FRONTIER METHODOLOGIES FOR THE DETERMINATION OF EFFICIENCIES IN DISTRIBUTION COSTS

The ef ciency of electric distribution companies is assessed under a productive ef ciency criterion viewpoint. The analysis methodology consists of the estimation of frontier production costs functions; a parametric and a non parametric function. Two individual technical ef ciency measurements for the distribution business are obtained from these functions. The assessment con rms actions and strategies that result in cost reductions in distribution companies. An application within the framework of the last Chilean regulatory process is illustrated in this work.


INTRODUCTION
The electric distribution sector is a fundamental element in the basic infrastructure of any country.Thus, governments worldwide are looking at regulations that assure an economic development of what is a monopolistic activity, so that demand growth is supplied in an ef cient manner.Benchmark regulations are being used, de ning growing ef ciency requirements to the distribution companies.This has been the case in the Latin-American electric sector, where after an in depth restructuring, distribution companies have had to make important efforts to adapt to a new environment and compete with the benchmarks, facing the challenge of reaching specified levels of ef ciency.
Thus, with the current regulatory policies, the maximization of the companies' returns is subject to price policies and quality of service requirements, which respond to the services needed by the customers.However, behind these benchmark regulations there are many matters that are not clearly de ned and that are key to the development of the sector.Among these, one stands out and it is the de nition of the price for the distribution service.The regulator establishes different methodological options that do not necessarily consider the particular structural and revenue characteristics of the distribution activity.To establish an adequate price system, the regulator must make distribution companies operate with reduced production costs.For that purpose, it must know if the electric distribution companies are, or are not, operating with the maximum technical and scale ef ciency.

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Ingeniare.Revista chilena de ingeniería, vol.15 Nº 3, 2007 Thus, it becomes of paramount importance to estimate and analyze the technical ef ciency of the distribution activity, as represented not only by a single company, but by a group of companies.The results thus obtained would be valid orientation to determine if it is possible to reduce production costs, while providing the same service with identical quality conditions.
A rst objective of this work is to review and compare the tools that allow evaluating the level of ef ciency of a group of electric distribution companies.A second objective is to analyze the variables that can determine the ef ciency levels reached, and attempt to clarify which are the optimal characteristics of an ef cient distribution business.A third objective is to contrast the relationship existing between the ef ciency and the unit cost.

FRONTIER METHODOLOGIES IN BENCHMARKING
When the operation of different regulated companies, whether electric distributions ones or not, is compared in a benchmark process, it is common to do it contrasting if they are more or less efficient.This comparison necessarily has to consider the factors that govern the production function of the activity of the companies, determined through ef ciency and productivity indexes or measurements.
As a basic concept, productivity is relatively easy to de ne.It is the coef cient between the amount of product (output) and the amount of supplies (input) for the speci c production situation.The productivity results that can be obtained are in uenced by the differences in production technology, differences in the ef ciency of the productive process, and differences due to exogenous variables that affect production [1,2].The productivity component resulting from the differences in the ef ciency of the productive process is frequently called "productive ef ciency".
Productive ef ciency is de ned as the capacity of the company to produce a product at a minimum cost.To reach the minimum cost, the company must use its inputs ef ciently (technical ef ciency) and must choose the right input combination given their relative price (assignment ef ciency).In the former case, the technical ef ciency is de ned in function of the production possibility and is measured as a coef cient between the observed production and the maximum potential production that can be obtained given a group of inputs, or as the coef cient between the total minimum total potential input required and the effectively total used, given a group of outputs.In the latter case, assignment ef ciency, the ef ciency optimal can be de ned in terms of costs, revenues, pro ts or any other objective of the company that is subject to quantity and price restrictions; therefore this ef ciency component has a character that is purely economic.In this manner, productive ef ciency requires both components, the technical and assignment ef ciency.
The method to empirically measure productive ef ciency is based on the measurement of the distance to the frontier that represents the maximum ef ciency [3].As the best behavior is unknown, the best practice observed among the sample of companies under study is considered as an ef cient reference.Thus, ef ciency indexes are calculated for each company by comparing it with those that have a better economic behavior.In this manner, a measure of ef ciency that has a relative character is obtained, namely, it depends on the sample under study.In addition, the composite ef ciency of two components is considered; the technical ef ciency and the assignment ef ciency.The technical ef ciency re ects the ability to obtain the maximum amount of outputs, given the inputs, and the assignment ef ciency re ects the ability to use the inputs in the optimal proportions, given their respective prices [3].Speci cally, the productive ef ciency requires both components, the technical and ef ciency assignment.
To obtain the ef ciency measurements for a group of companies, through data analyses oriented to determine a maximum production or minimum cost entails a broad range of methods, among which there are econometric and mathematical programming techniques.

