1. Diego Prior Dpt. of Business
Productive efficiency and innovation
Lesson 2. Output and input measurement: Partial productivity measures and aggregation
Diego Prior
Dpt. of Business

2. Outline for today 2.1. Measuring outputs and inputs.
2.2. Partial productivity indicators.
2.3 TFP indices.
2.4. Applications.
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. 2.1. Measuring outputs and inputs.
What is Production?
Transformation of resources (inputs) into products and services (outputs).
Examples:
Agriculture: Grain farming
Inputs: land; labour; biochemical input (fertiliser, pesticides, seeds); capital (equipment, power and buildings).
Output: corn, soybeans, wheat.
Industry: Cement production
Inputs: energy in calories; blue and white-collar workers (man hours/year); book value of machinery and equipment; raw materials (especially: limestone).
Output: cement production per year (tonnes).
Services: Urban transport with busses
Inputs: staff; fuel; vehicles.
Output: vehicle kilometres.

4. Problems:
Sometimes there are positive or negative external effects in production.
Example: Electricity generation using fossil fuels creates pollutants like sulphur dioxide and nitrogen oxide.
The internal organisation of production processes (including decision-making across different hierarchical levels, intermediate production stages, etc.) is ignored (black box models).
Problem of accurate measurement of inputs and outputs.
Example: Labour in agriculture should include farm family members.

5. Richard Murray (1992), Measuring Public-Sector Output:
Throughput
Measures workloads and may even come close to input (number of cases)
Output
Goods or services delivered (number of cases handled)
Outcome
The result from the point of view of the principal or the customers
Bradford, Malt and Oates (1969):
Intermediate outputs : hours of service delivered
D-output : direct output consumed
C-output : outcome indicator reflecting the degree to which the direct outputs translate into welfare improvements as perceived by consumers

7. Measuring the inputs:

8. 2.2. Partial productivity indicators.
Source: Prior, D. Vergés, J. and Vilardell, I. (1993), La evaluación de la eficiencia en el sector privado y en el sector público, Instituto de Estudios Fiscales. Madrid

11. 2.3. TFP indices (American Productivity Center proposal):

12. 2.3. TFP indices.
When we have just one input and output the TFP change between period 1 and 2 is:
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
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.

13. Index number formulae
When we have more than one input or output we need to find an aggregation method
Four most popular indices are:
Laspeyres (prices dependent)
Paasche (prices dependent)
Fisher (prices dependent)
Divisia/Tornqvist (good property: connected with production theory)
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.

14. Price Index Numbers
Measure changes (or levels) in prices of a set of commodities.
Let pmj and qmj represent prices and quantities (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.

15. 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
Very popular 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.

16. 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.

17. 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)
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.

18. 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.

19. Quantity Index Numbers
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.

20. 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...

21. 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)
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

22. 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.

23. Properties of index numbers
Used to evaluate index numbers
Economic theory
Axioms
Both suggest Tornqvist and Fisher best for TFP calculations

24. 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)
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