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Illegal migration and policy enforcement
Economics Letters 148 (2016) 83–86
Contents lists available at ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
Illegal migration and policy enforcement Sephorah Mangin a , Yves Zenou a,b,∗ a
Monash University, Australia
b
IFN, Sweden
article
info
Article history: Received 15 September 2016 Accepted 18 September 2016 Available online 28 September 2016 JEL classification: D8 F22 J61 O12
abstract Workers from a source country consider whether or not to illegally migrate to a host country. This decision depends crucially on the proportion of workers who migrate and on whether or not the host country government decides to regularize them. When there is no uncertainty about economic conditions in the host country, there are multiple equilibria where workers may migrate or not and the government may regularize them or not. When uncertainty is introduced, we show that there exists a unique equilibrium where each state of the world gives rise to a unique outcome in terms of migration and regularization. © 2016 Elsevier B.V. All rights reserved.
Keywords: Illegal migration International migration Global games
1. Introduction More than a million migrants and refugees crossed into Europe in 2015, compared with just 280,000 the year before. The scale of the crisis continues, with more than 135,000 people arriving in the first two months of 2016. The vast majority of these migrants, who have predominantly fled the Middle East and Africa, are illegal. The reactions from European countries have been very different. Some countries, such as Germany and Sweden, initially promised to regularize them if they came from war-torn countries such as Syria. Other countries, such as Poland and Hungary, took a strong stance against such migrants and committed to never regularize them. The aim of this paper is to analyze these issues using a simple framework where both illegal migration and the regularization policy of the host country are explicitly modeled. In our model, workers from a source country first decide whether or not to migrate illegally to a host country. Next, the government of the host country decides whether to regularize all migrants or none. Before the migration decision, workers do not know whether they will be regularized or not. However, workers know they are more likely to be regularized when more workers
migrate to the host country. In other words, from the potential migrant’s viewpoint, we have a game with strategic complements since the optimal action of each worker (migrating or not) is increasing in the average action of the other workers. We first consider the case where there is no uncertainty regarding the fundamentals of the host country’s economy, which are captured by the state of the economy θ . We show that, if θ does not take extreme values, then there are multiple equilibria. Indeed, if all workers believe that other workers will migrate and therefore that they will be regularized, then these beliefs are selffulfilling in equilibrium since the government finds it optimal to regularize them once they have migrated. On the other hand, if workers believe that nobody will migrate, then these beliefs are also self-fulfilling in equilibrium. Given this multiplicity of equilibria, no definitive prediction can be made as to whether workers will migrate or not. Using the tools of global games (Carlsson and van Damme, 1993; Morris and Shin, 1998),1 we introduce a small degree of uncertainty about the state of the economy θ in the host country. We show that common knowledge about this uncertainty leads to a unique equilibrium where each state of the world θ gives rise to a unique outcome in terms of migration and regularization. There is a large empirical and theoretical literature on illegal migration that analyzes the impact of immigration (both legal and
∗
Corresponding author. E-mail addresses: [email protected] (S. Mangin), [email protected] (Y. Zenou). http://dx.doi.org/10.1016/j.econlet.2016.09.012 0165-1765/© 2016 Elsevier B.V. All rights reserved.
1 For an overview on global games, see Morris and Shin (2003).
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S. Mangin, Y. Zenou / Economics Letters 148 (2016) 83–86
illegal) on employment outcomes in the host country, especially in the United States (see e.g. Ethier, 1986; Epstein and Weiss, 2011; Mayr et al., 2012; Chassamboulli and Peri, 2015; Miyagiwa and Sato, 2015). Western countries have spent significant resources on limiting the number of immigrants via both border controls and internal enforcement (such as employer penalties). Despite these efforts, however, many illegal immigrants have found a way to enter these countries. To deal with these issues, these countries have in return periodically granted amnesty to any worker who can demonstrate that he fulfills certain requirements. Epstein and Weiss (2011) show that the number of illegal workers regularized has been quite significant (see their Table 1). In this paper, we propose a new and different perspective on illegal migration. We do not study the mechanisms under which a government reduces illegal migration but rather focus on how the individual’s decision to illegally migrate is affected by the same decision from other workers from the same country and by the regularization policy of the host country. The rest of the paper unfolds as follows. In the next section, we describe the model. Section 3 deals with the complete information case while Section 4 solves the incomplete information case. Section 5 concludes. All proofs can be found in the Appendix. 2. Model We consider the strategic interaction between a set of potential migrants from a source country and the government of the host country. The set of potential migrants is a mass one of workers who must decide whether or not to migrate. This is a {0, 1} decision. Their payoff if they migrate depends on the migration policy enforced by the host country government. This government has two options: she can choose to regularize all migrants or none. A worker’s payoff from migrating is ωr (θ ) if he is regularized and ωi (θ ) if he is not, where θ is a random variable that characterizes the economic conditions in the host country. For example, a high θ indicates positive economic conditions such as low unemployment and a high growth rate. We assume that θ is uniformly distributed over [θ , θ¯ ]. The utility of a worker in the source country is normalized to zero and hence the payoffs ωr (θ ) and ωi (θ ) represent net gains from migrating for a regularized and an illegal migrant, respectively. Since they incorporate both the costs and benefits of migration, they can be either positive or negative. We make the following assumptions on the payoffs of potential migrants. Assumption 1. The functions ωr (θ ) and ωi (θ ) are continuous, increasing, and satisfy 1. For all θ , ωr (θ ) > ωi (θ ), i.e. the payoff of a legal migrant is always higher than the payoff of an illegal migrant; 2. ωr (θ) < 0 < ωi (θ¯ ), i.e. it is never profitable to migrate when economic conditions are ‘‘worst’’, but it is always profitable to do so when economic conditions are ‘‘best’’; and 3. The function 1ω(θ ) = ωr (θ ) − ωi (θ ) is weakly increasing. The government in the host country reacts to the migration decisions of workers by deciding whether or not to regularize migrants. Again, this is a {0, 1} decision. We assume that the government perfectly observes both the state of the economy θ and the fraction of potential migrants who decided to migrate illegally, denoted by s ∈ [0, 1]. The payoff for the government is γ r (θ , s) if she regularizes all migrants and γ i (θ , s) if she regularizes none. These payoffs reflect the interaction between various political economy factors such as the strength of syndicates, lobbies, the median voter’s preferences, etc. For notational simplicity, we denote 1γ (θ , s) = γ r (θ , s) − γ i (θ , s). We make the following assumptions regarding the payoff of the government.