Stochastic Parametric Frontiers
The Stochastic Parametric Frontiers [4,5] are econometric methods that arise from the idea that the deviations from the frontier are not necessarily under the full control of the analyzed company.In this approach, the frontier is de ned as a function of an ef cient production to which two disturbances are added: v, a symmetrical disturbance that includes the random noise and u, a biased disturbance that is originated by technical inef ciencies.In this manner, external events that affect the production function are normally distributed, (0, v 2 ), affecting the company by favorable or unfavorable external conditions.However, for u, the inef ciency term, various distributions have been proposed; normal mean distribution [4], exponential distribution [5], normal truncated distribution [6] and Gamma distribution [7].There is no a priori reason to prefer any speci c type of distribution on the errors.However, the different simulation exercises made in [7] Libro INGENIERIA.indb 221 8/1/08 16:42:45 Ingeniare.Revista chilena de ingeniería, vol.15 Nº 3, 2007 indicate that the simplest model, from the econometric viewpoint, is the normal mean distribution [4].
The problem that arises in these methods is that the u component ca not be observed and must be inferred from the composite error term, v + u.The separation of noise and inef ciency components is made from the conditional expectation of u given [8].

The Data Envelopment Analysis
The Data Envelopment Analysis (DEA) is an optimization technique built to measure the relative ef ciency of a group of organizational units, Decision Making Units (DMUs) where the presence of multiple supplies (inputs) and products (outputs) make it dif cult to compare their performance.DEA provides a method to compare ef ciency without knowing the production function, namely, without needing to know a functional relationship between inputs and outputs.
Assuming there are n DMUs, each one with m inputs and s outputs, the relative ef ciency result of a test DMU 0 is obtained solving the following linear programming model [9]., , In this problem, the DMU 0 under evaluation is compared to all the DMUs or the linear combination of inputs or outputs producing the same or more than the DMU 0 , consuming less or the same as the DMU 0 , respectively.In this manner, if the test DMU is ef cient, the programming model has not found among the DMUs a combination of amounts of inputs and outputs with which the same or more than the DMU 0 is produced.Therefore, it assigns the value of one to 0 .On the contrary, if the test DMU is inef cient, the programming model has identi ed a group of DMUs which inputs and outputs create a ctitious DMU that produces the same or less than the DMU 0 .
In this procedure the ef ciency measurement can result from the comparison of different scale units, a possibility that in practice can be inadequate.To solve this problem, it is possible to propose a model that considers the possibility of having inef ciencies due to the differences in the operational scales of each DMU [10], in which case it is necessary to add to the problem the restriction of having j equal to one.This restriction ensures that the model evaluates the pure technical ef ciency without includin scale considerations.

