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CHAPTER 4 PRICE INDEX THEORY
b) the Törnqvist price index, which is a weighted geometric mean of the price relatives where the weights are the average shares of total values in the two periods, that is:
The Fisher Ideal and Törnqvist indexes are often described as symmetrically weighted indexes because they treat the weights from the two periods equally.
4.23 The Laspeyres and Paasche formulae are expressed above in terms of quantities and prices. However, in practice, quantities might not be observable or meaningful (e.g. consider the quantity dimension of legal services, public transport, and education). Thus in practice, the Laspeyres formula is typically estimated using expenditure shares to weight price relatives - this is numerically equivalent to the formula (4.4) above.
4.24 To derive the price relatives form of the Laspeyres index, multiply the numerator of equation 4.4 by and rearrange to obtain:
where the term in parentheses represents the expenditure share of item i in the reference (or, more commonly labelled, base) period. Let:
then the Laspeyres formula may be expressed as:
where is the price relative for the ith item.
4.25 In a similar manner, the Paasche index may be constructed using expenditure weights. In equation 4.5, multiply the denominator by and rearrange terms to obtain:
which may be expressed as:
which is the inverse of a ‘backward’ Laspeyres index (i.e. a Laspeyres index going from period t to period 0 using period t expenditure weights).(footnote 5)
4.26 The important point to note here is that if price relatives are used, then value (or expenditure) weights must also be used. On the other hand, if prices are used directly rather than in their relative form, then the weights must be quantities.
4.27 An example of creating index numbers using the above formulae is presented in Table 4.1. For the purposes of this exercise, a limited range of the types of commodities households might purchase is used. The quantities that these items would typically be measured in may vary. There are likely to be differences in price behaviour of the commodities over time. Further, the quantities of these items households purchase may vary over time in response to changes in prices (of both the item and other items) and household incomes.
4.28 Differences that might arise in price changes (and, by implication expenditure patterns) are illustrated by the following:
4.29 Price changes influence, to varying degrees, the quantities of items households purchase. For some items, such as basic food stuffs, the quantities purchased may show little change in response to price changes. For other items, the quantities households purchase may change by a smaller or greater proportionate amount than the price change.(footnote 6)
4.30 The examples in Table 4.1 reflect some of these possibilities.
4.31 In Table 4.2 the different index formulae produce different index numbers, and thus different estimates of the price movements. Typically the Laspeyres formula will produce a higher index number than the Paasche formula in periods after the base period, with the Fisher Ideal and the Törnqvist of similar magnitude falling between the index numbers produced by the other two formulae. In other words the Laspeyres index will generally produce a higher (lower) measure of price increase (decrease) than the other formulae and the Paasche index a lower (higher) measure of price increase (decrease) in periods after the base period.(footnote 7)
Generating index series over more than two periods
4.32 Most users of price indexes require a continuous series of index numbers at specific time intervals. There are two options for applying the above formulae when compiling a price index series.
(i) Select one period as the base and separately calculate the movement between that period and the required period. This is called a fixed base or direct index.
(ii) Calculate the period-to-period movements and chain these (i.e. calculate the movement from the first period to the second, the second to the third with the movement from the first period to the third obtained as the product of these two movements).
4.33 The calculation of direct and chained indexes over three periods (0, 1, and 2) using observations on three items, is shown below. The procedures can be extended to cover many periods.
In order to have expenditure weights summing exactly to unity, the weight for clothing has been derived as a residual. The following illustrate the index number calculations:
(0.4310 x 1.1000) + (0.1293 x 1.3333) + (0.0776 x 1.4444) + (0.2586 x 0.6667) + (0.1035 x 1.1000) x 100 = 104.5
1/((0.4532 x 1.1000) + (0.1483 x 1.3333) + (0.0696 x 1.4444) + (0.2472 x 0.6667) + (0.0817 / 1.1000)) x 100 = 98.4
Törnqvist best calculated by first taking the logs of the index formula
(1/2) x (0.4310+0.4532) x ln(1.1000)
+ (1/2) x (0.1293 + 0.1483) x ln(1.3333)
+ (1/2) x (0.0776 + 0.0696) x ln(1.4444)
+ (1/2) x (0.2586 + 0.2472) x ln(0.6667)
+ (1/2) x (0.1035 + 0.0817) x ln(1.1000)
and then taking the exponent multiplied by 100 = 101.6.
