Abstract
This paper studies herd behavior, bubbles and social interactions in financial markets through the asset pricing models with heterogeneous interacting agents. The relationship between social interactions, herd behavior and bubbles is examined. It is found that herd behavior arises naturally when there are strong enough social interactions among individual investors. In addition, an extremely small bubble may cause a sufficiently large number of traders to engage in herd behavior when the social interactions among traders are strong.
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See also Blume and Easley (1992, 1993).
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It is possible that rational arbitrageurs are risk-averse, and noise traders are risk-loving. Therefore, as pointed out by one of the referees, the assumption of homogeneous risk-aversion is unreasonable in the current paper. By relaxing the assumptions of homogeneous degrees of risk-aversion and homogeneous expected conditional variance, Chiarella and He (2002) test the robustness of the results of Brock and Hommes (1998). They also investigate the effects of different memory lengths on the dynamics. Chiarella and He (2002) found that Brock and Hommes’s (1998) results are robust to this generalization. However, they do point out that the resulting dynamic behavior with this generalization is considerably enriched and exhibits some significant differences. Thus, how to incorporate the heterogeneous beliefs and risk in a simple asset pricing model with social interactions in order to investigate the resulting herd behavior and bubbles in financial markets will be an important extension of the current paper.
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The belief type of the noise trader used in this paper is equivalent to the term “trend chaser” with a belief bias in Brock and Hommes (1998). Notice that the term “noise trader” may have a different scope than a “trend chaser” with a belief bias in Brock and Hommes (1998). See for example, DeLong et al. (1990).
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The belief type of arbitrageurs used in this paper is equivalent to the term “contrarian” with a belief bias in Brock and Hommes (1998). Notice that the term “arbitrageurs” may be used differently in the finance literature. For example, in Hull (2011), “arbitrageurs” is used to denote the traders who are involved in “locking in a risk-less profit by simultaneously entering into transactions in two or more markets.”
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Here, S(‧) represents proportional spillover-type social interactions; see Brock and Durlauf (2001a,b).
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See Brock and Durlauf (2001a,b) and Anderson, Palma and Thisse (1996) for a derivation.
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Notice that the sufficient condition for |km*|<R is k<R.
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Friedman (1953, p. 175) argues, “People who argue that speculation is generally destabilizing seldom realize that this is largely equivalent to saying that speculators lose money, since speculation can be destabilizing in general only if speculators on average sell when currency is low and buy when it is high.”
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That is,
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Blume and Easley (1992, 1993) find that the most fit behavior in the financial market is that which maximizes the expected growth rate of wealth share accumulation.
I gratefully acknowledge research support from the National Science Council of Taiwan (NSC 99-2410-H-002-050). I would like also to thank Bruce Mizrach and two anonymous referees for their useful comments and suggestions. All errors which remain are mine.
Appendix
Proof of Proposition 1
Proof. (1) When βJ<1, since bR>0 and R>1, then Jb<0. Since βJ<1, βJm<1. Then there exists a unique steady state (0, 0).
(2) When βJ≥1, since bR>0 and R>1, then Jb<0, and there exists a threshold HR, (which depends on k): (a) If βJm<1, there exists only one steady state (0, 0). (b) If βJm<1 and R>HR, there exist three steady states. One of these roots is (0, 0), one root is
and one root is where m+>0 and m–<0. (c) If βJm>1 and R<HR, there exist five steady states. One of these roots is (0, 0), three roots are and and one root is where and m–<0. □Proof of Proposition 2
Proof. The profit of each type of trader at the steady state is
Thus, π*(ωit=1)=π*(ωit=–1)=0 when x*=0. Furthermore,
Then, when x*>0, kx*+b>0 since k, b≥0. Thus π*(ωit=1)–π*(ωit=–1)<0 since R>1. Therefore, rational arbitrageurs (ωit=–1) make more profit than noise traders (ωit=1) at the steady state if the steady state asset price is above its fundamental value (i.e., x*>0).
On the other hand, when the steady state asset price is below its fundamental value, that is, x*>0, then kx*+b>0 if
kx*+b<0 if and kx*+b=0 if Thus, rational arbitrageurs make more profit than noise traders if rational arbitrageurs earn less profit than noise traders if and rational arbitrageurs make the same profit as noise traders ifProof of Proposition 3
Proof. Since all arbitrageurs are rational arbitrageurs, all local dynamic characteristics are discussed around steady states:
When βJ<1, (m*, x*)=(0, 0) is a unique steady state based on Proposition 1. Also tanh′(0)=1. Therefore, βJm tanh′(0)<1 since βJ<1 and Jb≤0. Thus (m*, x*)=(0, 0) is a unique locally stable steady state.
When βJ≥1: (i) When b=0: according to Proposition 3, there are 3 steady states (the same situations as bR=0) and Jb=0. Since tanh′(0)=1, therefore βJm tanh′(0)=βJ tanh′(0)>1 and so (m*, x*)=(0, 0) is locally unstable. On the other hand, the other two steady states will be locally stable if |km*|<R and locally unstable steady states if |km*|<R, since βJ tanh′ (βJm*)<1 for the other two steady states. See the proof arguments in Brock and Durlauf (2001b). (ii) When b≠0: since
where According to Proposition 3, (m*, x*)=(0, 0) is a unique steady state if βJm<1. Furthermore, (0, 0) is locally stable when (0, 0) is a unique steady state since βJm tanh′(0)≥1. On the other hand, if there is more than one steady state in the economy, then βJm≥1 and (0, 0) is a locally unstable steady state since βJm tanh′(0)≥1. The other steady states will be locally stable if evaluated at m* is <1 and |km*|<R. Otherwise, they will be locally unstable steady states. □
Proof of Proposition 4
Proof. Based on Proposition 1, there is a unique steady state (m*, x*)=(0, 0) if βJ<1. Moreover, (0, 0) is also a locally stable steady state. Therefore, according to the definition of herd behavior, it is impossible to have herd behavior at the new steady state since m*=0. On the other hand, based on Proposition 3, there exist one or three or five steady states with strong exogenous social interactions (βJ>1). Furthermore, (m*, x*)=(0, 0) is a locally unstable steady state if there is more than one steady state in the economy, according to Proposition 3. |km*|<R since k<R, and thus the other two or four steady states are the locally stable ones if
evaluated at m* is <1. Therefore, at least one new steady state will be m* which is different from zero when there is more than one steady state in the economy. In other words, herd behavior occurs at the new steady state with some positive probability. Furthermore, with β fixed, |m*|→1 if J→∞ and |m*|→0 if Therefore, the scale of herd behavior depends on the strength of the social interactions. Moreover, the sign of the new steady state value m* is also determined by the sign of the expectation deviation.Proof of Proposition 5
Proof. (1) According to Proposition 1, (0, 0) is the unique stable steady state if βJ<1. Therefore, the bubble crashes in finite time and goes back to its fundamental value. (2) When βJ>1, (0, 0) will be a locally unstable steady state with some positive probability, according to the proof of Proposition 1. Then if there is an extremely small bubble at the initial locally unstable steady state (0, 0), it will cause an extremely small expectation deviation from the steady state (0, 0). Thus, the new steady state will be
with some positive probability where m*≠0, according to Propositions 3 and 4. In other words, with some positive probability, the traders will engage in herd behavior and the price bubble will stay at the new steady state when the exogenous social interactions are strong. □References
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