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# Price Indices

SOME WELL-KNOWN PRICE INDICES

THE TRUE COST OF LIVING INDEX AND SUBSTITUTION BIAS

SUPERLATIVE INDEXES

CHAIN-WEIGHTING

EMPIRICAL ILLUSTRATION

BIBLIOGRAPHY

In 1927 Irving Fisher provided a simple definition of price indices in his classic book on index numbers, The Making of Index Numbers: A Study of Their Varieties, Tests, and Reliability :

An index number of prices, then, shows the average percentage change of prices from one period to another. The percentage change in the price of a single commodity from one time to another is, of course, found by dividing its price at the second time by its price at the first time. The ratio between these two prices is called the price relative of that one particular commodity in relation to those two particular times. An index number of the prices of a number of commodities is an average of their price relatives (p. 3).

To begin a discussion on price indices, some notation is important: Let be a vector of prices for n goods in period 0. Let be a vector of prices for the same n goods in period 1. Thus, the price relative of any good i is . A simple way of measuring the average change in all n prices from period 0 to period 1 is to take an average of the price relatives across all of the goods. For example, the arithmetic mean of the price relatives is and the geometric mean of them is

As noted by E. Antony Selvanathan and D. S. Prasada Rao in their book Index Numbers: A Stochastic Approach, these simple measures:

tend to consider price changes for all commodities to be equally important. However, in practice, movements of prices in essential items are considered to be more important and it is expected that any meaningful price index should accord weights to different price relatives. This principle leads to a class of weighted averages, where the weights are usually based on the value shares of each commodity (1994, pp. 1920).

## SOME WELL-KNOWN PRICE INDICES

Let and q 1 = (q 11, ,q 1n) represent quantity vectors for the n goods in periods 0 and 1 respectively. Total expenditure in the two periods is the sum (across all n goods) of the prices multiplied by the corresponding quantities: and . Thus the expenditure shares for each good i are given by and for periods 0 and 1 respectively.

A price index is a function of the price and quantity vectors in two periods, which measures the change in the prices of the n goods between them. As in the previous section, the notation, P 01 is used to denote a price index between periods 0 and 1. A price index can be interpreted in the same way as an individual price relative. If it is greater than one, it means that prices are increasing from period 0 to 1. If it is less than one, prices are decreasing. The rate of increase or decrease of an individual price can be computed by either subtracting one from the price relative, giving the growth rate , or by taking natural logarithms of the price relative, giving . Similarly, the rate of increase or decrease in the prices of all n goods can be measured as either (P01-1) or ln(P01).

Given a price index, P01, one can always define a quantity index, Q 01, implicitly from the following equation: P01 Q01 = Y1 /Y0 which means that the product of the price and quantity indices equals the ratio of total expenditure in the two periods.

A number of well-known price indices can be obtained by taking the weighted arithmetic or geometric means of the price relatives, where the weights depend on the expenditure shares. The Laspeyres price index, is obtained by taking a weighted arithmetic mean of the price relatives, where the weights are the expenditure shares for period 0. As explained by Selvanathan and Rao, the Paasche price index, can be obtained in a similar fashion using hypothetical expenditure shares derived from period 0 prices and period 1 quantities. Although mathematically equivalent, the Laspeyres and Paasche price indices are usually defined as follows: and Fisher s ideal price index, is the geometric mean of the Laspeyres and Paasche price indices:

The Tornqvist-Theil price index, , is the weighted geometric mean of the price relatives with weights that are the average of the expenditure shares from the two periods: An attractive property of this index is that its natural log is a weighted sum of the natural logs of the n component price relatives:

## THE TRUE COST OF LIVING INDEX AND SUBSTITUTION BIAS

In his 1992 article Economic Theory and the BEAs Alternative Quantity and Price Indexes Jack E. Triplett discusses the connection between a price index for a set of consumer goods and services and a cost of living index:

a consumption price index should measure the change in the cost of maintaining a fixed, or constant, standard of living. If the price index holds the standard of living constant, then any increase in per capita consumption expenditures that exceeds the increase in the price index can be interpreted as an increase in the standard of living. Thus, from the standard-of-living orientation, the price index measures the changing cost of a constant standard of living, and the quantity index measures increases or decreases in the standard of living (p. 49).

