3.1 Dynamical system

To derive the dynamical system of the deterministic model, we first drop all random variables, i.e., we set $\sigma _{\nu }=\sigma _{\eta }=\sigma _{u}=0$ . Moreover, we follow De Grauwe and Ji (Reference De Grauwe and Ji2019b) and normalize the model by expressing inflation rates and interest rates as deviations from the inflation target $\pi ^T$ . Thus, we introduce $\hat {\pi }_t=\pi _t-\pi ^T$ , $\hat {r}_t=r_t-\pi ^T$ and $\tilde {E}_t \hat {\pi }_{t+1}=\tilde {E}_t \pi _{t+1}-\pi ^T$ . Equations (1)–(3) then become

(16) \begin{align} y_t=a_1 \tilde {E}_t y_{t+1}+(1-a_1)y_{t-1}+a_2(\hat {r}_t-\tilde {E}_t \hat {\pi }_{t+1}),\\[-2pc] \nonumber \end{align}

(17) \begin{align} \hat {\pi }_t=b_1 \tilde {E}_t \hat {\pi }_{t+1}+(1-b_1)\hat {\pi }_{t-1}+b_2 y_t,\\[6pt] \nonumber \end{align}

and

(18) \begin{equation} \hat {r}_t=c_1 \hat {\pi }_t+c_2 y_t+c_3 \hat {r}_{t-1}, \end{equation}

where $c_1=d_1 d_3$ , $c_2=d_2 d_3$ , and $c_3=1-d_3$ . Furthermore, we get

(19) \begin{equation} \tilde {E}_t^{f} \hat {\pi }_{t+1}=0 \end{equation}

and

(20) \begin{equation} \tilde {E}_t^{e} \hat {\pi }_{t+1}=\pi _{t-1}-\pi ^T, \end{equation}

yielding

(21) \begin{equation} \tilde {E}_t \hat {\pi }_{t+1}=\beta _{e,t} \hat {\pi }_{t-1} \end{equation}

and

(22) \begin{equation} A_{j,t}=\rho A_{j,t-1}-\rho (1-\rho )(\hat {\pi }_{t-1}-\tilde {E}_{t-2}^{j} \hat {\pi }_{t-1}),\,\,\,j=f,e. \end{equation}

Taking into account also equations (6), (8), (9), and (14), and by introducing the auxiliary variables $z_t=y_{t-1}$ , $x_t=z_{t-1}$ , $p_t=\hat {\pi }_{t-1}$ , and $q_t=p_{t-1}$ , we are able to summarize the model by De Grauwe and Ji (Reference De Grauwe and Ji2020) by the following 11-D nonlinear discrete dynamical system

(23) \begin{equation} S:\begin{cases} y_t=\dfrac {a_1 \alpha _{e,t} y_{t-1}+(1-a_1) y_{t-1}-a_2\beta _{e,t}\hat {\pi }_{t-1}(1-c_1 b_1)+a_2 c_1(1-b_1)\hat {\pi }_{t-1}+a_2 c_3 \hat {r}_{t-1}}{1-a_2 c_2-a_2 c_1 b_2}\\[5pt] \hat {\pi }_t=b_1 \beta _{e,t} \hat {\pi }_{t-1}+(1-b_1)\hat {\pi }_{t-1}+b_2 y_t\\ \hat {r}_t=c_1 \hat {\pi }_t+c_2 y_t+c_3 \hat {r}_{t-1}\\ U_{e,t}=\rho U_{e,t-1}-\rho (1-\rho )(y_{t-1}-x_{t-1})^2\\ U_{f,t}=\rho U_{f,t-1}-\rho (1-\rho )(y_{t-1})^2\\ A_{e,t}=\rho A_{e,t-1}-\rho (1-\rho )(\hat {\pi }_{t-1}-q_{t-1})^2\\ A_{f,t}=\rho A_{f,t-1}-\rho (1-\rho )(\hat {\pi }_{t-1})^2\\ z_t=y_{t-1}\\ x_t=z_{t-1}\\ p_t=\hat {\pi }_{t-1}\\ q_t=p_{t-1} \end{cases}, \end{equation}

where

\begin{equation*} \alpha _{e,t}=\frac {\exp\!(\gamma U_{e,t})}{\exp\!(\gamma U_{f,t})+\exp\!(\gamma U_{e,t})} \end{equation*}

and

\begin{equation*} \beta _{e,t}=\frac {\exp\!(\gamma A_{e,t})}{\exp\!(\gamma A_{f,t})+\exp\!(\gamma A_{e,t})}. \end{equation*}

