Immigration Policy and Self-Selecting Migrants

We build a simple model of self-selection into migration and immigration policy determination. We first show that the effect of any immigration policy can be decomposed into a size and a composition effect. We then explore how the optimal policy may change once the latter effect is considered.


Introduction
It is commonly understood that several e¤ects of immigration in receiving countries depend crucially on immigrants' characteristics. 1 At the same time, the relation between immigration policy and immigrants'skill composition remains largely unexplored. In the economics literature as in policy debates, the demand and supply sides of immigration tend to be considered separately. 2 As a …rst step towards …lling this gap, we develop a simple model in which both immigration policy and immigrants'skill composition are determined in equilibrium. Whether an individual with a given skill decides to migrate depends on the immigration policy implemented and at the same time that policy depends on whether high or low skilled individuals are more likely to migrate. We are interested in exploring how the optimal immigration policy changes once it is taken into account that any such policy a¤ects immigrants'skill composition.
It is important to look at both the supply and demand sides in a uni…ed framework because of a well known fact. Migrants are not a random sample of their home country population. Incentives to migrate and resources to pay the migration costs vary with skills. Given immigrants' self-selection, any immigration policy a¤ects not only the size but also the skill composition of the migration ‡ow. Since self-selection determines how di¤erent potential migrants respond to a policy change, understanding what drives such selection becomes crucial for optimal policy design.
We develop this argument in a setting with two countries. In the sending country, individuals, called foreigners, are endowed with di¤erent skills and wealth, and depending on their endowment they decide whether to work at home or migrate to the receiving country. In the receiving country, individuals, called natives, support an immigration policy that maximizes their equilibrium wages. In particular, high skilled natives aim at increasing the supply of low skilled immigrants, while low skilled natives push for increasing the supply of high skilled immigrants. According to these preferences and to the weight attached to di¤erent groups in the population, i.e. low vs. high skilled and immigrants vs. natives, the receiving country government sets the immigration restrictions.
In our main analysis, we focus on immigration restrictions which a¤ect the cost migrants have to pay to enter and work in the receiving country, such as direct fees or bureaucratic requirements that increase the time and money needed to comply. These restrictions are assumed to be the same 1 Con…ning attention to the economics literature, see Borjas (1994), Friedberg and Hunt (1995), Chiswick, Lee and Miller (2005) on the labor market e¤ects, and Storesletten (2000), Lee and Miller (2000), Chojnicki, Docquier and Ragot (2011) on the e¤ects on public …nance.
2 Some notable exceptions are mentioned below.
for all immigrants. This allows us to emphasize that even in this case the policy a¤ect immigrants in di¤erent ways, thereby determining their selfselection. 3 In fact, on the one hand, the restrictions imply that only the richest foreigners can migrate, and these tend to be the high skilled. On the other, the restrictions induce only those with the most to gain to migrate. If returns to skills are higher in the sending country, these migrants tend to be the low skilled. Hence, depending on whether immigration is driven by incentives or wealth constraints, and on whether returns to skills are higher in the sending or in the destination country, restrictions may improve or worsen immigrants'skill composition. We then decompose the e¤ect of immigration policy in the receiving country as an e¤ect on the size and an e¤ect on the composition of the migration ‡ow. We show that these e¤ects typically work in opposing directions: the size e¤ect, whereby the number of immigrants is varied, while keeping their skill composition …xed, hits hardest those foreigners with a higher propensity to migrate; conversely, the composition e¤ect tends to be stronger on those who migrate less. Moreover, the composition e¤ect may dominate the size e¤ect, especially at lower levels of restrictions: those foreigners with the lowest propensity to migrate may be, in absolute terms, the most sensitive to a policy change. Hence, the composition e¤ect may reverse the immigration policy outcomes as predicted by the size e¤ect alone.
We illustrate some implications of this result by considering the optimal policy design by a utilitarian government maximizing natives'total income. In this setting, if immigrants'skill composition was exogenous, the government would not restrict immigration. Open immigration would maximize the bene…ts arising from skill complementarities between natives and immigrants. However, the fact that restrictions in ‡uence immigrants' skills implies that even this utilitarian government may optimally implement positive immigration restrictions. The reason is that restrictions help in inducing the optimal skill mix of immigrants. This observation may provide a rationale for positive immigration restrictions even absent any redistributive concerns or political economy distortions. As we show, what is needed is that immigrants receive a lower weight than natives in the government's welfare function.
We also sketch some implications of our results for the political economy of immigration, and in particular for the determination of natives' preferences concerning immigration policy. The composition e¤ect implies that such preferences may not be fully de…ned in terms of current immigrants' skill composition. For example, some natives may support a more restrictive policy even though current immigrants are not detrimental to them, since tighter restrictions would change immigrants'skill composition in their favor.
The present paper builds on two broad streams of literature. On the supply side of migration, we model the migration decision as a basic human capital investment (Sjaastad, 1962), in which self-selection may be driven both by cross-country returns to skills (as in Borjas, 1987) and by wealth constraints (as in Lopez and Schi¤, 1998, Chiquiar and Hanson, 2005, and Friebel and Guriev, 2006. Unlike most of this literature, the emphasis is in how immigration policies may a¤ect immigrants'self-selection. Second, we contribute to the literature on the determination of immigration policies. 4 Apart for stressing the interaction with the supply side, our approach is novel in that we consider migration cost as the policy variable. 5 This variable seems important as any restriction to immigration entails, at least indirectly, monetary costs. Moreover, the exercise appears useful even if one considers our policy variable literally as a tax on immigrants. Such a tax has recently received attention in policy debates (see Freeman, 2006 andLegrain, 2007), but its e¤ects have received little attention in formal models. 6 As stressed, the interaction between demand and supply appears underemphasized in this literature. Two notable exceptions are Bellettini and Ceroni (2007) and Giordani and Ruta (2008). While similar in spirit, both their modeling approaches and their results are di¤erent. Bellettini and Ceroni (2007) assume that immigrants are positively self-selected and argue in favor of a high immigration quota: by reducing wages in the receiving country, this would increase immigrant quality and maximize national income. Giordani and Ruta (2008) focus instead on how immigration prejudices can be self-ful…lling: restrictive policies tend to attract low skilled immigrants and these immigrants are a drain on welfare in the receiving country, which in turn sustains anti-immigration attitudes.
The rest of the paper is organized as follows. Section 2 presents the basic model; Section 3 derives the main results. In Section 4, we discuss the robustness of our results to alternative assumptions and propose some avenues for extensions. Section 5 concludes by suggesting some policy implications. Omitted proofs are provided in the Appendix. 4 See for example Benhabib (1996), who explores how the median voter determines minimal capital requirements for admission and Epstein and Nitzan (2006) and Facchini and Willman (2005), who use a lobbying model to explain the formation of immigration quotas. 5 See Myers and Papageorgiou (2002) for a comparison of di¤erent immigration policies when the sending and receiving countries have homogeneous populations. 6 From an historical viewpoint, it is also interesting to notice that the …rst interventions to limit and select immigration ‡ows in the U.S. and Canada acted on costs rather than on quantities (see Timmer and Williamson, 1998 for a detailed account).

