buy or sell short the same amount of stocks. Thus traders with less wealth are prepared to take greater relative risks than traders with more wealth. It should be noted that in the case of zero supply of outside stocks, the model developed in this chapter reduces to the Brock and Hommes (1998) model.
In nonlinear dynamical models it is in general impossible to obtain explicit analytic expressions for the periodic and chaotic solutions. Therefore in applied nonlinear dynamics it is common practice to use a mixture of theoretical and numerical methods to analyze the dynamics. We perform a bifurcation analysis of the steady state by using numerical tools, such as delay and phase diagrams, bifurcation diagrams and the computation of Lyapunov exponents. In particular we show analytically that the fundamental steady state may become unstable due to a Hopf bifurcation.
In section 6.2 the Brock and Hommes (1998) financial market model with adaptively learning agents is reviewed. Thereafter, in section 6.3, the heterogeneous agents model with fundamentalists versus moving average traders, resulting in an eight dimensional nonlinear dynamical system, is derived. In section 6.4 a procedure is developed to determine trading volume. Section 6.5 presents an analytical stability analysis of the fundamental steady state. The eigenvalues of the linearized system are computed and it is examined which kind of bifurcations can occur. In section 6.6 numerical simulations are used to study the dynamical behavior of the model, especially when the steady state is locally unstable. Finally section 6.7 summarizes and concludes.