Hi guys! I want to construct an RBC model in which there are two regions and people can transfer to any region where they can get a higher expected utility levels. I know it will be very difficult to do so for the reason that agents in this model will not only decide traditional RBC variables, like consumption, investment and labor supply, but also decide whether to stay put or pull stakes, which is a discrete variable and I do not know how to get its first order condition. I also considered about some alternative models, like overlapping generation(OLG) model in which people live only two periods and the migration decision will be easier. How could I solve this problem? Thanks a lot!!!
P.S. I reconsidered the modeling and came up with an idea that there is a representative household which contains a family planner and a continuum family memebers with one inelastic labor supply in each region. Family planner decide how many members in his family live in local region and how many live in other region. By this approach, I can change the original discrete migration decision into a continuous variable by family planner who cares about the aggregate expected utiltiy level of the the whole family. Is this approach reasonable?