Conditions for some balances of economic system: An input–output analysis using the spectral theory of nonnegative matrices

Twelve kinds of special semipositive matrices and their basic characters are presented. Employing these matrices and the previous results in Zeng (2008), we research the conditions for the balances between final output values and values-added, and between input multipliers and output multipliers in an economy. A necessary and sufficient condition that (i) there exists a unique vector of output adjustment coefficients such that (a) all sectoral final output values equal their respective sectoral values-added in the new output system, or (b) all sectoral input multipliers redefined by the new output system equal their respective sectoral output multipliers; or (ii) there exists a unique vector of price adjustment coefficients such that (a) all sectoral values-added equal their respective sectoral final output values in the new price system, or (b) all sectoral output multipliers redefined by the new price system equal their respective sectoral input multipliers; is the irreducibility of the matrix of intermediate output (or input) coefficients. A necessary and sufficient condition that (i) there is no vector of output adjustment coefficients which enables all sectoral final output values (or input multipliers) to equal their respective sectoral values-added (or output multipliers), or (ii) there is no vector of price adjustment coefficients which enables all sectoral values-added (or output multipliers) to equal their respective sectoral final output values (or input multipliers); is that the matrix of intermediate output (or input) coefficients has at least one non-final (or non-initial) class. These class relations and their equivalent conditions are summarized. The elaborate examples verify the main conclusions thoroughly.