1. Efficiency and Productivity Measurement: Index Numbers
D.S. Prasada Rao
School 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
Fare et al 1994, OECD and MPI (regional concept)
Coelli and Rao (2005) Ag Economics 95 countries Ag productivity using MPI
technical efficiency, which reflects the ability of a firm to obtain maximal output from a given set of inputs.

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
PIN can be used for TFP measurement - and also for constructing aggregate input and output variables for use in SFA and DEA.
Laspeyres, Paasche, Fisher, Tornqvist are the most commonly used index number formula (eg. CPI, etc.).
Selection of best index number formula often based on economic theory or axiomatic arguments (or practical issues).
Tornqvist is most often used for TFP measurement.
Simple numerical example from the railways data in Coelli, Estache, Perelman and Trujillo (2000).
Indirect quantity indices = value / price index.
Chaining used to accumulate changes over time.
Transitivity (consistency) required in spatial comparisons.
Computer example uses TFPIP program - can also use Excel, Shazam, etc.

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:
Also, productivity = using less inputs to produce a particular level of output.
Productivity is a relative concept - between two time periods or two firms, etc.
The TFP change between period 1 and 2 is equal to the TFP index in year 2 over the TFP index in year 1 - or the ratio of the output change index over the input change index.
See the example on the next slide.
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
PIN can be used for TFP measurement - and also for constructing aggregate input and output variables for use in SFA and DEA.
Laspeyres, Paasche, Fisher, Tornqvist are the most commonly used index number formula (eg. CPI, etc.).
Selection of best index number formula often based on economic theory or axiomatic arguments (or practical issues).
Tornqvist is most often used for TFP measurement.
Simple numerical example from the railways data in Coelli, Estache, Perelman and Trujillo (2000).
Indirect quantity indices = value / price index.
Chaining used to accumulate changes over time.
Transitivity (consistency) required in spatial comparisons.
Computer example uses TFPIP program - can also use Excel, Shazam, etc.

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.
Fare et al 1994, OECD and MPI (regional concept)
Coelli and Rao (2005) Ag Economics 95 countries Ag productivity using MPI
technical efficiency, which reflects the ability of a firm to obtain maximal output from a given set of inputs.

7. Laspeyres price index numbers
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
Best to start with price indices because we are familiar with them - eg. the consumer price index (CPI).
Price change from period s to period t.
Note: p=price, q=quantity, =value share.
An unweighted index will not be independent of units of measurement and may not be representative - see bread and beef example on next slide.
Laspeyres uses base quantities as weights, while Paasche uses comparison period quantities. These indices can be interpreted as the change in the cost of a particular basket of goods.
Many countries use a fixed-base Laspeyres index to measure the CPI because it is expensive to re-estimate the shares every period - this is usually done every 5 years or so - but quantity weights can quickly change (eg. computers, travel)
When price changes induce demand changes the Laspeyres will overstate the price change while the Pasche will understate it - the true index lies between the two extremes.
Fisher is geometric average- and hence may be a good approximation.
The Tornqvist uses share weights from both periods - often expressed in log-change form for calculation.
Tornqvist is geometric weighted average, while Laspeyres and Paasche are arithmetic and harmonic averages, respectively.

8. Paasche price index numbers
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)
Best to start with price indices because we are familiar with them - eg. the consumer price index (CPI).
Price change from period s to period t.
Note: p=price, q=quantity, =value share.
An unweighted index will not be independent of units of measurement and may not be representative - see bread and beef example on next slide.
Laspeyres uses base quantities as weights, while Paasche uses comparison period quantities. These indices can be interpreted as the change in the cost of a particular basket of goods.
Many countries use a fixed-base Laspeyres index to measure the CPI because it is expensive to re-estimate the shares every period - this is usually done every 5 years or so - but quantity weights can quickly change (eg. computers, travel)
When price changes induce demand changes the Laspeyres will overstate the price change while the Pasche will understate it - the true index lies between the two extremes.
Fisher is geometric average- and hence may be a good approximation.
The Tornqvist uses share weights from both periods - often expressed in log-change form for calculation.
Tornqvist is geometric weighted average, while Laspeyres and Paasche are arithmetic and harmonic averages, respectively.

9. Fisher price index numbers
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
Best to start with price indices because we are familiar with them - eg. the consumer price index (CPI).
Price change from period s to period t.
Note: p=price, q=quantity, =value share.
An unweighted index will not be independent of units of measurement and may not be representative - see bread and beef example on next slide.
Laspeyres uses base quantities as weights, while Paasche uses comparison period quantities. These indices can be interpreted as the change in the cost of a particular basket of goods.
Many countries use a fixed-base Laspeyres index to measure the CPI because it is expensive to re-estimate the shares every period - this is usually done every 5 years or so - but quantity weights can quickly change (eg. computers, travel)
When price changes induce demand changes the Laspeyres will overstate the price change while the Pasche will understate it - the true index lies between the two extremes.
Fisher is geometric average- and hence may be a good approximation.
The Tornqvist uses share weights from both periods - often expressed in log-change form for calculation.
Tornqvist is geometric weighted average, while Laspeyres and Paasche are arithmetic and harmonic averages, respectively.