Accordingly, the model dynamics is driven by the iteration of map $S$ , which describes the state of the system at time $t$ , defined by $y_t$ , $\hat {\pi }_t$ , $\hat {r}_t$ , $U_{e,t}$ , $U_{f,t}$ , $A_{e,t}$ , $A_{f,t}$ , $z_t$ , $x_t$ , $p_t$ , and $q_t$ , as a function of the state of the system at time $t-1$ , i.e., $y_{t-1}$ , $\hat {\pi }_{t-1}$ , $\hat {r}_{t-1}$ , $U_{e,t-1}$ , $U_{f,t-1}$ , $A_{e,t-1}$ , $A_{f,t-1}$ , $z_{t-1}$ , $x_{t-1}$ , $p_{t-1}$ , and $q_{t-1}$ .Footnote ²

3.2 Steady state and local asymptotic stability

We are now ready to explore the existence of a steady state, which can be done by looking for a constant solution to the dynamical system $S$ . Proposition1 states that a steady state exists at which we have $\pi ^*=\pi ^T$ . Since the central bank aims to keep inflation at its target rate, we call it the fundamental steady state.

Proof. When imposing the fact that expectations are realized at the steady state, i.e., $\tilde {E} y^*=y^*$ and $\tilde {E} \pi ^*=\pi ^*$ , it follows from (16) and (17) that a steady state must satisfy

\begin{equation*} y^*=a_1 y^*+(1-a_1)y^*+a_2(\hat {r}^*-\hat {\pi }^*) \end{equation*}

and

\begin{equation*} \hat {\pi }^*=b_1 \hat {\pi }^*+(1-b_1)\hat {\pi }^*+b_2 y^*, \end{equation*}

yielding $\hat {r}^*=\hat {\pi }^*$ and $y^*=0$ , respectively. Then, the Taylor rule

\begin{equation*} \hat {r}^*=c_1 \hat {\pi }^*+c_2 y^*+c_3 \hat {r}^* \end{equation*}

can be solved for $\hat {\pi }^*=\hat {r}^*=0$ , implying that $\pi ^*=\pi ^T$ . It follows that $p^*=q^*=0$ , $z^*=x^*=0$ , $U_e^*=A_e^*=0$ , $U_f^*=-\rho y^{*2}=0$ , and $A_f^*=-\rho \hat {\pi }^{*2}=0$ . A steady-state solution is therefore given by $\boldsymbol{s}^*=(y^*,\hat {\pi }^*,\hat {r}^*, U_e^*, U_f^*, A_e^*, A_f^*,z^*,x^*,p^*,q^*)=(0,0,0,0,0,0,0,0,0,0,0)=0$ .

Note that $\pi ^*=\pi ^T$ implies that the steady state coincides with expectations of the fundamental rule. However, the naive heuristic also generates perfect forecasts at the steady state. Thus, both predictors yield the same forecast, and the difference in forecasting performance is zero. Consequently, half of the agents rely on the fundamental heuristic and the other half on the naive heuristic to forecast inflation, i.e., $\beta _{f}^*=\beta _{e}^*=0.5$ . The same is true for the two output forecasting rules. Since $y^*=0$ , both the fundamental and the naive expectation rule yield a perfect forecast, resulting again in the same forecasting performance, and therefore in a uniform distribution among agents, i.e., $\alpha _{f}^*=\alpha _{e}^*=0.5$ . However, the dynamical system (23) may also give rise to further nonfundamental steady states at which $\tilde {E} y^*\neq y^*$ and $\tilde {E} \pi ^*\neq \pi ^*$ .Footnote ³

A key contribution of our paper is to provide a (complete) local stability analysis of the fundamental steady state for the model by De Grauwe and Ji (Reference De Grauwe and Ji2020). The results are summarized by the following proposition.

