Return migration can be considered part of a lifetime utility maximisation plan with a given budget (and liquidity) constraints (Borjas and Bratsberg, 1996). In the international migration literature, the return motives notably include location preferences with a higher marginal utility of consumption in the area of origin (Djajic and Milbourne, 1988), a higher purchasing power of the destination area currency at home (Djajic, 1989; Stark et al., 1997), and higher returns to human capital accumulated in the destination area at home (Dustmann, 2001; Dustmann et al., 2011). However, as highlighted by Dustmann (2003) and Djajic (2008), the decision to return and the optimal time of return can also be influenced by altruistic motives of parents towards their offspring in the household. Hence, the migration behaviour, and the decision to return, may be driven not only by individual life-cycle considerations but also by dynastic motives such as offspring’s welfare in the future13. Emphasising the family unit rather than the individual migrant makes sense in rural China, where family ties are strong and may be important components in explaining individual decisions. Moreover, such an approach seems the most relevant in a context where migration patterns are shaped by the household registration system (hukou), which does not entitle rural migrants to urban benefits and leaves most children behind. In their study of a sample of migrants living in Beijing, Fan et al. (2011) argue that the desire to be near left-behind children is an important reason for a migrant’s return.
The simple model presented below is meant to be illustrative of the conjectured influence of left-behind children on return migration. It builds on Dustmann (2003) and includes a number of alterations to account for specific Chinese features. First, we assume that the parent migrates alone and leaves behind her child. Second, because we are interested in school-aged or pre-school children in the home village, we also assume that the child does not work in the second period. Given these two assumptions, the proposed model captures the situation of a family unit composed of a worker engaged in migration (the parent migrant) and a left-behind child.
We consider two periods. In period 1, the parent works and lives in a city, while her child lives in the countryside and is subsidised by the parent. In period 2, the parent may decide to return or to stay in the city. The parent decides her own and her child’s consumption in periods 1 and 2. Because the child is not assumed to work in period 2, the altruism of the parent takes place through income transfer to the child in period 1 and through daily care (in case of return) or income transfer (in case of settlement in city) in period 2. As in Dustmann (2003), the return decision is taken by simply comparing lifetime welfare in the two locations.
The utility functions of the parent are supposed to take the usual logarithmic form. Period 1’s utility function U
1 is given by:
$$ {U}^1\left({c}^1,{k}^1\right)=1\mathrm{n}\left({c}^1\right)+\gamma 1\mathrm{n}\left({k}^1\right), $$
(1)
where c
1 is the consumption of the migrant parent, k
1 is the consumption of the left-behind child, and the parameter γ > 0 is the altruism weight.
Period 2’s utility function U
2j depends on the location choice of the migrant, whether settled in the city (j = M) or returned home (j = R), and is given by:
$$ {U}^{2j}\left({c}^{2j},{k}^{2j}\right)=1\mathrm{n}\left({c}^{2j}{a}^j\right)+\gamma 1\mathrm{n}\left({k}^{2j}{b}^j\right), $$
(2)
where a
j and b
j are preference parameters. In particular, a
R > a
M and b
R > b
M reflect a location preference of the migrant for her home village in terms of both her own consumption (a) and her offspring’s consumption (b).
Under the simplifying assumption of no discounting, the total utility function U of the parent can be simply expressed as follows:
$$ U=1\mathrm{n}\left({c}^1\right)+\gamma 1\mathrm{n}\left({k}^1\right)+\left(1-h\right)\left[1\mathrm{n}\left({c}^{2M}{a}^M\right)+\gamma 1\mathrm{n}\left({k}^{2M}{b}^M\right)\right]+h\left[1\mathrm{n}\left({c}^{2R}{a}^R\right)+\gamma 1\mathrm{n}\left({k}^{2R}{b}^R\right)\right], $$
(3)
where the parameter h stands for the return decision. At h = 1, the migrant decides to return; at h = 0, she settles in the city.
