In this chapter we present a financial market application of the heterogeneous expectations evolutionary switching framework of Brock and Hommes (1997a). This application has been termed adaptive belief systems (ABS), with beliefs and prices coevolving over time, and was introduced in Brock and Hommes (1998). An ABS is in fact a standard discounted value asset pricing model derived from mean-variance maximization, extended to the case of heterogeneous expectations.

The chapter is organized as follows. After introducing some basic assumptions of the model, we discuss the benchmark case of rational expectations and derive the fundamental price. In the following three sections, we then introduce heterogeneous beliefs, evolutionary selection of beliefs and the forecasting strategies. Section 6.5 discusses three typical few type examples: a 2-type example with costly fundamentalists versus chartists, a 3-type example with fundamentalists versus optimists and pessimists and a 4-type example with fundamentalists, biased traders and trend followers. All examples exhibit rational routes to randomness, that is, bifurcation routes to cycles and/or chaos as traders become more sensitive to differences in strategy performance. All examples illustrate how non-fundamental traders may survive evolutionary competition and that markets are characterized by bubble and crash dynamics. Section 6.6 discusses another 2-type example with fundamentalists versus conditional trend followers, who condition their trend-following strategy upon deviations from the fundamental. This example exhibits coexisting attractors, as a stable fundamental steady state may coexist with a stable limit cycle or even a strange attractor. In the presence of small noise, the model then exhibits clustered volatility, with irregular switching between close to the fundamental price fluctuations with fundamentalists dominating the market and phases of large price swings amplified by trend followers. Finally, Section 6.7 discusses the case with many different trading strategies and presents the notion of large type limit, a simple low-dimensional system approximating an evolutionary market with many trader types.

Preface

This book has a long history. It grew out of courses on Nonlinear Economic Dynamics (NED), which I have been teaching in the past 20 years at the University of Amsterdam (UvA) and various other places. The NED course has been part of the MSc Econometrics program of the Amsterdam School of Economics, University of Amsterdam since I started at UvA in 1992. I have also taught a condensed version of NED bi-annually between 1996 and 2004 in the Network Algemene en Kwantitatieve Economie (NAKE), a quantitative network of economics PhD courses in the Netherlands. Since 2004 the NED course has been part of the Graduate Program of the Tinbergen Institute, the Graduate school in Economics, Econometrics and Finance in Amsterdam and Rotterdam. More recently, much of the material in this book has been taught at various summerschools and lecture series, in particular the Advanced School on Nonlinear Dynamical Systems in Economics, Udine, Italy, June 2004, the Lecture Series on Heterogeneous Agent Models, Pisa, Italy, June 2006, the Trento Summerschool on Agent-based Finance, Trento, Italy, July 2007 and the International School on Multidisciplinary approaches to Economic and Social Complex Systems, Siena, Italy, June 2010.

I am grateful to many colleagues and friends for inspiration and help over more than two decades. My main PhD thesis advisor at the University of Groningen, Helena Nusse, raised my enthusiasm for chaos and complexity. In Groningen, Floris Takens further deepened my knowledge of nonlinear dynamics and strange attractors, and Ad Pikkemaat taught me the first lessons in mathematical economics. At the University of Amsterdam, this role was taken over by Claus Weddepohl, who was one of the first mathematical economists in the Netherlands and Europe recognizing the importance of nonlinear dynamics and complexity for economics.

The economy is a complex system with nonlinear interactions and feedback loops. Early traces of this view date back, for example, to Schumpeter and Hayek, and to Simon. The complexity modeling paradigm has been strongly advocated since the 1980s by economists and multidisciplinary scientists from various fields, such as physics, computer science and biology, linked to the Santa Fe Institute. More recently the complexity view has also drawn the attention of policy makers, who are faced with complex phenomena, irregular fluctuations and sudden, unpredictable market transitions. For example, the chairman of the FED, Ben Bernanke, noted that the 1000-point collapse of the Dow Jones Industrial Average on the afternoon of May 6, 2010, reflected the complexity of financial-market systems:

The recent financial-economic crisis is a dramatic example of large movements, similar to critical transitions that are so characteristic for complex evolving systems. These large changes of global financial markets can hardly be viewed as a rational response to news about economic fundamentals and cannot be explained by traditional representative rational agent macro-finance models.

In the previous chapter we have focused on a representative behaviorally rational agent in the nonlinear cobweb model. But why would all agents have the same individual expectations? Agents are heterogeneous and laboratory experiments have shown that, even when facing the same information, individuals may disagree and take different strategic consumption, production or investment decisions. In a complex market environment it seems more appropriate to model agents as boundedly rational and heterogeneous, using different types of strategies. In this chapter, therefore, we study the cobweb model with heterogeneous expectations. The cobweb model provides a simple framework to study the interactions of heterogeneous expectation rules and the aggregate behavior resulting from these interactions.

Models with heterogeneous agents are becoming increasingly popular. In particular, in finance models with fundamentalists and chartists have received much attention. Examples include Zeeman (1974), Frankel and Froot (1986), Day and Huang (1990), DeLong et al. (1990a and b), Kirman (1991), Lux (1995), Brock and Hommes (1998) and DeGrauwe and Grimaldi (2006); see the reviews of Hommes (2006) and LeBaron (2006) with many more references.

