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The Dual Approach to Recursive Optimization: Theory and Examples

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  • Nicola Pavoni
  • Christopher Sleet
  • Matthias Messner

Abstract

We develop a recursive dual method for solving dynamic economic problems. The method uses a Lagrangian to pair a dynamic recursive economic problem with a dual problem. We show that such dual problems can be recursively decomposed with costates (i.e., Lagrange multipliers on laws of motion) functioning as state variables. In dynamic contracting and policy settings, the method often replaces an endogenous state space of forward†looking utilities with an exogenously given state space of costates. We provide a principle of optimality for dual problems and give conditions under which the dual Bellman operator is a contraction with the optimal dual value function its unique fixed point. We relate economic problems to their duals, address computational issues, and give examples.

Suggested Citation

  • Nicola Pavoni & Christopher Sleet & Matthias Messner, 2018. "The Dual Approach to Recursive Optimization: Theory and Examples," Econometrica, Econometric Society, vol. 86(1), pages 133-172, January.
    • Handle: RePEc:wly:emetrp:v:86:y:2018:i:1:p:133-172
    DOI: 10.3982/ECTA11905
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    Cited by:

    1. Marimon, Ramon & Werner, Jan, 2021. "The envelope theorem, Euler and Bellman equations, without differentiability," Journal of Economic Theory, Elsevier, vol. 196(C).
    2. Fabio Blasutto & Egor Kozlov, 2020. "(Changing) Marriage and Cohabitation Patterns in the US: do Divorce Laws Matter?," 2020 Papers pbl245, Job Market Papers.
    3. Felipe Brugués, 2026. "Take the Goods and Run: Contracting Frictions and Market Power in Supply Chains," American Economic Review, American Economic Association, vol. 116(2), pages 582-626, February.
    4. Gaetano Bloise, 2020. "Unique Markov Equilibrium Under Limited Commitment," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 61(2), pages 721-751, May.
    5. Gaetano Bloise & Paolo Siconolfi, 2022. "A Negishi Approach to Recursive Contracts," Econometrica, Econometric Society, vol. 90(6), pages 2821-2855, November.
    6. Clayton, Christopher & Schaab, Andreas, 2022. "A Theory of Dynamic Inflation Targets," TSE Working Papers 22-1389, Toulouse School of Economics (TSE).
    7. François Le Grand & Xavier Ragot, 2022. "Managing Inequality Over Business Cycles: Optimal Policies With Heterogeneous Agents And Aggregate Shocks," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 63(1), pages 511-540, February.
    8. Fabio Blasutto, 2024. "Cohabitation vs. Marriage: Mating Strategies by Education in The USA," Journal of the European Economic Association, European Economic Association, vol. 22(4), pages 1723-1761.
    9. François Le Grand & Xavier Ragot, 2022. "Managing Inequality Over Business Cycles: Optimal Policies With Heterogeneous Agents And Aggregate Shocks," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 63(1), pages 511-540, February.
    10. Yu, Meng & Zhang, Junnan, 2019. "Equilibrium in production chains with multiple upstream partners," Journal of Mathematical Economics, Elsevier, vol. 83(C), pages 1-10.
    11. Meng Yu & Junnan Zhang, 2019. "Equilibrium in Production Chains with Multiple Upstream Partners," Papers 1908.08208, arXiv.org.
    12. Lilia Maliar & Serguei Maliar, 2016. "Ruling Out Multiplicity of Smooth Equilibria in Dynamic Games: A Hyperbolic Discounting Example," Dynamic Games and Applications, Springer, vol. 6(2), pages 243-261, June.
    13. Can Urgun, 2021. "Restless Contracting," Working Papers 2021-88, Princeton University. Economics Department..
    14. Jean Guillaume Forand & Jan Zapal, 2017. "The Demand and Supply of Favours in Dynamic Relationships," Working Papers 1705, University of Waterloo, Department of Economics, revised Sep 2017.
    15. Golosov, M. & Tsyvinski, A. & Werquin, N., 2016. "Recursive Contracts and Endogenously Incomplete Markets," Handbook of Macroeconomics, in: J. B. Taylor & Harald Uhlig (ed.), Handbook of Macroeconomics, edition 1, volume 2, chapter 0, pages 725-841, Elsevier.
    16. Citanna, A. & Tvede, M., 2024. "Do taxspots matter?," Journal of Economic Theory, Elsevier, vol. 220(C).

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