# Computational Economic Modeling of Migration

## Abstract and Keywords

In this chapter an agent-based model of endogenously evolving migrant networks is developed to find and estimate the size of determinants of migration and return decisions. Individuals are connected by links, the strength of which declines over time and distance. Methodologically speaking, this chapter combines parameterization using data from the Mexican Migration Project with calibration. It is shown that expected earnings, an idiosyncratic home bias, network ties to other migrants, strength of links to the home country, and age have a significant impact on circular migration patterns over time. The model can reproduce spatial patterns of migration as well as the distribution of the number of trips of migrants. It can also be used for computational experiments and policy analysis.

Keywords: return migration, circular migration, temporary migration, agent-based modeling, social networks, calibration, distributions

19.1 Introduction

Fiftynine percent of Mexican migrants to the United States surveyed in the Mexican Migration Project (MMP128, described below) makemore than one move; that is, after returning to Mexico they go back to the United States at least once. The phenomenon makes it difficult to forecast stocks of migrants in the United States at any point in time and to make estimates of where they are likely to go and when, if at all, they are going to return. So far, research on so-called circular migration has mostly been empirical, using multinomial logit, count data models, duration models, or Markov transition matrices to estimate migration and return probabilities controlling for characteristics of individuals and of the home or host country. Examples are Constant and Zimmermann (2003, 2011), Bijwaard (2010), Vadean and Piracha (2009), Reyes (2001) and, using the MMP, Massey and Espinosa (1997). Hill (1987) is an attempt at formalizing duration of stay and frequency of trips in a life-cycle model. A more recent theoretical model of circular migration is Vergalli (2011), who studies the phenomenon in a real option framework.

When developing a model that is sufficiently realistic to be used for policy analysis or, eventually, forecasts and that is empirically founded, one has to take into account some important aspects of the issue at hand.

First, a migrant’s decision is not independent of that of other migrants and potential migrants. Other migrants support the newly arrived in their job search, and home-communitymembers help return migrants to reintegrate into the home country’s labor market. The role of social networks in migration decisions has been the subject of substantial research; Radu (2008) provides a survey of the literature. Networks are (p. 560) often thought to be the reason migration is concentrated in a certain number of places and that people from one neighborhood tend to go to the same few places.

Since a migrant expands his or her network with every migration move and network ties possibly become weaker over time, different parts of the migration cycle should not be seen separately.

An individual’s decision to move creates two externalities: one on the network at the location of origin, and the other one on the network in the destination country. If one migrant leaves, others might leave as well. Whether or not the migrant returns, the size of the destination country network will have changed because of his or her move. Thus when he or she considers migrating again, the conditions have changed from those of the previous move, in part because of his or her own behavior. Hence, there is a recursive process, withthenetworkinfluencingthemigrant, the migrant influencing the network, and the new network influencing the migrant. This has been dubbed the “reflection problem” by Manski (1993). This chapter investigates how large the effect of networks is on both migration and return decisions, and what other possible determinants of circular migration exist.

In order to approach this question and to create a space for policy experiments related to (circular) migration, an agent-based model is proposed that allows for the necessary modeling flexibility and for the spatial dimension of the problem. Its central component is the role of networks that evolve endogenously from migration decisions. Links decay over time and physical distance. The migration behavior of one generation of heads of household is modeled over a period of thirtythree years.

There are some rather simple, uncalibrated agent-basedmodels concerning different aspects of migration (Silveira et al. 2006; Espíndola et al. 2006; Biondo et al. 2012; Barbosa Filho et al. 2011). The present model, in turn, is one of the few examples of completely calibrated and empirically founded agent-based models that deal with migration. Related empirical models includeda Fonseca Feitosa (2010) on urban segregation, Sun and Manson (2010) on the search for housing in Minnesota, Haase et al. (2010) and Fontaine and Rounsevell (2009) on residential mobility, and Mena et al. (2011) and Entwisle et al. (2008), who model changes inland use. A recent paper by Kniveton et al. (2011) replicates climate-induced regional migration flows in Burkina Faso using an agent-based model with networks as information transmission mechanisms. Rehm (2012) provides a very sophisticated agent-based model to study remittances of Ecuadorian rural-urban and international migrants. A different computational approach to empirically founded models of Mexican circular migration has been introduced recently in which discrete choice dynamic programming models are estimated using Maximum Likelihood (Lessem 2011) or the Simulated Method of Moments (Thom 2010).

In the present model the Mexican Migration Project (MMP) and other data sources were used for parameterization. Parameters that cannot be found easily in econometric models owing to endogeneity problems and the spatial dimension are calibrated such that parameter values are found that create a close match between simulated and observed data. By proceeding in this way a common criticism of agent-based models, namely many degrees of freedom and the resulting possibility of creating almost any (p. 561) desired output, is avoided. All of the parameters but except four are fixed. The remaining four—the distribution of number of trips of migrants, the distribution of migrants across U.S. cities, and the time series of percent age of agents migrating and returning per year—are calibrated indirectly by matching the simulated data to real data. It is then possible to perform experiments with the model.

The chapter is structured as follows: Section 19.2 describes the methodology used and the main data set. Section 19.3 introduces three stylized facts about circular migration that the model should match. Section 19.4 derives and tests hypotheses about behavioral motives to include in the model. Section 19.5 describes the model, which is parameterized in section 19.6. The indirect calibration procedure is described in section 19.7. Section 19.8 describes the model, which is an example of how to use the model for policy experiments. Section 19.9 concludes.

