**3. Fundamentalists versus Contrarians**

320 Nonlinearity, Bifurcation and Chaos – Theory and Applications

dynamics.

is taken into account.

by using parameter

(1998), and in several other contributions.

discrete choice probability by a multinomial logit model:

,

*n*

intensity of choice approaches infinity (the neoclassical limit).

*h t H*

function:

enter the adaptive belief system before the equilibrium price is observed. This is suitable for analyzing the asset pricing model as an explicit nonlinear difference equation. Even though nonlinear asset pricing dynamics can be modelled either as a deterministic or a stochastic process, only the latter enables investigation of the effects of noise upon the asset pricing

The share of past fitness in the performance measure is expressed by the parameter *w*; 0 *w* 1, called memory strength. When the value of this parameter is zero (*w* = 0), the fitness is given by most recent net realized profit. Due to analytical tractability this is at present, y for the most part, the case in the existing literature on asset pricing models with heterogeneous agents, though not in this chapter. The main contribution of this chapter is that it analyzes the case of nonzero memory in the fitness measure. When the memory strength parameter takes a positive value, some share of current realized profits in any given period is taken into account when calculating the performance measure in the next time period. If the value of memory strength parameter amounts to one then of course the entire accumulated wealth

The expression (14) for the fitness function is somewhat different that the one used in Brock and Hommes (1998), where the coefficient of the current realized profits was fixed to 1.

to be a specific number of time periods, we obtain the following expression for the fitness

, ,1 , 1 1 <sup>1</sup> *U U h t ht ht T T*

which is equivalent to taking the last *T* observations into account with equal weight (as benchmark). When *T* approaches infinity, the memory parameter approaches 1 and the entire accumulated wealth is taken into account. We thus believe the expression (14) to be a more suitable formulation of the fitness measure than the one used in Brock and Hommes

Finally, we can express fractions of belief types, *nh,t*, which are updated in each period, as a

exp

economic agents switch between different prediction strategies; if the value of intensity of choice is zero, then all trader types have equal weight and the mass of traders distributes itself evenly across the set of available strategies, while on the other hand the entire mass of traders tends to use the best predictor, i.e. the strategy with the highest fitness, when the

 

exp

, 1

*U*

, determining the intensity of choice. The latter measures how fast

*h t*

1 , 1

*i t i*

*U*

  *w* 1

, (16)

*<sup>T</sup>* , where *T* is considered

, (17)

Namely, if we rewrite the memory strength parameter as <sup>1</sup>

The first case we are going to examine is a two-type heterogeneous agents model with fundamentalists and contrarians as market participants. Fundamentalists exhibit deterministic function of the form:

$$f\_{1,t} \equiv 0\tag{18}$$

and have some positive information gathering costs *C*, i.e. *C* > 0. Contrarians exhibit a deterministic function:

$$f\_{2,t} = \text{gx}\_{t-1}; \quad \text{g} < 0 \tag{19}$$

and zero information gathering costs. It is thus a case of fundamentalists versus pure contrarians. We have the following fractions of belief types equation:

$$m\_{h,t} = \frac{\exp\left[\beta \mathcal{U}\_{h,t-1}\right]}{\exp\left[\beta \mathcal{U}\_{1,t-1}\right] + \exp\left[\beta \mathcal{U}\_{2,t-1}\right]}; \quad h = 1, 2 \cdot \tag{20}$$

For convenience we shall also introduce a difference in fractions *mt*:

$$m\_t = n\_{1,t} - n\_{2,t} = \frac{\exp\left[\beta \mathcal{U}\_{1,t-1}\right] - \exp\left[\beta \mathcal{U}\_{2,t-1}\right]}{\exp\left[\beta \mathcal{U}\_{1,t-1}\right] + \exp\left[\beta \mathcal{U}\_{2,t-1}\right]} = \tanh\left[\frac{\beta}{2} \left(\mathcal{U}\_{1,t-1} - \mathcal{U}\_{2,t-1}\right)\right].\tag{21}$$

Finally, we have the fitness measure equation of each type:

$$\mathbf{U}\_{1,t} = w \mathbf{U}\_{1,t-1} + (\mathbf{1} - w) \left[ -k \mathbf{R} \mathbf{x}\_{t-1} \left( \mathbf{x}\_t - \mathbf{R} \mathbf{x}\_{t-1} \right) - \mathbf{C} \right],\tag{22}$$

