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 $\alpha$ times as much wealth as the average peasant, we have $x_n=\alpha x_p$.

So to solve for the highest tax that the nobles are willing to pay, we want to find $a>0$ such that $u(x_n,x_n/\alpha )=u(x_n-a,x_n/\alpha +\frac{N_n}{N_p}a)$.

This is $\log x_n+\log (x_n/\alpha )=\log (x_n-a)+\log (x_n/\alpha +\frac{N_n}{N_p}a)$ which reduces to $x_n(x_n/\alpha )=(x_n-a)(x_n/\alpha +\frac{N_n}{N_p}a)$, which is quadratic in $a$. Using the quadratic formula and simplifying, the variable $x_n$ actually drops out, and we get $a=1-\frac{1}{\alpha }\cdot \frac{N_p}{N_n}$ as the non-zero solution (the other solution is always zero).