Tsukuba Economics Working Papers
No. 2015-002

Immigration as a Policy Tool for the Double Burden Problem of Prefunding Pay-as-you-go Social Security System
byHisahiro Naito
April 2015

UNIVERSITY OF TSUKUBA
Graduate School of Humanities and Social Sciences
1-1-1 Tennodai
Tsukuba, Ibaraki 305-8571
JAPANImmigration as a Policy Tool for the Double Burden
Problem of Pre-funding Pay-as-you-go Social
Security Systems
Hisahiro Naito ∗†Department of Economics
Graduate School of Humanities and Social Sciences
University of Tsukuba, Japan
April 5, 2015∗

Email: naito@dpipe.tsukuba.ac.jp
I appreciate the comments provided by 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,
and the Public Finance Study Group in Tokyo. I also appreciate the comments made by Kosuke
Aoki, Masayoshi Hayashi, Toshihiro Ihori, Yasushi Iwamoto, Ryo Kato, Shinichi Nishiyama, Pierre
Pestieau, and Herbert Schuetze. Finally, I record my thanks to Pierre Pestieau for teaching the related
literature.
†Abstract
The effect of accepting more immigrants on welfare in the presence of a pay-asyou-go social security system is analyzed theoretically and quantitatively in thisstudy. First, it is shown that if intergenerational government transfers initiallyexist 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, by using thecomputational overlapping generation model, I calculate both the welfare gainof increasing the PITN from 15.5 percent to 25.5 percent in 80 years and theyears needed to reach the (modified) golden rule level in a Pareto-improving way
in a model economy. The simulation results show that the present discounted
value of the Pareto-improving welfare gain of increasing the PITN is 23 percent
of initial GDP. It takes 112 years for the model economy to reach the golden rulelevel in a Pareto-improving way.

1

Introduction

Pre-funding a pay-as-you-go (PYGO) security system or transforming it into a fundedsystem is not easy. When the government transforms a PYGO social security system
into a funded system or pre-funds a PYGO social security system, some generations
must bear the so-called “double burden,”namely that the young generation must pay
the social security tax twice (Geanakoplos, Mitchell and Zeldes (1998), Belan andPestieau (1999)). Thus, although pre-funding a PYGO social security system is desirable from the point of view of capital accumulation, it is difficult to implement in a
Pareto-improving way.
In the face of the fiscal problems caused by negative demographic shocks, policymakers have recently shown increased interest in accepting more immigrants. For
example, the IMF (2012) report on the economy in Japan, which suffers from an extremely low fertility rate and practical restrictions on the inflow of immigrants, statesthat facilitating a modest level of immigration could create significant benefits for theJapanese economy. However, although using immigrants to solve the fiscal problem of
PYGO social security systems in the short run is attractive, its desirability in the longrun is unclear.
Increasing immigrants has two economic effects in the presence of a PYGO social
system. First, increasing the number of immigrants that arrive each year will decrease
the dependency ratio. Given that the capital–labor ratio and factor prices are fixed, thisdecreasing dependency ratio will bring about a welfare gain in the presence of a PYGO
social security system. Second, increasing the number of immigrants implies increasing
the population growth rate. With a neoclassical production function that exhibits thediminishing marginal product of capital (MPK), starting from a dynamically efficient
initial steady state, the neoclassical growth model (Solow (1964)) predicts that such ahigher population growth rate leads to lower capital stock per capita, wages, and GDP
per capita (the capital shallowing effect). Thus, given those two conflicting effects, it
is not clear how such an increase in immigrants can help solve the problem of a PYGO
social security system in the long run.

1

The literature also presents an unclear picture. In the literature on public finance,
by using the computational overlapping generation model (Auerbach and Kotlikoff(1998)), Storesletten (2000) argues that accepting a particular type of immigrant(skilled immigrants who are of an age such that they will not be able to claim social
security benefit because they do not satisfy the minimum requirement of the durationof social security tax payments) will increase social welfare in the presence of the retirement of the Baby Boomer generation. By contrast, Fehr, Jokisch, and Kotlikoff(2004) show that no such welfare gain will ensue. They rather argue that the economic
force that leads to a lower capital–labor ratio and lower GDP per capita is very strong;hence, accepting more immigrants cannot bring about a welfare gain. Feldstein (2006)
analyzes the effect of immigration in Spain and concludes that immigration does notlead to a welfare gain, while Collado, Iturbe-Ormaetxeand, and Valera (2004) argue
that accepting more immigrants brings about a positive welfare gain to Spain.
Given the mixed results in the literature regarding the effect of accepting more
immigrants on social welfare, a natural question arises as to whether, from a theoreticalstandpoint, accepting more immigrants Pareto-improves welfare in the presence of a
PYGO social security system. The capital shallowing effect seems to be very strong,
as emphasized by Fehr, Jokisch, and Kotlikoff (2004), 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 previous studies have arrived at such different results
regarding the effect on welfare of accepting more immigrants.
Motivated by this question, I analyze theoretically and qualitatively the effect onwelfare 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 the 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
1

The Matlab code used for this simulation is available from the journal’s website and from theauthor.

2

distorting taxes, it is Pareto-improving to increase the percentage of immigrants tonatives (PITN) if there exist upward intergenerational transfers in the sense that the
pre-tax labor income, which is equal to labor supply times the marginal product oflabor, of a young individual 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 tofuture periods (the marginal product of labor (MPL) condition). In the presence of a
PYGO social security system, some of the pre-tax labor income of a young individualin period t is used for the consumption of the old in period t. Thus, the above MPL
condition is likely to be satisfied. In addition, the presence of government debt will
make the MPL condition more likely. 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 Pareto-improving way within a finite time by putting the government
budget surplus, which is obtained by increasing the PITN, into savings. Note that whenthe economy reaches the golden rule level, the problem of the under-accumulation ofcapital caused by PYGO social security is solved for all practical purposes. Thirdly,
I quantify the Pareto-improving welfare gain 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 in the PITN, such that the PITN starts to increase from 15.5
percent, reaches 25.5 percent in the 80th year, and remains constant at 25.5 percent inlater years.2
With this rate of increase in the PITN and in the model that mimics the importantdimensions of the US economy, my simulation shows it takes a minimum of 112 years
for the model economy to reach the golden rule level in a Pareto-improving way. On
the new balanced growth path, capital stock per efficient unit of labor increases by
102 percent and publicly provided private goods per capita increase by 36 percent.
When the target capital stock is set at the modified golden rule level with a 3 percent
2

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

3

intergenerational discount rate, it takes 65 years to reach the modified golden rule levelin a Pareto-improving way and capital stock per efficient unit of labor increases by 18
percent. The present discounted value (PDV) of Pareto-improved utility, measured by
the expenditure function, of natives and their descendants, which does not include theincreased utility of immigrants and their descendants, comprises 23 percent of initial
GDP.3 When the time to reach the target PITN is shortened to 42 years, the economy
reaches the modified golden rule level in the 59th year and the PDV of the Pareto
improvement comprises 28 percent of initial GDP.
Finally, I conduct robustness checks by changing a number of the parameter values,for example, the share of the surplus for the government savings, replacement rate, time
preference rate, risk aversion, initial government debt (asset) level, level of immigrants’earnings, and the consumption of public services by immigrants. Those robustness
checks show that the results of the simulation do not change substantially in magnitude
for different parameter values. Both the theoretical results and the computational
results suggest the robustness of the welfare gain of increasing the PITN in the presenceof a PYGO social security system.
The intuition of the above results can be understood as follows. Consider a situation where the government has a PYGO social security system initially and accepts
immigrants at a fixed ratio to the number of natives. Assume that the economy is at
the steady state initially with this constant growth rate of immigrants and that the
total amount of social security tax revenue is equal to the social security benefit paid
to the old in every period. Now, assume that the government starts to accept additional immigrants, in addition to the initially planned quota of immigrants, in periodt. When the government collects the social security tax from those additional immigrants in period t, two channels affect the welfare of the residents of the host country.
First, the government does not need to use this collected tax revenue to pay for thesocial security benefit of the old in period t. Since the social security tax and benefits
3

If I include the increased utility of immigrants and their descendants, the welfare gain of accepting
immigrants becomes bigger.Thus, my estimate is likely to underestimate the welfare gain of accepting
more immigrants. I calculate this number to analyze to what extent accepting more immigrants isPareto-improving from the point of the natives.

4are balanced at the initial steady state, the collected revenue from this social security
tax is “free money”for the government. The government can invest this social security
tax revenue in capital. Second, in period t + 1, the government does not need to use
the return from investment made in period t. Since the social security benefit of theadditionally accepted immigrants in period t can be financed by the social security taxof the children of those additionally accepted immigrants, the government can reinvest the return from investment in period t + 1. Starting from a dynamically efficient
allocation, this investment will grow at the rate of interest, which is larger than the
population growth rate.
Note that in the case of the standard pre-funding argument, the double burden
problem happens because the government does not have this kind of “free money.”For
the government to have money to invest, the young cohort needs to pay two taxes:
one for the benefit of the old and one for investment. However, when the government
accepts additional immigrants, the government can use the available “free money” tosolve the double burden problem of pre-funding social security.
There are several questions about the above intuition, however. First, it is not clear
how to Pareto-improve welfare in the absence of lump-sum taxes and transfers. Whenimmigrants are accepted, factor prices are changed. In the absence of lump-sum taxes
and transfers, it seems to be difficult to Pareto-improve welfare. Second, it might takequite a long time for the economy to increase capital stock per capita, although accepting additional immigrants can raise welfare in the long run. The following theoreticaland simulation analysis answers those questions.
The remainder of the paper is organized as follows. Section 2 reviews the related
literature and clarifies the research contributions. Section 3 describes the theoreticalanalysis. Section 3.1 obtains the MPL condition and explains the underlying intuitionof the MPL conditon by analyzing a case where the government cannot use the lumpsum tax and transfer and where immigrants and natives have identical productivities
and preferences. The MPL condtion plays a critical role in other cases. Section 3.2
analyzes a case where the government cannot use the lump-sum tax and transfer.
Section 3.3 analyzes a case where immigrants and natives have different productivities
5and preferences. Section 4 presents a simulation-based analysis using the computational
overlapping generation model. Section 5 concludes.
2

Literature review and contributions to the researchThis study is related to several strands of the literature. First, the theoretical results presented in previous studies of social security pre-funding show that it is notpossible to Pareto-improve all generations by pre-funding or transiting from a PYGO
social security system to a funded system (Geanakoplos, Mitchell and Zeldes (1998),
Belan and Pestieau (1999)). As a result, previous authors have focused on steady-state
welfare-maximizing social security reform or policies that aim to balance the social security burden across cohorts in the presence of negative demographic shocks (Feldsteinand Samwick (1998), Kotlikoff, Smetters and Walliser (2007), Nishiyama and Smetters(2007)).
Second, this study is also related to the public finance literature that analyzes theeffect of immigration in the presence of social security in a dynamic general equilibrium model. Storesletten (1995, 2000) examines the effect of accepting immigrants ina dynamic general equilibrium model with and without aggregate uncertainty, respectively. He argues, in the US context, that by selecting particular types of immigrants,
increased immigration can positively affect the welfare of natives. In contrast, Fehr,
Jokisch, and Kotlikoff (2004) find no such welfare gain.4
Third, a large body of research has calculated the costs and benefits of accepting
immigrants. The first generation of this strand of research (i.e., static cost/benefit
analyses) does not take account of the fact that social security benefits are partiallypaid by the children of immigrants ( Borjas (1994), Passel (1994), Simon (1984), Akbari(1989), Storesletten (2003)), whereas the second generation considers that the childrenof immigrants also contribute to the PYGO social security system (Lee and Miller(1998), Auerbach and Oreopoulos (1999)).5
4

For studying other countries, see Collado, Iturbe-Ormaetxe, and Valera (2004) for Spain andBonin, Raffelhüschen and Walliser (2000) for Germany.
5
In addition, a small number of studies have theoretically analyzed the effect of immigration in thepresence of social security. For instance, Razin and Sadka (2000) show that accepting immigrants is

6

This study contributes to the body of knowledge on this topic in several ways.
First, I identify the conditions under which it is Pareto-improving to accept more
immigrants (the MPL condition). Second, I show that the government can lead theeconomy to the (modified) golden rule level by accepting more immigrants within afinite time in a Pareto-improving way. This finding is in sharp contrast to those ofprevious studies of social security reform, which argue that increasing capital stockby pre-funding social security in a Pareto-improving way is difficult (Geanakoplos,
Mitchell and Zeldes (1998)). Third, I develop and present a method that shows that
accepting more immigrants Pareto-improves all generations in the presence of distorting
taxes and changing factor prices in a dynamic economy. This technique was originallydeveloped 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)).6 Fourth, I quantify the welfare gain of the Pareto improvement predicted by
the theoretical model. Consistent with the results of Storesletten (2000), I show a nontrivial welfare gain from increasing immigration into the United States and demonstratethat my results are robust for different parameter values. The theoretical results androbustness of the simulation results show that a policy of increasing immigration into
the United States should be considered to be an important policy option.7
3

The model

The model proposed in this study uses the standard overlapping generation model witha 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.” When they are in the second period, they are retired and are called “old.” I
assume that immigrants arrive in the host country only when they are young and thatthe host government prohibits inward migration when immigrants are old. I define
unlikely to improve welfare when capital accumulation is endogenous.
6
To the best of my knowledge, this is the first paper that applies the technique developed by Dixit
and Norman (1980).
7This paper does not take political factors, which was analyzed by Razin,Sadka and Suwankiri(2011), into consideration. Including those political factors into the analysis is an important agenda
for future research.

7

individuals born at the beginning of period t in the host country as natives of cohortt, regardless of the nationality of their parents. Immigrants who move to the hostcountry at the beginning of period t are considered to be immigrants of cohort t. Let
j be the index indicating nationality. If an individual is a native, j = n and if he or
she is an immigrant, j = m. Let Ntj be the number of the young of type j in periodt. Let (cy,jco,jtt+1 ) be 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 services for the young,for each young individual of type j, which is consumed in period t. Let gto,j be theamount of publicly provided private goods such as Medicaid and publicly provided
nursing homes for each old individual of type j in period t. Let gtind,j be the amount of
age-independent publicly provided private goods consumed by a young individual andan old individual of type j.8 I assume that the utility function of the 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
Further, I assume that uij (ci,j) (i = y, o; j = n, m) are strictly
t , lt ) and v (gt , gt

increasing and concave functions. Moreover, I assume the additive separability of
publicly provided private goods to ensure that the provision of such goods does notaffect the consumption and saving decisions of individuals. This assumption simplifies
the analysis because I also assume that the government redistributes the welfare gainof accepting more immigrants in the form of increased publicly provided private goods
to individuals.9
On the production side, let F (Lt , Kt ) be a production function, where Lt and Kt
8
In this study, I ignore non-rival public goods, whose presence would favor immigration because
accepting immigrants means that the costs of such goods are shared by more individuals without
decreasing their consumption.
9
There are several ways in which the government redistributes the welfare gain to individuals. I
use this method to simplify the analysis. The main conclusion does not change when other ways ofredistributing the welfare gain are used.