The model
Consider a world with two countries, a sending and a receiving country. We are interested in the interaction between workers in the sending country, who may decide to migrate, and the receiving country's government, which sets the immigration policy.

The sending country
The sending country is populated by a continuum n of workers, called foreigners. Foreigners are heterogeneous in three respects: skill, migration cost, and initial wealth. Let n denote the mass of foreigners with skill , where 2 fH; Lg: 7 A foreigner i with skill may migrate to the receiving country and earn the endogenous wage w , or he can work in the sending country for an exogenous wage w : 8 If he migrates, each foreigner has to incur a monetary cost . This cost has to be paid up-front, and no borrowing is possible, so migration may be limited by wealth constraints. Speci…cally, foreigners are endowed with some wealth, drawn by a distribution with continuous density ! : When is interpreted as an observable skill (like education), it is assumed that the high skilled tend to be wealthier than the low skilled (see Filmer and Pritchett, 1999;and Piketty, 2000). 9 Formally, this writes (1) for every 2 R + : 10 In addition, the migration cost includes an individual-speci…c psychological cost " i ; which may re ‡ect individual characteristics like age, family ties, access to networks in the origin and destination country. 11 Speci…cally, " i is assumed to be a random variable following a log-concave 7 We consider the sending country skill composition as exogenous in order to emphasize that immigration policy induces a composition e¤ect (described in Section 3.3) even when the pool of foreigners is given. For a complementary approach, see the recent literature on endogenous skill acquisition and migration (e.g. Mountford, 1997;Stark, Helmenstein and Prskawetz, 1997;Vidal, 1998;Beine, Docquier and Rapoport, 2001). 8 While wages in origin countries may be a¤ected by emigration (see Hanson, 2005 andMishra, 2007), we focus here on the e¤ects in the receiving country. 9 We notice in Section 3.3.1 how this framework can be applied to selection on unobservable characteristics.
1 0 Equation (1) assumes conditional stochastic dominance, which is slightly stronger than …rst order stochastic dominance and weaker than the standard assumption of monotone likelihood ratio (see Krishna, 2002).
1 1 In our formulation, these elements are not systematically correlated with the skill : One may instead assume that the low-skilled have higher migration costs, since for example they can hardly give up the support of their community in terms of access to credit (as in Banerjee andNewman, 1998 andMunshi andRosenzweig, 2009) or unemployment insurance (as in Cuecuecha, 2005). This would be qualitatively similar to our framework, in which wealth constraints are more likely to be binding for the low-skilled. cumulative distribution with continuous density . 12 This assumption implies that the ratio is decreasing. (2)

