Conceptual Foundations for Price and Cost-of-Living Indexes

For much of its life, the Consumer Price Index (CPI) was based on the idea of comparing the costs of a fixed bundle or “basket” of goods; this concept leads to what we call a “basket price index” or “cost-of-goods index” (COGI). However, beginning with the Stigler commission report (1961), there has been an increasing emphasis on thinking about the CPI as a cost-of-living index (COLI). Indeed, the “overarching recommendation” of the Boskin et al. (1996) report was that the Bureau of Labor Statistics (BLS) should try to make the CPI approximate a COLI as closely as possible.

This chapter lays out the theory behind both the COLI and the COGI in a form that will serve as a basis for the discussion of specific topics in the chapters that follow. We start from the underlying ideas in their simplest form and then work through a series of practical and conceptual issues, many of which are covered in detail in subsequent chapters. Consideration of each of these issues serves to sharpen and elaborate the concepts in ways that are necessary if either a COGI or a COLI is to fulfill the many sometimes conflicting demands that are placed on a measure like the CPI.

The basket price and cost-of-living approaches to index construction are conceptually quite different. Nonetheless, for many (perhaps even most) purposes, the distinctions are less important than they might seem. In particular, for most of the issues that we discuss in this chapter and in the report more broadly, there are close parallels between the two approaches. In consequence, the two approaches have always drawn on one another, so that a COGI has often been modified to make it more like a COLI and vice versa. Even when the CPI was

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At What Price?: Conceptualizing and Measuring Cost-of-Living and Price Indexes
2
Conceptual Foundations for Price and Cost-of-Living Indexes
For much of its life, the Consumer Price Index (CPI) was based on the idea of comparing the costs of a fixed bundle or “basket” of goods; this concept leads to what we call a “basket price index” or “cost-of-goods index” (COGI). However, beginning with the Stigler commission report (1961), there has been an increasing emphasis on thinking about the CPI as a cost-of-living index (COLI). Indeed, the “overarching recommendation” of the Boskin et al. (1996) report was that the Bureau of Labor Statistics (BLS) should try to make the CPI approximate a COLI as closely as possible.
This chapter lays out the theory behind both the COLI and the COGI in a form that will serve as a basis for the discussion of specific topics in the chapters that follow. We start from the underlying ideas in their simplest form and then work through a series of practical and conceptual issues, many of which are covered in detail in subsequent chapters. Consideration of each of these issues serves to sharpen and elaborate the concepts in ways that are necessary if either a COGI or a COLI is to fulfill the many sometimes conflicting demands that are placed on a measure like the CPI.
The basket price and cost-of-living approaches to index construction are conceptually quite different. Nonetheless, for many (perhaps even most) purposes, the distinctions are less important than they might seem. In particular, for most of the issues that we discuss in this chapter and in the report more broadly, there are close parallels between the two approaches. In consequence, the two approaches have always drawn on one another, so that a COGI has often been modified to make it more like a COLI and vice versa. Even when the CPI was

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At What Price?: Conceptualizing and Measuring Cost-of-Living and Price Indexes
defined in terms of the basket approach, the BLS kept the cost-of-living concept in mind when making decisions about index methodology. Similarly, and as we show in this chapter, there are strong arguments for replacing a pure cost-of-living index by what is known as a “conditional” cost-of-living index which, in some respects, brings the COLI concept closer to a basket price index. In consequence of this two-way traffic, sharp differences in operating practices are uncommon. Most practical indexes or procedures for computing the CPI can be justified in terms of both approaches, though the arguments will often differ.
Still, the distinctions are important. Indexes derived from the cost-of-living approach allow for the fact that, when relative prices change, consumers tend to substitute toward the relatively cheaper items. Basket price indexes simply measure the cost of a fixed bundle of goods and are not designed with substitution in mind, notwithstanding the fact that a suitable choice of basket sometimes allows them to be interpreted as cost-of-living index numbers. The language is also important, at least in the eyes of policy makers and the public, even if those who make the index know that the formulas are the same. How the CPI is labeled affects the way that people think about it and may influence the credibility of the measure in the view of those who are affected by it. A useful analogy is perhaps the social security system, whose legitimacy in the eyes of many is enhanced by the perception that it is a fund, “the social security trust fund,” out of which they draw during retirement the contributions and interest on savings they made when working. Note too that the words “price index” and “cost of living” do not have the same connotation in common speech. The coincidence of the two ideas is relatively recent even among economists. The relationship between the “cost of things” and the “cost of living” needs to be thought about seriously. The argument that, under some circumstances, they are the same thing needs to be carefully argued and clearly laid out. The possibility that under other circumstances they are not the same thing also needs to be kept in mind.
A clear conception of what one is trying to measure also serves as a touch-stone to help resolve the many practical issues of price index construction that come up as an economy changes. Theory is the authority to which index designers appeal when it is hard to choose among alternative practical procedures or deal with new developments. Theory can be thought of as a constitution whose wise (if occasionally rather general or even delphic) principles can be applied to settle questions and disputes. A recent practical example is the adoption by the BLS of a new “geometric means” procedure for combining prices at the most detailed level of commodity disaggregation. It seems unlikely that this change would have been adopted without the shift of conceptual basis toward a COLI that followed the Boskin et al. (1996) commission report. An even more important and more difficult issue, which pervades the present report, is how to allow for quality change in the CPI. Here again, the cost-of-living framework has promise for helping design good practical procedures.

