Advertisement

Market exchange models and geometric programming

  • Marianna Eisenberg-Nagy
  • Tibor IllésEmail author
  • Gábor Lovics
Original Paper

Abstract

Finding the equilibrium solution of the Market Exchange Models is an interesting topic. Here we discuss some relationship of the Fisher type Homogenous Market Exchange Model and the Geometric Programming. We also discuss that the equilibrium solution of the Linear and Cobb–Douglas Market Exchange Model as two special cases of the Homogenous Market Exchange Model can be found by Geometric Programming in polynomial time.

Keywords

Market exchange models Homogeneous concave utility functions Geometric programming Fisher type homogeneous market exchange models Linear and Cobb-Douglas utility functions 

Notes

Acknowledgements

This research has been partially supported by the Hungarian Research Fund, OTKA (Grant No. NKFIH 125700). The research of T. Illés and M. Eisenberg-Nagy has been partially supported by the Higher Education Excellence Program of the Ministry of Human Capacities in the frame of Artificial Intelligence research area of Budapest University of Technology and Economics (BME FIKP-MI/FM). Tibor Illés acknowledges the research support obtained as a part time John Anderson Research Lecturer from the Management Science Department, Strathclyde University, Glasgow, UK.

References

  1. Arrow KJ, Debreu G (1954) Existence of an equilibrium for a competitive economy. Econometrica 22:265–290CrossRefGoogle Scholar
  2. Csizmadia Z (2007) New pivot based methods in linear optimization, and an application in petroleum industry. Ph.D. thesis, Eötvös Loránd University of Sciences, Budapest, HungaryGoogle Scholar
  3. Csizmadia Z, Illés T (2006) New criss-cross type algorithms for linear complementarity problems with sufficient matrices. Optim Methods Softw 21(2):247–266CrossRefGoogle Scholar
  4. Csizmadia Z, Illés T, Nagy A (2013) The s-monotone index selection rule for criss-cross algorithms of linear complementarity problems. Acta Universitatis Sapientiae - Informatica 5(1):103–139CrossRefGoogle Scholar
  5. Csizmadia A, Csizmadia Z, Illés T (2018) Finiteness of the primal quadratic simplex method when s-monotone index selection rules are applied. Cent Eur J Oper Res.  https://doi.org/10.1007/s10100-018-0523-1 Google Scholar
  6. Devanur NR, Papadimitriou CH, Saber A, Vazirani VV (2008) Market equilibrium via a primal-dual algorithm for a convex program. J ACM 55:22–40CrossRefGoogle Scholar
  7. Devanur NR, Garg J, Végh AL (2016) A rational convex program for linear Arrow–Debreu markets. ACM Trans Econ Comput 5(1):13 article 6CrossRefGoogle Scholar
  8. Duan R, Mehlhorn K (2015) A combinatorial polynomial algorithm for the linear Arrow–Debreu market. Inf Comput 243:112–132CrossRefGoogle Scholar
  9. Duan R, Garg J, Mehlhorn K (2016) An improved combinatorial polynomial algorithm for the linear Arrow–Debreu market. In: Proceedings of the twenty-seventh annual ACM-SIAM symposium on discrete algorithms, pp 90–106Google Scholar
  10. Eaves BC (1985) Finite solution of pure trade markets with Cobb–Douglas utilites. In: Economic equilibrium: model formulation and solution. Matehemathical programing studies. North Holland, AmsterdamGoogle Scholar
  11. Eisenberg E (1961) Aggregation of utility function. Manag Sci 4:337–350CrossRefGoogle Scholar
  12. Eisenberg E, Gale D (1959) Consensus of subjective probabilities: the pari-mutuel method. Ann Math Stat 30:165–168CrossRefGoogle Scholar
  13. Gale D (1960) The theory of linear economic models. McGraw-Hill Book Company, New YorkGoogle Scholar
  14. Grag J, Mehta R, Sohonoi M, Vishnoi NK (2013) Towards polynomial simplex-like algorithm for market equilibria. In: Proceedings of the twenty-fourth annual ACM-SIAM symposium on discrete algorithms, p 17Google Scholar
  15. Grag J, Mehta R, Sohonoi M, Vazirani VV (2015) A complementary pivot algorithm for market equilibrium under separable, piecewise-linear concave utilities. SIAM J Comput 44(6):1820–1847CrossRefGoogle Scholar
  16. Hardy G, Littlewood JE, Pólya G (1939) Inequalites. Cambridge University Press, LondonGoogle Scholar
  17. Hertog D, Jarre F, Roos C, Terlaky T (1995) A sufficient condition for self-concordance, with application to some classes of structured convex programming problems. Math Progr 69:75–88Google Scholar
  18. Jain K (2007) A polynomial time algorithm for computing an Arrow–Debreu market equilibrium for linear utilities. SIAM J Comput 37(1):303–318CrossRefGoogle Scholar
  19. Klafszky E (1976) Geometric Programming. IIASA Systems Analisys and Related Topics, 11Google Scholar
  20. Klafszky E (1981) The determinantion of equilibrium prices of linear exchange models by geometric programming. Alkalmazott Matematikai Lapok 7:139–157 (in Hungarian)Google Scholar
  21. Luenberger DG, Ye Y (2008) Linear and nonlinear progrmaing, 3rd edn. Springer, New YorkCrossRefGoogle Scholar
  22. Mas-Colell A, Whinston MD, Green JR (1995) Microeconomic theory. Oxford University Press, New YorkGoogle Scholar
  23. McKenzie L (1954) On equilibrium in Graham’s model of world trade and other competitive systems. Econometrica 22:147–161CrossRefGoogle Scholar
  24. Nagy A (2014) On the theory and applications of flexible anti-cycling index selection rules for linear optimization problems. Ph.D. thesis, Eötvös Loránd University of Sciences, Budapest, HungaryGoogle Scholar
  25. Nissan N, Roughgarden T, Tardos E, Vazirani V (2007) Algorithmic game theory. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  26. Orlin JB (2010) Improved algorithms for computing Fisher’s market celaring prices. In: Proceedings of STOC. ACM, pp 291–300Google Scholar
  27. Shmyrev VI (2009) An algorythm for finding equibrium in the linear exchnage model with fixed bugets. J Appl Ind Math 3(4):505–518CrossRefGoogle Scholar
  28. Sydsaeter K, Hammond P (1995) Mathematics for economic analysis. Prentice Hall, Upper Saddle RiverGoogle Scholar
  29. Sydsaeter K, Hammond P, Seierstad A, Strom A (2008) Further mathematics for economic analysis, 3rd edn. Pearson Education Limited, HarlowGoogle Scholar
  30. Vazirani VV, Yannakakis M (2011) Market equilibrium under separable, piecewise-linear, concave utilities. J ACM 58(3):10CrossRefGoogle Scholar
  31. Végh AL (2014) Concave generalized flows with applications to market equilibria. Math Oper Res 39(2):573–596CrossRefGoogle Scholar
  32. Végh AL (2016) A strongly polynomial algorithm for a class of minimum-cost flow problems with separable convex objectives. SIAM J. Comput 45(5):1729–1761CrossRefGoogle Scholar
  33. Walras L (1874) Éléments d’Économie Politique Pure; ou, Théorie de la Richesse Sociale. Lausanne, RougeGoogle Scholar
  34. Ye Y (2006) A Path to the Arrow\(-\)Debreu competitive market equilibrium. Math. Program. Ser. B 111:315–348Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Budapest University of Technology and Economics, Institute of MathematicsBudapestHungary

Personalised recommendations