The invisible hand of Laplace: The role of market structure in price convergence and oscillation

The “invisible hand” of the free market is remarkably effective at producing near-equilibrium prices. This is difficult to quantify, however, in the absence of an agreed model for out-of-equilibrium trade. Short of a fully reductionist model, a useful substitute would be a scaling law relating equilibration time and other market parameters. Even this, however, is missing in the literature. We make progress in this direction. We examine a class of Arrow–Debreu markets with price signaling driven by continuous-time proportional-tâtonnement. We show that the connectivity among the participants in the market determines quite accurately a scaling law for convergence time of the market to equilibrium, and thus determines the effectiveness of the price signaling. To our knowledge this is the first characterization of price stability in terms of market connectivity. At a technical level, we show how convergence in our class of markets is determined by a market-dependent Laplacian matrix. If a market is not isolated but, rather, subject to external noise, equilibrium theory has predictive value only to the extent to which that noise is counterbalanced by the price equilibration process. Our model quantifies this predictive value by providing a scaling law that relates the connectivity of the market with the variance of its prices.