1. Background

In the last few years, the number of people seeking protection from political violence in other countries has increased drastically. While most displaced persons are displaced internally (i.e., within the country they lived in before), the number of internationally displaced persons has increased from 17 million to 34 million in the last two decades. Three notable recent displacement events are the European refugee crisis of 2015/16 during which Syrian and other nationals filed about 2.4 million asylum applications in Europe and the exodus of almost 4 million Venezuelans and 6 million Ukrainians to neighboring countries.Footnote ⁹

The legal origin of the current refugee regime is the 1948 Universal Declaration of Human Rights, which recognizes the right of any person to seek asylum from political persecution in other countries. Building on this declaration, the 1951 Refugee Convention (or Geneva Convention of July 28, 1951) defines a refugee as a person who, “owing to well-founded fear of being persecuted for reasons of race, religion, nationality, membership of a particular social group or political opinion, is outside the country of his nationality and is unable or, owing to such fear, is unwilling to avail himself of the protection of that country.” Some international regimes, such as the European Union, also recognize a need for (temporary) protection due to war even in the absence of personal persecution.Footnote ¹⁰

In some circumstances, countries may award protection to all persons with a particular nationality—as the European Union did with respect to Ukrainians after the Russian invasion of 2022—but, typically, asylum applications are evaluated on a case-by-case basis. To determine eligibility for protection, most countries maintain specialized bureaucracies that receive and evaluate each asylum application individually. These applications are costly because they typically entail migrating to the destination country, filing paperwork, and being subjected to the asylum procedure (which, for example, could limit applicants’ freedom of movement).Footnote ¹¹ If someone qualifies for protection, the person is officially recognized as a refugee. If an application is denied, the applicant may be forcefully deported unless the individual leaves the country voluntarily.

National laws typically regulate where and how applicants can file an application, applicants’ rights and responsibilities during the review process, and applicants’ access to the legal system to appeal rejections. While these differences are clearly important, our model focuses on the common core features of most national procedures: foreign nationals have to apply for asylum and bureaucrats decide on their cases.

2. A model of the asylum process

The players are a foreign national, E, and a bureaucrat, B. The bureaucrat could be an officer from a border protection agency tasked with making an initial assessment about the foreign national's asylum case. Alternatively, the bureaucrat could be an immigration judge who is in charge of making a final decision regarding the applicant's asylum case.Footnote ¹²

We denote the foreign national's (eventual) location by L ∈ {0,1}, where L = 1 means that the foreign national is admitted as a refugee and in a safe destination country while L = 0 means that the foreign national is in the home country. E chooses to apply for asylum (e = 1) or not (e = 0). If E chooses to apply, B chooses to grant asylum (a = 1) or not (a = 0). Thus, the foreign national is in the safe destination country if and only if the foreign national applies and is granted asylum, i.e., L = ea.

The foreign national can be interpreted as a representative citizen of a foreign country. In this polity, each citizen faces an expected threat of future violence denoted by v. We take the quantity v to measure the level of individual persecution.Footnote ¹³ We assume that v is private information to the foreign national (and, therefore, unknown to the asylum-granting bureaucrat) and drawn from a commonly known distribution F. The justification for this assumption is that v is difficult to verify as it depends on other actors’ (future) behavior. E's utility function is

$$U_E = Lw_1 + ( 1-L) w_0( v) -ec,\;$$

where c is the cost of applying for asylum (which might include the costs of emigrating to the destination country as discussed above) and w _L is the wage—or more generally the quality of life—when the foreign national is living in L. Importantly, we assume that, when remaining in the home polity, the foreign national's payoff is strictly decreasing in the level of individual persecution.

The bureaucrat's utility function, described in Table 1, depends on κ, which parametrizes the bureaucrat's concern for Type-I and Type-II errors. The bureaucrat wishes to make the correct decision. Given an asylum standard $\overline {v}$, the foreign national is eligible for asylum if the level of persecution is larger than $\overline {v}$. For reasons that will become apparent below, we interpret lower values of κ as representing a more lenient bureaucrat. One interpretation of κ could be that there are some bureaucrats that care much less about someone's eligibility for asylum, but rather prioritize their own understanding of deservingness. We assume that κ is the bureaucrat's private information, drawn from a distribution with cumulative distribution function H and associated density h. Thus, we assume that the foreign national does not know the exact payoff function of the bureaucrat. We sometimes refer to H as the agency's reputation.

Table 1.

