consider three cases in which it takes 80, 62 and 42 years for the PITN
to reach the target PITN. Column (3) of Table 1 shows the share of the government
surplus that is put into savings. Columns (4)-(9) are values that are calculated within
the simulation. Column (4) shows how many years it takes for the economy to reachthe (modified) golden rule level. When it does not reach the (modified) golden rule
within 300 years, this is indicated by * or **. * indicates that the capital-to-labor ratio(eﬃcient unit) at the 300th year is higher than at the initial balanced growth path,that it is increasing at the 300th year, but that it does not reach the (modified) golden
rule level at the 300th year. ** indicates that the capital-to-labor ratio at the 300th
year is lower than at the initial balanced growth path. For example, when V = 100%,
it takes 112 years to reach the golden rule level in the slow acceptance case(row (16)).
Column (5) shows how much the capital stock per eﬃcient unit of labor increases at
the golden rule compared with the level at the initial balanced growth path. WhenV = 100%, it shows that the capital stock per eﬃcient unit of labor increases by
102 percent. Column (6) shows how much the publicly provided private goods will
increase compared with the level at the initial balanced growth path. At V = 100%,37

column (6) shows that the publicly provided private goods will increase by about
36 percent. To calculate columns (5) and (6), I evaluate at the year at which theeconomy reached the golden rule level if it reaches that level within 300 years, andat the 300th year otherwise. Column (7) shows the percentage by which the utility,
measured by the expenditure function, of the cohort that is born when the economy
reaches the golden rule level increases compared with the utility of the same cohort
if it were on the initial balanced growth path. When the economy does not reach thegolden rule level, I calculate the utility of the cohort that is born at the 300th year.
In the expenditure function, the price vector on the initial balanced growth pathis used for evaluating the utility. Note that all welfare gain is distributed throughan increase in publicly provided private goods. Column (8) shows the share of thepresent discounted value (PDV) of the increased publicly provided private goods inthe initial GDP. For example, in row (16), the PDV of the increased publicly provided
private goods is 12 percent of the initial GDP. For discounting the increased publicly
provided private goods in future periods, I apply a 5 percent discount rate instead ofthe equilibrium interest rate, so that I can eliminate the eﬀect of the discount rate
when comparing diﬀerent cases. Column (9) measures the welfare gain of natives
and their descendants, not that of immigrants or their descendants. It measures how
much the utility of natives and their descendants, not including immigrants and theirdescendants, is Pareto-improved by accepting more immigrants compared with theutility level at the initial balanced growth path. It evaluates the utility by using theexpenditure function. I apply the price vector at the initial balanced growth pathfor the expenditure function and the equivalent variation to measure the diﬀerenceof the levels of utility in two cases. To discount the welfare gain of future cohorts, I
again use a 5 percent discount rate. Note that in all cases in Table 1, I assume thatthe government does not discriminate between immigrants and natives with respect
to publicly provided private goods. Both natives and immigrant consume publicly
provided private goods equally. Column (9) in row (16) shows that the PDV of thePareto-improving welfare gain of natives and their descendants comprises 11 percentof the initial GDP.
38

In rows (2)-(3), (5)-(6), and (8)-(9)..., I shorten the years needed to reach the target
PITN and increase the speed with which the PITN increases. When the number ofyears needed to reach the target PITN is reduced to 42, instead of 80, and V=100,
the PDV of the welfare gain of increasing the PITN comprises 13 percent of the initial
GDP (row (18)).
Table 1 shows that the number of years taken to reach the golden rule level decreases as V increases, because the government saves more for future cohorts. On
the other hand, the PDV of the increased utility, measured as the share of the initial
GDP, increases as V decreases as long as the V is greater than or equal to 50%. The
PDV of the increased utility, measured as the share of the initial GDP, is highest
when V=50 %. In this case, the quantified Pareto improvement ranges from 21% to
26% of the initial GDP.
In Table 1, I set the intergenerational discount rate for the modified golden rule
at 0% and set the target level of the capital stock at the golden rule level. However,
the targeting capital stock at the golden rule level does not necessarily maximize the
PDV of the welfare gain. In Table A2, I examine the eﬀect on welfare of increasing
the PITN for diﬀerent target levels of capital stock by changing the value of theintergenerational discount rate for the modified golden rule when V=100%. Table A2
shows that the welfare gain is maximized when the intergenerational discount rate is
set at 3 percent.
In Table 2, which I consider as the representative case of my simulation, I recalculate the all rows of Table 1 by setting the intergenerational discount rate for themodified golden rule at 3 percent. In Table 2, the Pareto-improving welfare gain of
increasing the PITN is more than 20 percent of the initial GDP and the capital stock
per eﬃcient unit of labor increases by 18 percent as long as V is greater than or equal
to 50 percent.
Figure 2 and Figure A6 show the marginal product of capital and the capital to
output ratio over time for diﬀerent values of V when the target capital stock is themodified golden rule level (3% intergenerational discount rate). The marginal productof capital increases initially, due to the acceptance of more immigrants. However, as
39

the government savings balance increases, the capital stock per eﬃcient unit of labor
starts to increases and the marginal product of capital continues to decrease until the
economy reaches the golden rule level. The capital to output ratio also displays asimilar pattern.
Figure 3 compares the utility level of all cohorts on the initial balanced growth pathwith the utility level of all cohorts for diﬀerent values of the share for the government
savings (V) when the target capital stock is set at the golden rule level. For example,
when V=100%, all the surplus is put into savings until the economy reaches the golden
rule level and the surplus is distributed to individuals only after the economy reaches
the golden rule level. This implies that the utility of the 65th cohort, which dies at
the 65th year, starts to experience higher utility than at the initial balanced growthpath.
In all cases considered in Tables 1,2,3 and 4, the results of simulations show that
all cohorts are Pareto-improved; this confirms my theoretical results.22
Figure 4 shows the importance of government savings balance at the new equilibrium path. It calculates the ratio between the interest income from the government
savings balance and the social security benefit payment in each period. When theeconomy reaches the golden rule level, the interest income from the government savings balance comprises 70 percent of the social security benefit payments. Thus, atthe new equilibrium path, the interest income from the government savings balanceis contributing substantial amount.
22

The results of this simulation diﬀer from those of some of the previous studies. For example,
Kotlikoﬀ et.al reports that the welfare gain of doubling immigrants is very small. Lee and Miller
argues that the fiscal impact of accepting additional 100,000 immigrants is very small. Several
factors in those studies generate diﬀerent results. Kotlikoﬀ et.al analyze immigration policy in an
open economy setting, whereas I use a closed economy setting. In an open economy setting, theeﬀect of additional government savings is oﬀset by the mobility of capital. In my simulation, thegovernment can use the budget surplus, which is obtained by accepting more immigrants, for savingsand the amount of savings aﬀects the over all welfare. Lee and Miller consider a much smaller increasein the number of immigrants than I do. On the other hand, my simulation results are consistent
with Auerbach Oreopoulos (2000). They find that if the initial fiscal imbalance is not adjusted, then
the welfare loss of halving the number of new immigrants is substantial. This is consistent with mytheoretical and simulation results.

40

4.4

Robustness Checks

Table 3 and 4 present results of robustness checks of Table 2. Rows (1)-(6) in Table 3
check whether the results in Table 2 are sensitive to the initial government debt (asset)
level. As I argued above, diﬀerent authors assume diﬀerent levels for the government
debt (asset) at the initial balanced growth path. In rows (1)-(3) in Table 5, I set theinitial government debt to 10 % of the private capital instead of 0 %. In rows (4)-(6),I assume that the initial government debt level is -10% of the private capital.
Rows (7)-(12) in Table 3 check whether the results in Table 2 are sensitive tothe replacement rate. The theoretical analysis implies that higher intergenerational
redistribution will result in the acceptance of more immigrants yielding a higher welfare gain. Thus, it is predicted that as the replacement rate decreases, so does thewelfare gain of accepting more immigrants. Rows (7)-(12) confirms this prediction.
Decreasing the replacement rate 10 percentage points decreases the welfare gain by 7
percentage point in terms of the percentage of the initial GDP.
Rows (13)-(21) in Table 3 conduct sensitivity checks on the immigrants’ level of
earnings. Following Storesletten (1995), I set the immigrant wage rate at 84.3 percentof that of natives. In my calculation using the CPS 2000 June supplement, I foundthat immigrants’ earning are 91 percent of natives’ earnings. Rows (13)-(15) assume
that the wage rate of immigrants is 89.3 percent of that of natives, rather than 84.3
percent. Rows (16)-(18) assume that the wage rate of immigrants is 79.3 percent ofthat of natives. Rows (19)-(21) assume that the wage rate of the immigrants is 75
percent of that of natives. The results in rows (19)-(21) show that if immigrants earn
25 percent lower than natives, the welfare gain of accepting more immigrant is 17
percent of the initial GDP, instead of 23 percent of the initial GDP.
Rows (22)-(27) in Table 3 conduct sensitivity checks on the consumption by immigrants of publicly provided private goods. In Table 1 and 2, I assumes that immigrants
and native consume publicly provided private goods equally as the previous empirical studies indicate. In rows (22)-(24), I assume that young immigrants consume 20
percent more publicly provided private goods than young natives. The PDV of thewelfare gain comprises 17 percent of the initial GDP instead of 23 percent. Rows
41

(25)-(27) assume that immigrants of all ages consume 20 percent more publicly provided private goods than natives. In this case, the PDV of the welfare gain comprises
15 percent of the initial GDP.
In Table 4, I conduct robustness checks by changing the values of parametersof the utility function and the return rate of immigrants. Rows (1)-(6) of Table
4 examine the sensitivity of the results regarding the values of the parameters forconstant relative risk aversion (CRRA). In Table 1-3, I assume that the CRRA isequal to 3. Auerbach and Kotlikoﬀ, and Storesletten assume that the CRRA=4,
while Nishiyama and Smetter assume that the CRRA is 2. Rows (1)-(3) assume that
the CRRA is equal to 4 and rows (5)-(6) assume that it is 2. The results presented
on rows (1)-(6) show that the results presented in Table 2 is not sensitive to thevalue of CRRA. Rows (7)-(12) examine the sensitivity of the results to the time
preference rate. Although it is quite common to assume that the time preferences
rate is greater than 1, readers might think that the results presented in Table 1 and 2
can be sensitive with respect to the assumption that the time preference rate is greaterthan 1. Theoretically, lowering the time preference rate will result in lowering savings
and will decrease the capital to labor ratio at the initial balanced growth path. Since
increasing the number of immigrants will increase the capital stock and the production
function exhibits diminishing marginal product of capital, lowering the initial capitalstock as a result of assuming a lower time preference rate will increase the welfare
gain of increasing the PITN. Rows (7)-(12) confirm this theoretical prediction, butthey show that the magnitude of those changes is very small. For example, changing
the time preference rate from 1.011 to 0.99 increase the PDV of the welfare gain by
only 1 percentage point.
Rows (13)-(15) in Table 4 conduct robustness checks by assuming that the return
rate of immigrants is equal to 0. Rows (13)-(15) show that the results presented inTable 2 do not change so much and that they are robust regarding the return rate ofimmigrants.

42

4.5

Summary of the Simulation Results

The simulation results presented in Tables 1-4 can be summarized as follows. In thecase of slow acceptance of more immigrants, in which the PITN increases from 15.5
percent and reaches 25.5 percent at the 80th year, the PDV of the increased utilityof the natives and their descendants, measured by the expenditure function, is 23
percent of the initial GDP when the target capital stock is the modified golden rulelevel (3 % intergenerational discount rate). The economy reaches the modified golden
rule level at 65th year and the capital stock per eﬃcient unit of labor increases by 18
percent. In the fast acceptance case, in which the PITN reaches 25.5 percent at the42th year, the PDV of the welfare gain comprises 28 percent of the initial GDP and
the economy reaches the modified golden rule level at 59th year.
For robustness checks, I conduct sensitivity checks for changing the values of thefollowing parameters: initial government debt (asset) level, the replacement rate,level of earnings of immigrants, immigrants’ consumption of publicly provided private
goods, the CRRA, the time preference rate and probability of the return to homecountry by immigrants. The sensitivity checks show that the results are quite robust.

5

Conclusion

I have examined, both theoretically and quantitatively, the eﬀect on welfare of increasing the PITN in the presence of a PYGO social security system. The results
for both the theoretical quantitative analysis show that the welfare gain of accepting
more immigrants is robust and non-trivial. The PDV of the welfare gain of increasing
the PITN from 15.5 to 25.5 percent amounts to about 20 percent of the initial GDP.
In the shortest case, the economy reaches the golden rule level at the 112th year in a
Pareto-improving way. The analysis suggests that accepting more immigrants can be
an important tool for policy makers when addressing the economic problems caused
by a PYGO social security system.

