Inflation Targeting under Heterogeneous Expectations
November 4, 2013
by Joep Lustenhouwer
BSc. University of Amsterdam, 2008 Student number: 5898587
Thesis Submitted in Partial Fullfilment of the Requirements for the Degree of
Master of Science
Supervisor: Prof. Dr. C.H. Hommes Second marker: Dr. D. Massaro
1
Introduction
Many central banks (CB’s) have recently switched to some form of inflation targeting. Some CB’s do this by claiming to set the interest rate such that, if the interest rate were to be kept constant, the inflation in some specified future period is expected to equal the target value. Other CB’s1 include predictions about the future path of the interest rate in their expecta-tions. Both forms of inflation targeting can be described as ”inflation forecast targeting”.
Besides setting the interest rate optimally, important aspects of inflation targeting are communication and managing expectations. This is for example stressed by Woodford (2004). For the inflation targeting to be effective it is important both that the target is perfectly communicated and that the CB is credible. If the private sector does not believe that inflation will be equal to the target, or if they do not know what the target is, the realized value of inflation will likely not be equal to its target. Salle et al (2013) investigate this issue.
Inflation targeting is modeled in various ways in the literature. A common way to model inflation dynamics is with a New Keynesian Philips curve and an IS curve, together with a Taylor type interest rate rule. With such an interest rate rule, the CB adjusts the interest rate in response to inflation and output gap in order to steer inflation toward a long term target. There is however little consideration about what the optimal paths of inflation and output gap are, and when the long term target should be reached. Forecast targeting is also modeled in the literature, but this is a form of strict inflation targeting, where no output considerations are allowed. While CB’s claim to set the interest rate to target inflation, they also take the consequences on output into account. With forecast targeting, a CB furthermore only tries to optimize one specific future period, and again has no considerations for the whole future paths of inflation and output gap.
An inflation targeting model that tries to optimize the future paths of inflation and output gap, uses a loss function including the expectations of all future deviations of inflation from target and all future output gaps. The first order conditions of this loss function give the CB a relation between inflation and output gap that it tries to achieve every period, using the interest rate as an instrument. Using this relation, the model can be solved for an implicit instrumental rule that depends on expectations of inflation and output gap. This rule could be solved for the model parameters if rational expectations are assumed. Rational
tions are however not very realistic when inflation forecasts by price setters (i.e. the private sector) are concerned. It is more likely that expectations are formed by an adaptive rule, or other simple heuristics. Mankiw et al. (2004) furthermore show that there is considerable heterogeneity in inflation expectations. Replacing rational expectations with another form of homogeneous expectations, hence does not solve the problem.
In this study we investigate the dynamics that result from inflation targeting using a loss function with both inflation and output consideration, without the assumptions of homoge-neous and rational expectations. Instead a heuristic switching model is used. This model was introduced by Brock and Hommes (1997), and has since been used to model heteroge-neous expectations in finance and macroeconomics. To provide some robustness to the results derived in this thesis, two sets of heuristics are considered.
The most important heuristic that is used in this thesis can be described as ”Trust the central bank”. Followers of this heuristic believe inflation to be equal to the target that is communicated by the central bank and believe output gap to be zero. Followers of this heuristic are called fundamentalists. In the first set of heuristics fundamentalists compete with naive predictors who believe inflation and output gap to be equal to their values in the previous period. In the second set of heuristic fundamentalists compete with optimists and pessimists, who believe the variables to be respectively above and below their fundamental values. The first set of heuristics is very similar to those used by Branch and McGough (2010) and Cornea et al. (2012). The second set is similar to the one used by de Grauwe (2011).
The first part of this thesis analytically analyzes the above two heuristic switching models. The main research question here is what policy parameters lead to desirable dynamics when expectations are heterogeneous and non-rational. It is shown that in the first set of heuristics letting the interest rate react less than point for point to inflation leads to the existence of multiple steady states. This reduces the region of convergence to the fundamental steady state. As the Central bank responds even weaker to inflation, this region gets smaller and eventually the fundamental steady state becomes locally unstable. In contrast with rational expectation models, reacting too strongly to inflation can in our model lead to an unstable steady state as well.
In the second set of heuristics, stability of the fundamental steady state not only depends on the policy parameters, but also on the bias of optimists and pessimist and on the intensity
of choice parameter of the heuristic switching model. It is furthermore shown that when the intensity of choice parameter goes to infinity, meaning that agents immediately switch to the best performing heuristic, nine different steady states can coexist.
Finally, we find that under the policy parameters that minimize the central banks loss function, the fundamental steady state is unique under the first set of heuristics, and locally stable under both sets of heuristics considered in this thesis. The policy that minimizes the loss function can therefore indeed be considered optimal under heterogeneous expectations.
In the second part of this thesis we conduct simulations. Here it is shown that when the target of the central bank is not perfectly communicated to the private sector, coordination on a specific heuristic for longer periods of time is less likely. For the first set of heuristics this implies that periods where inflation drifts away due to dominating naive expectations occur less often. For the second heuristic periods of optimism and pessimism are shorter when communication of the target is imperfect. Simulations furthermore show that more weight on output gap in the loss function leads to more stable output gap dynamics and less controlled inflation dynamics.
The remainder of this thesis is organized as follows. In Section 2 the inflation targeting model that is used in this thesis is described. Section 3 conducts the above mentioned analyt-ical analysis, while simulations are presented in Section 4. In Section 5 a sensitivity analysis is performed and section 6 concludes.
2
Inflation targeting model
In this Section the inflation targeting model that is used in this thesis is outlined.
2.1 General model
We use a New Keynesian model in line with Woodford (2003). The New Keynesian Philips curve and IS curve, describing inflation and output gap respectively, are given by
πt = βEtπt+1+ κxt+ et (1)
xt= Etxt+1+ 1
Here β is the discount factor, et and ut are shocks to respectively inflation and output gap, and
κ = (σ + η)(1 − ω)(1 − βω)
ω , (3)
with σ and η the inverses of respectively the elasticity of intertemporal substitution and the elasticity of labor supply. (1 − ω) is the fraction of firms that can adjust their price in a given period, and it is the nominal interest rate, which can be freely chosen by the central bank.
We assume that the central bank wants to minimize the discounted sum of all future deviations of inflation from the target the central bank has set for itself. It is furthermore assumed that the central bank also wants to minimize squared output gap, but that it does not necessarily assign equal weight to the minimization of both variables. Let µ be the relative weight that the central bank assigns to the minimization of squared output gap compared to squared deviation of inflation from target. The loss function is then given by
Et ∞ X i=0 βi(πt+i− πT)2+ µ(xt+i)2 (4) There are two ways the CB can minimize this loss function. If the CB optimizes under discretion, it chooses πt and xt to minimize the loss function in every period with the current information. If the CB optimizes under commitment it commits to a policy rule now, and does not reconsider this rule in future periods. This way it can influence future private sector expectations and will therefore ultimately be better off. The main problem with this approach is that the central bank will be tempted to re-optimize in every period. Commitment is only better for the CB because of the effect on private sector expectations. When those expectations have been formed the CB would be better off to renege on its commitment. However, the CB would than lose its credibility, so that we would be back in the discretion case.2 In this thesis it is assumed the CB optimizes under discretion. In this case the first order conditions that are obtained from minimizing (19) result in the following optimal targeting rule.
πt− πT = − µ
κxt (5)
The optimal policy rule that does not assume rational expectations and implements the above condition is derived by Evans and Honkapohja (2003). The same rule in slightly different
settings is derived by Berardi and Duffy (2007), Gomes (2006) and Froyen and Guender (2012). Evans and Honkapohja (2003) study optimal monetary policy under learning, with non-rational, but homogeneous expectations. They find that this rule leads to convergence to the optimum under discretion even if expectations are not rational. Branch and Evans (2011) find that the rule also performs well under heterogeneous expectations. The rule is given by it= ψ0+ ψ1Etπt+1+ ψ2Etxt+1+ ψ3ut+ ψ4et (6) ψ0 = − σκ µ + κ2π T (7) ψ1 = 1 + σκβ µ + κ2 (8) ψ2 = ψ3 = σ (9) ψ4 = σκ µ + κ2 (10)
We assume that the central bank can perfectly observe private sector expectations, as is amongst others done by Branch and MCGough (2010), Berardi and Duffy (2007), Branch and Evans (2011) and Evans and Honkapohja (2003). Although the CB can respond to current period expectations, those expectations are assumed to be based on past information. It is furthermore assumed that the central bank cannot respond to current period shocks. This way agents are not able to influence current period variables. Svensson (2003) strongly argues in favor of this. The interest rate rule that can actually be followed by the central bank is of the form
it = ψ0+ ψ1Etπt+1+ ψ2Etxt+1 (11)
As long as the expected values of shocks equal zero, this rule implements optimal policy in expectation when the coefficients are chosen optimally. The model is then described by equations (1), (2) and (11).
