Directory UMM :Data Elmu:jurnal:J-a:Journal Of Economic Dynamics And Control:Vol24.Issue2.Feb2000:
Journal of Economic Dynamics & Control
24 (2000) 307}323
The dynamics of migration in the presence of
chains
Christian Helmenstein!,*, Yury Yegorov"
! Department of Finance, Institute for Advanced Studies, Stumpergasse 56, 1060 Vienna, Austria
" Department of Economics, Universitat Pompeu Fabra, Ramon Trias Fargas 25-27,
Barcelona 08005, Spain
Received 1 May 1996; accepted 1 November 1998
Abstract
The experience of several host countries shows that small initial in#ows of migrants
have a perpetuating e!ect on further migration from the same country of origin. This
paper studies the interplay of factors that determine the transitional dynamics of
migration #ows in the presence of social relationships between migrants. We fully specify
and formally solve a stochastic two-country model of chain migration with one sending
and one receiving country in which initial immigration patterns matter for future
immigration patterns. Numerical calculations illustrate the crucial role of the wage
elasticity of labor demand in counteracting the potential explosiveness of chain migration. ( 2000 Elsevier Science B.V. All rights reserved.
JEL classixcation: F22; J61; J63
Keywords: International chain migration; Migrant networks; Migration dynamics;
Push and pull factors
1. Introduction
The experience of several host countries has shown that small initial #ows
of migrants from a particular source country have a perpetuating e!ect on
further migration from that country. While the consequences of social
* Corresponding author. Tel. #43-1-59991-143; fax: #43-1-59991-163; e-mail: helmen@
ihs.ac.at.
0165-1889/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 5 - 1 8 8 9 ( 9 8 ) 0 0 0 6 9 - 4
308 C. Helmenstein, Y. Yegorov / Journal of Economic Dynamics & Control 24 (2000) 307}323
relationships in prompting self-perpetuating migration #ows have been
subject to extensive study, not much is known yet about the interplay of
the underlying factors in determining the dynamics of such migration #ows. It is
the primary objective of the present paper to study the transitional
migration dynamics during periods when the stock of migrants is small
compared to the stock of natives. We elaborate conditions under which immigration has a self-enhancing e!ect on further immigration, and conversely,
we specify factors which may counteract the possible explosiveness of chain
migration.
The economically oriented, theoretical literature on labor migration is
largely based on the assumption that migrants form a homogeneous entity
(Grubel and Scott, 1966; Kemp, 1969; Rivera-Batiz, 1982). But a given migrant
is di!erent from the previous migrant by the very nature of his or her order.
In particular, the central role of linkages across family members in shaping
the migration pattern has been studied by Mincer (1978) and Borjas and
Bronars (1991). At the national level, &gravity-type' models of migration rest
on the hypotheses that migration is positively related to the size of the
relevant origin and destination populations and inversely related to distance
(Greenwood, 1975). In addition to the before-mentioned concepts, in this
paper we draw upon the concept of chain migration (MacDonald and
MacDonald, 1964; Gurak and Caces, 1992) to di!erentiate between migrants
conditional on the existence of pre-migration links towards the country of
destination. A review of recent theoretical work on chain migration is included
in Massey et al. (1993) while the empirical strand of research on chain migration
is pursued, among others, by Jasso and Rosenzweig (1986, 1989) as well as Boyd
(1989).
Chain migrants exhibit features that essentially di!er from those of single
migrants. Contrary to single migrants, chain migrants enjoy the advantage of
receiving information about labor market opportunities from their network
abroad as well as assistance in "nding a job. A chain migrant can also reduce
accommodation expenses by sharing housing with members of the network. The
income equivalent of such bene"ts diminishes the expenses necessary to maintain a baseline consumption level, that is the required expenses to reach a certain
utility level are likely to be lower for chain migrants than for single migrants. It
follows that faced with a choice of destination countries, an individual who has
a network in one country but none in another may choose the former even if
wage rates in the latter are higher.
The next section introduces a benchmark model of chain migration with
a deterministic labor market framework. In Section 3 we proceed with a model
of chain migration under stochastic labor market conditions. The results of
numerical calculations are discussed in Section 4. Concluding remarks complete
the paper.
C. Helmenstein, Y. Yegorov / Journal of Economic Dynamics & Control 24 (2000) 307}323
309
2. A deterministic benchmark model
Consider the case of two countries, one sending and one receiving. Wages are
assumed to be higher in the foreign country (F) than in the home country (H),
and thus there is an incentive to migrate from H to F. We abstract from growth
in F arising from technological progress, capital accumulation, or population
growth but assume that population growth in H just compensates the out#ow.1
Home country inhabitants are in the state of potential migration from the
moment of their decision to migrate up to the end of the job searching process in
the immigration country. From these, only those who "nd a job stay and
constitute the net in#ow of migrants. This assumption is in line with the practice
of immigration authorities not to count individuals from foreign countries who
spend less than three months in the destination country, as migrants.2
The #ows of potential migrants result from the interplay of both push and pull
factors (Lee, 1966). Every period a constant fraction of H inhabitants, a, migrates
from H to F because of exogenously given push factors, such as poor living conditions or ethnical and/or political con#icts. This implies a "xed number of potential
single migrants per period. Each migrant who resides in F represents a center of
attraction for further migration, exhibiting a constant pull capacity per person, b.
All migrants who are attracted by such a process are called chain migrants. Since the
periodic out#ow of migrants is supposed to be small relative to the population size
of H, we do not consider the interaction between the two migration-prompting
mechanisms, that is the same person's decision to migrate is induced either by the
push factor or by the pull factor but not by both factors together.
Single migrants, C , di!er from chain migrants, C , in that the baseline
4
)
consumption expenses are higher for single migrants than for chain migrants,
C 'C . If the wage o!ered at destination is su$cient to cover the baseline
4
)
consumption expenses and is higher than the wage at origin, the migrant accepts
the o!er. Having no assets, otherwise he or she returns (or, for that matter,
migrates elsewhere) since the time span available upon entry for searching a job
that generates a su$cient income is limited for legislative reasons and borrowing against future earnings is not possible.
Assuming that the slope of the labor demand curve at destination is negative,
the equilibrium wage declines with the size of the in#ow.3 Without migration,
1 This assumption reduces the technical load but could be relaxed without changing the core of
the model.
2 This period may vary from three to twelve months depending on the country of destination.
3 Several authors pointed out that the elasticity of labor demand is higher for low-skill labor than
for high-skill labor (Elmeskov, 1993; Snessens, 1993; Hamermesh, 1986; Johnson and Layard, 1986).
In the context of this paper, however, heterogeneity with respect to migrants' skills is irrelevant since
they are unobservable upon entry. Hence, employers view migrants as a pooled group with average
skills (Stark, 1995) and o!er the same wage independent of their true skill level. Later employers may
decipher migrants' true skill level and adjust wages accordingly, but this does not impinge on the
immigration dynamics that has been determined earlier.
310 C. Helmenstein, Y. Yegorov / Journal of Economic Dynamics & Control 24 (2000) 307}323
the equilibrium wage level in the host country is determined by the function
w (t). Immigration pushes the equilibrium wage downwards so that the new
0
equilibrium wage w (t), from a linear approximation of a Taylor expansion, is
1
given by
dw(¸ )
0 n(t),
w (t)"w (t)#
1
0
d¸
(1)
where n(t) denotes the in#ow of migrants in period t, and ¸ is equal to the
0
amount of labor employed in equilibrium at the wage w (t).4
0
If we ignore growth arising from technological progress or capital accumulation, the production function and the labor demand function are constant
through time. The parameter h denotes the slope of the labor demand curve in
the destination country,
(2)
w "!h,
L
where w "dw/d¸. For the vector (w "1, ¸ "1), which can be obtained from
L
0
0
any initial values by normalization, we also have h"!d¸/dw(w/¸). The
inverse expression, 1/h, can be interpreted as the wage elasticity of labor demand
in the pre-migration labor market equilibrium of two countries of equal population size.5 In this framework there exists a single equilibrium wage in the
economy which clears the labor market.
