Topic 5.1

ECON 377/477

Topic 5.2

Index Numbers (continued)

Outline

• Economic-theoretic approach• Simple numerical example

3ECON377/477 Topic 5.2

Economic-theoretic approach

• The economic-theoretic approach to index numbers postulates a functional relationship between observed prices and quantities for inputs and outputs

• In the case of productivity measurement, the microeconomic theory of the firm is especially relevant

• We consider the general case involving M outputs and N inputs



• Let:o s and t represent two time periods or firmso pms and pmt represent output prices for the m-th commodity in

periods s and t, respectivelyo qms and qmt represent output quantities in periods s and t,

respectively (m = 1,2,...,M)o wns and wnt represent input prices in periods s and t, respectively

o xns and xnt represent input quantities in periods s and t, respectively (n = 1,2,...,N)



• Further, let pt, ps, qt, qs, ws, wt, xt and xs represent vectors of non-negative real numbers of appropriate dimensions

• Let Ss and St represent the production technologies in periods s and t, respectively

• In deriving various price and quantity index numbers for inputs and outputs, we make use of revenue and cost functions, and input and output distance functions



• The economic-theoretic approach to index numbers assumes that the firms observed in periods s and t are both technically and allocatively efficient

• This means that observed output and input data are assumed to represent optimising behaviour involving revenue maximisation and cost minimisation, or, in some cases, constrained optimisation involving revenue maximisation with cost constraints


Economic-theoretic approach: output price indices

• For a given level of inputs, x, let the (maximum) revenue function be defined, for technology in period-t, as

• {pq: (x,q) is feasible in St}

• The point of tangency between the production possibility curve and the isorevenue line indicates the combination of the two outputs (q1 and q2) that maximise revenue, given the input vector, x, the output price vector, pt, and the technology, St


qxp max),( tR



y2

y1

St(y,x)

0

revenue maximisation given Pt

slope =-(p2/p1)


• The output price index based on period-t technology, is defined as:

• This index is the ratio of the maximum revenues possible with the two price vectors, ps and pt, using a fixed level of inputs, x, and period-t technology

• The revenue-maximising points for the price vectors, pt and ps are shown on the next slide

10ECON377/477 Topic 5.2

x,p

x,px,p,p

st

tt

tsto

R

RP


11ECON377/477 Topic 5.2

y2

y1

slope =-(ps2/ps1)

revenue maximisation given pt

St(y,x)

0

slope =-(pt2/pt1)

revenue maximisation given ps


• The output price index can also be defined using period-s technology leading to

• These two price indices depend on whether it is the period-t or period-s technology, and then on the input vector, x, at which the index is calculated

• Under what conditions are the indices independent of these two factors?

12ECON377/477 Topic 5.2

x,p

x,px,p,p

ss

ts

tsso

R

RP


• These indices are independent of x if and only if the technology is output-homothetic

• A production technology is output-homothetic if the output sets P(x) depend upon the output set for the unit input vector (input quantities equal to one for all inputs) and a real-valued function, G(x), of x

• In simple terms, the production possibility curves for different input vectors, x, are all parallel shifts of the production possibility curve for the unit-input vector

13ECON377/477 Topic 5.2


• In a similar vein, it can be shown that if the technology exhibits implicit output neutrality, the indices are independent of which period’s technology is used in the derivation

• The output price index numbers satisfy the properties of monotonicity, linear homogeneity, identity, proportionality, independence of units of measurement, transitivity for fixed t and x, and time-reversal properties

14ECON377/477 Topic 5.2


• Since xt and xs are the actual input levels used in periods t and s, we can define the indices using the actual input levels, leading to two natural output price index numbers:

15ECON377/477 Topic 5.2

tst

ttt

ttsto

R

RP

xp

xpxpp

,

,),,(

tss

tts

ttsso

R

RP

xp

xpxpp

,

,),,(


• We can get close to the above theoretically defined index numbers in equations in a number of ways

• Under the assumptions of allocative and technical efficiency, and regularity conditions on the production technologies, the two index numbers are, respectively, bounded from above and below by the Laspeyres and Paasche indices

• A reasonable approximation to the geometric mean of the two indices is provided by the Fisher output price index number

16ECON377/477 Topic 5.2


• An assumption that the revenue functions have the translog form is in line with the fact that the translog function is a flexible form and provides a second-order approximation to the unknown revenue function

• The translog revenue function is given by

17ECON377/477 Topic 5.2

K

k

M

mmkkmt

M

m

M

jjmmjt

K

k

M

m

K

k

K

jjkkjtmmtkktt

t

pxpp

xxpxR

1 121

1 121

1 1 1 121

0

lnlnlnln

lnlnlnln),(ln

px


• We can represent the revenue functions for periods s and t by translog functions, with second-order coefficients being equal for periods s and t (kjt = kjs, mjt = mjs,, kmt = kms)

• The geometric mean of the two natural output price index numbers is equal to the Törnqvist output price index

