Anmol on Income and happiness
If you go to school with me, you already know what this article is about, due to our heated discussion in Eco class about it. However, if you weren’t present, this is solely about the correlation between Income and happiness, and the optimal income distribution for net happiness. Also, this article discusses the best income distribution in a society, and what government should do to achieve that distribution. Note that this assumes ceterus paribus: all other things being equal, it also assumes that income and wealth are synonyms, and interchangeable, feel free to make the formula more complex later on.
To understand the link between income and happiness, we must first accept three things. Firstly, that income isn’t the only factor, but a major factor affecting happiness. Secondly, that income has a positive correlation with happiness. Thirdly, that the relationship between income and happiness is less than linear, my assets total less than $10,000, while Warren Buffet has some $50,000,000,000, but he isn’t 5 million times happier than I am. I personally believe that there is no maximum happiness, although many people might dispute this claim.
I think the relationship between income and happiness is logarithmic, i.e. H=k*log I.
It seems to me that someone with 100 times my wealth will be as much happier than I am, as someone 100 times wealthier than him, will be happier than he is. This system seems plausible to me, though we will deal with more complex equations later. With the current formula, let’s assume that there are 2 people in the world (this can be extrapolated to much larger numbers). Their incomes are P and O, and P+O=I. Net happiness= log P+log O, Net happiness= log PO, therefore the point of maximization of PO, is the point of maximization of happiness. This is when P=O, i.e. when income are equal. Equality of income is the point at which happiness is maximized, in this situation.
Assuming you have a function for income vs happiness, to determine that the optimal distribution is when incomes are equal, we must make sure that when incomes are equal, i.e. if one person 1$, and the other person loses 1$, net happiness goes down. Change in happiness for each person ~ gradient of the curve (dy/dx). If gradient above is less than gradient below, in the happiness versus income curve, then it’s best to split wealth evenly. Therefore, the gradient should decrease as the value of x (in this case income), increases, or d^2y/dx^2 should be negative. For all less than linear equations, d^2y/dx^2 is negative, because dy/dx of x^-n, where n is a positive constant, will always be negative, and dy/dx of x^a, where a is between 0 and 1, will always result in a polynomial of the form kx^-n. Therefore, when a is between 0 and 1, when the equation is less than linear, it’s best to concentrate wealth towards the centre. When d^2y/dx^2 is 0, the distribution of wealth is irrelevant to net happiness, and when it is positive, it’s best to split the wealth unevenly.
It would be extremely unusual if net happiness could be perfectly graphed by a function. Surely, doubling one’s wealth while poor is worth significantly more than doubling it while rich. Assuming statements such as the one are true, the gradient of the curve still continues to decrease, so d^2y/dx^2 remains less than 0, and it remains better to split the wealth evenly. The gradient can only reduce, as the curve continues. Therefore, we can conclude that net happiness is maximized when income is equally distributed.
I’m sure the above statement makes me seem like a socialist. In fact, I strongly believe in communism capitalism. I wanted to make it clear that the objective of a tax is not to blindly maximize net happiness. All of us know that socialism doesn’t work, because it disinsentivizes innovation, the engine for economic growth. However, lack of economic growth is only felt in the long run, while the positive attributes of redistribution are felt immediately, which is one of the reasons why socialist revolutions started off so well, and ended, well, not so well.
The first implication of this article is that progressive taxes are fair, although the extent to which they’re in place should be debated. I believe that in the long run, an increase in net wealth, leads to a total increase in societal wealth, for all classes, regardless of how the wealth is distributed initially. Many of the richest families 2,000 years ago, are normal families now, and many of the richest families now were normal 2,000 years ago, but we’ve all benefited from long run economic growth, despite the clear economic biases towards the rich in that era (flat taxes, feudalism, selective voting rights, etc.). However, you could argue that government shouldn’t care about the long long run, which makes sense if you believe that government is there to get re-elected, or serve it’s own purpose in some other way, in which case, changes which take more than 10 years are generally ignored regardless of their benefits. The first question which needs to be answered is “how much should a government base their decisions on long term good, and how far in the long term should they look?”
The next question which needs to be answered is “how progressive should progressive taxes be?”, any measure which can be used to derive, or calculate a perfect tax plan will be helpful.
The first implication of this concept was macroeconomic. The second, has to do with microeconomics. This is a good way to measure whether or not to take certain bets. The first thing this shows is that betting on a lottery, even with no rake, is completely irrational, because losing that 1 dollar, or other small amount of money, isn’t worth a one in a million chance at a million dollars, because the satisfaction of a million dollars isn’t a million times the satisfaction of 1 dollar. The second is that when bets are a very small percentage of your net worth, what probability dictates is generally your correct decision, however, when it’s a large percentage of your wealth, it differs largely from probability. For example, if you bet half your wealth, you need at least 1.71:1. The answer is probably more than that, so the curve is probably less than a logarithmic curve. In a survey, people were asked the minimum odds they needed on a coinflip to risk $100, they said 2.1:1. This would’ve been correct if they were betting half their wealth, but the correct answer (assuming $100,000 of total wealth), was 1.001:1. Another case of people misunderstanding statistics. If we manage to link happiness to income effectively, we can understand whether or not to take bets which are a certain percentage of our wealth, or other financial risks, such as betting on the stock market. Such a formula would be very valuable for academic purposes, although it would probably be ignored in real life.
