Bubble and Depression in Dynamic Global Games

Transcription

1 Bubble and Depression in Dynamic Global Games Huanhuan Zheng Tel: Fax: Institute of Global Economics and Finance The Chinese University of Hong Kong and Deparment of Economics and Related Studies University of York, UK August, 2013

2 Abstract Using insights from global game and heterogenous agents modeling, we extend the asset pricing model to account for strategic coordination and dynamic interaction between heterogenous trading activities and asset prices. Such a model gives rise to not only self-fullling beliefs and strategic complementarities but also self-destructive beliefs and strategic substitutions. Theoretical derivation implies that (i) bubble and depression can persist as a result of the noise in the information aggregation process and the presence of irrational agents; (ii) information transparency magnies the magnitude of both bubble and depression; (iii) in the presence of signicant external shock, the bubble may be transformed to depression and vice versa. Keywords: bubble, depression, crisis, global game, nonlinear dynamics, heterogenous agents JEL: G12, D53, D83

3 1 Introduction The canonical global game features a static framework, in which many agents simultaneously and strategically choose one of the two available actions conditional on some noisy and private signals, and such strategic coordination among agents lead to binary outcomes. It has been widely applied in studying speculative attacks in credit, currency, debt markets and banking sector, where the market can be characterized by two meaningful exclusive states (i.e., normal and failure) and the market participants' behavior can be easily classied into action and inaction (i.e., to run or not to run a bank). It is however rarely used to analyze the equity market that embraces multiple and possibly controversial and innite phases in asset prices. Moreover, it is not realistic to capture the complicated trading activities of equity market participants with binary actions, given the various hierarchies of trading orders. Extending the global game to account for multiple actions and outcomes is however complex to analyze (Frankel, et al.,2003, Sakovics and Steiner,2012). We overcome these difculties by using insights from heterogeneous agent models developed in Frankel and Froot (1986, 1990), Day and Huang (1990), Brock and Hommes (1998), Branch and Evans (2010) and Huang, et al. (2012). These papers show that nite heterogeneous strategies are able to capture various market-wide phenomena and the aggregate actions of all agents are important determinants of asset prices. Based on these ndings, we focus on two strategies, one rational and the other irrational, to describe a variety of trading behavior in a market with incomplete information. As the same strategy can lead to different trading orders due to the dispersion of private information and beliefs, the two strategies essentially incorporate multiple actions differentiated by the direction and size of trading orders. We endogenize the price impact of a marginal change in the aggregate actions to account for the dynamic interaction between aggregate actions of all agents and the asset price. In particular, the market maker adjusts the price up or down after observing the aggregate actions of all agents, which have feedback effects on the magnitude of price adjustment. The update price in turn affects agents' trading decisions in the next period. 1

4 Note that we not only model whether the price will go up or down but also how much the price will change, this setup essentially incorporates multiple outcomes corresponding to various price levels. We present a dynamic global game model with many agents trading one risky asset with a market maker over multiple periods. Our approach highlights two elements: strategic coordination among agents and dynamic interaction between trading strategies and asset prices. In the model, each agent receives an independent and noisy private signal about the fundamental value of the risky asset, where the information is imperfect and asymmetric. Agents, who are risk-neutral, seek to maximize their expected payoff by choosing one of the two strategies - the rational strategy represented by the fundamental strategy, which bets the price to converge towards its fundamental value in the next period; and the irrational strategy represented by chartist strategy, which expects the price to deviate further away from certain reference price, i.e. the supporting or resistance price as in technical analysis. All agents simultaneously place trading orders by adopting either fundamental strategy or chartist strategy, depending on which one yields better expected payoff. The expected payoffs of different are conditional on agents' strategic beliefs about the actions of others, which are determined by the private information and the latest asset price. The dispersion of private information leads to heterogeneous beliefs about payoffs, which may result in different trading strategies. These trading strategies affect the aggregate demand of the risky asset that inuences the market maker's decision on how to quote the subsequent price, which in turn has an effect on agents' trading strategies. We introduce a price impact function to capture the feedback effects of the agents' aggregate actions on the asset price, and therefore endogenize the price determination in a dynamic global game. In particular, the market maker adjusts the price up in the subsequent period if the aggregate trading actions result in more buying than selling (i.e., positive aggregate demand) and devaluates the price if there is more selling than buying (i.e., negative aggregate demand). If the trading order is in balance, market maker is indifferent between increasing or decreasing the price in the next period and will choose to keep the price unchanged. Our model suggests that in the presence of heterogeneous trading strategies, coordination game 2

5 may lead to strategic complementary but also strategic substitution. Moreover, agents' beliefs can be either self-fullling or self-destructive, depending on the state of fundamental and the implemented strategy. Extending the asset pricing model to account for the strategic coordination among agents as well as the dynamic interaction between trading strategies and asset price, allows us to understand the speculative bubbles and depression from a new angle. The model implies that: (i) bubble and depression may emerge and even persist as agents strategically cluster to the chartist strategy; (ii) increasing information transparency enlarges the size of bubbles and depression as small noise adds to non-fundmanetal volatility through increasing the sensitivity of the equilibrium; and (iii) the bubble may be transmitted into depression or vice versa in the presence of external shocks, such as spiicate deterioration or rebound in fundamentals. Literature Review The backbone of this model relates to the global game literature such as Carlsson and Damme (1993), Morris and Shin (1998, 2002, 2004), Goldstein and Pauzner (2005), Bebchuk and Goldstein (2011). While these works focus on speculative attacks that lead to currency crisis, debt default, credit freeze and bank runs, we analyze the evolution of asset price in equity market. The most commonly applied global game setup consists of binary actions (of whether to act or not, i.e., whether to attack the currency or not) played by many agents that share the same payoff function. We differ from this traditional setup from two perspectives. First, even though agents in our model choose between two strategies, due to the incomplete information, the same strategy can lead to various trading behavior that corresponds to to multiple actions. Second, the payoff function in our model is heterogeneous instead of homogeneous. Introducing heterogeneity makes the analysis of the global game relatively difcult (Frankel, et al., 2003, Sakovics and Steiner, 2012). We address this issue by endogenizing the price determination process. The two new features of this model embed the existing 22 global game setup. Our model relates to the work of Angeletos and Werning (2006) and Ozdenoren and Yuan (2008) that study the asset price in a coordination game with endogenous signal. While they emphasize the relation between asset price and volatility in a static model without committing to a 3

