Analyzing Heterogeneous Agents Models: Avoiding the Explicit Discretization of Stiff Equations

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Baumann, Michael Heinrich ; Baumann, Michaela ; Grüne, Lars ; Herz, Bernhard:
Analyzing Heterogeneous Agents Models: Avoiding the Explicit Discretization of Stiff Equations.
2019
Veranstaltung: Second Behavioral Macroeconomics Workshop: Heterogeneity and Expectations in Macroeconomics and Finance , June 13-15, 2019 , Otto-Friedrich-Universität Bamberg.
(Veranstaltungsbeitrag: Workshop , Vortrag )

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Abstract

One method to explain several stylized facts in financial markets is to model prices via market maker models with heterogeneous agents, usually fundamentalists and chartists. A basic approach going back to Day and Huang (1990) and Huang and Day (1993) is to assume the log-prices to be linear in the demand of the agents (cf. Brock and Hommes (1997, 1998), Lux (1995)). The demand function of the fundamentalists is assumed to be linear in the deviation of the log-price from the log-fundamental and the demand function of the chartists is assumed to be linear in the trend of the log-price.

When assuming the log-fundamentals as well as the number of the traders to be constant, we get a linear recurrence relation for the log-prices, where the log-fundamental is the unique equilibrium. Such models are used to determine the instability threshold of the fundamentalists to chartists ratio. We show that this model is an explicit Euler discretization of a properly chosen differential equation.

It turns out that even when there are no chartists, the equilibrium is unstable if there are "too many" fundamentalists: The price jumps in each point in time over the fundamental and the distance are even increasing. It is imaginable that when there are too many fundamentalists, the price jumps beyond the fundamental, however, in a long run, the price should converge to the fundamental value in the absence of chartists. That means, when this market becomes unstable it is not clear if too many chartists cause the instability or if the instability is a numerical artifact. The diverging behavior caused solely by the fundamentalists cannot be observed on real markets. This leads us to the question how to model heterogenous agents without these effects, which are well known from technical problems and are called stiff equations. We use an implicit Euler discretization for the ordinary differential equation mentioned above and observe that the model is stable in absence of chartists for arbitrarily many fundamentalists.