Proper quantity indexes for coherent productivity comparisons

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.