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Existence of Optimal Quantities and Shadow Prices with No Duality Gap

In: The Short-Run Approach to Long-Run Equilibrium in Competitive Markets

Author

Listed:
  • Anthony Horsley
  • Andrew J. Wrobel

Abstract

Once a pair of solutions (to a primal-dual programme pair) is found, a direct comparison of their values will show whether there is a duality gap. But there is also a method of checking for a gap at the outset—before solving the programmes. Namely, absence of a duality gap is equivalent to Type One semicontinuity of either optimal value, primal or dual (i.e., to semicontinuity of the primal value w.r.t. the primal parameters, or of the dual value w.r.t. the dual parameters). This well-known result—given in, e.g., [44, Theorem 15] and [36, 7.3.2]—is next stated for the SRP, LRC and SRC optimization programmes.

Suggested Citation

  • Anthony Horsley & Andrew J. Wrobel, 2016. "Existence of Optimal Quantities and Shadow Prices with No Duality Gap," Lecture Notes in Economics and Mathematical Systems, in: The Short-Run Approach to Long-Run Equilibrium in Competitive Markets, chapter 0, pages 119-135, Springer.
    • Handle: RePEc:spr:lnechp:978-3-319-33398-4_6
    DOI: 10.1007/978-3-319-33398-4_6
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