Tsukuba Economics Working Papers
No. 2013-004	
 Pareto-improving Immigration and Its Effect on Capital Accumulation in the
Presence of Social Security
byHisahiro Naito	 August 2013

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

Pareto-improving Immigration and Its Effect on Capital
Accumulation in the Presence of Social Security
Hisahiro Naito ∗†Department of Economics
University of Tsukuba, Japan
August 5, 2013

Abstract
The effect of accepting more immigrants on welfare in the presence of a pay-as-yougo social security system is analyzed qualitatively and quantitatively. First, it is shownthat if initially there exist intergenerational government transfers from the young to theold, the government can lead an economy to the (modified) golden rule level within afinite time in a Pareto-improving way by increasing the percentage of immigrants tonatives (PITN). Second, using the computational overlapping generation model, thewelfare gain is calculated of increasing the PITN from 15.5 percent to 25.5 percent andyears needed to reach the (modified) golden rule level in a Pareto-improving way ina model economy. The simulation shows that the present value of the welfare gain of
increasing the PITN comprises 23 percent of the initial GDP. It takes 112 years for themodel economy to reach the golden rule level in a Pareto-improving way.

∗†

Email:naito@dpipe.tsukuba.ac.jp
I appreciate comments from the participants of the Canadian Public Economics Meeting(Kingston), theNorth American Summer Econometric Society Meeting (St. Louis), the European Econometrics Society
Meeting (Oslo), Macroeconomics workshop at the University of Tokyo. I appreciate comments from Kosuke
Aoki, Masayoshi Hayashi and Shinichi Nishiyama. I especially appreciate comments from Pierre Pestieau
for pointing out the literature of the Serendipity theorem.

1

Introduction

Transforming a pay-as-you-go(PYGO) social security system into a funded system is noteasy. When the PYGO social security system is changed to a funded system, some generations must bear the so called “the double burden”, such that a young generation needs to pay
the social security tax twice. Thus, although the transition from a PYGO social security to afunded system is desirable since a PYGO social security system causes under-accumulationof capital, it is difficult to transit in a Pareto-improving way.
On the other hand, in the face of the fiscal problems caused by negative demographic
shocks in the presence of PYGO social security, policy makers have recently shown increased
interests in accepting more immigrants. For example, the IMF report on the Japanese
economy (2012), which is suffering as a result of an extremely low fertility rate and practical
restrictions on inflow of immigrants, states that facilitating a modest level of immigration
could have a large payoff for the Japanese economy. However, although using immigrants tosolve the fiscal problem of PYGO social security in the short run is attractive, its desirability
in the long run is unclear. Increasing the percentage of immigrants to natives implies ahigher population growth rate in a host country given the higher fertility rate of immigrants.
With a neoclassical production function that exhibits the diminishing marginal product of
capital, the standard growth model (Ramsey (1928) and Solow (1964)) predicts that sucha higher population growth rate leads to both lower level of capital stock per capita andincome per capita, starting from a dynamically efficient initial steady state. Thus, it is notclear how such an increase of immigrants can help to solve the problem of PYGO social
security in the long run. The literature also presents an unclear picture. In the literature
on public finance, there is increasing interest in the effect on the social welfare of accepting
more immigrants. By using the computational overlapping generation model (Auerbach andKotlikoff model(Auerbach and Kotlikoff (1998)), Storesletten (2000) argues that accepting
a particular type of immigrants, (skilled who are of an age such that they will not be able
to claim the social security benefit because they will not be able to satisfy the minimum
requirement of the duration of social security tax payments) will increase the social welfare
in the presence of the retirement of the baby boom generation. In contrast, Fehr, Jokischand Kotlikoff (2004) argue that there will be no such a welfare gain. Feldstein (2006)
analyzes the effect of immigration in Spain and concludes that immigration does not bringa welfare gain while Collado, Iturbe-Ormaetxeand and Valera (2004) argue that accepting
more immigrants brings a positive welfare gain to Spain.
2

Given the mixed results in the literature regarding the effect of accepting more immigrants on the social welfare and the theoretical prediction of the standard growth model, anatural question arises whether or not, from a theoretical standpoint, accepting more immigrants Pareto-improves welfare in the presence of PYGO social security. With a neoclassicalproduction function that exhibits diminishing marginal product of capital, a decrease of income per capita at the steady state seems inevitable when the economy experiences a higher
population growth rate due to an increased inflow of immigrants. Yet, if this is so, one may
wonder why the previous studies have arrived at such different results regarding the effecton welfare of accepting more immigrants.
Motivated by those questions, 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 underaccumulation of capital that is caused by implementing a PYGO social security system.1
More specifically, firstly, using the overlapping generation model developed by Diamond(1965), I show that in an economy with or without distorting taxes it is Pareto-improving
to increase the percentage of immigrants to natives (PITN) if there exit intergenerationaltransfers, in the sense that the marginal product of labor of a young individual is greaterthan the amount of resources that she receives while she is young, which is equal to thesum of the after-tax income when she is young and the amount of publicly provided private
goods for the young (the MPL condition). Note that the amount of resources an individual
receives when she is young does not include the social security benefit that she receives
when she reaches retirement age. Thus, in the presence of PYGO social security, thisMPL condition is likely to be satisfied. Secondly, I show analytically that when this MPL
condition is satisfied, the government can lead the economy to the (modified) golden rulelevel in a Pareto-improving way within a finite time by putting in savings the government
budget surplus, which is obtained by increasing the PITN. Note that when the economy
reaches the golden rule level, the problem of under-accumulation of capital caused by the
PYGO social security is solved for all practical purposes. Third, I quantify this Paretoimproving welfare gain that is yielded by increasing the PITN in the presence of a PYGO
social security system and calculate the year needed to reach the (modified) golden rule1

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

3

level in a Pareto-improving way by using the computational overlapping generation model
developed by Auerbach and Kotlikoff (1987). I consider a moderate increase of the PITN,
such that the PITN starts to increase from 15.5 percent, reaches 25.5 percent at the 80th
year and remains constant at 25.5 percent in later years.2 With this speed of increaseof the PITN and in the model that mimics important dimensions of the US economy, mysimulation 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, the capital stock
per efficient unit of labor increases by 102 percent and the publicly provided private goods
per capita increases by 36 percent. When the target capital stock is set at the modified
golden rule level with 3 percent of intergenerational discount rate, it takes 65 years to reachthe modified golden rule level in a Pareto-improving way and the capital stock per efficient
unit of labor increases by 18 percent. The present discounted value (PDV) of the Paretoimproved utility, measured by the expenditure function, of natives and their descendants,
which does not include the increased utility of immigrants and their descendants, comprises
23 percent of the initial GDP. When the time to reach the target PITN is shortened to 42
years, the economy reaches the modified golden rule at the 59th year and the PDV of thePareto improvement comprises 28 percent of the initial GDP. Finally, I conduct robustness
checks by changing a number of parameter values, for example, the share of the surplus forthe government savings, the replacement rate, the time preference rate, the risk aversion, theinitial government debt(asset) level, the level of immigrants’ earnings and the consumptionof public services by immigrants. Those robustness checks show that the results of thesimulation do not change substantially in magnitude for different parameter values. Boththeoretical results and the computational results suggest the robustness of the welfare gainof increasing the PITN in the presence of PYGO social security.
The remainder of the paper is organized as follows. Section 2 reviews the related literature. Section 3 presents a theoretical analysis. Section 4 presents a simulation-based
analysis using the computational overlapping generation model. Section 5 concludes.

2

Literature

In the theoretical literature on the effect of accepting immigrants, Razin (1999) is, to mybest knowledge, the first paper that shows that accepting immigrants improves welfare.
2

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

4.2 for more detailed discussion.

4

He shows that in a small open economy model in which factor prices are fixed, accepting
more immigrants can improve welfare in the presence of PYGO social security. However,in a subsequent paper, Razin and Sadka (2000) show that in an closed economy model inwhich capital accumulation and factor prices are endogenous, the result obtained under theassumption of a small open economy is not likely to hold. Subsequently, Razin, Sadka and
Suwankiri (2011) analyzes a political economy model in which there is a conflict of interests
among agents on accepting immigrants. They obtain results regarding on political behavior
on immigration policy.3
In the empirical literature, there are three types of studies. The first generation of
those studies conducts analysis with a short horizon which does not take into account thecontribution of immigrants’ children to the social security and tax revenue, and the costburden which they impose to public services (Huddle(1993), Borjas(1994), Passel (1994),
Simon (1984) and Akbari (1989)). The second generation of those studies conducts analysis
with a long horizon (Generational Accounting) which takes into account the contributionof immigrants’ children( and grand children, etc...) to the social security and tax revenue,
and the cost burden that they impose to public services (Lee and Miller (1997), Auerbachand Preopoulous (1999), Storesletten (2003), and, Collado and Valera (2004)).4
The third generation of those studies conducts analysis with a dynamic general equilibrium model (Storesletten(1995, 2000), Canova and Ravn (2000), and Fehr, Joisch andKotlikoff (2004)). Storesletten (1995) is the first paper to analyze the effect of accepting
immigrants in a dynamic general equilibrium mode with aggregate uncertainty and Storesletten (2000) analyzes without aggregate uncertainty. He argues, in the US context, that byselecting a particular type of immigrants, the acceptance of immigrants can have a positiveeffect on the native population.
3Fehr, Joisch and Kotlikoff (2004) argue that there is no

In the theoretical literature, there is another strand that analyzes the optimal population growth in thegrowth model; this was started by Samuelson(1975). He analyzes the welfare maximizing optimal population
growth rate in the standard neoclassical growth model. The subsequent analysis shows that the optimal
population growth rate in many cases does not, in many cases, maximize the welfare; rather it minimizes
welfare(Deardorfff 1976). For discussion, see Michel and Pestieau(1999). Due to the concern of the nonoptimality of the first order condition, I do not herein attempt to identify the optimal population growthrate or the optimal immigration rate. Rather, I check whether a small change in the population growth rate
will Pareto-improve welfare or not.
4
To my knowledge, Lee and Miller (1996) is the first paper that calculates the long-run fiscal impact ofaccepting immigrants. They calculate the tax revenue from and the cost to public services of, not only thecurrent immigrants, but also the descendant of immigrants. They find that accepting immigrant bring a
positive net present value to the government.

5such welfare gain, or, even if there were to be a gain, it would be very small.5
This paper contributes to the existing literature in several ways. For the theoretical part,I show first that even in a closed economy in which capital accumulations and factor prices
are endogenous, it is Pareto-improving to accept more immigrants in the presence of a PYGO
social security system if the MPL condition is satisfied. This is contrary to the findings ofRazin and Sadka (2000), but is consistent with the results of Razin (1999) and extends them
to a closed economy. The use of the MPL condition shows why typical measures for welfare
analysis that calculate the life-time taxes and expenditure are misleading.6 Second, I showthat the government can lead the economy to the (modified) golden rule level by accepting
more immigrants within a finite time in a Pareto-improving way. This is in sharp contrast to
the literature on the social security reform, in which it is argued that it is difficult to increase
the capital stock through social security reform in a Pareto-improving way (Geanakoplos,
Mitchell and Zeldes (1998)).
For the computational analysis, I add two important factors to the existing literature.
The first is the possibility of Pareto improvement. When more immigrants are accepted, thewage rate falls and the interest rate increases initially. This generates winners and losers
in the economy. In my calculation, I consider an immigration policy that Pareto-improves
all generations. Taking Pareto improvement into consideration is important because if
Pareto improvement is impossible to achieve, policy reform becomes less feasible even if
it increases the over-all efficiency of an economy. The second factor is that I assume thatthe government surplus obtained by accepting more immigrants can be put into savings.
When the government accepts more immigrants, the capital to labor ratio will decreaseat first. However, due to the fact that the government can put its surplus into savings,
the government can increase the capital to labor ratio in the long run. This will solve theproblem of the under-accumulation of capital that results from operating a PYGO social
security system.

3

The model

The model uses the standard overlapping generation model with a neoclassical production
function developed by Diamond (1965). Each individual lives for two periods. When individuals are in the first period, they work and are called ”young”. When they are in the
5
6

Shimasawa and Oguro (2010) apply the model of Hehr, Joisch and Kotlikoff to the Japanese economy.
To see why, see the discussion about Proposition 1

6second period, they are retired and are called ”old”. I assume that immigrants come tothe host country only when they are young and that the government of the host country
prohibits the immigrants from immigrating when they are already old. Each individual of
the native and the immigrant populations supplies one unit of labor inelastically in the hostcountry when they are young.7 I define individuals who are born at the beginning of period
t in the host country as natives of cohort t, regardless of the nationality of the parents.
Immigrants who move to the host country at the beginning of period t are considered as
immigrants of cohort t. Let j be the index indicating nationality. If an individual is anative, j = n and if she or he is an immigrant, j = m. Let Ntj be the number of the youngof type j in period t. Let (cy,jco,jtt+1 ) be the consumption in the young period and the oldperiod of a type j (j = n, m) individual of cohort t. Let gty,j be the amount of publicly
provided private goods, such as education and government provided health care service forthe young, for each young individual of type j which is consumed at period t. Let gto,jbe the amount of publicly provided private goods, such as medicaid and publicly provided
nursing home, for each old individual of type j at period t. Let gtind,j be the amount of
age-independent publicly provided private goods that are consumed by a young individual
and an old individual of type j.8 I assume that the utility function of cohort of type j isy ind o,n
y y,j ind,j
o
ind
y y,j
)
U (cy,nt , gt , gt , ct+1 , gt+1 , gt+1 ) = u (ct ) + v (gt , gt

+

1
ind,j
o o,j
[uo (co,j
t+1 ) + v (gt+1 , gt+1 )] .
1+ρ

(1)

i i,j ind,j ) (i = y, o) are strictly increasing and concave
I assume that ui (ci,jt ) and v (gt , gt

functions. I assume additive separability of publicly provided private goods so that theprovision of publicly provided private goods does not affect the consumption and saving
decisions of individuals. This assumption simplifies the analysis because I also assume thatthe government redistributes the welfare gain of accepting more immigrants in the form of
increased publicly provided private goods to individuals.9
For the production side, let F (Lt , Kt ) be a production function where Lt and Kt are thetotal amount of labor and the total capital stock used at period t. Let δ be the depreciation
7

In section 3.3, I show that the main results of this paper hold even when the labor supply is elastic. I

use the assumption of an inelastic labor supply here to simplify the analysis in this section.
8
In this paper, I ignore non-rivalry public goods. Note that the presence of non-rivalry public goods will
favor immigration because accepting immigrants means that the cost of non-rivalry public goods will beshared by more individuals without decreasing their consumption.
9
There are several ways to redistribute the welfare gain to individuals. I use this method to simplify the
analysis. The main conclusion does not change when other way of redistributing the welfare gain are used.

