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3.30 Price changes influence, to varying degrees, the quantities of items households purchase. For some items such as basic foods, 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 11)
3.31 The scenarios presented in table 3.1 merely reflect some of these possibilities.
3.32 In table 3.2 the different index formulas 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 formulas. In other words the Laspeyres index will generally produce a higher (lower) measure of price increase (decrease) than the other formulas and the Paasche index a lower (higher) measure of price increase (decrease) in periods after the base period. (footnote 12)
Generating index series over more than two periods
3.33 Most users of price indexes require a continuous series of index numbers at specific time intervals. There are two options for applying the above formulas 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, which 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).
3.34 The calculation of direct and chained indexes over three periods (0, 1, and 2) using observations on three items, is shown in table 3.2. The procedures can be extended to cover many periods.
TABLE 3.1: COMPILING PRICE INDEXES OVER TWO PERIODS
TABLE 3.2: CONSTRUCTING PRICE INDEX SERIES
3.35 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 formulas in table 3.2 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. (footnote 13) In both these cases all the formulas will produce the same result.
3.36 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 held 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 equal-weight indexes
3.37 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 overall 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 equal-weights formula, such as an arithmetic mean or a geometric mean.
3.38 Suppose there are price observations for N items in period 0 and t. Then three approaches (footnote 14), (footnote 15) for constructing an equal weights index are:
(i) calculate the arithmetic mean of prices in both periods and obtain the relative of the second period’s average with respect to the first period’s average
(i.e. divide the second period’s average by the first period’s average). This is the relative of the arithmetic mean of prices (RAP) approach, also referred to as the Dutot formula:
(ii) for each item, calculate its price relative (i.e. divide the price in the second period by the price in the base period) and then take the arithmetic average of these relatives. This is the arithmetic mean of price relatives (APR) approach, also referred to as the Carli formula:
(iii) for each item, calculate its price relative and then take the geometric mean (footnote 16) of the relatives. This is the geometric mean (GM) approach, also referred to as the Jevons formula:
3.39 Although these formulas 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.(footnote 17)
3.40 The following are calculations of the equal weight indexes using the data in table 3.2. Setting period 0 as the base with a value of 100.0, the following index numbers are obtained in period t:
RAP formula: 113.5 =
APR formula: 113.9 =
GM formula: 113.8 =
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 18) Real life examples generally support this proposition, (footnote 19) although with a small sample as in the above example, substantially different rankings for the RAP formula are possible depending on the prices.
3.41 The behaviour of these formulas under chaining and direct estimation is shown in table 3.3 using the price data from table 3.2. It is noted that the RAP and GM formulas are transitive, but not the APR.
TABLE 3.3: LINKING PROPERTIES OF EQUAL WEIGHT INDEX FORMULAS
Unit values as prices
3.42 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 while the CPI index 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 20) 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 (homogenous) products.
3.43 For example, suppose outlet X sells chocolate bars in weights of 50 g, 80 g and 100 g. 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.
3.44 The advent of scanner data is making the construction of unit values more feasible. To be successfully applied the information is required across all outlets. Scanners provide information on 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 it would remove any need for the unweighted index formulas discussed above (at least for those items where unit values are available).
DECOMPOSING EXPENDITURE AGGREGATES
3.45 Having introduced the major price index formulas, it is appropriate to re-examine the decomposition of an expenditure aggregate into price and quantity components introduced in equation (3.1). 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.
3.46 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 and/or quantities concerned. 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). For example, 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, decomposing that expenditure into its price and quantity components becomes more complicated.
3.47 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 or 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.
3.48 The change in an expenditure aggregate between period 0 and t may be expressed as
3.49 Multiplying the right-hand side of equation (3.16) 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 21) 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.
3.50 An alternative decomposition of the change in the expenditure aggregate is obtained by multiplying the right-hand side of (3.16) 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.