Assumption 2. The function 1γ (θ , s) is a continuous and differentiable function such that: 1. For all θ ∈ [θ , θ¯ ], 1γ (θ , 0) < 0 < 1γ (θ , 1), i.e. it is never profitable to regularize when there are no migrants while it is always profitable when there are a mass 1 of migrants; and 2. For all θ , the function 1γ (θ , s) is increasing in s. Assumption 2 implies that, for any given state of the economy
θ , there exists a unique a ∈ (0, 1) such that 1γ (θ, a) = 0. We can therefore introduce a function a(θ ) implicitly defined by 1γ (θ , a) = 0. Intuitively, a(θ ) is the mass of migrants for which the government is indifferent between regularizing or not regularizing when the state of the economy is θ . Assumption 3. The function a(θ ) is decreasing. Timing: The timing of the game is as follows. First, potential migrants simultaneously decide whether or not to migrate. Second, the government observes the mass of migrants and decides whether or not to regularize them. Observe that when a potential migrant decides whether or not to migrate, he does not know whether he will be regularized or not. Observe also that, from the potential migrant’s viewpoint, we have a game with strategic complements since the optimal action (migrating or not migrating) of each worker is increasing in the average action of the other workers. The more other workers migrate, the more likely that a given worker migrates, which increases her utility because the chance that the government will regularize them increases. We now consider two cases. In the complete information case, the potential migrant knows exactly the economic conditions θ of the host country before migrating. In the incomplete information case, he does not know the value of θ . 3. Complete information Suppose that θ is perfectly observed by migrants. In order to characterize the subgame perfect equilibria of the migration game, we first define two threshold values for the parameter θ . The threshold θ ∗ is the value of θ such that a migrant who knows that he will be regularized is indifferent between migrating or not, i.e. ωr (θ ∗ ) = 0. The threshold θ ∗∗ is the value of θ such that a migrant who knows that he will not be regularized is indifferent between migrating or not, i.e. ωi (θ ∗∗ ) = 0. Assumption 1 ensures that θ < θ ∗ < θ ∗∗ < θ¯ . Proposition 1. When θ < θ ∗ , the migration game possesses a unique subgame perfect equilibrium in which all potential migrants decide not to migrate. When θ > θ ∗∗ , the migration game possesses a unique subgame perfect equilibrium in which all potential migrants decide to migrate. When θ ∗ ≤ θ ≤ θ ∗∗ , the migration game possesses two subgame perfect equilibria: one in which all workers migrate, the other in which no worker migrates. As stated above, when a worker decides whether or not to migrate, he does not know whether or not he will be regularized but he knows the state of the economy θ in the host country. If he observes that θ < θ ∗ , he will clearly not migrate since this means, by Assumption 1 and the definition of θ ∗ , that, even if he will be regularized in the host country, his utility will be lower than staying at home. Since all individuals make the same calculations, nobody will migrate. The opposite is true when θ > θ ∗∗ since the economic conditions are sufficiently good in the host country that it is always profitable to migrate. When θ ∗ ≤ θ ≤ θ ∗∗ , then we have multiple (rational expectations) equilibria. If the workers anticipate that the government will regularize them, then they will all migrate and the government
S. Mangin, Y. Zenou / Economics Letters 148 (2016) 83–86
will indeed regularize them. If workers anticipate that the government will not regularize them, then they will not migrate and the government will not regularize them. In both cases, the expectations of the workers are self-fulfilling. These results may explain the huge illegal migration to Europe during the recent refugee crisis and why most migrants chose to stay in Germany and Sweden and not in Poland or Hungary. Given this multiplicity of equilibria, no definitive prediction can be made as to whether workers will migrate or not. Using the tools of global games, we will see in the next section that, when we introduce uncertainty about economic conditions in the host country, each state θ gives rise to a unique equilibrium outcome in terms of migration and regularization. 4. Incomplete information We now assume that potential migrants do not perfectly observe θ , the economic conditions in the host country. When the state of the economy is θ , each worker observes a noisy signal x, which is independently and uniformly distributed on [θ −ϵ, θ +ϵ]. In other words, conditional on observing the signal x, a potential migrant believes that the true parameter θ is uniformly distributed on [x − ϵ, x + ϵ]. For a given strategy profile for the workers, we denote by α(x) ∈ [0, 1] the proportion of workers who migrate when they receive a signal x, and s(θ , α) the proportion of workers who migrate when the state of the economy is θ . Since signals are uniformly distributed over [θ − ϵ, θ + ϵ ], at θ , we have: s(θ, α) =
1 2ϵ
θ+ϵ
θ−ϵ
α(x)dx.
