Milo Bianchi
email: bianchi@pse.ens.fr.

Abhijit Banerjee

Jess Benhabib

Guido Friebel

Sergei Izmalkov

Anna Maria Mayda

Sten Nyberg

Elena Paltseva

Jörgen Weibull

Immigration policy and self-selecting migrants

Keywords: J61, F22, O24, D78 Immigrant self-selection, political economy of immigration, immigration policy preferences Immigrant self-selection; immigration policy. JEL codes: F22, J61, O24

I build a simple model of self-selection into migration and immigration policy determination. I …rst show how immigration restrictions a¤ect not only the size but also the skill composition of the migration ‡ow. I then explore how the optimal policy may change once this e¤ect on immigrants' skill composition is considered. I show that the relation between immigrants' skill composition and immigration policies is governed by immigrants' self-selection, hence understanding what drives such selection becomes crucial for designing optimal immigration policies.

Introduction

Migrants are not a random sample of their home country population. Incentives to migrate, and resources to pay the migration costs, vary with skills. This paper builds on this well-known fact in order to explore the interaction between self-selection into migration and the determination of immigration policy in receiving countries. In fact, while it is commonly understood that various e¤ects of migration vary signi…cantly with immigrants' characteristics,1 the relation between immigration policy and immigrants' skill composition remains largely unexplored.

In what follows, I …rst show how immigration policy a¤ects not only the size but also the skill composition of the migration ‡ow. I then explore how the optimal policy may change once this e¤ect on immigrants' skill composition is taken into account. As it turns out, predicting the relation between immigrants'skill composition and immigration policy may be crucial for predicting the outcome of such policy in the receiving country. This in turns requires understanding the forces behind immigrants'self-selection, which determine how di¤erent potential migrants respond to policy changes.

The above argument is developed in a model with two countries. In the sending country, individuals, called foreigners, are endowed with di¤erent skills and wealth, and according to 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 the opposite. According to these preferences and to the weight attached to different groups in the population, i.e. low vs. high skilled and immigrants vs. natives, the receiving country government sets the immigration restrictions. In particular, these restrictions may a¤ect the cost migrants have to pay to enter and work in the receiving country. For example, the government may impose direct fees or bureaucratic requirements that increase the time and money needed to comply.

Even if common to all immigrants, these restrictions a¤ect immigrants in a di¤erent way. On the one hand, they allow only the richest foreigners to migrate, and these tend to be the high skilled. On the other, they induce only those with the most to gain to migrate. If returns to skills are higher in the sending country, these 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.

Understanding this composition e¤ect is crucial for the receiving country because such e¤ect may reverse the immigration policy outcomes, as predicted by the size e¤ect only. In fact, size and composition e¤ects have typically opposite directions. The size e¤ect, whereby one varies the number of immigrants while keeping their skill composition as …xed, is by de…nition random, and so it hits a group of foreigners proportionally to their propensity to migrate. In contrast, the composition e¤ect tends to be stronger on those who migrate less.

Moreover, the composition e¤ect may dominate the size e¤ect: the foreigners with the lowest propensity to migrate may be, in absolute terms, the most sensitive to a policy change. Finally, the strength of size and composition e¤ects depends on the level of restrictions. In particular, when the migration cost is so high that only one group of foreigners migrates, being they the richest or the most motivated, then by de…nition there is no composition e¤ect. As a result, immigration restrictions may have a non-monotone e¤ect on the receiving country.

It follows that it is generally misleading to view immigration restrictions as just selecting from a given pool of applicants, thereby acting independently from the migration decision. As an illustration, we highlight how the composition e¤ect may a¤ect natives'preferences over immigration policy and the government's optimal policy design. First, such e¤ect implies that natives' preferences over immigration policy depend not only on immigrants' skill composition but also on their self-selection. For example, some natives may support a more restrictive policy even though current immigrants are not harmful for them, since tighter restrictions would change immigrants'skill composition in their favor. Second, the composition e¤ect implies that even a utilitarian government which maximizes natives' total income may choose positive immigration restrictions. In fact, while free immigration would always be optimal if immigrants'skill composition were taken as given, in our setting restrictions may be imposed in order to select the optimal skill mix of immigrants.

Related literature

The present paper lies within two streams of literature: the supply side of immigration, dealing with the migration decision and immigrants' selfselection; and the demand side, dealing with citizens' preferences over immigration and immigration policy formation. At a general level, the major novelty of the paper is the focus on the interaction between demand and supply, in order to show that, by considering each side in isolation, one may draw erroneous conclusions both on self-selection and on the e¤ects of immigration policy in the receiving country. 2 More speci…cally, the migration decision is here viewed as a basic human capital investment (Sjaastad, 1962), in which self-selection may be driven both by cross-countries returns to skills (as in Borjas, 1987) and by wealth constraints. 3 In addition, I emphasize that immigration policies may be a signi…cant determinant of immigration costs, and then of immigrants 'skill composition. 4 This allows to explore in a simple way the interaction between demand and supply and to better match some recent empirical literature (see Section 3.3.2).

