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.
Similar content being viewed by others
References
Arrow, K.J.: Liquidity preference, Lecture IV in Lecture Notes for Economics 285, The Economics of Uncertainty, pp. 33-53. Stanford University (1963)
Aubin J.-P. (1984). Lipschitz behavior of solutions to convex minimization problems. Math. Oper. Res. 9: 87–111
Crouzeix J.P. (1977). A relationship between the second derivatives of a convex function and of its conjugate. Math. Program. 13: 364–365
de Finetti B. (1952). Sulla preferibilita. G. Econ. Ann. Econ. 11: 685–709
Dorf C., Svoboda J.A.: Introduction to Electric Circuits, Wiley, New York
Evstigneev I.V. and Flåm S.D. (2001). Sharing nonconvex cost. J. Glob. Optim. 20: 257–271
Fiacco A.V. and McCormick G.P. (1968). Nonlinear Programming: Sequential Unconstrained Minimization Techniques. Wiley, New York
Flåm, S.D., Godal, O.: Greenhouse gases, quota exchange and oligopolistic competition, In: Carraro, C., Fragnelli, V. (eds.) Game Practice and the Environment, Edward Elgar, Cheltenham, UK (2004)
Flåm S.D. and Jourani A. (2003). Strategic behavior and partial cost sharing. Games Econ. Behav. 43: 44–56
Gabszewicz, J.J.: Strategic Multilateral Exchange, Edward Elgar, Cheltenham, UK (2002)
Harville D.A. (1999). Matrix Algebra from a Statistician’s Perspective. Springer, New York
Henriet D. and Rochet J.-C. (1987). Some reflexions on insurance pricing. Eur. Econ. Rev. 31: 863–885
Henriet, D., Rochet, J.-C.: Microeconomie de l’assurance, Economica, Paris (1991)
Jongen H.Th., Jonker P. and Twilt F. (1986). One-parameter families of optimization problems: equality constraints. J. Optim. Theory Appl. 48: 141–161
Jongen H.Th., Möbert T. and Tammer K. (1986). On iterated optimization in nonconvex optimization. Math. Oper. Res. 11: 679–691
Jongen H.Th., Möbert T., Rückmann J. and Tammer K. (1987). On inertia and Schur complement in optimization. Linear Algebra Appl. 95: 97–109
Jongen H.Th., Meer K. and Triesch E. (2004). Optimization Theory. Kluwer, Boston
Klatte D. and Kummer B. (2002). Nonsmooth equations in optimization. Kluwer, Boston
Laurent P.-J. (1972). Approximation et optimisation. Hermann, Paris
Magill M. and Quinzii M. (1996). Theory of Incomplete Markets. The MIT Press, Cambridge
Pratt J.W. (1964). Risk aversion in the small and the large. Econometrica 32: 122–136
Rockafellar R.T. and Wets J.-B. (1998). Variational Analysis. Springer, Berlin
Shapley L.S. and Shubik M. (1969). On market games. J. Econ. Theory 1: 9–25
Author information
Authors and Affiliations
Corresponding author
Additional information
S. D. Flåm gratefully acknowledges support from Ruhrgas.
O. Stein gratefully acknowledges support through a Heisenberg grant of the Deutsche Forschungsgemeinschaft.
Rights and permissions
About this article
Cite this article
Flåm, S.D., Jongen, H.T. & Stein, O. Slopes of shadow prices and Lagrange multipliers. Optimization Letters 2, 143–155 (2008). https://doi.org/10.1007/s11590-007-0048-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11590-007-0048-3