Figure 6.1.
X1/Y
X 2/Y
only) to produce a single output y, so that the production frontier
inefficiency of the firm, measures the proportion by which fJf1>x23 could be reduced (holding the input ratio x / x constant) without
be shown to be approximately equal to the sum of technical and allocative inefficiency.
Even if we relax the assumption of constant returns to scale, we can still think in terms of Figure (6.1), where the units of measurement on the axis become simply, x a and x^. Therefore, if the
* * * * *
firm uses ( x 1 >x 2 ^ to produce y , point P is simply (x ,x ). The measures of technical, allocative, and total efficiency are, as before, OQ/OP, OR/OQ, and OR/OP, respectively. This is suggested by Kopp (1981) amongst others.
Another potential problem, as pointed out by Forsund and Hjalmarsson (1974), is that unless technology is homothetic (a production function is said to be homothetic if f(x) can be written as M g ( x ) where h is monotonic and g is homogenous of degree 1), the breakdown between technical and allocative inefficiency requires some assumption about what the firm’s expected output was.
6.3. Approaches to Production Frontiers.
There is a fundamental difference between statistical and non-statistical approaches to production frontiers. A statistical approach depends on assumptions about the stochastic properties of the data, while a non-statistical approach does not.
Among non-statistical approaches, there is a further
distinction between those that are parametric and those that are non-parametric. Basically, parametric approaches assume a particular functional form(e.g., Cobb-Douglas, CES, translog) for the production or cost function, while non-parametric approaches do not.
6.4. The Pure Programming Approach.
6.4.1. Data Envelopment Analysis.
This is a non-statistical, non-parametric efficiency measurement technique known as "data envelopment analysis" (DEA). It was originally developed by Charnes, Cooper and Rhodes(1978) (see also Banker 1984, Banker, Charnes and Cooper 1984) as a new technique in operations research for measuring and comparing the relative efficiency of a set of decision-making units (DMUs).
The DEA approach utilises a sequence of linear programmes to construct a transformation frontier and to compute primal and dual efficiency relative to the frontier. It applies the basic concept of Pareto Optimality by stipulating that a given DMU is not relatively efficient, if it can be shown that some other DMU or, combination of DMUs can produce more of some outputs without producing less of any other and without utilising more of any input. This technique has been found very useful in measuring efficiency for various public sector DMUs and/or quasi-market or non-market agencies e.g., schools, recruitment and training programmes in defense industries,
hospitals, extension services and family planning programmes, where price data are mostly unavailable and there are multiple goals pursued. Sengupta (1988) provides a good overview of the current status of the DEA approach emphasising mainly its applied economic and econometric aspects.
6.4.2. Consistency Approach Through Data Adjustment.
Two major questions arise when the efficiency hypothesis characterised by a production frontier or, an efficient production set is set up. One is: how can we estimate the production frontier when the observed data have the property that they contain points not satisfying the efficiency hypothesis? A second question concerns the consistency of the data. A literature has developed (e.g., Afriat (1972), Hanoch and Rothschild (1972), Diewert and Parkan
(1983), Varian (1984)), on non-parametric tests of certain hypothesis i.e., one can test the consistency of the data with hypothesis such as (i) the existence of a production function; (ii) constant returns to scale; (iii) homotheticity; or (iv) cost minimisation, without assuming a functional form for the production or cost function.
These tests involve checking for the satisfaction of certain inequalities, often by the solution of some linear program. Varian (1984), for example, gives a condition which is necessary and sufficient for the existence of a production function which
'rationalises’ the data in the sense that the data could be generated by cost-minimising behaviour given that production function. Given that the condition holds, Varian goes on to derive the tightest possible inner and outer bounds for any (set of) production functions which rationalise the data. His inner bound is essentially identical to the (set of) production functions constructed by the DEA approach in the sense that their efficient subsets coincide.
Banker and Maindiratta (1988) extend Varian’s work to the case in which the data do not satisfy Varian’s rationalisability condition; that is, the data could not (all) have arisen from cost-minimising behaviour. Hence, they introduce the concept of subset rationalisation, in which they construct inner and outer bounds for all possible (set of) production functions that rationalise the rationalisable subset of observations. The inner bound is essentially the same as Varian’s, while the outer bound is the same as Varian’s except that it is computed only from the rationalisable subset of observations. By using this subset rationalisation criterion they have developed technical, allocative and aggregate efficiency measures, which are consistent with Farrell’s approach.
Another way to interpret the data consistency problem is in terms of the existence of suitable Lagrange multipliers, which is the approach of Diewert and Parkan. For a good resumee of this approach, see Sengupta (1988).
Perhaps the most advantageous characteristic of the pure programming approach is that the (set of) production functions it constructs are the smallest well-behaved set containing all the data. Such a set is piecewise linear, and the construction process achieves considerable flexibility because the breaks among the pieces are determined endogenously so as to fit the data as closely as possible, (see for example, Banker and Maindiratta (1986), and Charnes, Cooper, Seiford, and Stutz (1982,1983)).
