Index Numbers and Consistency in Aggregation: Part I

There are many formulae for the index numbers used to compute price and quantity indices, such as a consumer price index or a volume index of imports, in economics. The Laspeyres, Paasche, Divisia, and Fisher indices are the best known of these formulae. A body of theory examines the criteria that can be used to decide which formula to use in a particular application. One important property is consistent aggregation. Say that consumers purchase the following categories of goods and services: Education, health care, food, clothing. First compute price indices for goods using the data on quantities and prices of food and clothing and services using the data on quantities and prices of education and health care and then compute a consumer price index using the resulting goods and services price indices. If this index is the same as a consumer price index computed using the data on the four original commodities then the index formula is said to exhibit consistent aggregation.

Another important property is “Fisher’s factor reversal test”. Compute a price index for the ratio of the prices of a group of commodities relative to a base year as well as the corresponding quantity ratio index. Then if the product of these two indices equals the ratio of the total value or cost in the second period relative to the first the index formula is said to pass Fisher’s factor reversal test.

Vartia (1976) proposed a formula that passes both these tests dubbed the Vartia I index. The Vartia I index for a change in price between period 0 and period 1 is:



where superscripts refer to the two time periods, the pi are the prices and the xi are the quantities of each of the commodities indexed by i. p and x are the vectors of the prices and quantities. L() is a function defined by:



But Vartia’s index isn’t perfect. Another desirable property of index functions is that a quantity index for the ratio of aggregate quantities in the second period relative to the first should be equal to the ratio of production functions that use those quantities of inputs in the second period and the first.* Such an index is called an exact index.

Diewert (1978) shows that Vartia’s index is only exact for the Cobb-Douglas production function. This seems rather disappointing as the Cobb-Douglas function imposes an elasticity of substitution of one between each pair of inputs rather than letting the data speak. This seems rather restrictive. So maybe Vartia’s index isn’t as ideal as he thought?

References
Diewert, W. E. (1978) Superlative index numbers and consistency in aggregation, Econometrica 46(4): 883-900.
Vartia, Y. O. (1976) Ideal log-change index numbers, Scandinavian Journal of Statistics 3: 121-126.

* Similar relationships exist for the price index and the unit cost function and for utility functions etc.