topic 5.1

ECON 377/477

Topic 5.1

Index Numbers

Outline

• Introduction• Conceptual framework and notation• Formulae for price index numbers• Quantity index numbers• Properties of index numbers

3ECON377/477 Topic 5.1

Introduction

• Index numbers are the most commonly used instruments to measure changes in levels of various economic variables

• Measuring productivity changes necessarily involves measuring changes in the levels of output and the associated changes in the input usage

• Such changes are easy to measure in the case of a single input and a single output, but are more difficult when multi-input and multi-output cases are considered


Introduction

• Three principal areas in productivity measurement where index numbers play a major role are:o measuring changes in TFPo generating data for use in the application of DEA or in the

estimation of the stochastic frontierso handling panel data sets, with price and quantity data over time

and space to meet some basic consistency requirements such as ‘transitivity’ and ‘base invariance’


Introduction

• The principal aim of the first part of this topic is to familiarise you with the various index numbers

• The main indices we shall deal with are the Laspeyres, Paasche, Fisher and Törnqvist index numbers

• We then focus on the construction of price and quantity index numbers

• Quantity index number formulae are applied to input and output data that lead to quantity index numbers that are, in turn, used in defining the TFP index in the next part


Conceptual framework and notation

• An index number is defined as a real number that measures changes in a set of related variables

• Conceptually, index numbers may be used for comparisons over time or space or both

• Price index numbers may refer to consumer prices, input and output prices, export and import prices, etc.

• Quantity index numbers may measure changes in quantities of outputs produced or inputs used by a firm or industry over time or across firms



• Let pmj and qmj represent the price and quantity, respectively, of the m-th commodity in the M commodities being considered (m = 1,2,...,M) in the j-th period (j = s, t)

• Without loss of generality, s and t may refer to two firms instead of time periods, and quantities may refer to either input or output quantities

• All index numbers measure changes in the levels of a set of variables from a reference period



• The reference period is denoted at the ‘base period’

• The period for which the index is calculated is referred to as the ‘current period’

• Let Ist represent a general index number for current period, t, with s as the base period

• Similarly, let Vst, Pst and Qst represent value, price and quantity index numbers, respectively



• The value change from period s to t is the ratio of the value of commodities in periods s and t, valued at respective prices

• Thus,

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• The index, Vst, measures the change in the value of the basket of quantities of M commodities from period s to period t

• It is the result of changes in the two components, prices and quantities

• While Vst is easy to measure, it is more difficult to disentangle the effects of price and quantity changes

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• If we are operating in a single-commodity world, decomposition is simple to achieve

• We have

• The ratios, pt/ps and qt/qs, measure the relative price and quantity changes and there is no index number problem

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• In general, when we have M 2 commodities, we have a problem of aggregation

• The price relative, pmt/pms, measures the change in the price level of the m-th commodity, and the quantity relative, qmt/qms, measures the change in the quantity level of the m-th commodity (m = 1,2,...,M)

• The problem is one of combining the M different measures of price (quantity) changes, into a single real number, called a price (quantity) index

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Formulae for price index numbers

• The Laspeyres price index uses the base-period quantities as weights to define the index:

• The Paasche price index uses the current-period weights to define the index:

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• The value share of m-th commodity in the base period in the Laspeyres price index is:

• The value shares reflect the relative importance of each commodity in the set involved

• They refer to the base period

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• There are two alternative interpretations of the Laspeyres index:1. It is the ratio of two value aggregates resulting from the

valuation of the base-period quantities at current- and base-period prices

2. It is a value-share weighted average of the M price relatives

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• The Paasche index number, based on current-period quantities, is a natural alternative to the use of base-period quantities in the Laspeyres index

• The first part shows that the Paasche index is the ratio of the two value aggregates resulting from the valuation of period-t quantities at the prices prevailing in periods t and s

• The last part suggests it is a weighted harmonic mean of price relatives, with current-period value shares as weights

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• The Laspeyres and Paasche formulae represent two extremes, one formula placing emphasis on base-period quantities and the other on current-period quantities

• They coincide if the price relatives do not exhibit any variation, that is, if pmt/pms = c, but diverge when price relatives exhibit a large variation

