16. Basic Index Number Theory

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Author: International Monetary Fund

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Keywords:: M&G; index number formula; reference period; quantity vector; value share; rate of change; Price indexes; Price structures; Export fluctuations

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Abstract

The answer to the question what is the Mean of a given set of magnitudes cannot in general be found, unless there is given also the object for the sake of which a mean value is required. There are as many kinds of average as there are purposes; and we may almost say in the matter of prices as many purposes as writers. Hence much vain controversy between persons who are literally at cross purposes. (Edgeworth, 1888, p. 347)

A. Introduction

The answer to the question what is the Mean of a given set of magnitudes cannot in general be found, unless there is given also the object for the sake of which a mean value is required. There are as many kinds of average as there are purposes; and we may almost say in the matter of prices as many purposes as writers. Hence much vain controversy between persons who are literally at cross purposes. (Edgeworth, 1888, p. 347)

16.1 The number of physically distinct goods and unique types of services that consumers can purchase is in the millions. On the business or production side of the economy, there are even more products that are actively traded. The reason is that firms produce not only products for final consumption but also exports and intermediate products that are demanded by other producers. Firms collectively also use millions of imported goods and services, thousands of different types of labor services, and hundreds of thousands of specific types of capital. If we further distinguish physical products by their geographic location or by the season or time of day that they are produced or consumed, then there are billions of products that are traded within each year in any advanced economy. For many purposes, it is necessary to summarize this vast amount of price and quantity information into a much smaller set of numbers. The question that this chapter addresses is the following: How exactly should the microeconomic information involving possibly millions of prices and quantities be aggregated into a smaller number of price and quantity variables? This is the basic index number problem.

16.2 It is possible to pose the index number problem in the context of microeconomic theory; that is, given that we wish to implement some economic model based on producer or consumer theory, what is the best method for constructing a set of aggregates for the model? However, when constructing aggregate prices or quantities, other points of view (that do not rely on economics) are possible. Some of these alternative points of view are considered in this chapter and the next chapter. Economic approaches are pursued in Chapter 18.

16.3 The index number problem can be framed as the problem of decomposing the value of a well-defined set of transactions in a period of time into an aggregate price multiplied by an aggregate quantity term. It turns out that this approach to the index number problem does not lead to any useful solutions. Therefore, Section B addresses the problem of decomposing a value ratio pertaining to two periods of time into a component that measures the overall change in prices between the two periods (this is the price index) multiplied by a term that measures the overall change in quantities between the two periods (this is the quantity index). The simplest price index is a fixed-basket index. In this index, fixed amounts of the n quantities in the value aggregate are chosen, and then this fixed basket of quantities at the prices of period 0 and period 1 are calculated. The fixed-basket price index is simply the ratio of these two values, where the prices vary but the quantities are held fixed. Two natural choices for the fixed basket are the quantities transacted in the base period, period 0, or the quantities transacted in the current period, period 1. These two choices lead to the Laspeyres (1871) and Paasche (1874) price indices, respectively.

16.4 Unfortunately, the Paasche and Laspeyres measures of aggregate price change can differ, sometimes substantially. Thus Section C considers taking an average of these two indices to come up with a single measure of price change. Section C.1 argues that the best average to take is the geometric mean, which is Irving Fisher’s (1922) ideal price index. In Section C.2, instead of averaging the Paasche and Laspeyres measures of price change, taking an average of the two baskets is considered. This fixed-basket approach to index number theory leads to a price index advocated by Walsh (1901, 1921a). However, other fixed-basket approaches are also possible. Instead of choosing the basket of period 0 or 1 (or an average of these two baskets), it is possible to choose a basket that pertains to an entirely different period, say period b. In fact, it is typical statistical agency practice to pick a basket that pertains to an entire year (or even two years) of transactions in a year before period 0, which is usually a month. Indices of this type, where the weight reference period differs from the price reference period, were originally proposed by Joseph Lowe (1823), and in Section D indices of this type are studied. They are also evaluated from the axiomatic perspective in Chapter 17 and from the economic perspective in Chapter 18.¹

16.5 In Section E, another approach to the determination of the functional form or the formula for the price index is considered. This approach, introduced by the French economist, François Divisia (1926), is based on the assumption that price and quantity data are available as continuous functions of time. The theory of differentiation is used in order to decompose the rate of change of a continuous time value aggregate into two components that reflect aggregate price and quantity change. Although Divisia’s approach offers some insights,² it does not offer much guidance to statistical agencies in terms of leading to a definite choice of index number formula.

16.6 In Section F, the advantages and disadvantages of using a fixed-base period in the bilateral index number comparison are considered versus always comparing the current period with the previous period, which is called the chain system. In the chain system, a link is an index number comparison of one period with the previous period. These links are multiplied to make comparisons over many periods.

B. Decomposition of Value Aggregates into Price and Quantity Components

B.1 Decomposition of value aggregates and the product test

16.7 A price index is a measure or function that summarizes the change in the prices of many products from one situation 0 (a time period or place) to another situation 1. More specifically, for most practical purposes, a price index can be regarded as a weighted mean of the change in the relative prices of the products under consideration in the two situations. To determine a price index, it is necessary to know

(1) Which products or items to include in the index,
(2) How to determine the item prices,
(3) Which transactions that involve these items to include in the index,
(4) How to determine the weights and from which sources should these weights be drawn, and
(5) What formula or type of mean should be used to average the selected item relative prices.

All of the above price index definition questions except the last can be answered by appealing to the definition of the value aggregate to which the price index refers. A value aggregate V for a given collection of items and transactions is computed as

\begin{matrix} V = Σ_{i = 1}^{n} p_{i} q_{i}, & (16.1) \end{matrix}

where p_i represents the price of the ith item in national currency units, q_i represents the corresponding quantity transacted in the time period under consideration, and the subscript i identifies the ith elementary item in the group of n items that make up the chosen value aggregate V. Included in this definition of a value aggregate is the specification of the group of included products (which items to include) and of the economic agents engaging in transactions involving those products (which transactions to include), as well as the valuation and time of recording principles motivating the behavior of the economic agents undertaking the transactions (determination of prices). The included elementary items, their valuation (the p_i), the eligibility of the transactions, and the item weights (the q_i) are all within the domain of definition of the value aggregate. The precise determination of the p_i and q_i is discussed in more detail in Chapter 3 and other chapters.³

16.8 The value aggregate V defined by equation (16.1) referred to a certain set of transactions pertaining to a single (unspecified) time period. Now, consider the same value aggregate for two places or time periods, periods 0 and 1. For the sake of definiteness, period 0 is called the base period and period 1 is called the current period. Assume that observations on the base-period price and quantity vectors, p⁰ ≡ [p₁⁰, . . . , p_n⁰] and q⁰ ≡ [q₁⁰, . . . , q_n⁰], respectively, have been collected.⁴ The value aggregates in the two periods are defined in the obvious way as

\begin{matrix} V^{0} \equiv Σ_{i = 1}^{n} p_{i}^{0} q_{i}^{0}; V^{1} \equiv Σ_{i = 1}^{n} p_{i}^{1} q_{i}^{1} . & (16.2) \end{matrix}

16.9 In the previous paragraph, a price index was defined as a function or measure that summarizes the change in the prices of the n products in the value aggregate from situation 0 to situation 1. In this paragraph, a price index P(p⁰, p¹, q⁰, q¹) along with the corresponding quantity index (or volume index) Q(p⁰, p¹, q⁰, q¹) is defined as two functions of the 4n variables p⁰, p¹, q⁰, q¹ (these variables describe the prices and quantities pertaining to the value aggregate for periods 0 and 1), where these two functions satisfy the following equation:⁵

\begin{matrix} V^{1} / V^{0} = P (p^{0}, p^{1}, q^{0}, q^{1}) Q (p^{0}, p^{1}, q^{0}, q^{1}) . & (16.3) \end{matrix}

If there is only one item in the value aggregate, then the price index P should collapse to the single-price ratio p₁¹/p₁⁰ and the quantity index Q should collapse to the single-quantity ratio q₁¹/q₁⁰. In the case of many items, the price index P is to be interpreted as some sort of weighted average of the individual price ratios, p₁¹/p₁⁰, . . . , p_n¹/p_n⁰.

16.10 Thus, the first approach to index number theory can be regarded as the problem of decomposing the change in a value aggregate, V¹/V⁰, into the product of a part that is due to price change, P(p⁰, p¹, q⁰, q¹), and a part that is due to quantity change, Q(p⁰, p¹, q⁰, q¹). This approach to the determination of the price index is the approach taken in the national accounts, where a price index is used to deflate a value ratio to obtain an estimate of quantity change. Thus, in this approach to index number theory, the primary use for the price index is as a deflator. Note that once the functional form for the price index P(p⁰, p¹, q⁰, q¹) is known, then the corresponding quantity or volume index Q(p⁰, p¹, q⁰, q¹) is completely determined by P; that is, rearranging equation (16.3),

\begin{matrix} Q (p^{0}, p^{1}, q^{0}, q^{1}) = (V^{1} / V^{0}) / P (p^{0}, p^{1}, q^{0}, q^{1}) . & (16.4) \end{matrix}

Conversely, if the functional form for the quantity index Q(p⁰, p¹, q⁰, q¹) is known, then the corresponding price index P(p⁰, p¹, q⁰, q¹) is completely determined by Q. Thus, using this deflation approach to index number theory, separate theories for the determination of the price and quantity indices are not required: If either P or Q is determined, then the other function is implicitly determined by the product test equation (16.4).

16.11 In the next subsection, two concrete choices for the price index P(p⁰, p¹, q⁰, q¹) are considered, and the corresponding quantity indices Q(p⁰, p¹, q⁰, q¹) that result from using equation (16.4) are also calculated. These are the two choices used most frequently by national accountants.

B.2 Laspeyres and Paasche indices

16.12 One of the simplest approaches to determining the price index formula was described in great detail by Joseph Lowe (1823). His approach to measuring the price change between periods 0 and 1 was to specify an approximate representative product basket,⁶ which is a quantity vector q ≡ [q₁, . . . , q_n] that is representative of purchases made during the two periods under consideration, and then to calculate the level of prices in period 1 relative to period 0 as the ratio of the period 1 cost of the basket, $Σ_{i = 1}^{n} p_{i}^{1} q_{i}$ , to the period 0 cost of the basket, $Σ_{i = 1}^{n} p_{i}^{0} q_{i}$ . This fixed-basket approach to the determination of the price index leaves open the following question: How exactly is the fixed-basket vector q to be chosen?

16.13 As time passed, economists and price statisticians demanded a bit more precision with respect to the specification of the basket vector q. There are two natural choices for the reference basket: the base period 0 product vector q⁰ or the current period 1 product vector q¹. These two choices led to the Laspeyres (1871) price index⁷ P_L defined by equation (16.5) and the Paasche (1874) price index⁸ P_P defined by equation (16.6)⁹:

\begin{matrix} P_{L} (p^{0}, p^{1}, q^{0}, q^{1}) \equiv \frac{Σ_{i = 1}^{n} p_{i}^{1} q_{i}^{0}}{Σ_{i = 1}^{n} p_{i}^{0} q_{i}^{0}}; & (16.5) \end{matrix}

\begin{matrix} P_{P} (p^{0}, p^{1}, q^{0}, q^{1}) \equiv \frac{Σ_{i = 1}^{n} p_{i}^{1} q_{i}^{1}}{Σ_{i = 1}^{n} p_{i}^{0} q_{i}^{1}} . & (16.6) \end{matrix}

16.14 The above formulas can be rewritten in a manner that is more useful for statistical agencies. Define the period t value share on product i as follows:

\begin{array}{r} s_{i t}^{t} \equiv p_{i}^{t} q_{i}^{t} / Σ_{j = 1}^{n} p_{j}^{t} q_{j}^{t} for i = 1, . . ., n a n d t = 0, 1. \\ (16.7) \end{array}

Then, the Laspeyres index equation (16.5) can be rewritten as¹⁰

\begin{matrix} P_{L} (p^{0}, p^{1}, q^{0}, q^{1}) & = Σ_{i = 1}^{n} p_{i}^{1} q_{i}^{1} / Σ_{j = 1}^{n} p_{j}^{0} q_{j}^{0} \\ = Σ_{i = 1}^{n} (p_{i}^{1} / q_{i}^{0}) / Σ_{j = 1}^{n} p_{j}^{0} q_{j}^{0} \\ = Σ_{i = 1}^{n} (p_{i}^{1} / q_{i}^{0}) s_{i}^{0} (16.8) \end{matrix}

using definitions in equation (16.7). Thus, the Laspeyres price index P_L can be written as a base-period value share-weighted arithmetic average of the n price ratios, $p_{i}^{1} / p_{i}^{0}$ . The Laspeyres formula (until the very recent past) has been widely used as the intellectual base for export and import price indices (XMPIs) around the world. To implement it, a statistical agency needs only to collect information on trade shares s_n⁰ for the index domain of definition for the base period 0 and then collect information on item prices alone on an ongoing basis. Thus, the Laspeyres XMPI can be produced on a timely basis without current-period quantity information.

16.15 The Paasche index can also be written in value share and price ratio form as¹¹

\begin{matrix} P_{P} (p^{0}, p^{1}, q^{0}, q^{1}) & = 1 / {Σ_{i = 1}^{n} p_{i}^{0} q_{i}^{1} / Σ_{j = 1}^{n} p_{j}^{1} q_{j}^{1}} \\ = 1 / {Σ_{i = 1}^{n} (p_{i}^{0} / q_{i}^{1}) p_{i}^{1} q_{i}^{1} / Σ_{j = 1}^{n} p_{j}^{1} q_{j}^{1}} \\ = 1 / {Σ_{i = 1}^{n} {(p_{i}^{1} / q_{i}^{0})}^{- 1} s_{i}^{1}} \\ = {Σ_{i = 1}^{n} (p_{i}^{1} / q_{i}^{0})^{- 1} s_{i}^{1}}^{- 1}, (16.9) \end{matrix}

using definitions in equation (16.7). Thus, the Paasche price index P_P can be written as a period 1 (or currentperiod) trade share–weighted harmonic average of the n item price ratios $p_{i}^{1} / p_{i}^{0}$ .¹² The lack of information on current-period quantities prevents statistical agencies from producing Paasche indices on a timely basis.

16.16 The quantity index that corresponds to the Laspeyres price index using the product test, equation (16.3), is the Paasche quantity index; that is, if P in equation (16.4) is replaced by P_L defined by equation (16.5), then the following quantity index is obtained:

Q_{P} (p^{0}, p^{1}, q^{0}, q^{1}) \equiv \frac{Σ_{i = 1}^{n} p_{i}^{1} q_{i}^{1}}{Σ_{i = 1}^{n} p_{i}^{1} q_{i}^{0}} . (16.10)

Note that Q_P is the value of the period 1 quantity vector valued at the period 1 prices, $Σ_{i = 1}^{n} p_{i}^{1} q_{i}^{1}$ , divided by the (hypothetical) value of the period 0 quantity vector valued at the period 1 prices, $Σ_{i = 1}^{n} p_{i}^{1} q_{i}^{0}$ . Thus, the period 0 and 1 quantity vectors are valued at the same set of prices, the current-period prices, p¹.

16.17 The quantity index that corresponds to the Paasche price index using the product test, equation (16.3), is the Laspeyres quantity index; that is, if P in equation (16.4) is replaced by P_P defined by equation (16.6), then the following quantity index is obtained:

Q_{L} (p^{0}, p^{1}, q^{0}, q^{1}) \equiv \frac{Σ_{i = 1}^{n} p_{i}^{0} q_{i}^{1}}{Σ_{i = 1}^{n} p_{i}^{0} q_{i}^{0}} . (16.11)

Note that Q_L is the (hypothetical) value of the period 1 quantity vector valued at the period 0 prices, $Σ_{i = 1}^{n} p_{i}^{0} q_{i}^{1}$ , divided by the value of the period 0 quantity vector valued at the period 0 prices, $Σ_{i = 1}^{n} p_{i}^{0} q_{i}^{0}$ . Thus, the period 0 and 1 quantity vectors are valued at the same set of prices, the base-period prices, p⁰.

16.18 The problem with the Laspeyres and Paasche index number formulas is that they are equally plausible, but, in general, they will give different answers. For most purposes, it is not satisfactory for the statistical agency to provide two answers to this question:¹³ What is the best overall summary measure of price change for the value aggregate over the two periods in question? Thus, in the following section, it is considered how “best” averages of these two estimates of price change can be constructed. Before doing this, we ask, What is the normal relationship between the Paasche and Laspeyres indices? Under normal economic conditions, when the price ratios pertaining to the two situations under consideration are negatively correlated with the corresponding quantity ratios, it can be shown that the Laspeyres price index will be larger than the corresponding Paasche index.¹⁴

In Appendix 16.1 a precise statement of this result is presented.¹⁵ This divergence between P_L and P_P suggests that if a single estimate for the price change between the two periods is required, then some sort of evenly weighted average of the two indices should be taken as the final estimate of price change between periods 0 and 1. This strategy is pursued in the following section. For indices such as the consumer price index, producer price index, and services XMPIs, statistical agencies will not have information on current value weights and, hence, averages of Paasche and Laspeyres indices can be produced only on a delayed basis (perhaps using national accounts information) or not at all. Reasonably timely Paasche indices for the goods XMPIs, however, may be feasible because monthly trade weights are available on a timely basis in many countries.¹⁶

C. Symmetric Averages of Fixed-Basket Price Indices

C.1 Fisher index as an average of the Paasche and Laspeyres indices

16.19 As was mentioned in the previous paragraph, because the Paasche and Laspeyres price indices are equally plausible but often give different estimates of the amount of aggregate price change between periods 0 and 1, it is useful to consider taking an evenly weighted average of these fixed-basket price indices as a single estimator of price change between the two periods. Examples of such symmetric averages¹⁷ are the arithmetic mean, which leads to the Drobisch (1871b, p. 425), Sidgwick (1883, p. 68), Bowley (1901, p. 227)¹⁸ index, P_D ≡ (1/2)P_L + (1/2)P_P, and the geometric mean, which leads to the Irving Fisher¹⁹ (1922) ideal index, P_F, defined as

\begin{matrix} P_{F} (p^{0}, p^{1}, q^{0}, q^{1}) \equiv {P_{L} (p^{0}, p^{1}, q^{0}, q^{1}) \\ \times P_{P} (p^{0}, p^{1}, q^{0}, q^{1})}^{1 / 2} . (16.12) \end{matrix}

At this point, the fixed-basket approach to index number theory is transformed into the test approach to index number theory; that is, in order to determine which of these fixed-basket indices or which averages of them might be best, desirable criteria or tests or properties are needed for the price index. This topic is pursued in more detail in the next chapter, but an introduction to the test approach is provided in the present section because a test is used to determine which average of the Paasche and Laspeyres indices might be best.

