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Finance and Stochastics

, Volume 15, Issue 4, pp 607–633 | Cite as

On irreversible investment

  • Frank Riedel
  • Xia Su
Article

Abstract

This paper presents a new and general approach to the theory of irreversible investment. We show that the optimal policy is a base capacity policy and derive general monotone comparative statics results. When the operating profit function is supermodular, the base capacity increases monotonically with the exogenous shock; and firm size is decreasing in the user cost of capital. Last but not least, the paper provides a general existence theorem for optimal policies.

Keywords

Sequential irreversible investment Capacity expansion Singular control problem Lévy processes 

Mathematics Subject Classification (2000)

91B28 34H05 49J30 93E20 

JEL Classification

C61 D81 E22 G11 

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References

  1. 1.
    Abel, A.B., Eberly, J.C.: Optimal investment with costly reversibility. Rev. Econ. Stud. 63, 581–593 (1996) CrossRefMATHGoogle Scholar
  2. 2.
    Abel, A.B., Eberly, J.C.: An exact solution for the investment and value of a firm facing uncertainty, adjustment costs, and irreversibility. J. Econ. Dyn. Control 21, 831–852 (1997) CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Abel, A.B., Eberly, J.C.: The effects of irreversibility and uncertainty on capital accumulation. J. Mon. Econ. 44, 339–377 (1999) CrossRefGoogle Scholar
  4. 4.
    Arrow, K.J.: Optimal capital policy with irreversible investment. In: Wolfe, J.N. (ed.) Value, Capital, and Growth, Essays in Honor of Sir John Hicks, pp. 1–19. Edinburgh University Press, Edinburgh (1968) Google Scholar
  5. 5.
    Balder, E.: New sequential compactness results for spaces of scalarly integrable functions. J. Math. Anal. Appl. 151, 1–16 (1990) CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Baldursson, F.M., Karatzas, I.: Irreversible investment and industry equilibrium. Finance Stoch. 1, 69–89 (1997) CrossRefMATHGoogle Scholar
  7. 7.
    Bank, P.: Optimal control under a dynamic fuel constraint. SIAM J. Control Optim. 44, 1529–1541 (2005) CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Bank, P., El Karoui, N.: A stochastic representation theorem with applications to optimization and obstacle problems. Ann. Probab. 32, 1030–1067 (2004) CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Bank, P., Föllmer, H.: American options, multi-armed bandits, and optimal consumption plans: A unified view. In: Carmona, R., et al. (eds.) Paris–Princeton Lectures in Finance 2002. Lecture Notes in Mathematics, vol. 1814, pp. 1–42. Springer, New York (2003) Google Scholar
  10. 10.
    Bank, P., Riedel, F.: Optimal consumption choice under uncertainty with intertemporal substitution. Ann. Appl. Probab. 11, 750–788 (2001) CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Bertoin, J.: Lévy Processes. Cambridge University Press, Cambridge (1996) MATHGoogle Scholar
  12. 12.
    Bertola, G.: Irreversible investment. Res. Econ. 52, 3–37 (1998) CrossRefMATHGoogle Scholar
  13. 13.
    Bessembinder, H., Coughenour, J.F., Seguin, P.J., Monroe Smoller, M.: Mean reversion in equilibrium asset prices: Evidence from the futures term structure. J. Finance 50, 361–375 (1995) CrossRefGoogle Scholar
  14. 14.
    Boyarchenko, S.: Irreversible decisions and record setting news principles. Am. Econ. Rev. 94, 557–568 (2004) CrossRefGoogle Scholar
  15. 15.
    Boyarchenko, S., Levendorskiĭ, S.: General option exercise rules, with applications to embedded options and monopolistic expansion. B.E. J. Theor. Econ. 6(1), 2 (2006) Google Scholar
  16. 16.
    Carr, P., Geman, H., Madan, D.B., Yor, M.: The fine structure of asset returns: An empirical investigation. J. Bus. 75, 305–332 (2002) CrossRefGoogle Scholar