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Even so, it is important not to expect too much of any conceptual framework. In the words of Commissioner Abraham (Bureau of Labor Statistics, 1997c), “the cost-of-living index is a theoretical construct, however, not a single or straightforward index formula readily amenable to practical use.” The consumer price index means different things to different people, and it is used in many, possibly contradictory, ways. An index that is good for one purpose will not always be good for another. It is a lot to ask of any one measure that it provide a general indicator of the level of prices in the country as a whole, that it yield an accurate measure of how much Congress intended social security recipients to be compensated for price changes, that it should hold constant the “real” rate of income taxation, that it should be an appropriate escalator for the poverty line as well as for a host of government, business, and private contracts involving a wide range of people, and that it should be useful to the Federal Reserve Board for setting monetary policy. Each purpose leads to a somewhat different conceptual framework. And as the panel’s own discussions have made clear, some of the most difficult issues, such as what to do about quality change, particularly but not exclusively in the provision of medical care, do not seem to be adequately handled by any of the conceptual frameworks currently available, or at least not in a way that commands widespread assent.
The remainder of the chapter has four major sections and a technical note. The first section provides some preliminary definitions of what is meant by a cost-of-living index and by a basket price index. It also lays out some practical considerations that limit the usefulness of at least some of the concepts that might be attractive in theory. The second major section presents the theory of the cost-of-living index. The COLI is rooted in a simple economic theory of consumer behavior that is the workhorse for much practical economic discussion and policy making. For economists, the discussion in the first part of this section will be familiar, although as became apparent in the public discussions on the CPI, this theory is often only vaguely understood. It is often criticized for defects or praised for charms which it may or may not possess. But even when fully understood, the theory is not immune to criticism of its behavioral assumptions, its empirical predictions, nor its approach to well-being. These criticisms, many of which derive from the literature in psychology, are also reviewed in this section.
The next section presents a discussion of specific topics, such as how to relate price indexes for individuals or groups of people to indexes for the nation, how to choose the prices that are appropriately included in a consumer price index, how to use price indexes to compensate people or groups of people for price change, and how to adjust price indexes for changes in the quality of goods. For each topic, we show how the different conceptual approaches are relevant, and how concrete application leads to sharpening and redefinition of the concepts. Although almost all of the topics are dealt with again in subsequent chapters, they need to be covered here in order to develop the conceptual apparatus that will later be used. We present conclusions in the third major section.

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At What Price?: Conceptualizing and Measuring Cost-of-Living and Price Indexes
SETTING THE STAGE: WHAT ARE PRICE INDEXES?
In Chapter 1 we identified two distinct conceptual bases for the CPI that have dominated the public discussion and are the most relevant for the work of the panel: the fixed-basket approach, which was long the basis for the CPI in the United States, and the cost-of-living approach, which was strongly recommended in the Boskin report. In somewhat more detail, the two approaches are:
The fixed-basket approach. A basket of goods is priced in each period and the price index calculated as the cost of the basket in the comparison period divided by the cost of the basket in the reference period. Because the goods in the basket are fixed across the comparison, we call this a “cost-of-goods index” or COGI. The relevant basket for a national CPI is the set of all goods and services bought by consumers in the United States during a base period, and the prices are the market prices paid for those goods and services during the reference and comparison periods. The reference period often coincides with the base period but need not necessarily do so.
The cost-of-living approach, sometimes referred to as the “economic” approach. Prices in the comparison and reference period are compared using the ratio of the cost of living in the two periods. Instead of comparing the costs of two baskets of goods, the comparison is between the cost of maintaining the same standard of living in the comparison and reference periods. Exactly what is meant by the standard of living and the cost of living are matters that we discuss and, as we show in the next section, accurate evaluation of a COLI requires not only data on quantities and prices but also knowledge of how consumers respond to changes in incomes and prices. In practice, therefore, adopting the COLI as a conceptual basis does not imply using an exact COLI for the CPI but, instead, using one of a number of feasible approximations. The nature of these approximations, as well as their relationship to an exact COLI, is developed in the next section. But as is the case for basket price indexes, the calculation of the price index starts from a basket, or baskets, of goods and from lists of prices in the reference and comparison periods.
The research literature contains a number of other approaches to price indexes. One of the most important is the “test” approach associated with Irving Fisher. According to this, price indexes are judged according to a number of desirable “tests” that price indexes should ideally satisfy. For example, one test that is satisfied by all sensible price indexes is that, if all the prices going into the index are doubled, the index doubles too. Another framework is provided by the stochastic approach, in which it is assumed that there is some underlying but unobservable price level, around which the prices of individual goods and services are randomly distributed. For the purposes of this report and in the current historical situation, the COGI and COLI approaches are the obvious contenders

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to be the conceptual basis. Nevertheless, both the test and stochastic approaches have intuitive appeal, and they are often useful for illuminating the properties of specific indexes or for dealing with technical issues that are not otherwise easily addressed. Given the many purposes to which price indexes are put, it is often helpful to have more than one conceptual framework.
It is useful at the outset to put these concepts in context. Note first that general discussions of price index numbers are often cast in terms of two situations, usually labeled the reference and the comparison. The two situations might be geographical locations—Los Angeles versus New York, or the United States versus India—but in the case of the CPI, the two situations are different time periods, typically a reference period that is held fixed for a number of years, and a series of later periods ending with the “current” period, which in practice is a period in the recent past. The CPI is produced on a schedule that, together with the availability of the underlying data, puts constraints on what is possible. In particular, the BLS is able to collect data on prices with a much shorter lag than is possible for collecting data on the quantities of items purchased. The monthly CPI is published quickly: for example, the October 1999 CPI was published on November 17, 1999. However, the basket that was priced for this CPI came from Consumer Expenditure Surveys that collected data during 1993, 1994, and 1995 and was therefore a little more than 5 years old on average. In the past, the basket had been updated infrequently, only once a decade. Although the BLS has undertaken to shorten the time between updates, baskets available for pricing are always likely to be several years old, at least in the absence of some radical new technology, such as the extensive use of scanner data or automatic computer-based reporting of sales from retailers.
The availability of data places limits on what can be achieved within any given conceptual approach to the CPI. To stay with the current production schedule, a basket price index approach must use a base that is considerably earlier than the comparison period. The BLS can compute basket indexes relative to any base period in which quantity (or expenditure) data are available, but no later. Any methodology that requires a current basket to compute the current price index can generate price indexes only with a lag of 2-3 years. To see the implications of this, think about the two most familiar forms of the basket price index, the Laspeyres price index and the Paasche price index. In the Laspeyres, the base period basket is priced in both base and current periods—the base period is also the reference period—and the price index is the ratio of the basket’s cost at current prices and at base period prices. No information is required on the current basket. The Paasche index, by contrast, works with the current basket; it is defined as the ratio of the cost of the current basket at current and reference period prices. Because quantity information comes more slowly than price information, the production of a Paasche price index requires a longer time lag than does the production of a Laspeyres. A Laspeyres index can be thought of as an approximation to a COLI. Better approximations are possible using information on both the reference period and current period baskets. One such approximation is Fisher’s