Memory and Asset Pricing Models with Heterogeneous Beliefs 323

*<sup>g</sup> is always greater than 1. On the other hand,* 

*X FX t t* 1 1 , (30)

 

> 

, (31)

2, 1 1, 1 *tt t xx x* , (32)

. (28)

. (29)

Possible other (non-fundamental) steady states should satisfy:

Therefore we can state the following lemma.

obtain:

**Proof of Lemma 1:** 

*Since g < 0,* <sup>2</sup> <sup>0</sup> *<sup>R</sup>*

equation:

*g*

**3.2. Stability of the steady state** 

1 2 tanh ( ) (1 ) <sup>2</sup>

*steady state of the system. Memory does not affect the position of this steady state.*

*m*

*and thus also in expression (26), memory does not affect the position of this steady state.* 

as: 1, 1 1 : *t t x x* , 2, 1 2 : *t t x x* , 3, 1 3 : *t t x x* , 1, 1 1, 2 : *t t u U* and 2, 1 2, 2 : *t t u U* .

1, 1 1, 1

 

*t t*

We therefore obtain the following 5-dimensional first-order difference equation:

1 1 exp exp

 

 *holds and expression* \* <sup>2</sup> <sup>1</sup> *<sup>R</sup>*

\* \* \* \*\* \*

Since it can be derived that \* 2\* <sup>1</sup> *U kRx R C* 1 and \* 2\* <sup>2</sup> *U kx R g R* <sup>1</sup> , we finally

 \* 2\* tanh 1 2 *<sup>m</sup> kgx R C* 

**Lemma 1:** *The fundamental steady state in case of fundamentalists versus contrarians is a unique* 

*the value of the hyperbolic tangent function is by definition between –1 and 1. In fact, since k > 0, g < 0, R > 1, C > 0 and the variable x is squared, the right-hand side of expression (29) is always between –1 and 0. Expression (29) thus never gives a solution and the fundamental steady state* (0, ) *eq m is a unique steady state of the system. Since there is no memory strength parameter in expression (27)* 

In order to analyze stability of the steady state we shall rewrite our system as a difference

where *X xx xuu t t tttt* 1 1, 1 2, 1 3, 1 1, 1 2, 1 ,,,, is a vector of new variables, which are defined

1, 2, 1, 1 1 *tt tt x x n gx <sup>R</sup>*

*gx gx R R U U uu*

exp exp exp exp

2, 1 2,

*U u*

*t t*

1, 1 2, 1 1, 2,

*t t t t*

*<sup>m</sup> w U U w kgx x Rx C*

$$\mathbf{L}I\_{2,t} = \mathbf{w}\mathbf{L}I\_{2,t-1} + (\mathbf{1} - \mathbf{w}) \left[ k \left( \mathbf{x}\_t - \mathbf{R}\mathbf{x}\_{t-1} \right) \left( \mathbf{g}\mathbf{x}\_{t-2} - \mathbf{R}\mathbf{x}\_{t-1} \right) \right]. \tag{23}$$

In order to analyze memory in our heterogeneous asset pricing model, we shall first determine the position and stability of the steady state and the period two-cycle in relation to the memory strength parameter. We will also examine the possible qualitative changes in dynamics. Then we will perform some numerical simulations to combine global stability analysis with local stability analysis.

### **3.1. Position of the steady state**

In our two-type heterogeneous agents model of fundamentalists versus contrarians the equilibrium pricing equation has the following form:

$$\mathbf{R}\mathbf{x}\_t = \mathbf{n}\_{2,t}\mathbf{g}\mathbf{x}\_{t-1} = \frac{1-m\_t}{2}\mathbf{g}\mathbf{x}\_{t-1} \tag{24}$$

where 1, 2, *t tt nn m* and 1, 2, 1 *t t n n* . The difference in fractions of belief types equation, on the other hand, has the following form:

$$m\_t = \tanh\left[\frac{\beta}{2}\left(w\left(\mathcal{U}\_{1,t-2} - \mathcal{U}\_{2,t-2}\right) - \left(1 - w\right)\left(k\text{gx}\_{t-3}\left(\mathbf{x}\_{t-1} - \mathbf{R}\mathbf{x}\_{t-2}\right) + \mathcal{C}\right)\right)\right].\tag{25}$$