On the demand side, individual skills are related to preferences over immigration policy by a standard labor market interaction. 5 Di¤erently from the existing literature, in which immigrants'skill composition is taken as given, I model individual preferences whereby natives realize that such composition is a¤ected by immigration restrictions.

Finally, the present paper contributes to the relatively small literature on the determination of immigration policies. 6 Apart for stressing the interaction with the supply side, our approach is novel in that we consider migration cost as the policy variable. This variable seems important as any restriction to immigration entails, at least indirectly, monetary costs. Indeed, as we discuss in Section 4.1, this framework may be interpreted in more general terms. Moreover, the exercise appears useful even if one considers our policy variable literally as a tax on immigrants. Such tax has recently received attention in policy debates (see Freeman, 2006 and Legrain, 2007), but to high immigration quota. By reducing wages in the receiving country, this would increase immigrant quality and maximize national income. my knowledge its e¤ects have not been explored in a formal model.7

2 The model Consider a world with two countries, a sending and a receiving one. We are interested in the interaction between the workers in the sending country, who may decide to migrate, and the receiving country 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: A foreigner i with skill may migrate to the receiving country, and receive the endogenous wage w , or he can work in the sending country for an exogenous wage w :8

If he migrates, such foreigner has to incur the migration cost ( + " i ); which includes a common monetary cost and an individual-speci…c psychological cost " i .9 Speci…cally, " i is assumed to be a random variable following a log-concave cumulative distribution with continuous density . 10 This assumption implies that the ratio is decreasing.

(1)

Finally, foreigners are endowed with some wealth, drawn by a distribution with continuous density ! : For now, we interpret as an observable skill (like education), and we then assume that the high skilled are on average wealthier than the low skilled (see Filmer and Pritchett, 1999; and Piketty, 2000). 11 Formally, we assume that, for every 2 R + ;

! L 1 L ! H 1 H :
(2)

That is, the high skilled wealth distribution is more favorable than the low skilled one, in the sense of conditional stochastic dominance. 12 We also assume that migrants have to incur the cost up-front, and there is no credit market for them. Hence, migration may be limited by wealth constraints.

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: Natives are assumed to have a linear utility function which depends only on equilibrium wages w : 13 These wages are determined in a competitive labor market as

w = @F (N H ; N L ) @N ; (3)
where F (N H ; N L ) is the receiving country production function and N is the sum of natives and immigrants with skill

N = n + x : (4)
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, for most of the analysis, we simply let the production technology be a constant returns to scale Cobb-Douglas

F (N H ; N L ) = N H N 1 L ; (5)
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

W = H w H + L w L ;
where denotes the weight attached to group 's utility, as determined by the speci…c institutional setting. 14 Immigration policy acts on ; which is 1 1 As it will be noticed below, this framework can be also applied to analyze selection along unobservable characteristics.

1 2 This is slightly stronger than …rst order stochastic dominance, but weaker than the standard assumption of monotone likelihood ratio (see Krishna, 2002, Appendix B).

1 3 In Section 4.3, we discuss more general formulations. 1 4 We will mostly consider a utilitarian function with = n : Some extensions and possible ways to endogenize these weights are discussed in Section 4.2.

the cost foreigners have to incur to enter and work in the receiving country 15 , so the government's program can be written as

max 2R + H w H ( ) + L w L ( ): (6)
3 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, i.e. of immigrants'self-selection.

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 incur the migration cost : Thus, the supply of migrants with skill is de…ned by

x = q n ; (7)
where q is the fraction of foreigners with skill who can a¤ord and who are willing to move, i.e.

q = [1 ( )] [w w ]: (8)
We de…ne immigrants' skill composition as the ratio of high to low skilled migrants, i.e.

Q = q H q L ; (9)
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

w H = R 1 ; (10) and w L = (1 )R ; ( 11
)
where R is the ratio of high to low skilled workers

R = N H N L = n H + x H n L + x L : (12)
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 ( 6) as

max 2R + H R 1 + L (1 )R : (13)
Obviously, the optimal policy depends on the weights . The higher is H , the lower R will be induced by such policy.

