Assume you have two countries A and B, with a tax rates $T_A$ and $T_B$. The tax is redistributed to each people equally. Hence if you live in A and you make $I$ as income then you will finally receive

$$I*(1-T_A) + \overline{I}*T_A$$

where $\overline{I}$ is the average income in $A$.The country A wants to choose an optimal rate, in order to do it the decision is taken by the median income. But the people can migrate if the new rate makes them poorer than if they were living in $B$. Of course this migration to B as a cost $M$, hence if the median income choose as new rate $T$ the people in A such that

$$ I(1-T) + \overline{I}\ T < I\ (1-T_B) + \overline{I}\ T_B -M $$

will leave A to B. And symmetrically the people in B such that

$$I\ (1-T_B) + \overline{I}\ T_B < I\ (1-T) + \overline{I}\ T -M $$

will leave B to A. Which changes the configuration of incomes in A and hence the decision of the median income since his income depends on the average income.

My question is how can find the taxe rate which will optimize the income of the median income after migration?

I have think to a dynamical approach, but it looks hard to show that we converge to an equilibrium. Is there is general tools for this kind of problem?

I hope, i have been clear enough.

P.S: I have already ask this question on Math.stackexchange, but i think it is in fact a research problem since i have find nothing in the literature except a a paper of Stéphane Rossignol and Emmanuelle Taugourdeau :Asymmetric social protection systems with migration in J Popul Econ 19:481–505 (2006). But they study an asymmetric case.

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