Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic

Pod Vodárenskou věí 4, 182 08 Praha, Czech Republic

tel.: +420 266 052 411, +420 266 052 400, fax: +420 286 890 449

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Nonlinear model of closed economy is formulated on an extended and modified Kaldor's model of small closed economy with a simple structure and several nonlinearities. Nonlinear pattern of dependencies in this model is created by a logistic function. The model is realized by four differential equations. The first equation and second equation describe output and capital dynamics. A core of the output dynamics is formulated by investment and savings disequilibrium. Capital dynamics is formed by a net investment. The third equation describes price dynamics through disequilibrium powers on monetary markets. And the fourth equation describes expected inflation formulated as an adaptive expectation. Behavior analysis of this model is realized on numerical calibrated model. The considered model demonstrates chaotic solution of attractors by positive Lyapunov exponent. Kaldor's model is described in some classical macroeconomics.

A simple dynamic continuous nonlinear model for the description and investigate the possibility of more complex behaviour of such systems was chosen. The stability such systems was tested and Lyapunov exponents were computed using appropriate numerical methods.

Previous model with Worst Out Algorithm (WOA) in heterogeneous agents model on financial markets was modified. The important outcome of the simulations was a possibility of a prediction in the case with the normally distributed memory length of agents presented on the financial market (normal case). An interesting result concerning risk behavior was the fact that the WOA plays a stabilizing role in the normal case in a sense of decreasing variance in time. Conversely, the uniform case (uniformly distributed memory length of agents presented on the financial market) affects the financial market risk level negatively, i.e., rising variance in time.

Main attention was paid to the order driven markets with asynchronous trading. As a first step of his long term research, the markets with completely random behavior of the agents and with the unit order sizes were studied. Even if the order driven markets are regarded as analytically intractable, two closed form results were achieved in this area: the first one being the analytical formula for the conditional distribution of the limit order book given the history of the best quote process. The second one being an arbitrarily exact approximation of the joint distribution of the market price and the traded volume.

- Jan Kodera

nonlinear economic dynamics. - Martin míd

capital market theory. - Luká Vácha

heterogenous capital markets. - Miloslav Vovrda

theoretical economics, econometrics and econometrical modelling.

- Kodera J., Sladký K., Vovrda M.: Stability and Lyapunov exponents in keynesian and classical macroeconomic models. Mathematical Methods in Economics 2005. (Hana Skalská, ed.). Gaudeamus, Hradec Králové 2005, pp. 203--210.
- Kodera J., Vovrda M.: Product, capital, and price motion in a simple nonlinear model closed economy. Politická ekonomie (forthcomming).
- míd M.: Forecasting in continuous double auction. (Research Report No. 2128). ÚTIA AV ČR, Praha 2005, 19 pp.
- míd M.: Forecasting in continuous double auction. In: Proceedings of the 23rd International Conference Mathematical Methods in Economics 2005. (Skalská H. ed.). Gaudeamus, Hradec Králové 2005, pp. 358-363.
- Vácha L., Vovrda M.: Dynamical agents' strategies and the fractal market hypothesis. Prague Economic Papers 14 (2005), 2, 172--179.

**Economic Dynamics: Analytical and Computational Treatment of Macroeconomic Models**

Karel Sladký, grant No. A 7075202 of the Grant Agency of the Czech Academy of Science.**Nonlinear Economic Dynamics**

Miloslav Vovrda, grant No. 402/03/H057 of the Czech Science Foundation, with University of Economics and Charles University in Prague.