Formally, for migration starting from the zero level we have the di!erential
equation
dN
F"aN #bN , N (0)"0, w'C 'C ,
H
F
F
4
)
dt
(3)
where the H population, N , has been normalized to 1. Eq. (3) states that the
H
periodic in#ow of migrants consists of a constant component which represents
immigration due to the push mechanism, and of a component that evolves
proportionally to the stock of migrants already residing in F, N .
F
Due to immigration, F wages decline and potential single migrants will no
longer "nd it advantageous to migrate, and hence migration to F is governed by
dN
F"bN ,
F
dt
N (0)"0, C (w(C .
F
)
4
(4)
4 This wage}employment mechanism is valid only if native and immigrant labor are assumed to
be substitutes.
5 There are no problems with deriving a similar equation for a constant elasticity labor demand
curve.
C. Helmenstein, Y. Yegorov / Journal of Economic Dynamics & Control 24 (2000) 307}323
311
As migration and the decline of wages continues, wages will eventually fall to the
subsistence level of chain migrants and thus immigration will terminate,
dN /dt"0.
F
(5)
If the stock of the domestic population in F is normalized to 1 and F and H are
of the same size, the range of admissible values for Eq. (3) is given by
N ((1!C )/h, for Eq. (4) by (1!C )/h(N ((1!C )/h, and for Eq. (5)
F
4
4
F
)
N '(1!C )/h.
F
)
The benchmark model (3)}(5) has the explicit solution
1!C
a
4,
N (t)"N (ebt!1), 0(t(¹ , N (¹ )"
1
F 1
F
Hb
h
(6)
1!C
),
N (t)"N (¹ )eb(t~T1), ¹ (t(¹ , N (¹ )"
1
2
F 2
F
F 1
h
(7)
N (t)"N (¹ ),
F
F 2
(8)
t'¹ .
2
Eqs. (6)}(8) specify the dynamics of migration through time. In pushing wages
below the subsistence levels of chain and single migrants, ongoing migration
brings about the endogenous transition between the three labor market regimes,
and migration eventually ceases.
3. Chain migration in a stochastic labor market framework
As it stands, from an empirical point of view the deterministic model has an
important shortcoming. With the deterministic labor market speci"cation assumed in the benchmark model there is absolutely no possibility for new
migrants to "nd a job once all vacancies have been occupied by earlier migrants.
Based on the intuition that newly arriving migrants face a certain probability
distribution of wages across vacancies when on job search, however, the probability to "nd a job which generates an income above the subsistence level,
though being negatively related to the number of migrants already present in F,
should not be zero.
In order to implement this notion, subsequently we model the dynamics of
chain migration for a stochastic labor market framework with exponentially
distributed wages across vacancies. Our decision to apply this particular distribution rests on two considerations. First, it nicely captures the concept that
wages are inversely related to the number of available vacancies. Second, if the
probability of a new vacancy with given wage to appear is proportional to the
number of currently existing vacancies and the reservation wages of job seekers
312 C. Helmenstein, Y. Yegorov / Journal of Economic Dynamics & Control 24 (2000) 307}323
are exponentially distributed, then the steady state for the matching process is
exactly described by an exponential distribution.6
We assume that during the search period every migrant receives only one o!er
with a wage above w (t). The wage w (t) represents a constant percentage of the
1
1
average wage of all vacancies and equals the wage of the lowest-paid vacancy
that implies zero search costs. In order to fully specify the model, we extend the
assumptions of the benchmark model as appropriate:
(a) Each migrant receives only one job o!er which he or she accepts if the
associated draw from the wage distribution exceeds the subsistence level. The
unsuccessful migrants return. According to the central limit theorem, for a continuum of migrants the random matching process between vacancies and
migrants returns a certain percentage of migrants who "nd employment.
(b) We assume a continuum of labor markets with similar labor demand
characteristics such that all distributions of wages and vacancies at any time
represent the same family of functions and di!er from each other only by the
variable scale parameter w (t).7 The multiplicity of labor markets is due to the
1
heterogeneity of employers which stems from the application of di!erent production technologies.
For a continuum of reservation wages across migrants at a given skill level,
high-wage vacancies are occupied with a higher probability than low-wage
vacancies. Each migrant's opportunity to "nd a job with a wage w higher than
w is represented by the exponential probability distribution
1
j
for w'w (t),
e~j((w@w1(t))~1)
1
(9)
p(w, t)" w1 (t)
0
for w(w (t)
1
G
with j determining the steepness of the distribution.8 As we abstract from
transportation costs, only those migrants who obtain jobs with wages
w'C (i"h, s) stay while the others return.
i
Subsequently, the cumulative probabilities that a single and a chain migrant
"nd employment with a wage higher than the subsistence level are de"ned by
p and p , respectively. Three cases of the level of the survival minimum abroad
4
)
relative to the wage distribution across vacancies can be distinguished. In case
(a) the minimum payment for vacancies exceeds the survival minimum for both
single migrants and chain migrants. Thus, both categories of migrants are able
6 Furthermore, exponentially distributed wages over vacancies exhibit desirable technical properties. Since the wage function is continuously di!erentiable, it renders the derivation of exact
solutions possible. In addition, exponentially distributed wages represent a density function for all
wages in the relevant interval from 0 to in"nity and yet their cumulated probability amounts to 1.
7 The function w (t) for any given t is a scalar. Distributions are considered for given t only.
1
8 Note that w (t) evolves over time proportionally to the average wage in the economy.
1
C. Helmenstein, Y. Yegorov / Journal of Economic Dynamics & Control 24 (2000) 307}323
313
to earn wages that are su$cient for survival with probability 1. In case (b) the
lower border w falls between the survival minimum of a chain migrant and
1
a single migrant, C (w (C . In this case chain migrants always receive an
)
1
4
income which exceeds their survival minimum while single migrants not necessarily so. In case (c) for both categories of migrants there is no guarantee to "nd
a job with an income su$cient to reach the survival minimum (Fig. 1).
Viewed intertemporally, this model provides a quasistationary framework in
the sense that wage o!ers evolve in such a way that at any time wage distributions are exponential and normalized to the lowest wage. Wage distributions
di!er only with respect to a parameter w which is proportional to the average
1
wage in the economy.
For all three cases the probabilities for a migration to turn out as a success are
given by the following equations:
(a) p "1, p "1,
4
)
=
(b) p " p(w, t) dw"e~j((C4@w1)~1), p "1,
)
4
C4
(10)
(c) p "e~j((C4@w1)~1), p "e~j((C)@w1)~1).
)
4
In analogy to Eqs. (3)}(5), using Eqs. (10a)}(10c) it would be possible to derive
a set of three corresponding di!erential equations. Further on, we will con"ne
the analysis of the stochastic model to case (c) since this is the case that di!ers
most from the benchmark model. While in the latter model migration ceases
once wages fall below C , the stochastic nature of the matching process now
)
P
Fig. 1. Wage distribution across vacancies that are available for migrants (case (c)).
314 C. Helmenstein, Y. Yegorov / Journal of Economic Dynamics & Control 24 (2000) 307}323
implies that migration continues, though at an ever decreasing magnitude. We
believe this characteristic of the stochastic model to be a favorable approximation to reality. In addition, after a su$ciently long period of immigration case (c)
represents the only relevant labor market regime for the F economy irrespective
of the initial setting.