8

are the total amount of labor and the total capital stock used in period t. Let δ be thecapital depreciation rate. I assume that F (Lt , Kt ) displays constant returns to scale
and that both the MPL and MPK are diminishing. I also assume that the standard
Inada condition is satisfied.
The economy is at the steady state initially and the initial economy is dynamicallyefficient.10 Furthermore, for the welfare analysis of accepting more immigrants, I makethe 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 a one-unit supply of labor by a native, φn efficient units of labor are supplied.
For a one-unit supply of labor by an immigrant, φm efficient units of labor are supplied
where φm ≤ φn . For normalization, I assume that φn = 1.
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 rate ofnatives.
AS7: If immigrants and natives have the same productivities, then the government
treats them in the same way in the tax and social security system.
I need AS2 to assess the effect of increasing the number of immigrants in thepresence of a PYGO social security system. AS5 is relaxed in the simulation analysis.
AS7 needs more discussion. Clearly, if the government can treat immigrants in a
discriminatory way in the tax and public pension systems, the utility of both natives
and immigrants could be increased. The wage rate of immigrants in their countries oforigin is typically lower than that in the host country. Thus, it is possible to Paretoimprove the welfare of both natives and immigrants if (a) the host government places
10

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 study’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)).

9

a high tax burden on incoming immigrants in such a way that the net wage rate ofimmigrants in the host country is still higher than that in their countries of origin and(b) the government redistributes to natives the tax revenue collected from immigrants.
AS7 precludes such an obvious case from occurring.
Let FK and FL be the partial derivative of the production function with respect tocapital and labor. Let wtj and rt be the wage rate of an individual of type j and theinterest rate in period t. Let sjt and at be the amount of savings made by cohort t of
type j and the total amount of government savings divided by the number of cohorts
t. Then, wtj and rt are determined as follows:
wtn = FL (Lt , Kt ), wtm = φm FL (Lt , Kt ), rt = FK (Lt , Kt ) − δ
X
X j
X j
j
where Lt =φj ltj Ntj and Kt =st−1 Nt−1+ at−1
Nt−1 .
j=n,m

j=n,m

(2)

j=n,m

The resource constraint in period t is as follows:=

X

F (Lt , Kt ) + (1 − δ)Kt
X j
j
y.j
o.j
{cy,j
+ g ind.j + at }Ntj +Nt−1 {co,j+ g ind.j }.
t + st + g
t +g

j=n,m

(3)

j=n,m

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

(4)

Let αt be the immigrants-to-natives ratio (INR) of cohort t.11 The immigration policy
is expressed in terms of αt . For example, a one-time increase in the INR means thatα0 = α ∗ , α1 = αe, and αt = α∗ for t ≥ 2, where α
e > α∗ . Permanently increasing the
INR means that α0 = α∗ and αt = αe for all t ≥ 1, where α
e > α∗ . The sum of natives
11

Note that INR of cohort t is the ratio of the number of immigrants of cohort t over the numberof natives of cohort t while PITN at period t is the percentage of the total number of immigrants atperiod t over the total number of natives at period t. In the simulation analysis, I use PITN because
calculating PITN involves less measurement errors.

10and immigrants in cohort t isX

Ntj = Ntn × (1 + αt )

j=n,m
m
n
= {(1 + πm ) × Nt−1
+ (1 + πn ) × Nt−1
} × (1 + αt )n(1 + αt ).
= {(1 + πm ) × αt−1 + (1 + πn )} × Nt−1(5)n(1 + αt−1 ). Thus, at the steady-state
Note that the total size of cohort t − 1 is Nt−1

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

(6)

At the golden rule level, the government sets the MPK equal to the population
growth rate and depreciation rate. Thus, I assume thatR(αt−1 ) − 1 + δ > 0.

(7)

By using R(αt−1 ), Ntn and Ntm can be denoted asnn
Ntn = R(αt−1 )Nt−1
and Ntm = R(αt−1 )Nt−1
αt .

3.1

(8)

An Economy with Lump-sum Tax and Identical Productivities andPreferences: Deriving the MPL Condition

In this subsection, we derive the MPL condition. This MPL condition plays an important role when we analyze an economy where there are with distorting taxes and11natives and immigrants have different productivities and preferences. To clarify the
economic meaning of the MPL condition, in this subsection I assume that the government can use the lump-sum tax and transfer and that natives and immigrants haveidentical productivities and preferences. Those two assumptions are not needed for myresults, but they help illustrate the underlying intuition of the MPL condition. Readers
can skip this subsection and go directly to subsections 3.2–3.4 to find the analysis of
the economy where immigrants and natives show different productivities, consumptionof public services, and preferences and where the lump-sum tax is not available.
Let bjt+1 be the social security benefit for type j of cohort t in period t + 1 andτtj the lump-sum tax in period t for type j of cohort t. The assumption of identical productivities and preferences for natives and immigrants and AS7 imply thatoj j jjj
φj , wtj , cyj
t , ct+1 lt , st , bt+1 and τt do not change for different values of j. Thus, we elim-

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

(9)

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

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

j=n,m

X

j
Nt−1
= 0.

(10)

j=n,m

By using the individual budget constraint, the homogeneity of the production function, and equation (2), it is straightforward to show that the government budget con-12

straint is equivalent to the following resource constraint:12
F (Lt , Kt ) + (1 − δ)Kt ≥X j
X j{cyt + st + g y + g ind + at }Nt + {cot + g o + g ind }
Nt−1
j=n,m

where Lt =X

lt Ntj and Kt = (st−1 + at−1 )

j=n,m

j=n,m

X

j
Nt−1
.

(11)

j=n,m

From (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 ). (12)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 initial steadystate, respectively, where the immigration policy at the initial steady state is α∗ for allt. Let s∗ and l∗ be the amount of savings of each individual and labor supply at theinitial steady state, while a∗ and b∗ represent government savings (or debt if this value
is negative) divided by the number of young individuals and the social security benefit
at the initial steady state. The government will choose the steady-state lump-sum tax
policy τ ∗ so that it satisfies its budget constraint. This approach implies that at b∗ andτ ∗ , the resource constraint must be satisfied. Conversely, when the resource constraint
is satisfied, then the government budget constraint is also satisfied. cy∗ and co∗ denotethe steady-state consumption of a young individual and an old individual, respectively.
The initial steady-state economy with the steady-state immigration policy α∗ is
12
By using the homogeneity of the production function, we have F (Lt , Kt ) + (1 − δ)Kt = Fk Kt +FL Lt + (1 − δ)Kt . Then, from (2), F (Lt , Kt ) = (rt + δ)Kt + wt Lt + (1 − δ)Kt . By using the definitionof Kt and Lt and the individual budget constraint, we have (11).

13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
(13)

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∗ )

(14)
(15)

F (K ∗ , L∗ ) + (1 − δ)K ∗ = {cy∗ + s∗ + g y + g ind + a∗ }R(α∗ )N0n∗ (1 + α∗ )+{co∗ + g o + g ind }N0n∗ (1 + α∗ )(16)cy∗ = w∗ l∗ − τ ∗ − s∗ and co∗ = (1 + r∗ )s∗ + b∗(17)N0n∗ is some positive number.
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

(18)

Welfare Effect of Increasing the INR

In this subsection, I examine, starting from period 1, whether increasing the INR
permanently Pareto-improves 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 (CMP), which is a function of α:

14

CMP :

V (α) =

s.t. uy (cyt , lt ) + v y (g y , g ind ) +max{cyt ,cot ,st , at |t=1,2,..}

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, ... (19)1+ρF (Lt , Kt ) + (1 − δ)Kt ≥nn(1 + αt−1 ) for t = 1, 2, ...
R(αt−1 )(1 + αt ) + {cot + g o + g ind } Nt−1
{cyt + st + g y + g ind + at }Nt−1(20)n(1 + αt−1 ) for t = 2, ... and s0 = s∗ and a0 = a∗
Kt = (st−1 + at−1 )Nt−1n
Lt = lt Nt−1
R(αt−1 )(1 + αt ) and (8)(21)α and α∗ are given.
The above CMP deserves several comments. First, V (α) is the utility of cohort 0 inperiod 1 when the government accepts immigrants at a constant ratio α∗ at the initialsteady state and starts to accept immigrants at the ratio α from period 1. Because theconsumption of cohort 0 in the young period is already determined in period 1, it is
excluded from the above programming problem. Second, the first constraint is related
to Pareto improvement and this requires that all cohorts except cohort 0 must have atleast the same utility as they would have at the initial steady state. Note that in thefirst constraint, there are α and α∗ . From period 1, the immigration policy in period
0 is predetermined. Such a variable is denoted as α∗ , while the policy determined inperiod 1 or later is denoted as α.
nnnNote that Ntn is determined by α and Nt−1
, but Nt−1
is also affected by α and Nt−2.

This fact implies that a change in immigration policy α affects all Ntn for t = 1, 2, ...
n
To simplify the calculation, it is useful to divide the resource constraint by Nt−1
when

15

n
(1 + α) when t = 2, 3, 4....13 Then, (20) becomes as follows:t = 1 and by Nt−1

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(22)

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, ...

(23)

Let γt and λt be the Lagrangian multipliers of the minimum utility constraint (19)and the resource constraints (22) and (23). Let γt∗ and λ∗t be the Lagrangian multipliers
when α = α∗ . Then, we have the following observation.
Observation 1
When α = α∗ the solution of CMP 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 =

(24)
(25)
(26)

For the proof of Observation 1, see Appendix B1.
Observation 1 implies that when the INR α is fixed at α∗ , the initial steady-state
allocation is Pareto-efficient and that it is not possible to have Pareto improvementfrom the initial steady-state holding α = α∗ . Now, suppose that the government
increases the INR from α∗ . Whether such an increase in the INR Pareto-improves
welfare can be analyzed by calculating dV /dα and evaluating it at α = α∗ because weput the minimum utility constraint (19) into the programming problem and because
the utility level of cohort 0 is u∗ at α∗ . According to the envelope theorem, dV /dα|α=α∗
is equal tonn, not by Nt−1
(1 + α), because the old
We divide the resource constraint in period 1 by Nt−1n∗npopulation in period 1 is Nt−1 (1 + α ), not Nt−1 (1 + α).
13

16

(
R(α∗ )λ∗1 +∞X)
λ∗t R0 (α∗ )× FL (R(α∗ ), s∗ + a∗ )l∗ − (cy∗ + s∗ + g y + g ind + a∗ ) ,

t=2

(27)
where R(α) = 1 + πn + α(1 + πm ).
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 MPL multiplied bylabor supply, which represents an individual’s contribution to the economy when he or
she is young at the initial steady state. It is also the pre-tax labor income of a young
individual. When a young individual contributes FL l∗ to the economy, the government
has three choices when deciding how to distribute this contribution. The first choice isto let the young individual consume this contribution. The second choice is to transfer
this contribution to future periods and let this young individual or his/her descendants
consume in future periods. The third choice is to transfer this contribution to old individuals. cy∗ + g y + g ind is the amount of the resource consumed by the current young
individual when he or she is young. s∗ is the amount of the resource consumed by thisindividual when he or she becomes old. a∗ is the amount of the resource transferred tofuture 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 old individuals at the initial steady state. We call this amount the “upward intergenerationaltransfer.”
Definition: When FL l∗ − cy∗ − g y − g ind − s∗ − a∗ is positive, the MPL conditionis satisfied and the amount FL l∗ − cy∗ − g y − g ind − s∗ − a∗ represents the upward
intergenerational transfer.
Proposition 1 (MPL condition version) If there exist upward intergenerationaltransfers, in the sense that the pre-tax labor income of a young individual is greater than
the sum of the resources that he or she consumes when young and the amount of resources transferred to future periods, then accepting more immigrants Pareto-improves17the welfare of all generations.14
Because cy∗ + s∗ is the after-tax income of the young by definition, FL l∗ − cy∗ − s∗is the amount of tax paid by a young individual. Moreover, g y + g ind + a∗ is the sumof the government resources provided to a current young individual and the amountof resources transferred to future cohorts per young individual. Thus, we have thefollowing corollary:
Corollary (Tax expenditure version) Alternatively, if the amount of tax paidby the young is greater than the amount of government resources provided to a current
young individual and the amount of resources transferred to future cohorts per young
individual at the initial steady state, accepting more immigrants Pareto-improves welfare.
This MPL condition plays a critical role in the analysis not only of the case where thegovernment has access to the lump-sum tax but also of the case where the government
has no such access. In addition, in the simulation analysis, this MPL condition is
crucial. The intuition of Proposition 1 is as follows. When the MPL condition issatisfied, upward intergenerational transfers are made at the initial steady state. FL l∗ −cy∗ − g y − g ind − s∗ − a∗ is transferred from a young individual to an old individual at
the initial steady state. However, when additional immigrants are accepted in periodt, FL l∗ −cy∗ − g y − g ind − s∗ − a∗ will become“free money” for the government since
the public pension granted to this group of additional immigrants is paid for by theirchildren. Thus, the government can invest this free money and use it later to increase
the welfare of all later generations.
The tax expenditure condition also has an important implication for the cost/benefit
analysis of accepting immigrants. Note that for the government expenditure part, thesocial security benefit and publicly provided private goods for the old are not includedin the tax expenditure condition. Only the tax that young immigrants pay and the14
Note that this condition does not change even in the presence of public goods because an increasein the number of immigrants does not affect the consumption of public goods owing to their “public”
nature.

18

resources used for the young or future cohorts should be included.1516 Graphically,
Proposition 1 is explained in Appendix A1.
3.2

Presence of Distorting Taxes and Implementation of Pareto Improvement

In the preceding subsection, I assumed that the lump-sum tax is available. Under aneoclassical production function that displays a diminishing marginal product, whenmore immigrants are accepted, the pre-tax wage will decrease and pre-tax interest
rate will increase. In the absence of the lump-sum transfer, it is unclear whether it ispossible to Pareto-improve all generations.
In this subsection, I show that it is possible to Pareto-improve welfare withoutchanging the incentives of individuals, by increasing the INR in the absence of thelump-sum transfer as long as the MPL condition is satisfied. In addition, I show thata relatively simple adjustment of taxes (wage tax and interest tax) and social security
benefit achieves this Pareto improvement when the government increases the INR. The
technique presented in this subsection was developed by Dixit and Norman (1980) toshow the superiority of free trade over autarky. I hence apply their technique to a
dynamic economic model.
For the analysis, let α
e be a time-invariant new immigration policy from period 1
where αe > α∗ . As in the previous section, I assume that the economy is dynamically15

When immigrants and natives have different preferences and productivities, then the sum of theredistribution from immigrants to natives and the MPL condition become important. See section 3.3.
16
Traditional studies of the fiscal effect of accepting more immigrants calculate the present value ofgovernment expenditure such as the consumption of publicly provided private goods and the social
security benefit that immigrants receive and tax revenue (including income tax and social securitytax) they pay. However, if the social security benefit that retired immigrants receive is included in the
cost/benefit calculation, then the social security tax that their children pay should also be included
because the social security benefits are financed by the social security tax that the children of natives
and immigrants pay. Of course, if the social 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 benefit the children of immigrants receive is included, then thesocial security tax that the grandchildren of immigrants receive should be included. Note that in the
PYGO social security system, the social security benefit that immigrants receive is roughly balancedby the social security tax paid by their children. This fact implies that in the cost/benefit calculation,
the social security benefit that immigrants receive is roughly canceled out by the social security tax
their children pay. Thus, in the cost/benefit calculation, only the tax that young immigrants pay andthe publicly provided private goods for young immigrants should be included.

19

efficient at the initial steady state. This fact implies that
FK (R(α∗ ), s∗ + a∗ ) > δ + R(α∗ ) − 1.

