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Envelope theorems for non-smooth and non-concave optimization

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  • Andrew Clausen
  • Carlo Strub

Abstract

We study general dynamic programming problems with continuous and discrete choices and general constraints. The value functions may have kinks arising (1) at indifference points between discrete choices and (2) at constraint boundaries. Nevertheless, we establish a general envelope theorem: first-order conditions are necessary at interior optimal choices. We only assume differentiability of the utility function with respect to the continuous choices. The continuous choice may be from any Banach space and the discrete choice from any non-empty set.

Suggested Citation

  • Andrew Clausen & Carlo Strub, 2012. "Envelope theorems for non-smooth and non-concave optimization," ECON - Working Papers 062, Department of Economics - University of Zurich.
  • Handle: RePEc:zur:econwp:062
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    Cited by:

    1. Andrew Clausen & Carlo Strub, 2016. "Money Cycles," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 57, pages 1279-1298, November.
    2. Menzio, Guido & Shi, Shouyong & Sun, Hongfei, 2013. "A monetary theory with non-degenerate distributions," Journal of Economic Theory, Elsevier, vol. 148(6), pages 2266-2312.
    3. Pavel Sevcik & Rui Castro, 2013. "Occupational Choice, Human Capital, and Financing Constraints," 2013 Meeting Papers 1321, Society for Economic Dynamics.
    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. Renou, Ludovic & Schlag, Karl H., 2014. "Ordients: Optimization and comparative statics without utility functions," Journal of Economic Theory, Elsevier, vol. 154(C), pages 612-632.
    6. Clausen, Andrew, 2013. "Moral Hazard with Counterfeit Signals," SIRE Discussion Papers 2013-13, Scottish Institute for Research in Economics (SIRE).
    7. Florian Oswald, 2015. "Regional Shocks, Migration and Homeownership," Sciences Po publications info:hdl:2441/n1d9kd7k48k, Sciences Po.
    8. Chen, Daphne & Qi, Shi & Schlagenhauf, Don E., 2017. "Corporate Income Tax, Legal Form of Organization, and Employment," Working Papers 2017-21, Federal Reserve Bank of St. Louis.
    9. Fedor Iskhakov & Michael Keane, 2018. "Effects of Taxes and Safety Net Pensions on life-cycle Labor Supply, Savings and Human Capital: the Case of Australia," ANU Working Papers in Economics and Econometrics 2018-661, Australian National University, College of Business and Economics, School of Economics.
    10. Florian Oswald, 2015. "Regional Shocks, Migration and Homeownership," 2015 Meeting Papers 759, Society for Economic Dynamics.
    11. Gregory Cox, 2018. "Almost Sure Uniqueness of a Global Minimum Without Convexity," Papers 1803.02415, arXiv.org, revised May 2018.
    12. Yuntong Wang, 2014. "Envelope Theorem without Differentiability," Working Papers 1404, University of Windsor, Department of Economics.
    13. Fedor Iskhakov & Thomas Høgholm Jørgensen & John Rust & Bertel Schjerning, 2015. "Estimating Discrete-Continuous Choice Models: The Endogenous Grid Method with Taste Shocks," Discussion Papers 15-19, University of Copenhagen. Department of Economics.
    14. Joost Röttger, 2014. "Monetary and Fiscal Policy with Sovereign Default," Working Paper Series in Economics 74, University of Cologne, Department of Economics.
    15. Robert Kirkby, 2017. "Convergence of Discretized Value Function Iteration," Computational Economics, Springer;Society for Computational Economics, vol. 49(1), pages 117-153, January.
    16. Jan Werner & Ramon Marimon, 2015. "Envelope Theorem, Euler, and Bellman Equations without Differentiability," 2015 Meeting Papers 1415, Society for Economic Dynamics.
    17. Wei Cui, 2017. "Macroeconomic Effects of Delayed Capital Liquidation," Discussion Papers 1719, Centre for Macroeconomics (CFM).
    18. Christoph Görtz & Afrasiab Mirza, 2014. "On the Applicability of Global Approximation Methods for Models with Jump Discontinuities in Policy Functions," CESifo Working Paper Series 4837, CESifo Group Munich.
    19. Wellschmied, Felix, 2015. "The Welfare Effects of Asset Means-Testing Income Support," IZA Discussion Papers 8838, Institute for the Study of Labor (IZA).
    20. Jeffrey T. Denning & Benjamin M. Marx & Lesley J. Turner, 2017. "ProPelled: The Effects of Grants on Graduation, Earnings, and Welfare," Upjohn Working Papers and Journal Articles 17-280, W.E. Upjohn Institute for Employment Research.
    21. Giulio Fella, 2011. "A Generalized Endogenous Grid Method for Non-concave Problems," Working Papers 677, Queen Mary University of London, School of Economics and Finance.
    22. Chen, Daphne & Qi, Shi & Schlagenhauf, Don E., 2014. "Corporate income tax, legal form of organization, and employment," Working Papers 2014-18, Federal Reserve Bank of St. Louis.
    23. Sebastian Dyrda & Benjamin Pugsley, 2018. "Taxes, Regulations of Businesses and Evolution of Income Inequality in the US," 2018 Meeting Papers 318, Society for Economic Dynamics.
    24. Bommier, Antoine & Harenberg, Daniel & Le Grand, François, 2017. "Household Finance and the Value of Life," Annual Conference 2017 (Vienna): Alternative Structures for Money and Banking 168189, Verein für Socialpolitik / German Economic Association.
    25. Youngsoo Jang, 2016. "Income Inequality, Medical Conditions, and Household Bankruptcy," Proceedings of Economics and Finance Conferences 4206835, International Institute of Social and Economic Sciences.

    More about this item

    Keywords

    Envelope theorem; differentiability; dynamic programming; discrete choice; non-smooth analysis;

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • E20 - Macroeconomics and Monetary Economics - - Consumption, Saving, Production, Employment, and Investment - - - General (includes Measurement and Data)

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