A partnership with academia

Building knowledge for trade and development

Vi Digital Library - Text Preview

Coordination Failures in Immigration Policy

Working paper by Giordani , Paolo E., Ruta, Michele, 2011

Download original document (English)

We propose a theoretical framework for analyzing the problems associated to unilateral immigration policy in receiving countries and for evaluating the grounds for reform of international institutions governing immigration. We build a model with multiple destination countries and show that immigration policy in one country is influenced by measures adopted abroad as migrants choose where to locate (in part) in response to differences in immigration policy. This interdependence gives rise to a leakage effect of immigration policy, an international externality well documented in the empirical literature. In this environment, immigration policy becomes strategic and unilateral behavior may lead to coordination failures, where receiving countries are stuck in welfare inferior equilibria. We then study the conditions under which a coordination failure is more likely to emerge and argue that multilateral institutions that help receiving countries make immigration policy commitments would address this inefficiency.

Staff Working Paper ERSD-2011-02 Date: 26 January 2011







World Trade Organization
Economic Research and Statistics Division










Coordination Failures in Immigration Policy







Paolo E. Giordani
LUISS "Guido Carli" University





Michele Ruta
World Trade Organization



Manuscript date: December 2010












Disclaimer: This is a working paper, and hence it represents research in progress. This paper
represents the opinions of the authors, and is the product of professional research. It is not meant to
represent the position or opinions of the WTO or its Members, nor the official position of any staff
members. Any errors are the fault of the author. Copies of working papers can be requested from the
divisional secretariat by writing to: Economic Research and Statistics Division, World Trade
Organization, Rue de Lausanne 154, CH 1211 Geneva 21, Switzerland. Please request papers by
number and title.




Coordination Failures in Immigration Policy


Paolo E. Giordaniy


LUISS "Guido Carli" University


Michele Rutaz


World Trade Organization


December 2010


Abstract


We propose a theoretical framework for analyzing the problems associated to unilat-


eral immigration policy in receiving countries and for evaluating the grounds for reform of


international institutions governing immigration. We build a model with multiple destina-


tion countries and show that immigration policy in one country is in‡uenced by measures


adopted abroad as migrants choose where to locate (in part) in response to di¤erences


in immigration policy. This interdependence gives rise to a leakage e¤ect of immigration


policy, an international externality well documented in the empirical literature. In this


environment, immigration policy becomes strategic and unilateral behavior may lead to


coordination failures, where receiving countries are stuck in welfare inferior equilibria.


We then study the conditions under which a coordination failure is more likely to emerge


and argue that multilateral institutions that help receiving countries make immigration


policy commitments would address this ine¢ ciency.


Keywords: Immigration policy, cross-border externalities, coordination failures, mul-


tilateral institutions.


JEL Classi…cation: F02, F22, J61


Acknowledgments: We would like to thank G. De Arcangelis, C. Devillanova, G. Facchini, F. Robert-
Nicoud, R. Staiger and seminar participants at the Geneva Trade and Development Workshop, the European
Trade Study Group Annual Meeting and the SIE Conference for comments and suggestions. Remaining errors
are our responsibility. Disclaimer: The opinions expressed in this paper should be attributed to the authors.
They are not meant to represent the positions or opinions of the WTO and its Members and are without
prejudice to Members’rights and obligations under the WTO.


yDepartment of Economics and Business, LUISS "Guido Carli" University, Viale Romania 32, 00197 Roma,
Italy (e-mail: pgiordani@luiss.it).


zEconomics Research Division, World Trade Organization, Rue de Lausanne 154, 1211 Geneva, Switzerland
(e-mail: michele.ruta@wto.org)


1




1 Introduction


What type of multilateral institutions do countries need to govern international migrations?


Several economists have recently raised this question (among others, Bhagwati, 2003, Hat-


ton, 2007, and Hanson, 2009). In particular, Hatton (2007) examines whether the basic


principles governing the World Trade Organization could improve international cooperation


on migration between sending and receiving countries.


The present work aims at contributing to this debate but takes a somewhat di¤erent ap-


proach for two reasons. In our view, a prerequisite for a precise answer to the above question


is the identi…cation of the international externalities associated with unilateral policy-making


in migration policy. In some sense, this is a key lesson that can be inferred from the economic


literature on the multilateral trading system. As Bagwell and Staiger (1999 and 2002) show,


the GATT/WTO system has e¤ectively improved international trade policy cooperation pre-


cisely because it provides a framework to neutralize a key cross-border spillover associated


with unilateral policy-making in the trade domain, the terms-of-trade externality. Second,


the scope of our analysis is di¤erent -and possibly more limited- compared to Hatton (2007).


Rather than looking at the problems of international cooperation between host and sending


economies, we focus on the interaction of immigration measures implemented by countries


that are on the receiving end of immigration. Our goal is to clearly identify the externality


associated with immigration policy in this set of countries and to investigate the welfare


implications of this economic interdependence.1


A large body of empirical literature has recently studied the long-run determinants of


immigration policy and found four key (and somehow interrelated) channels: distributional,


political economy, non-economic and international determinants. Distributional factors in-


clude the e¤ect of immigration on the labor market and on welfare systems (Borjas, 1994 and


2003, Boeri et al., 2002, Razin et al., 2002). In turn, distributional determinants are chan-


nelled into government policies through voting and/or lobbying activity by interest groups


that stand to lose or gain from immigration (Goldin, 1993, Facchini et al., 2008). Non-


economic forces, such as racism or xenophobia, may in‡uence voters’attitudes -and, hence,


1While we explicitly model the migration decision of foreign workers (as further discussed below, the
migratory decisions are the key transmission mechanism of policy spillovers), the welfare e¤ects of emigration
on the sending region are not analyzed in this work.


2




immigration policy (Dustmann and Preston, 2007, and O’Rourke and Sinnott, 2006). Finally,


and crucially for the present work, immigration policy abroad is a determinant of immigra-


tion policy at home (Timmer and Williamson, 1998, Boeri and Brücker, 2005, Hatton and


Williamson, 2005). The positive correlation between domestic and foreign measures suggests


that countries aim at anticipating an externality associated with the immigration policy of


other destination countries.


As these international determinants are a key concern of this paper, we brie‡y review the


available evidence. In their historical account of migratory ‡ows and immigration policy in the


NewWorld in the late 19th and early 20th century, Timmer and Williamson (1998) argue that


countries in the New World must have paid close attention to each others’policies as migrants


were pulled from and pushed toward one country in response to less or more restrictive policies


in others.2 In particular, they …nd that "Australia’s openness decreased ‡ows to Canada,


Brazil’s pro-immigrants subsidies reduced ‡ows to Australia, and Argentina saw an increased


share of the immigrant pie as the United States closed its doors" (Timmer and Williamson,


1998, p. 756). A second study that documents the immigration policy spillover is Boeri


and Brücker (2005) who adopt a di¤erent methodology and look at a di¤erent immigration


episode. In January 2004, the European Union enlarged to ten new member states from


Eastern and Central Europe. Transitional arrangements allowed individual EU countries to


temporarily breach the principle of free movement of people inside the Union and to impose


restrictions on immigration from the new member states. Boeri and Brücker (2005) …nd that


these arrangements a¤ected the geographical orientation of migrants from the new member


states and resulted in substantial diversion of migration ‡ows from countries closing their


borders to countries with more open rules.


Contrasting with these developments in the empirical literature, there have been few


attempts to integrate these factors into formal models of immigration policy formation. In


particular, most existing models, such as Benhabib (1996), de Melo et al. (2001), Dolmas


and Hu¤man (2004), Ortega (2005) and Facchini and Willman (2005), incorporate some


form of domestic factors, but are silent about international determinants.3 The main reason


is that standard theory focuses on the e¤ects of immigration on a single receiving economy


2See also Hatton and Williamson (2005).
3There are some recent noticeable exceptions that are discussed at the end of the Introduction.


3




and considers as exogenous the migratory decision of foreign workers (see Borjas, 1995).4


These features, by construction, shut down any possible cross-border spillover created by


immigration policy. The present work contributes to …lling this gap in the formal literature


by providing a simple and tractable model of immigration policy interdependence.


In our model immigration policy and migration choices are endogenous. The set up con-


siders two regions. The receiving region is formed of a set of identical countries that choose


independently immigration policy. In order to make a convincing case of the mechanism


discussed in this paper, we rely on a general model that is broadly consistent with distrib-


utional, political economy and non-economic determinants of immigration policy (a speci…c


model is presented in Appendix A). Immigration is assumed to have bene…ts and costs on


host economies, so that there is an optimal number of foreign workers for each receiving


country. The sending region is populated by a set of workers who can choose whether to


migrate or not and -to a certain extent- in which country to move to. Migratory decisions


depend on the economic incentives that foreign workers face and on the policy regulating


migratory ‡ows enacted in the receiving countries.


If the world had only a single receiving country, a host government could easily select a


policy that supports the e¢ cient level of immigration -that is, the level that optimally trades


o¤ the costs and bene…ts of immigration. Governments, however, do not act in a vacuum:


immigration policy in one country alters the migratory choices of foreign workers and, hence,


the ‡ows of migrants into other destinations (the immigration policy spillover). Note


that this externality is created by the international mobility of prospective migrants. When


foreign workers choose, not only whether to migrate or not, but also where to migrate (i.e. the


destination country), policy restrictions (liberalizations) in one country increase (decrease)


migratory ‡ows in other receiving economies, as a larger number of migrants will target the


country with lower restrictions. In other words, the costs and bene…ts of immigration in any


host economy are, in part, determined by the policy stance of other receiving countries. This


international externality lowers the ability of national governments to optimally manage their


immigration policy.