The bureaucrat's preferences

The bureaucrat's utility function depends on the agency's mission and individual characteristics. Generally speaking, for signatory states of the 1951 Refugee Convention, the agency's mission is to accept eligible individuals and reject non-eligible ones—but the exact standard ($\overline {v}$) may vary across polities. The bureaucrat may have internalized this standard, and mistakes may be differentially punished through diminished career prospects. Importantly, we assume that mistakes (as parametrized by κ) depend only on eligibility, not on the exact distance between the applicant's v and the standard $\overline {v}$. The empirical literature has studied bureaucrats’ decisions, showing that they are affected by the media environment (Spirig, Reference Spirig2021), their moral schemas (Shiff, Reference Shiff2021), and anti-Muslim biases (Emeriau, Reference Emeriau2021). To some extent, we capture these considerations with different specifications of H. For notational simplicity, we do not add a separate parameter that represents the bureaucrat's bias. However, as long as this parameter is not too large, this does not affect our substantive conclusions.

Summarizing, the sequence of moves by the players is as follows:

1. 1. Nature draws the type of the foreign national v from the distribution F.

2. 2. The foreign national applies for asylum (e = 1) or not (e = 0).

3. 3. Nature draws the concern for incorrect decisions κ from the distribution H.

4. 4. If there is an application, the bureaucrat grants asylum (a = 1) or not (a = 0).

To facilitate the analysis, we impose several assumptions on the primitives. We assume that the functions F, H, and w ₀ are continuously differentiable, with F having full support on $\mathbb{R}$ (v < 0 may be the perpetrators of violence) and H having full support on [0,1]. The density h is assumed to have a single interior peak and satisfies h(1) < 1. Furthermore, we assume that the function w ₀ is strictly decreasing in v and that parameter values are such that lim_v→−∞ w ₀(v) > w ₁ − c, which means that there are some types of the foreign national who do not have an incentive to apply even if the probability of admission is 1. However, we do assume that $w_0( \overline {v}) < w_1-c$, which means that the type who is just eligible, $\overline {v}$, finds it attractive to apply.

Our model is deliberately stylized. Both the foreign national and the bureaucrat make binary choices. This implies that the bureaucrat learns only from application decisions—it is impossible to gather further information about the foreign national's type through, for example, an interview. We abstract away from interviews (and other procedures designed to gather additional information) to focus on the essential choices of the players. However, our results are robust to allow the revelation of additional information (via an exogenous signal about eligibility) as long as communication is not too precise.

We also abstract away from demographic variables such as skill, education, and religion. We could incorporate them by indexing all the parameters of the model with a specific subset of demographic variables. For example, c, w ₁, and w ₀(v) might all depend on the foreign national's skill level. However, because our focus is on dynamics that hold for all demographic strata, we do not include these characteristics.

Here, we assume that the eligibility threshold is common knowledge. In the SM, we consider a variation in which there is uncertainty about the asylum eligibility standard. In this version, foreign nationals have to average over potential eligibility thresholds, and the bureaucrat's inferences become somewhat weaker, but our substantive conclusions are unchanged. More generally, in the SM, we consider a number of alternative variations to the baseline model (see SM-G), including a complete information version and versions with one-sided incomplete information. The upshot from analyzing these variations is that restrictive asylum policy choices have an ambiguous effect whenever the bureaucrat is responsive to the foreign national's application strategy.

Our model resembles existing models in which individuals are hired by or join a firm or organization and are assessed for some characteristic (e.g., Bolton and Dewatripont, Reference Bolton and Dewatripont2004; Bueno De Mesquita, Reference Bueno De Mesquita2005; Spaniel, Reference Spaniel2018). Our contribution is to consider the political persecution variable v and how it affects the foreign national's payoffs, i.e., that higher levels of it decrease the utility of not applying. By contrast, ideology or skill may (positively) affect the utility of being admitted as well. These differences in payoff can result in equilibria with different properties. In addition, our extensions incorporate additional outcomes that are closely linked to the politics of asylum. Nevertheless, it is useful to begin with a model that does not incorporate these outcomes to study the players’ fundamental incentives.

3. Equilibrium of baseline model

Our solution concept is Perfect Bayesian Equilibrium. Specifically, we look for an equilibrium in which both players employ interior threshold rules. In such an equilibrium, the foreign national applies if and only if $v \geq \hat {v}^\ast$, and the bureaucrat grants asylum if and only if $\kappa \leq \hat {\kappa }^\ast$, and both $\hat {v}^\ast$ and $\hat {\kappa }^\ast$ are real numbers. Such a semi-separating equilibrium is empirically plausible; however, it is worth pointing out that our model also admits a pooling equilibrium in which no type applies for asylum. As we detail in the SM, this equilibrium requires that the bureaucrat's belief about eligibility is relatively low. Finally, there is no pooling equilibrium in which all types apply for asylum because there are some types that do not wish to leave their home polity even if the probability of admission is 1.

