The Pontryagin Maximum Principle and Optimal Economic Growth Problems

S. M. Aseev ; A. V. Kryazhimskii

Trudy Matematicheskogo Instituta imeni V.A. Steklova, The Pontryagin maximum principle and optimal economic growth problems, Tome 257 (2007), pp. 3-271.

Résumé

This monograph is devoted to the theory of the Pontryagin maximum principle as applied to a special class of optimal control problems that arise in economics when studying economic growth processes. The first chapter presents a new approximation approach that leads to a complete set of necessary optimality conditions in the form of the Pontryagin maximum principle. The attention is focused on the characterization of the behavior of the adjoint variable and the Hamiltonian of a problem at infinity. In the second chapter, the approach proposed is applied to a problem of optimal dynamical allocation of labor resources in the endogenous economic growth theory. The monograph is addressed to a wide circle of scientists, postgraduates, and students who are interested in the theory of the Pontryagin maximum principle and its applications in economics.

Bibliographie

[1] Aleksandrov V. V., Boltyanskii V. G., Lemak S. S., Parusnikov N. A., Tikhomirov V. M., Optimizatsiya dinamiki upravlyaemykh sistem, MGU, M., 2000

[2] Arnold V. I., Matematicheskie metody klassicheskoi mekhaniki, Nauka, M., 1979 | MR

[3] Arnold V. I., “Optimizatsiya v srednem i fazovye perekhody v upravlyaemykh dinamicheskikh sistemakh”, Funkts. analiz i ego pril., 36:2 (2002), 1–11 | MR

[4] Arutyunov A. V., “Vozmuscheniya ekstremalnykh zadach s ogranicheniyami i neobkhodimye usloviya optimalnosti”, Itogi nauki i tekhniki. Matematicheskii analiz, 27, VINITI, M., 1989, 147–235 | MR

[5] Arutyunov A. V., Usloviya ekstremuma. Anormalnye i vyrozhdennye zadachi, Faktorial, M., 1997 | MR | Zbl

[6] Aseev S. M., “Metod gladkikh approksimatsii v teorii neobkhodimykh uslovii optimalnosti dlya differentsialnykh vklyuchenii”, Izv. RAN. Ser. mat., 61:2 (1997), 3–26 | MR | Zbl

[7] Aseev S. M., “Ekstremalnye zadachi dlya differentsialnykh vklyuchenii s fazovymi ogranicheniyami”, Tr. MIAN, 233, 2001, 5–70 | MR

[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., Kryazhimskii A. V., Tarasev A. M., “Printsip maksimuma Pontryagina i usloviya transversalnosti dlya odnoi zadachi optimalnogo upravleniya na beskonechnom intervale”, Tr. MIAN, 233, 2001, 71–88 | MR | Zbl

[10] Aseev S. M., Smirnov A. I., “Neobkhodimye usloviya optimalnosti pervogo poryadka dlya zadachi optimalnogo prokhozhdeniya cherez zadannuyu oblast”, Nelineinaya dinamika i upravlenie, no. 4, eds. S. V. Emelyanov, S. K. Korovin, Fizmatlit, M., 2004, 179–204

[11] Bobylev N. A., Emelyanov S. V., Korovin S. K., Geometricheskie metody v variatsionnykh zadachakh, Magistr, M., 1998

[12] Varga Dzh., Optimalnoe upravlenie differentsialnymi i funktsionalnymi uravneniyami, Nauka, M., 1977 | MR

[13] Gnedenko B. V., Kurs teorii veroyatnostei, URSS, M., 2001

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

[15] Ioffe A. D., Tikhomirov V. M., Teoriya ekstremalnykh zadach, Nauka, M., 1974 | MR | Zbl

[16] Kolmogorov A. N., Fomin S. V., Elementy teorii funktsii i funktsionalnogo analiza, Nauka, M., 1976 | MR

[17] Mordukhovich B. Sh., Metody approksimatsii v zadachakh optimizatsii i upravleniya, Nauka, M., 1988 | MR

[18] Pontryagin L. S., Obyknovennye differentsialnye uravneniya, Nauka, M., 1974 | MR

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

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

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

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

[23] Arrow K. J., “Applications of control theory to economic growth”, Mathematics of the Decision Sciences, Pt. 2, Amer. Math. Soc., Providence, RI, 1968, 85–119 | MR

[24] Arrow K. J., Kurz M., Public investment, the rate of return, and optimal fiscal policy, J. Hopkins Univ. Press, Baltimore, MD, 1970

[25] Arutyunov A. V., Aseev S. M., “Investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints”, SIAM J. Control and Optim., 35 (1997), 930–952 | DOI | MR | Zbl

[26] Aseev S. M., “Methods of regularization in nonsmooth problems of dynamic optimization”, J. Math. Sci., 94:3 (1999), 1366–1393 | DOI | MR | Zbl

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

[28] Aseev S., Hutschenreiter G., Kryazhimskii A., Optimal investment in R with international knowledge spillovers, WIFO Working Paper No 175, 2002