The receiving country
The receiving country is populated by a continuum n of workers, here called natives, who are heterogeneous in skill 2 fH; Lg: 13 Natives are assumed to have a linear utility function which depends only on equilibrium wages w : 14 These wages are determined in a competitive labor market as where F (N H ; N L ) is the receiving country production function and N is the sum of natives and immigrants with skill We focus on purely redistributive e¤ects of immigration, whereby immigrants compete with similarly skilled natives and complement natives with di¤erent skills. In particular, we simply let the production technology be a constant returns to scale Cobb-Douglas function where 2 (0; 1): The receiving country government is interested in regulating the in ‡ows of immigrants as these in ‡uence natives'utility. Its goal is to maximize the welfare function where denotes the weight attached to group 's utility. We will mostly consider a utilitarian function with = n : 15 Immigration policy acts on ; which is the cost foreigners have to incur to enter and work in the receiving country. This policy is common to all immigrants. As mentioned in the Introduction, and further discussed in Section 4.1, we want to emphasize that even such a policy a¤ects their self-selection. Hence, we write the government's program as 1 2 Log-concavity is satis…ed by basically all "named" distribution functions (see Bagnoli and Bergstrom, 2005).
1 3 While one may model the interaction between immigration policy and human capital formation in the receiving country, we wish here to make our point in the simplest setting and so take the receiving country skill distribution as given.
1 4 We discuss more general formulations in Section 4.3. 1 5 Some extensions and possible ways to endogenize these weights are discussed in Section 4.2.

Analysis
We now show that, in order to set the optimal policy, the receiving country government has to predict the e¤ects of such policy on immigrants' skill composition. This in turn requires an understanding of the forces driving the decision to migrate.

The migration decision
A foreigner i with skill prefers to migrate if w ( + " i ) w ; so for each skill there exists a cut-o¤ value " w w such that any individual with skill and a cost " i lower than " would like to migrate. In addition, this individual must be su¢ ciently wealthy to pay the migration cost : Thus, the supply of migrants with skill is de…ned by where q is the fraction of foreigners with skill who can a¤ord and who are willing to move, i.e.
We de…ne immigrants' skill composition as the ratio of high to low skilled migrants, i.e.
and we say that immigrants are positively self-selected if and only if Q 1:

Optimal immigration restrictions
According to equations (3) and (5), equilibrium wages in the receiving country can be written as and where R is the ratio of high to low skilled workers Hence, the receiving country skill distribution and equilibrium wages depend on migration ‡ows, and then on the immigration policy . 16 We can write the government's program in equation (7) as The low skilled bene…t from a high R, i.e. a large in ‡ow of high skilled immigrants, and the high skilled bene…t from the opposite. Given these preferences, the optimal policy depends on the weights . The higher H , the lower the R induced by such a policy. 17 For now, we ignore redistributive concerns or other political economy distortions, and consider a purely utilitarian setting in which each group is valued by its size. In this setting, no immigration restrictions are imposed if immigrants are given the same weight as natives. In fact, if then the welfare function W in (6) does not depend on R; i.e. on high vs. low skilled wages, but only on total production. Hence, W is maximized by setting = 0. A preference for high or for low skilled workers instead arises when immigrants receive a lower weight than natives. In this case, the government sets its policy so as to bene…t the group of workers with the lowest proportion of immigrants. Suppose the government cares only about natives, then = n ; and we have that In this case, the welfare function W is convex in R and it has a minimum at R = n H =n L : E¢ ciency gains from immigration are minimized when immigrants have the same skill composition as the native population, i.e. when x H =x L = n H =n L . Since the government maximizes e¢ ciency, i.e. natives' total income, it aims at optimizing the skill ratio R. As a benchmark for our following analysis, suppose that immigrants'skill composition was exogenous. In that case, setting the immigration policy and achieving the optimal skill ratio would be easy. We would have and so imposing no immigration restriction would be optimal. We can show this in the following Proposition.
Proposition 1 If immigrants' skill composition is taken as given, a utilitarian government imposes no immigration restrictions.
However, as is clear from (9), immigrants'skill composition does depend on the immigration policy. Hence, solving the government's program in equation (17) requires an understanding of the forces driving the migration ‡ows, which is the issue we now address.

