%Paper: ewp-urb/9511001
%From: tmitch@siu.edu
%Date: Wed, 1 Nov 95 14:02:13 CST

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\vspace*{1.1in}
\begin{center}
{\bf The Pattern of Migration with Variable Migration
Cost}\footnote{Forthcoming in the {\it Journal of Regional
Science.}}\footnote{I thank M. Ali Khan, M. Patricia
Fern\`{a}ndez-Kelly, Oded Stark and two anonymous referees for helpful
discussions and comments on an earlier draft of this paper.} \\
\vspace{2.5em}
Nancy H. Chau\footnote{Department of Economics, Southern Illinois 
University at Carbondale, IL62901.}
\vskip .75em
This version September, 1995.\\
\vskip 1.0em
\end{center}
{\bf Abstract:} In this paper, we examine the role of
migrant network in determining patterns of outmigration.
Conditions under which migration equilibrium may permit multiple
steady states are identified. Our analysis discusses instances
where migration generates its own demand and explains differences in
migration propensities across potential sources of outmigration.\\

\noindent {\bf JEL Classification:} F22 International migration,
R23  Regional labor migration.
\end{titlepage}

\setcounter{footnote}{0}
\section{Introduction} 
Determinants of  the movements of people across national borders
have been one of the central themes of recent research on labor
migration. To this end, transnational migrant network is shown
to play a significant role in the promotion of new and
repeated migration. These  networks are sets of interpersonal
ties that connect migrants, former migrants and workers in the
source country, interwoven within patterns of institutions such
as the family, friendship, work relationships and shared
community origin. In a series of studies pertaining to
outmigration behavior of Mexican communities \fn{See Massey-Goldring (1992),
Massey-Espa\~{n}a (1987), de Janvry-Mines (1982) 
and the references therein.}, 
the concept of social network is applied to account for 
differential migration propensities of otherwise similar locales.
The key differentiating factor across these communities is
attributed to the timing of the first trips abroad by pioneering
migrants. The success of these pioneers in turn renders
outmigration a feasible option for the 
set of non-migrants to which the pioneers are linked through a
variety of social ties. Various other studies\fn{See 
de Janvry-Mines {\it op. cit.}, Gregory (1986), Lomnitz (1977),
Portes-Borocz (1989), Portes-Sensenbrenner (1993) and Taylor
(1986). } also confirm that undocumented migrants in particular, are
able to secure their footholds in the host countries primarily
through family and work ties established through prior contacts.
The existence of such networks may facilitate migrant flows
through the reduction in expenses involved in the initial stages of
settlement,  assistance in job search, as well as alleviating the
psychological toll of leaving friends and family behind.\\

Despite mounting empirical evidence, attempts to incorporate the
concept of migrant network in  theoretical analysis of labor
migration remain scanty.  In this paper, we shall explore the
cost-saving aspect of migrant
networks and accordingly characterize the resulting pattern of
migration in a dynamic setting\fn{For an analysis of the {\it
informational} role of social networks in employment selection
and wage setting  decisions, see Rees (1966) and Montgomery
(1991).}. Our findings explore the existence of
migration  thresholds which suggests that extensive cost-reducing
migrant networks offer a plausible explanation for the
apparent link between past experiences of recruitment from
certain source countries and the extent of both documented and
undocumented migration from the same countries. Indeed, if
simple cost benefit calculation is to be employed to explain
labor movements across countries, there should be no reason to
believe that selectivity of migration propensities across geographic regions
should in any way be related to initial conditions or the foreign
employment history of the source
country's workers.\\

The theoretical framework constructed in this paper discusses
instances where migration generates its own demand and explains
differences in international migration propensities across
potential sources of outmigration. We distinguish between the
stock and flow concepts of migration and emphasize the
relationship between the stock of migrants in the destination country and the
cost of
migration  for subsequent chains of migrant flows. To keep our analysis
simple we postulate a world with perfect capital and labor
markets with no institutional constraints on voluntary labor
migration. We wish to underscore this set of assumptions, since
as a matter of fact, contemporary migration
from less developed to developed countries are seldom free and
the presence of credit market imperfection necessarily
constrains voluntary labor movements. Including elements of
market imperfections in our
model may, however, suggest that freeing markets from restrictive
practices is a feasible policy option.
To demonstrate the contrary, we shall show that even in such a
flawless world, equilibrium migration may nevertheless be
inefficient.\\

This paper is organized as follows: Section 2 examines the role
of migrant networks, the individual's decision to migrate is
discussed and the cost reducing nature of migrant networks is
explicitly accounted for. Accordingly, we work out possible
configurations of migration equilibrium that are consistent with
the patterns of migration discussed above where each individual's decision to
migrate depend crucially on the migration decisions of every one
else within the community. Finally in section 3, we discuss some
possible extensions of the model presented. 