16.20 What is the best symmetric average of P_L and P_P to use as a point estimate for the theoretical costof-living index? It is very desirable for a price index formula that depends on the price and quantity vectors pertaining to the two periods under consideration to satisfy the time reversal test.²⁰ An index number formula P(p⁰, p¹, q⁰, q¹) satisfies this test if

\begin{matrix} P (p^{1}, p^{0}, q^{1}, q^{0}) \equiv 1 / P (p^{0}, p^{1}, q^{0}, q^{1}); & (16.13) \end{matrix}

that is, if the period 0 and period 1 price and quantity data are interchanged and the index number formula is evaluated, then this new index P(p¹, p⁰, q¹, q⁰) is equal to the reciprocal of the original index P(p⁰, p¹, q⁰, q¹). It seems desirable that the measure of aggregate price change should satisfy this property so that it does not matter which period is chosen as the base period. Put another way, the index number comparison between any two points of time should not depend on the choice of which period we regard as the base period: If the other period is chosen as the base period, then the new index number should simply equal the reciprocal of the original index. It should be noted that the Laspeyres and Paasche price indices do not satisfy this time reversal property.

16.21 Having defined what it means for a price index P to satisfy the time reversal test, then it is possible to establish the following result:²¹ The Fisher ideal price index defined by equation (16.12) above is the only index that is a homogeneous²² symmetric average of the Laspeyres and Paasche price indices, P_L and P_P, and satisfies the time reversal test in equation (16.13). Thus the Fisher ideal price index emerges as perhaps the best evenly weighted average of the Paasche and Laspeyres price indices.

16.22 It is interesting to note that this symmetric basket approach to index number theory dates back to one of the early pioneers of index number theory, Arthur L. Bowley, as the following quotations indicate:

If [the Paasche index] and [the Laspeyres index] lie close together there is no further difficulty; if they differ by much they may be regarded as inferior and superior limits of the index number, which may be estimated as their arithmetic mean . . . as a first approximation. (Bowley, 1901, p. 227)
When estimating the factor necessary for the correction of a change found in money wages to obtain the change in real wages, statisticians have not been content to follow Method II only [to calculate a Laspeyres price index], but have worked the problem backwards [to calculate a Paasche price index] as well as forwards. . . . They have then taken the arithmetic, geometric or harmonic mean of the two numbers so found. (Bowley, 1919, p. 348)²³

16.23 The quantity index that corresponds to the Fisher price index using the product test, equation (16.3), is the Fisher quantity index; that is, if P in equation (16.4) is replaced by P_F defined by (16.12), the following quantity index is obtained:

\begin{matrix} Q_{F} (p^{0}, p^{1}, q^{0}, q^{1}) \equiv {Q_{L} (p^{0}, p^{1}, q^{0}, q^{1}) \\ \times Q_{P} (p^{0}, p^{1}, q^{0}, q^{1})}^{1 / 2} . (16.14) \end{matrix}

Thus, the Fisher quantity index is equal to the square root of the product of the Laspeyres and Paasche quantity indices. It should also be noted that Q_F(p⁰, p¹, q⁰, q¹) = P_F(q⁰, q¹, p⁰, p¹); that is, if the role of prices and quantities is interchanged in the Fisher price index formula, then the Fisher quantity index is obtained.²⁴

16.24 Rather than take a symmetric average of the two basic fixed-basket price indices pertaining to two situations, P_L and P_P, it is also possible to return to Lowe’s basic formulation and choose the basket vector q to be a symmetric average of the base- and current-period basket vectors, q⁰ and q¹. The following subsection pursues this approach to index number theory.

C.2 Walsh index and theory of “pure” price index

16.25 Price statisticians tend to be very comfortable with a concept of the price index based on pricing out a constant representative basket of products, q ≡ (q₁, q₂, . . . , q_n), at the prices of period 0 and 1, p⁰ ≡ (p₁⁰, p₂⁰, . . . , p_n⁰) and p¹ ≡ (p₁¹, p₂¹, . . . , p_n¹), respectively. Price statisticians refer to this type of index as a fixed-basket index or a pure price index,²⁵ and it corresponds to Knibbs’ (1924, p. 43) unequivocal price index.²⁶ Because Joseph Lowe (1823) was the first person to describe systematically this type of index, it is referred to as a Lowe index. Thus, the general functional form for the Lowe price index is

\begin{array}{r} P_{L o} (p^{0}, p^{1}, q) \equiv Σ_{i = 1}^{n} p_{i}^{1} q_{i} / Σ_{i = 1}^{n} p_{i}^{0} q_{i} = Σ_{i = 1}^{n} s_{i} (p_{i}^{1} / p_{i}^{0}), \\ (16.15) \end{array}

where the (hypothetical) hybrid value shares s_i²⁷ corresponding to the quantity weights vector q are defined by

\begin{matrix} s_{i} \equiv p_{i}^{0} q_{i} / Σ_{j = 1}^{n} p_{j}^{0} q_{j} f o r i = 1, 2, . . ., n . & (16.16) \end{matrix}

16.26 The main reason price statisticians might prefer a member of the family of Lowe or fixed-basket price indices defined by equation (16.15) is that the fixed-basket concept is easy to explain to the public. Note that the Laspeyres and Paasche indices are special cases of the pure price concept if we choose q = q⁰ (which leads to the Laspeyres index) or if we choose q = q¹ (which leads to the Paasche index).²⁸ The practical problem of picking q remains to be resolved, and that is the problem addressed in this section.

16.27 It should be noted that Walsh (1901, p. 105; 1921a) also saw the price index number problem in the above framework:

Commodities are to be weighted according to their importance, or their full values. But the problem of axiometry always involves at least two periods. There is a first period, and there is a second period which is compared with it. Price variations have taken place between the two, and these are to be averaged to get the amount of their variation as a whole. But the weights of the commodities at the second period are apt to be different from their weights at the first period. Which weights, then, are the right ones—those of the first period? Or those of the second? Or should there be a combination of the two sets? There is no reason for preferring either the first or the second. Then the combination of both would seem to be the proper answer. And this combination itself involves an averaging of the weights of the two periods. (Walsh, 1921a, p. 90)

Walsh’s suggestion will be followed, and thus the ith quantity weight, q_i, is restricted to be an average or mean of the base-period quantity q_i⁰ and the currentperiod quantity for product i q_i¹, say m(q_i⁰,q_i¹), for i = 1, 2, . . . , n.²⁹ Under this assumption, the Lowe price index (16.15) becomes

\begin{matrix} P_{L o} (p^{0}, p^{1}, q^{0}, q^{1}) \equiv \frac{Σ_{i = 1}^{n} p_{i}^{1} m (q_{i}^{0}, q_{i}^{1})}{Σ_{j = 1}^{n} p_{j}^{0} m (q_{j}^{0}, q_{j}^{1})} . & (16.17) \end{matrix}

16.28 In order to determine the functional form for the mean function m, it is necessary to impose some tests or axioms on the pure price index defined by equation (16.17). As in Section C.1, we ask that P_Lo satisfy the time reversal test, equation (16.13) above. Under this hypothesis, it is immediately obvious that the mean function m must be a symmetric mean;³⁰ that is, m must satisfy the following property: m(a,b) = m(b,a) for all a > 0 and b > 0. This assumption still does not pin down the functional form for the pure price index defined by equation (16.17) above. For example, the function m(a,b) could be the arithmetic mean, (1/2)a + (1/2)b, in which case equation (16.17) reduces to the Marshall Edgeworth (Marshall, 1887; Edgeworth, 1925) price index P_ME, which was the pure price index preferred by Knibbs (1924, p. 56):

\begin{matrix} P_{M E} (p^{0}, p^{1}, q^{0}, q^{1}) \equiv \frac{Σ_{i = 1}^{n} p_{i}^{1} {(q_{i}^{0} + q_{i}^{1}) / 2}}{Σ_{j = 1}^{n} p_{j}^{0} {(q_{j}^{0}, q_{j}^{1}) / 2}} . & (16.18) \end{matrix}

16.29 On the other hand, the function m(a,b) could be the geometric mean, (ab)^1/2, in which case equation (16.17) reduces to the Walsh (1901, p. 398; 1921a, p. 97) price index, P_W³¹:

\begin{matrix} P_{W} (p^{0}, p^{1}, q^{0}, q^{1}) \equiv \frac{Σ_{i = 1}^{n} p_{i}^{1} \sqrt{q_{i}^{0} q_{i}^{1}}}{Σ_{j = 1}^{n} p_{j}^{0} \sqrt{q_{j}^{0} q_{j}^{1}}} . & (16.19) \end{matrix}

16.30 There are many other possibilities for the mean function m, including the mean of order r, [(1/2) a^r + (1/2)b^r]^1/r for r ≠ 0. To completely determine the functional form for the pure price index P_Lo, it is necessary to impose at least one additional test or axiom on P_Lo(p⁰, p¹, q⁰, q¹).

16.31 There is a potential problem with the use of the Edgeworth-Marshall price index, equation (16.18), that has been noticed in the context of using the formula to make international comparisons of prices. If the price levels of a very large country are compared with the price levels of a small country using equation (16.18), then the quantity vector of the large country may totally overwhelm the influence of the quantity vector corresponding to the small country.³² In technical terms, the Edgeworth-Marshall formula is not homogeneous of degree 0 in the components of both q⁰ and q¹. To prevent this problem from occurring in the use of the pure price index P_K(p⁰, p¹, q⁰, q¹) defined by equation (16.17), it is asked that P_Lo satisfy the following invariance to proportional changes in current quantities test:³³

\begin{matrix} P_{L o} (p^{0}, p^{1}, q^{0}, λ q^{1}) = P_{L o} (p^{0}, p^{1}, q^{0}, q^{1}) \\ for all p^{0}, p^{1}, q^{0}, q^{1} and all λ > 0. & (16.20) \end{matrix}

The two tests, the time reversal test in equation (16.13) and the invariance test in equation (16.20), enable one to determine the precise functional form for the pure price index P_Lo defined by equation (16.17) above: The pure price index P_K must be the Walsh index P_W defined by equation (16.19).³⁴

16.32 To be of practical use by statistical agencies, an index number formula must be able to be expressed as a function of the base-period value shares, $s_{i}^{0}$ ; the current-period value shares, s_i¹; and the n price ratios, $p_{i}^{1} / p_{i}^{0}$ . The Walsh price index defined by equation (16.19) above can be rewritten in this format:

\begin{matrix} P_{w} (p^{0}, p^{1}, q^{0}, q^{1}) & \equiv \frac{Σ_{i = 1}^{n} p_{i}^{1} \sqrt{q_{i}^{0} q_{i}^{1}}}{Σ_{j = 1}^{n} p_{j}^{0} \sqrt{q_{j}^{0} q_{j}^{1}}} \\ = \frac{Σ_{i = 1}^{n} (p_{i}^{1} / \sqrt{p_{i}^{0} p_{i}^{1}}) \sqrt{s_{i}^{0} s_{i}^{1}}}{Σ_{j = 1}^{n} (p_{j}^{0} / \sqrt{p_{j}^{0} p_{j}^{1}}) \sqrt{s_{j}^{0} s_{j}^{1}}} \end{matrix}

\begin{matrix} = \frac{Σ_{i = 1}^{n} \sqrt{s_{i}^{0} s_{i}^{1}} \sqrt{p_{i}^{1} / p_{i}^{0}}}{Σ_{j = 1}^{n} \sqrt{s_{j}^{0} s_{j}^{1}} \sqrt{p_{j}^{0} / p_{j}^{1}}} . & (16.21) \end{matrix}

C.3 Conclusion

16.33 The approach taken to index number theory in this section was to consider averages of various fixed-basket price indices. The first approach was to take an evenhanded average of the two primary fixed-basket indices: the Laspeyres and Paasche price indices. These two primary indices are based on pricing out the baskets that pertain to the two periods (or locations) under consideration. Taking an average of them led to the Fisher ideal price index P_F defined by equation (16.12) above. The second approach was to average the basket quantity weights and then price out this average basket at the prices pertaining to the two situations under consideration. This approach led to the Walsh price index P_W defined by equation (16.19) above. Both these indices can be written as a function of the base-period value shares, $s_{i}^{0}$ ; the current-period value shares, s_i¹; and the n price ratios, $p_{i}^{1} / p_{i}^{0}$ . Assuming that the statistical agency has information on these three sets of variables, which index should be used? Experience with normal timeseries data has shown that these two indices will not differ substantially, and thus it is a matter of choice which of these indices is used in practice.³⁵ Both these indices are examples of “superlative indices.”³⁶ However, note that both treat the data pertaining to the two situations in a symmetric manner. Peter Hill (1993; see also Hill, 1988) commented on superlative price indices and the importance of a symmetric treatment of the data as follows:

Thus economic theory suggests that, in general, a symmetric index that assigns equal weight to the two situations being compared is to be preferred to either the Laspeyres or Paasche indices on their own. The precise choice of superlative index—whether Fisher, Törnqvist, or other superlative index—may be of only secondary importance as all the symmetric indices are likely to approximate each other, and the underlying theoretic index fairly closely, at least when the index number spread between the Laspeyres and Paasche is not very great. (P. Hill, 1993, p. 384)

D. Annual Weights and Monthly Price Indices

D.1 Lowe index with monthly prices and annual base-year quantities

16.34 It is now necessary to discuss a major practical problem with the theory of basket-type indices. Up to now, it has been assumed that the quantity vector q ≡ (q₁, q₂, . . . , q_n) that appeared in the definition of the Lowe index, P_Lo(p⁰, p¹, q) defined by equation (16.15), is either the base-period quantity vector q⁰ or the current-period quantity vector q¹ or an average of the two. In fact, in terms of actual statistical agency practice, the quantity vector q is usually taken to be an annual quantity vector that refers to a base year b, say, the year before the base period for the prices, period 0. Typically, a statistical agency will produce XMPIs at a monthly or quarterly frequency but, for the sake of definiteness, a monthly frequency will be assumed in what follows. Thus, a typical price index will have the form P_Lo(p⁰, p^t, q^b), where p⁰ is the price vector pertaining to the base-period month for prices, month 0, p^t is the price vector pertaining to the current-period month for prices, month t, say, and q^b is a reference basket quantity vector that refers to the base year b, which is equal to or before month 0.³⁷ Note that this Lowe index P_Lo(p⁰, p^t, q^b) is not a true Laspeyres index (because the annual quantity vector q^b is not equal to the monthly quantity vector q⁰ in general).³⁸

16.35 The question is this: Why do statistical agencies not pick the reference quantity vector q in the Lowe formula to be the monthly quantity vector q⁰ that pertains to transactions in month 0 (so that the index would reduce to an ordinary Laspeyres price index)? There are two main reasons:

Most economies are subject to seasonal fluctuations, and so picking the quantity vector of month 0 as the reference quantity vector for all months of the year would not be representative of transactions made throughout the year.
Monthly household quantity or value weights are usually collected by the statistical agency using an establishment survey with a relatively small sample. Hence, the resulting weights are usually subject to very large sampling errors, and so standard practice is to average these monthly value or quantity weights over an entire year (or in some cases, over several years), in an attempt to reduce these sampling errors. In other instances, where an establishment census is used, the reported value weights are for an annual period.

The index number problems that are caused by seasonal monthly weights are studied in more detail in Chapter 23. For now, it can be argued that the use of annual weights in a monthly index number formula is simply a method for dealing with the seasonality problem.³⁹

16.36 One problem with using annual weights corresponding to a perhaps distant year in the context of monthly XMPIs must be noted at this point. If there are systematic (but divergent) trends in product prices, and consumers or businesses increase their purchases of products that decline (relatively) in price and decrease their purchases of products that increase (relatively) in price, then the use of distant quantity weights will tend to lead to an upward bias in this Lowe index compared with one that uses more current weights, as is shown below. This observation suggests that statistical agencies should get up-to-date weights on an ongoing basis.

16.37 It is useful to explain how the annual quantity vector q^b could be obtained from monthly values on each product during the chosen base year b. Let the month m value of the reference population in the base year b for product i be $v_{i}^{b, m}$ , and let the corresponding price and quantity be $p_{i}^{b, m}$ and $q_{i}^{b, m}$ , respectively. Value, price, and quantity for each product are related by the following equations:

\begin{matrix} v_{i}^{b, m} = p_{i}^{b, m} q_{i}^{b, m}; i = 1, . . ., n; m = 1, . . ., 12. & (16.22) \end{matrix}

For each product i, the annual total $q_{i}^{b}$ can be obtained by price-deflating monthly values and summing over months in the base year b as follows:

\begin{matrix} p_{i}^{b} = Σ_{m = 1}^{12} \frac{v_{i}^{b, m}}{p_{i}^{b, m}} = Σ_{m = 1}^{12} q_{i}^{b, m}; i = 1, . . ., n, & (16.23) \end{matrix}

where equation (16.22) was used to derive equation (16.23). In practice, the above equations will be evaluated using aggregate values over closely related products, and the price $p_{i}^{b, m}$ will be the month m price index for this elementary product group i in year b relative to the first month of year b.