Size and composition e¤ects
In what follows, we provide an intuitive discussion of how R depends on the immigration policy ; a more formal derivation can be found in the Appendix (Section 6.2). By totally di¤erentiating R in (13) where @x =@ are partial derivatives (describing the direct e¤ect of immigration policy on immigration ‡ows). The relation in (18) can be decomposed in the product of two forces. If the composition of immigrants were independent of , that is dQ=d = 0; then If instead x H =x L = n H =n L ; then Equation (19) describes a size e¤ ect, i.e. what happens to the skill ratio R when one varies the number of immigrants, while keeping their skill composition …xed. According to equation (19), increasing the cost increases the ratio R if and only if immigrants are less skilled than natives. As shown in Proposition 1, open immigration would maximize total welfare if immigration restrictions had only a size e¤ect. However, as described by equation (20), any immigration policy also changes the average skill of immigrants. This represents a composition e¤ ect: higher restrictions increase the skill ratio R if and only if they increase immigrants'skill composition Q. Before turning to the rest of our analysis, in which we investigate what drives such a composition e¤ect and what its implications are for optimal policy design, we state the following Proposition.
Proposition 2 Immigration policy a¤ ects the receiving country's skill ratio by changing the size and the composition of the migration ‡ow, as described respectively by equations (19) and (20).

The composition e¤ect
Standard discussions about immigration policies ignore the composition effect. However, such disregard may be misleading: this e¤ect may reverse the predictions based on the size e¤ect alone. In fact, as we now show, there are situations in which the tension between the two e¤ects is inescapable, since immigrants'skill composition Q increases in if and only if immigrants are more skilled than natives. In addition, the composition e¤ect may be stronger than the size e¤ect. Self-selection is less likely to be an issue if the skill compositions of the two countries are very di¤erent. If for example the sending country has a very poor skill composition, all else being equal, a more restrictive policy is likely to have a larger impact on low skilled foreigners, thereby increasing the ratio R. This e¤ect being clear, we now concentrate on selection issues and so consider the case in which the skill composition between the sending and the receiving country is similar. 18 In particular, we let n H = n H and n L = n L : The relation between R and then depends on how the policy a¤ects the propensity of low and high skilled foreigners to migrate, i.e. on the elasticities of q H and q L with respect to . Predicting such a relation requires an understanding of the forces behind immigrants'self-selection, as we now consider.

The simplest case: no wealth constraints
To illustrate our argument in the cleanest way, we …rst ignore wealth constraints. Besides being simple, this way of modeling the migration decision emphasizes cross-country wage di¤erentials, as in the classic self-selection literature. Moreover, this may be the most natural setting if one is interested in selection along non-observable dimensions, which need not be systematically correlated with wealth. In this case, immigrants' self-selection is driven only by the incentives that foreigners face according to their skills, and immigrants'skill composition in equation (10) can be written simply as where w = w H w L and w = w H w L : Accordingly, when condition (23) holds, we say that returns to skills are higher in the receiving country. Wage di¤erentials also drive the relation between Q and immigration restrictions. In fact, simply di¤erentiating (22), we have Equation (24) describes an incentive e¤ ect. From equation (2), changing costs has a relatively higher impact on the foreigners with lower gains from migration. When wage dispersion is higher in the receiving country, these foreigners tend to be low skilled. Hence, further restrictions improve immigrants'skill composition if and only if w w ; that is if immigrants are more skilled than natives. This implies a tension between size and composition e¤ects. The reason for this is intuitive: the size e¤ect by de…nition hits a group of foreigners proportionally to their propensity to migrate, while the composition e¤ect tends to be stronger on the least represented group.
To see the e¤ects on the receiving country, we notice that the composition e¤ect can be stronger than the size e¤ect. A marginal increase in the cost may decrease (increase) R despite immigrants being less (more) skilled than natives. As we show in the Appendix, this is the case if the ratio =(1 + ) is decreasing, which in turn is more likely to happen when is low. As a result, the relation between the skill ratio and immigration restrictions may be non-monotone. We summarize these observations in the following Proposition. d) The relation between R and may be non-monotone, with the composition e¤ ect being stronger for low levels of .
The fact that the composition e¤ect may reverse the policy outcome as predicted by the size e¤ect alone, has a number of surprising implications. First, in this setting, even a utilitarian government may impose positive immigration restrictions. In fact, as discussed after equation (17), a government with weights = n aims at optimizing the skill ratio. If the relation between R and is non-monotone, however, this requires setting a positive : Restrictions here are not due to distributional concerns, or other departures from pure e¢ ciency, but they are a way to screen immigrants by a¤ecting their self-selection. We can then state the following Corollary.
Corollary 1 Immigration restrictions may be optimal even for a utilitarian government that cares only about natives' total income.
A second implication of the composition e¤ect is that some natives may support further immigration restrictions even if immigrants are not detrimental to them. Suppose for example that immigrants are positively selfselected and they improve the receiving country's skill ratio. In this case, low skilled natives may push for a higher even if immigration increases their wage, since restrictions would further improve immigrants'skill composition, the receiving country's skill ratio and so low skilled wages. Hence, individual preferences over immigration policy should consider immigrants' self-selection in addition to their skill composition. We summarize this in the following Corollary.
Corollary 2 When the composition e¤ ect prevails, some natives may support further restrictions even if immigrants are not detrimental to them.

