Many migrants move to the destination for economic opportunities rather than family reunion. Their family members usually wait in the home country because of the higher living costs in the destination, especially when migrants plan to return. As migrants accumulate human capital and host country currency, they have to deal with the psychic costs associated with leaving their families at the same time, which decrease their utility in the host country. Therefore, they face a trade-off between offering financial support to the family and avoiding psychic costs.

### 2.1 Optimal duration

A simple model can be developed to illustrate the relationships among psychic costs, family composition, and optimal migration duration. Following Dustmann (2003b), assume migrants enter a more developed country at time point 0 and standardize their life span to 1. They decide to stay in the host country for a time period, *t*, to maximize their lifetime utility, *U*. I bring family members to the framework and consider a family unit consisting of a migrant and *N* other family members.^{1} Suffering psychic costs from being separated from family members, the migrant’s utility in the host country discounts at a discounting factor *s*(*N*). It also denotes the percentage of utility that the migrant finally obtains; 0≤*s*(*N*)≤1. As more family members present in the household, the percentage decreases: *s*
^{′}(*N*)<0. The lifetime utility becomes:

$$ U= t\nu(C_{f}, \alpha)s(N)+(1-t)\nu(C_{h}, \beta) $$

((1))

where *ν* is the utility function for migrants. *C*
_{
f
} indicates the migrant’s consumption in the host country (foreign country), and *C*
_{
h
} is the migrant’s consumption in the home country. *α* and *β* give the preference of this migrant for consumption in the host country and home country, respectively. It is assumed that migrants enjoy consumption in the home country more: *β*>*α*. Marginal utilities are positive: *ν*
*f*′(*C*
_{
f
},*α*)>0 and *ν*
*h*′(*C*
_{
h
},*β*)>0. Diminishing marginal utility suggests that *ν*
*f*″(*C*
_{
f
},*α*)<0 and *ν*
*h*″(*C*
_{
h
},*β*)<0 (Dustmann 2003b).

Migrants maximize their lifetime utility with respect to *C*
_{
f
}, *C*
_{
h
}, and *t*, subject to the inter-temporal budget constraint:

$$ tw_{f}+(1-t)w_{h}-tC_{f}-(1-t)pC_{h}-p\hat{C}_{h}N=0 $$

((2))

where *w*
_{
f
} and *w*
_{
h
} indicate wage rates in the host country and home country, respectively; *p* denotes the price for consumption in the home country, relative to the host country. If consumption in the foreign country is more costly than consumption in the home country, *p*<1. Migrants give a fixed amount of money to each family member, \(\hat {C}_{h}\). For example, if a migrant has several children in the home country, the costs of raising each child could be roughly fixed, although later we will see that the costs of raising children change by their age and gender.

Solving the utility maximization problem with the budget constraint, the First Order Conditions (FOCs) are

$$ \nu(C_{f}, \alpha)s(N)-\nu(C_{h}, \beta)+\lambda(w_{f}-w_{h}-C_{f}+pC_{h})=0 $$

((3))

$$ \nu'(C_{f}, \alpha)s(N)-\lambda=0 $$

((4))

$$ \nu'(C_{h}, \beta)-p\lambda=0 $$

((5))

where *λ* denotes the marginal utility of wealth: *λ*>0.

Equation (3) is the equilibrium condition, which determines the optimal migration duration. *w*
_{
f
}−*w*
_{
h
}−*C*
_{
f
}+*p*
*C*
_{
h
}>0, because an additional unit of time in the host country increases a migrant’s lifetime wealth (Dustmann 2003b). *ν*(*C*
_{
h
},*β*)−*ν*(*C*
_{
f
},*α*)*s*(*N*)>0, this suggests that the forgone utility of staying a further unit of time abroad is positive because migrants cannot consume goods in the home country (*β*>*α*) and they miss family members. Combining these FOCs with Eq. (2), the optimal duration in the host country can be solved.

