The Demand for Immigrants in an Overlapping Generations Economy

The Demand for Immigrants
in an Overlapping Generations Economy
Rolando M. Guzman∗
January 2004
ABSTRACT
This paper develops a novel framework to analyze the dynamic interaction of
capital accumulation, population growth, and immigration policy. This frame-
work combines two different features which seem to prevail in several countries:
a descentralized behavior of the agents in factors and goods markets, and a coali-
tional attitude with respect to the immigration process. Our model considers
an overlapping generations economy in which agents make lump-sum contribu-
tions to an “immigration agency,” which uses its proceeds to finance a program
of immigration subsidies. The agency seeks to maximize the lifetime utility
of the contemporaneous agents, taking into account the posterior descentral-
ized reactions of the individuals to its own decisions. The model is then used
to study, through tentative simulations, the welfare implications of a policy of
immigrants-subsidy in the United States. It is found that the long-run effect is
nearly nil, because the immediate benefits from immigrants are partially coun-
tervailed by a negative effect on capital accumulation.
KEYWORDS : Immigration; Descentralized Equilibrium; Welfare.
JEL Clasification: F22, F43, O40.
∗ Department of Economics, Instituto Tecnologico de Santo Domingo (INTEC) and Ver-
izon Dominicana, POBox 1377, Santo Domingo, Dominican Republic.
E-mail address:
gcpareto@codetel.net.do. Fax: (809)567 4791. I am indebted to Bart Taub, Charles M. Kahn
(University of Illinois at Urbana-Champaign) and Charles D. Kolstad (University of California
at Santa Barbara) for helpful comments on the first version of this paper. I also acknowledge
valuable contributions from Magdalena Lizardo (INTEC), and Gustavo Ventura (Penn State
University). Shortcomings and omissions are mine.
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Introduction
Under standard assumptions, economic theory prescribes that free mobility
of labor represents a source of potential economic gains.
Moreover, recent
empirical studies suggest that the benefit to the world economy from remov-
ing barriers on productive factors might be significant, and might even ex-
ceed the benefit from removing restrictions on goods trade.1
The logic be-
hind such a claim should be apparent: given differences in endowments, resi-
dents of capital-intensive countries could improve by employing immigrant la-
bor, and residents of labor-intensive countries could improve by offering their
labor abroad. It seems quite natural to conclude that the typical immigration
policy should not impose too many obstacles to the entry of potential immi-
grants.
In practice, however, different groups inside the host economy might be
unequally affected by the immigration process: that is, some groups might
be benefited by the entry of new hordes of immigrants, while others might
be harmed, either in a short or a long-run perspective. Indeed, the economic
prescription in favor of free flow of labor implicitly presumes the existence of
redistribution mechanisms, so that the losses of some groups are compensated
by the greater gains of some others. Since those mechanisms are usually hard to
implement —specially if the cost and benefits of the process are spread over long
periods of time— the immigration policy is a matter of political economy, and
it depends on the relative strengths of different forces within the host economy.2
The appropriate analysis of these issues requires the explicit introduction
of some notion of heterogeneity among the set of economic players. The most
common approach in that direction relies on a static separation of the agents
in accordance with endowments or income sources. Not surprisingly, it is then
found that capital owners should be favorable to immigration, while labor or-
ganizations should oppose it. Although this framework is perhaps convenient
for short run analysis, it can hardly be useful to describe an evolutive economy.
For one thing, in a dynamic economy people accumulate capital over time, and
the division between capitalists and workers is not conceptually straightfor-
ward. Moreover, a relevant problem in the study of immigration processes is
the contribution of immigrants to the accumulation of capital inside the host
economy, an important feature which cannot be appropriately addressed in a
static capital-labor environment.3
1 Hamilton and Whalley (1984) estimated that global income would double in the absence
of immigration restrictions, and conclude that the immigration issue was “one of the (and,
perhaps, the) most important issues facing the global economy.”
2 In that sense, Grether et al. (2000) report that polls conducted in US and Europe show
that a significant majority of those interviewed believe that ’immigrants are too many’. These
authors then conclude that ”on economic ground, at least, one might be inclined to expect a
more positive attitude towards immigration than those expressed in recen attitude surveys.”
3 A case in point is the discussion concerning the social security in several OECD countries,
where the solvency of the system in the near future might require a significant increase in the
working population. For instance, population projections based on current trends suggest
that the flow of immigrants in the EU as a whole should reach 13.5 million a year to keep
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On the other hand, an overlapping generations framework seems to be a
very natural tool: it provides a simple world in which different types of agents
interact with each other over time. In that simple world, the economic problem
of individual agents could be easily posed and frequently analitically solved,
so that the resulting evolution of capital and population could be examined in
a tractable way. Following that intuition, this paper presents an overlapping
generations model to study the dynamics of an economy with immigration.