In this chapter we discuss price dynamics in nonlinear versions of the cobweb or the “hog cycle” model under homogeneous price expectations. The main purpose is to illustrate the role of expectations feedback and bounded rationality in a simple nonlinear economic environment. To keep the analysis as simple as possible, we will work with a linear demand curve, whereas the supply curve will be nonlinear, but monotonically increasing most of the time; in some examples the supply curve will be non-monotonic, backward-bending. We use the nonlinear cobweb model as a didactic tool, because it is one of the simplest dynamic models in economics. It has served as a benchmark model since its introduction in the 1930s and, in particular, it has served as a simple economic environment to discuss expectations feedback. For example, Ezekiel (1938) studied the cobweb model under naive expectations, Goodwin (1947) under linear trend-following and contrarian expectations rules and Nerlove (1958) under adaptive expectations. Muth (1961) used the cobweb framework in his seminal paper introducing rational expectations. In this chapter we study the nonlinear cobweb model with homogeneous expectations, i.e., all producers have the same price expectations. Our main purpose is to illustrate how simple but commonly used expectations rules and bounded rationality in a nonlinear environment may lead to irregular, possibly chaotic price fluctuations. In Chapter 5 we will consider the cobweb model with heterogeneous expectations.

The cobweb model

The classical cobweb model is a partial equilibrium model describing commodity price fluctuations of a non-storable good, such as corn or hogs, that takes one time period to produce. It is one of the simplest benchmark models in economic dynamics and can be found in many standard textbooks (e.g., Nicholson 1995, pp. 590–594). Producers form price expectations one period ahead and derive their optimal production decision from expected profit maximization.

In the previous chapters we have seen a number of stylized complexity models with heterogeneous expectations and heterogeneous trading strategies. What is the empirical relevance of heterogeneous expectations complexity models? In the last two chapters we discuss the empirical validation of heterogeneous expectations. In this chapter we discuss the empirical validity of the asset pricing model with heterogeneous beliefs compared to financial time series data, while the final chapter discusses the empirical relevance, both at the micro and at the macro level, of cobweb and asset pricing heuristic switching models in laboratory experiments with human subjects.

There is already a large literature on heterogeneous agent models (HAMs) replicating important stylized facts of financial time series on short time scales (say daily or higher frequency), such as fat tails and long memory in the returns distribution and clustered volatility. In fact, an important motivation to develop behavioral heterogeneous agent models has been the dissatisfaction with standard rational representative agent models to match stylized facts of financial time series. Many examples of heterogeneous agent models replicating stylized facts of financial markets have appeared in the literature, e.g., Brock and LeBaron (1996), Arthur et al. (1997b), Brock and Hommes (1997b), Youssefmir and Huberman (1997), LeBaron et al. (1999), Lux and Marchesi (1999, 2000), Farmer and Joshi (2002), Kirman and Teyssière (2002), Hommes (2002), Iori (2002), Cont and Bouchaud (2000) and Gaunersdorfer and Hommes (2007). The recent survey by Lux (2009) contains an extensive and stimulating survey of behavioral interacting agent models mimicking the stylized facts of asset returns with many more references; see also DeGrauwe and Grimaldi (2006) for an extensive discussion and applications in exchange rate modeling. In the stylized models discussed in this book, we have seen examples of simple heterogeneous agent models mimicking temporary bubbles and crashes and we have also discussed an endogenous mechanism for clustered volatility (either through intermittent chaos or through a co-existing stable steady state with a limit cycle or a more complicated attractor; see Subsection 6.6.3). In this chapter our focus will be whether observed bubbles and crashes in real markets can be explained by heterogeneous expectations models.

In Chapter 6 we have seen many examples of the simple asset pricing model with heterogeneous beliefs exhibiting complex dynamics characterized by temporary bubbles and crashes, triggered by exogenous news about economic fundamentals and strongly amplified by self-fulfilling expectations and trend-following trading strategies. While the previous chapter focused on the empirical relevance of the heterogeneous expectations switching model by comparing it to real financial time series, the current chapter confronts the heterogeneous expectations switching model with laboratory experimental data, both at the individual micro and at the aggregate macro level. Laboratory experiments with human subjects are well suited to discipline the class of behavioral modes or individual heuristics that boundedly rational subjects may use in economic decision making. This chapter discusses a number of “learning-to-forecast experiments” in the cobweb and asset pricing framework, where human subjects must forecast the price of an asset or a commodity, whose realized market price is an aggregation of individual expectations. These experiments may be seen as a testing of the heterogeneous expectations hypothesis in the lab. This chapter is based on the survey of laboratory experiments in Hommes (2011), with some recent updates.

In real markets, it is hard to obtain detailed information about individual expectations. One way to collect data on individual expectations is by survey data analysis; see, e.g., Pesaran and Weale (2006) for a stimulating overview. Survey data analysis can focus exclusively on the expectations-generating process, avoiding the dilemma of testing joint hypotheses. There is already quite some evidence on forecasting heterogeneity in survey data. Frankel and Froot (1987, 1990a and b), Allen and Taylor (1990) and Taylor and Allen (1992), for example, found that financial experts use different forecasting strategies to predict exchange rates. They tend to use trend extrapolating rules at short horizons (up to 3 months) and mean-reverting fundamentalists rules at longer horizons (6 months to 1 year). Moreover, the weight given to different forecasting techniques changes over time. Shiller (1987, 2000) and Vissing-Jorgensen (2003) present evidence of heterogeneity and time variation in the sentiment of investors in stock market data. For example, Shiller (2000) finds evidence that investor's sentiment changes over time, with both institutions and individual investors becoming more optimistic in response to recent significant increases of the stock market in the late 1990s. Mankiw et al. (2003), Capistran and Timmermann (2009) and Pfajfar and Santoro (2010) find evidence for heterogeneity and time variation in inflation expectations.