We prove that Pareto theory of circulation of elites results from our wealth evolution model, Kelly criterion for optimal betting and Keynes' observation of "animal spirits" that drive the economy and cause that human financial decisions are prone to excess risk-taking.

We determine Kelly criterion for a game with variable pay-off. The Kelly fraction satisfies a fundamental integral equation and is smaller than the classical Kelly fraction for the same game with the constant average pay-off.

We propose a simple dynamical model of wealth evolution. The invariant distributions are of Pareto type and are dynamically stable as conjectured by Pareto.