One simple way to model a return migration is to assume that migrants have a preference for consumption in their home country. In such a setting, migrants emigrate, because that increases their lifetime wealth (and, therefore, their lifetime consumption). At the same time, consumption abroad creates less pleasure than consumption at home. Under plausible assumptions, it is straightforward to show within this model that benefits of migration decrease over the migration cycle, while costs are positive, and may increase. This may lead eventually to a return migration. Below we provide a more formal discussion.
No migration, and permanent migration are special cases of this model. For a migration to take place, benefits must initially be higher than costs. Accordingly, if, despite a large wage differential, preferences for consumption in the home country (relative to the host country) are strong, no migration will take place. Permanent migration occurs if at the end of the migrant’s lifetime, benefits are still higher than costs. Again, this may depend on the preference of the immigrant for home country consumption.