6 specic form of the price generating process, we are interested in the strategic price dynamics that lead to bubbles and depressions. More important, we differ from these studies by quantifying the price as a function of the aggregate demand, which is endogenously determined by agents' investment strategies. In their models, an agent follows a cutoff strategy by buying the asset if his private signal exceeds a cutoff value and does nothing otherwise. In our model, the price endogenization introduces nonlinearity to agents' decision-making process, which gives rise to not only strategic complementarities that arise in pure coordination games but also strategic substitutions that are hardly addressed in current global game literature. This study also relates to the heterogeneous agent model literature such as Day and Huang (1990), Lux (1995), Brock and Hommes (1997, 1998), He and Westerhoff (2005), Chiarella, et al. (2007), Huang, et al. (2010, 2012). Based on simulation results, these papers demonstrate that, the dynamic interaction between representative agents with heterogeneous beliefs, such as fundamentalists and chartists, well replicate nancial bubbles, crashes and various stylized facts that are commonly observed in nancial market, such as non-stationarity in price, volatility clustering in returns, etc. Recently Alfarano, et al. (2008) and Huang and Zheng (2012) show analytically how these commonly observed features of nancial markets arise from the heterogeneous trading behavior of nite representative agents. These evidences enable us to focus on limited heterogeneous strategies while studying the broad price dynamics in a general nancial market. The remainder of this paper is organized as the follows. Section 2 introduces the model that utilize insights from global game and heterogeneous agent modeling. Section 3 discuss the theoretical implications. Section 4 analyzes how coordinations among market participants may contribute to the emergence and persistence of bubbles and depression based on a simple example. Section 5 concludes. 4

7 2 The Model There is a continuum of [0;1] agents trading on one risky asset. At period t, each agent i 2 [0;1] observes a private signal p i f ;t about the real fundamental value of the risky asset p f ;t. The actual value of p f ;t is determined exogenously by the nature of the entity underlying this risky asset. The private signal p i f ;t received by agent i is assumed to draw uniformly and independently from the interval p f ;t ε; p f ;t + ε, for some small ε > 0. After observing the private signal, we assume that agent i naively extrapolates the signals received by the others to fall uniformly in the interval of h i p i f ;t ε; p i f ;t + ε 1. Based on the desire to maximize the expected payoff, an agent chooses either fundamental strategy based on the belief that the price will converge to its fundamental value, or chartist strategy based on the belief that the price will deviate further away from a reference price p c;t. The expected payoff depends on his private signal as well as signals he expects the others to received (and therefore actions he expects the others to take). Each period, all agents submit their trading orders simultaneously to the market maker based on the strategies that maximize their expected payoff. Market maker observes the total orders of all agents and updates the price in the subsequent period accordingly. 2.1 Choice of Trading Strategy If agent i chooses fundamental strategy, his individual demand function follows: d i f ;t = α(pi f ;t p t ), (1) where α measures the trading intensity of agent i 2. We call agents choosing fundamental strategy as fundamentalist hereafter. The higher the value of α is, the more intensively the fundamentali 1 The Bayesian extrapolation of the signals received by the other agents falls uniformly in hp i f ;t 2ε; p i f ;t + 2ε. Adopting such posterior extrapolation rather than this simple assumption does not affect the qulitative implications of this model. It however complicates the numberical analysis. We therefore base our discussions on the naive extrapolation. 2 The specications of fundamentalists and chartists follows from Day and Huang (1990), Huang et al (2010, 2012), to list a few. 5

8 ists act on their beliefs that the price will converge to the fundamental. Fundamentalists engage in rational arbitrage to push the price towards revealing the fundamental value. Without loss of generality, the value of α is assumed to be common for all fundamentalists. As the private signal about the fundamental value of the risky asset p i f ;t varies across agents, fundamentalists do not share the same demand function at period t. If agent i chooses chartist strategy, his individual demand function is given by: d c;t = β (p t p c;t ), (2) where p c;t is the reference price (i.e., the supporting price or resistance price as in technical analysis) that agent i believes the price will deviate further away from, and β measures trading intensity. We call agents choosing chartist strategy as chartist henceforth. The higher the value of β is, the more intensively chartists act on their beliefs that the price will deviate further away from the reference price. The chartist represents the type of behavioral trader subject to `irrational exuberance', who believes that the price will perpetually deviates from a reference point. For simplicity, we assume β < α and 4β jp t p c;t j=(3α) > ε. As both p c;t and β are common knowledge, all chartists have the same demand function at period t. That is why we dropped the superx i from the individual demand function of chartist. Denote the payoffs of fundamental and chartist strategy at period t as π f ;t and π c;t respectively. Assume no dividend payments for the risky asset, agent i's payoff for each trading strategy is essentially the product of the demand (either d i f ;t or d c;t) and the price change p t+1 = p t+1 p t. Agent i's expected payoffs on the two trading strategies are characterized by: E i π f ;t = d i f ;t E i ( p t+1 ), and E i (π c;t ) = d c;t E i ( p t+1 ), where E i () denotes agent i' expectation on the variable in the parenthesis. Agent i's expected 6