7

rate of the capital. I assume that F (Lt , Kt ) exhibits constant returns to scale and thatboth the marginal product of labor and capital are diminishing. I assume that the standard
Inada condition is satisfied.
I assume that the economy is at the steady state initially and that the initial economy
is dynamically efficient. In the literature, it is well-known that if the market interest rate islower than the population growth rate, it is possible to Pareto-improve welfare (dynamic inefficiency). Since this paper’s interest is not such a dynamic inefficiency problem, I postulatethat at the initial steady state, the market interest rate is higher than the population growthrate (Cass(1972)). Furthermore, for the welfare analysis of accepting more immigrants, I
make the following additional assumptions.
AS1: The amount of publicly provided private goods per person, (gty , gto , gtind ) is constantat 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: A immigrants has φ × 100 percent efficient unit of labor in comparison with a native.
AS4: The government imposes the same flat wage tax rate on immigrants as on natives.
This wage tax includes both social security tax and income tax. For the benefit of the social
security, the government pays φ × 100 percent of each native’s benefit to each immigrant.
AS5: Immigrants and natives have the same preferences
AS6: The descendants of immigrants integrate with the native population and earn thesame income as natives.
AS 7: Immigrants and their children stay permanently in the host country.
AS 8: The fertility rate of immigrants is equal to or higher than the fertility of the native.
AS 9: At the initial steady state, the government does not impose a distorting tax, such as
capital income tax.
I use AS1 to focus on the issue of immigration, rather than on issues of public expenditure. AS2 is used to analyze the relationship between immigration and PYGO social
security. AS 4 needs more discussion. Clearly, if the government can treat the immigrants
in a discriminating way in the tax and the public pension systems, there is a way to increase
the utility of both natives and immigrants. Normally, the wage rate of the immigrants intheir country of origin is lower than the wage rate in their host country. Thus, it is possible
to Pareto-improve the welfare of both natives and immigrants if (a) the government in thehost country sets a high tax rate on incoming immigrants in such a way that the net wage
rate of immigrants in their host country is still higher than their net wage rate of their8country of origin, (b) the government redistributes to natives the tax revenue collected from
immigrants. AS4 precludes such an obvious case from occurring. AS5 is not important
for the analysis. In section 3.3, AS5 is relaxed. AS 6-8 are not critical and are used tosimplify the analysis. As for AS9, if there are already distorting taxes at the initial steadystate in the economy, clearly the government can Pareto-improve the initial steady state
by correcting those taxes. Because the labor supply is inelastic, the wage tax is optimaland the use of capital income tax is sub-optimal. I use AS9 because the focus of this paper
is the first order effect of accepting immigrants, not tax reform. Of course, in reality, thegovernments of most developed countries impose capital income taxes. In the section 3.2,
I show that the main conclusion does not change even in the presence of distorting taxes
including capital income tax.
Let sjt ,wti and bjt be the saving, the pre-tax wage rate and the social security benefit of
the old of type j of cohort t, respectively. Let rt , τwt and τrt be the pre-tax interest rate,
the wage tax rate and the tax rate on interest income, respectively, at period t. Cohort t
maximizes the above utility function subject to the budget constraint. The budget constraintof the cohort t of type j is
j
o,j
j
j
wtj (1 − τwt ) = cy,jt + st and bt+1 + (1 + (1 − τrt )rt )st = ct+1 .

(2)

Now consider the government budget constraint at period t. At this point, consider thegovernment budget constraint with the capital income tax, even though in this section it isassumed that the capital income tax is not used at the initial steady state. The government
budget constraint at period t is
τwt+1

X

j
j
wt+1
Nt+1
+ τrt+1 rt+1

j=n,m

X

sjt Ntj −j=n,mNtj (bjt + g o,j + g ind.j − (1 + rt )at )

X
j=n,m

−(g y.j + g ind,j − at+1 )

X

j
Nt+1
=0.

(3)

j=n,m

where at is the government saving (debt) balance divided by the total number of population
at t. Let FK and FL be the partial derivative of the production function with respect tojcapital and labor. Then, wt+1
and rt+1 are determined as follows:

X j
X j
X j j
X j
m
n
st Nti + atNt ), wt+1
= φFL (Nt+1+ φNtm ,st Nt + at
Nt )
X j j
X j
n
and rt+1 = FK (Nt+1+ φNtm ,st Nt + atNt ) − δ
(4)
n
n
wt+1
= FL (Nt+1+ φNtm ,Substituting the individual budget constraints into the government budget constraint, we
9have the following resource constraint:
X j j
X j
X j j
X j
st Nt + atNt ) + (1 − δ){st Nt + at
Nt }
X
X y,j
X
j
j
o.j
=
{ct+1 + sjt+1 + g y.j + g ind.j }Nt+1+ at+1
Nt+1+
Ntj {co,j+ g ind.j } (5)t+1 + gnm
F (Nt+1+ φNt+1,

Let the fertility rate of natives and immigrants be πn and πm , respectively. Let αt be theimmigrant-to-native ratio (INR). The immigration policy is expressed in terms of αt . For
example, a one-time increase of the immigrant-to-native ratio means that α0 = α∗ , α1 = αeand αt = α∗ for t ≥ 2 where α
e > α∗ . Permanently increasing the immigrant-to-native
ratio means that α0 = α∗ and αt = αe for all t ≥ 1 where α
e > α∗ . The sum of natives andn {(1 + π ) × α
immigrants of cohort t is Nt−1mt−1 + (1 + πn )} × (1 + αt ). At the steady state,

the population growth rate is (1 + πm ) × α∗ + (1 + πn ). To avoid a situation in which thetotal population of period t − 1 becomes zero or negative, I assume(1 + πm ) × αt−1 + (1 + πn ) > 0 for αt−1 ≥ 0 .

3.1

(6)

The Effect of Increasing the INR through Intergenerational Channel

When more immigrants are accepted, they affect the welfare of natives in two channels. The
first is the intergenerational channel. Immigrants will bring an inflow of labor that generates
additional funding for PYGO social security. The second channel is the intra-redistribution
effect. When the average earnings of immigrants are lower than the average earnings ofnatives but an immigrant consumes the same (or higher) amount of publicly provided private
goods such as public education and health services as a native, the inflow of immigrants
imposes a fiscal burden on the government budget and this tighter government budget islikely to reduce the welfare of natives. This section focuses on the inter-generational effectand its long run implication on capital accumulation. To see this intergenerational effectin a transparent way, I assume that the efficiency unit of labor of incoming immigrants isthe same as that for natives and, that natives and immigrants consume publicly provided
private goods in the same way. This implies that φ = 1 and g i,j = g i in our model. The casethat includes both the inter-generational and the intra-redistribution effects is analyzed in
section 3.5.
The assumptions, AS3-5, φ = 1 and g i,j = g i imply that consumption and labor supplyof natives and immigrants are the same. It also implies that bn = bm . Thus, the resource

10constraint becomes as follows:
F ({(1 + πm ) × αt + (1 + πn )} × (1 + αt+1 ), (snt + at )(1 + αt )) + (1 − δ)(snt + at ) ≥
n
y
ind
o
ind
{cy,n
+ at+1 }{1 + πm ) × αt + (1 + πn )} × (1 + αt+1 ) + {co,n}(1 + αt )t+1 + st+1 + g + g
t+1 + g + g

Before analyzing the effect of accepting more immigrants, we characterize the initialsteady state. Let w∗ and r∗ be the pre-tax wage rate and the interest rate at the initialsteady state respectively, where the immigration policy αt is equal to α∗ for all t. Let s∗
and N0n be the savings of each individual and the number of old people at period 0. Let τw∗
and τr∗ be the wage tax rate and capital income tax rate at the initial steady state. Let b∗
be the social security benefit at the initial steady state.
The initial steady-state economy with zero capital income tax and α∗ is characterized
as follows:s∗ = arg max
uy (w∗ (1 − τw∗ ) − snt ) + v y (g y , g ind ) +nst1[uo ((1 + r∗ )s + b∗ ) + v o (g o , g ind )]
1+ρ
(7)

w∗ =

∂F ((1 + πm )α∗ + 1 + πn , s∗ )∂F ((1 + πm )α∗ + 1 + πn , s∗ ); r∗ + δ =∂L∂K(τw∗ w∗ − g y − g ind ) × {(1 + πm )N0n α∗ + (1 + πn )N0n } × (1 + α∗ ) − (b∗ + g o ) × N0n (1 + α∗ ) = 0
where τw∗ , g y , g o ,g ind , b∗ are given.
The steady state level of the consumption and the utility with zero capital income tax
are defined as follows:cy∗ ≡ w∗ (1−τw∗ )−s∗ ; co∗ ≡ (1+r∗ )s∗ +b∗ ; u∗ ≡ uy (cy∗ )+v y (g y , g ind )+

3.1.1

1
[uo (co∗ )+v o (g o , g ind )] .
1+ρ(8)

Welfare Effect of Increasing the INR

In this sub-subsection, I examine whether or not increasing the INR to a higher level permanently will Pareto-improve welfare. Increasing the INR permanently is defined as α0 = α∗and αt = αe where α
e > α for t ≥ 1. For the analysis, consider the following constrained
maximization problem:11

V (α) =

max

o,n n
{cy,n
t ,ct ,st , at |t=1,2,..}y y inds.t. uy (cy,n)+
t ) + v (g , g

1oind[uo (co,n)]1 ) + v (go , g
1+ρ

1
o o ind[uo (co,n)] ≥ u∗ for t = 1, 2, ...
t+1 ) + v (g , g
1+ρ

(9)

F ({(1 + πm )α∗ + (1 + πn )} × (1 + α), s∗ × (1 + α∗ )) + (1 − δ)s∗ × (1 + α∗ ) ≥
n
y
ind
o
ind
{cy,n
+ a1 }{(1 + πm ) × α∗ + (1 + πn )} × (1 + α) + {co,n} (1 + α∗ )1 + s1 + g + g
1 +g +g(10)
F ((1 + πm )α + (1 + πn ), (snt + at )) + (1 − δ)(snt + at ) ≥
n
y
ind
o
ind
{cy,n
+ at+1 } × {(1 + πm ) × α + (1 + πn )} + {co,n} for t = 1, 2, ...
t+1 + st+1 + g + g
t+1 + g + g(11)
The above programming problem deserves several comments. First, V (α) is the utilityof cohort 0 at period 1 when the government accepts immigrants with a constant ratio α∗ at
the initial steady state and starts to accept immigrants with ratio α from period 1. Second,
the first constraint is related to Pareto improvement and requires that all cohorts except
cohort 0 need to have at least as the same utility as they would have at the initial steadystate. Third, (10) and (11) are the resource constraints of period 1 and later periods.10 I
use the resource constraint, rather than the government budget constraint, because the twoare equivalent.11
Let γt be the Lagrangian multiplier of (9). Let L and λt be the Lagrangian function
and Lagrangian multiplier of the resource constraints of (10) and (11), respectively. Let γt∗and λ∗t be the Lagrangian multiplier of the utility constraint and the resource constraint at
period t when α = α∗ . Then, we have the following observation.
Observation 1
When α = α∗ the solution of MPP iscy,n= cy∗ , co,n
= co∗ , st = s∗ , at = 0 for t = 1, 2, ...
t0)α∗ + 1 + π(1 + πm1n ∗u0o (co1 ) and λ∗t+1 =λt
1+ρ
1 + r∗
1γt∗ = 0 y∗ λ∗t {(1 + πm )α∗ + 1 + πn } and for t = 1, 2, ...
uy (c )

λ∗1 =

(12)
(13)
(14)

For the proof of observation 1, see appendix B1.
1011In (11), (1 + α) is multiplied on both sides of the original resource constraint. Thus, it is eliminated.
In these resource constraints, tax and social security are not defined. However, once the consumption

and saving level are defined, the tax and social security benefit are calculated implicitly.

12

Observation 1 implies that when α = α∗ , the initial steady state allocation is Paretoefficient and it is not possible to have Pareto improvement from the initial steady state. Now
suppose that the government increases the INR from α∗ . Whether or not such an increaseof the INR Pareto-improves welfare can be analyzed by calculating V 0 (α) and evaluating it
at α = α∗ . From the envelope theorem, dV /dα|α=α∗ is equal to({(1 + πm )α∗+ 1 + πn } λ∗1 + (1 + πm )

∞
X

)
λ∗t

n
o
× FL ((1 + πm )α∗ + 1 + πn , s∗ ) − cy∗ − s∗ − g y − g ind .

t=2

(15)
The first bracket is positive because the Lagrangian multiplier of the resource constraint
is positive. In the second bracket, the first term is the marginal product of labor, which iswhat an individual contributes to the economy when he or she is young at the initial steadystate. cy∗ + s∗ is equal to the after tax income of the young individual. g y + g ind are theprivate goods that are provided by the government for the young individual. Therefore,cy∗ + s∗ + g y + g ind is what an individual receives while he or she is young at the initialsteady state. Therefore, the bracket yields the amount of an intergenerational transfer fromthe young to the old. For example, if the inside of the second bracket is equal to zero,it implies that what the young earn is allocated fully to the young (by including publicly
provided private goods), whereas values above zero entail that there is an intergenerationaltransfer.
Because the marginal product of labor of the young is the pre-tax wage of the youngat the initial steady state and because cy∗ + s∗ is the after-tax income of the young by
the definition, FL (1 + πn , s∗ ) − cy∗ − s∗ is the amount of tax paid by the young. Thisimplies that if the amount of tax paid by the young is greater than the goods provided
by the government to the young at the initial steady state, accepting more immigrants
Pareto-improves welfare.12
Proposition1. (MPL condition) If there exist intergenerational transfers from theyoung to the old at the initial steady state in the sense that the marginal product of labor of
the young is greater than the amount of resources which they receives while they are young(after tax income plus publicly provided private goods), then accepting more immigrants
Pareto-improves the welfare of all generations. Alternatively, if the amount of tax thatthe young pay is greater than the amount of publicly provided private goods for the young,
12

Note that this condition does not change even in the presence of public goods because an increase of thenumber of immigrants does not affect the consumption of public goods by the nature of public goods.

13accepting more immigrants Pareto-improves the welfare of all generations.
It should be noted that this MPL condition is important for measuring the real effect onwelfare of accepting more immigrants. In a typical study of assessing the effect on welfare of
accepting more immigrants, it calculates the present value of the government expenditureand the tax revenue that a typical immigrant brings. The above MPL condition showsthat such a welfare index is quite misleading. The above MPL condition suggests that thedifference of taxes and expenditure while immigrants are young is more important than thepresent value of taxes and expenditures over the life cycle.13
Graphically, Proposition 1 can be explained as follows. Consider a simple case in which
the depreciation rate is 100% (δ = 1) and there are no immigrants at the initial steady state(α∗ = 0). Assume that the population of the old is equal to one. At period 0, the following
resource constraint must hold:F (1 + πn , s∗ ) − (co∗ + g o + g ind ) − (1 + πn )(cy∗ + s∗ + g y + g ind ) = 0.