Comparative statistics with respect to the model parameters are the following:

$$ \frac{\partial t}{\partial N}=\frac{-s'(N)}{\Delta^{2}}\left[\frac{t\nu(C_{f},\alpha)+t\nu'(C_{f},\alpha)\Delta}{s(N)\nu^{\prime\prime}(C_{f}, \alpha)}+\frac{(1-t)p^{2}\nu(C_{f},\alpha)}{\nu^{\prime\prime}(C_{h}, \beta)}\right]+\frac{p\hat{C}_{h}}{\Delta} $$

((6))

$$ \frac{\partial t}{\partial w_{f}}=-\frac{\lambda}{\Delta^{2}}\left[\frac{t}{s(N)\nu^{\prime\prime}(C_{f}, \alpha)}+\frac{(1-t)p^{2}}{\nu^{\prime\prime}(C_{h}, \beta)}\right]-\frac{t}{\Delta} $$

((7))

$$ \frac{\partial t}{\partial w_{h}}=\frac{\lambda}{\Delta^{2}}\left[\frac{t}{s(N)\nu^{\prime\prime}(C_{f}, \alpha)}+\frac{(1-t)p^{2}}{\nu^{\prime\prime}(C_{h}, \beta)}\right]+\frac{(t-1)}{\Delta} $$

((8))

where *Δ*=*w*
_{
f
}−*w*
_{
h
}−*C*
_{
f
}+*p*
*C*
_{
h
}>0.

Then \(\frac {\partial t}{\partial w_{h}}<0\), the higher the wage rates in the home country, the shorter the migrants would stay in the host country. But the sign of \(\frac {\partial t}{\partial w_{f}}\) is ambiguous. If the wage rates in the host country are higher, the income effect encourages migrants to return sooner, but the substitution effect suggests migrants to stay longer in the host country to enjoy the wealth (Dustmann 2003b).

The focus is on the sign of \(\frac {\partial t}{\partial N}\). The first part on the right hand side of Eq. (6) is negative, since *s*
^{′}(*N*)<0 and *Δ*>0, while the formula in brackets is negative because of the diminishing marginal utility assumption. The second part, \(\frac {p\hat {C}_{h}}{\Delta }\), is positive. If the financial costs of supporting family members in the home country, \(\hat {C}_{h}\), are sufficiently high, \(\frac {\partial t}{\partial N}\) could be positive, migrants would stay longer in the host country to accumulate money for family members’ consumption. When the absolute value of marginal discounting factor, *s*
^{′}(*N*), is high enough, \(\frac {\partial t}{\partial N}\) could be negative: an extra family member leads to a great decrease in the entire utility in the host country, migrants return sooner.

In the above model, these family members refer to children appropriately, because they are assumed to have zero income. In fact, changing the setup of the model does not change the implication greatly as I include adult family members’ income.

Assume family members have an income, *w*
_{
m
}, in the home country (*m* is short for member). Relax the assumption of a fixed amount of financial support to family members. Assume family members’ consumption at home, *C*
_{
m
}, affects migrants’ lifetime utility. Then migrants make decisions over *C*
_{
m
}, as well as *t*, *C*
_{
f
}, and *C*
_{
h
}, to maximize the utility function:

$$ U= t\nu(C_{f}, \alpha)s(N)+(1-t)\nu(C_{h}, \beta)+\nu(C_{m}, \gamma)N $$

((9))

with the budget constraint:

$$ tw_{f}+(1-t)w_{h}+w_{m}N-tC_{f}-(1-t)pC_{h}-pC_{m}N=0, $$

((10))

then,

$$ {\footnotesize{} \begin{aligned} \frac{\partial t}{\partial N}=\frac{-s'(N)}{\Delta^{2}}\left[\frac{t\nu(C_{f},\alpha)+t\nu'(C_{f},\alpha)\Delta}{s(N)\nu^{\prime\prime}(C_{f}, \alpha)}+\frac{(1-t)p^{2}\nu(C_{f},\alpha)}{\nu^{\prime\prime}(C_{h}, \beta)}+\frac{p^{2}\nu(C_{f},\alpha)N}{\nu^{\prime\prime}(C_{m}, \gamma)}\right]+\frac{pC_{m}-w_{m}}{\Delta}. \end{aligned}} $$

((11))

Again, the term in brackets is negative and the sign of \(\frac {\partial t}{\partial N}\) is ambiguous. If these family members can fully financially support themselves, *w*
_{
m
}−*p*
*C*
_{
m
}>0, then \(\frac {\partial t}{\partial N}<0\), migrants with more family members have shorter stays. Once the migrants need to offer financial support to these family members, *w*
_{
m
}−*p*
*C*
_{
m
}<0, migrants with more family members may have to stay longer, suffering the psychic costs, if \(\frac {\partial t}{\partial N}>0\).