9 payoff difference between fundamental and chartist strategy at period t is given by: E i ( π t ) = E i π f ;t E i (π c;t ) = d i f ;t d c;t E i ( p t+1 ). (3) When E i ( π t ) = 0, the two strategies yields the same expected payoff and agent i is indifferent between fundamental and chartist strategy. For simplicity, we assume that agent i chooses fundamentalist strategy if E i ( π t ) = 0. If E i ( π t ) 6= 0, agent i chooses the strategy that yields higher expected payoff. Under this setup, the optimal strategy of agent i at period t, denoted as st, i is to adopt fundamental strategy if the expected payoff difference is greater than or equal to 0, and to take chartist strategy otherwise: 8 >< st i 1, if E i ( π t ) > 0 = >: 0, if E i ( π t ) < 0. (4) Based on the decision to maximize the expected payoff, agent i's actual demand of the risky asset d i t is given by: 8 >< dt i d i f ;t =, if Ei ( π t ) > 0 >: d c;t, if E i ( π t ) < 0. Let F t be the set of agents that actually adopt fundamental strategy at period t: F t = ije i ( π t ) > 0. The fraction of agents taking fundamental strategy at period t (i.e. i 2 F t ), denoted as n f ;t 3, is dened as n f ;t = Z 1 0 Z st k dk = 1dk, F t where n f ;t 2 [0;1]. Given that there are only two types of strategies, the fraction of chartists equals to 1 n f ;t. 3 Note that the n f ;t is endogenously determined by the fraction of agents who expect the fundamental strategy to yield a higher payoff. We keep it as a variable for now and solve it later. 7

10 2.2 Price Dynamics After selecting the trading strategies that maximize their expected payoff, all agents submit their orders to the market maker simultaneously. The aggregate demand of all agents, denoted as D t, is essentially the sum of all orders, which can be decomposed into the aggregate demand of fundamentalist (D f ;t ) and the aggregate demand of chartists (D c;t ): D t = Z 1 0 d k t dk = D f ;t + D c;t. (5) The aggregate demand of fundamentalist can be written as: Z Z D f ;t = d k f ;t dk = α(p k f ;t p t )dk = α p f ;t p t n f ;t, (6) F t F t where the second equation is obtained by substituting d k f ;t with Eq.(1) and p f ;t is the average private signal received by agents that adopt the fundamentalist strategy at period t: p f ;t = 1 Z p k f ;tdk. (7) n f ;t F t As agents choosing chartist strategy at period t shares the same demand function, the expression for the aggregate demand of chartist is: D c;t = 1 n f ;t dc;t = 1 n f ;t β (pt p c;t ), (8) where the second line is obtained by substituting d c;t with Eq.(2) and 1 n f ;t is the fraction of chartists. After observing all trading orders, the market maker updates the price of the risky asset at 8

11 period t + 1 according to the aggregate demand D t : p t+1 = p t+1 p t = γd t = γ α p f ;t p t n f ;t + β (p t p c;t ) 1 n f ;t, (9) where γ measures price impact of the marginal change in aggregate demand and the second line is obtained by substituting Eq.(6) and (8) into Eq.(5). This price impact function not only allows us to analyze two broad outcomes, i.e., p t+1 > 0 and p t+1 < 0, but also various specic price levels that shape agents' trading decisions in the next period. 2.3 Strategic Expectation Aware of the price determination process in Eq.(9), agent i's expected price change E i ( p t+1 ) depends not only on his private signal but also on his expectation of the others' actions such that E i ( p t+1 ) = E i γα p f ;t p t n f ;t + β (p t p c;t ) 1 n f ;t (10) = γ α E i p f ;t p t β (p t p c;t ) E i n f ;t + β (pt p c;t ). (11) As both E i p f ;t and E i n f ;t is a function of p i f ;t, the expected price change E i ( p t+1 ) is uniquely determined conditional on p i f ;t. Conditional on the private signal pi f ;t, we assume that i agent i extrapolates the real fundamental value p f ;t to fall uniformly within the interval of hp i f ;t ε; p i f ;t + ε and that E i p f ;t = p i f ;t. Note that a fundamentalist's demand is monotonically increasing with his private signal while a chartist's demand is independent of his signal (see Eq. (1) and (2)), an agent's expected aggregate demand is either increasing with or independent of his private signal. Given the expected price change is essentially a multiplier of the expected aggregate demand such that E i ( p t+1 ) = γe i (D t ), it is easy to show that E i ( p t+1 ) is essentially an nondecreasing function of p i f ;t. 9

12 3 Theoretical Implications 3.1 Strategic Complementarity and Substitution Substituting Eq.(1), (2) and (10) into Eq.(3) yields h E i ( π t ) = γ α p i f ;t p t i α β (p t p c;t ) E i p f ;t p t β (p t p c;t ) E i n f ;t + β (pt p c;t ). It is easy to show that E i ( π t ) is an increasing function of E i n f ;t if max p i f ;t ;Ei p f ;t < (α + β) p t p c;t or min p i f ;t α ;Ei p f ;t > (α + β) p t p c;t. In this case strategic complementarity arises: the greater the expected proportion of fundamentalists (chartists), the more likely agent α i will adopt fundamental (chartist) strategy. When min p i f ;t ;Ei p f ;t < (α + β) p t p c;t < α max p i f ;t ;Ei p f ;t, the expected payoff difference E i ( π t ) is a decreasing function of E i n f ;t. In this case, the greater the expected proportion of fundamentalists (chartists), the less likely agent i will choose fundamental (chartist) strategy, which give rise to strategic substitution. When p c;t p i f ;t = (α + β) p t or E i (α + β) p t p c;t p f ;t =, E i ( π t ) is independent of E i n f ;t. In α α this case, the expected fraction of fundamentalists does not affect agents' choice of strategy. The results suggest that how agents taken into account of the others' strategies while choosing their own strategy is regime-dependent. They may not necessarily comply strategic complementarity. Strategic complementarity that arises from the traditional global game setup is only a special case in our model. 3.2 Equilibrium Analysis In the commonly applied global game, agents follow a threshold strategy by taking action (i.e., attack a currency) if and only if their private signals exceed (or fall below) certain threshold value and do nothing otherwise. The presence of nonlinearity in agents' decision making process as presented in the previous subsection makes it inappropriate to focus on monotone equilibria. Close 10