(1)
Given a strategy profile characterized by the function α , we denote by A(α) the set of states θ in which the government will decide to regularize all migrants, i.e. A(α) = {θ|1γ (θ , s(θ , α)) ≥ 0} .
(2)
Let us denote by θ (α) the state of the economy such that s(θ(α), α) = a(θ )
(3)
where a(θ ) is the mass of migrants for which the government is indifferent between regularizing or not. Given α , Eq. (3) defines θ(α) as the minimum value of θ for which the government regularizes all migrants. Since α(x) is weakly increasing in x, θ (α) exists and is unique by Assumption 3. We can therefore write: ¯ . As a result, the payoff of a worker who decides A(α) = [θ (α), θ] to migrate when the state of the economy is θ is given by:
r ω (θ ) ωi (θ )
θ ∈ A(α), θ ̸∈ A(α).
if if
(4)
The potential migrant does not observe θ directly. As a result, the expected utility of migrating must be calculated from the posterior distribution over the states conditional on the signal x, i.e. it can be calculated by taking the expectation of (4) conditional on x. This expected utility can be written as: u(x, α) =
1 2ϵ
x+ϵ
ωi (θ )dθ + x−ϵ
1 2ϵ
A(α)∩[x−ϵ,x+ϵ]
1ω(θ )dθ .
(5)
We now provide three lemmas which will help us derive our main result. Lemma 1. If α(x) ≥ α ′ (x), for all x, then u(x, α) ≥ u(x, α ′ ). This lemma states that, for any signal x, the higher is the fraction of workers who migrate, the higher is the expected utility of migrating. This means that migration decisions are strategic complements. Now, consider the strategy profile according to
85
which a worker migrates if and only if the signal he receives is greater than some fixed number k. The corresponding proportion of migrants is then given by the following indicator function Ik : Ik (x) =
1 0
if if
x ≥ k, x < k.
Notice that Ik (x) is weakly increasing in x. When migrants follow this strategy, we have: Lemma 2. The function u(k, Ik ) is continuous and strictly increasing in k. This lemma shows that the expected utility for a worker of migrating, given that he received signal k and the proportion of migrants is given by Ik (x), is increasing in k. In other words, when the fundamentals of the economy are stronger, the payoff from migrating is higher for a worker on the margin of switching from non-migrating to migrating. Lemma 3. There is a unique x˜ such that, in any Bayesian equilibrium of the game, a worker decides to migrate if and only if x ≥ x˜ . Once the equilibrium strategies are determined, it is possible to compute the equilibrium enforcement policy. Proposition 2. There is a unique θ˜ such that, in any (perfect Bayesian) equilibrium of the game with incomplete information, the government regularizes migrants if and only if θ ≥ θ˜ . This is a very strong result, which shows that, when potential migrants face a small amount of uncertainty concerning the fundamentals of the economy, there is a unique equilibrium such that, if the state of the economy θ is sufficiently high, then all workers migrate and the government regularizes all of them. If θ is sufficiently low, then no workers migrate and the government does not regularize any. This result suggests that some information events, for example, when Germany and Sweden announced that they will regularize all refugees from Syria, can trigger a huge inflow of migrants into these countries, even though these events do not convey any real information about the state of the economy. 5. Conclusion In this paper, we model workers’ decisions to migrate illegally and the government’s decision to regularize migrants. When there is no uncertainty about the economic conditions of the host country, we show that there are multiple equilibria where workers may migrate or not and the government may regularize them or not. When we introduce some uncertainty about economic conditions, there is a unique equilibrium where each state of the world gives rise to a unique equilibrium outcome in terms of migration and regularization. Appendix. Proofs of all results Proof of Proposition 1. When θ < θ ∗ , by Assumption 1 and the definition of θ ∗ , it is a dominant strategy for workers not to migrate. Similarly, when θ ∗∗ < θ , by Assumption 1 and the definition of θ ∗∗ , it is a dominant strategy for workers to migrate. When θ ∗ ≤ θ ≤ θ ∗∗ , workers migrate if they anticipate that the government will regularize them and do not migrate if they anticipate the government will not regularize them. In turn, the government regularizes the migrants if there is a mass 1 of migrants but does not regularize them if there is a mass 0 of migrants, by Assumption 2.