Combining Eqs. (6) and (11), the number of family members can be divided into *i* different groups, *g*
_{
i
}, based on their income and consumption. The sign of \(\frac {\partial t}{\partial N_{g_{i}}}\) is decided by the change in utility caused by psychic costs and the financial conditions of family members in each group. Regarding the family composition, migrants face a trade-off between avoiding high psychic costs and improving household consumption.

### 2.2 Marital status

Hypothesis 1: Unmarried migrants stay longer in the destination country than married migrants.

In Eq. (11), when the migrants are unmarried, *N*=0 and *w*
_{
m
}−*p*
*C*
_{
m
}=0, then \(\frac {\partial t}{\partial N}|_{N=0}<0\). Married migrants whose spouses earn more than their consumption face *w*
_{
m
}−*p*
*C*
_{
m
}>0, then \(\frac {\partial t}{\partial N}|_{N=1}<0\). Married migrants (*N*=1) will stay shorter than unmarried migrants (*N*=0).

If married migrants’ spouses cannot afford to live on their own, *w*
_{
m
}−*p*
*C*
_{
m
}<0, the sign of \(\frac {\partial t}{\partial N}|_{N=1}\) is ambiguous. If it is negative, hypothesis 1 holds. If it is positive, there is a *N*
^{∗}∈(0,1) satisfying \(\frac {\partial t}{\partial N}|_{N=N^{*}}=0\), then the effect of marriage on the length of duration is an empirical question.

The fact that most migrants are males rather than females, especially in Mexico-US migration, is consistent with the model to some degree. Traditionally, men support the family financially while women do most of the housework (Becker 1985). This labor division in a family suggests a low labor market participation of women. Married women with their husbands supporting the family financially may not migrate (*w*
_{
m
}−*p*
*C*
_{
m
}>0,*t*=0), while married men, compared to women, are more willing to migrate to accumulate human capital and wealth (*w*
_{
m
}−*p*
*C*
_{
m
}<0,*t*>0).

### 2.3 Children

#### 2.3.1 Number of children

Hypothesis 2: With more children in the home country, migrants have shorter migration stays than those with fewer children do.

Migrants with no child face zero psychic costs from being concerned about their children: in Eq. (6), \(\frac {\partial t}{\partial N}|_{N=0}<0\). When *N*>0, the sign of \(\frac {\partial t}{\partial N}\) is ambiguous. The higher the costs of raising a child, the greater \(\frac {p\hat {C}_{h}}{\Delta }\) is. Once these costs are high enough to make \(\frac {\partial t}{\partial N}>0\), one more child in the family implies a longer migration duration of the parent.

However, the costs of raising children are actually not fixed because of the economies of scale in raising children. The marginal money costs and time costs of raising children are diminishing (Holmes and Tiefenthaler 1997; McClements 1977): \(\frac {\partial (p\hat {C}_{h})}{\partial N}<0\). Then the greater the *N*, the more likely that \(\frac {\partial t}{\partial N}\) is negative.

#### 2.3.2 Age of children

Hypothesis 3: Migrants who are parents of younger children (babies or primary-school-age children) return sooner than migrants whose children are older.

The money and time costs of raising children change by children’s age. Adult children are supposed to be financially independent: *w*
_{
m
}−*p*
*C*
_{
m
}=0 in Eq. (11). They are not the main reason for the psychic costs of migrants: *s*
^{′}(*N*)=0 when *N* represents adult children. Then \(\frac {\partial t}{\partial N\mathrm {(Adult \;Children)}}=0\), these adult children would not affect their parents’ migration duration theoretically. In reality, the duration will be affected due to parents’ concern for adult children and the money flows between them. If adult children help to take care of migrants’ younger children, migrants may stay abroad longer because of the reduced psychic costs.