(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 in periodt. Then, the individual budget constraint (9) 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∗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 interest rate increases
initially. To achieve Pareto improvement by increasing the INR, first I assume thatthe government sets the tax rates such that the after-tax wage and interest rate after
an increase in the INR are equal to the after-tax wage rate and interest rate at theinitial steady state. Then, the wage tax rate and 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 individual
receives the same benefit when the wage rate, the wage tax rate, and labor supply are20

at the same level as at the initial steady state. This fact 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 andlabor supply behavior does not change because the budget constraint of a consumer
in any period t is the same as at the initial steady state. This fact implies that theequilibrium social security benefit in any period t is the same as at the initial steadystate. If the government provides at least the same level of publicly provided private
goods, the utility levels of all cohorts are at least the same as at the initial steady state.
As for the extent of the increased immigration, 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 result presented in the preceding subsection shows that a marginal increasein the number of immigrants Pareto-improves welfare if the MPL condition is satisfied. However, this result does not imply that an unlimited acceptance of immigrants
always Pareto-improves welfare. I impose two conditions on the extent to which immigrants are accepted. The first condition is regarding the MPL condition when the newimmigration policy αe is implemented. I assume that when αe is operative and when
government savings per young individual are held constant, the marginal increase in
output due to a one-unit increase in the population is greater than or equal to theresources an individual receives when he or she is young at the initial steady state.
This fact implies thatnmnml∗ FL (l∗ (1+eα)((1+πn )Nt−1+(1+πm )Nt−1), (s∗ +a∗ )(Nt−1+Nt−1)) ≥ cy∗ +g y +g ind +s∗ +a∗

(33)
m
n
for t = 1, 2, 3.... Note that (1 + πm ) × Nt−1
+ (1 + πn )Nt−1
is the population of young

21

natives in period t. For t = 2, 3, 4..., by using the homogeneity of the productionnm, (33) can be written as=αeNt−1function and Nt−1l∗ FL (l∗ (R(eα), s∗ + a∗ ) ≥ cy∗ + s∗ + g y + g ind + a∗ for t = 2, 3, 4...

(34)

The second condition is regarding the golden rule level. Note that we have assumed
that at the initial steady state, (28) holds. As R in (28) increases, both the right handside and the left hand side of (28) increase. However, the golden rule may be binding
even if the balance of government savings per young individual is held constant. Thisbinding is more likely when the MPK increases to a lower extent than the increase of
the population growth rate holding capital stock per capita constant. In this case, thefollowing golden rule is satisfied at αe:
FK (R(eα), s∗ + a∗ ) = δ + R(eα) − 1.

(35)

On other other hand, when 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. In this case,the following condition holds at αe when the balance of government savings per young
individual is held at a∗ :FK (R(eα), s∗ + a∗ ) > δ + R(eα) − 1.

(36)

If (35) holds, then the government does not need to increase its savings to reach thegolden rule and the analysis becomes trivial. Thus, in the following analysis, I assume
that (36) holds instead of (35).
When the government accepts more immigrants, it can increase its savings balance
as shown below, implying that, at some point, the MPK may become equal to thegolden rule level of capital stock per capita as at increases from a∗ . However, when
the MPK equals the golden rule level, it is clearly better to use the entire government
surplus to increase the supply of publicly provided private goods rather than to increase

22

the balance of government savings. Thus, I assume that as long as the MPK is higher
than the golden rule level, the government uses some of its budget surplus to increase
its savings and the remainder to increase the supply of publicly provided private goods.
When the MPK reaches the golden rule level, the government uses the entire surplus
to increase the supply of publicly provided private goods. Thus, we have the following
MPK condition:
FK (R(eα), s∗ + at ) ≥ δ + R(eα) − 1(37)

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

N1j +(r1 τr1 s∗ −b∗ −g o −g ind )

j=n,m

X
j=n,m

N0j +(1+r1 )a∗

X
j=n,m

(38)
Note that τw1 and τr1 are defined in (30). Further, I assume that the government willsave at least the same amount of government savings per young individual as at theinitial steady state. By substituting τw1 and τr1 into SP1 and using the homogeneityof the production function, we have (see Appendix B2)
SP1 = N1nZ 1+eα1+α∗FL (N1n l∗ z, (s∗ + a∗ )N0n (1 + α∗ ))l∗ − cy∗ − s∗ − g y − g ind − a∗ ]dz, (39)where N1n = N0n R(α∗ ).
Thus, from the MPL conditions (32) and (33), the inside of the integration ispositive for z ∈ [1 + α∗ , 1 + αe]. This fact means that this tax plan is feasible in period
1. The government can use some of its budget surplus to increase the supply of publicly
provided private goods and save the remainder. Let a1 be the balance of government
savings per young individual at the end of period 1, where a1 > a∗ . What will be the

23

N0j .

net budget surplus in period 2, SP2 ?
Note that the natives and immigrants in cohort 1 will save the same amount as
those in the cohort at the initial steady state, because under the proposed tax policy
they face the same after-tax wage rate and interest rate as at the initial steady state.
This fact implies that s1 = s∗ for both natives and immigrants in cohort 1. Assume
that in period 2, the government will save at least a∗ per young individual. Thus, SP2
becomes
SP2 =X

N2j ×(w2 τw2 −g y −g ind −a∗ )+

j=n,m

X

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

j=n,m

X

N1j .

j=n,m

(40)
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.
(41)
Again, the government sets τw2 and τr2 such that the 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 + αe)+ (1 + αe)N1nZ s∗ +a1[FK (l∗ R(eα), z) + 1 − δ]dzs∗ +a∗Z αe

R0 (α) [FL (l∗ R(z), (s∗ + a∗ ))l∗α∗−{cy∗ + s∗ + g y + g ind + a∗ } dz.

(42)

The first term of (42) measures the welfare gain that arises from the additional savings
that the government accumulates at the end of period 1. The second term measures
the welfare gain that arises from the increased population growth rate in the presenceof the PYGO social security system. From the MPK condition (37) and the conditionon the population growth rate, the inside of the first integration is positive. Moreover,from (34), the inside of the second integration is positive. Thus, SP2 is positive and24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 allocate the rest to increase the balance of government
savings. Similarly, the government surplus for period t becomesn
SPt = Nt−1
(1 + αe)n+ (1 + αe)Nt−1Z s∗ +at−1[FK (l∗ R(eα), z) + 1 − δ]dzs∗ +a∗Z αe

R0 (α)[FL (l∗ R(z), (s∗ + a∗ ))l∗α∗

− {cy∗ + s∗ + g y + g ind + a∗ }]dz,(43)where at−1 is the balance of government savings per young individual at the end of
period t − 1. This fact implies that SPt > 0 for all t = 1, 2, .... Thus, we have thefollowing 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 path of capital stock and the balance of government
savings when the government increases the INR. To examine the government savings
path, I need to specify how much of the government surplus, SPt , is placed into theadditional government savings. For the analysis, I assume that the surplus that arises
from the increased balance of government savings in period t − 1, which is the first
integration of SPt , is put into the additional government savings added to a∗ in periodt. Note that the government could use some part of the second integration in SPt ,the surplus generated directly from the increased immigration. Thus, my assumption
is a conservative value for government savings. However, I show that even with thisconservative level, the economy reaches the golden rule level of capital stock per capita

25

within a finite time in a Pareto-improving way. Note that the total number of youngnR(eα)(1 + αe). Thus, the balance of government savings
individuals in period t is Nt−1

per young individual at the end of period t minus a∗ for t ≥ 2 becomesZ s∗ +at−1
nNt−1(1 + αe)at − a = n[FK (l∗ R(eα), z) + 1 − δ]dz
α)(1 + αe) s∗ +a∗Nt−1 R(eZ s∗ +at−1
1=[FK (l∗ R(eα), z) + 1 − δ]dz
R(eα) s∗ +a∗∗(44)

≡ Q(at−1 ).
Note that from (37), we haveN n (1 + αe)at − a ≥ n t−1
Nt−1 R(eα)(1 + αe)∗Z s∗ +at−1
R(eα)dzs∗ +a∗= at−1 − a∗ .

(45)

Thus, at is increasing over time as long as it is determined according to equations (44)and (37). Now, consider the graph of at = Q(at−1 ) + a∗ , where at is measured on thevertical axis and at−1 on the horizontal axis. Q(at−1 ) + a∗ is equal to a∗ at at−1 = a∗ .
Q0 (at−1 ) is
Q0 (at−1 ) =1(FK (l∗ R(eα), s∗ + at−1 ) + 1 − δ) .
R(eα)

Thus, Q(at−1 ) is increasing and concave due to the diminishing MPK. The slope ofQ(at−1 ) + a∗ at at−1 = a∗ is
Q0 (a∗ ) =1(FK (l∗ R(eα), s∗ + a∗ ) + 1 − δ) .
R(e
α)

(46)

Because of the assumption regarding the golden rule (36), Q0 (at−1 ) at at−1 = a∗ isstrictly greater than 1. On the other hand, owing to the diminishing MPK, Fk approaches zero as at−1 rises from a∗ . Thus, at = Q(at−1 ) + a∗ and the 45-degree line
intersect at at−1 = a∗ and a∗∗∗ , where a∗∗∗ > a∗ . Let a∗∗ be the point where Q0 (a∗∗ )26

= 1, implying that at a∗∗ ,∂F (l∗ R(eα), s∗ + a∗∗ )+ 1 − δ = R(e
α).
∂K

(47)

In other words, at a∗∗ , the golden rule is satisfied. Note that the government canchoose a1 so that a1 > a∗ because the surplus in period 1 is strictly positive. Fromthe graph of at = Q(at−1 ) + a∗ , we see that at keeps increasing from a small a1 > a∗ .
Before it reaches a∗∗∗ , it reaches a∗∗ within a finite time, suggesting that the economy
achieves 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 ,namely the surplus that arises from the increased balance of government savings inperiod t − 1, is placed into government savings in period t. However, the second
integration of SPt , namely the surplus that arises directly from the increase in immigration, can also be placed into government savings. Thus, the government can shorten
the time to reach the golden rule level by adding the surplus that arises directly fromincreased immigration into government savings. Second, we note that the government
can induce the economy to reach the modified golden rule level within a finite time in a
Pareto-improving way because the modified golden rule level is lower than the golden
rule level.
Proposition 3. (Reaching the (modified) golden rule level) Suppose that a
PYGO social security system is used initially and that the MPL condition is satisfied.
Then, by accepting more immigrants, the government can induce the economy to reachthe (modified) golden rule level within a finite time in a Pareto-improving way.
3.3

Intra-redistributional Channel and Difference in Productivities andPreferences

When immigrants earn less or consume more publicly provided private goods thannatives, accepting more immigrants could decrease the welfare of natives, because itmeans that more resources are taken from natives and used by immigrants. This istermed the intra-redistributional channel of accepting more immigrants. A similar re-27

distribution could also occur when the preferences of immigrants and natives differ
and when the labor supply or savings of immigrants differ from those of natives. To
analyze this redistributional channel of accepting more immigrants, reconsider a permanent change to the immigration policy such that α
e > α∗ . Let (cy,j∗, , co,j∗ , sj∗ , lj∗ )be consumption in the young period, consumption in the old period, and the savingsand labor supply of type j(j = n, m), respectively at the initial steady state. Since I
assume that the preferences and productivities of natives and immigrants differ, I use
a superscript j for the variables in the steady-state situation. Further, let g i,j be thepublicly provided private goods of type i for type j nationality, where i = y, o, indand j = n, m. To guarantee that accepting more immigrants Pareto-improves welfare,I assume that the following condition is satisfied:
Modified MPL condition=R0 (α)[+R0 (α)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+intra-redistribution(α) for all α ∈ [α∗ , α
e] :

(48)

where intra-redistribution(α)= R(α)[∂F m m∗φ l − (cy,m∗ + sm∗ + g y,m + g ind,m + a∗ )]∂L∂F+ (1 − δ))(sm∗ + a∗ ) − (co,m∗ + g o,m + g ind,m )]∂Kn+ αsm∗ )∂F (R(α)ln + φm J(α)lm , sn∗ Nt−1
∂F
and
=
∂L
∂L
∂F
∂F (R(α)ln + φm J(α)lm , sn∗ + αsm∗ )
,
=
.
∂K
∂K

+[(

In the first and second lines, the inside of the bracket represents the upward intergenerational transfer made by a native and an immigrant, respectively. Thus, thesum of the first and second lines is essentially the same as the MPL condition in thepreceding section. The intra-redistribution channel captures the intra-redistribution
between immigrants and natives. The first bracket of the intra-redistribution is theamount a young immigrant earns minus the sum of the amount a young immigrant
receives and the government saving for a young immigrant. The second bracket is the28amount an old immigrant earns plus the return from the government savings for young
immigrant minus the amount an old immigrant receives. When the productivities andpreferences of immigrants and natives are the same, the intra-redistribution is equal tozero and this is equivalent to the MPL condition. Thus, the modified MPL condition
above states that the total upward intergenerational transfer is large enough to offset
the intra-redistribution from immigrants to natives. From (48), we therefore have thefollowing propositions.17
Proposition 4 Assume that natives and immigrants do not have the same productivities and preferences. If the modified MPL condition is satisfied, the welfare of
all generations may be Pareto-improved 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 2 show that accepting more immigrants can Pareto-improve thewelfare of all generations if there are upward intergenerational transfers. Furthermore,
Proposition 3 shows that if the government can place some of the welfare gain into
its savings, it may induce the economy to reach the golden rule level of capital ina Pareto-improving way within a finite time. This result is in sharp contrast to thefindings in the literature on social security reform that one generation must bear adouble burden in order to increase the capital stock of the economy in the presence ofa PYGO social security system (Geanakoplos, Mitchell and Zeldes (1998)). In addition,
Propositions 4 and 5 show that even if an immigrant earns less than a native, accepting
more immigrants Pareto-improves welfare if the intra-redistribution is lower than theintergenerational transfer from the young to the old.
17

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

29

A number of issues arise regarding accepting Propositions 1–5, however. First, these
propositions are based on a two-period overlapping generation model. In a realistic
multi-period overlapping generation model, an economy might not be able to reachthe (modified) golden rule level of capital stock per capita within a finite time in a
Pareto-improving way by accepting more immigrants. Second, although Proposition 3
shows that the economy reaches the (modified) golden rule level of capital stock per
capita within a finite time, in practice it might take a long time, as long as 1000 years,
to do so. Third, Propositions 1–5 are silent on the quantitative effect on welfare. Given
that it might take quite a long time to reach the golden rule level, the welfare gainof accepting more immigrants may be very small. Finally, Propositions 4 and 5 say
nothing about the degree to which the difference in productivity between immigrants
and natives is allowed with respect to Pareto improvement when more immigrants are
accepted.
This section addresses these four issues by using the computational overlapping
generation model developed by Auerbach and Kotlikoff (1987). I assume that themodel economy consists of overlapping generations in which each generation lives for
80 periods and that the probability of death increases with each passing period. I also
assume that the model economy is similar to the US economy in several dimensions.18
In the analysis, I assume both that the model economy is initially on the balanced
growth path and that the PITN is similar to the percentage indicated by data takenfrom the US Census in 2000. Then, I examine whether the welfare of all the generationsin the model economy can be Pareto-improved by increasing the PITN by a reasonableamount. In addition, I investigate how long it takes for the model economy to reachthe (modified) golden rule level in a Pareto-improving way and quantify the Paretoimproving welfare gain. To check the robustness of the results, I also recalculate themodel by changing the following parameter values: the replacement rate, initial government debt (assets) level, level of immigrants’ earnings, immigrants’ consumption of
publicly provided private goods, probability of immigrants returning to their countries
18

However, the model economy is different in several important dimensions as well. For example,
it does not include certain aspects of open economies such as international trade and capital mobility
and does not incorporate the accumulation of human capital for either natives or immigrants.