2.2 Simplifying the model
Assumption 1. β = 1
This is done for two reasons. First of all, when all beliefs are rational and β = 1, the system has a steady state with zero output gap and inflation equal to target. As is shown in appendix A, for β < 1 a steady state with inflation different from the target and non-zero output gap arises. This is less intuitive because it makes sense that under optimal policy and without shocks, a fully credible central bank would be able to obtain zero output gap and inflation equal to target. The CB is however, only able to achieve this when either the inflation target is normalized to zero or, as mentioned above, β = 1. Since β is usually calibrated to be around 0.99, Assumption 1 seems less restrictive than normalizing the inflation target to zero. Orphanides and Williams (2005), de Grauwe (2011) and Leitemo (2008) furthermore investigate inflation targeting using a hybrid Philips Curve that, when derived from micro foundations, implicitly assumes the discount factor is one.3 In Section 5 a sensitivity analysis for β not equal to one is performed.
The second reason to assume β = 1 is that under this assumption the optimal interest rate rule can be captured in the two parameter family of expectation based Taylor type interest rate rules. Taylor (1993) proposed a simple interest rate rule, where the interest rate is based on the long term inflation target and only responds to deviations of inflation from this target, and to output gap. When β = 1, equation (11) can be written as
it = πT + φ1(Etπt+1− πT) + φ2Etxt+1 (12)
The coefficients for optimal monetary policy are then given by
φ1 = 1 +
σκ
µ + κ2 (13)
φ2 = σ (14)
When β is close to 1, this rule can approximate optimal policy. The general rule given in equation (12), is very similar to a normal Taylor rule, but replaces past realizations of inflation and output gap with their expectations of next period.
In order to get an idea of the magnitude of the optimal policy parameters, the model needs to be calibrated. Table 1 gives the calibrations of σ and κ of Woodford (1999), Clarida et al.
Table 1: Calibrations Author(s) σ = φ2 κ φOpt1 φP F1 φP D1 W 0.157 0.024 1.015 -5.54 20.63 CGG 1 0.3 1.882 -2.33 11 MN 1/0.164 0.3 6.38 -19.33 61.98 TR 0.5 0.3 1.441 -0.67 6
(2000) and McCallum and Nelson (1999). Column four of Table 1 states the corresponding values of φ1. Here the weight on output gap, µ, is set to 0.25, as is done by McCallum and Nelson (2004) and Walsh (2003). Since under optimal policy φ2 = σ, the different values of φ2 can be read from column two of Table 1.
The Calibrations of Woodford (1999) and Clarida et al. (2000) result in reasonable optimal policy parameters, while the McCallum and Nelson (1999) calibration gives rise to what seems to be unrealistically aggressive monetary policy.
When σ and thus φ2 is set to 0.5 and κ to 0.3 the corresponding value of φ1 is 1.441. Optimal policy now is very close to the classical Taylor rule with φ1 = 1.5 and φ2 = 0.5. When µ is lowered to 0.21 those coefficients are even reached exactly. This calibration lies between the Woodford and Clarida et al. calibrations, and hence does not seem unreasonable.
Abstracting from shocks and plugging (12) into (2), gives the following model xt= (1 − φ2 σ )Etxt+1− φ1− 1 σ (Etπt+1− π T) (15) πt = Etπt+1+ κxt (16)
In this thesis it is assumed that σ, κ > 0. In addition, the following is assumed. Assumption 2. φ1≥ 0, φ2 ≥ 0
This assumption is made because it does not make sense for the CB to let the interest rate respond negatively to expected deviation of inflation from target, or to expected output gap.
3
Analytical analysis with Heuristic Switching model
In this Section a heuristic switching model is used to analytically analyse the dynamics of output gap and inflation when expectations are non-rational and heterogeneous. In a heuristic switching model as in Brock and Hommes (1997), beliefs are formed by a set of simple prediction rules, or heuristics. The population consists of agents that can switch between those heuristics. As a heuristic preforms better through time, the fraction of the population that follows that prediction rule increases. This happens according to the following discrete choice model with multinomial logit probabilities (see Manski and McFadden (1981)):
nh,t=
ebUh,t−1
PH
h=1ebUh,t−1
(17) Here nh,tis the fraction of agents that follows heuristic h in period t, Uh,tis the fitness measure of heuristic h in period t, and b is the intensity of choice. The higher the intensity of choice, the more sensitive agents become with respect to relative performance of the heuristics.
In this thesis two different sets of heuristics are considered. In Section 3.1 the case with fundamentalists and naive predictors is analyzed. In Section 3.2 the case with optimists, pessimist and fundamentalists is considered. In both sections we derive policy implications from the analytical results.
3.1 Fundamentalists versus naive
In this Section private sector beliefs are formed by two heuristics: fundamentalistic and naive. Followers of the fundamentalistic heuristic believe inflation or output gap to be equal to its fundamental value: xt= 0 and πt= πT. This heuristic can thus be interpreted as trusting the central bank. Whichever value of inflation the central bank targets, fundamentalist believe future inflation to be equal to this value. Followers of the naive heuristic believe future inflation or output gap to be equal to to its value in the previous period.
3.1.1 Dynamical system
Let a fraction nz
t be fundamentalists and a fraction 1 − nzt naive predictors, with z = x, π. These fractions are determined according to discrete choice model (17) with fitness measure
Uz
The belief fractions of inflation can thus differ from those of output gap. Now define the difference in fractions, mzt = nzt − (1 − nzt) (19) So that nzt = (1 + m z t) 2 (20)
The model is then given by xt= (1 − φ2 σ ) (1 − mx t) 2 xt−1− φ1− 1 σ (1 − mπ t) 2 (πt−1− π T) (21) πt = (1 + mπ t) 2 π T +(1 − mπt) 2 πt−1+ κxt (22) mxt+1 = tanh(b 2((xt− xt−2) 2− x2 t)) = tanh( b 2At) (23) At= x2t−2− 2xtxt−2= x2t−2− (1 − φ2 σ ) (1 − mx t) 2 xt−1xt−2+ φ1− 1 σ (1 − mπ t) 2 (πt−1− π T)x t−2(24) mπt+1 = tanh(b 2((πt− πt−2) 2) − (π t− πT)2) = tanh( b 2Bt) (25) Bt = π2t−2− (πT)2− 2(πt−2− πT)πt (26) Bt= πt−22 − (πT)2− 2(πt−2− πT) (1 + mπ t) 2 π T − 2(π t−2− πT) (1 − mπ t) 2 πt−1 −2(πt−2− πT)κ(1 − φ2 σ ) (1 − mx t) 2 xt−1+ 2(πt−2− π T)κφ1− 1 σ (1 − mπ t) 2 (πt−1− π T), (27)
The state vector of this dynamical system is
xt πt xt−1 πt−1 mxt+1 mπt+1
(28) Proposition 1 says which steady states exist in this system. Its proof is given in Appendix B.2.
Proposition 1. The fundamental steady state (x∗ = 0, π∗ = πT, mx∗ = 0, mπ∗ = 0) exists for all parameter settings. Two additional non-fundamental steady states exist if and only if
1 −(σ + φ2)
3.1.2 Stability of the fundamental steady state
The Jacobian of the dynamical system defined in the previous section, is given in Appendix B.1. In the fundamental steady state, the Jacobian reduces to
1 2(1 − φ2 σ) − 1 2 φ1−1 σ 0 0 0 0 κ 2(1 − φ2 σ ) 1 2− κ 2 φ1−1 σ 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
This is a lower triangular block matrix, and hence has four eigenvalues 0. The other eigen-values are the eigeneigen-values of the upper left 2x2 block. The characteristic polynomial equals
λ2− λ1 2(2 − φ2 σ − κ φ1− 1 σ ) + 1 4(1 − φ2 σ ) = 0 (30)
This has solutions λ1 = 1 4 (2 − φ2 σ − κ φ1− 1 σ ) + s 2 −φ2 σ − κ φ1− 1 σ 2 − 4(1 − φ2 σ ) (31) and λ2 = 1 4 (2 − φ2 σ − κ φ1− 1 σ ) − s 2 −φ2 σ − κ φ1− 1 σ 2 − 4(1 − φ2 σ ) (32)
Using these eigenvalues, the dynamics of the system around the fundamental steady state can be analyzed. The key parameters in this analysis are the policy parameters φ1 and φ2. Below, we investigate which policy parameter settings lead to local instability and compare this with benchmark results from rational expectation models.
