# Comment on W. Erwin Diewert's, "Index Number Theory Using Differences Rather Than Ratios"

## Article excerpt

I

Fisher

A SUBSTANTIAL FRACTION of what we know about how to construct measurements that are good approximations to the change in the cost of living owes to Irving Fisher. His Ideal price index was invented to address the vexing, although fairly narrowly defined, question of how to measure changes in the purchasing power of money. Fisher's thinking about index numbers is based on economics rather than accounting, based on prices and quantities rather than prices alone, and based, perhaps most critically, on multiplication rather than addition. Fisher's index is becoming incorporated in many aspects of the statistical system.

* As of the last major benchmark revision of the National Income and Product Accounts, the Bureau of Economic Analysis has adopted a feasible approximation of Fisher's Ideal index for measuring both the change in price and quantity. This major switch away from double deflation means that--at least for broad aggregates--the national statistics correctly account for the reaction of supply and demand to relative prices.

* Fisher's ideas about how to construct elementary price indexes, in other words, how to average individual observations on the price of bread to obtain a price index for bread, are also gaining currency. Despite stern warnings in Fisher's writings, the Bureau of Labor Statistics has used an arithmetic rather than a geometric average. In the face of growing evidence supporting Fisher's views, the Bureau of Labor Statistics (BLS) experimented with a geometric index and announced its intention to use it for parts of the Consumer Price Index.

Effective January 1999, the BLS began using the geometric formula to aggregate elementary price observations for approximately two-thirds of the items in the CPI. This change should reduce the rate of growth in the CPI by approximately 0.2 percentage points per year.

The Fisherian program for economic measurement does have some features that only an economist could love. For example, C + I + G does not equal Y in quantity or real terms in the National Income and Product Accounts. For some components of demand, the arithmetic is even more startling. For example, the implied level of real spending on computers actually exceeds the total of the subcomponent of investment including computers for some time periods. The Fisherian procedure alerts us to the effect of dramatic changes in relative prices that get swept under the rug by more mechanical procedures.

Fisher's method for calculating price indexes is in the tradition of the Econometric Society and the Cowles Foundation. Researchers in the program of the Econometric Society and Cowles have been willing to push the methodological frontier far ahead of what is practical for current analysis owing to limitations of computing power or data availability. In the case of Fisher's Ideal index and its superlative cousins, the indexes depend on current and historical data on both price and expenditure. The use of base-period expenditure weights in the CPI, for example, is a pragmatic response to having infrequent data on expenditures. More frequent expenditure data are now available. The statistical system has been slow to incorporate these data, especially in the CPI. (1) Moreover, with the growing amount of data available electronically--from scanners or other computerized transactions--there is the possibility for computing Fisher Ideal indexes virtually in real time.

II

Diewert

WHILE IT SHOULD NOT surprise us, it is not apparent on its face that Fisher's Ideal index performs far better than a number of seemingly plausible alternatives. We owe this knowledge to the extremely important work of Erwin Diewert (1976), who proved that under suitable assumptions, it was a second-order approximation to the true cost of living. Since his seminal paper in 1976, Diewert has continued an important program of expanding and systematizing what we know about index numbers. …