Social science addresses systems in which the individual actions of participants interacting in complex, non-additive ways through institutional structures determine social outcomes. In many cases, the institutions incorporate enough negative feedback to stabilize the resulting outcome as an equilibrium. We study a particular type of such equilibria, quantal response statistical equilibrium (QRSE) using the tools of constrained maximum entropy modeling developed by E. T. Jaynes. We use Adam Smith's theory of profit rate maximization through competition of freely mobile capitals as an example. Even in many cases where key model variables are unobserved, it is possible to infer the parameters characterizing the equilibrium through Bayesian methods. We apply this method to the Smithian theory of competition using data where firms' profit rates are observed but the entry and exit decisions that determine the distribution of profit rates is unobserved, and confirm Smith's prediction of the emergence of an average rate of profit, along with a characterization of equilibrium statistical fluctuations of individual rates of profit.

Motivated by classical political economy we detail a probabilistic, statistical equilibrium approach to explaining why even in equilibrium, the equalization of profit rates leads to a non-degenerate distribution. Based on this approach we investigate the empirical content of the profit rate distribution for previously unexamined annual firm level data comprising over 24,000 publicly listed North American firms for the period 1962-2014. We find strong evidence for a structural organization and equalization of profit rates on a relatively short time scale both at the economy wide and one-and two-digit SIC industry levels into a Laplace or double exponential distribution. We show that the statistical equilibrium approach is consistent with economic theorizing about profit rates and discuss research questions emerging from this novel look at profit rate distributions. We also highlight the applicability of the underlying principle of maximum entropy for inference in a wide range of economic topics. . We are grateful to two anonymous referees for their helpful feedback on an earlier version.