For now, we abstract from redistributive concerns or other political economy distortions, and consider a purely utilitarian setting in which each group is valued according to its size. Notice …rst that, in this setting, no immigration restrictions are imposed if immigrants are given the same weight as natives. In fact, if = N ;

then the welfare function W does not depend on R; i.e. on high vs. low skilled wages, but only on total production. Hence, W would be 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 in order to bene…t the group of workers in which the proportion of immigrants is lower. Suppose the government cares only about natives, then

= n ;

and we have that

dW d = (1 )w H [ n L x H n H x L n H + x H ] dR d : (14)
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 . 17 Since the government maximizes e¢ ciency, i.e. natives' total income, it aims at optimizing the skill ratio R. In particular, if immigrants are less skilled than natives, the optimal policy is the one preferred by high skilled natives, which is the one minimizing the ratio R; and vice-versa if immigrants are more skilled than natives.

Size and composition e¤ects

As expressed in equation ( 14), the government's program needs to account for how immigration policy a¤ects the skill ratio R: As we now show, this in turn requires an understanding of the forces driving the migration ‡ows. Notice …rst that, di¤erentiating equation ( 12) and rearranging terms, we have that

dR d N L 2 @x H @w H @w H @R N L + @x L @w L @w L @R N H = @x H @ N L @x L @ N H :
Since @w H =@R < 0 and @w L =@R > 0; the term in parentheses is positive, so the ratio R increases in if and only if

@x H @ N L @x L @ N H : (15)
Notice that @x =@ are partial derivatives, i.e. they describe the direct e¤ect of immigration policy on immigration ‡ows, abstracting from the e¤ect on equilibrium wages.

In order to highlight the role of self-selection, one can multiply both sides of equation ( 15) by x L x H and see that the ratio R increases in if and only if

x H n H + x H @x H @ x L x L n L + x L @x L @ x H : (16)
Equation ( 16) can be decomposed as the product of two forces:

x H n H + x H x L n L + x L ; which is equivalent to x H x L n H n L ; (17) and @x H @ x L @x L @ x H ; which is equivalent to @Q @ 0: (18)
Equation ( 17) 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 as …xed. If immigration restrictions had no other e¤ect, then welfare would be maximized with free immigration. In fact, equation ( 17) tells that increasing the cost increases the ratio R if and only if immigrants are less skilled than natives. According to equation ( 14), we would then have

dW d 0:
We summarize this result in the following Proposition.

Proposition 1 If immigrants'skill composition was taken as given, a utilitarian government would impose no immigration restrictions.

However, as described by equation ( 18), any immigration policy also changes the average skill of immigrants. This represents a composition effect: 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 composition e¤ect and which are its implications for the optimal policy design, we state the following Proposition.

Proposition 2 The e¤ ect of immigration policy on the receiving country skill ratio can be decomposed in an e¤ ect on the size and an e¤ ect on the composition of the migration ‡ow, described respectively by equations ( 17) and ( 18).

The composition e¤ect

Standard discussions about immigration policies abstract from the composition e¤ect. However, such abstraction may be misleading, since this e¤ect may sometimes reverse the predictions based on the size e¤ect only. As we now show, there are situations in which the tension between the two e¤ects is inescapable, i.e.

@Q @ 0 if and only if x H x L n H n L ; (19)
and, in addition, the composition e¤ect may be stronger. To show this, we …rst rewrite condition (15) 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: (20)
From the last equation, we see that the composition e¤ect is less likely to be an issue if the skill compositions of the two countries are very di¤erent. Suppose for example that the sending country has a very poor skill composition, i.e. n H n L is much lower than n L n H . All else equals, 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. In particular, we let

n H = n H and n L = n L ; (21)
and so we write condition (20) as @q H @ @q L @ + @q H @ q L @q L @ q H 0:

The last equation emphasizes that the relation between R and depends on how the policy a¤ects the propensity to migrate of low and high skilled foreigners. Predicting such relation then 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 abstract from wealth constraints. Besides being simple, this way of modeling the migration decision emphasizes cross-countries wage di¤erentials, as in the classic selfselection literature. 18 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 ( 9) writes simply as

Q = (w H w H ) (w L w L ) : (23)
Therefore, immigrants are positively self-selected if and only if absolute gains from migration increase with skills, i.e. if (w H w H ) (w L w L ): Alternatively, this condition can be rearranged in terms of wage di¤erentials in the sending vs. receiving country, de…ned respectively as w = w H w L and w = w H w L ; as w w :

Accordingly, when condition (24) holds, we say that returns to skills are higher in the receiving country. Wage di¤erentials drive also the relation between Q and immigration restrictions. In fact, simply di¤erentiating (23), we have

@Q @ 0 () (" L ) (" H ) (" H ) (" L ) 0 () w w : (25)
Equation ( 25) describes an incentive e¤ ect. As expressed in condition (1), changing costs has a relatively higher impact on the foreigners with lower gains from migration. These foreigners tend to be low skilled, and so restrictions and skill composition are positively related, if and only if wage dispersion is higher in the receiving country.