In rational expectations models, instead of a stable fundamental steady state, the central bank wants to achieve determinacy of the fundamental equilibrium. If determinacy is not achieved the fundamental equilibrium will not be unique and sunspot equilibria exist. It will in this case typically not be possible to predict at which equilibrium the system will arrive, while under determinacy convergence to the equilibrium is assured. Determinacy can there-fore be compared with both stability and uniqueness of an equilibrium in the heterogeneous expectations model of this thesis.
It follows from Proposition 1 that a sufficient condition for uniqueness of the fundamental steady state is that the central bank lets the interest rate react more than point for point to inflation. This result is very similar to the Taylor principle, which says that under rational expectations determinacy of the equilibrium can be achieved by imposing exactly that condi-tion. Stability results of the heterogeneous expectations model are however less in line with the Taylor principle. It turns out that the fundamental steady state can be locally stable when the inflation coefficient is less than one and that it can be unstable for values larger than one.
Proposition 2 states that reacting to weakly to inflation leads to instability of the funda-mental state, but that this happens for values of the inflation coefficient strictly smaller than one. Its proof is given in Appendix B.3.
Proposition 2. When
φ1< φP F1 = 1 −
(σ + φ2)
2κ , (33)
the fundamental steady state is unstable due to a subcritical pitchfork bifurcation withλ1 = +1. Under the assumption that φ2 ≥ 0 (Assumption 2), this expression is strictly smaller than one. Reacting more than point for point to inflation therefore is not the critical boundary for local stability.
The generalized Taylor principle says that determinacy of the rational expectations equi-librium is reached when the coefficient of inflation is larger than one minus the coefficient of output gap times the long run derivative of inflation to output gap. This has similarities with the expression found in Proposition 2. However, the long run derivative of inflation to output gap is calibrated very close to zero when there is no trend inflation, and can be negative when there is trend inflation.4 In contrast, the expression in Proposition 2 does not depend on the inflation target, and will be significantly smaller than one for any reasonable calibration.
Proposition 3 says that when the CB reacts too strongly to inflation and/or output gap, the fundamental steady state is unstable. The proof of proposition 3 is given in Appendix B.3.
Proposition 3. When the CB chooses φ1 > φP D1 = 1 +
3(3σ − φ2)
2κ , (34)
or equivalently
φ2 > φP D2 = 3σ −
2κ(φ1− 1)
3 , (35)
the fundamental steady state is unstable. This is so due to a period doubling bifurcation, with λ2 = −1, that occurs at this combination of parameters. This bifurcation is super critical if φ2 < 3σ and subcritical if φ2> 3σ.
Under rational expectations the problems of reacting too strongly to inflation or output gap that follow from proposition 3, do not arise. The classical Taylor rule with φ = 1.5 and φ2 = 0.5 furthermore always leads to stability under rational expectations, while in our model this parameter setting can lead to instability of the steady state, depending on the values of σ and κ.
Figure 1 shows the bifurcation diagram of φ1 implied by Proposition 1, Proposition 2 and Proposition 3 in the case that the period doubling bifurcation is subcritical. To the left of φP F
1 and to the right of φP D
1 the fundametnal steady state is unstable; between those two values the fundamental steady state is locally stable. This is depicted by a dotted line and solid line respectively. The dotted curves between φP F
1 and 1, represent the non fundamental unstable steady states from Proposition 1, that are created in the subcritical pitchfork bifurcation. The doted lines that start at φP D
1 represent an unstable 2-cycle that is created in the subcritical period doubling bifurcation.
3.1.3 Optimal policy
Under optimal policy, φ1 is larger than one. It therefore follows from Proposition 2 that the system will not be unstable because policy reacts too weakly to inflation when the parameter values from equations (13) and (14) are chosen. It could however be that the system is unstable because the central bank reacts too strongly to inflation and output gap with these parameter values. Proposition 4 states that this is not the case.
Proposition 4. When the central bank implements optimal policy by choosing φ1 and φ2 to satisfy (13) and (14), the fundamental steady state is locally stable.
Proof. As stated above, inequality (33) trivially does not hold under optimal policy. It then follows from equation (34) that the fundamental steady state is locally stable under optimal
Figure 1: Bifurcation diagram of the inflation policy parameter
monetary policy if and only if φ1 = 1 + σκ µ + κ2 ≤ 1 + 3(3σ − σ) 2κ = 3σ κ (36) σκ2 ≤ 3σ(µ + κ2) (37) −2κ 2 3 ≤ µ (38)
This inequality always holds, since the relative weight on output gap in the loss function (µ) cannot be negative.
It is of interest to know if the bifurcations from Proposition 2 and 3 happen for reasonable policy parameters, so that small deviations from optimal policy can lead to instability. To investigate this we look at the calibrations discussed in Section 2. Column 4 and 5 of Table 1 state the values of the pitchfork bifurcation and the period doubling bifurcation for the 4 different calibrations.
From column 4 it follows that the period doubling bifurcation occurs at negative values for all calibrations. This means that under these calibrations the fundamental steady state is
Table 2: Sign of Eigenvalues φ2 φ1 λ1 λ2 φ2< σ φ1≤2σ−φκ 2 − 2 q (σ2−σφ 2) κ2 + + 2σ−φ2 κ − 2 q (σ2−σφ 2) κ2 < φ1<2σ−φκ 2 + 2 q (σ2−σφ 2) κ2 complex complex φ1≥2σ−φκ 2 + 2 q (σ2−σφ2) κ2 - -φ2= σ φ1< σκ + 0 φ1= σκ 0 0 φ1> σκ 0 -φ2> σ +
-locally stable for monetary policy that reacts weakly to inflation (0 < φ1 < 1) as long as φ2 is chosen optimally. Furthermore, even when φ2 is chosen equal to zero, the steady state is locally stable with weak inflation policy under the calibrations of Woodford (1999), Clarida et al. (2000) and McCallum and Nelson (1999).
The values in column 5 of Table 1 are all unrealistically high. This means that when φ2 is chosen optimally, reacting too strongly to inflation will not be a problem for any reasonable value of φ1. The dependence of this result on the optimality of the output gap coefficient drastically differs over the three calibrations. Under the Clarida et al. (2000) calibration φ2 = 2.5 results in instability for φ1 > 3.5. While these values do not seem to be impossible, both coefficients are then 2 points higher than the classical Taylor rule, so this is still very high. Under the Woodford (1999) calibration, an output gap parameter of 0.45 implies that the system is unstable for inflation parameters above approximately 2.31. However, when φ2 = 0.5 the steady state is unstable for φ1 > −0.81. This means that under this calibration values of the output gap parameter higher or equal to its classical Taylor rule value, the fundamental steady state is unstable for all positive values of the inflation parameter.
Under optimal policy the system converges to the steady state in a way that optimally balances inflation and output gap considerations. As the policy parameters are chosen further away from their optimal values, the system will exhibit less optimal convergence to steady state in a way that depends on the eigenvalues. Table 2 summarizes when the eigenvalues are positive, negative, zero, or complex.
gap in the loss function (µ = 0), the optimal value of φ1 reduces to σκ and all eigenvalues are zero, so that convergence happens immediately under optimal policy. When there is positive weight on output gap, one eigenvalue is positive and all other eigenvalues are equal to zero. In this case choosing φ1 below optimal results in slower convergence, while choosing this parameter above optimal implies that convergence to the steady state happens more quickly, but that output gap becomes too high or too low during this convergence. Furthermore, when φ2 is chosen above optimal, the product of the eigenvalues is negative, implying slower oscillatory convergence to the steady state.
3.2 Optimists, pessimists and fundamentalists
The second system of heuristics that is used in this thesis, is a 3-type example: optimists, pessimists and fundamentalists. Fundamentalists are defined in the same manner as in Section 3, optimists believe xt = d or πt = πT + d and pessimists believe xt = −d or πt = πT − d. The higher the constant d, the higher the magnitude of the bias of optimists and pessimists.
3.2.1 Dynamical system
The fractions of optimists are given by nz,optt and the fractions of pessimists by nz,pest , with z = x, π. The fraction of fundamentalists equal 1 − nz,optt − nz,pest . Note that again the belief fractions of output gap can be different from those of inflation.