30

of origin, constant relative risk aversion (CRRA), and time preference rate.
4.1

The Model Economy: Auerbach–Kotlikoff Model with Immigration

Agents appear in the model from age 1, which 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 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 age i, given that he or she
is alive until age i − 1. The lack of data forces the assumption that pi is the same fornatives and immigrants. To simplify the notation, I assume that p1 = 1.19 An agent
who enters the model in period t maximizes the following utility function:

max

45
X

i
Y
i
β
pq

i=1

q=1

(

) 80
(
)
i
i,j
i,j
ζ
1−ζ 1−γ
ζ(1−γ)X Y[(ci,j
]
[c
]
i,j
i,j
t−1+i ) (1 − lt−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 ofi,ja type j agent of age i in period t − 1 + i. gt−1+i
is the amount of publicly provided

private goods used by a type j agent of age i in period t − 1 + i. I assume that thei,jamount of gt−1+i
is chosen by the government. In this formulation, as in Storesletten(2000), I postulate that immigrants assume they will stay in the host country until theend of their lives.20 β is the time preference rate and γ is the coefficient of constantrelative risk aversion. Let si,jt be the savings of type j agents of age i at time t. The
budget constraint of an 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,
19

(51)
(52)

Infant and child mortality is defined in (55).
Alternatively, we could postulate that immigrants enjoy the same wage level in their home countries as in the host country if they decide to return. The two assumptions generate the same results.
20

31

where H i,j is the efficient unit of human capital of type j agents 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 laborin period t. I assume that an individual cannot have a negative savings balance. Oncean individual dies, the government imposes a 100 percent inheritance tax.21

bi,j
t−1+i

is the social security benefit for type j agents of age i given in period t − 1 + i. Fori,ji ≤ 45, bi,jt−1+i = 0 and for i ≥ 46, bt−1+i is determined as follows:

j
j
bi,j
t−1+i = 12 × RR × AIM E (t) and AIM E (t) =

P45

i,j
45−i
wt−1+i lt−1+i
H i,ji=1 (1 + µ)

45 × 12,

where RR is the replacement rate and AIM E j (t) is the average income monthly index
of the cohort who become age 1 at time t.
Next, 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 the income
per capita growth rate, while Lt is the efficient unit of labor supply in period t.
Let 1 − pbi be the probability that an immigrant returns to his or her home country
at the beginning of age i, given that he or she stays in the host country at age i − 1
for i ≥ 2. To simplify the notation, I assume that pb1 = 1. Let Ntij be the number of
agents of age i of type j at time t. Then, Ntim = pi pbi Nti−1,m and Ntin = pi Nti−1,n . Lt isdefined as follows:

Lt =

45 X
X

H i,j Nti,j lti,j .

(54)

i=1 j=n,m

I assume that all immigrants arrive in the host country at age 1. Let σn be thegrowth rate of the number of natives of age 1 at the steady state and σm be the same
21

This assumption have little effect on my results. As long as the inheritance is distributed equally
among those who are alive, the effect of changing the inheritance tax rate affects the balance of totalsavings in the model economy and the effect is similar to the effect of changing the time preference
rate. As my robustness checks show in section 4.4, changing the time preference rate has little effecton my results.

32

1,n P80in
i=1 (1 +i=1 Nt = NtP−(i−1)The total number of immigrants is Nt1,m 80
×i=1 (1 + σm )growth rate for immigrants. The total number of natives isσn )−(i−1) × Πiq=1 pq .

P80

Πiq=1 pq pbq . This fact implies that the PITN is constant if and only if the growth ratesof Nt1,n and Nt1,m are the same. Thus, to calculate the steady state, I assume that
σn = σm = σ and that the children of immigrants become natives if their parents stayin the host country until the children attain adulthood. Then, Nt1,n is determined by
the fertility rates of both immigrants and natives and the return rate of the former. Letηji,j be the age-specific fertility rate of type j agents at age i. The number of children
born to type j parents of age i is Nti,j η i,,j . Let d be the infant and child mortality rate.
1,nThen, the number of natives of age 1 at t + 20, Nt+20, is determined as follows:22

1,n
Nt+20
= (1 − d) ×( 80
Xη i,m Nti,m Π20
bi+x +
x=1 p

i=1

80
X

)
η i,n Nti,n

.

(55)

i=1

The PITN at the steady state (see Appendix A6) becomes
P80
P80
1
1
i,n
i,m
−
× Πiq=1 pq
N
i=1 (1+σ)i+19 η
1−d
i=1 t
P80
i,n × 100 = P80
1i,m × Πibi+xbq Π20x=1 p
q=1 pq p
i=1 Nti=1 (1+σ)i+19 η
P80(1 + σ)−(i−1) × Πiq=1 pq pbq
× Pi=1× 100.
80−(i−1) × Πi
q=1 pqi=1 (1 + σ)

(56)

Equation (56) says that the steady-state PITN is determined once the age 1 population
growth rate, σ, the return rate of immigrants, and the fertility rates of natives andimmigrants are set. Conversely, we can choose σ so that the resulting PITN is consistent
with the data.23
The capital stock in period t is the sum of the balance of individual and government
savings. Let at−1 be the balance of government assets (or debt if this is negative) per
capita at the end of period t − 1. Then, the total capital stock in period t is

Kt =

80 X
X

i,j
Nt−1,isi,jt−1 + at−1

i=1 j=n,m
22
23

80 X
X

i,j
Nt−1
.

(57)

i=1 j=n,mThis comes from the assumption that the age 1 in our model corresponds to age 20 in real life.
The RHS is an increasing function of σ at the parameter values consistent with the US data.

33

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−θ
t

rt = θKtθ−1 Et1−θ L1−θ
.
t

and

(58)

Now, consider the initial balanced growth path where the capital/labor ratio (inefficient units) stays constant. Let w∗ (1 + µ)t and r∗ be the wage rate and interest rate
in period t on the initial balanced growth path. s∗i,j (1 + µ)t and b∗i,j (1 + µ)t are thesavings and social security benefit for type j agents of age i in period t on the initialbalanced growth path. Let a∗ (1 + µ)t and g ∗i,j (1 + µ)t be the government assets (debt)
per capita and publicly provided private goods for type j agents of age i in period t on
the initial balanced growth path. Because immigrants may use more public services(e.g., children’s education), here I use a superscript j. Let w∗ and r∗ be the efficiency
unit wage rate and interest rate on the initial balanced growth path determined from(57) and (58). Then, the government budget constraint on the initial balanced growth
path is((1 + µ)t−1∗

τwt w (1 + µ)Lt + τrt r

∗

80 X
X

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

80
X

X

Nti,j + (1 + r∗ )a∗i=1 j=n,m

−

80
X

X

i=46 j=n,mb∗i,j (1 + µ) × Nti,j − κ

80
X

80 X
X

i−1,j ∗i−1,j(1 − pi )Nt−1
st−1

i=2 j=n,mX

i=1 j=n,m
80 X
45
X

i,j
Nt−1
−

80
X

X

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

= 0,

(59)

i=46 x=11

where τw∗ and τr∗ are the wage tax rate and capital income tax rate on the initialbalanced growth path. b∗i,rm,x and Nti,rm,x are, respectively, the social security benefit
and 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 that actually claim social security benefits. US social security does not
require residence as long as the individual is eligible for benefits. I assume that animmigrant is eligible to receive social security benefits as long as he or she pays social

34

security contributions for at least 10 years. The first term of (59) is the revenue
from the wage tax and the second term is the revenue from the capital income tax.
The third term is the revenue from the inheritance tax. In my simulation, as noted
earlier, I assume that the government imposes a 100 percent inheritance tax rate. The
fourth term is the revenue from issuing government bonds and the fifth term is theexpenditure on the principal and interest of these bonds. The sixth and seventh terms
are the expenditure on the publicly provided private goods and social security benefitsof natives and immigrants who reside in the host country. The last term is the social
security benefits of eligible immigrants who returned to the home country.
When the government increases the PITN, the wage rate decreases and 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 balanced growth path. This fact implies thatthe 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) =

P45

i,j
45−i
wt−1+i (1 − τwt )lt−1+i
H i,ji=1 (1 + µ).
(1 − τw∗ ) × 45 × 12

(61)
When the tax and social security benefit formulae are adjusted according to (60)and (61), the individual budget constraint is the same as on the initial balanced growthpath.24 Thus, consumption, labor supply, and savings do not change. Then, a government budget surplus would exist even if the government were to spend the same amountof publicly provided private goods per person as on the initial balanced growth path,
as discussed in the previous section. The government can use this surplus to increase
the balance of government savings or increase the level of publicly provided private
24

Note that when wt (1 − τwt ) = wt∗ (1 − τw∗ ), AIM E j (t) =35i,j45−i ∗w (1+µ)t−1+i lt−1+i
H i,ji=1 (1+µ)

P45

45×12

.

goods. Let V be the distributional parameter that indicates what percentage of thebudget surplus is saved. The government can keep increasing its savings balance until
the economy reaches the golden rule level or modified golden rule level. As long as
the MPK is greater than or equal to the (modified) golden rule level, the balance ofgovernment savings in period t is determined from the following equation:t(1 + µ) at

80 X
X

Nti,j = V × SPt

i=1 j=n,m, where SPt is the government budget surplus in period t. If the MPK is equal to the(modified) golden rule level, V becomes 0. SPt is defined as follows:(t−1

SPt = τwt wt Lt + (1 + µ)

τrt rt80 X
X

i−1,j ∗i−1,jpi Nt−1s+ (1 + rt )i=1 j=n,m(1 + rt ) × at−1

80
X

X

−

X

b

∗i,j

i−1,j ∗i−1,j(1 − pi )Nt−1s

i=1 j=n,m

i,j
Nt−1
−

i=1 j=n,m
80
X

80 X
X

80
X

X

Nti,j × g ∗,i,j (1 + µ)i=1 j=n,m(1 + µ) × Nti,j − κ

80 X
45
X

)
b

∗i,rm,x(1 + µ) × Nti,rm,x

.

(62)

i=46 x=11

i=46 j=n,m

The rest of the surplus available for increasing publicly provided private goods is(1 − V ) × SPt , which is distributed equally across the entire population (including
immigrants) in period t.25 Then, after increasing the PITN, the amount of publicly
provided private goods, gtij , becomes(1 − V ) × SPt
gtij = g ∗i,j (1 + µ)t + P80
i,n
i,m .
i=1 (Nt + Nt )(63)

For the (modified) golden rule level, given the intergenerational discount rate forthe modified golden rule level, we say that the economy reaches the (modified) golden
rule level if the MPK equals the sum of the growth rate of the efficiency unit of labor,
the depreciation rate, and the intergenerational discount rate for the modified golden
rule level. In the case of the golden rule level, the intergenerational discount rate for
25
Because the government could distribute a greater share of this surplus to natives than to immigrants, this assumption underestimates the welfare gain to the former.

36

the modified golden rule level is set to zero.
4.2

Policy Experiments and Parameter Values for the Simulation

Policy Experiments
To calculate the PITN, I take data from the US Census in 2000. These data showthat the PITN above the age of 20 is 15.5 percent. Thus, I assume that the PITN
at the initial balanced growth path is 15.5 percent. The target PITN is derived byexamining past US Census data, which show a ratio of 5 percent in 1970, increasing to18.3 percent in 2010.26 Since the PITN increased by more than 10 percentage points
in the 40 years from 1970 to 2010, I assume that a 10 percentage point increase in
the PITN over the next 80 years is tolerable. Therefore, I set the target PITN to 25.5
percent. In the slow benchmark case, the PITN reaches its target in the 80th year andremains at that level thereafter. By contrast, in the intermediate and fast cases, the
PITN hits its target level in the 62nd and 42nd years, respectively.27 Finally, I assume
that all immigrants arrive at age 1. Figure 1 illustrates the PITN over time.28
Fertility, Mortality, the Return Rate of Immigrants, and Other Parameter
ValuesParameter values regarding fertility, mortality, the return rate of immigrants, government expenditure, government debt, taxes, preferences, and the production function
26
Data in 1970 are derived from the US Census, while those in 2010 are taken from the CPS. Since
2000, data on foreign-born US residents have only been provided by the American Community Survey
and the CPS, not the Census, which raises a concern about the comparability of the CPS and Census
data. However, for 2000, when both Census and CPS data are available, the PITN in the former is12.42 percent compared with 11.54 percent in the latter, a minor difference. Schmidley and Robinson(2003) also conclude that the difference between estimates based on Census and CPS data are trivial.
27
In the fast and intermediate cases, the PITN keeps increasing even after reaching its target levelto allow for a smooth transition. See Figure 1 for more details.
28
More specifically, each of the three cases is calculated as follows. Let f15.5 be the steady-state age
1 INR when the PITN is 15.5 percent. Define f25.5 in a similar fashion. Then, the age 1 INR in periodt)t, ft , in the slow case is ft = f25.5 for all t. For the intermediate case, ft = f25.5 + 0.1 × f25.5 × ( 15
for 1 ≤ t ≤ 15, ft = 1.1 × f25.5 − 0.1 × f25.5 × (t − 15) for 16 ≤ t ≤ 30 and ft = f25.5 for 31 ≤ t. Fortthe fast case, ft = f25.5 + 0.4 × f25.5 × ( 15
) for 1 ≤ 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 INR is defined in the first term on the RHS of equation (40). Thus,
once the PITN is determined, the steady-state age 1 INR is calculated from equation (40).

37are standard. To save space, I discuss them in Appendix A2.
4.3

Results

Figures A3–A5 respectively show the balance of assets, consumption, and leisure for the
lifecycle of an individual on the initial balanced growth path in the benchmark analysis.
At the age of 46, leisure consumption becomes 1 because of mandatory retirement. On
the initial balanced growth path, the capital/output ratio is 2.98, which is higher thanthe values used in Storesletten (2000) (2.4) and Nishiyama and Smetters (2007) (2.7),
but lower than the value (3.2) used in standard business cycle research (Cooley and
Prescott (1995)). To check how my results are affected by the capital/output ratio, Ichange the time preference rate and examine how the results change in the robustness
checks presented in section 4.4.
Table 1 shows the parameter values and welfare effect of increasing the PITN for
different values of V in equation (62), which is the share of the government surplus
placed into savings (see column (3)). With respect to the rate at which immigrants
are accepted, I consider the three benchmark cases introduced above (i.e., it takes 80,
62, and 42 years for the PITN to reach its target level).
Columns (4)–(9) present the values calculated within the simulation. Column (4)shows how many years it takes for the economy to reach the (modified) golden rulelevel. When it does not reach the (modified) golden rule level within 300 years, thisis indicated by * or **. * indicates that the capital/labor ratio (efficient unit) in the300th year is higher than that on the initial balanced growth path and that it is still
increasing. ** indicates that the capital/labor ratio in the 300th year is lower thanthat on 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 level compared with the level on the initial balanced growth path.
When V = 100%, the capital stock per efficient unit of labor increases by 102 percent.
Column (6) shows how much publicly provided private goods increase comparedwith the level on the initial balanced growth path. At V = 100%, publicly provided
38

private goods increase by about 36 percent. To calculate columns (5) and (6), I evaluate
at the year in which the economy reaches the golden rule level if it reaches within 300
years, and at the 300th year otherwise.
Column (7) shows the rate at which the utility, measured by the expenditure function, of the cohort 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 growthpath. When the economy does not reach the golden rule level, I calculate the utilityof the cohort born in the 300th year. In the expenditure function, the price vector on
the initial balanced growth path is used to evaluate utility. Note that all welfare gain
is distributed through an increase in publicly provided private goods.
Column (8) shows the share of the PDV of the increased publicly provided private
goods relative to initial GDP. For example, in row (16), the PDV of the increased
publicly provided private goods is 12 percent of initial GDP. To discount the increased
publicly provided private goods in future periods, I apply a 5 percent discount rateinstead of the equilibrium interest rate in order to eliminate the effect of the discountrate when comparing different cases.
Column (9) measures the welfare gain of natives and their descendants rather thatof immigrants and their descendants. Specifically, it measures the degree to which theutility of natives and their descendants, not including immigrants and their descendants, is Pareto-improved by accepting more immigrants compared with the utility
level on the initial balanced growth path. As before, it evaluates utility by using theexpenditure function. I apply the price vector on the initial balanced growth pathfor the expenditure function and the equivalent variation to measure the difference inthe utility levels for these 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 the distribution of the welfare gain. The welfare gain is distributed equally among
immigrants and natives in the forms of publicly provided private goods. Row (16) incolumn (9) shows that the PDV of the Pareto-improving welfare gain of natives andtheir descendants comprises 11 percent of initial GDP.
39

