Abstract
A competitive firm is considered to be inefficient when its observed profit falls short of the maximum possible profit at the applicable prices of inputs and outputs. Two alternative measures of performance of the firm are the ratio of the actual to the maximum profit and the difference between the maximum and the actual profit. The ratio measures its profit efficiency and is naturally bounded between 0 and 1 so long as the actual profit is strictly positive. However, the possibility of negative actual profit and zero profit at the maximum has prompted many researchers to opt for the difference measure of profit inefficiency, which is necessarily non-negative. For a meaningful comparison of performance across firms, however, the difference needs to be appropriately normalized to take account of differences in the scale of operation of firms. Three common variables used for normalization are the observed cost, the observed revenue, and the sum of revenue and cost. It is a common practice to measure the separate contributions of technical and allocative efficiencies to the overall profit efficiency of the firm. When the firm is not operating on the frontier of the production possibility set, there are many ways to project it on to the frontier. This leads to different decomposition of profit efficiency into technical and allocative components. In this paper, we consider McFadden’s gauge function and an endogenous projection based on the overall efficiency of the firm along with the usual input- or output-oriented distance functions, the graph hyperbolic distance function, and a slack-based Pareto Koopmans efficiency measure for technical efficiency. We show that in a multiplicative decomposition of profit efficiency, the technical efficiency component of profit efficiency is independent of input-output prices only from the projection based on the gauge function. Also, an empirical application using data from Indian banks shows that the alternative normalized difference measures of inefficiency generate different performance ranking of the firms. Thus, comparative evaluation of performance of firms depends on the analyst’s preference for a specific type of normalization.
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Notes
CCF (1998) describes this as the size of the firm.
Aparicio, Zofío and Pastor (2023) consider alternative measures of profit (in)efficiency and their additive and multiplicative decomposition. However, they require that the technically efficient projection of any inefficient unit should lie in the strongly efficient subset of the frontier. This often excludes radial projections. Besides, they select the profit-efficient bundle itself as the technically efficient projection of an inefficient input-output bundle. This leads to their claim that all profit inefficiency is either technical or allocative.
Although with free exit long-run profit will be non-negative, the actual profit may be negative in a specific case.
We address this point again in Section 5.2 again
By contrast, (ray) average productivity increases when outputs are radially scaled up or inputs are radially scaled down.
Thus, \(H\left( {x,\,y} \right) = \gamma = \frac{1}{{1 + \beta }}\).
This amounts to explicitly imposing a non-negativity restriction on βy. But if θ exceeds 1, βx in (22) can still be negative.
The endogenous efficient projection cannot be shown directly in the same diagram. Also, in the 1-input 1-output example, there is no input or output slack in this diagram and every point on the frontier is Pareto Koopmans efficient.
FHLZ (2019) provides a multiplicative decomposition of their output-oriented profit efficiency into three separate components that relate to potential for a radial expansion of the observed output vector y0 without any change in the input x0, allocative efficiency component of the revenue efficiency of \(R_0 = p^\prime y_0\) from input x0, and a residual factor. Their output-oriented profit efficiency (PEo) is different from the ratio measure profit efficiency considered above and more like a difference measure normalized by the actual revenue of the firm.
This example is not an in-depth study of the performance of Indian banks, which is beyond the scope of this paper.
We aim to distinguish technical efficiency and technical inefficiency in this table. Unlike other measures, DDF Pure Technical Inefficiency (TIE_n) and Endogenous DDF Pure Technical Inefficiency (TIE_r) quantify the level of inefficiency. Therefore, in contrast with other measures, a bigger number of technical inefficiency indicates lower efficiency.
Note that when the DDF takes the value 0, all the difference between the maximum and the actual profit is ascribed to allocative inefficiency.
It has lower Nerlovian inefficiency than 7 banks. When the difference is normalized by either actual revenue or actual cost, it is found to perform better than 11 banks.
\(1 - \frac{{\pi _0}}{{\pi ^ \ast }} = \frac{{\pi ^ \ast - \pi _0}}{{\pi ^ \ast }} = \frac{{\Delta}}{{\pi ^ \ast }}\).
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Ray, S.C., Yang, L. Measurement and decomposition of profit efficiency under alternative definitions in nonparametric models. J Prod Anal (2024). https://doi.org/10.1007/s11123-024-00720-8
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DOI: https://doi.org/10.1007/s11123-024-00720-8