Skip to main content
SpringerLink
Log in

Closed-form solutions of Lucas–Uzawa model with externalities via partial Hamiltonian approach

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we establish multiple closed-form solutions for all the variables in the Lucas–Uzawa model with externalities for the case with no parameter restrictions as well as for cases with specific parameter restrictions. These multiple solutions are derived with the help of the results derived in Naz et al. (Commun Nonlinear Sci Numer Simul 30(1):299–306, 2016) and Naz and Chaudhry (Math Modell Anal 22(4):464–483, 2017). This multiplicity of solutions is new to the economic growth literature on Lucas–Uzawa model with externalities. After finding solutions for the Lucas–Uzawa model with externalities, we use these solutions to derive the growth rates of all the variables in the system which enables us to fully describe the dynamics of the model. The multiple solutions can potentially explain why some countries economically overtake other countries even though they start from the same initial conditions. We have provided results of the numerical simulations procedure for \(\sigma =\beta \).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  • Ali Z, Naeem I, Husnine SM (2015) Group classification and exact solutions of generalized modified Boussinesq equation. Appl Anal 94(7):1397–1404

    Article  MathSciNet  MATH  Google Scholar 

  • Benhabib J, Perli R (1994) Uniqueness and indeterminacy: on the dynamics of endogenous growth. J Econ Theory 63(1):113–142

    Article  MATH  Google Scholar 

  • Boucekkine R, Ruiz-Tamarit JR (2008) Special functions for the study of economic dynamics: the case of the Lucas–Uzawa model. J Math Econ 44:33–54

    Article  MathSciNet  MATH  Google Scholar 

  • Bozhkov Y, Freire IL, Ibragimov NH (2014) Group analysis of the Novikov equation. Comput Appl Math 33(1):193–202

    Article  MathSciNet  MATH  Google Scholar 

  • Caballé J, Santos MS (1993) On endogenous growth with physical and human capital. J Political Econ 101:1042–1067

    Article  Google Scholar 

  • Hiraguchi R (2009) A note on the closed-form solution to the Lucas–Uzawa model with externality. J Econ Dyn Control 33:1757–1760

    Article  MathSciNet  MATH  Google Scholar 

  • Kara AH, Mahomed FM, Naeem I, Wafo Soh C (2007) Partial Noether operators and first integrals via partial Lagrangians. Math Methods Appl Sci 30(16):2079–2089

    Article  MathSciNet  MATH  Google Scholar 

  • Lucas R (1988) On the mechanics of economic development. J Monet Econ 22:3–42

    Article  Google Scholar 

  • Magan AB, Mason DP, Mahomed FM (2017) Analytical solution in parametric form for the two-dimensional liquid jet of a power-law fluid. Int J Non-Linear Mech 93:53–64

    Article  Google Scholar 

  • Mahomed FM, Roberts JAG (2015) Characterization of Hamiltonian symmetries and their first integrals. Int J Non-Linear Mech 74:84–91

    Article  Google Scholar 

  • Mankiw NG, Romer D, Weil DN (1992) A contribution to the empirics of economic growth. Q J Econ 107:407–437

    Article  MATH  Google Scholar 

  • Mulligan CB, Sala-i-Martin X (1993) Transitional dynamics in two-sector models of endogenous growth. Q J Econ 108:739–773

    Article  Google Scholar 

  • Naeem I, Mahomed FM (2008) Noether, partial Noether operators and first integrals for a linear system. J Math Anal Appl 342(1):70–82

    Article  MathSciNet  MATH  Google Scholar 

  • Naz R (2016) The applications of the partial Hamiltonian approach to mechanics and other areas. Int J Non-Linear Mech 86:1–6

    Article  Google Scholar 

  • Naz R, Chaudhry A (2017) Comparison of closed-form solutions for the Lucas–Uzawa model via the partial Hamiltonian approach and the classical approach. Math Model Anal 22(4):464–483

    Article  MathSciNet  Google Scholar 

  • Naz R, Mahomed FM, Chaudhry A (2014) A partial Hamiltonian approach for current value Hamiltonian systems. Commun Nonlinear Sci Numer Simul 19:3600–3610

    Article  MathSciNet  Google Scholar 

  • Naz R, Chaudhry A, Mahomed FM (2016) Closed-form solutions for the Lucas–Uzawa model of economic growth via the partial Hamiltonian approach. Commun Nonlinear Sci Numer Simul 30(1):299–306

    Article  MathSciNet  Google Scholar 

  • Romer PM (1997) Endogenous technological change. J Political Econ 98(5)(Part 2):S71–S102

  • Romer PM (1986) Increasing returns and long-run growth. J Political Econ 94:1002–1037

    Article  Google Scholar 

  • Ruiz-Tamarit JR (2008) The closed-form solution for a family of four-dimension nonlinear MHDS. J Econ Dyn Control 32(3):1000–1014

    Article  MathSciNet  MATH  Google Scholar 

  • Ruscica M, Tracin R (2015) Group classification of an energy transport model for semiconductors with crystal heating. Comput Appl Math 34(3):1167–1174

    Article  MathSciNet  MATH  Google Scholar 

  • Tamura R (1991) Income convergence in an endogeneous growth model. J Political Econ 99:522–540

    Article  Google Scholar 

  • Uzawa H (1965) Optimum technical change in an aggregative model of economic growth. Int Econ Rev 6:18–31

    Article  MATH  Google Scholar 

  • Xie D (1994) Divergence in economic performance: transitional dynamics with multiple equilibria. J Econ Theory 63(1):97–112

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centre for Mathematics and Statistical Sciences, Lahore School of Economics, Lahore, 53200, Pakistan

    R. Naz

  2. Department of Economics, Lahore School of Economics, Lahore, 53200, Pakistan

    Azam Chaudhry

Authors
  1. R. Naz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Azam Chaudhry
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. Naz.

Additional information

Communicated by Delfim F. M. Torres.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Naz, R., Chaudhry, A. Closed-form solutions of Lucas–Uzawa model with externalities via partial Hamiltonian approach. Comp. Appl. Math. 37, 5146–5161 (2018). https://doi.org/10.1007/s40314-018-0622-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40314-018-0622-6

Keywords

Mathematics Subject Classification

Search

Navigation