13 examination of the model suggests that there are piecewise linear equilibria that vary across the state of the fundamental value. We summarize these regime-dependent strategies in the following four lemmas. Lemma 1 Conditional on p t > p c;t and p f ;t < β (p t p c;t )=α + p t ε (Bubble Regime), (i) there exists a unique Bayesian Nash Equilibrium xt such that agent i chooses fundamental strategy if p i f ;t 6 x t and chartist strategy if p i f ;t > x t, where the equilibrium xt is characterized by the following equation: xt = β (p c;t p t )=α + p t + ε=2; (12) and (ii) the price at the next period t + 1 increases if p f ;t > x t, decreases if p f ;t < x t and remains unchanged if p f ;t = x t. Proof. see appendix. The condition p f ;t < β (p t p c;t )=α + p t ε is equivalent to α p f ;t + ε p t < β (pt p c;t ). It means that the demand of any individual fundamentalist, including the one that could possibly receive the highest private signal p f ;t + ε and therefore has the largest demand among all fundamentalists, is smaller than the demand of the chartist. Therefore agents that seek to maximize expected payoff choose chartist strategy if they expect the price to increase and fundamental strategy otherwise (see Eq.(3)). Agent i that receives a equilibrium signal (p i f ;t = x t ) expects the price to remain constant such that E i ( p t+1 ) = 0. Given that E i ( p t+1 ) is an increasing function of p i f ;t, agent i chooses chartist strategy if his private signal is higher than the equilibrium signal (p i f ;t > x t ) in the expectation that the price will increase in the next period (E i ( p t+1 ) > 0). Similarly, agent i takes fundamental strategy if p i f ;t 6 x t as he expects the price to decrease or remain constant subsequently (E i ( p t+1 ) 6 0). The condition p t > p c;t suggests that the aggregate demand of chartists D c;t is always positive (see Eq.(8)). Given the price impact function in Eq.(9), the price can only decline if the aggregate demand of fundamentalists D f ;t is sufciently negative so that D t < 0. Conditional on the actual 11

14 fundamental value p f ;t, the private signals are uniformly distributed over p f ;t ε; p f ;t + ε. Agent i adopts fundamental strategy if p i f ;t 2 p f ;t ε;xt and chartist strategy if p i f ;t 2 xt ; p f ;t + ε. This suggests that the stronger the fundamental is, the greater the proportion of chartists and the smaller the proportion of fundamentalists. Note that the individual demand of a chartist is greater than that of a fundamentalist, stronger fundamental essentially shifts agents from towards chartists strategy and therefore increases the aggregate demand. At p f ;t = xt, the the buying force of chartists is completely offset by selling force of fundamentalists so that D t = 0, the price will remain the same in the next period. If the real fundamental is sufciently weak, i.e p f ;t < xt, the selling force of fundamentalists will dominates such that D t < 0, the price will decline in the next period. In the regime dened by p t > p c;t and p f ;t < β (p t p c;t )=α + p t ε, it is possible for the price to increase further even if p t > p f ;t, which may lead to a bubble. We therefore call it bubble regime denoted as R bubble. In the bubble regime, an agent's belief can be either self-fullling or selfdestructive depending on the state of the real fundamental value p f ;t. Agents taking fundamental strategy expects the price to be nondecreasing while those taking chartists strategy expects the price to increase. In reality, the price drops or remains constant if p f ;t 6 xt and increases if p f ;t > xt. This implies that (i) if p f ;t 6 xt, fundamentalists' beliefs are self-fullling while chartists' beliefs are self-destructive; and (ii) if p f ;t > xt, fundamentalists' beliefs are self-destructive while chartists' beliefs are self-fullling. In the bubble regime, investors' choices of strategies cluster. Consider an agent i that observes a private signal p i f ;t. He extrapolates the signals received by the others to be uniformly distributed i over the interval hp i f ;t ε; p i f ;t + ε and expects the other agents to adopt chartist strategy if their private signals fall below xt and fundamental strategy otherwise. Therefore his expected fraction of fundamentalists is given by E i n f ;t = x t p i f ;t + ε =(2ε). 12

15 It is easy to show that E i n f ;t > 1=2 if p i f ;t 6 xt and E i n f ;t < 1=2 if p i f ;t > xt. Based on Lemma 1, agent i adopts fundamental strategy if E i n f ;t > 1=2 and chartist strategy if 1 E i n f ;t > 1=2. These results imply that agents cluster to the strategy that he expects to be taken by the majority of the others. Lemma 2 Conditional on p t > p c;t and p f ;t > β (p t p c;t )=α + p t ε (Recovery Regime), (i) there exists a unique Bayesian Nash Equilibrium x 0 t such that agent i chooses fundamental strategy if p i f ;t > x0 t, and chartist strategy if p i f ;t < x t, where the equilibrium x t is characterized by the following equation: and x 0 t = β (p t p c;t )=α + p t ; (ii) the price at the next period t + 1 always increases such that p t+1 > 0. Proof. see appendix. The condition p f ;t > β (p t p c;t )=α + p t ε suggests that E i p f ;t > β (pt p c;t )=α + p t 2ε > p t for any agent i. Therefore agents acting as fundamentalists will buy in the risky asset. Note that p t > p c;t, agents acting as chartists would also buy in the asset. As all agents, regardless of their strategies, will buy in the asset, the price of the asset in the next period will denitely go up. As the price dynamics in the regime dened by p t > p c;t and p f ;t > β (p t p c;t )=α + p t ε is characterized by price undervaluation (p t < p f ;t ) and consistent price increment ( p t+1 > 0), we call it recovery regime R recovery. Recall from that Eq.(10), all agents have a common belief that the price will increase in the next period. Such beliefs are self-fullling in the recovery regime. Moreover, it is easy to show that agents cluster to the strategy that is expected to be taken by the majority of others. Since E i ( p t+1 ) > 0 for any i, agents will choose the strategy with higher demand in order to maximize their expected payoff (seeeq.(3)). If agent i receives a sufciently high signal, i.e. p i f ;t > x0 t, his demand based on fundamental strategy is not lower than that based on chartist strategy (d i f ;t > d c;t), therefore he will choose fundamental strategy. Otherwise if the signal is low 13

