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Reverse Calculus and nested optimization

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  • Clausen, Andrew
  • Strub, Carlo

Abstract

Nested optimization problems arise when an agent must take into account the effect of their decisions on their own future behaviour, or the behaviour of others. In these problems, calculating marginal costs and benefits involves differentiating the solutions to nested problems. But are these solutions differentiable functions? We develop a tool called Reverse Calculus, and establish first-order conditions for (i) a Stackelberg leader considering the follower's best response function, (ii) a sovereign borrower considering its own future default policy, and (iii) non-convex dynamic programming problems.

Suggested Citation

  • Clausen, Andrew & Strub, Carlo, 2020. "Reverse Calculus and nested optimization," Journal of Economic Theory, Elsevier, vol. 187(C).
    • Handle: RePEc:eee:jetheo:v:187:y:2020:i:c:s0022053120300247
    DOI: 10.1016/j.jet.2020.105019
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    1. Mark Aguiar & Manuel Amador & Hugo Hopenhayn & Iván Werning, 2019. "Take the Short Route: Equilibrium Default and Debt Maturity," Econometrica, Econometric Society, vol. 87(2), pages 423-462, March.
    2. Benveniste, L M & Scheinkman, J A, 1979. "On the Differentiability of the Value Function in Dynamic Models of Economics," Econometrica, Econometric Society, vol. 47(3), pages 727-732, May.
    3. Santos, Manuel S, 1991. "Smoothness of the Policy Function in Discrete Time Economic Models," Econometrica, Econometric Society, vol. 59(5), pages 1365-1382, September.
    4. Giulio Fella, 2014. "A generalized endogenous grid method for non-smooth and non-concave problems," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 17(2), pages 329-344, April.
    5. Druedahl, Jeppe & Jørgensen, Thomas Høgholm, 2017. "A general endogenous grid method for multi-dimensional models with non-convexities and constraints," Journal of Economic Dynamics and Control, Elsevier, vol. 74(C), pages 87-107.
    6. Santos, Manuel S. & Vila, Jean-Luc, 1991. "Smoothness of the policy function in continuous-time economic models : The one-dimensional case," Journal of Economic Dynamics and Control, Elsevier, vol. 15(4), pages 741-753, October.
    7. Hatchondo, Juan Carlos & Martinez, Leonardo, 2009. "Long-duration bonds and sovereign defaults," Journal of International Economics, Elsevier, vol. 79(1), pages 117-125, September.
    8. S. Rao Aiyagari, 1994. "Uninsured Idiosyncratic Risk and Aggregate Saving," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 109(3), pages 659-684.
    9. Cristina Arellano, 2008. "Default Risk and Income Fluctuations in Emerging Economies," American Economic Review, American Economic Association, vol. 98(3), pages 690-712, June.
    10. Amir, Rabah & Mirman, Leonard J & Perkins, William R, 1991. "One-Sector Nonclassical Optimal Growth: Optimality Conditions and Comparative Dynamics," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 32(3), pages 625-644, August.
    11. Satyajit Chatterjee & Dean Corbae & Makoto Nakajima & José-Víctor Ríos-Rull, 2007. "A Quantitative Theory of Unsecured Consumer Credit with Risk of Default," Econometrica, Econometric Society, vol. 75(6), pages 1525-1589, November.
    12. Cristina Arellano & Ananth Ramanarayanan, 2012. "Default and the Maturity Structure in Sovereign Bonds," Journal of Political Economy, University of Chicago Press, vol. 120(2), pages 187-232.
    13. Jonathan Eaton & Mark Gersovitz, 1981. "Debt with Potential Repudiation: Theoretical and Empirical Analysis," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 48(2), pages 289-309.
    14. Andreas Müller & Kjetil Storesletten & Fabrizio Zilibotti, 2019. "Sovereign Debt and Structural Reforms," American Economic Review, American Economic Association, vol. 109(12), pages 4220-4259, December.
    15. Ljungqvist, Lars & Sargent, Thomas J., 2012. "Recursive Macroeconomic Theory, Third Edition," MIT Press Books, The MIT Press, edition 3, volume 1, number 0262018748, December.
    16. Araujo, A & Scheinkman, Jose A, 1977. "Smoothness, Comparative Dynamics, and the Turnpike Property," Econometrica, Econometric Society, vol. 45(3), pages 601-620, April.
    17. Paul Milgrom & Ilya Segal, 2002. "Envelope Theorems for Arbitrary Choice Sets," Econometrica, Econometric Society, vol. 70(2), pages 583-601, March.
    18. Mirman, Leonard J. & Zilcha, Itzhak, 1975. "On optimal growth under uncertainty," Journal of Economic Theory, Elsevier, vol. 11(3), pages 329-339, December.
    19. Aguiar, Mark & Gopinath, Gita, 2006. "Defaultable debt, interest rates and the current account," Journal of International Economics, Elsevier, vol. 69(1), pages 64-83, June.
    20. Bulow, Jeremy & Rogoff, Kenneth, 1989. "Sovereign Debt: Is to Forgive to Forget?," American Economic Review, American Economic Association, vol. 79(1), pages 43-50, March.
    21. Charalambos D. Aliprantis & Kim C. Border, 2006. "Infinite Dimensional Analysis," Springer Books, Springer, edition 0, number 978-3-540-29587-7, October.
    22. Kydland, Finn E & Prescott, Edward C, 1977. "Rules Rather Than Discretion: The Inconsistency of Optimal Plans," Journal of Political Economy, University of Chicago Press, vol. 85(3), pages 473-491, June.
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    3. Damián Pierri & Hernán D. Seoane, 2022. "An Ergodic Theory of Sovereign Default," Working Papers 206, Red Nacional de Investigadores en Economía (RedNIE).
    4. Jang, Youngsoo & Lee, Soyoung, 2019. "A Generalized Endogenous Grid Method for Models with the Option to Default," MPRA Paper 95721, University Library of Munich, Germany.
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    6. Youngsoo Jang, 2023. "Credit, Default, And Optimal Health Insurance," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 64(3), pages 943-977, August.
    7. Wicht, Adrien, 2025. "Seniority and sovereign default: The role of official multilateral lenders," Journal of International Economics, Elsevier, vol. 155(C).
    8. Youngsoo Jang & Soyoung Lee, 2021. "A Generalized Endogenous Grid Method for Default Risk Models," Staff Working Papers 21-11, Bank of Canada.
    9. Rui Castro & Pavel Ševčík, 2024. "Occupational choice, human capital and financial constraints," Canadian Journal of Economics/Revue canadienne d'économique, John Wiley & Sons, vol. 57(3), pages 674-703, August.
    10. Schneider, Ulrich & Groneck, Max, 2022. "Pension Reforms, Labor supply and Savings. The Importance of Natural Experiments for Structural Estimation," VfS Annual Conference 2022 (Basel): Big Data in Economics 264059, Verein für Socialpolitik / German Economic Association.
    11. Ludvig Sinander, 2022. "The Converse Envelope Theorem," Econometrica, Econometric Society, vol. 90(6), pages 2795-2819, November.
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    Keywords

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    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • F34 - International Economics - - International Finance - - - International Lending and Debt Problems
    • L13 - Industrial Organization - - Market Structure, Firm Strategy, and Market Performance - - - Oligopoly and Other Imperfect Markets

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