(16)

Note that F (1 + πn , s∗ ) is the GDP per one old individual when the population growth rateis equal to πn . Now consider the graph of (y, x) where the vertical axis measures the GDP
per one old individual and the horizontal axis measures one plus the population growthrate. Thus, x = 1 + πn , y = F (x, s∗ ). Next, draw the line of (y, x) where y is defined as
y = (cy∗ + s∗ + g y + g ind )x. At x = 1 + πn , the vertical distance of this line represents thetotal amount of resources used for the young divided by the number of old individuals when
the population growth rate is equal to πn . Thus, the difference between y = F (x, s∗ ) andy = (cy∗ + s∗ + g y + g ind )x at x = 1 + πn represents the amount of resources used for one
old individual.
Now suppose that a social planner increases the population growth rate by accepting
immigrants permanently. This implies that x moves to the right from x = 1 + πn . If
the slope of y = F (x, s∗ ) at the x = 1 + πn is greater than cy∗ + s∗ + g y + g ind , thesocial planner can maintain the same allocation of resources per each 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 thisconstitutes Pareto improvement. Note that when the government accepts immigrants there
is a surplus that is equal to ∂F (1 + πn , s∗ )/∂L − (cy∗ + s∗ + g y + g ind ) in every period. Also
13

When immigrants ana natives are different, then the sum of the redistribution from immigrants to natives

and the difference of MPL and what an individual receives while she is young becomes important.See section
3.3.

14

note thatP

λ∗t is the present discounted value of the increased amount of resources for all

periods. Therefore, (15) presents the present discounted value of the surplus of all periods
that is obtained by increasing the population growth rate(see Figure A1).

3.2

Elastic labor Supply, Distorting Taxes and Implementation of Pareto
improvement

In the preceding subsection, it was shown that it is Pareto-improving to accept more immigrants when there are intergenerational transfers from the young to the old in the sensethat the marginal product of labor of the young is greater than the resources that are allocated to the young. With a neoclassical production function that exhibits diminishingmarginal product, the pre-tax wage decreases and the pre-tax interest rate increases whenmore immigrants are accepted. To Pareto-improve the welfare of all generations, taxes must
be adjusted or other policies must be implemented. However, the preceding analysis does
not show how taxes or other polices are used while more immigrants are being accepted.
In addition, in the preceding subsection, I assumed that the labor supply is inelastic andthat there is no capital income tax at the initial steady state. When the labor supply iselastic, the wage tax rate or zero capital income tax rate can be suboptimal at the initialsteady state. When the wage tax or capital income tax is sub-optimal. The analysis of thepreceding subsection can not be used because the earlier analysis assumed that the initialsteady state is efficient.14
In this subsection, I will show that, as long as the MPL condition is satisfied, increasing
the INR Pareto-improves welfare without relying on changing incentives of individuals even
if the labor supply is elastic and even if there are distorting taxes (wage tax rate or capitalincome rate or both) at the initial steady state.15 In addition, I show that a relatively simple
adjustment of taxes and social security benefit achieves this Pareto improvement while thegovernment is increasing the INR.
For the analysis, let α
e be a time invariant new immigration policy from period 1 whereαe > α∗ . I assume that the labor supply is elastic and that the government uses a capitalincome tax and a wage tax at the initial steady state. I assume that those taxes do not
14

If we assume that the wage tax rate and capital income tax rate are the second-best, we can apply

the analysis of the preceding subsection to the case in which labor supply is endogenous. However such anassumption is very restrictive.
15
In the presence of non-optimal taxation, it is obvious that it is possible to Pareto-improve welfare if thegovernment can induce changes in incentives of individuals.

15

need to be the second best optimal. With the assumption of the elastic labor supply, theindividual budget constraint (2) is modified as follows:16
j
j
j
o,j
wtj ltj (1 − τwt ) = cy,jt + st and bt+1 + (1 + (1 − τrt )rt )st = ct+1

Let l∗ be the labor supply of a young native or immigrant when he or she faces the samewage rate and tax rate as at the initial steady state. I assume that at the initial steadystate the social security benefit is proportional to pre-tax earnings:b∗ = Ω × w∗ l∗(17)

When the government increases the INR, the wage rate falls and the interest rate increases
initially. To achieve Pareto improvement by increasing the INR, first I assume that the
government sets the tax rates such that after-tax wage and interest rate after an increaseof the INR are equal to the after-tax wage rate and interest rate at the initial steady state.
Then, the wage tax rate and the interest tax rate for period t are set as follows:
wt (1 − τwt ) = w∗ (1 − τw∗ ) and rt (1 − τrt ) = r∗ (1 − τr∗ )(18)

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 thesame benefit when wage rate, wage tax rate and labor supply are at the same level as at
the initial steady state. This implies thatbjt+1 =Ω
wt (1 − τwt )ltj∗
1 − τw(19)

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

(20)

In the following budget constraint, I put superscript j to apply the current analysis to other cases

although it is assumed that natives and immigrants behave in the same way since they have the same
preferences and wage levels.

16

The LHS of the above equation is a marginal increase of the output due to a one unit
increase of the population. The RHS is the amount of resources that an individual receives
when he or she is young.
The result that was generated in the preceding subsection shows that a marginal increasein the number of immigrants Pareto-improves welfare if the MPL condition is satisfied.
However, that result does not imply that an unlimited acceptance of immigrants always
Pareto-improves welfare. I assume that when the new immigration policy αe is operative,
the marginal increase of the output due to a one unit increase of the population is greateror equal to the amount of resources that an individual receives when he or she is young at
the initial steady state. This implies thatl∗n + (1 + π ) × N m }, (N n + N m )s∗ )∂F (l∗ × (1 + αe) × {(1 + πn )Nt−1
m
t−1
t−1
t−1
≥ cy∗ +s∗ +g y +g ind .
∂L(21)

m + (1 + π )N n
Note that (1 + πm ) × Nt−1nt−1 is the population of young native at period t.
m = αn , (21) can be written
Using the homogeneity of the production function and Nt−1
eNt−1

as
l∗

∂F (l∗ × {(1 + πn ) + (1 + πm ) × αe}, s∗ )≥ cy∗ + s∗ + g y + g ind .
∂L

(22)

When the government accepts more immigrants according to (20) and (21), and it adjusts
the wage tax rate and capital income tax rate so that the after-tax wage rate and after-tax
interest rate are the same as at the initials steady state as defined in (18), the government
can accumulate savings as I show below. This implies that it is possible that, at some point,
the marginal product of capital (MPK) becomes equal to the golden level of capital stock
per capita. But when the MPK becomes equal to the golden rule level, it is clearly better to
use the entire government surplus to increase the supply of publicly provided private goods
rather than to increase the government savings balance. Thus, I assume that as long as
the MPK is higher than the golden rule level, the government uses some of the government
budget surplus to increase the government saving balance and the remainder to increase thesupply of publicly provided private goods. When the MPK reaches the golden rule level, thegovernment uses all of the surplus to increase the publicly provided private goods. Thus,
we have the following MPK condition:∂F≥ δ + (1 + πm )e
α + πn∂K(23)

Now we examine whether social security benefit and taxes determined by (19) and (18)17are feasible from the point of the government budget constraint. To check the feasibility ofsuch taxes, consider the net government budget surplus for period 1, SP1 .
SP1 = w1 τw1 l∗

X
j=n,m

N1j + r1 τr1 s∗

X

N0j − (b∗ + g o + g ind )

j=n,m

X

N0j − (g y + g ind )

j=n,m

N1j

X
j=n,m

(24)
Note that τw1 and τr1 are defined in (18). By substituting τw1 and τr1 into SP1 and using
the homogeneity of the production function, we have (see appendix B2)
Z ∗ n
1 l N1 (1+eα) ∗ ∂F (z, sN0n (1 + α∗ ))SP1 = ∗
[l− cy∗ + s∗ − (g y + g ind )]dz
l l∗ N1n (1+α∗ )
∂L

(25)

where N1n = N0n (1 + πn ) + N0n (1 + α∗ )(1 + πm ). Thus, from the MPL conditions (20) ande)]. This means(21), the inside of the integration is positive for z ∈ [l∗ N1n (1 + α∗ ), l∗ N1n (1 + αthat this tax plan is feasible in period 1. The government can use some of the surplus of
the budget to increase the supply of publicly provided private goods and put remainder insavings. Let a1 > 0 be the balance of the government savings per each young individual at
the end of period 1. What will be the net budget surplus in period 2, SP2 ?
Note that the natives and immigrants of cohort 1 will save the same as the natives andimmigrants at the initial steady state, because they face the same after-tax wage rate andinterest rate under the proposed tax policy as at the initial steady state. This implies thatP
sn1 = s∗ for both groups of cohort 1. The total capital stock for period 2 is j N1j (s∗ + a1 ).
Thus, SP2 becomes
SP2 =

X

N2j ×(w2 τw2 −g y −g ind )+

j=n,m

X

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

j=n,m

X

N1j .

j=n,m

(26)
Note that the pre-tax wage for period 2, w2 , and the pre-tax interest rate for period 2,
r2 , are equal tow2 ≡∂F (l∗ (N2n + N2m ), (N1n + N1m )(s∗ + a1 ))∂F (l∗ (N2n + N2m ), (N1n + N1m )(s∗ + a1 ))and r2 ≡
−δ
∂L
∂K
(27)

Again the government sets τw2 and τr2 such that after-tax wage rate and after-tax interest
rate become the same as at the initial steady state. Thus, SP2 becomes (see appendix B3)
Z (s∗ +a1 )N n (1+eα)
1∂F (l∗ N2n (1 + αe), z){SP2 =+ (1 − δ)}dz∂Ks∗ N1n (1+eα)
Z αe∂F ((l∗ {1 + πn + (1 + πm )z}, s∗ ) ∗n+ (1 + αe)N1 (1 + πm )[l − (1 − τw∗ )w∗ − (g y + g ind )]dz
∂L
α∗
(28)
18

The first term of (28) measures the welfare gain that arises from the additional savings thatthe government accumulated at the end of period 1. The second term measures the welfare
gain that arises from the increased population growth rate in the presence of the PYGO
social security. From the MPK condition (23), the inside of the first integration is positive.
From (22), the inside of the second integration is positive. Thus, SP2 is positive and thegovernment can implement the proposed tax policy. Again, at the end of period 2, thegovernment can use some of the above surplus to increase the supply of publicly provided
private goods and put the rest to increase the balance of government savings. Similarly, thegovernment surplus for period t becomesZ (s∗ +at−1 )N n (1+eα)
t−1

SPt =

{

n (1+es∗ Nt−1α)+ (1 + αe)Ntn (1 + πm )∂F (l∗ Ntn (1 + αe), z)+ (1 − δ)}dz∂K∂F ((l∗ {1 + πn + (1 + πm )z}, s∗ ) ∗l − (1 − τw∗ )w∗ − (g y + g ind )]dz
∂L
α∗

Z αe

[

(29)
where at is the balance of government saving per each young individual at the end of periodt−1. Again the government uses some of the surplus for increasing publicly supplied private
goods and puts the rest in savings. This implies that SPt > 0 for all t = 1, 2, .... Thus, wehave the following proposition:
Proposition 2.

Consider an economy in which the labor supply is elastic and the(optimal or non-optimal) wage and interest taxes are used at the initial steady state. If the
MPL condition is satisfied at the initial steady state, accepting more immigrants with tax
rule (18) and social security benefit rule (19) Pareto-improves the welfare of all generations.
3.2.1

Government Savings and the Golden RuleIn this subsection, I examine the capital stock path and government saving path when thegovernment increases the INR. To examine the government saving path, I need to specify
how much of the government surplus, SPt , is put into savings. To simplify the argument, I
assume that the surplus that arises from the increased government savings at period t − 1,which is the first integration of SPt , is put into the government savings at period t. Notethat the government could use some part of the second integration in SPt , the surplus that is
generated directly from the increased immigration. Thus, my assumption is a conservativevalue for the government savings. I will show that, even with this conservative level, theeconomy reaches the golden rule level of capital stock per capita within a finite time in a19

Pareto-improving way. The government savings per each young individual at the end of
period t for t ≥ 2, at , becomes1
at = nNt (1 + αe)

Z (s∗ +at−1 )N n (1+eα)t−1{n (1+es∗ Nt−1α)∂F (l∗ Ntn (1 + αe), z)+ (1 − δ)}dz
∂K

(30)

Note that from (23), we have1at ≥ nNt (1 + αe)

Z (s∗ +at−1 )N n (1+eα)
t−1[(1 + πm )e
α + 1 + πn ]dz = at−1

n (1+es∗ Nt−1α)Thus, at is increasing over time as long as at is determined according to equation (30) and(23). By rewriting the above equation, we haveat =1{F (l∗ {1 + πn + αe(1 + πm )}, s∗ + at−1 )1 + πn + αe(1 + πm )−F (l∗ {1 + πn + αe(1 + πm )}, s∗ ) + (1 − δ)at−1 } ≡ Q(at−1 )Now, consider the graph of at = Q(at−1 ). Q(at−1 ) is zero at at−1 = 0 and Q(at−1 ) is concave
due to the diminishing marginal product of capital. The slope of Q(at−1 ) at at−1 = 0 is
Q0 (0) =1∂F (l∗ {1 + πn + αe(1 + πm )}, s∗ )(+ 1 − δ)1 + πn + αe(1 + πm )
∂K

(31)

Because of assumption (23), Q0 (at−1 ) at at−1 = 0 is greater than 1. Thus, the at = Q(at−1 )and 45 degree line intersect at at−1 = 0 and at a∗ where a∗ > 0. Let a∗∗ be the point where
Q0 (a∗∗ ) = 1. This implies that at a∗∗∂F (l∗ {1 + πn + αe(1 + πm )}, s∗ + a∗∗ )+ 1 − δ = 1 + πn + αe(1 + πm ).
∂K
In other words, at a∗∗ the golden rule is satisfied. Note that the government can choose
a1 because the surplus at period 1 is strictly positive. From the graph of at = Q(at−1 ), at
keeps increasing starting from a small a1 > 0. Before it reaches a∗ , it reaches a∗∗ within afinite time. This implies that the economy reaches the golden rule level of capital stock per
capita within a finite time.
Note first that in this analysis, I assume that only the first integration of SPt , i.e.
the surplus that arises from the increased government savings at period t − 1, is put intosavings at period t. However, the second integration of SPt , i.e. the surplus which arises
directly from increased immigrants, also can be put into savings. Thus, the government canshorten the time to reach the golden rule level by putting the surplus that arises directlyfrom increased immigrants. Secondly, we can note that the government can induce the20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 (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 and using the proposed tax rule, social security benefit rule andgovernment saving policy, the government can induce the economy to reach the (modified)golden rule level within in a finite time in a Pareto-improving way.