In rows (2) and (3), (5) and (6), and (8) and (9), I shorten the years needed toreach the target PITN and increase the rate at which the PITN increases. When thenumber of years 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 ofinitial GDP (row (18)).
Table 1 shows that the number of years needed to reach the golden rule level
decreases as V increases because the government saves more for future cohorts. In
contrast, the PDV of increased utility, measured as the share of initial GDP, increases
as V decreases as long as V is greater than or equal to 50 percent. The PDV of
increased utility, measured as the share of initial GDP, is highest when V=50. In thiscase, the quantified Pareto improvement ranges from 21 to 26 percent of initial GDP.
In Table 1, I set the intergenerational discount rate for the modified golden rulelevel to 0 percent and the target level of capital stock to the golden rule level. However,targeting capital stock at the golden rule level does not necessarily maximize the PDVof 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 the intergenerational
discount rate for the modified golden rule level when V=100. Table A2 shows that thewelfare gain is maximized when the intergenerational discount rate is 3 percent.
In Table 2, which I consider to be the representative case of my simulation, I
recalculate all the rows of Table 1 by setting the intergenerational discount rate forthe modified golden rule level to 3 percent. In this table, the Pareto-improving welfare
gain of increasing the PITN is more than 20 percent of initial GDP and the capitalstock per efficient unit of labor increases by 18 percent as long as V is greater than orequal to 50 percent.
Figures 2 and A6 show the MPK and capital/output ratio over time for different
values of V when the target capital stock is the modified golden rule level (3 percent
intergenerational discount rate). The MPK increases initially because of the acceptanceof more immigrants; however, as the government savings balance increases, the capitalstock per efficient unit of labor begins to rise and the MPK continues to decrease until
the economy reaches the golden rule level. The capital/output ratio also displays a40

consistent pattern.
Figure 3 compares the utility levels of all cohorts on the initial balanced growth
path with those of all cohorts for different values of the share for government savings(V) when the target capital stock is set at the golden rule level. For example, when
V=100, all the surplus is placed into savings until the economy reaches the golden
rule level and only distributed to individuals after the economy reaches the golden rulelevel. This fact implies that the utility of the 65th cohort, which dies in the 65th year,
starts to experience higher utility than that on the initial balanced growth path. In
all the cases considered in Tables 1–4, the simulation results show that all cohorts are
Pareto-improved; this finding confirms my theoretical results.29
Figure 4 shows the importance of the government savings balance for the new equilibrium path. It presents 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, on
the new equilibrium path, the interest income from the government savings balance
contributes a substantial amount.
4.4

Robustness Checks

Tables 3 and 4 present the robustness checks of the results in Table 2. Rows (1)–(6) inTable 3 check whether the results in Table 2 are sensitive to the initial government debt
(assets) level. As argued in A2, different authors assume different levels of government
debt (assets) on the initial balanced growth path. In rows (1)–(3) in Table 5, I set
29

The presented simulation results differ from those of some previous studies. For example, Fehr,
Jokisch and Kotlikoff (2004) report that the welfare gain of doubling immigrants is very small. Lee
and Miller (1998) similarly argue that the fiscal impact of accepting an additional 100,000 immigrants
is very small. Several factors in those studies generate different results. Fehr, Jokisch and Kotlikoff(2004) analyze immigration policy in an open economy setting, whereas I use a closed economy setting.
In an open economy setting, the effect of additional government savings is offset by the mobility ofcapital. In my simulation, the government can use the budget surplus, which is obtained by accepting
more immigrants, for savings and the amount of savings affects overall welfare. Likewise, Lee andMiller (1997) consider a much smaller increase in the number of immigrants than I do. By contrast,
my simulation results are consistent with those presented by Auerbach and Oreopoulos (1999). Theyfind that if the initial fiscal imbalance is not adjusted, then the welfare loss of halving the number of
new immigrants is substantial. This finding is consistent with my theoretical and simulation results.

41

the initial government debt to 10 percent of private capital instead of 0 percent. In
rows (4)–(6), I assume that the initial government debt level is -10 percent of 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 the welfare
gain of accepting more immigrants. Rows (7)–(12) confirm this prediction. Decreasing
the replacement rate by 10 percentage points decreases the welfare gain by 7 percentage
points in terms of the percentage of initial GDP.
Rows (13)–(21) in Table 3 present sensitivity checks on immigrants’ earnings. Following Storesletten (1995), I set the immigrant wage rate to 84.3 percent of that ofnatives. By deriving data from the CPS 2000 June supplement, I find that immigrants’earnings are 91 percent of those of natives. Rows (13)–(15) assume that the wage rateof immigrants is 89.3 percent of that of natives, rather than 84.3 percent. Rows (16)–(18) and (19)–(21) assume that the wage rate of immigrants is 79.3 and 75 percent ofthat of natives. The results in rows (19)–(21) show that if immigrants earn 25 percent
less than natives, the welfare gain of accepting more immigrants is 17 percent of initial
GDP instead of 23 percent.
Rows (22)–(27) in Table 3 show the sensitivity checks on the consumption of publicly
provided private goods by immigrants. In Tables 1 and 2, I assumed that immigrants
and natives consume publicly provided private goods equally. In contrast, in rows (22)–(24) of Table 3, I assume that young immigrants consume 20 percent more publicly
provided private goods than young natives. The PDV of the welfare gain comprises 17
percent of initial GDP instead of 23 percent. Rows (25)–(27) assume that immigrantsof all ages consume 20 percent more publicly provided private goods than natives. In
this case, the PDV of the welfare gain is 15 percent of initial GDP.
In Table 4, I present robustness checks by changing the parameter values of the utility function and the return rate of immigrants. Rows (1)–(6) examine the sensitivityof the results regarding the parameter values for CRRA. In Tables 1–3, I assumed that42

the CRRA is 3. Auerbach and Kotlikoff (1987) and Storesletten (2000) assume that
the CRRA is 4, while Nishiyama and Smetters (2007) assume that it is 2. Rows (1)–(3)in Table 4 assume that the CRRA is equal to 4 and rows (5) and (6) assume that it is
2. The results presented in rows (1)–(6) show that those in Table 2 are not sensitive
to the CRRA value. Rows (7)–(12) examine the sensitivity of the results to the time
preference rate. Although it is common to assume that the time preference rate is
greater than 1, readers might think that the results in Tables 1 and 2 are sensitive tothe assumption that the time preference rate is greater than 1. Theoretically, lowering the time preference rate would result in lowering savings and would decrease thecapital/labor ratio on the initial balanced growth path. Since increasing the numberof immigrants would increase capital stock and the production function displays thediminishing MPK, lowering the initial capital stock as a result of assuming a lower time
preference rate would increase the welfare gain of increasing the PITN. The results inrows (7)–(12) confirm this theoretical prediction, but show that the magnitude of thosechanges is very small. For example, changing the time preference rate from 1.011 to
0.99 increases the PDV of the welfare gain from 23.14 to 23.52 percent of the initial
GDP.
Finally, rows (13)–(15) in Table 4 present robustness checks by assuming that thereturn rate of immigrants is 0. They show that the results presented in Table 2 change
little and are robust.
5

Conclusion

This study examined both theoretically and quantitatively the effect on welfare ofincreasing the PITN in the presence of a PYGO social security system. The results
for both the theoretical and the quantitative analysis show that the welfare gain of
accepting more immigrants is robust and non-trivial. If intergenerational governmenttransfers exist from the young to the old, the government can lead an economy tothe (modified) golden rule level within a finite time in a Pareto-improving way byincreasing the PITN. The PDV of the welfare gain of increasing the PITN from 15.5
to 25.5 percent amounts to about 20 percent of initial GDP. In the shortest case, the43

economy reaches the golden rule level in the 112th year in a Pareto-improving way.
The presented analysis suggests that accepting more immigrants may be an important
tool for policymakers when addressing the economic problems caused by the existenceof a PYGO social security system.
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[3] Auerbach. Alan and Oreopoulos, Philip, “Analyzing the Fiscal Impact of U.S.
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[4] Belan, Pascal and Pestieau, Pierre, “Privatizing Social Security: A Critical Assessment”, The Geneva Papers on Risk and Insurance Vol. 24, pp. 114–130, 1999.
[5] Bonin, H., Raffelhüschen,B. and Walliser, J. “Can Immigration Alleviate the Demographic Burden?” FinanzArchiv 57, pp1-21, 2000.
[6] Borjas, George, “The Economics of Immigration”, Journal of Economic LiteratureVol. 32, pp. 1667–1717, 1994.
[7] Canova, Fabio and Ravn, Morten, “The Macroeconomic Effects of German Unification: Real Adjustments and the Welfare State”, Review of Economic Dynamics,
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[8] Cass, D. (1972), ”On Capital Overaccumulation in the Aggregative NeoclassicalModel of Economic Growth: A Complete Characterization”, Journal of Economic
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[9] Centers for Disease Control and Prevention, Vital Statistics of the United States,
1993, http://www.cdc.gov/nchs/products/vsus.htm, xxxx.
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[10] Collado, M. Dolores, Iturbe-Ormaetxe, Inigo, and Valera, Guadalupe, “Quantifying the Impact of Immigration on the Spanish Welfare State”, International Tax
and Public Finance, Vol. 11, pp. 335–353, 2004.
[11] Cooley, Thomas and Prescott, Edward, “Economic Growth and Business Cycles”,
in Frontiers of Business Cycle Research, Cooley, Thomas (ed.), Princeton University Press, New Jersey, 1995.
[12] Diamond, Peter, “National Debt in a Neoclassical Growth Model”, American
Economic Review, Vol. 55, pp. 1126–1150, 1965.
[13] Dixit, Avinash and Norman, Victor, Theory of International Trade: A Dual,
General Equilibrium Approach, Cambridge University Press, Cambridge, 1980.
[14] Fehr, Hans, Jokisch, Sabine, and Kotlikoff, Laurence, “The Role of Immigrationin Dealing with the Developed World’s Demographic Transition”, FinancArchiv,
Vol. 60, pp. 296–324, 2004.
[15] Feldstein, Martin, “The Effects of the Ageing European Population on Economic
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[16] Feldstein, Martin, and Samwick, Andrew, “The Transition Path in Privatizing
Social Security”, in Privatizing Social Security, Feldstein, Martin (ed.), Universityof Chicago Press, Chicago, 1998.
[17] Fix, Michael, Passel, Jeffrey, and Zimmermann, Wendy, “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, 1996.
[18] Geanakoplos, John, Mitchell, Olivia, and Zeldes, Stephen, “Would a PrivatizedSocial Security System Really Pay a Higher Rate of Return?”, in Framing the Social Security Debate. Values, Economics, and Politics, Arnold, R. Douglas, Graetz,

45Michael, and Munnell, Alicia (eds.), Brookings Institution Press, Washington DC,
1998.
[19] Hurd, Michael, “Mortality Risk and Bequests”, Econometrica, Vol. 57, pp. 779–
813, 1989.
[20] IMF, Japan 2012 Article IV Consultation 2012[21] Kotlikoff, Laurence, Smetters, Kent, and Walliser, Jan, “Mitigating America’s Demographic Dilemma by Pre-funding Social Security”, Journal of Monetary Economics, Vol. 54, pp. 247–266, 2007.
[22] Lee, Ronald D. and Timothy W. Miller, Chapter 5 “The Current Fiscal Impact of
Immigrants and Their Descendants: Beyond the Immigrant Household”, in The
Immigration Debate: Studies on the EconomicDemographic and Fiscal Effects of
Immigration, The National Academy of Sciences, 1998[23] Michel, Philippe and Pestieau, Pierre, “Population Growth and Optimality. When
Does Serendipity Hold?”, Journal of Population Economics, Vol. 6, pp. 353–362,
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[24] Nishiyama, Shinichi and Smetters, Kent, “Does Social Security Privatization Produce Efficiency Gains?”, Quarterly Journal of Economics, Vol. 122, pp. 1677–1719,
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[25] Passel, Jeffrey, “Immigrants and Taxes: A Reappraisal of Huddle’s ‘The cost of
Immigrants”’, Manuscript for the Washington Urban Institute, 1994.
[26] Ramsey, Frank, “A Mathematical Theory of Saving”, The Economic Journal, Vol.
38, pp. 543–559, 1928.
[27] Razin, Assaf and Sadka, Efraim, “Unskilled Migration: A Burden or a Boon for
the Welfare State?”, Scandinavian Journal of Economics, Vol. 102, pp. 463–479,
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46

[28] Razin, Assaf, Efraim Sadka, and Benjjarong Suwankiri, Migration and the WelfareState: Political-Economy Policy Formulation, MIT Press, Cambridge, 2011.
[29] Schmidley, A. Dianne and Robinson, J. Gregory, “Measuring the Foreign-born
Population in the United States with the Current Population Survey: 1994-2002”,
Population Division Working Paper No. 73, Bureau of the Census, 2003.
[30] Shimasawa, Manabu and Oguro, Kazumasa, “Impact of Immigration on theJapanese Economy: A Multi-country Simulation Model”, Journal of the Japaneseand International Economies, Vol. 24, pp. 586–602, 2010.
[31] Simon, Julian, “Immigrants, Taxes, and Welfare in the United States”, Population
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[32] Solow, Robert M., “A Contribution to the Theory of Economic Growth”, The
Quarterly Journal of Economics, Vol. 70, No. 1, pp. 65-94, 1956
[33] Storesletten, Kjetil, On the Economics of Immigration, Ph.D. Dissertation,
Carnegie Mellon University, 1995.
[34] Storesletten, Kjetil, “Sustaining Fiscal Policy through Immigration”, Journal of
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Calculation”, Scandinavian Journal of Economics, Vol. 105, pp. 487–506, 2003

47

28
standard caseintermediate
fast

26

PITN(%)

24

22

20

18

16

14

0

50

100

150

200

250

300

year

Figure 1: The PITN over time. The initial PITN is set at 15.5 percent and the target PITN
is set at 25.5 percent.

V=100%
V=90%
V=70%
V=50%
V=30%
V=0%
Modified Golden Rule

0.12

Marginal product of capital

0.115

0.11

0.105

0.1

0.095

50

100

150

200

250

300

year

Figure 2: The MPK over time for different values of the share of the surplus used forgovernment savings (V). The target capital stock is the modified golden rule level with a 3
percent intergenerational discount rate.

48

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 rule levelwith a 3 percent intergenerational discount rate for the modified golden rule level.

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 andthe social security benefit payment in each period. It is assumed that the PITN reaches its
target level in the 80th year. The intergenerational discount rate for the modified golden rulelevel is set at 3 percent.

49

(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 thesum of thesum of PDV% increase%change ofyears
of capital% change of
PDV ofof welfare
publicly
share of theyear
needed tostock perwelfare ofincreased
gain of all
surplus for reachingprovidedcohort born
publicly
natives andreach theefficient
the gov. the golden
private goods
targetunit laborat the golden
providedtheir
per capita at thesavings (V)
rule
PITN
at therule
private goods descendants ingolden 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*
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.

50

(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 ofincreased
welfare gain
share of theyear
capitalpubliclyneeded towelfare of
surplus for reaching stock perprovidedpubliclyof all natives
reach thecohort born
the gov. the golden efficientprivate goods
provided
and theirtargetat the golden
private goods descendants insavings (V)rule
unit labor per capita at the
PITN
rule
the initialat thegolden rulein 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.