\section{The Model}  

\noindent{\bf The two-region setup}\\
\noindent Consider a discrete time two region settting
(respectively the home and the foreign region). The home region
is endowed with a stock of inelastically supplied labor
normalized to unity.  At any time $t$, let
the share of home country workers who are employed in the foreign
region be $m_t $. Employment at home guarantees a per
period wage $w(1 - m_t)$ where, by full employment, $1 - m_t$ is
just the share of home country workers who are employed
domestically. The foreign host country pays a per period wage
$w^*(m_t)$. $w^*(\cdot)$ and $w(\cdot)$ are taken to
be continuously differentiable with $w^{* \prime}(\cdot)$ $\leq$
$0$ and $w^{\prime}(\cdot)$ $\leq$ $0$\fn{A prime ($ ^\prime$)
denotes the first derivative with respective to $m$.}.\\

\noindent{\bf Preferences}\\
\noindent The preferences of workers over income sequences
$\{y_{t+\ta}\}_{\ta =0}^{\infty}$ at any time $t$ is given by
$Y_t = \sum_{\ta =0}^{\infty} \beta^{t + \ta} y_{t+\ta}$. At the
beginning of each
period, a worker has two options. He can remain in the
home country and receive the domestic wage. Otherwise, he may
migrate abroad. Individual workers differ only in terms of
their propensity to migrate ( $a$)  which 
takes on any value between $0$ and $1$.  The proportion of
workers in this economy  endowed
with an $a$ $\leq$ $\bar{a}$ is $F(\bar{a})$. $F(\cdot)$ is
therefore a cumulative distribution function which has both its
range and domain as the closed unit interval with an associated density
function $f(\cdot)$\fn{For a discussion of how such propensities
to migrate can be a result of labor and credit market structure
of  the source region, see Section 4.}. For tractability, 
we assume that $F(\cdot)$
is continuous and the associated density function $f(\cdot)$ is
strictly positive.\\

\noindent{\bf  The Cost of migration  and Network Externality}\\
\noindent Migration involves cost. Such cost of migration may be
due to the physical cost of transportation and the initial
expenditure needed for settlement in the host country. Other perceived costs
of migration may well be purely psychological
such as the toll of leaving
friends and families behind. In view of these considerations and
to incorporate the cost reducing aspect of migrant networks, we
shall denote the cost of migration at time $t$ for any individual worker with
migration propensity $a$ from the home to the foreign region as
$C(a, m_{t - 1}) + \bar{C}$ $\geq$ $0$, where $m_{t - 1}$
measures the stock of migrant in the foreign region in the
previous period.  The cost function
$C(\cdot, \cdot)$ exhibits the following properties: 
(A1) $C(\cdot, \cdot)$ $\geq$ $0$ for all
$a$, $m_{t - 1}$, (A2) $C(\cdot, \cdot)$ is taken to be
continuously differentiable, increasing in $a$ and decreasing in $m_{t - 1}$.
\\

As such, the cost of migration, can be perceived as composed of a
fixed and a variable component. The first component $\bar{C}$,
which may be interpreted as the physical cost of migration, is
constant and equal across individuals. Condition (A1)
requires that the variable component of the cost of migration 
be positive and condition (A2) requires that the cost of
migration is decreasing in the stock of migrants in
 the foreign region $m_{t - 1}$\fn{ In
addition, the marginal cost of migration  may decrease 
(increase) as $m_{t - 1}$ increases, implying that the larger the stock of
veteran migrants, the more (less) effective is the migrant
network in reducing the cost of migration. While the precise
specification of the sign of the second derivative of the
migration cost function with respect to $m_{t - 1}$ does not
affect the results of our analysis in the sequel, see figure 1
for a graphical illustration of the case where migrant network
reduces the migration costs for future migrants more than
proportionately only in the initial stages of migration flows.}.
Finally, everything else equal, a worker with a higher
value of $a$ faces a higher cost of migration. These individuals
are therefore {\it followers} in adopting migration as an income
augmenting innovation. Those with low values of $a$, on the other
hand are {\it leaders}, and as we shall elaborate
below, with a given level of net benefit offered to potential
migrants, individuals with low $a$'s  always initiate
migration flows. Finally, the cost of return migration is taken
to involve only the physical cost of migration,  $\bar{C}$. \\

Denote the expected lifetime utility of a worker in the home and
the foreign region at any time $t$  as $Y^h_t(a, m_{t - 1})$ and
$Y^f_t(a, m_{t - 1})$ respectively, we have, 
\eqn
Y^h_t(a, m_{t - 1}) & = & E_t(\max\{ w(m_t) + \beta Y^h_{t+1}(a,
m_{t}), \\\nonumber & & w^*(m_t) + \beta Y^f_{t+1}(a, m_{t}) -
\bar{C}
- C(a, m_{t - 1})\}|m_{t - 1}),\\\nonumber
Y^f_t(a, m_{t - 1}) & = & E_t(\max\{ w(m_t) + \beta Y^h_{t+1}(a,
m_{t}) - \bar{C}, w^*(m_t) + \beta Y^f_{t+1}(a, m_{t})\}|m_{t -
1}).
\eqnn
where $E_t$ is the expectation operator. Migration in this
framework depends, therefore, not only on the income sequences of
the two regions, but also the stock of  migrants in
the foreign region as well as the way in which the expectation
of future income flows is formed. In this section, we shall
impose two assumptions which allows us to completely characterize
the relationship between $Y^h_t$ and $Y^f_t$ and thus the
dynamics of migration which ensue\fn{We shall relax both of
these assumptions in subsequent sections and discuss their
significance.}: (A3) $ E_t(m_t | m_{t - 1}) = m_{t - 1}$, (A4)
$w^*(m_t) - w(m_t)  - \bar{C} >0$ for any $m_t$.\\