4.34 An index formula is said to be 'transitive' if the index number derived directly is identical to the number derived by chaining. In general, no weighted index formula will be transitive because period-to-period calculation of the index involves changing the weights for each calculation. The index formulae in Table 4.2 will only result in transitivity if there is no change in the quantity of each item in each period or all prices show the same movement. In both these cases, all the formulae will produce the same result.
4.35 The direct Laspeyres formula has the advantage that the index can be extended to include another period's price observations when available, as the weights are fixed at some earlier base period. On the other hand, the direct Paasche formula requires both current period price observations and current period weights before the index can be extended.
Unweighted, or equally weighted indexes
4.36 In some situations, it is not possible or meaningful to derive weights in either quantity or expenditure terms for each price observation. This is typically so for a narrowly defined commodity grouping in which there might be many sellers (or producers). Information might not be available on the total volume of sales of the item or for the individual sellers or producers from whom the sample of price observations is taken. In these cases, it seems appropriate not to weight, or more correctly to assign an equal weight, to each price observation. It is a common practice in the CPI in many countries that the price indexes at the lowest level (where prices enter the index) are calculated using an equally weighted formula, such as an arithmetic mean or a geometric mean.
4.37 Suppose there are price observations for n items in period 0 and period t. Then three approaches(footnote 8) for constructing an equally weighted index are as follows.
4.38 Although these formulae apply equal weights, the basis of the weights differs. The geometric mean applies weights such that the expenditure shares of each observation are the same in each period. In other words, it is assumed that as an item becomes more (less) expensive relative to other items in the sample the quantity declines (increases) with the percentage change in the quantity offsetting the percentage change in the price. The RAP formula assumes equal quantities in both periods. That is, the RAP assumes there is no change in the quantity of an item purchased regardless of either its price movement or that of other items in the sample. The APR assumes equal expenditures in the first period with quantities being inversely proportional to first period prices.
4.39 The following are calculations of the equal weight indexes using the data in Table 4.2. Setting period 0 as the base with a value of 100.0, the following index numbers are obtained in period t:
4.40 Theory suggests that the APR formula will produce the largest estimate of price change, the GM the least and the RAP a little larger but close to the GM.(footnote 10) Real life examples generally support this proposition,(footnote 11) although with a small sample as in the example above, substantially different rankings for the RAP formula are possible depending on the prices.
4.41 The behaviour of these formulae under chaining and direct estimation is shown in Table 4.3 using the price data from Table 4.2. The RAP and GM formulae are transitive, but not the APR.
Unit values as prices
4.42 A common problem confronted by index compilers is how to measure the price of items in the index whose price may change several times during an index compilation period. For example, in Australia petrol prices change almost daily at many outlets, but the CPI is quarterly. Taking more frequent price readings and calculating an average is one approach to deriving an average quarterly price. A more desirable approach, data permitting, would be to calculate unit values and use these as price measures.(footnote 12) Unit values are obtained by dividing a value by a quantity (e.g. the total value of petrol sold in a particular period divided by the number of litres sold will give a unit value per litre for the price of petrol over the period). Unit values can be used to measure price changes only when the values are for similar (homogeneous) products.
4.43 For example, suppose outlet X sells chocolate bars in weights of 50g, 80g and 100g. Further, suppose the outlet keeps records of the value of sales of these chocolate bars in aggregate and the number of each size of chocolate bar sold. It is then possible to calculate the total quantity of chocolate sold in grams. Dividing the value of expenditure on chocolate by the total quantity in grams produces a unit value that could be used as the price measure for chocolate.
4.44 The advent of scanner data from retail outlets is making the construction of unit values more feasible. To be successfully applied, the information is required across all outlets. Scanners provide information about both values and quantities at the point of sale, and so enable the collection of a large number of unit values at fine levels. In effect, these data would remove any need for the unweighted index formulae discussed above (at least for those items where unit values are available).
RESOLVING EXPENDITURE AGGREGATES
4.45 It is appropriate at this point to re-examine the decomposition of an expenditure aggregate into price and quantity components introduced in equation 4.1. It is important to know the form of the quantity index when a particular form of the price index is used (and vice versa) to ensure the accurate decomposition of the value change.
4.46 A value is the product of a price and a quantity (in its simplest form, the price of a single item multiplied by 1 is the value of the item). It follows that changes in the value of expenditure on an item from period to period are the result of changes in the prices or quantities or both. If any two of the value, price or quantity are known, the third can be derived (i.e. E = P x Q, where E = expenditure, P = price and Q = quantity), e.g. Q=E/P. The calculation is straightforward when a single item is involved. However, in the case of an expenditure total that is the sum of several items, breaking up that expenditure into its price and quantity components becomes more complicated.
