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
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.
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.
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
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
“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
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
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
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 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
“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
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
of other COLIs, each associated with a different level of living. Just as with basket price indexes for which, in principle, one can think about using any basket as the base, so too can one use any level of living to construct the COLI. This multiplicity of possible COLIs is often inconvenient, so that it is natural to ask in what circumstances the multiple indexes are the same. This turns out to be the case if the consumer behaves in accord with what are known as homothetic preferences. This is also the condition that is necessary for the Laspeyres to be at least as large as the Paasche whatever the prices may be, a result first established by Frisch (1936). When preferences are not homothetic, there will always be at least one level of living, somewhere between the reference and comparison levels, for which the COLI lies between the Paasche and the Laspeyres (Konüs, 1924). Homotheticity in preferences implies that the way the consumer ranks different bundles of goods is the same no matter what her level of living so that, for example, the rate at which a person is prepared to trade food for tobacco, or baseball tickets for opera tickets, is the same whether the person is rich or poor. Homotheticity also implies that, as people become better off, they simply scale up their purchases without changing the pattern of consumption. However, such behavior is inconsistent with more than a century of empirical evidence dating back to Engel, who showed that the share of food in the budget diminishes at higher levels of income. Because homothetic preferences are not a reasonable description of reality, one must acknowledge a multiplicity of cost-of-living indexes.
So far, we have introduced the concept of a COLI and presented the classic results about the relationship between the Laspeyres and Paasche indexes and the associated COLIs. By themselves, these arguments are of limited practical application. Although they explain the limits of basket price indexes for thinking about cost-of-living indexes or compensation, they tell nothing about how to calculate a cost-of-living index more accurately. For example, one might argue that compensating social security recipients according to a Laspeyres-based CPI ignores their ability to substitute in response to changes in relative prices and therefore overcompensates them. But, from the discussion so far, it is not clear that it is possible to do better without a direct way of observing the standard of living.
One approach to constructing better cost-of-living indexes is to find out more about how consumers respond to changes in prices and income, something that in principle is directly observable. Between the 1950s and the late 1970s, economists worked out theoretical and empirical procedures for measuring the standard of living, given a knowledge of consumer demand functions, the relationships that tell us how purchases depend on prices and income. In particular, if the demand functions are known, cost-of-living indexes can be calculated exactly. Here then is a possible procedure. Econometric methods can be used to estimate the demand functions from market data on each individual’s purchases, prices, and income and the results used to calculate any cost-of-living index numbers that one wants. While it is useful to know that this is possible, there are serious drawbacks to recommending such procedures for routine use in national
statistical offices. Econometric modeling is often controversial because it relies heavily on judgment, and the assumptions needed to justify a given inference can often be challenged. This would make it difficult for the BLS to defend a CPI whose construction was crucially dependent on this sort of subjective work.
There is a less controversial approach that holds great promise for calculating good approximations to cost-of-living indexes. This uses what are known as superlative price indexes, which are better approximations to a COLI than the basket price indexes and can be calculated without knowing demand functions. Consider a concrete example. Using the “test” approach to price index construction, Fisher (1922) recommended what is known as Fisher’s ideal index, which is a geometric mean (the square root of the product) of the Paasche and the Laspeyres indexes. Although Fisher’s index was not derived from cost-of-living considerations, a natural question is whether it has a cost-of-living interpretation. This would be the case if there were consumer demand functions that led back to the Fisher index. This turns out to be true, as was demonstrated by the Russian mathematician Byushgens and economist Konüs in the 1920s (see Konüs and Byushgens, 1926). Indeed, the demand functions that do the trick are relatively general; although they are homothetic—which, as we have seen, is a considerable disadvantage—and have a specific functional form, they leave a large number of parameters unspecified. Subject to homotheticity, these parameters can be chosen to match any pattern of consumer substitution that is consistent with the theory. In consequence, the Fisher ideal index can be interpreted as a cost-of-living index without being specific about exactly how consumers substitute in response to changes in relative prices. Apart from the homotheticity (see below), this result comes close to squaring the circle. The statistical agency does not need to make potentially controversial estimates of demand functions. Instead, it can use the two basket price indexes, Paasche and Laspeyres, to calculate another index, the Fisher ideal index, that does what neither basket index can do by itself, namely, capture substitution behavior in a relatively general way.
In his work on superlative indexes, Diewert (1976) extended these results in important ways. First, he went beyond the Fisher ideal index and defined a whole class of superlative indexes whose members, like the Fisher ideal index, are capable of capturing general substitution responses. All of these, like the Fisher index, can be calculated from the same information that goes into basket price indexes—reference and comparison period prices and quantities. They also all require information on comparison baskets so that, like the Paasche index, they can only be produced as quickly as quantity data can be collected.
Diewert also addressed the homotheticity issue. He showed that when demand functions are not homothetic, so that there are different cost-of-living index numbers at different levels of living—and this is the relevant case in practice— superlative indexes can be interpreted as cost-of-living indexes for some level of living intermediate between those of the reference and current periods. If, over the interval of comparison, changes in the level of living are not very important
relative to substitution effects in explaining changes in purchases of goods, then cost-of-living indexes evaluated at different intermediate points between the reference and the comparison periods will not differ very much. In that case, the superlative index is a useful approximation to the change in the cost of living evaluated at the reference period of living. The cost of this extension is some loss of conceptual clarity. A COLI indexed on the standard of living in the reference period and a COLI indexed on the standard of living in the comparison period are two distinct concepts, and the superlative index yields neither one nor the other, but something in between. Superlative indexes must therefore be used with caution in situations in which the change in prices involves a substantial change in the level of living.
The apparatus is now almost complete, at least for the case in which the goods themselves remain constant over time. We have presented a theoretical concept for a cost-of-living index, described the intimate link between cost of living indexes and compensation, identified inequalities that link cost-of-living indexes to the Paasche and the Laspeyres indexes, and introduced a set of practical superlative indexes that can capture the consumer substitution effects missed by basket price indexes. But there remains one important step. Everything in this section has been presented for a single consumer, not for an aggregate or group of consumers for which price indexes are normally constructed. This step, from an individual to the aggregate or average, is far from straightforward, if only because the concept of standard of living, on which cost-of-living index numbers are based, has no immediate analog for an economy as a whole, or even for a group of consumers.
In some discussions of cost-of-living indexes, this problem is simply ignored, and all consumers together are treated as if their behavior was generated by a single “representative” consumer. This imaginary person has a living standard that is somehow supposed to represent a national level of living and for which a cost-of-living index number can sensibly be defined. Such fictions can be justified only under extremely implausible conditions (see “Technical Note 1” in Chapter 8). To pretend that the theory of living standards and of behavior makes sense at the national level is to do it such violence as to cast into doubt the value of constructing a theoretical basis in the first place. It is much better to construct a framework in which one can explicitly move from an individual to a group or the nation. To do so requires a conceptual basis for an aggregate cost-of-living index number.
The most frequently used theory was first suggested by Pollak (1980, 1981) and is known as the social cost-of-living index. It works as follows. As always, there is a reference period and a comparison period, each with its own set of prices. For each family or household in the economy one calculates the least amount of money needed in the comparison period to be as well off as it was in the base period. This amount, divided by expenditure in the base period, would give the family’s own base period COLI. But instead of doing the division, one
adds up all the required amounts over all families in the economy to get the total amount of money that would be needed to keep them all just as well off as before. The ratio of this total to the total amount of money spent in the base period is the social cost-of-living index.
An alternative to the social cost-of-living index would be to take the cost-of-living index number for each family and average those numbers over all families to get a national cost-of-living index. This democratic COLI is not the same as the social cost of living defined above, which is in fact the plutocratic COLI defined as a total expenditure weighted average of each family’s COLI. Indeed, the plutocratic and democratic COLIs bear exactly the same relationship to one another as do the plutocratic and democratic basket price indexes. The aggregation of index numbers over the population, or over groups, is not an issue that separates cost-of-living and basket price indexes.
Not surprisingly, the over- and underestimation results linking COLIs to the Paasche and Laspeyres indexes carry through to the social (and, indeed, to the democratic) cost-of-living index. If the CPI is computed as a plutocratic Laspeyres index, Pollak (1980) showed that the CPI is at least as large as the social cost-of-living index using each family’s base level of living. Similarly, if the CPI is a plutocratic Paasche, Diewert (1983) showed that the CPI is no larger than the social cost-of-living index using each family’s current cost of living. Once again, the aggregate (plutocratic or democratic) Laspeyres need not be larger than the (plutocratic or democratic) aggregate Paasche. But as Konüs (1924) showed for the individual consumer, there is at least one set of intermediate levels of living for which the (plutocratic or democratic) COLI lies between the (plutocratic or democratic) Paasche and Laspeyres indexes. Also as before, one can calculate aggregate superlative indexes, such as the Fisher ideal index (see Diewert, 2000a, for the precise arguments). These indexes will capture the effects of substitution in the aggregate and will provide closer approximations to one particular social cost-of-living index than either the Paasche or Laspeyres indexes. An aggregate superlative index of this kind is one candidate to supplement the Laspeyres-type CPI in the United States. But a superlative index cannot entirely replace the Laspeyres because it cannot be produced in as timely a manner.