The model is defined by xt= (1 − φ2 σ )n x,opt t d − (1 − φ2 σ )n x,pes t d − φ1− 1 σ n π,opt t d + φ1− 1 σ n π,pes t d (39) πt = πT + nπ,optt d − n π,pes t d + κxt (40) nx,optt = e −b(xt−1−d)2 e−b(xt−1−d)2 + e−b(xt−1+d)2 + e−bx2t−1 (41) nx,pest = e −b(xt−1+d)2 e−b(xt−1−d)2 + e−b(xt−1+d)2 + e−bx2t−1 (42) nπ,optt = e −b(πt−1−d−πT)2 e−b(πt−1+d−πT)2 + e−b(πt−1−d−πT)2 + e−b(πt−1−πT)2 (43)
nπ,pest = e
−b(πt−1+d−πT)2
e−b(πt−1+d−πT)2 + e−b(πt−1−d−πT)2 + e−b(πt−1−πT)2 (44) This is only a two dimensional system with state vector
xt πt
(45) A steady state with πt = πT and xt = 0 always exist. We again call this the fundamental steady states. The fractions of optimists and pessimists in this steady state are
e−bd2 2e−bd2
+ 1 =
1
2 + ebd2 (46)
When b or d go to infinity these fractions go to zero, so that everyone is fundamentalist in the fundamental steady state.
3.2.2 Stability of the fundamental steady state
As shown in Appendix C.1 the Jacobian evaluated at the fundamental steady state reduces to 4d2b 2 + ebd2 (1 − φ2 σ) − φ1−1 σ κ(1 −φ2 σ ) (1 − κ φ1−1 σ )
The characteristic equation is λ2− λ(2 −φ2 σ − κ φ1− 1 σ ) 4d2b 2 + ebd2 + 4d2b 2 + ebd2 2 (1 −φ2 σ ) (47)
And the eigenvalues are λ1 = 2d2b 2 + ebd2 (2 − φ2 σ − κ φ1− 1 σ ) + r (2 −φ2 σ − κ φ1− 1 σ ) 2− 4(1 − φ2 σ ) ! (48) λ2 = 2d2b 2 + ebd2 (2 − φ2 σ − κ φ1− 1 σ ) − r (2 −φ2 σ − κ φ1− 1 σ ) 2− 4(1 − φ2 σ ) ! (49) These eigenvalues are the same as the eigenvalues of the naive versus fundamental system, with 1
4 replaced by 2d2b
2+ebd2. In contrast with the fundamentalists versus naive case, the stability of the steady state now depends on the intensity of choice (b) and the belief bias (d). The reason for this is that in the fundamental steady state, optimists and pessimist make prediction errors, so that the steady state fractions depend on their bias and the intensity of choice.
In the fundamentalists versus naive model, both heuristics make perfect predictions in the fundamental steady state. This results in their fractions being equal to 12 independently of the intensity of choice.
2d2b
2+ebd2 is a function of of bd
2 and is plotted in Figure 1, together with 1
4. The function reaches a global maximum of
W 2 e ≈ 0.463, (50) at bd2= 1 + W 2 e ≈ 1.463, (51)
where W is the Lambert W function. For combinations of b and d around the global maximum the eigenvalues of this sections model are higher in absolute value than those of the 2-type example. Here the area of combinations of φ1 and φ2 that result in a stable fundamental steady state is smallest. When bd2 is chosen close to zero, or as bd2 gets higher, 2d2b
2+ebd2 tends to zero, so that the steady state becomes locally stable for almost all values of φ1 and φ2. As in Section 3.1, λ1 becomes +1 for high values of φ1 and φ2 and λ2 becomes −1 for low values of φ1 and φ2.
Proposition 5 states that the fundamental steady state is locally stable under optimal policy for all parameter settings.
Proposition 5. In the 3-type case with optimists, pessimists and fundamentalists, the fun-damental steady state is always locally stable under optimal monetary policy.
Proof. At the optimal policy values (equations (13) and (14)) the eigenvalues reduce to
λ1= 2d2b 2 + ebd2 (1 − κ2 µ + κ2) + s (1 − κ 2 µ + κ2)2 ! (52) λ2= 2d2b 2 + ebd2 (1 − κ2 µ + κ2) − s (1 − κ 2 µ + κ2)2 ! (53) Since µ is the relative weight on output gap in the loss function and thus either is zero or positive, λ2 reduces to zero and λ1 becomes
λ1 = 4d2b 2 + ebd2 µ µ + κ2 (54)
Figure 2: Comparison eigenvalues 2 4 6 8 10 b d 2 0.1 0.2 0.3 0.4
Both eigenvalues are thus inside the unit circle as long as λ1 ≤ 1, or 4d2b
2 + ebd2 ≤ 1 + κ2
µ (55)
The left-hand side of equation (55) has a global maximum of 2W 2
e
≈ 0.926 < 1 (56)
The right-hand side of equation (55) is always at least 1, so the inequality always holds. Therefore the steady state is always locally stable at the optimal policy values.
If φ1 and φ2 are chosen slightly different from their optimal values, the sign of the eigen-values are the same as those in Section 3. The conclusions given below Table 2 thus also apply to the 3-type case.
3.2.3 Intensity of choice to Infinity
In the previous section it was concluded that the higher the intensity of choice, the large the range of policy parameters for which the fundamental steady state is locally stable. In this section we investigate the limiting case, where b = +∞. In this special case all agents immediately switch to the heuristic that was most successful in the previous period. Since for
Table 3: Steady States
Believe xt Believe πt xt πt Existence
Fundament. Fundament. 0 πT Always
Optimistic Optimistic (1 −φ2 σ)d − φ1−1 σ d π T + d + κ((1 −φ2 σ)d − φ1−1 σ d) φ1+ φ2< 1 + σ 2 Pessimsitic Pessimistic −(1 −φ2 σ)d + φ1−1 σ d π T− d + κ(−(1 −φ2 σ)d + φ1−1 σ d) φ1+ φ2< 1 + σ 2 Optimisic Pessimistic (1 −φ2 σ)d + φ1−1 σ d π T− d + κ((1 −φ2 σ)d + φ1−1 σ d) −1 − σ( 1 2κ− 1) < −φ1+ φ2< σ 2− 1 Pessimisic Optimistic −(1 −φ2 σ)d − φ1−1 σ d π T + d + κ(−(1 −φ2 σ)d − φ1−1 σ d) −1 − σ( 1 2κ− 1) < −φ1+ φ2< σ 2− 1 Fundament. Optimistic −φ1−1 σ d π T + d − κφ1−1 σ d 1 − σ 2 < φ1< 1 + M in( σ 2, σ 2κ) Fundament. Pessimistic φ1−1 σ d π T− d + κφ1−1 σ d 1 − σ 2 < φ1< 1 + M in( σ 2, σ 2κ) Optimistic Fundament. (1 −φ2 σ)d π T + κ(1 −φ2 σ)d σ(1 − 1 2κ) < φ2< σ 2 Pessimistic Fundament. −(1 −φ2 σ)d π T− κ(1 −φ2 σ)d σ(1 − 1 2κ) < φ2< σ 2
this limiting case the eigenvalues of the fundamental steady state are zero, this steady state is locally stable. This does however not mean the system always converges to the fundamental steady state. It turns out that, for b = +∞, nine different stable steady states can coexist. This is stated in proposition 6. Its proof is given in Appendix C.2.
Proposition 6. When b = +∞ there are nine different stable steady states that each exist for some range of values of the policy parameters φ1 and φ2. The fundamental steady state is the only steady state that exists for all parameter settings.
The first four columns of Table 3 summarize all steady states that can exist when the intensity of choice goes to infinity. Column five gives the ranges of values of φ1 and φ2 for which the steady states exist. As shown in Appendix C.1 and C.2 there can also exist steady states that are not reported in Table 3, but they are always unstable.
Table 3 gives rise to the following conclusions. Steady states where everyone is either optimistic or pessimistic about both inflation and output gap only exist if the central bank does not react too strongly to both. Steady states where everyone is optimistic about either output gap or inflation and pessimistic about the other, exist when the CB responds about equally strong to output gap and inflation. Steady states where everyone has fundamentalistic beliefs about either inflation or output gap, and optimistic or pessimistic beliefs about the other, exist as long as the CB does not react too strongly to the variable where everyone is optimistic or pessimistic about.
in Table 3 when the intensity of choice is chosen large, but finite. This means that these steady states are relevant in this model and do not just arise in the, not very realistic, limiting case. An important question now is whether there exist non-fundamental steady states under optimal policy. To make the answer to this question insightful, the following assumption is made.