Young children need both financial support and companionship from parents. Generally, among children under 18, older children are more expensive than younger ones, excluding child labor in some poor countries. According to the annual report of Expenditures on Children by families from the United States Department of Agriculture (USDA), the annual expenditures overall in the United States on a child aged 12–17 years is about 10–20 %^{2} per year higher than that on a child aged 0–11 years.^{3} Specifically, the difference in consumption on food and health care captures the expenditure gap between younger children and older ones, while the expenditures on education do not change much by children’s age. However, in Mexico, the costs of secondary school are about 35 % higher than primary school (Wolff and Gurría 2005), thus the costs of raising a younger child may be much lower than an older child: \(p\hat {C}_{\mathrm {Little \; Children}}< p\hat {C}_{\mathrm {Older\; Children}}\).

In the meantime, migrants’ psychic costs change by the age of their children. Time costs are directly related with these psychic costs, since spending time with children is an efficient way for parents to build a close bond with them and reduce psychic costs. Parents’ time spent with their children usually decreases as their children grow up (Bittman 1999).^{4} Babies need to be taken care of all the time; primary school-aged children are not quite involved in heavy homework load compared with high school-aged children. They may have higher demand for parents’ time, resulting in lower discounts for migrants compared to older children, \(\frac {\partial (s'(N))}{\partial \mathrm {(Age\; of\; the\; Additional\; Child)}}>0\). Their parents return sooner.

In sum, raising younger children is time intensive, while raising children in secondary school is money intensive. Migrants with older children may stay longer in the host country than migrants who have younger children at home.

#### 2.3.3 Gender of children

Hypothesis 4: Children’s gender affects parents’ migration duration. The effects change by the home country.

Dustmann (2003a) finds that migrants traveling with daughters stay shorter abroad than migrants traveling with sons, because migrants may want daughters to preserve traditions in the home country while sons to pursue future economic careers in the host country. However, the story may vary by country because of different cultures. For example, with a dowry culture, the difference in the costs of raising a child by gender may highly depend on price and dower of the future bride.

Mexico has a patriarchal culture (Massey et al. 2006), thus parents may intend to spend more money and time on their sons than daughters. Their US duration may differ by the gender of children.

#### 2.3.4 Family migration decisions

The model above assumes that migrants travel alone, while family members may travel with them to the host country. Balancing the lower psychic costs and heavier financial burden due to migrating family members, how migrants make migration duration decisions is an empirical question.

In Mexico, males, who are more economically active, dominate the migration flow to the USA. Their wives usually travel to the USA for family reunion reasons, rather than economic motivations. In the meantime, to avoid the risky change in occupation, wives without a well-paid job in Mexico may have more incentives to travel because of the lower opportunity costs, compared to those with a good job. The negative selection and females’ lower labor force participation rate imply that the possible income decline from these spouses traveling may not be economically significant. When the increase in utility from the companionship is higher than the decrease in utility from increased living costs and declined income of their wives, migrants traveling with wives stay longer abroad than those with wives waiting in the home country. Once their wives can get a good job in the host country, the migrants would stay even longer.

Children, without an income for the family, would bring migrants a much heavier financial burden than migrating spouses if children live in the host country. The amount of money spent on children is supposed to be larger than that on spouses, because parents invest in their children, rather than just offer them basic financial supports. If the psychic costs associated with children are the same as the psychic costs associated with spouses, migrants traveling with children but without spouses may have a shorter stay than migrants traveling with spouses but without children, since supporting a child in the host country is more costly.

When migrants travel alone, to reduce psychic costs, they may make multiple trips, leading to a longer total duration. Furthermore, family composition may change by migrants’ experience in the host country.

In addition, people may make marital decisions, fertility decisions, and migration decisions simultaneously. If they decide to stay long in the host country, they may prefer to be single or have no children. The causality between family composition and migration duration is unclear.

