Queuing to leave: A new approach to immigration

This is denoted in Kendall's notation as an M/M/1 queue where the M stands for Markovian and I restrict attention to a single or centralized server in each channel for simplicity. The average arrival rate of migrants to the legal queue is the sum of the rate of initial entry (λ_o) and re-entry (λ_r) into the legal queue, i.e., λ_l = λ_o + λ_r. The corresponding average rate of arrivals to the illegal queue (λ_i) will be discussed later. This setup also assumes no balking, i.e., once a migrant decides to migrate, she does not leave the QS but only chooses among the two channels and the number of attempts to make in the legal queue.

This assumption does not change the main implications of the model because comparing the expected utility between legal and illegal migration to a third alternative (of not migrating) would not change the preferences between the first two. For further reference, Yue et al., 2009 relax this assumption and present model performance indicators using numerical simulations.

Now, I define P_n as the probability of there being n people in the QS. In the steady state, the rate up from n to n + 1 people in the queue must be equal to the rate down from n + 1 to n: (1) (λo+λr)pn=μlpn+1 (stability condition) \left( {{\lambda _o} + {\lambda _r}} \right){p_n} = {\mu _l}{p_{n + 1}}\,\,\, ({\rm stability\, condition}) where the rate of re-entry is λ_r = (1 − ν)μ_l − λ_i: i.e., the total proportion of migrants who are unsuccessful [(1 − ν)μ_l] minus the fraction who switch over to the illegal queue (λ_i). Queuing theory allows us to calculate standard performance indicators, such as the expected number of migrants in the legal QS (refer to Appendix A2). (2) E(Ql)=1+ρ−ν−λiμlν−ρ+λiμl(where ‘traffic’ is ρ=λoμl) \matrix{ {E\left( {{Q_l}} \right) = {{1 + \rho - \nu - {{{\lambda _i}} \over {{\mu _l}}}} \over {\nu - \rho + {{{\lambda _i}} \over {{\mu _l}}}}}} \hfill & {\left( {{\rm{where}}\,'{\rm{traffic}}'\,{\rm{is}}\,\rho = {{{\lambda _o}} \over {{\mu _l}}}} \right)} \hfill \cr }

Eq. (2) gives the expected length 〈Q〉 and we can then calculate the expected waiting time 〈W_l〉 of the queue for legal migration. (3) E(Ql)=(λo+λr)E(Wl)(using Little's law:〈Q〉=λ〈W〉) \matrix{ {E\left( {{Q_l}} \right) = \left( {{\lambda _o} + {\lambda _r}} \right)E\left( {{W_l}} \right)} \hfill & {\left( {{\rm{using}}\,{\rm{Little}}'{\rm{s}}\,{\rm{law}}:\left\langle Q \right\rangle = \lambda \left\langle W \right\rangle } \right)} \hfill \cr } (4) E(Wl)=1μl(ν−ρ+λiμl) E\left( {{W_l}} \right) = {1 \over {{\mu _l}\left( {\nu - \rho + {{{\lambda _i}} \over {{\mu _l}}}} \right)}}

This assumption follows from a technical requirement for the first moments of the queue content (Eq. (2)) and waiting time (Eq. (4)) to be positive and to ensure that the queue is stable, i.e., does not explode. When this is violated, we cannot conduct comparative statics deriving from the stability condition, as the queue increases infinitely. Intuitively, the condition imposes that on average, the traffic in the legal queue must be less than the rate at which migrants exit the QS by being accepted or by switching into the illegal queue.

Migrants have a planning horizon [0,T] and incur a cost C per unit of time that they remain in their origin countries. All migrants are taken to be exponential discounters with a discount rate α. The cost C may be thought of as an opportunity cost or the cost of facing negative circumstances in their home country (such as conflict, unemployment, famine, etc.). The cost is incurred in both legal and illegal queues while the migrant is “waiting” to leave. The cost and discount rate are the same for all migrants in the QS. Benefits from successfully exiting the legal and illegal QS are mR_l and R_i per unit of time respectively (R_l > R_i), where m is the maximum number of attempts made to migrate. The key idea behind this requirement is that “something” is gained even from unsuccessful attempts; it may be in the form of experience or know-how about the destination country, which yields higher rewards once the destination is reached. There is no rejection in the illegal QS and all agents are accepted for the journey in their first attempt. Benefits from exiting each queue successfully are the same for all migrants.

