A Simple Model of Temporary Migration

data contain very rich information about migration experience over the life cycle, which is valuable in exploring the return to migration experience. Zahniser and Green- wood 1998 use an early version of the MMP with fewer observations dominated by high- migration communities and fi nd a large positive return to migration experience. Alternately, using data from the 2000 Mexican census, Lacuesta 2010 fi nds that migrants tend to earn about 7–10 percent higher wages than nonmigrants upon return- ing. However, Lacuesta does not fi nd that wages grow with experience, and interprets this as evidence that the return migrant premium refl ects self- selection rather than skill upgrading. While the Census data has the advantage of being nationally representative, it only provides data on the duration of the last trip to the United States taken within the last fi ve years. We fi nd that these data limitations have substantial consequences for the estimation of returns to migration experience when we censor the MMP data in a similar way.

III. A Simple Model of Temporary Migration

Our empirical analysis centers on OLS regressions of Mexican earn- ings on accumulated migration experience. A natural concern is that the self- selection of return migrants and the endogeneity of accumulated migration experience bias the OLS estimates of a return to migration experience. Here we present a simple theoreti- cal model to help guide the interpretation of the observed relationship between earn- ings and migration experience given this concern. We argue below that the primary challenge to interpreting the results of such a regression comes not from extensive margin selection how return migrants compare to nonmigrants in terms of unobserved skill, but rather from intensive margin selection how migration experience is cor- related with unobserved skill within the group of return migrants. Under plausible parameterizations, the model predicts that total accumulated migration experience should be negatively correlated with skill conditional on migration. This implies that OLS regression coeffi cients will represent a lower bound on the true causal effect of migration experience on income. The model considered here builds on the temporary migration models of Dustmann 2003 and Dustmann and Weiss 2007. Consider the following environment. There are two countries: the home country h, and the foreign country f. Individuals are endowed with a unit of continuous time, and they start life in the home country. An individual can only migrate at time t = 0, and chooses some fraction of time τ to spend in the foreign country. If the individual chooses τ = 1, then the individual is a permanent migrant. If the individual chooses τ 1, the individual is a temporary migrant and returns home after τ units of time. Note that this model restricts immigrants to taking at most one migratory trip to the United States. In reality, many Mexican migrants engage in patterns of repeated circular migration involving multiple trips to the United States see Massey, Durand, and Malone 2003; Thom 2010; Rendon and Cuecuecha 2010. While the assumption of a one- trip structure is needed to keep the theoretical analysis tractable, the model presented here can be thought of as an approximation to a more general multiple- trip model in which individuals choose the overall fraction of time that they spend abroad. We assume that individuals differ in their level of permanent skill, s i . We assume that this refl ects anything responsible for permanent differences in the productivity of workers including innate ability and skills accumulated before migration. For the pur- poses of the model, s i can be thought of as subsuming several components of skill, some of which will be observed in the data such as schooling, and some of which will be unobserved. Skill is randomly distributed in the population, and we will allow some of the model parameters to depend on skill. We assume that wage rates in the home and foreign countries depend both on skill and on the amount of migration ex- perience that has been accumulated up to time t: w c = w c s i , τ t for c = h, f. There is an international wage gap, so w f – w h 0. Furthermore, we assume that wages are in- creasing in both skill and migration experience in both countries: ∂w c ∂s 0, ∂w c ∂τ 0. We allow the relationship between wages and migration experience to depend on baseline skill, and the nature of this dependence is theoretically ambiguous: ∂ 2 w c ∂τ∂s P 0. For example, consider the return to migration experience in the home country. It could be the case that less- skilled workers gain more from such ex- perience. This could occur if the job opportunities for migrants in the foreign country are skewed toward low- skilled occupations. Experience as a cab driver is unlikely to be useful for a migrant who is qualifi ed to work as an engineer in his home country. Additionally, work experience in the foreign country might serve a remedial function if it provides low- skilled workers with opportunities to gain skills that only more- skilled workers can gain at home. For example, if more advanced capital equipment in construction or manufacturing is scarce at home and only more- skilled workers are able to gain experience using it, then low- skilled workers might have more chances to gain such experience by working abroad. Alternately, the return to migration experi- ence might be higher for more- skilled