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

K. O. Besov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces and related problems of analysis, Tome 284 (2014), pp. 56-88.

Résumé

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.

Bibliographie

[1] Acemoglu D., Introduction to modern economic growth, Princeton Univ. Press, Princeton, NJ, 2008

[2] Aghion P., Howitt P., Endogenous growth theory, MIT Press, Cambridge, MA, 1998

[3] Aseev S., Besov K., Kaniovski S., “The problem of optimal endogenous growth with exhaustible resources revisited”, Green growth and sustainable development, Dyn. Model. Econometr. Econ. Finance, 14, Springer, Berlin, 2013, 3–30 | DOI | MR

[4] Aseev S.M., Besov K.O., Kryazhimskii A.V., “Zadachi optimalnogo upravleniya na beskonechnom intervale vremeni v ekonomike”, UMN, 67:2 (2012), 3–64 | DOI | MR | Zbl

[5] Aseev S., Besov K., Ollus S.-E., Palokangas T., “Optimal economic growth with a random environmental shock”, Dynamic systems, economic growth, and the environment, Dyn. Model. Econometr. Econ. Finance, 12, Springer, Berlin, 2010, 109–137 | DOI

[6] Aseev S., Besov K., Ollus S.-E., Palokangas T., “Optimal growth in a two-sector economy facing an expected random shock”, Tr. In-ta matematiki i mekhaniki UrO RAN, 17:2 (2011), 271–299

[7] Aseev S., Hutschenreiter G., Kryazhimskii A., A dynamic model of optimal allocation of resources to R, IIASA Interim Rep. IR-02-016, Laxenburg, 2002

[8] Aseev S.M., Kryazhimskii A.V., “Printsip maksimuma Pontryagina dlya zadachi optimalnogo upravleniya s funktsionalom, zadannym nesobstvennym integralom”, DAN., 394:5 (2004), 583–585 | MR | Zbl

[9] Aseev S.M., Kryazhimskiy A.V., “The Pontryagin maximum principle and transversality conditions for a class of optimal control problems with infinite time horizons”, SIAM J. Control Optim., 43 (2004), 1094–1119 | DOI | MR | Zbl

[10] Aseev S.M., Kryazhimskii A.V., Printsip maksimuma Pontryagina i zadachi optimalnogo ekonomicheskogo rosta, Tr. MIAN, 257, Nauka, M., 2007 | MR | Zbl

[11] Aseev S.M., Veliov V.M., “Maximum principle for infinite-horizon optimal control problems with dominating discount”, Dyn. Contin. Discrete Impuls. Syst. B: Appl. Algorithms, 19 (2012), 43–63 | MR | Zbl

[12] Aseev S.M., Veliov V.M., Needle variations in infinite-horizon optimal control, Res. Rep. 2012-04, Vienna Univ. Technol., Vienna, 2012

[13] Aubin J.P., Clarke F.H., “Shadow prices and duality for a class of optimal control problems”, SIAM J. Control Optim., 17 (1979), 567–586 | DOI | MR | Zbl

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

[15] Barro R.J., Sala-i-Martin X., Economic growth, McGraw Hill, New York, 1995

[16] Benveniste L.M., Scheinkman J.A., “Duality theory for dynamic optimization models of economics: the continuous time case”, J. Econ. Theory., 27 (1982), 1–19 | DOI | MR | Zbl

[17] Blagodatskikh V.I., Filippov A.F., “Differentsialnye vklyucheniya i optimalnoe upravlenie”, Tr. MIAN, 169, 1985, 194–252 | MR | Zbl

[18] Carlson D.A., Haurie A.B., Leizarowitz A., Infinite horizon optimal control. Deterministic and stochastic systems, Springer, Berlin, 1991 | Zbl

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

[20] Cesari L., Optimization — theory and applications. Problems with ordinary differential equations, Springer, New York, 1983 | MR | Zbl

[21] Chiang A.C., Elements of dynamic optimization, McGraw Hill, Singapore, 1992

[22] Demidovich B.P., Lektsii po matematicheskoi teorii ustoichivosti, Nauka, M., 1967 | MR

[23] Dmitruk A.V., Kuzkina N.V., “Teorema suschestvovaniya optimalnogo upravleniya na beskonechnom intervale vremeni”, Mat. zametki, 78:4 (2005), 503–518 ; “Письмо в редакцию”, Мат. заметки, 80:2 (2006), 320 | DOI | MR | Zbl | DOI | MR

[24] Ekeland I., “Some variational problems arising from mathematical economics”, Mathematical economics, Lect. Notes Math., 1330, Springer, Berlin, 1988, 1–18 | DOI | MR

[25] Filippov A.F., “O nekotorykh voprosakh teorii optimalnogo regulirovaniya”, Vestn. Mosk. un-ta. Ser. mat., mekh., astron., fiz., khim., 1959, no. 2, 25–32 | Zbl

[26] Filippov A.F., Differentsialnye uravneniya s razryvnoi pravoi chastyu, Nauka, M., 1985 | MR

[27] Gamkrelidze R.V., Principles of optimal control theory, Plenum Press, New York, 1978 | MR | Zbl

[28] Gamkrelidze R.V., “Skolzyaschie rezhimy v teorii optimalnogo upravleniya”, Tr. MIAN, 169, 1985, 180–193 | MR | Zbl

[29] Gantmakher F.R., Teoriya matrits, 2-e izd., Nauka, M., 1966 | MR

[30] Grass D., Caulkins J.P., Feichtinger G., Tragler G., Behrens D.A., Optimal control of nonlinear processes. With applications in drugs, corruption, and terror, Springer, Berlin, 2008 | MR

[31] Grossman G.M., Helpman E., Innovation and growth in the global economy, MIT Press, Cambridge, MA, 1991

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

[33] Khartman F., Obyknovennye differentsialnye uravneniya, Mir, M., 1970 | MR

[34] Himmelberg C.J., “Measurable relations”, Fundam. Math., 87 (1975), 53–72 | MR | Zbl

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

[36] Intriligator M.D., Mathematical optimization and economic theory, Prentice-Hall, Englewood Cliffs, NJ, 1971 | MR

[37] Kamihigashi T., “Necessity of transversality conditions for infinite horizon problems”, Econometrica, 69 (2001), 995–1012 | DOI | MR | Zbl

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

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

[40] Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mischenko E.F., Matematicheskaya teoriya optimalnykh protsessov, Fizmatgiz, M., 1961

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

[42] Rudin W., Real and complex analysis, McGraw-Hill, New York, 1987 | MR | Zbl

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

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

[45] Seierstad A., Sydsæter K., Optimal control theory with economic applications, North-Holland, Amsterdam, 1987 | MR | Zbl

[46] Sethi S.P., Thompson G.L., Optimal control theory: Applications to management science and economics, Kluwer, Dordrecht, 2000 | MR | Zbl

[47] Shell K., “Applications of Pontryagin's maximum principle to economics”, Mathematical systems theory and economics 1, Lect. Notes Oper. Res. Math. Econ., 11, Springer, Berlin, 1969, 241–292 | MR

[48] Smirnov G.V., “Transversality condition for infinite-horizon problems”, J. Optim. Theory Appl., 88:3 (1996), 671–688 | DOI | MR | Zbl

[49] Solow R.M., Growth theory: an exposition, Oxford Univ. Press, New York, 1970

[50] Weitzman M.L., Income, wealth, and the maximum principle, Harvard Univ. Press, Cambridge, MA, 2003

[51] Ye J.J., “Nonsmooth maximum principle for infinite-horizon problems”, J. Optim. Theory Appl., 76:3 (1993), 485–500 | DOI | MR | Zbl