[go: up one dir, main page]
More Web Proxy on the site http://driver.im/
IDEAS home Printed from https://ideas.repec.org/p/edn/sirdps/631.html
   My bibliography  Save this paper

A General and Intuitive Envelope Theorem

Author

Listed:
  • Clausen, Andrew
  • Strub, Carlo
Abstract
We present an envelope theorem for establishing first-order conditions in decision problems involving continuous and discrete choices. Our theorem accommodates general dynamic programming problems, even with unbounded marginal utilities. And, unlike classical envelope theorems that focus only on differentiating value functions, we accommodate other endogenous functions such as default probabilities and interest rates. Our main technical ingredient is how we establish the differentiability of a function at a point: we sandwich the function between two differentiable functions from above and below. Our theory is widely applicable. In unsecured credit models, neither interest rates nor continuation values are globally differentiable. Nevertheless, we establish an Euler equation involving marginal prices and values. In adjustment cost models, we show that first-order conditions apply universally, even if optimal policies are not (S,s). Finally, we incorporate indivisible choices into a classic dynamic insurance analysis.

Suggested Citation

  • Clausen, Andrew & Strub, Carlo, 2013. "A General and Intuitive Envelope Theorem," SIRE Discussion Papers 2015-43, Scottish Institute for Research in Economics (SIRE).
    • Handle: RePEc:edn:sirdps:631
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10943/631
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. 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.
    2. Ethan Ligon & Jonathan P. Thomas & Tim Worrall, 2002. "Informal Insurance Arrangements with Limited Commitment: Theory and Evidence from Village Economies," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 69(1), pages 209-244.
    3. 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.
    4. Ricardo J. Caballero & Eduardo M. R. A. Engel, 1999. "Explaining Investment Dynamics in U.S. Manufacturing: A Generalized (S,s) Approach," Econometrica, Econometric Society, vol. 67(4), pages 783-826, July.
    5. Santos, Manuel S, 1991. "Smoothness of the Policy Function in Discrete Time Economic Models," Econometrica, Econometric Society, vol. 59(5), pages 1365-1382, September.
    6. Mark Gertler & John Leahy, 2008. "A Phillips Curve with an Ss Foundation," Journal of Political Economy, University of Chicago Press, vol. 116(3), pages 533-572, June.
    7. J. Michael Harrison & Thomas M. Sellke & Allison J. Taylor, 1983. "Impulse Control of Brownian Motion," Mathematics of Operations Research, INFORMS, vol. 8(3), pages 454-466, August.
    8. 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.
    9. Townsend, Robert M, 1994. "Risk and Insurance in Village India," Econometrica, Econometric Society, vol. 62(3), pages 539-591, May.
    10. Hatchondo, Juan Carlos & Martinez, Leonardo, 2009. "Long-duration bonds and sovereign defaults," Journal of International Economics, Elsevier, vol. 79(1), pages 117-125, September.
    11. 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.
    12. Mikhail Golosov & Robert E. Lucas Jr., 2007. "Menu Costs and Phillips Curves," Journal of Political Economy, University of Chicago Press, vol. 115(2), pages 171-199.
    13. Cristina Arellano, 2008. "Default Risk and Income Fluctuations in Emerging Economies," American Economic Review, American Economic Association, vol. 98(3), pages 690-712, June.
    14. 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.
    15. 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.
    16. 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.
    17. Rincón-Zapatero, Juan Pablo & Santos, Manuel S., 2009. "Differentiability of the value function without interiority assumptions," Journal of Economic Theory, Elsevier, vol. 144(5), pages 1948-1964, September.
    18. 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.
    19. Ljungqvist, Lars & Sargent, Thomas J., 2012. "Recursive Macroeconomic Theory, Third Edition," MIT Press Books, The MIT Press, edition 3, volume 1, number 0262018748, April.
    20. Russell W. Cooper & John C. Haltiwanger, 2006. "On the Nature of Capital Adjustment Costs," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 73(3), pages 611-633.
    21. Araujo, A & Scheinkman, Jose A, 1977. "Smoothness, Comparative Dynamics, and the Turnpike Property," Econometrica, Econometric Society, vol. 45(3), pages 601-620, April.
    22. Jonathan Thomas & Tim Worrall, 1988. "Self-Enforcing Wage Contracts," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 55(4), pages 541-554.
    23. Paul Milgrom & Ilya Segal, 2002. "Envelope Theorems for Arbitrary Choice Sets," Econometrica, Econometric Society, vol. 70(2), pages 583-601, March.
    24. Dirk Krueger & Fabrizio Perri, 2006. "Does Income Inequality Lead to Consumption Inequality? Evidence and Theory -super-1," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 73(1), pages 163-193.
    25. Mirman, Leonard J. & Zilcha, Itzhak, 1975. "On optimal growth under uncertainty," Journal of Economic Theory, Elsevier, vol. 11(3), pages 329-339, December.
    26. Koeppl Thorsten V., 2006. "Differentiability of the Efficient Frontier when Commitment to Risk Sharing is Limited," The B.E. Journal of Macroeconomics, De Gruyter, vol. 6(1), pages 1-6, April.