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“ideal” index, which is the geometric mean of the Paasche and the Laspeyres. But the Fisher index cannot be produced any faster than its least timely component, so its production lag is as great as that of the Paasche index. As we show below, there are other indexes that may be more timely than the Fisher ideal, but that still do a better job of approximating a COLI than does the Laspeyres.
For constructing COLI price indexes, as for other economic statistics, there is a tradeoff between timeliness and accuracy. For some purposes, a longer wait is an acceptable price to pay for greater accuracy and closer conformity to a theoretically desirable concept. Moreover, technical and statistical innovations in data collection—such as scanner data—will likely reduce the lag in the future, at least for some components of the CPI. (Bar codes for rent, cars, haircuts, and medical care are still some way off!) As always, much depends on the purpose to which the CPI is to be put. Policy makers and many others value rapid availability, so the BLS puts a good deal of weight on timely production of the index. An index for compensating social security beneficiaries, or for adjusting income tax brackets, can presumably wait longer, though probably not 3 years.
THE THEORY OF PRICE INDEXES AND ITS CRITICS
There is a large literature in economics on the theory of price indexes. We present no more than is needed for use in this report. Much of the relevant literature makes free use of mathematics. While it is possible to give a useful verbal discussion of the main issues, clarity requires some use of formulas. We provide a verbal discussion in the main text and support the argument with a technical note that contains the most important equations. We begin with the basket price index because the ideas are more straightforward and because Laspeyres and Paasche indexes provide useful starting points for thinking about cost-of-living indexes.
Basket Price Indexes
A price index is needed because there are many goods and services in the economy, each with its own price, and each with its own rate of change in price. If all prices in the economy changed at the same rate, there would be no need to construct an index because the ratio of prices in the two periods would be the same for all goods, and any one would summarize all others. Price indexes are needed because prices do not move at the same rate. Because relative prices change over time, a way must be found to combine (or aggregate) all the changes into a reasonable measure of overall price change. This aggregation needs to take into account how much is spent on each good, so that price changes for goods on which more is spent get greater weight. One simple way to do so is to calculate a basket price index.
Beginning with a list of actual purchases in the base period, the total cost of this basket in the reference period can be calculated, as can its total cost in the

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current (or comparison) period. The ratio of these two costs is a basket price index or cost-of-goods index. If the basket is the list of goods actually purchased in the base period, this is a Laspeyres price index. If the current basket is used as the base and the price index is the ratio of the current to reference period costs of the current period basket, the result is a Paasche price index. Because the ratio of prices in the comparison to reference period differs from one good to another and because the baskets purchased in the two periods are generally different, the Laspeyres and the Paasche indexes are generally not the same. In principle, one could calculate a price index from any basket—for example, one at any point between the current and the base baskets. The relationship between various indexes cannot be known without information about how the baskets are generated and how quantity is related to price. In particular, it is not true, although it is often so claimed, that the Laspeyres must necessarily be greater than the Paasche, though this is usually the case in practice.
As we have noted, the Laspeyres index has an important practical advantage: once base quantities have been set, a Laspeyres index can be produced on the same schedule as prices are collected. The idea of continuously repricing a fixed basket is easily explained even to nonspecialists and corresponds well to what most people think of as a price index. The Laspeyres price index is the concept that is most frequently used by statistical offices around the world.
When the Laspeyres index is used to calculate a national CPI, the basket to be repriced is usually the total purchases of each good by all consumers in the country during the base period. But it is also possible to think about baskets purchased by various subsets of the population. Groups might be defined by region, to derive a regional price index; by age, to look at a price index for the elderly; or by income levels, to construct separate price indexes for the rich and the poor. Indeed, there is nothing in principle to stop us from thinking about a Laspeyres index for each individual in the economy. Different people spend their money in different ways, so that each is affected differently by changes in prices. For example, those who commute long distances to work are seriously affected by an increase in energy prices, while those who walk are not; smokers are affected by an increase in the price of cigarettes; nonsmokers are not.
Two important issues are raised by thinking about price indexes for groups or for individuals. First, not only do different people buy different baskets of goods, but different people often pay different prices for the same goods. Second, if one constructs (say) a national Laspeyres index and an individual Laspeyres index for each person in the country, how does one relate to the other? In particular, is the national price index an average of the individual price indexes? Both of these issues arise repeatedly throughout the report, so it is useful to discuss both at the outset.
The second issue, the aggregation of individual price indexes to get a national price index, is more easily dealt with if one assumes away the first issue and pretends that everyone in the economy pays the same price for everything. In

