Group Cost-of-Living Indexes
扩展生活成本指数概念到群体层面，讨论如何衡量价格变化对一组家庭福利的影响，并区分哪些问题需要群体指数、哪些不需要，对理解通胀和制定宏观经济政策有重要意义。
When households have different consumption patterns, whose cost of living should actual price represent? This issue was first raised by J. L. Nicholson and S. J. Prais in 1950's. Both made essentially same point: official price indexes give each household's consumption pattern an implicit weight proportional to its total (see Nicholson, p. 540). Prais calls such plutocratic, and both Nicholson and Prais suggest alternative democratic price index which gives all households equal weight. A cost-of-living index is that measures impact of price changes on welfare of a group or population of households. To define such requires explicit or implicit concept of the welfare of a group, and hence requires interpersonal comparison and distributional judgments. Since group indexes such as Consumer Price Index play important role in our perception of inflation and 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 number theorists. The theory of cost-of-living (CLI) provides a generally accepted framework for measuring impact of price changes on welfare of a particular household. This paper extends CLI concept to groups and discusses which questions require group indexes and which do not. I begin by introducing some notation and terminology in context of household CLIs. A household's CLI is ratio of expenditures required to attain a particular base indifference curve in two price situations. Suppose there are n goods and S households, and denote preference ordering of rth household by R r. The base indifference curve can be identified by a goods collection, Xro, which lies on it. The function, Er(P, xr, Rr), shows minimum expenditure required to attain base indifference curve at prices P. The CLI of rth household, Ir(Pa,pb,XroRr) is ratio of minimum expenditure required to attain base indifference curve at prices pa (comparison prices) to that required at prices pb (reference prices). Except in very special cases, value of CLI depends on base indifference curve at which it is evaluated; as successively higher base indifference curves are specified, one would expect prices of luxuries to become more important relative to prices of necessities.' Hence, it is convenient to regard CLI as a function of base indifference curve rather than as a single number corresponding to a particular base. Thus, instead of offering guidance in choosing appropriate base indifference curve, theory suggests that there is no need to choose. To construct exact CLI, investigator needs to know household's preferences. Lacking this knowledge, he rnust fall back on indexes which require less information and which are upper bounds on exact index. The Laspeyres index, Jr(papbXrb) is ratio of cost of purchasing reference period consumption basket at comparison prices to its cost