Our approach distinguishes itself by several novelties. A main contribution
is to combine two issues which seem to be observable in several countries —a de-
scentralized behavior of the individual agents in the factors and goods markets,
and a sort of collectively agreed-upon attitude with respect to immigration. The
underlying rationale for the latter feature is the basic perception that economic
agents similarly affected by the arrival of immigrants have an incentive to gather
into coalitions, and that such coalitions are important ingredients for the design
of the immigration policy of a country. Therefore, the paper offers a theoretical
framework to analyze the economic factors underlying immigration policies —
namely, the costs and benefits of immigration, their distribution among different
agents, and its evolution over time.
A second contribution is to explore the dynamic link between the state of
the host economy, as reflected in wages and interest rates, and the demand
for immigrants. In that sense, our model extends to a long run perspective
the commonly held idea that the attitude with respect to immigration is highly
depending on the phases of the business cycle.4 And, finally, we also incorporate
into the analysis the fact that, as immigrants settle down in the host economy,
their own preferences eventually begin to influence the prevailing immigration
policy. The recognition of this fact is a common place in informal discussion
of immigration, but its theoretical formalization has turned out to be somehow
elusive. Therefore, our treatment of that aspect represents a contribution to the
previous literature.5
The structure of this paper is as follows. In section 2, the most relevant
literature is briefly surveyed. In section 3, we set up the basic structure of
the model, and in section 4 we analyze the welfare implications of a policy of
subsidizing immigration. In particular, it is shown that the promotion of the
immigration process by part of a given generation increases the welfare of its
members, but it might also reduce the steady-state level of capital and utility. In
a steady ratio of pensioners to workers. Inasmuch as the survival of the system is mostly
convenient for current workers, those should be in favor of a greater number of immigrants,
rather than oppose them as the static model would predict. This highlights the limitations of
such a framework for the analysis of the political economy of immigration. See Drinkwater et
al. (2002), pg. 26 for further discussion on this issue.
4 See, for instance, Gador and Stark (1990), p. 463. In turn, Borjas (1996) has argued that
“simple economics and common sense suggest that the magic number (of admitted immigrants)
should not be an immutable constant regardless of economic conditions in the United States.”
5 For example, in a related vein, Myers and Papageorgiou (1997) emphasize that if current
trends continue, the recently immigrant groups “could acquire economic and political strength
sufficiently to implement deep changes in the overall way the historical mainstream of a
country organizes and conducts its affairs.” However, see Chang (1996) for a discussion of the
difficulties of incorporating this fact into the analysis.
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section 5, we extend the model to consider a two-country game in which migrants
can move in either direction, depending on the policies of both countries. The
analytical discussion of the model is followed in section 6 by a simulation which
roughly replicate the dynamics of capital accumulation, population growth and
immigration policy in the United States. The results suggest that the long-
run welfare effect of a policy of immigration-subsidy are nearly nil, because its
short-run benefits are offset by a decrease in the long-run level of capital. The
paper is closed with its main conclusions and suggestions for further research.
1
Background
In the recent past, several countries have experienced a remarkable inflow of
immigrants. By the late 1980’s, it was estimated that over 60 million people
—i.e., around 1.2 percent of world population— resided in a country where
they were not born. More recent estimations suggest that around 6% of the
population in France, 19% in Switzerland, 9% in Belgium, 4% in the United
Kingdom, and 8% in the United States is foreign-born.
Although a considerable part of the huge immigration process obeys to po-
litical reasons, it is widely believed that economic impulses are even stronger.
Thus, the rise of population mobility has been followed by a significant wave
of economic research. Most of the contributions have been empirical, and the
main general questions might be summarized as follows: (a) what is the impact
of immigrants on the employment opportunities of natives?, (b) how do immi-
grants perform as compared with native-born population?, and (c) what is the
net contribution of immigrants to the welfare state? Borjas (1999), Commander
et al. (2002), and Drinkwater et al. (2002) present update surveys of the liter-
ature on those issues, and an interesting historical perspectives can be found in
Goldin (1994) and Foreman-Peck (1992).
The theoretical analysis of immigration has not evolved with the same strength.