The ratio of the probabilities to "nd a job is given by
p
4 "e~j(CS~C))@w1(1,
(11)
p
)
that is, the probability for a single migrant to be successful is lower than that for
a chain migrant. The distribution
p(w)"e~j((w~w1)@w1), w5w
(12)
1
de"nes how the probability to obtain a job with a certain payment decreases
with an increasing wage. To illustrate, if j is equal to 1, the probability to get
a job with a wage that is double the minimal wage is p(w)"1/e.
With this labor market framework at hand, we are now ready to set up the
model. By de"nition, when the opportunity to migrate opens up "rst all
migrants are single migrants. We de"ne
m &flow of potential chain migrants in period k,
),k
m &flow of potential single migrants in period k,
(13)
4,k
n &flow of actual migrants who find a job in period k.
k
The pull mechanism, which determines the number of H individuals attracted by
a given migrant per period, is given by
m "bN .
(14)
),k
F
Here N is the stock of the foreign population in F which is equal to the sum of
F
net migrant in#ows in the K previous periods,
K
N "+ n.
F
k
k/1
The push mechanism is summarized by
(15)
m "aN ,
(16)
4,k
H
where a is the share of potential migrants among the individuals without
a network abroad. In a more general formulation, Eq. (16) should be replaced by
the expression
m "a(t)(N !N !bN ).
4,k
H
F
F
(17)
C. Helmenstein, Y. Yegorov / Journal of Economic Dynamics & Control 24 (2000) 307}323
315
Eq. (17) covers two aspects of the migration process. It links the dynamics of the
push factor a through time to the analysis and re#ects the decreasing size of the
home population due to migration in earlier periods, corrected by the number of
individuals who were induced to migrate because of the pull factor. Since Eq.
(17) asymptotically converges to Eq. (16) if a(t) is constant and migration #ows
are small compared to the native population (N @N ), for simplicity we will
F
H
employ formula (16) subsequently.9
According to Eqs. (10c) and (14), the in#ow of chain migrants who do not have
to return because their earnings exceed the survival minimum, equals p ) m .
) ),k
Analogously, p m represents the in#ow of single migrants who stay in the
4 4,k
destination country. This yields the net in#ow of actual migrants in period k as
n "p m #p m .
(18)
k
4 4,k
) ),k
Then for case (c) the following equation speci"es the dynamics of the migrant
in#ow:
n "aN e~j((C4@w1,k)~1)#bN e~j((C)@w1,k)~1).
(19)
F,k
k
H
The "rst term describes the net in#ow of single migrants, while the second term
represents chain migration. In a linear approximation of the labor demand
function, the wage decreases according to
w "w !hN .
(20)
1,k
0
F,k
With n de"ning the in#ow of migrants and N representing the stock of
k
F,k
the foreign population in F, the per period change in the stock of migrants is
given by
N
"N #n .
(21)
F,k`1
F,k
k`1
The share of single migrants in the current period in#ow, l , thus equals
k
aN e~jCs@w1,k
H
.
(22)
l"
k aN e~jC4@w1,k#bN e~jC)@w1,k
H
F,k
Eqs. (19)}(22) state the dynamics of migration in discrete time. The transformation of the discrete time model into a continuous time model is achieved by
9 International statistics document that the number of immigrants from a given country of origin
in a specixc country of destination is generally small relative to the total population of that country
of origin (OECD-SOPEMI, 1995). This does, of course, not imply that the total number of migrants,
cumulated over all countries of destination, from a given country of origin is small relative to the
total population of that country. Furthermore, sending countries are usually characterized by higher
natural population growth rates than receiving countries. The assumption of zero natural population growth in F and an o!setting growth in H is a straightforward way to capture this stylized fact.
316 C. Helmenstein, Y. Yegorov / Journal of Economic Dynamics & Control 24 (2000) 307}323
a standard limiting procedure. In continuous time the dynamic process of
migration is described by the following di!erential equation,
dN (t)
F "aN ej~jC4@(w0~hNF(t))#bN (t)ej~jC)@(w0~hNF(t)).
H
F
dt
(23)
Eq. (23) expresses the dynamics of the foreign population in the destination
country as a function of time and di!erent parameters. In order to solve
di!erential equation (23), we draw upon the transformation
1
w
N ! 0"! ,
F
y
h
(24)
with y as a new variable. The variables t and y can now be separated, and the
solution can be found as an inverse function analytically. The solution has the
form of an inde"nite integral,
P
y
dy
1
.
(25)
4y1@h#b((w /h)y2!y )ej~jC)y1@h
ej~jC
aN
y2
1
1
0
h@w0 H 1
Eq. (25) speci"es the relationship between time and the stock of immigrants.
From Eq. (23) it follows that dN (t)/dt'0, ∀t. As N (t)Pw /h, then
F
F
0
dN (t)/dtP0. Using Eq. (24), we conclude that yP#R, and the denominator
F
of Eq. (25) approaches zero from above. Thus the integrand is unboundedly
increasing but "nite for all "nite y and the integral takes all values strictly
between 0 and #R as h/w (y(#R. We conclude that yP#R as
0
tP#R. Hence from Eq. (24) we have N (t)Pw /h as tP#R, that is, the
F
0
stock of migrants asymptotically converges to w /h from below.10
0
t(y)"
4. Numerical experiments
Although the solution obtained from Eq. (25) is exact, it is necessary to resort
to numerical calculations to render a graphical presentation of the results
possible. In the subsequent chapter we therefore conduct various numerical
experiments to depict the e!ects of di!erent sets of parameters on the in#ow of
migrants, the equilibrium wage, and the share of single migrants in the in#ow.
10 By the very nature of the underlying assumptions, in particular that outmigration is small
relative to the size of the country of origin, the present model primarily applies to the transitional
dynamics of chain migration. Study of the long-run dynamics of chain migration would mandate to
endogenize variables (such as the push factor) that we assumed constant and would require
consideration of higher-order e!ects (such as the interaction between the two migration-prompting
mechanisms).
C. Helmenstein, Y. Yegorov / Journal of Economic Dynamics & Control 24 (2000) 307}323
317
The rather large number of free parameters (C , C , j, h, N , a, b) requires to
) 4
H
determine some of them in advance in order to isolate the e!ects of parameter
changes on the endogenous variables. If we normalize N to 1, then N denotes
H
F
the stock of migrants as a percentage of the total home population. C is also set
)
to 1 supposing that a chain migrant can initially (t"0) "nd a job with w'C
)
with probability 1, whereas later on he or she will face an increasing return
probability. With a growing number of immigrants, however, wages will be
negatively a!ected, and thus chain migrants might, along with single migrants,
face the necessity to return.
The following numerical calculations provide insights about the dynamics of
migration and the relative di$culty for an individual to "nd a job and thus to
migrate successfully under di!erent parameter constellations. The main results
are depicted in Figs. 2}6. For Figs. 2}4 the following set of parameters was
selected: C "1, C "1.5, j"1, N "1, a"0.05, b"0.5, h3M1, 1.5, 2N.
)
4
H
Fig. 2 reveals a dominant in#uence of the wage elasticity of labor demand,
1/h, on the dynamics of migration. If labor demand is inelastic, that is, for a
high value of h, it is more di$cult for a migrant to "nd a job with a
su$cient income to secure the baseline consumption level. Hence, fewer
migrants stay. Moreover, a higher number of returning potential migrants
implies a smaller number of nuclei for immigration chains. Along with the lower
pull capacity of chain migrants total migration is signi"cantly diminished. In
this model a relatively inelastic labor demand thus acts as an e!ective regulator
of immigration.