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Proof of Lemma 1. Since α(x) ≥ α ′ (x), Eq. (1) ensures that s(θ, α) ≥ s(θ , α ′ ), for all θ . Then, by Assumption 2, i.e. 1γ (θ , ·) increasing, and Eq. (2), we know that A(α) ⊇ A(α ′ ). Finally, because ωr (θ ) > ωi (θ ), for all θ , we can use Eq. (5) to establish that u(x, α) ≥ u(x, α ′ ). Proof of Lemma 2. First, let us compute s(θ , Ik ). Given Eq. (1) we have: s(θ, Ik ) =
0
1
2
+
1
1 2ϵ
(θ − k)
if
θ ≤ k − ϵ,
if
k − ϵ < θ < k + ϵ,
if
θ ≥ k + ϵ.
For all k, s(·, Ik ) is a weakly increasing function that takes values in [0, 1]. We now define θ (k) by the implicit equation: s(θ (k), Ik ) = a(θk ). By Assumption 3, we know that a(θ ) is decreasing in θ , takes value in [0, 1] and, therefore, that θk is uniquely defined. Let us denote ψ(k) = θk − k. We thus have 1 2
+
ψ(k) = a(k + ψ(k)). 2ϵ
ψ ′ (k) = a′ (k + ψ(k)) 1 + ψ ′ (k) , 2ϵ or, equivalently, a′ (k + ψ(k)) 1 2ϵ
−
a′ ( k
+ ψ(k))
,
from which we can deduce that ψ(k) is a continuous and weakly decreasing function in k. Moreover, we know by construction that A(Ik ) = [k + ψ(k), θ¯ ]. So we can write u(k, Ik ) =
k+ϵ
ωi (θ )dθ + k−ϵ
{A(Ix˜ ) ∩ [x − ϵ, x + ϵ]} ⊇ {A(Ix˜ ) ∩ [x′ − ϵ, x′ + ϵ]}. Therefore, from Assumption 1, it is easily seen in Eq. (5) that u(x, Ix˜ ) is strictly increasing. We thus have: u(x, Ix˜ ) ≥ 0 ⇔ x ≥ x˜ , and thus the strategies described in the lemma are equilibrium strategies. Second we show that there cannot be any other equilibrium. It is already clear that, if all workers play a symmetric threshold strategy according to which they migrate if and only if their signal is above a given level (the same for everyone), then this level must be x˜ . Suppose now that workers are not playing such an equilibrium strategy and follow some strategy profile summarized by the function α(x). The function α(x) then takes values different from 0 and 1 at least for some x. We then define: x = inf{x|α(x) > 0}, x¯ = sup{x|α(x) < 1}. By definition x¯ ≥ x. When α(x) > 0, some workers migrate and the net payoff to migration must be at least 0 for this to be consistent with equilibrium behavior. By continuity, at x, we must have: u(x, α) ≥ 0. Now, consider u(x, Ix ). By construction, Ix (x) ≥ α(x) for all x. So, from Lemma 1, we have u(x, Ix ) ≥ 0, which implies, by Lemma 2, that x ≥ x˜ . A similar argument shows that x¯ ≤ x˜ . But then we must have x = x˜ = x¯ .
By differentiating with respect to k, we obtain
ψ ′ (k) =
u(x, Ix˜ ). Notice that A(Ix˜ ) is of the form [θ (k), θ¯ ], so for all x > x′ , we have
Proof of Proposition 2. In any equilibrium, the net payoff of the government from regularizing all migrants is given by: 1γ (θ , s(θ , Ix˜ )). Because Ix˜ (x) is an increasing function of x, we ¯ , where θ˜ is the unique value of θ that know that A(Ix˜ ) = [θ˜ , θ] solves s(θ , Iθ˜ ) = a(θ ). References
k+ϵ k+ψ(k)
1ω(θ )dθ .
(6)
The first term in the right-hand side of (6) is strictly increasing in k. Because ψ(k) is weakly decreasing in k, Assumption 1 ensures that the second term in the right-hand side of (6) is also strictly increasing in k. Continuity comes from the fact that the limits of integration in Eq. (6) are continuous functions of k. Proof of Lemma 3. First, we show that there is a unique value x˜ of k for which u(k, Ik ) = 0. This is so because we know that u(θ − ϵ, Iθ−ϵ ) < 0 < u(θ¯ + ϵ, Iθ+ϵ ¯ ) and u(k, Ik ) is strictly increasing and continuous. When other workers are following the strategies summarized by Ix˜ , the net payoff of migration for a worker who receives signal x is given by
Carlsson, H., van Damme, E., 1993. Global games and equilibrium selection. Econometrica 61, 989–1018. Chassamboulli, A., Peri, G., 2015. The labor market effects of reducing the number of illegal immigrants. Rev. Econ. Dynam. 18, 792–821. Epstein, G.S., Weiss, A., 2011. The why, when and how of immigration amnesties. J. Popul. Econ. 24 (1), 285–316. Ethier, W.J., 1986. The illegal immigration: The host country problem. Amer. Econ. Rev. 76, 56–71. Mayr, K., Minter, S., Krieger, T., 2012. Policies on illegal immigration in a federation. Reg. Sci. Urban Econ. 42, 153–165. Morris, S., Shin, H.S., 1998. Unique equilibrium in a model of self-fulfilling currency attacks. Amer. Econ. Rev. 88, 587–597. Morris, S., Shin, H.S., 2003. Global games: Theory and applications. In: Dewatripont, M., Hansen, L., Turnovsky, S. (Eds.), Advances in Economics and Econometrics (Proceedings of the Eighth World Congress of the Econometric Society). Cambridge University Press, Cambridge, pp. 56–114. Miyagiwa, K., Sato, Y., 2015. Illegal immigration and multiple destinations, RIETI Discussion Paper Series 15-E-116.
