User:IssaRice/Moral public goods example

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working out the general optimal tax for the example given in https://www.greaterwrong.com/posts/pqKwra9rRYYMvySHc/moral-public-goods

each noble has utility function , where is the noble's own wealth and is the average wealth of a peasant (since the donation/tax is distributed equally among peasants, this is the same as saying the wealth of a single peasant).

a tax of amount shifts the current utility amount to , where is the number of nobles and is the number of peasants. This is because each noble loses , so there is to distribute, and this is divided by the number of peasants.

If each noble starts out with times as much wealth as the average peasant, we have .

So to solve for the highest tax that the nobles are willing to pay, we want to find such that .

This is which reduces to , which is quadratic in . Using the quadratic formula and simplifying, the variable actually drops out, and we get as the non-zero solution (the other solution is always zero).

Now put (the total wealth initially owned collectively by the nobles, normalized to one peasant owning 1) and (the total wealth initially owned collectively by the peasants). Then the tax is . We also know that the fraction of wealth the nobles initially control is . Can we find the optimal tax in terms of ? Yes. , so .