This paper considers a discrete-time model of a financial market with one risky asset and one risk-free asset, where the asset price and wealth dynamics are determined by the interaction of two groups of agents, fundamentalits and chartists. In each period each group allocates its wealth between the risky asset and the safe asset according to myopic expected utility maximization, but the two groups have heterogeneous beliefs about the price change over the next period: the chartists are trend extrapolators, while the fundamentalists expect that the price will return to the fundamental. We assume that investors have CRRA utility, so that their optimal demand for the risky asset depends on wealth. A market maker is assumed to adjust the price at the end of each trading period, on the basis of the excess demand and according to particular stabilization policies. The model results in a three-dimensional nonlinear discrete-time dynamical system, with growing price and wealth processes, but it is reduced to a stationary system in terms of asset returns and wealth shares of the two groups. It is shown that the long-run market dynamics are highly dependent on the parameters which characterize agents' behavior (in particular the risk aversion coefficient and the chartist extrapolation parameter) as well as on the initial condition (in particular the initial wealth shares of fundamentalists and chartists). It is also shown that the for wide ranges of the parameters a (locally) stable fundamental steady state may coexist with a stable "nonfundamental" steady state, where price grows faster than the fundamental and only chartists survive in the long-run. In such cases, the role played by the initial condition is analysed by means of numerical investigations and graphical representation of the basins of attraction. Other dynamic scenarios include limit cycles, periodic orbits or more complex attractors, where in general both types of agents survive in the long run, with time varying wealth fractions.
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Paper provided by Quantitative Finance Research Centre, University of Technology, Sydney in its series Research Paper Series with number
134.
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