To see how this e¤ect matters for the receiving country, we simplify further our analysis by assuming that the psychological cost of migration is uniformly distributed over some interval [a; b]: Hence, for " 2 [a; b]; (" L ) = (" H ) and, substituting into equation ( 22), we see that

@R @ 0 () (" H ) (" L ) 0 () w w : (26)
From equation ( 26), restrictions increase 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.

It follows that the composition e¤ect may prevail only when restrictions are not too high, so that both groups of natives have incentive to migrate. To see this, suppose for example that the low skilled have always higher gains from migration (i.e. w w for all ) and that, at the current level of restrictions, some high skilled is still willing to migrate (i.e. w H w H a >

). According to equation ( 26), a marginal increase in the cost decreases R despite that immigrants are less skilled than natives. This is due to the composition e¤ect, i.e. to the fact that an higher cost induces an even lower immigrants'skill composition. At some point, however, the cost becomes so high that no high skilled has incentive to move, so the composition e¤ect disappears. Such cost is implicitly de…ned by = w H ( ) w H a:

Beyond ; increasing the cost just decreases the number of immigrants, without a¤ecting their composition. Since immigrants are low skilled, this increases the skill ratio R: In sum, in this example, the composition e¤ect is stronger for ; while the size e¤ect dominates afterwards, and the relation between R and is U-shaped, with a minimum at = . 19 Even in such simple setting, we gain some fundamental insights on the relation between size and composition e¤ects. First, the composition e¤ect pushes the skill ratio into the opposite direction than the standard size e¤ect. By equations ( 23) and ( 25), further restrictions improve immigrants' skill composition if and only if immigrants are more skilled than natives, which is the tension de…ned in condition (19). The reason is intuitive: the size e¤ect is by de…nition random, so it 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.

Second, the relation between the skill ratio and immigration restrictions may be non-monotonic. 20 In particular, the composition e¤ect may dominate the size e¤ect at low levels of restrictions. We summarize these …ndings with the following Proposition. d) The relation between R and may be non-monotonic, with the composition e¤ ect being stronger for low levels of restrictions.

The fact that the composition e¤ect may reverse the policy outcome, as predicted by the size e¤ect only, has a number of counter-intuitive implications. First, some natives may support further immigration restrictions even if immigrants are not harmful for them. Suppose for example that w w ; so immigrants are positively self-selected and they improve the receiving country skill ratio. In this case, low skilled natives may push for an higher even if immigration increase their wage, since restrictions would further improve immigrants'skill composition and the receiving country skill ratio. Hence, individual preferences over immigration policy should consider immigrants'self-selection in addition to their skill composition.

Second, in this setting, even a utilitarian government may impose positive immigration restrictions. In fact, as discussed after equation ( 14), a government with weights

= n aims at maximizing or minimizing the skill ratio. If the relation between R and is non-monotonic, however, this requires setting a positive (in the example above, it would be = ): 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 summarize these observations in the following Corollaries.

Corollary 1 When the composition e¤ ect prevails, some natives may support further restrictions even if immigrants are not harmful for them.

Corollary 2 Immigration restrictions may be optimal even for a utilitarian government that cares only about natives' total income.

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, and so self-selection is driven also by di¤erent abilities to incur the migration cost. 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 highest resources to pay for it.

Besides being a more general formulation of the migration decision, this setting matches better with the empirical evidence. As implied by equation ( 2), wealth constraints are less severe for the high skilled. Hence, we have that [1 H ( )] [1 L ( )]; which pushes towards positive self-selection in terms of observables. 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, as in the case of Mexican immigrants to the U.S. considered in Chiquiar and Hanson (2005).

In this setting, we …rst notice that increasing immigration restrictions improves immigrant skill composition Q when

(" L ) (" H ) (" H ) (" L ) (" H ) (" L ) + ! L (1 H ) ! H (1 L ) (1 H )(1 L ) 0: (27)
The …rst term is the same incentive e¤ ect described in the previous Section.

The second term represents a wealth e¤ ect. By equation ( 2), this is always positive: by increasing the cost, one gets richer and more skilled migrants.

It is then clear that when only wealth constraints matter, or when w w ; migrants skill composition increases with migration costs. If instead w < w ; the e¤ect is ambiguous. As ! 0; the relation tends to be negative, since the wealth e¤ect is weak and incentives dominate. The shape of Q as the cost increases depends on the strength of the two e¤ects. Roughly, when (" H ) goes to zero faster than (1 L ), Q tends to zero as increases, since at some point a few high skilled are willing to migrate. When the opposite occurs, there exists a cost beyond which the wealth e¤ect takes over, so the relation is U-shaped.