Fig. 2. Dynamics of migration for di!erent h. Upper curve corresponds to h"1, middle to h"1.5,
lower to h"2.
318 C. Helmenstein, Y. Yegorov / Journal of Economic Dynamics & Control 24 (2000) 307}323
Fig. 3. Wage dynamics for di!erent h. Upper curve for h"2, lower curve for h"1.
Fig. 3 shows the wage dynamics for the same parameter values as before. Since
the in#ow of immigrant labor is positively related to the elasticity of labor
demand for the reasons provided above, the pace of the wage level decline is
slower if labor demand is inelastic.
The dynamics of the share of single migrants in total immigration is revealed
by Fig. 4. For high h (h"2) the share of single migrants remains relatively high
while for low h (h"1) it reaches almost zero already after a few periods. In this
case immigrants without chains are largely excluded from entering the foreign
labor market. Because of this process the host country may su!er from foregone
welfare gains if a considerable number of single immigrants have higher skills
than the native-born population.11
Fig. 5 shows the in#uence of simultaneous changes of the parameters h and
j on the in#ow of migrants. Recall that j determines the wage structure across
vacancies, with a high j implying that a large share of the vacancies is low-paid.
It turns out that parameter h has a considerably stronger in#uence on migration
than parameter j , that is the elasticity of labor demand is of primary importance for the migration dynamics.
In Fig. 6 we compare the case of pure single migration with the case of single
migration combined with chain migration. The parameters used are
11 See Funkhouser and Trejo (1995) for an analysis of labor market skills of newly arriving male
immigrants to the USA.
C. Helmenstein, Y. Yegorov / Journal of Economic Dynamics & Control 24 (2000) 307}323
319
Fig. 4. Dynamics of the share of single migrants in total immigration for di!erent h: upper curve for
h"2, lower curve for h"1.
Fig. 5. Dynamics of migration for di!erent h and j. Upper curve for h"1, j"2, lower curve for
h"2, j"1.
C "1, C "1.5, a"0.1 and j"1. For the upper curve b"0.2, for the lower
)
4
curve b"0. This implies that the pull mechanism is not e!ective in the second
case, that is all migration is single migration. It appears that even for pure single
320 C. Helmenstein, Y. Yegorov / Journal of Economic Dynamics & Control 24 (2000) 307}323
Fig. 6. Comparison of two cases: Single migration versus single migration combined with chain
migration. Upper curve for b"0.2, lower curve for b"0.
migration the dynamics is non-linear because of the diminishing equilibrium
wage. The di!erence between both paths becomes more and more pronounced
over time.
5. Conclusions
The model devised in this paper investigates the dynamics of migration #ows
in the presence of chains. Within the framework of this type of model, the
dynamics of migration in the presence of chains depends on the wage elasticity
of the demand for labor, the intensities of the pull capacity and the push
mechanism, the wage structure across vacancies, and the di!erences in the
baseline consumption levels between single and chain migrants. The non-linear
interaction of these factors gives rise to the phenomenon of hysteresis, wherein
factors with early or temporary occurrence persistently a!ect the future dynamics of migration. Since immigrants from a country with an established presence
in the host country are more likely to be successful, initial conditions matter, so
that initial immigration patterns reinforce future immigration patterns. This
implies that the growth of the immigrant population from that country will
exceed that of source countries with a smaller established presence. When
designing an immigration policy, potential host countries may therefore want to
pay special attention to migrants from countries with high pull capacities such
as high fertility rates and strong inter-family links.
C. Helmenstein, Y. Yegorov / Journal of Economic Dynamics & Control 24 (2000) 307}323
321
We fully specify a stochastic two-country model with one sending and one
receiving country and derive a non-linear di!erential equation as its solution.
The results of our model are outstandingly sensitive to changes in the wage
elasticity of labor demand. The relative importance of the wage elasticity is
stronger in our model than in a model with a homogenous group of migrants.
The reason is that the wage elasticity of labor demand governs the number of
"rst period single migrants who are absorbed by the host country, thereby
determining the strength of the multiplier e!ect if we take into account that the
second and higher-order periods in#ows are nearly proportional to the number
of migrants in the "rst period. In particular if migrants are entitled to but minor
social security bene"ts of the receiving economy, the labor market itself may be
regarded as an e!ective regulator of migration, and a fortiori so in the case of
illegal immigration.
The existence of chains towards a certain destination country reinforces the
attraction of this country compared to other potential destination countries.
The resulting comparative advantage might overcompensate other, disadvantageous factors in that speci"c destination country. This e!ect might well
explain large-scale migration #ows between countries with comparatively small
wage di!erentials, and its role becomes ever more pronounced with an increasing di!erence between the living costs of single migrants and chain migrants.
The analysis presented in this paper has a direct bearing for several contentious policy questions. Since the baseline consumption expenses of chain migrants are lower than those of single migrants, this approach has a distinct
implication for the skill composition of the migrants who stay in the country of
destination. According to Stark (1991) low-skill workers return while high-skill
workers stay in consequence of productivity-adjusted wages after the reinstatement of informational symmetry through monitoring. Contrary to the prediction of this approach, despite of individual identi"cation and removal from the
pooling process, low-skill workers might be able to draw on their &chainmigration capital' as a substitute for wage averaging and thus stay. To avoid
such an outcome of a progressively deteriorating performance of succeeding
cohorts of migrants, the country of destination, if it cares about the skill level of
the migrants who are there to stay, might wish to spend resources in order to
preselect migrants according to their skills (Markusen, 1988; Straubhaar and
Zimmermann, 1993). As appropriate, the receiving economy might also consider
to compensate high-skill migrants for the lack of &chain-migration capital'.
Furthermore, the model suggests that #ows of single and chain migrants also
di!er in their gender composition. Given that wages for female labor are lower
than for male labor even if di!erences in education are controlled for and the
minimum subsistence levels are approximately equal for both sexes, the likelihood for a female immigrant to "nd a job with a wage that is su$cient to cover
her baseline consumption expenses is lower than that for a male immigrant with
the same quali"cation. It follows that the typical single migrant would be male
322 C. Helmenstein, Y. Yegorov / Journal of Economic Dynamics & Control 24 (2000) 307}323
while women would predominantly engage in chain migration to enhance their
chances to bene"t from the potentially rewarding opportunity to migrate. In this
way women might counteract the e!ect of gender discrimination on economic
performance. Contrary to before, where the existence of chains turned out to
trigger a potentially disadvantageous skill composition for the receiving economy, a high ratio of chain migrants relative to single migrants facilitates access
to female labor resources and should in this case augment the bene"ts from
migration that accrue to the country of destination.
Finally, our model predicts that the composition of the migrant in#ow di!ers
conditional on the phase of the business cycle. During recessions a smaller share
of high-paid vacancies should be available for migrants since the competition for
these vacancies among natives increases, which implies that migrants face
a steepening wage distribution across vacancies. The testable hypothesis here is
that during periods of an economic downturn the share of chain migrants in
total immigration is higher than during phases of prosperity. This e!ect should
occur independently of an intervention of immigration authorities since it stems
directly from the optimizing behavior of labor market participants.
Acknowledgements
The authors are indebted to Vivek H. Dehejia, Bernhard Felderer, Oded
Stark, Andreas WoK rgoK tter, and an anonymous referee for insightful comments.
Partial "nancial support from the Austrian Science Foundation under contract
number P10967-S07 is gratefully acknowledged. Yury Yegorov's research was
"nancially supported by the Institute for Advanced Studies, Vienna.