3.3

Intra-redistributional Channel and Difference of Productivities andPreferences

When immigrants earn less than natives or immigrants consume more publicly provided
private goods than natives, accepting more immigrants could decrease the welfare of natives,because it means that more resources are taken from natives and used for immigrants. Thisis the intra-redistributional channel of accepting more immigrants. A similar redistribution
could also happen when the preferences of immigrants and natives are different and the labor
supply or savings of immigrants differs from those of natives. To analyze this redistributional
channel of accepting more immigrants, consider again a permanent change of immigration
policy such that α
e > α∗ . Let (cy,j∗, ,co,j∗ , sj∗ , lj∗ ), be the consumption at the young period,
the consumption at the old period, savings and labor supply of type j(j = n, m), respectively
when the wage rate and the interest rate are at the initial steady-state level and the tax
rate on interest and wage are also at the initial steady state level. Since I assume thatthe preferences and the productivities of the native and immigrant are different in thissection, I put superscript j even for the variables at the steady-state situation. Also, let
g i,j be the publicly provided private goods for goods type i and type j nationality wherei = y, o, ind and j = n, m. For conditions that guarantee that accepting more immigrants

21

Pareto-improves welfare, I assume that the following condition is satisfied:
Modified MPL condition=[∂F n∗l − cy,n∗ − sn∗ − g y,n − g ind,n ]
∂L

∂F n∗
φl − cy,m∗ − sm∗ − g y,m − g ind,m ]
∂L
∂F m
∂F
(1 + πn + (1 + πm )α) ×
φl + [+ (1 − δ)]sm∗
∂L
∂K
+α[

−(1 + πn + (1 + πm )α)[φ(1 − τw∗ )w∗ lm∗ + g y,m + g ind,m ]−(co,m∗ + g o,m + g ind,m ) > 0 for all α ∈ [α∗ , αe] :∂F ((1 + πn + α(1 + πm )ln + φJ(α)lm , sn∗ Ntn + sm∗ α)
∂F
=
,
∂L
∂L
∂F
∂F ((1 + πn + α(1 + πm )ln + φJ(α)lm , sn∗ Ntn + sm∗ α)
=
∂K
∂K

where

and J(α) = (1 + πn + α(1 + πm ))α(32)

The first line is the marginal product of labor of a native minus the amount of resourcesthat a young naive receives while he/she is young. The inside of the bracket of the secondline is the marginal product of labor of an immigrant minus the amount of resources thatan immigrant receives while he/she is young. Thus, the first and the second lines are theintergenerational effect. The third, fourth and fifth lines comprise what young and oldimmigrants earn minus what young and old immigrants receive. Thus, the third to fifthlines show the redistribution effect. If a native and an immigrant are completely identicalwith respect to human capital level, labor supply, savings and the consumption of publicly
provided private goods, the sum of the third to fifth line is equal to zero. (32) shows thatthe sum of intergenerational effect and redistributional effect should be greater than zero.
When immigrants and natives are identical with respect to human capital level, labor supply,savings and consumption of publicly provided private goods, (32) becomes (21). Given (32),
we have the following propositions.
Proposition 4 Assume that native and immigrant are not identical with respect tohuman capital level, labor supply, savings and consumption of publicly provided private
goods. If the Modified MPL condition is satisfied, it is possible to Pareto-improve thewelfare of all generations by accepting more immigrants. For the proof, see Appendix B4.
Proposition 5 If the Modified MPL condition is satisfied, the government can induce
the economy to reach the (modified) golden rule level within a finite time in a Paretoimproving way by accepting more immigrants. For the proof, see Appendix B5.

22

4

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

Propositions 1 and Proposition 2 above show that accepting more immigrants can Paretoimprove the welfare of all generations if there are intergenerational transfers in the sense thatthe marginal product of labor of a young individual is greater than what he or she receives
including publicly provided private goods while he or she is young (the MPL condition).
Furthermore, Proposition 3 shows that if the government can put some of the welfare gain
into savings, it is possible to induce the economy to reach the golden rule level of capitalin a Pareto-improving way within a finite time. This result is in sharp contrast to findingsin the literature on social security reform which shows that some generation must beara double burden to increase the capital stock of the economy in the presence of a PYGO
social security system (Geanakoplos, Mitchell and Zeldes 1998). In addition, the proposition
4 and 5 show that even if an immigrant earns less than a native, accepting more immigrants
Pareto-improves welfare when the intra-redistribution is not as large as the intergenerationaltransfer from the young to the old.
A number of issues arise as a result of accepting proposition 1-5. First, those propositions are based on a two-period overlapping generation model. In a realistic multi-period
overlapping generation model, it might not be possible for an economy to reach the (modified) golden rule level of capital stock per capita within a finite time in a Pareto-improving
way by accepting more immigrants. Second, although proposition 3 shows that the economy
reaches the (modified) golden rule level of capital stock per capita within a finite time, inpractice it might take a long time, as long as 1000 years, to reach the golden rule level.
Third, propositions 1-5 are silent on the quantitative effect on welfare. Given that it mighttake a quite long time to reach the golden rule level, the welfare gain of accepting more
immigrants can be very small. In addition, propositions 4 and 5 say nothing about thedegree to which the difference in productivity between immigrants and natives is allowed
for with respect to Pareto improvement when more immigrants are accepted.
This section addresses these issues. To address these issues, I use the computational
overlapping generation model developed by Auerbach and Kotlikoff (1987). I assume thatthe model 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 assume
that the model economy is similar to the US economy in several dimensions.17 In the17

However, the model economy is different in several important dimensions as well. For example, the23

analysis, I assume that initially the model economy is on the balanced growth path andthe percentage of immigrants to natives (PITN), which is defined as 100 times the numberof immigrants divided by the number of natives, is similar to that which the US 2000
census data indicates. Then, I will examine whether it is possible to Pareto-improve thewelfare of all the generations in the model economy by increasing the PITN by a reasonable
size. In addition, I examine how long it takes for the model economy to reach the (modified)golden-rule level in a Pareto-improving way and quantify the Pareto-improving welfare gain.
To check the robustness, I recalculate the model by changing the value of the following
parameters: the replacement, initial government debt (asset) level, the level of earningsof immigrants, immigrants’ consumption of publicly provided private goods, probability ofimmigrants returning to the country of origin, the CRRA and the time preference rate.

4.1

The Model Economy: Auerbach and Kotlikoff Model with Immigration

Agents will show up in the model from age 1. I assume that age 1 corresponds to age 20 inreal life. They work from age 1 until age 45. From the beginning of age 46, they retire. At
each age, they die with some probability and they can live until age 80. Let i be the index
of age. For i ≥ 2, let pi be the probability that an agent is alive at the age i, given thathe or she is alive until age i − 1. Due to the lack of data, I assume that pi is the same fornatives and immigrants. To simplify the notation, I assume that p1 = 1.18 An agent who
enters the model at period t maximizes the following utility function:

max

45
X
i=1

βi

i
Y

(
pq

q=1

) 80
( i,j
)
i,j
i
ζ
1−ζ ]1−γ
ζ(1−γ)X Y[c][(ci,jij
ij
t−1+i
t−1+i ) (1 − lt−1+i )β i pq+ gt−1+i ++ gt+1−i
1−γ
1−γ
i=46

q=1

(33)
i,j
where ci,jt−1+i and lt+1−i are the amounts of private consumption and labor supply of type jiagent of age i at period t − 1 + i. gt+1−i
is the amount of publicly provided private goodsij
to age i agent in period t + 1 − i. I assume that the amount of gt+1−i
is chosen by thegovernment. In this formulation, as in Storesletten (2000), I postulate that the immigrants
model economy does not include a certain aspect of open economy such as international trade and capital
mobility. The model economy does not incorporate the accumulation of human capital of the natives andimmigrants.
18
Infant and child mortality is defined in (39).

24

assume they will stay in the host country until the end of their lives.19 Let si,jt be thesavings of type j of age i at time t. The budget constraint of an agent at age i is
i−1,j
i,j
i,j
st−2+i(1 + rt−1+i (1 − τr,t−1+i )) + (1 − τw,t−1+i )wt−1+i H i,j × lt−1+i= ci,jt−1+i + st−1+i for 1 ≤ i ≤ 45

(34)
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 ≤ 80

(35)

i,j
si,j
t−1+i = 0 for i = 0 and st−1+i ≥ 0 for 1 ≤ i ≤ 80

(36)

where H i,j is the efficient unit of human capital of type j at age i and H i,j > 0 for 1 ≤ i ≤ 45
and H i,j = 0 for i ≥ 46. wt is the wage rate for one efficient unit of labor at period t. I
assume that an individual cannot have a negative savings balance. Once an individual dies,
the government imposes a 100 percent inheritance tax. bi,jt−1+i is the social security benefit
for type j of age i that is given at period t − 1 + i. For i ≤ 45, bi,jt−1+i = 0 and for i ≥ 46,bi,jt−1+i is determined as follows:jjbi,jt−1+i = 12 × RR × AIM E (t) and AIM E (t) =i,j45−i wi,jt−1+i lt−1+i H
i=1 (1 + µ)P4545 × 12

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

(37)

where θ is the capital share and Et represents the level of technology. µ is income per capita
growth rate. Lt is the efficient unit of labor supply at period t.
Let 1 − pbi be the probability that an immigrant returns to her/his home country at the
beginning of age i, given that she or he stays in the host country at the age i − 1 for i ≥ 2.
To simplify the notation, I assume that pb1 = 1. Let Ntij be the number of agent of age i oftype j at time t. Then, Ntim = pi pbi Nti−1,m and Ntin = pi Nti−1,n . Lt is defined as follows:
Lt =

45 X
X

H i,j Nti,j lti,j(38)

i=1 j=n,m19

Alternatively, we can assume that immigrants will enjoy the same wage level in their home country as

in the host country even if they return to their home countries. The two assumptions generate the same
results.

25

I assume that all immigrants will 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 let σm be the growthrate of the number of immigrants of age 1 at the steady state. The total number of natives
P80
P80−(i−1) × Πiin = N 1,nist
q=1 pq . The total number of immigrants is
i=1 (1 + σn )i=1 NtP−(i−1) × Πibq . This implies that the ratio of the total number of
Nt1,m 80q=1 pq p
i=1 (1 + σm )immigrants to the total number of natives is constant if and only if the growth rates of Nt1,nand Nt1,m are the same. Thus, for calculating the steady state, I assume that σn = σm = σ.
I assume that the children of immigrants become native if their parents stay in the hostcountry until the children attain adulthood. Then, Nt1,n is determined by the fertility rates
of both immigrants and natives and the return rate of immigrants. Let ηji,j be the agespecific fertility of type j at age i. The number of children born of type j parents of age i is
Nti,j η i,,j . Let d be the infant-child mortality. Then the number of natives of age 1 at t + 20,
1,n, is determined as follows:
Nt+20

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

(39)

i=1

The percentage of immigrants to natives (PITN) at the steady state (see Appendix A6)
becomesP80
1
1i,n × Πii,mq=1 pqi=1 (1+σ)i+19 η
1−d −i=1 Nt
×
100
=
P
P80
80
i,n
1
i
i,m × Πbq Π20bi+x
q=1 pq px=1 pi=1 (1+σ)i+19 η
i=1 Nt

P80

(40)

P80
×

−(i−1) × Πibq
q=1 pq p
i=1 (1 + σ)
× 100
P80
−(i−1) × Πi
q=1 pqi=1 (1 + σ)(41)(40) 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 and immigrants are set.
Conversely, we can choose σ so that the resulting PITN is consistent with the data20 .
The capital stock at period t is the sum of the balance of individual savings and government savings. Let at−1 be the balance of the government asset (or debt if it is negative) per capita at the end of period t − 1. Then, the total capital stock at period t is
Kt =

80 X
X

i,j
Nt−1,isi,jt−1 + at−1

i=1 j=n,m

80 X
X

i,j
Nt−1

(42)

i=1 j=n,m

The (efficient unit) wage rate at time t, wt and the pre-tax interest rate at time t, rt ,are determined as
wt = (1 − θ)Ktθ Et1−θ L−θ
t
20

and

rt = θKtθ−1 Et1−θ L1−θt

(43)

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

26

Now, consider the initial balanced growth path where the capital to labor ratio (inefficient units) stays constant. Let w∗ (1 + µ)t and r∗ be the wage rate and interest rate
at period t on the initial balanced growth path. Let s∗i,j (1 + µ)t and b∗i,j (1 + µ)t be thesavings and the social security benefit of type j of age i at period t on the initial balanced
growth path. Let a∗ (1 + µ)t and g ∗i,j (1 + µ)t be the government asset(debt) per capita andthe publicly provided private goods for type j of age i at period t on the initial balanced
growth path. It is possible that immigrants use more public services, for example, children’s
education. Thus, I put superscript j. Let w∗ and r∗ be the efficiency unit wage rate andthe interest rate on the initial balanced growth path determined from (42) and (43). Thenthe government budget constraint on the initial balanced growth path is

80 X
80 X

X
X
i−1,j ∗i−1,j
i−1,j ∗i−1,jst−1s+ (1 + r∗ )(1 − pi )Nt−1(1 + µ)t−1 τwt w∗ (1 + µ)Lt + τrt r∗pi Nt−1
i=2 j=n,mi=2 j=n,m−a∗ (1 + µ)

80 X
X

Nti,j + (1 + rt )a∗i=1 j=n,m

−

80 X
X

b∗i,j (1 + µ) × Nti,j − κ

i=46 j=n,m

80 X
X

i,j
−
Nt−1

i=1 j=n,m
80 X
45X
i=46 x=11

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




=0

(44)



where τw∗ and τr∗ are the wage tax rate and capital income tax rate at the initial balanced
growth path. b∗i,rm,x and Nti,rm,x are, respectively, the social security benefit and the numberof immigrants of age i who returned to the home country at age x, but who are still eligible
to claim social security benefits. κ is the share of those returned immigrants who actuallyclaim social security benefits. The US social security does not require the residence as longas the individual is eligible for benefits. I assume that an immigrant is eligible to receive
social security benefits as long as he or she pays social security contributions for at least
10 years. The first term of (44) is the revenue from wage tax and the second term is therevenue from capital income tax. The third term is the revenue from inheritance tax. In my
simulation, I assume that the government imposes a 100 percent inheritance tax. The fourthterm is the revenue from issuing government bonds and the fifth term is the expenditure on
the principal and interest of the bonds. The sixth and seventh terms are the expenditureon the publicly provided private goods and the expenditure on social security benefits ofnatives and immigrants who reside in the host country. The last term is the social security
benefits of immigrants who returned to the home country and who are eligible to receive
social security benefit.
When the government increases the PITN, the wage rate decreases and the interest rate27

increases. To Pareto-improve welfare, I assume that the wage tax rate, interest tax rate andsocial security benefit are adjusted as in the theoretical analysis. More specifically, I assume
that the government keeps the after-tax wage rate and interest rate at the same level as on
the initial balance growth path. This implies that the 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∗ )(45)

The social security benefit is adjusted as follows:jjbi,jt−1+i = 12×RR×AIM E (t) and AIM E (t) =i,j45−i wi,jt−1+i (1 − τwt )lt−1+i H
i=1 (1 + µ)
(1 − τw∗ ) × 45 × 12

