# The Output Side of the Firm: Direct Functions and Indices

## Abstract

The basic assumption in this chapter is that the firm maximizes revenue conditional on exogenously determined quantities of inputs. The first section reviews some well-known facts about the revenue function and the output efficiency measures. The second section introduces the output price index, which is based on the revenue function, discusses its properties and develops some nonparametric approximations. In section 4.3 we define the output quantity index, based on the output distance function, and in section 4.4 the relation between the output price and quantity index will be discussed. Then section 4.5 turns to the output based productivity indices. In order to derive nonparametric approximations it appears necessary to add an assumption about profit maximization. Finally, in section 4.6 we show under which condition the input based and the output based productivity indices coincide.

### Keywords

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### References

- 1.This elasticity was proposed as a measure of local returns to scale by Fire, Grosskopf and Lovell (1986).Google Scholar
- 2.See also Fisher and Shell (1998).Google Scholar
- 3.See Reece and Zieschang (1987) for the output price index of a regulated monopolist.
^{4}Diewert and Morrison (1986) consider also*conditional*output price indices. Suppose that the vector p is partitioned as*(pa*,*pb)*. Then*Po(p*_{a},*p*_{a},*pb*,*x) - R*^{t}*(x*,*pa*,*pb)/R*^{t}*(x*,*pâ*,*pb)*is a conditional output price index: given the input quantities*x*and the output prices*Pb*,the index compares the maximum revenue that can be obtained when the remaining output prices change from pâ to pa.Google Scholar - 5.Diewert ( 1992a, Theorem 7) proved this for the special case where
*D,(xt, yt) =*1. The generalization can be obtained from Appendix B (after interchanging*x*and y).Google Scholar - 6.One sees easily, by using (2.19), that output neutrality is equivalent to input neutrality if the period t’ and t technologies exhibit global CRS.Google Scholar
- 7.An empirical comparison was carried out by Bjurek, Forsund and Hjalmarsson (1998).Google Scholar
- 8.Färe, Grosskopf and Roos (1996) show that D°(x
**l y**^{l}**)/D°(xo****y**^{o}) equals Q2**(y1 yo xo)/Qo(xi****xo****yo)**if and only if the period 0 technology is inversely homothetic and exhibits global CRS. Färe and Primont (1995) show that, if (2.14) holds, inverse homotheticity is equivalent to simultaneous input and output homotheticity.Google Scholar