## Optimal tax Rate

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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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I don't think this is a research question in economics- and certainly not in mathematics. It is also underspecified. What is the overall income distribution? What is the initial income distribution? And there is nothing dynamic here. – Michael Greinecker May 31 2011 at 19:19
We can assume the income distribution is gaussian, for example. For me it looks like a non-cooperative game with $N$ player, isn't? and the question is to find the Nash equilibrium, which is a question of "applied" mathematics, no?. But I am not a specialist of game theory, that why i ask to be enlightened. – Raphael May 31 2011 at 19:30
I cannot figure out what median income you're trying to maximize. Country A currently contains x people. Then the tax rates are set, migration happens, and now Country A contains y people. Are we trying to maximize the median income of the the original population in Country A? Or of the final population in Country A? Or of that fraction of the original population of Country A that remains in Country A? Or perhaps of the total population of the two countries? And does "income" mean income net of moving costs? These are the first of many things I find unclear. – Steven Landsburg May 31 2011 at 20:35
There is three steps: first the median income choose the tax rate second people migrate third they pay their tax and receive the redistributed income. That's my problem, because the median income try to maximize is income but this one depends on the migration which depends on his choice. The fact that the choice is made by the median income is a consequence of the median voter theorem: en.wikipedia.org/wiki/Median_voter_theorem – Raphael May 31 2011 at 21:32
@Raphael: you may get more answers at the sister site devoted to math finance quant.stackexchange.com – Andrey Rekalo May 31 2011 at 21:34
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## 2 Answers

A response to the OP's comment than to the original question. The income distribution is definitely NOT gaussian -- a lot of thought has been devoted to figuring out what exactly it is, which thought has led to the creation of the "fractal" view of the world. Check out

The Misbehavior of Markets: A Fractal View of Financial Turbulence [Paperback] Benoit Mandelbrot (Author), Richard L. Hudson

The pivotal point in the book is Mandelbrodt's talk at Harvard, where he was about to talk about the distribution of incomes, but saw the curves from his talk on his host's blackboard (they came from some questions on variation of commodity prices). The distribution is a power law of some sort.

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Thanks for your answer. I will see this talk of Mandelbrot. But in fact we can choose whatever distribution to start. – Raphael May 31 2011 at 21:23

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For person $i$ with income $I_i$ let $w_i$ be 1 or -1 depending on whether said person lives in $A$ or $B$. Then the condition that the person is living in the right place is a linear constraint on $w_i$ and the other $w$. To make this tractable we can allow $w_i$ to be in the interval $[-1,1]$, since the optium will still be at the endpoints (you can also interpret this is a probability that a person with tht income woul dbe in $A$ or $B$. This gives a linear system for equilibrium. Unfortunately when $M>0$ there will not be a single equilibrium (as an extreme, when $M$ is sufficiently large any state is an equilibrium). If $M=0$ you get a unique equilibrium (or if you somehow pick a canonical equilibrium) and you could try to do a numerical search on $T$ (ternary search?) Another way is to pick a target median income, form a linear constraint that the median is at least this target, and do a binary search to find the best feasible median.

A discussion of the applicability of this to economics is beyond the scope of this post :)

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