# Imputing output prices for non-market production units: a comment

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## 1 Introduction

This note is a comment on Diewert (2011b), which is about productivity measurement for non-market production units. Diewert considered specifically the imputation of output prices when price and quantity data on output-specific inputs are available.

In this note I show that the two approaches offered by Diewert, the simple one in Sect. 2 and the seemingly more general one in the Appendix, are basically equivalent.

## 2 Two approaches

Consider a production unit producing \( y_{1}^{t} , \ldots ,y_{M}^{t} \) quantities of outputs (in the article called procedures) during period *t*. Each output *m* requires inputs from a set \( A_{m} \), the vector of quantities during period *t* being \( x_{m}^{t} \) (*m* = 1,…, *M*). The corresponding vectors of input prices are \( w_{m}^{t} \) (*m* = 1,…, *M*). It is assumed that output quantities as well as output-specific input quantities and prices can be observed. Note that this is a rather strong assumption because usually only *aggregate* input quantities and prices can be observed.

*m*(

*m*= 1,…,

*M*) and the dot denotes inner product. The production unit’s profitability, defined as its (total) revenue divided by its (total) cost, is then computed as

Substituting (1) into (2) leads immediately to the conclusion that profitability is identically equal to 1. *A fortiori*, when comparing two periods, profitability *change* is identically equal to 1. However, *productivity* change, defined as output quantity index divided by input quantity index (or, the quantity component of profitability change), is not necessarily equal to 1, unless the technical coefficients of the two periods are the same (\( a_{m}^{1} = a_{m}^{0} \), where 1 and 0 denote the two periods compared).

*m*during period

*t*is produced can be represented by a cost function \( C_{m}^{t} (w_{m} ,y_{m} ) \). Instead of (1), the output prices can then be imputed as unit costs by setting them equal to

## 3 Conclusion

*overall*technology exhibits CRS and the production unit acts cost minimizing, then it appears that

Then, of course, profitability and profitability change are identically equal to 1.^{1} Output quantity index divided by input quantity index, however, is not necessarily equal to 1, as in the simple situation considered before. However, expression (9) makes clear that for the computation of a productivity index knowledge of the cost function is essential.

## Footnotes

## Notes

### Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

### References

- Diewert WE (2011a) Measuring productivity in the public sector. J Prod Anal 36:177–191CrossRefGoogle Scholar
- Diewert WE (2011b) The measurement of productivity in the non-market sector. J Prod Anal. doi:10.1007/s11123-011-0247-x