Journal of Productivity Analysis

, Volume 37, Issue 3, pp 231–232

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

Open Access
Article

## 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.

Because the production unit is not operating on the market, there are no output prices. But given the input requirements, output prices can be imputed as being equal to unit costs; that is,
$$p_{m}^{t} \equiv w_{m}^{t} \cdot x_{m}^{t} /y_{m}^{t} = w_{m}^{t} \cdot a_{m}^{t} \quad \left( {m = 1, \ldots ,M} \right),$$
(1)
where $$a_{m}^{t} \equiv x_{m}^{t} /y_{m}^{t}$$ is the vector of input quantities per unit of output 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
$$\frac{{\sum\nolimits_{m = 1}^{M} {p_{m}^{t} y_{m}^{t} } }}{{\sum\nolimits_{m = 1}^{M} {w_{m}^{t} \cdot x_{m}^{t} } }}$$
(2)

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).

The above summarizes what is happening in Sect. 2 of Diewert’s article. The Appendix generalizes this by assuming that the technology with which output 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
$$p_{m}^{t} \equiv C_{m}^{t} (w_{m}^{t} ,1)\quad \left( {m = 1, \ldots ,M} \right).$$
(3)
Profitability is then computed as
$$\frac{{\sum\nolimits_{m = 1}^{M} {C_{m}^{t} (w_{m}^{t} ,1)y_{m}^{t} } }}{{\sum\nolimits_{m = 1}^{M} {w_{m}^{t} \cdot x_{m}^{t} } }},$$
(4)
which is not necessarily equal to 1. Now, if each technology exhibits constant returns to scale (CRS) and the production unit acts cost minimizing, then
$$C_{m}^{t} (w_{m}^{t} ,1)y_{m}^{t} = C_{m}^{t} (w_{m}^{t} ,y_{m}^{t} ) = w_{m}^{t} \cdot x_{m}^{t} \quad \left( {m = 1, \ldots ,M} \right),$$
(5)
which implies that profitability is identically equal to 1. Rewriting expression (5) we see that
$$C_{m}^{t} (w_{m}^{t} ,1) = w_{m}^{t} \cdot x_{m}^{t} /y_{m}^{t} \quad \left( {m = 1, \ldots ,M} \right),$$
(6)
which, when substituted into expression (3), leads to the same expression as (1). Thus the twin assumptions of CRS technologies and cost minimization essentially bring us back to the setup as discussed in Sect. 2. There is nothing gained here.

## 3 Conclusion

The basic assumption referred to in the sentence preceding expression (3) implies that the overall cost function has the following form
$$C_{{}}^{t} (w_{1} , \ldots ,w_{M} ,y_{1} , \ldots ,y_{M} ) = \sum\limits_{m = 1}^{M} {C_{m}^{t} (w_{m} ,y_{m} )} .$$
(7)
If this assumption is dropped and replaced by the twin assumptions that the overall technology exhibits CRS and the production unit acts cost minimizing, then it appears that
$$\sum\limits_{m = 1}^{M} {\frac{{\partial C_{{}}^{t} (w_{1}^{t}, \ldots ,w_{M}^{t} ,y_{1}^{t} , \ldots ,y_{M}^{t} )}}{{\partial y_{m} }} y_{m}^{t} = C_{{}}^{t} (w_{1}^{t} , \ldots ,w_{M}^{t},y_{1}^{t} , \ldots ,y_{M}^{t} ) = \sum\limits_{m = 1}^{M}{w_{m}^{t} \cdot x_{m}^{t} } }$$
(8)
Thus it makes sense to define output prices as marginal costs; that is,
$$p_{m}^{t} \equiv \frac{{\partial C_{{}}^{t} (w_{1}^{t} ,\ldots,w_{M}^{t} ,y_{1}^{t} ,\ldots,y_{M}^{t} )}}{{\partial y_{m} }}\quad \left( {m = 1, \ldots ,M} \right).$$
(9)

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

1. 1.

See also Diewert (2011a, Appendix), where the cost function additionally contains a vector of (output) quality characteristics.

## Notes

### References

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