Appendix to Transparency, Expectations Anchoring and Inflation Target Guido Ascari University of Oxford and University of Pavia
Anna Florio Politecnico di Milano
Alessandro Gobbi Universit`a Cattolica del Sacro Cuore
July 2016
The appendix contains three sections. In Section A we outline the structure of New Keynesian model with positive trend inflation adopted in the main article. In Section B we discuss the learning algorithm that agents use to form expectations, with focus on E-stability and speed of convergence. In Section C we provide analytical results about E-stability and speed of convergence for a simplified version of the model that can be derived under the assumption of zero trend inflation.
A
The Model
We here sketch the model by Ascari and Ropele (2009). Households.
Households live forever and their expected lifetime utility is: E0
∞ X
βt
t=0
N 1+σn log Ct − χ t 1 + σn
! ,
(A.1)
where β ∈ (0, 1) is the subjective rate of time preference, E0 is the expectation operator conditional on time t = 0 information, C is consumption, N is labour, χ and σn are parameters. The period budget constraint is given by: Pt Ct + Bt ≤ Pt wt Nt + (1 + it−1 ) Bt−1 + Ft
(A.2)
where Pt is the price of the final good, Bt represents holding of bonds offering a one-period nominal return it , wt is the real wage, and Ft are firms’ profits rebated to households. The households maximize (A.1) subject to the sequence of budget constraints (A.2), yielding the following first order conditions: wt = χNtσn Ct , 1 1 (1 + it ) Pt = βEt . Ct Ct+1 Pt+1
1
(A.3) (A.4)
Final Good Producers.
In each period, a final good Yt is produced by perfectly compet-
itive firms, using a continuum of intermediate inputs Yi,t indexed by i ∈ [0, 1] and a standard hR iθ/(θ−1) 1 (θ−1)/θ CES production function Yt = 0 Yi,t di , with θ > 1. The final good producer demand schedule for intermediate good quantities is Yi,t = (Pi,t /Pt )−θ Yt . The aggregate price hR i1/(1−θ) 1 1−θ index is Pt = 0 Pi,t . di Intermediate Goods Producers. Yi,t are produced by a continuum of firms indexed by i ∈ [0, 1] . The production function for intermediate input firms is: Yi,t = Ni,t . Intermediate goods producers sets prices according to the usual Calvo mechanism. In each period there is a ∗ . With probability fixed probability 1 − α that a firm can re-optimize its nominal price, i.e., Pi,t
α, instead, the firm keeps its nominal price unchanged. The first order condition of the problem is: ∗ Pi,t
Pt
=
θ θ−1
Et
P∞
j j=0 (αβ)
Et
P∞
j j=0 (αβ)
Ct Ct+j
Ct Ct+j
Pt Pt+j
−θ
Yt+j wt+j . 1−θ Pt Y t+j Pt+j
(A.5)
The aggregate price level evolves as h i1/(1−θ) ∗ 1−θ Pt = α (Pt−1 )1−θ + (1 − α) Pi,t .
(A.6)
As shown by Ascari and Ropele (2009), we will assume that for given parameter values of 0 6 α < 1, 0 < β < 1, and θ > 1, the positive level of trend inflation fulfils the restrictions: 1
1
16π ¯ < α 1−θ and 1 6 π ¯ < (αβ)− θ , to ensure existence of the steady state of the model. Relative price dispersion. At the level of intermediate firms, it holds true that (Pi,t /Pt )−θ Yt = Ni,t . R1 Aggregating this expression for all the firms i yields Yt st = Nt , where st ≡ 0 (Pi,t /Pt )−θ di R1 and Nt ≡ 0 Ni,t di. The variable st measures the relative price dispersion across intermediate firms. st is bounded below at one and it represents the resource costs (or inefficiency loss) due to relative price dispersion under the Calvo mechanism. st evolves as st = (1 − α)
∗ Pi,t Pt
−θ
+ αΠθt st−1 ,
(A.7)
where Πt = Pt /Pt−1 denotes the gross inflation rate. The variable st directly affects the real wage via the labour supply equation (A.3): wt = χYtσn sσt n Ct . Market clearing conditions. The market clearing conditions in the goods and labour R s = Y D = (P /P )−θ Y , ∀i; and N = 1 N di. markets are: Yt = Ct ; Yi,t i,t t t t i,t i,t 0 The model is closed with a Taylor rule that describes the central bank’s behaviour. Loglinearization leads to the five equations (1)-(5) in the main text (for all details, refer to Ascari
2
and Ropele, 2009).
