Estimating the Consumption Matrix from Inexact Data in the Leontief Model

The Leontief model, originally developed for describing an economic system in terms of mutually interrelated and structurally conditioned simultaneous flows of commodities and services, has important applications to wide ranging disciplines. A basic model assumes the linear form x = Tx+ d, where x represents the total output vector and d represents the final demand vector. The consumption matrix T plays the critical role of characterizing the entire input-output dynamics. Normally, T is determined by massive and arduous data gathering means which inadvertently bring in measurement noises. This paper considers the inverse problem of reconstructing the consumption matrix in the open Leontief model based on a sequence of inexact output vectors and demand vectors. Such a formulation might have two advantages: one is that no internal consumption measurements are required and the other is that inherent errors could be reduced by total least squares techniques. Several numerical methods are suggested. A comparison of performance and an application to real-world data are demonstrated.