Vol. 93 (2008), No. 3, pp. 223–265
DOI 10.1007/s00712-007-0299-6
Printed in The Netherlands
Journal of Economics
Migration dynamics
M. Moretto1 and S. Vergalli2
1
Department of Economics, University of Padua, Padua, Italy
Department of Economics, University of Brescia, Brescia, Italy
2
Received 8 September 2006; Accepted 7 August 2007;
Published online 28 February 2008
Springer-Verlag 2008
Most migration flows include observable jumps, a phenomenon that is in line
with migration irreversibility. We present a real option model where the migration
choice depends on both the wage differential between the host country and the
country of origin, and on the probability of full integration into the host country.
The optimal migration decision of an individual consists of waiting to migrate in
a (coordinated) mass of individuals. The size of the migration flow depends on
the behavioural characteristics of the ethnic groups: the more ‘‘sociable’’ they are,
the larger the wave and the lower the wage differential required. The second part
of the paper is devoted to calibrating the model and simulating migration flows
into Italy over the last decade. Our calibration can replicate the migration jumps
in the short term. In particular, the calibrated model is able to project the induced
labour demand elasticity level of the host country and the behavioural rationale of
the migrants.
Keywords: migration; real option; labour market; network effect.
JEL Classifications: F22; J61; O15; R23.
Correspondence: Sergio Vergalli, Department of Economics, University of
Brescia, via S. Faustino 74/b, Brescia 25122, Italy (E-mail: vergalli@eco.unibs.it)
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1. Introduction
Much economic research deals with mass migration inflows, observing
that migration dynamics are in general characterised by gradual waves at
the beginning of their processes, followed by suddenly increasing
migration rates (so-called ‘‘migration jumps’’ or ‘‘mass immigration’’)
and then again by constant entry rates. Thus, Angrist and Kugler (2003),
using descriptive statistics from the Eurostat labour force surveys for 18
EU and other EEA countries, observe that the late 1980s and early
1990s witnessed a ‘‘marked upturn’’. Moretti (1999), studying Italian
migration in the United States and Canada, between 1876 and 1913,
highlights a sharp increase in the migration flow after 1900. A
remarkable surge in immigration was also observable in the United
States (Ottaviano and Peri 2005; Peri 2006; Massey 1995), in the UK
(Jackman and Savouri 1992), in France (Thierry and Rogers 2004) and
in Europe (Maillat 1986).1 What could be the causes of these particular
dynamics? We try to answer this question by searching for an
endogenous explanation of migration jumps. We offer a model that
merges the real option approach of investment decision applied to
migration choice and the works on migration networks into a single
framework.
In the economic literature, the main variable that affects the migration
decision is the wage differential between the host country and the country
of origin (Todaro 1969; Langley 1974; Hart 1975; Borjas 1990, 1994).
Nevertheless, even if the wage differential is important, it is not sufficient
to totally explain migrant behaviour. Evidence seems to stress the focal
role of community networks in the migrant’s choice (Boyd 1989; Bauer
and Zimmermann 1997; Winters et al. 2001; Bauer et al. 2002; Coniglio
2003; Munshi 2001, 2003; Heitmueller 2003). Moretti (1999), for
example, with an alternative model to Todaro’s, found evidence that both
the timing and the destination of migration could be explained by the
presence of social networks in the host country.
1 The same evidence is found in Friedberg (2001), Hatton and Williamson
(2006), Pedersen et al. (2004), and Hartog and Winkelmann (2003).
Migration dynamics
225
Furthermore, the fact that the migration decision is in many cases at
least partially irreversible, is a third element that may help to explain the
presence of jumps in the migration flows. In this respect, Burda (1995),
following a real option approach, implemented Sjaastad’s assumption
(1962) that describes migration choice in terms of investment. Burda
showed that individuals prefer to wait before migrating, even if the
present value of the wage differential is positive, because of the
uncertainty and the sunk costs associated with migration.2 Subsequently
Khwaja (2002) and Anam et al. (2004) developed Burda’s approach by
describing the role of uncertainty in the migration decision. Another work
that uses real option in migration is Feist (1998), in which the author
analyses the option value of the low-skilled workers to escape to the
unofficial sector if welfare benefits come too close to the net wage in the
official sector.
Assimilating the decision of each individual to migrate to a new country
as a decision on an irreversible investment, we investigated the role
played by social networks to help immigrants integrate into the host
country, where an immigrant is completely integrated when his/her
economic and social status is no different from the natives’ status in the
host country. We did this by considering the opportunity that each
immigrant becomes a member of a network (a community) of homogeneous individuals, located in the host country. The community helps the
immigrants to obtain a higher wage or improve their working conditions
if there are strong ties among the members (‘‘positive network
externalities’’). The larger the community, the higher the number of ties,
the higher the flow of information on job opportunities, and therefore the
higher the probability of integrating.
Nevertheless, if the number of immigrants continues to increase, labour
competition as well as higher alienation3 among immigrants inside the
community may reduce their net benefits (‘‘negative network externalities’’).
2 Investment is defined as the act of incurring an immediate cost in the
expectation of future payoff. However, when the immediate cost is sunk (at least
partially) and there is uncertainty over future rewards, the timing of the
investment decision becomes crucial (Dixit and Pindyck 1994, p. 3).
3 This is the case in which the members of the incumbent population
discontinue their attraction of immigrants (see Heitmueller 2003).
226
M. Moretto and S. Vergalli
The struggle between these two forces is shown by an inverted
U-shaped benefit function which follows directly by modelling the
probability of each immigrant being totally integrated into the host
country à la Bass (1969).4 The Bass model5 describes the ‘‘behavioural
rationale’’ of migration flows well by focusing on the role played by
two kinds of immigrants: the innovators or individualists, and the
imitators. The innovators are those individuals that decide to migrate
independently of the decisions of others. The imitators are those
individuals influenced by the number of previous migrants: they share
information and tend to establish a network. The weight of each
different type of immigrant influences the timing of migration and then
the size of the community.6
On the one hand, the stronger the ties among individuals, the larger
the wave. On the other hand, the presence of congestion in the
community and/or strong competition among workers in the host
country delays entry.7
Finally, we calibrate the model and simulate some migration flows into
Italy in the period 1994–2000 by using the official national statistic
4 From a theoretical point of view, an U-shaped benefit function can be
derived as combination of a ‘‘herd behaviour’’ and a network effect (see Bauer
et al. 2002) or as an application of the theory of clubs (see Vergalli 2008).
5 The Bass model was originally built to study the diffusion of new durable
products and largely adopted in the marketing literature.
6 The distinction between innovators and imitators is reminiscent of ‘‘the
upper class theory of fashion’’ (Veblen 1924) as modeled by Matsuyama (1992).
In his model individuals belong to one of two groups, respectively, with
conformist and with anti-conformist preferences, and the equilibrium shows a
chase-flight pattern, with anti-conformist playing the role of fashion leaders and
conformist playing the role of fashion followers.
7 Similar results are showed by Corneo and Jeanne (1999). They describe a
continuous-time economy populated by two types of individuals, ‘‘desirable
type’’ (natives) and ‘‘undesirable type’’ (tourists), in which an action is
interpreted as a choice of location. Their model describes the dynamics of social
location, defining the conditions for the take-off. In particular, they show that if
an arbitrarily small number of individuals from a socially desirable group
innovate, a large wave of imitators will follow even when the new behaviour is
more costly than the old one.
Migration dynamics
227
Table 1. Average migration growth rates
Country
Inflow
Growth rate
Period 1
Growth rate
Period 2
Growth rate
Period 3
China
Nigeria
Syria
Thailand
Italy
Albania
China
Philippines
Romania
Netherlands Angola
China
Sudan
Suriname
Sweden
Chile
Ethiopia
Ireland
Somalia
1.28
1.17
1.09
0.76
0.11
0.10
0.06
0.19
1.83
1.33
1.67
1.41
0.88
0.81
1.83
0.25
1995–1999
1995–1999
1995–1998
1995–1999
1994–1996
1994–1996
1994–1996
1994–1996
1996–1999
1996–2000
1996–1997
1996–2000
1981–1985
1981–1985
1981–1988
1981–1990
3.21
2.09
1.52
1.11
0.89
0.63
0.24
0.32
4.48
3.24
2.43
2.70
2.71
4.01
5.20
3.84
2000–2002
2000–2003
1999–2003
2000–2003
1997
1997
1997
1997–2000
2000–2002
2001–2002
1998–2001
2001–2002
1986–1989
1986–1992
1989–1990
1991–1994
2.87
2003
0.12
0.07
@0.03
0.16
1998–2003
1998–2003
1998–2003
2001–2003
1.86
2002
0.46
0.93
1.47
1.43
1990–2001
1993–2001
1991–2001
1995–2001
UK
1.04
1.12
2.98
1.25
1992–1998
1992–1998
1992–1999
1992–1995
1.72
2.04
19.98
3.57
1999–2004
1999–2003 1.76
2000–2002 9.71
1996–2004
2004
2003–2004
Germany
Ghana
Pakistan
Somalia
Turkey
database (ISTAT)8. The results fit the theoretical approach and replicate
the observable migration jumps.
