Adam Krawiec , Aleksander Stachowski , Marek Szydłowski

(Polish) PDF


We consider economic growth models in the form of dynamical systems. We show a method of determining trajectories in a neighbourhood of a long-run equilibrium in some neoclassical models of exogenous economic growth. This method is applied primarily to these models which in general have no analytical solution. We propose the general method of finding solutions of arbitrarily dimensional dynamical system in the form of power series. We expand the state function in Taylor's series in the neighbourhood of the initial state. The coefficients of expansion represent the parameters of the variation of the state of the system and are calculated algebraically in Mathematica. We present the method of finding solutions for the Solow-Swan model and the Mankiw-Romer-Weil model. We use also the Padé aproximant method to obtain a better convergence of the power series. This method allows to obtain a solution in the form of a series for trajectories in a neighbourhood of a long-run equilibrium in two models of exogenous economic growth. We show that obtained solutions are a good approximation of time paths, along which the long-run equilibrium is reached. We show a possibility of estimation of model parameters for which solutions in the form of series are known.


economic growth, dynamical systems, power series solutions, Padé approximant


Ayres R. U., (1999), On Growth in Disequilibrium, preprint,

Baker G. A., Graves-Morris P., (1996), Padé Approximants, Cambridge University Press, Cambridge. Chang W. W., Smyth D. J., (1971), The Existence and Persistence of Cycles in a Non-linear Model: Kaldor’s 1940 Model Re-examined, Review of Economic Studies, 38 (1), 37–44.

Chiarella C., (1990), The Elements of a Nonlinear Theory of Economic Dynamics, Springer, Berlin.

Chiarella C., (1992), Developments in Nonlinear Economic Dynamics: Past, Present and Future, w: Hanusch H., (red.), Die Zukunf der Okonomischen Wissenschaft, Verlag Wirtschaft und Finanzen.

Frisch R., (1933), Propagation Problems and Impulse Problems in Dynamic Economics, w: Economic Essays in Honour of Gustav Cassel, 171–205, Allen & Unwin, London.

Goodwin R. M., (1967), A Growth Cycle, w: Feinstein C. H., (red.), Socialism, Capitalism, and Economic Growth, Cambridge University Press, Cambridge, 54–58.

Jones W. B., Thron, W. J., (1980), Continued Fractions: Analytic Theory and Applications, Addison- Wesley, Reading, MA.

Krawiec A., Szydłowski M., (2002), Własności dynamiki modeli nowej teorii wzrostu, Przegląd Statystyczny, 48 (1), 17–24.

Mankiw N., Romer D., Weil D., (1992), A Contribution to the Empirics of Economic Growth, Quarterly Journal of Economics, 107 (2), 407–437.

Medio A., (1992), Chaotic Dynamics. Theory and Applications to Economics, Cambridge University Press, Cambridge.

Nonneman W., Vanhoudt P., (1996), A Further Augmentation of the Solow Model and the Empirics of Economic Growth for OECD Countries, Quarterly Journal of Economics, 111 (3), 943–953.

Palczewski A., (2004), Równania różniczkowe zwyczajne, Wydawnictwo Naukowo-Techniczne, Warszawa.

Perko L., (2001), Differential Equations and Dynamical Systems, Springer, New York.

Robinson, J., (1962), Economic Philosophy, C. A. Watts, London.

Romer D., (2000), Makroekonomia dla zaawansowanych, PWN, Warszawa.

Schinasi G. J., (1981), A Nonlinear Dynamic Model of Short Run Fluctuations, Review of Economic Studies, 48 (4), 649-656.

Schinasi G. J., (1982), Fluctuations in a Dynamic, Intermediate-run IS-LM Model: Applications of the Poincaré-Bendixon Theorem, Journal of Economic Theory, 28 (2), 369–375.

Solow R., (1956), A Contribution to the Theory of Economic Growth, Quarterly Journal of Economics, 70 (1), 65-94.

Swan T. W., (1956), Economic Growth and Capital Accumulation, Economic Record, 32, 334–361.

Torre V., (1977), Existence of Limit Cycles and Control in Complete Keynesian Systems by Theory of Bifurcations, Econometrica, 45, 1457–1466.

Zawadzki H., (2015), Analiza dynamiki modeli wzrostu gospodarczego za pomocą środowiska obliczeniowego Mathematica, Zeszyty Naukowe UEK, 4 (490), 59–69.

Back to top
© 2019–2022 Copyright by Statistics Poland, some rights reserved. Creative Commons Attribution-ShareAlike 4.0 International Public License (CC BY-SA 4.0) Creative Commons — Attribution-ShareAlike 4.0 International — CC BY-SA 4.0