Effects of changes in outputs and in prices on the economic system: an input–output analysis using the spectral theory of nonnegative matrices

Abstract

The (physical) output adjustment model and the price adjustment model are presented. By the two models we quantitatively analyze the influences of alterations of one sectoral (physical) gross output and of one sectoral price on another sectoral (physical) gross output and on another sectoral price, respectively. Hence, a basic nature of the Ghosh inverse and a fundamental character of the monetary Leontief inverse are obtained. The proposition that a matrix of intermediate output (input) coefficients alters if the vector of output (price) adjustment coefficients is nontrivial holds, if and only if this matrix is C-irreducible. It is impossible that (i) the adaptation of output system causes all sectoral final output rates (or input multipliers) either to rise or to fall collectively, or (ii) an adjustment of price system causes all sectoral value-added rates (or output multipliers) either to increase or to decrease jointly. However, maybe (i) a change of output system enables some sectoral final output rates (or input multipliers) to rise (fall) and all others to be constant, and (ii) an alteration of price system enables some sectoral value-added rates (or output multipliers) to increase (decrease) and all others to be fixed, whose necessary and sufficient condition is that the matrix of intermediate output (or input) coefficients has at least one non-final (or non-initial) class. The proposition that the vector of final output rates (or input multipliers) changes if the vector of output adjustment coefficients is nontrivial is true, if and only if the matrix of intermediate output coefficients has only one final class. The proposition that the vector of value-added rates (or output multipliers) alters if the vector of price adjustment coefficients is nontrivial holds, if and only if the matrix of intermediate input coefficients has only one initial class. The necessary and sufficient conditions and the matching economic explanations for possibility and uniqueness of the economic adjustment that enables (i) all sectors to have a uniform final output rate (or input multiplier), and (ii) all sectors to have the same value-added rate (or output multiplier) are respectively given.