Assumption 3. κ < 1
Most calibrations of κ, like the ones presented in Table 1, are much lower than 1.5 The assumption therefore seems reasonable. Proposition 7 states which steady states exist under optimal policy for different weights on output gap in the loss function. The proof of the propositions is given in Appendix C.3.
Proposition 7. The fundamental steady state only is the unique stable steady state under optimal policy whenµ < κ2. Whenκ2< µ < 2κ−κ2the fundamental steady state coexists with two stable steady states where either all agents are optimistic about inflation and pessimistic about output gap, or the other way around. When µ > 2κ − κ2 the fundamental steady state coexists with two stable steady states where all agents are fundamentalistic about output gap and either all agents are optimistic or all agents are pessimistic about inflation.
As mentioned above, all steady steady states are locally stable, so when the weight on out-put gap is high enough, inflation and outout-put gap may either convergence to their fundamental values or to one of the other steady states, depending on initial conditions.
4
Simulations
In this section the results of simulations are presented. The two main heuristics of this thesis are analyzed in Section 4.1 and Section 4.3. In Section 4.2 the naive heuristic is replaced by trend extrapolators and contrarians, and in Section 4.4 the 3-type heuristic with fundamen-talists, optimists and pessimists is extended to include 12 different constant predictors.
For the simulations that follow, the Woodford calibration with κ = 0.024 and σ = 0.157 is used. However, as discussed in Section 2, we set β equal to 1 instead of 0.99. The intensity of choice is set to 10.000, as is done by de Grauwe (2010). We further set µ = 0.25, following McCallum and Nelson (2004) and Walsh (2003).
The shocks to inflation (et) and to output gap (ut), presented in equations (1) and (2) are reintroduced in this section. In all specifications below these shocks are defined as Gausian white noise with a standard deviation of 0.01.
In all simulations shocks are added to inflation and output gap, and in some subsection a shock to the communication of the central banks’s inflation target.
4.1 Fundamentalists versus naive
First consider the fundamentalist versus naive model, outlined in Section 3.1. When shocks are added to both equation (21) and (22) the following model is obtained.
xt= (1 − φ2 σ ) (1 − mx t) 2 xt−1− φ1− 1 σ (1 − mπ t) 2 (πt−1− π T) + u t (57) πt = (1 + mπ t) 2 π T +(1 − mπt) 2 πt−1+ κxt+ et, (58) with mx
t and mπt defined by equation (23) through (27).
Inflation and output gap now no longer exactly converge to a steady state, but fluctuate around it. In figure 3 time series of inflation (upper left panel) and output gap (upper right panel) are plotted, together with the differences in fractions of both inflation (lower left panel) and output gap (lower right panel). From figure 3 it can be seen that there are periods where inflation fluctuates around the target, and periods where inflation drifts away. In these periods naive predictors perform better than fundamentalists, and the difference in fractions here becomes −1. In order to stabilize rising inflation, the central bank raises the interest rate. This lowers output gap and causes more agents to switch to naive output gap expectations.
The intuition behind inflation that drifts away from its target is the following. When in-flation is above target for two consecutive periods, the fraction of naive predictors is relatively high. Agents that follow the naive heuristic furthermore expect inflation to be above target in this situation. This puts upward pressure on inflation and allows it to rise even higher. Meanwhile the central bank tries to control inflation with the interest rate and eventually succeeds to bring back inflation to its target. Note that in this model, downward drifting inflation is just as likely as upward drifting inflation.
Figure 3: Fundamentalist naive 0 200 400 600 800 1000 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 inflation t π EπT 0 200 400 600 800 1000 −0.06 −0.04 −0.02 0 0.02 0.04 Output gap t x 0 200 400 600 800 1000 −1 −0.5 0 0.5 1 Fractions Inflation t mπ 0 200 400 600 800 1000 −1 −0.5 0 0.5 1
Fractions Output gap
t
Figure 4: Fundamentalist naive with communication noise
0
200
400
600
800
1000
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
inflation
t
π EπT4.1.1 Noise in communication of the inflation target
When noise is added to the communication of the inflation target, followers of the ”trust the central bank” heuristic are no longer true fundamentalists in the sense that their expectations are no longer equal to the fundamental rational expectations steady state. Since this is solely due to a lack of information on their side, and not to a change in the way they form beliefs, we will keep calling them fundamentalists. Followers of this heuristic now believe that inflation will be equal to EπT
t = πT + ct, with ct a stochastic process that represents the noise in the CB’s communication. In what follows ct is specified as Gausian white noise with standard deviation 0.01. Output gap and inflation are now governed by
xt= (1 − φ2 σ ) (1 − mx t) 2 xt−1− φ1 σ (1 − mπ t) 2 (πt−1− π T) − φ1 σ (1 + mπ t) 2 (Eπ T t − πT) + ut (59) πt = (1 + mπ t) 2 Eπ T t + (1 − mπ t) 2 πt−1+ κxt+ et (60)
In figure 4 a similuated time series of inflation under this model specification is depicted by the blue line. The green line represents the expectation of fundamentalists EπT
t. In figure 5 a time series with the samel shock realizations in the model without noise is given. When
Figure 5: Fundamentalist naive without commmunication noise
0
200
400
600
800
1000
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
inflation
t
π EπTthe CB’s target is noisily communicated, coordination on inflation that drifts away from target occurs less frequent. This is explained by the following intuition. If, during a period of inflation above target, the noise in communication results in high fundamentalist expectations, in the subsequent period the fraction of fundamentalists will be relatively large compared to the case without communication noise. If in addition, next periods communication shock results in low fundamentalist expectations, this large fraction of fundamentalists will put downward pressure on inflation. Too large a drift of inflation is then prevented in a way that cannot occur without noise in the targets communication.
First order autocorrelation is reduced from 0.97 to 0.92 when noise to communication is added. The difference in fraction is increased from -.45 to -0.29. So clearly there are less periods where this difference is -1.
4.1.2 Weight on output gap
When more weight is put on output gap, the interest rate responds less strongly to fluctuations in inflation expectations. This leads to less fluctuations in output gap, but also less pressure on inflation to return to its target. In Figure 6 the situation with µ = 0.8 is plotted. When
the upper right panel of this figure is compared with the upper right panel of Figure 3 it is clear that output gap stays much closer to zero in periods where inflation drifts away.
Figure 6: Fundamentalists naive high µ
0 200 400 600 800 1000 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 inflation t π EπT 0 200 400 600 800 1000 −0.06 −0.04 −0.02 0 0.02 0.04 Output gap t x 0 200 400 600 800 1000 −1 −0.5 0 0.5 1 Fractions Inflation t mπ 0 200 400 600 800 1000 −1 −0.5 0 0.5 1
Fractions Output gap
t
mx
4.2 Fundamentalist versus trend followers or contrarians In this subsection the naive heuristic is replaced by the following belief.
Ezt= zt−1+ q(zt−1− zt−2) (61)
If q > 0, followers of this heuristic are called trend followers, if q < 0 they are called contrarians and if q = 0 the heuristics coincides with naive heuristic.
Output gap and inflation are now given by xt= (1 − φ2 σ ) (1 − mx t) 2 (1 + q)xt−1− (1 − φ2 σ ) (1 − mx t) 2 qxt−2− φ1 σ (1 − mπ t) 2 ((1 + q)πt−1− qπt−2− π T) + u t(62) πt = (1 + mπ t) 2 π T +(1 − mπt) 2 (1 + q)πt−1− (1 − mπ t) 2 qπt−2+ κxt+ et (63)
Figure 7: Fundamentalists weak trend followers 0 200 400 600 800 1000 −0.2 0 0.2 0.4 0.6 0.8 inflation t π EπT 0 200 400 600 800 1000 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 Output gap t x 0 200 400 600 800 1000 −1 −0.5 0 0.5 1 Fractions Inflation t mπ 0 200 400 600 800 1000 −1 −0.5 0 0.5 1
Fractions Output gap
t
mx
4.2.1 Trend followers
When inflation is above target, the trend following heuristic results in even higher inflation expectations than the naive heuristic. We would therefore expect more frequent and larger inflation drifts. In figure 7 the situation with q = 0.75 is plotted. In the upper left panel it can indeed be seen that the amplitude of the inflation drifts is much larger than in the previous section. Because of the resulting higher changes in interest rate, output gap fluctuates more as well, as can be seen in the upper right panel.
