Topic 5.3

ECON 377/477

Topic 5.3

Index Numbers (continued)

Outline

• Transitivity in multilateral comparisons• Measuring TFP change using index numbers• Conclusions

3ECON377/477 Topic 5.3

Transitivity in multilateral comparisons• The problem of deriving price and quantity index

numbers over space at a given point of time arises when making output, input and productivity comparisons across a number of countries, regions, firms, plants, etc.

• We are typically interested in all pairs of comparisons, such as comparisons across all pairs of firms

• To derive an index, Iij, for a pair of firms (i,j) using a formula of our choice, we consider all pairs (i,j) with i,j = 1,2,...,I where I represents the total number of firms


Transitivity in multilateral comparisons• Then we have a matrix of comparisons between

all pairs of firms:

• This matrix represents all multilateral comparisons involving I firms that we would like to be internally consistent, i.e., to satisfy the property of transitivity


IIII

I

I

III

III

III

21

22221

11211

Transitivity in multilateral comparisons• Internal consistency requires that a direct

comparison between any two firms i and j, should be the same as a possible indirect comparison between i and j through a third firm k

• Thus, we require, for any i, j and k,• For example, if a matrix of index numbers shows

that firm i produces 10 per cent more than firm k and firm k produces 20 per cent more than firm j, then we should always find that firm i produces 32 per cent (1.1 1.2 = 1.32) more than firm j


kjikij III

Transitivity in multilateral comparisons

• None of the index number formulae satisfies the transitivity property, but the Fisher and Törnqvist indices do satisfy the time-reversal test: Ist = 1/Its

• A simple solution to obtain consistent multilateral comparisons between firms is to generate transitive indices from a set of non-transitive multilateral comparisons using the EKS method

• The EKS method is to derive multilateral Törnqvist indices that are transitive



• We start with Törnqvist indices for all pairs i,j• Then, for all firms i and j, we use the EKS method

to convert the Törnqvist indices into multilateral (CCD) indices by calculating:

• These indices satisfy four properties:1. , for i,j = 1,2,...,I, are transitive

2. The new indices deviate least from the original Törnqvist indices in a least-squares sense


II

k

Tkj

Tik

CCDst III

1

1

CCDijI

After Caves, Christensen and Diewert


3. If we focus on quantity indices based on the Törnqvist formula, the CCD index in log-change form can be shown to be equal to:

where and


,lnln

2

1lnln

2

1

lnln1

ln

11

1

M

mmmjmmj

M

mmmjmmi

I

k

Tkj

Tik

CCDij

qqrrqqrr

QQI

Q

I

kmkm I

r1

1

I

kmkm q

Iq

1ln

1ln


4. The formula on the previous slide has an intuitive interpretation: a comparison between two firms is obtained by first comparing each firm with the average firm and then comparing the differences in firm levels relative to the average firm

10ECON377/477 Topic 5.3


• We now examine the main logic behind the CCD index

• Although equation (4.33) in CROB is the most popular form for a multilateral Törnqvist index, it is desirable to use the form in equation (4.32) as the root of the multilateral index

• Equation (4.32) provides an approach that can be applied to binary indices without detailed price and quantity data

• Suppose we have a matrix of binary Fisher or Törnqvist price and quantity indices

11ECON377/477 Topic 5.3


• Even if data are not available, it is feasible to apply equation (4.32) to derive multilateral comparisons that are transitive

• But it is not obvious from equation (4.32) how this procedure can be applied if the preferred index formula is different from the Törnqvist index

• Suppose, we are working with Fisher index numbers for output index numbers between firms

12ECON377/477 Topic 5.3


• Let QijF represent the Fisher index for firm i with

firm j as base

• Obviously, the QijFs for i,j = 1,2,...,I do not satisfy

transitivity• But the EKS procedure in equation (4.32) can

be applied to yield consistent indices as:

• The resulting quantity index numbers, QstF-EKS,

satisfy the transitivity property

13ECON377/477 Topic 5.3

II

k

Fkj

Fik

EKSFij QQQ

1

1


• The condition of transitivity is an operational constraint preserving internal consistency

• The imposition of the transitivity condition implies that a quantity (or price) comparison between two firms, s and t, is influenced by price and quantity data for not just these two firms but all the other firms in the analysis

• Hence, the addition of an extra firm to the sample necessitates the recalculation of all indices

14ECON377/477 Topic 5.3

TFP change measurement using index numbers

• The focus of this section is to describe the computational methods used to derive an index of TFP, either over time or across firms or enterprises.