Contents lists available at ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
Illegal migration and policy enforcement Sephorah Mangin a , Yves Zenou a,b,∗ a
Monash University, Australia
b
IFN, Sweden
article
info
Article history: Received 15 September 2016 Accepted 18 September 2016 Available online 28 September 2016 JEL classification: D8 F22 J61 O12
abstract Workers from a source country consider whether or not to illegally migrate to a host country. This decision depends crucially on the proportion of workers who migrate and on whether or not the host country government decides to regularize them. When there is no uncertainty about economic conditions in the host country, there are multiple equilibria where workers may migrate or not and the government may regularize them or not. When uncertainty is introduced, we show that there exists a unique equilibrium where each state of the world gives rise to a unique outcome in terms of migration and regularization. © 2016 Elsevier B.V. All rights reserved.
Keywords: Illegal migration International migration Global games
1. Introduction More than a million migrants and refugees crossed into Europe in 2015, compared with just 280,000 the year before. The scale of the crisis continues, with more than 135,000 people arriving in the first two months of 2016. The vast majority of these migrants, who have predominantly fled the Middle East and Africa, are illegal. The reactions from European countries have been very different. Some countries, such as Germany and Sweden, initially promised to regularize them if they came from war-torn countries such as Syria. Other countries, such as Poland and Hungary, took a strong stance against such migrants and committed to never regularize them. The aim of this paper is to analyze these issues using a simple framework where both illegal migration and the regularization policy of the host country are explicitly modeled. In our model, workers from a source country first decide whether or not to migrate illegally to a host country. Next, the government of the host country decides whether to regularize all migrants or none. Before the migration decision, workers do not know whether they will be regularized or not. However, workers know they are more likely to be regularized when more workers
migrate to the host country. In other words, from the potential migrant’s viewpoint, we have a game with strategic complements since the optimal action of each worker (migrating or not) is increasing in the average action of the other workers. We first consider the case where there is no uncertainty regarding the fundamentals of the host country’s economy, which are captured by the state of the economy θ . We show that, if θ does not take extreme values, then there are multiple equilibria. Indeed, if all workers believe that other workers will migrate and therefore that they will be regularized, then these beliefs are selffulfilling in equilibrium since the government finds it optimal to regularize them once they have migrated. On the other hand, if workers believe that nobody will migrate, then these beliefs are also self-fulfilling in equilibrium. Given this multiplicity of equilibria, no definitive prediction can be made as to whether workers will migrate or not. Using the tools of global games (Carlsson and van Damme, 1993; Morris and Shin, 1998),1 we introduce a small degree of uncertainty about the state of the economy θ in the host country. We show that common knowledge about this uncertainty leads to a unique equilibrium where each state of the world θ gives rise to a unique outcome in terms of migration and regularization. There is a large empirical and theoretical literature on illegal migration that analyzes the impact of immigration (both legal and
∗
Corresponding author. E-mail addresses: [email protected] (S. Mangin), [email protected] (Y. Zenou). http://dx.doi.org/10.1016/j.econlet.2016.09.012 0165-1765/© 2016 Elsevier B.V. All rights reserved.
