Efficiency Measurement Using the Stochastic Cost Frontier Model

A convincing and appropriate approach will be to briefly clarify the concept of efficiency and how the stochastic cost frontier could be used to measure it among other approaches already discussed in detail in chapter two. First, in the figure below, a distinction is made between the types of efficiency in a production function and how they are theoretically measured. Farrell (1957) clarifies the differences between allocative and technical efficiency (AE and TE).

Figure 4.1: Technical and Allocative Efficiency

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With SS’ as an isoquant line, any firm (Q) that produces using a combination of x1 x2 to produce y and found on the isoquant is technically efficient. Firms (P) that lie above the isoquant are technically inefficient because they use more input than required. With an iso- cost or budget line (AA’), firm Q’ is allocatively efficient since its inputs prices and quantities touch the cost line. A combination of technical efficiency and allocative efficiency estimated in figure 4.1 gives the level of Economic Efficiency (EE).

In order to estimate these efficiencies various techniques have been developed all geared towards the construction of an optimum or appropriate frontier where both parametric and non-parametric options have been explored.

One of such parametric approaches is the stochastic frontier model originally developed by Aigner, Lovell and Schmidt (1977) is famous for estimating efficiency of production units through either maximisation of output, revenue, profit or the minimisation of cost. Depending on the objective of the model and the choice of a particular consumer behaviour many types of models can be estimated. There are also other extensions that are extensively reviewed in chapter two. The stochastic cost frontier was however excluded in the review in order to be elucidated and applied in this chapter.

Therefore, this chapter focuses on the stochastic cost frontier methodology which assumes a cost minimisation behaviour to produce a particular output level, given input prices and the prevailing production or transformation technology. Due to the variation in managerial, supervisory capacities of managements of firms, it is implausible that all firms may operate on the optimum frontier. Not operating on the frontier may infer that there exists technical and allocative inefficiency in the operations of the firm in question.

In this investigation, a stochastic frontier cost function using panel data is considered. It commences by using the basic SFA panel cost function which is illustrated below as:

𝐶_𝑖𝑡 = 𝑋𝛽_𝑖𝑡+ 𝑈_𝑖 + 𝑉_𝑖𝑡 (4.1)

𝑈 ≥ 0, 𝑖 = 1,2,3, … , 𝑁 𝑎𝑛𝑑 𝑡 = 1,2,3 … , 𝑇

In this specification the error term is composed of two parts: the first, 𝑢𝑖 ; is a one-sided non- negative disturbance reflecting the effect of costs; the second, 𝑣𝑖𝑡, is a two-sided disturbance

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capturing the effect of noise. The statistical noise is assumed to follow a normal distribution, and the inefficiency term 𝑢𝑖, is generally assumed to follow a half-normal, truncated, exponential, Rayleigh or gamma distributions.

According to Pitt and Lee (1981) panel data is preferred in efficiency analysis for at least four reasons. First, it is possible to observe structural changes in a firms production function over time. Panel data approaches offer the opportunity to estimate the efficiency of individual firms from a single cross-section. A third reason is that the use of pooled data permits the comparison of pooled approach to the traditional analysis of covariance approach. Fourth, it permits us to investigate whether the inefficiency of firms is time variant or time invariant, and if it is time variant, whether or not it varies randomly or not. Therefore, panel data models provide information about the behaviour of firms over time which cannot be revealed from cross-sectional data.

The above features of panel models have consequently caused the stochastic frontier model to evolve from one form to another in order to offer more opportunities to analyse efficiency. The stochastic panel model with time-invariant inefficiency can be estimated under either the fixed effects’ or random effects’ framework as suggested by Greene (2005). These form avenues to explore heterogeneity within and between firms or units. The type of panel model to select depends on the level of relationship that is assumed between the inefficiency and the covariates of the model. Under the fixed effects framework correlation is allowed between 𝑋𝑖𝑡 and 𝑢𝑖 whereas under the random effects framework, no correlation is present between 𝑋𝑖𝑡 and 𝑢𝑖. The idea of the standard fixed and random effect models as applied in SFA is to produce a simple transformation and interpret the transformed individual effects as time-invariant inefficiency as opposed to pure firm heterogeneity. As such with the SFA panel models, it is possible to separate inefficiency from individual heterogeneity as opposed to the standard panel models that account for effects.

Many other models that separate firm heterogeneity from inefficiency have been developed in the stochastic frontier approach to allow both effects to be identified and estimated with various types of assumptions for the type and interactions of heterogeneity. Some of these models will be subsequently explained in detail to enhance the methods chosen in this investigation.

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A starting point with SFA panel will thus be the time-invariant model using a Cost frontier - Cobb-Douglas function specified as follows:

𝐶𝑖𝑡 = 𝛽0+ 𝑋𝛽𝑖𝑡+ 𝑉𝑖𝑡+ 𝑈𝑖 (4.2) Where 𝛽0 − 𝑈𝑖 ≡ α𝑖

α_𝑖 and 𝑋𝑖𝑡 by the way they behave and as to whether they are correlated or not defines whether it is a standard fixed effects’ or random effects’ model.