    27. Attanasio, Orazio & Rios-Rull, Jose-Victor, 2000. "Consumption smoothing in island economies: Can public insurance reduce welfare?," European Economic Review, Elsevier, vol. 44(7), pages 1225-1258, June.
    28. Melanie Morten, 2016. "Temporary Migration and Endogenous Risk Sharing in Village India," NBER Working Papers 22159, National Bureau of Economic Research, Inc.
    29. Narayana R. Kocherlakota, 1996. "Implications of Efficient Risk Sharing without Commitment," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 63(4), pages 595-609.
    30. Andrew Caplin & John Leahy, 2010. "Economic Theory and the World of Practice: A Celebration of the ( S , s ) Model," Journal of Economic Perspectives, American Economic Association, vol. 24(1), pages 183-202, Winter.
    31. Charalambos D. Aliprantis & Kim C. Border, 2006. "Infinite Dimensional Analysis," Springer Books, Springer, edition 0, number 978-3-540-29587-7, December.
    32. Bar-Ilan, Avner, 1990. "Trigger-Target Rules Need Not Be Optimal with Fixed Adjustment Costs: A Simple Comment on Optimal Money Holding under Uncertainty," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 31(1), pages 229-234, February.
    33. 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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Andrew Clausen & Carlo Strub, 2012. "Envelope theorems for non-smooth and non-concave optimization," ECON - Working Papers 062, Department of Economics - University of Zurich.
    2. Clausen, Andrew & Strub, Carlo, 2020. "Reverse Calculus and nested optimization," Journal of Economic Theory, Elsevier, vol. 187(C).
    3. 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.
    4. Árpád Ábrahám & Sarolta Laczó, 2018. "Efficient Risk Sharing with Limited Commitment and Storage," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 85(3), pages 1389-1424.
    5. Sergio Rebelo & Neng Wang & Jinqiang Yang, 2018. "Rare Disasters, Financial Development, and Sovereign Debt," NBER Working Papers 25031, National Bureau of Economic Research, Inc.
    6. Tessa Bold, 2009. "Implications of Endogenous Group Formation for Efficient Risk‐Sharing," Economic Journal, Royal Economic Society, vol. 119(536), pages 562-591, March.
    7. Lizarazo, Sandra Valentina, 2013. "Default risk and risk averse international investors," Journal of International Economics, Elsevier, vol. 89(2), pages 317-330.
    8. Niepelt, Dirk, 2014. "Debt maturity without commitment," Journal of Monetary Economics, Elsevier, vol. 68(S), pages 37-54.
    9. Rincón-Zapatero, Juan Pablo & Santos, Manuel S., 2009. "Differentiability of the value function without interiority assumptions," Journal of Economic Theory, Elsevier, vol. 144(5), pages 1948-1964, September.
    10. Sergio Rebelo & Neng Wang & Jinqiang Yang, 2022. "Rare Disasters, Financial Development, and Sovereign Debt," Journal of Finance, American Finance Association, vol. 77(5), pages 2719-2764, October.
    11. Bloise, Gaetano & Vailakis, Yiannis, 2024. "Sovereign debt crises and low interest rates," Journal of International Economics, Elsevier, vol. 150(C).
    12. Sandra Lizarazo, 2009. "Contagion of Financial Crises in Sovereing Debt Markets," Working Papers 0906, Centro de Investigacion Economica, ITAM.
    13. Mr. Michael Kumhof & Mr. Romain Ranciere & Pablo Winant, 2013. "Inequality, Leverage and Crises: The Case of Endogenous Default," IMF Working Papers 2013/249, International Monetary Fund.
    14. Eduardo Zilberman & Vinicius Carrasco & Pedro Hemsley, 2019. "Risk sharing contracts with private information and one-sided commitment," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 68(1), pages 53-81, July.
    15. Aguiar, M. & Chatterjee, S. & Cole, H. & Stangebye, Z., 2016. "Quantitative Models of Sovereign Debt Crises," Handbook of Macroeconomics, in: J. B. Taylor & Harald Uhlig (ed.), Handbook of Macroeconomics, edition 1, volume 2, chapter 0, pages 1697-1755, Elsevier.
    16. Miao, Jianjun & Zhang, Yuzhe, 2015. "A duality approach to continuous-time contracting problems with limited commitment," Journal of Economic Theory, Elsevier, vol. 159(PB), pages 929-988.
    17. Satyajit Chatterjee & Burcu Eyigungor, 2012. "Maturity, Indebtedness, and Default Risk," American Economic Review, American Economic Association, vol. 102(6), pages 2674-2699, October.
    18. Juan Carlos Conesa & Timothy J. Kehoe, 2017. "Gambling for redemption and self-fulfilling debt crises," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 64(4), pages 707-740, December.
    19. Juan Carlos Hatchondo & Leonardo Martinez & César Sosa-Padilla, 2016. "Debt Dilution and Sovereign Default Risk," Journal of Political Economy, University of Chicago Press, vol. 124(5), pages 1383-1422.
    20. xavier Ragot & Francois Le Grand, 2018. "Sovereign Default and Liquidity: The Case for a World Safe," 2018 Meeting Papers 889, Society for Economic Dynamics.

    More about this item

    Keywords

    First-order conditions; discrete choice; unsecured credit; adjustment costs; informal insurance arrangements;
    All these keywords.

    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)

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:edn:sirdps:631. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Research Office (email available below). General contact details of provider: https://edirc.repec.org/data/sireeuk.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.