Abstract
The standard assumption in the efficiency literature, that firms attempt to produce on the production frontier, may not hold in markets that are not perfectly competitive, where the production decisions of all firms will determine the market price, i.e., an increase in a firm’s output level leads to a lower market clearing price and potentially lower profits. This paper models both the production possibility set and the inverse demand function, and identifies a Nash equilibrium and improvement targets which may not be on the production frontier when some inputs or outputs are fixed. This behavior is referred to as rational inefficiency because the firm reduces its productivity levels in order to increase profits. For a general short-run multiple input/output production process, which allows a firm to adjust its output levels and variable input levels, the existence and the uniqueness of the Nash equilibrium is proven. The estimation of a production frontier extends standard market analysis by allowing benchmark performance to be identified. On-line supplementary materials include all proofs and two additional results; when changes in quantity have a significant influence on price and all input and outputs are adjustable, we observe more benchmark production plans on the increasing returns to scale portion of the frontier. Additionally, a direction for improvement toward the economic efficient production plan is estimated, thus providing a solution to the direction selection issue in a directional distance analysis.
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Notes
A production function is commonly defined as the maximum set of output(s) that can be produced with a given set of inputs. Thus, we will use the terms production function and production frontier interchangeable as is commonly done in the productivity and efficiency literature [2].
A firm is said to be rationally inefficient when they intentionally lower productivity levels to maximize revenues or profits, or alternatively, minimizes costs.
Firms will either expand their outputs, contract their inputs, or both, depending on the cost/price structure of inputs/outputs and adjustment costs associated with changing input levels. For now, we will assume input adjustment costs are very large and consider only output adjustment consistent with an output-oriented efficiency analysis in the efficiency literature [15]. This assumption is relaxed in Sect. 4.
Fig. 1 
Economic efficiency and production frontier
Fig. 2 
Change in supply and equilibrium price
This is consistent with an output-oriented efficiency analysis in the productivity literature [15].
This is consistent with a profit maximization model, given fixed input prices and levels.
The output level \(\bar{y}_r\) will be chosen by all firms. Clearly, the firms using input levels less than \(X_r\) will only be able to produce the output level defined by the production frontier. And firms using more than \(X_\mathrm{r}\) input cannot adjust their input levels by assumption, so they will produce \(\bar{y}_r\) with more than \(X_\mathrm{r}\) input.
Note that all output variables need to be normalized in data pre-processing to eliminate unit dependence.
For a discussion of the relationship among these properties see the weak, moderate, and strong dominance section in the on-line electronic supplementary material.
Note the exchange of \(q\) and \(h\).
This result is illustrated in Table 2, Case 2, DC 5.
In a special case, in which input markets are perfectly competitive \(\beta _{jl} =0\), the inverse supply function will be constant and the cost function becomes a linear function. This does not affect the optimality condition, i.e., the profit function is still a strictly concave function if the revenue function is strictly concave.
Weakly efficient frontier is defined as the portion of the input (output) isoquant along which one of the inputs (outputs) can be reduced (expanded) while holding all other netputs constant and remaining on the isoquant; see Färe and Lovell [34] for more details.
Bougnol and Dulá [35] propose a procedure to identify anchor points and show that, if a point is an anchor point, then increasing an input or decreasing an output generates a new point on the free-disposability portion of the production possibility set.
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Acknowledgments
The authors thank Steven Puller and Robin C. Sickles for providing constructive suggestions and helpful discussion. Any errors are the responsibilities of the authors. This research was partially funded by National Science Council, Taiwan (NSC101-2218-E-006-023), and Research Center for Energy Technology and Strategy at National Cheng Kung University, Taiwan.
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Communicated by Kaoru Tone.
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Lee, C.Y., Johnson, A.L. Measuring Efficiency in Imperfectly Competitive Markets: An Example of Rational Inefficiency. J Optim Theory Appl 164, 702–722 (2015). https://doi.org/10.1007/s10957-014-0557-z
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DOI: https://doi.org/10.1007/s10957-014-0557-z