A steady state price deviation *x* is a fixed point of the system, if it satisfies *x* = *f*(*x*) for mapping *f*(*x*). In our two-type heterogeneous agents model of fundamentalists versus contrarians we have:

$$R\mathbf{x} = \frac{1-m}{2}\mathbf{g}\mathbf{x} \tag{26}$$

where either 0 *eq x* , or \* 1 2 *<sup>m</sup> R g* and thus \* <sup>2</sup> <sup>1</sup> *<sup>R</sup> m <sup>g</sup>* . In the former case we get the fundamental steady state, where the price is equal to its fundamental value and the difference in fractions is:

$$m^{eq} = \tanh\left[\frac{\beta}{2}\left(w\left(U\_1^{eq} - U\_2^{eq}\right) - \left(1 - w\right)\mathbf{C}\right)\right].$$

Since it follows from expressions (22) and (23) that 1 *eq U C* and 2 <sup>0</sup> *eq <sup>U</sup>* when *<sup>w</sup>* 1, the steady state difference in fractions simplifies:

$$m^{eq} = \tanh\left[\frac{\beta}{2}(-w\mathcal{C} - (1 - \varpi)\mathcal{C})\right] = \tanh\left[-\frac{\beta\mathcal{C}}{2}\right].\tag{27}$$

Possible other (non-fundamental) steady states should satisfy:

$$\hat{m}^\* = \tanh\left[\frac{\beta}{2} \Big(\text{w}(\text{U}\_1^\* - \text{U}\_2^\*) - (1 - \text{w}) \Big(\text{kgx}^\* \left(\text{x}^\* - \text{Rx}^\*\right) + \text{C}\Big)\Big)\right].\tag{28}$$

Since it can be derived that \* 2\* <sup>1</sup> *U kRx R C* 1 and \* 2\* <sup>2</sup> *U kx R g R* <sup>1</sup> , we finally obtain:

$$\hat{\rho}m^\* = \tanh\left[-\frac{\beta}{2}\left(\log\mathbf{x}^{\*2}\left(1-R\right)+\mathbf{C}\right)\right].\tag{29}$$

Therefore we can state the following lemma.

**Lemma 1:** *The fundamental steady state in case of fundamentalists versus contrarians is a unique steady state of the system. Memory does not affect the position of this steady state.*

### **Proof of Lemma 1:**

322 Nonlinearity, Bifurcation and Chaos – Theory and Applications

analysis with local stability analysis.

**3.1. Position of the steady state** 

on the other hand, has the following form:

contrarians we have:

where either 0 *eq x* , or

difference in fractions is:

2

\* 1 2

Since it follows from expressions (22) and (23) that 1

steady state difference in fractions simplifies:

equilibrium pricing equation has the following form:

Finally, we have the fitness measure equation of each type:

1, 1, 1 <sup>1</sup> <sup>1</sup> (1 ) *U wU w kRx x Rx C t t tt t* , (22)

2, 2, 1 12 1 (1 ) *U wU w k x Rx gx Rx t t tt t t* . (23)

, (24)

*<sup>m</sup> Rx gx* , (26)

.

 

*<sup>g</sup>* . In the former case we get the

*eq U C* and 2 <sup>0</sup> *eq <sup>U</sup>* when *<sup>w</sup>* 1, the

. (27)

. (25)

In order to analyze memory in our heterogeneous asset pricing model, we shall first determine the position and stability of the steady state and the period two-cycle in relation to the memory strength parameter. We will also examine the possible qualitative changes in dynamics. Then we will perform some numerical simulations to combine global stability

In our two-type heterogeneous agents model of fundamentalists versus contrarians the

*t tt t <sup>m</sup> Rx n gx gx*

where 1, 2, *t tt nn m* and 1, 2, 1 *t t n n* . The difference in fractions of belief types equation,

1, 2 2, 2 31 2 tanh 1

A steady state price deviation *x* is a fixed point of the system, if it satisfies *x* = *f*(*x*) for mapping *f*(*x*). In our two-type heterogeneous agents model of fundamentalists versus

> 1 2

fundamental steady state, where the price is equal to its fundamental value and the

tanh 1 2 <sup>1</sup> <sup>2</sup> *eq eq eq <sup>m</sup> wU U wC*

tanh (1 ) tanh 2 2 *eq C*

*m*

*<sup>m</sup> R g* and thus \* <sup>2</sup> <sup>1</sup> *<sup>R</sup>*

*<sup>m</sup> wC w C* 

 