Condition (A3), which requires that expectations be static,
captures the notion that each individual takes the observed
foreign wage and the stock of established migrants as given at
the time of migration. Condition (A4) accounts
for the fact that the home region (a village, for instance) is
sufficiently small compared to the foreign region, so that the
wage equalization across these two regions does not occur
whatever be the degree of outmigration from the home
region\fn{That the size of the source region may play a
functional role in determining the pattern of outmigration is an
issue that we shall discuss in
subsequent sections. I am grateful to an anonymous referee for
pointing out this aspect of the model.}. \\

With these assumptions the lifetime income of a home and a
foreign worker at any time $t$ can be simplified as 
\eqn
Y^h_t(a, m_{t - 1}) & = & \max\{ \frac{1}{1 - \beta}w(m_{t - 1}),
\frac{1}{1 - \beta}w^*(m_{t-1})  - \bar{C} - C(a, m_{t -
1})\},\\\nonumber
Y^f_t(a, m_{t - 1}) & = & \max\{ \frac{1}{1 - \beta}w(m_{t - 1})
- \bar{C}, \frac{1}{1 - \beta}w^*(m_{t-1}) - \bar{C} - C(a, m_{t
- 1})\}.
\eqnn    

Figure 1 depicts three possible relationships between $Y^h_t$ and
$Y^f_t$. Curve $YY$ depicts the income differential across the
home and the foreign region net of the physical cost of
migration. Curve $CC$, on the other hand, denotes the variable
component of the cost of migration as a function of the extent of
outmigration. Note in particular that from assumption A2, both
$YY$ and $CC$ are downward sloping. Figure 1a depicts the
situation where the network effect dominates, namely, that the
reduction in migration cost as migration progresses is always
greater than the concurrent reduction in income differential. In
this particular case, migration is not feasible for all values of
$m < \bar{m}$ and thereafter, the benefits of migration is
strictly higher than the physical and the variable costs of migration
combined. \\

Figure 1b depicts the opposite scenario where the incentive
reduction effect dominates as the $CC$ curve is strictly flatter
than the $YY$ curve. In this case, migration is feasible for all
values of $m \leq \bar{m}$ although thereafter, the benefits of
migration no longer induce outmigration from the source country. 
Finally, figure 1c depicts the intermediate scenario where the
network effect dominates for small values of $m$ and this effect
is overtaken by the incentive reduction effect as migration
proceeds. As we shall elaborate in the sequel, each of these
cases represents benchmark situations which has an important
bearing on the dynamics of migration.\\

To see this, note that migration from the home to the foreign
region takes place at time $t$ if, for some $a$ and a given stock
of previous migrants, 
$$\frac{1}{1 - \beta}w(m_{t - 1}) - (\frac{1}{1 - \beta}w^*(m_{t-
1}) - \bar{C} - C(a, m_{t - 1})) \leq 0.$$

By assumption (A2), the monotonic relationship between $a$ and
the
cost of migration guarantees that for all $a'< a$, 
$$\frac{1}{1 - \beta}w(m_{t - 1}) - (\frac{1}{1 - \beta}w^*(m_{t-
1}) - \bar{C} - C(a^{\prime}, m_{t - 1})) < 0.$$

Finally, denote the propensity to migrate of a marginal migrant $a(m_{t - 1})$

as follows \fn{That $a(m_{t - 1})$ is well defined follows from A2 and the
continuity
of the wage and cost functions with respect to $m_{t - 1}$ and
$a$.}:
\eq
\frac{1}{1 - \beta}w(m_{t - 1}) - (\frac{1}{1 - \beta}w^*(m_{t-
1}) - \bar{C} - C(a(m_{t - 1}), m_{t - 1})) = 0.
\eqq

Accordingly, the degree of outmigration at any time $t$ given
$m_{t - 1}$ can be written as:
   
\eq
m_t = \max\{F(A(m_{t - 1})), m_{t - 1}\}.
\eqq
where $(A(m_{t - 1})) =min\{a(m_{t - 1})),\; 1\}$. 
The law of motion of the dynamic process of migration consists of
two elements. First, since the expected income differential
across the two regions at any time $t$ depends on the stock of
previous migrants both through the cost reduction effects of
migrant network and the role it plays on the expectation of
future income flows, $F(a(m_{t - 1}))$ is accordingly the measure of
home country workers with a propensity to migrate which is at
least as high as the marginal migrant. In addition,  since $m_{t - 1}$ is
itself determined by the degree of outmigration at time $t - 2$,
equation (4) depicts a cumulative process of migration in such a
way that each act of migration on the part of home country workers 
induces a chain of subsequent outmigration, which is otherwise impossible
had the pioneers remained in the village.\\

Secondly, while the marginal migrant  is just indifferent between migration or

staying put at the time of migration, the same may no longer hold once
migration at time $t$ is completed. Namely,  $\frac{1}{1 -
\beta}w(m_{t}) - (\frac{1}{1 -
\beta}w^*(F(m_{t}))$ $-$ $\bar{C}$ $-$ $C(a(m_{t - 1}),
m_{t}) $ may be of either positive or negative sign. Since
migration cost is a one time payment at the time of migration, we
obtain, by invoking (A4),   the second
term of equation (4)  which  requires that the
measure of migrants at any time $t$ be no less than that in the previous
period irrespective of the degree of outmigration.\\

It follows straightforwardly that a migration equilibrium
$\tilde{m}$ is a rest point to equation (4) with 
$$\tilde{m} \geq F(A(\tilde{m})).$$ 

