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

Revision as of 20:05, 26 January 2020

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 $x_{n} (x_{n} / α) = (x_{n} - a) (x_{n} / α + \frac{N_{n}}{N_{p}} a)$ , 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 $A = N_{n} α < m a t h > (t h e t o t a l w e a l t h i n i t i a l l y o w n e d c o l l e c t i v e l y b y t h e n o b l e s, n o r m a l i z e d t o o n e p e a s a n t o w n i n g 1) a n d < m a t h > B = N_{p}$ (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 $% t = 2 - \frac{1}{% n}$ .

Revision as of 19:56, 26 January 2020 (view source) IssaRice (talk \| contribs) No edit summary ← Older edit		Revision as of 20:05, 26 January 2020 (view source) IssaRice (talk \| contribs) No edit summary Newer edit →
Line 10:		Line 10:

	This is <math display=inline>\log x_n + \log(x_n/\alpha) = \log(x_n - a) + \log(x_n/\alpha + \frac{N_n}{N_p}a)</math> which reduces to <math display=inline>x_n(x_n/\alpha) = (x_n - a) (x_n/\alpha + \frac{N_n}{N_p}a)</math>, which is quadratic in <math>a</math>. Using the quadratic formula and simplifying, the variable <math>x_n</math> actually drops out, and we get <math display=inline>a = 1 - \frac1\alpha\cdot\frac{N_p}{N_n}</math> as the non-zero solution (the other solution is always zero).		This is <math display=inline>\log x_n + \log(x_n/\alpha) = \log(x_n - a) + \log(x_n/\alpha + \frac{N_n}{N_p}a)</math> which reduces to <math display=inline>x_n(x_n/\alpha) = (x_n - a) (x_n/\alpha + \frac{N_n}{N_p}a)</math>, which is quadratic in <math>a</math>. Using the quadratic formula and simplifying, the variable <math>x_n</math> actually drops out, and we get <math display=inline>a = 1 - \frac1\alpha\cdot\frac{N_p}{N_n}</math> as the non-zero solution (the other solution is always zero).

			Now put <math>A = N_n\alpha<math> (the total wealth initially owned collectively by the nobles, normalized to one peasant owning 1) and <math>B = N_p</math> (the total wealth initially owned collectively by the peasants). Then the tax is <math display=inline>\%t = 1 - \frac BA</math>. We also know that the fraction of wealth the nobles initially control is <math display=inline>\%n = \frac{N_n x_n}{N_n x_n + N_p x_p} = \frac{A}{A+B}</math>. Can we find the optimal tax in terms of <math>\%n</math>? Yes. <math display=inline>1 - \%t = \frac BA = \frac1{\%n} - 1</math>, so <math display=inline>\%t = 2 - \frac{1}{\%n}</math>.