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Discounting and Patience in Optimal Stopping and Control Problems


  • John K. -H Quah
  • Bruno Strulovici


This paper establishes that the optimal stopping time of virtually any optimal stopping problem is increasing in "patience," understood as a particular partial order on discount rate functions. With Markov dynamics, the result holds in a continuation- domain sense even if stopping is combined with an optimal control problem. Under intuitive additional assumptions, we obtain comparative statics on both the optimal control and optimal stopping time for one-dimensional di usions. We provide a simple example where, without these assumptions, increased patience can precipitate stopping. We also show that, with optimal stopping and control, a project's expected value is decreasing in the interest rate, generalizing analogous results in a deterministic context. All our results are robust to the presence of a salvage value. As an application we show that the internal rate of return of any endogenously-interrupted project is essentially unique, even if the project also involves a management problem until its interruption. We also apply our results to the theory of optimal growth and capital deepening and to optimal bankruptcy decisions.

Suggested Citation

  • John K. -H Quah & Bruno Strulovici, 2009. "Discounting and Patience in Optimal Stopping and Control Problems," Discussion Papers 1480, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  • Handle: RePEc:nwu:cmsems:1480

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    References listed on IDEAS

    1. John K.-H. Quah & Bruno Strulovici, 2009. "Comparative Statics, Informativeness, and the Interval Dominance Order," Econometrica, Econometric Society, vol. 77(6), pages 1949-1992, November.
    2. Godfrey Keller & Sven Rady & Martin Cripps, 2005. "Strategic Experimentation with Exponential Bandits," Econometrica, Econometric Society, vol. 73(1), pages 39-68, January.
    3. Gustavo Manso & Bruno Strulovici & Alexei Tchistyi, 2010. "Performance-Sensitive Debt," Review of Financial Studies, Society for Financial Studies, vol. 23(5), pages 1819-1854.
    4. Bruno Strulovici, 2010. "Learning While Voting: Determinants of Collective Experimentation," Econometrica, Econometric Society, vol. 78(3), pages 933-971, May.
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    Cited by:

    1. Manso, Gustavo, 2013. "Feedback effects of credit ratings," Journal of Financial Economics, Elsevier, vol. 109(2), pages 535-548.
    2. Gustavo Manso, 2011. "Feedback Effects of Credit Ratings," 2011 Meeting Papers 1338, Society for Economic Dynamics.

    More about this item


    capital growth; comparative statics; discounting; internal rate of return; optimal control; optimal stopping; patience; present value; project valuation;

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty
    • D90 - Microeconomics - - Micro-Based Behavioral Economics - - - General
    • E2 - Macroeconomics and Monetary Economics - - Consumption, Saving, Production, Employment, and Investment
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
    • G31 - Financial Economics - - Corporate Finance and Governance - - - Capital Budgeting; Fixed Investment and Inventory Studies


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