VARIATIONS ON THE THEME OF CONNINGa IN MATHEMATICAL ECONOMICS

Abstract The mathematization of economics is almost exclusively in terms of the mathematics of real analysis which, in turn, is founded on set theory (and the axiom of choice) and orthodox mathematical logic. In this paper I try to point out that this kind of mathematization is replete with economic infelicities. The attempt to extract these infelicities is in terms of three main examples and one general discussion: dynamics, policy and rational expectations and learning are the examples; a game theory without ‘subjectivism’, based on the axiom of determinateness, is discussed in general terms. The focus is on the role and reliance of standard fixed‐point theorems in orthodox mathematical economics.     

A statistical equilibrium model of competitive firms

We find that the empirical density of firm profit rates, measured as returns on assets, is markedly non-Gaussian and reasonably well described by an exponential power (or Subbotin) distribution. We start from a statistical equilibrium model that leads to a stationary Subbotin density in the presence of complex interactions among competitive heterogeneous firms. To investigate the dynamics of firm profitability, we construct a diffusion process that has the Subbotin distribution as its stationary probability density. This leads to a phenomenologically inspired interpretation of variations in the shape parameter of the Subbotin distribution, which essentially measures the competitive pressure in and across industries. Our findings have profound implications both for the previous literature on the ‘persistence of profits’ as well as for understanding competition as a dynamic process. Our main formal finding is that firms' idiosyncratic efforts and the tendency for competition to equalize profit rates are two sides of the same coin, and that a ratio of these two effects ultimately determines the dispersion of the equilibrium distribution.     

HETEROGENEITY IN ORDERED CHOICE MODELS: A REVIEW WITH APPLICATIONS TO SELF‐ASSESSED HEALTH

Discrete variables that have an inherent sense of ordering across outcomes are commonly found in large data sets available to many economists, and are often the focus of research. However, assumptions underlying the standard ordered probit (which is usually used to analyse such variables) are not always justified by the data. This study provides a review of the ways in which the ordered probit might be extended to account for additional heterogeneity. Differing from other reviews in scope, application and relevance in economic settings, a series of issues pertaining to choices of variables, and the economic assumptions underlying each model are discussed in the context of measuring the underlying health of respondents. The models are applied to a wave of the household, income and labour dynamics in Australia survey, in order to check the appropriateness of such assumptions in an applied context.     

Akaike-type criteria and the reliability of inference: Model selection versus statistical model specification

Since the 1990s, the Akaike Information Criterion (AIC) and its various modifications/extensions, including BIC, have found wide applicability in econometrics as objective procedures that can be used to select parsimonious statistical models. The aim of this paper is to argue that these model selection procedures invariably give rise to unreliable inferences, primarily because their choice within a prespecified family of models (a) assumes away the problem of model validation, and (b) ignores the relevant error probabilities. This paper argues for a return to the original statistical model specification problem, as envisaged by Fisher (1922), where the task is understood as one of selecting a statistical model in such a way as to render the particular data a truly typical realization of the stochastic process specified by the model in question. The key to addressing this problem is to replace trading goodness-of-fit against parsimony with statistical adequacy as the sole criterion for when a fitted model accounts for the regularities in the data.