The general case: incentive and wealth e¤ects
We now explore how the previous insights carry through in a setting where potential migrants face wealth constraints, which may also drive self-selection. For our purposes, this implies that it may not be su¢ cient to know whether immigrants are positively or negatively self-selected, but one needs to know also what drives self-selection. Those with the highest gain from migration, and then the highest willingness to pay for it, are not necessarily the ones with the greatest resources to pay for it.
Besides being a more general formulation of the migration decision, this setting matches better with the empirical evidence on self-selection in terms of observables. As implied by equation (1), wealth constraints are less severe for the high skilled, which pushes towards positive self-selection. As a result, immigrants may be positively self-selected even if returns to skills are higher in the source country and physical costs of migration are relatively small. 20 In this setting, we …rst notice that increasing immigration restrictions improves immigrant skill composition Q when The …rst term is the same incentive e¤ ect described in the previous Section. The second term represents a wealth e¤ ect. By equation (1), this is always positive: by increasing the cost, one gets richer and more skilled immigrants. In this setting, size and composition e¤ects have opposite directions whenever the relation between Q and is monotone, i.e. either w w or self-selection is driven only by wealth constraints or only by incentives (the reason being the same as in the previous analysis). As shown in the Appendix, in such cases, the composition e¤ect prevails when the foreigners with the lowest propensity to migrate are, in absolute terms, the most sensitive to a policy change. In turn, this is more likely to be the case when the cost is su¢ ciently small; hence, as in the previous analysis, the relation between R and need not be monotone. We state this more formally in the following Proposition.

Discussion and extensions
In this Section, we discuss the role of our main assumptions in the above analysis and propose some extensions of our framework.

Immigration policy
Taken literally, our model makes some important simpli…cations about immigration policy. We assume that restrictions a¤ect the migration cost alone, and that they act unconditionally on skills. We now discuss how our insights would be a¤ected by changing these assumptions.

Alternative policy instruments
First, immigration restrictions include several dimensions beside the monetary cost : Along these lines, one may view our framework as a starting point from which to complicate the policy space. For example, immigration is typically restricted via quotas. In our setting, however, changing the quota a¤ects immigrants'self-selection in a similar way as changing the cost . In fact, the quota a¤ects the probability that, upon submitting an application, a foreigner receives an entry visa. Suppose that making such an application entails a cost (either monetary or in terms of time). A foreigner applies for a visa only if the expected bene…ts, i.e. the wage di¤erential multiplied by the probability of getting the visa, exceeds the cost. Changing the quota then has a stronger impact on those with lower gains from migration, which is the same incentive e¤ect we described in the above analysis. Unlike in our analysis, the way in which di¤erent quotas interact with wealth constraints is less clear. In addition, immigration typically requires compliance with a signi…cant amount of bureaucracy. Bureaucracy too may a¤ect self-selection: it requires time, whose value may di¤er according to skills, or money (e.g. for paying for the assistance of speci…c private agencies). As a simple example, assume that each migrant has to invest some …xed amount of time in bureaucracy, and this time is worth w : Since in this case bureaucracy is more harmful for the high skilled, the conditions for positive self-selection become harder to satisfy. When only incentives matter, positive self-selection requires that w > (1 + ) w ; i.e. returns to skill in the receiving country are su¢ ciently high to compensate also for the greater waste of time. With respect to the case of no bureaucracy, then, an increase in restrictions (i.e. both and ) is more likely to reduce immigrants'skill composition.
There are several other ways in which the immigration policy space can be enriched. Nonetheless, the general theme stressed throughout this paper appears robust to the particular modeling choice. In order to predict the e¤ects of immigration policy, one needs to account for the (indirect) e¤ect on immigrants'skill composition, as determined by immigrants'self-selection.