References
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international migration: a review and appraisal. Population and Development Review 19,
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OECD-SOPEMI, 1995. Trends in international migration. Annual Report 1994, OECD, Paris.
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Snessens, H., 1993. The skill composition of labour demand and supply. Appendix to Drèze, J.H.,
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24 (2000) 307}323
The dynamics of migration in the presence of
chains
Christian Helmenstein!,*, Yury Yegorov"
! Department of Finance, Institute for Advanced Studies, Stumpergasse 56, 1060 Vienna, Austria
" Department of Economics, Universitat Pompeu Fabra, Ramon Trias Fargas 25-27,
Barcelona 08005, Spain
Received 1 May 1996; accepted 1 November 1998
Abstract
The experience of several host countries shows that small initial in#ows of migrants
have a perpetuating e!ect on further migration from the same country of origin. This
paper studies the interplay of factors that determine the transitional dynamics of
migration #ows in the presence of social relationships between migrants. We fully specify
and formally solve a stochastic two-country model of chain migration with one sending
and one receiving country in which initial immigration patterns matter for future
immigration patterns. Numerical calculations illustrate the crucial role of the wage
elasticity of labor demand in counteracting the potential explosiveness of chain migration. ( 2000 Elsevier Science B.V. All rights reserved.
JEL classixcation: F22; J61; J63
Keywords: International chain migration; Migrant networks; Migration dynamics;
Push and pull factors
1. Introduction
The experience of several host countries has shown that small initial #ows
of migrants from a particular source country have a perpetuating e!ect on
further migration from that country. While the consequences of social
* Corresponding author. Tel. #43-1-59991-143; fax: #43-1-59991-163; e-mail: helmen@
ihs.ac.at.
0165-1889/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 5 - 1 8 8 9 ( 9 8 ) 0 0 0 6 9 - 4
308 C. Helmenstein, Y. Yegorov / Journal of Economic Dynamics & Control 24 (2000) 307}323
relationships in prompting self-perpetuating migration #ows have been
subject to extensive study, not much is known yet about the interplay of
the underlying factors in determining the dynamics of such migration #ows. It is
the primary objective of the present paper to study the transitional
migration dynamics during periods when the stock of migrants is small
compared to the stock of natives. We elaborate conditions under which immigration has a self-enhancing e!ect on further immigration, and conversely,
we specify factors which may counteract the possible explosiveness of chain
migration.
The economically oriented, theoretical literature on labor migration is
largely based on the assumption that migrants form a homogeneous entity
(Grubel and Scott, 1966; Kemp, 1969; Rivera-Batiz, 1982). But a given migrant
is di!erent from the previous migrant by the very nature of his or her order.
In particular, the central role of linkages across family members in shaping
the migration pattern has been studied by Mincer (1978) and Borjas and
Bronars (1991). At the national level, &gravity-type' models of migration rest
on the hypotheses that migration is positively related to the size of the
relevant origin and destination populations and inversely related to distance
(Greenwood, 1975). In addition to the before-mentioned concepts, in this
paper we draw upon the concept of chain migration (MacDonald and
MacDonald, 1964; Gurak and Caces, 1992) to di!erentiate between migrants
conditional on the existence of pre-migration links towards the country of
destination. A review of recent theoretical work on chain migration is included
in Massey et al. (1993) while the empirical strand of research on chain migration
is pursued, among others, by Jasso and Rosenzweig (1986, 1989) as well as Boyd
(1989).
Chain migrants exhibit features that essentially di!er from those of single
migrants. Contrary to single migrants, chain migrants enjoy the advantage of
receiving information about labor market opportunities from their network
abroad as well as assistance in "nding a job. A chain migrant can also reduce
accommodation expenses by sharing housing with members of the network. The
income equivalent of such bene"ts diminishes the expenses necessary to maintain a baseline consumption level, that is the required expenses to reach a certain
utility level are likely to be lower for chain migrants than for single migrants. It
follows that faced with a choice of destination countries, an individual who has
a network in one country but none in another may choose the former even if
wage rates in the latter are higher.
The next section introduces a benchmark model of chain migration with
a deterministic labor market framework. In Section 3 we proceed with a model
of chain migration under stochastic labor market conditions. The results of
numerical calculations are discussed in Section 4. Concluding remarks complete
the paper.
C. Helmenstein, Y. Yegorov / Journal of Economic Dynamics & Control 24 (2000) 307}323
309
2. A deterministic benchmark model
Consider the case of two countries, one sending and one receiving. Wages are
assumed to be higher in the foreign country (F) than in the home country (H),
and thus there is an incentive to migrate from H to F. We abstract from growth
in F arising from technological progress, capital accumulation, or population
growth but assume that population growth in H just compensates the out#ow.1
Home country inhabitants are in the state of potential migration from the
moment of their decision to migrate up to the end of the job searching process in
the immigration country. From these, only those who "nd a job stay and
constitute the net in#ow of migrants. This assumption is in line with the practice
of immigration authorities not to count individuals from foreign countries who
spend less than three months in the destination country, as migrants.2
The #ows of potential migrants result from the interplay of both push and pull
factors (Lee, 1966). Every period a constant fraction of H inhabitants, a, migrates
from H to F because of exogenously given push factors, such as poor living conditions or ethnical and/or political con#icts. This implies a "xed number of potential
single migrants per period. Each migrant who resides in F represents a center of
attraction for further migration, exhibiting a constant pull capacity per person, b.
All migrants who are attracted by such a process are called chain migrants. Since the
periodic out#ow of migrants is supposed to be small relative to the population size
of H, we do not consider the interaction between the two migration-prompting
mechanisms, that is the same person's decision to migrate is induced either by the
push factor or by the pull factor but not by both factors together.
Single migrants, C , di!er from chain migrants, C , in that the baseline
4
)
consumption expenses are higher for single migrants than for chain migrants,
C 'C . If the wage o!ered at destination is su$cient to cover the baseline
4
)
consumption expenses and is higher than the wage at origin, the migrant accepts
the o!er. Having no assets, otherwise he or she returns (or, for that matter,
migrates elsewhere) since the time span available upon entry for searching a job
that generates a su$cient income is limited for legislative reasons and borrowing against future earnings is not possible.
Assuming that the slope of the labor demand curve at destination is negative,
the equilibrium wage declines with the size of the in#ow.3 Without migration,
1 This assumption reduces the technical load but could be relaxed without changing the core of
the model.
2 This period may vary from three to twelve months depending on the country of destination.
3 Several authors pointed out that the elasticity of labor demand is higher for low-skill labor than
for high-skill labor (Elmeskov, 1993; Snessens, 1993; Hamermesh, 1986; Johnson and Layard, 1986).
In the context of this paper, however, heterogeneity with respect to migrants' skills is irrelevant since
they are unobservable upon entry. Hence, employers view migrants as a pooled group with average
skills (Stark, 1995) and o!er the same wage independent of their true skill level. Later employers may
decipher migrants' true skill level and adjust wages accordingly, but this does not impinge on the
immigration dynamics that has been determined earlier.
310 C. Helmenstein, Y. Yegorov / Journal of Economic Dynamics & Control 24 (2000) 307}323
the equilibrium wage level in the host country is determined by the function
w (t). Immigration pushes the equilibrium wage downwards so that the new
0
equilibrium wage w (t), from a linear approximation of a Taylor expansion, is
1
given by
dw(¸ )
0 n(t),
w (t)"w (t)#
1
0
d¸
(1)
where n(t) denotes the in#ow of migrants in period t, and ¸ is equal to the
0
amount of labor employed in equilibrium at the wage w (t).4
0
If we ignore growth arising from technological progress or capital accumulation, the production function and the labor demand function are constant
through time. The parameter h denotes the slope of the labor demand curve in
the destination country,
(2)
w "!h,
L
where w "dw/d¸. For the vector (w "1, ¸ "1), which can be obtained from
L
0
0
any initial values by normalization, we also have h"!d¸/dw(w/¸). The
inverse expression, 1/h, can be interpreted as the wage elasticity of labor demand
in the pre-migration labor market equilibrium of two countries of equal population size.5 In this framework there exists a single equilibrium wage in the
economy which clears the labor market.