1 For an overview on global games, see Morris and Shin (2003).
84
S. Mangin, Y. Zenou / Economics Letters 148 (2016) 83–86
illegal) on employment outcomes in the host country, especially in the United States (see e.g. Ethier, 1986; Epstein and Weiss, 2011; Mayr et al., 2012; Chassamboulli and Peri, 2015; Miyagiwa and Sato, 2015). Western countries have spent significant resources on limiting the number of immigrants via both border controls and internal enforcement (such as employer penalties). Despite these efforts, however, many illegal immigrants have found a way to enter these countries. To deal with these issues, these countries have in return periodically granted amnesty to any worker who can demonstrate that he fulfills certain requirements. Epstein and Weiss (2011) show that the number of illegal workers regularized has been quite significant (see their Table 1). In this paper, we propose a new and different perspective on illegal migration. We do not study the mechanisms under which a government reduces illegal migration but rather focus on how the individual’s decision to illegally migrate is affected by the same decision from other workers from the same country and by the regularization policy of the host country. The rest of the paper unfolds as follows. In the next section, we describe the model. Section 3 deals with the complete information case while Section 4 solves the incomplete information case. Section 5 concludes. All proofs can be found in the Appendix. 2. Model We consider the strategic interaction between a set of potential migrants from a source country and the government of the host country. The set of potential migrants is a mass one of workers who must decide whether or not to migrate. This is a {0, 1} decision. Their payoff if they migrate depends on the migration policy enforced by the host country government. This government has two options: she can choose to regularize all migrants or none. A worker’s payoff from migrating is ωr (θ ) if he is regularized and ωi (θ ) if he is not, where θ is a random variable that characterizes the economic conditions in the host country. For example, a high θ indicates positive economic conditions such as low unemployment and a high growth rate. We assume that θ is uniformly distributed over [θ , θ¯ ]. The utility of a worker in the source country is normalized to zero and hence the payoffs ωr (θ ) and ωi (θ ) represent net gains from migrating for a regularized and an illegal migrant, respectively. Since they incorporate both the costs and benefits of migration, they can be either positive or negative. We make the following assumptions on the payoffs of potential migrants. Assumption 1. The functions ωr (θ ) and ωi (θ ) are continuous, increasing, and satisfy 1. For all θ , ωr (θ ) > ωi (θ ), i.e. the payoff of a legal migrant is always higher than the payoff of an illegal migrant; 2. ωr (θ) < 0 < ωi (θ¯ ), i.e. it is never profitable to migrate when economic conditions are ‘‘worst’’, but it is always profitable to do so when economic conditions are ‘‘best’’; and 3. The function 1ω(θ ) = ωr (θ ) − ωi (θ ) is weakly increasing. The government in the host country reacts to the migration decisions of workers by deciding whether or not to regularize migrants. Again, this is a {0, 1} decision. We assume that the government perfectly observes both the state of the economy θ and the fraction of potential migrants who decided to migrate illegally, denoted by s ∈ [0, 1]. The payoff for the government is γ r (θ , s) if she regularizes all migrants and γ i (θ , s) if she regularizes none. These payoffs reflect the interaction between various political economy factors such as the strength of syndicates, lobbies, the median voter’s preferences, etc. For notational simplicity, we denote 1γ (θ , s) = γ r (θ , s) − γ i (θ , s). We make the following assumptions regarding the payoff of the government.
Assumption 2. The function 1γ (θ , s) is a continuous and differentiable function such that: 1. For all θ ∈ [θ , θ¯ ], 1γ (θ , 0) < 0 < 1γ (θ , 1), i.e. it is never profitable to regularize when there are no migrants while it is always profitable when there are a mass 1 of migrants; and 2. For all θ , the function 1γ (θ , s) is increasing in s. Assumption 2 implies that, for any given state of the economy
θ , there exists a unique a ∈ (0, 1) such that 1γ (θ, a) = 0. We can therefore introduce a function a(θ ) implicitly defined by 1γ (θ , a) = 0. Intuitively, a(θ ) is the mass of migrants for which the government is indifferent between regularizing or not regularizing when the state of the economy is θ . Assumption 3. The function a(θ ) is decreasing. Timing: The timing of the game is as follows. First, potential migrants simultaneously decide whether or not to migrate. Second, the government observes the mass of migrants and decides whether or not to regularize them. Observe that when a potential migrant decides whether or not to migrate, he does not know whether he will be regularized or not. Observe also that, from the potential migrant’s viewpoint, we have a game with strategic complements since the optimal action (migrating or not migrating) of each worker is increasing in the average action of the other workers. The more other workers migrate, the more likely that a given worker migrates, which increases her utility because the chance that the government will regularize them increases. We now consider two cases. In the complete information case, the potential migrant knows exactly the economic conditions θ of the host country before migrating. In the incomplete information case, he does not know the value of θ . 3. Complete information Suppose that θ is perfectly observed by migrants. In order to characterize the subgame perfect equilibria of the migration game, we first define two threshold values for the parameter θ . The threshold θ ∗ is the value of θ such that a migrant who knows that he will be regularized is indifferent between migrating or not, i.e. ωr (θ ∗ ) = 0. The threshold θ ∗∗ is the value of θ such that a migrant who knows that he will not be regularized is indifferent between migrating or not, i.e. ωi (θ ∗∗ ) = 0. Assumption 1 ensures that θ < θ ∗ < θ ∗∗ < θ¯ . Proposition 1. When θ < θ ∗ , the migration game possesses a unique subgame perfect equilibrium in which all potential migrants decide not to migrate. When θ > θ ∗∗ , the migration game possesses a unique subgame perfect equilibrium in which all potential migrants decide to migrate. When θ ∗ ≤ θ ≤ θ ∗∗ , the migration game possesses two subgame perfect equilibria: one in which all workers migrate, the other in which no worker migrates. As stated above, when a worker decides whether or not to migrate, he does not know whether or not he will be regularized but he knows the state of the economy θ in the host country. If he observes that θ < θ ∗ , he will clearly not migrate since this means, by Assumption 1 and the definition of θ ∗ , that, even if he will be regularized in the host country, his utility will be lower than staying at home. Since all individuals make the same calculations, nobody will migrate. The opposite is true when θ > θ ∗∗ since the economic conditions are sufficiently good in the host country that it is always profitable to migrate. When θ ∗ ≤ θ ≤ θ ∗∗ , then we have multiple (rational expectations) equilibria. If the workers anticipate that the government will regularize them, then they will all migrate and the government
S. Mangin, Y. Zenou / Economics Letters 148 (2016) 83–86
will indeed regularize them. If workers anticipate that the government will not regularize them, then they will not migrate and the government will not regularize them. In both cases, the expectations of the workers are self-fulfilling. These results may explain the huge illegal migration to Europe during the recent refugee crisis and why most migrants chose to stay in Germany and Sweden and not in Poland or Hungary. Given this multiplicity of equilibria, no definitive prediction can be made as to whether workers will migrate or not. Using the tools of global games, we will see in the next section that, when we introduce uncertainty about economic conditions in the host country, each state θ gives rise to a unique equilibrium outcome in terms of migration and regularization. 4. Incomplete information We now assume that potential migrants do not perfectly observe θ , the economic conditions in the host country. When the state of the economy is θ , each worker observes a noisy signal x, which is independently and uniformly distributed on [θ −ϵ, θ +ϵ]. In other words, conditional on observing the signal x, a potential migrant believes that the true parameter θ is uniformly distributed on [x − ϵ, x + ϵ]. For a given strategy profile for the workers, we denote by α(x) ∈ [0, 1] the proportion of workers who migrate when they receive a signal x, and s(θ , α) the proportion of workers who migrate when the state of the economy is θ . Since signals are uniformly distributed over [θ − ϵ, θ + ϵ ], at θ , we have: s(θ, α) =
1 2ϵ
θ+ϵ
θ−ϵ
α(x)dx.