19.2 Methodology and Data

## 19.2.1 Methodology

The methodology used is the following. First, for a model to be adequate for policy analysis, it has to be “true” in the sense that it represents a plausible candidate for the true data-generating process of the phenomenon of interest. To find out whether this is the case, it is indispensable to have some empirical measure against which to check the model’s output, that is, some means of (external) validation. Therefore three stylized facts are introduced in section 19.3 that the model has to match in order to be considered useful, two of which are distributions of empirical data (number of migrants in each city and distribution of number of trips). Furthermore, the model will be matched against two time series of migration and return flows. For validation, this study follows largely Cirillo and Gallegati (2012) and Bianchi et al. (2008). It is assumed that migrants maximize a utility function that is implicit in the behavioral rules introduced in section 19.5, rather than explicitly stated. They use heuristics to cope with the high level of uncertainty they face in terms of future earnings, others’ migration behavior, and future levels of border control. In every period*t*, agent *i*’s payoff depends on the vector of players’ actions in that particular period and on the current (payoff-relevant) state of the system only (Maskin and Tirole 2001). Behavioral motives for migration and return are chosen from the literature as candidates to be included as behavioral heuristics in the model, similar to those in Rehm (2012). However, instead of systematically varying the behavioral parameters to calibrate the model so that it generates reasonable outputs as in Rehm’s model, the behavioral parameters here were directly estimated from microdata wherever possible. Comparable models in this respect are da Fonseca Feitosa (2010), Kniveton et al. (2011), and Entwisle et al. (2008). Sometimes, as in the case of the effect of the network, there were clear endogeneity problems, so these four parameters were calibrated later to match the stylized facts. Next, the model was built in NetLogo (Wilensky 1999), all parameters
(p. 562)
were set to fixed, empirically determined values, and the four remaining free parameters were set to reasonable values. After verifying that the model roughly matches most of the stylized facts for most of the settings of the free parameters, those were calibrated performing simple grid searches in the parameter space. The resulting match of model output and empirical data was considered satisfactory, given that this is a much simpler model than, for instance, Rehm (2012), and given that it has only four degrees of freedom. Finally, robustness checks were performed and was is demonstrated how the model can be used for policy analysis. The model code, all data files needed for running the model, the MATLAB code for estimating the properties of the power law distribution, and a full description are availabe on the “Open ABM platform” at http://www.openabm.org/model/3893/version/3/view.

## 19.2.2 Data

For estimating the behavioral rules and for setting most of the other model parameters, the MMP128 version of the Mexican Migration Project was used. It is a large event-history microsurvey data set of Mexican migrants and non-migrants from 128 different Mexican communities. Respondents were interviewed once in waves, starting in 1982 and ending (in the version used) in 2008. Both heads of housholds and spouses were asked to indicate their migration history (time spent in the United States) and labor market experience (employed or not and type of job) as well as family events for every year since they were born. Additional information is available for the time of the interview and for the first and last migration, such as whether it was a legal migration, the type of visa used, income, wealth, and health status. The full sample comprises 1.004.825 personyear observations. The simulation was run with 2,860 agents, the number of heads of household in the MMP128 data set born between the years 1955 and 1965 who—if they migrated—went to California and who were interviewed (or had lived, in the case of migrants) in the central and western Mexican states of Sinaloa, Durango, Zacatecas, Nayarit, Jalisco, Aguascalientes, Guanajuato, Colima, and Michoacán. Those states together form an area approximately the size of California and at the same time they comprise the most important states for out-migration. All population distribution measures refer to this subset of the data. The model therefore simulates migration behavior of one generation from one region over the course of thirtythree years.

19.3 Stylized Facts about Circular Migration

From the literature and the MMP128, three stylized facts about circular migration can be derived that the model should match. If it succeeds in re-creating these prominent (p. 563) characteristics of circular migration behavior, it is a plausible candidate for the true data-generating process.

## 19.3.1 The Distribution of Migrants Across Cities is HeavyTailed

In order to calibrate the model to the empirical distribution and to have a means to validate the model, the distribution of migrants across cities is determined.

One sees in the complete MMP128 sample that the distribution across cities is very similar to that of the western Mexico–California subsample. Therefore, both the subsample and the full sample are used in order to avoid bias in the estimates due to small sample size. The bulk of migration originates in a few places, and it concentrates on a fairly small number of places in the country of destination. In the case of Mexican migration to the United States, the communities with the highest percentage of adults with migrant experience are in the states of Guanajuato, Durango, Jalisco, and Michoacán (MMP128).The percentage varies from just above 1 percent to almost fifty percent across communities. Of the migrating heads of household surveyed in the MMP128, 20 percent went to the Los Angeles area on their last trip; this was by far the highest number, followed by the Chicago region (8 percent) and the San Diego region (5 percent).

Distributions that result from social interaction often follow a power law; that is,

(19.1)

Examples are Axtell (2001) for the distribution of size of cities, Redner (1998) for the distribution of scientific citations, and Liljeros et al. (2001) for the distribution of number of sexual partners. One of the generative mechanisms of power law distributions is preferential attachment: cases in which it is more likely for a new node in a network to attach to a node that already has many links to other nodes, rather than to a random node.Mitzenmacher (2004) provides an intuitive example that can explain the often-found power-law distribution of links to a Web site: If a new Web site appears and it attaches to other sites not completely at random, but, rather, links to a random site with probability *α<*1, and with probability 1−*α* it links to a page chosen in proportion to the number of links that already point to that site, then it can be shown that the resulting distribution of links to and from a Web site approaches a power law in the steady state. See Mitzenmacher (2004) for a simple derivation and Cooper and Frieze (2003) for a more general proof. There is a small number of cities that attract a very large proportion of migrants, and many cities attract only one migrant. Furthermore, the typical formation of migrant networks—joining friends and family in the host country—suggests a case of preferential attachment. Therefore, a power law is the first distributional candidate to check. The methodology is taken from Goldstein et al. (2004) and Clauset et al. (2007). TheMATLAB routines that were provided by that latter,
(p. 564)
include estimating a minimum value for *x* above which the power law applies. First, it is assumed that the distribution of migrants across cities does indeed follow a power law, and its parameters *γ* , the power law exponent, and *x _{min}*, the value above which the power law applies, are estimated. Then, it is checked for whether synthetic power-law distributions with the same exponent and the empirical distribution are likely to belong to the same distribution. The most commonly used power-law distribution for discrete data is the discrete Pareto distribution. It takes the form

(19.2)

where *x* is a positive integer measuring, in this case, number of migrants in a city, *p(x)* is the probability of observing the value *x*, *γ* is the power law exponent, *ζ(γ*,*x _{min})* is the Hurwitz or generalized zeta function defined as $\sum}_{n=0}^{\infty}{\left(n+{x}_{mon}\right)}^{-\gamma$, and

