Inner and Outer Approximations of Technology: A Shadow Profit Approach

In this paper we extend the taxonomy on inner and outer approximations to a technology by assuming that price data are not available. Mimicking Varian [Econometrica, 52(3) (1984) 579] we introduce a Weak Axiom of Shadow Profit Maximization (WASPM) to test if observed production plans are compatible with technically efficient behavior. If the test fails for an observed sample, we then characterize the maximal subset of observed production plans that meets WASPM and we derive lower and upper bounds on technical efficiency for production plans that are observed but not in this subset. We also derive linear programs to implement these bounds.