In this interdependent environment, coordination failures can materialize that lead to


4Few theoretical contributions have considered the interdependence between immigration policy in the host
economy and immigration decisions. See, in particular, Bellettini and Berti Ceroni (2007), Bianchi (2007) and
Giordani and Ruta (2010).


4




ine¢ cient equilibria. The choice of immigration policy is strategic and de…nes a symmetric,


simultaneous game among all destination countries from which multiple symmetric policy


equilibria emerge that can be Pareto-ranked. The "cooperative solution", that is, the immi-


gration policy associated with the optimal number of migrants for each country, is only one


in the continuum of Nash equilibria of this policy game. Coordination failures in immigra-


tion policy may arise because, for each policy maker, expectations on the behavior of the


governments of other destination economies are critical in the determination of the policy


outcome of the receiving region. For instance, if any one government expects that others will


strengthen immigration barriers, then it will …nd it convenient to restrict its policy stance


to neutralize the negative externality of an excessive in‡ux of migrants, thus triggering a


series of restrictive measures. Too little immigration will result relative to the e¢ cient level


for the overall destination region. Similarly, beliefs of immigration liberalizations by other


receiving economies will trigger a reduction in restrictions that will result in a Pareto-inferior


equilibrium characterized by too much immigration.


Once we identify the problem that characterizes immigration policy in this framework,


we discuss two further issues. First, we analyze the problem of equilibrium selection and


show that coordination failures in immigration policy are not only possible, but they are


also likely to emerge in presence of uncertainty on the policy strategy of other receiving


governments. The game-theoretic literature has proposed alternative equilibrium re…nements


for coordination games admitting a multiplicity of equilibria. These re…nements stress the


fact that players may coordinate on a strategy which is less risky, even if Pareto-dominated.5


In particular, we characterize the immigration policy equilibrium that is robust to strategic


uncertainty (Andersson et al., 2010) and show that the Pareto-e¢ cient equilibrium is not


robust -i.e. that the unilateral policy outcome may well support ine¢ ciently low or high


immigration.


The second issue that we investigate is how an increase in the international mobility of mi-


grants (for instance due to technological innovations, such as improvements in transportation


and communication means) a¤ects the "likelihood" of a coordination failure. We …nd that


an increase in migrants’mobility does not change the e¢ cient policy for the receiving region,


5The classic work is Harsanyi and Selten (1988) on the risk-dominant equilibrium. Cooper et al. (1990 and
1992) and van Huyck et al. (1990) …nd that coordination failures are likely to arise in experimental settings.
For a survey of the empirical literature see Cooper (1999), chapter 1.


5




but it expands the set of equilibria (a measure of the indeterminacy of equilibria) and alters


the robust equilibrium, as it increases each policy maker’s uncertainty about other govern-


ments’strategies. Intuitively, both …ndings can be rationalized as an increased international


mobility of migrants magni…es the cross-border externality associated with immigration pol-


icy. This suggests that the "globalization" may be amplifying the chances of coordination


failures across destination countries, thus augmenting the need for policy coordination in the


immigration domain.


While we leave a further discussion of the policy implications of our model to the conclu-


sion, some preliminary considerations can be put forward. First, while both trade policy and


immigration policy are characterized by a cross-border externality, the immigration policy


game has radically di¤erent features. Trade policy interactions determine a (terms-of-trade


driven) prisoner’s dilemma situation, while interactions in the domain of immigration pol-


icy lead to a trust dilemma, a coordination problem where governments achieve e¢ cient


policies only if they make mutually consistent decisions. Second, while the trade policy game


leads to too little trade, coordination failures in immigration policy may determine either too


little or too much immigration from the perspective of the receiving world. Third, multilat-


eral institutions should help countries escape ine¢ cient equilibria. This theory suggests that


immigration policy commitments (that can be credibly enforced) can provide a coordination


device to receiving countries.


Our work is related to several recent studies. The literature on asylum seeking has mod-


elled the spillover e¤ect in national refugee laws and emphasized that coordination problems


may emerge in this context (Hatton, 2004, Facchini, Lorz and Willmann, 2006). The paper


by Bubb, Kremer and Levine (2007), in particular, is closely related to ours. They show


that restricting refugee law in some host countries (i.e. increasing the standard of proof to


distinguish between refugees and migrants) may induce other host economies to do the same


and that this may lead to a multiplicity of equilibria in refugee law. Recent formal work


on immigration policy emphasizes di¤erent channels of international interdependence in this


domain. Brücker and Schröder (2010) build a model where a destination country’s e¤ort


to improve the skill composition of its immigration pool induces other host economies to


adopt similar immigration reforms. De la Croix and Docquier (2010) …nd positive external-


ities in immigration policy in a model where host economies have an aversion to the global


6




inequality created by barriers to international labor movements. Relative to these papers


there are two main innovations in our work. First, the immigration policy spillover is the


result of an endogenous response of potential migrants to di¤erences in immigration policy


in host economies. Second, we formally analyze the determinants of multiple equilibria in


immigration policy and study the issue of equilibrium selection.


The paper is organized as follows. Section 2 introduces the multiple-country framework.


In this setting, we formalize the immigration decision of foreign workers and the immigration


policy spillover. In Section 3 we prove the existence of a multiplicity of Nash equilibria and


carry out the comparative statics analysis. Section 4 studies the issue of equilibrium selection


under strategic uncertainty. A concluding section discusses the implications of this model for


the design of international institutions governing immigration.


2 A Multiple-Country Model of Immigration Policy


In this section we introduce a model of immigration policy with a sending region, populated


by F workers, and a receiving region composed of m countries, indexed by h = 1; :::;m. Each


host economy has identical fundamentals, but decides immigration policy independently of


the other destination countries. Foreign workers can choose whether to migrate or not and


where to locate in the receiving region. This setting is su¢ cient to determine the interna-


tional spillover characterizing immigration policy and, hence, the type of strategic problem


associated with unilateral immigration policy in the receiving world.


Immigration has bene…ts and costs for the host economies. De…ne the welfare of the


generic economy h in the receiving region as a continuous function in the number of immi-


grants in the country, Wh (Ih), and suppose that this function admits one and only one …nite


maximum at Ih = Î. This value of immigration is the one which optimally trades o¤ costs


and bene…ts of migrants for the host economy. While we are agnostic about the source of


these costs and bene…ts, a standard model of immigration policy that supports this structure


is presented in Appendix A.


7




2.1 Migratory Choices and the Policy Spillover


We now introduce the migratory choice of foreign workers. Immigration is a non-reversible


decision. Each migrant faces a psychological cost to leave her own country, i, which is


uniformly distributed in [0; ], where is normalized to 1. The government in h can set


up an immigration policy which is parametrized by a cost borne by immigrants once in the


new country, h 2 R+. This parameter can be interpreted in several ways, from the cost of
bureaucratic procedures that each immigrant faces in the host economy to laws that a¤ect


the life of immigrants in the host country, such as the number of years to obtain voting rights


or citizenship.


Migrants are internationally mobile, in the sense that in a world formed of several po-


tential host economies they have some freedom in choosing their destination. Clearly, the


international mobility of migrants is limited by a series of factors in addition to immigration


restrictions in the receiving world, including primarily geographical distance, but possibly


other factors such as technology (e.g. communication technologies) or cultural diversity (e.g.


adaptability to di¤erent cultures).6


We capture the limited international mobility of migrants by assuming that foreign work-


ers are of two kinds. A fraction F , with 2 (0; 1), can decide freely which receiving country
to move to in the set m ("free foreign workers"). The remaining fraction (1)F are instead
constrained in their choice ("constrained foreign workers"). For reasons of symmetry, we fur-


ther suppose that each receiving country can attract at most (1)F=m constrained foreign
workers; that is, potential migrants of "constrained type" are distributed uniformly across


the receiving region. The parameter captures the international mobility of migrants. A


higher value of , that is, an increase in the set of "free foreign workers", can be motivated


by several factors that reduce the (non-policy) constraints to the migrants’mobility, such as


an improvement in transportation or telecommunication technologies.


A constrained foreign worker in the pool (1 )F=m, indexed by i, will migrate to h if
and only if


bh (h) i 0; (1)
6See, among others, Belot and Hatton (2008) and Grogger and Hanson (2008).


8




where bh is the endogenous net bene…t that the foreign worker receives if she migrates to


country h, assumed to be a twice continuously di¤erentiable and decreasing function of h.
7


It is immediate to …nd the threshold value of the psychological cost (such that all those below


that value are willing to migrate) as


h = bh (h) :


Given that i is distributed uniformly in [0; 1], the number of constrained migrants to h will


be bh (h) (1)F=m.
The number of free foreign workers potentially entering each country h is instead given


by the whole pool of free foreign workers, F . In addition to satisfying condition (1), free


foreign workers will also compare the payo¤ obtained by migrating to country h to the one


obtained by migrating to any other receiving country (denoted by h).8 Free foreign workers
will target country h if


bh (h) > bh

h


() h < h
Therefore, policy di¤erences in the destination world a¤ect migration choices. Speci…cally,


the number of free foreign workers actually migrating to h is 0 if h > h (crowding out),


and bh (h)F if h < h (crowding in). Finally, if h = h, free migrants are indi¤erent


and distribute symmetrically across the receiving region, that is, bh (h)F=m for any h.