16.38 For some purposes, it is also useful to have annual prices by product to match up with the annual quantities defined by equation (16.23). Following national accounting conventions, a reasonable⁴⁰ price $p_{i}^{b}$ to match the annual quantity $q_{i}^{b}$ is the value of total value for product i in year b divided by $q_{i}^{b}$ . Thus, we have

\begin{matrix} p_{i}^{b} & = Σ_{m = 1}^{12} v_{i}^{b, m} / q_{i}^{b}; i = 1, . . ., n \\ = \frac{Σ_{m = 1}^{12} v_{i}^{b, m}}{Σ_{m = 1}^{12} v_{i}^{b, m} / p_{i}^{b, m}}; using equation (16.23) \\ = [Σ_{m = 1}^{12} s_{i}^{b, m} (p_{i}^{b, m})^{- 1}]^{- 1}, & (16.24) \end{matrix}

where the share of annual value on product i in month m of the base year is

\begin{matrix} s_{i}^{b, m} & = \frac{v_{i}^{b, m}}{Σ_{k = 1}^{12} v_{i}^{b, k}}; i = 1, . . ., n . & (16.25) \end{matrix}

Thus, the annual base-year price for product i, $p_{i}^{b}$ , turns out to be a monthly value weighted harmonic mean of the monthly prices for product i in the base year, $p_{i}^{b, 1}, p_{i}^{b, 2}, . . ., p_{i}^{b, 12}$ ¹².

16.39 Using the annual product prices for the base year defined by equation (16.24), a vector of these prices can be defined as p^b ≡ [p₁b, . . . , p_nb]. Using this definition, the Lowe index can be expressed as a ratio of two Laspeyres indices where the price vector p^b plays the role of base-period prices in each of the two Laspeyres indices:

\begin{matrix} P_{L o} (p^{0}, p^{t}, q^{b}) \equiv \frac{Σ_{i = 1}^{n} p_{i}^{t} q_{i}^{b}}{Σ_{i = 1}^{n} p_{i}^{0} q_{i}^{b}} \\ = \frac{Σ_{i = 1}^{n} p_{i}^{t} q_{i}^{b} / Σ_{i = 1}^{n} p_{i}^{b} q_{i}^{b}}{Σ_{i = 1}^{n} p_{i}^{0} q_{i}^{b} / Σ_{i = 1}^{n} p_{i}^{b} q_{i}^{b}} = \frac{Σ_{i = 1}^{n} s_{i}^{b} (p_{i}^{t} / q_{i}^{b})}{Σ_{i = 1}^{n} s_{i}^{b} (p_{i}^{0} / q_{i}^{b})} \\ = P_{L} (p^{b}, p^{t}, q^{b}) / P_{L} (p^{b}, p^{0}, q^{b}), & (16.26) \end{matrix}

where the Laspeyres formula P_L was defined by equation (16.5) above. Thus, the above equation shows that the Lowe monthly price index comparing the prices of month 0 with those of month t using the quantities of base year b as weights, P_Lo(p⁰, p^t, q^b), is equal to the Laspeyres index that compares the prices of month t with those of year b, P_L(p^b, p^t, q^b), divided by the Laspeyres index that compares the prices of month 0 with those of year b, P_L(p^b, p⁰, q^b). Note that the Laspeyres index in the numerator can be calculated if the base-year product value shares, $s_{i}^{b}$ , are known along with the price ratios that compare the prices of product i in month t, $p_{i}^{t}$ , with the corresponding annual average prices in the base year b, $p_{i}^{b}$ . The Laspeyres index in the denominator can be calculated if the base-year product value shares, $s_{i}^{b}$ , are known along with the price ratios that compare the prices of product i in month 0, $p_{i}^{0}$ , with the corresponding annual average prices in the base year b, $p_{i}^{b}$ .

16.40 Another convenient formula for evaluating the Lowe index, P_Lo(p⁰, p^t, q^b), uses the hybrid weights formula, equation (16.15). In the present context, the formula becomes

\begin{matrix} P_{L o} (p^{0}, p^{t}, q^{b}) & \equiv \frac{Σ_{i = 1}^{n} p_{i}^{t} q_{i}^{b}}{Σ_{i = 1}^{n} p_{i}^{0} q_{i}^{b}}, \\ = \frac{Σ_{i = 1}^{n} (p_{i}^{t} / p_{i}^{0}) p_{i}^{0} q_{i}^{b}}{Σ_{i = 1}^{n} p_{i}^{0} q_{i}^{b}} = Σ_{i = 1}^{n} (\frac{{Pi}^{t}}{{Pi}^{0}}) s_{i}^{0 b}, \\ (16.27) \end{matrix}

where the hybrid weights $s_{i}^{0b}$ using the prices of month 0 and the quantities of year b are defined by

\begin{matrix} s_{i}^{0 b} \equiv \frac{p_{i}^{0} q_{i}^{b}}{Σ_{j = 1}^{n} p_{j}^{0} q_{j}^{b}}; i = 1, . . ., n \\ = \frac{p_{i}^{b} q_{i}^{b} (p_{i}^{0} / p_{i}^{b})}{Σ_{j = 1}^{n} [p_{j}^{b} q_{j}^{b} (p_{j}^{0} / p_{j}^{b})]} . & (16.28) \end{matrix}

Equation (16.28) shows how the base-year values, $p_{i}^{b} q_{i}^{b}$ , can be multiplied by the product price indices, $p_{i}^{0} / p_{i}^{b}$ , to calculate the hybrid shares.

16.41 One additional formula for the Lowe index, P_Lo(p⁰, p^t, q^b), is exhibited. Note that the Laspeyres decomposition of the Lowe index defined by the third line in equation (16.26) involves the very long-term price relatives, $p_{i}^{t} / p_{i}^{b}$ , that compare the prices in month t, $p_{i}^{t}$ , with the possibly distant base-year prices, $p_{i}^{b}$ . Further, the hybrid share decomposition of the Lowe index defined by the third line in equation (16.27) involves the long-term monthly price relatives, $p_{i}^{t} / p_{i}^{0}$ , which compare the prices in month t, $p_{i}^{t}$ , with the base-month prices, $p_{i}^{0}$ . Both these formulas are not satisfactory in practice because of the problem of sample attrition: Each month, a substantial fraction of products disappears from the marketplace and thus it is useful to have a formula for updating the previous month’s price index using just month-over-month price relatives. In other words, long-term price relatives disappear at a rate that is too large in practice to base an index number formula on their use. The Lowe index for month t + 1, P_Lo(p⁰, p^t+1, q^b), can be written in terms of the Lowe index for month t, P_Lo(p⁰, p^t, q^b), and an updating factor as follows:

\begin{matrix} P_{L o} (p^{0}, p^{t + 1,} q^{b}) \equiv \frac{Σ_{i = 1}^{n} p_{i}^{t + 1} q_{i}^{b}}{Σ_{i = 1}^{n} p_{i}^{0} q_{i}^{b}} \\ = [\frac{Σ_{i = 1}^{n} p_{i}^{t} q_{i}^{b}}{Σ_{i = 1}^{n} p_{i}^{0} q_{i}^{b}}] [\frac{Σ_{i = 1}^{n} p_{i}^{t + 1} q_{i}^{b}}{Σ_{i = 1}^{n} p_{i}^{t} q_{i}^{b}}] \\ = P_{L o} (p^{0}, p^{t}, q^{b}) [\frac{Σ_{i = 1}^{n} p_{i}^{t + 1} q_{i}^{b}}{Σ_{i = 1}^{n} p_{i}^{t} q_{i}^{b}}] \\ = P_{L o} (p^{0}, p^{t}, q^{b}) [\frac{Σ_{i = 1}^{n} (\frac{p_{i}^{t + 1}}{p_{i}^{t}}) p_{i}^{t} q_{i}^{b}}{Σ_{i = 1}^{n} p_{i}^{t} q_{i}^{b}}] \\ = P_{L o} (p^{0}, p^{t}, q^{b}) [Σ_{i = 1}^{n} (\frac{p_{i}^{t + 1}}{p_{i}^{t}}) s_{i}^{t b}], & (16.29) \end{matrix}

where the hybrid weights $s_{i}^{tb}$ are defined by

\begin{matrix} s_{i}^{t b} \equiv \frac{p_{i}^{t + 1} q_{i}^{b}}{Σ_{j = 1}^{n} p_{j}^{t} q_{j}^{b}}; i = 1, . . ., n . & (16.30) \end{matrix}

Thus, the required updating factor, going from month t to month t + 1, is the chain link index $Σ_{i = 1}^{n} s_{i}^{tb} (p_{i}^{t + 1} / p_{i}^{t})$ , which uses the hybrid share weights $s_{i}^{b 0 t}$ corresponding to month t and base year b.

16.42 The Lowe index P_Lo(p⁰, p^t, q^b) can be regarded as an approximation to the ordinary Laspeyres index, P_L(p⁰, p^t, q⁰), that compares the prices of the base month 0, p⁰, to those of month t, p^t, using the quantity vector of month 0, q⁰, as weights. There is a relatively simple formula that relates these two indices. To explain this formula, it is first necessary to make a few definitions. Define the ith price relative between month 0 and month t as

\begin{matrix} r_{i} \equiv p_{i}^{t} / p_{i}^{0}; i = 1, . . ., n . & (16.31) \end{matrix}

The ordinary Laspeyres price index, going from month 0 to t, can be defined in terms of these price relatives as follows:

\begin{matrix} P_{L} (p^{0}, p^{t}, q^{0}) & \equiv \frac{Σ_{i = 1}^{n} p_{i}^{t} q_{i}^{0}}{Σ_{i = 1}^{n} p_{i}^{0} q_{i}^{0}} \\ = \frac{Σ_{i = 1}^{n} (\frac{p_{i}^{t}}{p_{i}^{0}}) p_{i}^{0} q_{i}^{0}}{Σ_{i = 1}^{n} p_{i}^{0} q_{i}^{0}} = Σ_{i = 1}^{n} (\frac{p_{i}^{t}}{p_{i}^{0}}) s_{i}^{0} \\ = Σ_{i = 1}^{n} s_{i}^{0} r_{i} \equiv r^{*}, (16.32) \end{matrix}

where the month 0 value shares $s_{i}^{0}$ are defined as follows:

\begin{matrix} s_{i}^{0} & \equiv \frac{p_{i}^{0} q_{i}^{0}}{Σ_{j = 1}^{n} p_{j}^{0} q_{j}^{0}}; i = 1, . . ., n . & (16.33) \end{matrix}

16.43 Define the ith quantity relative t_i as the ratio of the quantity of product i used in the base year b, $q_{i}^{b}$ , to the quantity used in month 0, $q_{i}^{0}$ , as follows:

\begin{matrix} t_{i} \equiv q_{i}^{b} / q_{i}^{0}; i = 1, . . ., n . & (16.34) \end{matrix}

The Laspeyres quantity index, Q_L(q⁰, q^b, p⁰), that compares quantities in year b, q^b, with the corresponding quantities in month 0, q⁰, using the prices of month 0, p⁰, as weights can be defined as a weighted average of the quantity ratios t_i as follows:

\begin{matrix} Q_{L} (q^{0}, q^{b}, p^{0}) & \equiv \frac{Σ_{i = 1}^{n} p_{i}^{0} q_{i}^{b}}{Σ_{i = 1}^{n} p_{i}^{0} q_{i}^{0}} \\ = \frac{Σ_{i = 1}^{n} (\frac{q_{i}^{b}}{q_{i}^{0}}) p_{i}^{0} q_{i}^{0}}{Σ_{i = 1}^{n} p_{i}^{0} q_{i}^{0}} \\ = Σ_{i = 1}^{n} (\frac{q_{i}^{b}}{q_{i}^{0}}) s_{i}^{0}; using equation (16.34) \\ = Σ_{i = 1}^{n} s_{i}^{0} t_{i} \equiv t^{*} . (16.35) \end{matrix}

16.44 Using equation (A16.2.4) in Appendix 16.2, the relationship between the Lowe index P_Lo(p⁰, p^t, q^b) that uses the quantities of year b as weights to compare the prices of month t to month 0 and the corresponding ordinary Laspeyres index P_L(p⁰, p^t, q⁰) that uses the quantities of month 0 as weights is defined as

\begin{matrix} P_{L o} (p^{0}, p^{t}, q^{b}) \equiv \frac{Σ_{i = 1}^{n} p_{i}^{t} q_{i}^{b}}{Σ_{i = 1}^{n} p_{i}^{0} q_{i}^{b}} \\ = \begin{matrix} P_{L} (p^{0}, p^{t}, q^{0}) \end{matrix} + \frac{Σ_{i = 1}^{n} (r_{i} - r *) (t_{i} - t *) s_{i}^{0}}{Q_{L} (q^{0}, q^{b} p^{0})} . & (16.36) \end{matrix}

Thus, the Lowe price index using the quantities of year b as weights, P_Lo(p⁰, p^t, q^b), is equal to the usual Laspeyres index using the quantities of month 0 as weights, P_L(p⁰, p^t, q⁰), plus a covariance term $Σ_{i = 1}^{n} (r_{i} - r^{*}) (t_{i} - t^{*}) s_{i}^{0}$ between the price relatives $r_{i} \equiv p_{i}^{t} / p_{i}^{0}$ and the quantity relatives $t_{i} \equiv q_{i}^{b} / q_{i}^{0}$ , divided by the Laspeyres quantity index Q_L(q⁰, q^b, p⁰) between month 0 and base year b.

16.45 Equation (16.36) shows that the Lowe price index will coincide with the Laspeyres price index if the covariance or correlation between the month 0 to t price relatives $r_{i} \equiv p_{i}^{t} / p_{i}^{0}$ and the month 0 to year b quantity relatives $t_{i} \equiv q_{i}^{b} / q_{i}^{0}$ is zero. Note that this covariance will be zero under three different sets of conditions:

If the month t prices are proportional to the month 0 prices so that all r_i – r^*,
If the base year b quantities are proportional to the month 0 quantities so that all t_i – t^*, and
If the distribution of the relative prices r_i is independent of the distribution of the relative quantities t_i.

The first two conditions are unlikely to hold empirically, but the third is possible, at least approximately, if purchasers do not systematically change their purchasing habits in response to changes in relative prices.

16.46 If this covariance in equation (16.36) is negative, then the Lowe index will be less than the Laspeyres and, finally, if the covariance is positive, then the Lowe index will be greater than the Laspeyres index. Although the sign and magnitude of the covariance term is ultimately an empirical matter, it is possible to make some reasonable conjectures about its likely sign. If the base year b precedes the price reference month 0 and there are long-term trends in prices, then it is likely that this covariance is positive, and hence that the Lowe index will exceed the corresponding Laspeyres price index;⁴¹ that is,

\begin{matrix} P_{L o} (p^{0}, p^{t}, q^{b}) > P_{L} (p^{0}, p^{t}, q^{0}) . & (16.37) \end{matrix}

To see why this covariance is likely to be positive, suppose that there is a long-term upward trend in the price of product i so that $r_{i} - r^{*} \equiv (p_{i}^{t} / p_{i}^{0}) - r^{*}$ is positive. With normal substitution responses,⁴² $q_{i}^{t} / q_{i}^{0}$ less an average quantity change of this type (t^*) is likely to be negative or, upon taking reciprocals, $q_{i}^{0} / q_{i}^{t}$ less an average quantity change of this (reciprocal) type is likely to be positive. But if the long-term upward trend in prices has persisted back to the base year b, then $t_{i} - t * \equiv (q_{i}^{b} / q_{i}^{0}) - t *$ is also likely to be positive. Hence, the covariance will be positive under these circumstances. Moreover, the more distant the weight reference year b is from the price reference month 0, the bigger the residuals t_i – t^* will likely be and the bigger will be the positive covariance. Similarly, the more distant is the current-period month t from the base-period month 0, the bigger the residuals r_i – r^* will likely be and the bigger will be the positive covariance. Thus, under the assumptions that there are long-term trends in prices and normal substitution responses, the Lowe index will normally be greater than the corresponding Laspeyres index.

16.47 Define the Paasche index between months 0 and t as follows:

\begin{matrix} P_{P} (p^{0}, p^{t}, q^{t}) \equiv \frac{Σ_{i = 1}^{n} p_{i}^{t} q_{i}^{t}}{Σ_{i = 1}^{n} p_{i}^{0} q_{i}^{t}} & (16.38) \end{matrix}

As was discussed in Section C.1 above, a reasonable target index to measure the price change going from month 0 to t is some sort of symmetric average of the Paasche index P_P(p⁰, p^t, q^t) defined by equation (16.38) and the corresponding Laspeyres index P_L(p⁰, p^t, q⁰) defined by equation (16.32). Adapting equation (A16.1.5) in Appendix 16.1, the relationship between the Paasche and Laspeyres indices can be written as follows:

\begin{matrix} P_{P} (p^{0}, p^{t}, q^{t}) = P_{L} (p^{0}, p^{t}, q^{0}) \\ + \frac{Σ_{i = 1}^{n} (r_{i} - r *) (u_{i} - u *) s_{i}^{0}}{Q_{L} (q^{0}, q^{t}, p^{0})}, & (16.39) \end{matrix}

where the price relatives $r_{i} \equiv p_{i}^{t} / p_{i}^{0}$ are defined by equation (16.31) and their share-weighted average r^* by equation (16.32) and the u_i, u^*, and Q_L are defined as follows:

\begin{matrix} u_{i} \equiv q_{i}^{t} / q_{i}^{0}; i = 1, . . ., n, & (16.40) \end{matrix}

\begin{matrix} u^{*} \equiv Σ_{i = 1}^{n} s_{i}^{0} u^{i} = Q_{L} (q^{0}, q^{t}, p^{0}), & (16.41) \end{matrix}

and the month 0 value shares $s_{i}^{0}$ are defined by equation (16.33). Thus, u^* is equal to the Laspeyres quantity index between months 0 and t. This means that the Paasche price index that uses the quantities of month t as weights, P_P(p⁰, p^t, q^t), is equal to the usual Laspeyres index using the quantities of month 0 as weights, P_L(p⁰, p^t, q⁰), plus a covariance term $Σ_{i = 1}^{n} (r_{i} - r^{*}) (u_{i} - u^{*}) s_{i}^{0}$ between the price relatives $r_{i} \equiv p_{i}^{t} / p_{i}^{0}$ and the quantity relatives $u_{i} \equiv q_{i}^{t} / q_{i}^{0}$ , divided by the Laspeyres quantity index Q_L(q⁰, q^t, p⁰) between month 0 and month t.