The benefits (R_l and R_i) can be conceived as expected benefits of reaching the destination country safely net of the traveling costs, inclusive of risk differentials. While legal travel is almost certain to deliver the migrant to the destination country, it is not so for illegal journeys.

The optimization problem involves choosing the maximum number of attempts (m) the migrant makes to travel legally. The total expected return to a migrant who makes m attempts to migrate legally is the expected return from failing all m times (first term of Eq. (5)) plus the expected return from failing (m − 1) times and succeeding in the m^th try (second term of Eq. (5)).

Migrants only make an additional attempt if they have failed in all previous attempts. Further, the ‘no balking’ assumption implies that λ_o is given. This—along with migrant homogeneity—rule out binding credit constraints for all entrants into the QS.

The first and second-order condition are given in Appendix A4 and evaluating the SOC at the optimum (m*) shows us that it is a local maximum as long as the benefits from migrating legally are high enough relative to the cost of waiting. However, if expected returns from migrating illegally are higher, migrants would switch until the indifference condition is met for the incoming entrant to the system: (6) E(Ul)|m=m*=E(Ui)⇒(1−ν)m*−1[νm*Rlα(e−αm*〈Wl〉−e−αT)−Cα(1−e−αm*〈Wl〉)]=Riα(e−α〈Wi〉−e−αT)−Cα(1−e−α〈Wi〉) \matrix{ {E\left( {{U_l}} \right)\left| {_{m = {m^*}} = E\left( {{U_i}} \right)} \right.} \hfill \cr { \Rightarrow {{\left( {1 - \nu } \right)}^{{m^*} - 1}}\left[ {{{\nu {m^*}{R_l}} \over \alpha }\left( {{e^{ - \alpha {m^*}\left\langle {{W_l}} \right\rangle }} - {e^{ - \alpha T}}} \right) - {C \over \alpha }\left( {1 - {e^{ - \alpha {m^*}\left\langle {{W_l}} \right\rangle }}} \right)} \right] = {{{R_i}} \over \alpha }\left( {{e^{ - \alpha \left\langle {{W_i}} \right\rangle }} - {e^{ - \alpha T}}} \right) - {C \over \alpha }\left( {1 - {e^{ - \alpha \left\langle {{W_i}} \right\rangle }}} \right)} \hfill \cr } where 〈W_l〉 and 〈W_i〉 are the expected waiting times in the legal and illegal QS, respectively, and are both functions of the average rate of arrivals to the illegal queue (λ_i). Using a queuing theory result, 〈W_i〉 can be calculated in the same manner as before using a similar stability condition specified in Assumption 1.

Some standard results from queuing theory are given in Appendix A1. For a more detailed discussion refer to Medhi (2002).

Next, I use Eq. (6) to examine the comparative statics of how changes in the parameters of the legal queue affect the rate of arrivals (λ_i) to the illegal queue and summarize them in the form of some results (proofs are in Appendix A5):

The proof is intuitive; an increase in traffic (ρ=λoμl) \left( {\rho = {{{\lambda _o}} \over {{\mu _l}}}} \right) arises either from more people willing to migrate from the origin country or slower rates of processing migrants legally, both of which increase the expected legal waiting time. Longer waiting periods associated with legal travel are accompanied by higher costs reducing the expected returns from legal migration E(Ul*) E\left( {U_l^*} \right) . To maintain the condition of indifference (Eq. (6)), more migrants would switch over to the illegal channel, increasing the waiting time in the illegal queue.

This follows from an increase in the expected benefits from legal migration E(Ul*) E\left( {U_l^*} \right) as the chances of getting accepted for travel (ν) increase and the inverse relation between E(Ul*) E\left( {U_l^*} \right) and λ_i is proved in Result 1. This result is relevant for the role of immigration policy in preventing dangerous and illegal journeys undertaken often with smugglers in periods of mass migration.

The detailed proof in Appendix A5 shows that Result 2 holds as long as the returns from legal entry relative to the cost of waiting is high enough and Figure 2 illustrates this graphically using specific parameter values.

