User:IssaRice/Moral public goods example - Revision history

IssaRice at 20:19, 26 January 2020

2020-01-26T20:19:02Z

← Older revision		Revision as of 20:19, 26 January 2020
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	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>a = 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 - a = \frac BA = \frac1{\%n} - 1</math>, so <math display=inline>a = 2 - \frac{1}{\%n}</math>.		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, using same normalization). Then the tax is <math display=inline>a = 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 - a = \frac BA = \frac1{\%n} - 1</math>, so <math display=inline>a = 2 - \frac{1}{\%n}</math>.

IssaRice at 20:13, 26 January 2020

2020-01-26T20:13:09Z

← Older revision		Revision as of 20:13, 26 January 2020
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	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>.		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>a = 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 - a = \frac BA = \frac1{\%n} - 1</math>, so <math display=inline>a = 2 - \frac{1}{\%n}</math>.

IssaRice at 20:10, 26 January 2020

2020-01-26T20:10:14Z

← Older revision		Revision as of 20:10, 26 January 2020
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	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		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 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.		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.

IssaRice at 20:09, 26 January 2020

2020-01-26T20:09:45Z

← Older revision		Revision as of 20:09, 26 January 2020
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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>.		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>.

IssaRice at 20:05, 26 January 2020

2020-01-26T20:05:42Z

← Older revision		Revision as of 20:05, 26 January 2020
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	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>.		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>.

IssaRice at 20:05, 26 January 2020

2020-01-26T20:05:23Z

← Older revision		Revision as of 20:05, 26 January 2020
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	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>.

IssaRice at 19:56, 26 January 2020

2020-01-26T19:56:10Z

← Older revision		Revision as of 19:56, 26 January 2020
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	So to solve for the highest tax that the nobles are willing to pay, we want to find <math>a > 0</math> such that <math display=inline>u(x_n, x_n/\alpha) = u(x_n - a, x_n/\alpha + \frac{N_n}{N_p}a)</math>.		So to solve for the highest tax that the nobles are willing to pay, we want to find <math>a > 0</math> such that <math display=inline>u(x_n, x_n/\alpha) = u(x_n - a, x_n/\alpha + \frac{N_n}{N_p}a)</math>.

	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 ~~the following quadratic in <math>a</math>:~~ <math display=inline>x_n(x_n/\alpha) = (x_n - a) (x_n/\alpha + \frac{N_n}{N_p}a)</math>.		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).

IssaRice at 19:53, 26 January 2020

2020-01-26T19:53:35Z

← Older revision		Revision as of 19:53, 26 January 2020
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	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>.		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>.

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

	This is <math display=inline>\log x_n + \log(~~x_p~~/\alpha) = \log(x_n - a) + \log(x_n/\alpha + \frac{N_n}{N_p}a)</math> which reduces to the following quadratic in <math>a</math>: <math display=inline>x_n(~~x_p~~/\alpha) = (x_n - a) (x_n/\alpha + \frac{N_n}{N_p}a)</math>.		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 the following quadratic in <math>a</math>: <math display=inline>x_n(x_n/\alpha) = (x_n - a) (x_n/\alpha + \frac{N_n}{N_p}a)</math>.

IssaRice at 19:52, 26 January 2020

2020-01-26T19:52:38Z

← Older revision		Revision as of 19:52, 26 January 2020
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	So to solve for the highest tax that the nobles are willing to pay, we want to find <math>a > 0</math> such that <math display=inline>u(x_n, x_p/\alpha) = u(x_n - a, x_n/\alpha + \frac{N_n}{N_p}a)</math>.		So to solve for the highest tax that the nobles are willing to pay, we want to find <math>a > 0</math> such that <math display=inline>u(x_n, x_p/\alpha) = u(x_n - a, x_n/\alpha + \frac{N_n}{N_p}a)</math>.

			This is <math display=inline>\log x_n + \log(x_p/\alpha) = \log(x_n - a) + \log(x_n/\alpha + \frac{N_n}{N_p}a)</math> which reduces to the following quadratic in <math>a</math>: <math display=inline>x_n(x_p/\alpha) = (x_n - a) (x_n/\alpha + \frac{N_n}{N_p}a)</math>.

IssaRice: Created page with "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 $u(x..."$

2020-01-26T19:50:33Z

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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 <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).

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>.

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

User:IssaRice/Moral public goods example - Revision history

IssaRice at 20:19, 26 January 2020

IssaRice at 20:13, 26 January 2020

IssaRice at 20:10, 26 January 2020

IssaRice at 20:09, 26 January 2020

IssaRice at 20:05, 26 January 2020

IssaRice at 20:05, 26 January 2020

IssaRice at 19:56, 26 January 2020

IssaRice at 19:53, 26 January 2020

IssaRice at 19:52, 26 January 2020

IssaRice: Created page with "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..."

IssaRice: Created page with "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 $u(x..."$