Appendix 2: Formulas and Meanings of Various Price Indexes

Let:

Then:

These forms are taken or adapted from those used by the Bureau of Labor Statistics.[1] They depart a little from the forms you’ll find in economics texts, partly because the BLS must work with data that are available (expenditure shares rather than actual quantities of items), and partly because, in practice, the CPI is built up in stages. The first stage calculates indexes for samples of "elementary items", and the later stages combine these into an overall index. In the second step, first-stage indexes are substituted for the price ratios shown above. That’s a needless complexity for the current discussion, so we show the formulas with price ratios.

In an economics text, the formulas generally look something like this:

Laspeyres = = = .
Paasche = = = .

These are mathematically equivalent to the BLS forms, but shed somewhat more light upon the economic meaning.

Imagine a time traveler planning a business trip to the future, from period 0 to period t. When asking for a travel advance, the savvy time traveler will multiply the cost of a period 0 market basket[2] by the Laspeyres index (communicated by a colleague from the future); the traveler can then buy at least that basket in period t, and may be able to improve upon it with substitutions.

A time traveler in the opposite direction, from period t to period 0, will make similar use of the Paasche index. Hopefully, he or she will notice that the index is upside down for this purpose, and divide rather than multiply the cost of a period t basket by it, in order to come up with an advance that will buy at least that good a basket in period 0.

With the Laspeyres index in form and Paasche in form , it is easy to work out the revealing inequalities expounded by Thomas Carroll[3]:

Index Inequalities which Suggest Consumer Preferences 

If Laspeyres , then , so consumers could have purchased either basket in period t. They must have preferred basket t, at least at that time.

If Paasche , then , so consumers could have purchased either basket in period 0. They must have preferred basket 0, at least at that time.

If Laspeyres , then , so the period 0 basket was no longer within consumers’ reach in period t. Maybe they would have preferred it by then, but nothing in their recorded behavior tells us so.

If Paasche , then , so the period t basket was out of consumers’ reach in period 0. Maybe they would have preferred it at that time, but they got no opportunity to show us so.

Putting all the foregoing observations together:

Laspeyres and Paasche both consumers could have purchased either basket in period 0, so evidently they preferred basket 0 originally. By period t, it was no longer within their reach, but we have no reason to suppose that their preferences changed, so there are strong reasons to believe that the switch to basket t was forced by adversity. That is, their real income fell.

Laspeyres and Paasche both consumers could have purchased either basket in period t, so evidently they preferred basket t by then. Originally, it was out of their reach, but we have no reason to suppose that their preferences changed, so there are strong reasons to believe that the switch to basket t was the result of improved opportunity. That is, their real income rose.

Laspeyres Paasche consumers could have purchased either basket in either period, so the fact that they switched must indicate a change of preferences. We can’t really judge whether their new attitude left them happier or unhappier (they might have become relatively disgruntled with all that life has to offer), though there’s a reasonable conjecture that more is merrier and newer is better.

Paasche Laspeyres in neither period could consumers have purchased the other period’s market basket, so nothing in their behavior tells us whether the switch was benign or adverse. We can infer nothing about the change in their real income, at least from this set of inequalities.

[Back to Example 3, if you digressed from there to review these inequalities.]

 

The Boskin Commission, whose report brought all this stuff before the public, argues that the Laspeyres index exaggerates inflation — essentially because the time traveller of our earlier discussion could make out like bandits if given a travel advance based on the Laspeyres index. (No, the Boskin Commission did not express itself in this frivolous manner. That’s my persona. But the charge is the same.)

Their preference is to use the Geometric Mean index when calculating price indexes for the elementary items sampled by the CPI. At this level, many of the items are close substitutes for each other (a fact which favors the use of the Geometric Mean). Then, when combining the groups into an overall index, what they’d really like to see would be the Tornqvist index, which uses weights averaged over the two periods.

The BLS, whose job it is to round up the data and make the calculations, has other preoccupations. It doesn’t quarrel with the theoretical apparatus of the Boskin Commission, but has to reconcile it with data availability, sampling procedures, and publication deadlines. In particular, any index which requires weights from the end period would come out later than is desirable. Data collection takes time. Moreover, it’s costly, and in all of the above indexes except Laspeyres and the (base-weighted) Geometric Mean, preparation must await the data on the later period’s expenditure shares. Moreover, these must be compiled for each successive period in which an index is prepared.

The Boskin Commission worked closely with the BLS, and was not unaware of its practical concerns. Because of the unavoidable lag in collecting data for a Tornqvist index, they recommended a "trailing Tornqvist index", which would average the weights of the last two or three past years. They remained unhappy, though, with existing budgetrary constraints. They would like to see weights revised every year (currently it’s every ten years), and urge that a revised CPI be published every year, "with a lag of a year or two", based on the new weights. In fact, they would like to see historical records revised continually, as new data become available.

Like every other government study that ever was or ever will be, this one recommended that the government should fund a permanent body to carry on its good work — "at the request of", but not part of, the BLS. The BLS would have quite enough new employment, keeping up with the additional data collection and massaging. (The commission, to be fair, did pin some of its hopes upon input from scanners at the nation’s checkout counters.)

As for the BLS, it is cautiously experimenting with the base-weighted Geometric Mean index and with experimental indexes which single out the purchasing behavior of particular groups such as seniors and the poor. Its budget is nowhere near that which would enthuse the Boskin Commission.

Now please return to the main thread of my discussion, which largely confirms the Boskin Commission’s preferences concerning the Laspeyres, Geometric Mean, and Tornqvist indexes, but turns up some entertaining quirks.

Depending how you got here:

[Back to first part of article.]

[Return to Example 1.]

[Back to Example 3.]