On necessary optimality conditions for infinite-horizon economic growth problems with locally unbounded instantaneous utility function

Article

Abstract

We consider a class of infinite-horizon optimal control problems that arise in studying models of optimal dynamic allocation of economic resources. In a typical problem of that kind the initial state is fixed, no constraints are imposed on the behavior of the admissible trajectories at infinity, and the objective functional is given by a discounted improper integral. Earlier, for such problems, S.M. Aseev and A.V. Kryazhimskiy in 2004–2007 and jointly with the author in 2012 developed a method of finite-horizon approximations and obtained variants of the Pontryagin maximum principle that guarantee normality of the problem and contain an explicit formula for the adjoint variable. In the present paper those results are extended to a more general situation where the instantaneous utility function need not be locally bounded from below. As an important illustrative example, we carry out a rigorous mathematical investigation of the transitional dynamics in the neoclassical model of optimal economic growth.

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References

  1. 1.
    D. Acemoglu, Introduction to Modern Economic Growth (Princeton Univ. Press, Princeton, NJ, 2008).Google Scholar
  2. 2.
    P. Aghion and P. Howitt, Endogenous Growth Theory (MIT Press, Cambridge, MA, 1998).Google Scholar
  3. 3.
    S. Aseev, K. Besov, and S. Kaniovski, “The problem of optimal endogenous growth with exhaustible resources revisited,” in Green Growth and Sustainable Development (Springer, Berlin, 2013), Dyn. Model. Econometr. Econ. Finance 14, pp. 3–30.CrossRefGoogle Scholar
  4. 4.
    S. M. Aseev, K. O. Besov, and A. V. Kryazhimskiy, “Infinite-horizon optimal control problems in economics,” Usp. Mat. Nauk 67(2), 3–64 (2012) [Russ. Math. Surv. 67, 195–253 (2012)].CrossRefGoogle Scholar
  5. 5.
    S. Aseev, K. Besov, S.-E. Ollus, and T. Palokangas, “Optimal economic growth with a random environmental shock,” in Dynamic Systems, Economic Growth, and the Environment (Springer, Berlin, 2010), Dyn. Model. Econometr. Econ. Finance 12, pp. 109–137.CrossRefGoogle Scholar
  6. 6.
    S. Aseev, K. Besov, S.-E. Ollus, and T. Palokangas, “Optimal growth in a two-sector economy facing an expected random shock,” Tr. Inst. Mat. Mekh., Ural. Otd. Ross. Akad. Nauk 17(2), 271–299 (2011) [Proc. Steklov Inst. Math. 276 (Suppl. 1), S4–S34 (2012)].Google Scholar
  7. 7.
    S. Aseev, G. Hutschenreiter, and A. Kryazhimskii, “A dynamic model of optimal allocation of resources to R&D,” IIASA Interim Rep. IR-02-016 (Laxenburg, 2002).Google Scholar
  8. 8.
    S. M. Aseev and A. V. Kryazhimskii, “The Pontryagin maximum principle for an optimal control problem with a functional specified by an improper integral,” Dokl. Akad. Nauk 394(5), 583–585 (2004) [Dokl. Math. 69 (1), 89–91 (2004)].MathSciNetGoogle Scholar
  9. 9.
    S. M. Aseev and A. V. Kryazhimskiy, “The Pontryagin maximum principle and transversality conditions for a class of optimal control problems with infinite time horizons,” SIAM J. Control Optim. 43, 1094–1119 (2004).CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    S. M. Aseev and A. V. Kryazhimskii, The Pontryagin Maximum Principle and Optimal Economic Growth Problems (Nauka, Moscow, 2007), Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 257 [Proc. Steklov Inst. Math. 257 (2007)].Google Scholar