*x*is a minimum value for

_{min}*k*above which the power law applies. The maximum likelihood estimator is derived by finding the zero of the derivative of the log-likelihood function, which comes down to solving

(19.3)

numerically for $\widehat{\gamma}$, with *x _{i}* as the number of migrants in city

*i*and

*n*as the total number of cities in the sample; see Goldstein et al. (2004) for the derivation. Usually, if empirical data follow a power-law distribution at all, they do so only for values larger than some minimum value (Clauset et al. 2007). This value should be estimated in order to not bias the estimated $\widehat{\gamma}$ by fitting a power law to data that are not actually power-law distributed. In accordance with Clauset et al. (2007), this

*x*is chosen so that the Kolmogorov-Smirnov (KS) statistic, which measures the maximum distance between two cumulative distribution functions (CDFs), is minimized. The KS statistic is

_{min}(19.4)

where *S (x)* is the CDF of the empirical observations with a value of at least *x _{min}* and

*P(x)*is the CDF of the estimated power-law distribution that best fits the data for the region

*x*≥

*x*. This yields a minimum value for

_{min}*x*of 34 and a scale parameter $\widehat{\gamma}$ of 1.9. Although visually the power law seems to be a good fit (not shown), it is checked to see whether the distribution might actually follow a power law above

*x*= 34. In order to do this, the KS statistic is computed for the distance between the empirical CDF and the best-fit power law. Then, a large number of artificial data sets distributed according to the power law with

_{min}*γ*= 1.9 and

*x*= 34 is created, a power-law model is fitted to each artificial data set again, and the KS statistic (the distance from that data set to its own power-law model) is computed. Then the proportion

_{min}*p*of artificial data sets is determined in which the KS statistic is larger than the one from the empirical distribution. If the proportion

*p*is such that

*p <*.1, a power law can be ruled out because extremely rarely the artificial data sets are a worse fit to a power-law distribution than (p. 565)

are the observed data. In the present case, however, *p* = .4250, so a power law seems a reasonable discription of the data.

The same procedure is followed for the smaller subsample that is used as a basis for the simulation. The results indicate that even for the small subsample the distribution might follow a power law for values larger than 15 with *γ* = 2.2 (see figure 19.1).

The *p*-value of .845 is even higher for the smaller sample, indicating that the artificial distributions are, on average, a worse fit to a power law than the empirical one. This result has to be taken with caution, however, owing to the small sample size.

For comparing the empirical to the simulated distribution at the end of the calibration procedure, the mean, standard deviation, and median of the two distributions are compared. Whether the simulated distribution resembles a power law, both for single runs and for the overall distribution after ten thousand model runs, will be checked (see section 19.7.2).

## 19.3.2 Tendency of People from one Neighborhood to Migrate to the Same Place

People tend to settle where people from the same region of origin have settled previously; for example, 65 percent of the migrant heads of household surveyed in the community with the highest percentage of migrants, a village in Michoacán, went to the (p. 566) Chicago region. Patterns in most other communities are very similar. For additional evidence see Munshi (2003) and Bauer et al. (2007). The reasons for this are positive network externalities.

## 19.3.3 Migration-Specific Capital

Several studies reveal the importance of migration-specific capital, that is, experience and knowledge that facilitate every subsequent move. This capital is closely related to migrant networks as well: with every move, migrants build up new links that facilitate job searches, (re)integration, and information flow (DaVanzo 1981).Therefore, once a move has taken place, migrants are more prone to move (again) than they were before their first move (Constant and Zimmermann 2011).

Because some of the individuals in the subsample were interviewed before the last year considered (2007) and therefore their migration histories are not complete, the full sample is used for measuring the distribution of number of trips. The total number of trips is measured at age fortyseven, which corresponds to the last year in the lives of the simulated agents. The distribution of number of trips displays overdispersion (mean = .964, standard deviation = 2.785) and “excess zeros” as compared to a Poisson distribution. The observed distribution fairly closely resembles a negative binomial (see figure 19.2). In fact, the null hypothesis that it is equal to a negative binomial one could not be rejected in a Kolmogorov-Smirnov test (p = .12).

The overdispersion and “excess zeros” could be due to either the heterogeneity of individuals or to the existence of two different data generating-mechanisms creating zero and nonzero counts of trips (Greene 2003, 744–752). Both explanations would be in line with the argument by DaVanzo (1981) and Constant and Zimmermann (2011): Migrants could have characteristics that distinguish them from nonmigrants, so the degree of heterogeneity between people who do not migrate at all (number of trips = 0) and people who make one trip would be much larger than that between migrants who make one trip and those who make two trips. Alternatively, the conditions for making the first trip are much different from those for subsequent trips owing to the above-mentioned build-up of migration-specific capital. Therefore, the mechanism “generating” 0 moves is different from the one generating a positive number of trips.

The model developed here is useful if it succeeds in re-creating these three stylized facts.

19.4 Selection of Behavioral Motives

Several behavioral motives can be found in the literature that might influence the migration or return decision. We determine which ones to include in the model by (p. 567)

running logit and probit regressions on the MMP128 data set for the probability of migrating and returning in a personyear. The full sample of individuals for the years 1970–2008 was used, thereby implicitly assuming that they are not systematically different from the western Mexican subsample that is used for the simulation. All of the hypotheses mentioned below are included in the regressions, as well as controls for family status, community of origin, profession, and current job. All results are displayed in table 19.1.

#### Hypothesis 1: Expected Earnings

The first hypothesis is as follows: Migrants are attracted by a higher expected income in the host country than in the home country, taking into account the unemployment rate (Harris and Todaro 1970). The higher the expected income as compared to the current income, the more likely someone is to migrate.