43

References[1] Akbari, Ather H. ” The Benefits of Immigrant to Canada:Evidence on Tax ad
Public Service,” Canadian Public Policy Vol 15, pp 424-435, 1989
[2] Auerbach, Alan and Kotlikoﬀ, Laurence, 1987, Dynamic Fiscal Policy, Cambridge University Press, Cambridge, 1987
[3] Auerbach Alan J. and Oreopoulos, Philip ”Analyzing the Fiscal Impact of US
Immigration”, American Economic Review paper and Proceeding 89, pp 170-180,
1999[4] Diamond, Peter A., ”National Debt in a Neoclassical Growth Model”, American
Economic Review, 1965, vol 55 pp 1126-1150[5] Borjas Georege J. ”The Economics of Immigration.’ Journal of Economic Literature vol 32, pp 1667-1717, 1997
[6] Center for Disease Control and Prevention, Vital Statistics of the United States,
1993 (http://www.cdc.gov/nchs/products/vsus.htm)[7] Canovaa, Fabio and Morten O. Ravn, ”The Macroeconomic Eﬀects of German
Unification: Real Adjustments and the Welfare State”, Review of Economic Dynamics, 2000, vol3, pp 423 460[8] Collado, M. Dolores, INIgiturbe-Ormaetxeand and Guadalupe Valera, ”Quantifying the Impact of Immigration on the Spanish Welfare State”, International
Tax and Public Finance, vol11, 335–353, 2004
[9] Cooley, Thomas and Edward Presecott, ”Economic Growth and Business Cycle”,
Chapter 1 in Frontier of Business Cycle Research, Cooley, Thomas (ed.), 1995,
Princeton University Press[10] Feldstein, Martin S. The Eﬀects of the Ageing European Population on Economic
Growth and Budgets: Implications for Immigration and Other Policies, 2006,
NBER Working Paper No. 12736
44[11] Michael E. Fix, Jeﬀrey S. Passel, Wendy Zimmermann, ”The Use of
SSI and Other Welfare Programs by Immigrants”,Testimony before theHouse of Representatives Ways and Means Committee, May 23, 1996,(http://www.urban.org/publications/900287.html[12] A. Dianne Schmidley and J. Gregory Robinson, ”Measuring The Foreign-born
Population in the United States with the Current Population Survey: 1994-2002,
Poulation Division Working Paper No73, Bureau of the Census[13] Nishiyama, Shinichi and Kent Smetters, . ”Does Social Security Privatization
Produce Eﬃciency Gains?,” The Quarterly Journal of Economics, vol. 122(4),pp 1677-1719, November, 2007.
[14] Fehr, Hans, Sabine Jokisch and Laurence J. Kotlikoﬀ, ”The Role of Immigrationin Dealing with the Developed World’s Demographic Transition,, FinancArchiv,
2004.
[15] Geanakoplos, John, Olivia S. Mitchell and Stephen P. Zeldes ”Would a PrivatizedSocial Security System Really Pay a Higher Rate of Return?” (with ). In R.D.
Arnold, M. Graetz and A. Munnell (eds.), Framing the Social Security Debate.
Values, Economics, and Politics. National Academy of Social Insurance, 1998,
pp. 137-157.
[16] Hurd, Michael D. ”Mortality Risk and Bequests.” Econometrica, vol 57 July,
pp779-813, 1989.
[17] Michel, Ph. and Pierre Pestieau, ”When Does Serendipity Hold?”, Journal of
Population Economics, Vol. 6, No. 4 Dec, pp. 353-362, 1999[18] Ramsey, Frank P. ”A Mathmatical Theory of Saving”, Economic Journal,38, No
152, pp543-559.
[19] Razin, Assaf, ”Migration and Pension with Internatioanl Capital Mobility”, Journal of Public Economics, vol 74(1),pp 141-15, 1999,45

[20] Razin, Assaf and Efraim Sadka ”Unskilled Migration: A Burden or a Boon for
the Welfare State? ”, Scandinavian Journal of Economics,2000, pp463-479[21] Razin, Assaf and Efraim Sadka and Benjjarong Suwankiri, Migraiton and the
Welfare State Political-Economy Policy Formulation,The MIT Press, Cambidge
MA, 2011[22] Passel, Jeﬀrey S. ” Immigrants and Taxes: A Reappraisal of Huddle’s The costof Immigrants ” Manuscript Washington Urban Institute, 1994[23] Shimasawa, Manabu and Kazumasa Oguro, ”Impact of immigration on theJapanese economy: A multi-country simulation model”, Journal of Japanese andInternational Economics, 2010, pp 5806-602.
[24] Simon, Jullian L. ” Immigrants, Taxes and Welfare in the United States”, Population and Development Review vol 10, pp 55-69, 1984[25] Storesletten, Kjetil, On the Economics of Immigration, Ph.D Dissertation,
Carnegie Mellon University, 1995
[26] Storesletten, Kjetil, ”Sustaining Fiscal Policy through Immigration,” Journal of
Political Economy, vol 108, pp 300-323, 2000,[27] Storesletten, Kjetil, ”Fiscal Implications of Immigration— A Net Present Value
Calculation,” Scandinavian Journal of Economics, vol105(3), pp 487-506, 2003,

46

Figure 1: PITN over time. The initial PITN is set at 15.5% and the target PITN is set at25.5%.

Figure 2: The marginal product of capital over time for diﬀerent values of the share of thesurplus used for government savings (V). The target capital stock is the modified golden
rule level with 3 % intergenerational discount rate.

47

2200
iniital balanced growth path
V=100%
V=50%

2000
1800

Utility level

1600
1400
1200
1000
800
600
400
200

0

20

40

60

80

100

120

140

cohortFigure 3: Utility level of diﬀerent cohorts for diﬀerent values of the share of the surplus
used for government savings (V). The target capital stock level is the modified golden rulelevel with 3% intergenerational discount rate for the modified golden rule.

0.8

interest income/ss benefit payment

0.7
0.6

V=100%
V=90%
V=70%
V=50%
V=30%

0.5
0.4
0.3
0.2
0.1
0
−0.1

0

50

100

150
year

200

250

300

Figure 4: The ratio between the interest income from the government savings balance
and the social security benefit payment in each period. It is assumed that the PITN
reaches the target PITN at 80th year. The inter-generational discount rate for themodified golden rule is set at 3%.

48

(1)

Row
No.

(2)

(3)

Table 1The effect of increasing the PITN(The target capital stock is the golden rule level)
(4)
(5)
(6)
(7)

(8)

(9)

share of theshare of the% increase
sum of thesum of PDV%change ofyears
of capital% change of
PDV ofof welfare
share of theyear
publiclyneeded tostock perwelfare ofincreased
gain of all
surplus for reachingprovidedreach theefficientcohort born
publicly
natives andthe gov. the golden
private goods
targetunit laborat the golden
providedtheirsavings (V)rule
per capita at the
PITN
at therule
private goods descendants ingolden rule
golden rulein the initial
the initial
GDP
GDP

1
2
3

80
62
42

0%
0%
0%

4
5
6

80
62
42

30%
30%
30%

7
8
9

80
62
42

50%
50%
50%

10
11
12

80
62
42

70%
70%
70%

13
14
15

80
62
42

90%
90%
90%

16
17
18
Notes

80
62
42

100%
100%
100%

share of the surplus for the gov. savings =0%300**-4.37%
5.00%
0.53%
300**
-4.37%
5.00%
0.53%
300**
-4.37%
5.00%
0.53%
share of the surplus for the gov. savings =30%300*
11.89%
14.69%
1.41%
300*
12.00%
14.76%
1.42%
300*
73.43%
30.82%
2.88%
share of the surplus for the gov. savings =50%300*
71.21%
30.27%
2.79%
300*
71.82%
30.42%
2.81%
300*
73.43%
30.82%
2.88%
share of the surplus for the gov. savings =70%
184
102.43%35.86%
3.77%
182
102.43%35.86%
3.77%
119
50.99%
30.14%
3.17%
share of the surplus for the gov. savings =90%127102.43%35.86%
3.77%
125
102.43%35.86%
3.77%
120
102.43%35.86%
3.77%
share of the surplus for the gov. savings =100%112
102.43%35.86%
3.77%
110
102.43%35.86%
3.77%
106
102.43%35.86%
3.77%

15.57%
16.74%
20.20%

17.72%
18.94%
22.47%

17.86%
19.20%
25.32%

19.59%
20.94%
26.09%

19.66%
21.10%
25.32%

20.79%
22.18%
26.09%

18.48%
19.75%
25.87%

18.92%
20.09%
25.56%

14.52%
15.47%
18.21%

13.75%
14.56%
16.80%

12.23%
13.04%
15.38%

10.69%
11.33%
13.14%

1. In all rows, the initial PITN is 15.5% and target PITN is 25.5%. The replacement rate is 60 %, CRRA=3 and thetime preference rate is 1.011. The equilibrium capital to output ratio on the initial balanced growth path is 2.98.
2. In all rows, wage rate of immigrants is 84.3 % of that of natives.
3. * indicates that the capital stock per efficient unit labor does not reach the golden rule level within 300 years. Itsvalue at the 300th year is higher than at the initial balanced growth path and keeps increasing at the 300th year. The
percent change of the capital stock per efficient unit of labor is evaluated at the 300th year.
4. ** indicates that capital stock per efficient unit labor does not reach the golden rule level within 300 years and thecapital stock per efficient unit labor at the 300th year is lower than at the initial balanced growth path. The percent
change of the capital stock per efficient unit of labor is evaluated at 300th year.

49

(1)

Row
No.

(2)

(3)

Table 2The effect of increasing the PITN for different values of V
(The target capital stock is the modified golden rule level)
(4)
(5)
(6)
(7)

(8)

(9)

% increase
share of theshare of theof the%change of
PV of
PDV ofyears% change ofshare of theyear
capitalpublicly
increased
welfare gain
needed towelfare of
surplus for reaching stock perprovidedpubliclyof all natives
reach thecohort born
the gov. the golden efficientprivate goods
provided
and theirtargetat the golden
savings (V)rule
unit labor per capita at theprivate goods descendants in
PITN
rule
at thegolden rulein the initial
the initial
golden rule
GDP
GDP

1
2
3

80
62
42

0%
0%
0%

4
5
6

80
62
42

30%
30%
30%

7
8
9

80
62
42

50%
50%
50%

10
11
12

80
62
42

70%
70%
70%

13
14
15

80
62
42

90%
90%
90%

16
17
18
Notes

80
62
42

100%
100%
100%

share of the surplus for the gov. savings =0%300**-4.37%
5.00%
0.53%
300**
-4.37%
5.00%
0.53%
300**
-4.37%
5.00%
0.53%
share of the surplus for the gov. savings =30%300*
11.89%
14.69%
1.41%
300*
12.00%
14.76%
1.42%
300*
12.32%
14.96%
1.45%
share of the surplus for the gov. savings =50%
127
18.19%
18.51%
1.95%
124
18.19%
18.51%
1.95%
116
18.19%
18.51%
1.95%
share of the surplus for the gov. savings =70%87
18.19%
18.51%
1.95%
78
18.19%
18.51%
1.95%
78
18.19%
18.51%
1.95%
share of the surplus for the gov. savings =90%70
18.19%
18.51%
1.95%
68
18.19%
18.51%
1.95%
63
18.19%
18.51%
1.95%
share of the surplus for the gov. savings =100%65
18.19%
18.51%
1.95%
63
18.19%
18.51%
1.95%
59
18.19%
18.51%
1.95%

15.57%
16.74%
20.20%

17.72%
18.94%
22.47%

17.86%
19.20%
23.18%

19.59%
20.94%
24.80%

19.86%
21.31%
25.58%

21.06%
22.47%
26.45%

21.31%
27.15%
27.15%

22.11%
27.49%
27.49%

22.32%
23.81%
28.14%

22.85%
24.24%
28.13%

22.71%
24.19%
28.51%

23.14%
24.51%
28.37%

1. In all rows, the initial PITN is 15.5% and target PITN is 25.5%. The replacement rate is 60 %, CRRA=3 and thetime preference rate is 1.011. The equilibrium capital to output ratio on the initial balanced growth path is 2.98.
2. In all rows, wage rate of immigrants is 84.3 % of that of natives.
3. The intergenerational discount rate for the modifed golden rule is 3%. At the modified golden rule level, themarginal product of capital is equal to the sum of the growth rate of efficient unit of labor, the depreciation rate and theintergenerational discount rate for the modified golden rule.
3. * indicates that the capital stock per efficient unit labor does not reach the modified golden rule level within 300
years. Tts value at the 300th year is higher than at the initial balanced growth path and keeps increasing at the 300th
year. The percent change of capital stock per efficient unit of labor is evaluated at the 300th year.
4. ** indicates that the capital stock per efficient unit labor does not reach the modified golden rule level within 300
years and the capital stock per efficient unit labor at the 300th year is lower than at the initial balanced growth path.
The percent change of the capital stock per efficient unit of labor is evaluated at the 300th year.

