Wealth-driven competition in a speculative financial market: examples with maximizing agents

Abstract

This paper demonstrates how both the quantitative and qualitative results of a general, analytically tractable asset-pricing model in which heterogeneous agents behave consistently with a constant relative risk-aversion assumption can be applied to the special case of optimizing behaviour. The analysis of the asymptotic properties of the market is performed using a geometric approach that allows the visualization of all possible equilibria by means of a simple one-dimensional Equilibrium Market Curve. The case of linear (particularly, mean–variance) investment functions is thoroughly analysed. This analysis highlights the features that are specific to linear investment functions. As a consequence, some previous contributions of the agent-based literature are generalized.

Keywords:

• Asset pricing model
• CRRA framework
• Equilibrium market curve
• Expected utility maximization
• Mean-variance optimization
• Linear investment functions

Acknowledgements

This paper is based on a chapter from my PhD dissertation and I would like to use this opportunity to thank Giulio Bottazzi for his unique supervision. This work would have been impossible without the many useful suggestions and helpful hints of Giulio. I am also very grateful to Amanda Brandellero, Christophe Deissenberg, Pietro Dindo, Cars Hommes, Valentyn Panchenko, Francesca Pancotto, Roald Ramer, the participants of the ACSEG-2005 meeting and the seminar at the University of Amsterdam, and three anonymous referees for many useful comments and discussions which allowed me to improve the paper. I am solely responsible for all remaining mistakes. This work was supported by Complex Markets E.U. STREP project 516446 under FP6-2003-NEST-PATH-1.

Notes

†For instance, DeLong et al. (1991) consider two types of investors, the model of Day and Huang (1990) is populated by three types of traders, while Brock and Hommes (1998) provide a number of examples with two, three and four different types. One recent exception from this rule is the model of Brock et al. (2005) where the low-dimensional Large Type Limit with the number of types converging to infinity is introduced.

†These are the conditions to guarantee that price is positive (Anufriev et al. 2006).

†Introduced by Anufriev et al. (2006) in a slightly different notation, this curve was first called the ‘Equilibrium Market Line’.

†The illustration corresponds to the case when r_f  < [ybar]. This is consistent with the real data collected by Robert Shiller and available at http://www.econ.yale.edu/∼shiller/data.htm. For the period from 1871 to 2005 the average real one-year interest rate was 0.029 and the average dividend yield was 0.044.

†At this point the reader is highly encouraged to follow the discussion drawing the different possible mutual locations of the linear investment function and the EMC.

†All plots in Chiarella and He (2001) which we mention here and below are just sketches obtained from the mixture of the analytic and numerical analysis. The advantage of our approach is that these qualitative sketches can be obtained from the EMC plot. Thus, on the one hand, they all become justified on an analytic basis. On the other hand, they also become clearer and, thus, can be easily generalized for the situations of three and more agents, and also corrected. For example, notice that, in this case, the return in equilibrium S_c does not approach the return in equilibrium S_f when [dbar] ₂  → 0.