All figures illustrating the comparative statics findings use parameter values comparable to Djajić (2014).

Figure 2

The intuition behind this proof is that as migrants become more impatient, they are more likely to switch over to the illegal queue. The higher gains from migrating legally would be worth less as they discount the future more. The formal proof given in Appendix A5 rests on precisely this condition that the absolute loss in expected utility from legal immigration E(Ul*) E\left( {U_l^*} \right) due to higher impatience is larger than that for the corresponding loss of expected utility from illegal immigration: (8) ∂E(Ul*)∂α<∂E(Ui)∂α {{\partial E\left( {U_l^*} \right)} \over {\partial \alpha }} < {{\partial E\left( {{U_i}} \right)} \over {\partial \alpha }}

Figure 3 illustrates that the optimal number of attempts (m*) increases as agents are more patient, i.e., have lower discount rates.

Figure 3

Cases II and III illustrate similar findings as Case I but by using different aspects of QS. The previous case considered immigration policy to be comprised of an arbitrary acceptance rate, which is not always realistic. This case expresses the stringency of legal immigration in terms of a minimum waiting time (Wlc) \left( {W_l^c} \right) and thereby a minimum cost that the migrant should incur. I assume that this is specified exogenously by the destination country. The waiting time criteria may be thought of as the due diligence towards acquiring the requisite documents, skills, networks, money, etc., necessary for the journey. Alternatively, in case the migrant is in a UNHCR resettlement program, it can be time spent literally waiting in a refugee camp. The limiting criteria, Wlc=0 W_l^c = 0 and Wlc=∞ W_l^c = \infty , lend themselves to immediate analysis. However, a finite Wlc W_l^c is not analytically feasible and I use simulations instead.

The expected return from legal migration after m attempts is: (9) E(Ul)=mRl|(〈Wl〉≥Wlc)∫m〈Wl〉Te−αtdt−C∫0m〈Wl〉e−αtdt E\left( {{U_l}} \right) = m{R_l}|\left( {\left\langle {{W_l}} \right\rangle \ge W_l^c} \right)\int\limits_{m\left\langle {{W_l}} \right\rangle }^T {{e^{ - \alpha t}}dt - C\int\limits_0^{m\left\langle {{W_l}} \right\rangle } {{e^{ - \alpha t}}dt} } where | is the indicator function equal to 1 if the migrant has “waited” more than the critical amount specified by the destination country and 0 otherwise. The corresponding expected returns from illegal migration, where there is no minimum criteria, is: (10) E(Ui)=Ri∫〈Wi〉Te−αtdt−C∫0〈Wi〉e−αtdt E\left( {{U_i}} \right) = {R_i}\int\limits_{\left\langle {{W_i}} \right\rangle }^T {{e^{ - \alpha t}}dt - C\int\limits_0^{\left\langle {{W_i}} \right\rangle } {{e^{ - \alpha t}}dt} } where mR_l, R_i and C are the homogeneous benefits and waiting cost per unit of time as described earlier, applicable to all migrants.

The limiting cases are of no legal criteria: Wlc=0 W_l^c = 0 , and infinite legal criteria: Wlc=∞ W_l^c = \infty . When there are no waiting criteria, success in the legal queue is certain and migrants do not have to make multiple attempts (m = 1). This case is analogous to countries which have open borders with each other (e.g., Schengen member countries). All migrants thus reach their destination legally and an illegal QS does not exist. Expected utility in this case is simply: (11) E(Ul)=Rl∫〈Wl〉Te−αtdt−C∫0〈Wl〉e−αtdt E\left( {{U_l}} \right) = {R_l}\int\limits_{\left\langle {{W_l}} \right\rangle }^T {{e^{ - \alpha t}}dt - C\int\limits_0^{\left\langle {{W_l}} \right\rangle } {{e^{ - \alpha t}}dt} } where 〈Wl〉=1μl−λo \left\langle {{W_l}} \right\rangle = {1 \over {{\mu _l} - {\lambda _o}}} . On the other hand, an infinite waiting period criterion makes it impossible for migrants to legally reach their destination country. A real world example of this is can be cases of “extreme vetting” of citizens from certain countries or immigration from Israel to certain Arab states. In this case all migration is cannibalized by illegal routes and there will be no legal immigration. Returns to illegal migration are as specified in Eq. (10) with the expected waiting time equal to that of a standard M/M/1 queue: (12) 〈Wi〉=1μl−λo \left\langle {{W_i}} \right\rangle = {1 \over {{\mu _l} - {\lambda _o}}}

