Infinite-horizon optimal control problems in economics

S. M. Aseev   ; K. O. Besov   ; A. V. Kryazhimskiy

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 67 (2012) no. 2, pp. 195-253

Résumé

This paper extends optimal control theory to a class of infinite-horizon problems that arise in studying models of optimal dynamic allocation of economic resources. In a typical problem of this sort the initial state is fixed, no constraints are imposed on the behaviour of the admissible trajectories at large times, and the objective functional is given by a discounted improper integral. We develop the method of finite-horizon approximations in a broad context and use it to derive complete versions of the Pontryagin maximum principle for such problems. We provide sufficient conditions for the normality of infinite-horizon optimal control problems and for the validity of the ‘standard’ limit transversality conditions with time going to infinity. As a meaningful example, we consider a new two-sector model of optimal economic growth subject to a random jump in prices. Bibliography: 53 titles.

Détail

Keywords: dynamic optimization, Pontryagin maximum principle, infinite horizon, transversality conditions at infinity, optimal economic growth.

Bibliographie

[1] D. Acemoglu, Introduction to modern economic growth, Princeton Univ. Press, Princeton, NJ, 2009, xviii+990 pp. | Zbl

[2] P. Aghion, P. Howitt, Endogenous growth theory, MIT Press, Cambridge, MA, 1998, 708 pp.

[3] K. J. Arrow, “Applications of control theory to economic growth”, Mathematics of the Decision Sciences, Part 2, Lectures Appl. Math., 12, Amer. Math. Soc., Providence, RI, 1968, 85–119 | MR | Zbl

[4] K. J. Arrow, M. Kurz, Public investment, the rate of return, and optimal fiscal policy, Johns Hopkins Press, Baltimore, MD, 1970, 218 pp.

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

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

[7] S. Aseev, G. Hutschenreiter, A. Kryazhimskii, A dynamic model of optimal allocation of resources to R, IIASA Interim Report IR-02-016, Laxenburg, Austria, 2002, 45 pp. http://www.iiasa.ac.at/Admin/PUB/Documents/IR-02-016.pdf

[8] S. M. Aseev, 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., 43:3 (2004), 1094–1119 | DOI | MR | Zbl

[9] S. M. Aseev, A. V. Kryazhimskiĭ, “The Pontryagin maximum principle for an optimal control problem with a functional specified by an improper integral”, Dokl. Math., 69:1 (2004), 89–91 | MR | Zbl

[10] S. M. Aseev, A. V. Kryazhimskii, “The Pontryagin maximum principle and problems of optimal economic growth”, Proc. Steklov Inst. Math., 257 (2007), 1–255 | DOI | MR | Zbl

[11] S. M. Aseev, A. V. Kryazhimskii, “On a class of optimal control problems arising in mathematical economics”, Proc. Steklov Inst. Math., 262 (2008), 10–25 | DOI | MR | Zbl

[12] S. Aseev, A. Kryazhimskii, A. Tarasyev, First order necessary optimality conditions for a class of infinite horizon optimal control problems, IIASA Interim Report IR-01-007, Laxenburg, Austria, 2001, 17 pp. http://www.iiasa.ac.at/Admin/PUB/Documents/IR-01-007.pdf

[13] S. M. Aseev, A. V. Kryazhimskij, A. M. Tarasyev, “The Pontryagin maximum principle and transversality conditions for an optimal control problem with infinite time interval”, Proc. Steklov Inst. Math., 233 (2001), 64–80 | MR | Zbl

[14] S. M. Aseev, V. M. Veliov, “Maximum principle for infinite-horizon optimal control problems with dominating discount”, Dynamics of Continuous, Discrete and Impulsive Systems, Ser. B: Applications Algorithms, 19 (2012), 43–63; Research Report 2011-06, ORCOS, Institute of Mathematical Methods in Economics, Vienna University of Technology, 2011, 21 pp. http://orcos.tuwien.ac.at/fileadmin/t/orcos/Research_Reports/2011-06.pdf

[15] J. P. Aubin, F. H. Clarke, “Shadow prices and duality for a class of optimal control problems”, SIAM J. Control Optim., 17:5 (1979), 567–586 | DOI | MR | Zbl

[16] E. J. Balder, “An existence result for optimal economic growth problems”, J. Math. Anal. Appl., 95:1 (1983), 195–213 | DOI | MR | Zbl

[17] R. J. Barro, X. Sala-i-Martin, Economic growth, New York, McGraw-Hill, 1995, 539 pp.

[18] Behavioral economics and its applications, eds. P. Diamond, H. Vartiainen, Princeton Univ. Press, Oxford, 2007, xvi+312 pp.

[19] R. Bellman, Dynamic programming, Princeton Univ. Press, Princeton, NJ, 1957, xxv+342 pp. | MR | Zbl

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

[21] V. I. Blagodatskikh, A. F. Filippov, “Differential inclusions and optimal control”, Proc. Steklov Inst. Math., 169 (1986), 199–259 | MR | Zbl | Zbl

[22] D. A. Carlson, A. B. Haurie, A. Leizarowitz, Infinite horizon optimal control. Deterministic and stochastic systems, Springer-Verlag, Berlin, 1991, xvi+332 pp. | Zbl

[23] D. Cass, “Optimum growth in an aggregative model of capital accumulation”, Rev. Econ. Stud., 32:3 (1965), 233–240 | DOI

[24] L. Cesari, Optimization – theory and applications. Problems with ordinary differential equations, Appl. Math. (N. Y.), 17, Springer-Verlag, New York, 1983, xiv+542 pp. | MR | Zbl

[25] A. C. Chiang, Elements of dynamic optimization, McGraw-Hill, Singapore, 1992, 327 pp.