1.1 Some supporting evidence
Table 1 shows the average growth rates of certain immigration inflow into
five European countries (Germany, Italy, The Netherlands, Sweden and
the UK) for different periods. The data for Germany were taken from the
Statistisches Bundesamt (Federal Statistical Office); for the Netherlands
8 ISTAT (Istituto Nazionale di Statistica) is the Official National Statistical
Institute and its database is based on data from the Ministry of the Interior,
www.istat.it.
228
M. Moretto and S. Vergalli
from the Statistics Netherlands (Centraal Bureau voor de Statistiek); for
Sweden from the Statistics Sweden (Statistiska Centralbyrån) and for the
UK from the Home Office, the British government. For Italy, the data
were taken from the ISTAT database for the years between 1994 and 2000
and from the Caritas report9 for the period between 2001 and 2003. Both
the ISTAT dataset and the Caritas data were supplied by the Ministry of
Internal Affairs, and were up-dated and revised by the official statistical
institute until 200010 (therefore, the two datasets overlap in the period
between 1994 and 2000).
We can see that the migration process does not proceed smoothly, but it
has sudden increases in the inflow growth rates. In some periods the
inflow growth rate doubles (like in Germany and Netherlands),
sometimes triples (like in Netherlands and UK) and sometimes increases
even more (Italy and Sweden or for some ethnic groups in the other
countries).
Looking at the immigration reforms11 (see Boeri and Bucker 2005) in
the countries and in the periods considered, we can see that they are not
homogeneous with respect to the generosity of the welfare system for the
immigrants. In two cases (UK and Netherlands) the reforms tightened the
condition for immigration, in one case (Sweden) the policy did not
substantially change over the years and in the last two cases (Germany
and Italy), favoured immigration. Furthermore in Germany, the reform
did not directly affect immigration. Because of the heterogeneity of
reforms as a homogeneous phenomenon it seems that, at first glance, the
9 Caritas Internationalis, however, ‘‘ is a confederation of 162 Catholic relief,
development and social service organisations working in over 200 countries and
territories’’ (Caritas 2003). The edition of the Caritas Statistic Immigration Report
is part of the project ‘‘The image of Migrants in Italy, Through Media, Civil
Society and the Labour Market’’, developed in the framework of the EU/EQUAL
Initiative, managed by the Italian Ministry of Welfare. The project has been
promoted by the International Organisation for Migration, Caritas of Rome and
the Archive of Immigration and involves other 19 partners, including both Italian
and immigrant associations. The first Caritas Report (‘‘Dossier Statistico
Immigrazione’’) was produced by the Caritas Organisation since 1991 and it
has now become an annual report on immigration.
10 For the lack of official revised data after year 2000 at the moment of our
submission, we have used Caritas dataset for the years between 2001 and 2003.
11 See Fondazione Rodolfo Benedetti Documentation Centre, http://
www.frdb.org.
Migration dynamics
229
140000
2,2
120000
100000
1,7
80000
60000
1,2
40000
20000
0
0,7
1994 1995 1996 1997 1998 1999 2000 2001 2002 2003
-20000
0,2
-40000
-60000
-0,3
-80000
Albania
Romania
Philippines growth
China
Albania growth
Romania growth
Philippines
China growth
Fig. 1. Migration jumps
immigration reforms were not the only variable affecting migration
choice. For this we looked for an additional endogenous explanation that
could explain migration jumps.
Finally, since we focus our analysis on Italy, in Fig. 1 we have shown
the four main foreign flows and their growth rates in Italy between 1994
and 2003: Albanians, Chinese, Filipinos and Romanians.12 The migration
flows have been depleted from the two important regularisations for
12 Description of the data:
– According to ISTAT (Istituto Nazionale di Statistica), ‘‘foreigners’’ in Italy are
persons with foreign citizenship. A child born to parents who are both foreign
citizens is considered to be a foreigner as well. A child born to an Italian and a
foreign parent is considered to be an Italian citizen. Once a foreigner acquires
Italian citizenship, they are not reported in official statistics as foreigners any
more.
– Data are based on the number of valid residence permits issued to foreigners as
of December 31 of each year. Children under 18 years old who are registered
on their parents’ permits are not counted.
230
M. Moretto and S. Vergalli
19000
18000
17000
Wages
16000
15000
14000
13000
12000
11000
10000
1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003
Years
Albania
China
Philippines
Romania
Italy
Fig. 2. Wage levels
illegal immigrants introduced in Italy in 1996 and 1998, and registrated
by the ISTAT database in subsequently years.13 For the sake of
completeness, Fig. 2 also shows the wage differentials in the same
period. These were obtained from the World Bank International
Comparison Programme database and were deflated using the Bank of
Italy14 deflator.
Figure 1 shows that, for all nationalities, the migration process was not
smooth. We observed ‘‘some substantial high increases in the inflow
growth rates’’ that we defined as ‘‘migration jumps’’. In particular, we can
see an important jump in 1997 after a certain number of years
characterised by low waves, as if a mass of individuals was waiting for
something to happen before deciding to migrate. Moreover, all nationalities showed heterogeneities in their behaviour after 1997: the Chineses
13 The expectation of regularisation programs foreseen by potential immigrants, can be interpreted as an endogenous cause for migration. Nevertheless the
political programs are not common knowledge. In fact in the period after 1991 in
Italy a quota system was imposed on the immigration flows and, therefore was
both extraordinary and had unpredictable results.
14 http://www.bancaditalia.it.
Migration dynamics
231
and the Filipinos had declining flows, whereas the Romanians had a
second jump in 2000.15
Another important aspect was that the wage differential did not seem to
be the main variable driving migration flows. In all cases (except,
partially, for Romania and Albania), the jumps did not occur together with
a steady rise in relative wage levels, as stressed by Moretti (1999). Then,
if policy choices do not completely explain migration dynamics, why do
potential migrants wait before taking their decision to migrate? What are
they waiting for? And why do they move on mass? We try to answer all
these questions by examining whether the migration investment characteristics and the role of ethnic groups, can explain the migration jumps
observed in Fig. 1. Although the phenomenon may be consistent with
various explanations, simple arguments have to do with logistics: it takes
time to decide and coordinate migration. This is consistent with the
progressive acceleration in migration flows: migration delays arise
because it is worth waiting to decide when certain fundamental
uncertainties are resolved over time and the decision is mostly
irreversible. A logistic curve also shows the fact that learning about the
host country’s labour market takes place sequentially and strongly
depends on the role of an ethnic network in the host country.
This paper is organised as follows. Section 3 presents the model and the
basic assumptions. Section 4 develops the theoretical framework that
combines real option theory and the network effects, namely the optimal
migration strategy in the presence of positive and negative externalities.
Section 5 calibrates the model and Sect. 6 makes some simulations which
confirm the theoretical results. Finally, Sect. 7 summarises the conclusions.
2. The model
We assume that an individual that move to another country is completely
integrated when his economic and social status is no different from the
native one. Nevertheless, the timing of the migrant’s integration suffers
from a phenomenon of attrition because of the lack of information about
15 The same phenomenon is also showed for five European countries in the
updated (2007) version of Vergalli (2008) at the link http://www.
sergiovergalli.it.
232
M. Moretto and S. Vergalli
the host country and its labour market. We also assume that in the host
country, a homogeneous group of people (a community/a network) exists
that can help each immigrant to increase integration. The larger the
community, the closer the ties among its members and then the higher the
integration probability. The number of ties also depends on idiosyncratic
characteristics of the immigrants, that we call ‘‘behavioural rationale’’
using the Bass terminology. That is, the more ‘‘sociable’’ an individual or
a group of individuals, the stronger and more ties they have.
2.1 The basic assumptions
Our main assumptions are the following:
(0) There exist two countries: the country of origin where each potential
migrant takes decisions and the host country.
(1) At any time t a risk-neutral16 individual is free to decide to migrate
to the host country discounting future benefits (the wage differential
between the host country and the country of origin) at the constant
interest rate q.
(2) When the migrant arrives in the host country, he/she receives only a
percentage n < 1 of the host wage as first entry wage.17 So defining
woi as the wage of her country of origin (where i is the country), we
are able to write the wage differential as a percentage of the wage of
the host country:
nw woi ½n woi =ww ¼ /0i w:
(3) In the host country there is a community of ethnically homogeneous
individuals that helps each member to integrate with the host labour
market (or to obtain a legal job if she is working on the illegal
market). When the immigrant is completely integrated, he/she gets
the difference between the legal host current market wage w and the
wage of the country of origin woi , i.e.,
w woi ½1 woi =ww ¼ /i w:
16 See Burda (1995), Khwaja (2002), and Locher (2002) for the use of this
assumption.
17 Empirical evidence shows that this is true whether the migrant finds a legal
or an illegal job (see Chiswick 1978; Borjas 1990; Massey 1987).