  17. 17.
    Chiarolla, M.B., Haussmann, U.B.: Explicit solution of a stochastic, irreversible investment problem and its moving threshold. Math. Oper. Res. 30, 91–108 (2005) CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Davis, M.H.A., Dempster, M.A.H., Sethi, S.P., Vermes, D.: Optimal capacity expansion under uncertainty. Adv. Appl. Probab. 19, 156–176 (1987) CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Dixit, A.: Investment and hysteresis. J. Econ. Perspect. 6, 107–132 (1992) Google Scholar
  20. 20.
    Dixit, A.K., Pindyck, R.S.: Investment Under Uncertainty. Princeton University Press, Princeton (1994) Google Scholar
  21. 21.
    Doob, J.L.: Measure Theory. Springer, Heidelberg (1998) Google Scholar
  22. 22.
    Eberlein, E., Keller, U.: Hyperbolic distributions in finance. Bernoulli 1, 281–299 (1995) CrossRefMATHGoogle Scholar
  23. 23.
    Fama, E., French, K.: Dividend yields and expected stock returns. J. Financ. Econ. 22, 3–27 (1988) CrossRefGoogle Scholar
  24. 24.
    Grenadier, S.R.: Option exercise games: An application to the equilibrium investment strategies of firms. Rev. Financ. Stud. 15, 691–721 (2002) CrossRefGoogle Scholar
  25. 25.
    Guo, X., Miao, J., Morellec, E.: Irreversible investment with regime shifts. J. Econ. Theory 122, 37–59 (2005) CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Hindy, A., Huang, C.-F.: Intertemporal preferences for uncertain consumption: A continuous-time approach. Econometrica 60, 781–801 (1992) CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Hindy, A., Huang, C.-F., Kreps, D.: On intertemporal preferences in continuous time. The case of certainty. J. Math. Econ. 21, 401–440 (1992) CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Jorgenson, D.W.: Capital theory and investment behavior. Am. Econ. Rev. 53, 247–257 (1963) Google Scholar
  29. 29.
    Kabanov, Y.: Hedging and liquidation under transaction costs in currency markets. Finance Stoch. 3, 237–248 (1999) CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Kobila, T.Ø.: A class of solvable stochastic investment problems involving singular controls. Stochastics 43, 29–63 (1993) MATHMathSciNetGoogle Scholar
  31. 31.
    Komlós, J.: A generalization of a problem of Steinhaus. Acta Math. Acad. Sci. Hung 18, 217–229 (1967) CrossRefMATHGoogle Scholar
  32. 32.
    Loéve, M.: Probability Theory II, 4th edn. Graduate Texts in Mathematics, vol. 46. Springer, New York (1978) MATHGoogle Scholar
  33. 33.
    McDonald, R., Siegel, D.: The value of waiting to invest. Q. J. Econ. 101, 707–727 (1986) CrossRefGoogle Scholar
  34. 34.
    Milgrom, P., Shannon, C.: Monotone comparative statics. Econometrica 62, 157–180 (1994) CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Øksendal, A.: Irreversible investment problems. Finance Stoch. 4, 223–250 (2000) CrossRefMathSciNetGoogle Scholar
  36. 36.
    Pindyck, R.S.: Irreversible investment, capacity choice, and the value of the firm. Am. Econ. Rev. 78, 969–985 (1988) Google Scholar
  37. 37.
    Porteus, E.L.: Foundations of Stochastic Inventory Theory. Stanford University Press, Stanford (1990) Google Scholar
  38. 38.
    Steg, J.H.: Irreversible investment in oligopoly. IMW Working Paper 415, Bielefeld University (2009). http://www.imw.uni-bielefeld.de/research/workingpapers.php
  39. 39.
    Topkis, D.M.: Minimizing a submodular function on a lattice. Oper. Res. 26, 305–321 (1978) CrossRefMATHMathSciNetGoogle Scholar
  40. 40.
    Wang, H.: Capacity expansion with exponential jump diffusion processes. Stoch. Stoch. Rep. 75, 259–274 (2003) MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Institute of Mathematical EconomicsBielefeld UniversityBielefeldGermany
  2. 2.GRA RMT Model Validation RatesCommerzbank AGFrankfurtGermany

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