18ECON377/477 Topic 5.2


• The importance of this result is that, even though the theoretical indices require knowledge of the parameters of the revenue function, their geometric mean is equal to the Törnqvist index and the index can be computed from the observed price and quantity data

• Knowledge of the parameters of the translog functions is therefore unnecessary

19ECON377/477 Topic 5.2


• The Törnqvist index is considered to be exact for the translog revenue function

• Also, it is considered superlative since the translog function is a flexible functional form

• That is, it provides a second-order approximation to any arbitrary function

• The Fisher index is exact for a quadratic function and, hence, is also superlative

20ECON377/477 Topic 5.2

Economic-theoretic approach: input price indices

• We can measure input price index numbers by comparing costs of producing a vector of outputs, given different input price vectors

• We need to define a cost function, associated with a given production technology, for a given output level, q, namely:

• The cost function, Ct(w,q), is the minimum cost of producing q, given period-t technology, using the input price vector, w

21ECON377/477 Topic 5.2

tt SC qxwxqw x ,min),(


• We can use the cost function to define input price index numbers

• Given the input prices, wt and ws in periods t and s, we can define the input price index as the ratio of the minimum costs of producing a given output vector q using an arbitrarily selected production technology, Sj (j = s,t)

• The index is given by

22ECON377/477 Topic 5.2

)S|,(C

)S,(C=),,(P

sj

tj

tsji

qw

qwqww


• The cost elements in the equation on the previous slide can be seen from the diagram on the next slide

• The isoquant under technology, Ss, for a given output level, q, is represented by Isoq(q)-Ss

• The sets of input prices, ws and wt, are represented by isocost lines AA and BB

• Minimum-cost combinations of inputs producing output vector, q, for these two input price vectors are given by the points, x* and x**

23ECON377/477 Topic 5.2


24ECON377/477 Topic 5.2

x2

x1

Isoq(q)-Ss

0

x*

x**

B

A

AB

a

a

b

b


• These points are obtained by shifting lines AA and BB to aa and bb, respectively, where they are tangential to Isoq(y)-Ss

• The input price index number for this two input case is then given by the ratio of the costs at points, x* and x**

• It satisfies many useful properties, including monotonicity, linear homogeneity in input prices independence of units of measurement, proportionality and transitivity (for a fixed q and technology)

25ECON377/477 Topic 5.2


• To compute the input price index, we need to specify the technology and also the output level, q, at which we wish to compute the index

• First, the price index is independent of which period technology we use if and only if the technology exhibits implicit Hicks input neutrality

• Second, the index, Pi(wt, ws, q), for a given technology is independent of the output level, q, if and only if the technology exhibits input homotheticity

26ECON377/477 Topic 5.2

MRS between any inputs is independent of the technology


• If the technology does not satisfy these conditions, we can define many input price index numbers using alternative specifications for technology, S, and the output vector, q

• Two natural specifications are to use the period-s and period-t technologies, along with the output vectors, qs and qt

27ECON377/477 Topic 5.2


• They result in the following input price index numbers:

• Assuming allocative and technical efficiency, the observed input costs, wsxs and wtxt, are equal to Cs(ws,qs) and Ct(wt,qt), respectively

28ECON377/477 Topic 5.2

sss

sts

stssi

C

CP

qw

qwqww

,

,),,(

tst

ttt

ttsti

wC

CP

q

qwqww

,

,),,(


• The Laspeyres and Paasche indices provide upper and lower bounds to the economic-theoretic index numbers in the equations on the previous slide

• The geometric mean of these indices can be approximated by the Fisher price index numbers for input prices

• Assume the technologies in periods t and s are represented by the translog cost function, with the additional assumption that the second-order coefficients are identical in these periods

29ECON377/477 Topic 5.2


• Under the assumption of technical and allocative efficiency, the geometric mean of the two input price index numbers in the above equations is given by the Törnqvist price index number applied to input prices and quantities

• That is,

where snt and sns are the input expenditure shares of n-th input in periods t and s, respectively

30ECON377/477 Topic 5.2

N

n

ss

ns

nttts

tists

si

nsnt

w

wPP

1

22/1,,,, qwwqww

Törnquist index


• These results imply that the Fisher and Törnqvist indices can be applied to measure changes in input prices and at the same time have a proper economic-theoretic framework to support their use

• They also illustrate that, under certain assumptions, it is not necessary to know the numerical values of the parameters of the cost or revenue function or the underlying production technology; it is sufficient to have the observed input price and quantity data to measure changes in input prices

31ECON377/477 Topic 5.2


• The Törnqvist input price index is exact for the geometric mean of the two theoretical indices when the underlying cost function is translog, and hence can also be considered superlative

• The Fisher input price index is exact for a quadratic cost function

32ECON377/477 Topic 5.2

Economic-theoretic approach: output quantity indices

• Unlike the case of price index numbers, three strategies can be followed in deriving theoretically sound quantity index numbers

• We focus on only one, the Malmquist index, which is defined using the distance function