Turning to the e¤ect of on the skill ratio R, we …rst notice that in this setting size and composition e¤ects need not go in opposite directions. In fact, increasing restriction may decrease the size of low skilled immigration and at the same time improve immigrants'self-selection by reinforcing the wealth e¤ect. 21 However, we concentrate in the more interesting case in which size and composition e¤ects have opposite directions. This occurs 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, and the reason is the same as in the previous analysis.

In such cases, given equation ( 21), a su¢ cient condition for the composition e¤ect to prevail is that 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 @ : (28)

In fact, when the relation between Q and is monotone, condition ( 27) holds if and only if x H x L . That is q L (@q H =@ ) q H (@q L =@ ) if and only if x H x L . Together with condition (28), this implies condition ( 22), that is R increases with despite that immigrants are more skilled than natives. 22 Moreover, as it can be noticed by equation ( 22), if q L and q H are small, condition ( 28) is almost necessary.

Moreover, 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 is su¢ ciently heterogeneous. If the cost is so high that only one group of foreigners migrates, being they the richest or the most motivated, then by de…nition there is no composition e¤ect. 23 It follows that the relation between R and need not be monotone. As discussed at the end of the previous Section, this implies that natives may support a more restrictive policy despite current immigrants are bene…cial for them; and that a utilitarian government may optimally impose positive immigration costs. We summarize this analysis in the following Proposition. d) The relation between R and may be non-monotonic, with the composition e¤ ect being stronger for low levels of restrictions;

e) Corollaries 1 and 2 still hold.

Discussion and extensions

In this Section, we discuss how our main assumptions a¤ect the above analysis and propose some extensions of our framework.

2 2 For example, this may occur when immigrants are mostly low skilled and illegal. Immigrants depress R but immigration restrictions are likely to worsen the situation by discouraging high skilled migrants, without a¤ecting low skilled (illegal) ones.

2 3 Similarly to the previous analysis, the maximum cost depends on the functional forms and one assumes (see Bianchi, 2006, for some examples).

Immigration policy

Taken literally, our model makes some important simpli…cations on the immigration policy space. We assume that restrictions only a¤ect the migration cost, and that they act unconditionally on skills. We now see how our analysis would change by relaxing these assumptions. First, immigration restrictions include several dimensions beside the monetary cost ; so our framework may be a starting point 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 than changing the cost . In fact, the quota a¤ects the probability that, upon submitting a demand, a foreigner receives an entry visa. Suppose that submitting such demand 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.

In addition, immigration typically requires spending a signi…cant amount of time in order to comply with bureaucracies. Since the value of time may di¤er according to skills, bureaucracies may also a¤ect self-selection. As a simple example, assume that each migrant has to invest some …xed amount of time in bureaucracies, and this time is worth w : Since in this case bureaucracies are more harmful for the high skilled, the conditions for positive self-selection become harder to satisfy. When only incentives matter, we need that w > (1 + ) w ; i.e. di¤erential returns to skill in the receiving country are su¢ ciently high to compensate also for the higher waste of time. Hence, with respect to the case of no bureaucracies, increasing restrictions (i.e. both and ) is now more likely to reduce immigrants'skill composition.

Finally, countries may try to impose di¤erent restrictions on di¤erent types of immigrants. Obviously, if the receiving country could perfectly contract on immigrant skills, it would directly select the desired size and type of immigration, and the interaction between policy and skill composition would be trivial. Still, there are reasons which suggest that our exercise may still be useful. First, many aspects of immigration policies, like bureaucracies, tend to be independent on skills. In this respect, we emphasize that even such policies have screening power. Second, while countries like Australia and Canada have implemented systems to directly screen immigrants according to their skills, several authors have stressed that such systems have very little ability to a¤ect immigrants' skills and long-term success in the receiving country (see e.g. Miller, 1999 on Australia; Antecol, Cobb-Clark and Trejo, 2003 and Jasso and Rosenzweig, 2008 on Canada and Australia vs. the U.S.). Third, immigrants' self-selection operates also along unob-servable dimensions, which may a¤ect immigrants' assimilation and so be of interest for receiving countries. These dimensions are by de…nition not contractible and hence they can be a¤ected only through indirect screening mechanisms.

Government' s preferences

Our analysis has concentrated on the case of a utilitarian government which values each group according to its size. There are many ways to extend our framework and make the process of aggregating natives' preferences more structured (and perhaps more realistic).

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 (see Bianchi, 2006, for some discussion along these lines). More generally, one could add to our model one stage in which the weights are determined. We have instead taken these weights as given and described how the government would change immigration costs in order to move towards the optimal skill ratio R. In this sense, much of the insights developed on size vs. composition e¤ects are robust to the speci…c way in which the weights are determined.