16.48 Although the sign and magnitude of the covariance term is again an empirical matter, it is possible to make a reasonable conjecture about its likely sign. If there are long-term trends in prices, and purchasers respond normally to price changes in their purchases, then it is likely that this covariance is negative, and hence the Paasche index will be less than the corresponding Laspeyres price index; that is,

\begin{matrix} P_{P} (p^{0}, p^{t}, q^{b}) < P_{L} (p^{0}, p^{t}, p^{0}) . & (16.42) \end{matrix}

To see why this covariance is likely to be negative, suppose that there is a long-term upward trend in the price of product i⁴³ so that $r_{i} - r^{*} \equiv (p_{i}^{t} / p_{i}^{0}) - r^{*}$ is positive. With normal substitution responses, ${q_{i}^{}}^{t} / {q_{i}^{}}^{0}$ less an average quantity change of this type (u^*) is likely to be negative. Hence, $u_{i} - u^{*} \equiv ({q_{i}^{}}^{t} / {q_{i}^{}}^{0}) - u^{*}$ is likely to be negative. Thus, the covariance will be negative under these circumstances. Moreover, the more distant the base month 0 is from the current month t, the bigger in magnitude the residuals u_i – u^* will likely be and the bigger in magnitude will be the negative covariance.⁴⁴ Similarly, the more distant the current-period month t is from the base-period month 0, the bigger the residuals r_i – r^* will likely be and the bigger in magnitude will be the covariance. Thus, under the assumptions that there are long-term trends in prices and normal substitution responses, the Laspeyres index will be greater than the corresponding Paasche index, with the divergence likely growing as month t becomes more distant from month 0.

16.49 Putting the arguments in the three previous paragraphs together, it can be seen that under the assumptions that there are long-term trends in prices and normal substitution responses, the Lowe price index between months 0 and t will exceed the corresponding Laspeyres price index, which in turn will exceed the corresponding Paasche price index; that is, under these hypotheses,

\begin{matrix} P_{L o} (p^{0}, p^{t}, q^{b}) > P_{L} (p^{0}, p^{t}, q^{0}) > P_{P} (p^{0}, p^{t}, q^{t}) . & (16.43) \end{matrix}

Thus, if the long-run target price index is an average of the Laspeyres and Paasche indices, it can be seen that the Laspeyres index will have an upward bias relative to this target index, and the Paasche index will have a downward bias. In addition, if the base year b is prior to the price reference month, month 0, then the Lowe index will also have an upward bias relative to the Laspeyres index and hence also to the target index.

D.2 Lowe index and midyear indices

16.50 The discussion in the previous paragraph assumed that the base year b for quantities preceded the base month for prices, month 0. However, if the currentperiod month t is quite distant from the base month 0, then it is possible to think of the base year b as referring to a year that lies between months 0 and t. If the year b does fall between months 0 and t, then the Lowe index becomes a midyear index.⁴⁵ The Lowe midyear index no longer has the upward biases indicated by the inequalities in equation (16.43) under the assumption of long-term trends in prices and normal substitution responses by quantities.

16.51 It is now assumed that the base-year quantity vector q^b corresponds to a year that lies between months 0 and t. Under the assumption of long-term trends in prices and normal substitution effects so that there are also long-term trends in quantities (in the opposite direction to the trends in prices so that if the ith product price is trending up, then the corresponding ith quantity is trending down), it is likely that the intermediate-year quantity vector will lie between the monthly quantity vectors q⁰ and q^t. The midyear Lowe index, P_Lo(p⁰, p^t, q^b), and the Laspeyres index going from month 0 to t, P_L(p⁰, p^t, q⁰), will still satisfy the exact relationship given by equation (16.36). Thus, P_Lo(p⁰, p^t, q^b) will equal P_L(p⁰, p^t, q⁰) plus the covariance term $Σ_{i = 1}^{n} (r_{i} - r^{*}) (t_{i} - t^{*}) s_{i}^{0} / Q_{L} (q^{0}, q^{b}, p^{0})$ , where Q_L(q⁰, q^b, p⁰) is the Laspeyres quantity index going from month 0 to t. This covariance term is likely to be negative, so that

\begin{matrix} P_{L} (p^{0}, p^{t}, q^{0}) > P_{Lo} (p^{0}, p^{t}, q^{b}) . & (16.44) \end{matrix}

To see why this covariance is likely to be negative, suppose that there is a long-term upward trend in the price of product i so that $r_{i} - r^{*} \equiv (p_{i}^{t} / p_{i}^{0}) - r^{*}$ is positive. With normal substitution responses, q_i will tend to decrease relatively over time, and because ${q_{i}^{}}^{b}$ is assumed to be between ${q_{i}^{}}^{0}$ and $q_{i}^{t}$ , $q_{i}^{b} / q_{i}^{0}$ less an average quantity change of this type, r^* is likely to be negative. Hence $u_{i} - u^{*} \equiv (q_{i}^{b} / q_{i}^{0}) - t^{*}$ is likely to be negative. Thus, the covariance is likely to be negative under these circumstances. Under the assumptions that the quantity base year falls between months 0 and t and that there are long-term trends in prices and normal substitution responses, the Laspeyres index will normally be larger than the corresponding Lowe midyear index, with the divergence likely growing as month t becomes more distant from month 0.

16.52 It can also be seen that under the above assumptions, the midyear Lowe index is likely to be greater than the Paasche index between months 0 and t; that is,

P_{L o} (p^{0}, p^{t}, q^{b}) > P_{P} (p^{0}, p^{t}, q^{t}) . (16.45)

To see why the above inequality is likely to hold, think of q^b starting at the month 0 quantity vector q⁰ and then trending smoothly to the month t quantity vector q^t. When q^b = q⁰, the Lowe index P_Lo(p⁰, p^t, q^b) becomes the Laspeyres index P_L(p⁰, p^t, q⁰). When q^b = q^t, the Lowe index P_Lo(p⁰, p^t, q^b) becomes the Paasche index P_P(p⁰, p^t, q^t). Under the assumption of trending prices and normal substitution responses to these trending prices, it was shown earlier that the Paasche index will be less than the corresponding Laspeyres price index; that is, that P_P(p⁰, p^t, q^t) was less than P_L(p⁰, p^t, q⁰); recall equation (16.42). Thus, under the assumption of smoothly trending prices and quantities between months 0 and t, and assuming that q^b is between q⁰ and q^t, we will have

\begin{matrix} P_{P} (p^{0}, p^{t}, q^{t}) < P_{L o} (p^{0}, p^{t}, q^{b}) < P_{L} (p^{0}, p^{t}, q^{0}) . \\ (16.46) \end{matrix}

Thus, if the base year for the Lowe index is chosen to be in between the base month for the prices, month 0, and the current month for prices, month t, and there are trends in prices with corresponding trends in quantities that correspond to normal substitution effects, then the resulting Lowe index is likely to lie between the Paasche and Laspeyres indices going from months 0 to t. If the trends in prices and quantities are smooth, then choosing the base year halfway between periods 0 and t should give a Lowe index that is approximately halfway between the Paasche and Laspeyres indices and hence will be very close to an ideal target index between months 0 and t. This basic idea has been implemented by Okamoto (2001) using Japanese consumer data and he found that the resulting midyear indices approximated the corresponding Fisher ideal indices very closely.

16.53 It should be noted that these midyear indices can be computed only on a retrospective basis; that is, they cannot be calculated in a timely fashion as can Lowe indices that use a base year before month 0. Thus, midyear indices cannot be used to replace the more timely Lowe indices. However, these timely Lowe indices are likely to have an upward bias even bigger than the usual Laspeyres upward bias compared with an ideal target index, which was taken to be an average of the Paasche and Laspeyres indices.

16.54 All of the inequalities derived in this section rest on the assumption of long-term trends in prices (and corresponding economic responses in quantities). If there are no systematic long-run trends in prices and only random fluctuations around a common trend in all prices, then the above inequalities are not valid and the Lowe index using a prior base year will probably provide a perfectly adequate approximation to both the Paasche and Laspeyres indices. However, there are some reasons for believing that some long-run trends in prices exist:

(1) The computer chip revolution of the past 40 years has led to strong downward trends in the prices of products that use these chips intensively. As new uses for chips are developed, the share of products that are chip-intensive grows, which implies that what used to be a relatively minor problem has become a more major problem.
(2) Other major scientific advances have had similar effects. For example, the invention of fiberoptic cable (and lasers) has led to a downward trend in telecommunications prices as obsolete technologies based on copper wire are gradually replaced.
(3) Since the end of World War II, a series of international trade agreements have dramatically reduced tariffs around the world. These reductions, combined with improvements in transportation technologies, have led to a rapid growth of international trade and remarkable improvements in international specialization. Manufacturing activities in the more developed economies have gradually been outsourced to lower-wage countries, leading to deflation in goods prices in most countries. However, many services cannot be readily outsourced,⁴⁶ and so on average the price of services trends upward while the price of goods trends downward.
(4) At the microeconomic level, there are tremendous differences in growth rates of firms. Successful firms expand their scale, lower their costs, and cause less successful competitors to wither away with their higher prices and lower volumes. This leads to a systematic negative correlation between changes in item prices and the corresponding changes in item volumes, which can be very large.

Thus, there is some a priori basis for assuming longrun divergent trends in prices and hence some basis for concern that a Lowe index that uses a base year for quantity weights that is prior to the base month for prices may be upward biased, compared to a more ideal target index.

D.3 Young index

16.55 Recall the definitions for the base-year quantities, ${q_{i}^{}}^{b}$ , and the base-year prices, ${p_{i}^{}}^{b}$ , given by equation (16.23) and equation (16.24). The baseyear value shares can be defined in the usual way as follows:

\begin{matrix} s_{i}^{b} \equiv \frac{p_{i}^{b} q_{i}^{b}}{Σ_{k = 1}^{n} p_{k}^{b} q_{k}^{b}}; i = 1, . . ., n . & (16.47) \end{matrix}

Define the vector of base-year value shares in the usual way as s^b ≡ [s₁^b, . . . , s_n^b]. These base-year value shares were used to provide an alternative formula for the base year b Lowe price index going from month 0 to t defined in equation (16.26) as $P_{Lo} (P^{0}, P^{t}, q^{b}) = \sum_{i = 1}^{n} {s_{i}}^{b} ({p_{i}}^{t} / {p_{i}}^{b}) / \sum_{i = 1}^{n} {s_{i}}^{b} ({p_{i}}^{0} / {p_{i}}^{b})$ . Rather than using this index as their short-run target index, many statistical agencies use the following closely related index:

\begin{matrix} P_{Y} (p^{0}, p^{t}, s^{b}) = Σ_{i = 1}^{n} s_{i}^{b} ({p_{i}^{}}^{t} / {p_{i}^{}}^{b}) . & (16.48) \end{matrix}

This type of index was first defined by the English economist Arthur Young (1812).⁴⁷ Note that there is a change in focus when the Young index is used compared with when the indices proposed earlier in this chapter are used. Up to this point, the indices proposed have been of the fixed-basket type (or averages of such indices), where a product basket that is somehow representative for the two periods being compared is chosen and then “purchased” at the prices of the two periods and the index is taken to be the ratio of these two costs. On the other hand, for the Young index, one instead chooses representative value shares that pertain to the two periods under consideration and then uses these shares to calculate the overall index as a share-weighted average of the individual price ratios, $p_{i}^{t} / p_{i}^{0}$ . Note that this share-weighted average of price ratios view of index number theory is a bit different from the view taken at the beginning of this chapter, which viewed the index number problem as the problem of decomposing a value ratio into the product of two terms, one of which expresses the amount of price change between the two periods and the other expresses the amount of quantity change.⁴⁸

16.56 Statistical agencies sometimes regard the Young index defined above as an approximation to the Laspeyres price index P_L(p⁰, p^t, q⁰). Hence, it is of interest to see how the two indices compare. Defining the longterm monthly price relatives going from month 0 to t as $r_{i} \equiv p_{i}^{t} / p_{i}^{0}$ and using equations (16.32) and (16.48),

\begin{matrix} P_{Y} (p^{0}, p^{t}, s^{b}) - P_{L} (p^{0}, p^{t}, q^{0}) \\ \equiv Σ_{i = 1}^{n} s_{i}^{b} (\frac{p_{i}^{t}}{p_{i}^{0}}) - Σ_{i = 1}^{n} s_{i}^{0} (\frac{p_{i}^{t}}{p_{i}^{0}}) \\ = Σ_{i = 1}^{n} [s_{i}^{b} - s_{i}^{0}] (\frac{p_{i}^{t}}{p_{i}^{0}}) = Σ_{i = 1}^{n} [s_{i}^{b} - s_{i}^{0}] r_{i} \\ = Σ_{i = 1}^{n} [s_{i}^{b} - s_{i}^{0}] [r_{i} - r^{*}] + r^{*} Σ_{i = 1}^{n} [s_{i}^{b} - s_{i}^{0}] \\ = Σ_{i = 1}^{n} [s_{i}^{b} - s_{i}^{0}] [r_{i} - r^{*}], & (16.49) \end{matrix}

because $\sum_{i = 1}^{n} s_{i}^{b} = \sum_{i = 1}^{n} s_{i}^{0} = 1$ and defining $\sum_{i = 1}^{n} s_{i}^{o} r_{i} = P_{L} (p^{0}, p^{t}, q^{0})$ . Thus, the Young index P_Y(p⁰, p^t, s^b) is equal to the Laspeyres index P_L(p⁰, p^t, q⁰) plus the covariance between the difference in the annual shares pertaining to year b and the month 0 shares, $s_{i}^{b}$ – $s_{i}^{0}$ , and the deviations of the relative prices from their mean, r_i – r^*.

16.57 It is no longer possible to guess the likely sign of the covariance term. The question is no longer whether the quantity demanded goes down as the price of product i goes up (the answer to this question is usually yes) but whether the share of traded value goes down as the price of product i goes up. The answer depends on the elasticity of demand/supply for the product. However, let us provisionally assume that, for, say, an import price index, there are long-run trends in product prices and if the trend in prices for product i is above the mean, then the value share for the product trends down (and vice versa). Thus, we are assuming high elasticities or very strong substitution effects. Assuming also that the base year b is before month 0, then under these conditions, suppose that there is a long-term upward trend in the price of product i so that $r_{i} - r^{*} \equiv (p_{i}^{t} / p_{i}^{0}) - r^{*}$ is positive. With the assumed very elastic purchaser substitution responses, s_i will tend to decrease relatively over time. Because $s_{i}^{b}$ is assumed to be before $s_{i}^{0}$ , $s_{i}^{0}$ is expected to be less than $s_{i}^{b}$ or, in other words, $s_{i}^{b}$ – $s_{i}^{0}$ will likely be positive. Thus, the covariance is likely to be positive under these circumstances. Hence with long-run trends in prices and very elastic responses of purchasers to price changes, the Young index is likely to be greater than the corresponding Laspeyres index.

16.58 Assume that there are long-run trends in product prices. If the trend in prices for product i is above the mean, then suppose that the value share for the product trends up (and vice versa). Thus, we are assuming low elasticities or very weak substitution effects. Assume also that the base year b is before month 0, and suppose that there is a long-term upward trend in the price of product i so that $r_{i} - r^{*} \equiv ({p_{i}^{}}^{t} / {p_{i}^{}}^{0}) - r^{*}$ is positive. With the assumed very inelastic substitution responses, s_i will tend to increase relatively over time and because $s_{i}^{b}$ is assumed to be before $s_{i}^{0}$ , we will have $s_{i}^{0}$ greater than $s_{i}^{b}$ or, in other words, $s_{i}^{b}$ – $s_{i}^{0}$ will likely be negative. Thus, the covariance is likely to be negative under these circumstances. Hence with long-run trends in prices and very inelastic responses of purchasers to price changes, the Young index is likely to be less than the corresponding Laspeyres index.

16.59 The previous two paragraphs indicate that a priori, it is not known what the likely difference between the Young index and the corresponding Laspeyres index will be. If elasticities of substitution are close to 1, then the two sets of value shares, $s_{i}^{b}$ and $s_{i}^{0}$ , will be close to each other and the difference between the two indices will be close to zero. However, if monthly value shares have strong seasonal components, then the annual shares $s_{i}^{b}$ could differ substantially from the monthly shares $s_{i}^{0}$ .