Migration dynamics
233
(4) For the sake of simplicity, we assume that the country-specific
0
0
percentages /i and /i (/i B /i) are constant over time.18
(5) Each individual enters a new country undertaking a single
irreversible investment which requires an initial sunk cost K.
(6) The size of the immigrant dn is infinitesimally small compared to
the total number of previous immigrants n.
(7) Finally, the inverse labour demand for immigrants in the host
country at time t is an isoelastic function of the total number of
previous immigrants n(t):
wðtÞ ¼ hðt Þnðt Þf ;
ð1Þ
where h is a labour-demand-specific shock, f < 0 is the elasticity
and w is the average wage of the host country.19
We introduce uncertainty into the model by assuming that:
(8) The labour-demand-specific shock h follows a Brownian motion:
dhðtÞ ¼ ahðtÞdt þ rhðtÞdW ðtÞ
ð2Þ
component
with h (t0) = h and a, r > 0 are constant over time. Thep
ffiffiffiffi
dW(t) is a Wiener disturbance defined as dW ðtÞ ¼ eðtÞ dt ; where
e(t) * N(0,1) is a white noise stochastic process (Cox and Miller
1965).
(9) The time taken to become perfectly integrated, say s, is stochastic
and depends on a distribution of probability defined as:
1 Fs ðtÞ Prðs [ tpt [ 0Þ
ð3Þ
and its corresponding hazard rate is:20
18 We calibrate them as the loss in Purchasing Power Parity with respect to
the initial year of our dataset. See Sect. 5 below.
19 There are two implicit assumptions beyond (1). Firstly, that all incumbent
immigrants have a job and that all future immigrants seek a job. Secondly, that w
refers to labour markets that are occupied mainly by immigrants so that we can
ignore the role of native employees (Heitmueller 2003).
20 ps(t) is the migrant’s conditioned probability of obtaining a better job at
time t ? dt, if he/she has worked at a low wage until t.
234
M. Moretto and S. Vergalli
ps ðt Þ
fs ðt Þ
;
1 Fs ðt Þ
ð4Þ
where fs(t) is the density function or the likelihood of being perfectly
integrated at t.
Each immigrant decides when to enter a new country maximising his/
her net benefit value defined as the expected discounted stream of wage
differentials over the planning horizon (taken infinite for simplicity)
minus the entry cost K.
By (1) and Assumptions 3–8 the benefits from being completely
integrated at s are given by:21
8 1
9
<Z
=
BðnðsÞ; hðsÞÞ ¼ Es
eqðtsÞ /wðtÞdt
:
;
s
8 1
9
<Z
=
Es
eqðtsÞ /hðt Þnðt Þf dt ;
ð5Þ
:
;
s
where B(•) accounts for the future evolution of the number of migrants
n(t), t C s. The expectation operator Es(•) is taken with respect to the
random variables s and h (t) [and then n(t)]. Next, taking into account the
benefits the immigrant may gain before integrating, we end up with a total
benefit value at the migration time zero as:
8 s
9
<Z
=
eqt /0 wðtÞdt þ eqs BðnðsÞ; hðsÞÞ ;
ð6Þ
V ðn; hÞ ¼ E0
:
;
0
where n(0) = n; h (0) = h. By using an indicator function J[s>t] that
assumes the value one or zero depending on whether the argument is true
or false, we can write (6) as:
21 If all immigrants face the same instantaneous probability of death k dt, we
can define q ¼ b
q þ k; where b
q is the market rate (Dixit and Pindyck 1994,
p. 200).
Migration dynamics
V ðn; hÞ ¼ E0
8 1
<Z
:
0
9
=
eqt J½s [ t /0 hðtÞnðt Þf dt þ eqs BðnðsÞ; hðsÞÞ :
;
Since E[J[s>t]] = 1 @ Fs(t), we can plug (3) in (7) to obtain:
8 1
<Z
V ðn; hÞ ¼ E0
eqt ½1 Fs ðtÞ/0 hðtÞnðt Þf dt
:
0
9
Z1
=
þ eqt fs ðt ÞBðnðt Þ; hðtÞÞdt ;
;
235
ð7Þ
ð8Þ
0
where the expectation is now taken only with respect to h(t) (and n(t)).
If the benefit value function V(•) is known, the optimal migration policy
implies that the return from migration must be at least equal to cost K at
the entry point. In other words, we need to find the curve h*(n(t)) (i.e., the
value of the labour demand shock) at which the n(t)th migrant is
indifferent between immediate entry or waiting:22
V ½nðt Þ; h ððnðt ÞÞ K ¼ 0:
ð9Þ
This is what we shall do in the next section.
2.2 The entry time s and the network effect
Before turning to the migrant’s optimal policy, we need to model the
probability of integrating (3). We have defined two different groups of
migrants:
– Innovators: those individuals who decide to migrate independently of
the decisions of other individuals in a social system. They are the
pioneers or the individualists: their decision depends on their intrinsic
characteristics.
– Imitators: those individuals influenced in the timing of migration by
the number of previous migrants. In particular, we mean the individuals
who follow the innovators. Their particular behavioural characteristic is
22 This condition is familiar in the real option theory with the name of
matching value condition (see Dixit and Pindyck 1994).
236
M. Moretto and S. Vergalli
their sociality: they have strong ties among themselves and tend to
establish a network.23
Following Bass (1969), the probability that perfect integration occurs at
t, given that no integration has yet occurred, is set as a linear function of
the size of the community, i.e.,
ps ðt Þ ¼ a þ bFs ðtÞ;
ð10Þ
where Fs(t) stands for the number of immigrants already entered; a is the
coefficient of innovation, the influence on entry regardless of the number
of previous members; b is the coefficient of imitation, the impact of
previous members on the probability of entry at time t. By using algebraic
operations (Bass 1969, p. 217), we get:
Fs ðtÞ ¼
m nðtÞ
;
m
ð11Þ
and the fraction of the total immigrants integrating at time t is:
fs ðt Þ ¼ a þ
ðb aÞ
b
nðt Þ 2 n2 ðtÞ;
m
m
ð12Þ
where m is the (fixed) total number of immigrants over the planning
horizon, which represents the critical ‘‘saturation’’ dimension of the
community.24 Finally, we get limn!m fs ðt Þ ¼ 0 and fs(t) is concave iff
b > 0.
23 A recent economic approach calls a similar phenomenon herd behaviour,
i.e., ‘‘I will go to where I have observed others go’’ (Bauer et al. 2002).
24 By observing that the cumulative function is a logistic curve, m
corresponds to the carrying capacity defined as ‘‘the number of individuals an
environment can support without significant negative impacts to the given
organism and its environment’’ (Vandermeer and Goldberg 2004). It corresponds
to the congestion level of the community. Since fs(t) is the likelihood of being
perfectly integrated at s, the total number of immigrants in (0, s) is (See Bass
1969, p. 217):
mFs ðtÞ ¼ m
Zs
0
fs ðtÞdt:
Migration dynamics
237
By plugging (12) and (11) into (8), we simplify (8) as:
V ðn; hÞ
8 1
9
"
#
2
<Z
=
ba
b
m nðtÞ 0 ½a þ m nðtÞ m2 nðtÞ
/ þ
/ hðt Þdt :
¼ E0
eqt nðtÞf
:
;
m
qa
0
ð13Þ
2.3 The benefit function
Network migration theory suggests that benefit is a positive function in
both wages and network size (Massey et al. 1993). However, by (13),
suppressing time for the sake of simplicity, we can write the benefit
function per unit of time as:
pðn; hÞ uðnÞh
ð14Þ
i
b 2
and BassðnÞ a þ ba
where uðnÞ nf mn
m / þ
m n m2 n :
h
0
BassðnÞ
qa /
Apart from shock h, each immigrant shares the same ‘‘utility’’ u(n). The
overall shape of u(n) is ambiguous: it depends heavily on the struggle
between the competitive effect (i.e., more immigrants reduce wages
depending on the magnitude of the elasticity f) and the network effect
[i.e., individuals gain ‘‘utility’’ by increasing the number of fellow
countrymen which in turn increases the probability of integration via the
Bass function Bass(n)]. According to the relative magnitude of these two
effects, we can observe three shapes of u(n) as in Fig. 3.
Let us analyse Fig. 3 from quadrant I to quadrant III for decreasing
levels of elasticity, ceteris paribus:
quadrant I: This is the general case for a not very low level of elasticity
f. A relative minimum in n@ and a relative maximum in n exist that divide
the function into three intervals:
(1) In the interval n [ (0, n@), the competition effect prevails over the
network effect: a new entry reduces the benefit more than the gain
caused by cooperation among members of the community.
(2) In the interval n 2 ðn}; nÞ the network effect prevails: the benefit
increases with n until the dimension of the network reaches level n:
238
M. Moretto and S. Vergalli
u(n)
I
II
u(n)=u(n’)
u(n)
u(n”)
u(n”)
n’ n”
n
n
n”
n
III
u(n)
n
n
Fig. 3. Peculiar shape of u(n)
(3) In the interval n 2 ðn; mÞ the competition effect prevails: the benefit
decreases with n until the dimension of the community hits the
saturation level m. Competition is coupled with a phenomenon of
congestion as n moves toward m.
