Abstract
A projection method is proposed for infinite-horizon economic growth problems. Exponentially discounted orthogonal Laguerre polynomials are used as the basis functions for the parameterization of the solution. The convergence of the method is studied numerically for integrable cases in the Ramsey model. It is shown that the best convergence of the method is achieved if the parameter in the exponent is chosen to be equal to the negative eigenvalue of the linearization matrix of the Hamiltonian system around a steady state at infinity. In the considered examples, the projection method leads to a system of equations with a small number of unknowns, in contrast to the methods using finite difference approximation.
Notes
In the paper [9], the inequality with the opposite sign is required to satisfy the balanced growth condition. In this case, there is no steady state \(k_{ss}\), and capital grows to infinity.
REFERENCES
J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd ed. (Dover, New York, 2001).
W. H. Press, S. A. Teukolsky, T. Vetterling, and B. P. Flannery, Numerical Recipes, 3rd ed. (Cambridge Univ. Press, New York, 2007).
K. L. Judd, “Projection methods for solving aggregate growth models,” J. Econ. Theory 58 (2), 410–452 (1992). https://doi.org/10.1016/0022-0531(92)90061-L
A. Miftakhova, K. Schmedders, and M. Schumacher, “Computing economic equilibria using projection methods,” Annu. Rev. Econ. 12, 317–353 (2020). https://doi.org/10.1146/annurev-economics-080218-025711
K. L. Judd, “The parametric path method: An alternative to Fair–Taylor and L-B-J for solving perfect foresight models,” J. Econ. Dyn. Control. 26, (9–10), 1557–1583 (2002). https://doi.org/10.1016/S0165-1889(01)00085-9
O. J. Blanchard and S. Fischer, Lectures on Macroeconomics (MIT Press, Cambridge, 1993).
R. J. Barro and X. Sala-i-Martin, Economic Growth, 2nd ed. (MIT Press, Cambridge, 1995).
S. M. Aseev and A. V. Kryazhimskii, “The Pontryagin maximum principle and optimal economic growth problems,” Proc. Steklov Inst. Math. 257, 1–255 (2007). https://doi.org/10.1134/S0081543807020010
W. T. Smith, “A closed form solution to the Ramsey model,” J. Macroecon. 6 (1), 1–27 (2006). https://doi.org/10.2202/1534-6005.1356
S. Lahiri, R. S. Eckaus, and M. Babiker, The Effects of Changing Consumption Patterns on the Costs of Emission Restrictions: Report No. 64 of the MIT Joint Program on the Science and Policy of Global Change (MIT Press, Cambridge, 2000).
N. B. Melnikov, B. C. O’Neill, and M. G. Dalton, “Accounting for household heterogeneity in general equilibrium economic growth models,” Energy Econ. 34 (5), 1475–1483 (2012). https://doi.org/10.1016/j.eneco.2012.06.010
N. B. Melnikov, A. P. Gruzdev, M. G. Dalton, M. Weitzel, and B. C. O’Neill, “Parallel extended path method for solving perfect foresight models,” Comput. Econ. 58 (2), 517–534 (2021). https://doi.org/10.1007/s10614-020-10044-y
R. Boháček and M. Kejak, Projection Methods for Economies with Heterogeneous Agents: CERGE-EI Working Paper No. 258 (Econ. Inst., Prague, 2005).
C. T. Kelley, Iterative Methods for Linear and Nonlinear Equations (SIAM, Philadelphia, 1995).
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The Mathematical Theory of Optimal Processes (Fizmatgiz, Moscow, 1961; Wiley, New York, 1962).
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1972).
J. B. Long and C. I. Plosser, “Real business cycles,” J. Polit. Econ. 91 (1), 39–69 (1983). https://doi.org/10.1086/261128
N. Stokey and R. Lucas, Recursive Methods in Economic Dynamics (Harvard Univ. Press, Cambridge, 1989).
F. Chang, “The inverse optimal problem: A dynamic programming approach,” Econometrica 56 (1), 147–172 (1988). https://doi.org/10.2307/1911845
B. M. Arystanbekov and N. B. Melnikov, “Generalized Galerkin method for an infinite time-horizon economic growth problem,” in Optimal Control Theory and Application: Proceedings of the International Conference, Yekaterinburg, Russia, 2022, pp. 281–285.
A. A. Krasovskii, P. D. Lebedev, and A. M. Tarasyev, “Bernoulli substitution in the Ramsey model: Optimal trajectories under control constraints,” Comput. Math. Math. Phys. 57 (5), 770–783 (2017). https://doi.org/10.1134/S0965542517050050
A. M. Oberman, “Convergent difference schemes for degenerate elliptic and parabolic equations: Hamilton–Jacobi equations and free boundary problems,” SIAM J. Numer.l Anal. 44 (2), 879–895 (2006). https://doi.org/10.1137/S0036142903435235
Y. Achdou, J. Han, J.-M. Lasry, P.-L. Lions, and B. Moll, “Income and wealth distribution in macroeconomics: A continuous-time approach,” Rev. Econ. Stud. 89 (1), 45–86 (2022). https://doi.org/10.1093/restud/rdab002
W. T. Smith, “Inspecting the mechanism exactly: A closed-form solution to a stochastic growth model,” J. Macroecon. 7 (1), 1–31 (2007). https://doi.org/10.2202/1935-1690.1524
Author information
Authors and Affiliations
Corresponding authors
Ethics declarations
The authors declare that they have no conflicts of interest.
Additional information
Translated from Trudy Instituta Matematiki i Mekhaniki UrO RAN, Vol. 28, No. 3, pp. 17 - 29, 2022 https://doi.org/10.21538/0134-4889-2022-28-3-17-29.
Translated by E. Vasil’eva
Rights and permissions
About this article
Cite this article
Arystanbekov, B.M., Melnikov, N.B. Projection Method for Infinite-Horizon Economic Growth Problems. Proc. Steklov Inst. Math. 319 (Suppl 1), S54–S65 (2022). https://doi.org/10.1134/S0081543822060062
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0081543822060062