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Consistency in Aggregation, Quasilinear Means and Index Numbers

In the calculation of economic aggregates it is often necessary to compute the value of these aggregates in some relevant subgroups as well as for the whole data. A method of calculation is said to be consistent in aggregation if it gives the same result regardless of whether it is applied directly to the whole data or to subaggregates calculated using the same method. The property has been well studied, yet a satisfactory general definition has been lacking. In this paper a definition of consistency in aggregation is given for a general class of aggregation rules. We show that any aggregation method satisfying this definition has a representation based on a commutative semigroup operation. Also, as an application we use algebraic methods to derive certain properties of index numbers satisfying consistency in aggregation. In particular, we show that any such index number formula is in fact a generalized weighted mean.