Ultimately, an assessment of the ability of a superlative index to approximate a measure of the ratio of expenditures required to maintain a consumer’s base period standard of living depends on a judgment about the extent to which changes in the pattern of quantities purchased are driven by changes in income and tastes or by substitution responses to changes in relative prices.
Criticisms of Cost-of-Living Indexes
One central insight of economics is that people respond to changes in prices by selecting away from relatively expensive goods toward relatively cheaper goods. More simply, demand curves slope down. Cost-of-living theory incorpo-
rates that insight into the construction of price indexes. Substitution effects are part of what separate cost-of-living indexes from basket price indexes. Yet it is important to remember that substitution in response to price is only a part of what determines purchasing behavior in the economy; other factors, though important, are sometimes neglected in cost-of-living discussions. The aggregate bundle of goods bought by consumers responds to many forces other than prices and incomes: the demographic composition of the population is constantly changing, by age and by ethnic group; tastes are not constant, nor is the distribution of income; technology changes the nature of goods and the way that people use them. Consequently, when one uses prices together with purchases in the reference and comparison periods to form Paasche, Laspeyres, and superlative indexes, the results may differ from what would be expected if only prices had changed, as in the simplified formulations above. In addition, it is not always clear how to interpret superlative indexes when many different forces affect the pattern of purchases. Diewert (2000b) has shown how superlative indexes can be defined in a way that recognizes changes in the environment. Just as a superlative index applies to levels of living that are intermediate between the base and comparison levels, so can other changes be dealt with, noting that the superlative will apply to intermediate values of environmental variables, such as demographic composition, tastes, or income distribution. Of course, the discussion has now come a long way from the straightforward concepts from which it began.
Theoretically, a COLI seeks to measure the amount of expenditure required for a consumer to be equally satisfied in one time period as in another, or “the minimum expenditure necessary to achieve a base period level of utility” (Boskin et al., 1998:5). In theory, conceptualizing a COLI in terms of satisfaction or utility has the potential to avoid many of the conceptual problems addressed in this report, from substitution to taste and quality changes. From this perspective, what exactly a consumer consumes is irrelevant; we are merely interested in the price tag associated with a given base level of satisfaction or utility, irrespective of the products and services from which this utility is derived. At present, such an approach is utopian because there are no appropriate measures of utility (see below). Instead, the economic theory of consumer behavior sidesteps the issue by assuming that consumers maximize utility by making the appropriate choices. Hence, consumers’ choices can be taken as indicating utility. One result of this theoretical decision is that the choice-based COLI is more similar to a price index than would be the case for a COLI based on other measures of utility.
Utility and Choice
The term utility was originally introduced by Bentham (1789) to refer to pleasures and pains, the “sovereign masters” that “point out what we ought to do, as well as determine what we shall do.” From this perspective, utility is an attribute of momentary experience, and a consumption episode that gives one
more pleasure has a higher utility than one that gives less pleasure. In principle, this experienced utility (Kahneman, 1999) can be assessed by measuring the degree to which a person is pleased or displeased at the time of the experience. Although common sense suggests that past experiences drive future behavior, the relationship is more complex. Individuals’ choices are based on predicted utility, that is, expected pleasure or pain, for which people draw on memories of previous experiences, that is, the event’s remembered utility. Unfortunately, experienced utility and remembered utility are not always closely related.
As an example, consider an experiment by Kahneman et al. (1993), who had participants go through two painful experiences of different duration. In a short trial, participants immersed one hand in painfully cold water (14°C) for 60 seconds. In a long trial, they went through the same experience, but kept their hand in the water for an additional 30 seconds, during which the water’s temperature increased from 14°C to 15°C, a temperature that is still within the range of pain, as confirmed by contemporaneous reports. Accordingly, the long trial entailed the same 60 seconds of intense pain as the short trial, plus an additional 30 seconds of less intense pain. Nevertheless, participants retrospectively evaluated the longer trial as less painful, exhibiting a bias known as “duration neglect.” This bias refers to the observation that people evaluate extended episodes by drawing primarily on two data points, the peak and the end, and largely neglect the overall duration of the episode. Hence, adding a better ending to the otherwise identical experience made the longer episode seem less unpleasant. In conceptual terms, the remembered (dis)utility (pain) of the longer trial is lower than its experienced (dis)utility. Following both experiences, participants were asked which one they wanted to repeat in a third trial. In contrast to common sense and the predictions of consumer behavior theory, a majority chose the longer trial, voluntarily exposing themselves to 60 seconds of intense pain plus 30 seconds of milder pain, instead of the merely 60 seconds of intense pain of the short trial. Apparently, their reliance on the peak and end of both pain episodes led them to prefer the episode with a milder ending, even though it entailed a longer exposure to painful stimulation. Aside from demonstrating that people learn from memories and not from experiences per se, findings of this type highlight the pitfalls of inferring utility from choice.
In contrast to core assumptions of the economic theory of consumer behavior, experimental research in psychology and decision making indicates that choice, or revealed preference, is at best an imperfect measure of experienced utility. Choices are often based on erroneous assumptions, always dependent on the given context, and frequently fail to increase experienced utility even when the consumer has abundant experience with the product of choice (for a review, see Loewenstein and Schkade, 1999). Hence, decision utility, that is, the weight assigned to the desirability of an outcome in the context of a specific decision, is only weakly related to experienced utility in the Bentham sense.
By adopting the economic theory of consumer behavior as the conceptual framework for a COLI, one endorses decision utility, rather than experienced utility, as the crucial measure of utility. Unfortunately, there is little empirical support for the assumption that decision utility is an appropriate measure of the pleasure and displeasure consumers may derive from their choices. Hence, a choice-based COLI may be unlikely to measure what the theoretical definition promises, namely the price tag of a given level of utility or satisfaction.
The COLI literature often uses utility and satisfaction as interchangeable terms. However, much as choice is poorly related to experienced utility, so is satisfaction. The same modest pleasure of the taste buds can leave a person very satisfied when he knew from the outset that he picked a local “greasy spoon,” but very dissatisfied when the person expected a fancy French restaurant. In short, satisfaction is a function of experience relative to a standard, not a function of the experience per se. Hence, the same objective situation can result in very different satisfaction judgments, which has important conceptual implications.
Every year hundreds of thousands of survey respondents around the world are asked to report how satisfied they are with their lives as a whole or with specific life domains. The answers they provide are mostly based on global evaluations of their living conditions relative to some standard, such as their own past, their current expectations, or the situation of others. Which aspects of their lives they consider, and against which standard they evaluate them, is highly context dependent (for a review, see Schwarz and Strack, 1999). More important for the present purposes, research into the determinants of life satisfaction provides little support for the assumption that improvements in the standard of living will result in corresponding improvements in subjective satisfaction, at least in industrialized nations.
All industrialized nations, and certainly the United States, have experienced enormous improvements in the objective standard of living over the last five decades. Nevertheless, reported life satisfaction has essentially remained flat. In contrast to the assumptions of the economic theory of consumer behavior, access to more and better goods apparently fails to increase consumers’ life satisfaction. Cross-national comparisons suggest a similar conclusion. Although the wealth of nations is strongly related to average life satisfaction at low levels of GDP (gross domestic product) per capita, this relationship levels off once basic needs are met (see, e.g., Easterlin, 1967, 1974; Inglehart, 1997; for a recent review of these literatures, see Diener and Suh, 2000). Consistent with these observations, income is only weakly related to life satisfaction within developed nations, accounting for 1 to 2 percent of the variance in reported satisfaction. Overall, the
available data indicate that the objective standard of living plays a more limited role in consumers’ subjective satisfaction than the theory of consumer behavior would lead us to expect, a finding that poses severe difficulty for a procedure, such as the COLI, that relies on attaching a price tag to satisfaction. Even in the more supportive case of poor nations, the causal nature of the observed relationship remains ambiguous because the wealth of nations is highly correlated with human rights, democracy, and a predominance of individualistic values.
One interpretation of these findings holds that nothing can keep people happy or unhappy for long. According to variants of this hedonic treadmill hypothesis, expectations quickly adapt to new circumstances. If these circumstances are characterized by continuous improvement, ever-increasing amounts of goods are required to maintain the same level of satisfaction (see Brickman and Campbell, 1971; Campbell, 1981). Conversely, deteriorating circumstances would make people unhappy for some time, but only until expectations are back in line with reality, as long as basic needs are met. From this perspective, a satisfaction-based COLI would always show inflation in times of economic improvement because more goods are needed to maintain the base level of satisfaction, and it would show deflation in times of economic hardship once adaptation has set in. More troublesome still, a satisfaction-based COLI may show inflation as well as deflation in the absence of any changes in the price of products, hardly a desirable feature for most practical purposes. Moreover, a satisfaction-based COLI could change in the absence of any actual changes in the standard of living. Given the highly comparative nature of “satisfaction,” a satisfaction-based COLI might, for example, indicate housing inflation once a luxurious new mansion is built in the neighborhood, by making the existing houses seem less satisfying by comparison.
An alternative interpretation of the weak relationship between the standard of living and consumers’ satisfaction suggests that subjective well-being and life satisfaction may be largely a function of people’s temperament and genetic endowment. Twin studies suggest, for example, that “the reported well-being of one’s identical twin, either now or 10 years earlier, is a far better predictor of one’s self-rated happiness than is one’s educational achievement, income, or status” (Lykken and Tellegen, 1996:189). From this perspective, the subjective well-being of consumers is largely independent of the level of material well-being, which is consistent with the available survey data.