50

Table 3
Robustness checks
The role of the initial government debt level, the replacement rate, immigrants earnings and the use of public
services by immigrants(The target capital stock is the modified golden rule level)
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
%change% increase% change
share of theof publicly
of capitalshare of the sum
capital toof welfare
years taken
sum of PDV
providedstock perof the PDV ofyears
outputof theto reachof increased
privateefficientwelfare gain ofneeded to ratio at thecohortthepubliclyRow
goods perunit labor
all natives andreach theinitialborn at themodified
providedNo.
capita at
at the new
theirtargetbalancednewgoldrenprivate goods
the newbalanced
descendants in
PITN
growthbalanced
rule levelin the initialbalanced
growthinitial GDP
path
growthGDP
growth
path
path
path
initial government debt ratio (% of private capital)= 10 %
1
80
2.91
63
22.39%
21.87%
2.29%
28.48%
27.22%
262
2.91
61
22.39%
21.87%
2.29%
30.31%
28.82%
342
2.91
57
22.39%
21.87%
2.29%
35.66%
33.30%
initial government debt ratio(% of private capital)= -10 %
4
80
3.04
67
14.71%
15.75%
1.67%
18.30%
19.68%
562
3.04
65
14.71%
15.75%
1.67%
19.51%
20.87%
6
42
3.04
61
14.71%
15.75%
1.67%
23.01%
24.16%
replacement rate =0.55
7
80
3.03
68
14.87%
15.79%
1.67%
18.05%
19.32%
8
62
3.03
66
14.87%
15.79%
1.67%
19.24%
20.48%
942
3.03
61
14.87%
15.79%
1.67%
22.67%
23.70%
replacement rate =0.5
10
80
3.1
71
11.51%
13.07%
1.39%
13.68%
15.38%
11
62
3.1
69
11.51%
13.07%
1.39%
14.572%16.30%
1242
3.1
65
11.51%
13.07%
1.39%
17.159%18.84%
level of human capital of immigrants is 89.3 % of that of natives
13
80
2.98
62
18.20%
18.96%
2.01%
25.22%
25.93%
14
62
2.98
60
18.20%
18.96%
2.01%
26.871%27.48%
15
42
2.98
56
18.20%
18.96%
2.01%
31.702%31.86%
level of human capital of immigrants is 79.3 % of that of natives
16
80
2.98
69
18.18%
18.05%
1.89%
20.09%
20.24%
17
62
2.98
67
18.18%
18.05%
1.89%
21.392%21.43%
18
42
2.98
62
18.18%
18.05%
1.89%
25.165%24.74%
level of human capital of immigrants is 74.3 % of that of natives
19
80
2.98
73
18.18%
17.58%
1.83%
17.32%
17.20%
20
62
2.98
71
18.18%
17.58%
1.83%
18.430%18.20%
21
42
2.98
67
18.18%
17.58%
1.83%
21.629%20.96%
young adult immigrants consume publicly provided private goods 20% higher than young adlut natives
22
80
2.97
74
18.46%
17.70%
1.90%
17.27%
17.01%
23
62
2.97
72
18.46%
17.70%
1.90%
18.343%17.96%
24
42
2.97
68
18.46%
17.70%
1.90%
21.409%20.56%
immigrants of all ages consume publicly provided private goods 20% higher than natives
25
80
2.96
77
18.84%
16.89%
1.86%
15.71%
15.25%
26
62
2.96
75
18.84%
16.89%
1.86%
16.674%16.09%
27
42
2.96
71
18.84%
16.89%
1.86%
19.410%18.36%
Notes
1. In all rows, V is 100 percent and the inter-generational social discount factor for the modified golden rule is set
at 3%. At the modified golden rule level, the marginal product of capital is equal to the sum of the growth rate ofefficient unit of labor, the depreciation rate and the inter-generational discount rate for the modified golden rule.
2. In all rows, the initial PITN is 15.5% and the target PITN is 25.5%. CRRA=3 and the time preference
4. The wage rate of immigrants is 84.3 % of that of natives in all rows except in rows (13) to (18).
5. Immigrants consume the same amount of publicly provided private goods as natives in rows (22) to (27).
6. The replacement is 0.6 in all rows except rows (7) to (12).

51Table 4
Robustness checks (2)The role of the CRRA, the time preference rate and the return rate of immigrant
(1)

Row
No.

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15

(The target capital stock is the modified golden rule level)
(4)
(5)
(6)
(7)
(8)
(9)
% increase %changeshare of theof theof publicly % change
sum of the share of sum ofcapital toyears taken capitalprovided of welfare
PDV ofthe PDV of
outputto reachof cohort
years
stock perprivateincreased
welfare gain ofratio at thetheneeded toefficient goods per born at the
publicly
all natives andinitialmodified unit labor capita at
newreach theprovidedtheirbalancedgoldren at the new the newbalancedtarget INR
private goods descendants ingrowthrule level balanced balanced
growthin the initial
initial GDP
path
path
growth
growth
GDP
path
path
CRRA=480
2.634
74
41.87% 33.40%
3.28%
32.25%
22.18%
62
2.634
72
41.87%
33.40%
3.28%
34.04%
23.26%
422.634
69
41.87%
33.40%
3.28%
39.10%
26.15%
CRRA=280
3.44
63
11.52%
12.70%
1.47%
16.26%
25.82%
62
3.44
61
11.52%
12.70%
1.47%
17.52%
27.70%
42
3.44
56
11.52%
12.70%
1.47%
21.32%
33.23%
the time prefernece rate=1
80
2.67
72
39.41%
32.00%
3.17%
32.35%
23.15%
62
2.67
71
39.41%
32.00%
3.17%
34.184%24.31%
42
2.67
67
39.41%
32.00%
3.17%
39.396%27.43%
the time preference rate=0.99
80
2.42
74
61.43%
46.18%
4.38%
43.96%
23.52%
62
2.42
73
61.43%
46.18%
4.38%
46.130%24.51%
42
2.42
70
61.43%
46.18%
4.38%
52.171%27.09%
the retrun rate of immigrant is 0%
80
2.97
63
17.04%
17.86%
1.87%
24.00%
24.25%
62
2.97
61
17.04%
17.86%
1.87%
25.56%
25.68%
42
2.97
61
14.87%
15.79%
1.67%
22.67%
23.70%
(2)

(3)

Notes
1. In all rows, V is 100 percent. The parameter values of the wage rate of immigrants, the replacement rate, and theconsumpiton of publcly provided private goods by immigrants are the same as in Table 2.
2. In all rows, the initial PITN is 15.5% and the target PITN is 25.5%.
3. The inter-generational discount rate for the modifed golden rule is 3% except in rows (4) to (6). In rows (4) to(6), the economy's capital stock is already above the modified golden rule level with a 3% inter-generational
discount rate. In rows (4) to (6), I set the inter-generational discount rate at 2% instead of 3%. At the modified
golden rule level, the marginal product of capital is equal to the sum of the growth rate of efficient unit of labor, thedepreciation rate and the inter-generational discount rate for the modified golden rule.

52

Appendices A (The following appendices are put on
the Journal’s webpage as supplement material, but
not included in the main paper. )Appendix A1Consider a simple case in which the depreciation rate is 100% (δ = 1) and immigration
policy α is implemented at the initial steady state. Assume that the population of
the old is equal to one. At period 0, the following resource constraint must hold:F (R(α∗ )l∗ , s∗ + a∗ ) − (co∗ + g o + g ind ) − R(α∗ )(cy∗ + s∗ + a∗ + g y + g ind ) = 0. (64)Note that F (R(α∗ )l∗ , s∗ + a∗ ) is the GDP per one old individual when one plus the
population growth rate is equal to R(α∗ ). Now consider the graph of (y, R(α)) wherethe vertical axis measures the GDP per one old individual and the horizontal axis
measures one plus the population growth rate which is R(α). This implies thaty = F (R(α)l∗ , s∗ + a∗ ). Next, draw another graph defined by (y, R(α)) where y =(cy∗ +s∗ +a∗ +g y +g ind )R(α). This is the line that passes through the origin and whose
slope is cy∗ + s∗ + a∗ + g y + g ind . At R(α) = R(α∗ ), the vertical distance of this line
represents the total amount of resources used for the young divided by the numberof old individuals when one plus the population growth rate is equal to R(α∗ ). Thus,
the diﬀerence between y = F (R(α)l∗ , s∗ + a∗ ) and y = (cy∗ + s∗ + a∗ + g y + g ind )R(α)
at R(α) = R(α∗ ) represents the amount of resources used for one old individual at
the initial steady state.
Now suppose that a social planner increases the population growth rate by accepting more immigrants permanently. This implies that R(α) increases by R� (α∗ )from R(α∗ ). If the slope of y = F (R(α), s∗ + a∗ ) at the R(α) = R(α∗ ) is greaterthan cy∗ + s∗ + a∗ + g y + g ind , the social planner can maintain the same allocationof resources per each young individual (private consumption, savings, government
provided private goods and age-independent public goods) and increase the alloca-

1tion of resources to each old individual. Clearly this constitutes Pareto improvement.
Note that when the government accepts immigrants there is a surplus that is equal toR� (α∗ ){l∗ FL ((R(α)l∗ , s∗ + a∗ ) − (cy∗ + s∗ + a∗ + g y + g ind )} in every period(see Figure
A1).

Appendix A2
Fertility, Mortality and Return Rate of Immigrants
For the age-nativity specific fertility, I use the CPS 2000 June supplement, which thecensus bureau also uses for calculating on the age-nativity specific fertility. Figure A2
in the appendices shows the average of the total number of births by each woman’s
age and it shows that immigrant women have a greater total number of births thannative women at all ages. From this figure, I calculate η i,n and η i,m , the age-specificbirth rate for native and immigrant women. This is shown in Table A1. For the adultmortality profile, pi , I take the values from Nishiyama and Smetter (2007). I set d,the sum of infant and child mortality, 1.7 percent, using the Vital Statistics of the US
for 1993. For the return rate of immigrants to their home countries, I use the oﬃcial
census estimate that was conducted by Ahmed and Robinson (1994). They estimatethat for the first 10 years, second 10 years and the third 10 years, the return rate ofimmigrants is 19 percent, 9 percent and 7 percent, respectively. For annual rate, thosenumbers correspond to 2.085 percent, 0.938 percent and 0.723 percent, respectively.
Thus, I set p�i = 0.9715 for 2 ≤ i < 10, p�i = 0.99062 for 11 ≤ i ≤ 20, p�i = 0.9927 for
21 ≤ i ≤ 30 and p�i = 1 for i ≥ 31.

Once the age-nativity specific fertility rate, infant-child mortality and the initial

PITN are set, then the annual population growth rate is calculated automaticallyaccording to equation (56) with the assumption that the PITN is at the steady state.
With the estimated age-nativity specific fertility, the infant-child mortality and an
initial PITN of 15.5 percent, the annual growth rate of the population of age 1 becomes
0.39 percent.23 This implies that at the initial steady state, the government accepts
23

The annual CPS data on immigrants and natives from 1995 to 2010 shows that the medianannual growth rate of the total population(sum of natives and immigrants) aged from 20 to 40 is2

immigrants such that the annual growth rate of immigrant of age 1 becomes 0.39
percent.
Age-nativity Specific Government Expenditure
I assume that the age-specific government expenditure, g ∗i,j is the same for natives
and immigrants. Empirical studies show that there is no systematic diﬀerence in theuse of public services by the two groups.24 Thus, I assume that for 1 ≤ i ≤ 24,g ∗i,j = g y , for 25 ≤ i ≤ 44, g ∗i,j = gm and for 45 ≤ i ≤ 80, g ∗i,j = go . I assume thatg y , g m and g o are 24.5%,13.4% and 23.2 % of GDP per capita at the initial balanced

growth path following Storesletten (1995) and Auerbach, Kotlikoﬀ Hagememann andNicoletti (1989). On the other hand, to check the robustness of my results, I also
assume that immigrants consume 20 percent more publicly provided private goods
than natives in the robustness checks.
Utility Function, Production Function and Human Capital Profile
A precise estimate for the coeﬃcient for relative risk aversion, γ, has not yet beenfound in the literature. Auerbach and Kotlikoﬀ (1986) and Storesletten (2000) assumed that γ is 4. Nishiyama and Smetter (2007) set γ equal to 2. I assume thatγ = 3 and check the robustness with γ = 2 and γ = 4. For the time preference rate,β, following Hurd (1989) and Storesletten (2000), I assume that β = 1.011. Higher β
implies higher savings and a higher capital to output ratio. To check the sensitivityof my results, I also calculate with β = 1 and β = 0.99. I assume that the leisureshare in the utility function, ζ, is 0.33.
0.12% while the median growth rate of immigrants aged from 20 to 40 is 1.85%. In the theoreticalmodel, the model assumes that the age 1 population growth rate of immigrants and natives are
the same. Thus, the theoretically predicted growth rate of age 1 population, 0.39%, is between thegrowth rate of immigrants aged 20-40 and the growth rate of the total population aged 20-40.
24
Borjas and Hilton (1996) show that immigrants have a higher participation rate in welfare programs than natives. Fix, Passel, and Zimmermann (1996) show that these diﬀerences are explainedby the higher participation rate in welfare programs among refugees and retired immigrants andthat there is no diﬀerences among labor immigrants. Thus, for the theoretical part, I assume thatg i is independent of the place of birth. For the computational part, I relax this assumption for therobustness checks.