*mt w U U w kgx x Rx C t t tt t*

2, 1 1 1 2 *t*

*Since g < 0,* <sup>2</sup> <sup>0</sup> *<sup>R</sup> g holds and expression* \* <sup>2</sup> <sup>1</sup> *<sup>R</sup> m <sup>g</sup> is always greater than 1. On the other hand, the value of the hyperbolic tangent function is by definition between –1 and 1. In fact, since k > 0, g < 0, R > 1, C > 0 and the variable x is squared, the right-hand side of expression (29) is always between –1 and 0. Expression (29) thus never gives a solution and the fundamental steady state* (0, ) *eq m is a unique steady state of the system. Since there is no memory strength parameter in expression (27) and thus also in expression (26), memory does not affect the position of this steady state.* 

### **3.2. Stability of the steady state**

In order to analyze stability of the steady state we shall rewrite our system as a difference equation:

$$X\_t = F\_1(X\_{t-1}) \, \prime \tag{30}$$

where *X xx xuu t t tttt* 1 1, 1 2, 1 3, 1 1, 1 2, 1 ,,,, is a vector of new variables, which are defined as: 1, 1 1 : *t t x x* , 2, 1 2 : *t t x x* , 3, 1 3 : *t t x x* , 1, 1 1, 2 : *t t u U* and 2, 1 2, 2 : *t t u U* .

We therefore obtain the following 5-dimensional first-order difference equation:

$$\mathbf{x}\_{1,t} = \mathbf{x}\_t = \frac{1}{R} \boldsymbol{\eta}\_{2,t} \mathbf{g} \mathbf{x}\_{1,t-1} =$$

$$\mathbf{x} = \frac{1}{R} \mathbf{g} \mathbf{x}\_{1,t-1} \frac{\exp\left[\boldsymbol{\beta} \mathbf{U}\_{2,t-1}\right]}{\exp\left[\boldsymbol{\beta} \mathbf{U}\_{1,t-1}\right] + \exp\left[\boldsymbol{\beta} \mathbf{U}\_{2,t-1}\right]} = \frac{1}{R} \mathbf{g} \mathbf{x}\_{1,t-1} \frac{\exp\left[\boldsymbol{\beta} \mathbf{u}\_{2,t}\right]}{\exp\left[\boldsymbol{\beta} \mathbf{u}\_{1,t}\right] + \exp\left[\boldsymbol{\beta} \mathbf{u}\_{2,t}\right]},\tag{31}$$

$$\mathbf{x}\_{2,t} = \mathbf{x}\_{t-1} = \mathbf{x}\_{1,t-1}.\tag{32}$$

$$\mathbf{x}\_{\mathbf{3},t} = \mathbf{x}\_{t-2} = \mathbf{x}\_{\mathbf{2},t-1} \,\prime \tag{33}$$

Memory and Asset Pricing Models with Heterogeneous Beliefs 325

1, which we are in our case interested in, we can observe

<sup>1</sup> <sup>1</sup> *eq n g <sup>R</sup>* (37)

<sup>1</sup> <sup>1</sup> *eq n g <sup>R</sup>* (38)

<sup>2</sup> *U kx R g R* 1 , a period two-cycle occurs


, (39)

into expression (38) and solving

. (40)

*1 of the characteristic equation (36) that the* 

 *< 1, the expression (40) has two* 

If we take a look at the eigenvalue

which has been computed by plugging 2

*<sup>m</sup> R g* , and thus \* <sup>2</sup> <sup>1</sup> *<sup>R</sup>*

Therefore we can state the following lemma.

