User:IssaRice/Moral public goods example

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 $u (x_{n}, x_{p}) = \log x_{n} + \log x_{p}$ , where $x_{n}$ is the noble's own wealth and $x_{p}$ 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 $a$ shifts the current utility amount $u (x_{n}, x_{p})$ to $u (x_{n} - a, x_{p} + \frac{N_{n}}{N_{p}} a)$ , where $N_{n}$ is the number of nobles and $N_{p}$ is the number of peasants. This is because each noble loses $a$ , so there is $N_{n} a$ to distribute, and this is divided by the number of peasants.

If each noble starts out with $α$ times as much wealth as the average peasant, we have $x_{n} = α x_{p}$ .

So to solve for the highest tax that the nobles are willing to pay, we want to find $a > 0$ such that $u (x_{n}, x_{n} / α) = u (x_{n} - a, x_{n} / α + \frac{N_{n}}{N_{p}} a)$ .

This is $\log x_{n} + \log (x_{n} / α) = \log (x_{n} - a) + \log (x_{n} / α + \frac{N_{n}}{N_{p}} a)$ which reduces to the following quadratic in $a$ : $x_{n} (x_{n} / α) = (x_{n} - a) (x_{n} / α + \frac{N_{n}}{N_{p}} a)$ .