1

Efficiency and Productivity Measurement:Index Numbers

D.S. Prasada RaoSchool of Economics

The University of Queensland, Australia

2

Index number methods

• Index numbers are used in measuring changes in a set of related variables:– consumer prices; – stock market prices; – quantities produced; etc.

• Index numbers can also be used in comparing levels of a set of related variables across space or firms:– Agricultural output in two different countries;– Input indexes across two farms or firms;– Price levels in different countries; etc.

• In general we compute– Price Index Numbers– Quantity Index Numbers– Decompose Value ratios into Price and Quantity index numbers

3

Outline

• A simple TFP index example• Price index numbers• Quantity index numbers• Tornqvist TFP index• A small empirical example • Properties of index numbers• Additional issues

– Indirect index numbers

– Chaining index numbers

– Transitive index numbers

4

TFP growth• Productivity growth means getting more output from a

particular level of inputs• When we have just one input and output the TFP change

between period 1 and 2 is:

2 1 2 212

2 1 1 1

q q q xTFP

x x q x

• When we have more inputs and outputs we must aggregate using index numbers

• A basic property of any TFP index is that when q2=aq1 and x2=bx1, then TFP12=a/b

5

Index number formulae

• When we have more than one input or output we need to find an aggregation method

• Four most popular index number formulae are:– Laspeyres– Paasche– Fisher– Tornqvist

• We will look at price indices first - they are more familiar

6

Price Index Numbers

• Measure changes (or levels) in prices of a set of commodities.

• Let pmj and qmj represent prices and quantiies (m-th commodity; m = 1,2,...,M and j-th period or firm j = s, t).

• The index number poblem is to decompose value change into price and quantity change components.

IndexQuantity Index Price

stst

iisis

iitit

s

t QP

qp

qp

V

V

7

Laspeyres price index numbers

1

1

1

N

it is NL i it

st isNisi

is isi

p qp

Pp

p q

, where 1

N

is is is is isi

p q p q

• Price change index for N goods from period s to period t

• pit = price of i-th good in t-th period, qit = quantity

• Uses base-period (period s) quantity weights

• Share-weighted sum of individual price indices

• Often used in CPI calculations

8

Paasche price index numbers

1

1

1

N

it itP i

st Nis

itis ititii

p q

Pp

p qp

• Uses current-period (period t) quantity weights

• Share-weighted harmonic mean of individual price indices

• Paasche Laspeyres - when people respond to relative price changes by adjusting mix of goods purchased (in periods of inflation)

9

Fisher price index numbers

F L Pst st stP P P

• Fisher index is the geometric mean of the Laspeyres and Paasche index numbers

• Paasche Fisher Laspeyres - when consumers respond to relative price changes by adjusting mix of goods purchased (in periods of inflation)

• Paasche and Fisher more data intensive and costly because we need to obtain expenditure weights in each period

10

Tornqvist price index numbers

2

1

is itN

T itst

isi

pP

p

,

1

ln ln ln2

NT is itst it is

i

P p p

• Share-weighted geometric mean of individual price indices

• Uses average of value share from period t and period s

• Log form is commonly used in calculations - has an approximate percentage change interpretation

11

Quantity Index Numbers

There are three approaches to the compilation of quantity index numbers.

1. Simply use the same formulae as in the case of price index numbers – simply interchange prices and quantities.

2. Use the index number identity:

iisis

sti

itit

st

iisis

iitit

st

stst

qp

Pqp

P

qp

qp

P

VQ

/

/

12

Quantity Index Numbers

( tan , )

( )stvalue in period t at cons t prices in period s

Qvalue in period s at prices in period s

3. Compute quantity index directly• Malmquist approach• Using distance functions defined before• Economic theoretic approach

Comments• All the three approaches have some common elements• Fisher index can be derived using all the three approaches• Tornqvist index can be derived using the first and the last approaches•Fisher index is known as the “ideal” index.

13

Four quantity index numbers

Laspeyres = 1

1

N

is itL ist N

is isi

p q

Q

p q

, Paasche = 1

1

N

it itP ist N

it isi

p q

Q

p q

Fisher = F L Pst st stQ Q Q

Tornqvist = 2

1

is itN

T itst

isi

qQ

q

,

1

ln ln ln2

NT is itst it is

i

Q q q

To obtain the corresponding quantity index numbers we interchange prices and quantities:

14

Which index is best for use in TFP studies?