• The extent of divergence also depends on quantity relatives and the statistical correlation between price and quantity relatives

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• The Fisher index is:

• It possesses a number of desirable statistical and economic theoretic properties

• It is also known as the Fisher ideal index

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• The Törnqvist price index is a weighted geometric average of the price relatives, with weights that are a simple average of the value shares in periods s and t:

• It is usually presented and applied in its log-change form

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• In log-change form, the Törnqvist index is a weighted average of logarithmic price changes

• The log-change in the price of the m-th commodity given by

represents the percentage change in the price of the m-th commodity

• It provides an indication of the overall growth rate in prices (inflation rate)

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Quantity index numbers: direct approach

• Two approaches can be used in measuring quantity changes

• The first approach is a direct approach, using a formula that measures overall quantity changes from individual commodity-specific quantity changes, measured by qmt/qms

• The Laspeyres, Paasche, Fisher and Törnqvist indices can be applied directly to quantity relatives

• These formulae may be defined using price index numbers by simply interchanging prices and quantities

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• The formulae described above yield:

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• The Törnqvist quantity index, in its multiplicative and additive (log-change) forms, is:

• The log-change form of the Törnqvist index is generally used for computational purposes

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Quantity index numbers: indirect approach

• The second approach is an indirect approach in which price and quantity changes comprise the value change over periods s and t

• If price changes are measured directly using the formulae in the previous section, quantity changes can be indirectly obtained after deflating the value change for the price change

• This approach is commonly used for purposes of quantity comparisons over time on the premise that Vst = Pst Qst

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• Since Vst is defined from data directly as the ratio of values in periods t and s, Qst can be obtained as a function of Pst

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• The numerator in this expression corresponds to the constant price series

• This approach states that quantity indices can be obtained from ratios of values, aggregated after removing the effect of price movements over the period under consideration

• Some features and applications of indirect quantity comparisons are discussed below

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Constant price aggregates and quantities

• Time series and cross-section data on price aggregates are often used as data series to estimate least-squares econometric production models and stochastic frontiers, and in DEA calculations, where it is necessary to reduce the dimensions of the output and input vectors

• This means that, even if index number methods are not used to measure productivity changes directly, they are regularly used to create intermediate data series

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Self-duality of formulae

• The Laspeyres price index and the Paasche quantity index form a pair that exactly decompose the value change

• In that sense, the Paasche quantity index can be considered as the dual of the Laspeyres price index

• The Paasche price index and Laspeyres quantity index decompose the value index, and therefore are dual to each other

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Self-duality of formulae

• The Fisher index for prices and the Fisher index for quantities form a dual pair

• This implies that the direct quantity index obtained using the Fisher formula is identical to the indirect quantity index derived by deflating the value change by the Fisher price index

• This property is sometimes referred to as the factor reversal test (see below)

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Direct versus indirect quantity comparisons

• The choice between direct and indirect measurements of quantity changes depends on:

o the type of data availableo the variability in the price and quantity relatives (see next slide)o the theoretical framework used in the quantity comparisons (see

below)

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• Since an index number is a scalar-valued representation of changes that are observed for different commodities, the reliability of such a representation depends upon the variabilities that are observed in the price and quantity changes for the different commodities

• Uniform price changes over different commodities mean the price index provides a reliable measure of the price changes

• A similar conclusion can be drawn for quantity index numbers

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• The relative variability in the price and quantity ratios, pmt/pms and qmt/qms (m = 1, 2, ...,M) provides a useful clue as to which index is more reliable

• If the price ratios exhibit less variability than the quantity ratios, an indirect quantity index is advocated

• If quantity relatives show less variability, a direct quantity index is preferred

• Price changes over time tend to be more uniform across commodities than quantity changes

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• If price (quantity) ratios exhibit little variability, most index number formulae lead to very similar measures of price (quantity) change

• There is more concurrence of results arising out of different formulae, and, therefore, the choice of a formula has less impact on the measure of price (quantity) change derived

• Direct quantity comparisons may offer theoretically more meaningful indices because they use the constraints underlying the production technologies

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• Direct input and output quantity indices based on the Törnqvist index formula are theoretically superior under certain conditions

• The Fisher index performs well with respect to both theoretical and test properties

• Also, it is self-dual in that it satisfies the factor-reversal test (see below)

