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A Note on the Underestimation and Overestimation of the Leontief Inverse

A. Simonovits

Abstract

Suppose that the coefficients of an input-output matrix, A, are random variables but that we have ascertained their expected values, EA. What will be the relation of the Leontief inverse of EA, (I - EA)^-1, to the expected value of the inverse, E(I - A)^-1? Will one or the other be uniformly greater? We will show that if all coefficients of A are independent, then the expected value of the inverse is uniformly greater than or equal to the inverse of the expected value. If, on the other hand, the column and row sums of the coefficient matrix are fixed, and smaller than one, so that the variables are not independent, then, in the two-by-two case, the opposite is true of the off-diagonal elements.