Exploring near-linearities in price-rate of profit trajectories and the concept of effective rank in input-output matrices

In recent years, there has been a resurgence of interest in the controversies surrounding capital theory. At the heart of these debates are the empirically observed near-linearities in the price-rate of profit and wage rate of profit curves. This article posits that these near-linearities can be attributed to the low effective rank property inherent in the economy's system matrices of input-output coefficients. This suggests that a comprehensive representation of how prices evolve in response to changes in income distribution can be achieved with only a few eigenvalues and their respective eigenvectors. Furthermore, this low-dimensional system possesses the capability to capture the majority of distinctive features that characterize the input-output structure of the economy in relation to price movements.