• Two methods are used to assess the suitability of index number formulae:

– economic theory or functional approach• Exact and superlative index numbers

– axiomatic or test approach• Index numbers that satisfy a number of desirable

properties

• Both approaches suggest that the Fisher and Tornqvist are best (Diewert)

• We outline these arguments later in this session

15

Tornqvist TFP index

ln ln

ln ln

stst

st

st st

Output IndexTFP Index

Input Index

Output Index Input Index

1 1

1 1ln ln ln ln

2 2

M K

is it it is js jt jt jsi j

r r q q s s x x

The Tornqvist has been the most popular TFP index

This approach is also know as the Hicks-Moorsteen Approach – defines productivity index simply as the ratio of output and input index numbers.

16

Example

• Recall our example in session 2• Two firms producing t-shirts using labour

and capital (machines)• Let us now assume that they face different

input prices

firm labour capital cost output

x1 w1 x2 w2 q

A 2 80 2 100 360 200

B 4 90 1 120 480 200

17

• In this example we compare productivity across 2 firms (instead of 2 periods)

• First we calculate the input cost shares

• Labour share for firm A

= (280)/(280+2100) = 0.44

• Labour share for firm B

= (490)/(490+1120) = 0.75

• Thus the capital shares are (1-0.44)=0.56 and (1-0.75)=0.25, respectively

18

Ln Output index = ln(200)-ln(200) = 0.0

Ln Input index = [0.5(0.44+0.75)(ln(2)-ln(4)) +0.5(0.56+0.25)(ln(2)-ln(1))]

= -0.13

ln TFP Index = 0.0-(-0.13) = 0.13

TFP Index = exp(0.13)=1.139

ie. firm A is 14% more productive than firm B

19

Properties of index numbers

• Used to evaluate index numbers– Economic theory– Axioms

• Both suggest Tornqvist and Fisher best for TFP calculations

20

Economic theory arguments

• Laspeyres and Paasche imply simplistic linear production structures

• Fisher is exact for quadratic - Tornqvist is exact for translog - both are 2nd-order flexible forms - thus “superlative” indices

• If we assume technical efficiency, allocative efficiency and CRS, then Tornqvist and Fisher indices can be interpreted as production function shift (technical change)

• Read more in text…

21

The Test or Axiomatic Approach

• Basically we postulate a number of desirable properties of index numbers in the form of axioms and tests and see which index number satisfy these properties

• Fisher (1922) provided a list of these tests.

22

• 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.

• Units invariance: The price (quantity) index must be independent of the units of measurement of quantities (prices).

23

• Time-reversal test: For two periods s and t: Ist=1/Its.

• 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 (PQ=V).

24

• Factor Test: The product of the price and quantity index numbers should be equal to the value index.

• Circularity test (transitivity): For any three periods, s, t and r, this test requires that: Ist=IsrIrt. That is, a direct comparison between s and t yields the same index as an indirect comparison through r (we provide an example later).

25

How many tests are satisfied?

• Diewert (1992) looks at 22 tests for TFP indices - Tornqvist fails factor reversal and transitivity - Fisher fails transitivity.

• Factor reversal is not greatly important - transitivity is important when making spatial comparisons - Tornqvist and Fisher indices often produce identical numbers (to 2 or 3 significant digits).

26

Chaining indices

• Example: 4 time periods

• Calculate indices between adjacent years (I12, I23, I34)=(1.03, 1.04, 0.98)

• Then form the chained index:

C1=1.00

C2=C1×I12=1.00×1.03=1.03

C3=C2×I23=1.03×1.05=1.08

C4=C3×I34=1.08×0.98=1.06

• Advantage is that weights change regularly

27

Transitivity

• Example: 3 firms

• Calculate all direct comparisons: (I12, I23, I13)=(1.10, 1.10, 1.15)

• These are not consistent (ie. transitive) because 1.10×1.10=1.211.15

• The EKS method is used to convert non-transitive indices into transitive indices:

N

1N

1rrtsr

transitivest III

28

• EKS is minimum sum of squares deviation from original index number series

• Original indices:

(I12, I23, I13)=(1.10, 1.10, 1.15)

• Transitive indices:

TI12=(I11×I12)1/3×(I12×I22)1/3×(I13×I32)1/3=1.0815

TI23=(I21×I13)1/3×(I22×I23)1/3×(I23×I33)1/3 =1.0815

TI13=(I11×I13)1/3×(I12×I23)1/3×(I13×I33)1/3 =1.1697

• Note that TI12×TI23 = 1.0815×1.0851

= 1.1697 = TI13

29

Transitive Tornqvist TFP index

ln ln

ln ln

stst

st

st st

Output IndexTFP Index

Input Index

Output Index Input Index

1 1

1 1ln ln ln ln

2 2

M K

is it it is js jt jt jsi j

r r q q s s x x

Recall that the binary TFP Index using Tornqvist formula is given by:

