Abstract
Determining the profit maximizing input–output bundle of a firm requires data on prices. This paper shows how endogenously determined shadow prices can be used in place of actual prices to obtain the optimal input–output bundle where the firm’s shadow profit is maximized. This approach amounts to an application of the Weak Axiom of Profit Maximization (WAPM) formulated by Varian [(1984) The Non-parametric approach to production analysis. Econometrica 52:3 (May) 579–597] based on shadow prices rather than actual prices. At these shadow prices, the shadow profit of a firm is zero. The maximum shadow profit that could have been attained at some other input–output bundle is shown to be a measure of the inefficiency of the firm. Because the benchmark input–output bundle is always an observed bundle from the data, it can be determined without having to solve any elaborate programming problem.
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Notes
Restricting inputs to decrease and/or outputs to increase avoids any trade off between input conservation and output expansion. This may be a valid approach when no behavioral assumption is made.
Pastor et al. (1997) introduced an extended Russell measure of efficiency defined as \({\Upgamma =\min \frac{\frac{1}{n}\sum_1^n {\theta_i}}{\frac{1}{m}\sum_1^m {\varphi_r}}}\). Ray (1998) and Ray and Jeon (2003) suggest using a linear approximation to the objective function \({{\Upgamma}}\) at all \({\phi_{r}}\) and \({\theta_{i}}\) set equal to unity. This amounts to minimizing \({\frac{1}{n}\sum_1^n {\theta_i} -\frac{1}{m}\sum_1^m {\varphi_r}}\). With this approximation, minimizing \({{\Upgamma}}\) would amount to maximizing the objective function in (16) with \({\alpha _r =\frac{1}{m}}\) for each output r and \({\gamma_i =\frac{1}{n}}\) for each input i.
This point was also noted by Zhu (2003).
This may, of course be true for multiple observations.
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This paper has benefited from very insightful comments from an anonymous referee on an earlier version of the manuscript. The usual disclaimer about responsibility for errors applies.
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Ray, S.C. Shadow profit maximization and a measure of overall inefficiency. J Prod Anal 27, 231–236 (2007). https://doi.org/10.1007/s11123-007-0036-8
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DOI: https://doi.org/10.1007/s11123-007-0036-8