• The Malmquist output index, based on technology in period-t, is defined as:

for an arbitrarily selected input vector, x

33ECON377/477 Topic 5.2

sto

tto

tsto

d

dQ

qx

qxxqq

,

,,,


• A similar Malmquist index can be defined using period-s technology

• The index defined on the previous slide is independent of the technology involved if and only if the technology exhibits Hicks output neutrality

• The quantity index is independent of the input level, x, if and only if the technology is output homothetic

34ECON377/477 Topic 5.2

MRT between any inputs is independent of the technology


• Even in the cases where these assumptions hold, we still need to know the functional form of the distance function as well as numerical values of all the parameters involved

• The index number approach attempts to bypass this problem by providing approximations to the index when we are unsure of the functional form, or do not have adequate information to estimate its parameters even when we know the functional form

35ECON377/477 Topic 5.2


• Consider output quantity indices based on technology in periods s and t, along with the inputs used in these periods

• We have two possible measures of output change, given by Qo

s(qt, qs, xs) and Qot(qt, qs, xt)

• There are many standard results that relate these indices to the standard Laspeyres and Paasche quantity index numbers

• A result of particular interest is that the Fisher index provides an approximation to the geometric average of these two indices

36ECON377/477 Topic 5.2


• The following result establishes the economic-theoretic properties of the Törnqvist output index and shows why the index is considered to be an exact and a superlative index

• If the distance functions for periods s and t are both represented by translog functions with identical second-order parameters, a geometric average of the Malmquist output indices, based on technologies of periods s and t, with corresponding input vectors xs and xt, is equivalent to the Törnqvist output quantity index

37ECON377/477 Topic 5.2


• That is, the Törnqvist output index equals:

• This result implies that the Törnqvist index is exact for the geometric mean of the period-t and period-s theoretical output index numbers when the technology is represented by a translog output distance function

• Since the translog functional form is flexible, the Törnqvist index is also considered to be superlative

38ECON377/477 Topic 5.2

2/1,,,, tts

tosts

so QQ xqqxqq

Economic-theoretic approach: input quantity indices

• We describe the input quantity index number, derived using the Malmquist distance measure

• Using the concept of the input distance function, we can now define the input quantity index along the same lines as the output index

• We can compare the levels of input vectors xt and xs, by measuring their respective distances from a given output vector for a given state of the production technology

39ECON377/477 Topic 5.2


• The input quantity index based on the Malmquist input distance function is defined for input vectors, xs and xt, with base period-s and using period-t technology:

• It satisfies monotonocity and linear homogeneity in the input vector xt, is invariant to scalar multiplication of the input vectors, and is independent of units of measurement

40ECON377/477 Topic 5.2

qx

qxqxx

,

,,,

sti

tti

tsti

d

dQ


• Following the same approach as in the previous sections, we note that the input quantity index depends upon the output level, q, we choose, as well as the production technology

• If we use period-s technology in defining the input distance functions, we get the following index:

• This index and the index on the previous slide are independent of the reference output vector, q, if and only if the technology exhibits input homotheticity

41ECON377/477 Topic 5.2

qx

qxqxx

,

,,,

ssi

tsi

tssi

d

dQ


• Our main purpose is to relate this Malmquist input quantity index number to the input index number derived using some of the formulae above

• The input index defined using base period-s technology is bounded from above by the Laspeyres quantity index

• The index defined on current period-t technology, is bounded from below by the Paasche quantity index

42ECON377/477 Topic 5.2


• Therefore, the Fisher input quantity index provides an approximation to the geometric mean of the indices, Qi

s(xt,xs,q) and Qit(xt,xs,q)

• If we assume a quadratic function, then the Fisher input quantity index can be shown to be equal to the geometric average of the two indices

43ECON377/477 Topic 5.2


• If the distance functions are of the translog form, the distance functions in periods t and s have identical second-order coefficients satisfying the usual restrictions on the parameters of the translog form, and the assumptions of allocative and technical efficiency holds, then the Törnquist input quantity index equals

44ECON377/477 Topic 5.2

2/1,,,, qxxqxx ts

tots

si QQ

2/)(

1

nsntN

n ns

nt

x

x


• In this equation, snt and sns are input cost-shares in periods t and s, respectively

• This result shows that the Törnqvist index is exact and superlative for the geometric mean of Malmquist input index numbers based on the technologies of periods t and s

• Refer to CROB for a simple numerical example involving index numbers on pages 113-115

45ECON377/477 Topic 5.2


• It is easy to see how the index number literature is closely connected with productivity measurement

• For example, the Hicks-Moorsteen TFP index, defined as a ratio of the output and input quantity index numbers, can be made operational using the results in this section

• Similarly, if we wish to use profitability ratios and adjust them for price level differences, we need to make use of appropriate output and input price index numbers

46ECON377/477 Topic 5.2

Topic 5.1

Business

output price vector

output price indicesy2slope

output price indicesfor

output price indicesin

output price indiceswe

output quantities

output prices

output price indicessince