Efficiency and Productivity Measurement:
Index Numbers
D.S. Prasada Rao
School of Economics
The University of Queensland, Australia
1
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
2
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
3
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:
T F P12 
q2
q1
x2
x1

q2
x2
q1
x1
• 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
4
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
5
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.
Vt
Vs
p

it q it
 Pst  Q st  Price Index  Quantity Index
i
p
is q is
i
6
Laspeyres price index numbers
N

L
Pst 
p it q is
i 1
N

N

p is q is

i 1
p it
p is
N
  is , where  is  p is q is

p is q is
i 1
i 1
• 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
7
Paasche price index numbers
N
 p it q it
1
P
Ps t 
i 1
N

i 1

p is q it

i
p is
p it
  it
• 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) 8
Fisher price index numbers
F
Pst

L
Pst

P
Pst
• 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
9
Tornqvist price index numbers
 is   it
N
T
Pst


i 1
 p it 


p
 is 
  is   it 
 
  ln p it  ln p is 
2

i 1 
N
2
, ln
T
Pst
• 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
10
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:
Q st 
V st
Pst








p it q it
i

i
p is q is



 / Pst 




p it q it / Pst
i

p is q is
i
11
Quantity Index Numbers
 Q st 
value in period t ( at cons tan t prices , in period s )
value 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.
12
Four quantity index numbers
To obtain the corresponding quantity index
numbers we interchange prices and quantities:
N

Laspeyres = Q stL 
N
p is q it
i 1
N

Paasche = Q stP 
,
p is q is
i 1

p it q it

p it q is
i 1
N
i 1
Fisher = Q stF  Q stL  Q stP
 is   it
N
Tornqvist =
T
Q st


i 1
 q it 


q
 is 
  is   it 
 
  ln q it  ln q is 
2

i 1 
N
2
, ln
T
Q st
13
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
14
Tornqvist TFP index
The Tornqvist has been the most popular TFP
index
ln T F P Index st  ln
O utput Index st
Input Index st
 ln O utput Index st  ln Input Index st

1
M
 ris  rit   ln

2
i 1
q it  ln q is  
K
s js  s jt  ln


2
1
x jt  ln x js

j 1
This approach is also know as the Hicks-Moorsteen Approach –
defines productivity index simply as the ratio of output and input
index numbers.
15
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
A
B
labour
x1
w1
2
80
4
90
capital cost output
x2 w2
q
2 100 360
200
1 120 480
200
16
• 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
= (280)/(280+2100) = 0.44
• Labour share for firm B
= (490)/(490+1120) = 0.75
• Thus the capital shares are (1-0.44)=0.56
and (1-0.75)=0.25, respectively
17
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
18
Properties of index numbers
• Used to evaluate index numbers
– Economic theory
– Axioms
• Both suggest Tornqvist and Fisher best for
TFP calculations
19
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…
20
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.
21
• 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).
22
• 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 (PQ=V).
23
• 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=IsrIrt. That is, a direct comparison between
s and t yields the same index as an indirect
comparison through r (we provide an example
later).
24
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).
25
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
26
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=1.211.15
• The EKS method is used to convert nontransitive indices into transitive indices:
1
 I
N
I
transitive
st

r 1
sr
 I rt

N
27
• 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
28
Transitive Tornqvist TFP index
Recall that the binary TFP Index using
Tornqvist formula is given by:
ln T F P Index st  ln
O utput Index st
Input Index st
 ln O utput Index st  ln Input Index st

1
M
 ris  rit   ln

2
i 1
q it  ln q is  
K
s js  s jt  ln


2
1
x jt  ln x js
j 1
We note that this index is not transitive
29

Transitive Tornqvist TFP Index
If we apply the EKS method and generate
transitive index numbers, we can show that
transitive
ln T F Pst
1 M
  2  rit  r i
 i 1

M


1
2
 ln q it  ln q i 
  ris  r i 
i 1
 K
1
2
 j  1

ln q is  ln q i 

  s jt  s j  ln

x jt  ln x j





K

1
2
  s js  s j  ln
j 1
x js  ln x j
30
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
31
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.
32
Output Data for the Australian National Railways Example
Mainland
Freight (1,000
NTKs)
5235000
5331000
5356000
4967000
5511000
5867000
6679000
6445000
7192000
7618000
7699000
7420000
Quantities
Tasrail
Freight (1,000
NTKs)
383000
420000
375000
381000
401000
403000
402000
429000
455000
459000
413000
369000
Passenger
(1,000
PTKs)
2924
3057
2992
2395
2355
2188
2486
2381
2439
2397
2316
1664
Mainland
Freight
($/NTK)
0.02
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
Prices
Tasrail
Freight
($/NTK)
0.07
0.07
0.08
0.08
0.08
0.08
0.09
0.09
0.09
0.08
0.11
0.12
Passenger
($/PTK)
10
12
14
18
20
22
23
23
23
26
32
47
33
Other
inputs
Capital
inputs
Labour
(persons)
10481
10071
9941
9575
9252
8799
8127
7838
7198
6648
6432
5965
Quantities
Fuel (1,000
litres)
77380
80148
77105
72129
85868
89706
96312
92519
96435
101327
98874
96016
Other Inputs
($1,000 )a
119113
112939
108263
110210
109292
97594
93178
80054
77716
74147
80826
73172
Land, Building
and Perway
($1,000 )a
Quantities
Plant and
Equipment
($1,000 )a
Rolling
Stock
($1,000)a
1858038
2101035
2059365
2118357
2117625
2095680
2069494
2034867
2017626
1998345
2011753
2018802
94057
93927
89764
93271
91837
90120
89617
88773
89653
98762
100495
107654
332307
308491
285626
269265
275134
261495
251588
239736
235834
252514
251850
242662
Labour
($/person)
13097
14730
16692
18651
20166
21307
24990
26412
28572
32617
34565
35646
Land,
Building and
Perway
(index)b
10
20
30
30
70
70
50
70
80
80
80
130
Prices
Fuel ($/litre)
0.18
0.26
0.28
0.37
0.37
0.39
0.41
0.42
0.43
0.39
0.43
0.46
Other Inputs
(index)
0.45
0.50
0.56
0.62
0.66
0.70
0.75
0.81
0.87
0.94
1.00
1.04
Prices
Plant and
Equipment
(index)b
Rolling
Stock
(index)b
50
80
120
100
140
160
90
120
200
240
190
200
50
80
120
100
140
160
90
120
200
240
190
200
34
Output, Input and TFP index numbers
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
• 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.
35