*are two possibilities: (1) if* \* <sup>2</sup> <sup>1</sup> *<sup>R</sup> eq m m*

*g*

*For g < –2R it is clear from the expression for eigenvalue* 

*fundamental steady state is unstable. Furthermore, since 0 < m\**

*while (2) if* \* <sup>2</sup> <sup>1</sup> *<sup>R</sup> eq m m*

**Proof of Lemma 3:** 

for the memory strength parameter

\* 1 2

when

that a saddle-node bifurcation can never occur. Namely, the expression:

always negative for *g* < 0, *R* > 0 and 2 <sup>0</sup> *eq <sup>n</sup>* . On the other hand, the expression:

(primary) period doubling bifurcation may occur in our model for the following

*eq n*

*g* satisfies:

.

account that \* 2\* <sup>1</sup> *U kRx R C* <sup>1</sup> and \* 2\*

*m*

*g*

*cycle*  \*\* \*\* ( , ),( , ) *xm xm . Memory does not affect the position of the period two-cycle.* 

2

2

1 exp 1

*<sup>C</sup>*

 *then the steady state (0, meq) is unstable and there exists a period two-*

 *then (0, meq) is the unique, globally stable steady state,* 

can never hold, since the left-hand side is a positive constant and the right-hand side is

may be satisfied for 2 <sup>0</sup> *eq <sup>n</sup>* , since both sides of the expression are then negative. Thus a

<sup>1</sup> \* ln *<sup>R</sup> C Rg*

Now we can check the existence of a period two-cycle \*\* \*\* ( , ),( , ) *xm xm* . Taking into

 \* 2\* tanh 1 2 *<sup>m</sup> kgx R C* 

**Lemma 3:** *In case of fundamentalists versus contrarians the fundamental steady state (0, meq) is unstable for g < –2R and there exists a period two-cycle*  \*\* \*\* ( , ),( , ) *xm xm . For –2R < g < –R there* 

$$\mathbf{u}\_{1,t} = \mathbf{U}\_{1,t-1} = w\mathbf{u}\_{1,t-1} + (1 - w) \left[ -kR\mathbf{x}\_{2,t-1} \left( \mathbf{x}\_{1,t-1} - R\mathbf{x}\_{2,t-1} \right) - \mathbf{C} \right],\tag{34}$$

$$\mu\_{2,t} = \mathcal{U}\_{2,t-1} = wu\_{2,t-1} + (1 - w) \left[ k \left( \mathbf{x}\_{1,t-1} - \mathbf{R} \mathbf{x}\_{2,t-1} \right) \left( \mathbf{g} \mathbf{x}\_{3,t-1} - \mathbf{R} \mathbf{x}\_{2,t-1} \right) \right]. \tag{35}$$

The local stability of a steady state is determined by the eigenvalues of the Jacobian matrix, which we do not present here due to the spatial limitations. We then compute the Jacobian matrix of the 5-dimensional map. At the fundamental steady state *Xeq* = (0, 0, 0, –*C*, 0) we obtain the new Jacobian matrix. A straightforward computation shows that the characteristic equation is in our case given by:

$$\log(\mathcal{A}) = \left(\frac{1}{R} n\_2^{eq} \mathcal{g} - \mathcal{\lambda}\right) \mathcal{\lambda}^2 \left(w - \mathcal{\lambda}\right)^2 = 0 \,\,\,\,\tag{36}$$

with solutions (eigenvalues): 1 2 <sup>1</sup> *eq n g <sup>R</sup>* , 2,3 0 and 4,5 *w* . The steady state *Xeq* is stable for 1 ; therefore in cases 2 *eq R gn R* and 1 *w* .

Thus we can state the following lemma.

**Lemma 2:** *The fundamental steady state in case of fundamentalists versus contrarians is globally stable for –R < g < 0. Memory does not affect the stability of this steady state.*

### **Proof of Lemma 2:**

*From the characteristic equation (36) we can observe three eigenvalues, where two of them are in fact double eigenvalues. The first eigenvalue assures stability when* <sup>2</sup> *eq R gn R , while the second and third (double) eigenvalue always assure stability. The fundamental steady state is stable for*  2 2 *eq eq R R g n n , but since* <sup>2</sup> *eq n depends on other parameters of the system and g < 0, stability is* 

*(more conveniently) guaranteed at least for R g* 0 *. Since the memory strength parameter is represented (only) by the third (double) eigenvalue, memory does not affect the stability of the steady state, as has been shown by the reduced system.* 

### **3.3. Bifurcations and the Period Two-cycle**

A bifurcation is a qualitative change of the dynamical behaviour that occurs when parameters are varied (Brock and Hommes, 1998). A specific type of bifurcation that occurs when one parameter is varied is called a co-dimension one bifurcation. There are several types of such bifurcations, *viz*. period doubling, saddle-node and Hopf bifurcations. The first type has eigenvalue –1 of the Jacobian matrix, the second type has eigenvalue 1 and the third type has complex eigenvalues on the unit circle.