As a result, immigration ‡ows to country h are a function of h’s immigration policy as


well as of the measures imposed in the rest of the destination countries. The total number


of (constrained plus free) migrants to country h can then be described as Ih

h; h



=


bh (h)Fh, where


Fh =


8>><>>:
F [(1) =m+] F if h < h
(1)F=m F if h > h
F=m ~F if h = h:


(2)


This e¤ect of immigration policy abroad on the ‡ow of migrants into the host economy


is the key cross-border externality in this model and the mechanism of economic interdepen-


7The fact that a tightenening of immigration policy reduces the foreign workers’bene…ts from immigration
is a desirable feature for any reasonable model of immigration policy. The model developed in appendix has
this feature.


8The absence of asymmetric equilibria (which will be proven in Appendix C) allows us to simplify the
notation: bh and h denote the (identical) bene…t and policy set in all m 1 countries other than h.


9




dence that we highlight. Importantly, the theory closely captures the essential international


policy spillover emphasized in the empirical literature discussed in the Introduction.


Two related considerations seem relevant. The …rst is on the interpretation of parameter


in the model. If is equal to zero (i.e. no international mobility of foreign workers),


then ~F = F = F and there is no policy spillover. As increases, di¤erences in immigration


policies among destination countries have a larger e¤ect on the ‡ow of migrants. In other


words, can be interpreted as an elasticity -i.e. the responsiveness of migrants to policy


di¤erences. Factors such as improvements in transportation and communication technologies


or proximity are likely to increase this elasticity and hence magnify the size of the immigration


policy spillover.


The second consideration relates to the size of this international externality. In their


study, Timmer and Williamson (1998) …nd that the e¤ect of the immigration policy spillover


is statistically signi…cant but small, while Boeri and Brücker (2005) show that policy dif-


ferences increased by up to …ve times immigration to more open EU members compared to


the counterfactual of free mobility in the EU. The two studies need not be in contradiction


as they are consistent with di¤erent values of in the model. The international mobility


of migrants from the Old to the New World in the 19th century was limited by distance


and technological factors compared to modern immigration from Eastern to Western Europe.


This is consistent with a higher value of in the latter immigration episode and, hence, with


a stronger policy externality.


3 Multiple Policy Equilibria and Coordination Failures


Given the international externality created by the migratory behavior of foreign workers, we


now characterize the equilibrium immigration policies by studying the strategic interaction


among receiving countries. Formally, this interaction can be represented as a symmetric


coordination game among the governments of the m destination countries, each deciding


its own immigration policy in a non-cooperative fashion. As we will see, this game admits a


continuum of symmetric, Pareto-rankable, Nash equilibria. In particular, we prove that there


exists an interval of immigration policies

;



such that, if all countries but h select any


policy in that interval, country h will …nd it best to do the same. We also show that there


10




exists a pay-o¤ dominant equilibrium belonging to that interval, and that such equilibrium


is associated with policy ̂ which, if implemented by all host countries, is able to "attract"


the optimal number of migrants, Î, for all of them. This policy is the one solving equation


Î = bh (̂) ~F . All other equilibria around this optimal policy equilibrium are instead sub-


optimal and represent a coordination failure among the receiving countries driven by the


immigration policy spillover.


A coordination failure arises in this game because immigration policies across receiving


countries are strategic complements. To give an intuition, start from the optimal policy


equilibrium, ̂. If all other countries but h restrict their policy above ̂, country h is better


o¤ following this restriction rather than su¤ering the “crowding in”of migrants that would


result from sticking to ̂. This incentive continues up to policy . Symmetrically, if all


other countries but h loosen up their policy below ̂, country h is better o¤ by implementing


this softer policy stance rather than su¤ering a “crowding out”of migrants. This incentive


continues up to policy . A strategic complementarity across host countries is thus responsible


for the positive co-movement of immigration policies documented in the data.


Let us now de…ne the payo¤ function of the government of generic country h as a func-


tion of its immigration policy, h, and of the policy strategy followed by all other receiv-


ing countries, h, h

h; h



. This payo¤ function can be found by substituting for


Ih

h; h



= bh (h)Fh (where Fh is given in (2)) into the welfare function, Wh (Ih). Intu-


itively, this payo¤ function is a step function which depends on whether h is higher, lower


or equal to h. If it is lower, country h will experience a crowding in (Fh = F ), and the


payo¤ function is obtained by substituting for Ih = bh (h)F into Wh (Ih). If it is higher,


country h will experience a crowding out (Fh = F ), and expression Ih = bh (h)F is instead


substituted into Wh (Ih). Finally, if it is equal, migrants distribute equally across the host


region (Fh = ~F ), and the payo¤ function is obtained by substituting for Ih = bh (h) ~F into


11




Wh (Ih). We can then write9


h

h; h



=


8>><>>:
Wh (h; F ) if h > h
Wh



h; ~F



if h = h


Wh

h; F



if h < h:


(3)


This payo¤ function is drawn in Figure 1. The solid curve in Figure 1 represents country


h’s welfare when its immigration policy is equal to the one implemented in the rest of the


receiving region (Wh

h; ~F



). The dashed curve captures h’s welfare when its policy stance


is more restrictive than abroad (Wh (h; F )), while the dotted curve represents the opposite


case (Wh

h; F



). Each of them is assumed to be twice continuously di¤erentiable and


strictly concave in h (in the standard model of immigration policy developed in appendix


we study the conditions for which this is the case -see Appendix B). Note that the optimal


number of migrants for country h is unambiguously given by Î and, hence, the three functions


have the same maximum. However, the policy delivering this level of immigration depends


on whether this policy is higher, lower or equal to the one implemented abroad. In Figure 1,


we have called these policy values respectively ̂,̂, ̂.10


INSERT FIGURE 1 HERE


Before proving the existence of the continuum of policy equilibria, we provide a simple


intuition of this result. Along the interval

;



it is Wh



h; ~F



Wh (h; F ) ;Wh



h; F



.


Assume that all other countries set up a policy h 2

;



. Then, it is easy to show


that, for country h, any payo¤ associated with h 6= h is lower than the one associated
with h = h. In fact, suppose country h sets up a policy h lower than h. Then, for


9The pay-o¤ function is not continuously di¤erentiable, which prevents us from using the standard tools of
di¤erential calculus to …nd the best-response functions and the Nash equilibria of the game. Note also that,
albeit more complicated, this function resembles the pay-o¤ function of a Bertrand competition game with
homogeneous goods, in which each …rm’s pro…t depends on whether its price is higher, lower or equal to the
one set up by its rivals (see for instance Tirole, 1988, pp. 209-211). In particular, the policy game described
in this paper shares many features with price competition games where …rms’costs are assumed to be convex
(Dastidar, 1995, Weibull, 2006).


10 It is easy to prove that ̂ < ̂ < ̂. Moreover, de…ning "open door" policy (od) and "closed door" policy


(cd) as the policies which induce, respectively, all foreign workers and no foreign worker to emigrate to h,


it is possible to prove that Wh

cd; ~F



= Wh



cd; F



= Wh



cd; F



, while Wh



od; F



> Wh



od; ~F



=


Wh

od; F



(as depicted in Figure 1). These proofs are available upon request from the authors.


12




any h < h, function Wh

h; F



lies uniformly below Wh



h; ~F



, that is to say, any


h < h is associated with lower welfare than h = h. On the other hand, suppose


country h chooses a policy h higher than h. Then, for any h > h, function Wh (h; F )


lies uniformly below Wh

h; ~F



. As a result, whatever h 2



;



, country h’s best


response is h = h.11


Policy ̂ belongs to the interval

;



and is a Nash equilibrium. Indeed, it is the pay-


o¤ dominant Nash equilibrium in that, if all other countries set up ̂, country h is able to


attract the optimal number of migrants Î by adopting the same policy, h = ̂. Equilibria


surrounding the optimal policy equilibrium are Pareto-inferior outcomes which result from


a coordination failure driven by the international policy spillover associated with migrants’


mobility across the receiving region. A graphical representation of the set of equilibria is


provided in Figure 2. We can enunciate the following


INSERT FIGURE 2 HERE


Proposition 1 There exist a lower and an upper threshold, and , such that any symmetric


con…guration of immigration policies, (1; :::; m) = (
; :::; ), for which 2 ; , is a


Nash equilibrium of the game. The optimal policy equilibrium h = ̂ 8h belongs to the set
of symmetric Nash equlibria. All other equilibria are sub-optimal and are Pareto-ranked by


the distance from ̂.


Proof. In Appendix C


The logic of the proof is simple and consists of exploiting some of the properties of the three


welfare functions Wh

h; ~F



,Wh (h; F ) ;Wh



h; F



(such as continuity and strict concav-


ity) to prove that, for any h 2

;



and for any h, it isWh



h; ~F



Wh (h; F ) ;Wh



h; F



.


11This reasoning only applies along the interval

;



. Suppose for instance h > . In this case, country


h’s best response would be to slightly undercut policy h. This softer policy (implying a crowding in country


h), would be associated with a higher welfare, that is, Wh

h; F



> Wh



h;


~F

. An analogous reasoning


applies to policy values below .


