Exact Relationships Between Fisher Indexes and Theoretical Indexes

Abstract

We develop exact relationships between empirical Fisher indexes and their theoretical Malmquist and Konus counterparts. We begin by using implicit Malmquist price and price recovery indexes to establish exact relationships between empirical Fisher quantity and productivity indexes and theoretical Malmquist quantity and productivity indexes. We then show that Malmquist quantity and productivity indexes and Fisher price and price recovery indexes “almost” satisfy the product test with the relevant value change, and we derive a quantity mix function that ensures satisfaction of the product test. We next use implicit Konus quantity and productivity indexes to establish exact relationships between empirical Fisher price and price recovery indexes and theoretical Konus price and price recovery indexes. We then show that Konus price and price recovery indexes and Fisher quantity and productivity indexes “almost” satisfy the product test with the relevant value change, we derive a price mix function that ensures satisfaction of the product test, and we show that this price mix function differs fundamentally from the quantity mix function relating Malmquist and Fisher indexes.