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Positive Solutions of Real Homogeneous Algebraic Inequalities

  • Alexander Shananin
  • Sergey TarasovEmail author
Conference paper
Part of the Smart Innovation, Systems and Technologies book series (SIST, volume 133)

Abstract

We show a simple and explicit reduction of the problem of the existence of positive solution of a system of real algebraic homogeneous inequalities in \( R_{ + }^{n} \) to solvability of a standard concave programming problem, which in turn is equivalent to checking whether a particular constraint of a finite system of convex inequalities is redundant. We illustrate this result on two well-known problems in mathematical economics: the weak separability problem and the collective consumption behavior for the homogeneous utilities.

Keywords

Convexity Redundant constraint Posinomials Utility function Economic indices theory Collective axiom of revealed preference Weak separability property 

Notes

Acknowledgements

The first author is partially supported by Russian Foundation for Basic Research, project No. 17-51-150001. The second author is partially supported by Russian Foundation for Basic Research, project No. 17-0700300.

References

  1. 1.
    Kondrakov, I., Pospelova, L., Shananin, A.: Generalized nonparametric method. Applications to the analysis of commodity markets. Proc. MIPT. 2, 32–45 (2010)Google Scholar
  2. 2.
    Petrov, A., Shananin, A.: Integrability conditions, income distribution, and social structures. In: Tangyan, A., Gruber, J. (eds.) Constructing Scalar-valued Utility Functions. LNEMS. vol. 453, pp. 271–288. Springer (1998)Google Scholar
  3. 3.
    Klemashev, N., Shananin, A.: Inverse problems of demand analysis and their applications to computation of positively-homogeneous Konus–Divisia indices and forecasting. J. Inverse Ill-Posed Prob. 24(4) (2015).  https://doi.org/10.1515/jiip-2015-0015
  4. 4.
    Afriat, S.: The construction of utility functions from expenditure data. Int. Econ. Rev. 8(1), 66–77 (1967)CrossRefGoogle Scholar
  5. 5.
    Varian, H.R.: Non-parametric tests of consumer behaviour. Rev. Econ. Stud. 60(1), 99–110 (1983)CrossRefGoogle Scholar
  6. 6.
    Blackorby, C., Primont, D., Russel, R.: Duality, Separability, and Functional Structure: Theory and Economic Application. North-Holland, Amsterdam (1979)Google Scholar
  7. 7.
    Cherchye, L., Demuynk, T., De Rock, B., Hjertstrand, P.: Revealed preference tests for weak separability: an integer programming approach. J. Econom. 186(1), 129–141 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Fleissig, A., Whitney, G.A.: A nonparametric test of weak separability and consumer preferences. J. Econom. 147(2), 275–281 (2008)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Swofford, J.L., Whitney, G.A.: A revealed preference test for weakly separable utility maximization with incomplete adjustment. J. Econom. 60(1–2), 235–249 (1994)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Echenique, F.: Testing for separability is hard. arXiv:1401.4499 [ cs.GT]
  11. 11.
    Chiappori, P.: Rational household labor supply. Econometrica 56(1), 63–90 (1988)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Talla Nobibon, F., Cherchye, L., Crama, F., Demuynk, T., De Rock, B., Spieksma, F.: Revealed preference tests of collectively rational consumption behavior: formulations and algorithms. Oper. Res. 64(6), 1197–1216 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Smeulders, B., Cherchye, L., De Rock, B., Spieksma, F.C.R., Talla Nobibon, F.: Complexity results for the weak axiom of revealed preference for collective consumption models. J. Math. Econ. 58, 82–91 (2010)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Talla Nobibon, F., Spieksma, F.: On the complexity of testing the collective axiom of revealed preference. Math. Soc. Sci. 60(2), 123–136 (2010)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Moscow Institute of Physics and Technology (MIPT)Dolgoprudny, Moscow RegionRussian Federation

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