With this in mind, a simple application of Brouwer's fixed point
theorem\fn{See Debreu (1959).}
gives part (1) of the following proposition: 
\prp 
(1) If $F(A(\cdot))$ is continuous, a migration equilibrium
exists.\\

\noindent (2) A migration equilibrium $m^*$ is locally stable if
and only if $F(A(m^*))\geq m^*$ and\fn{$C_a$ and $C_m$
denote respectively the first derivative of migration cost with
respect to the first and second arguments.} 
 
\eq
f(A(m^*))\phi(m^*) < 1 
\eqq
 where $\phi(m^*) = (w^{*\prime}(m^*) + w^{*\prime}(1 - m^*) - (1 -
\beta) C_m(A(m^*), m^*))/( (1 - \beta) C_a(A(m^*), m^*)).$
\prpp
Proof: See Appendix.\\

Figure 2 graphically depicts the dynamic process of migration as
in equation (4). The horizontal and the vertical axis
respectively denote the degree of outmigration at time $t-1$ and $t$. The
$45^o$ line is the set of all possible steady state solutions to
equation (4). The slope of curve $MM$ on figure 2a is given by:

\eq
\frac{d m_t}{d m_{t-1}}\left|_{MM}  \right. = 
\left\{\begin{array}{ll} 
f(A(m_{t - 1})) \phi(m_{t - 1})  & \mbox{if $F(A(m_{t - 1}))\geq m_{t -
1}$; }\\
& \\
1 & \mbox{otherwise.}
\end{array}
\right.
\eqq

Accordingly, curve $MM$ is respectively upward and downward
sloping depending precisely on whether the network effect or the
incentive reduction effect dominates. Furthermore, the frequency
distribution $f(\cdot)$ also affects the slope of the $MM$ curve
in that the smaller the frequency of individuals for any
particular value of $A(m_{t - 1})$ and thus the smaller the
density of the marginal migrant, the flatter the $MM$ curve.
Finally, the larger the discount rate (the smaller is $\beta$),
the flatter the $MM$ curve. \\

Let us begin with the case where the network effect dominates\fn{Equivalently,
we have
$\phi(\cdot) >0$}. One such scenario is depicted in figure 2a. As shown, the
$MM$ is
always upward sloping and in this particular scenario, it
intersects the $45^o$ line twice. There is a continuum of steady
states for all initial rates of outmigration where $m_0 \leq m^*$
with $m_0 = m^*$. That this should be the case is due to the
fact that voluntary migration cannot take place prior to the
point $m^*$. Any exogenously determined initial rates of
outmigration which satisfy this criterion is itself a steady
state due to the migration decision calculus described above.
$m^*$ is a equilibrium migration threshold whereby for any
initial rates of outmigration greater than $m^*$, migration takes
on its own momentum as network externality gives rise to 
sufficient reductions in the cost of migration for subsequent
rounds of migrants. Voluntary migration will proceed until the
stable equilibrium $m^{**}$ is reached. \\

In fact, it can be shown that if the migration equilibrium nearest
to the origin is locally unstable and that the 
same condition guaranteeing instability is
satisfied for any $m$ between zero and $m^*$, there exists an
$\bar{m}$ with $0 < \bar{m} < m^*$ such that $A(\bar{m})$ $=$
0\fn{See Appendix II.}. Put another way, the existence of a
migration equilibrium $m^*$ with strictly positive rate of
outmigration, does not necessarily imply that for all levels of
outmigration less than $m^*$, migration is a feasible option.
Notice that starting from a
point like $m$ $=$ $0$, equilibrium is one 
where no migration takes place at all. In other words, 
outmigration becomes feasible only when the stock of
established migrants exceeds a threshold level (point $m^*$).
Migration then perpetuates due to equation (4) since the 
cost saving role of migrant
networks outweighs the reduction in net benefits as migration
proceeds. The pattern of migration, therefore depends crucially
on initial conditions.\\

Our previous discussions on employer recruitment as well as
those immigration policies which encourage the reunion of
families and relatives thus play an active role in determining
these initial conditions. As a result, even in those
circumstances where wages and income are similar
in potential source communities, the pattern of outmigration may
turn out to be drastically different. Notable
examples include the deliberate migrant recruitment which led to
the onset of the Irish labor migration to the United States
in the mid-nineteenth century as well as from Brazil, Argentina
and Mexico to the same destination country later in the
century\fn{See for instance, Piore (1975), pp. 19-27. In the
context of  Turkish migration to West Germany, see also Waldorf et. al 
(1990) which  gives evidence for the importance 
of prior migrant contacts through  information acquisition,
assimilation of newcomers and family reunification .}. It bears emphasis that
while the chain migration phenomenon and nationwide selectivity in migration
flows are widely noted, evidence also exists for differences in 
migration propensity on a regional level where rates of
outmigration are markedly different across otherwise equally poor
regions or villages within the same source country \fn{See also
Cornelius (1993), Gregory {\it op. cit.} and Hugo (1981) for
discussions of differential migration propensities from various
Mexican municipals to the U.S.}. Along a similar vein, Greenwood (1970) shows
that
failure to include the stock of migration variable conceal the
true effect of other migration determinants in the context of US
interstate migration.\\