Proposition 2. The fundamental steady state $\boldsymbol{s}^*=0$ is locally asymptotically stable if and only if

\begin{equation*} c_1\gt \frac {a_1 b_1(1-c_3)+2a_2(b_2(1-c_3)-b_1 c_2)}{4 a_2 b_2} \end{equation*}

and

\begin{align*} \begin{split} &1-\frac {(-2+a_1)(-2+b_1)(-4+a_1+b_1+a_2 b_2+2 a_2 c_2-a_2 b_1 c_2-2 c_3)c_3}{8(-1+a_2(b_2 c_1+c_2))^2}\\ &\quad{}\gt \frac {(-2+a_1)^2(-2+b_1)^2 c_3^2}{(4-4a_2(b_2 c_1+c_2))^2}-\frac {(-2+a_1)(-2+b_1)-2(-4+a_1+b_1+a_2 b_2)c_3}{-4+4a_2(b_2 c_1+c_2)} \end{split} \end{align*}

are simultaneously satisfied.

Proof. We take the dynamic variables in the same order as they appear in (23), and find that the Jacobian matrix, computed at the fundamental steady state, has the following block structure

(24) \begin{equation} J(\boldsymbol{s}^*)= \begin{bmatrix} H_1 & \quad 0_{(3,4)} & \quad 0_{(3 ,4)} \\ 0_{(4,3)} & \quad \rho I_4& \quad 0_{(4,4)} \\ H_2 & \quad 0_{(4,4)}& \quad H_3 \end{bmatrix}, \end{equation}

where matrix $H_1$ is given by

\begin{equation*} H_1=\begin{bmatrix} \dfrac {-2+a_1}{-2+2a_2(b_2 c_1+c_2)} & \quad \dfrac {a_2+a_2(-2+b_1)c_1}{-2+2a_2(b_2 c_1+c_2)} & \quad -\dfrac {a_2 c_3}{-1+a_2(b_2 c_1+c_2)} \\[12pt]\dfrac {(-2+a_1)b_2}{-2+2a_2(b_2c_1+c_2)} & \quad \dfrac {-2+b_1+a_2 b_2-a_2(-2+b_1)c_2}{-2+2 a_2(b_2 c_1+c_2)} & \quad -\dfrac {a_2 b_2 c_3}{-1+a_2(b_2 c_1+c_2)} \\[12pt] \dfrac {(-2+a_1)(b_2 c_1+c_2)}{-2+2a_2(b_2 c_1+c_2)} & \quad \dfrac {(-2+b_1+a_2 b_2)c_1+a_2 c_2}{-2+2a_2(b_2 c_1+c_2)} & \quad -\dfrac {c_3}{-1+a_2(b_2 c_1+c_2)} \end{bmatrix}, \end{equation*}

$0_{(m,n)}$ denotes the null $(m,n)$ matrix, $p I_4$ is the 4-D identity matrix multiplied by the memory parameter, i.e.,

\begin{equation*} \rho I_4= \begin{bmatrix} \rho & \quad 0 & \quad 0 & \quad 0 \\ 0 & \quad \rho & \quad 0 & \quad 0 \\ 0 & \quad 0 & \quad \rho & \quad 0\\ 0& \quad 0& \quad 0& \quad \rho \end{bmatrix}, \end{equation*}

and

\begin{equation*}H_2= \begin{bmatrix} 1 & \quad 0 & \quad 0 \\ 0 & \quad 0& \quad 0 \\ 0 & \quad 1& \quad 0\\ 0 & \quad 0& \quad 0 \end{bmatrix} \,\,\,\,H_3= \begin{bmatrix} 0 & \quad 0 & \quad 0 & \quad 0 \\ 1 & \quad 0& \quad 0& \quad 0 \\ 0 & \quad 0& \quad 0& \quad 0\\ 0 & \quad 0& \quad 1& \quad 0 \end{bmatrix}\text{.}\end{equation*}

Since $J(\boldsymbol{s}^*)$ is a lower triangular block matrix, the eigenvalues can be determined by computing the eigenvalues of blocks $H_3$ , $\rho I_4$ , and $H_1$ separately. It immediately follows that four eigenvalues are equal to $0$ , and four are given by $\rho$ . Due to $0\lt \rho \lt 1$ , they are smaller than one in absolute value. The remaining three eigenvalues, say $\lambda _1$ , $\lambda _2$ , and $\lambda _3$ , are the ones of block $H_1$ , from which we get the characteristic polynomial $P_{H_1}(\lambda )=\lambda ^3+\kappa _1 \lambda ^2+\kappa _2 \lambda +\kappa _3$ , where $\kappa _1=-\frac {-4+a_1+b_1+a_2 b_2+2a_2 c_2-a_2 b_1 c_2-2 c_3}{-2+2a_2(b_2 c_1+c_2)}$ , $\kappa _2=\frac {-((-2+a_1)(-2+b_1))+2(-4+a_1+b_1+a_2 b_2)c_3}{-4+4a_2(b_2 c_1+c_2)}$ and $\kappa _3=\frac {(-2+a_1)(-2+b_1)c_3}{-4+4a_2(b_2 c_1+c_2)}$ . A set of necessary and sufficient conditions to have all roots of a cubic polynomial inside the unit circle is given by (see Gardini et al. (Reference Gardini, Schmitt, Sushko, Tramontana and Westerhoff2021)):

