Eigenvalue distribution, matrix size and the linearity of wage-profit curves

Brody (1997), while experimenting with random matrices, conjectured that the relative size of the second eigenvalue with respect to the first tended to fall as a random matrix got larger. Bidard and Schatteman (2001) proved that in a random matrix with independently and identically distributed entries the speed of convergence increases with the size of the matrix because the relative size of all subdominant eigenvalues tends to zero as the matrix size approaches infinity. Schefold (2010) then showed that zero subdominant eigenvalues imply linear wage-profit curves for any given numeraire. Our concern is with actual input-output matrices. We successively aggregate the US 2002 matrix from 403 to 10 industries and observe the distribution of the moduli of eigenvalues at each level of aggregation. The random matrix hypothesis predicts that both the size of ratio of the modulus of second eigenvalue to the first and the average size of all moduli will fall toward zero as matrix size increases. At an empirical level, we find that the eigenvalue ratio rises while the average size of eigenvalue moduli falls towards a positive constant. These findings do not support the applicability of Brody’s conjecture for real input-output tables, and by implication do not support the hypothesis that wage-profit curves will become strictly linear in the limit. It is still possible to reconcile our findings with empirically observed near-linear wage-profit curves.