On the circularity of the Malmquist productivity index

Circularity is a desirable property of a productivity index seldom satisfied in available bilateral indices, such as the Malmquist index. Within a setting of micro units belonging to groups with group-specific frontier technology, the bilateral Malmquist productivity index is investigated. Our setting can be interpreted as representing cross section data, but also cross section, time series by identifying groups as time periods. A general proposition giving the condition for the Malmquist index to be circular is presented. When the condition is not met, ways of making the index transitive are explored. Four strategies adopted from the literature is followed; using one group as base, taking an average over possible bases, developing a multilateral index, and chain-linking. An expression of the Malmquist multilateral index is developed giving the difference between the Malmquist bilateral- and multilateral indexes.