However, a word of caution is in order. As noted above, reported satisfaction is not closely related to experienced utility. At present, for example, it is not known if a new mansion next door would actually make the neighbors feel less comfortable in their old homes on a moment-to-moment basis or if it would only reduce their global evaluations of their homes, as expressed in satisfaction judgments (see Kahneman, 1999, for a discussion). It is conceivable that a higher standard of living may actually result in more positive moment-to-moment experiences with life which, however, may not show up in global satisfaction judg-
ments made relative to a higher standard. These issues are the topic of current research on the measurement of experienced utility, which is arguably the most adequate measure for assessing the benefits of the material conditions of life.
As this selective discussion of the complexities of utility and satisfaction illustrates, the conceptualization of a COLI in economic theory is based on a very specific definition of utility, namely the decision utility revealed in choice. Decision utility, however, is a poor measure of utility as pleasure and pain, as conceptualized by Bentham. Yet little may be gained for the purpose of an index system by adopting a broader conceptualization of utility. Although measures of experienced utility would provide the most meaningful assessment of consumers’ quality of life, these measures would most likely have properties that make them undesirable for an index system. Most importantly, they may indicate inflation or deflation in the absence of any changes in the price of products.
One difference between the COLI and COGI approaches is how much theory is built into each. The fixed-basket index (COGI) uses theory to choose the weights for the price index, but it takes very little theoretical background to explain to an intelligent but untrained bystander that a consumer price index ought to price the things that consumers buy. If one thinks of a COGI as the cost of things, one needs to know which things—to which the answer is, the things that people buy. Of course, this still leaves unanswered the questions of which people, when did they buy them, and what varieties of goods. The “which people” question is about the group for which the index is designed, whether one is concerned with individuals or aggregates, with a democratic or plutocratic measure. The question of when leads to questions of a Paasche (the comparison bundle) or a Laspeyres (the reference bundle) index. As already noted, the Paasche index is not feasible for a real-time CPI. If it is acceptable to produce indexes with a delay of (perhaps) 3 or more years, when data on the comparison period purchases will be available, and if there is little to choose between the two indexes on other grounds, it makes good sense to compute superlative indexes tailored to offer approximations to the COLI concept. The third question, about the varieties of goods, raises issues of quality, of whether a good is a simple irreducible “atom” on its own, or should be thought of as a bundle of characteristics. This question is perhaps the hardest to answer, and attempts to do so require a good deal of theoretical structure, in any framework. Indeed, current BLS practice, within a broad COGI framework, requires continuous judgments, many of which are based on theories of how consumers behave and of the relationship between quality and price.
In contrast to a COGI, a COLI framework builds on the economic theory of consumer behavior. Many would argue that this theory is so simple and obvious that there is little reason not to adopt it, but as we have shown in the previous section, this is not clear to everyone. Still, there are great benefits from using the theory. The idea that, as prices change, people substitute toward relatively cheaper goods strikes most people as reasonable, and the theory provides the apparatus to deal with this idea in a formal way. Given the idea, one has a good basis for the further idea that someone with enough money to buy the original bundle in the new situation is at least as well off as before and may be better off, which is the idea of substitution bias (though there is still the further step, that a measure of compensation is the same as a measure of the change in the price level; see below). Many (although not all) price statisticians around the world have at least muted their original suspicions of the COLI approach in the face of its usefulness for thinking about important practical issues, such as the problems with constructing lower-level price indexes. As Triplett (1999) argues, economists use the theory of consumer behavior for all sorts of purposes, and their very success as public policy practitioners attests to the usefulness of their approach, even if they do not believe its literal truth. However, skeptics might attribute the same success to economists’ willingness to rely on deductive reasoning even in the face of contrary evidence (such as that reviewed above).
Public Perception and Understanding
When it comes to public understanding, basket price indexes have an advantage over cost-of-living indexes: they are simple and can be explained in seconds to almost anyone. Against this must be noted that, in practice, the actual (modified) Laspeyres indexes used by the BLS are quite complex, so that much of the clarity is lost in day-to-day practice. However, this is true of almost any complex measure or, indeed, any complex object. One might understand very well in a general way why an airplane flies, and the knowledge probably makes people feel better when flying. But the detailed construction of a modern airliner is certainly unknown to almost everyone.
The cost-of-living index is a good deal harder to explain. The basic concept of comparing how much it costs to live at different prices is relatively straightforward, but making the concept practical or precise is quite difficult, sometimes even for those who support the concept. Part of the problem may be less a lack of understanding than a genuine intellectual resistance to the approach. For example, there is nothing incoherent in a position that accepts the argument about substitution bias and accepts that fixed-bundle compensation is overcompensation, but does not accept that the price level and the level of compensation are the same thing, or that the cost of things is the same as the cost of living. According to this view, the COLI is the right framework for calculating compensation, but not necessarily for calculating the price index.
Nevertheless, there is some disagreement, including among members of the panel, on how difficult it is to understand the COLI concept. Some argue that the basic idea of a cost-of-living index is straightforward and that the difficulties come with the detailed implementation. According to this view, there is no great difference between a COLI and a COGI, since the implementation of the latter also requires much that is complex and difficult to understand. Others challenge the comprehensibility of even the basic concepts underlying the COLI. The idea of holding constant the standard of living requires some notion of what is meant by a standard of living and whether this is “happiness,” “satisfaction,” “utility,” or something else. As we have shown, it is indeed something else, and explication of it takes a good deal of space. Finally, there is room for argument about the importance of public perception and understanding and how much weight it should be given in the construction of an index number for which there will inevitably be a great deal of technical detail.
The basket price index is just that, the cost of a basket of goods relative to the reference. The issue of substitution bias does not arise, at least if one is happy with the choice of basket. But if one has several reasonable choices of baskets and if they give different price indexes, one has to choose between them and recognize that at least part of the difference between the indexes, say the Paasche and Laspeyres, comes from consumers substituting in response to changes in relative prices. Once again, though, substitution is not the only reason—nor even necessarily the most important—for changes in the bundle; changes in tastes, in quality, and in the sociodemographic composition of the population also have their effects. If one chooses to maintain the distinction between the cost of living and the cost of things, with the former relevant for compensation and the latter for a price index, then recognizing the existence of substitution does not necessarily involve recognizing the existence of substitution bias. A fixed-basket index is biased as a measure of the cost of living but not necessarily as a measure of the price level itself.
From the COLI point of view, which does see the price level in terms of the cost of living, the Laspeyres index is at best an approximation that overstates the change in the price level between the reference and comparison periods. The degree of overstatement is the substitution bias, and it will tend to be larger when the difference in relative prices is large and when consumers’ ability and willingness to substitute one good for another are high. But it is not necessarily true, as is sometimes supposed, that the overstatement of the COLI by the Laspeyres becomes increasingly severe simply as the time between the base period and the (current) comparison period increases (see “Technical Note” at the end of this chapter). Indeed, there is some empirical evidence that, for recent U.S. history,
price indexes that allow for substitution do not rise faster than the CPI (see Shapiro and Wilcox, 1997).
Two points are noteworthy in this context. First, even if one is committed to a COGI approach for the CPI, there is nothing to stop one from using a COLI for compensation purposes. Indeed, if Congress mandated that social security benefits be indexed to hold constant the living standards of social security recipients who have no other income, the COLI would certainly be the appropriate index for escalation (at least subject to the issues related to compensation; see below). In practice, this might mean retaining the Laspeyres approach for the CPI, while recognizing that CPI-based compensation is overgenerous, and compensating people by the growth in the CPI less some modest amount in recognition of substitution bias. More sophisticated compensation schemes could be implemented using superlative indexes, albeit with a lag (discussed further in Chapter 7).
One alternative to waiting for superlative indexes is to use other indexes that make a somewhat less exact allowance for substitution but that can be produced on the same schedule as the CPI. There are a number of possibilities. One is a constant-elasticity-of-substitution (CES) price index, suggested by Lloyd (1975) and recently evaluated by Shapiro and Wilcox (1997). A CES index starts from the price relatives for each good, the ratios of the price in the current period to the price in the base period. In the Laspeyres index, these relatives are averaged, using as weights the share of the budget devoted to each good. In a CES index, each price relative is raised to a power (for example, 0.5) before being weighted and added up. The final index is then obtained by raising this weighted sum to the power not of the exponent but of its reciprocal (see “Technical Note”). If the exponent is 0.5, one is weighting together the square roots of the price relatives and squaring the result. If the exponent is 1.0 (unity), one would be reproducing the Laspeyres; at the other extreme, with an exponent of zero, one would have the expenditure-weighted geometric mean of the price relatives. The smaller the exponent, the more goods are substitutable for one another. Indeed, if one subtracts the exponent from unity, the result is the measure known as the elasticity of substitution.
The constant elasticity of substitution index is exactly equal to the cost-of-living index number if preferences are homothetic and if all goods are equally substitutable for one another. In practice, historical data could be used to choose the exponent that brings the CES index as close as possible to some superlative index, such as Fisher’s ideal index. And although the assumptions of homotheticity and equal substitution are not realistic, such an index will nevertheless capture substitution bias in a way that the Laspeyres does not, and it will do so without requiring data on current purchases. Thus, if substitution is the main concern, the CES index has attractions as a basis for the CPI.