To begin with, the bureaucrat's expected utility of choosing a = 1 is $-\kappa [ 1-\Pr ( v\geq \overline {v}\vert e = 1) ]$, where $\Pr ( v\geq \overline {v}\vert e = 1)$ is the bureaucrat's belief that the foreign national is eligible, conditional on having received an asylum application. Moreover, the bureaucrat's expected utility of choosing a = 0 is $-( 1-\kappa ) \Pr ( v\geq \overline {v}\vert e = 1)$. Thus, the bureaucrat grants admission if

$$\Pr( v \geq \overline{v}\vert e = 1) \geq \kappa.$$

Intuitively, for the bureaucrat to grant asylum, the probability of eligibility needs to be sufficiently large relative to the concern about granting admission when the applicant is not eligible, κ. Now, suppose that the foreign national chooses to apply if and only if $v\geq \hat {v}$, where $\hat {v}$ is a finite threshold to be determined in equilibrium. Then:

$$\Pr( v \geq \overline{v}\vert e = 1) \equiv q( \hat{v}) = \left\{\matrix{ {1-F\left(\overline{v}\right)\over 1-F\left(\hat{v}\right)} \hfill & \text{if} \ \hat{v} < \overline{v} \hfill \cr 1 \hfill & \text{if} \ \hat{v}> \overline{v}. \hfill }\right.$$

Using this notation, the bureaucrat grants asylum if

(1)$$q\left(\hat{v}\right)\geq \kappa.$$

An important effect concerns how the bureaucrat's posterior belief reacts to an increase in the threshold employed for applying. Formally, when $\hat {v}< \overline {v}$:

$${\partial q\over \partial \hat{v}} = -{\left[1-F\left(\overline{v}\right)\right]\over \left[1-F\left(\hat{v}\right)\right]^2}\left[-f\left(\hat{v}\right)\right] = q\left(\hat{v}\right){\,f\left(\hat{v}\right)\over 1-F\left(\hat{v}\right)}> 0.$$

Thus, when fewer types of the foreign national apply, the posterior belief of eligibility increases. Moreover, the strength of this effect depends on the existing posterior belief.

Now consider the foreign national's decision rule, and let p be the probability that the bureaucrat grants asylum (to be determined in equilibrium). Then, the foreign national chooses to apply if pw ₁ + (1 − p)w ₀(v) − c ≥ w ₀(v), or

(2)$$p \geq {c\over w_1-w_0( v) }.$$

The right-hand side is decreasing in v, confirming that the foreign national chooses to apply if and only if $v \geq \hat {v}$. Examining the foreign national's decision rule, it is useful to define v as the solution to

$$1 = {c\over w_1-w_0( \underline{v}) }.$$

Here, v is the type that is indifferent between applying or not when the bureaucrat grants asylum for sure. It represents the demand for asylum. Note that $\underline {v}< \overline {v}$, i.e., the type that is just indifferent is not eligible.

An equilibrium involves a consistency condition of Expressions 1 and 2. From the foreign national's perspective, the probability of achieving asylum is $H( q( \hat {v}) )$, which is increasing in $\hat {v}$ because both H and q are increasing. The equilibrium threshold is then determined by the following condition:

(3)$$H\left(q( \hat{v}^\ast) \right) = {c\over w_1-w_0( \hat{v}^\ast) }.$$

The left-hand side is increasing in $\hat {v}$ while the right-hand is decreasing in $\hat {v}$. Thus, if there is a solution to the expression, it will be unique (all proofs are included in the SM):

Figure 1 illustrates the equilibrium. The left panel displays the interpretation of foreign applicant types in equilibrium. The right panel illustrates the equilibrium condition as a function of potential thresholds for applying. The gray line is the probability with which the bureaucrat grants admission as a function of different thresholds for applying (the left-hand side of Expression 3). The black line represents the costs of applying for asylum adjusted by the stakes of admission, i.e., the change of living standard when the applicant is in the safe destination polity rather than the unsafe home polity (the right-hand side of Expression 3). The intersection of these lines determines the equilibrium threshold for applying, $\hat {v}^\ast$.

Figure 1.

Overview of the equilibrium. Parameter values: w ₁ = 1, w ₀(v) = 1.1 − v, c = 1.5, $F = {\cal N}( 2,\; \, 1)$, and $H = {\cal B}( 2,\; \, 5)$. (a) Types of the Foreign Nationals. (b) Finding the Equilibrium Threshold $\hat {v}^\ast$.

Before continuing the analysis, we briefly relate our model to existing empirical findings. Reviewing the literature on push and pull factors behind asylum applications to industrialized countries, Hatton (Reference Hatton2020) concludes that the most important origin-country variables are political terror and lack of civil liberties, while there is weaker evidence that declines in origin-country income per capita lead to more asylum applications. In the SM, we show that a reduction in origin-country wages increases the probability that the foreign national applies, which is consistent with the empirical correlation between per capita income and the number of asylum applications in destination countries (see SM-B). We also show that intensifying violence does not necessarily increase the probability that the foreign national will apply for asylum. Suppose the level of political persecution is on average higher. Keeping the threshold of an asylum application fixed, the probability of an application is higher because more types are threatened. However, in equilibrium, the threshold may increase or decrease when violence intensifies. The reason is that intensifying violence increases both the prior probability of eligibility and the probability of an application (for a fixed candidate threshold), inducing competing effects in the bureaucrat's posterior belief of eligibility (which is the ratio of these two probabilities). However, intensifying violence is more likely to increase applications if the prior probability of eligibility is relatively low.

