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

Bródy (1997) found that in randomly generated input-output matrices the modulus of the second eigenvalue (relative to the first) falls, hence the convergence speed of an initial vector towards equilibrium increases with matrix size. As the matrix size approached infinity, the convergence approaches a one-step process. On the quantity side, this means that an initial output vector can be transformed into a balanced growth one in a single step. On the price side, this implies that an initial price vector can be transformed into competitive price vector in a single step – in which case Marx’s ‘transformation procedure’ would be exactly correct. Bidard and Schatteman (2001) proved that for random matrices the relative size of all subdominant eigenvalues tends to zero as matrix size approaches infinity. We construct 295 matrices by successive aggregation of the 2002 and 2007 US benchmark matrices, and find that the relative moduli of all subdominant eigenvalues rise with matrix size. Strikingly, the limiting rank-size distribution of the moduli of 2002, 2007 and five additional 1977-1997 benchmark tables approaches a unique power-law function consistent with a Weibull probability distribution of the eigenvalue moduli. These findings do not support Bródy’s conjecture, or the notion of linear limiting Sraffian wage-profit curves, or the Samuelson surrogate (pseudo-)production function.