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Econometric models to estimate allocative and technical inefficiency include stochastic shadow distance frontiers, shadow cost frontiers, and shadow profit frontiers. In these models, the cost savings from eliminating both sources of inefficiency is often reported in total and then decomposed into the contribution of each source. This paper shows that this calculation, as formalized in Kumbhakar (1997) and used extensively in empirical applications over the last 25 years, is non-unique without additional information that is typically not available. The same results are obtained for shadow profit and shadow distance systems. The decomposition of cost (profit) savings is underidentified, since it is conditional on the arbitrarily normalized value for one shadow input (input or output) price when the shadow cost (profit) system is estimated. When the normalized value is rescaled, estimates of shadow costs will change by the scale factor, but the new model is observationally equivalent to the original one. We provide theoretical proofs and empirical examples of the restrictions (additional information) necessary to identify cost savings. Several methods of satisfying this decomposition requirement are discussed.


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