13




As a result, along that interval enacting the same policy as the rest of the region will be better


than enacting any other policy above or below that policy.12


Another way to look at this set of Nash equilibria is by drawing the reaction curves of


the host countries in the immigration policy game. The reaction function of generic country


h is drawn as the black line in space h; h in Figure 3. For any h 2

;



country h’s


best response is h = h. Hence, along that interval, the reaction curve is a 45 degree line


(as in Bryant’s (1983) game). For any h < ̂ (h > ̂), country h’s best response is to


set up ̂ (̂) -as that policy allows country h to attract the optimal number of migrants, Î.


Hence, the reaction curve is a horizontal line along that policy value. When h is any value


inside the interval [̂; ), country h’s best response is to set up a slightly (by a however small


") tougher immigration policy, and the best response is drawn as the solid black line slightly


above the 45 degree line. Finally, when h 2 [; ̂), country h’s best response is to set up a
slightly (by a however small ") softer immigration policy, and the best response is drawn as


the solid black line slightly below the 45 degree line.13 The best-response function of country


h can then be written as


brh

h



=


8>>>>>>><>>>>>>>:


̂ if h < ̂


h + " if h 2 [̂; )
h if h 2



;



h " if h 2 (; ̂]
̂ if h > ̂


(4)


The reaction curve of country h is the mirror image of the one of country h and is depicted as
the light grey line in Figure 3. They overlap along the interval



;



, which then constitutes


the measure of equilibria, while no intersection occurs when h is lower than or higher than


.


12Note that removing the assumption of symmetric fundamentals would not alter the logic of this result.
Speci…cally, if countries had asymmetric fundamentals, the policy attracting the optimal number of migrants
would di¤er across host economies. However, starting from this optimal policy con…guration, strategic com-
plementarities still characterize immigration policy.


13 In rigourous mathematical terms the best response function is not de…ned when h belongs to [̂; ) or


to [; ̂), the reason being that we have de…ned the policy variable as a continuous variable. With an abuse
of notation we write h " instead of ? in the expression for the best-response function (4), as if variable
were de…ned as a discrete variable which could only take multiple values of an indivisible ". This is because
we here privilege intuition to rigour. Of course, nothing substantial changes.


14




INSERT FIGURE 3 HERE


The above discussion illustrates the key problem associated with immigration policy when


the receiving economy is formed by multiple countries: coordination failures can arise in this


environment. The economy can be stuck in an inferior Nash equilibrium where restrictions


to immigration are either ine¢ ciently high (h 2 (̂; ] 8h) or ine¢ ciently low (h 2 [; ̂)
8h), and hence destination countries fail to attract the "right" number of foreign workers.
The reason for this ine¢ ciency is the international spillover created by immigration policy,


which in turn results from the international mobility of migrants (i.e. their ability to choose


their destination in addition to whether they want to migrate or not).


Starting at an ine¢ cient equilibrium, no country can improve its welfare with unilateral


immigration policy initiatives, but all receiving economies could be made better o¤ under an


agreement that called for mutual policy adjustments. In this respect, immigration policy has


much to learn from trade policy, even if the structure of the immigration and the trade policy


game is quite di¤erent. In particular, most authors consider current immigration policy in


advanced economies as too restrictive.14 As this model shows, excessive restrictions can be


the result of a coordination failure among receiving countries. No country would unilaterally


choose to loosen up its policy stance as this would result in an in‡ux of migrants beyond its


e¢ cient point. While each government, acting independently, is powerless to coordinate the


policy choices of the others, an international agreement could provide governments with an


avenue to commit to a reduction of immigration restrictions and escape from a coordination


failure.


However, the model also makes quite clear the di¤erences in the economic problem facing


trade and immigration policy makers. Di¤erently from the trade context, in the immigration


policy game among receiving countries the key element is con…dence rather than con‡ict. As


it is well known from the trade literature (Bagwell and Staiger, 1999 and 2002), excessive


trade restrictions can be the result of a terms-of-trade driven prisoner’s dilemma situation.


In contrast, the analysis of this section shows that receiving countries face a trust dilemma,


a coordination problem which leads to multiple equilibria. While in both situations govern-


14 In particular, several authors …nd that there would be global gains from lowering immigration restrictions
that limit the movement of workers from low-income to high-income countries. See Clemens, Montenegro and
Pritchett (2008), Hanson (2008) and Rosenzweig (2007).


15




ments can be stuck at an ine¢ cient equilibrium, a key issue of the immigration policy game is


equilibrium selection (an issue that does not emerge in a prisoner’s dilemma situation, where


there is only one equilibrium). Governments may coordinate on the ine¢ cient equilibrium as


this is the one that is associated with policy choices that are less "risky", an issue that will


be addressed in Section 4.


3.1 Migrants’International Mobility and Coordination Failures


An important question is how the set of equilibria is a¤ected by the underlying parameters


of the model. In particular, in this subsection we study the e¤ect on the receiving countries


of a change in the international mobility of foreign workers (). As discussed above, this


parameter captures the responsiveness of migrants to di¤erences in the policy stance and is


determined by factors, such as technology, that are likely to change over time.


We begin by stating the following


Proposition 2 An increase in international mobility of foreign workers () expands the set


of symmetric Nash equilibria, while it does not a¤ect the Pareto dominant equilibrium. That


is,
d


d
< 0,


d


d
> 0 and




d
= 0 :


Proof. In Appendix D.


To grasp the intuition, recall that function Wh

h; ~F



is not a¤ected by changes in


as the adoption of the same policy across the receiving region neutralizes the spillover e¤ect.


Therefore, the solid curve in Figure 2 does not move as varies. On the other hand, following


an increase in , functionWh (h; F ) shifts leftward and functionWh

h; F



shifts rightward.


Intuitively, workers’ international mobility is responsible for the cross-border externality,


which is the source of the equilibrium multiplicity. An increase in international mobility


implies a more powerful externality and an ever expanding measure of policy equilibria.


Taken together, Propositions 1 and 2 have two implications. First, higher realizations


of parameter , by expanding the set of equilibria, worsen the problem of coordination


failure and indeterminacy. Second, as the "new" equilibria are more distant from the optimal


policy, they are associated with lower welfare for the receiving region. To put it di¤erently,


16




this result suggests that the new wave of globalization, driven by a fall in transportation and


communication costs, may be exacerbating coordination failures and increasing the gains


from immigration policy coordination for all receiving countries.


4 Selection of Equilibria under Strategic Uncertainty


The previous section illustrates the possibility of coordination failures in immigration policy


due to the presence of multiple Nash equilibria in the immigration policy game. Whether


coordination failures actually occur depends on which equilibrium policy makers coordinate.


As shown above, the "payo¤-dominant" equilibrium, the one associated with policy h = ̂


8h, is in the set of equilibria. This, however, is not necessarily the equilibrium that players
select.


Experimental evidence on coordination games quite convincingly rejects the view that


coordination problems will not occur in simple strategic interactions (Cooper et al., 1992, Van


Huyck et al., 1990). One possible rationalization of this evidence is that payo¤-dominance is


not the only basis for coordination, and that players may converge towards other equilibria


which present alternative salient features. An alternative proposed in this literature is that


players coordinate towards the "risk-dominant" equilibrium (Harsanyi and Selten, 1988), the


key insight being that a strategy may be preferred over the other if it is less risky in the face


of strategic uncertainty.15


An evolution of this equilibrium selection criterion, which applies to games characterized


by a continuous space of strategies (as the immigration policy game introduced in Section 3),


is the robustness to strategic uncertainty (Andersson et al., 2010). Whenever a game admits


a continuum of equilibria, even the slightest uncertainty about the opponents’ strategies


might lead each player to deviate from any given policy equilibrium. It is then "arguably


reasonable to require equilibria to be robust to small amounts of uncertainty about other


players’strategies" (Andersson et al., 2010, p.2).16


15Whether players are more likely to coordinate towards payo¤-dominant or risk-dominant equilibria is the
focus of empirical literature (for a survey see Cooper, 1999). The experimental evidence in Cooper et al.
(1992) shows that risk-dominance can provide a better guide to equilibrium selection than payo¤-dominance.


16Speci…cally, Andersson et al. (2010) show that there is only one equilibrium surviving the robustness
test in a price competition game with a continuous strategy space and admitting a continuum of equilibria
(Dastidar, 1995). Abbink and Brandts (2008) and Argenton and Muller (2009) provide experimental evidence
in favor of this "robust equilibrium".


17




In this subsection we prove that there is a unique equilibrium which is robust to strate-


gic uncertainty and show that the robust equilibrium is di¤erent from the payo¤-dominant


equilibrium. This result reveals that coordination failures in immigration policy are not only


possible but also likely to emerge.


We …rst formally characterize strategic uncertainty in the immigration policy game. Fol-


lowing Andersson et al. (2010), we model strategic uncertainty by assuming that the proba-


bilistic belief of policy maker h about the action of any other government j in the receiving


economy is given by:


~hj = j + t"hj ;


where t 2 R+ and "hj hj are statistically independent noise terms. The distribution hj
belongs to an arbitrary family of probability distributions with non decreasing hazard rate


function.