3

I assume that the depreciation rate of capital, δ, is equal to 0.047. For the capitalshare in the production function, θ, I set θ = 0.4. For technological progress, I assume
that income per capita growth rate, µ, is 0.015.
For the human capital profile of natives, His , I take the value from Auerbach andKotlikoﬀ.
H i,n = exp(4.47 + 0.033 × i − 0.00067 × i2 )H i,n = 0

for 1 ≤ i ≤ 45

(65)

for 46 ≤ i(66)

For the human capital profile of the immigrants, Storesletten (1995) showed that
immigrants’ earnings are, on average, 15.7 percent lower than those of natives.25 Similarly, using the CPS June 2000 supplement, my calculations show that immigrants’earnings are 10 percent lower than those of natives. Using these estimates as a basis,
I assume that the eﬃcient unit of human capital of immigrants is 84.3 percent of thatof natives and that H i,m = 0.843 × Hin in the benchmark calculation. To examine
the robustness of my results, I change the level of human capital from 84.3 percent to89.3 percent, 79.3 percent or 74.3 percent and re-check the results.
Taxes and Government Debt
For capital income tax, I take the value from Nishiyama and Smetter (2007) andassume that τk = 0.28. For the level of social security benefit, a higher replacement
rate means that greater intergenerational redistribution of income, which leads togreater welfare gain as a result of increasing the immigrant population. Following
Auerbach and Kotlikoﬀ, in the benchmark case I set the replacement 0.6 and checkthe robustness of my results by varying it from 0.6 to 0.55 and 0.5.
As for the initial level of government debt or assets, diﬀerent authors set diﬀerent
levels. Storesletten considered only government debt and assumed that the initiallevel is 50 percent of the initial GDP. With his estimate of the initial capital to output
25

Figure 2.2 of Storesletten (1995) shows that at age 20, 25, 30, 35, 40, 45, the wage rate ofimmigrants is lower than that of natives by 15%, 20%, 17.8%, 16.4%, 12% and 13% respectively. By
averaging those rates, I obtained a working value of 15.7%.

4

ratio being 2.4, his assumption implies that government debt is about 20 percent ofprivate capital. Nishiyama and Smetter considered not only government debt, but also
government assets by using the BEA information on the government’s fixed capital.
They assumed that at the initial steady state, the government has positive net assetsof 10 percent of the total private capital. This naturally leads to a higher capitalto output ratio at the initial balanced growth path. Herein, as a benchmark case,I assume that initial level of government debt or assets is 0 percent of the private
capital and I experiment with the assumption that the government debt is 10 percent
or minus 10 percent of the private capital.

5

Figure A1– The one plus population growth rate and the resources used for one old person.
The curve through the origin is y = F (R(α), s∗ + a∗ )and the straight line is y = (cy∗ +s∗ + g y + g ind + a∗ )R(α). The vertical distance of the straight line through the origin
measures the total resources used for young divided the number of old people. The diﬀerence
between the curved line and the straight line measures the resource used per one old personat R(α). On the above graph, increasing R(α) will increase the resources available per
one old person without decreasing the resources used for the young.

6

Total number of birth by person’s age
Total number of birth by person’s age
.5
1
1.5
2
2.5

CPS 2000 June

20

25

30
person’s age
Foreign born

35

40

NativeFigure A2–Total number of births by each age. The data source is the CPS 2000 supplement.
The total number of births by each age is regressed on a sixth-order polynomial function ofage separately for natives and immigrants. The predicted values are plotted.

7

400
350

saving balance

300
250
200
150
100
50
0

0

10

20

30

40
age

50

60

70

80

Figure A3–Balance of asset over the life cycle at the initial balanced growth path

8

6500
6000
5500

Consumption

5000
4500
4000
3500
3000
2500
2000
1500

0

10

20

30

40
Age

50

60

70

80

Figure A4–Life cycle consumption path at the initial balanced growth path.

9

1
0.95
0.9

Leisure

0.85
0.8
0.75
0.7
0.65

0

10

20

30

40
age

50

60

70

80

Figure A5–Life cycle consumption of leisure at the initial balanced growth path.

10Figure A6–Capital to output ratio over time for diﬀerent values of the share of the surplus
used for government savings (V). In those simulations, the number of years needed for the
PITN to reach the target level is set at 80. The target capital stock is the modified golden
rule level. The intergenerational discount rate for the modified golden rule is set at 3%.

11

Table A1 Number of birth at each age for native andimmigrnt in the simulation
age
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23-80

number of birth ofimmigrant

number of birth ofnative0.1934279
0.0520106
0.0550358
0.0569373
0.0578227
0.0577998
0.0569765
0.0554604
0.0533594
0.0507811
0.0478333
0.044624
0.0412605
0.0378511
0.0345031
0.0313245
0.0284231
0.0259064
0.0238824
0.0224589
0.0217435
0.0218439
00.1743087
0.0474751
0.0505587
0.0527624
0.0541410.0547487
0.0546402
0.0538698
0.052492
0.0505614
0.0481323
0.0452592
0.0419967
0.0383992
0.034521
0.0304168
0.026141
0.0217481
0.0172925
0.0128286
0.0084112
0.0040944
0Note
1. The calculaton is based on Figure 3. Let TB(i,j) be thevertical axis of group j of Figure 3 where j is native or
immigrant. Then, the number of births of age i of group jin the model is calcluated as follows. When i=1, thenumber of births of age i of group j in the model is
TB(20,j)/2. When 2<=i<=22, the number of births is(TB(19+i,j)-TB(18+i,j))/2.

12

(1)

Row No.

1
2
3
4
5
6
7
8
9

Table A2
The effect of increasing the PITN for different inter-generational discount rates
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
% increase %changeshare of share of theinterof theof publicly % change the PDV of PDV of
years
year
generation
capitalprovided of welfare increased
welfareneeded toreaching
al discountof thestock perprivatepublicly gain of allreach thetherate for thecohortprovided natives andtargetmodified efficient goods per
modifedborn
at the private
their
PITN
golden rule unit labor capita at
golden ruleat thethe golden golden rule goods in descendants
80
80
80
80
80
80
80
80
80

0.0%
0.5%
1.0%
1.5%
2.0%
2.5%
3.0%
3.5%
4.0%

112
103
95
87
80
73
65
56
45

102.43%82.39%
65.46%
50.99%
38.52%
27.69%
18.19%
9.82%
2.40%

35.86%
35.07%
33.04%
30.14%
26.63%
22.71%
18.51%
14.12%
9.64%

3.77%
3.69%
3.48%
3.17%
2.80%
2.39%
1.95%
1.49%
1.01%

12.23%
15.39%
18.10%
20.26%
21.80%
22.64%
22.71%
21.86%
19.80%

10.69%
13.87%
16.75%
19.22%
21.17%
22.51%
23.14%
22.89%
21.43%

㻺㼛㼠㼑
1.At the modified golden rule level, the marginal product of capital is equal to the sum of the growth rate ofefficient unit of labor, the depreciation rate and the inter-generational discount rate for the modified golden
rule.

13

Appendices B(The following appendices are for the
purpose of refereeing. )Appendix B1
Notice that in the programming problem, the objective function is concave and theconstrained set is convex. Thus, if some allocation satisfies the first-order condition, it
is also the solution of the programming problem. Now set up the Lagrangian function
as follows:1[uo (co1 ) + v o (g o , q)]
1+ρ
∞
�
+
γt {uy (cyt , lt ) + v y (g y , g ind ) +
L=

t=1

+

∞
�
t=1

1
[uo (cot+1 ) + v o (g o , g ind ) − u∗ }1+ρλt {F (R(α∗ )lt , (st−1 + at−1 ) + (1 − δ)(st−1 + at−1 )− (cot + g o + g ind ) − R(α∗ ) × (cyt + st + at + g y + g ind ) }where a0 = a∗

(67)

The first order conditions are:11uo� (co1 ) = λ1 ; cot+1 : γtuo� (cot+1 ) = λt+1 ;
1+ρ
1+ρ
∂uy (cyt , lt )∂uy (cyt , lt )∂Fy∗ct : γ t= λt R(α ); lt : γt= λtR(α∗ )
y
∂ct
∂lt
∂L
1
γt : uy (cyt , lt ) + v y (g y , g ind ) +[uo (cot+1 ) + v o (g o , g ind ) − u∗ = 0;
1+ρ
�
�
∂F
st , at : λt+1
+ 1 − δ = λt R(α∗ )
∂K
co1 :

(68)

λt : F (R(α∗ ), st−1 + at−1 ) + (1 − δ)(st−1 + at−1 )−(cot + g o + q) − R(α∗ )(cyt + st + at + g o + q) = 0

1

(69)

Those first order conditions imply that�� � y y�∂uy (cyt , lt )∂u (ct , lt )
∂F
/
=
y
∂lt
∂ct
∂L
� y y
� �
�
∂F
∂u (ct , lt )1o� o+1−δ =/u (ct+1 )
∂K
∂cyt
1+ρ
On the other hand, at the initial steady state, the initial steady state allocation,(cy∗ , co∗ , s∗ , l∗ , a∗ ) satisfy the following allocation:�� � y y∗ ∗ �∂uy (cy∗ , l∗ )∂u (c , l )∂F (R(α∗ )l∗ , s∗ + a∗ )
∗
−
/
=
w
=
∂l
∂cy
∂L
� y y∗ ∗ � �
�
∗ ∗ ∗
∗
∂F (R(α )l , s + a )∂u (c , l )
1
∗
o� o∗
+1−δ =1+r =/u (c )
∂K
∂cyt
1+ρ
1
uy (cy∗ , l∗ ) + v y (g y , g ind ) +[uo (co∗ ) + v o (g o , g ind ) = u∗
1+ρ∗F (R(α∗ )l∗ , s∗ + a∗ ) + (1 − δ)(s∗ + a∗ ) = R(α∗ )(cy + s∗ + a∗ + g o + g ind ) + (co∗ + g o + g
Now, we set cyt , cot+1 , lt , st , at , λt , γt as follows

cot = co∗ ; cyt = cy∗ ; st = s∗ ; lt = l∗ ; at = a∗ ; λ1 =λt+1 = λt

1uo� (co∗ )1+ρ

R(α∗ )1;γuo� (co∗ ) = λt+1t
1 + r∗ 1 + ρ

If cyt , cot+1 , st , at , λt , γt are set in this way, it clearly satisfies the first-order conditions of
the programming problem. Thus, the initial steady state allocation is Pareto-eﬃcient.
Q.E.D.

Appendix B2Using the definitions of τwt and τrt , we havewt τwt = wt − (1 − τw∗ )w∗
rt τrt = rt − (1 − τr∗ )r∗2

Then, the government budget surplus at period 1 isSP1 = (w1 − (1 − τw∗ )w∗ )l∗− (b∗ + g o + g ind )

�

j=n,m

�

j=n,m

N1j + (r1 − (1 − τr∗ )r∗ )s∗N0j − (g y + g ind + a∗ )

�

�

N0j

j=n,m

N1j + a∗ (1 + r1 )

j=n,m

�

N0j

(70)

j=n,m

Note that N1m = N1n α
� where α
� > α∗ and that N1n is pre-determined where N1n =N10 (1 + πn ) + N0m (1 + πm ) and N0m = N0n α∗ .

SP1 = w1 l∗

�

j=n,m

N1j − w∗ (1 − τw∗ )l∗

�

N1j

j=n,m

+ r1 s∗ N0n (1 + α∗ ) − r∗ s∗ (1 − τr∗ )N0n (1 + α∗ ) − N0n (1 + α∗ )(b∗ + g o + g ind )− (g y + g ind + a∗ ){N1n (1 + α∗ ) + N1n (�α − α∗ )} + N0n (1 + α∗ )a∗ (1 + r1 )Also notice that

�

(71)

�
j
n
�) = N1n (1 + α∗ + α
� − α∗ ) and j=n,m N0j =j=n,m N1 = N1 (1+ αN0n (1 + α∗ ). Thus, SP1 becomes

SP1 = w1 l∗

�

j=n,m

N1j − w∗ (1 − τw∗ )l∗ {N1n (1 + α∗ ) + N1n (�α − α∗ )}+ r1 s∗ N0n (1 + α∗ ) − r∗ s∗ (1 − τr∗ )N0n (1 + α∗ ) − N0n (1 + α∗ )(b∗ + g o + g ind )− (g y + g ind + a∗ ){N1n (1 + α∗ ) + N1n (�α − α∗ )} + N0n (1 + α∗ )a∗ (1 + r1 )(72)

At the steady state, as for the government budget constraint, we have(τw∗ w∗ l∗ −g y −g ind −a∗ )N1n (1+α∗ )+(τr∗ r∗ s∗ −b∗ −g o −g ind )N0n (1+α∗ )+N0n (1+α∗ )a∗ (1+r∗ ) = 0(73)
By using the government budget constraint at the initial steady state, we can3

rewrite SP1 as follows:
SP1 = w1 l∗

�

j=n,m

N1j − w∗ l1∗ N1n (1 + α∗ ) − w∗ l∗ (1 − τw∗ )N1n (�α − α∗ ) − N0n (1 + α∗ )a∗ (1 + r∗ )+ r1 s∗ N0n (1 + α∗ ) − r∗ s∗ N0n (1 + α∗ ) − (g y + g ind + a∗ )N1n (�α − α∗ ) + N0n (1 + α∗ )a∗ (1 + r1 )N0n (1 + α∗ )a∗ can be canceled out in the above equation. Thus we have

SP1 = w1 l∗

�

j=n,m

N1j − w∗ l1∗ N1n (1 + α∗ ) − w∗ l∗ (1 − τw∗ )N1n (�α − α∗ )+ r1 s∗ N0n (1 + α∗ ) − r∗ s∗ N0n (1 + α∗ ) − (g y + g ind + a∗ )N1n (�α − α∗ ) + N0n (1 + α∗ )a∗ (r1 − r∗ )From the homogeneity of the production function and Euler’s theorem, we have
w1 l ∗