As shown in Fig. 3, within the interval (0, n@) a level n0 exists such that
uðn0 Þ ¼ uðnÞ: Further, for n > n0 each immigrant earns benefit lower than
u(n0 ) until the community size reaches the relative maximum n: Then each
immigrant receives a lower benefit if he/she enters with a community
population n 2 ðn0 ; nÞ:
Since the critical level of n0 depends on the relative influence of the
competition and network effects, for different levels of elasticity we can
observe the following:
Migration dynamics
239
quadrant II: Even if for low value of n competition prevails over the
network effect, the latter dominates any other effect as n increases. This
implies that for each individual, it is expedient to wait for the maximum
benefit uðnÞ before entering.
quadrant III: Since f ? 0 implies that n0 ? 0, for very low levels of
elasticity, the benefit function simply assumes an inverse U-shape.
3. Migration dynamics
Applying Itô’s Lemma to (14) and substituting (2) to eliminate dh, we get
an expression for the rate of change of p in terms of the shock and the
network size:
dp ¼ lðnÞpdn þ apdt þ rpdw;
0
with
p0 uðn0 Þh0 ¼ p:
ð15Þ
In (15) the first term l(n) : u (n)/u(n) shows the direct effect of
migration flows. Migration influences the level of benefits through its
effect on the labour market equilibrium depending on the dimension of
the community. In particular, given any value of the shock h, more
immigrants imply a higher or lower equilibrium level of benefits
depending on the presence of positive l(n) > 0 or negative l(n) < 0
network externalities, respectively.
3.1 Optimal migration policy for n [ n (and <n0 )
If the initial size of the community is n n (or n B n0 ), we can expect
migration to work in the following way. For any fixed n, the benefits per
unit of time move according to the above stochastic process with
l(n)dn = 0. If they climb to a certain level p = u(n)h (n), migration
becomes feasible, the network size increases from n to n ? dn and the
benefits go downward along the function u(n). Benefits will then
continue to move stochastically without the term l(n)dn, until another
entry episode occurs.25 This can be summarised by the following
proposition:
25 In technical terms, the threshold p becomes an upper reflecting barrier on
the benefit process (see Harrison 1985).
240
M. Moretto and S. Vergalli
Proposition 1: If n nðor n n0 Þ; the optimal migration policy is
described by the following upward-sloping curve (Fig. 4):
b1
K
b1
; with
ðq aÞ
[ 1;
ð16Þ
h ðnÞ
uðnÞ
b1 1
b1 1
where q > a and b1 > 1 is the positive root of the auxiliary quadratic
equation WðbÞ ¼ 12 r2 bðb 1Þ þ ab q ¼ 0:
Proof : See Leahy (1993) and the Appendix.
h
In the area above the curve, it is optimal to migrate: a wave of migrants
will enter in a lump to move the benefit level immediately to the threshold
curve. In the region below the curve the optimal policy is inaction. The
individual waits until the stochastic process h moves it vertically to h*(n)
and then again a flow of migrants will jump into the host country just
enough not to cross the threshold.
The ‘‘utility’’ threshold that triggers migration by individual immigrants is identical to that of the individual that correctly anticipates the
other immigrants’ strategies. This property, discovered by Leahy
(1993), has an important operative implication: the optimal migration
policy of each individual need not take account of the effect of rivals’
entry. She/he can behave competitively as if he/she is the last to
enter.26
26 In other words, when an individual decides to enter, by pretending to be the
last to migrate, he/she is ignoring two things: (1) he/she is thinking that his/her
benefit flow is given by u(n)h, with n held fixed forever. Thus, as u0 (n) <0, he/she
is ignoring that future entry by other members, in response to a higher value of h,
will reduce ‘‘utility’’. All other things being equal, this would make entry more
attractive for the migrant that behaves myopically. (2) He/she is unaware that the
prospect of future entry by competitors reduces the option value of waiting. That
is, pretending to be the last to migrate, the individual also believes he/she still has
a valuable option of waiting before making an irreversible decision. All things
being equal, this makes the decision to enter less attractive. The two effects offset
each other, allowing the migrant to act as in isolation (see Dixit and Pindyck
1994, p. 291).
Migration dynamics
241
Triggers, −0,02
6,5
Albania
China
Philippines
Romania
6
5,5
Theta (n)
5
4,5
4
3,5
3
2,5
2
1
15 29 43 57 71 85 99 113 127 141 155 169 183 197 211 225 239 253 267 281 295 309 323 337 351 365 379 393
n
Fig. 4. Optimal triggers level for f2 = @0.02
3.2 Optimal migration policy for n0 \n\n
For n 2 ðn0 ; nÞ; the network benefit prevails over the competitive effect
and then we expect the timing of an individual’s entry is influenced by the
entry decisions of others.
Intuition suggests that Leahy’s result cannot be extended to cover this
case. Since there are positive externalities, the higher the number of
members in the community the greater the advantage in terms of benefit
flow. This is evident in the case of an U-shaped benefit function (quadrant
III in Fig. 3) but it also works for the general case as uðn0 Þ ¼ uðnÞ and the
‘‘utility’’ is lower in within (quadrant I in Fig. 3). Therefore, although
entering may be profitable, it is more expensive to do so first than to enter
later on, when others have already done so. This makes the trigger
p = u(n)h (n) no longer optimal: each migrant can do better by delaying
entry.27
However, as all individuals are subject to the same labour demand
stochastic shock, two equilibrium patterns are possible: either the
community remains locked-in at the initial size n0 \n\n; sustained by
27 The decision problem involved here resembles one of war of attrition
where each agent waits for rivals to concede (Moretto 2000).
242
M. Moretto and S. Vergalli
self-fulfilling pessimistic expectations (infinite delay), or a mass of
individuals simultaneously rushes to enter. Excluding the former,28 we
can expect entry to work in the following way: for a fixed size of the
network, p moves according to the process (15) with l(n)dn = 0. If
benefits climb to p = u(n)h (n), it will trigger an entry of discrete size
n n that raises the dimension of the community instantaneously by a
jump. The exact form of the trigger h is given in the following
proposition.
Proposition 2: If 0 n0 \n\n; the optimal migration policy for a
mass of individuals n n is described by the following flat curve
(Fig. 4):
h ðnÞ ¼ h ðn0 Þ ¼ h ðnÞ
b1
K
ðq aÞ
;
uðnÞ
b1 1
Proof : See Moretto (2003, 2007) and the Appendix.
ð17Þ
h
Thus starting at n, if the initial shock is below the known trigger h ðnÞ;
all the migrants wait until h rises to this level, and then coordinate their
entry to bring the size to the optimal level n. Working back towards n0 , it
is verified for every n, as long as h (n0 ) is equal to h ðnÞ: In fact, if it were
h ðn0 Þ [ h ðnÞ; it could be convenient to delay entry until h ðnÞ; because
of a higher obtainable benefit. Once the optimal size is reached and to the
right of
n; further decision to enter proceeds as explained in the previous
section without externalities. Intuitively, starting at any n0 \n\n;
Proposition 2 locates the optimal entry threshold so as to maximise the
total benefits of the incremental number of members that enter ðn nÞ:
The shock value h ðnÞ that triggers this individual’s competitive run29 is
the same threshold that justifies further marginal entry under decreasing
benefits.
28 We exclude the former by using the subgame-perfectness arguments (see
Moretto 2003, 2007).
29 The term competitive run refers to Bartolini’s definition (1993).
Migration dynamics
243
4. Calibration
To simulate the optimal migration policy we need values for the
variables and parameters in Eq. (14). We could then calculate (16) and
(17) and then solve for n . To perform this calibration we used the
migration flows for Albanians, Filipinos, Chinese and Romanians and
the wage levels (deflated using the Bank of Italy deflator) obtained from
the ISTAT database.30 As we show below, determining values for most
of the model’s inputs is reasonably straightforward. Estimating the
coefficients of the labour demand’s stochastic process h and of the Bass
probability of integration a and b is more complex as will be discussed
below.
4.1 Basic inputs
The parameters to be calibrated are listed in Table 2: for the discount rate
we have used a basic level q2 = 0.03 (Nordhaus 1996) and a higher level
q1 = 0.05.31 We also add a mortality rate k = 0.001 calculated by the
Istituto Superiore della Sanità32 on ISTAT data.
According to Assumptions 2 and 3, the differential wage is assumed to
be a constant percentage of the wage of the host country and varies
whether the immigrant is completely integrated in the host country or not:
0
/i and /i, respectively. The percentage for complete integration /i has
been calibrated considering the GDP per capita based on the Purchasing
Power Parity of the initial year 1993, as listed in the International
Comparison Programme database of the World Bank. If the immigrant is
not integrated, he/she earns only a fraction n of the wage. We have
0
calibrated n and then the corresponding percentage /i referring to the
30 For the robustness of our analysis we have calibrated our parameters by
using only the official data for the years 1994–2000.