3.51 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 (3.17) and (3.18)) it can be shown that value change is the product of Fisher Ideal price and quantity indexes.
SOME PRACTICAL ISSUES IN PRICE INDEX CONSTRUCTION
To chain or not to chain
3.52 The use of fixed weights (as in a Laspeyres type formula) over an extended period of time is obviously not a sound index construction practice. For example, weights in a consumer price index have to be changed to reflect changes in consumption patterns over time. Consumption patterns change in response to longer-term movements in relative prices, changes in preference orderings and the introduction of new goods (and the displacement of other goods).
3.53 There are two options in these situations if a fixed-weight index is used. One is to hold the weights constant over as long a period as seems reasonable, starting a new index each time the weights are changed. This means that a longer-term series is not available. The second is to update the weights more frequently and to chain, as discussed above, to produce a long-term series. The latter is the more common practice.
3.54 The behaviour of the various formulas under chaining was discussed briefly above. The behaviours are further explored below in table 3.4 by adding two more periods. In period 3, prices and quantities are returned to their base period values and in period 4 the base period prices and quantities are ‘ shuffled’ between items. The period 3 situation is sometimes described as time reversal and the period 4 situation as ‘price bouncing’. (footnote 22)
3.55 Under the three formulas, the index number under direct estimation returns to 100.0 when prices and quantities of each item return to their base period levels. However, the chained index numbers do not (although the chained Fisher Ideal index might generally be expected to perform better than the chained Laspeyres or Paasche).
3.56 This situation poses a quandary for prices statisticians when using a fixed-weight index. There are obvious attractions in frequent chaining. However, chaining in a fixed-weight index can sometimes lead to biased estimates. This can occur if there is seasonality or cycles in the price and chaining coincides with the top and bottom of each cycle. For this reason it is generally accepted that indexes should not be chained at intervals less than annual. In effect, the conceptual underpinning of chaining is that the traditionally expected inverse relationship between prices and quantities actually applies in practice (i.e. growth in quantities is higher for those items whose prices increase less in relative terms). The System of National Accounts, 1993 describes the practical situations in which chaining works best. (footnote 23)
TABLE 3.4: A CLOSER LOOK AT CHAINING
Handling changes in price samples
3.57 All the index formulas 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 reweight the price observations. Examples of changes in a price sample include: a respondent goes out of business; or the sample needs to be updated to reflect changes in the market shares of respondents; to introduce a new respondent; or to include a new item.
3.58 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 called ‘splicing’. Splicing is similar to chaining except that it is carried out at the price sample level. An example of handling a sample change is shown in table 3.5, for equal weighted indexes assuming a new respondent is introduced in period t. A price is also observed for the new respondent in period t-1. The inclusion of the new respondent causes the geometric mean to fall from $5.94 to $5.83. We do 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.
3.5 A CHANGE IN SAMPLE - INTRODUCING A NEW RESPONDENT
3.59 In the case of the APR and GM formulas, the process involves:
3.60 For these two formulas, 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.
3.61 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:
3.62 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).
3.63 There are various options available to handle 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
(iii) use the price movement from another price sample.
3.64 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.
3.65 An example of imputing using the first two approaches for the equal weighted formula is provided in table 3.6. The example assumes that there is no price observation from respondent B in period 2.
TABLE 3.6: IMPUTATION OF MISSING PRICE OBSERVATIONS
CHOOSING AN INDEX NUMBER FORMULA
3.66 As different index number formulas will produce different results, there is a need for some ground rules to determine which formulas are more appropriate. Two main approaches have been used:
3.67 The use of tests to assess index number performance is a useful guide. However, a number of practical issues need to be considered. These include: how relevant are the tests for the application at hand, are some tests more important than others and, even if an index formula fails a test, how close in practice is the index likely to be to the 'best' measure.