workers if jobs in the foreign country provide learning opportunities that are only useful for the skilled. A physician, engineer, or Ph.D. trained academic from a developing country might profi t tremendously from the interactions with similar workers in a high- income country. Given this wage structure, individuals choose an optimal duration of stay in the foreign country and a life- time consumption profi le, where we assume that the instan- taneous utility from consumption is given by uc t = logc t . Migration enables higher consumption over the life cycle, but there exists a tradeoff because living abroad in- curs disutility of η per unit of time. In addition, if an individual migrates, he or she must pay a fi xed migration cost of λs i . As argued by Chiquiar and Hanson 2005, the costs of migration might be lower for high- ability people if such individuals are more capable or ambitious. If there is a correlation between baseline skill and the cost of migration, we would thus expect it to be negative: ∂λ ∂s 0. Given these assumptions, the individual’s decision problem becomes: τ,c t max V = 1 ∫ logc t e −δt dt − τ ∫ ηe −δt dt s.t. 1 ∫ c t e −rt dt = τ ∫ w f e −rt dt + τ 1 ∫ w h e −rt dt − λ1τ 0 ≤ τ ≤ 1 For simplicity, we follow Dustmann 2003 and assume that individuals do not discount future utility, and that the individuals can freely borrow and save at a real in- terest rate of 0 δ = r = 0. In such an environment, consumption is fi xed at a constant rate, c, over the life cycle. Furthermore, we also assume that wages in both countries have the following log- linear form: 1 logw c s i , τ = γ c s i + γ 1 c τ + γ 2 c τs i . Where γ c s i is an increasing function of s i . We refer to the derivative of log- wages with respect to migration experience as the return to migration experience. This is given by γ 1 c + γ 2 c s for the wages of someone with baseline skill level s in country c. Under these particular assumptions, the individual’s decision problem becomes: 2 max τ logc − ητ s.t. c = 1 γ 1 f + γ 2 f s [w f s, τ − w f s, 0] + w h s, τ1 − τ − λs1τ 0. The First Order Condition for an interior solution implies ∂c ∂τ − ηc = 0. Implicit differentiation of this condition allows us to derive the relationship between optimal migration duration and baseline skill: 3 ∂τ ∂s = − ∂w f ∂s − ∂w h ∂s ⎛ ⎝⎜ ⎞ ⎠⎟ + 1 − τ ∂ 2 w h ∂s∂τ − η ∂ c ∂s ∂w f ∂τ − 2 ∂w h ∂τ ⎛ ⎝⎜ ⎞ ⎠⎟ + 1 − τ ∂ 2 w h ∂τ 2 − η ∂ c ∂τ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ A set of plausible assumptions on the model parameters allows us to predict that the sign of this partial derivative should be negative. First note that the derivative of con- sumption with respect to baseline skill is positive for everyone ∂c ∂s 0 as long as migration costs do not increase with skill. Also, lifetime consumption should be increasing in τ for anyone who migrates ∂c ∂τ 0 evaluated at τ, or otherwise τ is not optimal. These conditions allow us to sign the terms involving η disutility of staying in the foreign country as negative. Neglecting the second order terms for a moment, ∂τ ∂s will be negative at the optimum if these two suffi cient conditions are satisfi ed. The fi rst condition is that [ ∂w f ∂s – ∂w h ∂s] 0 at the optimal duration, making the numerator of Equa- tion 3 negative. Since s includes both observed and unobserved components of an in- dividual’s skill, it is diffi cult to verify that this condition holds on the basis of existing empirical evidence. However, the condition does seems to hold for the case of Mexico if one looks at important observable components of skill, such as schooling. 4 For ex- ample, hourly wage calculations presented in Hanson 2006 suggest that the US- Mexico wage gap is generally declining with education, at least for individuals with less than 16 years of schooling. Evidence for an international earnings gap that de- clines with skill is also found by Ambrosini and Peri 2012 . The second condition is that the denominator of Equation 3 must be negative. Under the assumption that the optimal migration duration is small approximately zero, the following condition ensures that the denominator of Equation 3 will be negative for an individual of skill s: 4. See Hanson 2006, pp. 893-94, Chiquiar 2003. 4 γ 1 f + γ 2 f s 1 + θ θ ⎡ ⎣⎢ ⎤ ⎦⎥ γ 1 h + γ 2 h s + θ − 1 2 θ Here θ = w f w h 1 . A full derivation of this condition is offered in the appendix. This condition essentially requires that the return to migration experience in the United States not be too much larger than the return to migration experience in Mexico. In- deed, for plausible parameter values, this will only be violated for extremely high rates of return to migration experience in the United States. 