51

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).

52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
the PDV of
PDV ofoutputto 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.

53

Appendices A (The following appendices are placed on the journal’s website
as supplementary material, but not included in the main paper.)Appendix A1Consider a simple case in which the depreciation rate is 100 percent (δ = 1) andimmigration policy α is implemented at the initial steady state. Assume that the oldpopulation is equal to 1. In 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 old individual when one plus the population growth rate is equal to R(α∗ ). Now, consider the graph of (y, R(α)), where the vertical axis represents the GDP per old individual and horizontal axis represents one plus
the population growth rate, which is R(α). This implies that y = 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).
The slope of this line, which passes through the origin, is cy∗ + s∗ + a∗ + g y + g ind .
At R(α) = R(α∗ ), the vertical distance of this line represents the total resources used
for the young divided by the number of 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 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. Hence, R(α) increases by R0 (α∗ ) from R(α∗ ). If theslope of y = F (R(α), s∗ +a∗ ) at R(α) = R(α∗ ) is greater than cy∗ +s∗ +a∗ +g y +g ind , thesocial planner can maintain the same allocation of resources per young individual (private consumption, savings, government-provided private goods, and age-independent
public goods) and increase the allocation of resources to each old individual. Clearly,
this constitutes Pareto improvement. Note that when the government accepts immigrants, the surplus is equal to R0 (α∗ ){l∗ FL ((R(α)l∗ , s∗ + a∗ ) − (cy∗ + s∗ + a∗ + g y + g ind )}in every period (see Figure A1).

1

Appendix A2
Fertility, Mortality, and the Return Rate of Immigrants
To ascertain age-specific fertility rates for natives and immigrants, I use data takenfrom the CPS 2000 June supplement, which the Census bureau also uses to calculate
its fertility information. Figure A2 illustrates average births by age, showing that
immigrant women have a greater number of births than native women at all age levels.
From this figure, I calculate η i,n and η i,m , the age-specific birth rates for native andimmigrant women (see Table A1). For the adult mortality profile, pi , I take the values
from Nishiyama and Smetters (2007). I set d, the sum of infant and child mortality,
to be 1.7 percent, by using data derived from the Vital Statistics of the United Statesfor 1993 (see Centers for Disease Control and Prevention, 1993). For the return rateof immigrants to their home countries, I use the official Census estimate provided by
Ahmed and Robinson (1994). They estimate that the return rate of immigrants forthe first 10 years, second 10 years, and third 10 years is 19 percent, 9 percent, and7 percent, respectively. Those return proportions correspond to annual rates of 2.09
percent, 0.94 percent, and 0.72 percent, respectively. Thus, I set pbi = 0.9715 for2 ≤ i < 10, pbi = 0.99062 for 11 ≤ i ≤ 20, pbi = 0.9927 for 21 ≤ i ≤ 30, and pbi = 1 fori ≥ 31.
Setting the age-specific fertility rate, infant and child mortality rate, and initial
PITN (at 15.5 percent) allows us to calculate the annual population growth rate according to equation (56) under the assumption that the PITN is at the steady state.
The annual growth rate of the population of age 1 is thus 0.39 percent.30 This finding
implies that at the initial steady state, the government accepts immigrants such thatthe annual growth rate of immigrants of age 1 becomes 0.39 percent.
30

Annual CPS data on immigrants and natives from 1995 to 2010 show that the median annual
growth rate of the total population (immigrant population) aged from 20 to 40 is 0.12 (1.85) percent.
In the theoretical model, the model assumes that the age 1 population growth rates of immigrants
and natives are the same. Thus, the theoretically predicted growth rate of the age 1 population, 0.39
percent, is between the growth rate of immigrants aged 20–40 and that of the total population aged
20–40.

2

Age-specific Government Expenditure
I assume that age-specific government expenditure, g ∗i,j , is the same for natives andimmigrants. Empirical studies show no systematic difference in the use of public services by these two groups.31 Thus, I assume that for 1 ≤ i ≤ 24, g ∗i,j = gy , for 25 ≤i ≤ 44, g ∗i,j = gm , and for 45 ≤ i ≤ 80, g ∗i,j = go . Further, I assume that gy , gm,
and g0 are 24.5, 13.4, and 23.2 percent of GDP per capita on the initial balanced
growth path following Storesletten (1995) and Auerbach, Kotlikoff, Hagememann, andNicoletti (1989). Moreover, 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
Previous studies have not thus far estimated precisely the coefficient for relative risk
aversion, γ. Auerbach and Kotlikoff (1987) and Storesletten (2000) assume that γ is4, while Nishiyama and Smetters (2007) set γ equal to 2. In the presented analysis,
I assume that γ = 3 and check the robustness with γ = 2 and γ = 4. For thetime preference rate, β, following Hurd (1989) and Storesletten (2000), I assume thatβ = 1.011. A higher β implies higher savings and a higher capital/output ratio. To
check the sensitivity of my results, I also set β = 1 and β = 0.99. I assume that theleisure share in the utility function, ζ, is 0.33.
Further, I assume that the depreciation rate of capital, δ, is equal to 0.047, while
the capital share in the production function, θ, is set to θ = 0.4 and for technologicalprogress, I assume that the income per capita growth rate, µ, is 0.015.
For the human capital profile of natives, His , I take the value from Auerbach and31

Borjas and Hilton (1996) show that immigrants have a higher participation rate in welfare programs than natives. Fix, Passel, and Zimmermann (1996) find that these differences are explainedby the higher participation rate in welfare programs among refugees and retired immigrants and that
there is no difference for working immigrants. Thus, for the theoretical part of the analysis, I assume that g i is independent of birthplace. For the computational part of the analysis, I relax thisassumption for the robustness checks.

3

Kotlikoff (1987):
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 immigrants, Storesletten (1995) shows that immigrants’earnings are, on average, 15.7 percent lower than those of natives.32 Similarly, datataken from the CPS June 2000 supplement show that immigrants’ earnings are 10
percent lower than those of natives. These estimates allow me to assume that theefficient unit of human capital of immigrants is 84.3 percent of that of natives and that
H i,m = 0.843 × Hin in the benchmark calculation. To examine the robustness of myresults, I change the level of human capital from 84.3 percent to 89.3 percent, 79.3
percent, or 74.3 percent and recheck the results.
Taxes and Government Debt
For the capital income tax, I take the value from Nishiyama and Smetters (2007) andassume that τk = 0.28. For the level of social security benefit, a higher replacement
rate means a greater intergenerational redistribution of income, which leads to greater
welfare gain as a result of increasing the immigrant population. Following Auerbachand Kotlikoff (1987), in the benchmark case I set the replacement to 0.6 and check therobustness of my results by varying it to 0.55 and 0.5.
As for the initial level of government debt or assets, different authors set different
levels. Storesletten (xxxx) considers only government debt and assumes that the initiallevel is 50 percent of initial GDP. Given his estimated initial capital/output ratio of 2.4,
this implies that government debt is about 20 percent of private capital. Nishiyama
and Smetters (2007) consider not only government debt, but also government assets
by using BEA information on the government’s fixed capital. They assume that at
the initial steady state, the government has positive net assets of 10 percent of total
32

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

4

private capital. This naturally leads to a higher capital/output ratio on the initialbalanced growth path. Herein, as a benchmark case, I assume that the initial level ofgovernment debt or assets is 0 percent of private capital and that government debt is
10 percent or -10 percent of private capital.

5yy=F(x,s* +a*)

*

y*

C

+S

y

ind

+ g + g+a*

Figure A1: The one plus population growth rate and resources used for one old person. The
horizontal axis measures one plus population growth rate, R(α). The curve through theorigin 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 the young divided the number of old people. The curved line measures the total
output divided the number of the old. Thus, the difference between the curved line and thestraight line measures the resources used per one old person at R(α). This figure shows that
increasing R(α) will increase the resources available per old person without decreasing theresources 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 births by age. The data source is the CPS June 2000 supplement. Total
births by age are regressed on a sixth-order polynomial function of age separately for natives
and immigrants. The predicted values are then 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 assets over the lifecycle on 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: Lifecycle consumption path on 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: Lifecycle leisure consumption on the initial balanced growth path.

10

3.35
V=100%
V=90%
V=70%
V=50%
V=30%
V=0%

3.3
3.25

Capital output ratio

3.2
3.15
3.1
3.05
3
2.95
2.9
2.85

0

50

100

150

200

250

300

year

Figure A6: Capital/output ratio over time for different shares of the surplus used for government savings (V). In those simulations, the number of years needed for the PITN to reachits target level is set at 80. The target capital stock is the modified golden rule level. Theintergenerational discount rate for the modified golden rule level is set at 3 percent.

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

0.1743087
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

0

0

Note
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%

Note
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
∞
X
+
γt {uy (cyt , lt ) + v y (g y , g ind ) +
L=

t=1

+

∞
X

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 )

t=1− (cot + g o + g ind ) − R(α∗ ) × (cyt + st + at + g y + g ind ) }where a0 = a∗ and s0 = s∗

(67)

The first order conditions are:11
uo0 (co1 ) = λ1 ; cot+1 : γtuo0 (cot+1 ) = λt+1 ;
1+ρ
1+ρ
y y
∂uy (cyt , lt )∂F∂u (ct , lt )y∗
= λt R(α ); lt : γt= λtR(α∗ )
ct : γt
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 :

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

1

(68)

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 )1o0 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∗ ∗ ∂u (c , l )∂F (R(α∗ )l∗ , s∗ + a∗ )∂uy (cy∗ , l∗ )
∗
/
=
w
=
−
∂l
∂cy
∂L
 y y∗ ∗  

∗ ∗ ∗
∗
∂F (R(α )l , s + a )∂u (c , l )
1
∗
o0 o∗
+1−δ =1+r =u (c )
/
∂K
∂cyt
1+ρ
1
[uo (co∗ ) + v o (g o , g ind ) = u∗uy (cy∗ , l∗ ) + v y (g y , g ind ) +
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 ind )(69)
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 = λtR(α∗ )1;γ
uo0 (co∗ ) = λt+1
t
∗
1+r
1+ρ

1
uo0 (co∗ )
1+ρ
(70)

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.

2

Appendix B2Using the definitions of τwt and τrt , we havewt τwt = wt − (1 − τw∗ )w∗
rt τrt = rt − (1 − τr∗ )r∗Then, the government budget surplus at period 1 isSP1 = (w1 − (1 − τw∗ )w∗ )l∗X

N1j + (r1 − (1 − τr∗ )r∗ )s∗j=n,m− (b∗ + g o + g ind )

X

X

N0j

j=n,m

N0j − (g y + g ind + a∗ )

j=n,m

X

N1j + a∗ (1 + r1 )

j=n,m

X

N0j

(71)

j=n,m

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

SP1 = w1 l∗

X

N1j − w∗ (1 − τw∗ )l∗

j=n,m

X

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 (eα − α∗ )} + N0n (1 + α∗ )a∗ (1 + r1 )Also notice that

(72)

P
j
n
e) = N1n (1 + α∗ + α
e − α∗ ) and j=n,m N0j =j=n,m N1 = N1 (1+ αP

N0n (1 + α∗ ). Thus, SP1 becomes

SP1 = w1 l∗

X

N1j − w∗ (1 − τw∗ )l∗ {N1n (1 + α∗ ) + N1n (eα − α∗ )}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 (eα − α∗ )} + N0n (1 + α∗ )a∗ (1 + r1 )

3

(73)

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(74)
By using the government budget constraint at the initial steady state, we can rewrite
SP1 as follows:
SP1 = w1 l∗Xα − α∗ ) − N0n (1 + α∗ )a∗ (1 + r∗ )N1j − w∗ l1∗ N1n (1 + α∗ ) − w∗ l∗ (1 − τw∗ )N1n (ej=n,m+ r1 s∗ N0n (1 + α∗ ) − r∗ s∗ N0n (1 + α∗ ) − (g y + g ind + a∗ )N1n (eα − 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∗

X

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

X

N1j +r1 (s∗ +a∗ )N0n (1+α∗ ) = F (l∗

j=n,m

X

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∗ )4Thus, SP1 becomesX

SP1 = F (l∗N1j , (s∗ + a∗ )N0n (1 + α∗ )) − δ(s∗ + a∗ )N0n (1 + α∗ )j=n,m

− {F (l1∗ N1n (1 + α∗ ), (s∗ + a∗ )N0n (1 + α∗ ))α − α∗ ) − N1n (g y + g ind + a∗ )(eα − α∗ )− δ(s∗ + a∗ )N0n (1 + α∗ )} − w∗ l∗ (1 − τw∗ )N1n (e(75)
eN1n . Thus,
Note that N1m = α
SP1 = F (l∗ (N1n (1 + αe), (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 (eα − α∗ )Z 1+eα=[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

Z 1+eα1+α∗FL (N1n l∗ z, (s∗ + a∗ )N0n (1 + α∗ ))l∗ − w∗ l∗ (1 − τw ) − (g y + g ind + a∗ )]dz
(76)Note that w∗ l∗ (1 − tw ) = cy∗ + s∗ . Thus, we have
= N1n

Z 1+eα1+α∗FL (N1n l∗ z, (s∗ + a∗ )N0n (1 + α∗ ))l∗ − cy∗ − s∗ − g y − g ind − a∗ )]dz

(77)

Note that N1n = R(α∗ )N0n and using the homogeneity of FL , we have
Z 1+eαSP1 = N1

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

1+α∗

5

Appendix B3
Note thatjne) andj=n,m N2 = N2 (1 + αPjne). Thus, SP2
j=n,m N1 = N1 (1 + α

Pbecomes

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

(78)

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

(79)

By changing the order in the above equation, SP2 becomes
SP2 = w2 l∗ N2n (1 + αe) + r2 s∗ N1n (1 + αe)− w∗ l∗ N2n (1 + αe) − r∗ s∗ N1n (1 + αe)τw∗ w∗ l∗ N2n (1 + αe) + τr∗ r∗ s∗ N1n (1 + αe) − N1n (1 + αe)(b + g o + g ind )− N2n (1 + αe)(g y + g ind + a∗ ) + (1 + r2 )a1 N1n (1 + αe)= w2 l∗ N2n (1 + αe) + r2 s∗ N1n (1 + αe) + (1 + r2 )a1 N1n (1 + αe)− w∗ l∗ N2n (1 + αe) − r∗ s∗ N1n (1 + αe)+ (1 + αe){τw∗ w∗ l∗ N2n + τr∗ r∗ s∗ N1n − N1n (b + g o + g ind ) − N2n (g y + g ind + a∗ )}(80)
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 )e
α + 1 + πn ) = N1n ((1 + πm )α∗ + 1 + πn + (1 + πm )eα − (1 + πm )α∗ ).