16 enough, i.e., p i f ;t < x0 t, the agent will choose chartist strategy. Lemma 3 Conditional on p t < p c;t and p f ;t > β (p t p c;t )=α + p t + ε (Depression Regime), (i) there exists a unique Bayesian Nash Equilibrium yt such that agent i chooses fundamentalist strategy if p i f ;t > y t and chartist strategy if p i f ;t < y t, where the equilibrium yt is characterized by the following equation: yt = β (p c;t p t )=α + p t ε=2. (13) (ii) the price increases if p f ;t > y t, decreases if p f ;t < y t and remains unchanged if p f ;t = y t. Proof. see appendix. As p f ;t > β (p t p c;t )=α + p t + ε, we have α p f ;t ε p t > β (pt p c;t ), which suggests that even the lowest possible demand of a fundamentalist is greater than that of a chartist. Therefore an agent will choose fundamental strategy if and only if he expects the price to increase or remain constant. Given that the expected price change is increasing with the private signal and E i ( p t+1 ) = 0 when p i f ;t = y t, agent i expects the price to increase such that E i ( p t+1 ) > 0 if p i f ;t > y t and E i ( p t+1 ) < 0 if p i f ;t < y t. Therefore agent i adopts fundamental strategy if p i f ;t > y t and chartist strategy otherwise. Note that α (yt p t ) = β (p c;t p t ) ε > 0 and p i f ;t > y t if agent i acts as fundamentalist, the individual demand of each fundamentalist is always positive, suggesting all fundamentalists are buying the risky asset. On the other hand, chartists are selling the asset as p t < p c;t. If the real fundamental is sufciently high (low) such that p f ;t > yt (p f ;t < yt ), the aggregate buying force of fundamentalists dominates (subordinates) the aggregate selling force of chartists and the price will increase (drop) in the next period. Otherwise, if the aggregate buying force of fundamentalists is completely offset by the aggregate selling force of chartists, the price in the next period will remain constant. We call the regime dened by p t < p c;t and p f ;t > β (p t p c;t )=α + p t + ε depression regime R depression as the price dynamics may lead to nancial distress - p t may decline further even if p t < p f ;t. In depression regime, if p f ;t > y t, the price will increase as expected by fundamen- 14

17 talists; and if p f ;t < yt, the price will decline as expected by chartists. Therefore we have (i) fundamentalists' beliefs are self-fullling while chartists' beliefs are self-constructive if p f ;t > yt ; and (ii) fundamentalists' beliefs are self-destructive while chartists' beliefs are self-fullling if p f ;t < yt. It can also be shown that agents' trading strategies cluster in the depression regime. Lemma 4 Conditional on p t < p c;t and p f ;t 6 β (p t p c;t )=α + p t + ε (Crash Regime), (i) there exists an unique Bayesian Nash Equilibrium y 0 t such that agent i chooses fundamentalist strategy if p i f ;t 6 y0 t by the following equation: and chartist strategy if p i f ;t > y0 t, where the equilibrium yt 0 is characterized y 0 t = β (p t p c;t )=α + p t (ii) the price at period t + 1 decreases such that p t+1 < 0. Proof. see appendix. The condition p t < p c;t and p f ;t 6 β (p t p c;t )=α + p t + ε suggests that, all agents, whether acting as chartists or fundamentalists, will sell the risky asset. Therefore the price will denitely drop in the next period. Based on Eq.(10), it is straightforward to show that all agents expect the price to decline, that is E i ( p t+1 ) < 0 for any i. It suggests that all agents' beliefs on the future price are self-fullling in this regime. Given E i ( p t+1 ) < 0, agents will choose the strategy that yields lower individual demand in order to maximize expected payoff. Note that the demand of fundamentalist is increasing with his signal while the demand of chartists is independent of the signal, an agent will adopt fundamental strategy if his private signal is sufciently low, i.e. p i f ;t 6 y 0 t, and chartist strategy otherwise. As the price dynamics in the regime dened by p t < p c;t and p f ;t 6 β (p t p c;t )=α + p t + ε is characterized by continuous price decline ( p t+1 < 0) and price overvaluation (p t > p f ;t ), we call it crash regime R crash. In the crash regime, agents also cluster to the strategy that is expected to be popular. 15

18 3.3 Comparative Dynamics of Regime-dependent Equilibria Table 1 summarizes relations between the equilibrium signals and the variables/parameters, provided that the dynamics remain in the same regime. Sensitivity to the Price Level The four regime-dependent equilibria are all path-dependent, sensitive to the change in p t. Ceteris paribus, the higher the price is, the lower the equilibrium signal in bubble and depression regime. According to Lemma 1 and 3, this suggests that, in the bubble and depression regime, the required fundamental threshold for the asset price to increase is lower when the price is relatively high. In another words, it is more likely for the price to increase when the price is higher when the price falls in the bubble or depression regime. The equilibrium signals in recovery and crash regime are increasing with the price level. According to 1 and 4, this suggests that agents required a higher signal to choose fundamental (chartist) strategy in the recovery (crash) regime while the price is relatively high. Sensitivity to Chartist's Reference Price When chartists upgrade their reference price p c;t, the equilibrium signals in bubble and depression regime increase while those in the recovery and crash regime fall. This suggests that agents switch from chartist strategy to fundamental strategy in bubble and recovery regime when the reference price increases. As a result, the market power of chartists declines and the price is less likely to deviate substantially above its fundamental. Similarly, when the reference price increases, agents switch from fundamental strategy to chartist strategy in the depression and crash regime. In this case, the market power of chartists is enhanced and the price is more likely to deviate substantially below its fundamental. The Role of Information Noise Increasing the information noise ε shifts up the equilibrium signal x t in the bubble regime while reduce the equilibrium signal y t in depression regime. When x t increases, the following three scenarios can happen. First, if the x t 2 p f ;t ε; p f ;t + ε, the fraction of fundamentalists increases based on Lemma 1, ceteris paribus. Given that the demand of a fundamentalist is smaller 16