\begin{align*} \begin{split} ({\rm i})\ &1+\kappa _1+\kappa _2+\kappa _3\gt 0 \\ ({\rm ii})\ &1-\kappa _1+\kappa _2-\kappa _3\gt 0\\ ({\rm iii})\ &1-\kappa _2+\kappa _1 \kappa _3-\kappa _3^2\gt 0\\ ({\rm iv})\ &|\kappa _3|\lt 1 \end{split} \end{align*}

Since we have $0\lt a_1,b_1\lt 1$ , $a_2\lt 0$ , $b_2,c_1(=d_1 d_3),c_2(=d_2 d_3)\gt 0$ and $0\lt c_3(=1-d_3)\lt 1$ , the denominator of $\kappa _3$ is in absolute terms always greater than 4, while its numerator is always smaller than $4$ . Condition (iv) therefore always holds, and we consider only the three remaining conditions. From condition $(ii)$ , however, we obtain $-16-16 c_3-a_1 b_1-a_1 b_1 c_3+4 a_1+4 b_1+ 4a_1 c_3+ 4b_1 c_3+2 a_2 b_2(1+c_3)+2 a_2 c_2 (4-b_1)+4a_2b_2c_1\lt 0$ , which, given the model’s parameter restrictions, is always fulfilled. This can be easily seen as $a_2\lt 0$ , $-16-16 c_3-a_1 b_1-a_1 b_1 c_3\lt -16$ and $4 a_1+4 b_1+ 4a_1 c_3+ 4b_1 c_3\lt 16$ . Thus, the fundamental steady state $\boldsymbol{s}^*=0$ is locally asymptotically stable if $(i)$ and $(iii)$ hold simultaneously, i.e., if and only if

(25) \begin{equation} c_1\gt \frac {a_1 b_1(1-c_3)+2a_2(b_2(1-c_3)-b_1 c_2)}{4 a_2 b_2} \end{equation}

and

(26) \begin{equation} \begin{aligned} &1-\frac {(-2+a_1)(-2+b_1)(-4+a_1+b_1+a_2 b_2+2 a_2 c_2-a_2 b_1 c_2-2 c_3)c_3}{8(-1+a_2(b_2 c_1+c_2))^2}\\ &\quad{}\gt\frac {(-2+a_1)^2(-2+b_1)^2 c_3^2}{(4-4a_2(b_2 c_1+c_2))^2}-\frac {(-2+a_1)(-2+b_1)-2(-4+a_1+b_1+a_2 b_2)c_3}{-4+4a_2(b_2 c_1+c_2)}. \end{aligned} \end{equation}

As shown above, the local stability analysis is performed by determining the eigenvalues of the Jacobian matrix, evaluated at the fundamental steady state. A sufficient condition for the local stability of a fixed point is that all eigenvalues are inside the unit circle in the complex plane, i.e., smaller than one in modulus. Due to the particular structure of (24), eight out of the 11 eigenvalues can be determined immediately. Among these eight eigenvalues, four are equal to zero, while the remaining four are given by $0\lt \rho \lt 1$ . If stability conditions (25) and (26) are simultaneously satisfied, the remaining three eigenvalues also lie inside the unit circle, and the fundamental steady state is locally asymptotically stable. However, as soon as one of the two conditions is violated, at least one eigenvalue crosses the unit circle, and the steady state becomes unstable. If inequality (26) is violated, while (25) is satisfied, we have one real eigenvalue inside the unit circle and a pair of complex eigenvalues that is larger than one in absolute value (see Gardini et al. (Reference Gardini, Schmitt, Sushko, Tramontana and Westerhoff2021)). In such a situation, model dynamics may explode cyclically.Footnote ⁴ When (25) gets violated and (26) holds, we have one real eigenvalue that becomes larger than $+1$ , while the other two (complex or real) lie inside the unit circle. In this case, model dynamics may explode monotonically.Footnote ⁵ Now suppose the central bank faces a situation in which at least one of these conditions gets violated. Then, it can try to re-establish the stability of the steady state by adjusting the parameters of its monetary policy rule. As demonstrated in the next section, achieving steady-state stability depends on dampening self-fulfilling waves of optimism and pessimism that drive the economy away from equilibrium.