(1)
Given a strategy profile characterized by the function α , we denote by A(α) the set of states θ in which the government will decide to regularize all migrants, i.e. A(α) = {θ|1γ (θ , s(θ , α)) ≥ 0} .
(2)
Let us denote by θ (α) the state of the economy such that s(θ(α), α) = a(θ )
(3)
where a(θ ) is the mass of migrants for which the government is indifferent between regularizing or not. Given α , Eq. (3) defines θ(α) as the minimum value of θ for which the government regularizes all migrants. Since α(x) is weakly increasing in x, θ (α) exists and is unique by Assumption 3. We can therefore write: ¯ . As a result, the payoff of a worker who decides A(α) = [θ (α), θ] to migrate when the state of the economy is θ is given by:
r ω (θ ) ωi (θ )
θ ∈ A(α), θ ̸∈ A(α).
if if
(4)
The potential migrant does not observe θ directly. As a result, the expected utility of migrating must be calculated from the posterior distribution over the states conditional on the signal x, i.e. it can be calculated by taking the expectation of (4) conditional on x. This expected utility can be written as: u(x, α) =
1 2ϵ
x+ϵ
ωi (θ )dθ + x−ϵ
1 2ϵ
A(α)∩[x−ϵ,x+ϵ]
1ω(θ )dθ .
(5)
We now provide three lemmas which will help us derive our main result. Lemma 1. If α(x) ≥ α ′ (x), for all x, then u(x, α) ≥ u(x, α ′ ). This lemma states that, for any signal x, the higher is the fraction of workers who migrate, the higher is the expected utility of migrating. This means that migration decisions are strategic complements. Now, consider the strategy profile according to
85
which a worker migrates if and only if the signal he receives is greater than some fixed number k. The corresponding proportion of migrants is then given by the following indicator function Ik : Ik (x) =
1 0
if if
x ≥ k, x < k.
Notice that Ik (x) is weakly increasing in x. When migrants follow this strategy, we have: Lemma 2. The function u(k, Ik ) is continuous and strictly increasing in k. This lemma shows that the expected utility for a worker of migrating, given that he received signal k and the proportion of migrants is given by Ik (x), is increasing in k. In other words, when the fundamentals of the economy are stronger, the payoff from migrating is higher for a worker on the margin of switching from non-migrating to migrating. Lemma 3. There is a unique x˜ such that, in any Bayesian equilibrium of the game, a worker decides to migrate if and only if x ≥ x˜ . Once the equilibrium strategies are determined, it is possible to compute the equilibrium enforcement policy. Proposition 2. There is a unique θ˜ such that, in any (perfect Bayesian) equilibrium of the game with incomplete information, the government regularizes migrants if and only if θ ≥ θ˜ . This is a very strong result, which shows that, when potential migrants face a small amount of uncertainty concerning the fundamentals of the economy, there is a unique equilibrium such that, if the state of the economy θ is sufficiently high, then all workers migrate and the government regularizes all of them. If θ is sufficiently low, then no workers migrate and the government does not regularize any. This result suggests that some information events, for example, when Germany and Sweden announced that they will regularize all refugees from Syria, can trigger a huge inflow of migrants into these countries, even though these events do not convey any real information about the state of the economy. 5. Conclusion In this paper, we model workers’ decisions to migrate illegally and the government’s decision to regularize migrants. When there is no uncertainty about the economic conditions of the host country, we show that there are multiple equilibria where workers may migrate or not and the government may regularize them or not. When we introduce some uncertainty about economic conditions, there is a unique equilibrium where each state of the world gives rise to a unique equilibrium outcome in terms of migration and regularization. Appendix. Proofs of all results Proof of Proposition 1. When θ < θ ∗ , by Assumption 1 and the definition of θ ∗ , it is a dominant strategy for workers not to migrate. Similarly, when θ ∗∗ < θ , by Assumption 1 and the definition of θ ∗∗ , it is a dominant strategy for workers to migrate. When θ ∗ ≤ θ ≤ θ ∗∗ , workers migrate if they anticipate that the government will regularize them and do not migrate if they anticipate the government will not regularize them. In turn, the government regularizes the migrants if there is a mass 1 of migrants but does not regularize them if there is a mass 0 of migrants, by Assumption 2.