4. The effect of restrictive asylum policies

The above model considers a situation in which foreign nationals and bureaucrats interact without interference by politicians. But empirical research highlights that politicians have various opportunities to interfere (see, e.g., Hatton, Reference Hatton2017). In this section, we introduce a new variable, denoted by $t \in \left[\underline{t},\overline{t}\right]$, which represents specific asylum policies. We interpret larger values of t as a more restrictive policy choice that can take different forms.

We distinguish between two distinct types of policies. On the one hand, policies can manipulate the costs of applying for asylum (c) and/or the foreign national's wages in the home and destination polity (w ₀ and w ₁, respectively). We refer to these types of policies as demand-oriented asylum policies. Such policies may take the form of (i) reduced wages due to employment restrictions in the destination country (w ₁(t) is decreasing in t); (ii) increased wages in the origin country through foreign aid (w ₀(v,  t) is increasing in t); or (iii) increased costs of applying because of a border wall (c(t) is increasing in t). We assume that, in all cases, the relevant function is continuously differentiable. On the other hand, policies can manipulate the standards and procedures of the bureaucracy. Such supply-oriented policies can directly alter the asylum standard or aim to affect bureaucrats’ preferences through the hiring process.Footnote ¹⁴

Our typology differs from the typology in Hatton (Reference Hatton2017), who groups asylum policies based on the stage of the asylum process that they affect: rules and policies related to border and the ease of filing an application (access rules), policies related to the standards and procedures of the asylum application review (processing rules), and policies related to the rights and benefits of applicants and refugees (rights and benefit rules). In our model-driven typology, processing rules are supply-side policies while access and rights and benefit rules are demand-side policies.

In our main model, we focus on demand-oriented policies (we present a brief analysis of supply-oriented policies in SM-H). Regardless of the exact specification, when it comes to demand-oriented policies, we can show the following:

The first part of the Proposition confirms the conventional logic of asylum-seeker deterrence: employment bans, border walls, and development policy all decrease the probability that the foreign national applies for asylum by making this choice less attractive. However, we also point out that, due to informational effects, conditional on an application, the bureaucrat is more likely to grant admission.

From an ex ante perspective, for the semi-separating equilibrium derived above, the probability of admission is given by

$$\underbrace{\left[1-F\left(\hat{v}^\ast\right)\right]}_{\text{Application}}\underbrace{H\left(q( \hat{v}^\ast) \right)}_{\text{Admission}}.$$

We investigate how an exogenous increase in the variable t affects the equilibrium probability of admission. Differentiating with respect to t yields

$$-\underbrace{\,f\left(\hat{v}^\ast\right){\partial \hat{v}^\ast\over \partial t}H( q( \hat{v}^\ast) ) }_{\text{Deterrence}} + \underbrace{\left[1-F\left(\hat{v}^\ast( t) \right)\right]h( q( \hat{v}^\ast) ) {\partial q\over \partial \hat{v}}{\partial \hat{v}^\ast\over \partial t}}_{\text{Credibility}}.$$

Thus, a more restrictive asylum policy choice has competing effects on the admission probability. On the one hand, some types are deterred, reducing the probability of the foreign national applying for asylum. This is the deterrence effect. On the other hand, for those who do apply, the bureaucrat is less skeptical about their eligibility. This is the credibility effect. Due to these competing effects, the relationship between restrictive asylum policies t and the number of admitted refugees is ambiguous in general.

The proof of Proposition 3 reveals that the effect is negative if the equilibrium probability of eligibility is high. Deterrence is relatively more important here, as the foreign national already enjoys high credibility. This can be the case, for example, when the costs of applying for asylum are relatively high, so that fewer types apply. Figure 2 provides an example. In both panels, we plot, as a function of an application threshold $\hat {v}$, the probability of an application $1-F( \hat {v})$, the conditional probability of an admission given an application $H( q( \hat {v}) )$, and the unconditional probability of an admission, which is the product of the two preceding expressions. Note that the last function is non-monotone in the application threshold $\hat {v}$. We then consider the changes induced by a binary asylum policy, t, which can be permissive or restrictive, resulting in a wage w ₁ that is high or low. In both panels, implementing the more restrictive policy pushes the equilibrium application threshold up. However, in the left panel, the costs of applying are high and so the equilibrium probability of admission goes down. By contrast, in the right panel, the costs of applying are low and the equilibrium probability of admission goes up after choosing the more restrictive policy. An implication of Proposition 3 is that a politician who is interested in reducing the number of admitted refugees may not choose the most restrictive asylum policy. This is further explored below.

Figure 2.