But there are also arguments against the use of a CES price index. Goods are
not equally substitutable for one another. If prices change in a way that highlights the assumption to the contrary, a CES index can be quite misleading. Suppose, for example, that the dollar depreciates relative to the currencies of East Asian electronic and automobile producers, so that the imported versions of these products become more expensive. This will hurt American consumers, though the damage will be offset by substitution away from Asian to European and domestic substitutes. If some such substitution is built into its construction, the CPI will rise by less than the increase in the prices of the imports, which is exactly what ought to happen. Now imagine instead that an oil shock increases the price of gasoline and home heating products. In this case, there is much less scope for substitution, and the increase in the CPI ought to be much closer to the increase in the price of fuels: BLS cannot tell consumers that the CPI has not risen by much because they should drive their cars on milk or on orange juice! A superlative index can capture the difference between the two cases because it uses information on purchases after the price change and is sensitive to the fact that the shock induces a much smaller decline in demand for fuels than for imported electronics. But a CES price index treats both identically and assumes that fuels are just as substitutable as imported electronics. In one case or the other, the CPI will be wrong and possibly quite wrong.
Of course, there is no guarantee that a Laspeyres-based CPI will do better as an approximation to a COLI. Indeed, since a Laspeyres is itself a CES with an exponent of unity, a good choice of exponent will certainly lead to an index that does at least as well as the Laspeyres. It is also conceivable that more elaborate CES indexes—such as a two-stage CES, which has the same substitution elasticity between broad groups of goods, with different substitution elasticities within each group—could do even better and remedy the equal substitution problem of a simple CES.
A CES index has an advantage over a superlative index in its timeliness, but it is otherwise inferior. For example, there would be no point in using a CES index instead of a superlative for looking at long-run trends in inflation or for other historical analyses. With some improvement in data collection, much of which is already under way, a superlative index could be produced with a delay of only 1-2 years. Given data up to that time on both a Laspeyres-based CPI and the superlative, it would be possible to make an informed estimate of what the superlative is likely to be, even in advance of its calculation. In this situation, it is not clear that a real-time CES index adds very much. Compensation, such as social security compensation, could ultimately be tied to a superlative index. Interim payments could be made from a forecast of the superlative, with forecast errors rolled into subsequent cost-of-living adjustments (see Chapter 7).
One area in which a COLI concept has already entered BLS practice is the treatment of lower-level price aggregation. This is the procedure whereby the BLS combines prices of the most finely defined goods, such as different varieties
of apples or different brands of VCRs. Common sense suggests that consumers are much more willing and able to substitute between two kinds of apples than between an apple and an orange or between two kinds of VCRs than between a VCR and a stereo system. Some lower-level categories like medical supplies or sporting equipment are clearly different: a left-leg prosthesis is not a very good substitute for a right-leg prosthesis, nor a golf club for a tennis racket! Conversely, some goods within an item category may be more substitutable for each other than is explicitly assumed by the BLS procedure for combining prices at the lower level.
At this disaggregated level, the COLI perspective has a distinct advantage because substitution is clearly important and because the COLI recognizes it. It is probably easy to explain to consumers that an increase in the price of a Gala apple is not so serious as (i.e., requires less compensation than) an increase in the price of all apples, even if they usually buy the Galas. And the COLI approach gets this right. In response to this sort of argument and after publication of the Boskin commission’s report, the BLS switched from its previous Laspeyres approach to a weighted geometric means procedure (though not for all categories, including artificial limbs). This geomean method will give the right answer under rather specific assumptions about the degree of substitution between goods. Although the specific assumptions are unlikely to be exactly true, most observers regard the change in procedure as an improvement. The change also marks the BLS’s own change of perspective from a COGI conceptual basis (informed by COLI considerations) to an explicit COLI basis. It is probably the BLS’s first attempt to build substitution effects into the CPI itself, rather than into an experimental index.
We also note that, even prior to the introduction of the geomean procedure, the BLS used a seriously modified Laspeyres index that incorporated a procedure called seasoning, by which the weights come from a different period than either the reference or the comparison periods. The seasoned Laspeyres (including other modifications) used by the BLS was a long way from the simple fixed-basket approach, so that much of the original simplicity of the concept had already been lost. Analysis of the effects of seasoning (for example, by Shapiro and Wilcox, 1996, following the analysis of Reinsdorf and Moulton, 1995) shows that a seasoned index has different biases than an “unseasoned” index, so that it is not clear that the geomean index will always be practically superior to the seasoned Laspeyres (see “Technical Note” for more discussion).
Quality adjustment is possibly the area in which the COLI has the greatest advantage over the COGI approach. For the COGI, there are both practical and conceptual difficulties in trying to work with a fixed basket of goods when the functions and even definitions of goods change. When qualities are changing
rapidly, it can be difficult even to find the original basket. Where would one find a 16 MHz computer, let alone an electric calculating machine? And even if one could do so, the results would provide no clue to the effective reduction in price that comes from quality improvements. At worst, a fixed-basket methodology is impossible for such cases, and at best it is irrelevant. The COLI approach is more helpful: when the qualities and even definitions of goods are changing, it makes more sense to try to work with a constant standard of living than with a constant bundle of goods. With the COLI approach, one has conceptual clarity—one can compare the cost of living in two situations in which both prices and qualities of goods are different—which is not possible in the COGI framework.
As we have already seen in other contexts, however, the differences between basket and cost-of-living approaches are rarely so clear cut, and the same is true for quality. Basket approaches can sometimes be modified in sensible ways to deal with quality change, and the COLI approach, while conceptually clear, can sometimes fail to give solid practical guidance. We referred above to the results in the economics literature that show how to construct cost-of-living indexes from demand functions linking consumer purchases to prices and incomes, but these results were derived for situations in which quality is constant. There is no comparable body of theory that allows construction of cost-of-living indexes when prices, income, and quality are all changing. To make progress, one must know more about how quality is changing; and indeed there is a growing, albeit experimental, literature looking at ways of modeling quality change in theory and in specific empirical situations (see Feenstra, 1994, 1995; Berry et al., 1995). But as we argue in Chapter 4, knowledge of quality change can often also be used to adjust the definition of goods in a basket price index approach. Thus, once again, the distinctions between the two approaches are blurred. Indeed, in the judgment of the panel, the central issue in constructing quality-corrected price indexes is not the distinction between COLI and COGI approaches, but the measurement of quality change itself.
In some cases, it is relatively straightforward to see how quality should be adjusted. For example, when coffee now comes in a 12-ounce pack instead of a 16-ounce pack, an obvious (if not necessarily precisely correct) procedure would be to add a third to the price of the new pack before comparing it with the price of the old pack. Such adjustments are routinely made in repricing a basket of goods. In some cases, quality can be thought of as a special form of repackaging. Razor blades might give 10 shaves instead of 5, or a gasoline with an additive might give 25 miles per gallon instead of 20. In both cases, it would make sense to price the cost per shave or the cost per mile. None of this differentiates between basket and cost-of-living approaches, though the latter would make an allowance for substitution toward the improved commodity when cost per shave or cost per mile fell.
Unfortunately, quality changes are seldom easily converted to changes in effective quantity. A computer may run at 500 MHz instead of 200 but have the same infuriatingly slow connection to the Internet. A quality correction requires some way of knowing how to value an improvement in one characteristic while another is held constant. Some new goods, such as cellular telephones, home video machines, or digital cameras, mix characteristics of old goods with wholly new characteristics. The valuation of new goods with entirely new characteristics poses even more difficult problems than does the valuation of characteristics that previously existed but are brought to market in different combinations.
Once again, econometric analysis can provide some insights into the behavior of consumers and producers, including how consumers respond to new goods and how they value the underlying characteristics of goods. Hausman’s (1997) work on breakfast cereals provides a widely cited example of an econometric analysis of a new product. But the identification of such relationships is rarely uncontroversial, and we believe it would be unwise for statistical agencies to condition important data on the validity of specific econometric models. Pricing the underlying characteristics of goods is the aim of the hedonic technique whereby the market prices of goods are related to the amounts of each characteristic that they contain. The method requires prior knowledge of what characteristics consumers value, and its application often raises questions of interpretation and econometric technique. Even so, hedonic methods are probably the best hope for improving the way in which quality adjustments are made.
There are fundamental questions as to whether it is possible, even in principle, to measure certain kinds of quality change. In particular, taste change is sometimes indistinguishable from quality change. For example, becoming a vegetarian allows one to obtain the same nutrition from less food expenditure, just as would an improvement in the quality of fruits and vegetables. If all goods suddenly became twice as good, everyone would be better off, but there would not necessarily be any changes in consumers’ purchases. One might imagine an economy in which everyone agrees that the “goodness” of goods has increased but where the proponents of the “new goodness economy” say it has increased twofold, while skeptics say it increased by only 50 percent. There is no way of inferring from consumers’ behavior which is right, nor of making the corrections to their cost of living that such a change would presumably require. Unless a great deal is known about the nature of the quality change—for example, what the goodness of a good is and how much of it there is, perhaps from the manufacturer’s engineering specifications—it is generally not possible to infer quality from examining what has happened to consumer purchases. Yet if someone becomes better at using a good and so gets more out of the same purchase (say, a golf club)—something that is a taste change, not a quality improvement—the associated behavior will look the same as if the manufacturer had increased quality by improving, or putting more goodness into, the good.