19 than that of a chartist in the bubble regime, this will leads to a shrink in aggregate demand. Second, if x t < p f ;t ε, the fraction of fundamentalists increases if ε increases sufciently, otherwise the fraction of fundamentalists will remain to be 0. As a result, the aggregate demand will either decline or remain constant. Third, if xt > p f ;t + ε, the fraction of fundamentalists remain to be 1. Note that xt < p t, this implies that p t > p f ;t. As all agents are acting as fundamentalists are selling the risky asset, the price will decline on the path of revealing its fundamental value. To summarize, as xt increases, agents tend to switch from chartist strategy to fundamental strategy, which results in constant or lower aggregate demand in the bubble regime. Similarly, when yt decreases, agents tend to switch from chartist strategy to fundamental strategy, resulting in constant or higher aggregate demand in the depression regime. In both bubble and depression regime, more information noise tends to attract more fundamentalists, which makes it more likely for the price to move towards the fundamental value. In recovery and crash regime, the information noise does not affect the equilibrium signal. Trading Intensity of Fundamentalists As the trading intensity of fundamentalist α increases, the equilibrium signals in bubble and crash regime increase while those in recovery and depression regime decline. According to Lemma 1-4, this suggests that higher value of α increases the attractiveness of fundamentalists and makes it more likely for the price to reveal its value across four regimes. As a result, the price is more likely to reveal its fundamental value when α is relatively high. Trading Intensity of Chartists The trading intensity of chartists β decreases the equilibrium signals in bubble and crash regime while increases those in recovery and depression regime. This suggests that higher value of β increases the popularity of chartists across four regimes. As a result, the price is more likely to deviates from its fundamental when β is relatively high. 17

20 Table 1: Comparative dynamics of regime-dependent equilibria. Bubble Regime Recovery Regime Depression Regime Crash Regime xt = β (p c;t p t )=α + p t + ε=2 xt 0 = β (p t p c;t )=α + p t yt = β (p c;t p t )=α + p t ε=2 yt 0 = β (p t p c;t )=α + p t p t & % & % p c;t % & % & p t & % & % ε % & α % & & % β & % % & 3.4 The Role of Fundamental Although equilibrium signals are independent of the fundamental, changing fundamental may shift the equilibrium from one regime to another or shape the price dynamic patterns within the regime. We discuss these roles fundamental values and their interaction with other factors in this section. Based on the Lemma 1-4, the realized fraction of fundamentalists n f ;t and the average belief of fundamentalists p f ;t varies with the state of the real fundamental value p f ;t. To understand the strategic price dynamics and their implications for nancial market across different stages of the business cycle, we solve the model and study the role of fundamental value in this section Conditional on p t > p c;t Based on Lemma 1 and 2, the relation between n f ;t and p f ;t conditional on p t > p c;t can be summarized in the following equations: 8 >< n f ;t = >: 1 if p f ;t 2 (0;xt ε] [ [xt 0 + ε;+ ) x t p f ;t + ε =(2ε) if p f ;t 2 (x t ε;x t + ε) p f ;t xt 0 + ε =(2ε) if p f ;t 2 (xt 0 ε;x 0 t + ε) 0 if p f ;t 2 [x t + ε;x 0 t ε]. (14) As illustrated in Fig.1, the relation between n f ;t and p f ;t is piecewise linear, depending on the state of p f ;t. When p f ;t 6 x t ε, all agents take fundamental strategy as all private signals falls on or below the equilibrium signal x t. When p f ;t 2 (x t ε;x t + ε), the fraction of fundamentalists 18

21 Figure 1: The realized fraction of fundamentalists conditional on p t > p c;t. is decreasing monotonically with p f ;t - as p f ;t increases, fewer agents receive signals below x t, which reduces the fraction of agents that favor fundamental strategy. In particular, at p f ;t = x t, we have n f ;t = 1=2. When p f ;t 2 [x t + ε;x 0 t ε], all private signal falls above x t and below x 0 t, driving all agents to take chartist strategy, that is n f ;t = 0. When p f ;t 2 [x 0 t ε;xt 0 + ε), the relation between n f ;t and p f ;t switch to another state - strong fundamental leads to greater proportion of fundamentalists. In particular, we have n f ;t = 1=2 at p f ;t = x 0 t. When p f ;t > x 0 t + ε, all signals are above x 0 t, leading all agents to adopt fundamental strategy, that is n f ;t = 1. Overall, fundamentalists dominate the market (n f ;t > 1=2) when the risky asset is sufciently overvalued (p f ;t < x t = β (p c;t p t )=α + p t + ε=2) or undervalued (p f ;t > x 0 t = β (p t p c;t )=α + p t ). Given Lemma 1 and 2, the average signal received by fundamentalists conditional on p t > p c;t is uniquely determined by: 8 >< p f ;t = >: p f ;t if p f ;t 2 (0;xt ε] [ [xt 0 + ε;+ ) x t + p f ;t ε =2 if p f ;t 2 (x t ε;x t + ε) xt 0 + p f ;t + ε =2 if p f ;t 2 (xt 0 ε;x 0 t + ε) 0 if p f ;t 2 [x t + ε;x 0 t ε]. (15) After specifying n f ;t and p f ;t, the price impact function in Eq.(9) is essentially a nonlinear 19