Thus, two decades ago it was recognized that migration was a central feature of
the international economy, but “it has never received more than a small frac-
tion of the attention lavished on the theory of international capital movement.”6
However, while the theoretical study of immigration is still noticeably narrow,
more recent developments have corrected in part such unpleasant neglect, and
we will highlight those closest to our work. The earliest relevant work can be
traced to Ethier (1985, 1986) and Bhagwati and Srinivasan (1983). The first
two papers addressed the effects of alternative immigration policies on the level
of employment and national income, and the third one compared the welfare
implications of restrictions on capital and labor mobility.7 Those models were
essentially static and, in particular, the process of capital accumulation was
ignored. Moreover, the analyses relied on exogenous demand functions which
6 Ethier (1985).
7 A relevant antecedent for the general investigation of international migration is the classic
paper by Harris and Todaro (1970), where the aggregate effects of migration from rural into
urban areas of underdeveloped countries was first examined.
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were not explicitly linked to the utility-maximizing behavior of economic agents.
These two features limit the relevance of those models for long run dynamics.
The first consideration also applies to a static model developed by Myers and
Papageorgiou (1997), in which an active governmental control of immigration
is considered.
The model in Djajic and Milbourne (1988) partially overcomes the limita-
tions in the preceding work. In contrast with earlier studies, it is based on the
notion that migrants are utility-maximizing individuals with finite working lives,
and the asset accumulation of a typical worker is viewed as part of a solution to
an intertemporal optimization problem. Those ingredients are mixed to study
the general equilibrium of the source-economy, but the effects of immigration
on the host-economy is not approached. Braun (1993) introduced immigration
into a standard neoclassical growth model, and further variations of his models
are presented in Barro and Sala-i-Martin (1995). Since those models evolve in
the context of a representative-agent economy, the distributive effects of im-
migration cannot be possibly considered. Moreover, it is a well-known (and
intuitive) fact that, in such a context, the welfare of the native population must
necessarily improve with the arrival of immigrants, and then the discussion of
welfare issues in that world happens to be uninteresting.8 Steineck (1996), how-
ever, shows different models in which immigrants might have negative effects on
native welfare because they slow down the rate of techonological progress and
economic growth.
The theoretical approach of immigration is particularly in need of further
work on the political economy aspects of the immigration process. In that di-
rection, Benhabib (1996) studies how immigration policies that impose capital
and skill requirements would be determined under majority voting when natives
differ in their wealth holding and vote to maximize their income. Chang (1996)
offers a rich, although less formalized analysis of normative and positive aspects
related to distributive effects of immigration. In turn, Ghatak et al. (1996)
present an extension of the Harris-Todaro model to analyze the welfare impli-
cations of government intervention in the form of employment subsidies. Schiff
(1998) highlights the negative impact of immigration on the social capital of
the host country, in order to illustrate why free trade might be promoted at the
same time that free migration is surprisingly opposed. Other authors, such as
Levine et al. (2002) have embedded the immigration process in an endogenous
growth framework, in which the immigrants have welfare implications through
size effects and human capital dilution. Finally, Sempere (2000) explore the
dificulties to design an appropriate redistribution mechanism to obtain a Pareto
gain from freeing migration.
The present paper can be related to some additional work. Gador (1986) has
studied the welfare implications of international immigration in an overlapping
generations framework. The notion of “agency,” which we shall introduce later,
is related to the “council” used by Kotlikoff, Persson and Svensson (1987) for
different purposes. Lastly, John and Pecchenino (1994, fn. 10) informally de-
8 See Berry and Soligo (1969).
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scribe a problem to some extent similar to ours in the context of environmental
economics.
2
The Model
2.1
The Environment
Consider an infinitely-lived economy where agents live for two periods. At each
period, a new group of people is born and a group of young immigrants arrives.
A generation is defined as the set of native-born individuals in a given period and
the immigrants who arrived at the same time. Each agent (either a native born
or an immigrant) is endowed with one unit of labor during his first period of
life, and he supplies this endowment inelastically to receive a competitive wage.
The wage income is used for three different purposes: (i) to finance current
consumption, (ii) to provide the savings for retirement, and (iii) to make a
compulsory lump-sum contribution to an economic club, hereinafter identified
as the “agency.”
Each generation forms its own agency. An agency is a political institution
which represents the special interest of contemporaneous individuals. Its eco-
nomic role is to collect the contributions from its members when they are still
young, and to use the proceeds to finance the implementation of a convenient
immigration policy. More specifically, the objective of the agency formed at
period t is to select the contribution fee and corresponding immigration policy
—defined as a level of effort against or pro immigration— in such a way that
the discounted utility of a representative agent of generation t is maximized.
Having implemented its program, the agency formed at time t exits the model
at time t + 1, and is then replaced by a new agency which represent the special
interest of the generation (t + 1).
