# Probability distribution of income

I create an agent-based simulation of some economic phoenomenon, and I need to assign each citizen a random income level. I want to select the income level using a probability distribution that approximates the realistic distribution of income in a modern economy. What distribution should I use? Does it make sense to use exponential distribution?

NOTE: I am not looking for actual census data of a certain economy, but for a more general approximation.

• One big issue when using income in models is how to treat the unemployed. – Michael Bishop Nov 19 '12 at 17:07

Typically lognormal distributions or sometimes pareto distributions are used to model the distribution of income. Here you can find information how well these distrubtions fit real data for Germany, UK and the US:

Here is a proposal to use a generalized lognormal distribution

http://home.arcor.de/polecat/docs/kleiber@isi2003.pdf

• I agree, log-normal often seems to be a surprisingly good approximation for all sorts of economic variables. Definitely in this case it would be my first choice. Exponential istribution woudl rarely be a good fit. – Peter Ellis Jul 10 '12 at 10:20
• The real income distributions have (both) tails heavier than lognormal, or may have some intermediate bumps. But I think most income distribution economists (myself certainly included, dx.doi.org/10.1111/j.1467-9361.2005.00262.x) have committed a sin of using the lognormal as a very convenient model. – StasK Jul 10 '12 at 15:10
• +1. Welcome, Arne, and thank you for contributing a well-supported answer. – whuber Jul 10 '12 at 23:28
• Also, from here: en.wikipedia.org/wiki/Log-normal_distribution#Properties and here: en.wikipedia.org/wiki/Gini_coefficient it seems that the mu and sigma paramters of the lognormal can be estimated from the mean wage and the gini index, since: * mean = exp(mu + sigma^2/2) * gini = erf(sigma) – Erel Segal-Halevi Jul 12 '12 at 6:45
• @StasK: What about using a log-t distribution, to get both tails heavier than lognormal? – kjetil b halvorsen Aug 16 '15 at 12:31

If you want to include heavy tails while maintaining most of the remaining features of the lognormal, might I suggest the log-Cauchy or, if you need finite moments, the log-Student?

• Is there any evidence that such distributions are reasonable descriptions or approximations of actual income distributions? (Or at least better than a lognormal...) – whuber Jul 10 '12 at 21:28
• Not that I know of; I was just pointing out that there do exist distributions like lognormal but with the heavy tails StasK mentioned. I couldn't reply to his comment, as I don't have the rep on stats.SE yet. – Ghillie Dhu Jul 10 '12 at 22:29
• Well, then, +1 at least for the idea and to give you a start on that rep (you need 50 points I recall). BTW, welcome to our site! – whuber Jul 10 '12 at 23:26