General Spatial Price Equilibria: Sensitivity Analysis for Variational Inequality and Nonlinear Complementarity Formulations

  • Roger L. Tobin
Conference paper
Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 249)


General spatial price equilibrium models are formulated as variational inequalities and as nonlinear complementarity problems. Sensitivity analysis results recently developed for variational inequalities and nonlinear complementarity problems are reviewed which give conditions for existence and equations for calculating the derivatives of solution variables with respect to perturbation parameters. These results are applied to the formulations of general spatial price equilibria; derivatives of prices, supplies, demands, and flows with respect to perturbations of supply functions, demand functions, and transportation cost functions are calculated.


Variational Inequality Equilibrium Problem Linear Complementarity Problem Supply Function Solution Variable 
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  1. Chao, G.S. and T.L. Friez. “Spatial Price Equilibrium Sensitivity-Analysis.” Transportation Research 18B (1984), 423–440.Google Scholar
  2. Dafermos, S. “An Iterative Scheme for Variational Inequalities.” Mathematical Programming 26 (1983), 40–47.CrossRefGoogle Scholar
  3. Dafermos, S. and A. Nagurney. “Sensitivity Analysis for the General Spatial Economic Equilibrium Problem.” Operations Research 32 (1984), 1069–1086.CrossRefGoogle Scholar
  4. Fiacco, A.V. Introduction to Sensitivity and Stability Analysis in Nonlinear Programming. New York: Academic Press, 1983.Google Scholar
  5. Florian, M. and M. Los. “A New Look at Static Spatial Price Equilibrium Models.” Regional Science and Urban Economics 12 (1982), 579–597.CrossRefGoogle Scholar
  6. Friesz, T.L., R.L. Tobin, and P.T. Harker. “Variational Inequalities and Convergence of Diagonalization Methods for Derived Demand Network Equilibrium Problems.” Report CUE-FNEM-1981-10-1, Department of Civil and Urban Engineering, University of Pennsylvania, 1981.Google Scholar
  7. Friesz, T.L., R.L. Tobin, T.E. Smith, and P.T. Harker. “A Nonlinear Complementarity Formulation and Solution Procedure for the General Derived Demand Network Equilibrium Problem.” Journal of Regional Science 23 (1983), 337–359.CrossRefGoogle Scholar
  8. Friesz, T.L., P.T. Harker, and R.L. Tobin. “Alternative Algorithms for the General Network Spatial Price Equilibrium Problem. Journal of Regional Science 24 (1984), 475–507.CrossRefGoogle Scholar
  9. Irwin, CL. and C.W. Yang. “Iteration and Sensitivity for a Spatial Equilibrium Problem with Linear Supply and Demand Functions. Operations Research 30 (1982), 319–335.CrossRefGoogle Scholar
  10. Nagurney, A.B. “Stability, Sensitivity Analysis and Computation of Competitive Network Equilibria, PhD. dissertation. Division of Applied Mathematics, Brown University, Providence, R.I., 1983.Google Scholar
  11. Pang, J. and D. Chan. “Iterative Methods for Variational and Complementarity Problems.” Mathematical Programming 24 (1982), 284–313.CrossRefGoogle Scholar
  12. Samuelson, P.A. “Spatial Price Equilibrium and Linear Programming.” American Economic Review 42 (1952), 283–303.Google Scholar
  13. Takayama, T. and G.G. Judge. “Equilibrium Among Spatially Separated Markets: A Reformulation.” Econometrica 32 (1964), 510–524.CrossRefGoogle Scholar
  14. Takayama, T. and G.G. Judge. 1971. Spatial and Temporal Price and Allocation Models. Amsterdam: North Holland, 1971.Google Scholar
  15. Tobin, R.L. “Sensitivity Analysis for Variational Inequalities.” Journal of Optimization Theory and Applications, forthcoming.Google Scholar
  16. Tobin, R.L. and T.L. Friesz. “Formulating and Solving the Spatial Price Equilibrium Problem with Transshipment in Terms of Arc Variables.” Journal of Regional Science 23 (1983), 187–198.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1985

Authors and Affiliations

  • Roger L. Tobin
    • 1
  1. 1.Environmental Research DivisionArgonne National LaboratoryArgonneUSA

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