�

N1j +r1 (s∗ +a∗ )N0n (1+α∗ ) = F (l∗

j=n,m

�

N1j , (s∗ +a∗ )N0n (1+α∗ ))−δ(s∗ +a∗ )N0n (1+α∗ )j=n,mAt the initial steady-state, we also havew∗ l∗ N1n (1+α∗ )+r∗ (s∗ +a∗ )N0n (1+α∗ ) = F (l∗ N1n (1+α∗ ), (s∗ +a∗ )N0n (1+α∗ ))−δ(s∗ +a∗ )N0n (1+α∗ )Thus, SP1 becomes
SP1 = F (l∗

�

j=n,m

N1j , (s∗ + a∗ )N0n (1 + α∗ )) − δ(s∗ + a∗ )N0n (1 + α∗ )− {F (l1∗ N1n (1 + α∗ ), (s∗ + a∗ )N0n (1 + α∗ ))− δ(s∗ + a∗ )N0n (1 + α∗ )} − w∗ l∗ (1 − τw∗ )N1n (�α − α∗ ) − N1n (g y + g ind + a∗ )(�
α − α∗ )(74)

4

Note that N1m = α
�N1n . Thus,

SP1 = F (l∗ (N1n (1 + α�), (s∗ + a∗ )N0n (1 + α∗ )) − F (l∗ {N1n (1 + α∗ )}, (s∗ + a∗ )N0n (1 + α∗ ))− {w∗ l∗ (1 − τw∗ ) − (g y + g ind + a∗ )}N1n (�α − α∗ )� 1+�α=[FL (N1n l∗ z, (s∗ + a∗ )N0n (1 + α∗ ))N1n l∗ − w∗ (1 − τw )N1n l∗ − (g y + g ind + a∗ )N1n ]dz
1+α∗

= N1n

� 1+�α1+α∗FL (N1n l∗ z, (s∗ + a∗ )N0n (1 + α∗ ))l∗ − w∗ l∗ (1 − τw ) − (g y + g ind + a∗ )]dz
(75)Note that w∗ l∗ (1 − tw ) = cy∗ + s∗ . Thus, we have
= N1n

� 1+�α1+α∗FL (N1n l∗ z, (s∗ + a∗ )N0n (1 + α∗ ))l∗ − cy∗ − s∗ − g y − g ind − a∗ )]dz

(76)Note that N1n = R(α∗ )N0n and using the homogeneity of FL , we have
SP1 = N1

� 1+�α1+α∗FL (R(α∗ )l∗ z, (s∗ + a∗ )(1 + α∗ ))l∗ − cy∗ − s∗ − g y − g ind − a∗ )]dz

Appendix B3
Note thatas follows:�jn�) andj=n,m N2 = N2 (1 + α�jn�). Thus, SP2
j=n,m N1 = N1 (1 + α

becomes

SP2 = w2 τw2 l∗ N2n (1 + α�) + r2 τr2 s∗ N1n (1 + α�) − N1n (1 + α�) × (b∗ + g o + g ind )− N2n (1 + α�) × (g y + g ind + a∗ ) + (1 + r2 )a1 N1n (1 + α
�)

5

(77)

Using the definitions of τw2 and τr2 , we have
SP2 = w2 l∗ N2n (1 + α�) − w∗ (1 − τw∗ )l∗ N2n (1 + α�)+ r2 s∗ N1n (1 + α�) − r∗ (1 − τr∗ )s∗ N1n (1 + α�) − N1n (1 + α�)(b + g o + g ind )− N2n (1 + α�)(g y + g ind + a∗ ) + (1 + r2 )a1 N1n (1 + α
�)

(78)

By changing the order in the above equation, SP2 becomes
SP2 = w2 l∗ N2n (1 + α�) + r2 s∗ N1n (1 + α�)− w∗ l∗ N2n (1 + α�) − r∗ s∗ N1n (1 + α�)τw∗ w∗ l∗ N2n (1 + α�) + τr∗ r∗ s∗ N1n (1 + α�) − N1n (1 + α�)(b + g o + g ind )− N2n (1 + α�)(g y + g ind + a∗ ) + (1 + r2 )a1 N1n (1 + α�)= w2 l∗ N2n (1 + α�) + r2 s∗ N1n (1 + α�) + (1 + r2 )a1 N1n (1 + α�)− w∗ l∗ N2n (1 + α�) − r∗ s∗ N1n (1 + α�)+ (1 + α�){τw∗ w∗ l∗ N2n + τr∗ r∗ s∗ N1n − N1n (b + g o + g ind ) − N2n (g y + g ind + a∗ )}(79)

Now we calculate τw∗ w∗ l∗ N2n + τr∗ r∗ s∗ N1n − N1n (b + g o + g ind ) − N2n (g y + g ind + a∗ ). Notethat N2n = N1n ((1+πm )�
α +1+πn ) = N1n ((1+πm )α∗ +1+πn +(1+πm )�
α −(1+πm )α∗ ).
Thus,
τw∗ w∗ l∗ N2n + τr∗ r∗ s∗ N1n − N1n (b + g o + g ind ) − (g y + g ind + a∗ )N2n
= τw∗ w∗ l∗ N1n (R(α∗ ) + (1 + πm )(�
α − α∗ ))+ τr∗ r∗ s∗ N1n − N1n (b + g o + g ind )− (g y + g ind + a∗ )N1n (R(α∗ ) + (1 + πm )(�
α − α∗ ))6(80)

At the initial steady state, we have(τw∗ w∗ l∗ −g y −g ind −a∗ )R(α∗ )N0n (1+α∗ )+(τr∗ r∗ s∗ +(1+r∗ )a∗ −b∗ −g o )N0n (1+α∗ ) = 0(81)
By dividing by N0n (1 + α∗ ) , we have(τw∗ w∗ l∗ − g y − g ind − a∗ )R(α∗ ) + (τr∗ r∗ s∗ + (1 + r∗ )a∗ − b∗ − g o ) = 0(82)

Thus, (80) becomes= τw∗ w∗ l∗ N1n (1+πm )(�α −α∗ )−(g y +g ind +a∗ )N1n (1+πm )(�α −α∗ )−(1+r∗ )a∗ N1n (83)Therefore, SP2 becomes
SP2 = w2 l∗ N2n (1 + α�) + r2 s∗ N1n (1 + α�) + (1 + r2 )a1 N1n (1 + α�)− w∗ l∗ N2n (1 + α�) − r∗ s∗ N1n (1 + α�)+ (1 + α�){τw∗ w∗ l∗ N1n (1 + πm )(�
α − α∗ )− (g y + g ind + a∗ )N1n (1 + πm )(�
α − α∗ ) − (1 + r∗ )a∗ N1n }(84)

Now, we decompose w∗ l∗ N2n (1 + α�) in the second line in the above equitation. Since
N2n = N1n R(�α) = N1n (R(α∗ ) + (1 + πm )(�
α − α∗ )), SP2 becomes
SP2 = w2 l∗ N2n (1 + α�) + r2 s∗ N1n (1 + α�) + (1 + r2 )a1 N1n (1 + α�)− (1 + α�)w∗ l∗ N1n {R(α∗ ) + (1 + πm )(�
α − α∗ )} − r∗ s∗ N1n (1 + α�)+ (1 + α�){τw∗ w∗ l∗ N1n (1 + πm )(�
α − α∗ ) − (g y + g ind + a∗ )N1n (1 + πm )(�
α − α∗ ) − (1 + r∗ )a∗ N1n }Re-arranging the second line in the above equation, we have

SP2 = w2 l∗ N2n (1 + α�) + r2 s∗ N1n (1 + α�) + (1 + r2 )a1 N1n (1 + α�)− (1 + α�)w∗ l∗ N1n R(α∗ ) − (1 + α�)w∗ l∗ N1n (1 + πm )(�
α − α∗ ) − r∗ s∗ N1n (1 + α�)+ (1 + α�){τw∗ w∗ l∗ N1n (1 + πm )(�
α − α∗ ) − (g y + g ind + a∗ )N1n (1 + πm )(�
α − α∗ ) − (1 + r∗ )a∗ N1n }7

Next, we re-arrange r2 s∗ N1n (1 + α�) + (1 + r2 )a1 N1n (1 + α�) and (1 + r∗ )a∗ N1n . Then,
we have

SP2 = w2 l∗ N2n (1 + α�) + r2 (s∗ + a1 )N1n (1 + α�) + a1 N1n (1 + α�)− (1 + α�)w∗ l∗ N1n R(α∗ ) − r∗ (s∗ + a∗ )N1n (1 + α�) − (1 + α�)w∗ l∗ N1n (1 + πm )(�
α − α∗ )+ (1 + α�){τw∗ w∗ l∗ N1n (1 + πm )(�
α − α∗ ) − (g y + g ind + a∗ )N1n (1 + πm )(�
α − α∗ ) − a∗ N1n }Using the homogeneity of the production function, we have

w2 l∗ N2n (1 + α�) + r2 (s∗ + a1 )N1n (1 + α�) = F (l∗ N2n (1 + α�), (s∗ + a1 )N1n (1 + α�)) − δ(s∗ + a1 )N1n (1 + α�)andw∗ l∗ (1 + α∗ )N1n R(α∗ ) + r∗ (s∗ + a∗ )N1n (1 + α∗ )= F (l∗ (1 + α∗ )N1n R(α∗ ), (s∗ + a∗ )N1n (1 + α∗ ))−δ(s∗ + a∗ )N1n (1 + α∗ )Thus, SP2 becomes
SP2 = F (l∗ N2n (1 + α�), (s∗ + a1 )N1n (1 + α�)) − δ(s∗ + a1 )N1n (1 + α�) + a1 N1n (1 + α
�)
−

1+α
�
{F (l∗ (1 + α∗ )N1n R(α∗ ), (s∗ + a∗ )N1n (1 + α∗ )) − δ(s∗ + a∗ )N1n (1 + α∗ )}1 + α∗−(1 + α�)w∗ l∗ N1n (1 + πm )(�
α − α∗ )+(1 + α�){τw∗ w∗ l∗ (1 + πm )(�
α − α∗ )N1n − (g y + g ind + a∗ )N1n (1 + πm )(�
α − α∗ ) − a∗ N1n }(85)

Combining the third line and fourth line, we have8

SP2 = F (l∗ N2n (1 + α�), (s∗ + a1 )N1n (1 + α�)) − δ(s∗ + a1 )N1n (1 + α�) + a1 N1n (1 + α�)−F (l∗ (1 + α�)N1n R(α∗ ), (s∗ + a∗ )N1n (1 + α�)) + δ(s∗ + a∗ )N1n (1 + α�)+(1 + α�)N1n {−(1 − τw )w∗ l∗ (1 + πm )(�
α − α∗ ) − (g y + g ind + a∗ )(1 + πm )(�
α − α∗ ) − a∗ N1n }(86)

Notice that δs∗ N1n (1 + α�) is canceled out in the above equation. Rearranging the term(a1 − a∗ )N1n (1 + α�), we have

SP2 = F (l∗ N2n (1 + α�), (s∗ + a1 )N1n (1 + α�)) + (a1 − a∗ )N1n (1 + α�) − δ(a1 − a∗ )N1n (1 + α�)− F (l∗ (1 + α�)N1n R(α∗ ), (s∗ + a∗ )N1n (1 + α�))+ (1 + α�)N1n {−(1 − τw )w∗ l∗ (1 + πm )(�
α − α∗ ) − (g y + g ind + a∗ )(1 + πm )(�
α − α∗ )}Subtracting and adding F (l∗ N2n (1 + α�), (s∗ + a∗ )N1n (1 + α�)), SP2 becomes
SP2 = F (l∗ N2n (1 + α�), (s∗ + a1 )N1n (1 + α�)) + (1 − δ)(a1 − a∗ )N1n (1 + α�)− F (l∗ N2n (1 + α�), (s∗ + a∗ )N1n (1 + α�))+ F (l∗ N2n (1 + α�), (s∗ + a∗ )N1n (1 + α�)) − F (l∗ (1 + α�)N1n R(α∗ ), (s∗ + a∗ )N1n (1 + α�))+ (1 + α�)N1n {−(1 − τw )w∗ l∗ (1 + πm )(�
α − α∗ ) − (g y + g ind + a∗ )(1 + πm )(�
α − α∗ )}(87)

For the first and the second line in the above equation, it can be re-written as

9= F (l∗ N1n R(�α)((1 + α�), (s∗ + a1 )N1n (1 + α�)) + (1 − δ)(a1 − a∗ )N1n (1 + α�)− F (l∗∗ N1n R(�α)(1 + α�), (s∗ + a∗ )N1n (1 + α�))= N1n (1 + α�){F (l∗ R(�α), s∗ + a1 ) + (1 − δ)(a1 − a∗ )− F (l∗∗ (1 + α�), s∗ + a∗ )}� s∗ +a1n= N1 (1 + α
�)[FK (l∗ R(�α), z) + (1 − δ)] dzs∗ +a∗