### 2.4 Human capital

The above model also suggests that human capital, which mainly affects migrants’ earnings, would be an important determinant of migration duration. The higher the education level (*E*), the greater the wage rates: \(\frac {\partial w_{i}}{\partial E}>0, i=f \;or\; h\). The effect of education level on the migration duration can be expressed as

$$ \frac{d\, t}{d\, E}=\frac{\partial t}{\partial w_{f}}\frac{\partial w_{f}}{\partial E}+\frac{\partial t}{\partial w_{h}}\frac{\partial w_{h}}{\partial E}. $$

((12))

The sign of \(\frac {d\, t}{d\, E}\) is ambiguous, because \(\frac {\partial t}{\partial w_{h}}<0\) in Eq. (8) while the sign of \(\frac {\partial t}{\partial w_{f}}\) in Eq. (7) is uncertain. When \(\frac {\partial t}{\partial w_{f}}<0\), \(\frac {d\, t}{d\, E}<0\). Migrants with more years of schooling return to their home country sooner.

When \(\frac {\partial t}{\partial w_{f}}>0\), the sign of \(\frac {d\, t}{d\, E}\) depends on the comparison of the absolute values of the two components on the right-hand side of Eq. (12). The value of \(\frac {\partial w_{f}}{\partial E}\) may be highly related to the matching of migrants’ job search in the host country. Low-educated migrants are usually undereducated for their jobs compared to native workers, while high-educated migrants are often overeducated (Chiswick and Miller 2008).^{5} An extra year of schooling improves the matching in labor market for undereducated migrants, while it impairs the matching for overeducated migrants. Also, Mincer Equation implies that \(\frac {\partial ^{2} w_{f}}{\partial E^{2}}<0\) (Mincer 1974) if *E* represents labor market experience. Then it is quite possible that \(\frac {\partial w_{f}}{\partial E}|\text {undereducated} > \frac {\partial w_{f}}{\partial E} |\text {overeducated}\) or \(\frac {\partial w_{f}}{\partial E}|\text {low}\;E > \frac {\partial w_{f}}{\partial E} |\text {high}\; E\). When \(\frac {d\, t}{d\, E}|\text {low}\; E>0>\frac {d\, t}{d\, E}|\text {high}\; E\), there is probably a threshold education level for migrants. The length of migration increases as education levels rise up from low values to the threshold and then declines as education levels climb to values beyond the threshold. However, if \(\frac {d\, t}{d\, E}|\text {low}\; E<0<\frac {d\, t}{d\, E}|\text {high}\; E\), the threshold may still exist, but the story on each side may change.

Human capital would also be associated with migrants’ psychic costs. For example, better host country language skills, which raise migrants’ earnings in the host country (Chiswick 1998; Dustmann and Van Soest 2002), help migrants to adapt to the new environment at the same time, probably leading to lower psychic costs.

### 2.5 Other determinants

The higher the costs of migration are, the longer time period in the host country is needed to achieve a positive net present value of migration (Chiswick 1999). A longer distance between the origin and destination, usually suggesting more expensive trips in time if not also in money, leads to fewer trips and a longer duration for each trip. In addition, the legal status of migrants is highly correlated with migration costs and economic opportunities in the host country, influencing the migration stay.

Other determinants of Mexican migrants’ US duration include migrants’ characteristics, exchange rates, economic conditions, and immigration policies. How migrants get a job abroad reflects their abilities and social connections. The exchange rates have both an income effect and substitution effect: if US Dollars are more valuable compared with Mexico Pesos, the income effect of higher exchange rates suggests migrants to return sooner, since the US Dollars they earn in the host country give more pesos in Mexico; however, the substitution effect attracts migrants to stay longer for more expensive currency. Furthermore, if the unemployment rates are high in the US but low in Mexico, migrants may return sooner because finding a job may be more difficult in the US than in Mexico. Regarding the US immigration policy, Immigration Reform and Control Act (IRCA), which was enacted in 1986 in the US, reformed the US immigration law. Requiring employers to attest to their employees’ migration status and making it illegal to knowingly hire or recruit illegal migrants, the IRCA affects migrants’ job opportunities in the US.