If you go to school with me, you already know what this article is about, due to our heated discussion in Eco class about it. However, if you weren’t present, this is solely about the correlation between Income and happiness, and the optimal income distribution for net happiness. Also, this article discusses the best income distribution in a society, and what government should do to achieve that distribution. Note that this assumes ceterus paribus: all other things being equal, it also assumes that income and wealth are synonyms, and interchangeable, feel free to make the formula more complex later on.
To understand the link between income and happiness, we must first accept three things. Firstly, that income isn’t the only factor, but a major factor affecting happiness. Secondly, that income has a positive correlation with happiness. Thirdly, that the relationship between income and happiness is less than linear, my assets total less than $10,000, while Warren Buffet has some $50,000,000,000, but he isn’t 5 million times happier than I am. I personally believe that there is no maximum happiness, although many people might dispute this claim.
I think the relationship between income and happiness is logarithmic, i.e. H=k*log I.
It seems to me that someone with 100 times my wealth will be as much happier than I am, as someone 100 times wealthier than him, will be happier than he is. This system seems plausible to me, though we will deal with more complex equations later. With the current formula, let’s assume that there are 2 people in the world (this can be extrapolated to much larger numbers). Their incomes are P and O, and P+O=I. Net happiness= log P+log O, Net happiness= log PO, therefore the point of maximization of PO, is the point of maximization of happiness. This is when P=O, i.e. when income are equal. Equality of income is the point at which happiness is maximized, in this situation.
Assuming you have a function for income vs happiness, to determine that the optimal distribution is when incomes are equal, we must make sure that when incomes are equal, i.e. if one person 1$, and the other person loses 1$, net happiness goes down. Change in happiness for each person ~ gradient of the curve (dy/dx). If gradient above is less than gradient below, in the happiness versus income curve, then it’s best to split wealth evenly. Therefore, the gradient should decrease as the value of x (in this case income), increases, or d^2y/dx^2 should be negative. For all less than linear equations, d^2y/dx^2 is negative, because dy/dx of x^-n, where n is a positive constant, will always be negative, and dy/dx of x^a, where a is between 0 and 1, will always result in a polynomial of the form kx^-n. Therefore, when a is between 0 and 1, when the equation is less than linear, it’s best to concentrate wealth towards the centre. When d^2y/dx^2 is 0, the distribution of wealth is irrelevant to net happiness, and when it is positive, it’s best to split the wealth unevenly.
It would be extremely unusual if net happiness could be perfectly graphed by a function. Surely, doubling one’s wealth while poor is worth significantly more than doubling it while rich. Assuming statements such as the one are true, the gradient of the curve still continues to decrease, so d^2y/dx^2 remains less than 0, and it remains better to split the wealth evenly. The gradient can only reduce, as the curve continues. Therefore, we can conclude that net happiness is maximized when income is equally distributed.
I’m sure the above statement makes me seem like a socialist. In fact, I strongly believe in communism capitalism. I wanted to make it clear that the objective of a tax is not to blindly maximize net happiness. All of us know that socialism doesn’t work, because it disinsentivizes innovation, the engine for economic growth. However, lack of economic growth is only felt in the long run, while the positive attributes of redistribution are felt immediately, which is one of the reasons why socialist revolutions started off so well, and ended, well, not so well.
The first implication of this article is that progressive taxes are fair, although the extent to which they’re in place should be debated. I believe that in the long run, an increase in net wealth, leads to a total increase in societal wealth, for all classes, regardless of how the wealth is distributed initially. Many of the richest families 2,000 years ago, are normal families now, and many of the richest families now were normal 2,000 years ago, but we’ve all benefited from long run economic growth, despite the clear economic biases towards the rich in that era (flat taxes, feudalism, selective voting rights, etc.). However, you could argue that government shouldn’t care about the long long run, which makes sense if you believe that government is there to get re-elected, or serve it’s own purpose in some other way, in which case, changes which take more than 10 years are generally ignored regardless of their benefits. The first question which needs to be answered is “how much should a government base their decisions on long term good, and how far in the long term should they look?”
The next question which needs to be answered is “how progressive should progressive taxes be?”, any measure which can be used to derive, or calculate a perfect tax plan will be helpful.
The first implication of this concept was macroeconomic. The second, has to do with microeconomics. This is a good way to measure whether or not to take certain bets. The first thing this shows is that betting on a lottery, even with no rake, is completely irrational, because losing that 1 dollar, or other small amount of money, isn’t worth a one in a million chance at a million dollars, because the satisfaction of a million dollars isn’t a million times the satisfaction of 1 dollar. The second is that when bets are a very small percentage of your net worth, what probability dictates is generally your correct decision, however, when it’s a large percentage of your wealth, it differs largely from probability. For example, if you bet half your wealth, you need at least 1.71:1. The answer is probably more than that, so the curve is probably less than a logarithmic curve. In a survey, people were asked the minimum odds they needed on a coinflip to risk $100, they said 2.1:1. This would’ve been correct if they were betting half their wealth, but the correct answer (assuming $100,000 of total wealth), was 1.001:1. Another case of people misunderstanding statistics. If we manage to link happiness to income effectively, we can understand whether or not to take bets which are a certain percentage of our wealth, or other financial risks, such as betting on the stock market. Such a formula would be very valuable for academic purposes, although it would probably be ignored in real life.
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