Skill-dependent policy
Turning to the second issue, receiving countries may indeed try to impose di¤erent restrictions on di¤erent types of immigrants. Even under this lens, however, there are several reasons which make the above analysis of some value. First, and perhaps most importantly, this analysis emphasizes that even policies independent of immigrants'skills have a screening power. This is important as, in all receiving countries, many signi…cant aspects of immigration policy tend to be independent of skills. For example, these countries regulate the total number of immigrants, the amount of bureaucracy needed to get a visa, the way immigrants are treated once in the country (say in terms of access to welfare), and so on. Policy discussions around these issues tend to neglect the indirect e¤ect on the composition of immigrants, and this paper makes precise a sense in which such omission may be misleading.
Second, systems which directly screen immigrants according to their skills may be very complicated to implement and not necessarily very effective. Indeed, in countries where such systems are in place, like Australia and Canada, they do not appear to e¤ectively in ‡uence immigrants' skills and long-term success in the receiving country (as argued e.g. in Miller, 1999 on Australia;in Antecol, Cobb-Clark andTrejo, 2003 andJasso andRosenzweig, 2008 on Canada andAustralia). In other words, direct selection of immigrants appears to be a di¢ cult task since not all the desirable characteristics can be precisely described and veri…ed. Moreover, the assimilation of immigrants depends also on unobservable dimensions, which are by de…nition not contractible and as such they can be a¤ected only through indirect screening mechanisms.
Last, our analysis would be of use even if one took the extreme view that the receiving country can perfectly well implement a policy conditional on immigrant skill, and so impose di¤erent costs on low and on high skilled immigrants. In such an ideal world, our analysis would show the policy dimension along which the skill ratio R is most sensitive. Suppose for example that the government wishes to increase R: This could be done either by increasing the cost for low skilled immigrants L or by decreasing the cost for high skilled immigrants H : The government may be interested in which is the most e¢ cient way to go, i.e. in whether a larger e¤ect on R would be induced by changing L or H . This would require computing From these relations, we see that Condition (26) is equivalent to condition (18), from where our analysis started. 21 In other words, even in this setting, determining the optimal policy may require accounting for the same size and composition e¤ects emphasized above.

Government' s preferences
We focused on the case of a utilitarian government which values each group according to its size. We view the weights = n as a useful starting point as they allow us to identify how the composition e¤ect changes the optimal policy program. Distributional concerns and political economy considerations may provide additional reasons to impose restrictions. Instead, as shown in Proposition 1, in a setting with = n any immigration restriction is driven by the composition e¤ect.
There are however many ways in which our framework can be extended. First, while considering a linear utility function simpli…es our analysis, one may introduce a more general form for natives' utility. Each group would then receive a weight which depends on and on the group's marginal utility, and this would induce a greater concern for the low skilled (who have higher marginal utility). Second, one could model the process of aggregating natives'preferences in a more structured (and perhaps more realistic) way. For example, one could think of a majoritarian democracy where only the largest group of natives gets positive weight. If these are low skilled, the government would aim at maximizing the skill ratio R. Alternatively, one could introduce lobbying activities whereby each group may bid for protection and try to increase its weight in the government's program. In this case, the government may trade o¤ contributions and social welfare, and aim at some intermediate R. More generally, one could add to our model a stage in which the weights are determined. These weights then determine the optimal R and the ensuing optimal immigration cost. In the above analysis, we have taken the weights as given and described how the government would set its policy in order to move towards the optimal skill ratio R. In this sense, the insights developed on size vs. composition e¤ects are robust to the speci…c way in which the weights are determined. A di¤erent line of extension would be to include in the government objective function the direct costs and bene…ts of implementing a given immigration policy. In our main analysis, we have not considered the potential revenues associated with immigration costs (as instead emphasized by proponents of entry taxes, see the references in the Introduction). As detailed in Section 4.1, our policy need not be interpreted as an entry tax: a significant share of migration costs depends on immigration restrictions without being pocketed by the receiving country government (e.g. immigrants'expenses for legal and consulting services needed to comply with bureaucracy). At the same time, we have not considered any cost of implementing the immigration policy (as instead stressed e.g. in Beine, Docquier and Özden, 2009). In order to emphasize these costs and bene…ts, the government objective function in (6) can be rewritten as where T ( ) = (x H ( )+x L ( )) are the revenues and C( ) are the costs associated with implementing the policy . Denote the cost which maximizes labor market bene…ts H w H ( ) + L w L ( ) we have considered so far, and the cost which maximizes net revenues T ( ) C( ) (assuming the problem has a unique solution). The government would then typically choose a cost between and : That is, in this world, the government would have to balance the e¤ects of the implementation of immigration policy on …scal balances with its e¤ects on natives' wages. Still, in order to estimate the latter, it would have to take into account the composition e¤ect we stressed in our main analysis. 22