The introduction of this noise de…nes a new, "perturbed", game. Intuitively, the robust


equilibrium is an equilibrium of this perturbed game when the noise tends to zero. More


formally, if an equilibrium strategy pro…le (r; :::; r) is the unique limit to any sequence of


equilibria indexed by t as t ! 0, that strategy pro…le is robust to strategic uncertainty. In
the next proposition we prove that such limit exists and is unique.


Proposition 3 There exists a unique equilibrium which is robust to strategic uncertainty.


This equilibrium, (r; :::; r), is de…ned by Wh (r; F ) =Wh

r; F


8h and is Pareto-inferior
to the payo¤ dominant equilibrium (̂; :::; ̂).


Proof. In Appendix E.


Policy r is the one for which the incentives to restrict or loosen the immigration policy


stance for each strategically uncertain government in the receiving region exactly o¤set each


other. As shown in Figure 4, the robust equilibrium of the immigration policy game corre-


sponds to the point where the functions Wh (h; F ) (the dashed curve) and Wh

h; F



(the


dotted curve) intersect. In this point, denoted by A, the expected welfare loss associated


with a policy higher or lower than the rest of the host region tends to zero.


18




INSERT FIGURE 4 HERE


As there is a continuous strategy space, for any policy h, government h’s subjective


probability that any other government will choose exactly the same policy is zero. Hence,


with probability one, policy h will either be the lowest or not. In the …rst case, country h


will experience a crowding in, in the second it will experience a crowding out. In Figure 4, for


any policy h 2 (r; ], a government facing uncertainty on the strategies of other receiving
governments has an incentive to lower its immigration restrictions. The reason being that


the welfare if other countries’policies are less stringent (Wh (h; F )) is lower than the welfare


if other receiving countries’immigration policies are more restrictive than the one set up in


h (Wh

h; F



). Conversely, for h 2



; r



, every government has an incentive to raise


restrictions as the expected welfare under a crowding in (Wh

h; F



) is lower than under a


crowding out of migrants (Wh (h; F )). Only for policy
r welfare under crowding in and


crowding out are equal and a government has no incentive to alter its policy stance.


A comparison of the robust and Pareto-dominant equilibria sheds light on two issues.


First, under strategic uncertainty, the immigration policy equilibrium is distinct from the


one that maximizes welfare for the entire host region. The policy strategy robust to strategic


uncertainty can be more or less stringent than the optimal policy depending on the funda-


mentals of the economy (which determine the shapes of the two curves in Figure 4). In this


model, where immigration has both bene…ts and costs for the host economy, the presence of


an immigration policy spillover may, therefore, induce countries to select an excessively re-


strictive or loose policy. In other words, this model suggests that coordination failures driven


by the immigration policy spillover can give rise to both a "race to the top" and a "race to


the bottom" in immigration restrictions in receiving countries (see, for instance, Boeri and


Brücker, 2005).


Second, an increase in the international mobility of migrants () does not a¤ect the


optimal immigration policy but alters the robust equilibrium. Intuitively, the expected welfare


loss associated with both a crowding in and a crowding out increases with the size of the


immigration policy spillover (the dotted and the dashed curves in Figure 4 move further


apart, while the position of the solid curve is not a¤ected by changes in , see Proposition


2). Therefore, globalization, in the sense of an increase in the international mobility of


19




migrants, exacerbates the strategic uncertainty by increasing the set of Nash equilibria, and


has an ambiguous e¤ect on the robust equilibrium on which governments coordinate.


5 Conclusions and Policy Implications


This paper has examined receiving countries’motives in setting immigration policy and how


the institutional framework, particularly the absence of e¤ective coordination mechanisms,


translates these motives into policy outcomes. The analysis shows that policy at home is


in‡uenced by measures adopted abroad. The reason is that migrants choose where to locate,


in part in response to immigration policies in host economies. In the model, the international


mobility of migrants gives rise to a policy spillover e¤ect which rationalizes the evidence in


recent empirical studies on immigration. In this interdependent environment, immigration


policy becomes strategic and unilateral behavior may well lead to coordination failures, where


receiving countries are stuck in a welfare inferior equilibrium. The theory also shows that


ine¢ cient policy equilibria are more likely to emerge when governments are uncertain about


the immigration policy of other receiving countries and when the international mobility of


migrants is stronger.


In the rest of this section, we discuss some implications of this model and come back to the


initial question of this paper on the economic rationale for international institutions governing


immigration. A …rst implication of the model is that the type of coordination problem facing


immigration policy makers is di¤erent from the one facing trade policy makers. While both


unilateral trade and immigration policies may lead to an ine¢ cient equilibrium, the nature


of this equilibrium in the two cases is not the same. In the trade policy game, the …rst-best


policy outcome (i.e. trade openness) is not an equilibrium of the game as each government


has an incentive to impose restrictions when the others choose free trade. Instead, the key


element of coordination failures in the immigration policy game is the lack of con…dence in


the policy choice of other governments, not an inherent policy con‡ict as in trade policy. In


other words, it is the inability of policy makers to commit to the e¢ cient immigration policy


vis-à-vis other countries that constrains e¢ cient outcomes in this domain.


A second implication of the model is that, di¤erently from other contributions in the


literature (Hatton, 2007, Hanson, 2009), it provides a clear economic rationale for interna-


20




tional institutions dealing with immigration. Starting from an ine¢ cient policy equilibrium,


no receiving country could improve its welfare through unilateral policy actions. Speci…cally,


if restrictions in advanced economies are excessively high (as most experts believe), a looser


policy in any single country would result in a large in‡ux of migrants that would lower its


welfare. An international agreement that called for mutual policy adjustments in receiving


economies (i.e. a joint reduction of restrictions) could improve upon the initial ine¢ ciency


by neutralizing the immigration policy spillover. While not the only possible rationale, this


provides a strong case for the need of multilateral institutions in the immigration domain.


In facts, several international agreements and organizations aim at coordinating immigra-


tion policy. For instance, the stated objective of the International Organization for Migration


is "to promote international cooperation on migration issues". Moreover, a forum for dialogue


on migration and immigration policy is also provided by other international institutions, such


as the OECD. While these arrangements help coordination through dialogue and the dissem-


ination of information among receiving countries, they do generally not envisage an e¤ective


enforcement mechanism. This implies that the uncertainty on other governments’strategies


still characterizes policy makers’decisions, which can lead to coordination failures.


An exception is Mode 4 of the General Agreement on Trade in Services (GATS), which


provides an opportunity to WTO Members to take on commitments regarding the tempo-


rary presence of "natural persons" from a di¤erent Member who supply a service.17 While


GATS Mode 4 has a limited scope, the binding nature of commitments within the WTO,


backed up by the enforcement mechanism provided by its dispute settlement system, is an


appealing feature of this system.18 In this sense, expanding the scope of Mode 4 may be in


the interest of receiving countries. However, one should be aware of the di¢ culties of this


process. Immigration policy is not limited to border measures, but includes a large number


17Natural persons falling within the scope of Mode 4 include independent contractual service suppliers and
natural persons employed by service suppliers (WTO, 2004). Speci…cally, Mode 4 concerns a narrow (and not
clearly de…ned) subset of temporary migration, as it excludes coverage of access to labour market, citizenship
and employment on a permanent basis (see WTO Annex on Movement of Natural Persons).


18The size and scope of Mode 4 movements are an issue of current debate and negotiation. While a number
of WTO Members have undertaken Mode 4 commitments that cover short-term employees (the US binding
of 65.000 H-1B visas is a noteworthy example), the overall degree of Mode 4 commitments are low. WTO
Members have generally granted access to selected categories of highly skilled persons linked to a commercial
presence, such as managers, executives and specialists. The Hong Kong Ministerial declaration in December
2005 called for a new impetus on Mode 4 commitments (e.g. an extension of the categories of natural persons
included in the commitments and of the permitted duration of stay), but improvements in the ongoing Doha
negotiations have been so far slow to materialize (see Carzaniga, 2009).


21




of behind-the-border measures that a¤ect the welfare of foreign workers in the host economy.


As the trade experience shows, regulating this policy can be extremely challenging.


Another implication of this analysis is that the extent of the coordination problem depends


on the magnitude of the policy spillover e¤ect. In the model this is captured by the size of the


parameter -i.e. the international mobility of migrants. While in the paper we emphasized


technology as a determinant of this parameter, other factors can in‡uence the responsiveness


of foreign workers to immigration policy di¤erences. In particular, receiving countries that are


more strongly interconnected, because of geographic proximity, common cultural background,


or because they have formed an economic union, will experience stronger immigration policy


spillovers and are, therefore, more likely victims of coordination failures. Institutions that


allow for e¤ective coordination (or, the creation of a single immigration policy) are more


valuable in these circumstances. This provides formal support to the frequent calls in the


policy debate for a single immigration policy in an integrated area such as the European


Union (Boeri and Brücker, 2005). Similarly, a greater involvement of the States of the US


(and, hence, a more limited role of the federal government) in immigration policy -implicit


in the law passed in the State of Arizona in 2010- may lead to welfare reducing coordination


failures within the US, as the choice of one State will inevitably a¤ect others through location


decisions of foreign workers within the US and trigger a series of policy responses in other


States.


References


[1] Abbink, K. and J. Brandts (2008). "24. Pricing in Bertrand Competition with Increasing


Marginal Costs". Games and Economics Behavior, 63, 1-31.