The true cost of living index was originally proposed in a 1924 paper by A. A. Konüs (an English translation was published in 1939). The true cost of living index is defined as In the definition, the expenditure function, e (p, u ), gives the minimum expenditure needed to obtain a particular standard of living, u, when prices are p. The utility function, U (q ), in turn, gives the utility, or standard of living, associated with the quantity vector, q. Thus, the true cost of living index measures the change in the minimum expenditure needed to maintain the standard of living associated with a particular vector of quantities, q, as prices change from p 0 to p 1. (See W. E. Diewerts 1981 article, The Economic Theory of Index Numbers: A Survey, for an extensive discussion of these concepts.)

The true cost of living index can be evaluated using any quantity vector. It could, for example, be evaluated using the quantities q 0, from period 0. In the following discussion, it will be assumed that which means that the quantities q 0 are optimal given prices p 0. It would cost to purchase the same vector of quantities in period 1 and doing so would ensure that the standard of living is unchanged. If relative prices have changed, however, it may be possible to achieve the same standard of living at a lower cost by substituting away from goods whose relative prices have increased toward goods whose relative prices have decreased. This reasoning implies that

It follows that the Laspeyres index provides an upper bound on the true cost of living index when the standard of living is based on the quantities from period 0, since . The upward bias in the Laspeyres index, relative to the true cost of living index, is a type of substitution bias. The same reasoning also implies that the Paasche index provides a lower bound on the true cost of living index when the standard of living is based on the period 1 quantities, since

In general the true cost of living index depends on the choice of the quantity vector, so it would not be correct to say that the true cost of living index is bounded above and below by the Laspeyres and Paasce price indices respectively, since those upper and lower bounds are based on different quantities. A stronger result can be obtained if preferences are homothetic. As explained by Angus Deaton and John Muellbauer in their 1980 book, Economics and Consumer Behavior, homotheticity means that for some normalization of the utility function, doubling quantities doubles utility. Under homotheticity, it can be shown that the true cost of living index does not depend on q and is, consequently, bounded above and below by the Laspeyres and Paasche price indices respectively (see Diewert 1981). Nevertheless, homotheticity is a very strong assumption, since it implies that expenditure shares of the goods are independent of the level of total expenditure, which is contradicted by most empirical evidence (see Deaton and Muellbauer 1980).

## SUPERLATIVE INDEXES

In his 1976 article, Exact and Superlative Index Numbers, Diewert provides a strong rationale for preferring certain price and quantity indices, which he termed superlative indexes. The Tornqvist-Theil price index and Fishers ideal price index are both superlative. Without going into precise technical details, essentially a price index is superlative if it can provide a good approximation to the true cost of living index even though the functional form of the expenditure function is not known (see Diewerts 1976 and 1981 articles for further discussion, including the role of homotheticity in the theory of superlative indices).

The Laspeyres and Paasche price indices are not superlative, because they just measure the change in the cost of purchasing a fixed bundle of goods and, therefore, ignore substitution. As explained by Triplett:

Diewert showed that the Fisher Ideal index and the Tornqvist index are theoretically better measures of the cost of living than the traditional fixed-weighted Paasche or Laspeyres indexes. The superlative indexes accommodate substitution in consumer spending while holding living standards constant, something the Paasche and Laspeyres indexes do not do. From the view of theory, the Fisher Ideal formula and the Tornqvist formula are equally good; therefore, one can choose between the two on pragmatic grounds (1992, p. 50).

In fact, there are many other superlative indices besides the two that have been discussed.

