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
.