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Necessity of Transversality Conditions for Stochastic Problems

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  • Takashi Kamihigashi

    (Research Institute for Economics & Business Administration (RIEB), Kobe University, Japan)

Abstract

This paper establishes (i) an extension of Michel's (1990, Theorem 1) necessity result to an abstract reduced-form model, (ii) a generalization of the results of Weitzman (1973) and Ekeland and Scheinkman (1986), and (iii) a new result that is useful particularly in the case of homogeneous returns. These results are shown for an extremely general discrete-time reduced-form model that does not assume differentiability, continuity, or concavity, and that imposes virtually no restriction on the state spaces. The three results are futher extended to a stochastic reduced-form model. The stochastic extensions are easily accomplished since our deterministic model is so general that the stochastic model is in fact a special case of the deterministic model. We apply our stochastic results to a stochastic reduced-form model with homogeneous returns and a general type of stochastic growth model with CRRA utility.

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File URL: http://www.rieb.kobe-u.ac.jp/academic/ra/dp/English/dp115.pdf
File Function: First version, 2000
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Bibliographic Info

Paper provided by Research Institute for Economics & Business Administration, Kobe University in its series Discussion Paper Series with number 115.

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Length: 25 pages
Date of creation: Nov 2000
Date of revision:
Handle: RePEc:kob:dpaper:115

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Keywords: Transversality condition; Stochastic optimization; Stochastic reduced-form model; Homogeneous returns; Stochastic growth model;

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References

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  1. Montrucchio, Luigi & Privileggi, Fabio, 2001. "On Fragility of Bubbles in Equilibrium Asset Pricing Models of Lucas-Type," Journal of Economic Theory, Elsevier, vol. 101(1), pages 158-188, November.
  2. Takashi Kamihigashi, 2002. "Necessity of Transversality Conditions for Stochastic Problems," Discussion Paper Series 128, Research Institute for Economics & Business Administration, Kobe University.
  3. Lucas, Robert E, Jr, 1978. "Asset Prices in an Exchange Economy," Econometrica, Econometric Society, vol. 46(6), pages 1429-45, November.
  4. Brock, William A. & Mirman, Leonard J., 1972. "Optimal economic growth and uncertainty: The discounted case," Journal of Economic Theory, Elsevier, vol. 4(3), pages 479-513, June.
  5. Martin L. Weitzman, 1973. "Duality Theory for Infinite Horizon Convex Models," Management Science, INFORMS, vol. 19(7), pages 783-789, March.
  6. Lucas, Robert Jr., 1988. "On the mechanics of economic development," Journal of Monetary Economics, Elsevier, vol. 22(1), pages 3-42, July.
  7. Yano, Makoto, 1989. "Comparative statics in dynamic stochastic models : Differential analysis of a stochastic modified golden rule state in a banach space," Journal of Mathematical Economics, Elsevier, vol. 18(2), pages 169-185, April.
  8. Zilcha, Itzhak, 1976. "Characterization by prices of optimal programs under uncertainty," Journal of Mathematical Economics, Elsevier, vol. 3(2), pages 173-183, July.
  9. Peleg, Bezalel & Ryder, Harl E, Jr, 1972. "On Optimal Consumption Plans in a Multi-sector Economy," Review of Economic Studies, Wiley Blackwell, vol. 39(2), pages 159-69, April.
  10. Sergio Rebelo, 1999. "Long Run Policy Analysis and Long Run Growth," Levine's Working Paper Archive 2114, David K. Levine.
  11. Brock, William A, 1970. "On Existence of Weakly Maximal Programmes in a Multi-Sector Economy," Review of Economic Studies, Wiley Blackwell, vol. 37(2), pages 275-80, April.
  12. Takekuma, Shin-Ichi, 1992. "Optimal Growth under Uncertainty: A Complete Characterization of Weakly Maximal Programs," Hitotsubashi Journal of Economics, Hitotsubashi University, vol. 33(2), pages 169-182, December.
  13. Alvarez, Fernando & Stokey, Nancy L., 1998. "Dynamic Programming with Homogeneous Functions," Journal of Economic Theory, Elsevier, vol. 82(1), pages 167-189, September.
  14. Halkin, Hubert, 1974. "Necessary Conditions for Optimal Control Problems with Infinite Horizons," Econometrica, Econometric Society, vol. 42(2), pages 267-72, March.
  15. Manuel S. Santos & Michael Woodford, 1993. "Rational Asset Pricing Bubbles," Working Papers 9304, Centro de Investigacion Economica, ITAM.
  16. King, Robert G. & Plosser, Charles I. & Rebelo, Sergio T., 1988. "Production, growth and business cycles : I. The basic neoclassical model," Journal of Monetary Economics, Elsevier, vol. 21(2-3), pages 195-232.
  17. Takashi Kamihigashi, 2000. "A simple proof of Ekeland and Scheinkman's result on the necessity of a transversality condition," Economic Theory, Springer, vol. 15(2), pages 463-468.
  18. Takashi Kamihigashi, 1998. "Uniqueness of asset prices in an exchange economy with unbounded utility," Economic Theory, Springer, vol. 12(1), pages 103-122.
  19. Michel, Philippe, 1990. "Some Clarifications on the Transversality Condition," Econometrica, Econometric Society, vol. 58(3), pages 705-23, May.
  20. Takashi Kamihigashi, 2001. "A Simple Proof of the Necessity of the Transversality Condition," Discussion Paper Series 116, Research Institute for Economics & Business Administration, Kobe University.
  21. Kamihigashi, Takashi, 2001. "Necessity of Transversality Conditions for Infinite Horizon Problems," Econometrica, Econometric Society, vol. 69(4), pages 995-1012, July.
  22. Zilcha, Itzhak, 1978. "Transversality Condition in a Multi-Sector Economy under Uncertainty," Econometrica, Econometric Society, vol. 46(3), pages 515-25, May.
  23. Takashi Kamihigashi, 2002. "A simple proof of the necessity of the transversality condition," Economic Theory, Springer, vol. 20(2), pages 427-433.
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Citations