4.47 Price indexes provide a means of removing the effects of price changes from changes in expenditure so that the underlying changes in quantity can be identified. In the Australian National Accounts, price indexes are widely used in the process of estimating changes in volumes of expenditure, production etc. The process of using price indexes in this way is known as price deflation, with the index termed a deflator. The form of price index (current or fixed weighted) will determine the resulting index of quantity change.
4.48 The change in an expenditure aggregate between period 0 and t may be expressed as:
4.49 Multiplying the right-hand side of equation (4.16) by allows the equation to be expressed as:
where the first term on the right-hand side of the equals sign is a Laspeyres price index and the second is a Paasche volume index.(footnote 13) This is referred to as the Laspeyres decomposition. In other words, if an index of value change is deflated by a base- period-weighted price index, then the index of quantity change is a current-period-weighted quantity index.
4.50 An alternative decomposition of the change in the expenditure aggregate is obtained by multiplying the right-hand side of (4.16) by which produces:
where the first term on the right-hand side of the equals sign is a Paasche price index and the second is a Laspeyres volume index. This is referred to as the Paasche decomposition. In other words, if an index of value change is deflated by a current-period-weighted price index, then the index of quantity change is a base-period-weighted quantity index.
4.51 A similar decomposition can also be undertaken for the Fisher Ideal index. By taking the geometric average of the alternative Laspeyres and Paasche decompositions of value change (right-hand sides of equations (4.17) and (4.18)) it can be shown that value change is the product of Fisher Ideal price and quantity indexes.
SOME PRACTICAL ISSUES IN PRICE INDEX CONSTRUCTION
Handling changes in price samples
4.52 All the index formulae discussed above require observations on the same items in each period. In some situations it may be necessary to change the items or outlets included in the price sample or, if weights are used, to re-weight the price observations. Examples of changes in a price sample include:
4.53 It is important that changes in price samples are introduced without distorting the level of the index for the price sample. This usually involves a process commonly referred to as splicing. Splicing is similar to chaining except that it is carried out at the level of the price sample. An example of handling a sample change is shown in the table below, for equally weighted indexes assuming a new respondent is introduced in period t. A price is also observed for the new respondent in the previous period t-1. The inclusion of the new respondent causes the geometric mean to fall from $5.94 to $5.83. The ABS does not want this price change to be reflected in the index, but we do want to capture the effect of respondent 4’s price movement between period t-1 and t.
4.54 In the case of the APR and GM formulae, the process involves:
4.55 For these two formulae, the average of the price relatives is effectively the index number, so the GM index for period t-1 is 132.6 and for period t is 141.6.
4.56 In the case of the RAP formula, the method is similar, but prices are used instead of price relatives. The RAP formula uses the arithmetic mean of prices (not the arithmetic mean of the price relatives). The index for RAP can be calculated from the period-to-period price movements:
Temporarily missing price observations
4.57 In any period, an event may occur that makes it impossible to obtain a price measure for an item. For example, an item could be temporarily out of stock or the quality is not up to standard (as may occur with fresh fruit and vegetables because of climatic conditions).
4.58 There are a few options available to deal with temporarily missing observations. These include:
(i) repeat the previous period’s price of the item;
(ii) impute a movement for the item based on the price movement for all other items in the sample; or
(iii) use the price movement from another price sample.
4.59 Approach (ii) is equivalent to excluding the item, for which a price is unavailable in one period, from both periods involved in the index calculation. It strictly maintains the matched sample concept.
4.60 An example of imputing using the first two approaches for the equally weighted formula is provided below. The example assumes that there is no price observation from respondent B in period 2.
HANDLING CHANGES IN GOODS AND SERVICES
4.61 A price index by definition measures what can be described as pure price change; that is, it is not distorted by changes in quality. The concept of a good or service within a price index is important in determining whether an item is new or a modification (change in quality) of a previous item. Under the usual index compilation practices, if the change in price of the item fully or partly reflects a change in quality, then for index purposes an adjustment is necessary to account for that quality change. If it is a new item, then that item must be introduced into the index by linking (or splicing).