Illustration of Proposition 3 with a binary policy. Parameter values: $F = {\cal N}( 2,\; \, 2)$, $H = {\cal B}( 2,\; \, 2)$, $\overline {v} = 5$, $w_0( v) = 1-\Phi ( {v\over 5})$, where Φ is the CDF of the standard normal distribution. Further, w ₁(t) = 1 for the restrictive policy and 1.2 for the permissive policy. In the left panel, c = 0.83. In the right panel, c = 0.5. The equilibrium thresholds $\hat {v}^\ast _R$ (for the restrictive policy) and $\hat {v}^\ast _P$ (for the permissive policy) are computed from equation 3. (a) High application costs. (b) Low application costs.

We also briefly examine the implications of restrictive policies for the welfare of the foreign national in SM-C. For concreteness, consider the case of an employment ban, so that w ₁(t) is decreasing in t. We show that an employment ban can either increase or decrease the welfare of the foreign national. The intuition is that an employment ban has two competing effects on the foreign national's welfare. First, the bureaucrat has increased confidence that the foreign national is eligible, which increases the probability of admission. Second, conditional on in fact receiving admission, the employment ban hurts the foreign national because of the expected wage decline.

5. Extending the model

In our baseline model, we examined the strategic interaction between the bureaucrat and a foreign national in a frictionless setting, highlighting why restrictive asylum policies have an ambiguous effect on admitted refugee numbers. Next, we extend the model, exploring scenarios in which some rejected applicants are not deported; applicants have the opportunity to enter without authorization; and an asylum application might never reach the bureaucrat. Here, we focus on substantive insights, relegating much of the formal analysis to SM-D.

5.1 Imperfect agency enforcement

As previewed above, many governments struggle to return foreign nationals that remain without authorization in their territory (Wong, Reference Wong2015; de Haas et al., Reference de Haas, Castles and Miller2020). We now incorporate this possibility into the baseline model. We assume that once located in the safe destination country, the foreign national either can be formally admitted as a refugee and, hence, can work in the regular, formal sector (S = F), or may be there without formal refugee status and, hence, is able to work only in the informal sector (S = I). We denote the wage in sector S by $w^S_1$ and assume that the formal sector offers higher wages, i.e., $w^F_1> w^I_1$. Furthermore, we also assume that the bureaucrat's preferences depend only on the final location of the applicant, i.e., on L but not on S.

The sequence of moves is the same as before; the only difference is that the bureaucrat's decision is imperfectly enforced. Specifically, when the bureaucrat grants asylum (a = 1), then the outcome is (L,  S) = (1,  F). But when the bureaucrat rejects the application (a = 0), then (L,  S) = (1,  I) with probability 1 − λ, and L = 0 with probability λ. Thus, λ is the exogenous probability of deportation (we assumed λ = 1 above). We discuss possible micro-foundations for this probability below.

The bureaucrat's decision rule is unchanged; the bureaucrat admits the applicant as a refugee if the probability that the foreign national is eligible is larger than the relative concern for admitting an ineligible foreign national, i.e., $\Pr ( v \geq \overline {v}\vert e = 1) = q( \hat {v}) \geq \kappa$. Intuitively, while the bureaucrat recognizes the risk that the decision may not be enforced, the bureaucrat has no reason to change the decision calculus. The foreign national anticipates the bureaucrat's behavior and is aware of the possibility of remaining in the safe polity, even if the bureaucrat rejects the application for asylum. Following the reasoning for the baseline model, the equilibrium threshold is, thus, determined by the following equality:

(4)$$H\left(q( \hat{v}^\ast) \right) = {c - ( 1-\lambda) \left[w^I_1-w_0( \hat{v}^\ast) \right]\over w^F_1-w^I_1 + \lambda( w^I_1-w_0\left(\hat{v}^\ast) \right)}.$$

We show in the SM that there is a unique interior equilibrium threshold, which is characterized by the preceding expression (see SM-D).

In equilibrium, the effect of t on the admission probability is equal to:

$$\lambda\left[-\underbrace{\,f( \hat{v}^\ast) {\partial \hat{v}^\ast\over \partial t}H( q( \hat{v}^\ast) ) }_{\text{Deterrence I}} + \underbrace{( 1-F( \hat{v}^\ast) ) h( q( \hat{v}^\ast) ) {\partial q\over \partial \hat{v}}{\partial \hat{v}^\ast\over \partial t}}_{\text{Credibility}}\right]-\overbrace{( 1-\lambda) f( \hat{v}^\ast) {\partial \hat{v}^\ast\over \partial t}}^{\text{Deterrence II}}.$$

This generalizes the corresponding expression from the baseline analysis—which featured only the expression in square brackets—and shows that the effect is still ambiguous, although there is an additional term, labeled “Deterrence II,” which is negative. Intuitively, in the event of no enforcement (which happens with probability 1 − λ), decreasing the probability with which the foreign national applies is all that matters for reducing the probability of having a refugee admitted. This demonstrates that the parameter λ moderates the strength of the deterrence effect: if λ is sufficiently low, only deterrence matters because the bureaucrat cannot fulfill their gatekeeping role. However, if λ is relatively large, the analysis is qualitatively identical to the baseline case. This suggests that countries that have less enforcement capacity (i.e., λ is smaller) are more likely to rely on deterrence policies all else being equal.

