Two kinds of Pareto improvements of the economic system: An input–output analysis using the nonnegative matrices theory

• Define the Pareto improvement of final output rate and input multiplier.
• Define the Pareto improvement of value-added rate and output multiplier.
• Give a sufficient condition for Pareto improvement of final output rate and input multiplier.
• Give a sufficient condition for Pareto improvement of value-added rate and output multiplier.
• Define the partitional row and column indicator sets for a reducible nonnegative matrix.

This paper clearly reveals the economic meanings of final output rate, input multiplier, value-added rate, and output multiplier. Applying the output adjustment model and the price adjustment model as well as nonnegative matrices theory we find that if the matrix of intermediate output (or input) coefficients has at least one non-final (or non-initial) class then (i) an adjustment of output system enables the final output rates of whole or part sectors corresponding to the non-final classes to rise, the input multipliers of whole or part sectors corresponding to the non-final classes to decrease, and all other sectoral final output rates and input multipliers to be fixed, which is one kind of Pareto improvement of the economic system; (ii) the adaptation of price system enables the value-added rates of whole or part sectors corresponding to the non-initial classes to rise, the output multipliers of whole or part sectors corresponding to the non-initial classes to decrease, and all other sectoral value-added rates and output multipliers to be unchanged, which is another kind of Pareto improvement of the economic system. We respectively give a sufficient condition for the two kinds of Pareto improvements. A numerical example verifies these results.