But, the major problem with the pure programming approach lies in the fact that the sample data are enveloped by a deterministic frontier. Consequently the entire deviation of an observation from the frontier is attributed to inefficiency. Since the frontier is non-stochastic, no accommodation is made for environmental heterogeneity, random external shocks, noise in the data, measurement error, omitted variables etc.. All sorts of influences, favourable and unfavourable, beyond the control of the firm are combined together with inefficiency and called inefficiency.
Furthermore, since the approach is non-stochastic, there is no way of making probability statements about the shape and placement of the frontier, or about the computed inefficiencies relative to the f rontier.
6.5. The Modified Programming Approach.
This approach also uses a sequence of linear or quadratic
programming techniques to construct a transformation frontier and to compute primal and dual efficiency relative to the frontier. The only difference between the modified and pure programming approaches is that the frontier constructed by the modified programming approach is parametric. The modified programming approach was first suggested by Farrell (1957), and has been refined and extended by Ainger and Chu (1968), Forsund and Jansen (1977), and Forsund and Hjalmarsson (1979a,b) among others.
The modified programming approach has two drawbacks that limit its appeal. The first is that the approach, like the pure programming approach, is entirely deterministic, with no allowance made for noise, measurement error, etc.. Since the computed frontier is supported by a subset of the data, its shape and placement are highly sensitive to outliers. It is this deficiency that led to the development of probabilistic production frontiers by Timmer (1971), in which he eliminated a certain percentage of the total observations. Such a selection procedure, however, is not based on statistical theory, making hypothesis testing impossible.
Thus the neglect of the statistical error is a serious disadvantage of this method.
The second drawback of the modified programming approach is its inability to deal easily with multiple outputs. A remedy draws on a proposal of Kopp and Diewert (1982) and Zieschang (1983) to compute primal and dual efficiency relative to the dual cost frontier.
6.6. The Deterministic Statistical Frontier Approach.
This approach, in contrast to the two programming approaches, uses statistical techniques to estimate a transformation frontier and to estimate primal and dual efficiency relative to the estimated frontier. The technique was first proposed by Afriat (1972) and has been extended by Richmond (1974) and Green (1980a,b), among others.
A one-sided (non-positive) disturbance is explicitly assumed, of some particular form (e.g., truncated normal, exponential or gamma).
As in both programming approaches, the data are enveloped by a deterministic frontier. As in the modified programming approach, the deterministic frontier is parametric. In contrast to both programming approaches, the deterministic frontier is estimated rather than computed.
Schmidt (1976) showed that the Ainger-Chu linear programming
"estimator" is the maximum likelihood estimator (MLE) if the errors are exponential, while their quadratic programming "estimator" is the MLE if the errors are half-normal. However, since the regularity conditions for the consistency and asymptotic normality of MLEs are violated by this specification, (namely, that the range of the random variable should not depend on the parameters), estimation of frontiers is not completely straightforward since the properties of the MLEs are, in general, uncertain. This range problem which was pointed out by Schmidt (1976), was partially solved by Green (1980a), who found sufficient conditions on the distribution of the error term such that maximum likelihood is consistent and
asymptotically efficient. A gamma distribution for the error term satisfies these conditions, for example.
Since a deterministic statistical primal or dual frontier and their related efficiencies are estimated by statistical techniques, a large sample size is required. Furthermore, it is a disadvantage to have to specify a distribution for technical efficiency if a production frontier is estimated, or for allocative efficiency if a cost frontier is estimated. Ideally the specification would be based on a knowledge of the forces, economic or otherwise, that generate that inefficiency. However such information is rarely available.
There being no a priori arguments for a particular distribution, choice is typically based on analytical manageability.
Unfortunately, estimates of the parameters of the exogenous variables and of the magnitude of efficiency are not invariant with respect to the specification of a distribution for the efficiency term. Specification tests to evaluate half-normal and truncated normal distributions have been developed by Lee (1983a) for stochastic frontier models. These can be applied to deterministic frontier models as well. The advantage of a statistical approach is the possibility of statistical inference based on the results, although such inference is conditional on the specified distribution being the true distribution.
6.7. The Stochatic Frontier Approach.
Like the deterministic statistical frontier approach, the stochastic frontier approach uses statistical techniques to estimate a transformation frontier and to estimate efficiency relative to the estimated frontier. In contrast to the deterministic statistical frontier approach, but in accordance with the typical nonfrontier approach to the estimation of economic relationships, this allows the frontier to be stochastic.
6.7.1. Technical Efficiency Only.
The technique was first proposed by Ainger, Lovell, and Schmidt (1977), and Meeusen and van den Broek (1977). Their approach takes into account statistical error and uses a production model with a composed error. This filters out the statistical error and calculates a less biased efficiency measure. Composed error distributions which have been employed in the literature include:
the half-normal and exponential distributions proposed by Ainger et al. (1977) (among others), the truncated normal proposed by Stevenson (1980), and the two- parameter Gamma distribution proposed by Greene (1990). Tests of the appropriateness of these distributions can be made using Lagrange multiplier techniques proposed by Lee (1983) and Schmidt and Lin (1984).
To illustrate the basic econometric approach to estimating technical efficiency using a stochastic frontier, consider a