• Being defined using the Laspeyres and Paasche indices, the Fisher index is easier to understand and is capable of handling zero quantities in the data set

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• But, under the assumption of behaviour under revenue constraints, productivity indices are best computed using indirect quantity measures

• Given these results, from a theoretical point of view the choice between direct and indirect quantity (input or output) comparisons should be based on the assumptions about the behaviour of the decision-making unit

• A decision needs to be made on pragmatic considerations as well as on pure analytical grounds

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• Tests can be used to choose a formula to construct price and quantity index numbers

• An alternative (yet closely related) framework is to state a number of properties, in the form of axioms, and find an index number that satisfies a given set of axioms

• This approach is known as the axiomatic approach to index numbers

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Properties of index numbers

• Let Pst and Qst represent price and quantity index numbers, which are both real-valued functions of the prices and quantities (M commodities) observed in periods s and t, denoted by M-dimensional column vectors, ps, pt, qs and qt

• Some of the basic and commonly used axioms are listed on the next two slides

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• Positivity: The index (price or quantity) should be everywhere positive

• Continuity: The index is a continuous function of the prices and quantities

• Proportionality: If all prices (quantities) increase by the same proportion then Pst (Qst) should increase by that proportion

• Commensurability or dimensional invariance: The price (quantity) index must be independent of the units of measurement of quantities (prices)

• Time-reversal test: For two periods s and t:

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• Mean-value test: The price (or quantity) index must lie between the respective minimum and maximum changes at the commodity level

• Factor-reversal test: A formula is said to satisfy this test if the same formula is used for direct price and quantity indices and the product of the resulting indices is equal to the value ratio

• Circularity test (transitivity): For any three periods, s, t and r, this test requires that:

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rtsrst PPP

• The following two results describe the properties of the Fisher and Törnqvist indices, and thus offer justification for the common use of these indices in the context of productivity measurement:1. The Fisher index satisfies all the properties listed above, with the

exception of the circularity test (transitivity)2. The Törnqvist index satisfies all the tests listed above with the

exception of the factor-reversal and circularity tests

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• In the case of temporal comparisons for productivity measurement, we are usually interested in comparing each year with the previous year

• Annual changes in productivity are then combined to measure changes over a given period

• The index constructed using this procedure is known as a chain index

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• Let I(t, t+1) define an index of interest for period t+1 with t as the base period

• The index can be applied to a time series with t = 0, 1, 2, ..., T

• A comparison between period t and a fixed base period, 0, can be made using the following chained index of comparisons for consecutive periods:

• I(0, t) = I(0, 1)I(1, 2)...I(t-1, t)

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• As an alternative to the chain-base index, it is possible to compare period 0 with period t using any one of the formulae described earlier

• This index is known as the fixed-base index• From a practical angle, especially with respect to

productivity measurement, a chain index is more suitable than a fixed-base index

• It involves comparing consecutive periods and is measuring smaller changes, so some of the approximations involved in deriving theoretically meaningful indices are more likely to hold

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• Another advantage is that comparisons over consecutive periods mean that the Laspeyres-Paasche spread is likely to be small, indicating that most index number formulae result in indices that are very similar in magnitude

• A drawback is that the weights used in a chain index need to be revised every year

• Also, the use of a chained index does not result in transitive index numbers

• Although transitivity is not essential for temporal comparisons, it is needed for multilateral comparisons (more on this in Topic 5.2)

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• The foregoing discussion indicates that the choice of formula is essentially between the Fisher and Törnqvist indices

• Both possess important properties and satisfy a number of axioms

• But it is likely that Laspeyres or Paasche indices are used in published aggregated data

• If the indices are being computed for periods that are not far apart, differences in the numerical values of the Fisher and the Törnqvist indices are likely to be minimal

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Which index number to choose?

• Further, both of these indices also have important theoretical properties

• In practice, the Törnqvist index seems to be preferred

• But use of the Fisher index is recommended because of its additional self-dual property and its ability to accommodate zeros in the data

47ECON377/477 Topic 5.1

Which index number to choose?

topic 5.1

Education

quantity index

trnqvist index

index number

chained index

aggregates

quantity ratios

quantity relatives

trnqvist indices