22 dynamic function of p t. Since the price change is a multiplier of the aggregate demand D t, which is also a nonlinear function of p t, we focus our discussion on aggregate demand below. Figure 2 illustrates the aggregate demand D t conditional on p t > p c;t as well as its two components - the aggregate demand of fundamentalists (D ft ) and the aggregate demand of chartists (D ct ). Since p t > p c;t, all chartists are buying regardless of the value of p f ;t which leads to D c;t > 0 (Eq.(8)). The total size of their buying measured by D c;t increases with p f ;t as a result of the declining n f ;t, which means that the aggregate buying force of chartists expands as the fundamental becomes stronger. Whether fundamentalists are buying or selling depend on the state of the fundamental. When p f ;t < β (p t p c;t )=α + p t ε, because all agents that adopt fundamental strategy observe signals not greater than xt (see Lemma 1) and xt < p t, all fundamentalists sell the risky asset which results in D f ;t < 0 (Eq.(6)). In this scenario, as the real fundamental p f ;t strengthens, the aggregate demand of fundamentalist D f ;t increases, that is, the aggregate selling force of fundamentalist shrinks. Both the declining n f ;t and the smaller individual demand of fundamentalist (p f ;t moves closer to p t ) contribute to the shrinking selling force of fundamentalists. The aggregate selling force of fundamentalists is completely offset by the buying force of chartists at the point p f ;t = xt. When p f ;t > β (p t p c;t )=α + p t ε, all fundamentalists buy in the asset. As chartists always buy in the asset, the aggregate demand is always positive. Overall, we observe from Figure 2 that D t is generally increasing with p f ;t conditional on p t > p c;t. Moreover D t > 0 if p f ;t > xt and D t < 0 otherwise. Given that p t+1 = γ D t, we have p t+1 > 0 if p f ;t > xt and p t+1 < 0 if p f ;t < xt. Why stronger fundamental lead to price decline if the asset is sufciently overvalued i.e., p f ;t < β (p c;t p t )=α + p t + ε=2? This is because, the selling force of fundamentalists, even though mitigated by stronger fundamental, still dominates the aggregate buying of chartists, which drive the price down towards its value. The results shall not be confused with the positive relation between p t+1 (instead of p t ) and p f ;t. Previous section briey discusses whether agents' believes are self-fullling or self-destructive. Figure 2 illustrates these results further. Conditional on p t > p c;t, except for p f ;t 2 (x t ε;x t + ε), we have (i) D c;t > 0 and D f ;t > 0 when D t > 0; and (ii) D c;t 6 0 and D f ;t 6 0 when D t < 0. 20

23 Aggregate Demand 0 Figure 2: The aggregate demand conditional on p t > p c;t. As D t > 0 (D t < 0) is equivalent to p t+1 > 0 ( p t+1 < 0), these ndings suggest that, for any p f ;t =2 (x t ε;x t + ε), both fundamentalists and chartists buy (sell) in expectation that the price in the next period will increase (decrease) and such beliefs turn out to be self-fullling. If p f ;t 2 [xt ;xt + ε), agents that receive signals on or below xt adopt fundamental strategy as they expect the price to stabilize or decline; while agents that receive signals above xt take chartist strategy because they expect the price to decline. Given that D t > 0 and p t+1 > 0 if p f ;t 2 [xt ;xt + ε), the beliefs of agents acting as fundamentalists are self-destructive while the beliefs of agents acting as chartist are self-fullling. Similarly, if p f ;t 2 (x t ε;x t ], the beliefs of fundamentalists are selffullling while those of chartists are self-destructive. Agents' beliefs can be self-destructive when p f ;t 2 (x t ε;x t + ε) because their private signals and the real fundamental fall on the opposite side of the decision reference threshold xt, i.e. p i f ;t > x t > p f ;t or p f ;t > xt > p i f ;t, which misleads them to choose the wrong strategy. 21

24 Figure 3: The fraction of fundamentalists conditional on p t < p c;t Conditional on p t < p c;t Similarly, according to Lemma 3 and 4, conditional on p t < p c;t, both n f ;t and p f ;t are linear functions of p f ;t : 8 >< n f ;t = >: 1 if p f ;t 2 (0;y 0 t y 0 t p f ;t + ε =(2ε) if p f ;t 2 (y 0 t ε] [ [y t + ε;+ ) ε;y 0 t + ε) p f ;t y t + ε =(2ε) if p f ;t 2 (y t ε;y t + ε) 0 if p f ;t 2 [y 0 t ε;y t ε], and 8 >< p f ;t = p f ;t if p f ;t 2 (0;y 0 t yt 0 + p f ;t ε =2 if p f ;t 2 (yt 0 ε] [ [y t + ε;+ ) ε;y 0 t + ε) y t + p f ;t + ε =2 if p f ;t 2 (y t ε;y t + ε). >: 0 if p f ;t 2 [y 0 t + ε;y t ε] As shown in Figure 3, fundamentalists dominate the market if the price deviates sufciently from the fundamental value, i.e. p f ;t > y t or p f ;t < y 0 t, consistent with the result drawn from the condition of p t > p c;t. Conditional on p t > p c;t, the aggregate demand D t as well as its two components - the aggregate 22

25 Aggregate Demand 0 Figure 4: The aggregate demand conditional on p t < p c;t. demand of fundamentalists (D ft ) and the aggregate demand of chartists (D ct ) are illustrated in Figure 4. It is shown that D t is generally increasing with p f ;t. Moreover D t > 0 if p f ;t > yt and D t < 0 otherwise. It implies that p t+1 > 0 if p f ;t > yt and p t+1 < 0 if p f ;t < yt. When p f ;t > yt, the asset is signicantly undervalued. In this case, stronger fundamental always leads to higher price. Otherwise if p f ;t < yt, stronger fundamental results in lower price as p t+1 < 0. It is also shown in Figure 4 that, agents' beliefs, regardless of their choice of strategy, are selffullling if p f ;t =2 (y t ε;y t + ε). If p f ;t 2 [y t ;y t + ε), the beliefs of agents choosing fundamental strategy are self-fullling while the beliefs of agents choosing chartist strategy are self-destructive. If p f ;t 2 (y t ε;y t ], the beliefs of agents choosing fundamental strategy are self-destructive while that of agents choosing fundamental strategy are self-destructive. Agents' beliefs can be selfdestructive when p f ;t 2 (y t ε;y t + ε) because their private signals and the real fundamental fall on the opposite side of the decision reference threshold y t, i.e. p i f ;t > y t > p f ;t or p f ;t > y t > p i f ;t, which misleads them to choose the wrong strategy. To summarize, whether p t > p c;t or p t < p c;t, the following results hold: (i) fundamentalists dominate the market when the price deviates substantially from the fundamental value; 23