APPLICATION
The formulated methodology is applied to assess the Chilean regulatory process, which determined the distribution tariffs for the 2000-2004 period.The data used in the analysis corresponds to the information submitted by the Chilean regulator for the tariff process.
The analyzed sample is a cross section, with the year 2000 as reference, and corresponds to 35 companies that cover all the distribution zones in the country.
To estimate the parametric frontier, it has been assumed that it corresponds to a trans-logarithmic function.Once developed and taking into account the variables considered in the application, the function to be estimated is: In this approach, the variables that determine the cost function are the ones that are regularly used in literature [11,12].They are the total distribution costs (CDs), composed by the capital and business costs, the inclusion of price variables such as the work price (PL) that has been de ned here through the labor coef cient (REMUN) and the number of workers (NTRAB).The capital price (PK) is represented through a coef cient that relates the VNR (replacement value of installations) and the total lines' length, value that clearly acts as a proxy for the capital price.In addition, there are other input variables such as EVEND, the energy sales as the primary activity executed by each company, KMT, the network size measured through the total line high and low voltage kilometers.This last variable captures the size of the distribution system managed by the companies and ensures that, for example, a large rural distribution company is not penalized in the ef ciency evaluation when compared to a distribution company that renders the service in a city.
Together with the number of transformers, this variable Ingeniare.Revista chilena de ingeniería, vol.15 Nº 3, 2007 is also used to represent the capital cost [11].All these variables are xed in the short term and they ef ciently describe the service, the system and up to a certain level, the environment faced by distribution companies.Table 1 shows the distribution companies' descriptive statistics.KWT is the maximum demand (a proxy of the transformation capacity required), CLTST is the total number of clients (a proxy for the number of connection points), and CEXPLT are the operation costs (high and low voltage).Chilean pesos are used.
The sample is formed by different types of companies, with predominance from distribution companies that have a high percentage of non-urban consumption.The data indicates that there are important differences in size and activities executed by the companies.One of the smallest companies does not sell more than two thousand GWh per year, while one of the largest one sells more than six million GWh.The component that re ects the inef ciency in the frontier is the non-negative random variable u and the disturbance term is v that accounts for the effect of the measurement errors and unobserved random errors in the variables.
In line with the common practice, the random term v is distributed independently among the companies according to a normal with a mean of zero and variance of v 2 , (0, v 2 ).For the inef ciency term, u follows a non-negative normal mean, whose mean depends on the variables that explain +(mu, u 2 ), where mu is the linear speci cation of the technical inef ciency, to be able to capture external effects that may in uence the companies' ef ciencies and that are not directly controllable by them.In this manner, the percentage of high voltage clients with respect to the total amount of clients (RCLT) and the percentage of high voltage lines' length with respect to the total line length (RKML) are considered.
The maximum likelihood estimation, with deviations with respect to the model's mean, was made with a cross-section single stage estimation, using the STATA software [13].Table 2 shows the representative rst-order parameters for the frontier (with constant de ned by the logarithmic model).
Table 2 shows the estimation results that include the coef cients obtained when using the least squares estimator.These coef cients are a linear estimation of the cost function and do not allow any ef ciency prediction.These coef cients are included here with the sole purpose of comparing them with the stochastic frontier model coef cients.In the stochastic frontier model (2), the estimation considered a normal mean distribution for the inef ciency term.Ingeniare.Revista chilena de ingeniería, vol.15 Nº 3, 2007 In general, the frontier's parameters are highly signi cant, except for the ones associated to the ln (PL) and ln (PK) variables, which are less signi cant in the study model.It is also observed that the linear estimation of the inef ciency presents a high relevance with respect to the coef cient associated to the line's length in kilometers.
Once the cost frontier is estimated, it is possible to calculate the ef ciency for each company.However, as we intend to compare these results with the ones obtained through a non-parametric frontier, we calculated it as exp (-E(u i |e i )), because the function is expressed in logarithms.

Non-parametric frontier
The ef ciency results of each unit are obtained solving the linear programming problem (1), and these results can be corrected using a bootstrap procedure in order to give a statistical character to these results [14].The differences in size become an important factor that requires establishing a separation between the companies in terms of their tariff treatment.In fact, previous to the execution of the studies, the regulator recognized the existence of economies of scale in this activity in the 2000 process, which led to classify the companies in six service areas [17].This is the means to handle the different economies of scale in the model rm benchmarking process.Thus, the classi cation of each company within a service area becomes a crucial factor, as it de nes the comparison pattern.

