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Multiple Solutions in Systems of Functional Differential Equations

Author

Listed:
  • d'Albis, Hippolyte
  • Augeraud-Véron, Emmanuelle
  • Hupkes, Herman Jan
Abstract
This paper proposes conditions for the existence and uniqueness of solutions to systems of linear differential or algebraic equations with delays or advances, in which some variables may be non-predetermined. These conditions represent the counterpart to the Blanchard and Kahn conditions for the functional equations under consideration. To illustrate the mathematical results, applications to an overlapping generations model and a time-to-build model are developed.

Suggested Citation

  • d'Albis, Hippolyte & Augeraud-Véron, Emmanuelle & Hupkes, Herman Jan, 2014. "Multiple Solutions in Systems of Functional Differential Equations," MPRA Paper 54777, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:54777
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    References listed on IDEAS

    as
    1. Asea, Patrick K. & Zak, Paul J., 1999. "Time-to-build and cycles," Journal of Economic Dynamics and Control, Elsevier, vol. 23(8), pages 1155-1175, August.
    2. Hippolyte d'Albis & Emmanuelle Augeraud-Véron & Hermen Jan Hupkes, 2012. "Discontinuous Initial Value Problems for Funtional Differential-Algebraic Equations of Mixed Type," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00717412, HAL.
    3. Faggian, Silvia & Gozzi, Fausto, 2010. "Optimal investment models with vintage capital: Dynamic programming approach," Journal of Mathematical Economics, Elsevier, vol. 46(4), pages 416-437, July.
    4. Boucekkine, Raouf & Licandro, Omar & Puch, Luis A. & del Rio, Fernando, 2005. "Vintage capital and the dynamics of the AK model," Journal of Economic Theory, Elsevier, vol. 120(1), pages 39-72, January.
    5. d'Albis, Hippolyte & Augeraud-Véron, Emmanuelle & Hupkes, Hermen Jan, 2014. "Bounded interest rate feedback rules in continuous-time," Journal of Economic Dynamics and Control, Elsevier, vol. 39(C), pages 227-236.
    6. M. Bambi & G. Fabbri & F. Gozzi, 2012. "Optimal policy and consumption smoothing effects in the time-to-build AK model," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 50(3), pages 635-669, August.
    7. Bambi, Mauro & Gozzi, Fausto & Licandro, Omar, 2014. "Endogenous growth and wave-like business fluctuations," Journal of Economic Theory, Elsevier, vol. 154(C), pages 68-111.
    8. d’Albis, Hippolyte & Augeraud-Veron, Emmanuelle & Venditti, Alain, 2012. "Business cycle fluctuations and learning-by-doing externalities in a one-sector model," Journal of Mathematical Economics, Elsevier, vol. 48(5), pages 295-308.
    9. Blanchard, Olivier Jean & Kahn, Charles M, 1980. "The Solution of Linear Difference Models under Rational Expectations," Econometrica, Econometric Society, vol. 48(5), pages 1305-1311, July.
    10. Raouf Boucekkine & David Croix & Omar Licandro, 2004. "MODELLING VINTAGE STRUCTURES WITH DDEs: PRINCIPLES AND APPLICATIONS," Mathematical Population Studies, Taylor & Francis Journals, vol. 11(3-4), pages 151-179.
    11. Boucekkine, R. & Fabbri, G. & Gozzi, F., 2010. "Maintenance and investment: Complements or substitutes? A reappraisal," Journal of Economic Dynamics and Control, Elsevier, vol. 34(12), pages 2420-2439, December.
    12. Boucekkine, Raouf & de la Croix, David & Licandro, Omar, 2002. "Vintage Human Capital, Demographic Trends, and Endogenous Growth," Journal of Economic Theory, Elsevier, vol. 104(2), pages 340-375, June.
    13. Azariadis, Costas, 1981. "Self-fulfilling prophecies," Journal of Economic Theory, Elsevier, vol. 25(3), pages 380-396, December.
    14. Buiter, Willem H, 1984. "Saddlepoint Problems in Continuous Time Rational Expectations Models: A General Method and Some Macroeconomic Examples," Econometrica, Econometric Society, vol. 52(3), pages 665-680, May.
    15. Gray, Malcolm R & Turnovsky, Stephen J, 1979. "Expectational Consistency, Informational Lags, and the Formulation of Expectations in Continuous Time Models," Econometrica, Econometric Society, vol. 47(6), pages 1457-1474, November.
    16. Boucekkine, Raouf & Licandro, Omar & Paul, Christopher, 1997. "Differential-difference equations in economics: On the numerical solution of vintage capital growth models," Journal of Economic Dynamics and Control, Elsevier, vol. 21(2-3), pages 347-362.
    17. Mauro Bambi, 2006. "Endogenous Growth and Time-to-Build: the AK Case," Economics Working Papers ECO2006/17, European University Institute.
    18. Benhabib, Jess, 2004. "Interest Rate Policy in Continuous Time with Discrete Delays," Journal of Money, Credit and Banking, Blackwell Publishing, vol. 36(1), pages 1-15, February.
    19. repec:hal:pseose:hal-01015388 is not listed on IDEAS
    20. Jovanovic, Boyan & Yatsenko, Yuri, 2012. "Investment in vintage capital," Journal of Economic Theory, Elsevier, vol. 147(2), pages 551-569.
    21. Fabbri, Giorgio & Gozzi, Fausto, 2008. "Solving optimal growth models with vintage capital: The dynamic programming approach," Journal of Economic Theory, Elsevier, vol. 143(1), pages 331-373, November.
    22. Benhabib, Jess & Farmer, Roger E.A., 1999. "Indeterminacy and sunspots in macroeconomics," Handbook of Macroeconomics, in: J. B. Taylor & M. Woodford (ed.), Handbook of Macroeconomics, edition 1, volume 1, chapter 6, pages 387-448, Elsevier.
    23. El-Hodiri, Mohamed A & Loehman, Edna & Whinston, Andrew B, 1972. "An Optimal Growth Model with Time Lags," Econometrica, Econometric Society, vol. 40(6), pages 1137-1146, November.
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    1. d'Albis, Hippolyte & Augeraud-Véron, Emmanuelle & Hupkes, Hermen Jan, 2014. "Bounded interest rate feedback rules in continuous-time," Journal of Economic Dynamics and Control, Elsevier, vol. 39(C), pages 227-236.
    2. d’Albis, Hippolyte & Augeraud-Véron, Emmanuelle & Hupkes, Hermen Jan, 2014. "Stability and determinacy conditions for mixed-type functional differential equations," Journal of Mathematical Economics, Elsevier, vol. 53(C), pages 119-129.
    3. Emmanuelle Augeraud-Véron & Arnaud Ducrot, 2019. "Spatial externality and indeterminacy," Post-Print hal-02306568, HAL.
    4. Augeraud-Veron, Emmanuelle & Boucekkine, Raouf & Gozzi, Fausto & Venditti, Alain & Zou, Benteng, 2024. "Fifty years of mathematical growth theory: Classical topics and new trends," Journal of Mathematical Economics, Elsevier, vol. 111(C).
    5. Eugene Bravyi & Vladimir Maksimov & Pyotr Simonov, 2020. "Some Economic Dynamics Problems for Hybrid Models with Aftereffect," Mathematics, MDPI, vol. 8(10), pages 1-30, October.

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    More about this item

    Keywords

    Delay Differential Equations; Advance Differential Equations; Existence; Indeterminacy;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • E13 - Macroeconomics and Monetary Economics - - General Aggregative Models - - - Neoclassical

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