The Domain of the Index: Conditional and Unconditional Cost-of-Living Indexes
Any price index, whether derived from a COGI or COLI approach, needs a list of goods that are covered and a list of goods that are not. In its discussions the panel came to call this the domain issue. There is general agreement, within the panel if not universally, that the domain should follow current practice, including market goods and excluding nonmarket goods (e.g., public goods, the environment, crime, life expectancy). In addition, only current goods and services should be covered, not leisure, nor goods and services in past or future periods, even though consumers currently get part of their well-being from consuming time, as well as from the contemplation of past and future purchases. These conclusions are consistent with either a basket or cost-of-living approach to index number construction, though the arguments are different. As we shall see, the definition of a cost-of-living index needs to be modified to become a conditional cost-of-living index (for more technical discussions, see Caves et al., 1982; Pollak, 1989; Diewert 2000a). This modification is somewhat controversial, and it has important implications for the application of the cost-of-living framework in other contexts.
For a COGI, the domain can be anything that is thought to be suitable. For example, one can select what people think ought to be in a price index, recognizing that they will certainly need some guidance on how to handle such matters as interest rates or durable goods. Such a procedure would almost certainly lead to the inclusion of the prices of market goods and services. People recognize that it is a good thing when life expectancy goes up, when crime goes down, or when a new product (cell phones or Viagra) makes life more enjoyable, but they seldom think that such improvements reduce the level of prices.
The COLI approach can get to the same place but requires more steps, some of which would be resisted by those who take a comprehensive approach to cost-of-living indexes. A good place to start is with the example of a local government raising sales taxes to build a bridge. Some local taxpayers would prefer to keep their money while others would prefer the bridge, so that the tax to fund the bridge will make some people better off and some worse off. Suppose that, on average and taking into account the taxes, people are about as well off after completion of the bridge as they were before. What has happened to the cost of living? According to the comprehensive approach, nothing. Although prices of goods are higher, the bridge brings benefits which, by assumption, exactly offset the increased cost of goods. So consumers need no compensation, and the cost of living has not changed. By contrast, the COGI approach says that prices have gone up, which means that the CPI has gone up. It is not that the bridge is irrelevant to people’s welfare or is not worth anything, but simply that the existence of the bridge seems irrelevant to the measurement of the price level. This seems like an excellent example of a case in which the price index and the cost of
living are different things in practice, not just in theory. And, indeed, many of those who argue for the adoption of a comprehensive COLI approach see its treatment of such cases, not how it handles substitution, as the main advantage of COLI over COGI approaches.
One can, however, make the COLI give the same answer as the COGI by constructing a “conditional” cost-of-living measure, defined as the minimum expenditure on market goods needed to attain a given standard of living when the provision of nonmarket goods is at some specified level. In this way, the conditional COLI changes only when prices change. Without changes in prices, the conditional COLI is constant: it cannot be altered by changes in nonmarket goods or changes in the environment (such as the provision of the bridge) or by an increase in life expectancy.
The conditional COLI can be used to hold constant not just the provision of public goods but anything that one does not want to affect the price index. A good example is temperature: in unusually cold winters or hot summers, families have to spend more money to attain the same level of comfort (the same temperature in their homes). Should the CPI rise because the winter is unusually cold, even if the price of heating fuels remain constant? For the panel, the answer is no. The cost of living has gone up, but prices have not. We prefer a price index that does not change in response to temperature changes alone. For this reason, the preferred choice for a cost-of-living index is not the comprehensive or unconditional cost-of-living index but a conditional cost-of-living index that holds constant all environmental nonprice factors that affect people’s well-being.
The conditional cost-of-living index can exclude those things that people believe should be excluded—such as fluctuations in winter temperature—leaving it somewhat more like a price index and somewhat less like a cost-of-living index. For most purposes, a conditional COLI is arguably the right concept. It responds to price changes as a price index should, and it takes into account consumer substitution. Nevertheless, a conditional COLI has problems in dealing with some issues and arguably gives the wrong answer in some of them. Some economists object to almost any exclusions. For them a conditional cost-of-living index is no longer a cost-of-living index. Thus, in the case of a sales tax for a bridge, they think the value of the bridge should be taken into account or, if that is impractical, the increase in sales tax should be excluded from the price index. Similarly, the price index should be decreased for an increase in health status or a reduction in the crime rate because both reduce the amount of money required to reach a given standard of living. Although the Boskin report does not formally recommend such a position, it contains a number of statements that are sympathetic to such a treatment.
Even if such arguments are not rejected in principle, there are practical examples for which the case for a conditional COLI is unpersuasive. One example is the construction of regional or city price indexes. Nothing in cost-of-living theory says the base and comparison situations cannot be different places, rather than different times, and there are many situations in which such cross-
place comparisons are needed; for example, the State Department needs to make cost-of-living adjustments for employees living in foreign countries or when an employer wants to adjust salaries for the cost of living in different U.S. cities. A cost-of-living index that compares Phoenix with San Francisco would surely recognize that homes in Phoenix require more air-conditioning than homes in San Francisco. It is hard to see the purpose of a conditional cost of living computed under the assumption that Phoenix has San Francisco’s climate, or vice versa. Yet it is precisely this assumption that is needed to prevent a COLI from changing with climate fluctuations over time.
A conditional COLI can also limit our ability to handle quality change. Most people would probably agree that general increases in life expectancy that are not caused by changes in medical care or other market goods should not reduce the CPI, but a conditional COLI should take account of changes in life expectancy due to improvements in the quality of medical care, such as better treatment of heart attacks. Yet it is not clear where to draw the line between general increases in life expectancy and more specific quality improvements, for example, in the treatment of depression or heart attacks or in cataract surgery. When new drugs make it easier to ameliorate an episode of depression or when new techniques reduce the cost of cataract surgery, most would probably want the change to be reflected in a cost-of-living index, and perhaps even in the price index. Some help comes from an appropriate redefinition of commodities, for example as the treatment for an illness, rather than the drugs and medical services themselves. But if quality improvement comes through new technology and if a conditional COLI treats technology as an environmental variable to be held constant, the contribution of quality change to effective price reduction may be ignored or at least understated. Indeed, if one thinks of a conditional COLI as designed to prevent changes in the index level when prices are constant, then it would seem to rule out quality adjustments to price. To capture the contribution of technological change to effective price reduction in the price index, one must remove technology from the conditioning variables in a conditional COLI. Yet as the example of life expectancy shows, such “unconditioning” must be selective; one must hold some technologies constant while others are allowed to change (see “Technical Note” for a more formal discussion of this point). The difficulty of deciding on what to condition is further aggravated by the difficulty in practice of separating changes in technology from changes in tastes, as when the BLS counted as a quality improvement not only the switch from cotton to synthetic shirts but also the subsequent switch in the opposite direction.
One more example is worth thinking about. The availability of a new drug like Viagra certainly makes many people better off. There is no obvious way of redefining one or more commodities so that this shows up as a price decrease. Indeed, many people—including most members of the panel—are quite uncomfortable with the idea that the introduction of Viagra should reduce the CPI. (Here we are considering specifically the “new goods” effect. If, hypothetically, there had been a CPI stratum “treatment for impotence” and the introduction of Viagra
had lowered the price of that treatment, then perhaps a price decline for the good, so defined, could be justified. See Chapter 6 for more discussion of this issue.) Yet, on average, consumers are better off than before, even if their incomes and the prices of all other goods are the same. If the price level is not adjusted, the benefits of the technological innovation are missed and show up nowhere in the accounting system. Many economists are concerned that these phenomena are pervasive in modern economies, where the growth in quality has replaced the growth in quantity as the main engine of increased well-being, and where the production of new commodities is as important for economic growth as is the more efficient production of old ones. Neither Laspeyres price indexes nor conditional cost-of-living indexes are likely to capture this progress; it would likely be better captured by an unconditional COLI.
At present, there is probably no alternative to a selective treatment of whether or not the state of technology should be a conditioning variable for a COLI. Such selective treatment of the same variable, technological advance, it not a very comfortable position. It leaves much room for discretion of a kind that will undoubtedly be a source of debate for years to come. More generally, the unacceptability of the unconditional cost-of-living index, together with the apparent impossibility of devising a general way of conditioning the conditional cost-of-living index, has brought several members of the panel to the view that there has probably been too little research on other conceptual approaches, such as the test or stochastic view of index numbers.
There is another somewhat more technical issue concerning conditional cost-of-living indexes. The conditional cost of living itself, and thus a conditional COLI associated with it, depends not just on prices and the level of living but also on the levels chosen for nonmarket goods and other conditioning variables. If the government provides public schools, a family’s cost of living is different than when it does not, and the way in which the family’s COLI responds to price changes—the price of books or the price of tutoring—will depend on what the public school provides. Similarly, the conditional COLI of someone with a house to heat may be unaffected by changes in the price of fuel when the average temperature is 70°F, but very sensitive to fuel prices when the average temperature is 50°F. A conditional COLI will be independent of the environment only when preferences for market goods are “separable” from nonmarket goods or from characteristics of the environment. Separability requires that the way a family spends its money on market goods must be independent of the provision of nonmarket goods or that a family’s choice between food and fuel is independent of the outside temperature. Such conditions are unlikely to hold. Indeed, some public goods may altogether supplant some private goods from a family’s budget. When separability does not hold, the conditional cost-of-living index will be different depending on the level at which one chooses to hold constant the provision of nonmarket goods. It is hard to imagine such effects being taken into account in any practical CPI.