Comparison of results for both frontiers
The efficiency results found out through the DEA methodology, bootstrap correction and stochastic parametric frontier are shown in table 3.This table also shows the frequency distribution and the basic descriptive statistics for these results.
The differences observed are mainly due to the characteristics of the methodology used.DEA assigns any deviation from the frontier as inef ciency, while the stochastic parametric methodology distinguishes if this deviation has been caused by the inef ciency or by an arbitrary disturbance.This fact explains the lower ef ciency index assigned by DEA to inef cient companies compared to the stochastic technique, and in the other extreme, a higher ef ciency index for ef cient companies compared with the stochastic methodology.
The level of ef ciency resulting from the stochastic parametric frontier presents a Pearson correlation coef cient with the ef ciency obtained with the bootstrap DEA methodology of 0.904 and the signi cant coef cient was 1%.Therefore, we can conclude that for the group of companies considered, the two types of analyses offer similar results.On the other hand, as the model's parameters are obtained as deviations from the mean, they become representative of the respective cost function elasticities.
Figure 1 shows the behavior of these ef ciency results for each of the companies ordered according to the DEA ef ciency in an ascending sequence.
Similarly, the analysis of the function estimated for the cost frontier allows studying the properties of the productive process.This is so as this is an ef cient costs frontier where the parameters are representative of the respective cost function elasticities for the average company.

Scale Performances
The scale efficiency for the total cost functions is calculated as the inverse of the sum of all logarithmic partial derivatives of the total cost regarding each relevant output, minus 1 [15].
According to the speci ed model, the scaled ef ciency measures the costs reaction in face of an increase that is proportionally equal in the output, as an increase of the sold energy and an increase in the service area, namely, variation that is equivalent to an expansion of the output variables.However, we must consider that these measurements of scaled partial economies perhaps are not as relevant within the cost functions, because their effect is divided among various outputs.In this manner, the scaled ef ciency -that measures the total effect of the expansion of energy sales, keeping KMT and CLTS constant-can be obtained from the elasticities of the average cost function through EEP + 1, namely: where VAD is the distribution unit cost.
For the group of distribution companies analyzed, the results found indicate the presence of increasing scaled performances, Table 3. Efficiency to scale.

Stochastic Parametric Frontier 1,121 1,159
The ef ciency on scale, evaluated by the model, Table 3, is similar to the obtained one by square minimums.Therefore, many of the companies of the sample are excessively small and have not reached a minimum scaled ef ciency.

CONCLUSIONS
This work is part of an effort being developed worldwide in the regulation at the distribution level.The aim is to consolidate a theoretical and methodological procedure to evaluate the ef ciency of distribution companies.
It is a contribution to research on the use of classical organizational theory criteria in the institutional evaluation of distribution companies, establishing new concepts and adapting them to the speci cations and characteristics of the activities of these institutions.
The regulatory framework based on ef cient companies has been applied for many years in many Latin-American countries and there is an increasing interest worldwide on following this path [16].However, there is a lack of quantitative evaluation procedures adequate for the speci cation and characterization of an ef cient company and used as a benchmark for distribution companies.The incentive is there to contribute with an approach to solve this problem.
In summary, and considering the requirements, this work applies a methodology that formulates, develops and uses frontier techniques to obtain indicators for the productive ef ciency, and to determine the distribution added value of groups of distribution companies that are subject to an ef cient-company regulatory scheme.The procedure developed is conceptually clear, technically correct, and operationally applicable; and simultaneously considers the various factors that determine the companies' activities and their interrelationships.In addition, it is consistent with the criteria used in the ef cient-company regulatory scheme and it is coherent with modern comparative ef ciency analysis techniques.
On the other hand, the frontier methodology can consider the complexity of multiple aspects of the distribution companies' activities, jointly considering the various factors that characterize the diversity of their objectives and investment plans.

Figure 1 .
Figure 1.Costs with different frontier models.

Table 1 .
Descriptive statistics of variables used.

Table 3 .
Reference companies and conformation of areas.