Formally, for migration starting from the zero level we have the di!erential
equation
dN
F"aN #bN , N (0)"0, w'C 'C ,
H
F
F
4
)
dt
(3)
where the H population, N , has been normalized to 1. Eq. (3) states that the
H
periodic in#ow of migrants consists of a constant component which represents
immigration due to the push mechanism, and of a component that evolves
proportionally to the stock of migrants already residing in F, N .
F
Due to immigration, F wages decline and potential single migrants will no
longer "nd it advantageous to migrate, and hence migration to F is governed by
dN
F"bN ,
F
dt
N (0)"0, C (w(C .
F
)
4
(4)
4 This wage}employment mechanism is valid only if native and immigrant labor are assumed to
be substitutes.
5 There are no problems with deriving a similar equation for a constant elasticity labor demand
curve.
C. Helmenstein, Y. Yegorov / Journal of Economic Dynamics & Control 24 (2000) 307}323
311
As migration and the decline of wages continues, wages will eventually fall to the
subsistence level of chain migrants and thus immigration will terminate,
dN /dt"0.
F
(5)
If the stock of the domestic population in F is normalized to 1 and F and H are
of the same size, the range of admissible values for Eq. (3) is given by
N ((1!C )/h, for Eq. (4) by (1!C )/h(N ((1!C )/h, and for Eq. (5)
F
4
4
F
)
N '(1!C )/h.
F
)
The benchmark model (3)}(5) has the explicit solution
1!C
a
4,
N (t)"N (ebt!1), 0(t(¹ , N (¹ )"
1
F 1
F
Hb
h
(6)
1!C
),
N (t)"N (¹ )eb(t~T1), ¹ (t(¹ , N (¹ )"
1
2
F 2
F
F 1
h
(7)
N (t)"N (¹ ),
F
F 2
(8)
t'¹ .
2
Eqs. (6)}(8) specify the dynamics of migration through time. In pushing wages
below the subsistence levels of chain and single migrants, ongoing migration
brings about the endogenous transition between the three labor market regimes,
and migration eventually ceases.
3. Chain migration in a stochastic labor market framework
As it stands, from an empirical point of view the deterministic model has an
important shortcoming. With the deterministic labor market speci"cation assumed in the benchmark model there is absolutely no possibility for new
migrants to "nd a job once all vacancies have been occupied by earlier migrants.
Based on the intuition that newly arriving migrants face a certain probability
distribution of wages across vacancies when on job search, however, the probability to "nd a job which generates an income above the subsistence level,
though being negatively related to the number of migrants already present in F,
should not be zero.
In order to implement this notion, subsequently we model the dynamics of
chain migration for a stochastic labor market framework with exponentially
distributed wages across vacancies. Our decision to apply this particular distribution rests on two considerations. First, it nicely captures the concept that
wages are inversely related to the number of available vacancies. Second, if the
probability of a new vacancy with given wage to appear is proportional to the
number of currently existing vacancies and the reservation wages of job seekers
312 C. Helmenstein, Y. Yegorov / Journal of Economic Dynamics & Control 24 (2000) 307}323
are exponentially distributed, then the steady state for the matching process is
exactly described by an exponential distribution.6
We assume that during the search period every migrant receives only one o!er
with a wage above w (t). The wage w (t) represents a constant percentage of the
1
1
average wage of all vacancies and equals the wage of the lowest-paid vacancy
that implies zero search costs. In order to fully specify the model, we extend the
assumptions of the benchmark model as appropriate:
(a) Each migrant receives only one job o!er which he or she accepts if the
associated draw from the wage distribution exceeds the subsistence level. The
unsuccessful migrants return. According to the central limit theorem, for a continuum of migrants the random matching process between vacancies and
migrants returns a certain percentage of migrants who "nd employment.
(b) We assume a continuum of labor markets with similar labor demand
characteristics such that all distributions of wages and vacancies at any time
represent the same family of functions and di!er from each other only by the
variable scale parameter w (t).7 The multiplicity of labor markets is due to the
1
heterogeneity of employers which stems from the application of di!erent production technologies.
For a continuum of reservation wages across migrants at a given skill level,
high-wage vacancies are occupied with a higher probability than low-wage
vacancies. Each migrant's opportunity to "nd a job with a wage w higher than
w is represented by the exponential probability distribution
1
j
for w'w (t),
e~j((w@w1(t))~1)
1
(9)
p(w, t)" w1 (t)
0
for w(w (t)
1
G
with j determining the steepness of the distribution.8 As we abstract from
transportation costs, only those migrants who obtain jobs with wages
w'C (i"h, s) stay while the others return.
i
Subsequently, the cumulative probabilities that a single and a chain migrant
"nd employment with a wage higher than the subsistence level are de"ned by
p and p , respectively. Three cases of the level of the survival minimum abroad
4
)
relative to the wage distribution across vacancies can be distinguished. In case
(a) the minimum payment for vacancies exceeds the survival minimum for both
single migrants and chain migrants. Thus, both categories of migrants are able
6 Furthermore, exponentially distributed wages over vacancies exhibit desirable technical properties. Since the wage function is continuously di!erentiable, it renders the derivation of exact
solutions possible. In addition, exponentially distributed wages represent a density function for all
wages in the relevant interval from 0 to in"nity and yet their cumulated probability amounts to 1.
7 The function w (t) for any given t is a scalar. Distributions are considered for given t only.
1
8 Note that w (t) evolves over time proportionally to the average wage in the economy.
1
C. Helmenstein, Y. Yegorov / Journal of Economic Dynamics & Control 24 (2000) 307}323
313
to earn wages that are su$cient for survival with probability 1. In case (b) the
lower border w falls between the survival minimum of a chain migrant and
1
a single migrant, C (w (C . In this case chain migrants always receive an
)
1
4
income which exceeds their survival minimum while single migrants not necessarily so. In case (c) for both categories of migrants there is no guarantee to "nd
a job with an income su$cient to reach the survival minimum (Fig. 1).
Viewed intertemporally, this model provides a quasistationary framework in
the sense that wage o!ers evolve in such a way that at any time wage distributions are exponential and normalized to the lowest wage. Wage distributions
di!er only with respect to a parameter w which is proportional to the average
1
wage in the economy.
For all three cases the probabilities for a migration to turn out as a success are
given by the following equations:
(a) p "1, p "1,
4
)
=
(b) p " p(w, t) dw"e~j((C4@w1)~1), p "1,
)
4
C4
(10)
(c) p "e~j((C4@w1)~1), p "e~j((C)@w1)~1).
)
4
In analogy to Eqs. (3)}(5), using Eqs. (10a)}(10c) it would be possible to derive
a set of three corresponding di!erential equations. Further on, we will con"ne
the analysis of the stochastic model to case (c) since this is the case that di!ers
most from the benchmark model. While in the latter model migration ceases
once wages fall below C , the stochastic nature of the matching process now
)
P
Fig. 1. Wage distribution across vacancies that are available for migrants (case (c)).
314 C. Helmenstein, Y. Yegorov / Journal of Economic Dynamics & Control 24 (2000) 307}323
implies that migration continues, though at an ever decreasing magnitude. We
believe this characteristic of the stochastic model to be a favorable approximation to reality. In addition, after a su$ciently long period of immigration case (c)
represents the only relevant labor market regime for the F economy irrespective
of the initial setting.