Changes in Tastes
A cost-of-living index compares the costs of equivalent standards of living under two different sets of prices. If someone becomes a vegetarian or decides that she prefers not to smoke, the cost of any given level of living will change, even with no change in prices. This can perhaps be dealt with in the same way as a change in the environment, regarding vegetarianism or nonvegetarianism as background variables that are subject to change. In this way, one can think about an unconditional cost-of-living index that calculates the change in the cost of living that comes about from both price and taste changes—becoming a vegetarian reduces the cost of living. In contrast, a conditional cost-of-living index calculates the change in costs, holding the original tastes fixed. It is not entirely clear how much of the original COLI concept is retained under either of these devices. The unconditional COLI essentially attaches values to different systems of tastes, something that most economists prefer to avoid. The conditional COLI is evaluating price changes according to tastes that are no longer valid, so that if tastes change rapidly and if the base period is held fixed for a long period of time, the conditional comparisons will become less and less relevant to consumers. But this is conceptually no different from the usual problems with selecting any base, and it is merely an argument for frequent updating of the base.
More serious are the practical questions, in particular, how to recognize taste change when it has taken place and, having done so, how to correct for it. Taste change is conceptually closely related to quality change. One is a change in the nature of goods, the other is a change in how goods are perceived. In general, it is not possible to distinguish one from the other by watching how consumers behave. Quality change is often directly observable from examination of the goods; observing taste change is much more difficult. Thus, one has little choice but to accept the conditional approach and to assume that tastes are constant.
In a COGI approach, where tastes are not mentioned, it might at first seem that taste change is not an issue. But the choice of a sensible basket is almost impossible if tastes are radically different in the two periods. This issue comes up forcefully when computing price indexes that compare price levels between two diverse countries such as the United States and India. Because the nature and pattern of consumer expenditures are so different in the two countries, with many goods that are bought in one not bought in the other, there is no comparable basket to price. Attempts to use one basket or the other can give absurd results if a staple in one country is not available in the other or is available only occasionally at an extremely high price.
Neither the COGI nor the COLI approach (nor any other we know of) is likely to do a very good job of constructing a CPI when there is a great deal of taste change. In this context, one might be seriously concerned about some of the psychological phenomena discussed above, that nothing makes people happy for long or the hedonic treadmill, which condemns a consumer to ever-increasing
expenditures to maintain a constant level of satisfaction. It is not clear what to make of COLIs in such an environment, though one line of approach is through consideration of the literature on habit formation, according to which the cost of living increases in response to previously increased consumption because of the “needs” induced by the earlier consumption, so that the unconditional cost of living will drift upward relative to the conditional cost of living. Once again, a conditional COLI seems to be the appropriate concept for measuring the price level.
Using Indexes for Compensation
Since a COLI is calculated by measuring compensation, it is the natural index to use for compensation and indexation purposes, though one might want different COLIs for different people and circumstances. But a COGI could be adjusted to make it more like a COLI, for example, by making an allowance for substitution bias. Similarly, one could adjust a COLI to make it more like a price index by narrowing the domain (conditioning). However, there are practical and conceptual issues that arise in cost-of-living adjustments when people are sellers as well as consumers of goods, as well as when people have incomes other than those which the index is intended to compensate.
In an industrialized economy, and putting aside the supply of labor which is outside the domain of our index, the most important group of consumers who supply goods are homeowners, who sell housing services to themselves. The issue can perhaps be most clearly seen if one considers the example of a farmer who grows beans for market and uses the proceeds to buy as much as he can afford of a fixed bundle of goods. Sales of beans are his only source of income, and he saves nothing. Suppose that all prices increase by 10 percent. The farmer needs no compensation: the cost of his bundle of goods has gone up by 10 percent, but so has his income. At the conceptual level, a COLI can once again be made to give the right answer but only if one separates the COLI from its basis as compensation. This can be done by adopting the (perhaps strained) device of separating the farmer into his production and consumption selves, so that the latter can be said to have suffered a 10 percent increase in prices. The farmer as producer gets the profits and is better off, while the farmer as consumer pays more for his consumption and is worse off. The “integrated” farmer is both 10 percent better off and 10 percent worse off; he has had no net change in real income. The compensation required by the farmer for the price increase is zero, though he would receive money from a social security or other benefits system that was indexed to a price index of goods. Such a system would therefore fail to hold the farmer at the same level of living as before the prices rose.
Exactly the same issue arise for homeowners, though because they sell only to themselves they cannot be made better off by an increase in the cost of housing. Yet an increase in the price index driven by an increase in rental costs has no
effect on their real incomes or their cost of living, and they need no compensation for it. Yet prices have risen, and a price index based on standard COGI or COLI procedures would recognize the fact. In such circumstances, though, compensation by such a COLI would not hold constant the level of living of homeowners whenever the rate of change of the price of housing is different from the rate of change of other prices. A COLI might still be useful as a price index in other contexts, and one might decide on other grounds not to treat homeowners differently from anyone else (or asset holders, for example), but the COLI would no longer be the correct cost-of-living index for homeowners.
Different problems arise when one seeks to compensate people for only part of their income. This issue arises most immediately for social security benefits. Many social security retirees have other income, in some cases substantially exceeding their social security benefits. When Congress legislated to protect these benefits, its intent was to protect the benefits themselves, not the total income of the recipients. It is not immediately obvious how to design a COLI for this purpose. In particular, one does not want an index that holds constant the standards of living of social security recipients supported by more than social security benefits. There are a number of possible approaches to this issue; perhaps the simplest is to define the COLI in terms of the costs of maintaining living standards for those who have no income other than social security benefits.
It seems quite unlikely that it would be worthwhile in practice to try to design price indexes that deal with homeowners separately from renters, or separately for social security recipients with or without other income. Nevertheless, this discussion highlights the fact that, even in the area for which it seems best suited—compensation—the cost-of-living index is not as obvious a choice as at first appears.
Stocks and Flows
Both basket and cost-of-living indexes are constructed from purchases and prices of goods. As we discussed above, the definition of goods cannot be taken for granted in a world of quality change. One property of a good, which can be thought of as an aspect of quality, is the length of time it lasts. For many goods, it is reasonable to use the convenient fiction that consumption happens at the moment of purchase. But for long-lived items like automobiles or houses, consumption is typically spread over several or many years. When computing a price index, it makes no sense to add together prices of durable and nondurable goods. Thus, one must use not the purchase price but the consumption price. For nondurable goods, they are the same thing, but for durable goods they differ. For durables, one needs an estimate of the cost of consuming the good for the same length of time for which one is looking at the consumption of nondurable goods. This concept is known as user cost. If the costs of buying and selling (the transactions costs) are ignored, it is calculated by finding out how much it would cost for someone to buy the good, use it for a year (or whatever is the specified
period), and then sell it. This cost has three components: the interest foregone on the purchase price, the depreciation due to the physical wear and tear on the good, and any capital loss or gain other than wear and tear. At the time of purchase, the last two components are not known, so that user cost necessarily contains a speculative or expectational element.
The prices of durable goods should be converted to user cost before being aggregated into a price index, whether a basket price index or a COLI. The quantity that is used for pricing, directly in the Laspeyres or through the various approximations for a COLI, should be the stock of the good, because user cost is the cost of holding that stock for the year. Note that user cost introduces the (nominal) rate of interest into the consumer price index; user costs are higher at higher interest rates, as are both the cost-of-living and (properly computed) basket price indexes. (If higher nominal interest rates are a product of general price inflation, they will be largely offset in user cost by the expected price increase of the durable good over the holding period.)
There are a number of practical issues to do with user cost. One is whether to calculate it directly and, if so, what interest rate to use and how to proxy the expected capital loss or gain. If a car rental company faces the same costs of owning the car as does a consumer, so that competition sets price equal to cost, car rental rates ought to be close to user costs, and the “rental equivalent” can be used to construct the price index. If transactions costs are important, or are different for people and for rental companies, or if renters treat rental cars differently than their own cars, the two measures will not be equivalent. Currently, BLS treats cars as nondurable and works with user cost only for the “price” of owner-occupied housing. (Nonowners pay rents, which can be used directly.) If user costs are to be constructed directly, there are a number of practical issues associated with the treatment of capital gains. One possibility is to ignore them, but this causes problems when inflation rates are high. Another possibility is to use actual ex-post price movements, but this runs the risk of incorporating volatility into the index, and indeed of imputing negative user costs, and potentially even a negative CPI. Using ex-post capital gains also induces a perverse inverse relationship between the price of the durable and its user cost, at least in the short run. A CPI index that falls when complaints about the unaffordability of housing are the loudest would have difficulty gaining public acceptance. It would also be possible to forecast capital gains, or more simply just smooth the ex-post price changes. As is the current practice with housing, we believe that using rental rates is probably the best option.