Natives'preferences

We now discuss our assumptions about natives'preferences. First, it should be noticed that the evidence on the labor market impact of immigration is quite controversial. While some studies …nd a rather small impact on natives'wages (Friedberg and Hunt, 1995 and Card, 2005), others like Borjas (2003) document that immigrants compete with similarly skilled natives and signi…cantly lower their equilibrium wages. In a political economy framework, it would su¢ ce that citizens' beliefs, rather than actual e¤ects, are consistent with our assumptions on labor market interactions. 24 Moreover, as argued below, our focus on the e¤ects on R may be useful to analyze several other issues. Second, 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). In this case, it need not be that any immigration restriction is driven by the composition e¤ect (as stated in Proposition 1), since distributional concerns would provide an additional reason to restrict low skilled immigration. But again, to the extent that the government program requires setting an optimal R; our analysis on size and composition e¤ects would be valid.

Finally, natives'preferences over immigration policy may include several dimensions we have abstracted from. However, focusing on the skill ratio R may provide a useful framework to analyze many such dimensions. Consider for example 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 less welfare bene…ts. Hence, high skilled natives would trade-o¤ the reduction in wages with the …scal ben-e…t of accepting high skilled immigrants. 25 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 one on the political equilibrium (like in Ortega, 2005). However, if immigrants oppose restrictions to immigration irrespective of their skills, 26 this political economy e¤ect has the same direction of the labor market e¤ect considered above.

Returns to skills

Many of our predictions on the relation between immigration restrictions and immigrants'skill composition depend on di¤erential returns to skills in sending vs. destination countries. We now discuss the extent to which the literature provides some general pattern for such di¤erentials.

Several reasons have been proposed to expect higher returns to skills in developed countries, like higher total factor productivity (Lucas, 1990), skill complementarities (Kremer, 1993), or skill biased technological change (Acemoglu, 1998); but also to expect the opposite, like the high supply of skills (Blau and Kahn, 1996) or the existence of labor market institutions which compress wages (Leuven, Oosterbeek and van Ophem, 2004). Empirical studies tends to report that returns to skills decrease with per-capita GDP (Bils and Klenow, 2000; Freeman and Oostendorp, 2000; Psacharopou-los and Patrinos, 2002; Caselli and Coleman, 2006), but di¤erences are not huge, and general patterns appear weak. 27 In the immigration literature, accordingly, there is no consensus. Various models simply assume that a worker with skill s in country j gets a wage k j s; which by construction implies that returns to skills are higher in more developed countries (Chiswick, 1999; Giannetti, 2003; Jasso and Rosenzweig, 2008). On the other hand, several studies, following Borjas (1987), stress that wage inequality may be higher in developing countries and so the low skilled may have the greatest incentives to migrate.

Instead of looking for a general pattern, it appears that sensible insights may be derived from speci…c microanalysis. Chiquiar and Hanson (2005), for example, estimate that real wage premia for Mexican immigrants to the U.S. decrease with education. Similar estimates can be found in the analysis of Palestinian immigrants to Israel (Yashiv, 2004). These studies con…rm our general point that self-selection is in general driven both by incentives and by constraints: those who can access migration are not necessarily those who have more to gain from it.

An equally important issue in estimating returns to skills regards the mapping from skills to jobs. Immigrants need not access the same spectrum of jobs and wages as natives, and in many instances they may be locked into low skilled occupations (see Munshi, 2003). Such mapping is di¢ cult to measure and to compare across countries, and indeed it has been typically overlooked by this literature (see Borjas, 1994), but it may drive self-selection and the impact of immigration in the receiving country. 28 Suppose that good jobs are harder to get for immigrants, perhaps due to their inability to assimilate or to a discriminatory labor market. In this case, it is easy to see in our framework that incentives would push towards negative self-selection, and so towards a negative relation between immigration restrictions and immigrants'skill composition. Moreover, irrespective of their skills, immigrants would be more likely to depress low skilled wages. 29

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 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. Indeed, a large part of the policy debate discusses how receiving countries may improve their ability to screen. In this respect, our results show that, given immigrants'self-selection, any policy a¤ects di¤erent migrants in a di¤erent way, and so it has some indirect screening power. Given that size and composition e¤ects tend to have opposite 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. In fact, since as mentioned in Section 4.1 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.

In this respect, however, the present model does not deliver absolute policy prescriptions. Instead, we have seen how things may change dramatically depending on the forces driving the decision to migrate. If those who migrate are simply those who can a¤ord it, increasing the migration cost, e.g. through a tax on entry, is likely to improve immigrants' skill composition. 30 Instead, as economic incentives become the main argument of the migration decision, the e¤ect of these policies depends on di¤erential returns to skills, and thus it may be more di¢ cult to predict. Discrimination and bureaucracies push towards negative self-selection and, in this case, a more restrictive policy is likely to lead to an even less skilled immigration (see Sections 4.1 and 4.4).