31 Policy uncertainty regulating immigrants’ flows can also explain the choice
of two different discount rates. In particular, policy uncertainty acts as a scale
factor on the optimal threshold and it can be modelised as a poisson process. See
Vergalli (2007) and Rodick (1991).
32 http://www.iss.it.
244
M. Moretto and S. Vergalli
Table 2. Parameters
Parameter
Description
Discount rate
Elasticity
Labour
demand
elasticity
Symbol
Source
q1 = 0.03
q1 = 0.05
Nordhaus (1996)
f1 = @0.2 Borjas (1990), De New and
f2 = @0.02 Zimmermann (1997),
Borjas (1994), and
Bauer (1997)
Wage differential /i = (1 @ w/w) /i
World Bank
Wage differential /i = (n @ w/w) /i
World Bank
Entry salary
Average level
n
Chiswick (1978)
Borjas (1990)
Massey (1987)
works of Massey (1987), Borjas (1990) and Chiswick (1978).33 The
0
resulting /i and /i are shown in Table 3.
4.2 Demand volatility
The calibration of Italian immigrants’ labour demand elasticity has two
problems: the lack of studies on Italy’s demand function for immigrants34
and the lack of work that isolates the effect of immigration inflow for
foreigners and not only all (or native) workers. This problem can be
overcome if we look at the EU and US work on the level of labour demand
elasticity and standard deviation of the stochastic shock h. For European
labour market (especially for Germany and France) some work shows
elasticity levels between @0.021 (Bauer 1997) and @0.24 (De New and
Zimmermann 1994), even if, also in these cases, some identification
33 Massey estimates that the illegal wage is 63% of the legal wage. Borjas and
Chiswick show that the entry wage for each immigrant is 79 or 85% of the native
one, respectively.
34 In this respect: Gavosto et al. (1999) show a positive elasticity for natives
(+0.01) that the author justifies as a short-term effect; Venturini (1999) finds a
long term elasticity in the non-regular labour market between @0.01 and @0.02;
Venturini (1997) calculates an elasticity level of @0.3 and @0.5 among all the
workers.
Migration dynamics
245
0
Table 3. /i and /i calibration
0
Country
/i
/i
Albania
China
Philippines
Romania
0.639
0.649
0.593
0.510
29/33
8/9
5/6
3/4
problems about immigrants elasticity, remain.35 For the US there are many
papers that try to estimate the peculiar effect of entering immigrants on
labour wages. All the US elasticity levels seem to converge towards two
representative values:36 @0.2 (Borjas 1990) @0.02 (Borjas 1994). On this
basis we use the following elasticities f1 = @0.2 and f2 = @0.02, that
seem to be representative both for the US and for the EU labour market. To
calculate an estimate of the variance of h, we used the boot-strap method,37
35 Furthermore, Peri (2005) shows an elasticity level of @0.4 among all
European workers; De New and Zimmermann (1994) find an elasticity level for
blue collar foreigners in Germany equal to @0.24; again in Germany, Bauer
(1997) has a value of @0.021; Hunt (1992) about elasticity with respect foreigners
share in occupation in France, shows a level between @0.139 and @0.08; Gang
and Rivera-Batiz (1994) have a level between @0.01 and @0.11, while Garson
et al. (1987) show a level included between @0.01 and @0.04 for the French
labour market.
36 Borjas (1994) reviewing the literature argues that the value of the elasticity
should be between @0.01 and @0.06. Dos Santos (2000) affirms that ‘‘from an
empirical point of view, many studies attempt to estimate the impact of immigration
on wages. The elasticity of wages with respect to the number of immigrants is
generally found to be between @0.01 and @0.02@. Garson (1987), using 1985 data,
coming from ISEE, finds a level of elasticity between @0.01 and @0.04. Borjas
(1990), using data from the US census of 1990, shows a level around @0.2 and in a
recent paper (Borjas 2003) he obtains an elasticity around @0.33. Antonji and Card
(1991), using data of the US census 1970–1980, finds a level of @0.3.
37 The bootstrap method is a computer-based method for assigning measures
of accuracy to sample estimates (Efron and Tibshirani 1994). This technique
allows estimation of the sample distribution of almost any statistic using simple
methods (Varian 2005), like resampling with replacement from the original
sample, most often with the purpose of deriving estimates of standard errors and
confidence intervals of a population parameter like a mean, median, proportion,
odds ratio, correlation coefficient or regression coefficient. In our case, by using
known parameters (elasticity level, wage level, number of immigrants),
generating errors 1000 times, we have obtained two unknown levels of variance
for each elaticity value.
246
M. Moretto and S. Vergalli
Table 4. The Bass parameters for 1996, 1997 and 1998
r1
r2
b
a
m
Albania
China
Philippines
Romania
0.063
0.055
0.973
0.110
0.274
0.061
0.055
0.850
0.117
0.138
0.057
0.056
0.648
0.141
0.256
0.047
0.054
0.828
0.123
0.115
obtaining two levels of variance r1 and r2 for each flow, corresponding,
respectively, to f1 = @0.2 and f1 = @0.02. These values are reported in
Table 4.
4.3 Bass parameters
Finally for the parameters of the Bass model (i.e., a, b, m), we employ the
recursive method proposed by Bass (1969, p. 224) using the years 1996,
1997 and 1998 as initial conditions.38 The results are described in
Table 4. Simple observation shows that in all cases, the coefficient b is
greater than a, which guarantees the concavity of the Bass function
Bass(n).
5. Results
To compare different migration inflows, we simulate the optimal trigger
levels Eqs. (16) and (17), for the four migration waves, in the case of
elasticity levels @0.02 and @0.2. Because of the difficulty of perfectly
quantifying the migration costs, we normalise K to the same arbitrary
38 By Considering (12) as the basic equation, we know that:
nðtÞ ¼ mf ðtÞ ¼ am þ ðb aÞnðtÞ mb n2 ðtÞ: In estimating the parameters from
discrete time series data we use the following analogue: n(t) = j ? vn(t
P@ 1)
? zn2(t @ 1), for t = 2, 3, . . ., where n(t) immigrants at t, and nðt 1Þ ¼ nðtÞ
cumulative immigrants through period t @ 1. Since j estimates am, v estimates
(b @ a), and z estimates @(b/m): @mz = b, j/m =
a. Thenffi (b @ a) = @mz @ j/
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
m = b, and zm2 ? vm ? j = 0, or m ¼ ðv v2 4zjÞ=2z; and the parameters a, b and m are identified. See Bass (p. 219) for further details.
Migration dynamics
247
Table 5. Main results
Parameters
0
n0 /m
n0
n /m
n
n, 1997
Albania
0.05; @0.2
0.05; @0.02
0.03; @0.2
0.03; @0.02
83.95
12.48
22.56
3.33
0.008
0.000
0.004
0.000
2,275
0
1,225
0
0.33
0.42
0.35
0.43
91,000
115,150
96,600
118,300
101,634
China
0.05;
0.05;
0.03;
0.03;
@0.2
@0.02
@0.2
@0.02
77.51
13.26
20.97
3.55
0.017
0.000
0.008
0.000
2,375
0
1,125
0
0.30
0.41
0.33
0.42
42,000
56,000
45,500
58,000
55,352
Philippines
0.05; @0.2
0.05; @0.02
0.03; @0.2
0.03; @0.02
98.81
16.19
27.58
4.38
0.000
0.000
0.066
0.000
0
0
16,975
0
0.10
0.36
0.23
0.38
25,900
91,700
59,850
96,250
93,837
Romania
0.05; @0.2
0.05; @0.02
0.03; @0.2
0.03; @0.02
85.85
15.73
23.17
4.21
0.023
0.000
0.012
0.000
2,625
0
1,375
0
0.29
0.40
0.32
0.41
34,000
46,250
37,000
47,750
44,413
level for all cases.39 The principal results are shown in Figs. 4 and 5 and
are displayed in Table 5.
Some remarks are in order:
(1) In all ethnic groups, the wave starts when the network size, nm ;
reaches 30 or 40% of the critical saturation level m,40 for f1 = @0.2
and f2 = @0.02, respectively. Yet, the lower the elasticity level the
bigger the wave, that is, as market competition increases the network
effect and the ties among immigrants reduce and they seem to be
unable to coordinate entry perfectly.
(2) The higher the elasticity, the higher the threshold level h and the
lower the migration flow. This fact depends on the sum of two effects:
(i) the labour market competition, increasing with the absolute level
of f; (ii) the network effect that depends on the probability of being
39 This permits comparison of the timing and the ‘‘behavioural rationale’’
among migration inflows.
40 The parameter m is described in note 23 and its values are in Table 4.
248
M. Moretto and S. Vergalli
completely integrated (i.e., Bass function). The combined effect
defines the magnitude of the benefit perceived by every migrant in the
host country. On the one hand, a high number of incumbent
immigrants increases the total benefit due to the network effect. On
the other hand, however, low wave dimensions require a high shock
to trigger entry.