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a well-integrated, low-transport-cost economy like that of the United States, the assumption works well for many consumer goods, but there are obvious exceptions, of which shelter and medical care are almost certainly the most important. Nevertheless, imagine an economy in which everyone faces the same prices, and differs only in the total amount they spend and in the how they divide it among different goods. In the individual Laspeyres indexes, prices are weighted in proportion to individual expenditures, while in the national Laspeyres index, prices are weighted by aggregate expenditures. It is useful to think of the Laspeyres index as a weighted average of the “price relatives,” which are the ratios of current to reference prices for each good. The Laspeyres weights are the shares of each good in total expenditure, whether for the individual family or the nation (for the equations, see the “Technical Note” at the end of the chapter). The national Laspeyres then differs from the individual Laspeyres only in the weights used: For the national index, the weights are the shares of each good in national total expenditure; for each individual family’s index, the weights are the shares of each good in the family’s total expenditure.
Is the national price index an average of the price indexes for each family? Yes, but it is a weighted average, not a simple average. Because the national index uses national expenditures as weights, and because families who spend more contribute more to the national expenditure than do families who spend less, those who spend more get a higher weight in the national index. Indeed, the national Laspeyres price index is a weighted average of the individual families’ Laspeyres price indexes, with weights equal to the total expenditure on all goods by each family. This weighting was termed plutocratic by Prais (1959); the rich—or at least the rich who consume more—get a higher weight in the price index than do the poor. The obvious alternative, in which each family makes an equal contribution to the index, is called the democratic price index and would be calculated from the individual price indexes by simple averaging. In general, the democratic and plutocratic price indexes differ, and they will move differently whenever the prices of goods consumed by different income groups change at different rates. A recent example is the price of cigarettes, which makes up a larger share of the budgets of people with lower incomes. Increases in cigarette prices increase a democratic price index by more than a plutocratic index.
The Laspeyres price indexes produced by statistical offices around the world are always plutocratic, not democratic, indexes. Elsewhere in the report, we argue that, were it possible to calculate a democratic price index at reasonable cost, it should be preferred to a plutocratic index for many purposes, especially those to do with compensation. But we also argue that there are real practical difficulties in constructing the democratic index. Those difficulties help explain the universal reliance on plutocratic indexes.
The relationship between national and individual price indexes is much murkier if one allows for the fact that different people often pay different prices

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for the same product (price heterogeneity). It is still straightforward to imagine a price index for each family; one could simply take a family’s basket in a base year and price it at the reference and current market prices paid by that family. The difficulty arises at the national level; an aggregate national bundle is priced, not at the specific prices that individuals actually pay, but at prices that are averaged over all the prices paid. But such an index is not related to the individual indexes in any predictable way; in particular, the national index is no longer a weighted average of the individual indexes. More generally, it is hard to derive any good rationale for the aggregate index when price heterogeneity is important. As always, one remedy is to assume away the problem, which, in effect, is what the BLS currently does. It is a good solution if price heterogeneity is not very important, except for a few goods such as shelter and medical care, both of which require special treatment in any case. If price heterogeneity is important, or if technical change (such as the Internet) allows even greater possibilities in the future than now for firms to charge different prices to different people, there is no good alternative to working at the individual level, at least conceptually. Price indexes would be calculated for each household, or at least for a random sample of households from the population, and averaged to obtain the national index. This radical departure from current practice has many attractions but is almost certainly not feasible given current technology for data collection. We explore these matters further in Chapter 8 on aggregating across households.
When thinking about aggregation from households to nations, it is also worth giving consideration to the opposite process, that of disaggregating households into their individual members. We have used the terms individuals, families, and households more or less interchangeably, contrasting them with national aggregates. Yet multiperson households are themselves collections of individuals whose interests do not always coincide. In the next subsection we deal with the textbook “consumer,” who is assumed to make consistent choices within the available opportunities. If such an account is applied to a family or household, it supposes a unity of purpose and preference that might not be the case in practice. Recent research in economics has gone some way to looking inside the household, thinking about ways to model and to recognize non-unitary behavior. Nevertheless, none of this work has been directed toward the construction of price indexes, and in this report we work within the older tradition of regarding households and families as the basic units of the economy.
Cost-of-Living Indexes
Cost-of-living indexes compare prices, not by looking at the cost of a basket at different sets of prices but at the cost of living at different sets of prices. Basket price indexes work with the cost of specific goods and services; cost-of-living indexes work with the cost of “living.” Measuring the cost of living requires one to compare different baskets of goods and to say when they yield the same

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“standard of living.” This is done by using the economic theory of consumer behavior. Consumers always think that more goods are better (or at least no worse) than less, and they can rank different bundles of goods consistently. Consumers’ choices are governed by preferences but constrained by the market prices of goods, as well as by the amount of money they have to spend. Subject to these constraints, each consumer chooses the best (most preferred) basket among all the baskets that are affordable. The standard of living is then a measure of the extent to which preferences are satisfied. Given a set of prices that remain constant over a number of periods, the standard of living can be measured by the amount of money spent or, essentially, by real income. More technically, one can measure living standards by the size of the budget at a reference set of prices. This concept of the standard of living is a narrow one, defined entirely in terms of consumption of goods and services. It makes no claim to capture broader aspects of well-being, such as health or happiness, even though consumer choice is often described, for largely historical reasons, as “maximizing utility” or “maximizing consumer satisfaction.”
Consider an individual who is behaving according to the theory. In the reference period, there is a set of (reference) prices, and the individual has a certain amount of money to spend. This, together with the prices of goods, sets her standard of living. Next, consider a new, comparison, situation, when the prices are different. How can we think about a cost-of-living index based on holding constant not the original bundle but the standard of living? Since the standard of living is not observed, one may appear to be facing a difficult, if not impossible, task. But there is one straightforward way to make at least a first approximation, which is to calculate the new cost of the reference period basket of goods. This is, of course, the Laspeyres procedure discussed above. The key insight is that, provided nothing else (such as the quality of goods) has changed, the new cost of the original basket is always sufficient to ensure that the individual can reach the original standard of living. If the consumer buys the same bundle, her standard of living is the same. But because relative prices have changed, there may be other bundles that are just as good for the consumer, that also maintain the original standard of living. At the new prices, some of these bundles may cost less than the original bundle. If so, it will be possible to maintain the original standard of living for an amount of money less than the new cost of the original basket. Since it is always possible to reach the original standard of living by buying the original basket, the Laspeyres price index sets an upper bound on the increase in the cost-of-living index based on the original (or base) standard of living.
The difference between changes in the cost of the base period basket and changes in the cost of the base period level of living plays an important part in cost-of-living index theory, as well as in this report. The size of this difference depends on the extent to which the consumer is able to rearrange her purchases to take advantage of the fact that some goods have become relatively cheaper and others relatively more expensive. This rearrangement of purchases is referred to