2.2.1

Finite legal criteria: Wlc=Wlc* \boldsymbol{W}_\boldsymbol{l}^\boldsymbol{c}\boldsymbol{ = W}_\boldsymbol{l}^{\boldsymbol{c*}}

This intermediate case of a finite waiting time criterion is perhaps most realistic and general. It places some “reasonable” bounds on the level of diligence required from the migrant. This diligence may be in the form of actual waiting time or even the time spent in acquiring specific skills which form the basis of immigration – such as for science, technology, engineering, and mathematics (STEM) field workers in the US.

https://www.nber.org/digest/nov16/immigrants-play-key-role-stem-fields (Accessed: March 25, 2021)

It is standard for many European countries to explicitly require immigrants to reside within the country and wait a stipulated number of years before getting permanent residency or becoming naturalized or citizens, for example Permit C holders in Switzerland. However, such waiting requirements for accepted asylum status are more nuanced or implicit. The Pew Research Center estimates that the average waiting times for Syrian refugees ranges from 3 months in Germany to more than 1 year in Norway (Connor, 2017). These wait times are not explicit requirements, but depend on processing capacity, origin country, and political pressure on authorities to either accelerate or slow the processing of asylum applications.

If a request for refugee status is rejected, there is an appeal process in the EU which would involve another round of waiting.

Similarly, in the US, prospective immigrants are faced with different waiting time criteria due to country limits, which cap the number of green cards for any nationality. Under this requirement, immigrants who are applying from China, India, Mexico, and the Philippines, which are at or nearing the country limits, face waiting periods equal to several times that for other nationalities, between 7 years and 23 years (Obinna, 2020).

To model immigration choices under the regime of waiting time criteria, I define a common threshold patience parameter that all migrants are endowed with P = f (α) such that f (α) is a monotonically decreasing function of the discount rate defined in Case 1. This patience parameter represents the maximum number of times a migrant is willing to re-attempt legal migration before switching to illegal routes.

This is similar to m* described in Case I but it will not have the same functional form.

In this case, it is analytically infeasible to calculate the moments of the queue content and waiting time in the parallel systems. However, a simulation illustrates how queue content varies with Wlc* W_l^{c*} and P.

The simulations (Figures 4 and 5) show that:

The relative magnitude of illegal migration increases as the minimum waiting criteria, Wlc* W_l^{c*} is increased (from 18 to 40) or the patience parameter, P is decreased (from 4 to 1). The converse is true for the relative share of legal migration.

The queue content exhibits cycles, accumulating and reaching a peak before periodically emptying out once enough potential migrants achieve the minimum waiting criteria.

Figure 6 illustrates that cycles of immigration flows exhibit such cycles over time. However, this must be interpreted with caution because cyclic behavior in the queue content induced by the waiting time criteria is not possible to distinguish in the data from changes in immigration policy, seasonality and changes in origin-country circumstances.

This behavior is similar to that of a syphon studied in detail by Filliger and Hongler (2005).

Figure 4

Figure 5

This case models the stringency of legal channels of immigration in the form of limited waiting area capacity (Figure 6). It is similar to quotas – e.g., those imposed yearly on residence permits and short term permits in Switzerland.

https://www.ge.ch/moe/uk/autorisation.asp (Accessed: September 12, 2018)

Figure 6

The analysis is simplified in this case since migrants do not really have a choice about which route to take.

Another matter of interest may be to find the optimal K for the destination country by considering the cost of border control. Since optimal immigration policy itself is not the subject of this paper, I do not attempt it here.

As expected, an increase in the waiting area capacity (K) decreases the rate of arrivals (λ_i) into the illegal QS. Refer to Appendix A3, Eq. (19).

Before moving on to the empirical section, I summarize the implications of the theory below:

Given an increase in the number of people attempting to migrate from an origin country and fixed parameters of the legal QS, there will be a spillover into illegal immigration.