16.60 It is useful to have a formula for updating the previous month’s Young price index using only month over-month price relatives. The Young index for month t + 1, P_Y(p⁰, p^t + 1, s^b), can be written in terms of the Lowe index for month t, P_Y(p⁰, p^t, s^b), and an updating factor as follows:

\begin{matrix} P_{Y} (p^{0}, p^{t + 1}, s^{b}) \equiv Σ_{i = 1}^{n} s_{i}^{b} (\frac{p_{i}^{t + 1}}{p_{i}^{0}}) \\ = P_{Y} (p^{0}, p^{t}, s^{b}) \frac{Σ_{i = 1}^{n} s_{i}^{b} (p_{i}^{t + 1} / p_{i}^{0})}{Σ_{i = 1}^{n} s_{i}^{b} (p_{i}^{t} / p_{i}^{0})} \\ = P_{Y} (p^{0}, p^{t}, s^{b}) \frac{Σ_{i = 1}^{n} p_{i}^{b} q_{i}^{b} (p_{i}^{t + 1} / p_{i}^{0})}{Σ_{i = 1}^{n} p_{i}^{b} q_{i}^{b} (p_{i}^{t} / p_{i}^{0})}; \\ using equation (16.47) \\ = P_{Y} (p^{0}, p^{t}, s^{b}) \frac{Σ_{i = 1}^{n} p_{i}^{b} q_{i}^{b} (\frac{p_{i}^{t}}{p_{i}^{0}}) (\frac{p_{i}^{t + 1}}{p_{i}^{0}})}{Σ_{i = 1}^{n} p_{i}^{b} q_{i}^{b} (p_{i}^{t} / p_{i}^{0})} \\ = P_{Y} (p^{0}, p^{t}, s^{b}) [Σ_{i = 1}^{n} s_{i}^{b 0 t} (p_{i}^{t + 1} / p_{i}^{t})], & (16.50) \end{matrix}

where the hybrid weights $s_{i}^{b 0 t}$ are defined by

\begin{matrix} s_{i}^{b 0 t} & \equiv \frac{p_{i}^{b} q_{i}^{b} (p_{i}^{t} / p_{i}^{0})}{Σ_{i = 1}^{n} p_{k}^{b} q_{k}^{b} (p_{k}^{t} / p_{k}^{0})} \\ = \frac{s_{i}^{b} (p_{i}^{t} / p_{i}^{0})}{Σ_{i = 1}^{n} s_{k}^{b} (p_{k}^{t} / p_{k}^{0})}; i = 1, . . ., n . & (16.51) \end{matrix}

Thus, the hybrid weights $s_{i}^{b 0 t}$ can be obtained from the base-year weights $s_{i}^{b}$ by updating them, that is, by multiplying them by the price relatives (or indices at higher levels of aggregation), $p_{i}^{t} / p_{i}^{0}$ . Thus, the required updating factor, going from month t to month t + 1, is the chain link index, $\sum_{i = 1}^{n} s_{i}^{b 0 t} (p_{i}^{t + 1} / p_{t}^{t})$ , which uses the hybrid share weights $s_{i}^{b 0 t}$ defined by equation (16.51).

16.61 Even if the Young index provides a close approximation to the corresponding Laspeyres index, it is difficult to recommend the use of the Young index as a final estimate of the change in prices going from period 0 to t, just as it was difficult to recommend the use of the Laspeyres index as the final estimate of inflation going from period 0 to t. Recall that the problem with the Laspeyres index was its lack of symmetry in the treatment of the two periods under consideration; that is, using the justification for the Laspeyres index as a good fixedbasket index, there was an identical justification for the use of the Paasche index as an equally good fixedbasket index to compare periods 0 and t. The Young index suffers from a similar lack of symmetry with respect to the treatment of the base period. The problem can be explained as follows. The Young index, P_Y(p⁰, p^t, s^b), defined by equation (16.48), calculates the price change between months 0 and t, treating month 0 as the base. But there is no particular reason other than convention to treat month 0 as the base month. Hence, if we treat month t as the base and use the same formula to measure the price change from month t back to month 0, the index $P_{Y} (p^{0}, p^{t}, s^{b}) = \sum_{i = 1}^{n} s_{i}^{b} (p_{i}^{0} / p_{i}^{t})$ would be appropriate. This estimate of price change can then be made comparable to the original Young index by taking its reciprocal, leading to the following rebased Young index,⁴⁹ P_Y^*(p⁰, p^t, s^b), defined as

\begin{matrix} {P_{Y}}^{*} (p^{0}, p^{t}, s^{b}) & \equiv 1 / Σ_{i = 1}^{n} s_{i}^{b} (p_{i}^{0} / p_{i}^{t}) \\ = {[Σ_{i = 1}^{n} s_{i}^{b} {(p_{i}^{t} / p_{i}^{0})}^{- 1}]}^{- 1} . & (16.52) \end{matrix}

Thus, the rebased Young index, P_Y^*(p0, p^t, s^b), that uses the current month as the initial base period is a shareweighted harmonic mean of the price relatives going from month 0 to month t, whereas the original Young index, P_Y(p⁰, p^t, s^b), is a share-weighted arithmetic mean of the same price relatives.

16.62 Fisher argued that an index number formula should give the same answer no matter which period was chosen as the base:

Either one of the two times may be taken as the “base”. Will it make a difference which is chosen? Certainly, it ought not and our Test 1 demands that it shall not. More fully expressed, the test is that the formula for calculating an index number should be such that it will give the same ratio between one point of comparison and the other point, no matter which of the two is taken as the base. (I. Fisher, 1922, p. 64)

16.63 The problem with the Young index is that not only does it not coincide with its rebased counterpart, but there is a definite inequality between the two indices, namely

\begin{matrix} P_{Y}^{*} (p^{0}, p^{t}, s^{b}) \leq P_{Y} (p^{0}, p^{t}, s^{b}), & (16.53) \end{matrix}

with a strict inequality provided that the period t price vector p^t is not proportional to the period 0 price vector p⁰.⁵⁰ Thus, a statistical agency that uses the direct Young index P_Y(p⁰, p^t, s^b) will generally show a higher inflation rate than will a statistical agency that uses the same raw data but uses the rebased Young index, P_Y^*(p⁰, p^t, s^b).

16.64 The inequality in equation (16.53) does not tell us by how much the Young index will exceed its rebased time antithesis. However, in Appendix 16.3, it is shown that to the accuracy of a certain second-order Taylor series approximation, the following relationship holds between the direct Young index and its time antithesis:

\begin{matrix} P_{Y} (p^{0}, p^{t}, s^{b}) \approx & P_{Y} * (p^{0}, p^{t}, s^{b}) \\ + P_{Y} (p^{0}, p^{t}, s^{b}) \times Var e, & (16.54) \end{matrix}

where Var e is defined as

\begin{matrix} Var e \equiv Σ_{i = 1}^{n} s_{i}^{b} {[e_{i} - e^{*}]}^{2} . & (16.55) \end{matrix}

The deviations e_i are defined by 1 + e_i = r_i / r^* for i = 1, . . . , n where the r_i and their weighted mean r^* are defined by

\begin{matrix} r_{i} \equiv p_{i}^{t} / p_{i}^{0}; i = 1, . . ., n, & (16.56) \end{matrix}

\begin{matrix} r^{*} \equiv Σ_{i = 1}^{n} s_{i}^{b} r_{i}, & (16.57) \end{matrix}

which turns out to equal the direct Young index, P_Y(p⁰, p^t, s^b). The weighted mean of the e_i is defined as

\begin{matrix} e^{*} \equiv Σ_{i = 1}^{n} s_{i}^{b} e_{i}, & (16.58) \end{matrix}

which turns out to equal 0. Hence, the more dispersion there is in the price relatives $p_{i}^{t} / p_{i}^{0}$ , to the accuracy of a second-order approximation, the more the direct Young index will exceed its counterpart that uses month t as the initial price reference period rather than month 0.

16.65 Given two a priori equally plausible index number formulas that give different answers, such as the Young index and its time antithesis, Irving Fisher (1922, p. 136) generally suggested taking the geometric average of the two indices.⁵¹ A benefit of this averaging is that the resulting formula will satisfy the time reversal test. Thus, rather than using either the base period 0 Young index, P_Y(p⁰, p^t, s^b), or the current period t Young index, P_Y^*(p⁰, p^t, s^b), which is always below the base period 0 Young index if there is any dispersion in relative prices, it seems preferable to use the following index, which is the geometric average of the two alternatively based Young indices:⁵²

\begin{matrix} {P_{Y}}^{* *} (p^{0}, p^{t}, s^{b}) \equiv [P_{Y} (p^{0}, p^{t}, s^{b}) {P_{Y}}^{*} (p^{0}, p^{t}, s^{b})]^{1 / 2} \\ (16.59) \end{matrix}

If the base-year shares $s_{i}^{b}$ happen to coincide with both the month 0 and month t shares, $s_{i}^{0}$ and $s_{i}^{t}$ respectively, the time-rectified Young index $P_{Y}^{* *} (p^{0}, p^{t}, s^{b})$ defined by (16.59) will coincide with the Fisher ideal price index between months 0 and t, P_F(p⁰, p^t, q⁰, q^t), which will also equal the Laspeyres and Paasche indices under these conditions. Note also that the index $P_{Y}^{* *}$ defined by equation (16.59) can be produced on a timely basis by a statistical agency.

E. Divisia Index and Discrete Approximations

E.1 Divisia price and quantity indices

16.66 The second broad approach to index number theory relies on the assumption that price and quantity data change in a more or less continuous way.

16.67 Suppose that the price and quantity data on the n products in the chosen domain of definition can be regarded as continuous functions of (continuous) time, say p_i(t) and q_i(t) for i 1, . . . , n. Suppose further that the traded value at time t is V(t) defined in the obvious way as

\begin{matrix} V (t) \equiv Σ_{i = 1}^{n} p_{i} (t) q_{i} (t) . & (16.60) \end{matrix}

16.68 Now suppose that the functions p_i(t) and q_i(t) are differentiable. Then both sides of equation (16.60) can be differentiated with respect to time to obtain

\begin{matrix} V^{'} (t) / V (t) = Σ_{i = 1}^{n} {p_{i}}^{'} (t) q_{i} (t) + Σ_{i = 1}^{n} p_{i} (t) {q_{i}}^{'} (t) . & (16.61) \end{matrix}

Divide both sides of equation (16.61) by V(t) and, using equation (16.60), obtain the following equation:

\begin{matrix} V^{'} (t) / V (t) & = \frac{Σ_{i = 1}^{n} {p_{i}}^{'} (t) q_{i} (t) + Σ_{i = 1}^{n} p_{i} (t) {q_{i}}^{'} (t)}{Σ_{j = 1}^{n} p_{j} (t) q_{j} (t)} \\ = Σ_{i = 1}^{n} \frac{{p_{i}}^{'} (t)}{p_{i} (t)} s_{i} (t) + Σ_{i = 1}^{n} \frac{{q_{i}}^{'} (t)}{q_{i} (t)} s_{i} (t), & (16.62) \end{matrix}

where the time t value share on product i, s_i(t), is defined as

\begin{matrix} s_{i} (t) \equiv \frac{p_{i} (t) q_{i} (t)}{Σ_{i = 1}^{n} p_{m} (t) q_{m} (t)} for i = 1, . . ., n . & (16.63) \end{matrix}

16.69 François Divisia (1926, p. 39) argued as follows: suppose the aggregate value at time t, V(t), can be written as the product of a time t price-level function, P(t), say, multiplied by a time I quantity-level function, Q(t), say; that is, we have

\begin{matrix} V (t) = P (t) Q (t) . & (16.64) \end{matrix}

Suppose, further, that the functions P(t) and Q(t) are differentiable. Then, differentiating (16.64) yields

\begin{matrix} V^{'} (t) = P^{'} (t) Q (t) + P (t) Q^{'} (t) . & (16.65) \end{matrix}

Dividing both sides of equation (16.65) by V(t) and using equation (16.64) leads to the following equation:

\begin{matrix} \frac{V^{'} (t)}{V (t)} = \frac{P^{'} (t)}{P t} + \frac{Q^{'} (t)}{Q (t)} . & (16.66) \end{matrix}

16.70 Divisia compared the two expressions for the logarithmic value derivative, V′(t)/V(t), given by equation (16.62) and equation (16.66). He simply defined the logarithmic rate of change of the aggregate price level, P′(t)/P(t), as the first set of terms on the right-hand side of equation (16.62), and he simply defined the logarithmic rate of change of the aggregate quantity level, Q′(t)/Q(t), as the second set of terms on the right-hand side of equation (16.62); that is, he made the following definitions:

\begin{matrix} \frac{P^{'} (t)}{P (t)} \equiv Σ_{i = 1}^{n} s_{i} (t) \frac{{P_{i}}^{'} (t)}{P_{i} (t)}; & (16.67) \end{matrix}

\begin{matrix} \frac{Q^{'} (t)}{Q (t)} \equiv Σ_{i = 1}^{n} s_{i} (t) \frac{{q_{i}}^{'} (t)}{q_{i} (t)}; & (16.68) \end{matrix}

16.71 Equations (16.67) and (16.68) are reasonable definitions for the proportional changes in the aggregate price and quantity (or quantity) levels, P(t) and Q(t).⁵³ The problem with these definitions is that economic data are not collected in continuous time; they are collected in discrete time. In other words, even though transactions can be thought of as occurring in continuous time, no producer records his or her purchases as they occur in continuous time; rather, purchases over a finite time period are cumulated and then recorded. A similar situation occurs for producers or sellers of products; firms cumulate their sales over discrete periods of time for accounting or analytical purposes. If one attempts to approximate continuous time by shorter and shorter discrete time intervals, empirical price and quantity data can be expected to become increasingly erratic, because consumers make purchases only at discrete points of time (and producers or sellers of products make sales only at discrete points of time). However, it is still of some interest to approximate the continuous time price and quantity levels, P(t) and Q(t) defined implicitly by equations (16.67) and (16.68), by discrete time approximations. This can be done in two ways. Either methods of numerical approximation can be used or assumptions about the path taken by the functions p_i(t) and q_i(t) (i = 1, . . . , n) through time can be made. The first strategy is used in the following section. For discussions of the second strategy, see Van Ijzeren (1987, pp. 8–12), Vogt (1977 and 1978), Vogt and Barta (1997), and Balk (2000a).

16.72 There is a connection between the Divisia price and quantity levels, P(t) and Q(t), and the economic approach to index number theory. However, this connection is best made after one has studied the economic approach to index number theory in Chapter 18.

E.2 Discrete approximations to continuous time Divisia index

16.73 To make operational the continuous time Divisia price and quantity levels, P(t) and Q(t) defined by the differential equations (16.67) and (16.68), it is necessary to convert to discrete time. Divisia (1926, p. 40) suggested a straightforward method for doing this conversion, which we now outline.

16.74 Define the following price and quantity (forward) differences:

\begin{matrix} Δ P \equiv P (1) - P (0); & (16.69) \end{matrix}

\begin{array}{l} Δ P_{i} \equiv P_{i} (1) - P_{i} (0); i = 1, . . ., n . & (16.70) \end{array}

Using the above definitions,

\begin{matrix} \frac{P (1)}{p (0)} & = \frac{P (0) + Δ P}{P (0)} \\ = 1 + \frac{Δ P}{P (0)} \approx 1 + \frac{Σ_{i = 1}^{n} Δ p_{i} q_{i} (0)}{Σ_{m = 1}^{n} p_{m} (0) q_{m} (0)}, & (16.71) \end{matrix}

using equation (16.67) when t = 0 and approximating p_i(0) by the difference ∆p_i:

\begin{array}{l} = \frac{Σ_{i = 1}^{n} {p_{i} (0) + Δ p_{i}} q_{i} (0)}{Σ_{m = 1}^{n} p_{m} (0) q_{m} (0)} \\ = \frac{Σ_{i = 1}^{n} p_{1} (1) q_{i} (0)}{Σ_{m = 1}^{n} p_{m} (0) q_{m} (0)} \\ = P_{L} (p^{0}, p^{1}, q^{0}, q^{1}), \end{array}

where p^t ≡ [p₁(t), . . . , p_n(t)] and q^t ≡ [q₁(t), . . . , q_n(t)] for t 0, 1. Thus, it can be seen that Divisia’s discrete approximation to his continuous time price index is just the Laspeyres price index, P_L, defined by equation (16.5).

16.75 But now a problem noted by Frisch (1936, p. 8) occurs: Instead of approximating the derivatives by the discrete (forward) differences defined by equation (16.69) and (16.70), other approximations could be used and a wide variety of discrete time approximations can be obtained. For example, instead of using forward differences and evaluating the index at time t = 0, one could use backward differences and evaluate the index at time t = 1. These backward differences are defined as

\begin{matrix} Δ_{b} p_{i} \equiv p_{i} (0) - p_{i} (1); i = 1, . . ., n . & (16.72) \end{matrix}

This use of backward differences leads to the following approximation for P(0)/P(1):

\begin{matrix} \frac{P (0)}{P (1)} & = \frac{P (1) + Δ_{b} P}{P (1)} \\ = 1 + \frac{Δ_{b} P}{P (1)} \approx 1 + \frac{Σ_{i = 1}^{n} Δ_{b} p_{i} q_{i} (1)}{Σ_{m = 1}^{n} p_{m} (1) q_{m} (1)}, & (16.73) \end{matrix}

using equation (16.67) when t = 1 and approximating p_i(1) by the difference ∆_bp_i:

\begin{array}{l} = \frac{Σ_{i = 1}^{n} {p_{i} (1) + Δ_{b} p_{i}} q_{i} (1)}{Σ_{m = 1}^{n} p_{m} (1) q_{m} (1)} \\ = \frac{Σ_{i = 1}^{n} p_{i} (0) q_{i} (1)}{Σ_{m = 1}^{n} p_{m} (1) q_{m} (1)} \\ = \frac{1}{P_{P} (p^{0}, P^{1}, q^{0}, q^{1})}, \end{array}

where P_P is the Paasche index defined by equation (16.6). Taking reciprocals of both sides of equation (16.73) leads to the following discrete approximation to P(1)/P(0):

\begin{matrix} \frac{P (1)}{P (0)} \approx P_{p} . & (16.74) \end{matrix}

16.76 Thus, as Frisch⁵⁴ noted, both the Paasche and Laspeyres indices can be regarded as (equally valid) approximations to the continuous time Divisia price index.⁵⁵ Because the Paasche and Laspeyres indices can differ considerably in some empirical applications, it can be seen that Divisia’s idea is not all that helpful in determining a unique discrete time index number formula.⁵⁶ What is useful about the Divisia indices is the idea that as the discrete unit of time gets smaller, discrete approximations to the Divisia indices can approach meaningful economic indices under certain conditions. Moreover, if the Divisia concept is accepted as the correct one for index number theory, then the corresponding correct discrete time counterpart might be taken as a weighted average of the chain price relatives pertaining to the adjacent periods under consideration, where the weights are somehow representative for the two periods under consideration.

