The Double Power Law in Income Distribution: Explanations and Evidence

At Yale, third year PhD students were required to write an empirical paper. When I was third year in 2010, my initial intention was to see whether income is exponentially distributed, which is an implication of my earlier paper Toda (2010). Upon inspecting CPS data, I found that the income distribution has both upper and lower Pareto tails. I thought about the rationalization and noticed that the stochastic differential equation used in Alfarano et al. (2012) (which was then a 2008 working paper) can rationalize it, so I applied this mechanism. My interest in power laws started around this time.

Appendix D contains a derivation of the closed-form solution to the Fokker-Planck equation corresponding to the stochastic differential equation. In 2009, I took a continuous-time finance class from Jonathan Ingersoll. When he explained the Black-Scholes model, he mentioned something like “if you ever solve a partial differential equation in closed-form, it has already been solved by somebody else”. Therefore when I solved the PDE in closed-form, I was quite proud of myself. However, Jonathan turned out to be correct: the solution I obtained was already known in physics (Chvosta & Reineker, 2003)!

An abridged version of this paper was published as Toda (2011).