[2] Amir, R. (1996). "Cournot Oligopoly and the Theory of Supermodular Games". Games


and Economic Behavior, 15, 132-148.


[3] Andersson, O., C. Argenton and J.W. Weibull (2010). "Robustness to Strategic Uncer-


tainty in Price Competition". SSE/EFI Workin Paper NO. 726.


22




[4] Argenton, C. and W. Müller (2009). "Collusion in Experimental Bertrand Duopolies


with Convex Costs: the Role of Information and Cost Asymmetry”. CentER Discussion


Paper 2009-87.


[5] Armenter, R. and F. Ortega (2010). "Credible Redistribution Policy and Skilled Migra-


tion". European Economic Review, Forthcoming.


[6] Bagwell, K. and R.W. Staiger (1999). "An Economic Theory of GATT". The American


Economic Review, Vol. 89, No. 1, pp. 215-248.


[7] Bagwell, K. and R.W. Staiger (2002). "The Economics of the World Trading System",


Boston, MA: MIT Press.


[8] Bellettini, G. and C. Berti Ceroni (2007). “Immigration Policy, Self-Selection, and the


Quality of Immigrants”. Review of International Economics, 15(5), 869-877.


[9] Benhabib, J. (1996). "On the Political Economy of Immigration". European Economic


Review, 40(9), 1737-1743.


[10] Belot, M. and Hatton, T.J (2008). "Immigrant Selection in the OECD", CEPR Working


Paper No. 6675.


[11] Bhagwati, J.N. (2003), “Borders Beyond Control”, Foreign A¤airs, 82, pp. 98-104.


[12] Bianchi, M. (2007). "Immigration Policy and Self-Selecting Migrants", mimeo, Paris


School of Economics.


[13] Boeri, T. and H. Brücker (2005). "Why Are European So Tough on Migrants?". Eco-


nomic Policy, 44, pp. 629-704.


[14] Boeri, T., G.H. Hanson and B. McCormick (2002). "Immigration Policy and The Welfare


System: A Report for the Fondazione Rodolfo Debenedetti", Oxford Press, Oxford.


[15] Borjas, G.J. (1994). “The Economics of Immigration”. Journal of Economic Literature,


pp. 1667-1717.


[16] Borjas, G. J. (1995). "The Economic Bene…ts from Immigration", Journal of Economic


Perspectives, 9(2), pp. 3-22.


23




[17] Borjas, G.J. (2003). "The Labor Demand Curve Is Downward Sloping: Reexamining


the Impact of Immigration on the Labor Market". The Quarterly Journal of Economics,


118(4), pp. 1335-1374.


[18] Brücker, H. and P.J.H. Schröder (2010). "Migration Regulation Contagion". Mimeo,


University of Aarhus.


[19] Bryant, J. (1983). "A Simple Rational Expectations Keynes-Type Model". The Quarterly


Journal of Economics, 98, pp. 525-528.


[20] Bubb R., M. Kremer and D. Levine (2007). “The Economics of International Refugee


Law". Forthcoming, Journal of Legal Studies.


[21] Carzaniga, A. (2009). "A Warmer Welcome? Access for Natural Persons Under PTAs",


in Juan Marchetti and Martin Roy (eds.), Opening Markets for Trade in Services, Cam-


bridge University Press, Cambridge, UK.


[22] Casella A. (2005). "Redistribution Policy: A European Model". Journal of Public Eco-


nomics, 89, 1305-1331.


[23] Clemens, M., C. Montenegro and L. Pritchett (2008). "The Great Discrimination: Bor-


ders as a Labor Market Barrier", mimeo, Center on Global Development.


[24] Cooper, R.W. (1999). "Coordination Games, Complementarities and Macroeconomics",


Cambridge University Press, Cambridge.


[25] Cooper, R.W., D. V. DeJong, R. Forsythe and T. W. Ross (1990). "Selection Criteria


in Coordination Games". The American Economic Review, 80, pp.218-233.


[26] Cooper R.W., D. V. DeJong, R. Forsythe and T. W. Ross (1992), "Communication in


Coordination Games". The Quarterly Journal of Economics, 107, pp. 739-771.


[27] Dastidar, K. G. (1995). "On the existence of pure strategy Bertrand equilibrium". Eco-


nomic Theory, 5, 19-32.


[28] De la Croix, D. and F. Docquier (2010). "An Incentive Mechanism to Break the Low


Skill Immigration Deadlock". Mimeo. UCL.


24




[29] De Melo, J., J. Grether and T. Müller (2001). "The Political Economy of International


Migration in a Ricardo-Viner Model". CEPR Discussion Paper No. 2714.


[30] Dolmas, J. and G.W. Hu¤man (2004). "On The Political Economy Of Immigration and


Income Redistribution". International Economic Review, 45(4), pp. 1129-1168.


[31] Dustmann, C. and I. Preston (2007). "Racial and Economic Factors in Attitudes to


Immigration," The B.E. Journal of Economic Analysis & Policy, Berkeley Electronic


Press, vol. 7(1).


[32] Ellison, G. (1995) . "Basins of Attraction and Long Run Equilibria,” mimeo. Massa-


chusetts Institute of Technology.


[33] Facchini, G., O. Lorz and G. Willmann (2006). "Asylum Seekers in Europe: The Warm


Glow of a Hot Potato", Journal of Population Economics, Vol. 19:2, pp. 411-430.


[34] Facchini, G., A. Mayda and P. Mishra (2008)."Do Interest Groups A¤ect US Immigration


Policy?", CEPR Discussion Paper no. 6898.


[35] Facchini, G. and G. Willmann (2005). "The Political Economy of International Factor


Mobility". Journal of International Economics, 67(1), pp. 201-219.


[36] Giordani, P. and M. Ruta (2010). "Prejudice and Immigration". Mimeograph Luiss and


WTO.


[37] Goldin, C. (1993), "The political Economy of Immigration Restriction in the United


States". NBER Working paper No. 4345.


[38] Grogger, J. and G. Hanson (2008). "Income Maximization and the Selection and Sorting


of International Migrants". Mimeograph UCSD.


[39] Hanson, G.H. (2008). "The Economic Consequences of the Interanational Migration of


Labor", Annual Review of Economics, forthcoming.


[40] Hanson, G.H. (2009). "The Governance of Migration Policy", Human Development Re-


search Paper (HDRP) Series , Vol. 02, No. 2009.


25




[41] Harsanyi, J.C. and R. Selten (1988). "A General Theory of Equilibrium Selection in


Games". Cambridge: The MIT press.


[42] Hatton, T.J. (2004). "Seeking asylum in Europe". Economic Policy, 38, pp. 5–62


[43] Hatton, T.J. (2007). "Should we have a WTO for International Migration?". Economic


Policy, 22(4), pp. 339-383.


[44] Hatton, T.J. and J. Williamson (2005). "Global Migration and the World Economy:


Two Centuries of Policy and Performance". Cambridge: The MIT press.


[45] O’Rourke, K.H and R. Sinnott (2006). "The Determinants of Individual Attitudes To-


wards Immigration". European Journal of Political Economy, 22, 838-861.


[46] Ortega, F. (2005). "Immigration Quotas and Skill Upgrading". Journal of Public Eco-


nomics, 89(9-10), pp 1841-1863.


[47] Razin, A., E. Sadka and P. Swagel (2002). "Tax Burden and Migration: A Political


Economy Theory and Evidence". Journal of Public Economics, 85, pp. 167-190.


[48] Rosenzweig, M. (2007). "Education and Migration: A Global Perspective", mimeo, Yale


University.


[49] Timmer, A.S. and J.G. Williamson (1998). "Immigration Policy Prior to the 1930s:


Labor Markets, Policy Interactions, and Globalization Backlash". Population and De-


velopment Review, 24(4), 739-771.


[50] Tirole, J. (1988). "The Theory of Industrial Organization". MIT Press.


[51] van Huyck, J., R. Battalio and R. Beil (1990). "Tacit Coordination Games, Strategic


Uncertainty, and Coordination Failure". American Economic Review, pp. 234-248.


[52] Weibull, J.W. (2006). "Price Competition and Convex Costs". SSE/EFI Working Paper


No. 622.


[53] WTO (2004), Trade and Migration: Background Paper, available at


http://www.wto.org/english/tratop_e/serv_e/sem_oct04_e/background_paper_e.pdf


26




A A Standard Model of Immigration Policy


This section introduces a speci…c model of immigration policy with standard features which


may help give more structure to the reasoning developed in the main text. In particular,


Subsection A.1 opens the "black box" of the welfare function we de…ned at the beginning of


Section 2, Wh (Ih). Subsection A.2 does the same for the migratory choice, by specifying the


foreign workers’bene…t from immigration.


