XVI. PRICE AND VOLUME MEASURES
C. Intertemporal index numbers of prices and volumes
1. Introduction
| 16.14. | A price index is an average of the proportionate changes in the prices of a specified set of goods and services between two periods of time. Similarly, a volume index is an average of proportionate changes in the quantities of a specified set of goods and services. As already emphasized, the price and quantity changes refer to individual goods or services as distinct from groups of similar products. Different qualities of the same kind of product must be treated as separate goods or services in this context. |
| 16.15. | In line with normal conventions, the period that serves as the reference point will be designated as period o and the period which is compared with it designated as period t. The two periods may be consecutive or be separated by intervening periods. The ratio of the price, or quantity, of a specific product in period t to the price, or quantity, of the same product in period o, is described as a price relative, or quantity relative: namely,  or  . Price and quantity relatives are pure numbers that are independent of units in which the quantities are measured and the prices are quoted. Most index numbers can be expressed as, or derived from, weighted averages of these price or quantity relatives, the various formulas differing from each other mainly in the weights which they attach to the individual price or quantity relatives and the particular form of averages used - arithmetic, geometric, harmonic etc. |
2. Laspeyres and Paasche indices
| 16.16. | The two most commonly used indices are the Laspeyres and Paasche indices. Both may be defined as weighted averages of price or quantity relatives, the weights being the values of the individual goods or services in one or other of the two periods being compared.
Let  : the value of the ith product in period j
The Laspeyres price index  is defined as a weighted arithmetic average of the price relatives using the values of the earlier period o as weights:
where the summation takes place over different goods and services. The Laspeyres volume index  is a similar weighted average of the quantity relatives, that is:
The period that provides the weights for an index is described as the "base" period. It usually (but not always) coincides with the reference period to which the comparisons relate. As the summation always takes place over the same set of goods and services it is possible to dispense with the subscript i in expressions such as (1) and (2). As  is equal to  by definition, it is also possible to substitute for in (1) and (2) to obtain:
and
Expressions (1) and (3) are algebraically identical with each other, as are (2) and (4). |
| 16.17. | Paasche price and volume indices are defined reciprocally to Laspeyres indices by using the values of the later period t as weights and a harmonic average of the relatives instead of an arithmetic average. A Paasche index  is defined as follows:
and
When a time series of Paasche indices is compiled, the weights therefore vary from one period to the next. |
| 16.18. | The Paasche index may also be interpreted as the reciprocal of a "backward looking" Laspeyres: that is, the reciprocal of a "Laspeyres" index for period o that uses period t as the base period. Because of this reciprocity between Laspeyres and Paasche indices there are important symmetries between them. In particular, the product of a Laspeyres price (volume) index and the corresponding Paasche volume (price) index is identical with the proportionate change in the total value of the flow of goods or services in question, that is:
and
This relationship can be exploited whenever the total values for both periods are known. When both  and  are known, one or the other out of a complementary pair of Laspeyres and Paasche indices can be derived indirectly. For example,
and
Thus, the Laspeyres volume index can be derived indirectly by dividing the proportionate change in values by the Paasche price index, a procedure described as price deflation. As it is usually easier, and less costly, to calculate direct price than direct volume indices, it is common to obtain volume measures indirectly both in national accounts and economic statistics generally. |
Values at constant prices
| 16.19. | Consider a time series of Laspeyres volume indices, namely:
Multiplying through the series by the common denominator  yields the constant price series:
The relative movements from period to period for this series are identical with those of the associated Laspeyres volume indices given by (11), the two series differing only by a scalar. Constant price series of the kind illustrated by (12) are easy to understand and used extensively in national accounts. The term volume "measure" is used to cover both time series of monetary values at constant prices and the corresponding series of volume index numbers. |
3. The relationship between Laspeyres and Paasche indices
| 16.20. | Before considering other possible formulas, it is necessary to establish the behaviour of Laspeyres and Paasche indices vis-a-vis each other. In general, a Laspeyres index tends to register a larger increase over time than a Paasche index:
that is, in general
It can be shown that relationship (13) 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 the passage of time. |
The economic theoretic approach to index numbers
| 16.21. | From the point of view of economic theory, the observed quantities may be assumed to be functions of the prices, as specified in some utility or production function. Assuming that a consumer's expenditures are related to an underlying utility function, a cost of living index may then be defined as the ratio of the minimum expenditures required to enable a consumer to attain the same level of utility under the two sets of prices. It is equal to the amount by which the money income of a consumer needs to be changed in order to leave the consumer as well off as before the price changes occurred. This amount depends not only on the consumer's preferences, or indifference map, but also on the initial level of income and expenditures of the consumer. The value of the theoretic index is not the same for different consumers with a different set of preferences, nor even for the same consumer starting from different income levels. |
| 16.22. | The following conclusions may be drawn about the relationships between Laspeyres, Paasche and the underlying theoretic cost of living indices:
(a) The Laspeyres index provides an upper bound to the theoretic index. Suppose the consumer's income were to be increased by the same proportion as the Laspeyres index. It follows that the consumer must be able to purchase the same quantities as in the base period and must therefore be at least as well off as before. However, by substituting products that have become relatively less expensive for ones that have become relatively more expensive the consumer should be able to obtain a higher level of utility. This substitution will set up a negative correlation between the price and quantity relatives. As the consumer can thereby attain a higher level of utility, the Laspeyres price index must exceed the theoretic index;
(b) Similarly, the Paasche index can be shown to provide a lower bound to the theoretic index based on the later period. The reasoning behind this runs along the same lines as that just used for the Laspeyres index. |
| 16.23. | While these conclusions show that the Laspeyres and Paasche indices provide upper and lower bounds to the corresponding theoretic indices, it must be noted that two theoretic indices are involved and not one. The theoretic index depends upon the situation in the base period and income level which are not the same in the two periods. However, if it can be assumed that the preferences of the consumer are homothetic - i.e., if each indifference curve is a uniform enlargement, or contraction, of each other - the two theoretic indices coincide. In this case, the Laspeyres and Paasche indices provide upper and lower bounds to the same underlying theoretic index. This is still not sufficient to identify the latter. In order to do this it is necessary to go one step further by specifying the precise functional form of the indifference curves. As early as 1925 it was proved that if the utility function can be represented by a homogeneous quadratic function (which is homothetic) Fisher's Ideal Index (F) is equal to the underlying theoretic index. Although a special case, this result has had a considerable influence on attitudes towards index numbers. |
| 16.24. | Fisher's Ideal Index (F) is defined as the geometric mean of the Laspeyres and Paasche indices, that is:
and
Fisher described this index as "ideal" because it satisfies various tests that he considered important, such as the "time reversal" and "factor reversal" tests. The time reversal test requires that the index for t based on o should be the reciprocal of that for o based on t. The factor reversal test requires that the product of the price index and the volume index should be equal to the proportionate change in the current values,  . Laspeyres and Paasche indices on their own do not pass either of these tests. On the contrary, assuming the relationships given in (13) hold, it follows from (7), (8) and (13) that:
while
so that neither index passes the factor reversal test. |
| 16.25. | The Fisher index therefore has a number of attractions that have led it to be extensively used in general economic statistics. However, it is worth noting that it also has some disadvantages, some practical, some conceptual:
(a) The Fisher index is demanding in its data requirements as both the Laspeyres and the Paasche indices have to be calculated, thereby not only increasing costs but also possibly leading to delays in calculation and publication;
(b) The Fisher index is not so easy to understand as Laspeyres or Paasche indices which can be interpreted simply as measuring the change in the value of a specified basket of goods and services;
(c) The particular preference function for which Fisher provides the exact measure of the underlying theoretic index is only a special case;
(d) The Fisher index is not additively consistent. As explained below, it cannot be used to create an additive set of "constant price" data. |
| 16.26. | Although the underlying theoretic index may be unknown, the Fisher index seems likely to provide a much closer approximation to it than either the Laspeyres or Paasche indices on their own. However, the Fisher index is not alone in this respect. It has been shown that any symmetric mean of the Laspeyres and Paasche indices is likely to approximate the theoretic index quite closely, the Fisher index being only one example of such a symmetric mean. |
| 16.27. | The notion of symmetry can be extended to describe any index that attaches equal weight or importance to the two situations being compared. Another important example of a symmetric index is the Tornqvist, or translog, index (T) the volume version of which is defined as follows:
where  and  denote the share of the total values  accounted for by each product in the two periods. The Tornqvist index is a weighted geometric average of the quantity relatives using arithmetic averages of the value shares in the two periods as weights. The Tornqvist price index is obtained by replacing the quantity relatives  in (18) by price relatives  . |
| 16.28. | The Tornqvist index is commonly used to measure volumes changes for purposes of productivity measurement. When the production possibilities being analysed can be represented by a homogeneous translog production function, it can be shown that the Tornqvist index provides an exact measure of the underlying theoretic volume index. Thus, the Tornqvist index, like the Fisher index, provides an exact measure under certain very specific circumstances. Both indices are examples of "superlative indices": i.e., indices that provide exact measures for some underlying functional form that is "flexible", the homogeneous quadratic and homogeneous translog functions being particular examples of such flexible functional forms. |
| 16.29. | The Tornqvist index, like the Fisher, utilizes information on the values in both periods for weighting purposes and attaches equal importance to the values in both periods. For this reason, its value may be expected to be close to that of an average of the Laspeyres and Paasche indices, such as the Fisher, especially if the index number spread between them is not very large. The difference between the numerical values of the Tornqvist and Fisher indices is likely to be small compared with the difference between either of them and the Laspeyres or Paasche indices. |
| 16.30. | 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 symmetric index - whether Fisher, Tornqvist 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. |
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