Optimal Control of Investments in the Closed-Form Dynamic Model of Three-Sector Economy: Mathematical Statement of the Problem and General Analysis Based on the Maximum Principle

A mathematical optimal control problem formulated on the basis of the closed-form dynamic problem of three-sector economy is studied. The system state is described by a set of functions of specific capital in each sector; the control parameter is the quantity characterizing a volume of specific investments of the fund-creating sector playing a key role in the economic system. The mathematical problem is formulated as a classical optimal control problem with a fixed time interval, with the fastened left and free right trajectory ends. Solving the stated problem is based on the Pontryagin maximum principle. A general control structure is determined that corresponds to the maximum principle. A method for further study of the stated problem is described, which consists in analytical determination of main and associated variables and in development of the procedure for finding the optimal control.

References

[1] Arrow K.J., Intriligator M.D., eds. Handbook of Mathematical Economics. Vol. 3. Amsterdam-New York, North-Holland Publishing Co., 1986.

[2] Belen’kiy V.Z. Optimizatsionnye modeli ekonomicheskoy dinamiki. Ponyatiynyy apparat. Odnomernye modeli [Optimization models of economic dynamics. Conceptual apparatus. One-dimensional models]. Moscow, Nauka Publ., 2007. 259 p.

[3] Kolemaev V.A. Matematicheskaya ekonomika [Mathematical economy]. Moscow, Yuniti-Dana Publ., 2002. 399 p.

[4] Kolemaev V.A. Optimal balanced growth of open three-sector economy. Prikladnaya ekonometrika [Applied econometrics], 2008, iss. 3, pp. 14-42 (in Russ.).

[5] Alekseev V.M., Tikhomirov V.M., Fomin S.V. Optimal’noe upravlenie [Optimal control]. Moscow, Fizmatlit Publ., 2007. 408 p.

[6] Arutyunov A.A., Magaril-Il’yaev G.G., Tikhomirov V.M. Printsip maksimuma Pontryagina [Pontriagin’s maximum principle]. Moscow, Faktorial Press Publ., 2006. 144 p.

[7] Ioffe A.D., Tikhomirov V.M. Teoriya ekstremal’nykh zadach [Extremal problems theory]. Moscow, Nauka Publ., 1974. 481 p.

[8] Van’ko V.I., Ermoshina O.V., Kuvyrkin G.N., Zarubin B.C., Krishchenko A.P., eds. Variatsionnoe ischislenie i optimal’noe upravlenie [Variational calculus. Optimum control]. Moscow, MGTU im. N.E. Baumana Publ., 2006. 488 p.

[9] Leonard D., Long N. Optimal control theory and static optimization in economics. Cambrige Univ. Press, 1992. 353 p.

[10] Sethi S.P., Thompson G.L. Optimal control theory: applications to management science and economics. Second Ed. New York, Springer, 2000. 504 p.

[11] Belen’kiy V.Z. A theorem on the stationary solution of the generalized Ramsey -Cass-Koopmans model. Analysis and simulation of economic processes. Sb. statey "Analiz i modelirovanie ekonomicheskikh protsessov" [Coll. Pap. "Analysis and simulation of economic processes"]. Moscow, TsEMI RAN Publ., 2004, iss. 1 (in Russ.).

[12] Matveenko V.D. Struktura optimal’nykh traektoriy v modelyakh ekonomicheskoy dinamiki. Diss. Dokt. Econ. Nauk [The structure of optimal trajectories in models of economic dynamics. Dr. econ. sci. diss.]. Moscow, TsEMI RAN Publ., 2004. 261 p.

[13] Koopmans T.C. On the concept of optimal economic growth. Ex Aedibus Academicis in Civitate Vaticana, 1965, no. 28, pp. 225-300.