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Proof of Lemma 1. Since α(x) ≥ α ′ (x), Eq. (1) ensures that s(θ, α) ≥ s(θ , α ′ ), for all θ . Then, by Assumption 2, i.e. 1γ (θ , ·) increasing, and Eq. (2), we know that A(α) ⊇ A(α ′ ). Finally, because ωr (θ ) > ωi (θ ), for all θ , we can use Eq. (5) to establish that u(x, α) ≥ u(x, α ′ ). Proof of Lemma 2. First, let us compute s(θ , Ik ). Given Eq. (1) we have: s(θ, Ik ) =
0
1
2
+
1
1 2ϵ
(θ − k)
if
θ ≤ k − ϵ,
if
k − ϵ < θ < k + ϵ,
if
θ ≥ k + ϵ.
For all k, s(·, Ik ) is a weakly increasing function that takes values in [0, 1]. We now define θ (k) by the implicit equation: s(θ (k), Ik ) = a(θk ). By Assumption 3, we know that a(θ ) is decreasing in θ , takes value in [0, 1] and, therefore, that θk is uniquely defined. Let us denote ψ(k) = θk − k. We thus have 1 2
+
ψ(k) = a(k + ψ(k)). 2ϵ
ψ ′ (k) = a′ (k + ψ(k)) 1 + ψ ′ (k) , 2ϵ or, equivalently, a′ (k + ψ(k)) 1 2ϵ
−
a′ ( k
+ ψ(k))
,
from which we can deduce that ψ(k) is a continuous and weakly decreasing function in k. Moreover, we know by construction that A(Ik ) = [k + ψ(k), θ¯ ]. So we can write u(k, Ik ) =
k+ϵ
ωi (θ )dθ + k−ϵ
{A(Ix˜ ) ∩ [x − ϵ, x + ϵ]} ⊇ {A(Ix˜ ) ∩ [x′ − ϵ, x′ + ϵ]}. Therefore, from Assumption 1, it is easily seen in Eq. (5) that u(x, Ix˜ ) is strictly increasing. We thus have: u(x, Ix˜ ) ≥ 0 ⇔ x ≥ x˜ , and thus the strategies described in the lemma are equilibrium strategies. Second we show that there cannot be any other equilibrium. It is already clear that, if all workers play a symmetric threshold strategy according to which they migrate if and only if their signal is above a given level (the same for everyone), then this level must be x˜ . Suppose now that workers are not playing such an equilibrium strategy and follow some strategy profile summarized by the function α(x). The function α(x) then takes values different from 0 and 1 at least for some x. We then define: x = inf{x|α(x) > 0}, x¯ = sup{x|α(x) < 1}. By definition x¯ ≥ x. When α(x) > 0, some workers migrate and the net payoff to migration must be at least 0 for this to be consistent with equilibrium behavior. By continuity, at x, we must have: u(x, α) ≥ 0. Now, consider u(x, Ix ). By construction, Ix (x) ≥ α(x) for all x. So, from Lemma 1, we have u(x, Ix ) ≥ 0, which implies, by Lemma 2, that x ≥ x˜ . A similar argument shows that x¯ ≤ x˜ . But then we must have x = x˜ = x¯ .
By differentiating with respect to k, we obtain
ψ ′ (k) =
u(x, Ix˜ ). Notice that A(Ix˜ ) is of the form [θ (k), θ¯ ], so for all x > x′ , we have
Proof of Proposition 2. In any equilibrium, the net payoff of the government from regularizing all migrants is given by: 1γ (θ , s(θ , Ix˜ )). Because Ix˜ (x) is an increasing function of x, we ¯ , where θ˜ is the unique value of θ that know that A(Ix˜ ) = [θ˜ , θ] solves s(θ , Iθ˜ ) = a(θ ). References
k+ϵ k+ψ(k)
1ω(θ )dθ .
(6)
The first term in the right-hand side of (6) is strictly increasing in k. Because ψ(k) is weakly decreasing in k, Assumption 1 ensures that the second term in the right-hand side of (6) is also strictly increasing in k. Continuity comes from the fact that the limits of integration in Eq. (6) are continuous functions of k. Proof of Lemma 3. First, we show that there is a unique value x˜ of k for which u(k, Ik ) = 0. This is so because we know that u(θ − ϵ, Iθ−ϵ ) < 0 < u(θ¯ + ϵ, Iθ+ϵ ¯ ) and u(k, Ik ) is strictly increasing and continuous. When other workers are following the strategies summarized by Ix˜ , the net payoff of migration for a worker who receives signal x is given by
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