A Robust Reference-Dependent Model For Speculative Bubbles

Mu Zhang

Tsinghua University – School of Economics & Management

Jie Zheng

Tsinghua University, School of Economics & Management

November 30, 2015

Abstract:

We present a robust model of speculative bubbles by introducing loss-averse reference-dependent preferences by Koszegi and Rabin (2006) into the framework of Allen, Morris and Postlewaite (1993), where in equilibrium, asymmetrically-informed rational investors buy overvalued assets, hoping to sell them to less informed agents before the crash occurs. With reference dependent preferences, the asset price may not necessarily be observable to agents when there is no trade. However, this is never the case with classical preferences, as shown in the paper. Incorporating the classical model as a special case, we generalize the notion of bubbles to allow for the analysis in the case of silent market with unobservable prices, and our model is able to generate strong bubbles robust to moderate perturbations in parameters with no need for stronger conditions as suggested in previous literature. Assuming for simplicity that dividends can only take on two values, we construct an example of a robust reference-dependent bubble which is not robust in the classical setting, and we also show that the positive results regarding of limit of bubble size and bubble frequency in the classical setting are preserved in our framework. Our main results and economic implications remain valid in more general settings.

A Robust Reference-Dependent Model For Speculative Bubbles – Introduction

The last two decades have witnessed at least two dramatic boom-and-bust episodes – the dot-com bubble (Ofek and Richardson, 2003) and the subprime crisis (Varadarajan, Christiano and Keho, 2008), which seem like replications of the stories in Kindleberger, Chalrles and Aliber (2011), including the Dutch tulip mania (1634-1637), the Mississippi bubble (1719-1720) and the South Sea bubble (1720).1 Similar phenomena have also been observed in the laboratory environment (Dufwenberg, Lindqvist and Moore, 2005; Moinas and Poufet, 2012; Lugovskyy et al,2014, among others) where bubbles occur with asymmetrically informed agents aware of the possibilities of both riding the bubble and getting stuck.

Despite its nearly unambiguous existence and prevalence in empirical studies, the phenomena of bubbles seem difficult to explain using classical economic theory. There is a large strand of literature trying to introduce the ideas of overlapping generations to rationalize bubbles (Tirole, 1985; Fahri and Tirole, 2012; Martin and Ventura, 2010). We refer to this type of bubbles as “investment bubbles” in the sense that the asset serves as a store of value and may grow slowly without bursting, or alternatively burst because of the insufficiency of cash (Caginalp and Ilieva, 2008). This can be regarded as a type of moderate-scale bubbles from the long-term perspective. However, the bubbles mentioned in the beginning of the paper typically involved an intense crash, calling for a distinct definition of bubbles from the short-term perspective. Following Conlon (2004) and Doblas-Madrid (2012), we characterize this type of bubbles as “speculative bubbles” where rational agents consciously buy the over-valued assets in the hope of selling them to a greater fool before the assets crash. In this paper we narrow down our focus to speculative bubbles.

The quotation by Warren Buffet aptly captures the logic behind speculative bubbles: Investors hold the over-priced asset in the expectation of getting a higher payoff by selling it to a “greater fool” and quitting the market just before the bubble bursts, but at the same time it is possible that they may stay too long to actually successfully ride the bubble. Allen, Morris and Postlewaite (1993, henceforth referred to as AMP) greatly captured this intuition in their finite-horizon bubble model with asymmetric information and short sale constraint. By their notion of “strong bubbles”, every trader knows that the asset is over-priced for sure, however, they still would like to hold the asset because there is uncertainty about other traders’ knowledge of this over-pricing phenomenon due to information asymmetry. The AMP framework has been well adopted in the literature on rational bubbles, given its success in providing great insight to explain the existence of bubbles from the perspective of information economics (Conlon, 2004; Zheng, 2014; Conlon 2015; Lien, Zhang and Zheng, 2015; among others). However, it has also been noted that the bubble equilibria in AMP model are fragile and not very robust to small perturbations in payoff or belief parameters (Zheng, 2014; Conlon and Zheng, 2013).

In order to take into account the main concern of the bubbles’ robustness issue, we extend the AMP framework to allow for a more general type of utilities – reference-dependent utilities in this paper, and show that the bubbles are no longer fragile when agents have such preferences with the feature of loss aversion.2 The ideas of reference dependence was firstly observed and formulated in the Kahneman and Tversky’s seminal paper on prospect theory (1976) and has been studied in various fields (for example, Ericson and Fuster, 2011; Humphreys and Zhou, 2015, among many others). Koszegi and Rabin (2006, 2007) study the loss aversion feature of reference-dependent preferences by introducing an extra gain-loss utility term into the traditional consumption utility function and set a consumer’s recent rational expectations about outcomes as her reference point. As for empirical justifications of using expectations as the reference point, it has been well observed that expectations influence the trading behavior in general (for example, List, 2003; Ericson and Fuster, 2011) and the bubble formation in particular (Hommes et al, 2008; Husler, Sornette and Hommes, 2013, among others) in the lab environment.3 Convinced by the empirical and experimental evidence, we follow Koszegi and Rabin (2006, 2007), adopt the loss aversion type of reference-dependent preferences, and assume rational expectations as the reference point for every trader in our model. Henceforth, for convenience, we refer to such a behaviorial approach to modeling preferences as the KR approach and the relevant preferences as the KR preferences.

Speculative Bubbles

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