P45

(46)

When the taxes and social security benefit formula are adjusted according to (45) and(46), the individual budget constraints is the same as on the initial balanced growth path21 .
Thus, consumption, labor supply and savings do not change. Then, there would be a surplus
for the government budget even if the government were to spend the same amount of publicly
provided private goods per person as at the initial balanced growth path, as analyzed in
the previous section. The government can use this surplus to increase savings or to increase
the level of publicly provided private goods. Let V be the distributional parameter thatindicates what percentage of the budget surplus is put into savings. The government cankeep increasing the government savings balance until the economy reaches the golden rulelevel or a modified golden rule level. As long as the MPK is greater or equal to the (modified)golden rule level, the balance of the government savings at period t is determined from thefollowing equation:t(1 + µ) at

80 X
X

Nti,j = V × SPt

i=1 j=n,mwhere SPt is the government budget surplus at period t. If the MPK is equal to the(modified) golden rule level, V becomes 0. SPt is defined as follows:

80 X
80 X

X
X
i−1,j ∗i−1,j
i−1,j ∗i−1,jt−1
SPt = τwt wt Lt + (1 + µ)
τrt rtpi Nt−1 s+ (1 + rt )(1 − pi )Nt−1s
i=1 j=n,m(1 + rt ) × at−1

80
X

X

i=1 j=n,m

i,j
Nt−1
−

i=1 j=n,m

−

80 X
X
i=46 j=n,m

21

80
X

X

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

80 X
45
X

b∗i,rm,x (1 + µ) × Nti,rm,x

i=46 x=11

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

28




i,j
45−i ∗w (1+µ)t−1+i lt−1+i H i,ji=1 (1+µ)

P45

45×12

(47)

The rest of the surplus that is available for increasing publicly provided private goods is(1 − V ) × SPt . This is distributed equally to the entire population (including immigrants)
at period t.22 Then, after increasing the PITN, the amount of publicly provided private
goods, gti , becomes(1 − V ) × SPt
gti = g ∗i,j (1 + µ)t + P80i,ni,mi=1 (Nt + Nt )(48)

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

4.2

Policy Experiments and Parameters Values for Simulation

Policy Experiments
For the percentage of immigrants to natives (PITN), I calculate the data from the US census
at 2000. US Census 2000 shows that the PITN above the age of 20 is 15.5 percent. Thus,
I assume that the PITN at the initial balanced growth path is 15.5 percent. For the target
PITN, I look at past US data. Historically, the PITN was 5 percent in 1970 and increased
to 18.3 percent in 2010.23 The PITN increased more than 10 percent within 40 years from1970 to 2010, so I assume that a 10 percentage point increase in the PITN over 80 years is
tolerable. Therefore, I set the target PITN at 25.5 percent. In the slow benchmark case, the
PITN starts to increase from 15.5 percent and reaches the target PITN at the 80th year.
After the 80th year, it remains on the same level. To see how accelerating the speed atwhich the PITN is increased affects my results, I also consider intermediate and fast cases,
22

The government could distribute this surplus more to the natives than to the immigrants. Thus, thisassumption underestimates the welfare gain to the natives.
23
Data in 1970 is from the US Census. Data in 2010 is from the CPS. After 2000, data on the foreign-born
are available only through the American Community Survey and the CPS, not the Census. For calculating
the PITN, one concern is the comparability of the CPS and the Census data. In 2000 when both the census
and the CPS are available, the PITN in the Census is 12.42% and 11.54% in the CPS. Thus, the difference
between the Census and CPS would seem to be very minor. This view is shared by the Census bureau.
Schmidley and Robinson (2003) conclude that the difference of the estimates based on the Census data and
CPS data are trivial.

29

in which the PITN hits the target PITN at the 62nd or 42nd year.24 I assume that all
immigrants arrive at age 1. The graph for the PITN over time is shown in Figure 1.25
Fertility, Mortality and Return Rate of Immigrants
For the age-nativity specific fertility, I use the CPS 2000 June supplement, which the census
bureau also uses for calculating on the age-nativity specific fertility. Figure A2 in the
appendices shows the average of the total number of births by each woman’s age and it
shows that immigrant women have a greater total number of births than native women at
all ages. From this figure, I calculate η i,n and η i,m , the age-specific birth rate for nativeand immigrant women. This is shown in Table A1. For the adult mortality profile, pi , I
take the values from Nishiyama and Smetter (2007). I set d, the sum of infant and childmortality, 1.7 percent, using the Vital Statistics of the US for 1993. For the return rate ofimmigrants to their home countries, I use the official census estimate that was conductedby Ahmed and Robinson (1994). They estimate that for the first 10 years, second 10 years
and the third 10 years, the return rate of immigrants is 19 percent, 9 percent and 7 percent,respectively. For annual rate, those numbers correspond to 2.085 percent, 0.938 percent
and 0.723 percent, respectively. Thus, I set pbi = 0.9715 for 2 ≤ i < 10, pbi = 0.99062 for
11 ≤ i ≤ 20, pbi = 0.9927 for 21 ≤ i ≤ 30 and pbi = 1 for i ≥ 31.
Once the age-nativity specific fertility rate, infant-child mortality and the initial PITN
are set, then the annual population growth rate is calculated automatically according toequation (40) with the assumption that the PITN is at the steady state. With the estimatedage-nativity specific fertility, the infant-child mortality and an initial PITN of 15.5 percent,the annual growth rate of the population of age 1 becomes 0.39 percent.26 This implies that24

In the fast and intermediate cases, to allow smooth transition of the PITN, the PITN keeps increasing

even after hitting the target PITN. See Figure 1.
25
More specifically, each of the three cases is calculated as follows. Let f15.5 be the steady-state age1
immigrant-native ratio when the PITN is 15.5 percent. Define f25.5 in a similar fashion.

Then, the age

1 immigrant-native ratio at period t, ft , in the slow case is ft = f25.5 for all t. For the intermediate case,t
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
t
for 31 ≤ t. For the 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 immigrant-native ratio is defined in the first term of the RHS in equation(40). Thus, once the PITN is determined, the steady state age 1 immigrant-native ratio is calculated fromequation (40).
26
The annual CPS data on immigrants and natives from 1995 to 2010 shows that the median annual growthrate of the total population(sum of natives and immigrants) aged from 20 to 40 is 0.12% while the median30

at the initial steady state, the government accepts immigrants such that the annual growthrate of immigrant of age 1 becomes 0.39 percent.
Age-nativity Specific Government Expenditure
I assume that the age-specific government expenditure, g ∗i,j is the same for natives andimmigrants. Empirical studies show that there is no systematic difference in the use of
public services by the two groups.27 Thus, I assume that for 1 ≤ i ≤ 24, g ∗i,j = g y , for
25 ≤ i ≤ 44, g ∗i,j = gm and for 45 ≤ i ≤ 80, g ∗i,j = go . I assume that g y , g m and g o are
24.5%,13.4% and 23.2 % of GDP per capita at the initial balanced growth path following
Storesletten (1995) and Auerbach, Kotlikoff Hagememann and Nicoletti (1989). On theother hand, to check the robustness of my results, I also assume that immigrants consume
20 percent more publicly provided private goods than natives in the robustness checks.
Utility Function, Production Function and Human Capital Profile
A precise estimate for the coefficient for relative risk aversion, γ, has not yet been foundin the literature. Auerbach and Kotlikoff (1986) and Storesletten (2000) assumed that γ
is 4. Nishiyama and Smetter (2007) set γ equal to 2. I assume that γ = 3 and check therobustness with γ = 2 and γ = 4. For the time preference rate, β, following Hurd (1989)
and Storesletten (2000), I assume that β = 1.011. Higher β implies higher savings and ahigher capital to output ratio. To check the sensitivity of my results, I also calculate with
β = 1 and β = 0.99. I assume that the leisure share in the utility function, ζ, is 0.33.
I assume that the depreciation rate of capital, δ, is equal to 0.047. For the capital sharein the production function, θ, I set θ = 0.4. For technological progress, I assume that income
per capita growth rate, µ, is 0.015.
For the human capital profile of natives, His , I take the value from Auerbach and Kotgrowth rate of immigrants aged from 20 to 40 is 1.85%. In the theoretical model, the model assumes thatthe age 1 population growth rate of immigrants and natives are the same. Thus, the theoretically predicted
growth rate of age 1 population, 0.39%, is between the growth rate of immigrants aged 20-40 and the growthrate of the total population aged 20-40.
27
Borjas and Hilton (1996) show that immigrants have a higher participation rate in welfare programs
than natives. Fix, Passel, and Zimmermann (1996) show that these differences are explained by the higher
participation rate in welfare programs among refugees and retired immigrants and that there is no differencesamong labor immigrants. Thus, for the theoretical part, I assume that g i is independent of the place of birth.
For the computational part, I relax this assumption for the robustness checks.

31likoff.
H i,n = exp(4.47 + 0.033 × i − 0.00067 × i2 )H i,n = 0

for 1 ≤ i ≤ 45

(49)

for 46 ≤ i(50)

For the human capital profile of the immigrants, Storesletten (1995) showed that immigrants’earnings are, on average, 15.7 percent lower than those of natives.28 Similarly, using the
CPS June 2000 supplement, my calculations show that immigrants’ earnings are 10 percent
lower than those of natives. Using these estimates as a basis, I assume that the efficient unitof human capital of immigrants is 84.3 percent of that of natives and that H i,m = 0.843×Hinin the benchmark calculation. To examine the robustness of my results, I change the level ofhuman capital from 84.3 percent to 89.3 percent, 79.3 percent or 74.3 percent and re-checkthe results.
Taxes and Government Debt
For capital income tax, I take the value from Nishiyama and Smetter (2007) and assume
that τk = 0.28. For the level of social security benefit, a higher replacement rate means that
greater intergenerational redistribution of income, which leads to greater welfare gain as aresult of increasing the immigrant population. Following Auerbach and Kotlikoff, in thebenchmark case I set the replacement 0.6 and check the robustness of my results by varying
it from 0.6 to 0.55 and 0.5.
As for the initial level of government debt or assets, different authors set different levels.
Storesletten considered only government debt and assumed that the initial level is 50 percentof the initial GDP. With his estimate of the initial capital to output ratio being 2.4, hisassumption implies that government debt is about 20 percent of private capital. Nishiyama
and Smetter considered not only government debt, but also government assets by using
the BEA information on the government’s fixed capital. They assumed that at the initialsteady state, the government has positive net assets of 10 percent of the total private capital.
This naturally leads to a higher capital to output ratio at the initial balanced growth path.
Herein, as a benchmark case, I assume that initial level of government debt or assets is 0
percent of the private capital and I experiment with the assumption that the government
debt is 10 percent or minus 10 percent of the private capital.
28

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

32

4.3

Results

Figures A3, A4 and A5 in the appendices show the balance of assets, consumption andleisure for the life cycle of an individual at the initial balanced growth path in the benchmark analysis. At the age of 46, the consumption of leisure becomes 1 due to mandatory
retirement. At the initial balanced growth path, the capital to output ratio is 2.98, which
is higher than the valued used in Storesletten (2.4) and Nishiyama and Smetter (2.7), but
lower than the value used in the standard business cycle research, which is 3.2 (Cooly and
Prescott(1995)). To check how my results are affected by the capital to output ratio, Ichange the time preference rate and examine how the results change for different values of
the time preference rate in the robustness checks.
Table 1 shows the parameter values and the welfare effect of increasing the PITN for
different values of V in equation (47), which is the share of the government surplus that is
put into savings. As mentioned above, the initial PITN is set at 15.5 percent and the target
INR at 25.5 percent. With respect to rate at which immigrants are accepted, I consider threecases in which it takes 80, 62 and 42 years for the PITN to reach the target PITN. Column(3) of Table 1 shows the share of the government surplus that is put into savings. Columns(4)-(9) are values that are calculated within the simulation. Column (4) shows how many
years it takes for the economy to reach the (modified) golden rule level. When it does notreach the (modified) golden rule within 300 years, this is indicated by * or **. * indicatesthat the capital-to-labor ratio (efficient unit) at the 300th year is higher than at the initialbalanced growth path, that it is increasing at the 300th year, but that it does not reach the(modified) golden rule level at the 300th year. ** indicates that the capital-to-labor ratioat the 300th year is lower than at the initial balanced growth path. For example, whenV = 100%, it takes 112 years to reach the golden rule level in the slow acceptance case(row(16)). Column (5) shows how much the capital stock per efficient unit of labor increases
at the golden rule compared with the level at the initial balanced growth path. WhenV = 100%, it shows that the capital stock per efficient unit of labor increases by 102 percent.
Column (6) shows how much the publicly provided private goods will increase comparedwith the level at the initial balanced growth path. At V = 100%, column (6) shows thatthe publicly provided private goods will increase by about 36 percent. To calculate columns(5) and (6), I evaluate at the year at which the economy reached the golden rule level if itreaches that level within 300 years, and at the 300th year otherwise. Column (7) shows thepercentage by which the utility, measured by the expenditure function, of the cohort that is33

born when the economy reaches the golden rule level increases compared with the utility ofthe same cohort if it were on the initial balanced growth path. When the economy does notreach the golden rule level, I calculate the utility of the cohort that is born at the 300th year.
In the expenditure function, the price vector on the initial balanced growth path is used forevaluating the utility. Note that all welfare gain is distributed through an increase in publicly
provided private goods. Column (8) shows the share of the present discounted value (PDV)of the increased publicly provided private goods in the initial GDP. For example, in row (16),the PDV of the increased publicly provided private goods is 12 percent of the initial GDP.
For discounting the increased publicly provided private goods in future periods, I apply
a 5 percent discount rate instead of the equilibrium interest rate, so that I can eliminate
the effect of the discount rate when comparing different cases. Column (9) measures thewelfare gain of natives and their descendants, not that of immigrants or their descendants. It
measures how much the utility of natives and their descendants, not including immigrants
and their descendants, is Pareto-improved by accepting more immigrants compared withthe utility level at the initial balanced growth path. It evaluates the utility by using theexpenditure function. I apply the price vector at the initial balanced growth path for the
expenditure function and the equivalent variation to measure the difference of the levelsof utility in two cases. To discount the welfare gain of future cohorts, I again use a 5
percent discount rate. Note that in all cases in Table 1, I assume that the government
does not discriminate between immigrants and natives with respect to publicly provided
private goods. Both natives and immigrant consume publicly provided private goods equally.
Column (9) in row (16) shows that the PDV of the Pareto-improving welfare gain of natives
and their descendants comprises 11 percent of the initial GDP.
In rows (2)-(3), (5)-(6), and (8)-(9)..., I shorten the years needed to reach the target
PITN and increase the speed with which the PITN increases. When the number 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 of the initial GDP (row (18)).
Table 1 shows that the number of years taken to reach the golden rule level decreases
as V increases, because the government saves more for future cohorts. On the other hand,
the PDV of the increased utility, measured as the share of the initial GDP, increases as
V decreases as long as the V is greater than or equal to 50%. The PDV of the increased
utility, measured as the share of the initial GDP, is highest when V=50 %. In this case, thequantified Pareto improvement ranges from 21% to 26% of the initial GDP.

