Group Cost-of-Living Indexes

When households have different consumption patterns, whose cost of living should an actual price index represent? This issue was first raised by J. L. Nicholson and S. J. Prais in the 1950's. Both made essentially the same point: official price indexes give each household's consumption pattern "an implicit weight proportional to its total expenditures" (see Nicholson, p. 540). Prais calls such an index "plutocratic," and both Nicholson and Prais suggest an alternative "democratic price index" which gives all households equal weight. A "group cost-of-living index" is an index that measures the impact of price changes on the welfare of a group or population of households. To define such an index requires an explicit or implicit concept of "the welfare of a group," and hence requires interpersonal comparison and distributional judgments. Since group indexes such as the Consumer Price Index play an important role in our perception of inflation and the formation of macro-economic policy and are used to escalate wages and Social Security benefits, they have significant effects on government decisions and economic welfare. Despite their intellectual interest and practical importance, however, until recently they have been virtually ignored by index number theorists. The theory of the cost-of-living index (CLI) provides a generally accepted framework for measuring the impact of price changes on the welfare of a particular household. This paper extends the CLI concept to groups and discusses which questions require group indexes and which do not. I begin by introducing some notation and terminology in the context of household CLIs. A household's CLI is the ratio of the expenditures required to attain a particular base indifference curve in two price situations. Suppose there are n goods and S households, and denote the preference ordering of the rth household by R r. The base indifference curve can be identified by a goods collection, Xro, which lies on it. The "expenditure function," Er(P, xr, Rr), shows the minimum expenditure required to attain the base indifference curve at prices P. The CLI of the rth household, Ir(Pa,pb,XroRr) is the ratio of the minimum expenditure required to attain the base indifference curve at prices pa (comparison prices) to that required at prices pb (reference prices). Except in very special cases, the value of the CLI depends on the base indifference curve at which it is evaluated; as successively higher base indifference curves are specified, one would expect the prices of "luxuries" to become more important relative to the prices of "necessities."' Hence, it is convenient to regard the CLI as a function of the base indifference curve rather than as a single number corresponding to a particular base. Thus, instead of offering guidance in choosing an appropriate base indifference curve, theory suggests that there is no need to choose. To construct the exact CLI, an investigator needs to know the household's preferences. Lacking this knowledge, he rnust fall back on indexes which require less information and which are upper bounds on the exact index. The "Laspeyres index," Jr(papbXrb) is the ratio of the cost of purchasing the reference period consumption basket at comparison prices to its cost