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A Duality Approach to Continuous- Time Contracting Problems with Limited Commitment

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  • Jianjun Miao
  • Yuzhe Zhang

Abstract

We propose a duality approach to solving contracting models with either one-sided or two-sided limited commitment in continuous time. We establish weak and strong duality theorems and provide a dynamic programming characterization of the dual problem. The dual problem gives a linear Hamilton-Jacobi-Bellman equation with a known state space subject to free-boundary conditions, making analysis much more tractable than the primal problem. We provide two explicitly solved examples of a consumption insurance problem. We characterize the optimal consumption allocation in terms of the marginal utility ratio. We find that neither autarky nor full risk sharing can be an optimal contract with two-sided limited commitment, unlike in discrete-time models. We also derive an explicit solution for the unique long-run stationary distribution of consumption relative to income.

Suggested Citation

  • Jianjun Miao & Yuzhe Zhang, 2013. "A Duality Approach to Continuous- Time Contracting Problems with Limited Commitment," Boston University - Department of Economics - Working Papers Series 2013-008, Boston University - Department of Economics.
  • Handle: RePEc:bos:wpaper:wp2013-008
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    as
    1. Grochulski, Borys & Zhang, Yuzhe, 2011. "Optimal risk sharing and borrowing constraints in a continuous-time model with limited commitment," Journal of Economic Theory, Elsevier, vol. 146(6), pages 2356-2388.
    2. Jonathan Thomas & Tim Worrall, 1988. "Self-Enforcing Wage Contracts," Review of Economic Studies, Oxford University Press, vol. 55(4), pages 541-554.
    3. Marcet, Albert & Marimon, Ramon, 1992. "Communication, commitment, and growth," Journal of Economic Theory, Elsevier, vol. 58(2), pages 219-249, December.
    4. He, Hua & Pages, Henri F, 1993. "Labor Income, Borrowing Constraints, and Equilibrium Asset Prices," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 3(4), pages 663-696, October.
    5. Kocherlakota, Narayana, 2008. "Injecting rational bubbles," Journal of Economic Theory, Elsevier, vol. 142(1), pages 218-232, September.
    6. Zhang, Yuzhe, 2013. "Characterization of a risk sharing contract with one-sided commitment," Journal of Economic Dynamics and Control, Elsevier, vol. 37(4), pages 794-809.
    7. Krueger, Dirk & Uhlig, Harald, 2006. "Competitive risk sharing contracts with one-sided commitment," Journal of Monetary Economics, Elsevier, vol. 53(7), pages 1661-1691, October.
    8. Thomas Cooley & Ramon Marimon & Vincenzo Quadrini, 2004. "Aggregate Consequences of Limited Contract Enforceability," Journal of Political Economy, University of Chicago Press, vol. 112(4), pages 817-847, August.
    9. Townsend, Robert M, 1994. "Risk and Insurance in Village India," Econometrica, Econometric Society, vol. 62(3), pages 539-591, May.
    10. Stephen E. Spear & Sanjay Srivastava, 1987. "On Repeated Moral Hazard with Discounting," Review of Economic Studies, Oxford University Press, vol. 54(4), pages 599-617.
    11. Krueger, Dirk & Perri, Fabrizio, 2011. "Public versus private risk sharing," Journal of Economic Theory, Elsevier, vol. 146(3), pages 920-956, May.
    12. Patrick J. Kehoe & Fabrizio Perri, 2002. "International Business Cycles with Endogenous Incomplete Markets," Econometrica, Econometric Society, vol. 70(3), pages 907-928, May.
    13. Matthias Messner & Nicola Pavoni & Christopher Sleet, 2012. "Recursive Methods for Incentive Problems," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 15(4), pages 501-525, October.
    14. Acemoglu, Daron & Golosov, Mikhail & Tsyvinski, Aleh, 2011. "Political economy of Ramsey taxation," Journal of Public Economics, Elsevier, vol. 95(7), pages 467-475.
    15. Ethan Ligon & Jonathan P. Thomas & Tim Worrall, 2002. "Informal Insurance Arrangements with Limited Commitment: Theory and Evidence from Village Economies," Review of Economic Studies, Oxford University Press, vol. 69(1), pages 209-244.
    16. Brian D. Wright & Kenneth M. Kletzer, 2000. "Sovereign Debt as Intertemporal Barter," American Economic Review, American Economic Association, vol. 90(3), pages 621-639, June.
    17. Kydland, Finn E. & Prescott, Edward C., 1980. "Dynamic optimal taxation, rational expectations and optimal control," Journal of Economic Dynamics and Control, Elsevier, vol. 2(1), pages 79-91, May.
    18. Christian Hellwig & Guido Lorenzoni, 2009. "Bubbles and Self-Enforcing Debt," Econometrica, Econometric Society, vol. 77(4), pages 1137-1164, July.
    19. Fernando Alvarez & Urban J. Jermann, 2000. "Efficiency, Equilibrium, and Asset Pricing with Risk of Default," Econometrica, Econometric Society, vol. 68(4), pages 775-798, July.
    20. Albert Marcet & Ramon Marimon, 1994. "Recursive contracts," Economics Working Papers 337, Department of Economics and Business, Universitat Pompeu Fabra, revised Oct 1998.
    21. 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.
    22. Timothy J. Kehoe & David K. Levine, 1993. "Debt-Constrained Asset Markets," Review of Economic Studies, Oxford University Press, vol. 60(4), pages 865-888.
    23. Dumas, Bernard, 1991. "Super contact and related optimality conditions," Journal of Economic Dynamics and Control, Elsevier, vol. 15(4), pages 675-685, October.
    24. PETER M. DeMARZO & YULIY SANNIKOV, 2006. "Optimal Security Design and Dynamic Capital Structure in a Continuous-Time Agency Model," Journal of Finance, American Finance Association, vol. 61(6), pages 2681-2724, December.
    25. Bruno Biais & Thomas Mariotti & Guillaume Plantin & Jean-Charles Rochet, 2007. "Dynamic Security Design: Convergence to Continuous Time and Asset Pricing Implications," Review of Economic Studies, Oxford University Press, vol. 74(2), pages 345-390.
    26. Noah Williams, 2011. "Persistent Private Information," Econometrica, Econometric Society, vol. 79(4), pages 1233-1275, July.
    27. Cochrane, John H, 1991. "A Simple Test of Consumption Insurance," Journal of Political Economy, University of Chicago Press, vol. 99(5), pages 957-976, October.
    28. Narayana R. Kocherlakota, 1996. "Implications of Efficient Risk Sharing without Commitment," Review of Economic Studies, Oxford University Press, vol. 63(4), pages 595-609.
    29. Mace, Barbara J, 1991. "Full Insurance in the Presence of Aggregate Uncertainty," Journal of Political Economy, University of Chicago Press, vol. 99(5), pages 928-956, October.
    30. Alvarez, Fernando & Jermann, Urban J, 2001. "Quantitative Asset Pricing Implications of Endogenous Solvency Constraints," Review of Financial Studies, Society for Financial Studies, vol. 14(4), pages 1117-1151.
    31. Alessandro Pavan & Ilya Segal & Juuso Toikka, 2014. "Dynamic Mechanism Design: A Myersonian Approach," Econometrica, Econometric Society, vol. 82(2), pages 601-653, March.
    32. V. V. Chari & Patrick J. Kehoe, 1993. "Sustainable Plans and Mutual Default," Review of Economic Studies, Oxford University Press, vol. 60(1), pages 175-195.
    33. Jianjun Miao & Pengfei Wang, 2012. "Bubbles and Total Factor Productivity," American Economic Review, American Economic Association, vol. 102(3), pages 82-87, May.
    34. Lars Ljungqvist & Thomas J. Sargent, 2004. "Recursive Macroeconomic Theory, 2nd Edition," MIT Press Books, The MIT Press, edition 2, volume 1, number 026212274x, January.
    35. Yuliy Sannikov, 2008. "A Continuous-Time Version of the Principal-Agent Problem," Review of Economic Studies, Oxford University Press, vol. 75(3), pages 957-984.
    36. Pengfei Wang & Jianjun Miao, 2011. "Bubbles and Credit Constraints," 2011 Meeting Papers 94, Society for Economic Dynamics.
    37. Abreu, Dilip & Pearce, David & Stacchetti, Ennio, 1990. "Toward a Theory of Discounted Repeated Games with Imperfect Monitoring," Econometrica, Econometric Society, vol. 58(5), pages 1041-1063, September.
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    Cited by:

    1. Bergemann, Dirk & Pavan, Alessandro, 2015. "Introduction to Symposium on Dynamic Contracts and Mechanism Design," Journal of Economic Theory, Elsevier, vol. 159(PB), pages 679-701.
    2. Rui Li & Dana Kiku & Hengjie Ai, 2014. "A Mechanism Design Model of Firm Dynamics: The Case of Limited Commitment," 2014 Meeting Papers 855, Society for Economic Dynamics.
    3. Dirk Bergemann & Alessandro Pavan, 2015. "Introduction to JET Symposium Issue on "Dynamic Contracts and Mechanism Design"," Cowles Foundation Discussion Papers 2016, Cowles Foundation for Research in Economics, Yale University.
    4. Patrick Bolton & Neng Wang & Jinqiang Yang, 2016. "Liquidity and Risk Management: Coordinating Investment and Compensation Policies," 2016 Meeting Papers 1703, Society for Economic Dynamics.
    5. repec:eee:dyncon:v:92:y:2018:i:c:p:153-182 is not listed on IDEAS
    6. Hackbarth, Dirk & Taub, Bart, 2018. "Does the Potential to Merge Reduce Competition?," CEPR Discussion Papers 12732, C.E.P.R. Discussion Papers.
    7. Golosov, M. & Tsyvinski, A. & Werquin, N., 2016. "Recursive Contracts and Endogenously Incomplete Markets," Handbook of Macroeconomics, Elsevier.
    8. Patrick Bolton & Neng Wang & Jinqiang Yang, 2015. "Liquidity and Risk Management, Executive Compensation, and Investment with Inalienable Human Capital: Optimal Contracting and Corporate Valuation," NBER Working Papers 20979, National Bureau of Economic Research, Inc.
    9. Felix Feng, 2018. "Dynamic Compensation under Uncertainty Shocks and Limited Commitment," 2018 Meeting Papers 159, Society for Economic Dynamics.

    More about this item

    Keywords

    continuous-time contracts; limited commitment; risk sharing; duality; dynamic programming; regulated Brownian motion;

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • D86 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Economics of Contract Law
    • D91 - Microeconomics - - Micro-Based Behavioral Economics - - - Role and Effects of Psychological, Emotional, Social, and Cognitive Factors on Decision Making
    • E21 - Macroeconomics and Monetary Economics - - Consumption, Saving, Production, Employment, and Investment - - - Consumption; Saving; Wealth

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