We investigate the resource allocation problem in a two-sector economic model with a Cobb-Douglas production function with different depreciation rates. The problem is considered on a finite time horizon with an integral type functional. Optimality of the extremum solution constructed by the Pontryagin maximum principle is established. When the planning horizon is sufficiently long, the optimal control has two or three switching points, contains one singular section, and vanishes on the terminal section. A transitional “calibration” regime exists between the singular section, where the motion is along a singular ray, and the terminal section. The solution of the maximum-principle boundary-value problem is presented in explicit form, accompanied by graphs based on numerical results.
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References
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, Mathematical Theory of Optimal Processes [in Russian], Nauka, Moscow (1961).
Yu. N. Kiselev, S. N. Avvakumov, and M. V. Orlov, Optimal Control. Linear Theory and Applications [in Russian], MAKS Press, Moscow (2007).
Yu. N. Kiselev, “Sufficient conditions of optimality in terms of Pontryagin maximum principle constructs,” Mathematical Models in Economics and Biology, Proceedings of Scientific Seminar, Planernoe, Moscow Oblast [in Russian], MAKS Press, Moscow (2003), pp. 57–67.
S. N. Avvakumov and Yu. N. Kiselev, “Some optimal control algorithms,” Trudy Inst. Matem. Mekhan. UrO RAN, Ekaterinburg, 12, No. 2, 3–17 (2006).
Yu. N. Kiselev and M. V. Orlov, “Resource allocation problem in a two-sector economic model with Cobb-Douglas production function,” Modern Methods in the Theory of Boundary-Value Problems: Pontryagin Readings XX [in Russian], Voronezh (2009), pp. 85–86.
Yu. N. Kiselev, “Construction of exact solutions for a nonlinear time-optimal problem of a special kind,” Fundament. i Prikl.Matem., 3, No. 3, 847–868 (1997).
S. A. Ashmanov, An Introduction to Mathematical Economics [n Russian], Nauka, Moscow (1984).
S. A. Ashmanov, Mathematical Models and Methods in Economics [in Russian], Izd. MGU, Moscow (1980).
S. N. Avvakumov, Yu. N. Kiselev, M. V. Orlov, and A. M. Taras’ev, “Profit maximization problem for Cobb-Douglas and CES production functions,” in: Nonlinear Dynamics and Control [in Russian], No. 5, Fizmatlit, Moscow (2007), pp. 309–350.
Yu. N. Kiselev, V. Yu. Reshetov, S. N. Avvakumov, and M. V. Orlov, “Construction of optimal solution and reachability sets in a resource allocation problem,” Probl. Dinam. Upravl., MGU, MAKS Press, Moscow, No. 2, 106–120 (2007).
Yu. N. Kiselev, S. N. Avvakumov, and M. V. Orlov, “Construction in analytical form of optimal solution and reachability sets in a resource allocation problem,” Prikl. Matem. Informat., MAKS Press, Moscow, No. 27, 80–99 (2007).
Yu. N. Kiselev, V. Yu. Reshetov, S. N. Avvakumov, andM. V. Orlov, “An investigation of a resource allocation problem,” Differential Equations and Topology: Int. Conf. honoring 100th Anniversary of L. S. Pontryagin, Abstracts of Papers [in Russian], MAKS Press, Moscow (2008), pp. 350–352.
Yu. N. Kiselev, S. N. Avvakumov, and M. V. Orlov, “Investigation of a two-sector economic model with possible singular regimes,” Probl. Dinam. Upravl., MGU, MAKS Press, Moscow, No. 3, 77–116 (2008).
Yu. N. Kiselev and M. V. Orlov, “An optimal resource allocation problem for a two-sector economic model with a Cobb-Douglas production function,” Diff. Uravn., 46, No. 12, 1749–1765 (2010).
M. S. Nikol’skii, “A simplified game for the interaction of two states,” Vestn. MGU, Ser. 15, Vychisl. Matem. Kibernet., No. 2, 14–20 (2009).
Yu. N. Kiselev and M. V. Orlov, “Optimal control problems with singular regimes for a model from microbiology,” Vestnik MGU, Ser. 15, Vychil. Matem. Kibernet., No. 3, 23–26 (1998).
H. A. van den Berg, Yu. V. Kiselev, S. A. L. M. Kooijman, and M. V. Orlov, “Optimal allocation between nutrient uptake and growth in a microbial trichome,” J. Math. Biology, 37, 28–48 (1998).
Yu. N. Kiselev and M. V. Orlov, Investigation of one-dimensional optimization models with an infinite horizon,” Diff. Uravn., 40, No. 12, 1615–1628 (2004).
Yu. N. Kiselev and M. V. Orlov, “An optimal resource allocation program for a two-sector economic model with a Cobb-Douglas production function with different depreciation rates,” Diff. Uravn., 48, No. 12, 1642–1657 (2012).
Yu. N. Kiselev, M. V. Orlov, and S. M. Orlov, “An investigation of a two-sector economic model with an integral functional,” Vestnik MGU, Ser. 15, Vychil. Matem. Kibernet., No. 4, 27–37 (2013).
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Translated from Prikladnaya Matematika i Informatika, No. 53, 2016, pp. 21–45.
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Kiselev, Y.N., Avvakumov, S.N., Orlov, M.V. et al. Optimal Resource Allocation in a Two-Sector Economic Model with an Integral Functional. Comput Math Model 28, 316–338 (2017). https://doi.org/10.1007/s10598-017-9367-0
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DOI: https://doi.org/10.1007/s10598-017-9367-0