A parametric approach to efficiency measurement using a flexible profit function.

• Author: Kumbhakar, Subal C.

1. Introduction

   Although the idea of measuring productive efficiency goes back to Farrell [8], the econometric estimation of it began with Aigner, Lovell, and Schmidt [1] and Meeusen and van den Broeck [17]. In Aigner, Lovell, and Schmidt [1] and Meeusen and van den Broeck [17], and their extensions which incorporate either cost minimizing or profit maximizing behavior, efficiency is measured relative to a frontier. An alternative to the frontier approach that started with Hopper [10] and Lau and Yotopoulos [15], is to measure efficiency (mostly allocative) without estimating any frontier. In the latter approach, allocative inefficiency (defined as the deviations of the first order conditions of profit maximization or cost minimization) is modeled through shadow (virtual) prices which are parametric functions of observed prices. The shadow price approach is extensively used in the literature on efficiency measurement because it has an advantage over the frontier approach, which requires distributional assumptions on the error terms--especially in cross sectional models.

   In empirical studies on the efficiency of regulated utilities, it has been argued that due to the presence of rate of return and other regulations, these firms optimize with respect to shadow prices instead of observed prices. These studies often fail to consider the possibility that the firms under question may be technically inefficient as well, which can affect allocation of inputs.(1) This issue is important in both cost minimization and profit maximization cases.

   This paper uses a profit maximization framework and develops a generalized profit function approach that accommodates both technical and allocative inefficiencies in the context of a panel data model. The relationship between production technical inefficiency (loss of output due to technical inefficiency) and profit technical inefficiency (profit loss due to technical inefficiency) is derived for a flexible production function. Presently, it is believed that the derivation of this relationship is only possible for the self-dual production functions. We show that if the profit function is translog, some popular approaches used in modeling technical and allocative inefficiency are incorrect. In particular, we show that: (i) models that fail to include technical inefficiency and consider only allocative inefficiency yield biased and inconsistent parameter estimates; (ii) if technical inefficiency is neglected, the measure of technical change will be biased. The generalized profit function developed here is capable of distinguishing between technical change and time-varying technical inefficiency. Such a distinction is not possible if, for example, one uses either the production or the cost function in estimation. We specify time-varying technical and allocative inefficiencies in a flexible manner, and these specifications are tested for time-invariant technical inefficiency and both firm- and time-invariant allocative inefficiencies. A panel data on ten investor-owned electric utilities in Texas is used to illustrate empirical implementation of our model.

   The rest of the paper is organized as follows. The profit maximizing model is developed in the next section. First, the model is developed under the assumption that firms are technically efficient. Then we extend it to accommodate both technical and allocative inefficiencies. Finally, the panel data model is outlined. The data are described in section III. Section IV is devoted to estimation and discussion of results. Section V concludes the paper.

2. Formulation of the Model

   Only Allocative Inefficiency

   Consider the case where firms are technically efficient. The economic problem to a firm is to

   max [Pi] = pY - w'X

   (1) s.t. Y = f(X, t)

   and

   [R.sub.s](X, Y|p, w) = 0, s = 1, . . ., S

   where [Pi] is profit, Y is output and p is output price. X is a vector of J inputs and w is the vector of input prices. f ([multiplied by]) is the production technology and t is the time variable which is introduced to capture exogenous technical change. [R.sub.s] ([multiplied by]) are S additional constraints faced by the producers [3; 4]. Some of these constraints might be qualitative in nature and are often difficult to express in formal mathematical form.

   The first order conditions of the above maximization problem are

   (2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

   where [[Lambda].sub.s] is the Lagrange multiplier associated with the sth constraint in [R.sub.s] ([multiplied by]) = 0. The marginal product of input j is denoted by [f.sub.j](X, t), and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the shadow price of input j which differs from the observed price [w.sub.j] because of the presence of the constraints [R.sub.s] ([multiplied by]) = 0.

   From the first order conditions in (2) one can, in principle, solve for [X.sub.j], the unconditional input demand functions, viz.,

   (3) [X.sub.j] = [X.sub.j](??, t), j = 1, . . ., J,

   where ?? = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. These input demand functions can be substituted into the production function to get the output supply function

   (4) Y = Y(??, t).

   The shadow profit function is then obtained as

   [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

   from which the normalized shadow profit function ??(??, t) = [[Phi].sup.*]([w.sup.*],p, t)/p can be expressed as

   (5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

   The foregoing analysis shows that input use and output production decisions are based on shadow prices. This means that the producers are maximizing shadow profit instead of actual profit where shadow prices, not actual (observed) prices, are relevant in input use and output supply decisions.

   Thus, if one starts from a normalized shadow profit function, then the input demand functions can be derived using the duality results, viz.,

   (6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

   where

   (7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

   is the normalized actual profit, and ?? = [w.sub.j]/p.

   The empirical studies use either the input demand and output supply functions [16], or the profit function and the associated profit share equations [19; 3; 9; 12]. The specification in the latter models requires a precise relationship between the normalized shadow profit function and the normalized actual profit function. This relationship can be established from the definitions of the two profit functions in (5) and (7), viz.,

   [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

   or

   (8a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

   where

   (8b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

   Equations in (8a) and (8b) establish the precise relationship between the normalized actual profit function with the normalized shadow profit function In ?? (??, t), which has to be specified as a parametric function of ?? and t. Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are unobserved it is also necessary to express them in terms of observed prices. Here we follow Atkinson and Halvorsen [3] and specify ?? as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [[Theta].sub.j](j = 1, . . ., J) are unknown parameters. Using this specification the profit function in (8a) can be written as

   (9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

   Thus, once a parametric form of ln ??(??,t) is chosen, everything appearing in equation (9) is either data or unknown parameters to be estimated. Atkinson and Halvorsen [3], Hollas and Stansell [9], Kumbhakar and Bhattacharyya [12], and others estimated the above profit function using a translog normalized shadow profit function ??(??, t).

   It is to be noted that the above models assume firms to be technically efficient. The implications of ignoring technical inefficiency, if any, is considered next.

   Both Technical and Allocative Inefficiencies

   We now assume that a firm is both technically and allocatively inefficient, in which case the production function can be written as

   (10) Y =f(X, t)A

   where A [is less than or equal to] 1 represents firm-specific production technical efficiency. If A [is less than] 1, the output is less than maximum (given X and t) and the firm is said to be technically inefficient. Technical inefficiency is then measured by lnA, which shows the percentage by which actual output (Y) falls short of maximum possible output (f(X, t)), given X and t. Technical inefficiency can vary across firms as well as over time.

   Now the first order conditions in (2) can be written as

   (11) [MATHEMATICAL...