Oskar Morgenstern and Max A. Woodbury · 1950
This conference-paper abstract within a journal meeting report examines how errors in observed input-output coefficients affect the inverses used in economic calculation. It moves from a small empirical comparison to an account of Woodbury’s mathematical results. Its central distinction is between the numerical stability of an inverse and the stability of the economy represented by the matrix: reliable inversion does not establish either the accuracy of the underlying observations or the resilience of economic activity.
INPUT-OUTPUT matrices are based on observations that are naturally afflicted with errors. It is important to investigate quantitatively the extent to which their inverses are affected by variations in the quality of the information.
The authors thus make observational uncertainty the starting point of their inquiry. Given limited experience with large matrices, they approach the question empirically by inverting two 18-by-18 matrices. The second differs from the first in only 42 entries, each altered by at most two per cent. This deliberately modest, uniform perturbation simplifies a problem whose actual errors, they acknowledge, are likely both larger and unevenly distributed across economic fields. The experiment therefore tests sensitivity under restricted conditions, rather than reproducing the full uncertainty of the data.
The inverses prove to be highly stable; this is attributed to the fact that the matrices differ but little from the identity matrix.
The explanation qualifies the encouraging result. Stability belongs to the particular structure of the matrices examined, whose proximity to the identity limits the effects of small changes. The authors question whether that structure adequately represents the extensive interdependence of economic activities. Better observations might reveal connections currently recorded as zeros, producing matrices less convenient for inversion. Improved data could therefore weaken an apparent numerical advantage while strengthening the economic representation. Stability is not offered as a reason to accept the existing coefficients without further scrutiny.
The stability of inverses must not be interpreted as proving stability of the economy. It is merely an indication how, and to what extent, errors in the given matrix carry over into its inverse.
This warning fixes the scope of the argument. The investigation concerns error transmission through a mathematical operation, not the economy’s response to disturbances. Its practical relevance lies in evaluating the prognostic value of input-output tables in linear programming and related applications. Even there, a stable inverse cannot compensate for deficient information: the calculation’s sensitivity and the model’s empirical adequacy remain separate questions.
The latter portion presents Woodbury’s mathematical contribution, identifying an output-input matrix as the inverse of an input-output matrix. When both remain moderately close to the unit matrix, changes in their inverses tend to occur in the same positions as changes in the original matrices, with approximately equal magnitudes but opposite signs. This gives a more specific account of the empirical stability than a general assurance that the inverses change little.
Some related results are also obtained, namely a bound for the ratios of the norms of the changes in the original and inverted matrices (thus related to the stability of the inverses of the matrices) and a method for obtaining the exact difference between the inverse of two matrices that differ only in certain rows.
The displayed bounds relate changes in the original matrices to changes in their inverses through measures of departure from the identity. The abstract also supplies an explicit inverse-update formula when the original matrices differ in just one row, identifying it as a generalization of a result by Sherman and Morrison. These results extend the numerical comparison toward quantitative bounds and exact calculations for localized revisions. The short report’s contribution is consequently both practical and cautionary: it offers tools for tracing observational error while refusing to equate computational convenience with a faithful account of economic interdependence.
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