It is not a straightforward matter to find the effect of the difference between expected earnings and current earnings on the migration and return decision with the data available from the MMP 128, for two reasons: First, the data do not contain information about earnings for every personyear, but only about the year of the survey and of the first and last migration. Second, it is unclear how to compute expected earnings without knowing how those expectations are formed. In section 19.5 it is suggested that they (p. 568)

are formed by averaging over network neighbors’ earnings in the host country. As a proxy for expected wage, we use the difference between real GDP per capita in Mexico, and United States GDP per capita multiplied by the United States employment rate. The coefficient of the expected annual wage difference between Mexico and the United States is positive and highly significant for the probability of making a trip. I checked whether the marginal effect of the wage difference on the probability of going on a trip differs by whether someone is a potential first-time migrant or has gone on at least one migration before, which is the case. The increase in probability of migrating in a personyear per thousand USD expected wage difference is .0011 for those who have never migrated before and .0035 for those who have.

The effect of the expected wage difference on the return decision should be opposite: The higher the expected wage difference, the lower the probability of return. Indeed, (p. 569) the coefficient for expected wage difference is negative, but only significant in the logit model and not in the probit model (see table 19.1). Therefore, it is not included as a behavioral parameter for the return decision.

#### Hypothesis 2: Number of Previous Migrants

Workers with a network are both less likely to be unemployed and to have higher wages (Munshi 2003). Therefore, migrants tend to go where they know somebody, as shown by Lindstrom and Lauster (2001), Floresyeffal and Aysa-Lastra (2011), and Massey and Aysa-Lastra (2011). Previous migrants have an incentive to help the newly arrived find jobs because this increases the flow of information and trade among migrants, as argued by Stark and Bloom (1985). The help of others decreases assimilation costs for new migrants, as shown for Mexican migrants by Massey and Riosmena (2010). Previous migrants influence potential migrants’ decisions through the policy channel as well: Immigration policy often includes a family reunification element that permits family members of migrants to immigrate as well. However, Beine et al. (2011) estimate the relative importance of the different channels for immigrants to the United States in a recent paper and find that the immigration policy channel is much less important than the assimilation cost channel and has decreased in importance since the 1980s.

In sum, the more previous migrants somebody knows, the more likely he or she is to migrate.

This seems to be true for the sample here as well; the coefficient for the number of family members in the United States is highly significant (table 19.1). The influence of the number of previous migrants on the migration decision is calibrated in section 19.7.

#### Hypothesis 3: Home Preference

Migrants are often assumed to have a preference for consuming home amenities (a home bias, as in Faini and Venturini 2008 and Hill 1987). Everything else held constant, utility is always higher if he or she is at home. The hypothesis is therefore: The stronger someone’s home preference, the less likely he or she is to migrate.

Assuming that people are heterogeneous in their home preference, each individual is assigned an idiosyncratic home preference parameter. Property ownership in Mexico before first migration is used as a proxy because people who consider it likely that they will spend their life in the home country are more likely to invest in property there than in the host country. Logit and probit regressions of the probability of ever migrating on property ownership, individual controls, and community fixed effects before first migration (table 19.2) show that property ownership before first migration is significantly negatively correlated with becoming a migrant. This confirms the findings by Massey and Espinosa (1997). (p. 570)

An index was created from hectares, properties, and businesses owned. The number of hectares owned is transformed to a logscale, then the values from the categories are added. The coefficient of the property index is also negative and significant.

The relative frequencies of the property index in the central and western Mexico subsample are used as relative frequencies for the home preference parameter *h _{i}*. The analysis is confined to values for the property index from 0 to 4, because the proportion of individuals with a property index larger than 4 is less than 1 percent. Of the subsample, 57.85 percent of individuals have a property index of 0, 29.96 percent
(p. 571)

have a value of 1, 7.83 percent have a value of 2, 2.57 percent have a value of 3, and 1.09 percent have a value of 4.

The probability of migrating in a personyear negatively depends on the property index, as can be seen in table 19.1. The average probability of migrating in a personyear was subsequently computed at every level of the property index (see table 19.3).

It is interesting that a property index of 4 increases the probability as compared to a property index of 3.

#### Hypothesis 4: Ties to Home

Constant and Zimmermann (2003) find that family is a driving force of repeat migration. Ties to the home country can be understood as relationship capital. It is helpful for the migrant’s reintegration into the home community on return. However, the longer a migrant is away from the home country, the stronger might be the depreciation of that form of capital because of physical distance. This phenomenon is studied analytically by McCann et al. (2010) and found to be empirically relevant for the return decision by de Haas and Fokkema (2011). This yields the following hypothesis: The more family and friends someone has at home, and the stronger the links are with them, the more likely someone is to return.

The decrease in likelihood of returning to the home country (for people in the host country) or of migrating again (for people in the home country) is measured for
(p. 572)
migrants with at least one trip to the host coutnry, taking time since last migration move as an explanatory variable. This illustrates the diminishing importance of ties across physical distance over time. A probit regression of the likelihood of making a move in a year on the number of years since the last move yields a negative coefficient that is significant at the 1 percent level for both the migration decision and the return decision (see table 19.1). The links connecting physically distant network neighbors are, therefore, assumed to become weaker in each period by an amount *a*. The probability of making a move in a personyear (migration or return) starts out at 3.3 percent when the last trip took place in the previous year. It decreases on average by 1.9 percent with each additional year that has passed since the last move.After thirtytwo years without a trip the probability is 1.8 percent. The relationship capital associated with links between physically distant neighbors is, therefore, assumed to decrease by 2 percent every year. The coefficient of the size of the effect of relationship capital in the home country on the probability to return home is calibrated in section 19.7.1.

#### Hypothesis 5: Purchasing Power

Migrants’s savings have a higher purchasing power in their home country than in the host country, as modeled by Dustmann (2001). This might be a motive to return. Lindstrom(1996) follows a similar argument: He tests whether Mexican migrants from areas that provide dynamic investment opportunities stay longer in the United States in order to accumulate more savings that they can put to productive use in their home country, and he finds some evidence in favor of his hypothesis. Reyes (2004) shows that devaluation of the peso relative to the dollar leads to more return migration, providing another piece of evidence in favor of the purchasing power motive. A related argument is brought forward by Berg (1961) and Hill (1987), who discuss the case in which migrants aim to achieve a level of lifetime income, and once that is achieved they return home because they have a preference for living there. Either argument yields the same conclusion: Holding everything else constant, the higher someone’s savings are, the more likely he or she is to return.

