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This paper is concerned with formulating directional distance functions assuming that firms operate subject to rate-of-return regulation. To this end, we consider two different contexts. First, we assume that input prices are known, which allows us to extend the rate of return regulated version of Farrell efficiency. Secondly, we assume that input prices are unknown, showing then that a specific reference direction arises as a natural choice for measuring efficiency with directional distance functions.

All over the world, most countries deal with the problem of monopoly by means of regulation. This type of solution is widespread in the case of natural monopolies: water, natural gas, and electric companies. These companies are not allowed to charge any price they want to. Instead, government agencies regulate their output prices.

One form of regulation is that of rate-of-return regulation. After the firm subtracts its operating expenses from gross revenues, the remaining net revenue should be just sufficient to compensate the firm for its investment in plant and equipment. In particular, the regulator authorizes the output price which, if anticipated future market conditions are realized, results in the firm earning a rate of return equal to the predetermined allowed level upon which the output price has been estimated. At a subsequent stage, if the obtained firm rate of return is less than the allowed level, the firm can request an increase in the output price.

It is well known that one disadvantage of rate-of-return regulation is that it may encourage inefficiency because the regulated firms have no incentive to decrease costs. For this reason, assessing the performance of regulated companies with respect to technical inefficiency is an important issue for government agencies.

Measuring inefficiency of firms subject to rate-of-return regulation has been yet studied previously in the literature (see [

We have an additional justification for introducing in the literature directional distance functions under rate-of-return regulation. Granderson [

The paper is organized as follows. In Section

We will work in the same theoretical framework as Färe and Logan [

Regarding mathematical notation, we denote a vector of inputs by

Firms subject to rate-of-return regulation face technology and regulatory constraints, where regulators set the allowed rate of return the firm can earn. To formalize the regulatory constraint let

Following Färe and Logan [

Next we show the general expression of input-oriented directional distance functions [

In this section we develop directional distance functions for firms subject to rate-of-return regulation, assuming that input prices are known.

By analogy with (

Next we need to prove a lemma that we will use later and that shows the relationship between (

First of all, thanks to Theorem 9 of McFadden [

The distance function defined by (

On one hand, if

On the other hand, if

Additionally, Lemma

Before focusing our analysis on proving a dual relationship between (

If

Since

As a direct consequence of Proposition

We now turn to duality and begin with a result that relates (

One has the following

(1) It is a direct consequence of Proposition

(2) Thanks to Lemma

To define the regulated directional input distance function, (

To finish this section, we would like to point out that any vector

The choice of the cost function as an economic criterion to select between alternative firms was already proposed by Farrell [

In this section we will show how to assess the overall performance of each firm in terms of technical and allocative inefficiency dimensions. To achieve this goal, we resort to the Nerlove’s definition of overall inefficiency [

Following Chambers et al. [

Equation (

Regarding the decomposition of the overall inefficiency measure, it is obvious that the technical inefficiency must correspond, following to Chambers et al. [

First of all, we note that in the electric utility industry, and in general in most regulated natural monopolies, it is really easy to find out the quantity of generated electricity, the price of electricity, the capital investment, and even other inputs via official reports (see [

In order to measure technical inefficiency with directional input distance functions, we will select a specific reference direction vector

We propose to work with

We now turn to the expression of the directional input distance function, (

Using directional input distance functions with the specific reference vector

Finally, we operationalize the approach by presenting a nonparametric DEA model and prove an interesting property of

Let us assume that we have a set of

An interesting property of

Let

By hypothesis,

As a consequence of Proposition

In this paper we proposed to introduce directional distance functions in the context of regulation. First, we defined this concept and showed several properties under the assumption that we knew input prices. We showed that the regulated directional input distance function collapses to the regulated Farrell efficiency measure when we consider a specific reference vector. Additionally, we defined an overall inefficiency measure to assess the performance of the firms subject to rate-of-return regulation, which can be decomposed in three terms. One related to the pure technical inefficiency, another one related to the regulation effect on the inefficiency, and a third component related to the allocative inefficiency of the firm.

Finally, we studied the case in which the researcher has no information about input prices. We showed that even in this case it is possible to measure inefficiency under regulation by means of directional distance function. To this aim, we resorted to the dual of the directional distance function, estimating a vector of input prices that satisfies the regulatory constraint. Such approach allowed us to suggest a specific reference direction vector for measuring technical inefficiency under regulation.

The authors are grateful to the Ministerio de Ciencia e Innovacion, Spain and to the Conselleria de Educacion, Generalitat Valenciana, for supporting this research with Grants nos. MTM2009-10479 and ACOMP/2012/144, respectively.