Pareto-Zipf, Gibrat’s Laws, Detailed-Balance and their Breakdown

By employing exhaustive lists of personal income and firms, we show that the upper-tail of the distribution of income and firm size has power-law (Pareto-Zipf law), and that in this region their growth rate is independent of the initial value of income or size (Gibrat’s law of proportionate effect). In addition, detailed balance holds in the power-law region; the empirical probability for an individual (a firm) to change its income (size) from a value to another is statistically the same as that for its reverse process in the ensemble. We prove that Pareto-Zipf law follows from Gibrat’s law under the condition of detailed balance. We also show that the distribution of growth rate possesses a non-trivial relation between the positive and negative sides of the distribution, through the value of Pareto index, as is confirmed empirically. Furthermore, we also show that these properties break down in the non power-law region of distribution, and can possibly do so temporally according to drastic change in financial or real economy.