Available data on the EU refer to migrants who have successfully, either legally or illegally, entered the EU. The data on the inflow of migrants with legal status are from Eurostat, the statistical office of the EU. These data are disaggregated by country of citizenship and are available yearly. The definition of a “legal migrant” varies from country to country.

Eurostat defines immigration as the action by which a person establishes his or her usual residence in the territory of a Member State for a period that is, or is expected to be, of at least 12 months, having previously been usually resident in another Member State or a third country. Illegal migration or asylum seekers without legal status are not included in this Eurostat immigration dataset.

Frontex records the number of detected IBCs via land and sea routes into the Schengen region. Disaggregated data by nationality are available from the 3rd quarter of 2007 and are updated monthly.

While there is bound to be some measurement error in the Frontex data as many entries into the Schengen region go undetected, the error is unlikely to be specific to any origin countries.

Figure 7 shows the evolution of IBCs from the six countries with the highest flows over 2010–2014 period. Pursuant to eyeballing the panels for Tunisia and Syria, we are presented with a sharp peak for the months immediately following their respective domestic conflicts.

Figure 7

For data on conflict, I use the Georeferenced Event Dataset (GED) from the Uppsala Conflict Data Program (UCDP version 17.1 2017). An event is an incident where armed force was used by an organized actor against another organized actor, or against civilians, resulting in at least one direct death at a specific location and a specific date. The UCDP dataset records three values for the number of conflict-related deaths: a low, high, and a best estimate. All estimates are sourced from publicly available and reliable information and are collected at the level of town and village for urban and rural areas respectively. For this paper, I collapse the data to the country level. The UCDP GED dataset does not yet include deaths from Syria as data collection is not complete at the granular level of towns and villages. I include Syria by using the conflict level data from the UCDP Battle-related Deaths (BRD) dataset.

The UCDP BRD dataset is an automatic filtering and aggregation of the UCDP GED and records deaths resulting from an armed conflict at multiple battle locations. I use only the BRDs which occurred solely within Syria. UCDP defines an armed conflict as a contested incompatibility that concerns government and/or territory over which the use of armed force between two parties, of which at least one is the government of a state, has resulted in at least 25 BRDs in 1 calendar year.

My sources for other relevant covariates are the World Development Indicators (WDI) for population density and World Governance Indicators (WGI) for institutional quality.

I use an estimate of rule of law from WGI which defines it as capturing perceptions of the extent to which agents have confidence in and abide by the rules of society, and in particular the quality of contract enforcement, property rights, the police, and the courts, as well as the likelihood of crime and violence. The estimate is a normalized indicator ranging from −2.5 to 2.5.

The parameters of interest are the β′s and γ′s of Eqs (15) and (16). Due to potential endogeneity and the dynamic panel setting, I use a GMM approach to estimate the coefficients of interest using internal as well as external instruments. The external instrument to consistently estimate the effect of conflict arises from the dataset itself. The UCDP dataset records three values for the number of battle related deaths: a low, high, and a best estimate. The estimates are collected from third parties such as news agencies, international and non-governmental organizations. In addition, the sources also include reports from the warring parties themselves (governments and opposing groups). Due to the difficulties in accessing accurate information during a violent conflict and the vested interests of governments or opposition parties in releasing accurate fatalities, there are discrepancies regarding the exact number of conflict-related deaths. This is even greater when conflicts are spread out geographically over a region, some locations of which are inaccessible. The discrepancies, measured by the difference between the high and the low estimate, are positively and strongly correlated with the level of the best estimate, thereby satisfying the criterion of instrument relevance. This should be unsurprising as it is harder for independent reporters or watchdogs to access locations during conflicts occurring at a massive scale (Themnér and Wallensteen, 2014).

Refer to Iraq Body Count studies: https://www.iraqbodycount.org/

The second criterion of the exclusion restriction is more complicated to support. The difference in the high and low estimates of death may actually be picking up the nature of institutions instead of the intensity of the conflict itself. To account for this, I control for the quality of institutions in the country by using the rule of law indicator. Further, if we consider the group of countries which have no discrepancies in the measurement of deaths in the presence of violent conflict, their mean score on the institutions indicator is negative (−0.61), which is clearly a counter to the argument that the instrument is picking up institutional quality.