The agency should not be confused with a conventional social planner. The
social planner would be long-lived, and it should be concerned with the welfare
of all generations; on the other hand, the agency is short-lived, and is only con-
cerned with the welfare of its contemporaneous generation. It would be more
natural to think of the agency as a government whose policy is decided by ma-
jority vote —assuming the current generation is larger than the previous one—,
but is politically constrained in its ability to impose excessive burden on older
generations to finance activities which are not beneficial to them. Alternatively,
it can be taken as a theoretical abstraction for the self-evident fact that agents
with economic affinities have a tendency to form coalitions in defense of their
common interest. The economic affinity in our model is given by the age of the
agents, which is in turn a surrogate of their wealth.
Each individual agent behaves in a competitive fashion, taking factor prices
as given. She also takes the contribution demanded by the agency as an ex-
ogenous parameter. The agency, however, recognizes that its actions affect the
future path of capital, population, and prices. Thus, it will be assumed that
the agency formed at period t implements a Ramsey policy: it selects the immi-
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gration fee and immigration policy so that the welfare of a representative agent
of generation t is maximized, taking into account the posterior behavior of the
economy once those parameters have been imposed.
The investment decisions of the agents at a period t are affected by the
expectation of the prices at period t + 1. On the other hand, the prices at
period t + 1 depends on the previous savings decisions of the agents. This yields
a typical interaction in which the expectations of future prices affect current
decisions, which in turn determine future prices. We will analyze a perfect
foresight equilibrium in which those relationships are mutually consistent, in
the sense that the expectations of individual agents are fulfilled through their
own optimizing decisions.
2.2
Household and Firms
Let us assume that preferences are the same for all agents in all generations.
Moreover, let us say that the preferences of an agent of generation t can be
represented by an additive and twice-differentiable utility function, described
over all nonnegative consumption bundles. That is,
U (c1t, c2t+1) = u(c1t) + β u(c2t+1),
0 < β < 1
(1)
where c1t denotes his consumption in the first period of life, and c2t+1 denotes
his consumption in the second. It is also assumed that u > 0 and u < 0, so
that u is increasing and strictly concave, and that
lim u (c) = ∞
c → 0
lim u (c) = 0
c → ∞
Then, the household’s problem is
max
u(c1t) + β u(c2t+1)
(2)
c1t,c2t+1
c2t+1
s.t c1t +
= wt − τt
(3)
Rt+1
where {τt, wt, Rt+1} are supposed to be known and exogenously given from the
individual agent’s perspective.9 Conventionally, wt represents the wage income
at time t and Rt+1 represents the (gross) interest rate at period t + 1. In turn,
τt denotes the contribution of the representative agent of generation t to the
immigration agency.10 The optimal behavior of the agency will be characterized
below.
From the continuity of u and the compactness of the constraint set, a solution
for the consumer problem exists for any finite interest rate. Moreover, from the
9 Alternatively, one might consider Rt+1 as the expected interest rate, and then endow
the agents with perfect foresight.
10 It is also possible to assume that the utility of the households depends on the received
transfer, τt−1. Since τt−1 is taken as given at period t, that does not represent any important
issue.
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strict concavity of u and the convexity of the constraint set, the solution is
single-valued. Standard conditions for the interior solution of this problem can
be immediately stated as:
u (c1t) = β Rt+1
(4)
u (c2t+1)
Combining (4) with the budget constraint in (3), we can derive a continuous
consumption and savings function
c1t
=
c1t(wt − τt, Rt+1)
(5)
st
=
st(wt − τt, Rt+1)
(6)
Introducing (5) and (6) into the objective, we obtain an indirect utility
function V (wt − τt, Rt+1). From the Maximum Theorem, V is a continuous
function for finite values of Rt+1. It is clearly increasing in wt and Rt+1, but
decreasing in τt.
From (4),
u (wt − τt − st) = β Rt+1 u (Rt+1 st)
(7)
Assuming that st is differentiable, we obtain the conventional results
u (c1t)
0 < sw = −sτ =
< 1
(8)
u (c1t) + β R2 u (c
t+1
2t+1)
and
≥
sr < 0.
(9)
On the other hand, the firm’s problem is:11
max
Π(Kt, Pt) ≡ F (Kt, Pt) − wt Pt − rt Kt
Kt,Pt
where Kt represents the amount of physical capital, Pt denotes the working
population, and
rt is the (net) interest rate at period t. F is a standard
neoclassical production function, so that the usual Inada conditions are assumed
to hold.
For convenience, we ignore depreciation and technical progress. Hence, under
perfect competition, the absence of arbitrage opportunities leads to
Rt = Fk(Kt, Pt) + 1,
(10)
and
wt = Fp(Kt, Pt)
(11)
11 Under constant returns to scale, the assumption of a single firm is unharmful. The results
are the same as with many identical competitive firms.