• A TFP index may be applied to:o binary comparisons, to compare two time periods or two cross-

sectional unitso a multilateral situation, where the TFP index is computed for

several cross-sectional units

15ECON377/477 Topic 5.3

TFP change measurement using index numbers: binary comparisons

• Consider first the Hicks-Moorsteen (HM) TFP index for two time periods (or enterprises), s and t

• The HM TFP index in its logarithmic form is

• We need to use one of the index number formulae to compute numerical values of this measure of TFP change

16ECON377/477 Topic 5.3

st

ststTFP

IndexInput

IndexOutputlnln


• The most obvious choice is to use the Fisher index or the Törnqvist index to compute the input and output indices from the observed price and quantity data on outputs and inputs

• Let qs and xs represent output and input

quantities, and rs and ss represent the revenue

shares of outputs and cost shares for inputs, respectively

• The Törnqvist index formula is commonly used to calculate output and input indices

17ECON377/477 Topic 5.3


• The Törnqvist TFP index is defined, in its logarithmic form, as

• where the first part of the right-hand side is the logarithmic form of the Törnqvist index applied to output data, and the second part is the input index, calculated using input quantities and the corresponding cost shares

18ECON377/477 Topic 5.3

ststst

ststTFP IndexInputlnIndexOutputln

IndexInput

IndexOutputlnln

N

nnsntntns

M

mmsmtitis xxssqqrr

11lnln

2

1lnln

2

1


• In many respects, the Fisher index is more intuitive than the Törnqvist index and, more importantly, it decomposes the value index exactly into price and quantity components

• The fact that it is in an additive format also makes the Fisher index more easily understood

• The Fisher TFP index is given by

19ECON377/477 Topic 5.3

)Fisher(IndexInput

)Fisher(IndexOutput

st

ststTFP


• The Fisher and Törnqvist index numbers provide reasonable approximations to the true output and input quantity indices in most practical applications involving time-series data

• They yield very similar numerical values for the TFP index

20ECON377/477 Topic 5.3


• The Malmquist TFP index can be approximated, under a set of conditions, by the ratio of an output quantity index to an input quantity index, where both indices are computed using the Törnqvist formula

• Assume the Malmquist output distance functions for periods s and t have a translog functional form with identical second-order terms, and assume technical and allocative efficiency of the firm in the two periods

21ECON377/477 Topic 5.3


• The geometric average of the two output-based Malmquist TFP productivity indices is given by

• where ; t and s are the

local returns-to-scale values at the observed input and output levels; and sn

s and snt are the

shares of n-th input in total input cost22ECON377/477 Topic 5.3

5.0,,,,,,,,, tsts

sotst

totstso mxmm xxqqxqqxxqq s

2/*

1indexinputTornqvist

indexoutputTornqvistnsN

n ns

nt

x

x

snstntn sss 11*


• Even when the exact nature of the output distance functions is unknown, we can define an exact measure of the geometric average of the Malmquist output-orientated productivity indices, based on the technologies of periods s and t

• If constant returns to scale prevail in both periods (t = s = 1), then

23ECON377/477 Topic 5.3

indexinputTornqvist

indexoutputTornqvistindexTFPMalmquist


• Under decreasing returns to scale, using duality results and a profit-maximisation assumption, the returns-to-scale parameters can be measured using the observed price and quantity data as:

• In the case of increasing returns to scale, observed costs and revenues cannot be used to compute returns–to-scale parameters