It is possible to imagine moving to a user cost basis, not only for housing and cars but for other durable goods, such as household appliances and furnishings, electronic equipment, and even clothing. The whole concept of user cost ignores the fact that, if some people cannot get loans, not everyone has access to these goods by paying the user costs. How far to extend the user cost approach remains an important issue for BLS.
The arguments of this chapter reappear in subsequent chapters of this report as we deal with specific topics. On the basis of our discussion in this chapter, we present two general conclusions, largely about the conceptual basis for price and cost-of-living indexes, which serve to guide our more detailed conclusions and recommendations in the rest of this report.
Conclusion 2-1: An unconditional cost-of-living index is an unsuitable conceptual basis for the CPI. While research aimed at better understanding the economic effects related to changes in such matters as life expectancy, crime rates, or the environment would be useful for evaluating various aspects of public policy, the CPI should not change in response to changes in such factors.
Conclusion 2-2: Within the general conceptual framework of cost-of-living indexes, the appropriate theoretical concept for the CPI is a conditional cost-of-living index that is restricted to private goods and services and in which environmental background factors are held constant.
On the broader issue of assessing the relative merits of COGI and COLI conceptual approaches as a guide for construction of the CPI, various members of the panel strike the balance differently. All panel members find it difficult to think about the definition of goods and about quality change without considering what it is that consumers value, and we agree that it is impossible to think about substitution behavior without the concept of a constant standard of living that allows price changes to be converted into a monetary equivalent. For all these issues, especially the last, the cost-of-living framework is central. However, some panel members are skeptical about our ability to define a constant standard of living in an economy in which the nature of goods and services is constantly changing. They are therefore concerned about BLS adopting a conceptual framework that is not always well defined in the presence of quality change. They are also concerned about the BLS adopting an approach that differs from that of many statistical agencies around the world. All panel members do agree that the COGI and the conditional COLI that the panel recommends share many common aspects. We also concur that neither conceptual approach, viewed in its pure form, can provide the single guide to index construction. Rather, each of them can make a contribution toward dealing with the various problems that arise in designing the CPI. Taking a pragmatic approach, the panel found that it could come, sometimes by different routes, to unanimous agreement on all of the specific recommendations in this report. But in its inability to achieve unanimity behind a recommendation that the cost-of-living framework be the sole appropriate basis for construction of the CPI, our panel differs from the Stigler committee and Boskin commission.
TECHNICAL NOTE: A MATHEMATICAL APPROACH TO PRICE INDEXES
Notation, Laspeyres, and Paasche Indexes
We start by introducing some notation for the variables that most concern us, prices and quantities. In each period t, there are N goods, each of which has a price, pn, and a quantity, qn, with the subscript n labeling the good and running from 1 to N. We shall also need to refer to these prices and quantities in different periods, typically a base or reference period, denoted 0, and a later comparison or current period, denoted t. (We will occasionally separate base and reference periods later.) Superscripts refer to these time periods, so that is the purchase of good n in period t. Sometimes we need to distinguish between purchases by different people, in which case we add another superscript h, for household. It is also occasionally useful to use vector notation, in which case subscripts are dropped; hence q is the (column) vector of N quantities and p the corresponding vector of N prices. Associated with the vector notation is the “dot” or inner product, p . q, which denotes the sum of the element by element product of the vectors, in this case the total amount of money spent on q when it is bought at prices p.
Armed with only this notation, we can introduce the two most important fixed-basket price indexes or cost-of-goods indexes, or COGIs. For the Laspeyres price index, there is a base set of quantities, which we can denote q0, which is repriced in successive periods. Hence, the Laspeyres price index for period t, which we denote is defined by the equation
In equation (1), the two sets of prices pt and p0 are compared using the base period quantities, q0, as weights. Note that the numerator and denominator of (1) are identical, except that the prices in the numerator are current prices pt, while those in the denominator are base period prices p0. A useful alternative way to write the Laspeyres index is to define a price relative for each good. We write for good n
which can be used to rewrite equation (1) in the form
where x0 = p0. q0 is the total amount of money spent on all goods in period 0 and is the expenditure share of commodity n in period 0. According to (3), the Laspeyres can be thought of as a weighted sum of the price relatives, where the weights are the shares of the base period budget devoted to each of the goods. This way of thinking about the price index is useful because it shows so clearly how the Laspeyres “solves” the problem of making a single index in a situation where the price of each good has changed in a different way. Each of the N goods has its own rate of inflation, represented by its price relative. The Laspeyres averages these price relatives, each weighted according to the good’s importance in the base period.
The period t Paasche price index is constructed in the same way as the Laspeyres but with the current basket replacing the base basket. Hence, replacing the base period quantities in (1) with the current period quantities, we have
The Paasche index can also be written in terms of the price relatives and the budget shares, though the formulas are not quite so intuitive. Nevertheless it is easy to show that, instead of (3), we have
so that the reciprocal of the Paasche is the current budget share weighted average of the reciprocals of the price relatives. If we take reciprocals of both sides of (5), we see that the Paasche index is a weighted harmonic mean of N price relatives, as opposed to the Laspeyres index in (3), which is a weighted arithmetic mean of the price relatives. If the price relatives are not all equal to one another, and under the special assumption that the expenditure shares in periods 0 and t are equal to one another, then a theorem of Hardy et al. (1934:26) implies that the Paasche index is strictly less than the Laspeyres.
The Paasche and Laspeyres price indexes are the two most familiar fixed-basket price indexes that can be used to measure price change going from period 0 to t. As we have presented them, there is no strong reason to prefer one over the other. However, from the viewpoint of statistical agency practice, there are strong reasons for preferring the Laspeyres. Both indexes require information on the
price relatives, and statistical agencies are quite successful in collecting information on prices in a timely manner. However, while the Laspeyres price index requires information on base period expenditure shares, the , the Paasche index requires information on current period expenditure shares, the With present methods of data collection, it is not possible to have accurate information on current period expenditure shares in a timely manner. Thus, from a practical point of view, the preferred fixed-basket price index is the Laspeyres price index since it can be evaluated in a timely manner. We note another advantage of the Laspeyres price index over its Paasche counterpart in the context of indexation of incomes below.
Averages of Fixed-Price Indexes
After a lag of about 2 years, it becomes feasible to evaluate the Paasche price index.1 Hence, in the context of making price comparisons over the long run, we have (at least) two different measures of price change between periods 0 and t: the Laspeyres estimate of price change, , and the Paasche estimate of price change, . These are conceptually different, because they price different bundles over time, and in some cases the distinction may be important, and statistical agencies might wish to make both of these indexes available to the public. However, suppose that for practical or political reasons we need a single estimate of price change between periods 0 and t; is there a “best” such estimate? Obviously, there are many possible approaches to answering this question. We consider two simple and intuitive approaches.
The first way of combining the Paasche and Laspeyres measures of price change is to take some sort of an average, which we write in the form so that we can write the new index as
We want this average to treat both price indexes symmetrically, to be positive, to be linearly homogeneous in both price indexes, and to be equal to either one when they are the same. In addition, we would like our new index to satisfy the time reversal test, which says that a price index from 0 to t should be the reciprocal of the price index from t to 0, so that
Equation (7) means that it does not matter which period we regard as the base period; we obtain essentially the same answer either way. Since the choice of which period to regard as the base is essentially arbitrary, other things being equal, we would like our price index to satisfy the time reversal test. (It is worth noting that neither the Laspeyres nor the Paasche price index satisfies the time reversal test.)
Diewert (1997:138) showed that only one average satisfies all the properties listed. This is the geometric mean (the square root of the product) of the Paasche and the Laspeyres
The price index defined by (8) is known as the Fisher (1922) ideal price index. The foregoing argument provides one justification for thinking of the Fisher price index as a “best” estimator of price change between periods 0 and t.
An alternative approach to combining the Paasche and the Laspeyres is to average not the indexes themselves but the two different baskets that go into them, an approach that was originated by Walsh (1901, 1921) and Knibbs (1924). If we use a geometric mean of the two baskets, we obtain the Walsh price index, PW, written as
If we replace the geometric mean in (9) with the simple arithmetic mean, we reach yet another index in the Walsh-Knibbs family, known as the Marshall Edgeworth price index (Marshall, 1887; Edgeworth, 1925).
Aggregation: Democratic and Plutocratic Indexes
We have been careful so far not to distinguish individual from aggregate quantities. Paasche and Laspeyres indexes can be equally well constructed using individual baskets or aggregate (or average) baskets. In this section, we consider the relationships between these various types of Laspeyres and Paasche indexes under the assumption that each household faces the same vector of prices in each period.
Suppose that there are H households in the economy. Household h’s period t Laspeyres index can be written following (3) but with the household superscript h in the form
where q0h is the vector of purchases for household h in the base year 0, x0h is its total expenditure in period 0, and is its share of total expenditures on good n in period 0. If, by contrast, we evaluate the national Laspeyres index using the aggregate bundle for all households, we would have
where the superscript A denotes “aggregate,” is the aggregate quantity defined as the sum of the individual quantities, X0 is aggregate expenditures on all goods and services, again the sum of the individual x0h, and is the share of aggregate expenditures on good n.