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as consumer substitution, and this substitution effect is one of the most important differences between basket price and cost-of-living indexes.
An important concept in this discussion is that of compensation. When one thinks about taking someone back to his original standard of living after prices have changed, one is asking how much that person must be compensated to make up for the price change. This compensation is the difference between the cost of obtaining the original standard of living at the old and new prices; it is known in the economics literature as the compensating variation. The cost-of-living index is the ratio of the same two costs. It is this close relationship between the compensating variation and the cost-of-living index that makes the latter a natural candidate for price indexes that are to be used for compensation purposes, such as for maintaining the standard of living of social security recipients. Note that there is nothing to stop the compensation from being negative if the price change reduces the cost of obtaining the original standard of living.
The discussion so far has been in terms of the cost-of-living index associated with the reference period level of living and with the corresponding Laspeyres price index, which uses the reference period basket of purchases as the base. In this case, the COLI holds constant the reference period level of living. But one can also construct a cost-of-living index associated with the comparison period level of living and compare the cost of this level of living at the prices in the reference and comparison periods. In this case, the COLI would use the comparison period level of living as the base. If one follows through exactly the same line of argument as above (or checks the equations in “Technical Note” at the end of this chapter), one finds that this current period cost-of-living index is always at least as large as the Paasche price index comparing the current period basket at the two sets of prices. Stating the two results together, for a consumer who behaves according to the theory, the Laspeyres price index is always at least as large as the cost-of-living index using the reference period level of living, and the cost-of-living index using the comparison period standard of living is at least as large as the Paasche price index. It is important to note that these two cost-of-living indexes, one using the reference period level of living as the base and the other using the comparison period level of living as the base, are conceptually different and will only coincide in very special circumstances. As is the case for basket price indexes for which the choice of basket matters, the choice of the base level of living will also generally matter. In consequence, it is not true, though it is often loosely claimed to be true, that the cost-of-living index lies between the Paasche and the Laspeyres. Indeed, it is perfectly possible, even for a consumer who obeys the theory, for the Paasche to exceed the Laspeyres.
The cost-of-living price index is sometimes referred to as the “true” cost-of-living index, a usage which suggests that it is unique. But as we have seen, this is not generally the case. For a consumer obeying the theory, a COLI using the reference period level of living as its base may differ from a COLI using the comparison level of living as its base, and there are potentially an infinite number

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(27)
Intuitively, for small changes in price, the effect of the cost of living of a price increase is equal to the amount of the good purchased; if one buys a hamburger every day, an increase of a cent in the price of a hamburger raises one’s weekly cost of living by seven cents (provided that we do not substitute hot dogs for hamburgers, something that will not be important for sufficiently small changes in price).
Comparing the demand functions (27) with Shephard’s Lemma (26), and noting that expenditure is equal to the cost of living, equation (13), we can write
(28)
Equations (28) is a set of n partial differential equations whose solution, given knowledge of the observable functions gn, gives the cost function which, in turn, can be used to construct the cost-of-living indexes. While not all such systems of partial differential equations have a solution at all, (28) will always have a solution if the demand functions from which we begin, equations (26), come from a consumer who is obeying the theory of consumer behavior.
Practical algorithms for calculating the cost function have been worked out in the literature; a simple example is given in Hausman (1981), while a more comprehensive treatment can be found in Vartia (1983). However, these methods cannot be recommended as a practical method for statistical agencies to construct cost-of-living index numbers. The functions (26) must be estimated, which involves estimating the derivatives of each demand function with respect to total expenditures and the prices of all goods, not to mention the other factors that condition consumer behavior. In practical price indexes, there are a large number of goods, so this is a formidable undertaking. Although the theory of consumer behavior provides some help in this task, estimation is not possible without a host of additional assumptions about the structure of preferences, as well as about econometric identification, many of which are not easy to defend. There is therefore a considerable payoff to any method that avoids altogether the need to obtain demand functions.
Superlative Indexes
It is useful to start by recalling the Fisher ideal index, , defined by equation (8). Fisher proposed his index because it passes a number of desirable tests not rooted in cost-of-living theory. But it turns out that the Fisher index is a cost-of-living index for a specific utility function, and its associated cost and demand functions. In particular, if the utility function takes the form

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(29)
for some matrix A = [amn] (which must be symmetric and have a single positive eigenvalue), then the Fisher ideal index (8) is exact in the sense that if we calculated the cost function associated with (29) and used it to calculate the COLIs (18), (19) or (24), we would obtain (8) (Byushgens, 1925). If the matrix A has an inverse B, say, the cost function associated with (29) takes the form
(30)
(If A is not invertible, (30) will still lead to the Fisher ideal index.) The demand functions associated with equation (30) can be written in the form
(31)
The remarkable thing about this result is not that it is possible to find a cost function and a set of demand functions that justify a given price index, but the fact that the result is so general. Although preferences (29) are homothetic—and indeed we can see directly from (30) that the cost function is the product of utility and a function of prices, or from (31) that the shares of the budget pnqn/x are independent of x—the matrices A and B are not specified, except that they must be symmetric and have a single nonnegative eigenvalue, a requirement that comes from the general theory of consumer demand and guarantees, among other things, that demand curves slope down. As a result, and always subject to homotheticity, the demand functions (31) allow the consumer to respond to price changes in a general way; the price elasticities of demand from (31) are unrestricted, except by the general restrictions of consumer theory. The Fisher ideal index is therefore exact for a set of preferences and demand functions that do not restrict substitution behavior in ways beyond that required for the theory. It therefore permits a way of computing a general cost-of-living index without having to estimate the demand functions.
Diewert (1976) extended and generalized these results. A particular specification of preferences, or of the cost function, is said to be a second-order flexible functional form if the utility (or cost) function can provide a second-order approximation to an arbitrary utility (or cost) function. A superlative price index is then one that is exact for some second-order flexible functional form for either the cost or utility function but with preferences restricted to be homothetic. Diewert showed that the utility and cost functions (29) and (30) are flexible for homothetic preferences, so that the Fisher ideal index is an example of a superlative index.

