User:IssaRice/Moral public goods example: Difference between revisions

Revision as of 20:09, 26 January 2020

working out the general optimal tax (i.e. highest tax a noble is willing to pay) 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). Each noble has no term in the utility function for the other nobles; as far as each noble is concerned, the other nobles are just bags of money that contribute to the tax.

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 $x_{n} (x_{n} / α) = (x_{n} - a) (x_{n} / α + \frac{N_{n}}{N_{p}} a)$ , which is quadratic in $a$ . Using the quadratic formula and simplifying, the variable $x_{n}$ actually drops out, and we get $a = 1 - \frac{1}{α} \cdot \frac{N_{p}}{N_{n}}$ as the non-zero solution (the other solution is always zero).

Now put $A = N_{n} α$ (the total wealth initially owned collectively by the nobles, normalized to one peasant owning 1) and $B = N_{p}$ (the total wealth initially owned collectively by the peasants). Then the tax is $% t = 1 - \frac{B}{A}$ . We also know that the fraction of wealth the nobles initially control is $% n = \frac{N_{n} x_{n}}{N_{n} x_{n} + N_{p} x_{p}} = \frac{A}{A + B}$ . Can we find the optimal tax in terms of $% n$ ? Yes. $1 - % t = \frac{B}{A} = \frac{1}{% n} - 1$ , so $% t = 2 - \frac{1}{% n}$ .

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-working out the general optimal tax for the example given in https://www.greaterwrong.com/posts/pqKwra9rRYYMvySHc/moral-public-goods
+working out the general optimal tax (i.e. highest tax a noble is willing to pay) for the example given in https://www.greaterwrong.com/posts/pqKwra9rRYYMvySHc/moral-public-goods
-each noble has utility function <math>u(x_n, x_p) = \log x_n + \log x_p</math>, where <math>x_n</math> is the noble's own wealth and <math>x_p</math> 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).
+each noble has utility function <math>u(x_n, x_p) = \log x_n + \log x_p</math>, where <math>x_n</math> is the noble's own wealth and <math>x_p</math> 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). Each noble has no term in the utility function for the other nobles; as far as each noble is concerned, the other nobles are just bags of money that contribute to the tax.
-a tax of amount <math>a</math> shifts the current utility amount <math>u(x_n,x_p)</math> to <math display=inline>u(x_n - a, x_p + \frac{N_n}{N_p}a)</math>, where <math>N_n</math> is the number of nobles and <math>N_p</math> is the number of peasants. This is because each noble loses <math>a</math>, so there is <math>N_na</math> to distribute, and this is divided by the number of peasants.
+A tax of amount <math>a</math> shifts the current utility amount <math>u(x_n,x_p)</math> to <math display=inline>u(x_n - a, x_p + \frac{N_n}{N_p}a)</math>, where <math>N_n</math> is the number of nobles and <math>N_p</math> is the number of peasants. This is because each noble loses <math>a</math>, so there is <math>N_na</math> to distribute, and this is divided by the number of peasants.
 If each noble starts out with <math>\alpha</math> times as much wealth as the average peasant, we have <math>x_n = \alpha x_p</math>.