24ECON377/477 Topic 5.3

j

jj periodininputofvalue

periodinoutputofvalue


• If the input use has not changed over the two periods, then returns-to-scale issues do not arise in productivity change calculations

• There is an economic-theoretic justification for using the standard measure of TFP, defined as a ratio of Törnqvist indices of output and inputs

• Such a justification holds when the underlying technologies exhibit constant returns to scale

25ECON377/477 Topic 5.3


• A final comment is needed to serve as a warning

• Where our data suggest that either the prices are not market prices or the behaviour of the firms is not optimal, use of the index number approach to TFP measurement may not be measuring the Malmquist TFP index

• Hence, no real economic-theoretic interpretation can be accorded to the input and output quantity index numbers

26ECON377/477 Topic 5.3

TFP change measurement using index numbers: multilateral comparisons

• In the case of productivity comparisons across a number of firms, it is necessary to impose the transitivity condition on the index numbers used

• In such cases, the TFP indices in the previous section should be computed using transitive EKS-type index numbers for measuring differences in the levels of inputs and outputs across firms

27ECON377/477 Topic 5.3


• If the Hicks-Moorsteen approach is used, it is necessary first to generate transitive output and input quantity index numbers, based on the Fisher or Törnqvist indices, and then a ratio of the transitive indices be used to measure TFP change

• A similar approach needs to be adopted in the case of the Malmquist TFP index

• Routine application of the Törnqvist formulae to multilateral comparisons leads to the problem of transitivity

28ECON377/477 Topic 5.3


• The following index is derived by applying the EKS approach to obtain a transitive index (ln TFPst

*) that is a multilateral generalisation of the

Törnqvist index:

29ECON377/477 Topic 5.3

M

mmmsmms

M

mmmtmmt yyrrqqrr

121

121 lnlnlnln

K

kniksnns

N

nnkjtnnt xxssxxss

121

121 lnlnlnln


• In this index:• is the arithmetic mean of the output shares• is the arithmetic mean of the input shares• =

• =

• All averages are taken over I enterprises or time periods or a combination of both

30ECON377/477 Topic 5.3

mr

ns

myln

I

1imiI

1 yln

nxln ∑I

1=iniI

1 xln


• We can define alternative TFP formulae by using transitive output and input indices in the general multilateral TFP index given by:

• It is feasible to use any output and input index numbers

• A suitable choice is the multilateral generalisation of the Fisher index derived using the EKS procedure

31ECON377/477 Topic 5.3

indexinputTransitive

indexoutputTransitive* stTFP

TFP change measurement using index numbers: numerical examples

• Refer to the Bus Company example in CROB (pages 124-127) and the empirical application to Australian National Railways (CROB, pages 128-131)

• These examples demonstrate that we can easily obtain Törnqvist and Fisher TFP indices

• These calculations could be performed using TFPIP Version 1.0, SHAZAM or spreadsheet software

32ECON377/477 Topic 5.3

Conclusions• This topic covers the use of index number

methods to compute various TFP change indices• Using very limited data, we can compute various

measures of TFP change and assess the underlying assumptions that make it possible

• Proper care should be taken in interpreting the resulting TFP measures

• If the assumptions do not hold, we can still compute the TFP index as a measure of TFP change but the interpretation is not straightforward

33ECON377/477 Topic 5.3

Conclusions• In such cases, we need to find a way of computing

all the distances involved in defining either the Malmquist TFP index or component-based measures of TFP change

• In both of these cases, we need data on a reasonable number of firms or cross-sections for the two periods under consideration, to use the techniques such as DEA or SFA

• The index number, DEA and SFA methods for measuring TFP change are compared in a later topic

34ECON377/477 Topic 5.3

Conclusions

• We have focused only on multiplicative index numbers, mainly due to the fact that all the efficiency and productivity measures are multiplicative

• In the recent years, there have been some applications of additive index numbers

• Additive index numbers have very interesting applications to the economic approach to decomposing profit change

35ECON377/477 Topic 5.3

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