Both individual and aggregate Laspeyres indexes are weighted averages of the same price relatives, and the formulas (10) and (11) differ only in the weights. The aggregate index (11) uses the shares in the national budget, while the individual index (10) uses the shares in the household’s budget. The two sets of weights can be related to one another by noting that
so that the shares in the national budget are the weighted average of the shares in each household’s budget, where the weights are each household’s total expenditure as a share of national total expenditure. People who spend a lot count more in the national weights than do people who spend a little. Given (12), the individual and national Laspeyres indexes are related by
Equation (13) is the reason why the aggregate Laspeyres is referred to as a plutocratic index; each household’s individual Laspeyres price index is weighted by the total amount of money that it spends in period 0. This is in contrast to a democratic Laspeyres index in which each household’s index is averaged to obtain the national index
Note that the democratic and plutocratic Laspeyres indexes will coincide if everyone has the same income, or if everyone spends their money in the same proportions over the different goods, or if all the price relatives are equal.
Note finally that, if we combine (13) and (14), we can write
where is the simple average over households of the budget shares in the base period. Equation (15) shows that the democratic Laspeyres can be estimated if we can calculate, in addition to the price relatives, the population average of household budget shares, something that can be estimated from a consumer expenditure survey, such as the Consumer Expenditure Survey (CEX). We note that the above average can be repeated for the period t Paasche index, though the calculations are not so straightforward. The national Paasche index, which uses the national aggregate bundle at time t to compare prices at 0 and t, turns out to be a weighted harmonic mean of the individual Paasche indexes. Parallel to the Laspeyres, the weights are plutocratic weights, now the ratios xth/Xt, the shares of each household in aggregate national expenditure on all commodities in the domain of the index but now in period t. We can also define a democratic Paasche index as the simple average of the individual Paasche indexes. However, there is no formula corresponding to (15) for the democratic Paasche index. In consequence, it cannot be calculated by weighting the price relatives by an average of the expenditure shares; instead, it must be calculated directly by averaging the individual Paasche indexes.
What are the merits and demerits of the plutocratic versus democratic price indexes? The democratic indexes, which give each individual an equal weight in the overall index, are the natural indexes for the analysis of welfare when we want each person to count the same rather than in proportion to his expenditure. By contrast, when we want every dollar to count the same, as, for example, when we are calculating the national accounts, the plutocratic indexes are the natural choice. Each family of indexes has its own justification.
Note finally that the arguments in the second section can be repeated in the present context leading, for example, to the use of the plutocratic Fisher ideal index as a good candidate for combining the information in the plutocratic Paasche and Laspeyres indexes into a single measure of the change in prices from 0 to t.
In the economic theory of consumer behavior, each household (or person) is assumed to spend their money so as to be as well off as possible. The way this is formalized is by writing down a utility function whereby the level of utility (or level of living) is determined by the vector of quantities consumed
u= f(q). (16)
The main role of the utility function is to codify consumer preferences; by inserting any quantity vector q into (16) we can test whether it is better than, the same as, or worse than any other quantity vector, and this ranking tells us the consumer’s preferences over goods. The value assigned to u itself is of no significance; provided higher u means a better bundle, it does not matter what particular
values are assigned to u. More important is the concept of an indifference curve or indifference surface; this is a collection of q’s all of which yield the same value of the utility function. They are therefore bundles between which the consumer is indifferent. Higher indifference curves are those with a higher value of u and correspond to a higher standard of living.
The most useful concept for cost-of-living theory is the cost or expenditure function, which measures the least amount of money that the consumer would have to pay at specified prices to reach a specified indifference curve. We write this function as c(u,p) where, as before, p is the vector of prices, and u is some arbitrary label that identifies the indifference curve. Given that the consumer has a total x to spend, and given the assumption that she spends that money to do as well as possible, we can write
x= c(u,p ).
Note that this function also can be thought of as defining u, the standard of living, in terms of the prices p, and total expenditure x.
Cost-of-living index numbers are defined directly from the cost function. Suppose that the base period level of living is u0. The cost-of-living index number using base period level of living is the ratio of the costs of reaching the indifference curve u0 at the two sets of prices, p0 and p1. Hence,
is the cost-of-living analog to the Laspeyres index (1). Both indexes compare the current prices in the numerator with the base prices in the denominator. Because c(u0,p0) = x0 = p0·q0, the denominators of (1) and (18) are the same. However, the numerator of the Laspeyres is the cost of the base basket q0 evaluated at period t prices pt, while the numerator of the cost-of-living index is the minimum cost of obtaining the base period indifference curve at prices pt. If instead of the base indifference curve in (18), we use the current indifference curve, we get the cost-of-living index corresponding to the Paasche index, which is
Each of the two cost-of-living indexes (18) and (19) involves a counterfactual cost; in (18) it is the minimum cost of reaching u0 at prices pt, while in (19) it is the minimum cost of reaching ut at p0. Although we do not immediately know what these counterfactuals are, we can set limits on them. In particular, since one way of reaching u0 is to buy the original bundle q0, the minimum cost of reaching u0 at pt can be no larger than the cost of that bundle at the current prices, which is q0. pt. Similarly, one way of reaching ut at the original prices is to buy the bundle qt, so that the minimum cost of ut at p0 can be no larger than qt. p0. Hence, if we go back to the definition of the base period cost-of-living index (18) and note that the minimum cost of u0 at prices p0 is the actual expenditure q0. p0, we have
so that the base period cost-of-living index is always no larger than the Laspeyres. For the current period true cost-of-living index, the hypothetical cost is in the denominator, so that replacing it by something larger will make the result smaller. Hence, using (20) and this inequality, we have
so that the current period cost-of-living index is always at least as large as the Paasche price index .
There is an immediate link between each of these cost-of-living indexes and a measure of compensation. The amount of money that the consumer needs to reach the base level of living at the current prices is simply c(u0,pt) so that the (possibly negative) compensation that the consumer requires to make up for the price change from p0 to pt is given by
CV=c(u0,pt) - c(u0,p0).
This quantity is known as the compensating variation. It is the difference between the same two costs whose ratio is the cost-of-living index for the base level of living . We can also construct the equivalent variation, defined as the maximum amount of money that the consumer would have been prepared to pay in the base situation to avoid the price change from p0 to pt. It is
EV = c(ut,pt) - c(ut, p0)
and bears the same relationship to the cost-of-living index for the current period level of living ut as does the compensating variation to the cost-of-living index for the base period level of living u0. To illustrate how these measures work with the cost-of-living indexes, suppose that a consumer’s base level of total expenditures is x0 = c(u0,p0) and that we escalate this by the base period cost-of-living index (20). The new escalated total expenditure will be c(u0,pt), so that the escalation pays the compensating variation (22) and exactly compensates the consumer for the change in prices. If in the absence of the cost-of-living index, we escalate by the Laspeyres price index , the consumer will have at least as much as needed to remain as well off. If the object of policy is to ensure that compensation is adequate, and if it is better to compensate too much than to compensate too little, this would be an argument for the use of the Laspeyres price index for escalation.
In principle, we can construct a cost-of-living index around any level of living. We might write this arbitrarily based cost-of-living index in the form
for some indifference curve u. From inspection of (24), it is clear that if we want cost-of-living indexes to be the same whatever the choice of u, the cost function must factor into two components, one containing only u and one containing only the prices p, so that we can write
c(u, p) =θ(u)γ(p).
This condition, known as homothetic preferences, implies that the pattern of demand, the way the budget is spread over goods, is the same at all levels of living, something that is not in accord with the empirical evidence. In general then, we can expect cost-of-living indexes to depend on the level of living on which they are based, so that the base period cost-of-living index will be different from the current period cost-of-living index, and cost-of-living indexes will be different for the poor and for the rich. It is only under homotheticity that it is possible to talk about the “true” cost-of-living index, since it is only then that it will be unique, and it is only then that it can be correctly asserted that the “true” cost-of-living index always lies between the Paasche and the Laspeyres (or even that the Laspeyres is always greater than the Paasche).
When preferences are not homothetic, we cannot calculate observable bounds for the two COLIs (20) and (21). However, Konüs (1924:20) proved that there exists a utility level u*, intermediate between u0 and u1, whose cost-of-living index (24) lies between the Paasche and the Laspeyres indexes. Hence, if the Paasche and Laspeyres indexes are not far apart, an average of them, such as the Fisher ideal index (8), is likely to be a good approximation to a COLI such as (24), whose reference standard of living is between the base and current period standards of living.
How can we recover the cost function and the associated cost-of-living index numbers from observable behavior in the market? What we typically see is the relationship between each period’s quantities purchased, the q’s, their prices p, and the incomes (or total outlays) of consumers. Suppose that we can do so for a single consumer and that we can recover, by experimentation or econometric analysis, the n functions, one for each good
According to (26), each of the N purchases is a function of total expenditure x and the vector of N prices p. The cost function is also directly linked to the quantities purchased, and the crucial result here, known as Shephard’s Lemma (Shephard, 1953:11), states that the quantities are the partial derivatives of the cost function with respect to prices
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
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.
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
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
(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
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.
There are many other superlative indexes, for example, the Törnqvist index defined by
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
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
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:
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
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
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
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
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
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
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,
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
some level different from what would have been chosen by weighting price against benefit. In this case, the budget constraint (39) becomes
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,
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
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-
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
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
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
when s is not unity or
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
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
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.