Oskar Morgenstern and Thomson M. Whitin, The Economics of Input-Output Relations (1955)

This conference discussion contribution reports research at Princeton on the reliability and aggregation of input-output models. Responding to issues raised by Carl Christ, W. Duane Evans, and Tjalling C. Koopmans, Morgenstern and Whitin connect observational error, matrix computation, and the economic meaning of simplified models. Their central finding is cautiously practical: highly aggregated systems may approximate selected results of much larger Leontief systems surprisingly well. Yet computational accuracy cannot establish economic validity independently of uncertain data and the intended application.

The opening discussion makes sensitivity to observational error a prerequisite for confidence in large-scale computation. Electronic computers increase the importance of this question rather than resolving it:

A thorough understanding of them is required before one can have a degree of confidence in the results of large-scale computations sufficient for the intended applications.

Here “them” refers to errors in economic observations and their effects on computed results. The authors distinguish mathematical sophistication from useful mathematical strength. Some advanced proofs establish comparatively weak inequalities for inverse matrices; Y. K. Wong’s elementary algebra yields stronger results. Likewise, his tests for nonsingularity avoid the computationally tedious requirement of establishing indecomposability through rearrangements of rows and columns. These examples make mathematical economy part of the research programme: a useful result should reduce computational burdens while improving what can be established about the model.

The discussion also connects computation to the treatment of small entries. Larger input-output tables contain many zeros, but replacing these with small numbers would greatly increase the noise involved in calculating the inverse. Greater detail therefore does not straightforwardly mean greater reliability. The structure of recorded transactions, including which flows are treated as absent, affects the practical stability of computation.

Aggregation occupies most of the contribution. In the absence of adequate theory, the Princeton group experiments by aggregating large matrices, inverting the smaller systems, and comparing corresponding results. Industries preserved as distinct sectors permit direct comparison of particular inverse coefficients. Their column sums also permit comparison of the total additional economic output required by a dollar’s increase in consumption in a specified industry. Preliminary results suggest that substantial aggregation can preserve economically relevant effects—a finding the authors regard as puzzling, not self-evident.

Their principal procedure retains two industries separately and combines the rest of the economy into one heterogeneous sector. A three-by-three inverse then approximates selected coefficients of the larger inverse without requiring its complete calculation. The authors explain that omitted indirect effects involve products of at least three coefficients and will usually be small. Where important interindustry connections make these effects larger, additional sectors can remain distinct. Aggregation is thus adapted to the particular relationship being investigated rather than imposed uniformly across the economy.

Fourteen three-by-three inversions are compared with corresponding coefficients from a BLS forty-four-by-forty-four inverse. The comparisons support approximation, but they also expose a deeper difficulty in defining its error:

Thus, there is no true matrix that could ever be inverted. We are always dealing with aggregates.

The larger matrix is itself an aggregation of finer classifications and ultimately of millions of economic relationships. It cannot simply serve as an unquestionable representation of reality. The authors therefore distinguish numerical disagreement between models from demonstrable departure from a known economic truth. Some percentage discrepancies reach roughly fifty per cent, but these generally concern small coefficients, making their absolute differences modest. Whether such discrepancies introduce an important bias depends on the purpose of the calculation. One larger discrepancy improves substantially when two additional industries are retained, illustrating how selective disaggregation can refine the method.

The computational advantages extend beyond smaller inversions. Researchers can calculate only the effects they need, exploit detailed data selectively, distribute independent coefficient calculations among multiple computers, and incorporate changes in transactions or technology more flexibly. Nevertheless, the authors explicitly restrict their empirical claim:

It is clear, of course, that these observations relating to 3-by-3 aggregations do not hold for arbitrary matrices.

Nor do they claim a rigorous theoretical explanation for the observed behaviour of these particular Leontief matrices. Their suggestion that exact inversion may offer little advantage rests on the imprecision of underlying economic data, not on a general equivalence between aggregated and disaggregated systems.

The closing sections turn from experiments to theoretical criteria for aggregation without error and to an alternative geometrical approach using oriented trees. Drawing on circuit theory, R. Bott connects the existence of a tree with a nonvanishing determinant. Interpreted as dollar flows, the tree establishes the relevant static equilibrium when every good receives consumer money, directly or through other industries.

The aggregation problem is unquestionably one of the most profound facing the economist.

This judgment frames the unresolved paradox: extremely heterogeneous aggregates perform better than the authors expected, perhaps because of an unidentified property of the model. They conclude by opening the question to alternative models, especially ones incorporating nonadditive phenomena such as monopoly and restrictions on competition. The contribution’s lasting relevance lies in its conjunction of computational ingenuity and epistemic restraint: useful approximations must be judged against data quality, economic purpose, and model structure, not numerical detail alone.