The most general conclusion of this exploration is that immigrants' self-selection matters, also for receiving countries, since the forces shaping self-selection a¤ect the way di¤erent potential migrants respond to policy changes. Nothing is terribly surprising in this statement. There is a huge and fundamental literature studying the response of di¤erent agents to changes in prices. 31 For some reason, the literature on immigration policy has generally overlooked this issue, and, under this perspective, this paper may be a step towards …lling the gap.

Proposition 3

 3 When migration is driven only by incentives, a) Immigrants are positively self-selected if and only if w w ; b) Immigration restrictions increase Q if and only if w w ; c) Under condition (21), size and composition e¤ ects have the opposite direction;

Proposition 4

 4 When migration is driven both by incentives and by wealth constraints, a) Immigrants are positively self-selected if w w or wealth constraints dominate; b) Immigration restrictions increase Q if w w or the wealth e¤ ect dominates; c) Size and composition e¤ ects may have the opposite direction;

Restricting the attention to the economics literature, see[START_REF] Borjas | The economics of immigration[END_REF],[START_REF] Friedberg | The impact of immigrants on host country wages, employment and growth[END_REF],[START_REF] Chiswick | Immigrant Earnings: A Longitudinal Analysis[END_REF] on the labor market e¤ects, and[START_REF] Storesletten | Sustaining …scal policy through immigration[END_REF],[START_REF] Lee | Immigration, social security and broader …scal impacts[END_REF],[START_REF] Chojnicki | Should the u.s. have locked the heaven's door? reassessing the bene…ts of the postwar immigration[END_REF] on the e¤ects on public …nance.

The only paper considering this interaction is, to my knowledge,[START_REF] Bellettini | Immigration Policy, Self-Selection, and the Quality of Immigrants[END_REF]. Assuming that immigrants are positively self-selected, they argue in favor of a

Wealth constraints have been relatively underemphasized in this literature, as pointed out by[START_REF] Hatton | International migration in the long-run: Positive selection, negative selection and policy[END_REF]. Exceptions are the theory of illegal migration in[START_REF] Friebel | Smuggling humans: A theory of debt-…nanced migration[END_REF] and the work by[START_REF] Lopez | Migration and the skill composition of the labor force: The impact of trade liberalization in ldcs[END_REF], who focus on the e¤ect of trade liberalization in the sending country in a modi…ed Hecksher-Ohlin model with heterogeneous labor force, migration costs and …nancing constraints.

Of course, migration costs have also exogenous components, like geography. However, notice that policies may become increasingly relevant, given the historical trend of decreasing transportation costs and increasing immigration restrictions[START_REF] Hatton | World Mass Migration: Two Centuries of Policy and Performance[END_REF]. Indeed, the fact that migration costs can partly be a policy variable is recognized also in[START_REF] Clark | Where do u.s. immigrants come from, and why?[END_REF], who assume that lower quotas indirectly imply higher costs for migrants. However their analysis, similarly to[START_REF] Mayda | International migration: A panel data analysis of economic and non-economic determinants[END_REF], is focused on the volume of immigration ‡ows and does not address the relation between policy and skills composition of immigrants.

 5 This approach is taken for example in[START_REF] Scheve | Labor market competition and individual preferences over immigration policy[END_REF] and[START_REF] Mayda | Who is against immigration? a cross-country investigation of individual attitudes toward immigrants[END_REF]. They document that, in developed countries, where immigrants tend to be less skilled than natives, individual education and support for liberal immigration policy are positively correlated. The typical interpretation is in terms of labor market interaction, which is supported by the fact that the correlation disappears once one looks at people outside the labor

force.6 See for example[START_REF] Benhabib | On the political economy of immigration[END_REF], who explores how the median voter determines minimal capital requirements for admission, and[START_REF] Epstein | The struggle over migration policy[END_REF] and[START_REF] Facchini | The political economy of international factor mobility[END_REF], who use a lobbying model to explain the formation of immigration quotas.

One may also be interested on such policy from an historical viewpoint, since the …rst interventions to limit and select immigration ‡ows in the U.S. and Canada acted on costs rather than on quantities. In the U.S., for example, a head tax of 50 cents per migrant was introduced in 1882, while the …rst quota restriction came in 1921 (see[START_REF] Timmer | Immigration policy prior to the thirties: Labor markets, policy interaction,and globalization backlash[END_REF] for a detailed account).

While wages in origin countries may be a¤ected by emigration (see[START_REF] Mishra | Emigration and wages in source countries: Evidence from mexico[END_REF][START_REF] Hanson | Emigration, labor supply, and earnings in mexico[END_REF], we here focus on the e¤ects in the receiving country.

