This rejoinder responds to Professor Diewert’s commentary on my papers concerning proper quantity indexes and productivity measurement. It clarifies three issues that seem to be central to the debate: the role of the transitivity axiom in coherent multilateral and multitemporal quantity measurement, the practical guidance provided by my measurement-theoretic framework, and the distinction between measuring productivity change and explaining its economic drivers.
This paper comments on Diewert’s discussion of proper quantity indexes. Several aspects of the theory and interpretation of these indexes are clarified, including the role of production-theoretic assumptions in generating quantity and price indexes that satisfy product rules, the interpretation of alternative proper index constructions, and the ability of proper indexes to preserve historical quantity comparisons when the representation or scope of a dataset changes. A robustness measure based on quantity data alone is proposed for assessing the sensitivity of quantity comparisons to the choice of proper index. The paper shows that Diewert’s SPQ index does not satisfy the weak monotonicity axiom and therefore is not proper in the sense defined by O’Donnell (2016).
The paper outlines O’Donnell’s axiomatic approach to index number theory that he applies to the measurement of output, input and productivity. His concept of a Proper Index is related to other approaches to index number theory in the producer context. The alternative approaches that are discussed are basket approaches, stochastic approaches, axiomatic approaches and economic approaches. O’Donnell’s approach and other approaches are illustrated by the computation of various indexes using US consumption data for the years 2000–2025. The paper also discusses two important problems that compilers of producer price and quantity indexes face: the lack of matching problem and the extension problem. An alternative axiomatic framework to O’Donnell’s axioms is discussed.
This study estimates the shadow prices of greenhouse gas emissions in global crop production by constructing multiple levels of production frontiers that account for cross-country heterogeneity. Traditional nonparametric models usually rely on a single frontier, which may mask differences in resource endowments, production scale, technologies, and input-use patterns across countries. To address this limitation, we adopt the context-dependent framework proposed by Seiford and Zhu (2003) that identifies several layers of performance frontiers representing different efficiency levels. Two alternative representations of production technology are considered: the by-production approach and the weak disposability approach. Using data for 157 countries over the period 2004–2022, we find that the by-production model produces a richer stratification structure with up to ten frontier levels, whereas the weak disposability model identifies up to five levels. The by-production framework yields higher and more dispersed shadow prices of greenhouse gas emissions, indicating greater heterogeneity in marginal abatement costs. In contrast, the weak disposability framework produces lower and more compressed shadow price distributions, suggesting that the joint weak disposability assumption provides a less differentiated representation of abatement difficulty. Regional comparisons reveal that Asia, Africa, and South America exhibit wider intervals and stronger fluctuations, Europe maintains low and stable values, and Oceania and North America face relatively higher marginal costs of emission reduction. Statistical tests confirm that the distributions of shadow prices differ significantly between the two technological representations. These findings highlight the importance of allowing for multiple production frontiers when assessing marginal abatement costs and designing global mitigation strategies.
This paper defines productivity as the ratio of an output quantity to an input quantity. Under this definition, a productivity index is the ratio of an output quantity index to an input quantity index. Coherent productivity comparisons across both time and space require quantity indexes that satisfy a small set of basic axioms. Quantity indexes satisfying these axioms are referred to as proper indexes. Such indexes can be constructed using a wide range of aggregator functions and weighting systems. There is no uniquely correct proper index. Coherent productivity comparisons require only that empirical aggregation choices preserve the axioms necessary for consistent quantity comparisons. Many index-number formulas widely used in applied work are designed for pairwise or one-dimensional comparisons (over time or across space) and generally fail to satisfy these axioms when comparisons extend across multiple periods or units. Because multi-period and multi-unit comparisons are ubiquitous in applied economics, the framework described in this paper has implications well beyond productivity analysis. The paper clarifies the conceptual foundations of productivity comparisons, reviews the principal classes of productivity measures used in practice, and explains why statistical agencies and applied researchers continue to rely on measures that are not proper.