3.68 The range of tests developed for index numbers has expanded over the years. Diewert (1992) describes 20 tests for weighted index formula, while Diewert (1995) provides 17 tests for equal weighted (or elementary) index formulas and attributes the tests to their original authors. It is beyond the scope of this manual to describe all the tests, but several important and relevant ones for current purposes will be discussed.
3.69 Many of the tests apply to both the equal and unequal weighted formulas. The tests include:
3.70 The Fisher Ideal index formula passes tests on time reversal, circularity, commensurability and factor reversal, whereas the Laspeyres and Paasche only pass the test of commensurability. (footnote 28)
3.71 In regard to the three equal weight price index formulas discussed earlier, the APR fails the first three tests, the RAP fails the commensurability test while the GM approach passes all tests. (footnote 29) In terms of Diewert's 17 tests for elementary index formulas, the RAP passes 15 tests and the GM 16 tests.
3.72 While the equal weighted GM appears to have considerable appeal as an elementary index formula, there are some situations in which it produces an undesirable result. The GM cannot handle zero prices, which might occur if the government introduced a policy to fully subsidise a good or service. In addition, the GM may not produce acceptable movements when a price falls sharply. For example, consider a price sample of two items, each selling for $10 in one period, with the price of one of the items falling to $1 in the second period. The GM produces an index of 31.6 for the second period (assuming it was 100 in the first period), a fall of around 68 per cent. Because the GM maintains equal expenditure shares in each period, it effectively gives a larger weight to lower prices. (footnote 30)
3.73 Only in recent years has the GM formula become more widely accepted in official circles for compiling consumer price indexes. For example, Canada switched to using geometric means in the late 1980s, the United States introduced the GM formula for items making up about 61% of the CPI in January 1999 and Australia began introducing the formula in December quarter 1998. The GM formula is prescribed by the European Union for calculation of price sample means in its Harmonised Indices of Consumer Prices (HICP). There appear to be two reasons for the slow adoption of the GM. One is that prior to the use of computers in compiling the official indexes, calculation of geometric means was a very laborious task. A second reason is the perceived difficulty in explaining the measure to users.
3.74 There is another aspect to indexes that is worth considering, although not rated as a test in the literature. In most countries the CPI is produced at various levels of aggregation, there typically being three or more levels between the lowest published level and the total all goods and services. In practice it is desirable that the same result is obtained whether the total index is compiled directly from the lowest level or in a staged way, using progressively higher levels of aggregation. Diewert (1978) shows that the fixed-weight Laspeyres and Paasche indexes have this 'consistency' in aggregation property, while the Fisher and Tornqvist indexes are (very) closely consistent. (footnote 31)
HANDLING CHANGES IN GOODS AND SERVICES
3.75 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 (quality change) 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 may be 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).
3.76 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 essentially 'breaks' actual commodities into their component characteristics or attributes which are valued by the consumer, which the consumer then combines to produce desired products. 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 32)
3.77 The characteristics approach provides a conceptual basis for describing quality change. In the sense 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 33) For example, in the case of apples, consumers will value them for nutritional content as well as taste and absence of blemishes/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).
3.78 The characteristics approach has not been used to date as the sole basis of constructing a consumer price index. However, it is the foundation of the so-called 'hedonic' technique for estimating pure prices for commodities (footnote 34) and the hedonic technique is now being used by some countries in their CPIs for certain consumer goods. (footnote 35) Essentially the hedonic approach involves estimating a relationship between a commodity’s price and the characteristics that it contains (e.g. in the case of 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 36)
3.79 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 37)
3.80 While intuitively appealing, the successful application of the hedonic technique is not a trivial exercise. It requires substantial amounts 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, is engine power or acceleration speed or some other parameter the 'best' measure of performance). In addition there are issues such as the functional form to be used and weighting. (footnote 38) Nevertheless, the hedonic technique does provide a tool that may assist in identifying the price influencing characteristics of commodities and provides a basis for adjusting for quality change.
3.81 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 captures the rapid quality change in these goods. This raises questions as to whether there is much to be gained by using a more complicated hedonic approach for some commodities.