This cost may re ‡ect individual characteristics like age, family ties, access to networks at origin and destination countries. In the present 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 inBanerjee and[START_REF] Banerjee | Information, the dual economy, and development[END_REF][START_REF] Munshi | Networks in the modern economy: Mexican migrants in the u.s. labor market[END_REF][START_REF] Munshi | Why is mobility in india so low? social insurance, inequality, and growth[END_REF] or unemployment insurance (as in[START_REF] Cuecuecha | The immigration of educated mexicans: The role of informal social insurance and migration costs[END_REF]. This would be qualitatively similar to our analysis below, in which wealth constraints make the migration cost more di¢ cult to pay for the low-skilled.1 0 Formally, log-concavity means that those at the tails of the distribution of cost are not too sensitive to a change in the returns of migration. This is a rather weak assumption, as basically all "named" distribution functions satisfy it (see[START_REF] Bagnoli | Log-concave probability and its applications[END_REF].

5 As discussed in Section 4.1, this cost need not be though only as an entry tax, and as such, we do not include its potential revenues in the welfare function. In fact, a signi…cant part of migration costs depends on immigration restrictions, but it is not pocketed by the receiving country government (e.g. immigrants'expenses for legal and consulting services needed to comply with bureaucracies).

6 Notice that wH ; wL and R are uniquely de…ned by equations (10), (11) and (12). In fact, the right hand sides of equations (10), (11) and (12) are continuous are decreasing in wH ; wL and R; respectively. Hence, the …xed point problems in such equations have always a unique solution.1 7 The result resembles a well known principle in international trade, where gains from trade are higher the greater the trading countries di¤er in their factor endowments. A similar point, in a more complicated setting, is made by[START_REF] Borjas | The economic bene…ts from immigration[END_REF].

8 This may also be the most natural setting to consider if one is interested in selection along non-observable dimensions, which need not be systematically correlated with wealth.

9 Such always exists. In fact, even if in this example wH …rst increases in ; wH is bounded so > wH w H a for su¢ ciently large.

0 This has been shown using the convenient special case of a uniformly distributed cost "i. As it is clear from equation (22), however, this is the case whenever the ratio =(1 + ) is decreasing.

1 Say for example that is very low and that w < w , so immigrants are negatively self-selected and they decrease R. By the size e¤ect, increasing restrictions increases R. Moreover, further restrictions may also improve immigrants' skill composition as higher costs make wealth constraints more binding, which in turn induces a further increase in R.

4 Considering the centrality of labor market competition in policy debates, and the evidence on attitudes towards immigration mentioned in footnote 6, this seems less controversial.

5 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 is documented in[START_REF] Hanson | Public …nance and individual preferences over globalization strategies[END_REF] and[START_REF] Facchini | Individual attitudes towards immigrants: Welfare-state determinants across countries[END_REF].2 6 Lowell, Bean and de la[START_REF] Lowell | Undocumented immigration: An analysis of the 1984 simpson-mazzoli vote[END_REF] and[START_REF] Goldin | The political economy of immigration restriction in the united states, 1890 to 1921[END_REF] report that immigrants lobbied and voted for pro-immigration policies, and a number of survey studies report that immigrant have more favorable attitudes towards immigration, irrespective of their economic condition (see[START_REF] Espenshade | Contemporary american attitudes toward u.s. immigration[END_REF][START_REF] Espenshade | Contemporary american attitudes toward u.s. immigration[END_REF] Scheve and[START_REF] Scheve | Labor market competition and individual preferences over immigration policy[END_REF].

7[START_REF] Banerjee | Growth theory through the lens of development economics[END_REF] argue that the common wisdom that returns to skills are higher in developing country is an artifact of low quality data.2 8 Assume that, in the sending country, workers with skill have probability p to get a good job and 1 p to get a bad one, and similarly for the destination country. The requirement that returns to skills are higher in the receiving country writes (pH pL) w > (p H p L ) w : For example, given that in Mexico wage inequality is higher but social mobility is lower than in the U.S.[START_REF] Dahan | Sibling correlations and intergenerational mobility in latin america[END_REF], who should be more likely to migrate? 2 9 To see this, one could introduce a parameter 2 [0; 1]; which measures barriers to entry into well-paid jobs. High skilled immigrants would expect to earn wL + w; and the skill ratio R would write (nH + xH )=(nL + xL + (1)xH ). Also, one may replicate our analysis by focusing on as a policy variable, as this is a¤ected e.g. by anti-discrimination laws, recognition of foreign education...

0 Of course, one concern is that restrictions tend to encourage illegal immigration, that is more attractive for the low skilled. However, the issue is common to any intervention directed to regulate legal migration, and it reveals once again that restricting entry cannot be the only dimension of a sound immigration policy.3 1 These agents being borrowers dealing with interest rates, workers with wages or policyholders with insurance premia (see e.g.[START_REF] Stiglitz | Credit rationing in markets with imperfect information[END_REF]