Tintner and Sondermann’s edited-volume essay examines how economic microrelations can be replaced by smaller systems without losing essential information. Its five sections move from optimal aggregation of linear systems through moments and principal components to input-output analysis and the Cobb-Douglas production function. The connecting problem is that aggregation depends not merely on adding quantities, but on the relationships, distributions, and criteria of approximation that make those quantities economically informative. Exact consistency is a demanding benchmark; statistical methods provide ways either to minimize departures from it or to preserve information that simple totals discard.
The first and most extensive section develops W. D. Fisher’s approach. Exogenous microvariables are mapped into macrovariables, a smaller model relates the macrovariables, and a disaggregation map reconstructs the endogenous microvariables. The authors distinguish the aggregated model itself from the “simplified” model obtained by combining all three operations. Information loss is measured by a quadratic function of the discrepancy between original and reconstructed outcomes. Its weighting matrix specifies which discrepancies matter, while the second-moment matrix of the exogenous variables determines how frequently different discrepancies enter expected loss.
Perfect aggregation reproduces the original outcomes for every possible input. Almost perfect aggregation requires only zero expected loss under the specified weighting and probability structure. Generalized, or Penrose, inverses allow these conditions to be formulated even when the relevant matrices are singular. Yet exact equivalence is not the practically decisive case:
However, it turns out that even if condition (1.11) is violated, the relation (1.13) remains relevant.
The conceptual move is to retain the same generalized-inverse construction as an approximation rule when the condition for almost perfect aggregation fails. With aggregation and disaggregation maps fixed, the resulting macro-model minimizes expected quadratic information loss. Optimality therefore means the best reconstruction under stated weights and moments, not an unconditional equivalence between microeconomics and macroeconomics.
Allowing the maps themselves to vary leads to a rank-constrained approximation problem. Eigenvectors associated with the largest eigenvalues identify the directions to retain, extending the logic of principal components. The authors nevertheless distinguish finding the best reduced-rank reconstruction from jointly determining an economically usable macro-model and its maps:
The problem of simultaneous optimal determination of $\bar{A}$, $T$ and $S$ is still unsolved.
They offer a conjectured joint construction rather than presenting this remaining problem as settled. This qualification marks the boundary between their demonstrated approximation results and the stronger aggregation solution they seek.
The second section asks what aggregate variables must contain when relations are heterogeneous or nonlinear. Consistent aggregation into simple sums requires restrictive linear relations with common slopes. When individual coefficients differ, covariance terms enter aggregate coefficients and disturbances: average behavior alone does not determine the macrorelation. Weighted moments offer another route. For quadratic microrelations, aggregates of individual variables and their cross-products can preserve the terms needed to express total output, provided their weights satisfy the stated coefficient restrictions.
An example involving individual consumption relations makes the role of heterogeneity explicit:
This example shows, how computed aggregates depend upon the probability structure of the model.
The distribution of individual responses and their association with the variables used to construct aggregate relations affect the resulting coefficients. A further example, drawn from Prais and Houthakker, aggregates consumption that depends logarithmically on income. The geometric mean becomes relevant, and the assumed lognormal income distribution connects aggregation to distributional moments. The larger point is that a macrorelation may require information about dispersion and composition, not simply an arithmetic total.
The short third section introduces principal components as a practical statistical reduction:
The largest root of (3.1) corresponds to the first principal component, which explains the largest part of the total variance of the random variables.
Retaining leading components preserves dominant variation rather than guaranteeing exact structural aggregation. The authors note proposals to use the first component as an aggregate and components of predetermined variables in simultaneous-equation estimation. This supplies a variance-based counterpart to the earlier loss-minimization framework.
The final sections examine more specific economic structures. In input-output analysis, consistent aggregation of sectors requires weights satisfying an eigenvector condition for the technical-coefficient matrix. The positive Frobenius root and its nonnegative eigenvector provide the proposed weights. Although the authors identify this procedure with principal components, its immediate derivation concerns consistency of sectoral balances.
The Cobb-Douglas discussion instead explains an aggregate functional form through a distribution of production requirements. Following Houthakker, the authors integrate output and factor use over techniques profitable at given prices. Assuming a multidimensional Pareto distribution and eliminating prices yields a Cobb-Douglas relation with constant elasticities. Together, these applications show why aggregation is a substantive modeling choice: reduced systems depend on the information retained, the criterion used to judge loss, and the distributional assumptions connecting heterogeneous units to aggregate behavior.
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