The budget constraint of the parent equalises intertemporal income and consumption:
$$ {c}^1+\left(1-h\right){c}^{2M}+h{c}^{2R}+{k}^1+\left(1-h\right){k}^{2M}+h{k}^{2R}={y}^1+\left(1-h\right){y}^{2M}+h{y}^{2R}, $$
(4)
where y
1, y
2M, and y
2R are the income of the parent in period 1, period 2 in the city, and period 2 at home, respectively.
The return decision of the migrant is given by the maximisation of her utility U with respect to her own consumption in periods 1 and 2 as well as to her left-behind child’s consumption in periods 1 and 2 under the budget constraint expressed above for two scenarios: settling in the city (h = 0) or returning to the countryside (h = 1). The intertemporal utility maximisation leads to the following results. The migrant parent will choose to return if:
$$ 2\left(1+\gamma \right)1\mathrm{n}\left(\frac{y^1+{y}^{2R}}{y^1+{y}^{2M}}\right)+1\mathrm{n}\left(\frac{a^R}{a^M}\right)+\gamma 1\mathrm{n}\left(\frac{b^R}{b^M}\right)>0. $$
(5)
As in Dustmann (2003), the first term illustrates the income impact of return on total utility: as earnings can be assumed to be lower at home (y
2R < y
2M), the decision to return will entail a loss in utility. The loss in utility is higher for altruistic parents (γ > 0) because their reduced earnings also affect the child’s outcomes. This may be the case, for instance, if the reduced earnings contribute to reduce opportunities for education or health care. This first term captures a standard money income effect. Moreover, if the migrant has no location preference (a
R = a
M and b
R = bM), her altruistic behaviour would reinforce the standard income effect towards a decision not to return. The second term shows the influence of the relative location preference of the migrant in terms of her own consumption. If a
R > a
M, her relative preference for her home village may partly compensate the income effect and logically reduce migration duration. The third term reflects the parent’s perception of the well-being of the left-behind child. If the child is perceived as suffering from parental absence in her daily life, then b
R > bM will give an incentive to the parent to return. This third term captures a time of the parent effect. Assuming no migrant parent location preference in her own consumption (a
R = a
M), the decision to return for an altruistic parent simply reduces to a comparison of the loss in utility due to lower income (and, possibly, a reduction in education opportunities) with the gain in utility thanks to a better-off child (through better quality day-to-day care, for instance).
In the vein of Dustmann (2003), this model illustrates the major trade-off a migrant parent faces when deciding to stay or to return: the consumption of the child is multidimensional and the various channels related to money and to parental time may work in opposite directions regarding the decision to return. In addition, the magnitude of each effect may vary with the age of the child: day-to-day care may be particularly valued for young children, while educational opportunities become important when the child is of school age. In a society with a strong tradition of preference for sons13, one may further expect that the return-decision outcome is also linked to the gender of the child, although the total child effect may remain ambiguous. Finally, in the Chinese context, an additional feature to consider is the high prevalence of multigenerational coresidence in rural areas and the potentially major role of coresident grandparents in their grandchildren’s lives (Zeng and Xie, 2014). Analysing rural data from the China Household Income Project (CHIP) survey 2002, Zeng and Xie (2014) show for instance that the effect of coresident grandparents’ education on grandchildren’s educational attainment is as large as that of parental education (while this is not the case for non-coresident grandparents). From a survey on migrants living in Beijing, Fan et al. (2011) find evidence that migrants with fewer parents in the home village are more likely to bring their children with them, and they argue that migrants prefer to leave their school-age children behind for easier access to education in the home village when their own parents are available to help. Hence, the presence of coresiding grandparents may attenuate the parental time effect through a (parents-to-grandparents) substitution effect for day-to-day care, and again, the magnitude of the effect may vary with the age of the child, and possibly with the gender of the migrant parent.
In summary, the return decision (h) of a migrant will depend on the expected income gap between the city and the hometown, the migrant’s preferences and altruism, her children’s characteristics (notably gender and age), and the availability of some parental substitutes (e.g., grandparents). The empirical analysis presented below aims to estimate this reduced-form relationship by focusing on the migrants’ length of stay in cities.