In this dynamic panel setting of lagged effects and in the absence of sufficient external instruments, I use a GMM approach to exploit all available moment conditions using internal instruments. Due to the possibility of persistence and the high relative share of variance in fixed effects (σα2σε2) \left( {{{\sigma _\alpha ^2} \over {\sigma _\varepsilon ^2}}} \right) , the first-differenced GMM estimates may suffer from poor finite sample properties, namely bias and imprecision, since lagged levels are weak instruments for first-differences (Blundell and Bond, 2000). Hence, I estimate the model using system GMM, which estimates the model using the equations in levels and differences to improve the strength of internal instruments. Lastly, another issue potentially compounding the endogeneity problem is the possibility of serially correlated errors. This is addressed by the common factor representation below given by Blundell and Bond, 2000 and also implemented by Arcand et al. (2008). (17) εit=θ1εit−1+eitlit=αi1(1−θ1)︸αi1*+(λt1−θ1λt1−1)︸λt1*+π1cit−1+π2cit−2+θ1lit−1+π3xit+π4xit−1+eit \matrix{ {{\varepsilon _{it}}} \hfill & = \hfill & {{\theta _1}{\varepsilon _{it - 1}} + {e_{it}}} \hfill \cr {{l_{it}}} \hfill & = \hfill & {\underbrace {{\alpha _{i1}}\left( {1 - {\theta _1}} \right)}_{\alpha _{i1}^*} + \underbrace {\left( {{\lambda _{t1}} - {\theta _1}{\lambda _t}1 - 1} \right)}_{\lambda _{t1}^*} + {\pi _1}{c_{it - 1}} + {\pi _2}{c_{it - 2}} + {\theta _1}{l_{it - 1}} + {\pi _3}{x_{it}} + {\pi _4}{x_{it - 1}} + {e_{it}}} \hfill \cr } (18) ηit=θ2ηit−1+νitkit=αi2(1−θ2)︸αi2*+(λt2−θ2λt2−1)︸λt2*+π^1cit−1+π^2cit−2+θ2kit−1+π^3xit+π^4xit−1+νit \matrix{ {{\eta _{it}}} \hfill & = \hfill & {{\theta _2}{\eta _{it - 1}} + {\nu _{it}}} \hfill \cr {{k_{it}}} \hfill & = \hfill & {\underbrace {{\alpha _{i2}}\left( {1 - {\theta _2}} \right)}_{\alpha _{i2}^*} + \underbrace {\left( {{\lambda _{t2}} - {\theta _2}{\lambda _t}2 - 1} \right)}_{\lambda _{t2}^*} + {{\hat \pi }_1}{c_{it - 1}} + {{\hat \pi }_2}{c_{it - 2}} + {\theta _2}{k_{it - 1}} + {{\hat \pi }_3}{x_{it}} + {{\hat \pi }_4}{x_{it - 1}} + {\nu _{it}}} \hfill \cr }

Results using the above estimation techniques are presented in the next section.

One drawback of the above results is that it does not account for the inherent differences between people attempting to migrate legally compared to those who do so illegally. Documented migration to Europe usually occurs under the following categories – remunerated activities, family reunification, education and humanitarian reasons. Differences within individuals from the same country would make some people more eligible to meet these criteria than others. Since I do not have data on individual level characteristics, I cannot control for them. However, the theory in Section 3 implies that given the same characteristics of individuals, migrating via the legal queue is likely to involve larger waiting costs that lead some people to switch over to the illegal channel. Looking at the effect of violent conflict on the flows of legally resettled people who might share similar characteristics within the origin countries gives evidence to these waiting costs.

“Resettlement is one tool to help displaced persons in need of protection reach Europe safely and legally, and receive protection for as long as necessary. It is a durable solution which includes selection and transfer of refugees from a country where they seek protection to another country” EPRS, 2017. The UNHCR has to determine whether an applicant is a refugee according to the 1951 Geneva Convention, and has to identify resettlement as the most appropriate solution. From Table 6 we see that the relationship between violent conflict and the magnitude of resettled persons is not very robust within the dataset available. Only the third lag of CD is significant using the within estimator and this significance becomes weaker after introducing other covariates. The same goes for results from 2SLS estimation using the difference in estimates of CD as an IV for CD.