The influential 1996 Boskin Commission report, Toward a More Accurate Measure of the Cost of Living, concluded that the consumer price index (CPI) overstated inflation in the United States by 1.1 percent per year in 19951996. The report further concluded that 0.4 percent of that bias could be attributed to upper and lower level substitution. The remainder of the bias was mostly attributed to quality change and new products. Robert J. Gordon, in his 2000 paper, The Boskin Commission Report and Its Aftermath, states:

It is noteworthy that few if any criticisms addressed the Commissions basic recommendations that the CPI should become a COL [cost of living] index and that substitution at both the upper and lower level should be addressed within the framework of superlative index numbers (2000, p. 24).

## CHAIN-WEIGHTING

Suppose there is price and quantity data for many periods denoted by pt and qt where t = 0, 1, , T (for example, suppose there is annual data for twenty years). One can then compute price indexes from each period to the successive

one, Pt - 1, t, for all t. A time series covering more than two periods can be produced using the concept of chain-weighting. A chain-weighted time series can be constructed as follows: The value of the series, It, for any period t is the previous value of the series, It-1, multiplied by the corresponding price index for the two periods, so that It = It-1 Pt-1,t for all t. Thus, the growth rate of a chain-weighted index between adjacent periods is (It - It-1)/It-1 = Pt-1 t-1. Any price index formula can be chain-weighted, although a superlative formula is preferable as previously discussed.

The Bureau of Economic Analysis (BEA) produces chain-type price indices for Personal Consumption Expenditures (PCE) and for Gross Domestic Product (GDP) for the United States, which are based on Fishers ideal formula (see A Guide to the National Income and Product Accounts of the United States published by BEA in 2006). See Tripletts 1992 article, as well as Allan H. Youngs Alternative Measures of Change in Real Output and Prices, Quarterly Estimates for 195992 (1993) for related discussion including alternatives to chain-weighting.

## EMPIRICAL ILLUSTRATION

In this section, the use of price indices to measure inflation is illustrated. Annualized quarterly percentage changes in the chain-type price indices for PCE and GDP for the United States were calculated using data from 1960Q1 to 2006Q3. These two inflation measures are graphed in Figure 1 (PCE is black and GDP is grey).

These measures indicate that inflation was relatively low and stable in the early 1960s, but was higher in the mid to late 1960s. In the 1970s inflation was higher and more volatile and, in particular, oil shocks in 1973 and 1974 and 1979 and 1980 were associated with high rates of inflation. Inflation declined in the early 1980s and has been relatively low and stable in the 1990s and beyond.

## BIBLIOGRAPHY

Boskin, Michael J., Ellen R. Dulberger, Robert J. Gordon, et al. 1996. Toward a More Accurate Measure of the Cost of Living. Final Report to the Senate Finance Committee from the Advisory Commission to Study the Consumer Price Index.

Bureau of Economic Analysis. A Guide to the National Income and Product Accounts, September 2006. http://bea.gov/bea/mp.htm.

Deaton, Angus, and John Muellbauer. 1980. Economics and Consumer Behavior. Cambridge, U.K.: Cambridge University Press.

Diewert, W. E. 1976. Exact and Superlative Index Numbers. Journal of Econometrics 4 (2): 115145.

Diewert, W. E. 1981. The Economic Theory of Index Numbers: A Survey. In Essays in the Theory and Measurement of Consumer Behaviour in Honour of Sir Richard Stone, ed. Angus Deaton, 163208. Cambridge, U.K.: Cambridge University Press.

Fisher, Irving. 1927. The Making of Index Numbers: A Study of Their Varieties, Tests, and Reliability. Boston: Houghton Mifflin.

Gordon, Robert J. 2000. The Boskin Commission Report and Its Aftermath. NBER Working Paper 7759.

Konüs, A. A. 1939. The Problem of the True Cost of Living. Econometrica 7: 1029.

Selvanathan, E. Antony, and D. S. Prasada Rao. 1994. Index Numbers: A Stochastic Approach. Ann Arbor: University of Michigan Press.

Triplett, Jack E. 1992. Economic Theory and the BEAs Alternative Quantity and Price Indexes. Survey of Current Business 72 (April): 4852.

Young, Allan H. 1993. Alternative Measures of Change in Real Output and Prices, Quarterly Estimates for 195992. Survey of Current Business (March): 3141.

Barry Jones