Gibrat’s law of proportionate growth is a well-established property of econometric time series,
from firm size to stock values. It suggests that these data can be modelled as multiplicative random
walks. However, such models lead to log-normal distributions, while longer tails and outliers are
observed empirically. Many solutions have been proposed to this paradox, often involving strong
assumptions on the micro-dynamics or time correlations in the data.
As a parsimonious alternative, empirical properties arise generically from multi-agent interactions: the minimal model is then a kinetic theory where agents undergo stochastic capital transfers.
If these transfers scale with both the sender and the receiver’s wealth so as to recover Gibrat’s
law, the zero-sum and irreversible nature of the dynamics generates a long tail, and even allows
macroscopic condensation into massive outliers. Such a kinetic theory shows excellent quantitative
agreement with data from the International Trade Network, Korean stock markets and U.S. firm
size distributions. We conjecture that this principle defines a universality class of models which are
all susceptible to generate the observed distributions, and that empirical differences between these
distributions reflect their “age”, that is, how far along they are in the condensation process.

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