4.1 Stochastic dynamics

For reasons of illustration, we follow De Grauwe and Ji (Reference De Grauwe and Ji2020) and first introduce an index of market sentiments, reflecting how optimistic or pessimistic agents’ forecasts are. These animal spirits are defined by

\begin{align*} S_t=\begin{cases} \alpha _{e,t}-\alpha _{f,t}=\alpha _{e,t}-(1-\alpha _{e,t})=2\alpha _{e,t}-1& \text{if}\,\,y_{t-1}\gt 0\\ -\alpha _{e,t}+\alpha _{f,t}=-\alpha _{e,t}+(1-\alpha _{e,t})=-2\alpha _{e,t}+1& \text{if}\,\,y_{t-1}\lt 0\end{cases}, \end{align*}

where $S_t \in [-1,1]$ . Let us assume the last period’s output gap is positive (negative). If this is the case, we have that agents who rely on the naive predictor forecast a positive (negative) output gap, while those using the fundamental heuristic make a pessimistic (optimistic) forecast. The market sentiment can be captured by subtracting the fraction of pessimists from the fraction of optimists, i.e., $\alpha _{e,t}-\alpha _{f,t}$ if $y_{t-1}\gt 0$ and $\alpha _{f,t}-\alpha _{e,t}$ if $y_{t-1}\lt 0$ . Of course, index $S_t$ becomes positive (negative) if the fraction of optimists (pessimists) exceeds the fraction of pessimists (optimists), and is equal to $0$ if $\alpha _{e,t}=\alpha _{f,t}=0.5$ .

Figure 1. Stochastic dynamics. The panels on the left-hand side show movements of the output gap and animal spirits for a sample of $200$ periods, while their frequency distributions are presented on the right-hand for the full $10,000$ periods. The parameters are $a_1=0.5$ , $a_2=-0.2$ , $b_1=0.5$ , $b_2=0.05$ , $d_1=1.5$ , $d_2=0.5$ , and $d_3=0.5$ , implying $c_1=d_1 d_3=0.75$ , $c_2=d_2 d_3=0.25$ and $c_3=1-d_3=0.5$ , $\gamma =2$ , $\rho =0.65$ , and $\sigma _\nu =\sigma _\eta =\sigma _u=0.5$ .

In Figure 1, we depict a representative simulation run based on $10,000$ observations. The panels on the left-hand side show movements of the output gap and animal spirits for a sample of $200$ periods, respectively. Since the model by De Grauwe and Ji (Reference De Grauwe and Ji2020) was calibrated such that time units can be considered to be quarters, $200$ periods correspond to a time span of about 50 years. As can be seen, the output gap shows a strong cyclical movement in the time domain. These movements of output are triggered by the waves of optimism and pessimism depicted in the lower left-hand side panel. If optimists dominate, we observe a positive output gap, while the output is below its equilibrium when pessimists dominate. Between observations $160$ and $180$ , we even observe a situation where all agents extrapolate a positive output gap, i.e., $S_t$ becomes equal to $+1$ . Obviously, the output gap shows extreme positive movements during this period. The high correlation between these two variables can be explained by the self-fulfilling nature of expectations. When a wave of optimism (pessimism) is set in motion, aggregate demand increases (decreases), and those who made a optimistic (pessimistic) forecast benefit as their forecasting rule performs better. As a result, more and more agents switch to being optimists (pessimists) by which, in turn, aggregate demand is stimulated (dampened) and a boom (bust) may be created.

The right-hand panels show the output gap and animal spirits in the frequency domain for the full $10,000$ periods. In the upper panel, we compare the frequency distribution of the output gap to the one of a normal distribution with identical mean and standard deviation, represented by the black line. As can be seen, the output gap is not normally distributed. De Grauwe and Ji (Reference De Grauwe and Ji2020) find a higher concentration around the mean, thinner shoulders, and more probability mass in the tails. This non-normality can be explained by the lower panel. While the distribution of the animal spirits shows a high concentration around 0, we also observe a concentration of observations at the extreme values of $+1$ and $-1$ . As extreme optimism (pessimism) creates extreme positive (negative) movements, the distribution of the output gap reveals fat tails.