4.62 There are two main approaches to treating goods and services for the purposes of compiling a price index. The conventional or goods approach is to treat each good and service as a separate item; for example, a distinction might be made between red and green apples. The alternative approach could be termed a characteristics approach that takes commodities and tries to identify the component characteristics or attributes which are valued by the consumer. For example, the characteristics of an apple which households value might be its nutritional content plus the ability to consume without having to perform any food preparation. The outcome is that consumers satisfy their hunger.(footnote 14)
4.63 The characteristics approach provides a conceptual basis for describing quality change. In the context of price indexes, quality can be thought of as embracing all those attributes or characteristics of an item on which the consumer places some value.(footnote 15) Take apples as an example. Consumers will value them for nutritional content as well as taste and absence of blemishes and bruising. The price index will be biased unless an apple of the same quality is priced each period. For some items quality change over time is not a major issue (e.g. the quality change in apples might only reflect differences in growing conditions between seasons), but for other items quality changes are very important (e.g. the increase in power and speed of personal computers, and changes in safety and ride quality of motor vehicles).
4.64 The characteristics approach has not been used so far as the sole basis for constructing a consumer price index. However, it is the foundation of the so-called hedonic technique for estimating pure prices for commodities.(footnote 16) The hedonic technique is now being used by some countries in their CPIs for some types of consumer goods.(footnote 17) Essentially the hedonic approach involves estimating a relationship between a commodity’s price and the characteristics that it contains (e.g. for personal computers, a relationship might be estimated between the price of the computer and its processing power (chip type and speed), amount of RAM, hard disk size, etc. over a range of computers). This effectively imputes a price for each characteristic that can be used to adjust prices as specifications change.(footnote 18)
4.65 Strict adherence to a goods approach would see frequent linking in response to any change in the specifications of individual items priced. Frequent linking is undesirable as each link is effectively a break in the series and can introduce bias. In the absence of the hedonic approach, quality adjustments must rely heavily on subjective methods. In a consumer price index these adjustments should be based, as far as possible, on the value of the quality change to the consumer (user value). In this respect, use of manufacturing cost (resource cost) data to value quality change can be misleading in many situations.(footnote 19)
4.66 Although intuitively appealing, the hedonic technique is difficult to apply in practice. It requires a lot of information and the careful selection of attributes that would be appropriate in a household utility function (e.g. if performance is one characteristic of a motor vehicle that consumers desire, would engine power or acceleration speed or some other parameter be the best measure of it). In addition, there are issues such as the functional form to be used and weighting.(footnote 20) Nevertheless, the hedonic technique does provide a tool that may assist in identifying the characteristics of commodities that influence their price, and it does provide a basis for adjusting for quality change.
4.67 Recent research by Aizcorbe et al. (2000) has indicated that for high technology goods such as computers, the use of matched models and a superlative index formula, for example the Fisher Ideal index, captures the rapid quality change in these goods. This raises questions as to whether there is much to be gained by using the more complicated hedonic approach.
4.68 It is not clear that prices should be adjusted for all changes in quality. An issue here is the appropriate treatment of mandated environmental measures (such as pollution-control hardware on automobiles) which increase the cost of items.(footnote 21) Mandated measures that (say) increase consumer safety can have a user value imputed to them, but the situation is not as simple for environmental measures. Indeed, Pollak (1998) argues that it is impractical to include environmental variables and produce meaningful price indexes.
4.69 Prices statisticians are often confronted with the problem of determining when a new item on the market is a new good for index construction purposes. A completely new good is not easily included in an existing price collection because there is no product category to which it can be readily classified. In these cases, it may eventually require its own separate recognition within the index rather than being a part of an existing product group.
4.70 The use of a hedonics or characteristics approach may assist in defining new goods. For example, the hedonics approach might suggest that DVDs are not actually new goods, but rather a better bundling of sound and images and other characteristics that people value (such as a more durable medium).
4.71 The difficulty of new goods is that they often show substantial falls in price once they gain market acceptance (sometimes after improvements in quality), and the supply of the good expands. There are two problems here. The first is that the traditional fixed-weighted index does not allow for the introduction of new goods until weights are updated. The second is that if the new good is not included until some time after establishing a significant market share, then the initial phase of falling prices is missed.
4.72 It has been suggested (Hicks (1940), and Fisher and Shell (1972)) that, in a cost-of-living framework, new goods should be valued at their demand reservation price. This price is the intercept of the demand curve with the price axis, essentially the price at which no units of the good would be sold. However, procedures to estimate reliably the demand reservation price have yet to be established.
BIAS IN PRICE INDEXES
4.73 Some of the issues about bias have already been covered in this manual. However, it is useful to bring these matters together to consider further some of the practical issues involving price indexes, especially considering a major inquiry into the issue was held in the United States in 1996.(footnote 22)
4.74 A price index may be described as biased if it produces estimates which depart from a notionally true or correct measure. In the case of consumer price indexes, the true measure is usually taken to be the cost-of-living index, as it allows for the substitutions in consumption that consumers make in response to changes in relative prices. As it is impractical to construct a true cost-of-living index, official agencies are forced into second-best solutions.