The ratio of the probabilities to "nd a job is given by
p
4 "e~j(CS~C))@w1(1,
(11)
p
)
that is, the probability for a single migrant to be successful is lower than that for
a chain migrant. The distribution
p(w)"e~j((w~w1)@w1), w5w
(12)
1
de"nes how the probability to obtain a job with a certain payment decreases
with an increasing wage. To illustrate, if j is equal to 1, the probability to get
a job with a wage that is double the minimal wage is p(w)"1/e.
With this labor market framework at hand, we are now ready to set up the
model. By de"nition, when the opportunity to migrate opens up "rst all
migrants are single migrants. We de"ne
m &flow of potential chain migrants in period k,
),k
m &flow of potential single migrants in period k,
(13)
4,k
n &flow of actual migrants who find a job in period k.
k
The pull mechanism, which determines the number of H individuals attracted by
a given migrant per period, is given by
m "bN .
(14)
),k
F
Here N is the stock of the foreign population in F which is equal to the sum of
F
net migrant in#ows in the K previous periods,
K
N "+ n.
F
k
k/1
The push mechanism is summarized by
(15)
m "aN ,
(16)
4,k
H
where a is the share of potential migrants among the individuals without
a network abroad. In a more general formulation, Eq. (16) should be replaced by
the expression
m "a(t)(N !N !bN ).
4,k
H
F
F
(17)
C. Helmenstein, Y. Yegorov / Journal of Economic Dynamics & Control 24 (2000) 307}323
315
Eq. (17) covers two aspects of the migration process. It links the dynamics of the
push factor a through time to the analysis and re#ects the decreasing size of the
home population due to migration in earlier periods, corrected by the number of
individuals who were induced to migrate because of the pull factor. Since Eq.
(17) asymptotically converges to Eq. (16) if a(t) is constant and migration #ows
are small compared to the native population (N @N ), for simplicity we will
F
H
employ formula (16) subsequently.9
According to Eqs. (10c) and (14), the in#ow of chain migrants who do not have
to return because their earnings exceed the survival minimum, equals p ) m .
) ),k
Analogously, p m represents the in#ow of single migrants who stay in the
4 4,k
destination country. This yields the net in#ow of actual migrants in period k as
n "p m #p m .
(18)
k
4 4,k
) ),k
Then for case (c) the following equation speci"es the dynamics of the migrant
in#ow:
n "aN e~j((C4@w1,k)~1)#bN e~j((C)@w1,k)~1).
(19)
F,k
k
H
The "rst term describes the net in#ow of single migrants, while the second term
represents chain migration. In a linear approximation of the labor demand
function, the wage decreases according to
w "w !hN .
(20)
1,k
0
F,k
With n de"ning the in#ow of migrants and N representing the stock of
k
F,k
the foreign population in F, the per period change in the stock of migrants is
given by
N
"N #n .
(21)
F,k`1
F,k
k`1
The share of single migrants in the current period in#ow, l , thus equals
k
aN e~jCs@w1,k
H
.
(22)
l"
k aN e~jC4@w1,k#bN e~jC)@w1,k
H
F,k
Eqs. (19)}(22) state the dynamics of migration in discrete time. The transformation of the discrete time model into a continuous time model is achieved by
9 International statistics document that the number of immigrants from a given country of origin
in a specixc country of destination is generally small relative to the total population of that country
of origin (OECD-SOPEMI, 1995). This does, of course, not imply that the total number of migrants,
cumulated over all countries of destination, from a given country of origin is small relative to the
total population of that country. Furthermore, sending countries are usually characterized by higher
natural population growth rates than receiving countries. The assumption of zero natural population growth in F and an o!setting growth in H is a straightforward way to capture this stylized fact.
316 C. Helmenstein, Y. Yegorov / Journal of Economic Dynamics & Control 24 (2000) 307}323
a standard limiting procedure. In continuous time the dynamic process of
migration is described by the following di!erential equation,
dN (t)
F "aN ej~jC4@(w0~hNF(t))#bN (t)ej~jC)@(w0~hNF(t)).
H
F
dt
(23)
Eq. (23) expresses the dynamics of the foreign population in the destination
country as a function of time and di!erent parameters. In order to solve
di!erential equation (23), we draw upon the transformation
1
w
N ! 0"! ,
F
y
h
(24)
with y as a new variable. The variables t and y can now be separated, and the
solution can be found as an inverse function analytically. The solution has the
form of an inde"nite integral,
P
y
dy
1
.
(25)
4y1@h#b((w /h)y2!y )ej~jC)y1@h
ej~jC
aN
y2
1
1
0
h@w0 H 1
Eq. (25) speci"es the relationship between time and the stock of immigrants.
From Eq. (23) it follows that dN (t)/dt'0, ∀t. As N (t)Pw /h, then
F
F
0
dN (t)/dtP0. Using Eq. (24), we conclude that yP#R, and the denominator
F
of Eq. (25) approaches zero from above. Thus the integrand is unboundedly
increasing but "nite for all "nite y and the integral takes all values strictly
between 0 and #R as h/w (y(#R. We conclude that yP#R as
0
tP#R. Hence from Eq. (24) we have N (t)Pw /h as tP#R, that is, the
F
0
stock of migrants asymptotically converges to w /h from below.10
0
t(y)"
4. Numerical experiments
Although the solution obtained from Eq. (25) is exact, it is necessary to resort
to numerical calculations to render a graphical presentation of the results
possible. In the subsequent chapter we therefore conduct various numerical
experiments to depict the e!ects of di!erent sets of parameters on the in#ow of
migrants, the equilibrium wage, and the share of single migrants in the in#ow.
10 By the very nature of the underlying assumptions, in particular that outmigration is small
relative to the size of the country of origin, the present model primarily applies to the transitional
dynamics of chain migration. Study of the long-run dynamics of chain migration would mandate to
endogenize variables (such as the push factor) that we assumed constant and would require
consideration of higher-order e!ects (such as the interaction between the two migration-prompting
mechanisms).
C. Helmenstein, Y. Yegorov / Journal of Economic Dynamics & Control 24 (2000) 307}323
317
The rather large number of free parameters (C , C , j, h, N , a, b) requires to
) 4
H
determine some of them in advance in order to isolate the e!ects of parameter
changes on the endogenous variables. If we normalize N to 1, then N denotes
H
F
the stock of migrants as a percentage of the total home population. C is also set
)
to 1 supposing that a chain migrant can initially (t"0) "nd a job with w'C
)
with probability 1, whereas later on he or she will face an increasing return
probability. With a growing number of immigrants, however, wages will be
negatively a!ected, and thus chain migrants might, along with single migrants,
face the necessity to return.
The following numerical calculations provide insights about the dynamics of
migration and the relative di$culty for an individual to "nd a job and thus to
migrate successfully under di!erent parameter constellations. The main results
are depicted in Figs. 2}6. For Figs. 2}4 the following set of parameters was
selected: C "1, C "1.5, j"1, N "1, a"0.05, b"0.5, h3M1, 1.5, 2N.
)
4
H
Fig. 2 reveals a dominant in#uence of the wage elasticity of labor demand,
1/h, on the dynamics of migration. If labor demand is inelastic, that is, for a
high value of h, it is more di$cult for a migrant to "nd a job with a
su$cient income to secure the baseline consumption level. Hence, fewer
migrants stay. Moreover, a higher number of returning potential migrants
implies a smaller number of nuclei for immigration chains. Along with the lower
pull capacity of chain migrants total migration is signi"cantly diminished. In
this model a relatively inelastic labor demand thus acts as an e!ective regulator
of immigration.