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There are many other superlative indexes, for example, the Törnqvist index defined by
(32)
which is exact for the translog cost function, in which the logarithm of costs is a quadratic form in the logarithms of prices. The Walsh price index (9) is exact for a utility function that is a quadratic form in the square roots of the quantities; it too is therefore a superlative index. Diewert (1978) shows that these three superlative price indexes approximate one another to the second order around any given price-quantity combination, so that the choice between them is unlikely to matter much in practice.
The Fisher ideal index is computed from both the Paasche and Laspeyres, and thus requires information on both base period and current baskets. The (logarithm of the) Törnqvist index (31) is a weighted average of logarithmic price relatives, with weights that are the average of current and base period patterns of demand. Indeed, superlative indexes always require both current and base period quantity information. Intuitively, their ability to capture the substitution effects of prices has to be based on observation of the effects of the price change, which requires data on demand both before and after the change.
The analysis so far has been entirely within the framework of homothetic preferences, something that is unattractive in practice. It is possible to accommodate nonhomotheticity at the price of interpreting the superlative index as the cost-of-living index for some specific intermediate level of living. For example, Diewert (1976:122) showed that the Törnqvist price index is exact at the level of utility that is the geometric mean of the utility in periods 0 and t.
Aggregation of Cost-of-Living Indexes
The analysis of the passage from individual to aggregate indexes is essentially identical to the same analysis for the basket price indexes in the second section of these notes. Nevertheless, it is worth defining Pollak’s (1980, 1981) social cost-of-living index which is the ratio of the aggregate cost of obtaining the base levels of living at current prices to the aggregate cost of obtaining the base levels of living at the base period prices. Hence, adding superscripts h to denote individual households
(33)
where ch(uh,p) is the cost function of household h—note that there is no requirement that different households have the same preferences—and u0h is the label

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for household’s h’s indifference curve in the base period. Following through the earlier analysis, it is easily seen that the social cost-of-living index (33) is a weighted average of the individual (base period) cost-of-living index numbers, with each household weighted by its total expenditure on goods and services:
(34)
The social cost-of-living index, like the aggregate Laspeyres, is a plutocratic index.
We will not work through the results here, but it is intuitively clear—and true—that we can define a social cost of living around current living standards, and that this too is a plutocratic average of the individual current period cost-of-living indexes. The inequalities between the Paasche and Laspeyres and their corresponding cost-of-living indexes all carry through to the corresponding aggregate and social cost-of-living indexes. We can also define superlative indexes from the social aggregate indexes, such as an aggregate Fisher ideal index, and show that they are exact for social cost-of-living indexes when individual consumers have preferences that are second-order flexible functional forms. For formal demonstrations of this material, see Diewert (2000a). Finally, the whole process can be repeated using democratic instead of plutocratic indexes.
Conditional COLIs, Quality Change, and Health
As we emphasize in the main text, the use of COLIs as price indexes often requires us to ensure that a COLI changes only when prices change, and not when there are changes in the myriad other factors that affect the cost of living. In the text, this is what we refer to as the “domain” issue, that the COLI be a function of the prices of the goods and services that people buy, and not change with such things as the provision of public goods, people’s tastes, their family composition, the crime rate, the ambient temperature, or the number of years that they can be expected to live. Yet all of these things affect people’s well-being, so that we must formally modify the theoretical framework to allow for their existence. We capture those nonmarket influences on living standards through a vector of “environmental” factors, labeled e, which differs from household to household, and we recognize their effect on utility by writing the utility function in the form uh = fh(qh,eh). The dependence on e carries through to the cost function, which becomes ch(uh,p,eh). We can then follow the example of Caves, Christensen, and Diewert (1982) and Pollak (1989) and define household h’s conditional cost-of-living index between periods 0 and t as
(35)

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The important thing to note here is that, not only is the level of utility held constant between the numerator and denominator of (35), but so also is the level of the environmental variables e. As a result, changes in e from 0 to t do not affect the index. For example, if the winter is colder in t than in 0, so that more fuel must be bought to keep living standards the same, (35) will not show an increase in the cost of living unless prices change. It is a price index that is conditional on the temperature or other environmental factors. If prices remain the same in the two periods, so that pt = p0, the price index will be equal to unity. As discussed in the text, these properties are just what we want in a price index; whether they are appropriate for a cost-of-living index is a more controversial question.
Two special cases of (35) are of particular interest: the Laspeyres-type conditional COLI, in which u and e are replaced by u0 and e0, and the Paasche-type conditional COLI, in which u and e are replaced by ut and et. It is a routine exercise to check that all of the results and apparatus developed so far apply to these concepts, including the bounding relationships, the construction of superlative indexes, and the aggregation of price indexes to the national level. The results that involve a utility level intermediate between u0 and ut, for example, for superlative indexes in the nonhomothetic case, now involve intermediate levels of both e and u.
One important use of a conditional COLI is to help us think about the difficult issue of quality change. For example, if a computer costs the same today as it did yesterday but works faster and has more features, a price index that did not control for quality would not capture the effective fall in price. By contrast, a conditional COLI, which treated quality as one of the environmental goods and held it constant from 0 to t, would give a better answer. As will be argued in Chapter 4, using a conditional COLI in this way is straightforward when we know what quality change is and can measure it. Matters become more complicated when quality is not readily observed, or when we do not know the source of quality improvement. In the rest of this section, we provide an example from the important case of health care. This example illustrates how conditional COLIs work in a concrete case, as well as showing that getting the adjustment right can be very difficult in practice.
We start from a utility function in which “health” h is one argument and the vector of other goods q is another, so that the (unconditional) utility function can be written
u= f[h,q].
(36)
where u denotes utility including health, not just the well-being from goods and services. The quantity h is a latent variable “health status,” which determines life, death, and morbidity. More of it is better. Consumers have budget x which has to cover health (or medical) purchases m at price pm as well as the vector of other goods q at price p. The budget constraint is then
x =p·q+pmm=.
(37)