34

In Table 1, I set the intergenerational discount rate for the modified golden rule at 0%
and set the target level of the capital stock at the golden rule level. However, the targeting
capital stock at the golden rule level does not necessarily maximize the PDV of the welfare
gain. In Table A2, I examine the effect on welfare of increasing the PITN for different target
levels of capital stock by changing the value of the intergenerational discount rate for themodified golden rule when V=100%. Table A2 shows that the welfare gain is maximized
when the intergenerational discount rate is set at 3 percent.
In Table 2, which I consider as the representative case of my simulation, I recalculate
the all rows of Table 1 by setting the intergenerational discount rate for the modified golden
rule at 3 percent. In Table 2, the Pareto-improving welfare gain of increasing the PITN ismore than 20 percent of the initial GDP and the capital stock per efficient unit of labor
increases by 18 percent as long as V is greater than or equal to 50 percent.
Figure 2 and Figure A6 show the marginal product of capital and the capital to output
ratio over time for different values of V when the target capital stock is the modified golden
rule level (3% intergenerational discount rate). The marginal product of capital increases
initially, due to the acceptance of more immigrants. However, as the government savingsbalance increases, the capital stock per efficient unit of labor starts to increases and themarginal product of capital continues to decrease until the economy reaches the golden rulelevel. The capital to output ratio also displays a similar pattern.
Figure 3 compares the utility level of all cohorts on the initial balanced growth pathwith the utility level of all cohorts for different values of the share for the government
savings (V) when the target capital stock is set at the golden rule level. For example, when
V=100%, all the surplus is put into savings until the economy reaches the golden rule level
and the surplus is distributed to individuals only after the economy reaches the golden rulelevel. This implies that the utility of the 65th cohort, which dies at the 65th year, starts toexperience higher utility than at the initial balanced growth path.
In all cases considered in Tables 1,2,3 and 4, the results of simulations show that all
cohorts are Pareto-improved; this confirms my theoretical results.29
29

The results of this simulation differ from those of some of the previous studies. For example, Kotlikoff

et.al reports that the welfare gain of doubling immigrants is very small. Lee and Miller argues that the fiscal
impact of accepting additional 100,000 immigrants is very small. Several factors in those studies generate
different results. Kotlikoff et.al analyze immigration policy in an open economy setting, whereas I use aclosed economy setting. In an open economy setting, the effect of additional government savings is offset by
the mobility of capital. In my simulation, the government can use the budget surplus, which is obtained by35

Figure 4 shows the importance of government savings balance at the new equilibrium
path. It calculates the ratio between the interest income from the government savingsbalance and the social security benefit payment in each period. When the economy reaches
the golden rule level, the interest income from the government savings balance comprises
70 percent of the social security benefit payments. Thus, at the new equilibrium path, theinterest income from the government savings balance is contributing substantial amount.

4.4

Robustness Checks

Table 3 and 4 present results of robustness checks of Table 2. Rows (1)-(6) in Table 3 check
whether the results in Table 2 are sensitive to the initial government debt (asset) level. As
I argued above, different authors assume different levels for the government debt (asset) at
the initial balanced growth path. In rows (1)-(3) in Table 5, I set the initial government
debt to 10 % of the private capital instead of 0 %. In rows (4)-(6), I assume that the initial
government debt level is -10% of the private capital.
Rows (7)-(12) in Table 3 check whether the results in Table 2 are sensitive to the 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) confirms this prediction. Decreasing the replacement
rate 10 percentage points decreases the welfare gain by 7 percentage point in terms of thepercentage of the initial GDP.
Rows (13)-(21) in Table 3 conduct sensitivity checks on the immigrants’ level of earnings.
Following Storesletten (1995), I set the immigrant wage rate at 84.3 percent of that ofnatives. In my calculation using the CPS 2000 June supplement, I found that immigrants’earning are 91 percent of natives’ earnings. Rows (13)-(15) assume that the wage rate ofimmigrants is 89.3 percent of that of natives, rather than 84.3 percent. Rows (16)-(18)assume that the wage rate of immigrants is 79.3 percent of that of natives. Rows (19)-(21)assume that the wage rate of the immigrants is 75 percent of that of natives. The results inrows (19)-(21) show that if immigrants earn 25 percent lower than natives, the welfare gain
accepting more immigrants, for savings and the amount of savings affects the over all welfare. Lee and Miller
consider a much smaller increase in the number of immigrants than I do. On the other hand, my simulation
results are consistent with Auerbach Oreopoulos (2000). They find that if the initial fiscal imbalance is not
adjusted, then the welfare loss of halving the number of new immigrants is substantial. This is consistent
with my theoretical and simulation results.

36

of accepting more immigrant is 17 percent of the initial GDP, instead of 23 percent of theinitial GDP.
Rows (22)-(27) in Table 3 conduct sensitivity checks on the consumption by immigrantsof publicly provided private goods. In Table 1 and 2, I assumes that immigrants andnative consume publicly provided private goods equally as the previous empirical studies
indicate. In rows (22)-(24), I assume that young immigrants consume 20 percent more
publicly provided private goods than young natives. The PDV of the welfare gain comprises
17 percent of the 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 thiscase, the PDV of the welfare gain comprises 15 percent of the initial GDP.
In Table 4, I conduct robustness checks by changing the values of parameters of the utility
function and the return rate of immigrants. Rows (1)-(6) of Table 4 examine the sensitivityof the results regarding the values of the parameters for constant relative risk aversion(CRRA). In Table 1-3, I assume that the CRRA is equal to 3. Auerbach and Kotlikoff,and Storesletten assume that the CRRA=4, while Nishiyama and Smetter assume that the
CRRA is 2. Rows (1)-(3) assume that the CRRA is equal to 4 and rows (5)-(6) assume that
it is 2. The results presented on rows (1)-(6) show that the results presented in Table 2 is
not sensitive to the value of CRRA. Rows (7)-(12) examine the sensitivity of the results tothe time preference rate. Although it is quite common to assume that the time preferences
rate is greater than 1, readers might think that the results presented in Table 1 and 2 canbe sensitive with respect to the assumption that the time preference rate is greater than
1. Theoretically, lowering the time preference rate will result in lowering savings and will
decrease the capital to labor ratio at the initial balanced growth path. Since increasing thenumber of immigrants will increase the capital stock and the production function exhibits
diminishing marginal product of capital, lowering the initial capital stock as a result of
assuming a lower time preference rate will increase the welfare gain of increasing the PITN.
Rows (7)-(12) confirm this theoretical prediction, but they show that the magnitude of thosechanges is very small. For example, changing the time preference rate from 1.011 to 0.99
increase the PDV of the welfare gain by only 1 percentage point.
Rows (13)-(15) in Table 4 conduct robustness checks by assuming that the return rateof immigrants is equal to 0. Rows (13)-(15) show that the results presented in Table 2 do
not change so much and that they are robust regarding the return rate of immigrants.

37

4.5

Summary of the Simulation Results

The simulation results presented in Tables 1-4 can be summarized as follows. In the caseof slow acceptance of more immigrants, in which the PITN increases from 15.5 percent
and reaches 25.5 percent at the 80th year, the PDV of the increased utility of the natives
and their descendants, measured by the expenditure function, is 23 percent of the initial
GDP when the target capital stock is the modified golden rule level (3 % intergenerational
discount rate). The economy reaches the modified golden rule level at 65th year and thecapital stock per efficient unit of labor increases by 18 percent. In the fast acceptance case,in which the PITN reaches 25.5 percent at the 42th year, the PDV of the welfare gain
comprises 28 percent of the initial GDP and the economy reaches the modified golden rulelevel at 59th year.
For robustness checks, I conduct sensitivity checks for changing the values of the following
parameters: initial government debt (asset) level, the replacement rate, level of earnings ofimmigrants, immigrants’ consumption of publicly provided private goods, the CRRA, thetime preference rate and probability of the return to home country by immigrants. The
sensitivity checks show that the results are quite robust.

5

Conclusion

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

References[1] Akbari, Ather H. ” The Benefits of Immigrant to Canada:Evidence on Tax ad PublicService,” Canadian Public Policy Vol 15, pp 424-435, 198938

[2] Auerbach, Alan and Kotlikoff, Laurence, 1987, Dynamic Fiscal Policy, Cambridge University Press, Cambridge, 1987
[3] Auerbach Alan J. and Oreopoulos, Philip ”Analyzing the Fiscal Impact of US Immigration”, American Economic Review paper and Proceeding 89, pp 170-180, 1999
[4] Diamond, Peter A., ”National Debt in a Neoclassical Growth Model”, American Economic Review, 1965, vol 55 pp 1126-1150[5] Borjas Georege J. ”The Economics of Immigration.’ Journal of Economic Literature
vol 32, pp 1667-1717, 1997
[6] Center for Disease Control and Prevention, Vital Statistics of the United States, 1993(http://www.cdc.gov/nchs/products/vsus.htm)[7] Canovaa, Fabio and Morten O. Ravn, ”The Macroeconomic Effects of German Unification: Real Adjustments and the Welfare State”, Review of Economic Dynamics, 2000,
vol3, pp 423 460[8] Collado, M. Dolores, INIgiturbe-Ormaetxeand and Guadalupe Valera, ”Quantifying
the Impact of Immigration on the Spanish Welfare State”, International Tax and Public
Finance, vol11, 335–353, 2004
[9] Cooley, Thomas and Edward Presecott, ”Economic Growth and Business Cycle”, Chapter 1 in Frontier of Business Cycle Research, Cooley, Thomas (ed.), 1995, Princeton
University Press[10] Feldstein, Martin S. The Effects of the Ageing European Population on Economic
Growth and Budgets: Implications for Immigration and Other Policies, 2006, NBER
Working Paper No. 12736[11] Michael
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Wendy
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Zimmermann,”The

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Means

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the

23,

1996,

(http://www.urban.org/publications/900287.html[12] A. Dianne Schmidley and J. Gregory Robinson, ”Measuring The Foreign-born Population in the United States with the Current Population Survey: 1994-2002, Poulation
Division Working Paper No73, Bureau of the Census
39[13] Nishiyama, Shinichi and Kent Smetters, . ”Does Social Security Privatization Produce
Efficiency Gains?,” The Quarterly Journal of Economics, vol. 122(4), pp 1677-1719,
November, 2007.
[14] Fehr, Hans, Sabine Jokisch and Laurence J. Kotlikoff, ”The Role of Immigration in
Dealing with the Developed World’s Demographic Transition,, FinancArchiv, 2004.
[15] Geanakoplos, John, Olivia S. Mitchell and Stephen P. Zeldes ”Would a Privatized Social
Security System Really Pay a Higher Rate of Return?” (with ). In R.D. Arnold, M.
Graetz and A. Munnell (eds.), Framing the Social Security Debate. Values, Economics,
and Politics. National Academy of Social Insurance, 1998, pp. 137-157.
[16] Hurd, Michael D. ”Mortality Risk and Bequests.” Econometrica, vol 57 July, pp779813, 1989.
[17] Michel, Ph. and Pierre Pestieau, ”When Does Serendipity Hold?”, Journal of Population Economics, Vol. 6, No. 4 Dec, pp. 353-362, 1999[18] Ramsey, Frank P. ”A Mathmatical Theory of Saving”, Economic Journal,38, No 152,
pp543-559.
[19] Razin, Assaf, ”Migration and Pension with Internatioanl Capital Mobility”, Journal of
Public Economics, vol 74(1),pp 141-15, 1999,[20] Razin, Assaf and Efraim Sadka ”Unskilled Migration: A Burden or a Boon for the
Welfare State? ”, Scandinavian Journal of Economics,2000, pp463-479[21] Razin, Assaf and Efraim Sadka and Benjjarong Suwankiri, Migraiton and the WelfareState Political-Economy Policy Formulation,The MIT Press, Cambidge MA, 2011[22] Passel, Jeffrey S. ” Immigrants and Taxes: A Reappraisal of Huddle’s The cost of
Immigrants ” Manuscript Washington Urban Institute, 1994[23] Shimasawa, Manabu and Kazumasa Oguro, ”Impact of immigration on the Japanese
economy: A multi-country simulation model”, Journal of Japanese and International
Economics, 2010, pp 5806-602.
[24] Simon, Jullian L. ” Immigrants, Taxes and Welfare in the United States”, Population
and Development Review vol 10, pp 55-69, 198440

[25] Storesletten, Kjetil, On the Economics of Immigration, Ph.D Dissertation, Carnegie
Mellon University, 1995
[26] Storesletten, Kjetil, ”Sustaining Fiscal Policy through Immigration,” Journal of Political Economy, vol 108, pp 300-323, 2000,[27] Storesletten, Kjetil, ”Fiscal Implications of Immigration— A Net Present Value Calculation,” Scandinavian Journal of Economics, vol105(3), pp 487-506, 2003,

28
standard caseintermediate
fast

26

PITN(%)

24

22

20

18

16

14

0

50

100

150

200

250

300

year

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

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

41

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 forgovernment savings (V). The target capital stock level is the modified golden rule level with 3%
intergenerational discount rate for the modified golden rule.

0.8

interest income/ss benefit payment

0.7
0.6

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

0.5
0.4
0.3
0.2
0.1
0
−0.1

0

50

100

150
year

200

250

300

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

42

(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.

43

(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.

44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).

45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.

46

Appendices (The following appendices are put on the Journal’s
webpage as supplement material, but not included in the maintext. )
Appendix Ayy=F(x,s*)

*

y*

C

+S

y

ind

+ g + gx*

Figure A1– The one plus population growth rate and the resources used for one old person. The
curve through the origin is y = F (x, s∗ )and the straight line is y = (cy∗ + s∗ + g y + g ind ) × x
where x is one plus the population growth rate. The vertical distance of the straight line through theorigin measures the total resources used for young divided the number of old people. The difference
between the curved line and the straight line measures the resource used per one old person at x. If

x is lower than 1 + πn∗ , increasing x will increase the resources available per one old person without
decreasing the resources used for the young.

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

CPS 2000 June

20

25

30
person’s age
Foreign born

35

40

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

2

400
350

saving balance

300
250
200
150
100
50
0

0

10

20

30

40
age

50

60

70

80

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

3

6500
6000
5500

Consumption

5000
4500
4000
3500
3000
2500
2000
1500

0

10

20

30

40
Age

50

60

70

80

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

4

1
0.95
0.9

Leisure

0.85
0.8
0.75
0.7
0.65

0

10

20

30

40
age

50

60

70

80

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

5

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

6

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.

7

(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.

