Scandinavian Working Papers in Economics

Working Papers in Economics,
University of Bergen, Department of Economics

No 05/07: Slopes of Shadow Prices and Lagrange Multipliers

Sjur Didrik Flåm (), Hubertus Th. Jongen () and Oliver Stein ()
Additional contact information
Sjur Didrik Flåm: University of Bergen, Department of Economics, Postal: Hermann Fossgt. 6, N-5007 Bergen
Hubertus Th. Jongen: RWTH Aachen University, Germany, Postal: Mathematics, Section C, Templergraben 55,, 52056 Aachen, Germany,
Oliver Stein: Karlsruhe Institute of Technology, Postal: Karlsruhe Institute of Technology, P.O. Box 3640, , 76021 Karlsruhe, Germany

Abstract: Many economic models and optimization problems generate (endogenous) shadow prices - alias dual variables or Lagrange multipliers. Frequently the “slopes” of resulting price curves - that is, multiplier derivatives - are of great interest. These objects relate to the Jacobian of the optimality conditions. That particular matrix often has block structure. So, we derive explicit formulas for the inverse of such matrices and, as a consequence, for the multiplier derivatives.

Keywords: Sensitivity; optimal value function; shadow price; Karush-Kuhn-Tucker system; matrix inversion.

JEL-codes: C60; C71

15 pages, February 28, 2007

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