1. Efficiency and Productivity Measurement: Index Numbers
D.S. Prasada Rao
School of Economics
The University of Queensland, Australia

2. 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.
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.

3. 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
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.

4. 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
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.

5. Output Price Index Numbers
The economic theoretic output-price index number is defined using period t technology as (Diewert, 1980, Fisher and Shell, 1972):
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.

6. Output Price Index Numbers
We can define a similar price index number using period s technology:
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.
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.

7. 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:
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
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.

8. 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
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.

9. 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.
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.

10. Input Price Index Numbers
We can derive similar results for input price index numbers by using the input cost function defined as:
The input price index numbers are then defined as ratio of two cost functions and is of the form:
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.
Then Tornqvist (Fisher) is exact for translog (quadratic) cost functions. It is also superlative.

11. Malmquist Output Quantity Index Numbers
The Malmquist output index numbers make use of the output distance functions defined given a technology.
Then the Malmquist output index defined using technologies periods t and s, respectively, are given by
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.
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.

12. 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.
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.

13. 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.
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.