10. Tornqvist price index numbers
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
Best to start with price indices because we are familiar with them - eg. the consumer price index (CPI).
Price change from period s to period t.
Note: p=price, q=quantity, =value share.
An unweighted index will not be independent of units of measurement and may not be representative - see bread and beef example on next slide.
Laspeyres uses base quantities as weights, while Paasche uses comparison period quantities. These indices can be interpreted as the change in the cost of a particular basket of goods.
Many countries use a fixed-base Laspeyres index to measure the CPI because it is expensive to re-estimate the shares every period - this is usually done every 5 years or so - but quantity weights can quickly change (eg. computers, travel)
When price changes induce demand changes the Laspeyres will overstate the price change while the Pasche will understate it - the true index lies between the two extremes.
Fisher is geometric average- and hence may be a good approximation.
The Tornqvist uses share weights from both periods - often expressed in log-change form for calculation.
Tornqvist is geometric weighted average, while Laspeyres and Paasche are arithmetic and harmonic averages, respectively.

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:
Fare et al 1994, OECD and MPI (regional concept)
Coelli and Rao (2005) Ag Economics 95 countries Ag productivity using MPI
technical efficiency, which reflects the ability of a firm to obtain maximal output from a given set of inputs.

12. Quantity Index Numbers
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.
Fare et al 1994, OECD and MPI (regional concept)
Coelli and Rao (2005) Ag Economics 95 countries Ag productivity using MPI
technical efficiency, which reflects the ability of a firm to obtain maximal output from a given set of inputs.

13. Four quantity index numbers
To obtain the corresponding quantity index numbers we interchange prices and quantities:
For quantity indices we simply interchange prices and quantities.
Best index for TFP measurement? - Diewert and others argue for Tornqvist and Fisher on the basis of economic theory and axiomatic arguments - we now discuss these...

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
Linear production functions imply infinite elasticities of substitution.
Most published work by Diewert.
Diewert argues for more emphasis on axiomatic arguments because above assumptions often do not apply - especially in regulated industries

15. Tornqvist TFP index The Tornqvist has been the most popular TFP index
The Tornqvist has been most commonly used for TFP calculations in the past 2 decades.
Usually expressed in log-change form for ease of calculation and interpretation.
Notation: M outputs and K inputs, y=output quantity, x=input quantity, =output revenue share, =input cost share.
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
Each firm produces 200 t-shirts per day
We ignore other possible inputs to keep the example simple

17. First we calculate the input cost shares Labour share for firm A
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
Here we suggest a linear aggregation function - there are many other options.
Note that this TFP measure is equal to the inverse of the unit cost measures, which are $1.80 and $2.10, respectively.
What do we do if the firms face different prices? – we could select one set of prices (toss a coin?) or we could use both sets of prices - eg. use a Tornqvist or Fisher index – see session 5.
What do we do if we have no price information? – we could use the shadow prices implicit in an estimated production technology - eg. SFA or DEA – more on this in later sessions.
Having no price information is a common issue in trying to construct aggregate output indices in public service sectors, such as police, education and health.

18. Ln Output index = ln(200)-ln(200)
= 0.0
Ln Input index = [0.5( )(ln(2)-ln(4))
+0.5( )(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…
Linear production functions imply infinite elasticities of substitution.
Most published work by Diewert.
Diewert argues for more emphasis on axiomatic arguments because above assumptions often do not apply - especially in regulated industries

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.
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).
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).
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.
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).

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 (PQ=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=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).

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
Note I12=1.03 implies a 3% increase in productivity (or whatever) from period 1 to period 2.
The other option is the calculation of direct comparisons ( eg. I14)
The chained indices involve smaller changes between adjacent periods and hence are likely to be better approximations to the unknown “true” index - and the different formula will give similar results.
In the case of CPI calculations the continuous updating of weights will mean the chained index is more representative - but the costs of data collection will be higher.
Note that the direct and chained index are likely to differ - because the Tornqvist, Fisher, Laspeyres and Paasche are not transitive.
There is a natural order in time series data for chaining - but in spatial data (eg.firm-level comparisons) there is not

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=1.211.15
The EKS method is used to convert non-transitive indices into transitive indices:
Firm 2 is 10% more productive than firm 1, etc.
EKS method constructs geometric mean of all indirect comparisons via the N firms in the sample.
Also applied to panel data.
EKS adjustment is minimum mean squared deviation from original index.
EKS can be applied to any index - Fisher, Tornqvist, etc.
Application to the Tornqvist TFP index produces the following index…..

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 = ×1.0851
= = TI13

29. Transitive Tornqvist TFP index
Recall that the binary TFP Index using Tornqvist formula is given by:
The Tornqvist has been most commonly used for TFP calculations in the past 2 decades.
Usually expressed in log-change form for ease of calculation and interpretation.
Notation: M outputs and K inputs, y=output quantity, x=input quantity, =output revenue share, =input cost share.
We note that this index is not transitive

30. Transitive Tornqvist TFP Index
If we apply the EKS method and generate transitive index numbers, we can show that
Note the typos in equation 4.29 on page 91 of CRB.
The bars refer to sample means.
This is quite similar to the normal Tornqvist TFP index, except we make an indirect comparison through the sample mean.
The Fisher, Laspeyres and Paasche do not produce a simplified formula like this when EKS is applied.

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
Note the typos in equation 4.29 on page 91 of CRB.
The bars refer to sample means.
This is quite similar to the normal Tornqvist TFP index, except we make an indirect comparison through the sample mean.
The Fisher, Laspeyres and Paasche do not produce a simplified formula like this when EKS is applied.

32. Productivity comparisons using index numbers
We note the following important properties:
Productivity measures can be computed using data on just two firms (i.e., very limited data);
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).
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.
Fare et al 1994, OECD and MPI (regional concept)
Coelli and Rao (2005) Ag Economics 95 countries Ag productivity using MPI
technical efficiency, which reflects the ability of a firm to obtain maximal output from a given set of inputs.

33. Output Data for the Australian National Railways Example

34. Other inputs
Capital inputs

35. 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.