4.75 The following types of bias, typically upwards, have been described by Diewert (1996).
(i) Elementary index bias, which results from the use of inappropriate formulae for compiling index numbers at the elementary aggregate level;
(ii) Substitution bias, which arises from using formulae at levels above the elementary aggregates which do not allow for substitution in response to changes in relative prices;
(iii) Outlet substitution bias, which occurs when consumers shift their purchases from higher cost outlets to lower cost outlets for the same commodity;
(iv) Quality adjustment bias, which arises from inadequate adjustment for quality changes; and
(v) New-goods bias, which arises largely from the failure to include new goods when first introduced into the market.
4.76 Although it is almost impossible to eliminate these sources of bias, some measures can be taken to minimise them.
(i) Use appropriate formulae in compiling elementary aggregate indexes, in particular use of the GM formula where appropriate or the RAP formula.
(ii) Use a superlative index formula rather than the Laspeyres, if current-period weighting data can be obtained on time. More frequent updating of weights in the Laspeyres formula is also suggested, although changing weights alone does not have a significant effect in the short to medium term unless the change in the weighting pattern is significant.(footnote 23) Other options might be to use formulae that allow substitution or assumptions about substitution between commodity groupings to be entered.
(iii) Closely monitor and update price samples to reflect changes in the outlets from which households purchase. For example, there is clearly a need to plan for the inclusion in consumer price indexes of purchases from outlets operating exclusively over the Internet.
(iv) Make greater use of the hedonic technique to adjust for quality change and to determine comparable items.
(v) Include new goods into the CPI as soon as possible. For a fixed-weighted index such as Laspeyres, there would also be a need to update the fixed weights to allow for the inclusion of the new goods if they are substituting for all goods in general, or to adjust the weights within a commodity grouping if the new good is substituting for specific items. For example, one could argue that CDs were a new good, but as they were substituting for records and tapes they could be introduced into the commodity grouping for records and tapes, and weights between these items adjusted accordingly.
4.77 Price index theory guides prices statisticians as to the best practices and formulae to use in compiling price indexes in order to produce reliable price measures. However, the highly desirable must be balanced against the practical. It would be highly desirable to use a superlative index formula such as the Fisher Ideal for all price indexes, but this is often not possible because of data problems and issues with timeliness.
4.78 There is much more to a price index than which formula to use. Also important is the determination of what items are to be included in the index, that is the index domain. This subject is covered in the next chapter.
1 For a detailed discussion of price index theory and internationally recommended practices, see Consumer Price Index Manual, Theory and Practice, 2004 (International Labour Office). <back
2 This is the terminology used by Pollak (1971). <back
3 In this example, the price relative shows the change in price between two times. If, instead of two different periods we looked at the price between two different markets in the same period, the price relative would show the difference between the prices in the two markets in the same period. <back
4 The use of the geometric mean of the Laspeyres and Paasche indexes was first proposed by Pigou in 1920, and given the title "ideal" by Fisher (1922). <back
5 For further discussion of forward and backward Laspeyres and Paasche price and quantity indexes, refer to Chapter 2 of Allen (1975). <back
6 Economists measure the change in the quantity of an item in response to a change in price (or income) by elasticities, which are measured as the ratio of the percentage change in the quantity to the percentage change in price (or income). An item is price inelastic if the percentage change in the quantity is less than the percentage change in price. It has unit elasticity if the percentage changes are the same, and is price elastic if the percentage change in the quantity is greater than the percentage change in price. If an item is price inelastic, the change in expenditure will be in the same direction as the change in price (i.e. if price increases, then expenditure also increases). If the item has unit elasticity, then expenditure is unchanged. If the item is price elastic, the change in expenditure will be in the opposite direction to the price change (i.e. if price increases, then expenditure decreases). <back
7 The relationship between the Laspeyres and Paasche indexes holds while there is a normal relationship (negative correlation) between prices and quantities; that is, quantity declines if price increases between the two periods, and vice versa. <back
8 The implicit weights applied by the three formulae are equal base–period quantities (RAP), equal base–period expenditures (quantities inversely proportional to base–period prices) (APR) and equal expenditure shares in both periods (GM). <back
9 The geometric mean of n numbers is the nth root of the product of the numbers. For example, the geometric mean of 4 and 9 is 6 (= ), but the arithmetic mean is 6.5 ( = (4+9)/2 ).
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