Gottfried Haberler, “The Construction of Index Numbers” (1929)

Gottfried Haberler’s review of Warren Milton Persons’s The Construction of Index Numbers (1928) combines appreciation of a technical contribution with criticism of its formal foundations. Following Persons from the choice of averaging formulas through comparisons across multiple periods to the divergence of fixed-base and chain indexes, Haberler asks whether mathematical consistency tests can establish economic validity.

Persons develops Irving Fisher’s approach by evaluating arithmetic, harmonic, and geometric means under alternative weighting systems. Haberler identifies the governing methodological commitment:

Economic considerations may determine the selection of the price data, but not the construction of the formula.

This separation frames the review’s central objection: the economic comparability of the underlying observations cannot ultimately be settled through formula selection. Haberler nevertheless carefully reconstructs Persons’s “weight correlation test.” Changing price weights can systematically change the resulting average; quantity weights can reinforce or offset that effect, depending on the correlation between price and quantity relatives. Haberler stresses that the relevant correlation concerns these relatives, not prices and quantities as levels. Persons combines compromises between competing weights to obtain eleven formulas, eliminates two through base reversal, and singles out Fisher’s “ideal” formula because it also satisfies factor reversal.

The move from two periods to several exposes a conflict between the tests. The circular test extends the logic of base reversal, but cannot be satisfied together with factor reversal. Persons responds by questioning the possibility of accurate comparisons when commodities and quantities have changed radically. Haberler finds this concession damaging to the formal approach: differences in the data remain a problem even when they are slight. He proposes a different resolution:

There are two ways out of this dilemma: Either drop the circular test (as Fisher does) or abandon the factor reversal test. I would prefer to do away with the latter.

His reason is substantive rather than merely technical. Factor reversal assumes that the same formula should average both prices and quantities, a requirement he regards as insufficiently justified. He goes further by questioning whether a quantity index must mathematically be an average of quantity relatives at all. The conceptual move is to challenge the authority of the test itself, rather than accept its implications as necessary properties of a valid index.

Haberler applies the same reasoning to Persons’s second major contribution, a formula describing the divergence between fixed-base and chain indexes. Under specified conditions frequently encountered in practice, Persons finds divergence cumulative in a geometric progression and concludes that chain indexes should be rejected. Haberler distinguishes the demonstration of divergence from the identification of error:

This decision seems to me to be unfounded, for Persons does not show why the error lies in the chain system and not in the fixed base series.

He instead favors chain indexes because comparisons between adjacent periods minimize the heterogeneity that increasingly compromises comparisons with a fixed base. The review thus connects its two principal criticisms: neither satisfaction of a formal test nor deviation from a competing series establishes an index’s economic superiority without examining the comparability of what it measures. Haberler closes by recognizing Persons’s achievement while judging the exposition difficult and the problem unresolved. The review’s enduring relevance lies in this distinction between mathematical properties of index formulas and economically defensible judgments about change.