We note that this index is not transitive

30

Transitive Tornqvist TFP Index

12

1

12

1

12

1

12

1

ln ln ln

ln ln

ln ln

ln ln

Mtransitive

ist it it ii

M

iis is ii

K

jjt jt jj

K

jjs js jj

TFP r r q q

r r q q

s s x x

s s x x

If we apply the EKS method and generate transitive index numbers, we can show that

31

Transitive Tornqvist TFP Index

• This can be interpreted as an indirect comparison through the sample mean

• The transitive Tornqvist can be calculated directly using this formula

• The Fisher index has no equivalent formula - one must calculate the Fisher indices first and then apply EKS

• Need to recalculate all when one new observation added

32

Productivity comparisons using index numbers

We note the following important properties:

1. Productivity measures can be computed using data on just two firms (i.e., very limited data);

2. If the data refers to the same firm over two periods and if the firm is technically and allocatively efficient in both periods, then under the assumption of constant returns to scale the productivity measures provided correspond to theoretical measures of productivity growth (Malmquist productivity index – to be discussed next week).

3. Since the TFP index is based only on two observations for s and t, the index is not transitive.

• If we have several firms, then we need to make the measures transitive.

• Normally the EKS (Elteto-Koves-Szulc) method is used for this purpose.

33

Output Data for the Australian National Railways Example

Quantities Prices Mainland

Freight (1,000 NTKs)

Tasrail Freight (1,000

NTKs)

Passenger (1,000 PTKs)

Mainland Freight ($/NTK)

Tasrail Freight ($/NTK)

Passenger ($/PTK)

5235000 383000 2924 0.02 0.07 10 5331000 420000 3057 0.03 0.07 12 5356000 375000 2992 0.03 0.08 14 4967000 381000 2395 0.03 0.08 18 5511000 401000 2355 0.03 0.08 20 5867000 403000 2188 0.03 0.08 22 6679000 402000 2486 0.03 0.09 23 6445000 429000 2381 0.03 0.09 23 7192000 455000 2439 0.03 0.09 23 7618000 459000 2397 0.03 0.08 26 7699000 413000 2316 0.03 0.11 32 7420000 369000 1664 0.03 0.12 47

34

Quantities Prices Labour

(persons) Fuel (1,000

litres) Other Inputs ($1,000 )a

Labour ($/person)

Fuel ($/litre) Other Inputs (index)

10481 77380 119113 13097 0.18 0.45 10071 80148 112939 14730 0.26 0.50 9941 77105 108263 16692 0.28 0.56 9575 72129 110210 18651 0.37 0.62 9252 85868 109292 20166 0.37 0.66 8799 89706 97594 21307 0.39 0.70 8127 96312 93178 24990 0.41 0.75 7838 92519 80054 26412 0.42 0.81 7198 96435 77716 28572 0.43 0.87 6648 101327 74147 32617 0.39 0.94 6432 98874 80826 34565 0.43 1.00 5965 96016 73172 35646 0.46 1.04

Quantities Prices

Land, Building and Perway

($1,000 )a

Plant and Equipment ($1,000 )a

Rolling Stock

($1,000)a

Land, Building and

Perway (index)b

Plant and Equipment

(index)b

Rolling Stock

(index)b

1858038 94057 332307 10 50 50 2101035 93927 308491 20 80 80 2059365 89764 285626 30 120 120 2118357 93271 269265 30 100 100 2117625 91837 275134 70 140 140 2095680 90120 261495 70 160 160 2069494 89617 251588 50 90 90 2034867 88773 239736 70 120 120 2017626 89653 235834 80 200 200 1998345 98762 252514 80 240 240 2011753 100495 251850 80 190 190 2018802 107654 242662 130 200 200

Capital inputs

Other inputs

35

Year Output Input TFP

79/80 1.0000 1.0000 1.0000

80/81 1.0343 0.9782 1.0573

81/82 1.0188 0.9515 1.0707

82/83 0.9304 0.9345 0.9956

83/84 1.0014 0.9316 1.0748

84/85 1.0311 0.8950 1.1521

85/86 1.1543 0.8596 1.3428

86/87 1.1268 0.8191 1.3756

87/88 1.2293 0.7885 1.5590

88/89 1.2766 0.7690 1.6600

89/90 1.2607 0.7684 1.6407

90/91 1.1283 0.7376 1.5296

Output, Input and TFP index numbers

• All the index numbers reported here are calculated using the Tornqvist index number formula.

• All the indices here are reported for the base year 79/80.

•While there is a steady increase in output over the years, the input index shows a secular decline resulting in TFP growth over time.