Unfortunately, the MMP128 does not provide information about savings. Therefore the supposed purchasing power effect is captured by including the last wage in the United States, multiplied by the exchange rate for that year and by the consumer price index from the Bank of Mexico, and a dummy that is 1 if property ownership was larger in *t* +1 than in *t* in the return regression (see table 19.1). An interaction term of the dummy and the last wage is also included. The ownership dummy is significant and negative, which implies that people who lived in the United States in year *t* and bought property the same or the following year are *less* likely to have returned that year than are people who did not buy property. That somewhat contradicts the hypothesis and indicates that people seem to buy property in the United States rather than in Mexico. The coefficient for the last wage in the United States is negative and significant for return,
(p. 573)
albeit the coefficient is extremely small in size. The interaction term has a positive and significant effect, in line with the hypothesis. This implies that if property ownership in *t*+1 is larger than in *t*, the probability of return increases with the wage. The size of the coefficient, however, is very small as well. For that reason and because the proxy for the purchasing power motive is imperfect, it is not included in the model.

#### Hypothesis 6: Education

Education and heterogeneity in skill levels have been found to be important determinants of self-selection of migrants in a wide range of theoretical papers originating from Borjas (1987) and in empirical studies (Brücker and Trübswetter 2007).

The evidence in the literature about skill selection of Mexican migrants, however, is mixed: Borjas and Katz (2007), Fernández-Huertas Moraga (2011), and Ibarraran and Lubotsky (2007) find that Mexican migrants to the United States are mostly from the lower tail of the Mexican earnings distribution. Other studies find that migrants tend to have a medium position in the country’s skill distribution because returns to skill are higher in Mexico, making migration less attractive for highly skilled individuals, while low-skilled individuals are likely to be more credit-restrained and not able to afford the moving costs (Chiquiar 2005; Lacuesta 2006; Orrenius and Zavodny 2005). There is, furthermore, evidence that there is a a self-selection process for migrants who move to a different region within Mexico (Michaelsen and Haisken-DeNew 2011) but not for international migrants between Mexico and the United States (Boucher et al. 2005).

In the simulation model it is difficult to take different levels of education and skills into account without significantly increasing the complexity of the problem. The fact that the individuals in the subsample are pre–dominantly low-skilled (81 percent of migrants born between 1955 and 1965 had completed nine years of schooling or less), in combination with the very mixed evidence in the literature, suggests that it does not seem to bias the results dramatically to leave out education and assume a uniform level of education across individuals. This path is chosen here.

#### Hypothesis 7: Age

All cohorts display a similar migration behavior during their life cycle (see figure 19.3). Migration behavior starts about age eighteen, reaches a peak between the ages of

twentyfive and thirty, and then decreases, with small peaks in both migration and return behavior, at about age seventy. Age might, therefore, have an effect on migration and return moves, independent of the other motives.

Age is significant in all regressions, except for the fifth age group in the return regression. All in all, the results confirm the inverted U-pattern shown in Figure 19.3. (p. 574)

Considering marginal probabilities, the probability of migrating increases by .8 percentage points when entering the age-group of 18 to 30, then decreases by 1.1 percentage points between the ages of thirtyone and fortyfive, and so on (see table 19.4).

The behavioral parameters that were included in the model are summarized in table 19.5.

19.5 The Model

The model assumes two types of agents—workers and firms—which are spread out randomly on a grid. Workers are heterogeneous only in a home preference parameter (fixed throughout the simulation) and in a savings parameter (time-specific). There are two countries: one with high productivity of labor (the host country) and one with low productivity of labor (the home country). Workers can move, but firms cannot.

The model is initiated via a setup procedure. During setup, the following happens:

• The world with two countries separated by a wall is initialized.

• Workers are created. A number that is equal to the initial percentage of workers in the home country is assigned a random spot in the home country. The remainder is assigned a random spot in the host country. (p. 576)

• Workers receive their individual values for the home preference and the savings parameters.

• Workers in the home country create links with other workers in their Moore neighborhood, whereas workers in the host country create links with all other workers within a radius

*s*.• Firms are created in both home and host country and assigned a random spot and a random initial wage.

In every step of a model run the following happens:

• Workers form links to all other workers in their Moore neighborhood (home country) or within a small radius of size

*s*(host country).(p. 577) • Links between workers that are not immediate neighbors get weaker by amount

*a*(relationship capital diminishing over time owing to physical distance). Through migration and renewed physical closeness, the relationship capital associated with those links can be replenished, as in McCann et al. (2010).• All other variables are updated.

• Workers consume their earnings of the previous period minus savings determined by their individual savings rate. Savings are added to wealth.

• Workers without earnings consume a minimum consumption.

• Workers use the information about earnings of network neighbors in the host country to compute their expected earnings in the host country:

(19.5)

$${w}_{\mathrm{exp},i,t}=\frac{1}{N}{\displaystyle \sum _{n=1}^{N}{w}_{n,t}}$$where

*n*= 1,*…*,*N*are all the worker’s network neighbors in the host country, measured at time*t*.• Migration is a three-step procedure. First, workers in the home country compute whether their wealth is larger than the moving costs and whether their expected earnings in the host country are larger than their current earnings. If so, they next compute their individual probability of moving. The probability of worker

*i*to migrate at time*t*is assumed to have the following functional form:(19.6)

$$\begin{array}{l}{p}_{i,t}\left(migrate|{K}_{i,t}>{m}_{1},{w}_{\mathrm{exp},i,t}>{w}_{i,t}\right)=\\ {p}_{\text{o}}+{p}_{1,i}\left({w}_{\mathrm{exp},i,t}-{w}_{i,t}\right)+{p}_{2}{N}_{i}-{p}_{3,i}+{p}_{4,t}\end{array}$$where

*K*,_{i}_{t}is the worker’s wealth in time*t*,*m*_{1}are the migration costs,*p*_{0}is the baseline migration probability,*p*_{1,i,t}is the behavioral parameter for the difference between expected and current earnings that depends on whether it is a first migration or not,*p*_{2}is the behavioral parameter for the number of network neighbors in the host country (*N*),_{i}*p*_{3,i}is the individual home preference parameter, and*p*_{4,t}is the age parameter. They draw a random number ∈ (0, 1). If this number is smaller than their individual probability, they migrate. Their wealth*K*decreases by the amount of moving costs*m*_{1}. In the last step, the probability that somebody who is willing to migrate can do so is determined by the level of border control.• Migrants become unemployed and decide where to go: If they have any network neighbors in the host country, they move to the network neighbor with the highest wage. If not, they move to a random spot in the host country.

