An Always Convergent Method for Approximating the Spectral Radius of a Non-Negative Matrix, With Particular Reference to a Leontief Input-Output System

The Leontief input-output model is used to forecast various effects that can occur in an industry as it interacts with other industries under changing conditions within an economy. It is because of this valuable ability to forecast various effects (interaction) that the model has become very widely used. Users of the model need to be aware of the conditions under which a unique solution exists for the system of input-output equations, and also when this system is ill-conditioned. The Hawkins-Simon and Spectral Radius conditions have been previously established and guarantee uniqueness. New proofs are presented for these two conditions and their equivalence is established. Several useful bounds are developed for the condition number of the system of input-output equations. An always convergent method for approximating the spectral radius of an input-output matrix is developed. The spectral radius of a matrix is the largest eigenvalue of the matrix in absolute value. The more general problem of approximating the spectral radius of a general non-negative matrix is considered and an always convergent method is developed for this problem.