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Cited by:
  1. Kamihigashi, Takashi & Roy, Santanu, 2007. "A nonsmooth, nonconvex model of optimal growth," Journal of Economic Theory, Elsevier, vol. 132(1), pages 435-460, January.
  2. Takashi Kamihigashi, 2001. "A Simple Proof of the Necessity of the Transversality Condition," Discussion Paper Series 116, Research Institute for Economics & Business Administration, Kobe University.
  3. Takashi Kamihigashi, 2010. "Recurrent Bubbles," Discussion Paper Series DP2010-27, Research Institute for Economics & Business Administration, Kobe University, revised Nov 2010.
  4. Takashi Kamihigashi, 2007. "The Spirit of Capitalism, Stock Market Bubbles, and Output Fluctuations," Discussion Paper Series 205, Research Institute for Economics & Business Administration, Kobe University, revised Oct 2007.
  5. Takashi Kamihigashi, 2006. "Transversality Conditions and Dynamic Economic Behavior," Discussion Paper Series 180, Research Institute for Economics & Business Administration, Kobe University.
  6. Travis D. Nesmith, 2005. "Solving stochastic money-in-the-utility-function models," Finance and Economics Discussion Series 2005-52, Board of Governors of the Federal Reserve System (U.S.).
  7. Takashi Kamihigashi, 2004. "Necessity of the Transversality Condition for Stochastic Models with Bounded or CRRA Utility," Discussion Paper Series 162, Research Institute for Economics & Business Administration, Kobe University, revised Oct 2004.
  8. Dapeng Cai & Takashi Gyoshin Nitta, 2012. "Transversality Conditions for Stochastic Higher-Order Optimality: Continuous and Discrete Time Problems," Papers 1203.3869, arXiv.org.
  9. Takashi Kamihigashi, 2001. "Necessity of Transversality Conditions for Stochastic Problems," Department of Economics Working Papers 01-02, Stony Brook University, Department of Economics.
  10. Takashi Kamihigashi, 2003. "Necessity of the Transversality Condition for Stochastic Models with CRRA Utility," Discussion Paper Series 137, Research Institute for Economics & Business Administration, Kobe University.
  11. Hanno Lustig & Christopher Sleet & Sevin Yeltekin, 2005. "Fiscal Hedging and the Yield Curve," NBER Working Papers 11687, National Bureau of Economic Research, Inc.

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