Quantal Response Statistical Equilibrium in Economic Interactions: Theory and Estimation

Abstract

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.

Information Theoretic Approaches in Economics

“…Second, the purposeful human behavior can be directly incorporated into the maximum entropy model as a constraint. For example, Foley and Scharfenaker's model of profit rate (Scharfenaker and Foley, ) introduces the pursuit of higher profit rates by capitalists in terms of the quantal response function as a constraint of the maximum entropy program. This approach has some advantages over other maximum entropy models in that it can simultaneously express the constraint on the purposeful human behavior and the other macroeconomic constraints in the same maximum entropy program.…”

“…We will discuss Scharfenaker and Foley () on the social interaction model of the profit rate as an example that shows how to utilize an economic theory as a constraint of the maximum entropy problem in the study of the profit rate. Their model is represented by an action variable of the entry–exit decision of firms in each subsector and the aggregate profit rate, x .…”

“…Unlike Alfarano et al . () and Scharfenaker and Semieniuk (), the constraints of the maximum entropy program of the profit rate in Scharfenaker and Foley () are formulated based on an economic theory, the Smithian competition theory, and the quantal response of the economic agent.…”

“…Therefore, we call this class of model SE models of individual decision makers . Two versions of this model is the mutual information‐constrained model under the name of RI (Sims, , ; Woodford, ; Luo and Young, , ; Yang, ) and the Shannon entropy constrained model under the name of quantal response model (Scharfenaker and Foley, ). In these models, individual decision makers are assumed to have a limited capacity to process market signals or they are exposed to a positive degree of uncertainty about the system.…”

“…Based on this basic framework, Scharfenaker and Foley () derive the quantal response function with a discrete action variable a and outcome variables x . This model, therefore, assumes that the economic agent has a positive degree of uncertainty of her action when she sees a particular state of aggregate economic variable.…”