If we take a look at the eigenvalue 1, which we are in our case interested in, we can observe that a saddle-node bifurcation can never occur. Namely, the expression:

$$1 = \frac{1}{R} n\_2^{eq} g$$

can never hold, since the left-hand side is a positive constant and the right-hand side is always negative for *g* < 0, *R* > 0 and 2 <sup>0</sup> *eq <sup>n</sup>* . On the other hand, the expression:

$$-1 = \frac{1}{R} n\_2^{eq} g$$

may be satisfied for 2 <sup>0</sup> *eq <sup>n</sup>* , since both sides of the expression are then negative. Thus a (primary) period doubling bifurcation may occur in our model for the following -value:

$$\mathcal{J}^\* = \frac{1}{C} \ln \left[ -\frac{R}{R+g} \right] \tag{39}$$

which has been computed by plugging 2 1 exp 1 *eq n <sup>C</sup>* into expression (38) and solving for the memory strength parameter .

Now we can check the existence of a period two-cycle \*\* \*\* ( , ),( , ) *xm xm* . Taking into account that \* 2\* <sup>1</sup> *U kRx R C* <sup>1</sup> and \* 2\* <sup>2</sup> *U kx R g R* 1 , a period two-cycle occurs when \* 1 2 *<sup>m</sup> R g* , and thus \* <sup>2</sup> <sup>1</sup> *<sup>R</sup> m g* satisfies: \* 2\* tanh 1 *<sup>m</sup> kgx R C* . (40)

2

Therefore we can state the following lemma.

**Lemma 3:** *In case of fundamentalists versus contrarians the fundamental steady state (0, meq) is unstable for g < –2R and there exists a period two-cycle*  \*\* \*\* ( , ),( , ) *xm xm . For –2R < g < –R there are two possibilities: (1) if* \* <sup>2</sup> <sup>1</sup> *<sup>R</sup> eq m m g then (0, meq) is the unique, globally stable steady state, while (2) if* \* <sup>2</sup> <sup>1</sup> *<sup>R</sup> eq m m g then the steady state (0, meq) is unstable and there exists a period twocycle*  \*\* \*\* ( , ),( , ) *xm xm . Memory does not affect the position of the period two-cycle.* 

### **Proof of Lemma 3:**

324 Nonlinearity, Bifurcation and Chaos – Theory and Applications

characteristic equation is in our case given by:

with solutions (eigenvalues): 1 2

Thus we can state the following lemma.

stable for

**Proof of Lemma 2:** 

2 2 *eq eq R R g n n*

*, but since* <sup>2</sup>

*state, as has been shown by the reduced system.* 

**3.3. Bifurcations and the Period Two-cycle** 

third type has complex eigenvalues on the unit circle.

3, 2 2, 1 *tt t xx x* , (33)

*u U wu w kRx x Rx C* 1, 1, 1 1, 1 *tt t* (1 ) 2, 1 1, 1 2, 1 *tt t* , (34)

*u U wu w k x Rx gx Rx* 2, 2, 1 2, 1 *tt t* (1 ) 1, 1 2, 1 3, 1 2, 1 *t tt t* . (35)

<sup>2</sup> <sup>2</sup>

0 and

 

, (36)

4,5 *w* . The steady state *Xeq* is

*eq R gn R , while the second and* 

*eq n depends on other parameters of the system and g < 0, stability is* 

The local stability of a steady state is determined by the eigenvalues of the Jacobian matrix, which we do not present here due to the spatial limitations. We then compute the Jacobian matrix of the 5-dimensional map. At the fundamental steady state *Xeq* = (0, 0, 0, –*C*, 0) we obtain the new Jacobian matrix. A straightforward computation shows that the

<sup>1</sup> ( ) <sup>0</sup> *eq g ng w <sup>R</sup>*

 

*eq R gn R* and 1 *w* .

