# Optimal tax rate

Suppose you have two countries A and B, with a tax rate $T_A$ and $T_B$, respectively. The tax is redistributed to all 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 this the decision is taken at 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 comes at 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 for 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 we find the tax rate which will optimize the income of the median income after migration?

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

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It seems to me that the optimum value of $T$ will depend on the distribution of incomes, $p(I)$. Do you have a reason to think that it won't? – Chris Taylor May 29 '11 at 19:53
Of course, since if all the income are the same, every taxe rat are optimal. Hence my question is : a distribution of incomes given, how to compute the optimal tax rate? – Paul May 30 '11 at 5:32
I suppose that $T_B$ is given as well, right? Otherwise this would seem like a game theory problem where we are looking for the Nash equilibrium values of the tax rates $T_A$ and $T_B$. (It's been a while since you posted this. Any progress?) – jmbejara Nov 4 '13 at 21:14