3.82 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, which increase the cost of items, such as pollution control hardware on automobiles. 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 (1989) argues that it is impractical to include environmental variables and produce meaningful price indexes. (footnote 39)
3.83 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 cannot easily be included in an existing price collection because there is no product category to which it can be readily classified. In such cases, it may eventually require its own separate recognition within the index rather than being a part of an existing product group.
3.84 The use of a hedonics or characteristics approach may assist in defining new goods. For example the hedonics approach might suggest that compact discs (CDs) were not actually new goods but rather a better bundling of musical recordings and other characteristics that people valued, such as a more durable media.
3.85 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-weight 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 price fall phase will be missed.
3.86 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. The demand reservation 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 reliably estimate the demand reservation price have yet to be established. (footnote 40)
BIAS IN PRICE INDEXES
3.87 Some of the issues on bias have already been covered above. However, it is useful to bring these matters together to further consider some practical issues involving price indexes, especially considering a major inquiry into the issue was held in the United States in 1996. (footnote 41)
3.88 A price index may be described as biased if it produces estimates, which depart from the '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 situations. However, if unwise practices are adopted, second best could turn out to be a very poor second best.
3.89 The following types of bias, typically upwards, have been described by Diewert (1996):
(i) elementary index bias, which results from the use of inappropriate formulas for compiling index numbers at the elementary aggregate level
(ii) substitution bias, arising from using formulas at levels above the elementary aggregates, which do not allow for substitution in response to changes in relative
(iii) outlet substitution bias, 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
(v) new goods bias, which arises largely from the failure to include new goods when first introduced into the market.
3.90 While it is almost impossible to eliminate these sources of bias, certain measures can be taken to minimise them. Some measures to minimise each of the sources of bias are:
(i) The use of appropriate formulas in compiling elementary aggregate indexes, in particular use of the GM formula where appropriate or the RAP formula.
(ii) Use of a superlative type 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 42) Other options might be to use formulas that allow substitution or assumptions about substitution between commodity groupings to be fed in.
(iii) Closely monitoring and updating price samples to reflect changes in the outlets from which households purchase. For example, looking ahead, there is clearly a need to plan for the inclusion in consumer price indexes of purchases from outlets operating exclusively over the internet.
(iv) Greater use of the hedonic technique to adjust for quality change and to determine comparable items.
(v) The inclusion of new goods into the CPI as soon as possible. In the case of a fixed-weight index such as Laspeyres, there would also be a need to update the fixed weights to allow 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 (e.g. 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).
3.91 Price index theory provides prices statisticians with guidance as to the best practices and formulas 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 timeliness issues and data availability preclude this.
3.92 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, i.e. the index domain.
1. The literature on price indexes is quite extensive. The intention of this Chapter is to present a broad overview of the theory drawing heavily on documents that are in many cases overviews themselves as well as to present some views of the ABS. This Chapter does not include any reference to the Divisia index as this index has data requirements that restrict its application in practice. For a detailed consolidation 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. By convention, the base period value for an index series is made equal to 100. < Back
4. In this example, the price relative shows the change in price between two points in time. 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 (if any) between the prices in the two markets in the same period. < Back
5. For a more lengthy discussion see Allen (1975). < Back
6. To quote Fisher (1922, p. 45). < Back
”… any index number implies two dates, and the values by which we are to weight the price ratios for those two dates will be different at the two dates. Constant weighting (the same weight for the same item in different years) is, therefore, a mere makeshift, never theoretically correct, and not even practically admissible when values change widely”. < Back
7. 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
8. See Diewert (1993) for a discussion of symmetrical averages. < Back