• Unemployed workers in both the host and the home country move to the network neighbor in the same country who is employed and has the highest wage. If they do not have any network neighbors, they move one step in a random direction in search of employment (but never across the border).

(p. 578) • Firms hire unemployed workers who are on their patch. All workers receive the firm’s current wage rate. In order to keep the model as simple as possible, firms are assumed to pay a fixed, uniform, idiosyncratic wage to all of their employees. At every time step the wage is adjusted exogenously to account for inflation.

• Analogous to the potential migrants, potential return migrants in the host country first determine whether their wealth is larger than the return costs and then decide to return according to their individual probability. The probability of worker

*i*to return at time*t*given that his or her wealth*K*is larger than the return costs*m*_{2}is thus assumed to have the following functional form:(19.7)

$${q}_{i,t}\left(return|{K}_{i,t}>{m}_{2}\right)={q}_{\text{o}}+{\displaystyle \sum _{r=1}^{R}\frac{{q}_{1}}{{a}_{r,t}}+{q}_{2,t}}$$where

*q*_{0}is the baseline return probability,*q*_{1}is the behavioral parameter for ties to the home country,*r*= 1,*…*,*R*are the worker’s network neighbors in the home country,*a*,_{r}_{t}is the age of a link, and*q*_{2,t}is the age parameter.• Return migrants’ wealth decreases by the amount of return costs

*m*_{2}. They become unemployed and return to the spot in the home country they were assigned in the setup procedure.• All measurements of model output take place.

The model is run for thirtythree time steps, with each step representing one year.

19.6 Parameterization of Nonbehavioral Parameters

All nonbehavioral parameters in the model were fixed to empirically determined values (summarized in table 19.6).

Parameters of the model were set to sample population parameters that were estimated using the MMP128, the Encuesta Nacional de Ingresos y Gastos de los Hogares (ENIGH) and the Encuesta Nacional de la Dinámica Demográfica (ENADID).

The number of firms in the home country is determined by dividing the number of workers initially in the home country (2,700) by the average firm size in Mexico, which, according to Laeven and Woodruff (2007), is 13.6 employees per firm. That yields 199 firms. For the host country, the number of firms is assumed to be 58, which is the number of counties in California. In this way, the distribution of migrants across cities can be measured conveniently (see section 19.7.2). Values reported in pesos are converted to USD using the annual average of the official exchange rate for the year the data were measured (reported by the World Bank). In order to obtain moving costs, an average of legal and illegal crossings weighted according to the proportions of legal and illegal crossings in the MMP128 data set was computed. Return costs are assumed (p. 579)

to be travel costs plus loss of one month’s American wages, which are determined by a weighted average of illegal immigrants’ and legal workers’ wages. For details, please refer to http://www.openabm.org/model/3893/version/3/view.

Firms’ wages are determined in the following way: In the setup procedure firms are assigned an idiosyncratic productivity parameter *α* ∼ *N(*0,*σ* ^{2}*)* for the host country. The standard deviation *σ* =.28 is the standard deviation of the average wage in a county as a percentage of the overall average per capita personal income in California in 2007 from the U.S. Census Bureau. For Mexico, the standard deviation of wages across states for the usual western and central states in 2001 from Chiquiar (2005) was used, which is 22 percent of the overall average wage. Accordingly, for each period, a firm’s wage is
(p. 580)
set in the following way:

(19.8)

where *α* ∼ *N(*0,*σ* ^{2}*)* and ${\overline{w}}_{t}$ is the time-specific average wage for the country. For this value, the time series are updated in each step of the model run. For the United States, data from 1975 to 2007 are taken from the average hours and earnings of production and nonsupervisory employees on private nonfarm payrolls by major industry sector data set from the Bureau of Labor Statistics. For Mexico, Gross National Income per capita in purchasing power parity, 1975–2007, from the World Bank was used because wage data for the subsample are not available for all years.

For minimum consumption in the United States, the average annual expenditure on food and housing of a household in the lowest income quintile of the population in 2010 from the Consumer Expenditure Survey was used, which is US $17,290.81 (in 2002 prices). For Mexico, the average annual overall expenditure of a household at the bottom income decile in 2006 from the ENIGH, which is US $4,819 (in 2002 prices) was chosen. For both cases the percentage of average income that this value constitutes is calculated for the respective year in which it was measured. Assuming that the relation between minimum consumption and average income remains constant over time, the minimum consumption is updated by multiplying the average wage each year by .3 for the home country and by .7 for the host country.

To determine the savings parameter, data from the 2008 ENIGH were used. The data set was restricted to 2,860 random observations from western Mexico, thereby assuming that the sample surveyed for the ENIGH is not different in relevant ways from the one surveyed for the MMP128.

Since only 17 percent of respondents make any deposits in saving and other accounts, the difference between current income and current expenditure is used as the measure for savings. The distribution of the saving rate in the population is approximately skew normal with parameters *ξ* = 0.616, *ω* = 0.721, and *α* =−7.5. This distribution is used for the simulation, drawing a savings rate for each worker in each period from this distribution.

A set of correlated border enforcement indicators were checked using a principal components analysis (line watch hours, probability of apprehension, visa accessibility, real border enforcement budget, and number of border patrol officers) for principal components in order to find a good proxy for the threshold of border control *b*, which is the probability of actually being successful when wanting to migrate. Three factors account for 87 percent of the variance. The border enforcement budget contributes the most to the first factor, which in turn accounts for 54 percent of the overall variance. The unique variance of the border enforcement budget is one of the lowest as well.Therefore, that variable is chosen as a proxy for border control. The annual values from 1975 to 2007 are normalized to [0,1] so that the probability that an agent who wants to migrate can do so is inversely proportional to the level of border enforcement in the respective year. Of course, there is a clear endogeneity problem here: if the level of border
(p. 581)
enforcement is low, a lot of people will decide to try their luck and migrate. That might increase border protection, which in turn influences whether migrants choose to try to cross the border. For this reason, the way this is modeled here—migration decision and independent random draw whether migration is permitted—is not realistic. Therefore, a baseline probability of migrating is estimated within the final calibration procedure (section 19.7.1) with the border enforcement in place as it is.