  11. 11.
    S. M. Aseev and V. M. Veliov, “Maximum principle for infinite-horizon optimal control problems with dominating discount,” Dyn. Contin. Discrete Impuls. Syst. B: Appl. Algorithms 19, 43–63 (2012).MATHMathSciNetGoogle Scholar
  12. 12.
    S. M. Aseev and V. M. Veliov, “Needle variations in infinite-horizon optimal control,” Res. Rep. 2012-04 (Vienna Univ. Technol., Vienna, 2012).Google Scholar
  13. 13.
    J. P. Aubin and F. H. Clarke, “Shadow prices and duality for a class of optimal control problems,” SIAM J. Control Optim. 17, 567–586 (1979).CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    E. J. Balder, “An existence result for optimal economic growth problems,” J. Math. Anal. Appl. 95, 195–213 (1983).CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    R. J. Barro and X. Sala-i-Martin, Economic Growth (McGraw Hill, New York, 1995).Google Scholar
  16. 16.
    L. M. Benveniste and J. A. Scheinkman, “Duality theory for dynamic optimization models of economics: The continuous time case,” J. Econ. Theory 27, 1–19 (1982).CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    V. I. Blagodatskikh and A. F. Filippov, “Differential inclusions and optimal control,” Tr. Mat. Inst. im. V.A. Steklova, Akad. Nauk SSSR 169, 194–252 (1985) [Proc. Steklov Inst. Math. 169, 199–259 (1986)].MATHMathSciNetGoogle Scholar
  18. 18.
    D. A. Carlson, A. B. Haurie, and A. Leizarowitz, Infinite Horizon Optimal Control. Deterministic and Stochastic Systems (Springer, Berlin, 1991).CrossRefMATHGoogle Scholar
  19. 19.
    D. Cass, “Optimum growth in an aggregative model of capital accumulation,” Rev. Econ. Stud. 32, 233–240 (1965).CrossRefGoogle Scholar
  20. 20.
    L. Cesari, Optimization—Theory and Applications. Problems with Ordinary Differential Equations (Springer, New York, 1983).MATHGoogle Scholar
  21. 21.
    A. C. Chiang, Elements of Dynamic Optimization (McGraw Hill, Singapore, 1992).Google Scholar
  22. 22.
    B. P. Demidovich, Lectures on the Mathematical Theory of Stability (Nauka, Moscow, 1967) [in Russian].Google Scholar
  23. 23.
    A. V. Dmitruk and N. V. Kuz’kina, “Existence theorem in the optimal control problem on an infinite time interval,” Mat. Zametki 78(4), 503–518 (2005) [Math. Notes 78, 466–480 (2005)]; “Letter to the editor,” Mat. Zametki 80 (2), 320 (2006) [Math. Notes 80, 309 (2006)].CrossRefMathSciNetGoogle Scholar
  24. 24.
    I. Ekeland, “Some variational problems arising from mathematical economics,” in Mathematical Economics (Springer, Berlin, 1988), Lect. Notes Math. 1330, pp. 1–18.CrossRefGoogle Scholar
  25. 25.
    A. F. Filippov, “On some problems in optimal control theory,” Vestn. Mosk. Univ., Ser. 1: Mat., Mekh., Astron., Fiz., Khim., No. 2, 25–32 (1959).Google Scholar
  26. 26.
    A. F. Filippov, Differential Equations with Discontinuous Right-Hand Sides (Nauka, Moscow, 1985; Kluwer, Dordrecht, 1988).Google Scholar
  27. 27.
    R. V. Gamkrelidze, Principles of Optimal Control Theory (Plenum Press, New York, 1978).CrossRefMATHGoogle Scholar
  28. 28.
    R. V. Gamkrelidze, “Sliding regimes in optimal control theory,” Tr. Mat. Inst. im. V.A. Steklova, Akad. Nauk SSSR 169, 180–193 (1985) [Proc. Steklov Inst. Math. 169, 185–198 (1986)].MATHMathSciNetGoogle Scholar
  29. 29.
    F. R. Gantmakher, Matrix Theory, 2nd ed. (Nauka, Moscow, 1966) [in Russian].Google Scholar
  30. 30.