Fig. 2. Dynamics of migration for di!erent h. Upper curve corresponds to h"1, middle to h"1.5,
lower to h"2.
318 C. Helmenstein, Y. Yegorov / Journal of Economic Dynamics & Control 24 (2000) 307}323
Fig. 3. Wage dynamics for di!erent h. Upper curve for h"2, lower curve for h"1.
Fig. 3 shows the wage dynamics for the same parameter values as before. Since
the in#ow of immigrant labor is positively related to the elasticity of labor
demand for the reasons provided above, the pace of the wage level decline is
slower if labor demand is inelastic.
The dynamics of the share of single migrants in total immigration is revealed
by Fig. 4. For high h (h"2) the share of single migrants remains relatively high
while for low h (h"1) it reaches almost zero already after a few periods. In this
case immigrants without chains are largely excluded from entering the foreign
labor market. Because of this process the host country may su!er from foregone
welfare gains if a considerable number of single immigrants have higher skills
than the native-born population.11
Fig. 5 shows the in#uence of simultaneous changes of the parameters h and
j on the in#ow of migrants. Recall that j determines the wage structure across
vacancies, with a high j implying that a large share of the vacancies is low-paid.
It turns out that parameter h has a considerably stronger in#uence on migration
than parameter j , that is the elasticity of labor demand is of primary importance for the migration dynamics.
In Fig. 6 we compare the case of pure single migration with the case of single
migration combined with chain migration. The parameters used are
11 See Funkhouser and Trejo (1995) for an analysis of labor market skills of newly arriving male
immigrants to the USA.
C. Helmenstein, Y. Yegorov / Journal of Economic Dynamics & Control 24 (2000) 307}323
319
Fig. 4. Dynamics of the share of single migrants in total immigration for di!erent h: upper curve for
h"2, lower curve for h"1.
Fig. 5. Dynamics of migration for di!erent h and j. Upper curve for h"1, j"2, lower curve for
h"2, j"1.
C "1, C "1.5, a"0.1 and j"1. For the upper curve b"0.2, for the lower
)
4
curve b"0. This implies that the pull mechanism is not e!ective in the second
case, that is all migration is single migration. It appears that even for pure single
320 C. Helmenstein, Y. Yegorov / Journal of Economic Dynamics & Control 24 (2000) 307}323
Fig. 6. Comparison of two cases: Single migration versus single migration combined with chain
migration. Upper curve for b"0.2, lower curve for b"0.
migration the dynamics is non-linear because of the diminishing equilibrium
wage. The di!erence between both paths becomes more and more pronounced
over time.
5. Conclusions
The model devised in this paper investigates the dynamics of migration #ows
in the presence of chains. Within the framework of this type of model, the
dynamics of migration in the presence of chains depends on the wage elasticity
of the demand for labor, the intensities of the pull capacity and the push
mechanism, the wage structure across vacancies, and the di!erences in the
baseline consumption levels between single and chain migrants. The non-linear
interaction of these factors gives rise to the phenomenon of hysteresis, wherein
factors with early or temporary occurrence persistently a!ect the future dynamics of migration. Since immigrants from a country with an established presence
in the host country are more likely to be successful, initial conditions matter, so
that initial immigration patterns reinforce future immigration patterns. This
implies that the growth of the immigrant population from that country will
exceed that of source countries with a smaller established presence. When
designing an immigration policy, potential host countries may therefore want to
pay special attention to migrants from countries with high pull capacities such
as high fertility rates and strong inter-family links.
C. Helmenstein, Y. Yegorov / Journal of Economic Dynamics & Control 24 (2000) 307}323
321
We fully specify a stochastic two-country model with one sending and one
receiving country and derive a non-linear di!erential equation as its solution.
The results of our model are outstandingly sensitive to changes in the wage
elasticity of labor demand. The relative importance of the wage elasticity is
stronger in our model than in a model with a homogenous group of migrants.
The reason is that the wage elasticity of labor demand governs the number of
"rst period single migrants who are absorbed by the host country, thereby
determining the strength of the multiplier e!ect if we take into account that the
second and higher-order periods in#ows are nearly proportional to the number
of migrants in the "rst period. In particular if migrants are entitled to but minor
social security bene"ts of the receiving economy, the labor market itself may be
regarded as an e!ective regulator of migration, and a fortiori so in the case of
illegal immigration.
The existence of chains towards a certain destination country reinforces the
attraction of this country compared to other potential destination countries.
The resulting comparative advantage might overcompensate other, disadvantageous factors in that speci"c destination country. This e!ect might well
explain large-scale migration #ows between countries with comparatively small
wage di!erentials, and its role becomes ever more pronounced with an increasing di!erence between the living costs of single migrants and chain migrants.
The analysis presented in this paper has a direct bearing for several contentious policy questions. Since the baseline consumption expenses of chain migrants are lower than those of single migrants, this approach has a distinct
implication for the skill composition of the migrants who stay in the country of
destination. According to Stark (1991) low-skill workers return while high-skill
workers stay in consequence of productivity-adjusted wages after the reinstatement of informational symmetry through monitoring. Contrary to the prediction of this approach, despite of individual identi"cation and removal from the
pooling process, low-skill workers might be able to draw on their &chainmigration capital' as a substitute for wage averaging and thus stay. To avoid
such an outcome of a progressively deteriorating performance of succeeding
cohorts of migrants, the country of destination, if it cares about the skill level of
the migrants who are there to stay, might wish to spend resources in order to
preselect migrants according to their skills (Markusen, 1988; Straubhaar and
Zimmermann, 1993). As appropriate, the receiving economy might also consider
to compensate high-skill migrants for the lack of &chain-migration capital'.
Furthermore, the model suggests that #ows of single and chain migrants also
di!er in their gender composition. Given that wages for female labor are lower
than for male labor even if di!erences in education are controlled for and the
minimum subsistence levels are approximately equal for both sexes, the likelihood for a female immigrant to "nd a job with a wage that is su$cient to cover
her baseline consumption expenses is lower than that for a male immigrant with
the same quali"cation. It follows that the typical single migrant would be male
322 C. Helmenstein, Y. Yegorov / Journal of Economic Dynamics & Control 24 (2000) 307}323
while women would predominantly engage in chain migration to enhance their
chances to bene"t from the potentially rewarding opportunity to migrate. In this
way women might counteract the e!ect of gender discrimination on economic
performance. Contrary to before, where the existence of chains turned out to
trigger a potentially disadvantageous skill composition for the receiving economy, a high ratio of chain migrants relative to single migrants facilitates access
to female labor resources and should in this case augment the bene"ts from
migration that accrue to the country of destination.
Finally, our model predicts that the composition of the migrant in#ow di!ers
conditional on the phase of the business cycle. During recessions a smaller share
of high-paid vacancies should be available for migrants since the competition for
these vacancies among natives increases, which implies that migrants face
a steepening wage distribution across vacancies. The testable hypothesis here is
that during periods of an economic downturn the share of chain migrants in
total immigration is higher than during phases of prosperity. This e!ect should
occur independently of an intervention of immigration authorities since it stems
directly from the optimizing behavior of labor market participants.
Acknowledgements
The authors are indebted to Vivek H. Dehejia, Bernhard Felderer, Oded
Stark, Andreas WoK rgoK tter, and an anonymous referee for insightful comments.
Partial "nancial support from the Austrian Science Foundation under contract
number P10967-S07 is gratefully acknowledged. Yury Yegorov's research was
"nancially supported by the Institute for Advanced Studies, Vienna.
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