19.7 Calibration and Match

## 19.7.1 Determination of Remaining Parameters

The first remaining parameter to be calibrated via simulation is the baseline probability of moving in any given year. This cannot be obtained from the data because the data set does not contain information about failed migration attempts of people who end up not migrating at all (only of those who, after failed attempts, finally succeed). The baseline return probability is also calibrated via simulation, as well as the two network-related parameters *p*_{2} and *q* _{1}. In order to find the best values for the remaining free parameters, 27,951 combinations of parameter were run; that is, every parameter was set to values between 0 and 1 (for *p*_{0}, *q*_{0}, and *p*_{2}) or between 0 and 2 (for *q*_{1}), in steps of .1. Using a simple grid search, the parameter combination is determined that is closest to fulfilling three criteria: causing an emergence of the mean, standard deviation, and median of the distribution of migrants across cities, causing the emergence of the negative binomial distribution of number of trips of migrants, and yielding a similar time series of flows of migrants and return migrants. For each of the three criteria, a distance function was minimized. For the flows of migrants, the function was

(19.9)

where *m _{t}*,

_{emp}is the empirical number of migrants at time t,

*m*,

_{t}_{sim}is the number of migrants in the simulation,

*r*,

_{t}_{emp}is the number of empirical return migrants at time

*t*, and

*r*,

_{t}_{sim}is the simulated number of return migrants. The four points

*t*

_{1}up to

*t*

_{4}represent the first, the twelfth, the fourteenth, and the thirtysecond years of the simulation. For the distribution of migrants across cities, the distance function to be minimized was

(19.10)

(p. 582)
where ${\overline{x}}_{emp}$ is the empirical average number of migrants in a city, ${\overline{x}}_{sim}$ is the simulated equivalent, *n _{emp}* is the total number of cities in the data,

*n*is the simulated equivalent,

_{sim}*x*,

_{i}_{emp}is the number of empirical migrants in city

*i*,

*x*,

_{i}_{sim}is the simulated equivalent, and ${\overline{x}}_{emp}$ and ${\overline{x}}_{sim}$ are the empirical and simulated median values. For the distribution of numbers of trips, a distance function very similar to the one above was minimized, however, this time without using the median. In a next step, the parameter combinations which were among the top decile of matches for all objective functions were selected. This was the case for two parameter combinations. The search was refined around those values in steps of .01, then the above procedure was repeated. The overall best match turned out to be

*p*

_{0}= .1,

*p*

_{2}= .2,

*q*

_{0}= .38, and

*q*

_{1}= .12 (details of derivation and sensitivity analysis are available from the author).

Subsequently ten thousand simulations were run with the best parameter combination found, using different random seeds each time, to see how much the resulting distributions and time series differed from one another and from the empirical ones. All of the following is based on these ten thousand runs with the combination listed above.

## 19.7.2 Stylized Facts Revisited: The Distribution of Migrants Across Cities

The mean, standard deviation, and median of the distributions of survey respondents’ last U.S. trip and of the last trip of the same number of computer agents were directly compared and checked to see whether the power law hypothesis can be rejected for the simulated data.

To determine the simulated distribution, all patches on the left-hand side of the grid containing at least one worker were brought in a random order. Then, in a radius of city size *s*, the number of workers who chose this radius as the destination for their final migration move was counted. I moved on to the next random patch until all workers who migrated at some point were counted. Finally, the distribution of number of migrants per radius of city size *s* was determined.

Some of the individual runs were extremely close to the empirically observed mean and standard deviation (e.g., mean = 17.6, standard deviation = 54.9, and median = 2 compared to mean = 17.5, standard deviation = 54, and median = 1 for the empirical observation). As with the empirical distribution of migrants across cities from the small sample, most of the simulated ones also seemed to follow a power law (see figure 19.4 and figure 19.1 for comparison).

As for the empirical distributions, however, one has to be cautious because of the small sample size. Furthermore, the overall distribution after ten thousand model runs had a mean of 27.2, standard deviation of 60.1, and median of 4, which are slightly too high.

The facts that not all simulated distributions follow a power law and that often the median is too high is because there are, on average, more medium-sized cities in the (p. 583)

simulation than in reality. The simulated distribution is not as skewed as the empirical one. This is probably because, for reasons of simplicity, the model does not take into account that some cities attract many more migrants than others, not just because of network effects but simply because they are larger and provide better job and other opportunities. Bauer et al. (2007) find that the probability that migrants choose a particular U.S. location increases with the total population in that location for almost all groups of migrants. Future versions of should take this model this factor into account.

## 19.7.3 Stylized Facts Revisited: Migration-Specific Capital

The distribution of the number of trips in the sample was negative binomial. The simulated distribution is not exactly negative binomial because even-numbered counts of trips are much more frequent than odd-numbered ones in the simulation, but not in reality. That is to say, moving to the host country and moving back at some point is more frequent in the simulation than in reality. That might be because survey respondents have more degrees of heterogeneity than do computer agents: The people (p. 584)

who stay in the United States are different from the ones who return, with respect to a set of characteristics that were not considered here. Furthermore, in reality, some of the migrants have family in the United States and others do not; this factor might fundamentally alter the psychic costs of separation (Lindstrom 1996). Therefore, their behavioral rules might also differ. In the simulation, everyone makes the same type of decision, albeit with different idiosyncratic parameter values such as the home bias *p*_{3,i}. To correct this inaccuracy of the model in a satisfactory way will be a subject of further research. When smoothing the distribution of number of trips by forming categories of two values each to correct for this inaccuracy (0, 1−2, 3−4, etc.), the distribution is very close to being negative binomial (see figure 19.5).