A contrasting situation is depicted in figure 2b. That curve $MM$
is downward sloping for all values of $m_{t - 1}$ is a result of
the fact that the incentive reduction effect dominates and hence, $\phi(\cdot)
\leq 0$.
Note in particular that  the $45^o$ line and the $MM$ curve may intersect
once only (at $m^*$) and thus for all initial rates of outmigration
$m_0 \leq m^*$,   $F(a(m_0)) \geq m_0$. As the  $MM$ curve is
downward sloping, $F(a(m_0))$ is a stable steady state. In fact, it can
be easily confirmed by figure 2b that the type of chain migration
described figure 2a simply does not occur and we are back to the
neoclassical world where any incentive to migrate from the home
to the foreign country is wiped out within one period in a
situation where network externality is insignificant. Finally,
figure 2c illustrates the third case where the relative dominance
of network and incentive reduction effect varies over the course
of migration. Migration equilibrium may be stable or unstable depending once
again on the initial stock of migrants from the home region.\\

The multiplicity of equilibria is possible, precisely due to the
favorable externalities engendered by the stock of established
migrants in the foreign country. Everything else constant, each
act of migration generates a {\it chain} of further movements of
labor in ways that spread migration within the source community.
Nevertheless, when migration decisions are based on
cost-benefit calculations undertaken by the individual worker,
the potential gain to the source community at large will not be
internalized when each agent maximizes his / her lifetime utility. 
Migrant networks, therefore, exhibit a public good
property. The next section is thus devoted to study the welfare
implications of migrant networks.\\

\section{Welfare Analysis}
Consider a social planner who may choose an optimal path of
outmigration in order to maximize discounted societal welfare of 
the home region. The planner's problem is thus

\eq
\max_{\{a_\ta \}_{\ta = 1}^{\infty}} \sum_{\ta = 1}^{\infty}  (\beta^\ta
w^*(F(a_\ta))F(a_\ta) + 
w(1 - F(a_\ta))(1 - F(a_\ta))  - \int_{a_{\ta -
1}}^{a_{\ta}}(\bar{C} +
C(a, F(a_{\ta-1})f(a)da.
\eqq
subject to $a_\ta \geq 0$. Denoting the optimal path as 
$\{\tilde{a}_\ta
\}_{\ta = 1}^{\infty} $, it follows that:

\eqn
w^*(\ F(\tilde{a}_t ) )+ w^{*\prime}( F(\tilde{a}_t))
F(\tilde{a}_t) + \beta(\bar{C} +
C(\tilde{a}_t, F(\tilde{a}_t) ) - \beta  \int_{\tilde{a}_{t -
1}}^{\tilde{a}_t
}(\bar{C} + C_m(a_,
F(a_{t-1})f(a)da = \\\nonumber
w(1 - F(\tilde{a}_t)) +  (w^{\prime}(1 - F(\tilde{a}_t))(1 -
F(\tilde{a}_t))  +
(\bar{C} + C(\tilde{a}_t, F(\tilde{a}_{t-1})).
\eqnn

Along the  optimal path of migration, $\tilde{a}_t $ equalizes
the marginal benefits and costs of migration where the marginal
benefits should take into account the change in foreign wage as
well as the reduction in the cost of migration as a result of an
incremental increase in the outmigration rate. The marginal cost
of migration, on the other hand, should take into account the
change in the home income as a result of outmigration as well as
any increase in the cost of migration as workers with higher $a$
migrates. \\

That this socially optimal path may not  coincide with the
competitive outcome should be clear by comparing equation (3) and
(5).  In particular, since at any time $t$, voluntary migration
takes place until
$$w^*(m_{t - 1}) - (1 -  \beta)(\bar{C} + C(a(m_{t - 1}), m_{t -
1}) =  w(1 - m_{t - 1}), $$ 
which states that migration equalizes the expected lifetime
income
across the two regions for the marginal migrant. The
deviation of this privately optimal rule of migration and
equation (8) depends on the rate at which domestic and foreign wages
change in response to migration flows, as well as the size of the
network effect $C_m(\cdot, \cdot)$. Ceteris paribus, the larger the
network effect, the larger will be the deviation of voluntary
migration from the social optimum.  The source of the suboptimality of
voluntary migration, 
is two-fold. Social optimum entails migration decision making in the form of
marginal as opposed to average income comparisons. Secondly, the socially
optimal rate of outmigration also take into 
account the network externality that each act of migration engenders. It can
also be confirmed
straightforwardly that if this source village is a ``price-taker" in the
domestic
as well as the foreign labor market and if  network externality
is absent, the privately optimal degree of outmigration
coincides with the socially optimal one. 
     
\section{Discussion}

In this section, we shall relax some of the assumptions made in
the previous section and examine the pattern of migration and its
welfare implications.
\begin{enumerate} 
\item The suboptimality of decentralized migration decisions
arises when the benefits of each act of migration extends to all
members of the source community
at large. Nevertheless, instances where such external benefits
can be internalized in
migration decision making need not be farfetched. For example,
when family decisions take on an
important role in migration behavior, a family member may well be
sent abroad even when costs
exceeds benefits - an oft noted puzzle in the migration
literature\fn{See, for instance,
Stark (1991).} - in anticipation of lower expenses for successive
outmigration of family
members\fn{See  Findley {\it op. cit.} for an illustrative survey
of family and community profiles with
regards to migration from the Philippines.}.