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Health is getting better over time in some disembodied way and is also improved by purchases of health goods m. We assume that the effectiveness of health goods in improving health also changes over time through an efficiency parameter θ. Taking these together, we can write health status at time t as
ht=δ1+θtm
(38)
where δt is the cumulated effects up to the beginning of t of the disembodied health progress, and θt is the efficiency of health goods and services m in producing health. Examples of δ would be improvements achieved through better childhood nutrition, lower pollution, or reductions in smoking. Combining (37) and (38), we can rewrite the budget constraint as
(39)
so that the disembodied technical progress δt acts like a gift of income (though because it works by reducing the need to purchase health care, its value is reduced the cheaper or more efficient health care is), and the “effective” price of health care is its quality-adjusted price pm/θt. In this set-up, the disembodied improvements in health status increase utility at any given set of prices and thus reduce the (unconditional) cost of living. Writing the budget constraint in the form of (39) allows us to see the consumer’s problem as a standard one; utility (36) is defined over q and h, and (39) gives their effective prices, p and pm/θt, as well as the effective budget available to fund them, x + pmδt/θt. Given this, we can immediately see that the unconditional cost function—the minimum cost of reaching u (including both health status and consumption) at prices pm and p can be written in the form
(40)
From (40) we see that (a) pm always appears deflated by the efficiency parameter θt, so that only the effective price matters, and (b) an increase in disembodied technical progress δt decreases the cost of living. The efficiency parameter reduces the price of health care, while the disembodied parameter effectively generates additional income.
Suppose that, in line with our discussion of the domain issue in the main text, we decide that the COLI price index should not fall in response to disembodied improvements in health status but should fall when new medical procedures or drugs mean that a given episode of illness can be treated at less cost. In the framework here, this decision can be implemented by including δt among the environmental variables, e, and holding it constant in cost-of-living comparisons while allowing θt to change in comparisons from 0 to t, so that we compare, not the prices and , but the quality-corrected prices, and /θt.The conditional cost function that we need to make this work is (40) with δt held constant,

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(41)
which no longer changes unless there is a change in price or, more precisely, a change in an “effective” or quality-adjusted price. Although it might seem odd to treat the two sources of technical progress asymmetrically, it can readily be defended as making the distinction between a price change and an income change. In our usual income accounting, we regularly treat income increases differently from price reductions, and that is exactly what is happening here. The part of technical progress that makes health care more efficient is properly counted as a price reduction, while the part that rains down from heaven (or at least is unconnected with current health care provision) is an increase in income; see again (39). Equation (41) is the conditional cost function that would be used to calculate the conditional COLI price index, by insertion into equation (35).
The problem with this approach is an empirical one, that it is very difficult to separate out the two kinds of technical progress. More people are surviving heart disease, and mortality rates are falling rapidly among the age groups most at risk. This outcome could result from better treatment, which is an increase in efficiency and which should rightly be counted as an increase in θ and as a decrease in the effective price of treating heart disease. But it could also be that people are surviving heart disease more frequently because of improvements in some background factor (e.g., they are smoking less or were better nourished in utero), without any increase in efficiency of care, even though its cost is increasing. The argument about causation, between background social factors on the one hand and technical change on the other, has been inconclusively debated in the literature for at least the past 30 years, so it is difficult to think that we can get the assignment right. If we get it wrong and attribute the effects of background factors to medical improvements, we will understate the increase in the price level. And because health status is not included in the National Income and Products Accounts, there is no offsetting effect in the underestimation of income. It is not hard to imagine a situation in which the costs of health care services are rising rapidly, driven by the introduction of new technologies and new drugs. And even if the innovations were not effective, mortality might be falling for other reasons, like the cessation of smoking or improvements in nutrition a long time ago. In this case, the price increase in medical care is real and quality correction would be the wrong thing to do, masking or eliminating the true increase in the price of health care.
The situation is complicated further by the fact that people rarely choose the quantity of their health care, setting price proportional to marginal benefits, but usually have it chosen for them, by a physician or by the combination of an insurer and a physician. Abstracting from the personal contribution to health status, through behavioral choices, we can imagine that health status is set at

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some level different from what would have been chosen by weighting price against benefit. In this case, the budget constraint (39) becomes
(42)
so that the fixed amount of health care is simply a charge on the budget for other goods. The conditional cost function with preset health care, sometimes referred to as the rationed cost function, written (U,pm,p,), can be linked to the conditional cost function with chosen health care by a linear approximation around the free choice,
(43)
where h* is the optimal health status for a consumer who is taking price into account and choosing for him- or herself, and is the shadow price (willingness to pay) for health care at the margin. When = h*, the shadow and actual prices coincide, but when more health care is provided than would have been chosen in the market, the shadow price is below the market price, so that the last term on the right-hand side of (43) is positive. According to this there is an additional element to the cost of living associated with “overconsumption” of health goods, for example, through point-of-purchase price being low or other considerations. This term is also not taken into account under any of the proposals we are considering and, if present, would further exacerbate the understatement of the cost of living through the sort of effects discussed in the previous paragraph.
Taylor Series Approximations to Cost-of-Living Indexes
Although superlative indexes are better approximations, the Laspeyres index is often itself a useful approximation to the base period cost-of-living index. This depends on a result that we already have, Shephard’s Lemma, that the derivatives of the cost function are the quantities, as well as on a result on substitution that we introduce here. If we differentiate Shephard’s Lemma (27) for good i with respect to the price of good j, we obtain
(44)
The N × N matrix of these sij(u,p) is denoted by S and is called the consumer’s substitution matrix (sometimes called the Slutsky matrix) and (44) shows that its i,jth element is equal to the derivative of the demand for good i with respect to the jth price when the consumer is held on the same indifference curve. Such price derivatives are called substitution effects and abstract from the income effects also associated with price changes. They are the key to the substitution behavior that differentiates between basket and cost-of-living price indexes. In what fol-

