A Generalized Input-Output Model of an Economy with Environmental Protection

Abstract

I NPUT-OUTPUT analysis can be extended to an economic system with antipollution measures to describe interdependency among economic sectors and antipollution sectors in the economy. Such a model was formulated by Professor Leontief (1970) as a system of equations extending his original input-output framework. This extended model has been extensively discussed by Leontief and Ford (1972), Chen (1973), and Leontief (1973). A linear programming problem of the extended model with substitutions was presented by Lowe (1979). However, counter-examples with unrealistic solutions to the extended model were presented by Flick (1974) and Lee (1975). Validity of Flick's counter-example was questioned by Leontief (1974) and Steenge (1978). There seems to be an unresolved question concerning the existence of a unique nonnegative set of output levels in the extended model for any given nonnegative levels of final demand and tolerated pollution. The structural matrix of Leontief's original input-output model possesses two important properties: (1) off-diagonal elements are nonpositive; (2) all principal minors are positive. These two properties, known as Leontief properties, guarantee the existence of a unique nonnegative solution to Leontief's original model for any given nonnegative final demand. The above-mentioned problem of feasibility arises when the original model is extended to endogenize antipollution sectors and to set the total amount of pollution generated equal to the antipollution output level plus the tolerable pollution level. In this paper, we present a complementarity formulation of the economic system with antipollution measures that overcomes the problem of determining the existence and uniqueness of a feasible solution under Leontief properties, and derive conditions under which the present formulation guarantees these desirable properties when the structural matrix has some positive offdiagonal elements.' We apply the model to solve examples with different final demands and tolerated pollution levels (including Flick's counterexample) to obtain realistic solutions.