8

Appendix B1
Notice that in the programming problem, the objective function is concave and the constrained 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 ) + v y (g y , g ind ) +
L=

t=1

+

∞
X

1
[uo (cot+1 ) + v o (g o , g ind ) − u∗ }1+ρλt {F (((1 + πm )α∗ + 1 + πn ), (st−1 + at−1 ) + (1 − δ)(st−1 + at−1 )

t=1− (cot + g o + g ind ) − {(1 + πm )α∗ + 1 + πn } × (cyt + st + at + g y + g ind ) }The first order conditions are:
1
1
co1 :
uo0 (co1 ) = λ1 ; cot+1 : γtuo0 (cot+1 ) = λt+1 ; cyt : γt uy0 (cyt ) = λt {(1 + πm )α∗ + 1 + πn };
1+ρ
1+ρ
1
[uo (cot+1 ) + v o (g o , g ind ) − u∗ = 0;γt : uy (cyt ) + v y (g y , g ind ) +1+ρ
∂F
st + at : λt+1 {+ 1 − δ} = λt {(1 + πm )α∗ + 1 + πn }∂K
λt : F ((1 + πm )α∗ + 1 + πn , st−1 + at−1 ) + (1 − δ)(st−1 + at−1 )−(cot + g o + q) −t {(1 + πm )α∗ + 1 + πn } × (cyt + st + at + g o + q) } = 0
Now set the cyt , cot−1 , st , at , λt , γt as follows:
cot = co∗ ; cyt = cy∗ ; st = s∗ ; at = 0; λ1 =λt+1 = λt1uo0 (co∗ )1+ρ

(1 + πm )α∗ + 1 + πn1
; γtuo0 (co∗ ) = λt+1
∗
1+r
1+ρ

When we set cyt , cot+1 , st , at , λt , γt in this way, it clearly satisfies the first-order conditions.
Thus, the initial allocation is Pareto-efficient.
Q.E.D.

Appendix B2Using the definitions of τw1 and τr1 , the government budget surplus at period 1 is
X

SP1 = w1 l∗

N1j − w∗ l∗

j=n,m+ r1 s∗Xj=n,mN0j − r∗ s∗

X

N1j (1 − τw∗ )

j=n,m

X
j=n,m

N0j (1 − t∗r ) −

X
j=n,m

N0j (b∗ + g o + g ind ) −X

N1j × (g y + g ind )

j=n,m

(1)
9

Note that N1m = N1n αe where α
e > α∗ and that N1n is pre-determined where N1n =N10 (1 + πn ) + N0n α∗ (1 + πm ). N0m = N0n α∗ . Thus, SP1 becomes

SP1 = w1 l∗

N1j − w∗ l1∗j (1 − τw∗ ){N1n (1 + α∗ ) + N1n (eα − α∗ )}X
j=n,m+ r1 s∗ N0n (1 + α∗ ) − r∗ s∗ N0n (1 + α∗ )(1 − t∗r ) − N0n (1 + α∗ )(b∗ + g o + g ind )− (g y + g ind ){N1n (1 + α∗ ) + N1n (eα − α∗ )}(2)

At the steady state, as for the government budget constraint, we have(τw∗ w∗ l∗ − g y − g ind ) × N1n × (1 + α∗ ) + (r∗ s∗ − b∗ − g o )N0n (1 + α∗ ) = 0(3)

Thus, SP1 becomes
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 )N1n (eα − α∗ )From the homogeneity of the production function and Euler’s theorem, we have
w1 l ∗

N1j + +r1 s∗ N0n (1 + α∗ ) = F (l∗XX

N1j ), s∗ N0n (1 + α∗ )) − δs∗ N0n (1 + α∗ )

j=n,m

j=n,m
∗ ∗

and w l {N1n (1 + α∗ )} + r∗ s∗ N0n (1 + α∗ ) = F (l∗ {N1n (1 + α∗ )}, s∗ N0n (1 + α∗ )) − δ ∗ s∗ N0n (1 + α∗ )Thus, SP1 becomes
SP1 = F (l∗X

N1j ), s∗ N0n (1 + α∗ )) − δs∗ N0n (1 + α∗ ) − {F (l1∗ {N1n (1 + α∗ )}, s∗ N0n (1 + α∗ ))

j=n,m
∗ ∗

− δ s N0n (1 + α∗ )} − w∗ l∗ (1 − τw∗ )N1n (eα − α∗ ) − N1n (g y + g ind )(eα − α∗ )(4)

Note that N1m = αeN1n . Thus,
SP1 = F (l∗ (N1n (1 + αe), s∗ N0n (1 + α∗ )) − F (l∗ {N1n (1 + α∗ )}, s∗ N0n (1 + α∗ ))− {w∗ l∗ (1 − τw∗ ) − (g y + g ind )}N1n (eα − α∗ )(5)
Z αe∂F (N1n (1 + z)l∗ , s∗ N0n (1 + α∗ )) n ∗N1 l − w∗ (1 − tw )N1n l∗ − (g y + g ind )N1n ]dz
=
[
∂L
∗
α
(6)
Z αe
∂F (N1n (1 + z)l∗ , s∗ N0n (1 + α∗ )) ∗= N1nl − w∗ l∗ (1 − tw ) − (g y + g ind )]dz
(7)
∂L
α∗
Note that w∗ l∗ (1 − tw ) = cy∗ + s∗ . Thus, we have equation (25).
10Appendix B3
Note that
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 )+ (1 + r2 )a1 N1n (1 + αe)(8)Using the definitions of τw2 and τr2 , we have
SP2 = w2 l∗ N2n (1 + αe) − w∗ l∗ N2n (1 + αe)(1 − τw∗ )+ r2 s∗ N1n (1 + αe) − r∗ s∗ N1n (1 + αe)(1 − τr∗ ) − N1n (1 + αe)(b + g o + g ind )− N2n (1 + αe)(g y + g ind ) + (1 + r2 )a1 N1n (1 + αe)(9)Thus, SP2 becomes
SP2 = w2 l∗ N2n (1 + αe) + r2 s∗ N1n (1 + αe)− w∗ l∗ N2n (1 + αe) − r∗ s∗ N1n (1 + αe)e) − N1n (1 + αe)(b + g o + g ind )τw∗ w∗ l∗ N2n (1 + αe) + τr∗ r∗ s∗ N1n (1 + α− N2n (1 + αe)(g y + g ind ) + (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 )}(10)

Now we calculate τw∗ w∗ l∗ N2n + τr∗ r∗ s∗ N1n − N1n (b + g o + g ind ) − (g y + g ind )N2n . Note thatN2n = N1n ((1 + πm )e
α + 1 + πn ) = N1n ((1 + πm )α∗ + 1 + πn + (1 + πm )eα − (1 + πm )α∗ ). Thus,
τw∗ w∗ l∗ N2n + τr∗ r∗ s∗ N1n − N1n (b + g o + g ind ) − (g y + g ind )N2n
= τw∗ w∗ l∗ N1n ((1 + πm )α∗ + 1 + πn + (1 + πm )(eα − α∗ ))+ τr∗ r∗ s∗ N1n − N1n (b + g o + g ind )− (g y + g ind )N1n ((1 + πm )α∗ + 1 + πn + (1 + πm )(eα − α∗ ))(11)

At the initial steady state, we have(τw∗ w∗ l∗ − g y − g ind )(1 + πn + α∗ (1 + πm )) × (1 + α∗ ) + (r∗ s∗ − b∗ − g o )(1 + α∗ ) = 0 (12)Thus, (11) becomes= τw∗ w∗ l∗ N1n (1 + πm )(eα − α∗ ) − (g y + g ind )N1n (1 + πm )(eα − α∗ )11(13)

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)+ (1 + αe){τw∗ w∗ l∗ N1n (1 + πm )(eα − α∗ ) − (g y + g ind )N1n (1 + πm )(eα − α∗ )}(14)

Since N2n = N1n ((1 + πn ) + αe(1 + πm )) = N1n {1 + πn + (1 + πm )α∗ + (1 + πm )(eα − α∗ )), SP2
becomes
SP2 = w2 l∗ N2n (1 + αe) + r2 s∗ N1n (1 + αe) + (1 + r2 )a1 N1n (1 + αe)− (1 + αe) {w∗ l∗ N1n {1 + πn + (1 + πm )α∗ + (1 + πm )(eα − α∗ )} + r∗ s∗ N1n }α − α∗ ) − (g y + g ind )(1 + πm )(eα − α∗ )}+ (1 + αe)N1n {τw∗ w∗ l∗ (1 + πm )(eUsing 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 {1 + πn + (1 + πm )α∗ } + r∗ s∗ N1n (1 + α∗ )= F (l∗ (1 + α∗ )N1n {1 + πn + (1 + πm )α∗ }, s∗ N1n (1 + α∗ ))−δs∗ N1n (1 + α∗ )Thus, SP2 becomes
SP2 = F (2 l∗ N2n (1 + αe), (s∗ + a1 )N1n (1 + αe)) − δ(s∗ + a1 )N1n (1 + αe)
1+αe{F (l∗ (1 + α∗ )N1n {1 + πn + (1 + πm )α∗ }, s∗ N1n (1 + α∗ )) − δs∗ N1n (1 + α∗ )}1 + α∗+(1 + αe)N1n {−w∗ l∗ (1 + πm )(eα − α∗ ) + τw∗ w∗ l∗ (1 + πm )(eα − α∗ ) − (g y + g ind )(1 + πm )(eα − α∗ )}−+a1 N1n (1 + α
e)

(15)

= F (2 l∗ N2n (1 + αe), (s∗ + a1 )N1n (1 + αe)) + (1 − δ)a1 N1n (1 + αe)−F (l∗ (1 + αe)N1n {1 + πn + (1 + πm )α∗ }, s∗ N1n (1 + αe))+(1 + αe)N1n {−w∗ l∗ (1 + πm )(eα − α∗ ) + τw∗ w∗ l∗ (1 + πm )(eα − α∗ ) − (g y + g ind )(1 + πm )(eα − α∗ )}(16)
Subtracting and adding F (l∗ N2n (1 + αe), s∗ N1n (1 + αe)), SP2 becomes
SP2 = F (l∗ N2n (1 + αe), (s∗ + a1 )N1n (1 + αe)) + (1 − δ)a1 N1n (1 + αe) − F (l∗ N2n (1 + αe), s∗ N1n (1 + αe))+ F (l∗ N2n (1 + αe), s∗ N1n (1 + αe)) − F (l∗ (1 + αe)N1n {1 + πn + (1 + πm )α∗ }, s∗ N1n (1 + αe))+ (1 + αe)N1n {−(1 − τw∗ )w∗ l∗ (1 + πm )(eα − α∗ ) − (g y + g ind )(1 + πm )(eα − α∗ )}

12

(17)

Note that N2n = N1n ((1 + πm )eα + (1 + πn )). Thus,
Z (s∗ +a1 )N n (1+eα)
1

SP2 =

s∗ N1n (1+eα){∂F (l∗ N2n (1 + αe), z)+ (1 − δ)}dz∂K+ (1 + αe)N1n {F (l∗ {1 + πn + (1 + πm )eα}, s∗ ) − F (l∗ {1 + πn + (1 + πm )α∗ }, s∗ )+ (1 + αe)N1n {−(1 − τw∗ )w∗ l∗ (1 + πm )(eα − α∗ ) − (g y + g ind )(1 + πm )(eα − α∗ )}Z (s∗ +a1 )N n (1+eα)
1∂F (l∗ N2n (1 + αe), z){=+ (1 − δ)}dzn∂K∗
s N1 (1+eα)
Z αe∂F ((l∗ {1 + πn + (1 + πm )z}, s∗ )n[
(1 + πm )l∗+ (1 + α
e)N1
∂L
∗
α
− (1 + πm ){(1 − τw∗ )w∗ + (g y + g ind )}]dz(18)
Z (s∗ +a1 )N n (1+eα)
1∂F (l∗ N2n (1 + αe), z)+ (1 − δ)}dz={∂Ks∗ N1n (1+eα)
Z αe∂F ((l∗ {1 + πn + (1 + πm )z}, s∗ ) ∗+ (1 + αe)N1n (1 + πm )[l − (1 − τw∗ )w∗ − (g y + g ind )]dz
∂L
∗
α
(19)
Since w∗ l∗ (1 − τw∗ ) = cy∗ + s∗ , we have equation (28)

Appendix B4
To save space, I will show that SPt+1 > 0 for t = 1, 2, ... For t = 0, the same proof is applied
as in the appendix A2.
We consider the same tax adjustment as in the preceding subsection.
(1 − τwt )wt = (1 − τw∗ )w∗ and (1 − τrt )rt = (1 − τr∗ )r∗(20)

Note thatnm
SPt+1 = wt+1 τwt+1 ln∗ Nt+1
+ φwt+1 τwt+1 lm∗ Nt+1
+ τrt+1 rt+1 sn∗ Ntn + τrt+1 rt+1 sm∗ Ntmnm− Nt+1× (g y,n + g ind,m ) + Nt+1
× (g y,m + g ind,m )+ Ntn (bn∗ + g o,n + g ind,n ) − Ntm × (bm∗ + g o,m + g ind,m )+ (1 + rt+1 )at (Ntn + Ntm )(21)

Using the definition of τwt+1 and τrt+1 , we have
τwt+1 = {wt+1 − (1 − τw∗ )w∗ }/wt+1
τrt = {rt+1 − (1 − τr∗ )r∗ }/rt+1

13

Thus, SPt+1 becomes
n
∗
n
n
m
m
m
SPt+1 = wt+1 ln∗ Nt+1
− wt+1ln∗ Nt+1+ τw∗ w∗ ln∗ Nt+1
+ φwt+1 lm∗ Nt+1
− φw∗ lm∗ Nt+1
+ φτw∗ w∗ lm∗ Nt+1

+ rt+1 sn∗ Ntn − r∗ sn∗ Ntn + τr∗ r∗ sn∗ Ntn ++ rt+1 sm∗ Ntm − r∗ sm∗ Ntm + τr∗ r∗ sm∗ Ntmnm− Nt+1× (g y,n + g ind,n ) − Nt+1
× (g y,m + g ind,m )− Ntn (bn∗ + g o,n + g ind,n ) − Ntm × (bm∗ + g o,m + g ind,m )+ (1 + rt+1 )at (Ntn + Ntm )(22)

Now we need to calculate the following term:nmτw∗ w∗ ln∗ Nt+1
+ φτw∗ w∗ lm∗ Nt+1
+ τr∗ r∗ sn∗ Ntn + τr∗ r∗ sm∗ Ntmnm− Nt+1× (g y,n + g ind,n ) − Nt+1
× (g y,m + g ind,m )− Ntn (bn∗ + g o,n + g ind,n ) − Ntm × (bm∗ + g o,m + g ind,m )(23)

n = N n (1 + π + α
m = Nn αNote that Nt+1e(1 + πm )) and Nt+1nt
t+1 e. Also at the initial steady
n
n
∗
m
n
state, Nt+1 = Nt (1 + πn + α (1 + πm ))and Nt+1 = Nt+1 α∗ . On the other hand, at theinitial steady state, the government budget constraint implies