Natives'preferences
In the above analysis, natives'skills determine preferences over immigration policy through standard labor market competition between immigrants and natives. Labor market interactions have certainly received the greatest attention in the economics literature on the e¤ects of immigration in receiving countries (see e.g. Borjas, 1994 andBauer andZimmermann, 2002 for surveys). Nonetheless, the evidence is quite controversial. Some studies …nd a rather small e¤ect on natives'wages (Friedberg andHunt, 1995 andCard, 2005), while others (e.g. Borjas, 2003) report that immigrants compete with similarly skilled natives and signi…cantly lower their equilibrium wages (see also Ottaviano and Peri, 2008 for a recent review of these estimates). The same labor market e¤ect has also been proposed as an explanation of natives'attitudes towards immigration policy (see Slaughter, 2001 andMayda, 2006). In countries where immigrants are less skilled than natives, more educated individuals tend to support more liberal immigration policies, and this correlation disappears once one considers people outside the labor force.
More generally, our focus on the e¤ects on R may be useful to analyze several other issues which we have left aside. For example, natives' preferences on immigration may be driven also by public …nance and political economy issues. From a …scal viewpoint, one may argue that high skilled immigrants are always preferred since they pay higher taxes and receive fewer welfare bene…ts. Hence, high skilled natives would trade-o¤ the reduction in wages with the …scal bene…t of accepting high skilled immigrants. 23 On political economy issues, if immigrants gain political power in the receiving country, then natives may trade-o¤ the e¤ect on their wages with the e¤ect 2 2 I acknowledge useful suggestions by one of the referees in shaping this discussion. 2 3 Suppose that the government collects tw and distributes the revenues with a lump sum transfer to every worker. Now high skilled utility is a convex combination with weight t of the wage wH ; which depends negatively on R; and the transfers, which depend positively on R. The e¤ects of these concerns on individual preferences over immigration are documented in Hanson, Scheve and Slaughter (2007) and Facchini and Mayda (2009). on the political equilibrium (as in Ortega, 2005). Depending on these trade-o¤s, natives would determine their preferred skill ratio R and push for any policy which moves towards such R. But again, our results on how R varies with do not depend on the speci…c way in which such a preferred R is determined.

Conclusion
In this paper, we have developed a simple framework for analyzing the interaction between immigrants' self-selection and the determination of immigration policy. We have shown that any immigration policy a¤ects the composition of the migration ‡ow and have explored some implications of this e¤ect for optimal policy design. We have carried out our analysis in a setting intended to be the simplest for conveying our insights on the composition e¤ect. This clearly leaves many avenues open to future research, some of which have been mentioned in Sections 2 and 4. Another interesting extension would be to consider a dynamic model in which current migration ‡ows depend also on past ‡ows, through migration networks or family reuni…cation. In such a model, current immigration policy a¤ects future migration ‡ows and so the optimal policy design should also consider the relation between the characteristics of initial and subsequent immigrants. 24 We wish to conclude by suggesting some possible policy implications of our results. As mentioned in the Introduction, the e¤ects of immigration largely depend on immigrants'composition, and as such receiving countries may have a great interest in improving their ability to screen. In this respect, our results show that, given immigrants'self-selection, any policy has some indirect screening power. Given that size and composition e¤ects tend to work in opposing directions, this may signi…cantly complicate the optimal policy design. On the other hand, such screening power may be viewed as an additional dimension to exploit. Since the e¤ectiveness of direct screening mechanisms appears limited, immigration policies may consider in ‡uencing self-selection ex-ante rather than imposing restrictions ex-post. As we have shown, this in turn requires understanding the forces shaping the decision to migrate. By a¤ecting the way di¤erent potential migrants respond to policy changes, immigrants' self-selection is then key also for receiving countries. Nothing is terribly surprising in this statement. There is a large and fundamental literature studying how di¤erent types of agents respond di¤erently to changes in prices. 25 For some reason, the literature on immigration policy has generally overlooked this issue, and, from this perspective, this paper may be a step towards …lling the gap.
6 Omitted Proofs 6.1 Proof of Proposition 1 If the composition of immigrants is independent of ; it must be that Hence, by (27), and so again by (27), recalling that @x =@ < 0; Hence, by equation (17), dW=d has the same sign as (n L x H n H x L )(n H x L n L x H ); which is negative. That is, W decreases in and so open immigration is optimal when the composition of immigrants is independent of .