Now consider the effect of λ on the equilibrium threshold. We have:

Remark 1. An increase in the enforcement probability λ increases the equilibrium threshold if $w_1^I> w_0( \hat {v}^\ast )$ holds, and decreases it if the reverse inequality holds.

This ambiguous effect is consistent with empirical work that finds no systematic relationship between the number of deportations and the number of applications (Wong, Reference Wong2015). The condition highlighted in Remark 1 is whether the type who is indifferent between applying or not has a home polity wage that is larger or smaller than the informal wage. To see that an increase in enforcement can actually lead to more applicants, suppose that $w_1^I$ is very small and so receiving it is unattractive even for types with a moderately high level of v. Hence, the outcome in which they are rejected but stay is unattractive and decreasing enforcement (a lower level of λ) would actually decrease the probability of an application.

We assume that the enforcement probability λ is exogenous here. There are two salient ways to endogenize it. First, an enforcement agency could be in charge of exerting effort to deport applicants that are rejected. Second, one can also imagine that political actors are trying to shape the enforcement probability through law-making and lobbying. One frequently mentioned argument is that some firms benefit from low enforcement because they benefit from the prevalence of irregular workers and their low wages (Freeman, Reference Freeman1995). We pursue both extensions in the SM, showing that increasing the funding for an enforcement agency does not always decrease applications and that firms have incentives to lobby for weak enforcement if the profits from inexpensive labor are relatively large (see SM-D).

5.2 Irregular migration

We next explore the consequences of allowing for irregular migration, which is a common occurrence in many polities (Hollifield et al., Reference Hollifield, Martin and Orrenius2014). Specifically, at the beginning of the game, the foreign national decides to apply for asylum (e = 1), to stay in the home polity (e = 0), or to attempt to enter the destination country without a formal process (e = −1). We assume that when the third option is chosen, with probability π, the foreign national earns an informal wage $w^I_{-1}< w^F_1$; with probability 1 − π, the foreign national is caught by border enforcement agents and sent back, in which case the wage w ₀(v) is earned. This assumption implies that the foreign national cannot apply for asylum after having decided to enter the destination country irregularly. We also assume that attempting to enter irregularly incurs costs k.

We search for an equilibrium in which there exist two thresholds, $\hat {v}_{-1}$ and $\hat {v}_{1}$, such that if $v< \hat {v}_{-1}$, then type v stays in the home polity; if $v \in [ \hat {v}_{-1},\; \, \hat {v}_{1})$, type v attempts irregular migration; and if $v \geq \hat {v}_1$, type v applies for asylum. Other equilibria may exist, but this is a natural form given our baseline analysis without irregular migration. In the SM, we impose conditions on the model's primitives such that this equilibrium exists. Note that in such an equilibrium, the bureaucrat's posterior belief is again given by ${1-F( \overline {v}) \over 1-F( \hat {v}_1) }\equiv q( \hat {v}_1)$, and the probability of admission is $H( q( \hat {v}_1) )$.

We can immediately pin down the equilibrium threshold value $\hat {v}_{-1}^\ast$, which is given by the solution to the following equality:

(5)$$\pi = {k\over w^I_{-1}-w_0( \hat{v}_{-1}^\ast) }.$$

Intuitively, the costs of irregular migration, adjusted for the wage difference, are equal to the probability of successfully migrating. Moreover, following the reasoning above, the second threshold, $\hat {v}_1^\ast$, is given by the solution to the following equality:

(6)$$H( q( \hat{v}_1^\ast) ) = {c-k + \pi\left[w^I_{-1}-w_0\left(\hat{v}_1^\ast\right)\right]\over w^F_1-w_0\left(\hat{v}_1^\ast\right)}.$$

Now consider the effect of asylum policy, t. The probability of having the foreign national in the destination polity (L = 1) is

$$\underbrace{\left(1-F( \hat{v}_{1}^\ast) \right)H( q( \hat{v}^\ast_1) ) }_{\text{Regular status}} + \underbrace{\pi\left[F( \hat{v}_{1}^\ast) -F( \hat{v}_{-1}^\ast) \right]}_{\text{Irregular status}}.$$

The derivative of the preceding expression with respect to asylum policy t is:

$$-\underbrace{\,f( \hat{v}_1^\ast) {\partial \hat{v}_{1}^\ast\over \partial t}H( q( \hat{v}_1^\ast) ) }_{\text{Deterrence}} + \underbrace{\left(1-F( \hat{v}^\ast_1) \right)h( q( \hat{v}_1^\ast) ) {\partial q\over \partial \hat{v}_1}{\partial \hat{v}_1^\ast\over \partial t}}_{\text{Credibility}} + \underbrace{\pi\left[\,f( \hat{v}_1^\ast) {\partial \hat{v}_{1}^\ast\over \partial t}-f( \hat{v}_{-1}^\ast) {\partial \hat{v}_{-1}^\ast\over \partial t}\right]}_{\text{Distributional Effects}}.$$

