Abstract
In an influential paper, Färe and Lovell (J Econ Theory 19:150–162, 1978) proposed an (input based) technical efficiency index designed to correct two fundamental inadequacies of the Debreu-Farrell index: its failure to satisfy (1) indication (the index is equal to 1 if and only if the input bundle is technically efficient) and (2) weak monotonicity (an increase in any one input quantity cannot increase the value of the index). Färe et al. (1985) extended the index to measure efficiency in the full space of input and output quantities. Unfortunately, this index fails to satisfy not only indication and monotonicity at the boundary (of output space), but also weak monotonicity. We show, however, that a simple modification of the index corrects these flaws. To demonstrate the tractability of our proposal, we apply it to baseball batting performance, in which zero outputs occur frequently.
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Notes
See Russell (1985).
Our modification is not needed to maintain indication and weak monotonicity of the input-based FL index at the boundary of input space.
All but free disposability of these conditions are necessary to guarantee that our efficiency indexes are well defined. Free disposability could be dispensed with (theoretically); the only change that would be needed in what follows would be to redefine the inefficiency indexes on the free-disposal hull of \(T, T+({\bf R}^n_{+} \times -{\bf R}^m_+),\) rather than on T itself (as in Russell 1987 for input-based efficiency indexes).
Vector notation: \(\bar{x}\ge x\) if \(\bar{x}_i\ge x_i\) for all \(i;\,\bar{x}> x\) if \(\bar{x}_i\ge x_i\) for all i and \(\bar{x}\ne x\); and \(\bar{x}\gg x\) if \(\bar{x}_i> x_i\) for all i.
FGL order coordinates so that the first k input quantities and first l output quantities are positive and then minimize only over the sum of these k + l coordinates, all of which have positive values. Our characterization does not require a permutation of the coordinates whenever a production vector (or a technology) is changed.
For the sake of symmetry, we weight each input-contraction factor by \(\delta(x_i),\,i=1,\ldots n\), but the index would be unaffected by omitting these weights, since the value of α i at the minimum is zero if x i = 0. In the initial specification of the FGL index (p. 154), α is restricted to be strictly positive, in which case their minimization problem has no solution if, for some i, x i > 0 and the minimum value of α i is zero; this case is illustrated in Fig. 2, where \(x^{\prime}\) would be contracted to \(\hat{x}\). In their characterization of the domain of the index (p. 153), however, they restrict α to be non-negative. We choose the latter formulation to ensure that the index is well-defined on the domain, which includes input and output quantities with zero values.
Note that we are able to use the min operator instead of inf even though the constraint set \(\Upomega(x,y,T)\) is not closed because, using inf, \({\mathop{\beta}\limits^{\ast}}_{j}=0\) only if δ(y j ) = 0, in which case β j does not appear in the objective function in (2.2).
Neither of the indexes we consider satisfies the stronger property of (strict) monotonicity. Nor does either satisfy continuity. See Russell and Schworm (2011) for details.
And replacing min with inf.
Note that we must use the infimum here, because ψ j (x, y, T) can be non-zero when y j = 0.
And, of course, in this formulation \({\mathop{\alpha}\limits^{\ast}}_i\) is an arbitrary selection from [0, 1] if δ(x i ) = 0. Note that, in addition to this arbitrariness, the “solution” values for α i or β j not associated with zero values of x i or y j need not be unique, since there can be ties in the optimization problem.
In fact, it will be lower, since the zeros in the optimal values of the numerator and denominator will be replaced by a zero in the numerator and a 1 in the denominator.
As noted above, the minimum is well defined (no attempt to divide by zero): since \(y_j^{d^{\prime}}\) is non-zero and the output constraint set is bounded, the solution value for each β j will be non-zero.
The programming codes are available at http://www.economics.ucr.edu/people/russell/index.html.
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Levkoff, S.B., Russell, R.R. & Schworm, W. Boundary problems with the “Russell” graph measure of technical efficiency: a refinement. J Prod Anal 37, 239–248 (2012). https://doi.org/10.1007/s11123-011-0241-3
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DOI: https://doi.org/10.1007/s11123-011-0241-3