Proof of Proposition 2
Totally di¤erentiating equation (13) and rearranging terms, we have that Since @w H =@R < 0 and @w L =@R > 0; the term in parentheses is positive, which is equation (18) Equation (28) can be decomposed into a size and a composition e¤ect, as described respectively by equations (19) and (20) in the main text.

Proof of Proposition 3
When immigrants' self-selection is determined by incentives alone, immigrants'skill composition in equation (10) can be written simply as that is point (a) in the Proposition. As mentioned in the text, point (b) follows by simply di¤erentiating (22) and using assumption (2). Turning to point (c), notice that substituting (21) into equation (19), we can write the size e¤ect as dR d 0 , q H q L ; while by assumption (2) the composition e¤ect in (20) writes as Hence under (21) size and composition e¤ects work in opposing directions. To see point (d), rewrite condition (18) as @q H @ n H n L @q L @ n L n H + @q H @ q L n H n L @q L @ q H n L n H 0; which under (21) writes as @q H @ @q L @ + @q H @ q L @q L @ q H 0: Hence, if the ratio =(1 + ) is decreasing, that is @q H @ @q L @ + @q H @ q L @q L @ q H 0 , q H q L ; then the composition e¤ect dominates. Further restrictions increase R if and only if they improve Q, which in our case requires w w . The ratio =(1 + ) is decreasing if for example the psychological cost of migration " i is uniformly distributed over some interval [a; b]: Hence, for " 2 [a; b]; (" L ) = (" H ) and, substituting into equation (30), we see that @R @ 0 () (" H ) (" L ) 0 () w w : From equation (31), restrictions increase the skill ratio R if and only if immigrants are more skilled than natives. Hence, as long as both thresholds " L and " H lie within the interval [a; b]; the composition e¤ect prevails. When instead one of the thresholds " L and " H lies outside the interval [a; b], the sign of the derivative is reversed, i.e. the size e¤ect prevails. This also shows the possibly non-monotone e¤ect of on R. If the cost becomes so high that no high skilled foreigner has any incentive to move, the composition e¤ect disappears. Such cost is implicitly de…ned by where always exists since w H is bounded so > w H w H a for su¢ ciently large. Hence, in this example, the composition e¤ect is stronger for ; while the size e¤ect dominates afterwards. As a result, the relation between R and is U-shaped, with a minimum at = .

Proof of Proposition 4
The logic follows the proof of Proposition 3 by noticing that now selection is driven both by incentives and wealth constraints, where the latter always push towards positive self-selection due to assumption (1). Hence, as stated in (a), if w w then immigrants are positively self-selected because both of incentive and wealth e¤ects. (If instead w < w ; the e¤ect is ambiguous. As ! 0; incentives dominate so the relation tends to be negative. As increases, the shape of Q depends on the strength of the two e¤ects: when (" H ) goes to zero faster than (1 L ), Q tends to zero as increases; when the opposite occurs, there exists a cost beyond which the wealth e¤ect takes over, so the relation is U-shaped.) If w w or the wealth constraint dominates, then the relation between Q and is monotone, and the analysis in Proposition 3 goes through (points b and c). In this case, given equation (21), a su¢ cient condition for the composition e¤ect to prevail is that, as mentioned in the text, the foreigners with the lowest propensity to migrate are, in absolute terms, the most sensitive to a policy change, that is x H x L if and only if @q H @ @q L @ : In fact, condition (25) holds if and only if x H x L; that is, @q H @ q L @q L @ q H , x H x L : Together with condition (32), (33) implies condition (30), that is R increases with despite immigrants being more skilled than natives. That is, the composition e¤ect prevails (point d). Similarly to the previous analysis, the composition e¤ect may prevail only when the cost is su¢ ciently small, so that the population of migrants who respond to policy changes is su¢ ciently heterogeneous. If the cost is so high that only one group of foreigners migrates, being the richest or the most motivated, then by de…nition there is no composition e¤ect. Given this possible non-monotonicity in the e¤ect of on R, the discussion after Proposition 3 still holds and so Corollaries 1 and 2 follow.