We study a class of finite horizon optimal economic growth problems with nonlinear utility functions and linear production functions. By using a maximum principle in the optimal control theory and employing the special structure of the problems, we are able to explicitly describe the unique solution via input parameters. Economic interpretations of the obtained results and an open problem about the case where the total factor productivity falls into a bounded open interval defined by the growth rate of labor force, the real interest rate, and the exponent of the utility function are also expressed.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
Data Availability Statement
This manuscript has no associated data.
Ramsey, F.P.: A mathematical theory of saving. Econ. J. 38, 543–559 (1928)
Harrod, R.F.: An essay in dynamic theory. Econ. J. 49, 14–33 (1939)
Domar, E.D.: Capital expansion, rate of growth, and employment. Econometrica 14, 137–147 (1946)
Solow, R.M.: A contribution to the theory of economic growth. Q. J. Econ. 70, 65–94 (1956)
Swan, T.W.: Economic growth and capital accumulation. Econ. Rec. 32, 334–361 (1956)
Cass, D.: Optimum growth in an aggregative model of capital accumulation. Rev. Econ. Stud. 32, 233–240 (1965)
Koopmans, T.C.: On the concept of optimal economic growth, pp. 225–295. In: The Econometric Approach to Development Planning. North-Holland, Amsterdam (1965)
Takayama, A.: Mathematical Economics. The Dryden Press, Hinsdale (1974)
Barro, R.J., Sala-i-Martin, X.: Economic Growth. MIT Press, Cambridge (2004)
Morimoto, H.: Optimal consumption models in economic growth. J. Math. Anal. Appl. 337, 480–492 (2008)
Morimoto, H., Zhou, X.Y.: Optimal consumption in a growth model with the Cobb–Douglas production function. SIAM J. Control Optim. 47, 2991–3006 (2008)
Acemoglu, D.: Introduction to Modern Economic Growth. Princeton University Press, Princeton (2009)
Adachi, T., Morimoto, H.: Optimal consumption of the finite time horizon Ramsey problem. J. Math. Anal. Appl. 358, 28–46 (2009)
Huang, H.-J., Khalili, S.: Optimal consumption in the stochastic Ramsey problem without boundedness constraints. SIAM J. Control Optim. 57, 783–809 (2019)
Huong, V.T.: Solution existence theorems for finite horizon optimal economic growth problems. arXiv:2001.03298. (Submitted)
Huong, V.T.: Optimal economic growth problems with high values of total factor productivity. Appl. Anal. (2020). https://doi.org/10.1080/00036811.2020.1779231
Huong, V.T., Yao, J.-C., Yen, N.D.: Optimal processes in a parametric optimal economic growth model. Taiwanese J. Math. 24, 1283–1306 (2020)
Vinter, R.: Optimal Control. Birkhäuser, Boston (2000)
Ioffe, A.D., Tihomirov, V.M.: Theory of Extremal Problems. North-Holland Publishing Company, Amsterdam (1979)
Pierre, N.V.T.: Introductory Optimization Dynamics. Optimal Control with Economics and Management Science Applications. Springer, Berlin (1984)
Chiang, A.C., Wainwright, K.: Fundamental Methods of Mathematical Economics, 4th edn. McGraw-Hill, New York (2005)
Cesari, L.: Optimization Theory and Applications. Springer, New York (1983)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation (Vol. I. Basic Theory, Vol. II. Applications). Springer, Berlin (2006)
Mordukhovich, B.S.: Variational Analysis and Applications. Springer, Berlin (2018)
Alekseev, V.M., Tikhomirov, V.M., Fomin, S.V.: Optimal Control. Consultants Bureau, New York (1987)
Ferreira, M.M.A., Vinter, R.B.: When is the maximum principle for state constrained problems nondegenerate? J. Math. Anal. Appl. 187, 438–467 (1994)
Frankowska, H.: Normality of the maximum principle for absolutely continuous solutions to Bolza problems under state constraints. Control Cybern. 38, 1327–1340 (2009)
Fontes, F.A.C.C., Frankowska, H.: Normality and nondegeneracy for optimal control problems with state constraints. J. Optim. Theory Appl. 166, 115–136 (2015)
Kolmogorov, A.N., Fomin, S.V.: Introductory Real Analysis. Dover Publications Inc, New York (1970)
Vu Thi Huong and Nguyen Dong Yen were supported by the project “Some qualitative properties of optimization problems and dynamical systems, and applications” (Code: ICRTM01\(\_\)2020.08) of the International Center for Research and Postgraduate Training in Mathematics (ICRTM) under the auspices of UNESCO of Institute of Mathematics, Vietnam Academy of Science and Technology. Jen-Chih Yao was supported by China Medical University, Taichung, Taiwan. We are grateful to the five anonymous referees, whose detailed comments and significant suggestions have helped us to improve the presentation of the obtained results.
Dedicated to Professor Pham Huu Sach on the occasion of his 80th birthday.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Communicated by Kok Lay Teo.
About this article
Cite this article
Huong, V.T., Yao, JC. & Yen, N.D. Optimal Economic Growth Models with Nonlinear Utility Functions. J Optim Theory Appl 188, 571–596 (2021). https://doi.org/10.1007/s10957-020-01797-5
- Optimal economic growth
- Optimal control
- Maximum principle
Mathematics Subject Classification