[29] Aseev S. M., Hutschenreiter G., Kryazhimskii A. V., “A dynamical model of optimal investment in R”, J. Math. Sci., 126:6 (2005), 1495–1535 | DOI | MR | Zbl

[30] Aseev S., Hutschenreiter G., Kryazhimskiy A., Lysenko A., “A dynamical model of optimal investment in research and development with international knowledge spillovers”, Math. and Comput. Model. Dyn. Syst., 11:2 (2005), 125–123 | DOI | MR

[31] Aseev S., Kryazhimskii A., The Pontryagin maximum principle for infinite-horizon optimal controls, IIASA Interim Rept. IR-03-013, Laxenburg, Austria, 2003

[32] 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 and Optim., 43 (2004), 1094–1119 | DOI | Zbl

[33] Aseev S., Kryazhimskii A., Tarasyev A., First order necessary optimality conditions for a class of infinite horizon optimal control problems, IIASA Interim Rept. IR-01-007, Laxenburg, Austria, 2001

[34] Aseev S. M., Kryazhimskii A. V., Tarasyev A. M., “The Pontryagin maximum principle and transversality conditions for a class of optimal economic growth problems”, Nonlinear Control Systems, 2001, Proc. 5th IFAC Symp., v. 1 (St.-Petersburg, Russia, July 4–6, 2001), Elsevier, New York, London, 2001, 71–76 | MR

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

[36] Bagliano F.-C., Bertola G., Models for dynamic macroeconomics, Oxford Univ. Press, New York, 2004 | MR | Zbl

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

[38] Baratchart L., Chyba M., Pomet J.-B., “A Grobman–Hartnman theorem for control systems”, J. Dyn. and Diff. Equat., 19:1 (2007), 75–107 | DOI | MR | Zbl

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

[40] Bellman R., Dynamic programming, Princeton Univ. Press, Princeton, NJ, 1957 | MR

[41] Benassy J.-P., “Is there always too little research in endogenous growth with expanding product variety?”, Eur. Econ. Rev., 42 (1998), 61–69 | DOI

[42] 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

[43] Blot J., Michel P., “First-order necessary conditions for infinite-horizon variational problems”, J. Optim. Theory and Appl., 88:2 (1996), 339–364 | DOI | MR | Zbl

[44] Borisov V. F., Hutschenreiter G., Kryazhimskii A. V., “Asymptotic growth rates in knowledge-exchanging economies”, Ann. Oper. Res., 89 (1999), 61–73 | DOI | MR | Zbl

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

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

[47] Cesari L., Asymptotic behavior and stability problems in ordinary differential equations, Springer, Berlin, 1959 | MR | Zbl

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

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

[50] Clarke F. H., Optimization and nonsmooth analysis, J. Wiley, New York, 1983 | MR

[51] Dixit A. K., Pyndyck R. S., Investment under uncertainty, Princeton Univ. Press, Princeton, NJ, 1994

[52] Dixit A. K., Stiglitz J., “Monopolistic competition and optimum product diversity”, Amer. Econ. Rev., 67 (1977), 297–308

[53] Dorfman R., “An economic interpretation of optimal control theory”, Amer. Econ. Rev., 59 (1969), 817–831

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

[55] Ethier W. J., “National and international returns to scale in the modern theory of international trade”, Amer. Econ. Rev., 72 (1982), 389–405

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

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

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

[59] Hartman P., Ordinary differential equations, J. Wiley Sons, New York, 1964 | MR | Zbl

[60] Hirsch M. W., Smale S., Differential equations, dynamical systems, and linear algebra, Acad. Press, London, 1974 | MR

[61] Hutschenreiter G., Kaniovski Yu. M., Kryazhimskii A. V., Endogenous growth, absorptive capacities, and international R spillovers, IIASA Work. Paper WP-95-092, Laxenburg, Austria, 1995

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

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

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

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

[66] Liapounoff A., Problème général de la stabilité du mouvement, Ann. Math. Stud., 17, Princeton Univ. Press, Princeton, NJ, 1947 | MR | Zbl

[67] Leung S. F., “Transversality condition and optimality in a class of infinite horizon continuous time economic models”, J. Econ. Theory, 54 (1991), 224–233 | DOI | MR | Zbl

[68] Lorenz H.-W., Nonlinear dynamical economics and chaotic motion, Lect. Notes Econ. and Math. Syst., 334, Springer, Berlin, 1989 | MR | Zbl

[69] Mangasarian O. L., “Sufficient conditions for the optimal control of nonlinear systems”, SIAM J. Control, 4 (1966), 139–152 | DOI | MR | Zbl

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

[71] Nordhaus W. D., Managing the global commons. The economics of climate change, MIT Press, Cambridge, MA, 1994

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

[73] Romer P. M., “Endogenous technological change”, J. Polit. Econ., 98:5 (1990), S71–S102, Pt. 2 | DOI

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

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

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

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

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

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

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

[81] Stern L. E., “Criteria of optimality in the infinite-time optimal control problem”, J. Optim. Theory and Appl., 44:3 (1984), 497–508 | DOI | MR | Zbl

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

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