## 19.7.4 Match of Empirical and Simulated Time Series

The observed empirical time series of migration and return are depicted in figure 19.6.

In order not to overcalibrate the model, the mean squared error between simulated and empirical data was minimized in four points only. The focus was on matching the overall pattern: an inverted U-shape. The results of the ten thousand Monte Carlo runs with the best parameter setting *p*_{0} = .1, *p*_{2} = .2, *q*_{0} = .38, and *q*_{1} = .12 are depicted in figures 19.7 and 19.8. The curves that indicate mean, standard deviation, and quantiles
(p. 585)

can be used to classify particular simulation results in the context of the conceptual population of simulated scenarios, similar to Voudouris et al. (2011).

Most of the simulation runs are within a fairly narrow range. The overall pattern—both migration and return movement behavior increase and then decrease over time—follows the pattern of the empirical data.

19.8 Robustness Checks and Policy Analysis

A robustness analysis of an agent-based model serves to check whether the model reacts as expected when parameter changes are introduced that should alter the results in an unambiguous way. By increasing the home-country wage relative to the host country wage it shall now be demonstrated that the model at hand passes a test for robustness. Afterwards it will be illustrated how the model can be used for policy analysis. It is shown how the effect of a tightening of border control depends on the level of foresight of potential return migrants.

Some potential migrants can probably not afford to migrate and would therefore be enabled to overcome a “poverty trap” if wages increased slightly (McKenzie and Rapoport 2007). A larger increase in home-country wages should decrease stocks of migrants in the host country. To check whether this result is produced by the model, the home-country wage is increased by multiplying each value in the time series by 1.1,1.2, (p. 587)

and so on up to 3.2 and running the model one thousand times for each treatment. An increase in average stocks is observed in early periods for increases in the average home wage. At some point every potential migrant has gathered sufficient wealth to overcome the poverty trap. The higher the home-country wage, the earlier that point is reached. Beyond that point, the higher the home-country wage, the lower are the average migrant stocks in the host country (see figure 19.9). At values larger than 3.2, migration ceases almost completely as the home-countrywages are, on average, as high as the host-country wages.

This is the effect that was expected, and it is reproduced by the model. Whether this estimate can be trusted quantitatively depends on whether one believes that the behavioral rules, and in particular, the impact of the wage difference on the migration decision, are stable if the wages increase substantially. Further research is needed to verify this assumption.

The next experiment that is performed concerns the level of border control. It is unclear whether increasing border protection increases or decreases the stock of migrants in a country. Kossoudji (1992) observes that tighter regulation increases stocks of migrants because it decreases out-migration. Angelucci (2005) finds an ambiguous answer: Tighter border control clearly decreases inflow but also decreases outflow. Thom (2010) does not find that stricter border control increases stocks of migrants. Clearly, the net effect depends on how far migrants are deterred from returning since they take into account the lower probability of being able to migrate again.

In order to test the impact on stocks, it is assumed that the level of border control increases by 10 percent. Figure 19.10 depicts the average stocks across a period of (p. 588)

thirtythree years at levels of baseline return probability of .38 and at lower levels, showing how stocks increase with a decrease in return probability.

The relation between average stocks and baseline return probability is almost linear. Average stocks increase by 3.58 individuals with every percentage point decrease in baseline return probability. It can be concluded that, on average, stocks increase after an increase in border control by 10 percent if, of one hundred migrants in the United States of whom thirtyeight would have returned in a given year, seven (or 18 percent) or more take into account the reduced migration probability and refrain from returning.

19.9 Conclusions

In this study the phenomenon of circular migration is analyzed in an agent-based model. To the author’s knowledge it is the first completely empirically founded and spatially explicit model of the phenomenon that is able to take account of the whole cycle of migration and the role of networks. Three stylized facts about circular migration are introduced that the model can match, despite its being fairly simple: Migration concentrates on a certain number of places, people from one neighborhood tend to (p. 589) go to the same few places, and migration-specific capital makes subsequent migration more likely. A set of hypotheses is derived from the literature concerning influential factors in the decision to migrate or to return in a given year. These hypotheses are tested using the Mexican Migration Project (MMP128). The behavioral motives that survived the empirical check are included in the model. It is found that expected earnings, an idiosyncratic home bias, network ties to other migrants, strength of links to the home country, and age have a highly significant impact on circular migration patterns over time. A model is presented that includes two countries with differing average wages, workers who search for employment, and firms. Workers can migrate and return according to probabilistic behavioral rules estimated from the MMP128. Four remaining parameters are calibrated by running Monte Carlo simulations. Thus, avoiding a common criticism of agent-based models, this model has only four degrees of freedom and yet is able to replicate two distributions and two time series from the data fairly well.

Computational experiments are performed in order to demonstrate the robustness of the model. Finally, it is shown how the model can be used to perform policy analysis. It has the potential to help answer the much-debated question whether increasing border protection increases or decreases the stock of migrants in a country. It is found that if 18 percent or more of migrants who would have returned at the lower level of border control take into account that they might not be able to migrate again and therefore refrain from returning, stocks increase. Otherwise, they decrease.

Promising avenues for future research are making the model spatially accurate using a geographic information system or introducing more sophisticated behavioral rules and additional degrees of heterogeneity to account for existing mismatches between data and simulation.

Moreover, with further calibration and sensitivity analysis, the model can be used for forecasting flows of migration and return in certain regions or cities, possibly by combining it with local border enforcement data, and to estimate the effect of labor market shocks or changes in immigration law.

Acknowledgments

The author is grateful for helpful comments and suggestions by Simone Alfarano, LukasHambach,Wolfgang Luhan,Maren Michaelsen,Matteo Richiardi,Michael Roos, Pietro Terna, KlausG. Troitzsch, Alessandra Venturini,Vlasios Voudouris, and Thomas Weitner; participants at the Eastern Economic Association Conference in New york in February 2011 and the 17th International Conference on Computing in Economics and Finance in San Francisco in June 2011; seminar participants at Ruhr University Bochum, Philipps-University Marburg, and Università di Torino; and two anonymous referees. She would like to acknowledge financial support by the RUB Research School.

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