As in the fundamentalist versus naive case, noise in communication of the CB stabilizes inflation somewhat. Higher weight on output gap again stabilizes output gap, but also allows for more fluctuations in inflation.
With strong trend followers (q > 1) the central bank is unable to provide enough downward pressure on inflation to prevent it from unboundedly growing. It does however keep trying to control inflation by setting the interest rate higher and higher. As a consequence output gap
Figure 8: Fundamentalists strong trend followers 0 10 20 30 40 50 0 20 40 60 80 100 120 inflation t π EπT 0 10 20 30 40 50 −10 −8 −6 −4 −2 0 2 Output gap t x 0 10 20 30 40 50 −1 −0.5 0 0.5 1 Fractions Inflation t mπ 0 10 20 30 40 50 −1 −0.5 0 0.5 1
Fractions Output gap
t
mx
grows unboundedly negative. This situation is depicted in figure 8 , where q is set to 1.25. Here only 50 periods are plotted to give an interesting figure. Initially the fractions of both inflation and output gap fluctuate in a range where fundamentalists dominate. At some point however, all agents become trend followers with respect to inflation. At this point inflation keeps rising an output soon starts to decline. When all agents become trend followers with respect to output gap as well, both variables move increasingly fast away from their steady state value.
4.2.2 Contrarians
Contrarians lead to more stable dynamics compared to the fundamentalist versus naive case. since contrarians believe an increase of inflation will be followed by a decrease, coordination on a continued rise in inflation happens less often. The stronger the contrarians the less likely inflation drifts become. Figure 9 shows that inflation and output fluctuate fairly stable
Figure 9: Fundamentalists contrarians 0 200 400 600 800 1000 −0.05 0 0.05 0.1 0.15 inflation t π EπT 0 200 400 600 800 1000 −0.04 −0.02 0 0.02 0.04 Output gap t x 0 200 400 600 800 1000 −1 −0.5 0 0.5 1 Fractions Inflation t mπ 0 200 400 600 800 1000 −1 −0.5 0 0.5 1
Fractions Output gap
t
mx
around their steady state values when q = −1. Although occasional spikes in inflation remain, inflation is never far away from target for a large amount of consecutive periods. Fundamen-talists furthermore seem to perform better than contrarians as can be seen from the bottom two panels of figure 9.
4.3 Optimists, pessimist and fundamentalists
Now consider the 3-type example from Section 3.2. With shock to inflation and output gap the model is specified by
xt= (1 − φ2 σ )n x,opt t d − (1 − φ2 σ )n x,pes t d − φ1− 1 σ n π,opt t d + φ1− 1 σ n π,pes t d + ut (64) πt = πT + nπ,optt d − n π,pes t d + κxt+ et (65)
In Figure 10 simulated time series of inflation and output gap are plotted, together with the fractions of optimists and pessimists of both inflation and output gap. In this simulation, d is set to 3%. For lower values of the bias, fluctuations between regimes are much less clear. From the upper left panel of Figure 10 it can be seen that inflation fluctuates between three main levels for short periods of time. The lower left panel shows that in periods with high inflation, the fraction of optimists (the blue line) is high, and in periods of low inflation the fraction of pessimists (green line) dominates. There thus are periods of optimism, pessimism and fundamentalism. As shown in the upper right panel, output gap rises or falls a little for some periods, but mainly fluctuates around its fundamental value. From the lower right panel of Figure 10 it can be seen that fractions of optimists and pessimists do occasionally shoot upward, but that both stay close to zero most of the time. The fraction of output gap fundamentalists thus dominates. From this we can conclude that periods of slightly higher and lower output gap are caused by the response of the interest rate to inflation and that there are no periods of optimism and pessimism with respect to output gap.
4.3.1 Noise in communication
In the case with optimists, pessimists, and fundamentalists, noise in communicating the inflation targets means that all agents base there expectations on wrong information. When all agents think the central banks inflation target is high, optimist think inflation will be even higher. The 3-type model with noise is given by
xt= (1 − φ2 σ )n x,opt t d − (1 − φ2 σ )n x,pes t d − φ1 σ n π,opt t d + φ1 σ n π,pes t d − ct+ ut (66) πt = πT + ct+ nπ,optt d − n π,pes t d + κxt+ et (67)
Figure 11 shows inflation, output gap and their belief fractions for a simulation with communication noise.
From comparing Figure 11 with Figure 10 it is clear that the added noise results in more switching between the three inflation levels. The time series of inflation looks more chaotic, and periods of optimism, pessimism or fundamentalism are less persistent. Fractions change more as well. In the output gap dynamics, not much seems to have changed. When µ is
Figure 10: Optimists pessimists fundamentalists 0 100 200 300 400 −0.05 0 0.05 0.1 0.15 inflation t π EπT 0 100 200 300 400 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 Output gap t x 0 100 200 300 400 0 0.2 0.4 0.6 0.8 1
fractions Output gap
t nxopt nxpes 0 100 200 300 400 0 0.2 0.4 0.6 0.8 1 fractions Inflation t nπopt nπpes
Figure 11: Optimists pessimists Fundamentalists with noise in target communication 0 100 200 300 400 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 inflation t π EπT 0 100 200 300 400 −0.04 −0.02 0 0.02 0.04 Output gap t x 0 100 200 300 400 0 0.2 0.4 0.6 0.8 1
fractions Output gap
t nxopt nxpes 0 100 200 300 400 0 0.2 0.4 0.6 0.8 1 fractions Inflation t nπopt nπpes
Figure 12: Twelve constant predictors 0 20 40 60 80 100 120 140 160 180 200 −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 inflation t π EπT
increased this seems to have little impact on output gap dynamics as well. This might be due to the fact that output gap is pretty stable in all these settings.
Again autocorrelation is lower with noise in communication. Without noise first order autocorrelation is 0.88 and with noise it is 0,73.
4.4 Many types
A time series that constantly switches between fluctuations around −1%, 2% and 5% does not seem very realistic. In practice no fixed values for optimism or pessimism exist, but there are different gradations. Inflation is furthermore more likely to rise far above the target than to fall far below it. In order to produce a more realistic inflation time series that captures these two ideas, we simulate a case with twelve different types. Like Anufriev et al. (2012), we use the following constant predictors: (0.01, 0.02 .... 0.11). This comes down to biases varying from -0.01 to 0.09.
In figure 12 a simulated time series of 200 quarters of inflation in the case of twelve constant predictors is plotted.
5
Sensitivity analysis
5.1 β ≤ 1In this Section, the assumption β = 1 is not made. Instead the general case with β ≤ 1 is considered. The policy rule that is used is the two parameter rule given in equation (12). Although this rule can only be optimal when β = 1, the rule is close to optimal when β is close to 1. In this section interest lies in the robustness of the model to the β = 1 assumption and not in the optimality of the assumed policy rule .
The model is now given by xt= (1 − φ2 σ ) (1 + mx t) 2 φ1 φ2 (1 − κ φ1 φ2 (1 − β + κφ1 φ2) )πT + (1 − φ2 σ ) (1 − mx t) 2 xt−1 −φ1− 1 σ (1 + mπ t) 2 ( κφ1 φ2 (1 − β + κφ1 φ2) πT − πT) −φ1− 1 σ (1 − mπ t) 2 (πt−1− π T) (68) πt = β (1 + mπ t) 2 κφ1 φ2 (1 − β + κφ1 φ2) πT + β(1 − mπt) 2 πt−1+ κxt (69) mx t+1= tanh b 2 (xt− xt−2)2− xt− φ1 φ2 (1 − κ φ1 φ2 (1 − β + κφ1 φ2) )πT !2 = tanh( b 2At) (70) At= x2t−2− φ1 φ2 (1 − κ φ1 φ2 (1 − β + κφ1 φ2) )πT !2 + 2 φ1 φ2 (1 − κ φ1 φ2 (1 − β + κφ1 φ2) )πT − xt−2 ! xt (71) mπt+1 = tanh(b 2((πt− πt−2) 2) − (π t− κφ1 φ2 (1 − β + κφ1 φ2) πT)2) = tanh(b 2Bt) (72) Bt = π2t−2− ( κφ1 φ2 (1 − β + κφ1 φ2) πT)2+ 2( κ φ1 φ2 (1 − β + κφ1 φ2) πT − π t−2)πt (73)
As shown in Appendix B.1, the characteristic equation now is λ2− λ1 2(1 + β − φ2 σ − κ φ1− 1 σ ) + β 4(1 − φ2 σ ) = 0 (74)
And the eigenvalues are λ1 = 1 4 (1 + β −φ2 σ − κ φ1− 1 σ ) + s 1 + β −φ2 σ − κ φ1− 1 σ 2 − 4(1 −φ2 σ )β (75)
Figure 13: Inpulse Response inflation λ2 = 1 4 (1 + β −φ2 σ − κ φ1− 1 σ ) − s 1 + β −φ2 σ − κ φ1− 1 σ 2 − 4(1 −φ2 σ )β (76)
As long as β is close to 1, these eigenvalues are closely approximated by the ones given in Section 3.1. The same bifurcations thus occur, but with bifurcation values depend on an extra parameter. In Appendix B.5 the bifurcation values with β included are provided.