4.2 Deterministic dynamics

As demonstrated above, the model proposed by De Grauwe and Ji (Reference De Grauwe and Ji2020) is able to replicate several key empirical statistical properties, and the simulated data closely resembles the dynamics observed in the real world. However, the model’s dynamics depends strongly on exogenous shocks affecting the economy. Specifically, three different shocks, each with a standard deviation of 0.5, hit the system in every period. Given the amplitude of the dynamics, these shocks are relatively large.

Figure 2. Deterministic dynamics for the base parameter setting. The panels show, from top to bottom, the evolution of the output gap, the inflation rate, and the interest rate for 50 periods, respectively. The parameters are $a_1=0.5$ , $a_2=-0.2$ , $b_1=0.5$ , $b_2=0.05$ , $d_1=1.5$ , $d_2=0.5$ , and $d_3=0.5$ , implying $c_1=d_1 d_3=0.75$ , $c_2=d_2 d_3=0.25$ and $c_3=1-d_3=0.5$ , $\gamma =2$ , $\rho =0.65$ , and $\sigma _\nu =\sigma _\eta =\sigma _u=0$ .

Now, we set $\sigma _\nu =\sigma _\nu =\sigma _u=0$ and illustrate the dynamics in the time domain in Figure 2. From top to bottom, the panels show the evolution of the output gap, the inflation rate, and the interest rate for 50 periods, respectively. At $t=0$ , the dynamics is hit by a single exogenous demand shock of 0.5. As can be seen, all three state variables quickly converge towards their equilibrium. The reason for this is provided by our analytical results. For the given parameter setting, the characteristic polynomial $P_{H_1}(\lambda )$ is given by $\lambda ^3-1.93144\lambda ^2+1.2435 \lambda -0.265957$ , from which we obtain $\lambda _1=0.55$ and $\lambda _{2,3}=0.69\pm 0.08 \mathrm{i}$ . Since $|\lambda _{1,2,3}|\lt 1$ , the fundamental steady state is locally stable. Moreover, we have two eigenvalues that are complex, which is why the adjustment towards the steady state is cyclical. If no further shock hits the dynamics, the economy remains at its steady state.

Now, suppose a series of random shocks hits the dynamics, initiating a wave of optimism or pessimism. Once such a wave is set in motion, the nonlinear features of the model kick in and amplify these shocks. Since the shocks are i.i.d, it is the model’s nonlinear forces due to which we find non-normality in the distribution of the output gap. In this way, the model transforms normally distributed shocks – without any boom-bust characteristics – into dynamics of booms and busts. However, without any of these exogenous shocks, the dynamics can be characterized by fixed-point dynamics.

Interestingly, the fundamental steady state would also be stable if the central bank were inactive, i.e., if $d_1=0$ and $d_2=0$ , implying $c_1=0$ and $c_2=0$ . In this case, the characteristic polynomial of matrix $H_1$ would be $P_{H_1}(\lambda )=\lambda ^3-2.005\lambda ^2+1.315\lambda -0.28125$ , yielding the eigenvalues $\lambda _1=0.5$ , $\lambda _2=0.69$ , and $\lambda _3=0.81$ . Since all eigenvalues lie inside the unit circle, the system would still converge to the steady state, even if there were no inflation targeting and no output stabilization. This result differs from the findings of standard DSGE models. Woodford (Reference Woodford2003) and Galí (Reference Galí2008), for instance, show that the inflation parameter $d_1$ must exceed one to ensure stability of the model. However, other New Keynesian models with heterogeneous expectations and boundedly rational agents – such as those proposed by Hommes and Lustenhouwer (Reference Hommes and Lustenhouwer2019) and Salle et al. (Reference Salle, Yıldızoğlu and Sénégas2013) – also suggest that convergence to the steady state can be ensured even if the Taylor Principle is not satisfied. Interestingly, Angeletos and Lian (Reference Angeletos and Lian2023) arrive at a similar result. They translate a dynamic game among consumers into the standard representative agents New Keynesian model and assume that social memory is imperfect. Under this perturbation, they demonstrate that the equilibrium is unique and given by the fundamental solution, regardless of monetary policy.

