Optimal Economic Growth Models with Nonlinear Utility Functions


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.

Fig. 1
Fig. 2
Fig. 3

Data Availability Statement

This manuscript has no associated data.


  1. 1.

    See, e.g., https://en.wikipedia.org/wiki/Total_factor_productivity.


  1. 1.

    Ramsey, F.P.: A mathematical theory of saving. Econ. J. 38, 543–559 (1928)

    Article  Google Scholar 

  2. 2.

    Harrod, R.F.: An essay in dynamic theory. Econ. J. 49, 14–33 (1939)

    Article  Google Scholar 

  3. 3.

    Domar, E.D.: Capital expansion, rate of growth, and employment. Econometrica 14, 137–147 (1946)

    Article  Google Scholar 

  4. 4.

    Solow, R.M.: A contribution to the theory of economic growth. Q. J. Econ. 70, 65–94 (1956)

    Article  Google Scholar 

  5. 5.

    Swan, T.W.: Economic growth and capital accumulation. Econ. Rec. 32, 334–361 (1956)

    Article  Google Scholar 

  6. 6.

    Cass, D.: Optimum growth in an aggregative model of capital accumulation. Rev. Econ. Stud. 32, 233–240 (1965)

    Article  Google Scholar 

  7. 7.

    Koopmans, T.C.: On the concept of optimal economic growth, pp. 225–295. In: The Econometric Approach to Development Planning. North-Holland, Amsterdam (1965)

  8. 8.

    Takayama, A.: Mathematical Economics. The Dryden Press, Hinsdale (1974)

    Google Scholar 

  9. 9.

    Barro, R.J., Sala-i-Martin, X.: Economic Growth. MIT Press, Cambridge (2004)

    Google Scholar 

  10. 10.

    Morimoto, H.: Optimal consumption models in economic growth. J. Math. Anal. Appl. 337, 480–492 (2008)

    MathSciNet  Article  Google Scholar 

  11. 11.

    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)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Acemoglu, D.: Introduction to Modern Economic Growth. Princeton University Press, Princeton (2009)

    Google Scholar 

  13. 13.

    Adachi, T., Morimoto, H.: Optimal consumption of the finite time horizon Ramsey problem. J. Math. Anal. Appl. 358, 28–46 (2009)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Huang, H.-J., Khalili, S.: Optimal consumption in the stochastic Ramsey problem without boundedness constraints. SIAM J. Control Optim. 57, 783–809 (2019)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Huong, V.T.: Solution existence theorems for finite horizon optimal economic growth problems. arXiv:2001.03298. (Submitted)

  16. 16.

    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

  17. 17.

    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)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Vinter, R.: Optimal Control. Birkhäuser, Boston (2000)

    Google Scholar 

  19. 19.

    Ioffe, A.D., Tihomirov, V.M.: Theory of Extremal Problems. North-Holland Publishing Company, Amsterdam (1979)

    Google Scholar 

  20. 20.

    Pierre, N.V.T.: Introductory Optimization Dynamics. Optimal Control with Economics and Management Science Applications. Springer, Berlin (1984)

  21. 21.

    Chiang, A.C., Wainwright, K.: Fundamental Methods of Mathematical Economics, 4th edn. McGraw-Hill, New York (2005)

    Google Scholar 

  22. 22.

    Cesari, L.: Optimization Theory and Applications. Springer, New York (1983)

    Google Scholar 

  23. 23.

    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation (Vol. I. Basic Theory, Vol. II. Applications). Springer, Berlin (2006)

  24. 24.

    Mordukhovich, B.S.: Variational Analysis and Applications. Springer, Berlin (2018)

    Google Scholar 

  25. 25.

    Alekseev, V.M., Tikhomirov, V.M., Fomin, S.V.: Optimal Control. Consultants Bureau, New York (1987)

    Google Scholar 

  26. 26.

    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)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Frankowska, H.: Normality of the maximum principle for absolutely continuous solutions to Bolza problems under state constraints. Control Cybern. 38, 1327–1340 (2009)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    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)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Kolmogorov, A.N., Fomin, S.V.: Introductory Real Analysis. Dover Publications Inc, New York (1970)

    Google Scholar 

Download references


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.

Author information



Corresponding author

Correspondence to Nguyen Dong Yen.

Additional information

Dedicated to Professor Pham Huu Sach on the occasion of his 80th birthday.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by Kok Lay Teo.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

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

Download citation


  • Optimal economic growth
  • Optimal control
  • Maximum principle

Mathematics Subject Classification

  • 91B62
  • 49J15
  • 37N40
  • 46N10
  • 91B55