5 In the model presented here, optimal migration duration is negatively associated with skill. The primary reason for this is that more- skilled individuals are assumed to face a smaller international wage gap. The opportunity cost of spending time abroad is higher for these individuals, and so they choose shorter durations. Under some al- ternate assumptions, this result becomes ambiguous. For example, if the disutility of spending time abroad is lower for more- skilled individuals ∂η ∂s 0, then it will not be possible to sign ∂τ ∂s with the assumptions made here. We have also neglected the second- order terms in Equation 3. The prediction that our model makes about the sign of ∂τ ∂s would become ambiguous if there are large increasing returns to U.S. labor market experience in Mexico ∂ 2 w h ∂τ 2 and large. In addition, we have to consider the cross- derivative ∂ 2 w h ∂s∂τ = w h s, τγ 2 h + γ h ′ + γ 2 h τγ 1 h + γ 2 h s . If there are large complementarities between skill and U.S. labor experience if γ 2 h and large, then this could make sign of ∂τ ∂s ambiguous as more- skilled workers would have an extra incentive to extend their stay. Note that this also implies that return migrants will be positively selected from the set of migrants, a pattern consistent with the empirical results of Ambrosini and Peri 2012 and Biavaschi 2010. We proceed by assuming that wages are log- linear in their arguments, and that the parameters are such that ∂τ ∂s 0 for an interior solution. In the population indi- viduals will choose to stay home, become permanent migrants, or become temporary migrants depending on their initial skill level. Figure 1 provides some numerical ex- amples that highlight how the optimal migration duration varies with baseline skill, which is drawn from the interval [0,1]. Panel A assumes some reasonable parameters that generate a pattern of behavior that is broadly consistent with the features of Mex- ican migration. Nonmigrants are drawn from both the lower and upper ends of the skill distribution. At the lower end, there is some group of individuals that either cannot afford to migrate so lifetime resources under migration are negative, or who can af- ford to migrate but are better off not paying the cost of migration. At the upper end, there is a group of individuals that do not have a wage incentive to migrate since the international wage gap is declining in skill. For those who migrate temporarily, the 5. To give some sense of the strictness of this condition, consider males aged 23-27 with fi ve to eight years of education, a group with one of the highest migration rates in Mexico. The calculations in Hanson 2006 indicate that the US-Mexico wage ratio for this group is equal to θ = 8.19 1.80 = 4.55. In this case, the condition in Equation 4 requires γ f s 1.2 γ h s + 2.77. Thus, this condition requires that the return to migra- tion experience in the United States not be very much greater than the returns in Mexico. Suppose that the return to migration experience is 2 percent per year in Mexico. Since τ represents a fraction of a working lifetime, a hypothetical 2 percent return in Mexico per year of migration experience corresponds to γ h = 1 assuming a 50 year working life. In order for the condition in Equation 4 to hold in this scenario, it would need to be true that γ f 3.97, or that the return to migration experience be less than 8 percent per year in the United States. The suffi cient condition presented here is only violated if there are extremely large returns to migration experience in the United States. amount of time spent abroad will vary negatively with skill as long as our maintained assumptions hold. There will generally be skill bounds S and S such that only indi- viduals with baseline skill levels in the interval s ∈ [S, S] will fi nd it optimal to mi- grate. The skill threshold S further separates the group of migrants who permanently migrate from those that temporarily migrate. Panel B shows what happens to behavior if the returns to migration experience are shut down in both countries. Although this change in parameters induces fewer individuals to migrate, the same basic qualitative patterns present in Panel A continue to hold. Panels C and D further highlight scenarios in which violations of our assumptions can result in a positive relationship between migration durations and baseline skill. 0.5 1 Baseline Skill Optimal Migration Duration 0.5 1 C γ 0 and large h 2 A Baseline Scenario B No Returns to Experience D η is declining in skill 0.5 1 Baseline Skill Optimal Migration Duration 0.5 1 S S S 0.5 1 Baseline Skill Optimal Migration Duration 0.5 1 S S S 0.5 1 Baseline Skill Optimal Migration Duration 0.5 1 Figure 1 Numerical Examples of the Model Solution Note: Unless otherwise noted, wages are parameterized as follows: w h s, τ = exp1.5 + 2s + 0.4τ, and w f s, τ = exp3.5 + 0.2s + 0.4 τ. This generates an international wage gap that decline with skill, with foreign wages being about six to seven times home wages at low levels of skill, about three times home wages at intermedi- ate levels, and about 1.15 times homes wages at high levels. It is assumed that monetary migration costs are declining in skill: λs = exp3.91 − 5s. The disutility of being away from home is set at η = 1. In Panel B, returns to migration experience both at home and abroad are set to 0. In Panel C, it is assumed that the returns to migration experience at home are increasing in skill, so that w h s, τ = exp1.5 + 2s + 4sτ. Additionally, the disutility of being abroad is increased in this example to η = 2, and migration costs are increased to λs = exp4.61 − 5s. In Panel D, it is assumed that η is declining in skill, so that ηs = exp1.61 − 3.25s. Ad- ditionally, migration costs are lowered in this panel so that λs = exp3.69 − 5s. Panel C describes a scenario in which the return to migration experience at home rises steeply with baseline skill. In this scenario, more- skilled individuals have an incentive to make longer temporary trips abroad. Likewise, if the disutility of spending time abroad is negatively related to skill, then the more- skilled will be observed completing longer stays abroad. Such a scenario is depicted in Panel D.

IV. Implications for Relationships in the Data