(3) A higher q magnifies the optimal trigger as expected.
(4) The highest flows observed in the data are consistent with the
predictions of the model (i.e., n with respect to n, 1997 in Table 5):
the real wave is between the upper and the lower simulated flow in
every case studied.
0
(5) The higher /i or /i, the lower the entry trigger h as expected.
The Albanian flow is the first to start in the case of low demand
elasticity and the second for high elasticity. This happens just behind the
Chinese flow (the second and the first, respectively), with wide jump
dimensions. Nevertheless, since the historical timing of the entries shows
that the Chinese flow is more recent than the Albanian one, the level of
elasticity on the labour market might be close to f2.41
The timing of the migration phenomenon also depends on the particular
ethnic characteristics summarised in the Bass parameters: the higher the
imitator’s parameter b, the earlier the migration starts. This is due to a
high network effect that offsets labour market competition with a larger
wave. Moreover, the higher the innovator’s parameter a, the lower the ties
among immigrants and the higher the number of first entries. This can
explain the differences in behaviour among the four migration inflows
observable in Figs. 4 and 5. In fact, the Filipinos, characterised by strong
individualist behaviour, showed a magnified first entry but a reduced
jump size; vice versa, the Chinese, the Albanians and the Romanians
were characterised by higher imitator parameters and a higher wave.
5.1 Entry costs
So far, we have compared different entry triggers based on normalised
sunk costs K. This normalisation allows the Bass model to describe the
41 We should remember that, since the labour demand shock is depicted as a
Brownian motion (2), the higher the threshold level, the longer the time elapsed.
Migration dynamics
249
Triggers; −0,2
35
Albania
China
Philippines
Romania
33
31
29
Theta (n)
27
25
23
21
19
17
15
1
15 29 43 57 71 85 99 113 127 141 155 169 183 197 211 225 239 253 267 281 295 309 323 337 351 365 379 393
n
Fig. 5. Optimal triggers level for f1 = @0.2
migration behaviour of the flows, by defining the percentage of
innovators and imitators in each flow, thus showing the implicit signals
that drive the waves.
We can now step back and following the theory, ‘‘quantify’’ the entry
costs faced by the four different ethnic groups by inverting (16) and (17)
and evaluating them at their minimum level.42 The results are shown in
Table 6 from which we derive two main points: (1) the geographical
distance is not the focal element of sunk costs, as generally stressed in the
economic literature. In fact, the Philippines and Romania face a K similar
to Albania and China. This implies that the sunk cost faced by the
immigrant must be a wider basket of socio-economic elements; (2) it is
important to stress that the sunk costs displayed correspond to the optimal
threshold. In all cases, since the migration occurred in the same year (i.e.,
42 In fact, if the jump started when the trigger reached the minimum level, we
can take the value of the observed flows (see the 7th column in the Table 4) and
the level of the wage in the year of the peak and substitute these values in the
following equation:
b 1 m
n 0 Bassð
nÞ
/i þ
/i :
K ¼ w 1
b1 ðq aÞ
m
ðq aÞ
250
M. Moretto and S. Vergalli
Table 6. Different relative entry costs
K*
Albania
China
Philippines
Romania
q1 = 0.05
q2 = 0.03
1.29
1.30
1.21
1.22
1.00
1.00
1.01
1.02
Table 7. Caritas report. Number of residents per million of inhabitants, 2004
Country
Number of residents
Albania
China
Philippines
Romania
0.234
0.100
0.074
0.239
1997), the migrants entered a labour market with the same shock
magnitude. This fact meant that all immigrants gained a similar labour
market benefit but faced different costs: for the same level of wages some
ethnic groups were able to face higher costs. Which element made the
difference? The answer is in the ‘‘behavioural rationale’’: high cooperative behaviour helped each individual to face a higher cost. Therefore,
the timing of the entry should be inversely related to the sunk cost in the
optimum, i.e., Albania first, China, Romania and then the Philippines.
Comparing this rank with Figs. 3 and 4, it appears that the true labour
demand elasticity should be nearer @0.02.
5.2 Saturation level
Although the simulations appear to be consistent with the ISTAT data
between 1994 and 2000, we wanted to check whether the model was also
consistent over time, by displaying the results of the 2004 CARITAS
migration report in Table 7. According to our model, the Romanian
community should be near saturation level, but this fact does not
correspond to current data by CARITAS that shows an increase in
Romanian immigration waves.
We suggest two explanations for this. First, our analysis uses the whole
national migration flow as a single community, and this surely
overestimates the alienation effect. We should consider single regional
homogeneous ethnic groups. Secondly, due to the particular method used
Migration dynamics
251
Table 8. The Bass parameters for 1996, 1997 and 1999
t1
Albania
China
Philippines
Romania
b
a
m
0.991
0.097
0.312
0.883
0.103
0.157
0.652
0.137
0.262
0.835
0.108
0.132
to calibrate the Bass parameters, the critical saturation level m is strongly
time-dependent. To overcome this problem we have calibrated the Bass
coefficients one and two steps ahead displaying the changes in
‘‘behavioural rationale’’ of the procedure.
5.2.1 Forward projection technique
In the Bass methodology, m depends on the years (initial condition) used
to calculate the parameters a and b. In particular we used 1996, 1997 and
1998. We then repeated the analysis by using 1996, 1997 and 1999 and
then 1996, 1997 and 2000. Values for a, b and m are reported in Tables 8
and 9, respectively.
In Fig. 6, we show three curves for the Albanian triggers. theta98 is the
benchmark case calibrated with the years 1996, 1997 and 1998, theta99
with 1996, 1997 and 1999 and finally theta00 with 1996, 1997 and 2000.
The same method applied in Fig. 7 for the Romanian flow.
Moving ahead, the last year in calibrating the Bass parameters caused a
substantial change in the shape of the entry trigger functions. In particular,
in both the figures, m(t) increases from theta98 to theta00 which implies,
ceteris paribus, an increase in the size of the jump. Yet, the network effect
is magnified, diluting the innovators’ weight (this is why h increases for
n? 0).
The higher the imitators’ coefficient, the greater the perceived saturation
dimension will be.43 Therefore, if an ethnic group has strong ties, its
community will probably increase more than other groups, ceteris
paribus. Another effect of the increasing imitator’s coefficient with time,
43 Comparing the Filipino flow to the Albanian flow we notice that the
Albanian growth rate in the saturation level is higher than the Filipino one (see
Tables 7 and 8).
252
M. Moretto and S. Vergalli
Theta (n)_Albania
45
theta 98
theta 99
theta 00
40
35
Theta*
30
25
20
15
10
5
96
10
2
10
8
11
4
12
0
12
6
13
2
13
8
14
4
15
0
15
6
16
2
16
8
17
4
18
0
18
6
19
2
19
8
78
84
90
66
72
54
60
36
42
48
24
30
6
12
18
0
0
n
Fig. 6. Forward projection technique: Albanian threshold
is that, the stronger the network effect, the lower the shock required to
migrate. This result is clearly shown in Fig. 6, but it does not emerge
from Fig. 7. The explanation of this odd result depends on the peculiar
Theta (n)_Romania
70
60
theta 98
theta 99
theta 00
Theta*
50
40
30
20
10
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81
n
Fig. 7. Forward projection technique: Romanian threshold
Migration dynamics
253
Table 9. The Bass parameters for 1996, 1997 and 2000
t2
Albania
China
Philippines
Romania
b
a
m
1.194
0.073
0.412
1.077
0.086
0.189
0.667
0.122
0.294
0.952
0.059
0.242
Romanian flows characterised by two jumps. In particular, moving ahead
to the last year, the Bass parameters incorporate even the second jump.
Finally, comparing Tables 4, 8 and 9, we can highlight how the
‘‘behavioural rationale’’ changes for the four flows: the Chinese,
Albanians and Romanians become more cooperative, whereas the
Filipinos seem to remain more individualists. This may explain why
some communities tend to explode and others increase at a constant
rate.
6. Conclusions
This paper has tried to explain why migration flows are characterised
by observable jumps. Real option theory suggests that migration may
be delayed beyond the Marshallian trigger since the option value of
waiting may be sufficiently positive in the face of uncertainty.
Possibly, waiting may resolve uncertainty and thus enable avoidance of
the downside risk of an irreversible investment. Burda (1995) was the
first to use real option theory to explain slow migration rates from East
to West Germany despite a large wage differential. Subsequent work
(Khwaja 2002; Anam et al. 2004) has developed this approach
describing the role of uncertainty in the migration decision. Recent
papers (Moretti 1998; Bauer et al. 2002) show, however, that the role
of the community is important in the migration decision. In this paper,
we have shown a real option model where the choice to migrate
depends on the differential wage and on the probability of being
integrated into a host country. The corresponding integration probability is modelled following the Bass model (1969) where the ‘‘behavioural rationale’’ of the migration flows is shown by two kinds of
immigrants: innovators or individualists and imitators. The weight of
254
M. Moretto and S. Vergalli
each different type influences the timing of migration and the size of
the community. The closer the ties among the individuals, the higher
the dimension of the wave and the higher the entry cost faced, ceteris
paribus.