26 (ii) the aggregate demand D t and the price change p t+1 are nondecreasing with p f ;t ; (iii) stronger fundamental may lead to lower asset price; (iv) agents' beliefs can be either self-fullling or self-destructive depending on the state of p f ;t ; (v) when p f ;t 2 (x t ε;x t + ε) or p f ;t 2 (y t ε;y t + ε), agents whose private signals and the real fundamental fall on the opposite side of the decision reference threshold have selfdestructive beliefs. 4 Bubble and Depression in a Simple Example In the short-term, the fundamental value of the risky asset is expected to be constant and the reference benchmark for chartists is relatively stable. We therefore focus on a simple case with p f ;t u and p c;t v to study the bubble and depression embedded in this model. The four price regimes can be rewritten as: 8 >< >: αu + αε + βv R bubble = max v; ;+ α + β αu + αε + βv R recovery = v; α + β αu αε + βv R depression = ;min v; α + β αu αε + βv R crash = ;v α + β Note that R recovery exists if and only if v < u+ε and R crash exists if and only if v > u ε. These four price regimes are mutually exclusive. Based on the regime-dependent dynamic patterns outlined in Lemma 1-4, we derive conditions of the existence and persistence of bubble and depression in this section. Note that the trading activities of fundamentalists always drive the asset price towards the fundamental value, bubble and depression can only arise when chartists trade on the opposite side of fundamentalists and their market power measured by trading volumes dominates the market. In the bubbly (depressed) equilibrium, agents shall strategically cluster to the irrational trading strategy and drives the price signicantly above (below) its fundamental value. 24

27 4.1 Bubble Proposition 5 Conditional on u v > bubble equilibrium price p bubble (8 γα + 3γβ)ε 4γβ Proof. See appendix. and γ () < 2. (3)ε, for any initial price p 0 2 (v;+ ), there exists a 4β αu αε=2 βv = and the bubble equilibrium is stable if u v < Regime-dependent dynamics that leads to bubble equilibrium (3)ε Conditional on u v >, the regime (v;+ ) is an union of the recovery regime R recovery 4β and bubble regime R bubble. To facilitate our analysis below, we decompose the bubble regime into αu + αε + βv three sub-regimes, namely the lower part of the bubble regime R L bubble = ; α + β αu αε=2 βv the equilibrium point and the upper part of the bubble regime R U bubble = αu αε βv ;+. Therefore, we can write (v;+ ) as the following: αu αε=2 βv, i (v;+ ) = v; αu+αε+βv α+β {z } R L bubble z } { n o z } { [ αu αε=2 βv [ αu αε=2 βv ;+ {z } [ αu+αε+βv αu αε=2 βv α+β ; R recovery R U bubble R bubble. According to lemma 2, the price dynamics always lead to higher price in the recovery regime. As long as p t 2 R recovery for any t > 0, we have p t+1 > p t. This suggests the price dynamics within αu + αε + βv regime R recovery is not stable. Eventually, the price will exceed the boundary and α + β switch up to the bubble regime. It is impossible to have an equilibrium within the recovery regime. Based on Lemma 1, the price dynamics results in price increment if the current price falls in the lower part of the bubble regime R L bubble 4. As long as p t 2 R L bubble, we have p t+1 > p t for any t > 0. Eventually at some period T the price will exceed the boundary of the regime R L bubble such αu αε=2 βv that p T >. Similarly, based on Lemma 1, the price dynamics within the upper 4 It is easy to show that u > x when p t 2 R L bubble, which leads to p t+1 > 0 According to Lemma 1. Similarly u < x when p t 2 R U bubble, which leads to p t+1 < 0. 25

28 part of the bubble regime R U bubble leads to price decline, that is, p t+1 < p t for any p t 2 R U bubble and t > 0. Eventually, the price will escape from the regime R U bubble. Therefore there cannot be any equilibrium in R L bubble or RU bubble. It is only when p αu αε=2 βv t =, the price in the subsequent period will remain the same such that p t+1 = p t for any t > 0. It is easy to show that the equilibrium αu αε=2 βv price characterizes the bubble state as p t = > u. For any initial price p 0 2 (v;+ ), the repeated price dynamics implies that p t+1 > 0 if p t < αu αε=2 βv αu αε=2 βv αu αε=2 βv, p t+1 < 0 if p t > and p t+1 = 0 if p t =. The price dynamics will eventually leads the price to escape from regime R recovery, R L bubble or RU bubble if αu αε=2 βv the price falls into any one of them and stabilize at the equilibrium point if p t = Stability of the bubble equilibrium If the bubble equilibrium is unstable, the price will diverge further and further away from the equilibrium price resulting in unrealistic price jumps and falls, i.e. p t < 0. When the bubble equilibrium is stable, the price either stabilize at the equilibrium point or moves innitely close to it. We will focus our discussion on the stable equilibrium. Dene Ω bubble as the necessary conditions for the emergence of a stable bubble so that Ω bubble = p 0 2 (v;+ ); (3)ε 4β < u v < (8 γα + 3γβ)ε 4γβ and γ () < 2 : Figure 5 illustrates two types of phase diagrams based on the price dynamics condition on Ω bubble. The phase line in Figure 5a is positively inclined with a slope less than unity when evaluated at the equilibrium point. For any initial price p 0 2 (v;+ ), the dynamics process leads the price consistently towards the equilibrium in a steady time path. The phase line in Figure 5b is negatively inclined around the equilibrium with a slope less than 1 in its absolute value. In this case, the price converges to the equilibrium in an oscillating time path. During the oscillation process, the price movs up and down disturbingly before reaching its steady state. This characterizes the period of nancial distress that precedes the burst of a bubble as described in Kindleberger et al (2005). 26