    D. Grass, J. P. Caulkins, G. Feichtinger, G. Tragler, and D. A. Behrens, Optimal Control of Nonlinear Processes. With Applications in Drugs, Corruption, and Terror (Springer, Berlin, 2008).CrossRefMATHGoogle Scholar
  31. 31.
    G. M. Grossman and E. Helpman, Innovation and Growth in the Global Economy (MIT Press, Cambridge, MA, 1991).Google Scholar
  32. 32.
    H. Halkin, “Necessary conditions for optimal control problems with infinite horizons,” Econometrica 42, 267–272 (1974).CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    P. Hartman, Ordinary Differential Equations (J. Wiley & Sons, New York, 1964).MATHGoogle Scholar
  34. 34.
    C. J. Himmelberg, “Measurable relations,” Fundam. Math. 87, 53–72 (1975).MATHMathSciNetGoogle Scholar
  35. 35.
    K. Inada, “On a two-sector model of economic growth: Comments and a generalization,” Rev. Econ. Stud. 30(2), 119–127 (1963).CrossRefGoogle Scholar
  36. 36.
    M. D. Intriligator, Mathematical Optimization and Economic Theory (Prentice-Hall, Englewood Cliffs, NJ, 1971).Google Scholar
  37. 37.
    T. Kamihigashi, “Necessity of transversality conditions for infinite horizon problems,” Econometrica 69, 995–1012 (2001).CrossRefMATHMathSciNetGoogle Scholar
  38. 38.
    T. C. Koopmans, “Objectives, constraints, and outcomes in optimal growth models,” Econometrica 35, 1–15 (1967).CrossRefMATHMathSciNetGoogle Scholar
  39. 39.
    P. Michel, “On the transversality condition in infinite horizon optimal problems,” Econometrica 50, 975–985 (1982).CrossRefMATHMathSciNetGoogle Scholar
  40. 40.
    L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The Mathematical Theory of Optimal Processes (Fizmatgiz, Moscow, 1961; Pergamon, Oxford, 1964).Google Scholar
  41. 41.
    F. P. Ramsey, “A mathematical theory of saving,” Econ. J. 38, 543–559 (1928).CrossRefGoogle Scholar
  42. 42.
    W. Rudin, Real and Complex Analysis (McGraw-Hill, New York, 1987).MATHGoogle Scholar
  43. 43.
    P. A. Samuelson, “Paul Douglas’s measurement of production functions and marginal productivities,” J. Polit. Econ. 87(5), 923–939 (1979).CrossRefGoogle Scholar
  44. 44.
    A. Seierstad, “Necessary conditions for nonsmooth, infinite-horizon, optimal control problems,” J. Optim. Theory Appl. 103(1), 201–229 (1999).CrossRefMATHMathSciNetGoogle Scholar
  45. 45.
    A. Seierstad and K. Sydsæter, Optimal Control Theory with Economic Applications (North-Holland, Amsterdam, 1987).MATHGoogle Scholar
  46. 46.
    S. P. Sethi and G. L. Thompson, Optimal Control Theory: Applications to Management Science and Economics (Kluwer, Dordrecht, 2000).Google Scholar
  47. 47.
    K. Shell, “Applications of Pontryagin’s maximum principle to economics,” in Mathematical Systems Theory and Economics 1 (Springer, Berlin, 1969), Lect. Notes Oper. Res. Math. Econ. 11, pp. 241–292.CrossRefGoogle Scholar
  48. 48.
    G. V. Smirnov, “Transversality condition for infinite-horizon problems,” J. Optim. Theory Appl. 88(3), 671–688 (1996).CrossRefMATHMathSciNetGoogle Scholar
  49. 49.
    R. M. Solow, Growth Theory: An Exposition (Oxford Univ. Press, New York, 1970).Google Scholar
  50. 50.
    M. L. Weitzman, Income, Wealth, and the Maximum Principle (Harvard Univ. Press, Cambridge, MA, 2003).MATHGoogle Scholar
  51. 51.
    J. J. Ye, “Nonsmooth maximum principle for infinite-horizon problems,” J. Optim. Theory Appl. 76(3), 485–500 (1993).CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

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