6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 )(eα − α∗ ))+ τr∗ r∗ s∗ N1n − N1n (b + g o + g ind )α − α∗ ))− (g y + g ind + a∗ )N1n (R(α∗ ) + (1 + πm )(e(81)

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(82)
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(83)

Thus, (81) becomes= τw∗ w∗ l∗ N1n (1 + πm )(eα − α∗ ) − (g y + g ind + a∗ )N1n (1 + πm )(eα − α∗ ) − (1 + r∗ )a∗ N1n (84)Therefore, SP2 becomes
SP2 = w2 l∗ N2n (1 + αe) + r2 s∗ N1n (1 + αe) + (1 + r2 )a1 N1n (1 + αe)− w∗ l∗ N2n (1 + αe) − r∗ s∗ N1n (1 + αe)
α − α∗ )+ (1 + αe){τw∗ w∗ l∗ N1n (1 + πm )(eα − α∗ ) − (1 + r∗ )a∗ N1n }− (g y + g ind + a∗ )N1n (1 + πm )(e(85)

Now, we decompose w∗ l∗ N2n (1 + αe) in the second line in the above equitation. Since7α − α∗ )), SP2 becomesα) = N1n (R(α∗ ) + (1 + πm )(eN2n = N1n R(eSP2 = w2 l∗ N2n (1 + αe) + r2 s∗ N1n (1 + αe) + (1 + r2 )a1 N1n (1 + αe)− (1 + αe)w∗ l∗ N1n {R(α∗ ) + (1 + πm )(eα − α∗ )} − r∗ s∗ N1n (1 + αe)
α − α∗ ) − (g y + g ind + a∗ )N1n (1 + πm )(eα − α∗ ) − (1 + r∗ )a∗ N1n }+ (1 + αe){τw∗ w∗ l∗ N1n (1 + πm )(e(86)
Re-arranging the second line in the above equation, we have
SP2 = w2 l∗ N2n (1 + αe) + r2 s∗ N1n (1 + αe) + (1 + r2 )a1 N1n (1 + αe)− (1 + αe)w∗ l∗ N1n R(α∗ ) − (1 + αe)w∗ l∗ N1n (1 + πm )(eα − α∗ ) − r∗ s∗ N1n (1 + αe)+ (1 + αe){τw∗ w∗ l∗ N1n (1 + πm )(eα − α∗ ) − (g y + g ind + a∗ )N1n (1 + πm )(eα − α∗ ) − (1 + r∗ )a∗ N1n }(87)
Next, we re-arrange r2 s∗ N1n (1 + αe) + (1 + r2 )a1 N1n (1 + αe) and (1 + r∗ )a∗ N1n . Then, we
have

SP2 = w2 l∗ N2n (1 + αe) + r2 (s∗ + a1 )N1n (1 + αe) + a1 N1n (1 + αe)− (1 + αe)w∗ l∗ N1n R(α∗ ) − r∗ (s∗ + a∗ )N1n (1 + αe) − (1 + αe)w∗ l∗ N1n (1 + πm )(eα − α∗ )+ (1 + αe){τw∗ w∗ l∗ N1n (1 + πm )(eα − α∗ ) − (g y + g ind + a∗ )N1n (1 + πm )(eα − α∗ ) − a∗ N1n }(88)
Using the homogeneity of the production function, we have
w2 l∗ N2n (1 + αe) + r2 (s∗ + a1 )N1n (1 + αe) = F (l∗ N2n (1 + αe), (s∗ + a1 )N1n (1 + αe)) − δ(s∗ + a1 )N1n (1 + αe)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∗ )

8

(89)

Thus, SP2 becomes
SP2 = F (l∗ N2n (1 + αe), (s∗ + a1 )N1n (1 + αe)) − δ(s∗ + a1 )N1n (1 + αe) + a1 N1n (1 + α
e)
−

1+α
e
{F (l∗ (1 + α∗ )N1n R(α∗ ), (s∗ + a∗ )N1n (1 + α∗ )) − δ(s∗ + a∗ )N1n (1 + α∗ )}∗1+α−(1 + αe)w∗ l∗ N1n (1 + πm )(eα − α∗ )+(1 + αe){τw∗ w∗ l∗ (1 + πm )(eα − α∗ )N1n − (g y + g ind + a∗ )N1n (1 + πm )(eα − α∗ ) − a∗ N1n }(90)
Combining the third line and fourth line, we have

SP2 = F (l∗ N2n (1 + αe), (s∗ + a1 )N1n (1 + αe)) − δ(s∗ + a1 )N1n (1 + αe) + a1 N1n (1 + αe)−F (l∗ (1 + αe)N1n R(α∗ ), (s∗ + a∗ )N1n (1 + αe)) + δ(s∗ + a∗ )N1n (1 + αe)+(1 + αe)N1n {−(1 − τw )w∗ l∗ (1 + πm )(eα − α∗ ) − (g y + g ind + a∗ )(1 + πm )(eα − α∗ ) − a∗ N1n }(91)
Notice that δs∗ N1n (1 + αe) is canceled out in the above equation. Rearranging the term(a1 − a∗ )N1n (1 + αe), we have
SP2 = F (l∗ N2n (1 + αe), (s∗ + a1 )N1n (1 + αe)) + (a1 − a∗ )N1n (1 + αe) − δ(a1 − a∗ )N1n (1 + αe)− F (l∗ (1 + αe)N1n R(α∗ ), (s∗ + a∗ )N1n (1 + αe))+ (1 + αe)N1n {−(1 − τw )w∗ l∗ (1 + πm )(eα − α∗ ) − (g y + g ind + a∗ )(1 + πm )(eα − α∗ )}(92)
Subtracting and adding F (l∗ N2n (1 + αe), (s∗ + a∗ )N1n (1 + αe)), SP2 becomes9

SP2 = F (l∗ N2n (1 + αe), (s∗ + a1 )N1n (1 + αe)) + (1 − δ)(a1 − a∗ )N1n (1 + αe)− F (l∗ N2n (1 + αe), (s∗ + a∗ )N1n (1 + αe))e))e)) − F (l∗ (1 + αe)N1n R(α∗ ), (s∗ + a∗ )N1n (1 + αe), (s∗ + a∗ )N1n (1 + α+ F (l∗ N2n (1 + α
α − α∗ ) − (g y + g ind + a∗ )(1 + πm )(eα − α∗ )}+ (1 + αe)N1n {−(1 − τw )w∗ l∗ (1 + πm )(e(93)
For the first and the second line in the above equation, it can be re-written as= F (l∗ N1n R(eα)((1 + αe), (s∗ + a1 )N1n (1 + αe)) + (1 − δ)(a1 − a∗ )N1n (1 + αe)− F (l∗∗ N1n R(eα)(1 + αe), (s∗ + a∗ )N1n (1 + αe))= N1n (1 + αe){F (l∗ R(eα), s∗ + a1 ) + (1 − δ)(a1 − a∗ )− F (l∗∗ (1 + αe), s∗ + a∗ )}Z s∗ +a1n= N1 (1 + αe)[FK (l∗ R(eα), z) + (1 − δ)] dz(94)s∗ +a∗α).
Next, we focus on the third and fourth lines of (93) . Note that N2n = N1n R(eThus, the third and fourth line of (93) can be re-written as
Z αeα∗[FL (l∗ (1 + αe)N1n R(z), (s∗ + a∗ )N1n (1 + αe))l∗ (1 + αe)N1n (1 + πm )− (1 + αe)N1n (1 + πm ){(1 − τw )w∗ l∗ + g y + g ind + a∗ }]dz
Z αen
= (1 + αe)N1 (1 + πm )[FL (l∗ (1 + αe)N1n R(z), (s∗ + a∗ )N1n (1 + α
e))l∗
α∗

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

10

(95)

Since theFL is homogenous degree of zero, the above equation becomes= (1 + αe)N1n (1 + πm )Z αe

[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 R0 (α) = 1 + πm . Therefore, SP2 becomes asfollows:

SP2 = N1n (1 + αe)(1 + αe)N1nZ s∗ +a1

Z αe[FK (l∗ R(eα), z)) + 1 − δ]dzs∗ +a∗R0 (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∗(96)

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 per

11each 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∗ )mn(bm∗ + g o,m + g ind,m )(bn∗ + g o,n + g ind,n ) − Nt−1
− Nt−1mn)+ Nt−1+ (1 + rt )at−1 (Nt−1(97)

Using the definition of τwt and τrt , we havewt τwt = wt − (1 − τw∗ )w∗(98)

rt τrt = rt+1 − (1 − τr∗ )r∗(99)

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
)

(100)

12

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−1mn)
+ Nt−1+ (1 + rt )at−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 )(101)

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 )

(102)

n
Note that Ntn = Nt−1R(eα) and Ntm = Ntn αe. Thus,the above equation becomes asfollows:nn
τw∗ w∗ ln∗ Nt−1R(eα) + φm τw∗ w∗ ln∗ Nt−1
J(e
α)
n
n
+ τr∗ r∗ sn∗ Nt−1
+ τr∗ r∗ sm∗ Nt−1αen
− Nt−1R(eα)(g y,n + g ind,n + a∗ )nJ(eα)(g y,m + g ind,m + a∗ )− Nt−1nn− Nt−1(bn∗ + g o,n + g ind,n ) − Nt−1αe(bm∗ + g o,m + g ind,m )(103)

where J(α) is defined as follows:
J(α) = (1 + πn + α(1 + πm ))αnn
On the other hand, at the initial steady state, Ntn = Nt−1
R(α∗ )and Ntm = Nt−1α∗13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(104)

Thus, (103) becomesnτw∗ w∗ ln∗ Nt−1(eα − α∗ )(1 + πm )n+φm τw∗ w∗ lm∗ Nt−1(J(eα) − J(α∗ ))n+τr∗ r∗ sm∗ Nt−1(eα − α∗ )n−Nt−1(g y,n + g ind,n + a∗ )(eα − α∗ )(1 + πm )n−Nt−1
(g y,m + g ind,m + a∗ )(J(eα) − J(α∗ ))n−Nt−1(bm∗ + g o,m + g ind,m )(eα − α∗ )n−(1 + r∗ )a∗ Nt−1
(1 + α∗ )

14

(105)

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

15

(106)

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

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)mnmn))+ Nt−1+ at−1 (Nt−1+ sm∗ Nt−1
−δ(sn∗ 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 )(eα − α∗ )n+φm τw∗ w∗ lm∗ Nt−1{J(eα) − J(α∗ )}n+τr∗ r∗ sm∗ Nt−1(eα − α∗ )n−Nt−1(g y,n + g ind,n + a∗ )(1 + πm )(eα − α∗ )n−Nt−1(g y,m + g ind,m + a∗ ){J(eα) − J(α∗ )}n−Nt−1(bm∗ + g o,m + g ind,m )(eα − α∗ )n−a∗ Nt−1(1 + α∗ )(107)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)) − δ(sn∗ Nt−1
+ sm∗ Nt−1)
+ (1 − δ)at−1 (Nt−1+ Nt−1

(108)

n
m
n
Next, 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(eα)N, Ntm =

16

n
m
n
α
e. Thus, we have
= Nt−1J(eα), Nt−1
Nt−1

n
m
n
w∗ 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∗ ))
n
n
J(e
α)
R(e
α) + φm w∗ lm∗ Nt−1
= w∗ ln∗ Nt−1nnn(1 + α∗ ))αe + r∗ a∗ Nt−1
+ r∗ sm∗ Nt−1
+ r∗ sn∗ Nt−1nnn
+ δ(sn∗ Nt−1
+ sm∗ α∗ Nt−1
+ a∗ Nt−1(1 + α∗ ))n= w∗ ln∗ Nt−1
{R(α∗ ) + (1 + πm )(eα − α∗ )}n m∗+ φm w∗ lm∗ Nt−1l (J(α∗ ) + J(eα) − J(α∗ ))nnn+ r∗ sn∗ Nt−1
+ r∗ sm∗ Nt−1(α∗ + αe − 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 )(eα − α∗ ) + φm w∗ lm∗ Nt−1(J(eα) − J(α∗ ))n+ r∗ sm∗ Nt−1(eα − α∗ )(109)

On the other hand, from the homogeneity of the production function and Euler’s
theorem, (109) 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 )(eα − α∗ )n m∗l (J(eα) − J(α∗ ))+ φm w∗ lm∗ Nt−1n+ r∗ sm∗ Nt−1(eα − α∗ )(110)

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

17

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

18

(111)

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 + αe) − δ(sn∗ Nt−1
+ sm∗ Nt−1
)
n
n
n
n
n
α∗ + a∗ Nt−1
(1 + α∗ ))+ sm∗ Nt−1
, sn∗ Nt−1
+ φm J(α∗ )lm∗ Nt−1
−F (R(α∗ )ln∗ Nt−1nnn(1 + α∗ ))) − (1 − δ)a∗ Nt−1
+ sm∗ α∗ Nt−1
+δ(sn∗ Nt−1n(1 + πm )(eα − α∗ )−w∗ ln∗ Nt−1n−φm w∗ lm∗ Nt−1{J(eα) − J(α∗ )}n−r∗ sm∗ Nt−1(eα − α∗ )n+τw∗ w∗ ln∗ Nt−1
(1 + πm )(eα − α∗ )n+φm τw∗ w∗ lm∗ Nt−1(J(eα) − J(α∗ ))n+τr∗ r∗ sm∗ Nt−1(eα − α∗ )n−Nt−1(1 + πm )(g y,n + g ind,n + a∗ )(eα − α∗ )n−Nt−1(g y,m + g ind,m + a∗ ){J(eα) − J(α∗ )}n−Nt−1(bm∗ + g o,m + g ind,m )(eα − α∗ )nnnNote that −(1−δ)a∗ Nt−1
(1+α∗ ) = −(1−δ)a∗ Nt−1
(1+ αe +α∗ − αe) = −(1−δ)a∗ Nt−1
(1+
n
n
α
e) + (1 − δ)Nt−1(e
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∗ + αe − α∗ ). Thus,19nα∗ is canceled out from the second and fourth line. Thus, we haveδsm∗ Nt−1nmnm
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
e)
n
n
n
n
n
n
α∗ + a∗ (Nt−1+ Nt−1α∗ ))+ sm∗ Nt−1
, sn∗ Nt−1
+ φm J(α∗ )lm∗ Nt−1
−F (R(α∗ )ln∗ Nt−1nn(eα − α∗ )(1 + αe) + (1 − δ)a∗ Nt−1−(1 − δ)a∗ Nt−1n−δsm∗ Nt−1(eα − α∗ )n−w∗ ln∗ Nt−1
(1 + πm )(eα − α∗ )n−φm w∗ lm∗ Nt−1{J(eα) − J(α∗ )}n−r∗ sm∗ Nt−1(eα − α∗ )n+τw∗ w∗ ln∗ Nt−1
(1 + πm )(eα − α∗ )n+φm τw∗ w∗ lm∗ Nt−1(J(eα) − J(α∗ ))n+τr∗ r∗ sm∗ Nt−1(eα − α∗ )n−Nt−1(1 + πm )(g y,n + g ind,n + a∗ )(eα − α∗ )n−Nt−1(g y,m + g ind,m + a∗ ){J(eα) − J(α∗ )}n−Nt−1(bm∗ + g o,m + g ind,m )(eα − α∗ )mnmn)) and+Nt−1+a∗ (Nt−1+sm∗ Nt−1
We subtract and add F (Ntn ln∗ +φm Ntm lm∗ , sn∗ Nt−1nnn(eα −α∗ )(eα −α∗ ), r∗ sm∗ Nt−1(eα −α∗ ) from and to SPt . We also combine sm∗ Nt−1
sm∗ Nt−120n(e
α − α∗ ). Then, SPt becomes as follows
and τr∗ sm∗ Nt−1nmnm
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 + αe)(1 + αe) − (1 − δ)a∗ Nt−1
+ (1 − δ)at−1 Nt−1nmnm
+ 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(eα − α∗ ) + (1 − δ)sm∗ Nt−1(eα − α∗ )− (1 − τw∗ )w∗ ln∗ Ntn (1 + πm )(eα − α∗ )− (1 − τw∗ )φm w∗ lm∗ Ntn {J(eα) − J(α∗ )}− (1 + (1 − τr∗ )r∗ )sm∗ Ntn (eα − α∗ )− Ntn (1 + πm )(g y,n + g ind,n + a∗ )(eα − α∗ )− Ntn (g y,m + g ind,m + a∗ ){J(eα) − J(α∗ )}− Ntn (bm∗ + g o,m + g ind,m )(eα − α∗ )

(112)

n
n
J(e
α). Then, the first three lines of (112)R(eα) and Ntm = Nt−1
Note that Ntn = Nt−1