Furthermore, we have highlighted two opposing forces that influence
entry: on the labour market side, strong competition among workers in the
host country delays entry; at the same time, the more immigrants, the
higher the network effect that reduces the optimal threshold and
anticipates entry.
Simulations of some migration flows into Italy over the last twenty
years fit the theoretical approach and replicate the observable migration
jumps at least in the short-term. The model is able to project the
induced labour demand elasticity level of the host country and the
‘‘behavioural rationale’’ of the migrants. Nevertheless, the use of
national flows, as a proxy for the size of the communities, probably
overestimates the results, suggesting future disaggregation of the ethnic
flows.
Appendix A
This appendix is dedicated to proving Propositions 1 and 2 in the text. To
do this we rely on the work of Leahy (1993), Bartolini (1993), Dixit and
Pindyck (1994), and Moretto (2003, 2007).
To determine the migrant’s optimal entry policy, the first thing to do is
to find his/her value of being perfectly integrated given each individual’s
optimal future entry policy. A solution for Eq. (8) can be obtained starting
within a time interval where no entry occurs (n, h < h ). By the typical
methodology of real options, we obtain the general solution for Eq. (8) as
(Dixit and Pindyck 1994, p. 181):
V ðn; hÞ ¼ A1 ðnÞhb1 þ A2 ðnÞhb2 þ vðn; hÞ;
ð18Þ
where 1 < b1 < q/a, b2 < 0 are, respectively, the positive and the negative
root of the characteristic equation WðbÞ ¼ 12 r2 bðb 1Þþ ab q ¼ 0;
and A1, A2 are two constants to be determined.
To keep V(n, h) finite as h becomes small, i.e., lim V ðn; hÞ ¼ 0; we
h!0
discard the term in the negative power of h setting A2 = 0. Moreover, the
Migration dynamics
255
boundary conditions also require limh!1 fV ðn; hÞ vðn; hÞg ¼ 0; where
the second term in the limit is the discounted present value of the benefit
flows over an infinite horizon starting from h with n fixed. By Eq. (13) we
get:
m n /0 hnf
ba
b
/hnf
þ aþ
:
ð19Þ
n 2 n2
vðn; hÞ ¼
m qa
m
m
ðq aÞ2
h
i
0
Bass
Remembering that uðnÞ ¼ nf mn
/
þ
/
and BassðnÞ ½aþ
m
qa
ba
m n
mb2 n2 ; the general solution of Eq. (18) becomes:
V ðn; hÞ ¼ A1 ðnÞhb1 þ
huðnÞ
:
qa
ð20Þ
It is worth noting that for a B b the function u(n), is shaped according
to Fig. 3. Since the last term represents the value of being in the
community in the absence of new entry, then A1 ðnÞhb1 must be the
correction due to the new entry, therefore A1(n) must be negative. To
determine this coefficient for each n, we need to impose suitable
boundary conditions. First of all, free entry requires the (idle) migrant to
expect zero benefits on entry. Then, indicating with h (n) the value of
the shock h at which the n-th individual is indifferent to immediate entry
or waiting for another opportunity, the condition (9) in the text
(matching value condition) becomes:
V ðn; h ðnÞÞ A1 ðnÞh ðnÞb1 þ
uðnÞh ðnÞ
¼ K:
qa
ð21Þ
Secondly, the number of migrants n affects V(n, h) depending on the sign
of h (n). Since hb1 is always positive, any change in n either raises or
lowers the whole function V(n, h), depending on whether the coefficient
A1(n) increases or decreases. Therefore, by totally differentiating Eq. (21)
with respect to n we obtain:
dV ðn; h ðnÞÞ
dh ðnÞ
¼ Vn ðn; h ðnÞÞ þ Vh ðn; h ðnÞÞ
¼0
dn
dn
¼ Vh ðn; h ðnÞÞ
dh ðnÞ
¼ 0;
dn
ð22Þ
ð23Þ
256
M. Moretto and S. Vergalli
where, (since each individual rationally forecasts the future path of new
entries by competitors), Vn(n, h (n)) = 0 (Bartolini 1993, Proposition
1).44
In conjunction with the Eq. (21), the above extended smooth pasting
condition states that either each migrant exercises his/her entry option
at the level of h at which the value is tangent to the entry cost, i.e.,
Vh(n, h*n)) = 0, or the optimal trigger h (n) does not change with n.
While the former means that the value function is smooth at entry and
the trigger is a continuous function of n,45 the latter case states that, if
this condition is not satisfied, an individual would benefit from
marginally anticipating or delaying the entry decision. In particular if
Vh(n, h*(n)) < 0, it means that the value of staying in the host country
is expected to increase if h falls (investing now will be expected to
lead to almost certain benefits), on the contrary if Vh(n, h (n)) > 0 it
means that a member would expect to make losses because of a
decrease in h. In both situations Eq. (23) is satisfied by imposing
dh ðnÞ
dn
¼ 0; therefore the same level of shock may either trigger entry by
a positive mass of migrants or lock-in the community at the initial
level of members.46
It should be noted that using Eqs. (20), (21) and (23), it is possible to
find the optimal threshold function. The solution depends on the
concavity of u(n). As we have seen in the previous part, a generic
representation of u(n) distinguishes three intervals for the particular shape
of the benefit function. Let us now solve the model backwards.
A.1 Proof of Proposition 1
For the case of n B n0 or n n we show two things: (i) the smooth pasting
condition (23) reduces to Vh(n, h (n)) = 0; (ii) the optimal trigger h (n) is
44 Note that this is a generalisation of the condition in Dixit (1993, p. 35). If
the migrant pretends to be unique or the last entering the host country, then
u0 (n) = A0 (n) = 0 and the first-order condition reduces to Vh(n, h (n)) = 0
45 Moreover, as we assumed that the individual’s size is infinitesimal, then the
trigger level h (n) is also a continuous function in n.
46 If this condition does not hold, the expected benefit gain or loss at h (n)
would be infinite due to the infinite variation property of the stochastic process h.
Migration dynamics
257
equivalent to that of an individual in isolation, that is of a migrant
pretending to be the last to immigrate.
For (i), let us consider the value of a migrant being in the host country
starting at point (n, h < h ), and subject to the possibility of new entries
when h hits h . Indicating with T the first time that h reaches the trigger
h , the optimal entry policy must then satisfy:
2 T
Z
4
V ðn;hÞ¼max
E
eqt ð1Fs ðtÞÞ/0 hðtÞnf þfs ðtÞBðn;hðtÞÞ dt
0
h
0
þ
Z1
T
3
n
o
eqT ð1Fs ðtÞÞ/0 hðtÞnðtÞf þfs ðtÞBðnðtÞ;hðtÞÞ dt5
¼max
E0
h
" ZT
eqt
mn 0
ba
b
hðtÞnf
dt:
/ hðtÞnf þ½aþ
n 2 n2 /
qa
m
m
m
0
þe
qT
#
V ðn;h ðnÞÞ ;
ð24Þ
where V(n, h (n)) represents the optimal continuation value of staying in
the host country. Because, by Eq. (21), the present value of benefits at T is
K, the above value can be written as:
3
2 T
3
Z
4uðnÞE0 4 eqt hðtÞdt 5 þ KE0 ½eqT 5
V ðn; hÞ ¼ max
h
2
0
or, after simplification (Moretto 2003, 2007):
"
#
uðnÞh
uðnÞh
h b1
V ðn; hÞ ¼ max
K
:
qa
h
qa
h
ð25Þ
The value of being perfectly integrated Eq. (25) is the difference between
the value of a migrant with a myopic strategy pretending to be the last to
have to migrate
uðnÞh
qa
and the value of an idle individual pretending to be
258
M. Moretto and S. Vergalli
h b1
: To choose optithe last to migrate as expressed by uðnÞh
h
qa K
mally h*, the first-order condition is:
oV
uðnÞ
K
h b1
b1
¼ ðb1 1Þ
¼0
ð26Þ
oh
qa
h
h
and the optimal threshold function takes the form:
h ðnÞ
b1
K
;
ðq aÞ
uðnÞ
b1 1
with
b1
[ 1:
b1 1
ð27Þ
Since u(n) decreases in the interval ½n; m; h*(n) increases. Moreover,
substituting Eq. (27) into Eq. (25) we can solve for A(n) which is negative
as required by Eq. (20):
AðnÞ ¼
½h ðnÞ1b1
\0:
b1 ðq aÞ
ð28Þ
Finally, substituting Eq. (28) into Eq. (25) and rearranging we obtain
Eq. (20):
V ðn; hÞ ¼ AðnÞhb1 þ
uðnÞh
½h ðnÞ1b1 b1 uðnÞh
h þ
b1 ðq aÞ
qa
qa
ð29Þ
from which it is easy to verify that Vn(n, h) = 0 within the interval
h < h (n) and zero at the boundary.