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lows, we will be evaluating S at the base level of utility and prices, u0 and p0; when we do so, we use the notation S0 to denote S(u0,p0).
The base period cost-of-living index number (18) uses the counterfactual cost of attaining the base period indifference curve at current prices, c(u0,pt). One way to approximate it is to take a second-order Taylor series approximation around the point u0,p0. Using Shephard’s Lemma (27) and (44), we can write this approximation as
(45)
Recall that c(u0,p0) = p0. q0 so that the first term on the right-hand side cancels with the second term in the first bracket so that, if we divide through both sides of (45) by c(u0,p0), we get the following approximate relationship between the base period COLI, , and the Laspeyres index
(46)
Thus, the difference between the base period cost-of-living index and the Laspeyres price index is zero to the first order so that the Laspeyres is a first-order approximation to the base period cost of living. The approximate difference between them, the right hand-side of (46), depends on how much substitution is possible, which is represented by the matrix S0 as well as by the size of the difference between the base and current price vectors. Note that, because the Slutsky matrix is a negative semidefinite matrix, the quadratic form on the right-hand side of (46) is nonpositive as we should expect, given that the Laspeyres is an upper bound for the base period cost-of-living index. Note also that p0 lies in the nullspace of S0, so that if period t prices are proportional to period 0 prices, the right-hand side of (46) will be zero. More generally, the right-hand side will be larger the more pt deviates from p0 in a nonproportionate manner.
We note that a similar approximation analysis can be carried out for the Paasche index and the current period cost-of-living index. We leave the details to the reader.
CES Price Indexes
Suppose that the consumer’s cost function takes the form
(47)
when s is not unity or

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(48)
when s = 1. This cost function represents homothetic preference, and the corresponding utility function is the constant elasticity of substitution (CES) utility function introduced into the economics literature by Arrow, Chenery, Minhas, and Solow (1961). The parameter s is the elasticity of substitution; when s = 0, the unit cost function defined by (47) is linear in prices and hence corresponds to a fixed-coefficients utility function with zero substitutability between all commodities. When s = 1, equation (48), the corresponding utility function is a Cobb-Douglas function. When s tends to infinity, the corresponding utility function approaches a linear utility function which exhibits infinite substitutability between all commodities. Even within the class of homothetic preferences, the CES cost function defined by (47) and (48) is not a fully flexible functional form (unless the number of commodities is two), but it is more flexible than the zero substitutability utility function that is exact for the Laspeyres and Paasche price indexes.
The base period cost-of-living index associated with (47) takes the form
(49)
Note that (49) is itself a CES function of the price relatives; in the mathematical literature, it is also known as the mean of order 1 - s. When s takes the value zero, (38) is the Laspeyres index; the Laspeyres is only a COLI when the consumer is unable (or unwilling) to substitute between goods, always consuming them in fixed proportions. As s tends to unity, (38) tends to the base period expenditure share weighted geometric mean. Provided not all the price relatives are the same, the CES index (49) is monotonically decreasing as the elasticity of substitution increases from 0 to infinity. If some consumers have an extreme aversion to substitution so that their elasticity of substitution is 0, then as relative prices change from period 0 to t, they will face a higher cost of living than consumers who substitute toward commodities that have decreased in relative price. Hence, if the elasticity of substitution s is positive and prices in period t are not proportional to prices in period 0, the Laspeyres price index, , will always b strictly greater than the corresponding CES price index, .
The CES cost-of-living index was first derived from CES preferences by Lloyd (1975), though it was Moulton (1996) who noted its usefulness for statistical agencies. In order to evaluate (50), the only requirements are information on the base period expenditure shares , the price relatives , and an estimate of the elasticity of substitution s. The first two requirements are met by the standard information that statistical agencies use to evaluate the Laspeyres price index. Hence, if the statistical agency is somehow able to estimate the elasticity of

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substitution s, the CES price index can be evaluated using the same information used to evaluate the usual Laspeyres index.
How might the statistical agency obtain an estimate for the substitution parameter s? Shapiro and Wilcox (1997:121-123) provide one method. They calculate superlative Törnqvist indexes for the United States for the years 1986-1995 and then the CES index for the same period using various values of s. They then chose the value of s (in this case 0.7) which caused the CES indexes to most closely approximate the corresponding Törnqvist indexes (which could be evaluated on a delayed basis).2 Assuming that the Törnqvist index is more or less free from substitution bias, it can be seen that the Shapiro and Wilcox procedure will generate a historical time series of CES index values which are largely free of substitution bias. Thus the CES price index, combined with a method for estimating the elasticity of substitution, could be used to provide a timely estimator for a superlative index, which can only be produced on a delayed basis. However, there are some risks associated with this methodology: namely, that past (average) movements in relative prices (which are used in order to obtain an estimator for the elasticity of substitution) are no guarantee for future (or present) movements in relative prices. It is also possible that the historical pattern of demand is determined by other factors not recognized in the analysis, such as changes in incomes, demographic factors, or tastes and technologies. Therefore a risk exists that the CES price index, based on a historical procedure for estimating s, could generate misleading advance estimates for a superlative index.
2
Essentially the same methodology was used by Alterman et al. (1999) in their study of U.S. import and export price indexes. For alternative methods for estimating, see Balk (2000).