21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 + αe)(1 + αe) − (1 − δ)a∗ Nt−1
+ (1 − δ)at−1 Nt−1
n
n
n
n
n
n
= F (Nt−1R(eα)ln∗ + φm Nt−1
J(eα)lm∗ , sn∗ Nt−1
+ sm∗ αeNt−1
+ at−1 (Nt−1
+α
eNt−1
))
n
n
n
n
n
n
− F (Nt−1R(eα)ln∗ + φm Nt−1
J(eα)lm∗ , sn∗ Nt−1
+ sm∗ αeNt−1+ a∗ (Nt−1
+α
eNt−1
))
n
n
+ (1 − δ)at−1 Nt−1
(1 + αe) − (1 − δ)a∗ Nt−1
(1 + αe)n= Nt−1α)lm∗ , sn∗ + sm∗ α
e + at−1 (1 + αe)){F (R(eα)ln∗ + φm J(e− F (R(eα)ln∗ + φm J(eα)lm∗ , sn∗ + sm∗ α
e + a∗ (1 + αe))+ (1 − δ)at−1 (1 + αe) − (1 − δ)a∗ (1 + αe)}
Z sn∗ +sm∗ αe+at−1 (1+eα)n= Nt−1[FK (R(eα)ln∗ + φm J(eα)lm∗ , z) + (1 − δ)]dz(113)

sn∗ +sm∗ αe+a∗ (1+eα))

The fourth line and the fifth line of (112)can be transformed as follows
n
n
n
n
n
n
F (Nt−1R(eα)ln∗ + φm Nt−1
J(eα)lm∗ , sn∗ Nt−1
+ sm∗ Nt−1αe + a∗ (Nt−1+ Nt−1
α
e))
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(eα)ln∗ + φm J(eα)lm∗ , sn∗ + sm∗ α
e + a∗ (1 + αe))− F (R(α∗ )ln∗ + φm J(α∗ )lm∗ , sn∗ + sm∗ α∗ + a∗ (1 + α∗ ))Z αen[FL (R(z)ln∗ + φm J(z)lm∗ , sn∗ + sm∗ z + a∗ (1 + z))(R0 (z)ln∗ + φm J 0 (z)lm∗ )= Nt−1α∗+ FK (R(z)ln∗ + φm J(z)lm∗ , sn∗ + sm∗ z + a∗ (1 + z))(sm∗ + a∗ )]dz
. Then, SPt becomes as follows:

22

(114)

n
SPt = Nt−1n
+ Nt−1Z sn∗ +sm∗ αe+at−1 (1+eα)[FK (R(eα)ln∗ + φm J(eα)lm∗ , z) + (1 − δ)]dz

sn∗ +sm∗ αe+a∗ (1+e
α))

Z αe

[FL (R(z)ln∗ + φm J(z)lm∗ , sn∗ + sm∗ z + a∗ (1 + z))(R0 (z)ln∗ + φm J 0 (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∗ )(eα − α∗ )− (1 − τw∗ )w∗ ln∗ Ntn (1 + πm )(eα − α∗ )α) − J(α∗ )}− (1 − τw∗ )φm w∗ lm∗ Ntn {J(e− (1 + (1 − τr∗ )r∗ )sm∗ Ntn (eα − α∗ )− Ntn (1 + πm )(g y,n + g ind,n )(eα − α∗ )− Ntn (g y,m + g ind,m ){J(eα) − J(α∗ )}− Ntn (bm∗ + g o,m + g ind,m )(eα − α∗ )(115)

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 becomes23

nSPt = Nt−1n
+ Nt−1Z sn∗ +sm∗ αe+at−1 (1+eα)[FK (R(eα)ln∗ + φm J(eα)lm∗ , z) + (1 − δ)]dz

sn∗ +sm∗ αe+a∗ (1+e
α))

Z αe
α∗

[FeL × (R0 (z)ln∗ + φm J 0 (z)lm∗ )+ FeK × (sm∗ + a∗ )]dzn
+ (1 − δ)Nt−1a∗ (eα − α∗ )n(1 − δ)sm∗ Nt−1(eα − α∗ )n(1 + πm )(eα − α∗ )− (1 − τw∗ )w∗ ln∗ Nt−1n− (1 − τw∗ )φm w∗ lm∗ Nt−1{J(eα) − J(α∗ )}n− (1 + (1 − τr∗ )r∗ )sm∗ Nt−1(eα − α∗ )n− Nt−1(1 + πm )(g y,n + g ind,n + a∗ )(eα − α∗ )n− Nt−1(g y,m + g ind,m + a∗ ){J(eα) − J(α∗ )}n− Nt−1(co,m∗ + g o,m + g ind,m )(eα − α∗ )(116)

where FeL = FL (R(z)ln∗ + φm J(z)lm∗ , sn∗ + sm∗ z + a∗ (1 + z))
FeK = 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

Z αe

{(1 − δ)(sm∗ + a∗ )α∗0

− R0 (α)(1 − τw∗ )w∗ ln∗ − (1 − τw∗ )φm w∗ lm∗ J (z)− R0 (α)(g y,n + g ind,n + a∗ )− (g y,m + g ind,m + a∗ )J 0 (z)− (co,m∗ + g o,m + g ind,m )}dz

24

(117)

Thus, SPt becomes as follows:n
SPt = Nt−1n
+ Nt−1Z sn∗ +sm∗ αe+at−1 (1+eα)[FK (R(eα)ln∗ + φm J(eα)lm∗ , z) + (1 − δ)]dz

sn∗ +sm∗ αe+a∗ (1+e
α))

Z αe
α∗

{FeL × (R0 (z)ln∗ + φm J 0 (z)lm∗ )+ (FeK × (sm∗ + a∗ )0

− R0 (α)(1 − τw∗ )w∗ ln∗ − (1 − τw∗ )φm w∗ lm∗ J (z)− R0 (α)(g y,n + g ind,n + a∗ ) − (g y,m + g ind,m + a∗ )J 0 (z) − (co,m∗ + g o,m + g ind,m )}dz
(118)
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−1Z sn∗ +sm∗ αe+at−1 (1+eα)[FK (R(eα)ln∗ + φm J(eα)lm∗ , z) + (1 − δ)]dz

sn∗ +sm∗ αe+a∗ (1+e
α))

Z αe
α∗

{R0 (z)[FeL ln∗ − (cy,n∗ + sn∗ + g y,n + g ind,n + a∗ )]+ R0 (z)z[FeL φm lm∗ − (cy,m∗ + sm∗ . + g y,m + g ind,m + a∗ )]+ R(z)[FeL φm lm∗ − (cy,m∗ + sm∗ . + g y,m + g ind,m + a∗ )](119)+ (FeK + 1 − δ)(sm∗ + a∗ ) − (co,m∗ + g o,m + g ind,m )}dz

(120)

where FeL = FL (R(z)ln∗ + φm J(z)lm∗ , sn∗ + sm∗ z + a∗ (1 + z))
FeK = FK (R(z)ln∗ + φm J(z)lm∗ , sn∗ + sm∗ z + a∗ (1 + z))
Note that J(z) = R(z)z and J 0 (z) = R(z) + R0 (z)z. In the above equation, thefirst 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 and firth lines measure the intra-redistributional effect.

25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 and immigrants
have the same preferences and productivities Thus, we haven
SPt = Nt−1n
+ Nt−1Z (sn∗ +at−1 )(1+eα)(sn∗ +a∗ )(1+e
α))

Z αe
α∗

[FK (R(eα)ln∗ (1 + αe), z) + (1 − δ)]dz

R0 (z)[FeL ln∗ − (cy,n∗ + sn∗ + g y,n + g ind,n + a∗ )]dz

Z αe

J 0 (z)[FeL ln∗ − (cy,n∗ + sn∗ + g y,n + g ind,n + a∗ )]J 0 (z)dz
α∗
Z αe
n
n∗
∗
+ Nt−1 × (s + a )[FeK + 1 − δ]dz
∗
α
Z αe
n
− Nt−1(co,n∗n + g o,n + g ind,n )dzn+ Nt−1α∗where FeL = FL (R(z)ln∗ + J(z)ln∗ , sn∗ + sn∗ z + a∗ (1 + z)) and
FeK = FK (R(z)ln∗ + J(z)ln∗ , sn∗ + sn∗ z + a∗ (1 + z))
The first line of (121) can be transformed as follows:n
Nt−1{F (R(eα)ln∗ (1 + αe), (sn∗ + at−1 )(1 + αe)) + (1 − δ)(sn∗ + at−1 )(1 + αe)− F (R(eα)ln∗ (1 + αe), (sn∗ + a∗ )(1 + αe)) − (1 − δ)(sn∗ + a∗ )(1 + αe)}n= Nt−1(1 + αe){{F (R(eα)ln∗ , (sn∗ + at−1 )) + (1 − δ)(sn∗ + at−1 )− F (R(eα)ln∗ , (sn∗ + a∗ )) − (1 − δ)(sn∗ + a∗ )}Z sn∗ +at−1ne)[FK (R(eα)ln∗ , z) + (1 − δ)]dz= Nt−1 (1 + α
sn∗ +a∗26(121)

Thus,2nd to 4th line of (121) can be transformed as follows:n+ Nt−1
F (R(eα)ln∗ + J(eα)ln∗ , sn∗ + sn∗ α
e + a∗ (1 + αe))n− Nt−1
F (R(α∗ )ln∗ + J(α∗ )ln∗ , sn∗ + sn∗ α∗ + a∗ (1 + α∗ ))}n(1 − δ)(sn∗ + a∗ )(eα − α∗ )+ Nt−1n(cy,n∗ + sn∗ + a∗ + g y + g ind )(eα − α∗ )− (1 + πm )Nt−1n− Nt−1(cy,n∗ + sn∗ + a∗ + g y + g ind )(J(eα) − J(α∗ ))n− Nt−1(co,n∗n + g o,n + g ind,n )(eα − 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(eα − α∗ ).

Then, we have∗nn
F (ln R(α∗ )Nt−1, (s∗ + a∗ )Nt−1 (eα − α∗ ) + (1 − δ)(sn∗ + a∗ )Nt−1(e
α − α∗ )= (1 + α∗ )R(α∗ )(cy,n∗ + sn∗ + a∗ + g y + g ind )(eα − α∗ )n+ (co,n∗ + g o + g ind )Nt−1(e
α − α∗ )27

n(e
α − α∗ ), we have
Solving for −(co,n∗ + g o + g ind )Nt−1∗nn−(co,n∗ + g o + g ind )Nt−1(e
α − α∗ ) = −F (ln R(α∗ )Nt−1, (s∗ + a∗ )Nt−1 )(eα − α∗ )n− (1 − δ)(sn∗ + a∗ )Nt−1(e
α − α∗ )nR(α∗ )(eα − α∗ )+ (cy,n∗ + sn∗ + a∗ + g y + g ind )Nt−1∗n{F (ln R(α∗ ), s∗ + a∗ )(eα − α∗ )= −Nt−1+ (1 − δ)(sn∗ + a∗ )(eα − α∗ )− (cy,n∗ + sn∗ + a∗ + g y + g ind )R(α∗ )(eα − α∗ )}Thus, SPt becomesn
SPt = Nt−1
(1 + αe)Z sn∗ +at−1[FK (R(eα)ln∗ , z) + (1 − δ)]dz

sn∗ +a∗n+ Nt−1{F (R(eα)ln∗ + J(eα)ln∗ , sn∗ + sn∗ α
e + a∗ (1 + αe))− F (R(α∗ )ln∗ + J(α∗ )ln∗ , sn∗ + sn∗ α∗ + a∗ (1 + α∗ ))∗− F (ln R(α∗ ), s∗ + a∗ )(eα − α∗ )− (1 + πm )(cy,n∗ + sn∗ + a∗ + g y + g ind )(eα − α∗ )− (cy,n∗ + sn∗ + a∗ + g y + g ind )(J(eα) − J(α∗ ))+ (cy,n∗ + sn∗ + a∗ + g y + g ind )R(α∗ )(eα − α∗ )}(122)

Note that (cy,n∗ + sn∗ + a∗ + g y + g ind )J(α∗ ) and (cy,n∗ + sn∗ + a∗ + g y + g ind )R(α∗ )α∗28

are canceled out. Thus, SPt becomesn
SPt = Nt−1
(1 + αe)Z sn∗ +at−1[FK (R(eα)ln∗ , z) + (1 − δ)]dz

sn∗ +a∗n+ Nt−1{F (R(eα)ln∗ + J(eα)ln∗ , sn∗ + sn∗ α
e + a∗ (1 + αe))− F (R(α∗ )ln∗ + J(α∗ )ln∗ , sn∗ + sn∗ α∗ + a∗ (1 + α∗ ))∗− F (ln R(α∗ ), s∗ + a∗ )(eα − α∗ )− (1 + πm )(cy,n∗ + sn∗ + a∗ + g y + g ind )(eα − α∗ )− (cy,n∗ + sn∗ + a∗ + g y + g ind )J(eα)+ (cy,n∗ + sn∗ + a∗ + g y + g ind )R(α∗ )e
α}

(123)

Note that −(cy,n∗ + sn∗ + a∗ + g y + g ind )J(eα) + (cy,n∗ + sn∗ + a∗ + g y + g ind )R(α∗ )eαbecome equal to−(cy,n∗ + sn∗ + a∗ + g y + g ind )(1 + πm )eα(eα − α∗ )Thus, SPt becomesn
SPt = Nt−1
(1 + αe)Z sn∗ +at−1[FK (R(eα)ln∗ , z) + (1 − δ)]dz

sn∗ +a∗n+ Nt−1{F (R(eα)ln∗ + J(eα)ln∗ , sn∗ + sn∗ α
e + a∗ (1 + αe))− F (R(α∗ )ln∗ + J(α∗ )ln∗ , sn∗ + sn∗ α∗ + a∗ (1 + α∗ ))∗− F (ln R(α∗ ), s∗ + a∗ )(eα − α∗ )− (1 + πm )(cy,n∗ + sn∗ + a∗ + g y + g ind )(eα − α∗ )− (cy,n∗ + sn∗ + a∗ + g y + g ind )(1 + πm )eα(eα − α∗ )}The second line of (124) becomesn
Nt−1(1 + αe){F (R(eα)ln∗ , sn∗ + a∗ )

29

(124)

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∗ )(eα − α∗ )n(1 + αe)F (R(α∗ )ln∗ , sn∗ + a∗ )= −Nt−1

The fifth and sixth line becomesn−(1 + πm )Nt−1(cy,n∗ + sn∗ + a∗ + g y + g ind )(1 + αe)(eα − α∗ )Thus, SPt becomesn
SPt = Nt−1
(1 + αe)Z sn∗ +at−1[FK (R(eα)ln∗ , z) + (1 − δ)]dz

sn∗ +a∗n∗+ Nt−1(1 + αe){F (R(eα)ln∗ , sn∗ + a∗ ) − F (R(α∗ )ln∗ , sn∗ + a∗ )− (1 + πm )(cy,n∗ + sn∗ + a∗ + g y + g ind )(eα − α∗ )}Z sn∗ +at−1n= Nt−1 (1 + αe)[FK (R(eα)ln∗ , z) + (1 − δ)]dz
sn∗ +a∗n+ Nt−1(1 + αe)n− Nt−1(1 + αe)Z αe(1 + πm )ln∗ FL (R(z), sn∗ + a∗ )dz

α∗

Z αe

(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 + αe)n+ Nt−1(1 + αe)Z sn∗ +at−1[FK (R(eα)ln∗ , z) + (1 − δ)]dz

sn∗ +a∗Z αe

R0 (z)[FL (R(z), sn∗ + a∗ )ln∗α∗

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

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

30