To illustrate how our model can be extended to this context and
the resulting dynamics of outmigration, 
consider a single family as the source region.  Let such a family
consists of  $L$ members. Family production in the migration
origin yields a per period output of $y(L - M_t)$ where $M_t$
denotes the stock of family members abroad. $y(\cdot)$ satisfies
all the conditions of a standard production function with
$y(\cdot)$ strictly increasing in its argument and exhibiting diminishing
marginal returns. 
The foreign country pays a wage
$w^*$ which this family takes as given\fn{An alternative
interpretation of this assumption is that the family unit is
sufficiently small so that the foreign wage is invariant to any
change in labor allocation of this household across the domestic
and foreign country.}. As before, preferences are represented by
the expectation of the discounted lifetime income of this
household over  the infinite horizon.  At the beginning of each time
period, the problem of the household involves the decision of how
many additional family members to send abroad  given the stock of
family members already employed  in the foreign region. The cost
of migration for one member of this household at time $t$ is
$\bar{C} + C(M_{t - 1})$ where $C(\cdot)$ is decreasing and
strictly positive for all possible values of $ M_{t - 1}$. \\

The problem of the household at any time $t$ is thus
\eq
\max_{\{M_t + \ta \}_{\ta = 0}^{\infty}} \sum_{\ta = 0}^{\infty} \beta^{t+\ta}
(w^*M_{t +\ta} + 
y(L - M_{t + \ta})  - (M_{t + \ta} - M_{t +
\ta - 1})(\bar{C} +
C( M_{t + \ta-1})).
\eqq
subject to $L \geq M _{t+\ta} \geq 0$. Denoting the optimal path of
outmigration from this family as $\{\tilde{M}_{t + \ta}\}_{\ta =
0}^\infty$. The first order condition of this problem is:
  
\eqn
w^* + \beta(\bar{C} + C(\tilde{M}_{t + \ta})) -
\beta(\tilde{M}_{t +\ta + 1 } -\tilde{M}_{t+ \ta})
C_m (\tilde{M}_{t + \ta}) =
\\\nonumber  y^{\prime} (L - \tilde{M}_{t + \ta}) + (\bar{C} +
C(\tilde{M}_{t + \ta - 1}).
\eqnn

Equation (10) thus specifies the relationship between the
degree of previous family migration and the degree of
outmigration in successive periods. Note in particular that
unlike the competitive outcome which suffers from the public good
property of migrant network as illustrated in the previous
section, the same no longer applies in this case. Along the
optimal path of outmigration, the degree of outmigration
depends not only on the gains and losses in household income for
the marginal migrant, family income maximization also guarantees
that the cost reductions for subsequent family outmigration are fully
taken into account.\\

A further manipulation of equation (10) also confirms that any
migration steady state ($m^*$) is locally stable for any
household with a sufficiently low rate of discount and a
sufficiently effective migrant network. 
To see this, note that the more patient households will be 
willing to sacrifice the payment of migration costs in the 
current period in exchange for an increase in the flow of future
benefits. On the other hand, such benefits would be worthwhile 
only if the effectiveness of the migrant network is sufficiently 
large.\fn{For a proof of this result, see part III of the appendix.}.

\item  Another important assumption we made in the previous
section pertains to the size of the home region, namely, that the
size of the migration origin is sufficiently small so that wage
equalization never occurs irrespective of the degree of
outmigration from the source region. As discussed, a migrant
network is a set of interpersonal ties which render the
psychological costs of migration variable with respect to the
stock of veteran migrants who indeed belong to the network.  To
incorporate the fact that the source region may indeed be large
without compromising the above characteristic of migrant
networks, the model may be modified in the following way.
Consider a source region which consists of a large (say $N$)
number of villages. Population and preference 
structure pertaining to the individual worker in each village
satisfy the all the assumptions made in the previous section. 
Clearly, there are a number of parameters in this model which may
be utilized to distinguish between heterogeneous villages, such
as the domestic wages, the distribution of migration propensities
($F(a)$)  and the rate of time preference. While a detailed
analysis of each of these cases is beyond the scope of this
paper, in this section, we shall only consider the latter.\\

Let individual villages differ from one another in terms of
their rate of time preference $\beta$. Further, let the fraction
of villages with rate of time preference less than or equal to
$\bar{\beta}$ and with an initial stock of migrant $m_0 \leq
\bar{m}$ as $G(\bar{\beta}, \bar{m})$. Workers are homogeneous in
terms of their productivity. It follows that per period foreign
wage payment at any time $t$ depends on the measure of the stock
of migrants in the foreign country, that is $w^*_t = w^* (M_t)$.
Internal migration across villages is assumed to be costless and
thus domestic wages also depend on the aggregate degree of
outmigration from the source region, or, 

$w_t = w (N - M_t)$. \\

The main element of this extended version of our model is that
the cost of migration for any individual belonging to a village
with rate of discount $\beta$ is decreasing in the stock of
migrants from this village alone. That is, if the stock of
migrants from a village at time $t - 1$ is $m_{t - 1}$, the cost
of migration for any individual with propensity to migration $a$
is just $\bar{C} + C(a, m_{t - 1})$. Accordingly, the lifetime
expected utility of such an individual currently residing
respectively in the home country ($Y^h_t(\beta, a, m_{t - 1}, M_{t -
1})$)
and the foreign country ($Y^f_t(\beta, a, m_{t - 1}, M_{t - 1})$) is
just 