A generic receiving country, denoted by h ("home"), is populated by Nh ("native") work-


ers, each of whom supplies one unit of labor inelastically, and by Kh capitalists, each of


whom is endowed with one unit of capital. A …nal good is produced competitively via a


constant-return-to-scale technology in labor and capital:


Yh = K

hL


1
h :


Lh is the sum of natives and immigrants working in country h, that is, Lh = Nh+Ih, where Ih


denotes the endogenous number of migrants. The …nal good is the numeraire in the receiving


economy, and its price is normalized to one. As the product market is competitive, input


factors are paid their marginal productivities:


wh = (1 )

Kh
Lh



and rh =



Kh
Lh


1
:


Country h has a welfare system that, de facto, redistributes income from capitalists to


workers. Speci…cally, the policy consists of a …xed lump-sum transfer h to (native and


foreign) workers which is …nanced through a proportional tax h 2 [0; 1] on the capital rent.
This simple formulation captures the idea that welfare spending in h depends on the number


of migrants.19 A balanced government budget implies


hrhKh = h (Nh + Ih) ;


and hence the tax rate on capital income, as a function of the number of immigrants is


19The welfare system is assumed to be pre-existent to immigration. This is reasonable when the size of the
migrant labor force is low relative to the size of the native population. When this is not the case, one can
think that welfare and immigration policy are jointly determined (see for instance Casella, 2005 and Armenter
and Ortega 2010).


27




h (Ih) =
h (Nh + Ih)


rh (Ih)Kh
: (5)


A.1 The Optimal Number of Immigrants


We introduce a general representation of the Home government preferences over immigrants,


which includes both the case where policy makers maximize the host economy’s national


welfare as well as the general possibility that governments are also motivated by the distrib-


utional e¤ects of immigration among natives. We assume that agents use their (disposable)


income to purchase the …nal good and have a linear utility function in consumption. Let us


de…ne the objective function of the government as a function of the number of immigrants as


Wh (Ih) [rh (Ih) Kh h (Nh + Ih)] + (1 ) [wh (Ih) + h]Nh; (6)


where we used the above balanced budget condition (5) to substitute for h and where


2 [0; 1] is the political bias (i.e. the weight on the utility of capitalists). This formulation
includes as a special case national income maximization for = 1=2.


The optimal number of migrants in country h, denoted by Î, is the one which maximizes


condition (6).20 The FOC of this problem is


@Wh
@Ih


= (1 )


Kh
Nh + Ih



(1 ) Nh


Nh + Ih



h = 0:


A number Î solving the FOC above is a maximum if the second derivative, evaluated in Î, is


strictly negative, that is, if


@2Wh
@I2h





= (1 )



Kh


Nh + Î


1
Nh + Î



(1 ) Nh


Nh + Î



(1 ) Nh


Nh + Î



< 0.


In Appendix B we provide a su¢ cient condition for Î to be the global maximum, that is, the


only politically optimal number of migrants for the host country.


20We refer to Î as the "optimal" number of migrants. Needless to say, this is the "politically-optimal" level of
immigration, as it maximizes the government’s objective function, and it corresponds to the "socially-optimal"
number of migrants only in the special case in which = 1=2.


28




A.2 The Migratory Choice


We can now be more precise about the net bene…ts from migration for foreign workers (bh)


and thus about their migratory choice. Bene…ts from migration are given by wh + h, that


is, the salary plus the welfare transfer. Costs are given by h + i, that is, the policy plus


the psychological cost. If we normalize the wage rate in the sending region (w) to zero, a


constrained foreign worker i will migrate to h if and only if


wh + h h i 0; (7)


from which we can determine the threshold value of the psychological cost as


h = wh + h h:


Thus the number of constrained migrants (as function of h) will be h(1)F=m.
On the other hand, a free foreign worker migrates to h if condition (7) holds, and if the


payo¤ in h is higher than any other payo¤ in the rest of the host region, that is, if21


wh + h h i > wh + h h i () h < h:


Immigration ‡ows to country h are function of h’s immigration policy as well as of the


measures imposed in the rest of the destination countries. In particular, the number of


free foreign workers actually migrating to h is 0 if h > h (crowding out), and hF if


h < h (crowding in). Finally, if h = h, free migrants distribute symmetrically across


the receiving region, that is, hF=m for any h.


The total number of migrants to country h can then be described as Ih = hFh, where


h = wh + h h and Fh is given in (2).
21Since countries are symmetric by assumption, it is h = h. Moreover, it is easy to show that in this


model dwh=dh 2 (0; 1). Thus the following condition holds true.


29




B Characterization of the Pay-O¤ Function


The payo¤ function, which is de…ned only implicitly in the main text (expression (3)), can


here be characterized explicitly as


h

h; h


"Kh Kh
Nh + hFh


1
h [Nh + hFh]


#
+ (8)


(1 )

(1 )



Kh


Nh + hFh



+ h



Nh;


where h and Fh are functions of both h and h, and Fh is de…ned in (2). Thus, we can


draw three distinct welfare functions depending on whether h is higher, lower or equal to


h. If h = h, the welfare function, Wh

h; ~F



, is given by (8) but where Fh = ~F . If


h > h (crowding out), the welfare function Wh (h; F ) is given by (8) but where Fh = F .


Finally, if h < h (crowding in), then the welfare function, Wh

h; F



, is still given by


(8) but where Fh = F . The next subsection invetigates the conditions under which the three


welfare functions are strictly concave.


B.1 Strict Concavity of the Welfare Functions


Consider welfare function


Wh (h; F ) =


"
Kh



Kh


Nh + hF


1
h (Nh + hF )


#
+


(1 )

(1 )



Kh


Nh + hF



+ h



Nh:


The …rst derivative can be written as


@Wh (h; F )


@h
=



(1 )



Kh
Lh



(1 ) Nh


Lh



h





~F


(1 )

Kh
Lh



F
Lh


+ 1
= 0;


30




where Lh = Nh + hF . It is now possible to calculate the second derivative as


@2Wh (h; F )


@2h
=



(1 )



Kh
Lh


1
Lh



(1 ) Nh


Lh



(1 ) Nh


Lh



0@ F


(1 )

Kh
Lh



F
Lh


+ 1


1A2 +

(1 )



Kh
Lh



(1 ) Nh


Lh



h



0@ F


(1 )

Kh
Lh



F
Lh


+ 1


1A2

1 2KhLh FL2h


(1 )

Kh
Lh



F
Lh


+ 1
;


where again Lh = Nh + hF . Simple algebra shows that the following condition on the


parameters of the model ensures that @2Wh (h; F ) =@
2
h is strictly lower than zero for any


value of h:


> H(F ) 1 +
(1 + )



1 + h


F
Nh+F



+ 2


KhF


N+1h


: (9)


We assume that this condition is satis…ed. Given that H(F ) is a decreasing function in F


(and given that F < ~F < F ), condition (9) also ensures the strict concavity of the other two


welfare functions, Wh

h; F



and Wh



h; ~F



.


C Proof of Proposition 1


Welfare function Wh (h; F ) admits a global maximum in ̂, and it is continuous, strictly


decreasing and strictly concave in the interval

̂; ̂



. Welfare function Wh



h; ~F



admits


a global maximum in ̂, and it is continuous, strictly increasing and strictly concave in the


same interval

̂; ̂



. Since Wh



̂;F



= Wh



̂; ~F



, and since ̂ < ̂, then the two curves


must cross once and only once in the interval

̂; ̂



. Denote this intersection point by .


Moreover, for exactly the same reasons, it must be that Wh

h; ~F



Wh (h; F ) for any


h 2

; ̂



. An entirely analogous reasoning holds for the other interval,



̂; ̂



: there


exists a unique value 2

̂; ̂



, such that Wh



; ~F



= Wh



; F



, and it must be that


Wh



h; ~F



Wh



h; F



for any h 2


h
̂; ̂


i
.


As a result, for any h 2

;



it is Wh



h; ~F



Wh



h; F



;Wh (h; F ), and hence


whatever h 2

;



, country h’s best response is h = h. Thus, any policy h 2



;



31




8h is a symmetric Nash equilibrium of the game.
Since < ̂ < , policy ̂ is a Nash equilibrium of the game. Moreover, it is immediate


to show that it is the optimal policy equilibrium: when all other countries h set up ̂, then
h = ̂ is the policy which maximizes welfare function Wh



h; ~F



, as it allows country h to


attract the optimal number of migrants Î. Moreover, since Wh

h; ~F



is strictly increasing


in h in the interval

; ̂



and strictly decreasing in the interval [̂; ], it is also immediate


to verify that the e¢ ciency loss is greater, the higher the distance from the optimal policy.


This proves that the equilibria are Pareto-ordered by the distance of from ̂.


Notice that any h < as well as any h > 8h are not Nash equilibria. If all other
countries set up policy h in the interval



̂;



, then Wh (h; F ) Wh



h; ~F



, and


country h’s best response is to set up a slightly tighter policy. As a result, there are no


equilibria below . If instead h < ̂, country h’s best response is simply h = ̂. The


same reasoning applies for h > .


Finally, a simple contradiction argument (drawn from Amir et al., 1996) proves that


asymmetric equilibria do not exist in this policy game. Let (1; 2; 3:::; h; :::; m) be an


asymmetric equilibrium (thus with at least two ’s being distinct). Assume then, w.l.o.g.,


that 1 = maxh fhg and 2 = minh fhg so that 1 > 2. Since the game is symmetric,
every permutation of (1; 2; 3:::; h; :::; m) is also an equilibrium. Consider for instance


(1; 2; 3:::; h; :::; m) and (2; 1; 3:::; h; :::; m). The fact that both of them are equilib-


ria implies that country 1 strictly weakens its immigration policy from 1 to 2 as the other


countries restrict theirs from (2; 3:::; h; :::; m) to (1; 3:::; h; :::; m), which contradicts


the fact that country 1’s best-response is nondecreasing (see expression (4)).