[26] B. P. Demidovich, Lektsii po matematicheskoi teorii ustoichivosti, Nauka, M., 1967, 472 pp. ; “РџРёСЃСЊРјРѕ РІ СЂРμдакцию”, МатРμРј. замРμтки, 80:2 (2006), 320 | MR | Zbl

[27] A. V. Dmitruk, N. V. Kuz'kina, “Existence theorem in the optimal control problem on an infinite time interval”, Math. Notes, 78:4 (2005), 466–480 | DOI | MR | Zbl

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

[29] I. Ekeland, “Some variational problems arising from mathematical economics”, Mathematical economics (Montecatini Terme, 1986), Lectures Notes in Math., 1330, Springer, Berlin, 1988, 1–18 | DOI | MR | Zbl

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

[31] A. F. Filippov, Differential equations with discontinuous right-hand sides, Mathematics and Its Applications: Soviet Series, 18, Kluwer, Dordrecht, 1988, x+304 pp. | MR | MR | Zbl | Zbl

[32] R. V. Gamkrelidze, “Optimal sliding states”, Soviet Math. Dokl., 3 (1962), 559–562 | MR | Zbl

[33] R. V. Gamkrelidze, Principles of optimal control theory, Math. Concepts Methods Sci. Engrg., 7, Plenum Press, New York–London, 1978, xii+175 pp. | MR | Zbl

[34] R. V. Gamkrelidze, “Skolzyaschie rezhimy v teorii optimalnogo upravleniya”, Topologiya, obyknovennye differentsialnye uravneniya, dinamicheskie sistemy, Sbornik obzornykh statei. 2. K 50-letiyu instituta, Tr. MIAN SSSR, 169, 1985, 180–193 | MR | Zbl

[35] F. R. Gantmacher, The theory of matrices, v. 1, 2, Chelsea, New York, 1959, x+374, ix+276 pp. | MR | MR | Zbl | Zbl

[36] B. V. Gnedenko, Kurs teorii veroyatnostei, URSS, M., 2001, 320 pp.

[37] D. Grass, J. P. Caulkins, G. Feichtinger, G. Tragler, D. A. Behrens, Optimal control of nonlinear processes. With applications in drugs, corruption, and terror, Springer-Verlag, Berlin, 2008, xx+551 pp. | DOI | MR | Zbl

[38] G. M. Grossman, E. Helpman, Innovation and growth in the global economy, MIT Press, Cambridge, MA, 1991, 375 pp.

[39] H. Halkin, “Necessary conditions for optimal control problems with infinite horizons”, Econometrica, 42:2 (1974), 267–272 | DOI | MR | Zbl

[40] Ph. Hartman, Ordinary differential equations, John Wiley Sons, Inc., New York–London–Sydney, 1964, xiv+612 pp. | MR | MR | Zbl | Zbl

[41] T. Kamihigashi, “Necessity of transversality conditions for infinite horizon problems”, Econometrica, 69:4 (2001), 995–1012 | DOI | MR | Zbl

[42] A. Liapounoff, Problème général de la stabilité du mouvement, Ann. of Math. Stud., 17, Princeton Univ. Press, Princeton, NJ; Oxford Univ. Press, London, 1947, iv+272 pp. ; A. Lyapunov, Obschaya zadacha ob ustoichivosti dvizheniya, Izd-vo Khark. matem. obsch., Kharkov, 1892, 250 pp.; Рђ. Ляпунов, “Рљ РІРѕРїСЂРѕСЃСѓ РѕР± устоРNoчивости РґРІРёР¶РμРЅРёСЏ”, РЎРѕРѕР±С‰. Харьк. матРμРј. РѕР±С‰., 3:6 (1893), 265–272, Р�Р·Рґ-РІРѕ Харьк. матРμРј. РѕР±С‰., Харьков | MR | Zbl

[43] Ph. Michel, “On the transversality condition in infinite horizon optimal problems”, Econometrica, 50:4 (1982), 975–985 | DOI | MR | Zbl

[44] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko, The mathematical theory of optimal processes, Interscience Publishers John Wiley Sons, Inc., New York–London, 1962, viii+360 pp. | MR | MR | Zbl | Zbl

[45] F. P. Ramsey, “A mathematical theory of saving”, Econ. J., 38:152 (1928), 543–559 | DOI

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

[47] A. Seierstad, K. Sydsæter, Optimal control theory with economic applications, Adv. Textbooks Econom., 24, North-Holland, Amsterdam, 1987, xvi+445 pp. | MR | Zbl

[48] S. P. Sethi, G. L. Thompson, Optimal control theory. Applications to management science and economics, 2nd ed., Kluwer, Boston, MA, 2000, xviii+505 pp. | MR | Zbl

[49] K. Shell, “Applications of Pontryagin's maximum principle to economics”, Mathematical systems theory and economics, I, Proc. Internat. Summer School (Varenna, 1967), Lect. Notes Oper. Res. Math. Econ., 11, Springer, Berlin, 1969, 241–292 | MR | Zbl

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

[51] J. Warga, Optimal control of differential and functional equations, Academic Press, New York–London, 1972, xiii+531 pp. | MR | MR | Zbl

[52] M. L. Weitzman, Income, wealth, and the maximum principle, Harvard Univ. Press, Cambridge, MA, 2003, viii+341 pp. | Zbl

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