The notable drawback of the standard fixed effects’ and random effect’ models has been reemphasised in Greene (2005) motivates further extension of these panel models. Illustrated in two folds, the first is the inability to distinguish individual heterogeneity from inefficiency making all time invariant heterogeneity to be confused with inefficiency and the second, the implausibility of the level of inefficiency of a firm to stay constant as T becomes large.

The model is thus further modified as seen below to include a time-variant efficiency term. 𝐶_𝑖𝑡 = 𝛽₀+ 𝑋𝛽_𝑖𝑡+ 𝑈_𝑖𝑡+ 𝑉_𝑖𝑡 (4.3)

Compared to standard panel models and the time invariant model discussed in (4.2), there is a modification of the inefficiency term (𝑈𝑖) to accommodate changes overtime and hence the term 𝑈𝑖𝑡 in equation (4.3). If one treats α𝑖 , i = 1, · · · , N, as a random variable that is correlated with 𝑋𝑖𝑡 but does not capture inefficiency, then the above model becomes what has been termed the ‘true fixed effects’ panel stochastic frontier model (Greene 2005). Unlike the initially developed frontier fixed effect model by Schmidt and Sickles (1984), Green shows that the true fixed effects model captures unobserved heterogeneity as the fixed effects. A model is considered as the ‘true random effects’ stochastic frontier model when α𝑖 is treated as uncorrelated with 𝑋𝑖𝑡 in addition to an introduced random constant assumed to be normally distributed and reconstitutes the constant term to be symmetrical with a finite variance.

In this chapter, the techniques employed include the time invariant Pitt and Lee (1981) model which is a random effects model with a time invariant inefficiency term (𝑈𝑖) and the Pooled SFA model which considers an inefficient term that is time varying. The True Random Effects’ Model (TREM) and True Fixed Effects’ Model (TFEM) are also used as proposed in Greene (2004). These two models allows for a separate stochastic term that encapsulates the time- invariant unobserved heterogeneity. The main difference as stated earlier is that the TFEM

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permits correlation with the regressors while in the TREM, 𝑈𝑖𝑡 is independent of 𝑋𝑖𝑡. As discussed earlier in chapters 2 and 3, the stochastic frontier model can assume different distributions. Thus the above models are estimated using various distribution assumptions of half-normal, truncated-normal and exponential distributions.

Regardless of the distribution chosen, for efficiency measurement and scores analyses, the composed error term needs to be decomposed and the efficiency score computed. Jondrow et al. (1982) showed that for the half-normal case, the expected value of 𝑢𝑖 conditional on the composed error term, 𝑒𝑖 is computed as:

𝐸[𝑢_𝑖|𝑒_𝑖] =_1+𝜆σλ₂[𝜙(𝑒𝑖𝜆 σ)⁄

𝛷−𝑒𝑖𝜆 σ⁄ −

𝑒𝑖𝜆

σ ] (4.4)

With 𝜙(. ) being the density of the standard normal distribution, 𝛷(. ) the cumulative density function,𝜆 = σ𝑢⁄ , 𝑒σ𝑣 𝑖 = 𝑦𝑖 − 𝜷𝑥𝑖 = 𝑣𝑖− 𝑢𝑖 and σ = (σ𝑢2 + σ𝑣2)1 2⁄

The technical efficiency estimates for each unit was computable once Jondrow et al. (1982) obtained the conditional estimates of 𝑢𝑖 found below as

𝑇𝐸_𝑖 = 1 − 𝐸[𝑢_𝑖|𝑒_𝑖] (4.5)

Thus, in a stochastic cost frontier setting, efficiency is measured as the ratio of actual costs to the least-cost of production and can simply be shown as:

𝐸𝐹𝐹 = 𝐸(𝐶|𝑢𝑖,𝑋𝑖)

(𝐸(𝐶|𝑢𝑖 0,𝑋𝑖) (4.6)

Cost efficiency (EFF) predictions are therefore computed using the following expression: 𝐸𝐹𝐹 =𝐸(𝑋𝛽𝑖𝑡+𝑢𝑖,)

(𝐸(𝑋𝛽𝑖𝑡) ≥ 1 (4.7)

There are other definitions suggested by other authors including Battese and Coelli (1988) Hjalmarsson, Kumbhakar and Heshmati (1996) and Bera and Sharma (1999) concerning point estimations of technical efficiency. However, the point estimator of Jondrow et al. (1982) also referred to as “JMLS point estimator”, is much popular and used in many SFA packages as a default estimator of efficiency.

In this section, the foundations of the Stochastic Cost Frontier Approach and its related models have been explained. This was a very important step towards achieving the objective of measuring the efficiency levels of the EDCs in West Africa which is the focus of the next section. The next section therefore presents the methodology and data used in estimating the cost efficiencies of the West African electricity distributions sector.

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