In Figure 13, an impulse response function of inflation is plotted. The calibration of Section 4 is used and the shock to inflation is 0.1. The red line represents the β = 1 case, while the blue line depicts the case with β = 0.99. Clearly the dynamics do not radically change when β is changed. There is a slight difference after two periods, but convergence to steady state occurs after 3 periods for both cases. The qualitative results obtained in Section 4 furthermore still hold when β is lowered to 0.99.
6
conclusion
In this thesis we use a New Keynesian model to study inflation targeting. Instead of assum-ing rational expectations, we allow expectations to be formed heterogeneously by usassum-ing two different heuristic switching models. In one model, the fundamentalist and the naive heuristic are used. In the other, fundamentalists compete with optimists and pessimists.
The main interest of this thesis lies in the stability of the fundamental steady state for different policy parameters. In this fundamental steady state inflation is equal to the target of the central bank and output gap equals zero. In addition it is of interest whether the optimal monetary policy rule derived from a loss function containing inflation and output gap, leads to a stable fundamental steady state. Our analytical analysis of the heuristic switching models shows that in both models the fundamental steady state is locally stable when optimal monetary policy is used.
When the central bank responds too weakly to inflation, the fundamental steady state is unstable and convergence to this steady state does not occur. This result is similar to the rational expectations Taylor principle, which says that the interest rate must respond more than one to one to inflation in order to have a determinate equilibrium. However, the critical value of the inflation coefficient below which the fundamental steady state is no longer stable is strictly smaller than one and for the calibrations considered in this thesis smaller than zero. In contrast with rational expectations models, reacting too strongly to inflation and output gap can results in unfavorable dynamics in our model as well.
In the Fundamentalist versus naive model, the fundamental steady state is the only steady state that can be locally stable. When the central bank responds less than point for point to inflation two unstable non fundamental steady states exist. The region of convergence to the fundamental steady state than is smaller. This is more in line with the Taylor principle. In the 3-type example with optimists, pessimist and fundamentalists, there are nine different stable steady states that can coexist in the limiting case where the intensity of choice of the heuristic switching model goes to infinity. In this limiting case all agents immediately switch to the best performing heuristic.
Under optimal policy the existence of non-fundamental steady states depends on the weight on output gap in the loss function. When the central bank is almost solely interested in targeting inflation, convergence to the fundamental steady state arises. When the central bank cares about minimizing output gap as well, two other steady states coexist with the fundamental steady state, so that the long run dynamics of the system depend on the initial conditions.
Another research question that this thesis tries to answer is what effect noise in commu-nication of the inflation target has on the dynamics of inflation and output gap. Simulations
show that when fundamentalists, who follow the inflation target as they perceive it, compete with naive predictors, noise in the target communication leads to more stable dynamics. This can be explained by the fact that with noise, a period of high inflation does not necessarily result in a high fraction of agents that use the naive heuristic, who believe inflation will remain high. Coordination on rising inflation is therefore less likely.
In the 3-type example noise in communication of the inflation target has the effect of disrupting periods of coordination on a specific heuristic as well. Without this noise there are periods where either optimists, pessimists or fundamentalists dominate. With noise in communication, these periods become shorter and inflation dynamics become more chaotic.
Simulations furthermore provide the intuitive result that putting more weight on output gap leads to less fluctuations in output gap dynamics. Inflation dynamics are however less stable when this is done.
Appendices
A
Fundamental steady state
Under rational expectations the model defined by equations (1), (2) and (11), can in steady state be rewritten as ψ2 σ xt= − ψ0 σ (βπt− π T) (A.1) πt = βπt− κ ψ0 ψ2 (βπt− πT) (A.2)
The steady state values thus are xt = ψ0 ψ2 (1 − κ ψ0 ψ2 (1 − β + βκψ0 ψ2) )πT (A.3) πt = κψ0 ψ2 (1 − β + βκψ0 ψ2) πT (A.4)
We call this steady state the fundamental steady state. The expectations of fundamentalists are equal to these values. The steady state reduces to xt= 0 and πt= πT when either β = 1 or πT = 0.
B
Fundamentalist versus Naive
B.1 Jacobian
In this subsection the general case with β ≤ 1 is considered. Results derived under Assumption 1 can be obtained by filling in β = 1. The model is given by equation (68) through (72). The Jacobian of this system equals
(1 −φ2 σ ) (1−mxt) 2 − φ1−1 σ (1−mπt) 2 0 0 f11 f12 κ(1 − φ2 σ) (1−mx t) 2 β (1−mπ t) 2 − κ φ1−1 σ (1−mπ t) 2 0 0 f21 f22 1 0 0 0 0 0 0 1 0 0 0 0 c11sA c12sA d11sA 0 e11sA e12sA c21sB c22sB 0 d22sB e21sB e22sB sA= 2bsech(2bA) sB= 2bsech(2bB) F = 1 2(1 − φ2 σ ) φ1 φ2(1 − κφ1 φ2 (1−β+κφ1 φ2) )πT − x t−1 φ1 2σ πt−1− κφ1 φ2 (1−β+˜κφ1 φ2) πT κ 2(1 − φ2 σ) φ1 φ2(1 − κφ1 φ2 (1−β+κφ1 φ2) )πT − x t−1 (κφ1 2σ − β 2) πt−1− κφ1 φ2 (1−β+κφ1 φ2) πT C = (1 −φ2 σ)(1 − m x t) φ1 φ2(1 − ˜ κφ1 φ2 (1−β+˜κφ1 φ2) )πT − x t−2 −φ1−1 σ (1 − m π t) φ1 φ2(1 − κφ1 φ2 (1−β+κφ1 φ2) )πT − x t−2 κ(1 − φ2 σ)(1 − m x t)( κφ1φ2 (1−β+κφ1φ2)π T − π t−2) (β − κφ1σ−1)(1 − mπt)( κφ1φ2 (1−β+κφ1φ2)π T − π t−2) d11= 2xt−2− (1 − φ2 σ )(1 + m x t) φ1 φ2 (1 − κ φ1 φ2 (1 − β + κφ1 φ2) )πTxt−2− (1 − φ2 σ )(1 − m x t)xt−1xt−2 +φ1− 1 σ (1 + m π t)( κφ1 φ2 (1 − β + κφ1 φ2) πT − πT)xt−2+ φ1− 1 σ (1 − m π t)(πt−1− πT)xt−2(B.1) d22= 2πt−2− 2πt With πt from equation (69) with equation (68) pluged in.
e11= (1 − φ2 σ ) φ1 φ2 (1 − κ φ1 φ2 (1 − β + ˜κφ1 φ2) )πT − xt−1 ! φ1 φ2 (1 − κ φ1 φ2 (1 − β + κφ1 φ2) )πT − xt−2 ! e12= φ1 σ πt−1− κφ1 φ2 (1 − β + κφ1 φ2) πT ! φ1 φ2 (1 − κ φ1 φ2 (1 − β + κφ1 φ2) )πT − xt−2 ! e21= κ(1 − φ2 σ ) φ1 φ2 (1 − κ φ1 φ2 (1 − β + κφ1 φ2) )πT − xt−1 ! ( κ φ1 φ2 (1 − β + κφ1 φ2) πT − πt−2) e22= ( κφ1 σ − β) πt−1− κφ1 φ2 (1 − β + κφ1 φ2) πT ! ( κ φ1 φ2 (1 − β + κφ1 φ2) πT − πt−2) In the fundamental steady state (where mx
t = mπt = 0) the Jacobian reduces to 1 2(1 − φ2 σ) − 1 2 φ1−1 σ 0 0 0 0 κ 2(1 − φ2 σ ) β 2 − κ 2 φ1−1 σ 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
The characteristic polynomial thus equals λ2− λ1 2(1 + β − φ2 σ − κ φ1− 1 σ ) + β 4(1 − φ2 σ ) = 0 (B.2) B.2 Steady states
In this Section a proof of Proposition 1 is given. Steady states are derived under the as-sumption β = 1 to simplify calculations. Equation (21) and (22) can combined to give the following equation that holds in steady state.