\eqn
Y^h_t(\beta, a, m_{t - 1}, M_{t - 1}) & = & \max\{ \frac{1}{1 -
\beta}w(N
- M_{t - 1}), \frac{1}{1 - \beta}w^*(M_{t-1}) \\\nonumber
 & & - \bar{C} - C(a, m_{t -
1})\}\\\nonumber
Y^f_t(\beta, a, m_{t - 1}, M_{t - 1}) & = & \max\{ \frac{1}{1 -
\beta}w(N
- M_{t - 1}) - \bar{C}, \frac{1}{1 - \beta}w^*(M_{t-1}) - \bar{C}
- C(a, m_{t - 1})\}.
\eqnn 

In the appendix, we shall derive the set of difference equations
which governs the dynamic process of migration in this case.
Several comments are in order. Our previous
analysis points out that in the presence of migration thresholds,
initial conditions in the form of the stock of existing migrants
from the source region determines the dynamics of outmigration
which ensue. In the present case, where domestic wages are
determined by the aggregate degree of outmigration from the host
country, it is only natural that migration thresholds is are
dependent on aggregate outmigration from the source region.
Referring to figure 3 which is just the analogue of figure 2 for the present
case, curve $YY$ and $Y'Y'$ depicts the dynamics of
migration for two different levels of the stock of migrants from the
rest of the villages. In fact, as the stock of migrants from
other villages increases, the migration threshold moves from $m^*$ 
a higher value,  $m^{**}$. Namely, the concomitant reduction in the wage
differential across the two regions now entails a much higher
initial stock of migrants in the foreign region in order for
voluntary migration to take place. It bears emphasis that the
selectivity of outmigration pattern across apparently identical
villages can be explained not only through the difference in the
initial stock of migrants, from figure 3, we also find that the {\it timing}
of initial outmigration relative to the rest of the potential
migrant pool is equally important. \\

As discussed in the previous section, a necessary and sufficient
condition which precludes the occurrence of return migration is
that, for any $m$,    $$w^*(m) - (1 -  \beta)\bar{C} \geq w(1 - m), $$
which can be justified on the grounds that the source region is
sufficiently small so that migratory movements does not give rise
to substantial fluctuations in regional wage differentials. While
this justification no longer applies in the present case, one can
no longer undermine the possibility of return migration either.
That this is so is due to the fact that when each act of
migration generates a set of favorable externalities on potential migrants
from the same village, it also engenders a negative incentive for
migrants originating from different source villages. In addition,
heterogeneity across villages in terms of the rate of time
preference leads to the possibility that return migration by some
villages may entail additional outmigration from others. In essence,
simultaneous two-way flow of migrants may occur whenever return
migration takes place.  Finally, welfare implications of migrant
networks we elaborated above still applies. Since private
migration decisions in this setting does not take into account
the positive (negative) externality it imposes on other members of
the source region, migration may entail both under and / or over
migration of labor, depending once again on the marginal
migration benefits and costs to the source region at large. 


\item In addition to reducing migration costs, having prior
contacts in a foreign country may also allow the potential
benefits of migration to increase.  The provision of assistance
during job search as well as accurate information about labor
market characteristics, opportunities and skill requirements
(Gibbs (1994))  are all important ingredients in guaranteeing
successful migration. One might also note the information
asymmetry which arises when foreign employers are imperfectly
informed about the true productivity of migrants. Consequently,
employers may favor those new workers with recommendation through
previous contacts (Montgomery (1991)). \\

The benefit enhancing effect of migrant networks can be easily
incorporated to our framework by appropriately modifying our
specification of $w^*$. Note, however,  that the qualitative
outcome of our analysis need not change. In particular, the three
cases we identify in the section 2 still depend both on the
combined effect of domestic and foreign wage as well as migration
cost changes with respect to the stock of migrants.  The stronger
the network effect, both through benefit enhancement and cost
reduction, the steeper the $MM$ curve and thus the more likely
that the network effect is the dominating factor in the dynamic
process of migration.

\item In a typical less developed country setting where
inequality in the distribution of wealth and heterogeneity of
preferences are rules rather than exceptions,\fn{See also Smith
(1979) for an analysis of the role of the distribution of risk averseness in
the context of  inter-regional migration.} selectivity in
migration propensities in response to a given set of
opportunities, prices, and the changes in these variables thereof, takes on
additional importance in determining the pattern of outmigration
from the region as a whole\fn{See Connell et al. (1976) and
Findley (1987).}. In Lipton's terms\fn{See Lipton (1982).}, migration may take
the form
of a ``leader-follower", ``innovator-imitator" relationship,
which are in turn synonyms for the young and the elderly, the
resourceful and the resourceless. Accordingly, those who have few
assets must rely on social contacts and the exchange of favors through
networks, in order for migration to become a feasible option.\\
 
To mention one such possible interpersonal heterogeneity,
ownership of assets, such as land, has two distinct roles in determining an
agent's propensity to migrate. In a setting where rural labor
market is completely absent \fn{See, for instance, Bhattacharyya
(1985).} or when transactions costs in
labor hiring activities are high,
outmigration from families of large landowners
becomes more expensive than say, the landless, since
the absence of a family
member can only be replaced at a cost higher than the prevailing
market wage. On the other
hand, imperfections in rural credit markets prohibit those who
are incapable offering family
owned assets as collateral from financing the cost of
migration\fn{There is by now a large
theoretical literature on the role of transaction costs on the
formation of agricultural
institutions. See, for instance, Jaynes (1984), Eswaran-Kotwal
(1986) and the references
therein. See also Danesh (1987) for a case study on the migration
patterns in Iran.}. In this
case, it is thus the poor who are more reliant on foreign
contacts and other cost-saving means of migration.\\