This raises the question: why does this result occur in the model we consider? To answer this, let us consider the example described above where $c_1=0$ and $c_2=0$ . The fundamental steady state is locally stable, implying that half of the agents rely on the fundamental predictor and the other half on the naive predictor, i.e.,  $\alpha _{f}^*=\alpha _{e}^*=0.5$ . Then, a positive demand shock occurs, causing the output gap to increase. As the output gap moves away from equilibrium, the naive heuristic, making a positive forecast, becomes more attractive. However, agents update their choice of heuristic with a time delay of three periods. Thus, at the time of the shock, the fraction of those relying on the fundamental predictor remains at 50 percent, and their mean-reverting power is strong enough to push the output gap back towards its equilibrium value. Once agents adjust their choices, the fraction of those making a positive forecast increases. However, by then, the output gap is already converging toward equilibrium, making the fundamental predictor attractive again. As a result, the fraction of agents using the naive predictor gradually decreases, while the fraction of those following the fundamental predictor increases. This process continues – without any central bank intervention – until equilibrium is reached, with both fractions equal to 0.5. Nonetheless, the central bank can accelerate this stabilization process by increasing its policy parameters. In doing so, it can reduce the amplitude of fluctuations – much like in Angeletos and Lian (Reference Angeletos and Lian2023) – and help the economy return to equilibrium more quickly. However, if the system is regularly hit by exogenous shocks, the volatility of the adjustment dynamics of the output gap and the inflation rate may become unacceptably high, forcing the central bank to intervene, even though the system is dynamically stable.

Figure 3. Deterministic dynamics for a different parameter setting. The panels show, from top to bottom, the evolution of the output gap, the inflation rate, and the interest rate for 500 periods, respectively. On the left-hand (right-hand) side, the parameters are $a_1=0.8$ , $a_2=-1$ , $b_1=0.8$ , $b_2=2$ , $c_1=0.2$ ( $c_1=0.23$ ), $c_2=0.2$ , $c_3=0.8$ , $\gamma =2$ , $\rho =0.65$ , and $\sigma _\nu =\sigma _\eta =\sigma _u=0$ .

Next, we illustrate that there are parameter settings within the defined parameter ranges for which the fundamental steady state is unstable. In Figure 3, we assume the following parameter values: $a_1=0.8$ , $a_2=-1$ , $b_1=0.8$ , $b_2=2$ , $c_1=0.2$ , $c_2=0.2$ , $c_3=0.8$ , $\gamma =2$ , $\rho =0.65$ , and $\sigma _\nu =\sigma _\eta =\sigma _u=0$ , and display the dynamics in the left-hand panels.Footnote ⁷ At $t=0$ , we assume an exogenous demand shock of $0.1$ . The panels show again, from top to bottom, the evolution of the output gap, the inflation rate, and the interest rate in the time domain (for 500 periods), respectively. As evident from the figure, the system generates cyclical dynamics. However, these boom-bust dynamics eventually explode and diverges to infinity. Applying our analytical results to this parameter setting reveals that the first stability condition is satisfied, while the second one is violated. From (26), we obtain, for instance, that parameter $c_1$ must exceed $0.207482$ to ensure stability of the fundamental steady state. Given that $c_1=0.2$ , the dynamics diverge whenever an exogenous shock disrupts the system.

However, the central bank can achieve stability of the steady state by increasing the intensity at which it does inflation targeting. As soon as parameter $c_1$ exceeds its critical value of $0.204782$ , the system may converge to its equilibrium. In the right-hand panels of Figure 3, we show how the dynamics change when the central bank increases its inflation parameter up to $c_1=0.23$ . The output gap, the inflation rate and the interest rate now appear to cyclically converge to their equilibrium values.

What economic mechanism leads to this stabilization? The key is to mitigate the intensity of self-fulfilling waves of optimism and pessimism that drive the economy away from equilibrium. When monetary policy responds strongly enough to deviations in inflation and output, it can counteract expectation-driven fluctuations, thereby reducing excessive belief-switching and, in turn, mitigating boom-bust cycles. For instance, suppose we observe a strong increase in aggregate demand. Those who made a positive output forecast will benefit, encouraging more agents to become optimistic, which further stimulates aggregate demand. If the central bank does not raise interest rates sufficiently, aggregate demand will continue to rise, potentially leading to a boom. However, if the central bank responds strongly enough, the increase in aggregate demand will be dampened, causing fewer agents to become optimists and thus reducing the boom.