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where subscripts k and p denote partial derivatives.
Let us say that F (., .) has constant returns to scale, so that we obtain:
wt
= ω(kt) ≡
f (kt) − kt f (kt)
(12)
Rt
= r(kt) ≡
f (kt) + 1
(13)
where kt ≡ Kt and f (k) ≡ F (k, 1). Of course, ω (k
P
t) = −kt f (kt) > 0, while
t
r (kt) < 0.
2.3
The Immigration Function
Let Mt be the number of new immigrants arriving at period t. It will be postu-
lated that
Mt+1 = Ω(wt+1, ˜wt+1) + φ(et), Ω1 > 0, φ > 0
(14)
Pt
≥
where ˜
wt+1 denotes the exogenous opportunity cost of immigrants, and et < 0
denotes the average effort (among t-generation’s agents) against or pro immi-
gration. It is clear that et satisfies:
|et| = τt ≤ wt.
(15)
The intuition behind the expression in (14) should be quite appealing. Its
first term is similar to the expression used by Barro and Sala-i-Martin (1995).
The effect of wage differential between the source and destination countries is
also emphasized in Djajic and Milbourne (1988) and Gador and Stark (1990).
For convenience, and without loss of generality, we shall further assume that
{ ˜
wt} is constant over time, and we shall normalize it to zero. A sufficient con-
dition for that feature is that the supply of workers form the rest of the world
is unlimited, and then foreign wages are not affected by level of immigration
Mt+1. Other alternatives for the specification of the function are of course pos-
sible . For example, it might be desirable to introduce the number of previous
immigrants as an argument of the function Ω, which is the approach in Guzm´
an
(1997). Similarly, individual’s decision to emigrate might be affected by the
prospective return on savings, rather than solely by the current wage, so that
the inclusion of Rt+2 as an additional argument might be appropriate.12 In this
paper, we will proceed in a simplified fashion, postponing those complicating
features until later extensions.
12 A technical reason to exclude the interest rate from Ω is that its inclusion would lead
to a third order difference equation for the dynamics of capital, which is hard to characterize
analytically. Economically, one might argue that prospective immigrants do not have enough
information about the future path of the foreign economy, so that the influence of future
interest rates on their decisions is weak. In any case, the inclusion of the feature is not quali-
tatively important: it basically increases the incentive of prospective immigrants to move into
the host economy, reducing the optimal amount of subsidy from the agencies. An empirical
exploration of the model might easily incorporate that effect.
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The function φ symbolizes the ability of the immigration agency to control
the immigration flow through its policy. It simply formalizes the common sense
observation that international migration depends not only on the decisions of
prospective migrants, but also on the policies of the host country.13 In turn,
the variable et is an indicator of the direction and intensity of the “immigration
policy” adopted at period t. A positive et might be interpreted as describing
the construction of hospitals, schools, public services, and facilities primarily
enjoyed by immigrants, as well as the use of resources for social programs to
which many immigrants are entitled. A negative et might be thought as the
average cost of police training, detection technology and infrastructure aimed
to control immigration, or the average cost of economic resources given to other
countries to improve their economic conditions and reduce the incentive of their
citizens to emigrate.14 It is important to point out that the effect of a particular
level of et operates at period t + 1, so that the old generation a any given
period is not affected by the current immigration policy of the current agency,
as implied by the political constraint of the t-agency to impose tax burden on
the old generation.
For simplicity, it will be assumed that the number of new born at any period
is equal to the number of agents getting old. Hence, without immigration, the
total working population in this economy would remain constant.15 Given the
arrival of immigrants, we have instead
Pt+1 = Pt + Mt+1.
(16)
It follows immediately that
Pt+1 = Pt(1 + φ(et) + Ω(wt+1)),
(17)
or, equivalently,
Pt+1 = 1 + nt,
(18)
Pt
where
nt ≡ φ(et) + Ω(wt+1)
(19)
In other words, the population grows at the variable rate nt, where nt de-
pends on the level of wages, and on the per capita level of effort against or pro
immigration.
13 Gador and Stark (1990) also remark this point, but they do not dwell on the idea.
14 In that respect, two relevant examples are the transfer of financial resources to Mexico
during the peso crisis and the US expenditures seeking to improve the economic conditions of
Haiti. Both efforts can be (and, indeed, they were) understood as efforts of the United States
to control the immigration flow from those countries by improving their internal economic
conditions.
15 The introduction of a positive natural rate of population growth is straightforward, but
it is not relevant for our purposes.
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