Next, we focus on the third and fourth lines of (87) . Note that N2n = N1n R(�α).
Thus, the third and fourth line of (87) can be re-written as� α�α∗[FL (l∗ (1 + α�)N1n R(z), (s∗ + a∗ )N1n (1 + α�))l∗ (1 + α�)N1n (1 + πm )− (1 + α�)N1n (1 + πm ){(1 − τw )w∗ l∗ + g y + g ind + a∗ }]dz� α�n= (1 + α�)N1 (1 + πm )[FL (l∗ (1 + α�)N1n R(z), (s∗ + a∗ )N1n (1 + α
�))l∗
α∗

− ((1 − τw )w∗ l∗ + g y + g ind + a∗ )]dz

Since theFL is homogenous degree of zero, the above equation becomes= (1 + α�)N1n (1 + πm )� α�[FL (l∗ R(z), (s∗ + a∗ ))l∗α∗− {(1 − τw )w∗ l∗ + g y + g ind + a∗ }]dz
Note that (1 − τw )w∗ l∗ = cy∗ + s∗ and R� (α) = 1 + πm . Therefore, SP2 becomes asfollows:

10

SP2 = N1n (1 + α�)(1 + α�)N1n� s∗ +a1� α�s∗ +a∗[FK (l∗ R(�α), z)) + 1 − δ]dz

R� (z) [FL (l∗ R(z), s∗ + a∗ )l∗α∗�−{cy∗ + s∗ + g y + g ind + a∗ } dz

Appendix B4
To save space, I will show that SPt > 0 for t = 2, 3, ... For t = 1, the same proof isapplied as in the appendix B2.
We assume the same tax adjustment as in the preceding subsection.
(1 − τwt )wt = (1 − τw∗ )w∗ and (1 − τrt )rt = (1 − τr∗ )r∗(88)

With this tax adjustment, labor supply and saving of each individual is the sameas at the initial steady state. As in the proof in the appendix B2 and B3, I assume
that the government will save at least the same amount of the government saving pereach young individual as at the initial steady state. Note thatnm
SPt = wt τwt ln∗ Ntn + φm wt τwt lm∗ Ntm + τrt rt sn∗ Nt−1
+ τrt rt sm∗ Nt−1

− Ntn (g y,n + g ind,m + a∗ ) − Ntm (g y,m + g ind,m + a∗ )nm
− Nt−1(bn∗ + g o,n + g ind,n ) − Nt−1(bm∗ + g o,m + g ind,m )nm+ (1 + rt )at−1 (Nt−1+ Nt−1)(89)

Using the definition of τwt and τrt , we havewt τwt = wt − (1 − τw∗ )w∗(90)

rt τrt = rt+1 − (1 − τr∗ )r∗

(91)

11

Thus, SPt becomes
SPt = wt ln∗ Ntn − w∗ ln∗ Ntn + τw∗ w∗ ln∗ Ntn + φm wt lm∗ Ntm − φm w∗ lm∗ Ntm + φm τw∗ w∗ lm∗ Ntmnnn+ rt sn∗ Nt−1
− r∗ sn∗ Nt−1
+ τr∗ r∗ sn∗ Nt−1+mmm
+ rt sm∗ Nt−1
− r∗ sm∗ Nt−1
+ τr∗ r∗ sm∗ Nt−1

− Ntn × (g y,n + g ind,n + a∗ ) − Ntm (g y,m + g ind,m + a∗ )nm
− Nt−1(bn∗ + g o,n + g ind,n ) − Nt−1(bm∗ + g o,m + g ind,m )nm+ (1 + rt )at−1 (Nt−1+ Nt−1)(92)

By changing the order of the above equation, we have
SPt = wt ln∗ Ntn − w∗ ln∗ Ntn + φm wt lm∗ Ntm − φm w∗ lm∗ Ntm
nnmm+ rt sn∗ Nt−1
− r∗ sn∗ Nt−1
+ rt sm∗ Nt−1
− r∗ sm∗ Nt−1nm
+ (1 + rt )at−1 (Nt−1+ Nt−1)+ τw∗ w∗ ln∗ Ntn + φm τw∗ w∗ lm∗ Ntmnm+ τr∗ r∗ sn∗ Nt−1
+ τr∗ r∗ sm∗ Nt−1

− Ntn × (g y,n + g ind,n + a∗ ) − Ntm (g y,m + g ind,m + a∗ )nm
− Nt−1(bn∗ + g o,n + g ind,n ) − Nt−1(bm∗ + g o,m + g ind,m )Now we need to calculate the fourth line to seventh line:nmτw∗ w∗ ln∗ Ntn + φm τw∗ w∗ lm∗ Ntm + τr∗ r∗ sn∗ Nt−1
+ τr∗ r∗ sm∗ Nt−1

− Ntn (g y,n + g ind,n + a∗ ) − Ntm (g y,m + g ind,m + a∗ )nm
− Nt−1(bn∗ + g o,n + g ind,n ) − Nt−1(bm∗ + g o,m + g ind,m )(93)nNote that Ntn = Nt−1R(�α) and Ntm = Ntn α
�. Thus,the above equation becomes as

12

follows:nnτw∗ w∗ ln∗ Nt−1R(�α) + φm τw∗ w∗ ln∗ Nt−1
J(�
α)
n
n
+ τr∗ r∗ sn∗ Nt−1
+ τr∗ r∗ sm∗ Nt−1α�n
− Nt−1R(�α)(g y,n + g ind,n + a∗ )n
− Nt−1J(�α)(g y,m + g ind,m + a∗ )nn
− Nt−1(bn∗ + g o,n + g ind,n ) − Nt−1α�(bm∗ + g o,m + g ind,m )

(94)

n
n
On the other hand, at the initial steady state, Ntn = Nt−1
R(α∗ )and Ntm = Nt−1α∗Thus, .the government budget constraint at the initial steady state implies
τw∗ w∗ ln∗ R(α∗ ) + φm τw∗ w∗ lm∗ J(α∗ )+ τr∗ r∗ sn∗ + τr∗ r∗ sm∗ α∗− R(α∗ )(g y,n + g ind,n + a∗ ) − J(α∗ )(g y,m + g ind,m + a∗ )− (bn∗ + g o,n + g ind,n ) − α∗ (bm∗ + g o,m + g ind,m ) + (1 + r∗ )a∗ (1 + α∗ ) = 0(95)

Thus, (94) becomesnτw∗ w∗ ln∗ Nt−1(�α − α∗ )(1 + πm )n+φm τw∗ w∗ lm∗ Nt−1(J(�α) − J(α∗ ))n+τr∗ r∗ sm∗ Nt−1(�α − α∗ )n−Nt−1(g y,n + g ind,n + a∗ )(�
α − α∗ )(1 + πm )n−Nt−1
(g y,m + g ind,m + a∗ )(J(�α) − J(α∗ ))n−Nt−1(bm∗ + g o,m + g ind,m )(�
α − α∗ )n−(1 + r∗ )a∗ Nt−1
(1 + α∗ )

13

(96)

Therefore, SPt becomes
SPt = wt ln∗ Ntn − w∗ ln∗ Ntn+φm wt lm∗ Ntm − φm w∗ lm∗ Ntmnn+rt sn∗ Nt−1
− r∗ sn∗ Nt−1mm
+rt sm∗ Nt−1
− r∗ sm∗ Nt−1nm+(1 + rt )at−1 (Nt−1+ Nt−1)n+τw∗ w∗ ln∗ Nt−1
(1 + πm )(�
α − α∗ )n+φm τw∗ w∗ lm∗ Nt−1{J(�α) − J(α∗ )}n+τr∗ r∗ sm∗ Nt−1(�α − α∗ )n−Nt−1(g y,n + g ind,n + a∗ )(1 + πm )(�
α − α∗ )n−Nt−1(g y,m + g ind,m + a∗ )(J(�α) − J(α∗ ))n−Nt−1(bm∗ + g o,m + g ind,m )(�
α − α∗ )n−(1 + r∗ )a∗ Nt−1
(1 + α∗ )= wt ln∗ Ntn + φm wt lm∗ Ntmnmnm+rt sn∗ Nt−1
+ rt sm∗ Nt−1
+ rt at−1 (Nt−1+ Nt−1)nm+at−1 (Nt−1+ Nt−1)nm−{w∗ ln∗ Ntn + φm w∗ lm∗ Ntm + r∗ sn∗ Nt−1
+ r∗ sm∗ Nt−1n+r∗ a∗ Nt−1
(1 + α∗ )}n−a∗ Nt−1(1 + α∗ )n+τw∗ w∗ ln∗ Nt−1
(1 + πm )(�
α − α∗ )n+φm τw∗ w∗ lm∗ Nt−1{J(�α) − J(α∗ )}n+τr∗ r∗ sm∗ Nt−1(�α − α∗ )n−Nt−1(g y,n + g ind,n + a∗ )(1 + πm )(�
α − α∗ )n−Nt−1(bm∗ + g o,m + g ind,m )(�
α − α∗ )n−Nt−1(g y,m + g ind,m + a∗ )(J(�α) − J(α∗ ))

14

n
m
n
m
We add and subtract δ(sn∗ Nt−1
+ sm∗ Nt−1
+ at−1 (Nt−1+ Nt−1)) to and from SPt .
nnn
We also subtract and δ(sn∗ Nt−1
+ sm∗ α∗ Nt−1
+ a∗ Nt−1(1 + α∗ ) to and from SPt . Then,

we have
SPt = wt ln∗ Ntn + φm wt lm∗ Ntm
n
m
n
m
n
m
+(rt + δ)(sn∗ Nt−1
+ sm∗ Nt−1
+ at−1 (Nt−1+ Nt−1)) + at−1 (Nt−1+ Nt−1)nmnm−δ(sn∗ Nt−1
+ sm∗ Nt−1
+ at−1 (Nt−1+ Nt−1))−{w∗ ln∗ Ntn + φm w∗ lm∗ Ntmnmn+r∗ sn∗ Nt−1
+ sm∗ Nt−1
+ a∗ Nt−1(1 + α∗ )}nnn−δ(sn∗ Nt−1
+ sm∗ α∗ Nt−1
+ a∗ (Nt−1(1 + α∗ ))nnn+δ(sn∗ Nt−1
+ sm∗ α∗ Nt−1
+ a∗ Nt−1(1 + α∗ ))n+τw∗ w∗ ln∗ Nt−1
(1 + πm )(�
α − α∗ )n+φm τw∗ w∗ lm∗ Nt−1{J(�α) − J(α∗ )}n+τr∗ r∗ sm∗ Nt−1(�α − α∗ )n−Nt−1(g y,n + g ind,n + a∗ )(1 + πm )(�
α − α∗ )n−Nt−1(g y,m + g ind,m + a∗ ){J(�α) − J(α∗ )}n−Nt−1(bm∗ + g o,m + g ind,m )(�
α − α∗ )n−a∗ Nt−1(1 + α∗ )(97)Note that the first three line of the above equation becomesnmnm
F (Ntn ln∗ + φm Ntm lm∗ , sn∗ Nt−1
+ sm∗ Nt−1
+ at−1 (Nt−1+ Nt−1))nmnm+ (1 − δ)at−1 (Nt−1+ Nt−1)) − δ(sn∗ Nt−1
+ sm∗ Nt−1)nmnNext, we focus on w∗ ln∗ Ntn + φm w∗ lm∗ Ntm + r∗ (sn∗ Nt−1
+ sm∗ Nt−1
+ r∗ a∗ (Nt−1
+
n
n
n
n
α∗ Nt−1)) + δ(sn∗ Nt−1
+ sm∗ α∗ Nt−1
+ a∗ Nt−1(1 + α∗ )). Note that Ntn = R(�α)N, Ntm =

15

n
m
n
Nt−1
J(�
α), Nt−1= Nt−1α�. Thus, we havenmnw∗ ln∗ Ntn + φm w∗ lm∗ Ntm + r∗ (sn∗ Nt−1
+ sm∗ Nt−1
+ a∗ Nt−1(1 + α∗ ))nnn+ δ(sn∗ Nt−1
+ sm∗ α∗ Nt−1
+ a∗ Nt−1(1 + α∗ ))nn= w∗ ln∗ Nt−1R(�α) + φm w∗ lm∗ Nt−1
J(�
α)
n
n
n
+ r∗ sn∗ Nt−1
+ r∗ sm∗ Nt−1α
� + r∗ a∗ Nt−1
(1 + α∗ ))nnn+ δ(sn∗ Nt−1
+ sm∗ α∗ Nt−1
+ a∗ Nt−1(1 + α∗ ))n= w∗ ln∗ Nt−1
{R(α∗ ) + (1 + πm )(�
α − α∗ )}n m∗+ φm w∗ lm∗ Nt−1l (J(α∗ ) + J(�α) − J(α∗ ))nnn+ r∗ sn∗ Nt−1
+ r∗ sm∗ Nt−1(α∗ + α� − α∗ ) + r∗ a∗ Nt−1
(1 + α∗ ))nnn+ δ(sn∗ Nt−1
+ sm∗ α∗ Nt−1
+ r∗ a∗ Nt−1
(1 + α∗ ))n= w∗ Nt−1[R(α∗ )ln∗ + φm J(α∗ )lm ]nnn+ (r∗ + δ)[sn∗ Nt−1
+ sm∗ Nt−1α∗ + a∗ Nt−1
(1 + α∗ )]nn+ w∗ ln∗ Nt−1
(1 + πm )(�
α − α∗ ) + φm w∗ lm∗ Nt−1(J(�α) − J(α∗ ))n+ r∗ sm∗ Nt−1(�α − α∗ )(98)

