Price and Quantity Trajectories: Second-order Dynamics

TL;DR

This paper extends a second-order price adjustment model to include quantity changes, revealing energy-like activity, stability properties, and implications for market dynamics and consumer preference degeneracy.

Contribution

It introduces a combined price and quantity adjustment model with novel insights into stability, energy analogues, and market trajectory behavior.

Findings

Energy analogue called 'activity' arises naturally

Trajectories end at local minima or circulate endlessly

Disturbances in substitutable commodities decay over time

Abstract

In two previous papers the author developed a second-order price adjustment (t\^atonnement) process. This paper extends the approach to include both quantity and price adjustments. We demonstrate three results: a analogue to physical energy, called "activity" arises naturally in the model, and is not conserved in general; price and quantity trajectories must either end at a local minimum of a scalar potential or circulate endlessly; and disturbances into a subspace of substitutable commodities decay over time. From this we argue, although we do not prove, that the model features global stability, combined with local instability, a characteristic of many real markets. Following these observations and a brief survey of empirical results for price-setting and consumption behavior in markets for "real" goods (as opposed to financial markets), we conjecture that Stigler and Becker's well-known theory of consumer preference opens the possibility of substantial degeneracy in commodity space, and therefore that price and quantity trajectories could lie on a relatively low-dimensional subspace within the full commodity space.