9. For further discussion of forward and backward Laspeyres and Paasche price and quantity indexes, refer to Chapter 2 of Allen (1975). < Back
which is not the Laspeyres formula in equation (3.4). Note that the summation is over n price observations so that the in the numerator and denominator in the last expression do not cancel out. < Back
11. 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 then 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
unless the quantities (and ) are constant for each item or the individual prices show the same proportional change between periods (the trivial case of this being where there is no change in the individual prices between periods). < Back
The harmonic mean is equal to or lower than the geometric mean. Fisher (1922) also discusses use of the median and mode. < Back
15. The implicit weights applied by the three formulas 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
17. The assumption underlying the equal weight APR can be illustrated with a simple example. Suppose there is a price sample of two items, selling for $5 and $4 in period 0. Suppose the prices in period t are double those in period 0. Assume expenditure on each item is $20 in period 0, giving quantities of 4 and 5 respectively. Then the average quantity weighted price in period 0 is $4.4444 ((4x5+5x4)/(4+5)) and $8.8889 in period t ((4x10+5x8)/(4+5)), giving an index of 200.0. This is the same result as taking the unweighted arithmetic average of the two price relatives ((1/2*(10/5+8/4)*100). < Back
23. Paragraph 16.48: “.......a chain index should be used when the relative prices in the first and last periods are very different from each other and chaining involves linking through intervening periods in which the relative prices and quantities are intermediate between those in the first and last periods. Relative prices and quantities are described as intermediate when they may be approximated by some average of those in the first and last periods. This will happen when the opening prices and quantities are transformed into those of the final period by the gradual accumulation of successive changes which tend to be in the same direction. In this case, the individual links in the chain are strong as they involve comparisons between situations that are very similar to each other.” < Back
27. Fisher's (1922) factor antithesis is obtained by interchanging prices and quantities in the formula and then dividing this expression into the 'value ratio'. Diewert (1992, p. 222) notes that various researchers have objected to this test and does not count it in his list of 20 tests. < Back
29. Fisher (1922) summarised the poor performance of the APR approach in the following terms: “... the simple arithmetic average (APR) produces one of the very worst of index numbers. And if this book has no other effect than to lead to the total abandonment of the simple arithmetic type of index number, it will have served a useful purpose.” (pp. 29–30). < Back
30. The RAP and APR formulas both produce an index of 55. < Back
31. The aggregation property of the Laspeyres and Paasche indexes allows them to be broken down into points contributions which is very useful for analysing the relative significance of items in the index and their contribution to changes in the aggregate index. However, Diewert (2000) has produced an approach for similarly decomposing superlative index formulas. < Back
32. The characteristics approach to goods is the basis of the so-called 'household production theory'. The development of this theory is generally attributed to Lancaster (1966), Muth (1966) and Becker (1965). Bresnahan and Gordon (1998) also provide a good example in terms of household light, tracing the development from whale oil lamps through to the electric light bulb, pointing out how the additional inputs required on the part of households (such as trimming wicks etc.) were an important part in the production of light. < Back
33. Pollak (1983) identifies two characteristics approaches, that of Lancaster (1966) and Houthakker (1952). The Lancaster approach assumes that characteristics are additive across items (e.g. protein from meat can be added to protein from bread) whereas the Houthakker approach assumes characteristics are commodity specific. < Back
37. This point and the use of characteristics space in compiling consumer and producer price indexes is explained in Triplett (1983). < Back
38. Current thinking as presented in Koskimaki and Vartia (2001) for example is that hedonic equations should have log price as the dependent variable and should be estimated for each period. The use of weighted regressions is also supported by researchers such as Diewert. < Back
39. In the United States the treatment of mandated anti-pollution devices has changed over the years. As from January 1999, modifications to goods and services solely to meet air quality standards have not been regarded as quality improvements, a practice that had previously applied since 1971. See Fixler (1998) for more information. In Australia, such modifications have always been regarded as price increases, not quality improvements. < Back
40. Hausman (1994) estimated that the US CPI price index for cereals was substantially overstated by not taking into account the gains in consumer surplus arising from the introduction of new varieties of breakfast cereals. < Back