Maximum entropy: a stochastic frontier approach for electricity distribution regulation

Abstract

The literature on incentive-based regulation in the electricity sector indicates that the size of this sector in a country constrains the choice of frontier methods as well as the model specification itself to measure economic efficiency of regulated firms. The aim of this study is to propose a stochastic frontier approach with maximum entropy estimation, which is designed to extract information from limited and noisy data with minimal statements on the data generation process. Stochastic frontier analysis with generalized maximum entropy and data envelopment analysis—the latter one has been widely used by national regulators—are applied to a cross-section data on thirteen European electricity distribution companies. Technical efficiency scores and rankings of the distribution companies generated by both approaches are sensitive to model specification. Nevertheless, the stochastic frontier analysis with generalized maximum entropy results indicate that technical efficiency scores have similar distributional properties and these scores as well as the rankings of the companies are not very sensitive to the prior information. In general, the same electricity distribution companies are found to be in the highest and lowest efficient groups, reflecting weak sensitivity to the prior information considered in the estimation procedure.

Appendices

Appendix A

See Table 9.

Table 9 International data on electricity distribution utilities.

Appendix B: DEA models

DEA is a non-parametric, mathematical programming-based method to generate the efficient frontier in a given data set and measure the efficiency of each firm relative to the frontier. It fully envelops the data and makes no accommodation for noise (Fried et al. 2008, chapter 1).

The DEA model, assuming CRS, to generate technical efficiency for each firm i, is given as:

$$ \begin{aligned} {\text{TE}}({\text{y}}^{{\rm i}} ,{\text{x}}^{{\rm i}} ) = (1/{\text{D}}({\text{y}}^{{\rm i}} ,{\text{x}}^{{\rm i}} )) &= \mathop {\hbox{min} }\limits_{{\uplambda,{{\rm z}}}} \left\{ {\uplambda:\uplambda{\text{x}}^{{\rm i}} \in {\text{V}}({\text{y}}^{{\rm i}} )} \right\} \\ &= \mathop {\hbox{min} }\limits_{{\uplambda,{{\rm z}}}} \left\{ {\uplambda:\sum\limits_{{{{\rm j}} = 1}}^{{\rm J}} {{\text{z}}^{{\rm j}} {\text{y}}_{{\rm m}}^{{\rm j}} \ge {\text{y}}_{{\rm m}}^{{\rm i}} ,\quad {\text{m}} = 1,\ldots,{\text{M}};} } \right. \\ &\quad \quad \quad \quad \sum\limits_{{{{\rm j}} = 1}}^{{\rm J}} {{\text{z}}^{{\rm j}} {\text{x}}_{{\rm n}}^{{\rm j}} \le\uplambda{\text{x}}_{{\rm n}}^{{\rm i}} ,\quad {\text{n}} = 1,\ldots,{\text{N}};} \\ &\quad \quad \quad \quad \left. {{\text{z}}^{{\rm j}} \ge 0,\quad {\text{j}} = 1,\ldots,{\text{J}}} \right\}, \\ \end{aligned} $$

where ${\text{y}}^{\text{i}}$ and ${\text{x}}^{\text{i}}$ are, respectively, the M-output vector and the N-input vector of firm i, z is a J × 1 intensity vector, where J is the total number of firms in the data set. λ is a scalar whose optimal value is the technical efficiency score of firm i, ${\text{TE}}({\text{y}}^{\text{i}} ,{\text{x}}^{\text{i}} )$, which, in turn, is equal to the inverse of the value of the radial distance function.

In the minimization problem, technical efficiency of firm i is assessed in terms of its ability to contract its input vector subject to the efficient frontier. If a radial contraction of the input vector is possible for firm i, its optimal λ < 1 (i.e., firm i is technically inefficient), while if the radial contraction is not possible, its optimal λ = 1 (i.e., firm i is technically efficient).

For model 1, M = 3 and N = 1. For model 2, M = 2 and N = 2, where network length is considered a fixed input. Models 3 and 4 are similar to, respectively, models 1 and 2, except that the former models assume VRS. This assumption is modeled by adding the convexity constraint $\sum\nolimits_{j} {z^{j} } = 1$ in the minimization problem, presented above.