A new empirical contribution to an old theoretical puzzle: what input–output matrix properties tells us about equilibrium prices and quantities

Abstract

Eigenvalues of input-output matrices have significant implications for the structures of equilibrium prices and quantities. According to the Bródy Conjecture (BC), all subdominant eigenvalues of matrix would approach zero as matrix size approached infinity. Thus, any given initial quantity or price vector would converge to the corresponding equilibrium one in a single step. This paper adds significant empirical evidence to this theoretical discussion. We create a database of 307 different sizes matrices ranging over 30 years. Contrary to BC, we find that: the coefficient of variation and the subdominant eigenvalue moduli rise with matrix size; there’s a universal rank-size curve of eigenvalue moduli, but it is smooth and convex rather than L-shaped; the distribution of eigenvalue moduli is best fit by a Weibull probability distribution; the Weibull quantile function in turn yields a power law for eigenvalue moduli which is a better fit than a previously proposed exponential function.