On the other hand, from the homogeneity of the production function and Euler’s
theorem, (98) becomesnnnn
F (R(α∗ )ln∗ + φm J(α∗ )lm∗ )Nt−1, sn∗ Nt−1
+ sm∗ Nt−1α∗ + a∗ Nt−1
(1 + α∗ ))n+ w∗ ln∗ Nt−1
(1 + πm )(�
α − α∗ )n m∗+ φm w∗ lm∗ Nt−1l (J(�
α) − J(α∗ ))n+ r∗ sm∗ Nt−1(�α − α∗ )(99)

Similarly, we have the following relationship. Thus, SPt becomes as follows:

16

n
m
n
m
SPt = F (Ntn ln∗ + φm Ntm lm∗ , sn∗ Nt−1
+ sm∗ Nt−1
+ at−1 (Nt−1+ Nt−1))nnm+(1 − δ)at−1 Nt−1
(1 + α�) − δ(sn∗ Nt−1
+ sm∗ Nt−1
)

n
n
n
n
n
−F (R(α∗ )ln∗ Nt−1
+ φm J(α∗ )lm∗ Nt−1
, sn∗ Nt−1
+ sm∗ Nt−1α∗ + a∗ Nt−1
(1 + α∗ ))nnn+δ(sn∗ Nt−1
+ sm∗ α∗ Nt−1
+ a∗ Nt−1(1 + α∗ ))n−w∗ ln∗ Nt−1
(1 + πm )(�
α − α∗ )n−φm w∗ lm∗ Nt−1{J(�α) − J(α∗ )}n−r∗ sm∗ Nt−1(�α − α∗ )n+τw∗ w∗ ln∗ Nt−1
(1 + πm )(�
α − α∗ )n+φm τw∗ w∗ lm∗ Nt−1(J(�α) − J(α∗ ))n+τr∗ r∗ sm∗ Nt−1(�α − α∗ )n−Nt−1(1 + πm )(g y,n + g ind,n + a∗ )(�
α − α∗ )n−Nt−1(g y,m + g ind,m + a∗ ){J(�α) − J(α∗ )}n−Nt−1(bm∗ + g o,m + g ind,m )(�
α − α∗ )n−a∗ Nt−1(1 + α∗ )

17

(100)

Combining 4th line and 14th line, we havenmnm
SPt = F (Ntn ln∗ + φm Ntm lm∗ , sn∗ Nt−1
+ sm∗ Nt−1
+ at−1 (Nt−1+ Nt−1))nnm+(1 − δ)at−1 Nt−1
(1 + α�) − δ(sn∗ Nt−1
+ sm∗ Nt−1
)

n
n
n
n
n
−F (R(α∗ )ln∗ Nt−1
+ φm J(α∗ )lm∗ Nt−1
, sn∗ Nt−1
+ sm∗ Nt−1α∗ + a∗ Nt−1
(1 + α∗ ))nnn+δ(sn∗ Nt−1
+ sm∗ α∗ Nt−1
) − (1 − δ)a∗ Nt−1
(1 + α∗ ))n−w∗ ln∗ Nt−1
(1 + πm )(�
α − α∗ )n−φm w∗ lm∗ Nt−1{J(�α) − J(α∗ )}n−r∗ sm∗ Nt−1(�α − α∗ )n+τw∗ w∗ ln∗ Nt−1
(1 + πm )(�
α − α∗ )n+φm τw∗ w∗ lm∗ Nt−1(J(�α) − J(α∗ ))n+τr∗ r∗ sm∗ Nt−1(�α − α∗ )n−Nt−1(1 + πm )(g y,n + g ind,n + a∗ )(�
α − α∗ )n−Nt−1(g y,m + g ind,m + a∗ ){J(�α) − J(α∗ )}n−Nt−1(bm∗ + g o,m + g ind,m )(�
α − α∗ )nnnNote that −(1−δ)a∗ Nt−1
(1+α∗ ) = −(1−δ)a∗ Nt−1
(1+�α+α∗ −�α) = −(1−δ)a∗ Nt−1
(1+
n
n
α
�) + (1 − δ)Nt−1(�α − α∗ ). Also note that δsn∗ Nt−1
is canceled out from the secondmnand fourth lines. Also note that in the second line Nt−1
= Nt−1(α∗ + α� − α∗ ). Thus,18

nδsm∗ Nt−1α∗ is canceled out from the second and fourth line. Thus, we havenmnm
SPt = F (Ntn ln∗ + φm Ntm lm∗ , sn∗ Nt−1
+ sm∗ Nt−1
+ at−1 (Nt−1+ Nt−1))n+(1 − δ)at−1 Nt−1
(1 + α
�)

n
n
n
n
n
n
−F (R(α∗ )ln∗ Nt−1
+ φm J(α∗ )lm∗ Nt−1
, sn∗ Nt−1
+ sm∗ Nt−1α∗ + a∗ (Nt−1+ Nt−1α∗ ))nn−(1 − δ)a∗ Nt−1
(1 + α�) + (1 − δ)a∗ Nt−1(�α − α∗ )n−δsm∗ Nt−1(�α − α∗ )n−w∗ ln∗ Nt−1
(1 + πm )(�
α − α∗ )n−φm w∗ lm∗ Nt−1{J(�α) − J(α∗ )}n−r∗ sm∗ Nt−1(�α − α∗ )n+τw∗ w∗ ln∗ Nt−1
(1 + πm )(�
α − α∗ )n+φm τw∗ w∗ lm∗ Nt−1(J(�α) − J(α∗ ))n+τr∗ r∗ sm∗ Nt−1(�α − α∗ )n−Nt−1(1 + πm )(g y,n + g ind,n + a∗ )(�
α − α∗ )n−Nt−1(g y,m + g ind,m + a∗ ){J(�α) − J(α∗ )}n−Nt−1(bm∗ + g o,m + g ind,m )(�
α − α∗ )nmnm+sm∗ Nt−1+a∗ (Nt−1+Nt−1)) andWe subtract and add F (Ntn ln∗ +φm Ntm lm∗ , sn∗ Nt−1nnnsm∗ Nt−1(�
α −α∗ ) from and to SPt . We also combine sm∗ Nt−1(�α −α∗ ), r∗ sm∗ Nt−1(�α −α∗ )

19n
and τr∗ sm∗ Nt−1(�α − α∗ ). Then, SPt becomes as followsnmnm
SPt = F (Ntn ln∗ + φm Ntm lm∗ , sn∗ Nt−1
+ sm∗ Nt−1
+ at−1 (Nt−1+ Nt−1))nmnm− F (Ntn ln∗ + φm Ntm lm∗ , sn∗ Nt−1
+ sm∗ Nt−1
+ a∗ (Nt−1+ Nt−1))nn+ (1 − δ)at−1 Nt−1
(1 + α�) − (1 − δ)a∗ Nt−1
(1 + α�)nmnm+ F (Ntn ln∗ + φm Ntm lm∗ , sn∗ Nt−1
+ sm∗ Nt−1
+ a∗ (Nt−1+ Nt−1))nn− F (Ntn R(α∗ )ln∗ + φm Ntn J(α∗ )lm∗ , sn∗ Ntn + sm∗ Ntn α∗ + a∗ (Nt−1+ α∗ Nt−1))nn+ (1 − δ)a∗ Nt−1(�α − α∗ ) + (1 − δ)sm∗ Nt−1(�α − α∗ )− (1 − τw∗ )w∗ ln∗ Ntn (1 + πm )(�
α − α∗ )− (1 − τw∗ )φm w∗ lm∗ Ntn {J(�α) − J(α∗ )}− (1 + (1 − τr∗ )r∗ )sm∗ Ntn (�α − α∗ )− Ntn (1 + πm )(g y,n + g ind,n + a∗ )(�
α − α∗ )− Ntn (g y,m + g ind,m + a∗ ){J(�α) − J(α∗ )}− Ntn (bm∗ + g o,m + g ind,m )(�
α − α∗ )

(101)

n
n
R(�
α) and Ntm = Nt−1
J(�α). Then, the first three lines ofNote that Ntn = Nt−1

20(101) can be re-written as follows:nmnm
F (Ntn ln∗ + φm Ntm lm∗ , sn∗ Nt−1
+ sm∗ Nt−1
+ at−1 (Nt−1+ Nt−1))nmnm− F (Ntn ln∗ + φm Ntm lm∗ , sn∗ Nt−1
+ sm∗ Nt−1
+ a∗ (Nt−1+ Nt−1))nn+ (1 − δ)at−1 Nt−1
(1 + α�) − (1 − δ)a∗ Nt−1
(1 + α
�)

n
n
n
n
n
n
= F (Nt−1R(�α)ln∗ + φm Nt−1J(�α)lm∗ , sn∗ Nt−1
+ sm∗ α�Nt−1
+ at−1 (Nt−1
+α
�Nt−1
))

n
n
n
n
n
n
− F (Nt−1R(�α)ln∗ + φm Nt−1J(�α)lm∗ , sn∗ Nt−1
+ sm∗ α�Nt−1+ a∗ (Nt−1
+α
�Nt−1
))
n
n
+ (1 − δ)at−1 Nt−1
(1 + α�) − (1 − δ)a∗ Nt−1
(1 + α�)n= Nt−1{F (R(�α)ln∗ + φm J(�α)lm∗ , sn∗ + sm∗ α� + at−1 (1 + α�))− F (R(�α)ln∗ + φm J(�α)lm∗ , sn∗ + sm∗ α� + a∗ (1 + α�))+ (1 − δ)at−1 (1 + α�) − (1 − δ)a∗ (1 + α�)}� sn∗ +sm∗ α�+at−1 (1+�α)n= Nt−1[FK (R(�α)ln∗ + φm J(�α)lm∗ , z) + (1 − δ)]dz(102)

sn∗ +sm∗ α�+a∗ (1+�α))The fourth line and the fifth line of (101)can be transformed as follows
n
n
n
n
n
n
F (Nt−1R(�α)ln∗ + φm Nt−1J(�α)lm∗ , sn∗ Nt−1
+ sm∗ Nt−1α
� + a∗ (Nt−1+ Nt−1
α
�))

n
n
n
n
n
− F (Nt−1
R(α∗ )ln∗ + φm Ntn J(α∗ )lm∗ , sn∗ Nt−1
+ sm∗ Nt−1α∗ + a∗ (Nt−1+ α∗ Nt−1))n= Nt−1{F (R(�α)ln∗ + φm J(�α)lm∗ , sn∗ + sm∗ α� + a∗ (1 + α�))− F (R(α∗ )ln∗ + φm J(α∗ )lm∗ , sn∗ + sm∗ α∗ + a∗ (1 + α∗ ))� α�n= Nt−1[FL (R(z)ln∗ + φm J(z)lm∗ , sn∗ + sm∗ z + a∗ (1 + z))(R� (z)ln∗ + φm J � (z)lm∗ )α∗+ FK (R(z)ln∗ + φm J(z)lm∗ , sn∗ + sm∗ z + a∗ (1 + z))(sm∗ + a∗ )]dz
. Then, SPt becomes as follows:

21

(103)

n
SPt = Nt−1n
+ Nt−1� sn∗ +sm∗ α�+at−1 (1+�α)sn∗ +sm∗ α�+a∗ (1+�
α))

� α�

[FK (R(�α)ln∗ + φm J(�α)lm∗ , z) + (1 − δ)]dz[FL (R(z)ln∗ + φm J(z)lm∗ , sn∗ + sm∗ z + a∗ (1 + z))(R� (z)ln∗ + φm J � (z)lm∗ )α∗+ FK (R(z)ln∗ + φm J(z)lm∗ , sn∗ + sm∗ z + a∗ (1 + z))(sm∗ + a∗ )]dzn
+ (1 − δ)Nt−1(a∗ + sm∗ )(�
α − α∗ )− (1 − τw∗ )w∗ ln∗ Ntn (1 + πm )(�
α − α∗ )− (1 − τw∗ )φm w∗ lm∗ Ntn {J(�α) − J(α∗ )}− (1 + (1 − τr∗ )r∗ )sm∗ Ntn (�α − α∗ )− Ntn (1 + πm )(g y,n + g ind,n )(�
α − α∗ )− Ntn (g y,m + g ind,m ){J(�α) − J(α∗ )}− Ntn (bm∗ + g o,m + g ind,m )(�
α − α∗ )(104)

Let co,m∗ be the consumption of old immigrants at the initial steady state. Fromthe individual budget constraint, co,m∗ = bm + (1 + (1 − τr∗ )r∗ )sm∗ . Thus, SPt becomes22

nSPt = Nt−1n
+ Nt−1� sn∗ +sm∗ α�+at−1 (1+�α)sn∗ +sm∗ α�+a∗ (1+�
α))

