# Nonlinearity in your wallet

Many human traits like height, IQ, and 50 metre dash times are very close to being normally distributed. The normal distribution (more technically the normal probability density function) or Gaussian function

$f(x) = \frac{1}{\sqrt{2\pi}\sigma}e^{-(x-\mu)^2/2\sigma^2}$

is the famous bell shaped curve that the histogram of class grades fall on. The shape of the Gaussian is specified by two parameters the mean $\mu$, which coincides with the peak of the bell, and the standard deviation $\sigma$, which is a measure of how wide the Gaussian is. Let’s take height as an example. There is a 68% chance that any person will be within one standard deviation of the mean and a little more than 95% that you will be within two standard deviations. The tallest one percent are about 2.3 standard deviations from the mean.

The fact that lots of things are normally distributed  is not an accident but a consequence of the central limit theorem (CLT), which may be the most important mathematical law in your life. The theorem says that the probability distribution of a sum of a large number of random things will be normal (i.e. a Gaussian). In the example of height, it suggests that there are perhaps hundreds or thousands of genetic and environmental factors that determine your height, each contributing a little amount. When you add them together you get your height and the distribution is normal.

Now, the one major thing in your life that bucks the normal trend is income and especially wealth distribution. Incomes are extremely non-normal. They have what are called fat tails, meaning that the income of the top earners are much higher than would be expected by a normal distribution. A general rule of thumb called the Pareto Principle is that 20% of the population controls 80% of the wealth. It may even be more skewed these days.

There are many theories as to why income and wealth is distributed the way it is and I won’t go into any of these. What I want to point out is that whatever it is that governs income and wealth, it is definitely nonlinear. The key ingredient in the CLT is that the factors add linearly. If there were some nonlinear combination of the variables then the result need not be normal. It has been argued that some amount of inequality is unavoidable given that we are born with unequal innate traits but the translation of those differences into  income inequality is a social choice to some degree. If we rewarded the contributors to income more linearly, then incomes would be distributed more normally (there would be some inherent skew because incomes must be positive). In some sense, the fact that some sectors of the economy seem to have much higher incomes than other sectors implies a market failure.

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## 3 thoughts on “Nonlinearity in your wallet”

1. Andy says:

Income looks pretty close to lognormal to me. See figure 1 here: http://fmwww.bc.edu/ec-p/wp671.pdf. With maybe a power-law tail. Couldn’t this just mean that the individual factors which contribute to income (conscientiousness, drive, fluid intelligence, etc) are normally distributed and that they act multiplicatively? It’s worth pointing out that others think that achievement is also log-normally distributed: http://www.johndcook.com/blog/2009/09/29/achievement-is-log-normal/.

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2. Lognormal is already a fat tailed distribution and the power law tail makes it even fatter. Multiplicative is nonlinear.

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3. […] Scientific Clearing House Carson C. Chow « Nonlinearity in your wallet […]

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