Within this framework, the distribution of willingness to migrate
may be appropriately translated as the distribution of assets.
The precise mechanism under which these two opposing effects of asset ownership
operate
and hence the sequence of migration of the
resourceless versus the resourceful will in turn depend on the
institutional framework one has in mind. These conclusions will, in turn,
determine the 
impact of migration on the income distribution across home region workers.
\end{enumerate}

\section{Conclusion}
To summarize, we have singled out an example where regional
migration propensities may differ even if the domestic
wages are identical throughout the source country. Conditions
under which migration may be suboptimal from the source 
communities point of view are
identified. These conclusions are
drawn by postulating that the cost of migration is
endogenously determined by the
degree of outmigration itself, and in addition, the stock of {\it
experienced} migrants
pertaining to a particular
region is a parameter specific to each region. In the case where
the cost-saving role of migrant networks dominates, we identify a
threshold level of migration such that once the number of
experienced migrants exceeds this level, voluntary movements of
labor, which is otherwise an infeasible endeavor, commence and
perpetuates.

\newpage
\begin{center}
{\bf Reference}
\end{center}
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\end{description}
\newpage
\begin{center}
{\bf Appendix}
\end{center}

\noindent I. {\bf Proof of Proposition 1:}
A migration equilibrium is locally stable if and only if
$dm_t
/ dm_{t - 1}$ $<$$1$, where
$$\frac{dm_t}{dm_{t - 1}} = 
f(A(m^*))[\frac{w^{*\prime}(m^*) +
w^{\prime}(1 - m^*) - (1 - \beta) C_m(A(m^*) ,  m^*)) }{(1 -
\beta) C_a(A(m^*) ,  m^*))}. \Box$$ 

\noindent II. In section 3, it is claimed that the existence of a
strictly positive rate of equilibrium migration $m^*$ does not 
necessarily imply that  for all levels of outmigration less than
$m^*$, migration is a feasible option. To show that this is indeed the case,
suppose otherwise.
Notice first of all that for any $m = m^* - \epsilon$,
\eq
f(A(m))[\frac{w^{*\prime}(m) + w^{\prime}(1 - m) - (1 - \beta)
C_m(A(m) ,  m) }{(1 - \beta) C_m(A(m) ,  m)}] > 1. 
\eqq
if and only if $F(A(m + \epsilon)) - F(A(m)) > \epsilon$ for any $\epsilon >0$
and $m^* = m + \epsilon$. It follows that 
\begin{eqnarray*}
F(A(m^*)) - F(A(m)) > \epsilon\\
m^* - \epsilon  > F(A(m))\\
m > F(A(m)).
\end{eqnarray*}
A contradiction. $\Box$\\

\noindent III. To determine the parameters which affect the
stability of household migration, note that from equation (10),
we have  
$$w^* - y^{\prime} (L - M_t) - (\bar{C} + C(M_{t - 1})) + \beta(
(\bar{C} + C(M_{t})) - \beta(M_{t + 1 } -M_t)C_m (M_t) = 0.$$
If follows that at any point along the optimal path of migration,


$$\frac{dM_t}{dM_{t - 1}} = \frac{w^{\prime \prime }(N - M_t) +  
2 \beta C_m (M_t) - \beta(M_{t + 1} - M_t) C_{mm}(M_t)}{C_m (M_{t
- 1})}.$$ 
which is greater than $1$ for sufficiently large $\beta$ and if 
$C_{mm}(\cdot) >0$ since $M_{t +1} \geq M_t$ for any $t$.$\Box$\\

\noindent IV. This section offers a characterization of the
dynamic process of
migration in section III.2. For any village with a rate of time
preference $\beta$, let $\psi(\beta)$ be the maximum degree of
outmigration from the home country without inducing return
migration from this village: 

$$w^*(\psi (\beta) ) - (1 -  \beta)\bar{C} =  w(1 - \psi(\beta)),$$ 
Since $w^*(\cdot)$ and $w(\cdot)$ are decreasing in their
arguments, 
it follows that $\psi(\beta)$ is decreasing in $\beta$.
Hence, given $m_{t - 1}$ and the total degree of outmigration
from the migrant source country, $M_{t - 1}$, we obtain, 

\eq
m_t(\beta,  m_{t-1}, M_{t - 1}) = \left\{ \begin{array}{ll} 
\max\{F(A(\beta, m_{t - 1}, M_{t - 1})), m_{t - 1}\})  
& \mbox{if $M_{t - 1} \leq \psi(\beta)$; }\\
& \\
0 & \mbox{otherwise.}
\end{array}
\right.
\eqq 
where $ A(\beta, m_{t - 1}, M_{t - 1}))$ is the analogue of
equation (3) and (4) in the text.  Aggregate outmigration at any
time $t$ from the source country is
thus $$\int_0^1 \int_0^1 N m_t(\beta,  m_{t-1}, M_{t - 1})g(m_{t
-
 1}, \beta) d\beta dm_{t - 1}$$
where $g(\cdot, \cdot)$ is the density function associated with
$G(\cdot, \cdot) \Box$ 

\end{document}