The first part of the expression is the same as in the baseline analysis, showing that our basic insights are robust to allowing for irregular migration in our model. The second part of the expression is new and details the additional costs and benefits when there is irregular migration. Depending on the exact instrument (t) and on the shape of the density f, it can be positive or negative. This demonstrates that subtle distributional effects may be present. For example, consider the case of a sectoral employment ban. When the sectoral employment ban restricts asylum seekers to working only in low-wage jobs, $w^F_1$ is decreasing in t. Suppose that t increases, decreasing $w^F_1$. This increases $\hat {v}_1^\ast$ but leaves $\hat {v}_{-1}^\ast$ unaffected. Thus, some types that previously applied for asylum now attempt to migrate through irregular means. All else equal, this increases the probability of having the foreign national in the polity (this effect is represented by the expression $\pi {\partial \hat {v}_1^\ast \over \partial t}f( \hat {v}_1^\ast )$ above).Footnote ¹⁵

Turning to the effect of the probability that irregular migration is successful, π, we show the following:

Remark 2. In an equilibrium with thresholds $\hat {v}_{-1}^\ast$ and $\hat {v}_{1}^\ast$ such that $\hat {v}_{-1}^\ast < \hat {v}_1^\ast$, holding fixed $\hat {v}_1^\ast$, the higher π, the more types attempt to enter through irregular means. Conversely, holding fixed $\hat {v}_{-1}^\ast$, the higher π, the fewer types attempt to enter by applying for asylum.

Substantively, this means that when a border tightening occurs, the probability of an irregular border crossing decreases but the probability of an application for asylum increases. Moreover, due to the bureaucrat's endogenous response, the probability of admission decreases. Thus, our model predicts that a tighter border decreases total attempted migration but may increase or decrease the number of admitted refugees.

5.3 Dangerous journeys

So far, we have assumed that if the applicant applies for asylum, e = 1, the application will reach the bureaucrat for sure—and the bureaucrat will make a decision on the case. However, when the foreign national has to engage in a long journey to file an asylum application, this assumption may be too restrictive. For example, the applicant might not be able to reach the territory of a country that provides opportunities to file an asylum application or might die while migrating, as in the case of migrants crossing the Mediterranean Sea. In the model, suppose that if the foreign national chooses e = 1, there are two initial outcomes. The foreign national may make it to the destination country and thus successfully apply for asylum. We denote this event by s = 1. However, there is also a chance that the attempt is unsuccessful. We denote this event by s = 0 and assume that the foreign national's payoff here is w. Since the foreign national is no longer in the home polity, this parameter is independent of v. We assume that this term is relatively low: $\underline {w}< w_0( \overline {v})$. Note that this includes the case w = 0, which might be interpreted as the applicant's death. Conditional on e = 1, the probability that s = 1 is ρ: $\Pr ( s = 1\vert e = 1) = \rho \in ( 0,\; \, 1)$. The bureaucrat decides to grant asylum or not only if s = 1.

In a profile with application threshold $\hat {v}$, if the attempt is successful (s = 1), the bureaucrat holds posterior of eligibility $q( \hat {v})$ and grants asylum if $q( \hat {v}) \geq \kappa$. As a result, similar to the baseline case, there is a unique equilibrium threshold characterized by

$$H\left(q( \hat{v}^\ast) \right) = {c + ( 1-\rho) \left[w_0( \hat{v}^\ast) -\underline{w}\right]\over \rho\left[w_1-w_0( \hat{v}^\ast) \right]}.$$

Given the equilibrium threshold, we compute the effect of the asylum policy t on the probability of an admitted refugee:

$$-\rho \underbrace{\,f( \hat{v}^\ast) {\partial \hat{v}^\ast\over \partial t}H\left(q( \hat{v}^\ast) \right)}_{\text{Deterrence}} + \rho\underbrace{\left(1-F( \hat{v}^\ast\right)h\left(q( \hat{v}^\ast) \right){\partial q\over \partial \hat{v}}{\partial \hat{v}^\ast\over \partial t}}_{\text{Credibility}}.$$

As in the baseline case, this is ambiguous, with the only difference being that both the deterrence and the credibility effects are scaled by the probability of a successful asylum application, ρ. Intuitively, only if the application reaches the bureaucrat is the foreign national ever admitted.

Turning to an analysis of the effect of ρ on the equilibrium application threshold, we can show:

Remark 3. An increase in the probability of a successful application ρ decreases the equilibrium threshold $\hat {v}^\ast$.