F. Fixed-Base Versus Chain Indices

16.77 This section⁵⁷ discusses the merits of using the chain system for constructing price indices in the timeseries context versus using the fixed-base system.⁵⁸

16.78 The chain system⁵⁹ measures the change in prices going from one period to another using a bilateral index number formula involving the prices and quantities pertaining to the two adjacent periods. These one-period rates of change (the links in the chain) are then cumulated to yield the relative levels of prices over the entire period under consideration. Thus, if the bilateral price index is P, the chain system generates the following pattern of price levels for the first three periods:

\begin{matrix} 1, P (p^{0}, p^{1}, q^{0}, q^{1}), P (p^{0}, p^{1}, q^{0}, q^{1}) P (p^{1}, p^{2}, q^{1}, q^{2}) . \\ (16.75) \end{matrix}

16.79 On the other hand, the fixed-base system of price levels using the same bilateral index number formula P simply computes the level of prices in period t relative to the base period 0 as P(p⁰, p^t, q⁰, q^t). Thus, the fixed-base pattern of price levels for periods 0, 1, and 2 is

\begin{matrix} 1, P (p^{0}, p^{1}, q^{0}, q^{1}), P (p^{0}, p^{2}, q^{0}, q^{2}) . & (16.76) \end{matrix}

16.80 Note that in both the chain system and the fixed-base system of price levels defined by equations (16.75) and (16.76), the base-period price level is equal to 1. The usual practice in statistical agencies is to set the base-period price level equal to 100. If this is done, then it is necessary to multiply each of the numbers in equations (16.75) and (16.76) by 100.

16.81 Because of the difficulties involved in obtaining current-period information on traded quantities (or, equivalently, on values), many statistical agencies loosely base their XMPIs on the Laspeyres formula in equation (16.5) and the fixed-base system. Therefore, it is of some interest to look at the possible problems associated with the use of fixed-base Laspeyres indices.

16.82 The main problem with the use of fixed-base Laspeyres indices is that the period 0 fixed basket of products that is being priced out in period t often can be quite different from the period t basket. Thus, if there are systematic trends in at least some of the prices and quantities⁶⁰ in the index basket, the fixed-base Laspeyres price index, P_L(p⁰, p^t, q⁰, q^t), can be quite different from the corresponding fixed-base Paasche price index, P_P(p⁰, p^t, q⁰, q^t).⁶¹ This means that both indices are likely to be an inadequate representation of the movement in average prices over the time period under consideration.

16.83 The fixed-base Laspeyres quantity index cannot be used forever; eventually, the base-period quantities q⁰ are so far removed from the current-period quantities q^t that the base must be changed. Chaining is merely the limiting case where the base is changed each period.⁶²

16.84 The main advantage of the chain system is that under normal conditions, chaining will reduce the spread between the Paasche and Laspeyres indices.⁶³ These indices provide an asymmetric perspective on the amount of price change that has occurred between the two periods under consideration, and it could be expected that a single point estimate of the aggregate price change should lie between these two estimates. Thus, the use of either a chained Paasche or Laspeyres index will usually lead to a smaller difference between the two and hence to estimates that are closer to the “truth.”⁶⁴

16.85 Peter Hill (1993, p. 388), drawing on his earlier research (1988, pp. 136–37) and that of Szulc (1983), noted that it is not appropriate to use the chain system when prices oscillate, or “bounce,” to use Szulc’s (1983, p. 548) term. This phenomenon can occur in the context of regular seasonal fluctuations or in the context of price wars. However, in the context of roughly monotonically changing prices and quantities, Peter Hill (1993, p. 389) recommended the use of chained symmetrically weighted indices (see Section C). The Fisher and Walsh indices are examples of symmetrically weighted indices.

16.86 It is possible to be a bit more precise regarding under which conditions one should or should not chain. Basically, one should chain if the prices and quantities pertaining to adjacent periods are more similar than the prices and quantities of more distant periods, because this strategy will lead to a narrowing of the spread between the Paasche and Laspeyres indices at each link.⁶⁵ One needs a measure of how similar are the prices and quantities pertaining to two periods. The similarity measures could be relative or absolute. In the case of absolute comparisons, two vectors of the same dimension are similar if they are identical and dissimilar otherwise. In the case of relative comparisons, two vectors are similar if they are proportional and dissimilar if they are nonproportional.⁶⁶ Once a similarity measure has been defined, the prices and quantities of each period can be compared using this measure, and a “tree” or path that links all of the observations can be constructed where the most similar observations are compared using a bilateral index number formula.⁶⁷ R.J. Hill (1995) defined the price structures between the two countries to be more dissimilar the bigger the spread is between P_L and P_P, that is, the bigger max {P_L/P_P, P_P/P_L}. The problem with this measure of dissimilarity in the price structures of the two countries is that it could be the case that P_L = P_P (so that the R.J. Hill measure would register a maximal degree of similarity), but p⁰ could be very different from p^t. Thus, there is a need for a more systematic study of similarity (or dissimilarity) measures to pick the best one that could be used as an input into R.J. Hill’s (1999a, 1999b, 2001) spanning tree algorithm for linking observations.

16.87 The method of linking observations explained in the previous paragraph based on the similarity of the price and quantity structures of any two observations may not be practical in a statistical agency context, because the addition of a new period may lead to a reordering of the previous links. However, the above scientific method for linking observations may be useful in deciding whether chaining is preferable or whether fixed-base indices should be used while making month-to-month comparisons within a year.

16.88 Some index number theorists have objected to the chain principle on the grounds that it has no counterpart in the spatial context:

They [chain indices] only apply to intertemporal comparisons, and in contrast to direct indices they are not applicable to cases in which no natural order or sequence exists. Thus, the idea of a chain index, for example, has no counterpart in interregional or international price comparisons, because countries cannot be sequenced in a “logical” or “natural” way (there is no k + 1 nor k – 1 country to be compared with country k). (von der Lippe, 2001, p. 12)⁶⁸

This is correct, but R.J. Hill’s approach does lead to a natural set of spatial links. Applying the same approach to the time-series context will lead to a set of links between periods that may not be month to month, but it will in many cases justify year-over-year linking of the data pertaining to the same month. This problem will be reconsidered in Chapter 23.

16.89 It is of some interest to determine if there are index number formulas that give the same answer when either the fixed-base or chain system is used. Comparing the sequence of chain indices defined by equation (16.75) above to the corresponding fixed-base indices, it can be seen that we will obtain the same answer in all three periods if the index number formula P satisfies the following functional equation for all price and quantity vectors:

\begin{matrix} P (p^{0}, p^{2}, q^{0}, q^{2}) = P (p^{0}, p^{1}, q^{0}, q^{1}) P (p^{1}, p^{2}, q^{1}, q^{2}) . & (16.77) \end{matrix}

If an index number formula P satisfies equation (16.77), then P satisfies the circularity test.⁶⁹

16.90 If it is assumed that the index number formula P satisfies certain properties or tests in addition to the circularity test above,⁷⁰ then Funke, Hacker, and Voeller (1979) showed that P must have the following functional form credited originally to Konüs and Byushgens⁷¹ (1926, pp. 163–66)⁷²:

\begin{matrix} P_{K B} (p^{0}, p^{1}, q^{0}, q^{1}) = Π_{i = 1}^{n} {(\frac{p_{i}^{1}}{p_{i}^{0}})}^{α_{i}}, & (16.78) \end{matrix}

where the n constants α_i satisfy the following restrictions:

\begin{matrix} Σ_{i = 1}^{n} α_{i} = 1 a n d α_{i} > 0 for i = 1, . . ., n . & (16.79) \end{matrix}

Thus, under very weak regularity conditions, the only price index satisfying the circularity test is a weighted geometric average of all the individual price ratios, the weights being constant through time.

16.91 An interesting special case of the family of indices defined by equation (16.78) occurs when the weights α_i are all equal. In this case, P_KB reduces to the Jevons (1865) index:

\begin{matrix} P_{J} (p^{0}, p^{1}, q^{0}, q^{1}) \equiv Π_{i = 1}^{n} (\frac{p_{i}^{1}}{p_{i}^{0}})^{\frac{1}{n}} . & (16.80) \end{matrix}

16.92 The problem with the indices defined by Konüs and Byushgens and Jevons is that the individual price ratios, $p_{i}^{1} / p_{i}^{0}$ , have weights (either α_i or 1/n) that are independent of the economic importance of product i in the two periods under consideration. Put another way, these price weights are independent of the quantities of product i or the values on product i traded during the two periods. Hence, these indices are not really suitable for use by statistical agencies at higher levels of aggregation when value share information is available.

16.93 The above results indicate that it is not useful to ask that the price index P satisfy the circularity test exactly. However, it is of some interest to find index number formulas that satisfy the circularity test to some degree of approximation, because the use of such an index number formula will lead to measures of aggregate price change that are more or less the same whether we use the chain or fixed-base systems. Irving Fisher (1922) found that deviations from circularity using his data set and the Fisher ideal price index P_F defined by equation (16.12) above were quite small.

This relatively high degree of correspondence between fixed-base and chain indices has been found to hold for other symmetrically weighted formulas like the Walsh index P_W defined by equation (16.19).⁷³ Thus, in most time-series applications of index number theory where the base year in fixed-base indices is changed every five years or so, it will not matter very much whether the statistical agency uses a fixed-base price index or a chain index, provided that a symmetrically weighted formula is used.⁷⁴ This, of course, depends on the length of the time series considered and the degree of variation in the prices and quantities as we go from period to period. The more prices and quantities are subject to large fluctuations (rather than smooth trends), the less the correspondence.⁷⁵

16.94 It is possible to give a theoretical explanation for the approximate satisfaction of the circularity test for symmetrically weighted index number formulas. Another symmetrically weighted formula is the Törnqvist index P_T.⁷⁶ The natural logarithm of this index is defined as follows:

\begin{matrix} In P_{T} (p^{0}, p^{1}, q^{0}, q^{1}) \equiv Σ_{i = 1}^{n} \frac{1}{2} (s_{i}^{0} + s_{i}^{1}) In (\frac{p_{i}^{1}}{p_{i}^{0}}), \\ (16.81) \end{matrix}

where the period t value shares $s_{i}^{t}$ are defined by equation (16.7) above. Alterman, Diewert, and Feenstra (1999, p. 61) showed that if the logarithmic price ratios ln ( $p_{i}^{t} / p_{i}^{t-1}$ ) trend linearly with time t, and the value shares $s_{i}^{t}$ also trend linearly with time, then the Törnqvist index P_T will satisfy the circularity test exactly.⁷⁷ Because many economic time series on prices and quantities satisfy these assumptions approximately, then the Törnqvist index will satisfy the circularity test approximately. As is shown in Chapter 20, generally the Törnqvist index closely approximates the symmetrically weighted Fisher and Walsh indices, so that for many economic time series (with smooth trends), all three of these symmetrically weighted indices will satisfy the circularity test to a high enough degree of approximation so that it will not matter whether we use the fixed-base or chain principle.

16.95 Walsh (1901, p. 401; 1921a, p. 98; and 1921b, p. 540) introduced the following useful variant of the circularity test:

\begin{matrix} 1 = & P (p^{0}, p^{1}, q^{0}, q^{1}) P (p^{1}, p^{2}, q^{1}, q^{2}) . . . \\ P (p^{T}, p^{0}, q^{T}, q^{0}) . & (16.82) \end{matrix}

The motivation for this test is the following. Use the bilateral index formula P(p⁰, p¹, q⁰, q¹) to calculate the change in prices going from period 0 to 1. Use the same formula evaluated at the data corresponding to periods 1 and 2, P(p¹, p², q¹, q²), to calculate the change in prices going from period 1 to 2, . . . , use P(p^T–1, p^T, q^T–1, q^T) to calculate the change in prices going from period T – 1 to T. Introduce an artificial period T – 1 that has exactly the price and quantity of the initial period 0 and use P(p^t, p⁰, q^t, q⁰) to calculate the change in prices going from period T to 0. Finally, multiply all these indices and, because we end up where we started, then the product of all of these indices should ideally be 1. Diewert (1993a, p. 40) called this test a multiperiod identity test.⁷⁸ Note that if T = 2 (so that the number of periods is 3 in total), then Walsh’s test reduces to Irving Fisher’s (1921, p. 534; 1922, p. 64) time reversal test.⁷⁹

16.96 Walsh (1901, pp. 423–33) showed how his circularity test could be used in order to evaluate the worth of a bilateral index number formula. He invented artificial price and quantity data for five periods and added a sixth period that had the data of the first period. He then evaluated the right-hand side of equation (16.82) for various formulas, P(p⁰, p¹, q⁰, q¹), and determined how far from unity the results were. His best formulas had products that were close to 1.⁸⁰

16.97 This same framework is often used to evaluate the efficacy of chained indices versus their direct counterparts. Thus, if the right-hand side of equation (16.82) turns out to be different from unity, the chained indices are said to suffer from “chain drift.” If a formula does suffer from chain drift, it is sometimes recommended that fixed-base indices be used in place of chained ones. However, this advice, if accepted, would always lead to the adoption of fixed-base indices, provided that the bilateral index formula satisfies the identity test, P(p⁰, p⁰, q⁰, q⁰) = 1. Thus, it is not recommended that Walsh’s circularity test be used to decide whether fixedbase or chained indices should be calculated. However, it is fair to use Walsh’s circularity test as he originally used it; that is, as an approximate method for deciding the utility of a particular index number formula. To decide whether to chain or use fixed-base indices, one should decide on the basis of how similar are the observations being compared and choose the method that will best link up the most similar observations.

16.98 Various properties, axioms, or tests that an index number formula could satisfy were introduced in this chapter. In the following chapter, the test approach to index number theory is studied in a more systematic manner.

Appendix 16.1 Relationship Between Paasche and Laspeyres Indices

16.99 Recall the notation used in Section B.2. Define the ith relative price or price relative r_i and the ith quantity relative t_i as follows:

\begin{matrix} r_{i} \equiv \frac{p_{i}^{1}}{p_{i}^{0}}; t_{i} \equiv \frac{q_{i}^{1}}{q_{i}^{0}}; i = 1, . . ., n . & (A 16.1.1) \end{matrix}

Using equation (16.8) above for the Laspeyres price index P_L and equation (A16.1.1), we have

P_{L} = Σ_{i = 1}^{n} r_{i} s_{i}^{0} \equiv r *; (A 16.1.2)

that is, we define the “average” price relative r^* as the base-period value share-weighted average of the individual price relatives, r_i.

16.100 Using equation (16.6) for the Paasche price index P_P, we have

\begin{matrix} P_{P} & \equiv \frac{Σ_{i = 1}^{n} p_{i}^{1} q_{i}^{1}}{Σ_{m = 1}^{n} p_{m}^{0} q_{m}^{1}} & (A 16.1.3) \\ = \frac{Σ_{i = 1}^{n} r_{i} t_{i} p_{i}^{0} q_{i}^{0}}{Σ_{m = 1}^{n} t_{m} p_{m}^{0} q_{m}^{1}} & using equation (A16.1.1) \\ = \frac{Σ_{i = 1}^{n} r_{i} t_{i} s_{i}^{0}}{Σ_{m = 1}^{n} t_{m} s_{m}^{0}} \\ = {\frac{1}{Σ_{m = 1}^{n} t_{m} s_{m}^{0}} Σ_{i = 1}^{n} (r_{i} - r^{*}) (t_{i} - t^{*}) s_{i}^{0}} + r^{*}, \end{matrix}

using equation (A16.1.2) and $\sum_{i = 1}^{n} s_{i}^{0} = 1$ and where the average quantity relative t^* is defined as

\begin{matrix} t^{*} = Σ_{i = 1}^{n} t_{i} s_{i}^{0} = Q_{L}, & (A16.1.4) \end{matrix}

where the last equality follows using equation (16.11), the definition of the Laspeyres quantity index Q_L.

16.101 Taking the difference between P_P and P_L and using equation (A16.1.2), equation (A16.1.4) yields

\begin{matrix} P_{P} - P_{L} - \frac{1}{Q_{L}} Σ_{i = 1}^{n} (r_{i} - r^{*}) (t_{i} - t^{*}) s_{i}^{0} . & (A16.1.5) \end{matrix}

16.102 Now let r and t be discrete random variables that take on the n values r_i and t_i, respectively. Let $s_{i}^{0}$ be the joint probability that r = r_i and t = t_i for i = 1, . . . , n, and let the joint probability be 0 if r = r_i and t = t_j where i ≠ j. It can be verified that the summation $\sum_{i = 1}^{n} (r_{i} - r^{*}) (t_{i} - t^{*}) s_{i}^{0}$ on the right-hand side of equation (A16.1.5) is the covariance between the price relatives r_i and the corresponding quantity relatives t_i. This covariance can be converted into a correlation coefficient.⁸¹ If this covariance is negative, which is the usual case in the consumer (import) context, then P_P will be less than P_L. If it is positive, which will occur in the situations where supply conditions are fixed (as in the fixed-input output price index), but demand is changing, then P_P will be greater than P_L.

