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

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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. D. Acemoglu, Introduction to Modern Economic Growth (Princeton Univ. Press, Princeton, NJ, 2008).

    Google Scholar 

  2. P. Aghion and P. Howitt, Endogenous Growth Theory (MIT Press, Cambridge, MA, 1998).

    Google Scholar 

  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.

    Chapter  Google Scholar 

  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)].

    Article  Google Scholar 

  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.

    Chapter  Google Scholar 

  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. 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. 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)].

    MathSciNet  Google Scholar 

  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).

    Article  MATH  MathSciNet  Google Scholar 

  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. 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).

    MATH  MathSciNet  Google Scholar 

  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. 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).

    Article  MATH  MathSciNet  Google Scholar 

  14. E. J. Balder, “An existence result for optimal economic growth problems,” J. Math. Anal. Appl. 95, 195–213 (1983).

    Article  MATH  MathSciNet  Google Scholar 

  15. R. J. Barro and X. Sala-i-Martin, Economic Growth (McGraw Hill, New York, 1995).

    Google Scholar 

  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).

    Article  MATH  MathSciNet  Google Scholar 

  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)].

    MATH  MathSciNet  Google Scholar 

  18. D. A. Carlson, A. B. Haurie, and A. Leizarowitz, Infinite Horizon Optimal Control. Deterministic and Stochastic Systems (Springer, Berlin, 1991).

    Book  MATH  Google Scholar 

  19. D. Cass, “Optimum growth in an aggregative model of capital accumulation,” Rev. Econ. Stud. 32, 233–240 (1965).

    Article  Google Scholar 

  20. L. Cesari, Optimization—Theory and Applications. Problems with Ordinary Differential Equations (Springer, New York, 1983).

    MATH  Google Scholar 

  21. A. C. Chiang, Elements of Dynamic Optimization (McGraw Hill, Singapore, 1992).

    Google Scholar 

  22. B. P. Demidovich, Lectures on the Mathematical Theory of Stability (Nauka, Moscow, 1967) [in Russian].

    Google Scholar 

  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)].

    Article  MathSciNet  Google Scholar 

  24. I. Ekeland, “Some variational problems arising from mathematical economics,” in Mathematical Economics (Springer, Berlin, 1988), Lect. Notes Math. 1330, pp. 1–18.

    Chapter  Google Scholar 

  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. A. F. Filippov, Differential Equations with Discontinuous Right-Hand Sides (Nauka, Moscow, 1985; Kluwer, Dordrecht, 1988).

    Google Scholar 

  27. R. V. Gamkrelidze, Principles of Optimal Control Theory (Plenum Press, New York, 1978).

    Book  MATH  Google Scholar 

  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)].

    MATH  MathSciNet  Google Scholar 

  29. F. R. Gantmakher, Matrix Theory, 2nd ed. (Nauka, Moscow, 1966) [in Russian].

    Google Scholar 

  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).

    Book  MATH  Google Scholar 

  31. G. M. Grossman and E. Helpman, Innovation and Growth in the Global Economy (MIT Press, Cambridge, MA, 1991).

    Google Scholar 

  32. H. Halkin, “Necessary conditions for optimal control problems with infinite horizons,” Econometrica 42, 267–272 (1974).

    Article  MATH  MathSciNet  Google Scholar 

  33. P. Hartman, Ordinary Differential Equations (J. Wiley & Sons, New York, 1964).

    MATH  Google Scholar 

  34. C. J. Himmelberg, “Measurable relations,” Fundam. Math. 87, 53–72 (1975).

    MATH  MathSciNet  Google Scholar 

  35. K. Inada, “On a two-sector model of economic growth: Comments and a generalization,” Rev. Econ. Stud. 30(2), 119–127 (1963).

    Article  Google Scholar 

  36. M. D. Intriligator, Mathematical Optimization and Economic Theory (Prentice-Hall, Englewood Cliffs, NJ, 1971).

    Google Scholar 

  37. T. Kamihigashi, “Necessity of transversality conditions for infinite horizon problems,” Econometrica 69, 995–1012 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  38. T. C. Koopmans, “Objectives, constraints, and outcomes in optimal growth models,” Econometrica 35, 1–15 (1967).

    Article  MATH  MathSciNet  Google Scholar 

  39. P. Michel, “On the transversality condition in infinite horizon optimal problems,” Econometrica 50, 975–985 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  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. F. P. Ramsey, “A mathematical theory of saving,” Econ. J. 38, 543–559 (1928).

    Article  Google Scholar 

  42. W. Rudin, Real and Complex Analysis (McGraw-Hill, New York, 1987).

    MATH  Google Scholar 

  43. P. A. Samuelson, “Paul Douglas’s measurement of production functions and marginal productivities,” J. Polit. Econ. 87(5), 923–939 (1979).

    Article  Google Scholar 

  44. A. Seierstad, “Necessary conditions for nonsmooth, infinite-horizon, optimal control problems,” J. Optim. Theory Appl. 103(1), 201–229 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  45. A. Seierstad and K. Sydsæter, Optimal Control Theory with Economic Applications (North-Holland, Amsterdam, 1987).

    MATH  Google Scholar 

  46. S. P. Sethi and G. L. Thompson, Optimal Control Theory: Applications to Management Science and Economics (Kluwer, Dordrecht, 2000).

    Google Scholar 

  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.

    Chapter  Google Scholar 

  48. G. V. Smirnov, “Transversality condition for infinite-horizon problems,” J. Optim. Theory Appl. 88(3), 671–688 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  49. R. M. Solow, Growth Theory: An Exposition (Oxford Univ. Press, New York, 1970).

    Google Scholar 

  50. M. L. Weitzman, Income, Wealth, and the Maximum Principle (Harvard Univ. Press, Cambridge, MA, 2003).

    MATH  Google Scholar 

  51. J. J. Ye, “Nonsmooth maximum principle for infinite-horizon problems,” J. Optim. Theory Appl. 76(3), 485–500 (1993).

    Article  MATH  MathSciNet  Google Scholar 

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  1. Steklov Mathematical Institute, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    K. O. Besov

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Correspondence to K. O. Besov.

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Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2014, Vol. 284, pp. 56–88.

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Besov, K.O. On necessary optimality conditions for infinite-horizon economic growth problems with locally unbounded instantaneous utility function. Proc. Steklov Inst. Math. 284, 50–80 (2014). https://doi.org/10.1134/S0081543814010040

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