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Efficiency and Productivity Measurement:Index Numbers

D.S. Prasada RaoSchool of Economics

The University of Queensland, Australia

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Index Numbers

• We use economic theoretic concepts to measure output and input price and quantity index numbers.

• Main assumption: Firms are technically and allocatively efficient.

• Cost minimising or revenue maximising behavious is assumed to result in the observed output and input quantities.

• Firms are assumed to be price takers and determine optimal quantities.

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Notation

• We consider the general case involving M outputs and K inputs. • Let s and t represent two time periods or firms; • pms and pmt represent output prices for the m-th commodity in

periods s and t, respectively; vectors ps and pt

• qms and qmt represent output quantities in periods s and t, respectively (m = 1,2,...,M); vectors qs and qt

• wms and wmt represent input prices in periods s and t, respectively; vectors ws and wt

• and xks and xkt represent input quantities in periods s and t, respectively (k = 1,2,...,K); vectors xs and xt

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Output Price Index Numbers

• Here we use the concept of a revenue function. This shows the maximum revenue a firm can derived with a given input vector, x, and output price vector, p, under the given technology

tt SqxqpMaxxpR ),(.),(

y1

y2

0

St(y,x)

slope = -p2/p1

revenue max. given pt

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Output Price Index Numbers

• The economic theoretic output-price index number is defined using period t technology as (Diewert, 1980, Fisher and Shell, 1972):

)(

)(),(

x,p

x,px,pp t

tss

t

tt

oR

RP

q1

q2

0

B

A

St(y,x) slope = -pt2/pt1

rev. max. given pt

rev. max. given ps

slope = -ps2/ps1

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Output Price Index Numbers

• We can define a similar price index number using period s technology:

ts t

(p ,x)(p ,p ,x)

(p ,x)

ss

o ss

RP

R

• We note that the output price index numbers defined depend upon the choice of the input vector, x.• The output price index is independent of choice of input vector, x, if the production technology is

output homothetic – production possibility curves for different x are all parallel shifts of the production possibility curve for the unit input vector.

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Output Price Index Numbers

• Suppose we choose the input vector of period s, xs. Then the output price index, using period s technology is given by:

ts t

(p ,x )(p ,p ,x )

(p ,x )

ss s

o s ss s

RP

R

• Observing optimal behaviour in period s, we have the numerator is equal to actual revenue: psqs.

• We can also define a similar index based on xt and period t technology. Then

1s t s

1

p ,p ,x

Mmt ms

smoM

ms msm

p qP

p q

1s t s

1

p ,p ,x

Mmt mt

tmoM

ms mtm

p qP

p q

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Output Price Index Numbers• We note that the two output price index numbers can depend upon the knowledge of the production technologies

in periods t and s. However, we have no knowledge of the actual technologies!

• Here we mention two powerful results due to Diewert.

• Result: If the revenue functions are quadratic in their arguments and they are different in the two periods, then under the assumption of technical and allocative efficiency, we have

[ Pot(ps, pt, xt) Po

s(ps, pt, xs) ]1/2

5.0

M

1mmtms

M

1mmtmt

M

1mmsms

M

1mmsmt

yp

yp

qp

qp

= Fisher Price Index.

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Output Price Index Numbers• Therefore the Fisher output price index number (which we can calculate from observed price and quantity data) is exact theoretical

index number when the revenue function is quadratic.• Since an unknown revenue function function can be approximated by a quadratic function, the Fisher price index is known as

superlative.• If the Revenue function is translog, then the Tornqvist price index number is exact for true index number. Further it is superlative since

translog approximates an unknown revenue function to the second order.

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Input Price Index Numbers

• We can derive similar results for input price index numbers by using the input cost function defined as:

tt SC )(min)( qx,w.xqw,

• The input price index numbers are then defined as ratio of two cost functions and is of the form:

ts t

s

(w ,q)(w ,w ,q) ;for ,

(w ,q)

jj

i j

CP j s t

C

• Then Tornqvist (Fisher) is exact for translog (quadratic) cost functions. It is also superlative.

K

1k

2

ss

kis

kt21ti

si

kskt

w

wPP

/,,,, ttssts qwwqww

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Malmquist Output Quantity Index Numbers

• The Malmquist output index numbers make use of the output distance functions defined given a technology.

)( and )( qx,qx, so

to dd

• Then the Malmquist output index defined using technologies periods t and s, respectively, are given by

s

tts

qx

qxxqq

,

,,,

to

tot

od

dQ

• We still need to specify the input vector x unless the production technology is output homothetic. Otherwise, we use xt and xs with periods t and s. This gives two indexes.

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Malmquist Output Quantity Index Numbers

• We note that we need to know the distance functions if we wish to compute the output quantity index numbers – this means we need to know the technology.

• We now have the following result, again due to Diewert (1976). If the distance function is of the translog (quadratic) form in both periods with the same second order terms and if the firms are technically efficient then the Tornqvist (Fisher) index is exact and superlative.

1/ 2 2

s t s s t t1

q ,q ,x q ,q ,x

w wit isNs t it

o oi is

qQ Q

q

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Comments

• These results illustrate how simple index numbers can be used in approximating economic theoretic index numbers.• The results are possible even when we do not know the actual parameters. All we need to know is that the form is translog or quadratic!• However, these results need the assumption that the firms are technically and allocatively efficient. • In the absence of these assumptions, the only properties we can attribute to Fisher and Tornqvist index numbers come from their

axiomatic properties.