The result explains the decrease in asylum applications after Australia implemented its “Pacific Solution,” in which boats transporting asylum seekers were turned around by its coast guard. It also explains some of the dilemmas of asylum politics more broadly: a low level of success deters potential applicants for asylum, but, conditional on an application, a low ρ means a higher likelihood of bad outcomes for the applicants. Moreover, conditional on the bureaucrat having received the application, a low level of ρ means that the bureaucrat is more likely to admit the applicant. While an analysis of an endogenous ρ is beyond the scope of this paper, our results provide a stepping stone for future work on this.

6. Endogenous asylum policies

What kinds of asylum policies do politicians actually choose? Suppose that, before the interaction between the bureaucrat and the foreign national unfolds, an incumbent politician can affect the admission process through a policy choice $t \in \left[\underline{t},\overline{t}\right]$, inducing the same changes in the environment as above. We focus on a populist, refugee-skeptic politician who receives a payoff of 0 if the foreign national applies and is granted asylum, i.e., if L = 1, and 1 in the event that L = 0. In the remainder, we refer to the politician as a as refugee-skeptic politician. The politician's payoff function is consistent with a policy-seeking politician who intrinsically values reducing the number of refugees admitted. Of course, a variety of electoral considerations may also motivate a politician to choose particular asylum policies. In this paper, we abstract away from electoral politics, but we discuss this issue further in the conclusion. Finally, we assume that the politician does not internalize the costs of implementing specific policies, even though the policy chosen may be costly to implement.

As a consequence of the results derived above, we can write the politician's optimization problem as follows:

$$\max\limits_{t \in \left[\underline{t}, \overline{t}\right]} \; 1- \underbrace{\left(1-F( \hat{v}^\ast( t) ) \right)H( q( \hat{v}^\ast( t) ) ) }_{\text{Probability Admitted}}$$

The derivative of the objective function is:

$$\underbrace{\,f\left(\hat{v}^\ast\right){\partial \hat{v}^\ast\over \partial t}H\left(q( \hat{v}^\ast( t) ) \right)}_{\text{Deterrence}} - \underbrace{\left(1-F( \hat{v}^\ast( t) ) \right)h( q( \hat{v}^\ast) ) {\partial q\over \partial \hat{v}}{\partial \hat{v}^\ast\over \partial t}}_{\text{Credibility}}.$$

This is precisely the change in the probability of the foreign national being admitted as a refugee. As we detailed before, this change is influenced by two effects: the deterrence effect, which pushes the politician to choose a more restrictive policy (t ↑), and the credibility effect, which pushes the politician to choose a less restrictive policy (t ↓). Recalling that ${\partial q\over \partial \hat {v}} = q( \hat {v}) {f( \hat {v}) \over 1-F( \hat {v}) }$, we can further rearrange the derivative to obtain the following:

(7)$$\underbrace{\,f\left(\hat{v}^\ast\right){\partial \hat{v}^\ast\over \partial t}\left[H\left(q( \hat{v}^\ast( t) ) \right)-h( q( \hat{v}^\ast( t) ) ) q( \hat{v}^\ast( t) ) \right]}_{\text{Deterrence} - \text{Credibility}}.$$

As seen in Proposition 3, the deterrence effect can be larger or smaller than the credibility effect. As a result, when considering a (marginally) more restrictive asylum policy, the change in the probability of the foreign national being admitted as a refugee can be positive or negative. Hence, due to the bureaucrat's changing beliefs, the politician here faces an endogenous cost when choosing a marginally higher level of asylum policy t.

We show in SM-E that the derivative of the politician's objective function is negative for low values of t and positive for large values of t. As a result, the politician's optimal choice is a corner solution: he chooses either the least restrictive policy t or the most restrictive policy $\overline {t}$.Footnote ¹⁶ Intuitively, by choosing $t^\ast = \underline {\rm t}$, the politician allows a high number of applicants, trusting that the bureaucrat assesses them as ineligible and rejects them (“gate-keeping”). Alternatively, by choosing $t^\ast = \overline {t}$, the politician focuses on “deterrence.” Here, the politician sharply limits access to asylum, accepting that the relatively few remaining applicants have a relatively high likelihood of being eligible and are, hence, likely to be admitted by the bureaucrat.

To state the politician's optimal choice, define the difference in utility between the most restrictive and the least restrictive policy:

$$\Delta \equiv 1- \left(1-F( \hat{v}^\ast( \overline{t}) ) \right)H( q( \hat{v}^\ast( \overline{t}) ) ) - \left[1- \left(1-F( \hat{v}^\ast( \underline{t}) ) \right)H( q( \hat{v}^\ast( \underline{t}) ) ) \right].$$

The politician's optimal choice is $t^\ast = \overline {t}$ if Δ ≥ 0 and $t^\ast = \underline {\rm t}$ otherwise. Below, we consider how the quantity Δ, which represents the incentives to choose the most restrictive policy, varies depending on the political context and on the politician's objectives.