Index Theory and the CPI

My previous post regarding government statistics elicited a lot of commentary, with a tremendous amount of vitriolic commentary directed at the current approach to calculating the CPI. Rather than provide more of my own thoughts on what constitutes an appropriate mix of theory and pragmatism, I will quote from the author whose work I had to read in graduate school, W. Erwin Diewert. From his entry in the 1998 Journal of Economic Perspectives which had a symposium on the Boskin commission report:

Defining a true cost of living index must begin on the household level, and then move to the social level. The Konuis (1939) true cost of living index for a single household is defined as the ratio of the minimum costs of achieving a certain reference utility level in a base period, given the prices prevailing at that time, and at a later “current” period, given whatever changes in prices had occurred in the interval. An appropriate generalization of the Konuis cost of living concept to the case of many households is Pollak’s (1981, p. 328) social cost of living index, which is the ratio of the total minimum cost or expenditure required to enable each of the households present in the two periods to attain their reference utility levels in both time periods.

As economists have long known, a Laspeyres index, which finds the cost of purchasing a fixed basket of goods representing the base period and then the cost of buying the same basket in the present, tends to overstate the rise in the cost of living by not allowing any substitution between goods to occur. Conversely, a Paasche index, which finds the cost of purchasing a fixed basket of goods representing the present and then the cost of buying that same basket in the past, tends to understate the rise in the cost of living. Diewert (1983, p. 191) showed that the (unobservable) Pollak-Konuis true cost of living index was between the (observable) Paasche and Laspeyres price indexes. An implication of this result is that some average of the Paasche and Laspeyres aggregate price indexes should provide a reasonably close approximation to the underlying true cost of living. Note that this argument does not rely on any particular assumption about the form of the house- hold preferences; in particular, it does not assume that indifference curves are homothetic (that is, shaped so that the slope of the indifference curves will be the same along the path of a ray extending out from the origin).

One strong candidate for an average of the Laspeyres and Paasche indexes is the Fisher (1922) ideal price index, which is the geometric average of the Laspeyres and Paasche indexes (that is, the square root of their product). This choice can be defended from at least four different perspectives. First, it is evident that the base period basket used in the Laspeyres index is just as valid as the current period basket used in the Paasche index. Hence it makes sense to take an even-handed average of the two. The geometric mean is more desirable than other simple averages, like the arithmetic mean, because it has a time reversibility property: using the Fisher formula, price change going from the current period to the base period is the reciprocal of the original price change (Diewert, 1997). Note that the Paasche and Laspeyres indexes also do not satisfy this time reversal test. This leads to a second justification for the Fisher formula: it satisfies more reasonable “tests” or “axioms” than any of its competitors (Diewert, 1992). The test approach to index number theory, initiated by Walsh (1901) and Fisher (1922), looks at an index number formula from the viewpoint of its mathematical properties. For example, if current period prices increase, does the price index increase? Does the price index lie between the Paasche and Laspeyres indexes? If current period prices increase by a common factor of proportionality, does the price index increase by that same factor of proportionality? These reasonable tests are all satisfied by the Fisher formula.’ A third justification for the use of the Fisher formula is the fact that it is exact for (that is, consistent with) a homothetic preference function that can approximate arbitrary homothetic preferences. Diewert (1976) calls index number formulae that have this property “superlative”. The Toernqvist index which is discussed by the Boskin Commission is an example of another superlative formula. A final justification for the use of the Fisher formula rests on its consistency with revealed preference theory (Diewert, 1976, p. 137).

Of course, in this article directed toward generalists, the specific rationales and arguments and proofs are omitted, so one might feel that one is taking too much on faith. Fortunately, Diewert has written a comprehensive review recently published in the New Palgrave Dictionary of Economics (co-edited by UW professor Steven Durlauf). This article is available in its pre-publication form as UBC Discussion Paper 07-02. In this review, Diewert lays out the criteria by which one might prefer one index over another, and what is true is that the Laspeyres (fixed base period weight) approach is not one of the preferred methods.


Originally published at Econbrowser and reproduced here with the author’s permission.Related RGE Content:1) Is U.S. Consumer Confidence Reflected in the Consumption Pattern?

2) U.S. Retail Sales Up Just 0.1% in June: Is the Impact of Tax Rebates on Private Consumption Fading?

3) Growing Income Inequality in the U.S: Resembling the Pre-Depression Era?