τw∗ w∗ Ntn (1 + πn + α∗ (1 + πm ))ln∗ + τw∗ φw∗ Ntn (1 + πn + α∗ (1 + πm ))lm∗ α∗+ τr∗ r∗ sn∗ Ntn + τr∗ r∗ sm∗ Ntm− Ntn (1 + πn + α∗ (1 + πm )) × (g y,n + g ind,n ) + Ntn (1 + πn + α∗ (1 + πm ))α∗ (g y,m + g ind,m )− Ntn (bn∗ + g o,n + g ind,n ) − Ntn α∗ (bm∗ + g o,m + g ind,m )(24)

Thus, (23) becomesnmτw∗ w∗ ln∗ Nt+1
+ φτw∗ w∗ lm∗ Nt+1
+ τr∗ r∗ sn∗ Ntn + τr∗ r∗ sm∗ Ntmnm− Nt+1× (g y,n + g ind,n ) − Nt+1
× (g y,m + g ind,m )− Ntn (bn∗ + g o,n + g ind,n ) − Ntm × (bm∗ + g o,m + g ind,m )= τw∗ w∗ ln∗ Ntn (eα − α∗ )(25)+ φτw∗ w∗ lm∗ Ntn {J(eα) − J(α∗ )}+ τr∗ r∗ sm∗ Ntn (eα − α∗ )− Ntn (1 + πm )[g y,n + g ind,n ](eα − α∗ )− Ntn (1 + πm )[g y,n + g ind,n ]{J(eα) − J(α∗ )}− Ntn (bm∗ + g o,m + g ind,m )(eα − α∗ )where J(z) = (1 + πn + z(1 + πm ))z
14

(26)

Therefore, SPt+1 becomesnm
SPt+1 = wt+1 ln∗ Nt+1
+ φwt+1 lm∗ Nt+1nm
− w∗ ln∗ Nt+1
− φw∗ lm∗ Nt+1

+ rt+1 sn∗ Ntn + rt+1 sm∗ Ntm + (1 + rt+1 )at (Ntn + Ntm )− r∗ sn∗ Ntn − r∗ sm∗ Ntm+ τw∗ w∗ ln∗ Ntn (1 + πm )(eα − α∗ )+ φτw∗ w∗ lm∗ Ntn {J(eα) − J(α∗ )}+ 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∗ )(27)

Note thatnmw∗ ln∗ Nt+1
+ φw∗ lm∗ Nt+1
+ r∗ sn∗ Ntn + r∗ sm∗ Ntm= w∗ ln∗ Ntn {1 + πn + α∗ (1 + πm ) + (1 + πm )(eα − α∗ )}+ φw∗ lm∗ Ntn lm∗ (J(α∗ ) + J(eα) − J(α∗ ))+ r∗ sn∗ Ntn + r∗ sm∗ Ntm (α∗ + α
e − α∗ )+ δ{sn∗ Ntn + sm∗ Ntm } − δ{sn∗ Ntn + sm∗ Ntm }= F ((1 + πn + (1 + πm )α∗ )ln∗ + φ(1 + πn + (1 + πm )α∗ )α∗ lm∗ , sn∗ Ntn + sm∗ Ntn α∗ )− δ{sn∗ Ntn + sm∗ Ntm }+ w∗ ln∗ Ntn (1 + πm )(eα − α∗ )+ φw∗ lm∗ Ntn lm∗ (J(eα) − J(α∗ ))+ r∗ sm∗ Ntm (eα − α∗ )(28)

Using the homogeneity of the production function and (28), SPt+1 becomes as follows:

15

n n∗
m m∗ n∗ n
SPt+1 = F (Nt+1
l + φNt+1
l , s Nt + sm∗ Ntm + at (Ntn + Ntm ))− δ{sn∗ Ntn + sm∗ Ntm + at (Ntn + Ntm )} + at (Ntn + Ntm ))− F ((1 + πn + (1 + πm )α∗ )ln∗ + φ(1 + πn + (1 + πm )α∗ )α∗ lm∗ , sn∗ Ntn + sm∗ Ntn α∗ )+δ{sn∗ Ntn + sm∗ Ntn α∗ }− w∗ ln∗ (1 + πm )(eα − α∗ )− φw∗ lm∗ {J(eα) − J(α∗ )}− r∗ sm∗ Ntn (eα − α∗ )+ τw∗ w∗ ln∗ Ntn (1 + πm )(eα − α∗ )+ φτw∗ w∗ lm∗ Ntn {J(eα) − J(α∗ )}+ 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∗ )(29)

n ln∗ + φN m lm∗ , sn∗ N n + sm∗ N m ), SP
By subtracting and adding F (Nt+1
t+1 becomesttt+1

Z sn∗ Ntn +sm∗ Ntm +at (Ntn +Ntm )SPt+1 ={
sn∗ Ntn +sm∗ Ntm

n ln∗ + φN m lm∗ , z)
∂F (Nt+1
t+1+ (1 − δ)}dz
∂K

n n
n
+ F (Nt+1
l + φNt+1αelm , sn∗ Ntn + sm∗ Ntn αe)− F (Ntn (1 + πn + α∗ (1 + πm ))ln∗ + φNtn (1 + πn + α∗ (1 + πm ))α∗ lm∗ , sn∗ Ntn + sm∗ Ntn α∗ )− δsm∗ Ntn (eα − α∗ )− (1 − τw∗ )w∗ ln∗ Ntn (1 + πm )(eα − α∗ )− (1 − τw∗ )φw∗ lm∗ Ntn {J(eα) − J(α∗ )}− (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∗ )(30)

Let co,m∗ be the consumption of old immigrants at the initial steady state. Then, co,m∗ =bm + (1 + (1 − τr∗ )r∗ )sm∗ . Thus, SPt+1 becomes16

Z sn∗ Ntn +sm∗ Ntm +at (Ntn +Ntm )SPt+1 ={
sn∗ Ntn +sm∗ Ntm

n ln∗ + φN m lm∗ , z)
∂F (Nt+1
t+1+ (1 − δ)}dz
∂K

n n∗
n
l + φNt+1αelm∗ , sn∗ Ntn + sm∗ Ntn αe)+ F (Nt+1− F (Ntn (1 + πn + α∗ (1 + πm ))ln∗ + φNtn (1 + πn + α∗ (1 + πm ))α∗ lm∗ , sn∗ Ntn + sm∗ Ntn α∗ )− (1 − δ)sm∗ Ntn (eα − α∗ )− (1 − τw∗ )w∗ ln∗ Ntn (1 + πm )(eα − α∗ )− (1 − τw∗ )φw∗ lm∗ Ntn {J(eα) − J(α∗ )}− (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 (co,m∗ + g o,m + g ind,m )(eα − α∗ )(31)

Note that the second line and the third line of the above equation can be transformed
as follows:n n∗nF (Nt+1l + φNt+1αelm∗ , sn∗ Ntn + sm∗ Ntn αe)− F (Ntn (1 + πn + α∗ (1 + πm ))ln∗ + φNtn (1 + πn + α∗ (1 + πm ))α∗ lm∗ , sn∗ Ntn + sm∗ Ntn α∗ )= F (Ntn (1 + πn + αe(1 + πm )) × (ln∗ + φeαlm∗ ), sn∗ Ntn + sm∗ Ntn αe)− F (Ntn (1 + πn + α∗ (1 + πm ))ln∗ + φNtn (1 + πn + α∗ (1 + πm ))α∗ lm∗ , sn∗ Ntn + sm∗ Ntn α∗ )Z αe ∂F (Ntn {(1 + πn + z(1 + πm )ln∗ + φJ(z)lm∗ }, Ntn {sn∗ + sm∗ z}) n[Nt (1 + πm )ln∗ + Ntn φJ 0 (z)lm∗ ]
=
∂L
∗
α

∂F (Ntn {(1 + πn + z(1 + πm )ln∗ + φJ(z)lm∗ }, Ntn {sn∗ + sm∗ z}) m∗ n+s Nt dz∂K
Using the homogeneity of ∂F/∂K and ∂F/∂L, we have17Z sn∗ Ntn +sm∗ Ntm +at (Ntn +Ntm )SPt+1 =
+
+

Z αe

{
sn∗ Ntn +sm∗ Ntm

n ln∗ + φN m lm∗ , z)
∂F (Nt+1
t+1+ (1 − δ)}dz∂K∂F ((1 + πn + z(1 + πm )ln∗ + φJ(z)lm∗ , sn∗ + sm∗ z)+ (1 − δ)]sm∗ Ntn dz
∂K
∗
α
[

∂F ((1 + πn + z(1 + πm )ln∗ + φJ(z)lm∗ , sn∗ + sm∗ α∗ ) n[Nt (1 + πm )ln∗ + Ntn φJ 0 (z)lm∗ ]dz
∂L
∗
α
Z αe
−
[(1 − τw∗ )w∗ ln∗ Ntn (1 + πm )(eα − α∗ )Z αe

α∗+(1 − τw∗ )φw∗ lm∗ Ntn J 0 (z)+(1 − τr∗ )r∗ sm∗ Ntn+Ntn (1 + πm )(g y,n + g ind,n )+Ntn (g y,m + g ind,m )J 0 (z)+Ntn (co,m∗ + g o,m + g ind,m )]dz

(32)

Note that J 0 (z) = (1 + πm )z + (1 + πn + (1 + πm )z). By rearranging the above equation,
we have
Z sn∗ Ntn +sm∗ Ntm +at (Ntn +Ntm )n ln∗ + φN m lm∗ , z)
∂F (Nt+1
t+1+ (1 − δ)}dz
SPt+1 =
{
n
m
∂K
n∗
m∗
s Nt +s Nt
Z αe∂F ((1 + πn + z(1 + πm )ln∗ + φJ(z)lm , sn∗ + sm∗ z) n∗n+(1 + πm )Nt
[
l
∂L
α∗
−(1 − τw∗ )w∗ ln∗ − g y,n − g ind,n ]dz+(1 + πm )Ntn

Z αe
α∗

z[

∂F ((1 + πn + z(1 + πm )ln∗ + φJ(z)lm∗ , sn∗ + sm∗ z) m∗l∂L−φ(1 − τw∗ )w∗ lm∗ − g y,m − g ind,m ]dz+Ntn

Z αe α∗+Ntn∂F ((1 + πn + z(1 + πm )ln∗ + φJ(z)lm∗ , sn∗ + sm∗ z) mφl∂Lo−φ(1 − τw∗ )w∗ lm∗ − g y,m − g ind,m ] − (co,m∗ + g o,m + g ind,m ) dz

(1 + πn + (1 + πm )z) × [Z αe∂F ((1 + πn + z(1 + πm )ln∗ + φJ(z)lm∗ , sn∗ + sm∗ z)+ (1 − δ)]sm∗ dz
∂K
∗
α
[

(33)

The first line is positive from the condition related with the golden rule. The second andthird lines are the MPL condition. The fourth and fifth lines are the redistribution effect.
The fourth line is the difference between what immigrants contribute minus the resources
used for immigrants. Note that when natives and immigrants are identical, we can showthat (33) becomes (28) by using the government budget constraint.
18

Appendix B5
From the discussion of A4, the government budget surplus at period t + 1 can be rewrittenasZ sn∗ Ntn +sm∗ Ntm +at (Ntn +Ntm ){SPt+1 =

sn∗ Ntn +sm∗ Ntm+(1 + πm )Ntn

n ln + φN m lm , z)
∂F (Nt+1
t+1+ (1 − δ)}dz∂K∂F (Ntn (1 + πn + z(1 + πm )ln∗ + φJ(z)lm∗ , sn∗ Ntn + sm∗ Ntn α∗ ) n∗
l
∂L
α∗

Z αe

[

−(1 − τw∗ )w∗ ln∗ − g y,n − g ind,n ]dz+(1 + πm )Ntn

Z αe

z[

α∗

∂F (Ntn (1 + πn + z(1 + πm )ln∗ + φJ(z)lm∗ , sn∗ Ntn + sm∗ Ntn α∗ ) m∗l∂L
−φ(1 − τw∗ )w∗ lm∗ − g y,m − g ind,m ]dz+Ntn

Z αe α∗∂F (Ntn (1 + πn + z(1 + πm )ln∗ + φJ(z)lm , sn∗ Ntn + sm∗ Ntn α∗ ) m∗φl∂Lo−φ(1 − τw∗ )w∗ lm∗ − g y,m − g ind,m ] − (co,m∗ + g o,m + g ind,m ) dz

(1 + πn + (1 + πm )z) × [+n ln∗ + φN n αm∗ , sn∗ N n + sm∗ N n z)
∂F (Nt+1tt
t+1 el+ (1 − δ)]sm∗ Ntn dz
∂K
α∗

Z αe

[

Since SP1 > 0 ,the government can accumulate savings. Thus, a1 > 0. For at (t ≥ 2),
assume that the government uses the surplus that arises from the first term. This implies
Z sn∗ Ntn +sm∗ Ntm +at (Ntn +Ntm )n ln∗ + φN m lm∗ , z)
∂F (Nt+1
1
t+1+ (1 − δ)}dz
at+1 = n{mnm
Nt+1 + Nt+1
∂Kn∗m∗s Nt +s Nt
Then, the proof is the same as for the proposition 3.

Appendix B6
Πiq=1 pq is the probability that a native is still alive at age i given that an agent is aliveat age 1 and Πiq=1 pq pbq is the probability that an immigrant is alive and stays in the host1n1mcountry at the age i. Then, Ntin = Nt−(i−1)×Πiq=1 pq and Ntim = Nt−(i−1)×Πiq=1 pq pbi . Thus,
1,n
Nt+20
/(1 − d) becomes as follows:
80
80
1,n
X
X
Nt+20
1,n
i,m 1,mi20=
η Nt−(i−1) × Πq=1 pq pbq Πx=1 pbi+x +Nt−(i−1)
η i,n × Πiq=1 pq
1−d
i=1

(34)

i=1

1,n
1,j
Note that, Nt+20= Nt1,n (1 + σ)20 and Nt−(i−1)= Nt1,j /(1 + σ)i−1 for j = i, m. Thus,(34) becomes
80

80

i=1

i=1

X N 1,nNt1,n (1 + σ)20 X Nt1,m
i,m
i
20
t
=
η
×
Π
p
p
b
Π
p
b
+
η i,n × Πiq=1 pq
q=1 q q x=1 i+x
1−d(1 + σ)i−1(1 + σ)i−1
19Let the steady state value of the age 1 immigrant native ratio, Nt1,m /Nt1,n be f . Solving for
f , we haveP80
1
1i,n × Πiq=1 pqi=1 (1+σ)i+19 η
1−d −f = P80(35)
1i,m × Πibq Π20bi+x
q=1 pq px=1 pi=1 (1+σ)i+19 ηThus, the ratio of immigrants to natives at the steady state becomes (40).

20