D Proof of Proposition 2


The lower threshold is by de…nition the immigration policy such thatWh

; ~F



=Wh



; F



,


where, remind, ~F F=m and F (1)F=m. Let us de…ne


G

;


Wh ; F Wh ; ~F = 0


32




as the implicit function of with respect to . It holds that


d


d
=


dG
d
dG
d


: (10)


We show that d=d < 0.


The numerator writes as


dG


d
=
dWh



; F



d



dWh



; ~F



d


:


The second term is null as Wh

; ~F



does not depend on . The …rst term can be shown


to be negative. In fact
dWh



; F



d


=
@Wh



; F



@I


@I


@F


@F


@
:


First note that @Wh

; F



=@I is positive since in point welfare increases with the number


of immigrants. Second, since I = b



F , it is @I=@F > 0. Finally, it holds @F=@ =


F=m < 0. Then it is dWh

; F



=d < 0 and hence dG=d < 0.


As regards the denominator in (10), it is


dG


dh





=
dWh



; F



dh



dWh



; ~F



dh


< 0


as in point it is dWh

; F



=dh < 0 and dWh



; ~F



=dh > 0. This proves that d=d <


0:


The proof that d=d > 0 is entirely analogous and is then omitted. Finally, it is trivial


to show that the optimal policy is not a¤ected by an increase in international mobility.


Notice that functionWh

h; ~F



does not depend on , and thus ̂, as a solution to equation


dWh



h; ~F



=dh = 0, will not depend on either.


E Proof of Proposition 3


The proof will follow closely the argument developed in Andersson et al. (2010) for the price


competition game. Let t 2 R+ and suppose that the government of each country h holds a


33




probabilistic belief about any other government j’s policy of the following form:


~hj = j + t"hj ; (11)


for some statistically independent noise terms "hj hj . Distribution hj belongs to an
arbitrary family of cumulative distribution functions, D : R ! [0; 1], characterized by
non-decreasing hazard rate function.22


Given that the support of random variable ~hj is

od; cd



, this variable distributes


according to the following cumulative distribution function (c.d.f.):


Dthj (x) =
hj



xj
t



hj



odj


t



hj



cdj


t



hj



odj


t



The introduction of uncertainty de…nes a new, "perturbed", game. For any t 2 R+,


a strategy pro…le (1; 2; 3:::; h; :::; m) is a t-equilibrium if, for each player h, the
strategy h maximizes h’s expected payo¤ under the probabilistic belief of the form given in


(11). Notice that a t equilibrium is simply a Nash equilibrium of this perturbed game.
A strategy pro…le r is (strictly) robust to strategic uncertainty if, for any collection of


c.d.f’s hj 2 , there exists a sequence of t equilibria,

tk
1
k=1


with tk ! 0, such that
tk ! r as k !1. We now apply these de…nitions to our policy game.


For any policy h that the policy maker of country h decides to implement, her subjective


probability that any other policy maker will choose exactly the same policy is zero. Hence,


with probability one, her policy will either be the lowest or not. In the …rst case, country


h will experience a "crowding in", in the second it will experience a "crowding out". Under


strategic uncertainty, country h will then select that policy which maximizes the following


expected payo¤ function:


th () =
Y
j 6=h


241 hjhjt hj

odj


t



hj



cdj


t



hj



odj


t



35 Wh h; F + (12)8<:1Y


j 6=h


241 hjhjt hj

odj


t



hj



cdj


t



hj



odj


t



359=; Wh (h; F ) :


22Many common distributions, such as the normal or the exponential distribution, present this feature. This
is the only assumption we impose on this arbitrary family of distribution functions, and it is useful to provide
easily a su¢ cient condition for the uniqueness of the robust equilibrium.


34




The expression above represents the expected payo¤ of country h when setting up policy


h. In particular, the …rst term is equal to the probability that h is lower than any other


policy j 8j times the payo¤ associated with the resulting "crowding in". The second term
is instead given by the probability that h is higher than at least one j times the payo¤


associated with the resulting "crowding out". The FOC for the maximization problem writes


as


@th()
@h


=
Y
j 6=h


24 hjcdjt hjhjt
hj



cdj


t



hj



od
h
j
t



3524@Wh(h;F)


@h
@Wh(h;F )@h



Wh(h;F)Wh(h;F )t


P
j 6=i


hj



hj


t



hj



cdj


t



hj



hj


t



35+


@Wh(h;F )
@h


= 0;


where hj () = 0hj (). It can be proven that the objective function (12) is strictly concave
along the interval



̂; ̂



(to make the argument developed here less burdensome, this proof


is provided separately in Subsection E1). Hence, every solution to the FOC above in that


interval is a t-equilibrium.


Consider any sequence htki1k=1 ! 0 and de…ne limk!1 kh h (for the Bolzano-Weierstrass
theorem this limit exists and belongs to the interval



od; cd



). We now investigate what


happens to the solution to the FOC when t tends to zero. In particular, we prove that


htki1k=1 ! 0 implies Wh

h; F



=Wh (



h; F ), whose solution is



h =


r 8h.
The FOC can be rearranged as follows:


tk


8><>:@Wh(kh;F)@kh
Y
j 6=h


264hj


cdkj
tk


!
hj



odh


k
j


tk


!


hj



cdk


j
tk


!
hj



k
h
k


j
tk


!
375+ @Wh(kh;F)@kh @Wh(kh;F)@kh


9>=>;
=


h
Wh



kh; F



Wh



kh; F


iX
j 6=i


hj



kh


k
j


tk


!


hj



cdk


j
tk


!
hj



k
h
k


j
tk


!


Suppose by contradiction that, when t tends to zero, it isWh

h; F


6=Wh (h; F ). Then,


35




in order for the FOC to be satis…ed when htki1k=1 ! 0, it must necessarily be that


X
j 6=i


hj



kh kj


tk


!
! 0:


The expression above says that, in the limit, the sum of the instantaneous probabilities that


any j is equal to h must tend to zero. This is true if

j 6= h 8j. We now show that this


is impossible.


The fact that Wh

h; F


6= Wh (h; F ) implies that, for the generic government h,
Wh



h; F



is either higher or lower than Wh (h; F ). Suppose for instance it is higher (the


reasoning under the opposite case in which Wh

h; F



< Wh (



h; F ) is entirely analogous


and is omitted). If that is the case then, in order for h to be a best response, it must


necessarily be that r < h 6 j for any j. But since j > r, then also for country j it


must be that Wj

j ; F



> Wj



j ; F



, and thus j 6 h for any h. The two implications


are true only when h =

j , which contradicts the above statement. As a result, in order for


the FOC to be true, it must necessarily be that Wh

h; F



= Wh (



h; F ) whose solution is


h =
r for any h. This completes the proof.


E.1 Concavity of the Objective Function


We here prove that a su¢ cient condition for function (12) to be strictly concave along the


interval

̂; ̂



is that distribution hj has non-decreasing hazard rate. The FOC can equiv-


alently be written as


@th()
@h


= @Wh(h;F )@h

0@1Y


j 6=h


24 hjcdjt hjhjt
hj



cdj


t



hj



od
h
j
t



351A+


Y
j 6=h


24 hjcdjt hjhjt
hj



cdj


t



hj



od
h
j
t



3524@Wh(h;F)


@h
Wh(h;F)Wh(h;F )t


P
j 6=i


hj



hj


t



hj



cdj


t



hj



hj


t



35 = 0:


36




It is now easy to show that function @th () =@h is strictly decreasing in h. Starting from


the …rst addend, its derivative has the following expression and is negative:


@Wh(h;F )@h
D
0@Y
j 6=h


24 hjcdjt hjhjt
hj



cdj


t



hj



od
h
j
t



351A




+


0@1Y
j 6=h


24 hjcdjt hjhjt
hj



cdj


t



hj



od
h
j
t



351A


+


@2Wh(h;F )
@2h


< 0


where D () stands for "derivative with respect to h". For practical reasons, the signs of the
four terms are denoted under each of them. In particular, while the signs of the last three


terms are self-apparent, the …rst term is negative whenever h > ̂.


Turning to the second addend, the expression under the product operator is decreasing


(as the cumulative distribution hj () is an increasing function of h). The derivative of
@Wh



h; F



=@h with respect to h is also strictly decreasing by assumption. The di¤erence


Wh

h; F


Wh (h; F ) is instead increasing in h, at least for any h < ̂. Finally, notice
that the function under the sum operator is non-decreasing to the extent that the hazard


rate, de…ned as


h



h j


t



=


hj



hj


t



1 hj



hj


t


:
is assumed to be non-decreasing. Given that


hj



h j


t



hj



cd j


t


!
1 8h


a decreasing hazard rate implies that the function under the sum operator is decreasing as


well.


37




hmm̂ m̂m̂
cdm


hW


( )FW hh ~,m ( )FW hh ,m( )FW hh ,m


Wod


Figure 1: The payo¤ function of receiving country h depending on whether h is higher, lower
or equal to h.


38




hmm̂
cdm


hW


The span of equilibria


m mW
od


Figure 2: The policy equilibria of the game.


39




m




hm


h-mm m m̂m̂


m




450


Figure 3: The best-response functions of country h (in black) and of country h (in grey).


40




hmm̂
cdm


hW


( )FW hh ~,m ( )FW hh ,m( )FW hh ,m


A


rmm mWod


Figure 4: The equilibrium robust to strategic uncertainty in point A.


41




Login