� α�
α∗

[FK (R(�α)ln∗ + φm J(�α)lm∗ , z) + (1 − δ)]dz[F�L × (R� (z)ln∗ + φm J � (z)lm∗ )+ F�K × (sm∗ + a∗ )]dzn
+ (1 − δ)Nt−1a∗ (�α − α∗ )n(1 − δ)sm∗ Nt−1(�α − α∗ )n− (1 − τw∗ )w∗ ln∗ Nt−1
(1 + πm )(�
α − α∗ )n− (1 − τw∗ )φm w∗ lm∗ Nt−1{J(�α) − J(α∗ )}n− (1 + (1 − τr∗ )r∗ )sm∗ Nt−1(�α − α∗ )n− Nt−1(1 + πm )(g y,n + g ind,n + a∗ )(�
α − α∗ )n− Nt−1(g y,m + g ind,m + a∗ ){J(�α) − J(α∗ )}n− Nt−1(co,m∗ + g o,m + g ind,m )(�
α − α∗ )(105)

where F�L = FL (R(z)ln∗ + φm J(z)lm∗ , sn∗ + sm∗ z + a∗ (1 + z))

F�K = FK (R(z)ln∗ + φm J(z)lm∗ , sn∗ + sm∗ z + a∗ (1 + z))

From fourth line to twelfth line, we can rearrange as follows:
n
Nt−1

� α�
α∗

{(1 − δ)(sm∗ + a∗ )�− R� (α)(1 − τw∗ )w∗ ln∗ − (1 − τw∗ )φm w∗ lm∗ J (z)− R� (α)(g y,n + g ind,n + a∗ )− (g y,m + g ind,m + a∗ )J � (z)− (co,m∗ + g o,m + g ind,m )}dz

23

(106)

Thus, SPt becomes as follows:n
SPt = Nt−1n
+ Nt−1� sn∗ +sm∗ α�+at−1 (1+�α)sn∗ +sm∗ α�+a∗ (1+�
α))

� α�
α∗

[FK (R(�α)ln∗ + φm J(�α)lm∗ , z) + (1 − δ)]dz{F�L × (R� (z)ln∗ + φm J � (z)lm∗ )+ (F�K × (sm∗ + a∗ )�− R� (α)(1 − τw∗ )w∗ ln∗ − (1 − τw∗ )φm w∗ lm∗ J (z)− R� (α)(g y,n + g ind,n + a∗ ) − (g y,m + g ind,m + a∗ )J � (z) − (co,m∗ + g o,m + g ind,m )}dz
(107)
Note that (1 − τw∗ )w∗ ln∗ is the after-tax income of the native when the native isyoung at the initial steady state. From the individual budget constraint, this is equal
to cy,n∗ + sn∗ . Similarly, (1 − τw∗ )φm w∗ lm∗ = cy,m∗ + sm∗ . Therefore, we haven
SPt = Nt−1n
+ Nt−1� sn∗ +sm∗ α�+at−1 (1+�α)sn∗ +sm∗ α�+a∗ (1+�
α))

� α�
α∗

[FK (R(�α)ln∗ + φm J(�α)lm∗ , z) + (1 − δ)]dz{R� (z)[F�L ln∗ − (cy,n∗ + sn∗ + g y,n + g ind,n + a∗ )]+ J � (z)[F�L φm lm∗ − (cy,m∗ + sm∗ . + g y,m + g ind,m + a∗ )]+ (F�K + 1 − δ)(sm∗ + a∗ ) − (co,m∗ + g o,m + g ind,m )}dz

(108)

where F�L = FL (R(z)ln∗ + φm J(z)lm∗ , sn∗ + sm∗ z + a∗ (1 + z))

F�K = FK (R(z)ln∗ + φm J(z)lm∗ , sn∗ + sm∗ z + a∗ (1 + z))

In the above equation, the first line is the eﬀect of increasing the government
savings. The second line is the MPL condition for the native. The third line is the
MPL condition for immigrants. The fourth line measure the intra-redistributionaleﬀect.

24

Appendix B5Now, to see the correctness of the above equation, check what will happen to (??)
when natives and immigrants have the same productivities and the same preferences.
First, note that ln = lm and g y,n = g y,m , sn∗ = sm∗ and φm = 1 when natives andimmigrants have the same preferences and productivities Thus, we haven
SPt = Nt−1n
+ Nt−1

� (sn∗ +at−1 )(1+�α)(sn∗ +a∗ )(1+�
α))

� α�
α∗

� α�

[FK (R(�α)ln∗ (1 + α�), z) + (1 − δ)]dz

R� (z)[F�L ln∗ − (cy,n∗ + sn∗ + g y,n + g ind,n + a∗ )]dz

J � (z)[F�L ln∗ − (cy,n∗ + sn∗ + g y,n + g ind,n + a∗ )]J � (z)dz
α∗
� α�
n
n∗
∗
+ Nt−1 × (s + a )[F�K + 1 − δ]dz
∗
α
� α�
n
− Nt−1(co,n∗n + g o,n + g ind,n )dzn+ Nt−1α∗where F�L = FL (R(z)ln∗ + J(z)ln∗ , sn∗ + sn∗ z + a∗ (1 + z)) andF�K = FK (R(z)ln∗ + J(z)ln∗ , sn∗ + sn∗ z + a∗ (1 + z))

The first line of (109) can be transformed as follows:n
Nt−1{F (R(�α)ln∗ (1 + α�), (sn∗ + at−1 )(1 + α�)) + (1 − δ)(sn∗ + at−1 )(1 + α�)− F (R(�α)ln∗ (1 + α�), (sn∗ + a∗ )(1 + α�)) − (1 − δ)(sn∗ + a∗ )(1 + α�)}n= Nt−1(1 + α�){{F (R(�α)ln∗ , (sn∗ + at−1 )) + (1 − δ)(sn∗ + at−1 )− F (R(�α)ln∗ , (sn∗ + a∗ )) − (1 − δ)(sn∗ + a∗ )}� sn∗ +at−1n= Nt−1 (1 + α
�)[FK (R(�α)ln∗ , z) + (1 − δ)]dz
sn∗ +a∗25(109)

Thus,2nd to 4th line of (109) can be transformed as follows:n+ Nt−1
F (R(�
α)ln∗ + J(�α)ln∗ , sn∗ + sn∗ α
� + a∗ (1 + α�))n− Nt−1
F (R(α∗ )ln∗ + J(α∗ )ln∗ , sn∗ + sn∗ α∗ + a∗ (1 + α∗ ))}n+ Nt−1(1 − δ)(sn∗ + a∗ )(�
α − α∗ )n− (1 + πm )Nt−1(cy,n∗ + sn∗ + a∗ + g y + g ind )(�
α − α∗ )n− Nt−1(cy,n∗ + sn∗ + a∗ + g y + g ind )(J(�α) − J(α∗ ))n− Nt−1(co,n∗n + g o,n + g ind,n )(�
α − α∗ )On the other hand, from the resource constraint at the initial steady state we have∗

F (ln R(α∗ )(1 + α∗ )N0 , (s∗ + a∗ )(1 + α∗ )N0 + (1 − δ)(sn∗ + a∗ )N0 (1 + α∗ )N0= (cy,n∗ + sn∗ + a∗ + g y + g ind )N0 R(α∗ )(1 + α∗ )+ (co,n∗ + g o + g ind )(1 + α∗ )nWe divide the above resource constraint by N0 (1 + α∗ ) and multiply Nt−1(�α − α∗ ).

Then, we have∗nn
F (ln R(α∗ )Nt−1, (s∗ + a∗ )Nt−1 (�α − α∗ ) + (1 − δ)(sn∗ + a∗ )Nt−1(�α − α∗ )= (1 + α∗ )R(α∗ )(cy,n∗ + sn∗ + a∗ + g y + g ind )(�
α − α∗ )n+ (co,n∗ + g o + g ind )Nt−1(�α − α∗ )26

nSolving for −(co,n∗ + g o + g ind )Nt−1(�α − α∗ ), we have∗nn−(co,n∗ + g o + g ind )Nt−1(�α − α∗ ) = −F (ln R(α∗ )Nt−1, (s∗ + a∗ )Nt−1 )(�
α − α∗ )n− (1 − δ)(sn∗ + a∗ )Nt−1(�α − α∗ )n+ (cy,n∗ + sn∗ + a∗ + g y + g ind )Nt−1R(α∗ )(�
α − α∗ )∗n= −Nt−1{F (ln R(α∗ ), s∗ + a∗ )(�
α − α∗ )+ (1 − δ)(sn∗ + a∗ )(�
α − α∗ )− (cy,n∗ + sn∗ + a∗ + g y + g ind )R(α∗ )(�
α − α∗ )}Thus, SPt becomesn
SPt = Nt−1
(1 + α�)� sn∗ +at−1
sn∗ +a∗[FK (R(�α)ln∗ , z) + (1 − δ)]dzn+ Nt−1{F (R(�α)ln∗ + J(�α)ln∗ , sn∗ + sn∗ α
� + a∗ (1 + α�))− F (R(α∗ )ln∗ + J(α∗ )ln∗ , sn∗ + sn∗ α∗ + a∗ (1 + α∗ ))∗− F (ln R(α∗ ), s∗ + a∗ )(�
α − α∗ )− (1 + πm )(cy,n∗ + sn∗ + a∗ + g y + g ind )(�
α − α∗ )− (cy,n∗ + sn∗ + a∗ + g y + g ind )(J(�α) − J(α∗ ))+ (cy,n∗ + sn∗ + a∗ + g y + g ind )R(α∗ )(�
α − α∗ )}(110)

Note that (cy,n∗ + sn∗ + a∗ + g y + g ind )J(α∗ ) and (cy,n∗ + sn∗ + a∗ + g y + g ind )R(α∗ )α∗27

are canceled out. Thus, SPt becomesn
SPt = Nt−1
(1 + α�)� sn∗ +at−1
sn∗ +a∗[FK (R(�α)ln∗ , z) + (1 − δ)]dzn+ Nt−1{F (R(�α)ln∗ + J(�α)ln∗ , sn∗ + sn∗ α
� + a∗ (1 + α�))− F (R(α∗ )ln∗ + J(α∗ )ln∗ , sn∗ + sn∗ α∗ + a∗ (1 + α∗ ))∗− F (ln R(α∗ ), s∗ + a∗ )(�
α − α∗ )− (1 + πm )(cy,n∗ + sn∗ + a∗ + g y + g ind )(�
α − α∗ )− (cy,n∗ + sn∗ + a∗ + g y + g ind )J(�α)
+ (cy,n∗ + sn∗ + a∗ + g y + g ind )R(α∗ )�
α}

(111)

Note that −(cy,n∗ + sn∗ + a∗ + g y + g ind )J(�α) + (cy,n∗ + sn∗ + a∗ + g y + g ind )R(α∗ )�
αbecome equal to−(cy,n∗ + sn∗ + a∗ + g y + g ind )(1 + πm )�
α(�α − α∗ )Thus, SPt becomesn
SPt = Nt−1
(1 + α�)� sn∗ +at−1
sn∗ +a∗[FK (R(�α)ln∗ , z) + (1 − δ)]dzn+ Nt−1{F (R(�α)ln∗ + J(�α)ln∗ , sn∗ + sn∗ α
� + a∗ (1 + α�))− F (R(α∗ )ln∗ + J(α∗ )ln∗ , sn∗ + sn∗ α∗ + a∗ (1 + α∗ ))∗− F (ln R(α∗ ), s∗ + a∗ )(�
α − α∗ )− (1 + πm )(cy,n∗ + sn∗ + a∗ + g y + g ind )(�
α − α∗ )− (cy,n∗ + sn∗ + a∗ + g y + g ind )(1 + πm )�
α(�α − α∗ )}The second line of (112) becomesn
Nt−1(1 + α�){F (R(�α)ln∗ , sn∗ + a∗ )

28

(112)

The third and fourth line of the above equations becomen− Nt−1(1 + α∗ )F (R(α∗ )ln∗ , sn∗ + a∗ )∗n
− Nt−1F (ln R(α∗ ), s∗ + a∗ )(�
α − α∗ )n= −Nt−1(1 + α�)F (R(α∗ )ln∗ , sn∗ + a∗ )The fifth and sixth line becomesn−(1 + πm )Nt−1(cy,n∗ + sn∗ + a∗ + g y + g ind )(1 + α
�)(�
α − α∗ )Thus, SPt becomesn
SPt = Nt−1
(1 + α�)� sn∗ +at−1
sn∗ +a∗[FK (R(�α)ln∗ , z) + (1 − δ)]dzn∗+ Nt−1(1 + α�){F (R(�α)ln∗ , sn∗ + a∗ ) − F (R(α∗ )ln∗ , sn∗ + a∗ )− (1 + πm )(cy,n∗ + sn∗ + a∗ + g y + g ind )(�
α − α∗ )}� sn∗ +at−1n= Nt−1 (1 + α
�)[FK (R(�α)ln∗ , z) + (1 − δ)]dz
sn∗ +a∗n+ Nt−1(1 + α�)n− Nt−1(1 + α�)� α�

(1 + πm )ln∗ FL (R(z), sn∗ + a∗ )dz

α∗

� α�

(1 + πm )(cy,n∗ + sn∗ + a∗ + g y + g ind )dz

α∗Combining the second and the third line of the above equation, we haven
SPt = Nt−1
(1 + α�)n+ Nt−1(1 + α�)� sn∗ +at−1
sn∗ +a∗� α�[FK (R(�α)ln∗ , z) + (1 − δ)]dz

R� (z)[FL (R(z), sn∗ + a∗ )ln∗α∗

− (cy,n∗ + sn∗ + a∗ + g y + g ind )]dz

(113)

This is SPt when immigrant and native have the same productivities and preferences.

29