Now for (ii), let us suppose that all individuals have decided to enter at
^
h; with h \^h: This cannot be a (Nash) equilibrium because a single
migrant can do better by entering at h . In fact, the flow of benefits that
each individual is able to obtain following the policy h is the best that
they can do, at least till T. However, by the principle of optimality, this
choice is also optimal for the rest of the period as (24) shows: if the
optimal policy of the single migrant calls for them to be active at ^h
tomorrow, it immediately follows that the optimal policy today is to enter
at h . As (24) is a continuous function in h , the limit as ^h ! h shows
that h is a Nash equilibrium (Leahy 1993, proposition 1).
If the elasticity is not too low we obtain an interval n [ (0, n0 ) where the
competitive effect prevails over the network effect. Therefore, with these
results, within the interval (0, n0 ) the optimal threshold is still given by
Eq. (27) until n0 . Finally, for f ? 0, n0 ? 0.
Migration dynamics
259
A.2 Proof of Proposition 2
For 0 n0 \n\n we have to show three things: (i) that an individual
cannot pretend to be the last to migrate and, therefore, the optimal
competitive trigger is no longer equivalent to that of a migrant in
isolation; (ii) that the candidate policy, described in the Proposition 2,
satisfies the necessary and sufficient conditions of optimality; (iii) that it is
a sub-game perfect equilibrium.47
Let us assume that u(n) is U-shaped as in the quadrant III of Fig. 3. For
(i) and (ii), let us begin with an idle individual that follows the optimal
policy h (n). Since h (n) is decreasing in the interval n\n: the higher the
number of members in the community the greater their entry value. In
other words, an idle migrant would maximise his/her entry option by
pretending always to be the last to migrate. In fact a migrant that pretends
to be the last to enter expects an inadmissible upward jump in benefits
following the policy h (n). To see this, consider an individual that pretends
to have been the last to enter at h = h (n); by Eq. (19) his/her value is
ðnÞ
simply V ðn; h ðnÞÞ vðn; h ðnÞÞ ¼ uðnÞh
qa : Then we can see that:
V ðn; h ðnÞÞ lim
V ðn; hÞ ¼
h!h ðnÞ
h ðnÞ
[ 0;
b1 ðq aÞ
ð30Þ
This contradicts the smooth pasting condition Vh(n, h (n)) = 0 and then
the optimality of h (n).
To verify that the necessary conditions are satisfied, let us calculate the
value of an (incumbent) immigrant in the host country starting at the point
(n, h), that would follow a policy defined by two parameters: wait until the
first instant T at which the process h rises to a level c > h, corresponding to
an immediate increase in the community size to b > n. Making use of
Eq. (24) the expected payoff V(n, h) from this policy is equal to:
2
3
ZT
V ðn; h; b; cÞ ¼ E0 4uðnÞ eqt ht dt þ eqT V ðb; cÞ5
0
¼
b1
uðnÞh
uðnÞc
h
V ðb; cÞ
:
qa
qa
c
47 See Moretto (2003) for a conjecture of how this can be proved.
ð31Þ
260
M. Moretto and S. Vergalli
If each individual were able to choose the best moment for the
community’s size as well as the dimension of the jump, the first-order
condition would be:
oV ðn; h; b; cÞ
uðnÞ
V ðb; cÞ oV ðb; cÞ h b1
¼ ðb1 1Þ
b1
þ
¼0
oc
qa
c
oc
c
oV ðn; h; b; cÞ oV ðb; cÞ h b1
¼
¼ 0:
ob
ob
c
When b and c are chosen according to the candidate policy so that b ¼ n
and c ¼ h ðnÞ the value function reduces to (20) and the matching value
condition requires V(b, c) = K. These properties verify that the candidate
policy satisfies these conditions.
Let the immigrant, as in Eq. (31), wait until the first time the process h
rises to the trigger level c : h (b), corresponding to an immediate
increase of the network size to b > n, and assume also that he/she expects
no more entry after b. Therefore the expected payoff V(b, h) from this
time onwards equals the discounted stream of benefits fixed at u(b), i.e.,
by Eq. (19):
V ðb; hÞ ¼
uðbÞh
:
qa
ð32Þ
Comparing Eq. (32) with Eq. (20) gives A1(b) = 0. Therefore to obtain
the constant A1(n), subject to the claim that beyond b no other immigrants
will enter, we substitute Eq. (20) into the condition Vn(n, h (n)) = 0 to
0
ðnÞ
¼ 0 resulting in:
get A01 ðnÞh ðnÞb1 þ u ðnÞh
qa
A01 ðnÞ ¼
h ðnÞ1b1 u0 ðnÞ
ðp Þ1b1 u0 ðnÞ
:
qa
q a uðnÞ1b1
ð33Þ
Integrating Eq. (33) between n and b gives:
Zb
n
A01 ðxÞdx
ðp Þ1b1
¼
qa
Zb
n
u0 ðxÞ
uðxÞ1b1
dx:
Taking account of the fact that A1(b) = 0, this integral gives the constant
A1(n) as:
Migration dynamics
A1 ðnÞ ¼
i
ðp Þ1b1 h
uðbÞb1 uðnÞb1 :
b1 ðq aÞ
261
ð34Þ
Substituting Eq. (34) into Eq. (20), which we rewrite to make its
dependence explicit on the end size b, gives:
V ðn; h; b; h ðbÞÞ ¼
i
ðp Þ1b1 h
uðnÞh
:
uðbÞb1 uðnÞb1 hb1 þ
qa
b1 ðq aÞ
ð35Þ
As long as u(b) > u(n) the first term in Eq. (35) is positive and it forecasts
the advantage the immigrant would experience by the entry of b @ n new
immigrants when h hits h*(b). That is, if he/she were able to choose the
optimal dimension of the jump, it would be b ! n which happens the first
time that h reaches h ðnÞ: Thus, as opposed to before non-sequential entry
are possible, the necessary conditions would coordinate an optimal
simultaneous entry by n n new immigrants. If u00 (n) < 0 the necessary
conditions are also sufficient. Furthermore, substituting Eq. (35) into the
extended smooth pasting condition (23) and letting b ! n; we obtain:
"
#
i
ðp Þ1b1 h
uðnÞ dh
b1
b1
b1
uðnÞ uðnÞ b1 h þ
¼ 0:
ð36Þ
q a dn
b1 ðq aÞ
The term inside square brackets is always positive (i.e., there is no value
n 2 ðn; nÞ that makes it nil), and Eq. (36) holds with dh
dn ¼ 0: That is, all
immigrants in the range ðn; nÞ must enter at h ¼ h ðnÞ:
In other words, as the stochastic process h is common knowledge, each
immigrant can foresee the benefit from the entry of others and observing
the realization of the state variable h instantaneously considers when to
enter by maximizing Eq. (35). Then, with simultaneous entry, the
immigrants’ optimal strategies are easy to find: each individual enters as if
he/she were the only person to enter but with the expectation of earning
all the network benefits, i.e., h ðnÞ is a (symmetric) Pareto-dominant Nash
equilibrium for all n\n (see Moretto 2003, 2007). In addition, as the
reaction lags are literally nonexistent, none have the incentive to deviate
from the entry strategy h ! h ðnÞ and b ! n given that the others do not
deviate. Finally, since h is a Markov process in levels (Harrison 1985, p.
5–6), the conditional expectation (31) is in fact a function solely of the
starting states so that, at each date t > 0, the immigrant’s values resemble
262
M. Moretto and S. Vergalli
those described in Eq. (35) which makes the equilibrium subgame
perfect.
Finally, we can to deal with the general case (quadrant I and II in
Fig. 3). If n [ (n0 , n@) we need first to find a network size n such that
u(n) = u(n) with u0 (n) > 0 and then to perform the same policy as in
Eq. (35) or Eq. (36). That is, the optimal entry would be of
ð
n n Þ þ ðn nÞ immigrants the first time that h reaches h ðnÞ:
where:
– Parameters: are, respectively, the discount factors (i.e., q1 = 0.05;
q2 = 0.03) and the elasticity levels (i.e., f1 = @0.2; f2 = @0.02);
– h : represents the optimal trigger level at which the migration wave
starts;
– n0 /m: is the critical level that ‘‘triggers’’ the network effect as a
percentage of the saturation dimension m;
– n /m: is the optimal dimension of the community in percentage of the
theoretic maximum dimension m;
– n : is the level of the community that triggers the migration flow;
– n_year (i.e., n_1997) is the empirical jump observed in our data (see
Fig. 1).
Acknowledgements
We acknowledge the financial support of the University of Brescia under the 60%
scheme and the two-years project, protocol N. 6413/2005. The authors wish to
thank Dennis Snower, Carlo Scarpa, Tito Boeri, Paolo Buonanno and Laura
Poddi for their many helpful comments and suggestions. We have also benefited
from the comments of Giuseppe Russo and participants of the 20th National
Conference of Labour Economics, Rome, September 22 and 23, 2005.
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