Malmquist Firm Productivity Index

Efficiency change (the catch-up term) measures whether a firm moved closer to or farther from the best-practice frontier of its own period — essentially whether management improved relative to peers. Technical change (the frontier-shift term) measures whether the frontier itself moved outward, reflecting new technology or knowledge that expands what is achievable from given inputs. Fare, Grosskopf, Norris and Zhang showed their product equals total factor productivity change, so a firm can grow productivity by catching up, by riding an advancing frontier, or both, and the decomposition tells you which.

A Malmquist index can be defined relative to the earlier period's technology or the later period's technology, and the two need not agree. Rather than arbitrarily pick one base period, Fare and colleagues take the geometric mean of the two, following the index-number tradition of Caves, Christensen and Diewert. This makes the measure symmetric and base-period-neutral, so the productivity comparison does not depend on which period you happen to treat as the reference.

In the Fare, Grosskopf, Norris and Zhang approach, the distance functions are estimated nonparametrically with DEA, so computing the index for one firm over one period pair requires solving four linear programs: two ordinary within-period DEA distances and two mixed cross-period distances that evaluate one period's data against another period's frontier. No prices or functional form are needed — only input and output quantities — and the LPs are solved for every firm and each adjacent pair of periods.