Appendix 16.2 Relationship Between Lowe and Laspeyres Indices

16.103 Recall the notation used in Section D.1. Define the ith relative price relating the price of product i of month t to month 0, r_i, and the ith quantity relative, t_i, relating quantity of product i in base year b to month 0, t_i, as follows:

\begin{matrix} r_{i} \equiv \frac{p_{i}^{t}}{p_{i}^{0}}; t_{i} \equiv \frac{q_{i}^{b}}{q_{i}^{0}}; i = 1, . . ., n . & (A16.2.1) \end{matrix}

16.104 As in Appendix 16.1, the Laspeyres price index P_L(p⁰, p^t, q⁰) can be defined as r^*, the month 0 expenditure share-weighted average of the individual price relatives r_i defined in equation (A16.2.1), except that the month t price, $p_{i}^{t}$ , now replaces period 1 price, $p_{i}^{1}$ , in the definition of the ith price relative r_i:

\begin{matrix} r^{*} \equiv Σ_{i = 1}^{n} r_{i} s_{i}^{0} = P_{L} . & (A16.2.2) \end{matrix}

16.105 The average quantity relative t^* relating the quantities of base year b to those of month 0 is defined as the month 0 expenditure share-weighted average of the individual quantity relatives t_i defined in equation (A16.2.1):

\begin{matrix} t^{*} \equiv Σ_{i = 1}^{n} t_{i} s_{i}^{0} = Q_{L}, & (A16.2.3) \end{matrix}

where Q_L Q_L(q⁰, q^b, p⁰) is the Laspeyres quantity index relating the quantities of month 0, q⁰, to those of the year b, q^b, using the prices of month 0, p⁰, as weights.

16.106 Using equation (16.26), the Lowe index comparing the prices in month t with those of month 0, using the quantity weights of the base year b, is equal to

\begin{matrix} P_{L 0} (p^{0}, p^{t}, p^{b}) \equiv \frac{Σ_{i = 1}^{n} p_{i}^{t} q_{i}^{b}}{Σ_{i = 1}^{n} p_{i}^{0} q_{i}^{b}} & (A16.2.4) \\ = \frac{Σ_{i = 1}^{n} p_{i}^{t} t_{i} q_{i}^{0}}{Σ_{i = 1}^{n} p_{i}^{0} t_{i} q_{i}^{0}} \\ = {\frac{Σ_{i = 1}^{n} p_{i}^{t} t_{i} q_{i}^{0}}{Σ_{i = 1}^{n} p_{i}^{0} q_{i}^{0}}} {\frac{Σ_{i = 1}^{n} p_{i}^{t} t_{i} q_{i}^{0}}{Σ_{i = 1}^{n} p_{i}^{0} q_{i}^{0}}}^{- 1} & using equation (A 16.2.1) \\ = {\frac{Σ_{i - 1}^{n} (\frac{p_{i}^{t}}{p_{i}^{0}}) t_{i} p_{i}^{0} q_{i}^{0}}{Σ_{i - 1}^{n} p_{i}^{0} q_{i}^{0}}} / t^{*} & using equation (A 16.2.3) \\ = {\frac{Σ_{i - 1}^{n} r_{i} t_{i} p_{i}^{0} q_{i}^{0}}{Σ_{i - 1}^{n} p_{i}^{0} q_{i}^{0}}} / t^{*} & using equation (A 16.2.1) \\ = \frac{Σ_{i = 1}^{n} r_{i} t_{i} s_{i}^{0}}{t *} = \frac{Σ_{i = 1}^{n} (r_{i} - r *) t_{i} s_{i}^{0}}{t *} + \frac{Σ_{i = 1}^{n} r_{i} t_{i} s_{i}^{0}}{t *} \\ = \frac{Σ_{i = 1}^{n} (r_{i} - r *) t_{i} s_{i}^{0}}{t *} + \frac{r * [Σ_{i = 1}^{n} r * t_{i} s_{i}^{0}]}{t *} \\ = \frac{Σ_{i = 1}^{n} (r_{i} - r *) t_{i} s_{i}^{0}}{t *} + \frac{r * [t *]}{t *} \\ using equation (A 16.2.3) \\ = \frac{Σ_{i = 1}^{n} (r_{i} - r *) (t_{i} - t *) s_{i}^{0}}{t *} + \frac{t * [Σ_{i = 1}^{n} r_{i} s_{i}^{0} - r *]}{t *} + r * \\ = \frac{Σ_{i = 1}^{n} (r_{i} - r *) (t_{i} - t *) s_{i}^{0}}{t *} + r * since Σ_{i = 1}^{n} r_{i} s_{i}^{0} = r * \\ = P_{L} (p^{0}, p^{t}, q^{0}) + \frac{Σ_{i = 1}^{n} (r_{i} - r *) (t_{i} - t *) s_{i}^{0}}{Q_{L} (q^{0}, q^{b}, p^{0})} \end{matrix}

because using equation (A16.2.2), r^* equals the Laspeyres price index, P_L(p⁰, p^t, q⁰), and using equation (A16.2.3), t^* equals the Laspeyres quantity index, Q_L(q⁰, q^b, p⁰). Thus, equation (A16.2.4) tells us that the Lowe price index using the quantities of year b as weights, P_Lo(p⁰, p^t, q^b), is equal to the usual Laspeyres index using the quantities of month 0 as weights, P_L(p⁰, p^t, q⁰), plus a covariance term $\sum_{i = 1}^{n} (r_{i} - r^{*}) (t_{i} - t^{*}) s_{i}^{0}$ between the price relatives $r_{i} \equiv p_{i}^{t} / p_{i}^{0}$ and the quantity relatives $t_{i} \equiv q_{i}^{b} / q_{i}^{0}$ , divided by the Laspeyres quantity index Q_L(q⁰, q^b, p⁰) between month 0 and base year b.

Appendix 16.3 Relationship Between Young Index and Its Time Antithesis

16.107 Recall that the direct Young index, P_Y(p⁰, p^t, s^b), was defined by equation (16.48) and its time antithesis, P_Y^*(p⁰, p^t, s^b), was defined by equation (16.52). Define the ith relative price between months 0 and t as

\begin{matrix} r_{i} \equiv p_{i}^{t} / p_{i}^{0}, i = 1, . . ., n, & (A16.3.1) \end{matrix}

and define the weighted average (using the base-year weights $s_{i}^{b}$ ) of the r_i as

\begin{matrix} r^{*} \equiv Σ_{i = 1}^{n} s_{i}^{b} r_{i}, & (A16.3.2) \end{matrix}

which turns out to equal the direct Young index, P_Y(p⁰, p^t, s^b). Define the deviation e_i of r_i from their weighted average r^* using the following equation:

\begin{matrix} r_{i} = r^{*} (1 + e_{i}); i = 1, . . ., n . & (A16.3.3) \end{matrix}

16.108 If equation (A16.3.3) is substituted into equation (A16.3.2), the following equations are obtained:

\begin{matrix} r^{*} & = Σ_{i = 1}^{n} s_{i}^{b} r^{*} (1 + e_{i}) & (A16.3.4) \\ = r^{*} + r^{*} Σ_{i = 1}^{n} s_{i}^{b} e_{i} since Σ_{i = 1}^{n} s_{i}^{b} = 1 \end{matrix}

\begin{matrix} e^{*} \end{matrix} \begin{matrix} = Σ_{i = 1}^{n} s_{i}^{b} e_{i} = 0. & (A16.3.5) \end{matrix}

Thus, the weighted mean e^* of the deviations e_i equals 0.

16.109 The direct Young index, P_Y(p⁰, p^t, s^b), and its time antithesis, P_Y^*(p⁰, p^t, s^b), can be written as functions of r^*, the weights $s_{i}^{b}$ , and the deviations of the price relatives e_i as follows:

\begin{matrix} P_{Y} (p^{0}, p^{t}, s^{b}) = r^{*}; & (A16.3.6) \end{matrix}

\begin{matrix} {P^{*}}_{Y} (p^{0}, p^{t}, s^{b}) & = [Σ_{i = 1}^{n} s_{i}^{b} {r^{*} (1 + e_{i})}^{- 1}]^{- 1} \\ = r^{*} [Σ_{i = 1}^{n} s_{i}^{b} {(1 + e_{i})}^{- 1}]^{- 1} . (A16.3.7) \end{matrix}

16.110 Now, regard P_Y^*(p⁰, p^t, s^b) as a function of the vector of deviations, e ≡ [e₁, . . . ,e_n], say P_Y^*(e). The second-order Taylor series approximation to P_Y^*(e) around the point e = 0_n is given by the following expression:⁸²

\begin{matrix} P_{Y}^{*} (e) \approx & r^{*} + r * Σ_{i = 1}^{n} s_{i}^{b} e_{i} + r^{*} Σ_{i = 1}^{n} Σ_{j = 1}^{n} s_{i}^{b} s_{j}^{b} e_{i} e_{j} \\ - r^{*} Σ_{i = 1}^{n} s_{i}^{b} {[e_{i}]}^{2} & (A16.3.8) \\ = & r^{*} + r^{*} 0 + r^{*} Σ_{i = 1}^{n} s_{i}^{b} [Σ_{i = 1}^{n} s_{j}^{b} e_{j}] e_{i} \\ - r^{*} Σ_{i = 1}^{n} s_{i}^{b} [e_{i} - e^{*}]^{2} using equation (A16.3.5) \\ = & r^{*} + r^{*} Σ_{i = 1}^{n} s_{i}^{b} [0] e_{i} \\ - r^{*} Σ_{i = 1}^{n} s_{i}^{b} [e_{i} - e^{*}]^{2} using equation (A16.3.5) \\ = & P_{Y} (p^{0}, p^{t}, s^{b}) - P_{Y} (p^{0}, p^{t}, s^{b}) \\ \times Σ_{i = 1}^{n} s_{i}^{b} [e_{i} - e^{*}]^{2} using equation (A16.3.6) \\ = & P_{Y} (p^{0}, p^{t}, s^{b}) - P_{Y} (p^{0}, p^{t}, s^{b}) Var e \end{matrix}

where the weighted sample variance of the vector e of price deviations is defined as

\begin{matrix} Var e \equiv Σ_{i = 1}^{n} s_{i}^{b} {[e_{i} - e^{*}]}^{2} . & (A16.3.9) \end{matrix}

16.111 Rearranging equation (A16.3.8) gives the following approximate relationship between the direct Young index P_Y(p⁰, p^t, s^b) and its time antithesis P_Y^*(p⁰, p^t, s^b), to the accuracy of a second-order Taylor series approximation about a price point where the month t price vector is proportional to the month 0 price vector:

\begin{array}{l} P_{Y} (p^{0}, p^{t} s^{b}) \approx P_{Y}^{*} (p^{0}, p^{t} s^{b}) + P_{Y} (P^{0}, p^{t} s^{b}) Var e . \\ (A16.3.10) \end{array}

Thus, to the accuracy of a second-order approximation, the direct Young index will exceed its time antithesis by a term equal to the direct Young index times the weighted variance of the deviations of the price relatives from their weighted mean. Thus, the bigger the dispersion in relative prices, the more the direct Young index will exceed its time antithesis.

Indices of this type do not appear in Chapter 19, where most of the index number formulas exhibited in Chapters 15–18 are illustrated using an artificial data set. However, indices where the weight reference period differs from the price reference period are illustrated numerically in Chapter 22, where the problem of seasonal products is discussed.

In particular, it can be used to justify the chain system of index numbers, which is discussed in Section E.

Turvey and others (1989) has noted that some values may be difficult to decompose into unambiguous price and quantity components. Some examples of values difficult to decompose are bank charges, gambling expenditures, and life insurance payments.

Note that it is assumed that there are no new or disappearing products in the value aggregates. Approaches to the “new goods problem” and the problem of accounting for quality change are discussed in Chapters 7, 8, and 21.

The first person to suggest that the price and quantity indices should be jointly determined to satisfy equation (16.3) was Irving Fisher (1911, p. 418). Frisch (1930, p. 399) called equation (16.3) the product test.

Joseph Lowe (1823, Appendix 95) suggested that the product basket vector q should be updated every five years. Lowe indices are studied in more detail in Section D.

This index was actually introduced and justified by Drobisch (1871a, p. 147) slightly earlier than by Laspeyres. Laspeyres (1871, p. 305) in fact explicitly acknowledged that Drobisch showed him the way forward. However, the contributions of Drobisch were forgotten for the most part by later writers because Drobisch aggressively pushed for the ratio of two unit values as being the best index number formula. Although this formula has some excellent properties, if all the n products being compared have the same unit of measurement, the formula is useless when, say, both goods and services are in the index basket.

Again, Drobisch (1871b, p. 424) appears to have been the first to explicitly define and justify this formula. However, he rejected this formula in favor of his preferred formula, the ratio of unit values, and so again he did not get any credit for his early suggestion of the Paasche formula.

Note that P_L(p⁰, p¹, q⁰, q¹) does not actually depend on q¹, and P_P(p⁰, p¹, q⁰, q¹) does not actually depend on q⁰. However, it does no harm to include these vectors, and the notation indicates that the reader is in the realm of bilateral index number theory; that is, the prices and quantities for a value aggregate pertaining to two periods are being compared.

This method of rewriting the Laspeyres index (or any fixed-basket index) as a share-weighted arithmetic average of price ratios is due to Irving Fisher (1897, p. 517; 1911, p. 397; 1922, p. 51) and Walsh (1901, p. 506; 1921a, p. 92).

This method of rewriting the Paasche index (or any fixed-basket index) as a share–weighted harmonic average of the price ratios is due to Walsh (1901, p. 511; 1921a, p. 93) and Irving Fisher (1911, pp. 397–98).

Note that the derivation in equation (16.9) shows how harmonic averages arise in index number theory in a very natural way.

In principle, instead of averaging the Paasche and Laspeyres indices, the statistical agency could think of providing both (the Paasche index on a delayed basis). This suggestion would lead to a matrix of price comparisons between every pair of periods instead of a time series of comparisons. Walsh (1901, p. 425) noted this possibility: “In fact, if we use such direct comparisons at all, we ought to use all possible ones.”

Peter Hill (1993, p. 383) summarized this inequality as follows: “It can be shown that relationship (13) [that is, that P_L is greater than P_P] holds whenever the price and quantity relatives (weighted by values) are negatively correlated. Such negative correlation is to be expected for price takers who react to changes in relative prices by substituting goods and services that have become relatively less expensive for those that have become relatively more expensive. In the vast majority of situations covered by index numbers, the price and quantity relatives turn out to be negatively correlated so that Laspeyres indices tend systematically to record greater increases than Paasche with the gap between them tending to widen with time.”

There is another way to see why P_P will often be less than P_L. If the period 0 value shares $s_{i}^{0}$ are exactly equal to the corresponding period 1 value shares s_i¹, then by Schlömilch’s (1858) Inequality (see Hardy, Littlewood, and Pólya, 1934, p. 26), it can be shown that a weighted harmonic mean of n numbers is equal to or less than the corresponding arithmetic mean of the n numbers and the inequality is strict if the n numbers are not all equal. If value shares are approximately constant across periods, then it follows that P_P will usually be less than P_L under these conditions; see Section D.3.

However, the monthly goods trade figures from customs services likely will be subject to revision, so, by implication, the Paasche goods trade price index also would be subject to revision, at least in principle. If revisions to trade by commodity and destination/source country usually are small, these revisions will be very small, however.

For a discussion of the properties of symmetric averages, see Diewert (1993c). Formally, an average m(a, b) of two numbers a and b is symmetric if m(a, b) = m(b, a). In other words, the numbers a and b are treated in the same manner in the average. An example of a nonsymmetric average of a and b is (1/4)a + (3/4)b. In general, Walsh (1901, p. 105) argued for a symmetric treatment if the two periods (or countries) under consideration were to be given equal importance.

Walsh (1901, p. 99) also suggested this index. See Diewert (1993a, p. 36) for additional references to the early history of index number theory.

Bowley (1899, p. 641) appears to have been the first to suggest the use of this index. Walsh (1901, pp. 428–29) also suggested this index while commenting on the big differences between the Laspeyres and Paasche indices in one of his numerical examples: “The figures in columns (2) [Laspeyres] and (3) [Paasche] are, singly, extravagant and absurd. But there is order in their extravagance; for the nearness of their means to the more truthful results shows that they straddle the true course, the one varying on the one side about as the other does on the other.”

See Diewert (1992a, p. 218) for early references to this test. If we want the price index to have the same property as a single price ratio, then it is important to satisfy the time reversal test. However, other points of view are possible. For example, we may want to use our price index for compensation purposes, in which case satisfaction of the time reversal test may not be so important.

This is formally derived in Diewert (1997) p. 138).

An average or mean of two numbers a and b, m(a, b), is homogeneous if, when both numbers a and b are multiplied by a positive number λ, then the mean is also multiplied by; that is, m satisfies the following property: m(λa, b) = λm(a, b).

Irving Fisher (1911, pp. 417–18; 1922) also considered the arithmetic, geometric, and harmonic averages of the Paasche and Laspeyres indices.

Irving Fisher (1922, p. 72) said that P and Q satisfied the factor reversal test if Q(p⁰, p¹, q⁰, q¹) P(q⁰, q¹, p⁰, p1) and P and Q satisfied the product test in equation (16.3) as well.

See Section 7 in Diewert (2001a).

“Suppose, however, that for each commodity, Q′ = Q, the fraction, ∑(P′Q) ∑(PQ), viz., the ratio of aggregate value for the second unit-period to the aggregate value for the first unit-period is no longer merely a ratio of totals; it also shows unequivocally the effect of the change in price. Thus, it is an unequivocal price index for the quantitatively unchanged complex of commodities, A, B, C, et cetera.

“It is obvious that if the quantities were different on the two occasions, and if at the same time the prices had been unchanged, the preceding formula would become ∑(PQ′) ∑(PQ). It would still be the ratio of the aggregate value for the second unit-period to the aggregate value for the first unit-period. But it would be also more than this. It would show in a generalized way the ratio of the quantities on the two occasions. Thus it is an unequivocal quantity index for the complex of commodities, unchanged as to price and differing only as to quantity.

