The Economic Approach to Index Number Theory: the Single-household Case

17.1 This chapter and the next cover the economic approach to index number theory. This chapter considers the case of a single household, while the following chapter deals with the case of many households. A brief outline of the contents of the present chapter follows. 17.2 In paragraphs 17.9 to 17.17, the theory of the cost of living index for a single consumer or household is presented. This theory was originally developed by the Russian economist, A.A. Konüs (1924). The relationship between the (unobservable) true cost of living index and the observable Laspeyres and Paasche indices will be explained. It should be noted that, in the economic approach to index number theory, it is assumed that households regard the observed price data as given, while the quantity data are regarded as solutions to various economic optimization problems. Many price statisticians find the assumptions made in the economic approach to be somewhat implausible. Perhaps the best way to regard the assumptions made in the economic approach is that these assumptions simply formalize the fact that consumers tend to purchase more of a commodity if its price falls relative to other prices. 17.3 In paragraphs 17.18 to 17.26, the preferences of the consumer are restricted compared to the completely general case treated in paragraphs 17.9 to 17.17. In paragraphs 17.18 to 17.26, it is assumed that the function that represents the consumer’s preferences over alternative combinations of commodities is homogeneous of degree one. This assumption means that each indifference surface (the set of commodity bundles that give the consumer the same satisfaction or utility) is a radial blow-up of a single indifference surface. With this extra assumption, the theory of the true cost of living simplifies, as will be seen. 17.4 In the sections starting with paragraphs 17.27, 17.33 and 17.44, it is shown that the Fisher, Walsh and Törnqvist price indices (which emerge as being ‘‘best’’ in the various non-economic approaches) are also among the ‘‘best’’ in the economic approach to index number theory. In these sections, the preference function of the single household will be further restricted compared to the assumptions on preferences made in the previous two sections. Specific functional forms for the consumer’s utility function are assumed and it turns out that, with each of these specific assumptions, the consumer’s true cost of living index can be exactly calculated using observable price and quantity data. Each of the three specific functional forms for the consumer’s utility function has the property that it can approximate an arbitrary linearly homogeneous function to the second order; i.e., in economics terminology, each of these three functional forms is flexible. Hence, using the terminology introduced by Diewert (1976), the Fisher, Walsh and Törnqvist price indices are examples of superlative index number formulae. 17.5 In paragraphs 17.50 to 17.54, it is shown that the Fisher, Walsh and Törnqvist price indices approximate each other very closely using ‘‘normal’’ time series data. This is a very convenient result since these three index number formulae repeatedly show up as being ‘‘best’’ in all the approaches to index number theory. Hence this approximation result implies that it normally will not matter which of these three indices is chosen as the preferred target index for a consumer price index (CPI). 17.6 The Paasche and Laspeyres price indices have a very convenient mathematical property: they are consistent in aggregation. For example, if the Laspeyres formula is used to construct sub-indices for, say, food or clothing, then these sub-index values can be treated as sub-aggregate price relatives and, using the expenditure shares on these sub-aggregates, the Laspeyres formula can be applied again to form a two-stage Laspeyres price index. Consistency in aggregation means that this twostage index is equal to the corresponding single-stage index. In paragraphs 17.55 to 17.60, it is shown that the superlative indices derived in the earlier sections are not exactly consistent in aggregation but are approximately consistent in aggregation. 17.7 In paragraphs 17.61 to 17.64, a very interesting index number formula is derived: the Lloyd (1975) and Moulton (1996a) price index. This index number formula makes use of the same information that is required in order to calculate a Laspeyres index (namely, base period expenditure shares, base period prices and current period prices), plus one other parameter (the elasticity of substitution between commodities). If information on this extra parameter can be obtained, then the resulting index can largely eliminate substitution bias and it can be calculated using basically the same information that is required to obtain the Laspeyres index. 17.8 The section starting with paragraph 17.65 considers the problem of defining a true cost of living index when the consumer has annual preferences over commodities but faces monthly (or quarterly) prices. This section attempts to provide an economic foundation for the Lowe index studied in Chapter 15. It also provides an introduction to the problems associated with the existence of seasonal commodities, which are considered at more length in Chapter 22. The final section deals