((1 + m π) 2 + κ φ1−1 σ (1−mπ) 2 (1 − (1 − φ2 σ ) (1−mx) 2 ) )π = ((1 + m π) 2 + κ, φ1−1 σ (1−mπ) 2 (1 − (1 − φ2 σ ) (1−mx) 2 ) )πT with mx= tanh(−b 2x 2) (B.3)
mπ = tanh(−b 2(π − π
T)2) (B.4)
This has as solution the fundamental steady state: x∗ = 0, π∗ = πT, mx∗ = 0, mπ∗ = 0. Other steady states might exist as solutions of
(1 + mπ) 2 + κ φ1−1 σ (1−mπ) 2 (1 − (1 − φ2 σ ) (1−mx) 2 ) = 0 (B.5) mπ(−1 + κ φ1−1 σ (1 − (1 − φ2 σ ) (1−mx) 2 ) ) = 1 + κ φ1−1 σ (1 − (1 − φ2 σ) (1−mx) 2 ) (B.6) mπ = (1 − (1 − φ2 σ ) (1−mx) 2 ) + κ φ1−1 σ −(1 − (1 − φ2 σ) (1−mx) 2 ) + κ φ1−1 σ (B.7) The steady state values of π then are
π∗ = πT ± v u u t− 2 btanh −1 (1 − (1 − φ2 σ ) (1−mx) 2 ) + κ φ1−1 σ −(1 − (1 −φ2 σ ) (1−mx) 2 ) + κ φ1−1 σ ! (B.8) When non-fundamental steady states exist, there thus are two steady states, symmetric around the fundamental value of πT.
Because in a non-fundamental steady states naive predictors perform better than funda-mentalists, non-fundamental steady states can only exist if
−1 ≤ mπ < 0 (B.9)
Since it is assumed that both σ and φ1 are non-negative we must have (1 − (1 − φ2
σ )
(1 − mx)
2 ) > 0 (B.10)
Using this and equation (B.7), the inequalities in (B.9) reduce to (1 − (1 − φ2 σ ) (1 − mx) 2 ) − κ φ1− 1 σ ≥ (1 − (1 − φ2 σ ) (1 − mx) 2 ) + κ φ1− 1 σ > 0, (B.11) or equivalently 1 ≥ φ1 > 1 − σ κ 1 − (1 −φ2 σ ) (1 − mx) 2 (B.12) From the equivalence of inequalities (B.9) and (B.12), it can be concluded that as φ1 gets close to its right-hand limit, mπ gets close to zero. This implies that mx , x and π also go
to their fundamental values as this happens. Using that mx goes to zero in the limit, we see from equation (B.12) that the limiting value of φ1 for which the non-fundamental steady state exist is φ1> 1 − σ κ(1 − (1 − φ2 σ ) 1 2) = 1 − φ2+ σ 2κ (B.13)
At this point both steady states coincide with the fundamental steady state. Equations (B.12) and (B.13) complete the proof of proposition 1.
B.3 Proof Propositon 2
This Section provides a proof of Proposition 2. First it is established that a subcritical pitchfork bifurcation occurs for small values of φ1, than it is concluded that this does not happen at φ1 = 1, as the Taylor principle implies.
From equation (B.23) we know that λ1 > 1 when 1 4 (2 −φ2 σ − κ φ1− 1 σ ) + s 2 −φ2 σ − κ φ1− 1 σ 2 − 4(1 − φ2 σ ) > 1 (B.14) s 2 −φ2 σ − κ φ1− 1 σ 2 − 4(1 −φ2 σ ) > 4 − (2 − φ2 σ − κ φ1− 1 σ ) (B.15) 2 −φ2 σ − κ φ1− 1 σ 2 − 4(1 − φ2 σ ) > 16 + (2 − φ2 σ − κ φ1− 1 σ ) 2− 8(2 − φ2 σ − κ φ1− 1 σ (B.16)) −4 + 4φ2 σ > 8 φ2 σ + 8κ φ1− 1 σ (B.17) −1 −φ2 σ > 2κ φ1− 1 σ (B.18) φ1 < 1 − φ2+ σ 2κ (B.19)
From Section B.4 we know that at the right hand side of equation (B.19), two non-fundamental steady states are created. Below this value the non-fundamental steady state is unique and unstable; above this value the fundamental steady state is unstable, and two other
steady states exist. This implies a subcritical pitchfork bifurcation. This in turn implies that the non-fundamental steady states are unstable.
Under Assumption 2 the right hand side of equation (B.19), is strictly smaller than one. Therefore, the fundamental steady state is always stable when φ1 is slightly below one, and thus when the Central bank reacts slightly less than point for point to inflation. This com-pletes the proof of Proposition 2.
B.4 2-cycle
In this section the proof op Proposition 3 is presented. λ2 is smaller than −1 when
1 4 (2 −φ2 σ − κ φ1− 1 σ ) − s 2 −φ2 σ − κ φ1− 1 σ 2 − 4(1 − φ2 σ ) < −1 (B.20) Rewriting this as s 2 −φ2 σ − κ φ1− 1 σ 2 − 4(1 −φ2 σ ) < 4 + (2 − φ2 σ − κ φ1− 1 σ ) (B.21) 2 −φ2 σ − κ φ1− 1 σ 2 − 4(1 − φ2 σ ) < 16 + (2 − φ2 σ − κ φ1− 1 σ ) 2+ 8(2 − φ2 σ − κ φ1− 1 σ ),(B.22) the condition reduces to
−3(1 −φ2
σ ) < 6 − 2κ φ1− 1
σ (B.23)
This inequality can easily be written in terms of φ1 or φ2, which results in the expressions given in the proposition.
When one eigenvalue becomes −1, a 2-cycle must exists either below or above the bifur-cation value. This makes the period doubling bifurbifur-cation either subcritical or supercritical. In what follows φ1 is treated as the bifurcation parameter. The value of φ2 then turns out to determine if the bifurcation is subcritical or supercritical.
The 2-cycle in question is symmetric around the fundamental steady state. We thus have x1= −x2 and (π1− πT) = −(π2− πT). Using this, equations (21) and (22) can be written as
x = x1 = (1 − φ2 σ ) (1 − mx) 2 x2− φ1− 1 σ (1 − mπ) 2 (π2− π T) (B.24)
(1 + (1 − φ2 σ ) (1 − mx) 2 )x = − φ1− 1 σ (1 − mπ) 2 (π2− π T) (B.25) π = π1 = (1 + mπ) 2 π T +(1 − mπ) 2 π2− κ φ1−1 σ (1−mπ) 2 (π2− π T) (1 + (1 − φ2 σ ) (1−mx) 2 ) (B.26) π = (π2− πT)( (1 − mπ) 2 − κ φ1−1 σ (1−mπ) 2 (1 + (1 − φ2 σ) (1−mx) 2 ) ) + πT (B.27) (π − πT)(1 +(1 − m π) 2 − κ φ1−1 σ (1−mπ) 2 (1 + (1 − φ2 σ) (1−mx) 2 ) ) = 0 (B.28) with mx= tanh(b 2x 2 1) = tanh(− b 2x 2 2) (B.29) mπ = tanh(b 2(π1− π T)2) = tanh(−b 2(π2− π T)2) (B.30)
So a 2-cycle must satisfy
(1 +(1 − m π) 2 − κ φ1−1 σ (1−mπ) 2 (1 + (1 − φ2 σ) (1−mx) 2 ) ) = 0 (B.31) 3 − κ φ1−1 σ (1 + (1 − φ2 σ ) (1−mx) 2 ) = mπ(1 − κ φ1−1 σ (1 + (1 − φ2 σ ) (1−mx) 2 ) ) (B.32) mπ = 3(1 + (1 − φ2 σ) (1−mx) 2 ) − κ φ1−1 σ (1 + (1 − φ2 σ) (1−mx) 2 ) − κ φ1−1 σ (B.33) Since in a two cycle around the fundamental steady state, fundamentalists perform better than naive predictors, a 2-cycle can only exist if
0 < mπt ≤ 1 (B.34)
In this case we must either have that numerator and denominator of equations (B.33) are negative with φ1 > 1 and
(1 + (1 −φ2 σ )
(1 − mx t)