**Lemma 2:** *The fundamental steady state in case of fundamentalists versus contrarians is globally* 

*From the characteristic equation (36) we can observe three eigenvalues, where two of them are in fact* 

*third (double) eigenvalue always assure stability. The fundamental steady state is stable for* 

*(more conveniently) guaranteed at least for R g* 0 *. Since the memory strength parameter is represented (only) by the third (double) eigenvalue, memory does not affect the stability of the steady* 

A bifurcation is a qualitative change of the dynamical behaviour that occurs when parameters are varied (Brock and Hommes, 1998). A specific type of bifurcation that occurs when one parameter is varied is called a co-dimension one bifurcation. There are several types of such bifurcations, *viz*. period doubling, saddle-node and Hopf bifurcations. The first type has eigenvalue –1 of the Jacobian matrix, the second type has eigenvalue 1 and the

2

<sup>1</sup> *eq n g <sup>R</sup>*

, 2,3

*stable for –R < g < 0. Memory does not affect the stability of this steady state.*

*double eigenvalues. The first eigenvalue assures stability when* <sup>2</sup>

1 ; therefore in cases 2

*For g < –2R it is clear from the expression for eigenvalue 1 of the characteristic equation (36) that the fundamental steady state is unstable. Furthermore, since 0 < m\* < 1, the expression (40) has two*  *solutions, x\* and –x\* . If expression (38) is satisfied, it then follows from expressions* \* <sup>2</sup> <sup>1</sup> *<sup>R</sup> m g and (40) that*  \*\* \*\* ( , ),( , ) *xm xm is a period two-cycle. Finally, for –2R < g < –R, the fundamental steady state is unstable and expression (40) has solutions* \* *x if and only if*  \* tanh 2 *eq C m m . Since the memory strength parameter does not affect the difference in*  Memory and Asset Pricing Models with Heterogeneous Beliefs 327

 = 

\*\*, as has been

proven in Lemma 3. As can be seen from the simulations, this bifurcation value is also independent of the memory. The stability of the steady state is thus unaffected by the

claimed in Section 3.3; the period two-cycle becomes unstable and an attractor appears consisting of two invariant circles around each of the two (unstable) period two-points, one lying above and the other one below the fundamental. It is a supercritical Hopf bifurcation, where the steady state gradually changes either into an unstable equilibrium or into an attractor (*cf*. Guckenheimer and Holmes, 1983; Frøyland, 1992; Kuznetsov, 1995). The position of the period two-cycle is independent of the memory, but it is not independent of the intensity of choice, as can be seen from expression (40). Numerical simulations suggest that the secondary bifurcation value also does not vary with changing memory strength

many (mostly higher order) stable cycles. Such a bifurcation route to chaos was also called the rational route to randomness (Brock and Hommes, 1997a), while the last part of it has

fundamental value (*x*) in the upper two diagrams and the value of the largest LCE in the lower two diagrams, respectively. The diagrams differ with respect to the memory strength parameter *w*; the left one corresponds to *w* = 0.3,

). Vertical axis represents deviations of the price from the

in case of fundamentalists versus contrarians

\*\* chaotic dynamic behaviour appears, which is interspersed with

increases further, indeed a (secondary) Hopf bifurcation occurs at

memory, as proven in Lemma 2.

 > 

**Notes:** Horizontal axis represents the intensity of choice (

**Figure 1.** Bifurcation diagrams and Largest LCE plots of

while the right one corresponds to *w* = 0.9.

been referred to as the breaking of an invariant circle.

parameter *w*. For

If 

*fractions of belief types, memory does not affect the position of the period two-cycle.* 

As in the paper of Brock and Hommes (1998), very strong contrarians with *g* < –2*R* may lead to the existence of a period two-cycle, even when there are no costs for fundamentalists (*C* = 0). When the fundamentalists' costs are positive (*C* > 0), strong contrarians with –2*R* < *g* < –*R* may lead to a period two-cycle. As the intensity of choice increases to = \*, a period doubling bifurcation occurs in which the fundamental steady state becomes unstable and a (stable) period two-cycle is created, with one point above and the other one below the fundamental.

When the intensity of choice further increases, we are likely to find a value = \*\*, for which the period two-cycle becomes unstable and a Hopf bifurcation of this period two-cycle occurs, as in Brock and Hommes (1998). The model would then get an attractor consisting of two invariant circles around each of the two (unstable) period two-points, one lying above and the other one below the fundamental. Immediately after such a Hopf bifurcation, the price dynamics is either periodic or quasi-periodic, jumping back and forth between the two circles. The proof of this phenomenon is not straightforward due to the non-zero period points, although the 5-dimensional system (31) – (35) is still symmetric with respect to the origin. We shall thus demonstrate the occurrence of the Hopf bifurcation and the emergence of the attractor numerically in the next section.