“Let it be noted that the mere algebraic form of these expressions shows at once the logic of the problem of finding these two indices is identical.” (Knibbs, 1924, pp. 43–44)

Irving Fisher (1922, p. 53) used the terminology “weighted by a hybrid value,” whereas Walsh (1932, p. 657) used the term “hybrid weights.”

Note that the ith share defined by equation (16.16) in this case is the hybrid share $s_{i} = p_{i}^{0} q_{i}^{1} / Σ_{i = 1}^{n} p_{i}^{0} q_{i}^{1}$ which uses the prices of period 0 and the quantities of period 1.

Note that the mean function m( $q_{i}^{0}$ , $q_{i}^{1}$ ) has been chosen to be the same for each item i. It is assumed that m(a,b) has the following two properties: m(a,b) is a positive and continuous function, defined for all positive numbers a and b, and m(a, a) a for all a > 0.

For more on symmetric means, see Diewert (1993c, p. 361).

Walsh endorsed P_W as being the best index number formula: “We have seen reason to believe formula 6 better than formula 7. Perhaps formula 9 is the best of the rest, but between it and Nos. 6 and 8 it would be difficult to decide with assurance” (Walsh, 1921a, p. 103). His formula 6 is P_W defined by equation (16.19) and his 9 is the Fisher ideal defined by equation (16.12) above. The Walsh quantity index, Q_W(p⁰, p¹, q⁰, q¹), is defined as P_W(q⁰, q¹, p⁰, p1); that is, prices and quantities in equation (16.19) are interchanged. If the Walsh quantity index is used to deflate the value ratio, an implicit price index is obtained, which is Walsh’s formula 8.

This is not likely to be a severe problem in the time-series context where the change in quantity vectors going from one period to the next is small.

This is the terminology used by Diewert (1992a, p. 216). Vogt (1980) was the first to propose this test.

See Section 7 in Diewert (2001a).

Diewert (1978, pp. 887–89) showed that these two indices will approximate each other to the second order around an equal price and quantity point. Thus, for normal time-series data where prices and quantities do not change much going from the base period to the current period, the indices will approximate each other quite closely.

Superlative indices are defined in Chapter 18.

Month 0 is called the price reference period, and year b is called the weight reference period.

Triplett (1981, p. 12) defined the Lowe index, calling it a Laspeyres index, and calling the index that has the weight reference period equal to the price reference period a pure Laspeyres index. Triplett also noted the hybrid share representation for the Lowe index defined by equation (16.15) and equation (16.16). Triplett noted that the ratio of two Lowe indices using the same quantity weights was also a Lowe index.

In fact, using the Lowe index P_Lo(p 0, p^t, q^b) in the context of seasonal products corresponds to Bean and Stine’s (1924, p. 31) Type A index number formula. Bean and Stine made three additional suggestions for price indices in the context of seasonal products. Their contributions are evaluated in Chapter 22.

Hence, these annual product prices are essentially unit value prices. Under conditions of high inflation, the annual prices defined by equation (16.24) may no longer be reasonable or representative of prices during the entire base year because the values in the final months of the high-inflation year will be somewhat artificially blown up by general inflation. Under these conditions, the annual prices and annual product value shares should be interpreted with caution. For more on dealing with situations where there is high inflation within a year, see Hill (1996).

It is also necessary to assume that purchasers have normal substitution effects in response to these long-term trends in prices; that is, if a product increases (relatively) in price, its quantity purchased will decline (relatively), and if a product decreases relatively in price, its quantity purchased will increase relatively. This reflects the normal “market equilibrium” response to changes in supply.

Walsh (1901, pp. 281–82) was well aware of substitution effects, as can be seen in the following comment which noted the basic problem with a fixed-basket index that uses the quantity weights of a single period: “The argument made by the arithmetic averagist supposes that we buy the same quantities of every class at both periods in spite of the variation in their prices, which we rarely, if ever, do. As a rough proposition, we—a community—generally spend more on articles that have risen in price and get less of them, and spend less on articles that have fallen in price and get more of them.”

The reader can carry through the argument if there is a long-term relative decline in the price of the ith product. The argument required to obtain a negative covariance requires that there be some differences in the long-term trends in prices; that is, if all prices grow (or fall) at the same rate, we have price proportionality, and the covariance will be zero.

However, Q_L = u^* may also be growing in magnitude, so the net effect on the divergence between P_L and P_P is ambiguous.

This concept can be traced to Peter Hill (1998, p. 46): “When inflation has to be measured over a specified sequence of years, such as a decade, a pragmatic solution to the problems raised above would be to take the middle year as the base year. This can be justified on the grounds that the basket of goods and services purchased in the middle year is likely to be much more representative of the pattern of consumption over the decade as a whole than baskets purchased in either the first or the last years. Moreover, choosing a more representative basket will also tend to reduce, or even eliminate, any bias in the rate of inflation over the decade as a whole as compared with the increase in the CoL [cost-of-living] index.” Thus, in addition to introducing the concept of a midyear index, Hill also introduced the idea of representativity bias. For additional material on midyear indices, see Schultz (1999) and Okamoto (2001). Note that the midyear index concept could be viewed as a close competitor to Walsh’s (1901, p. 431) multiyear fixed-basket index, where the quantity vector was chosen to be an arithmetic or geometric average of the quantity vectors in the period.

However, some services, such as call centers, computer programming, and airline maintenance, can be internationally outsourced.

Walsh (1901, p. 536; 1932, p. 657) attributed this formula to Young.

Irving Fisher’s 1922 book is famous for developing the value ratio decomposition approach to index number theory, but his introductory chapters took the share-weighted average point of view: “An index number of prices, then shows the average percentage change of prices from one point of time to another” (1922, p. 3). Fisher went on to note the importance of economic weighting: “The preceding calculation treats all the commodities as equally important; consequently, the average was called ‘simple’. If one commodity is more important than another, we may treat the more important as though it were two or three commodities, thus giving it two or three times as much ‘weight’ as the other commodity” (1922, p. 6). Walsh considered both approaches: “We can either (1) draw some average of the total money values of the classes during an epoch of years, and with weighting so determined employ the geometric average of the price variations [ratios]; or (2) draw some average of the mass quantities of the classes during the epoch, and apply to them Scrope’s method” (1901, pp. 430–31). Scrope’s method is the same as using the Lowe index. Walsh (1901, pp. 88–90) consistently stressed the importance of weighting price ratios by their economic importance (rather than using equally weighted averages of price relatives). Both the value ratio decomposition approach and the share-weighted average approach to index number theory are studied from the axiomatic perspective in the following chapter; see also Sections C and E in Chapter 16.

Using Irving Fisher’s (1922, p. 118) terminology, P_Y^*(p⁰, p^t, s^b) ≡ 1/[P_Y(p t, p⁰, s^b)] is the time antithesis of the original Young index, P_Y(p⁰, p^t, s^b).

These inequalities follow from the fact that a harmonic mean of M positive numbers is always equal to or less than the corresponding arithmetic mean; see Walsh (1901, p. 517) or Irving Fisher (1922, pp. 383–84). This inequality is a special case of Schlömilch’s (1858) Inequality; see Hardy, Littlewood, and Pólya (1934, p. 26). Walsh (1901, pp. 330–32) explicitly noted the inequality in equation (16.53) and also noted that the corresponding geometric average would fall between the harmonic and arithmetic averages. Walsh (1901, p. 432) computed some numerical examples of the Young index and found big differences between it and his best indices, even using weights that were representative for the periods being compared. Recall that the Lowe index becomes the Walsh index when geometric mean quantity weights are chosen, and so the Lowe index can perform well when representative weights are used. This is not necessarily the case for the Young index, even using representative weights. Walsh summed up his numerical experiments with the Young index as follows: “In fact, Young’s method, in every form, has been found to be bad” (1901, p. 433).

“We now come to a third use of these tests, namely, to ‘rectify’ formulae, i.e., to derive from any given formula which does not satisfy a test another formula which does satisfy it; . . . This is easily done by ‘crossing’, that is, by averaging antitheses. If a given formula fails to satisfy Test 1 [the time reversal test], its time antithesis will also fail to satisfy it; but the two will fail, as it were, in opposite ways, so that a cross between them (obtained by geometrical averaging) will give the golden mean which does satisfy” (Irving Fisher, 1922, p. 136). Actually, the basic idea behind Fisher’s rectification procedure was suggested by Walsh, who was a discussant for Fisher (1921), where Fisher gave a preview of his 1922 book: “We merely have to take any index number, find its antithesis in the way prescribed by Professor Fisher, and then draw the geometric mean between the two” (Walsh, 1921b, p. 542).

This index is a base-year weighted counterpart to an equally weighted index proposed by Carruthers, Sellwood, and Ward (1980, p. 25) and Dalén (1992, p. 140) in the context of elementary index number formulas. See Chapter 20 for further discussion of this unweighted index.

If these definitions are applied (approximately) to the Young index studied in the previous section, then it can be seen that for the Young price index to be consistent with the Divisia price index, the base-year shares should be chosen to be average shares that apply to the entire time period between months 0 and t.

“As the elementary formula of the chaining, we may get Laspeyres’ or Paasche’s or Edgeworth’s or nearly any other formula, according as we choose the approximation principle for the steps of the numerical integration” (Frisch, 1936, p. 8).

Diewert (1980, p. 444) also obtained the Paasche and Laspeyres approximations to the Divisia index using a somewhat different approximation argument. He also showed how several other popular discrete time index number formulas could be regarded as approximations to the continuous time Divisia index.

Trivedi (1981) systematically examined the problems involved in finding a best discrete time approximation to the Divisia indices using the techniques of numerical analysis. However, these numerical analysis techniques depend on the assumption that the true continuous time micro price functions, p_i(t), can be adequately represented by a polynomial approximation. Thus, we are led to the conclusion that the best discrete time approximation to the Divisia index depends on assumptions that are difficult to verify.

This section is based largely on the work of Peter Hill (1988; 1993, pp. 385–90).

The results in Appendix 17.1 provide some theoretical support for the use of chain indices in that it is shown that under certain conditions, the Divisia index will equal an economic index. Hence, any discrete approximation to the Divisia index will approach the economic index as the time period gets shorter. Thus, under certain conditions, chain indices will approach an underlying economic index.

The chain principle was introduced independently into the economics literature by Lehr (1885, pp. 45–46) and Marshall (1887, p. 373). Both authors observed that the chain system would mitigate the difficulties because of the introduction of new products into the economy, a point also mentioned by Peter Hill (1993, p. 388). Irving Fisher (1911, p. 203) introduced the term “chain system.”

Examples of rapidly downward trending prices and upward trending quantities can be found with computers, electronic equipment of all types, Internet access, and telecommunication charges.

Note that P_L(p⁰, p^t, q⁰, q^t) will equal P_P(p⁰, p^t, q⁰, q^t) if either the two quantity vectors q⁰ and q^t are proportional or the two price vectors p0 and p^t are proportional. Thus, to obtain a difference between the Paasche and Laspeyres indices, nonproportionality in both prices and quantities is required.

Regular seasonal fluctuations can cause monthly or quarterly data to “bounce,” using Szulc’s (1983) term and chaining bouncing data can lead to a considerable amount of index drift; that is, if after 12 months, prices and quantities return to their levels of a year earlier, then a chained monthly index will usually not return to unity. Hence, the use of chained indices for “noisy” monthly or quarterly data is not recommended without careful consideration.

See Diewert (1978, p. 895) and Peter Hill (1988; 1993, pp. 387–88).

This observation will be illustrated with an artificial data set in Chapter 19.

Walsh, in discussing whether fixed-base or chained index numbers should be constructed, took for granted that the precision of all reasonable bilateral index number formulas would improve, provided that the two periods or situations being compared were more similar and hence, for this reason, favored the use of chained indices: “The question is really, in which of the two courses [fixed-base or chained index numbers] are we likely to gain greater exactness in the comparisons actually made? Here the probability seems to incline in favor of the second course; for the conditions are likely to be less diverse between two contiguous periods than between two periods say fifty years apart” (Walsh, 1901, p. 206). Walsh (1921a, pp. 84–85) later reiterated his preference for chained index numbers. Fisher also made use of the idea that the chain system would usually make bilateral comparisons between price and quantity data that were more similar and hence the resulting comparisons would be more accurate: “The index numbers for 1909 and 1910 (each calculated in terms of 1867– 1877) are compared with each other. But direct comparison between 1909 and 1910 would give a different and more valuable result. To use a common base is like comparing the relative heights of two men by measuring the height of each above the floor, instead of putting them back to back and directly measuring the difference of level between the tops of their heads” (1911, p. 204). “It seems, therefore, advisable to compare each year with the next, or, in other words, to make each year the base year for the next. Such a procedure has been recommended by Marshall, Edgeworth and Flux. It largely meets the difficulty of non-uniform changes in the Q’s [quantities], for any inequalities for successive years are relatively small” (I. Fisher, 1911, pp. 423–24).

Diewert (2002b) took an axiomatic approach to defining various indices of absolute and relative dissimilarity.

Irving Fisher (1922, pp. 271–76) hinted at the possibility of using spatial linking; that is, of linking countries that are similar in structure. However, the modern literature has grown owing to the pioneering efforts of R.J. Hill (1995; 1999a; 1999b; and 2001). Hill (1995) used the spread between the Paasche and Laspeyres price indices as an indicator of similarity and showed that this criterion gives the same results as a criterion that looks at the spread between the Paasche and Laspeyres quantity indices.

It should be noted that von der Lippe (2001, pp. 56–58) is a vigorous critic of all index number tests based on symmetry in the timeseries context, although he is willing to accept symmetry in the context of making international comparisons. “But there are good reasons not to insist on such criteria in the intertemporal case. When no symmetry exists between 0 and t, there is no point in interchanging 0 and t” (von der Lippe, 2001, p. 58).

The test name is credited to Irving Fisher (1922, p. 413), and the concept was originally credited to Westergaard (1890, pp. 218–19).

The additional tests are (1) positivity and continuity of P(p⁰, p¹, q⁰, q¹) for all strictly positive price and quantity vectors p⁰, p¹, q⁰, q¹; (2) the identity test; (3) the commensurability test; (4) P(p⁰, p¹, q⁰, q¹) is positively homogeneous of degree 1 in the components of p1; and (5) P(p⁰, p¹, q⁰, q¹) is positively homogeneous of degree zero in the components of q¹.

Konüs and Byushgens (1926) show that the index defined by equation (16.78) is exact for Cobb and Douglas (1928) preferences; see also Pollak (1983, pp. 119–20). The concept of an exact index number formula is explained in Chapter 17.

This result can be derived using results in Eichhorn (1978, pp. 167–68) and Vogt and Barta (1997, p. 47). A simple proof can be found in Balk (1995). This result vindicates Irving Fisher’s intuition. He asserted that “the only formulae which conform perfectly to the circular test are index numbers which have constant weights” (1922, p. 274). Irving Fisher went on to assert; “But, clearly, constant weighting is not theoretically correct. If we compare 1913 with 1914, we need one set of weights; if we compare 1913 with 1915, we need, theoretically at least, another set of weights. . . . Similarly, turning from time to space, an index number for comparing the United States and England requires one set of weights, and an index number for comparing the United States and France requires, theoretically at least, another” (1922, p. 275).

See, for example, Diewert (1978, p. 894). Walsh (1901, pp. 424 and 429) found that his three preferred formulas all approximated each other very well, as did the Fisher ideal for his artificial data set.

More specifically, most superlative indices (which are symmetrically weighted) will satisfy the circularity test to a high degree of approximation in the time-series context. See Chapter 17 for the definition of a superlative index. It is worth stressing that fixed-base Paasche and Laspeyres indices are very likely to diverge considerably over a five-year period if computers (or any other product that has price and quantity trends different from the trends in the other products) are included in the value aggregate under consideration. See Chapter 19 for some empirical evidence on this topic.

Again, see Szulc (1983) and Peter Hill (1988).

This formula was implicitly introduced in Törnqvist (1936) and explicitly defined in Törnqvist and Törnqvist (1937).

This exactness result can be extended to cover the case when there are monthly proportional variations in prices and the value shares have constant seasonal effects in addition to linear trends; see Alterman, Diewert, and Feenstra (1999, p. 65).

Walsh (1921a, p. 98) called his test the circular test, but because Irving Fisher also used this term to describe his transitivity test defined earlier by equation (16.77), it seems best to stick to Fisher’s terminology because it is well established in the literature.

Walsh (1921b, pp. 540–541) noted that the time reversal test was a special case of his circularity test.

This is essentially a variant of the methodology that Irving Fisher (1922, p. 284) used to check how well various formulas corresponded to his version of the circularity test.

See Bortkiewicz (1923, pp. 374 –75) for the first application of this correlation coefficient decomposition technique.

This type of second-order approximation is credited to Dalén (1992, p. 143) for the case r^* = 1 and to Diewert (1995a, p. 29) for the case of a general r^*.

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Abstract

A. Introduction

B. Decomposition of Value Aggregates into Price and Quantity Components

B.1 Decomposition of value aggregates and the product test

B.2 Laspeyres and Paasche indices

C. Symmetric Averages of Fixed-Basket Price Indices

C.1 Fisher index as an average of the Paasche and Laspeyres indices

C.2 Walsh index and theory of “pure” price index

C.3 Conclusion

D. Annual Weights and Monthly Price Indices

D.1 Lowe index with monthly prices and annual base-year quantities

D.2 Lowe index and midyear indices

D.3 Young index

E. Divisia Index and Discrete Approximations

E.1 Divisia price and quantity indices

E.2 Discrete approximations to continuous time Divisia index

F. Fixed-Base Versus Chain Indices

Appendix 16.1 Relationship Between Paasche and Laspeyres Indices

Appendix 16.2 Relationship Between Lowe and Laspeyres Indices

Appendix 16.3 Relationship Between Young Index and Its Time Antithesis

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