B
Learning and speed of convergence
B.1
E-stability
The following discussion closely follows Evans and Honkapohja (2001, Chapter 10). Our model can be written in matrix form as: ∗ ∗ yt = β0 Et−1 yt + β1 Et−1 yt+1 + β2 yt−1 + kwt ,
(A.8)
wt = ϕwt−1 + et , where yt is the vector of endogenous variables, wt is the vector of (stationary) exogenous processes, and et is a white noise. Note that yt will depend on the assumptions regarding the size of trend inflation and the transparency (vs. opacity) of central bank communications. For instance, in case of positive trend inflation and opacity, yt = [ˆıt , sˆt , π ˆt , yˆt , φˆt ]0 . The minimal state variable (MSV) solution of the system is: ¯ t−1 + c ¯wt−1 + ket , yt = by
(A.9)
¯ and c ¯ are convolutions of the structural parameters. When the conditions where the matrices b for determinacy hold, there exists a unique stationary MSV solution. Under learning the agents are assumed to believe that the economy evolves according to a perceived law of motion (PLM) that has the same structure of the MSV solution: yt = byt−1 + cwt−1 + εt ,
(A.10)
where εt is the zero-mean innovation in yt relative to the information set available at time t − 1. The parameters ζ = [b, c]0 are updated through recursive least squares once variables yt have been determined: 0 0 ζt = ζt−1 + (1/t) Rt−1 zt−1 yt − ζt−1 zt−1 , 0 Rt = Rt−1 + (1/t) zt−1 zt−1 − Rt−1 ,
(A.11)
with zt = [yt0 , wt0 ]0 . By projecting forward the PLM and plugging it into the model (A.9), it is possible to obtain the actual law of motion (ALM) of the economy and the associated T -map: yt = Tb yt−1 + Tc wt−1 + ket , T (b, c) = (Tb , Tc ) = β1 b2 + β0 b + β2 , β0 c + β1 bc + β1 cϕ + kϕ . To study whether the learning algorithm converges to the rational expectation equilibrium, from the T -map it is convenient to derive the matrix differential equation d (b, c) = T (b, c) − (b, c) . dτ 3
(A.12)
Evans and Honkapohja (2001, Proposition 10.1, p. 232) provide the conditions for E-stability which correspond to the local asymptotic stability of the differential equation around the fixed ¯ c ¯ and ¯). In particular, a MSV solution is E-stable if all eigenvalues of matrices DTb (b) point (b, ¯ have real part less than 1, where DTc (b) ¯ =b ¯ 0 ⊗ β1 + I ⊗ β0 + β1 b ¯ , DTb b ¯ = ϕ0 ⊗ β1 + I ⊗ β0 + β1 b ¯ , DTc b
(A.13) (A.14)
are the Jacobian matrices of the T -map evaluated at the equilibrium. When the E-stability condition is satisfied, expectations under learning converge to rational expectations and the economy (A.8) behaves asymptotically according to the law of motion (A.9).
B.2
Speed of Convergence
Besides E-stability, the differential equation (A.12) allows to assess the speed at which the convergence of the learning algorithm takes place. The main theoretical result is represented by Benveniste et al. (1990, Theorem 3, p. 110) and is again related to the magnitude of the ¯ and DTc (b) ¯ are Jacobian of the T -map at the fixed point: when all eigenvalues of DTb (b) smaller than 0.5 in real part, the recursive least squares estimates converge to their limit point at speed root-t, i.e. D ¯ → t1/2 vec(ζt ) − vec(ζ) N (0, P ) , for some positive definite covariance matrix P . Therefore, root-t convergence is feasible only in a subset of the region where E-stability holds. As discussed in Marcet and Sargent (1995), when the largest eigenvalue of the Jacobian is larger than 0.5 but still smaller than 1, the recursive algorithm is able to converge but the effect of the initial condition is not washed out at an exponential rate, so that the speed of convergence is expected to be slower than root-t. However, the sole way to explore the rate of convergence in this parametric region is by means of numerical simulations of the ALM and the learning algorithm. In the paper we implement the procedure suggested by Marcet and Sargent (1995). Assume there is an unknown rate of convergence δ for which D
tδ xt → F, ¯ for some non-degenerate, well-defined distribution F with mean where xt ≡ vec(ζt ) − vec(ζ), zero and finite variance. The rate δ can be estimated by noting that
2δ
E t xt x0t
h i
→ 1, as t → ∞,
E (tk)2δ xtk x0 tk
which can be used to obtain " # 1 kE (xt x0t )k δ= log
E xtk x0 . 2 log k tk 4
To make last expression operational, expected values can be approximated by simulating a large number of independent realizations of length t and tk and computing the norm of the mean squared error across realizations. In Table A.1 we report additional estimates of the rate of convergence, which extend Table 2 in the main text.
C
Analytical results under zero trend inflation
We present some analytical results regarding E-stability and speed of convergence for the case of zero trend inflation. This is useful because it allows us to develop intuition for the more complex case of positive trend inflation. The analysis is similar in spirit to Bullard and Mitra (2002), but we add the discussion on the speed of convergence and the distinction between transparency and opacity. With zero trend inflation π ¯ is equal to one so that ηπ¯ = ξπ¯ = 0. In addition, the price dispersion is negligible around the steady state and the model reduces to a standard threeequation New Keynesian framework - equations (1), (2) and (6) in the main text. In matrix form, equation (A.8) rewrites as ∗ ∗ ˆıt 0 φπ φy 0 0 0 Et−1ˆıt+1 1 0 0 um,t Et−1ˆıt ∗ ∗ ˆt = 0 0 κ Et−1 π ˆt+1 + 0 κ 0 us,t , π ˆt + 0 β 0 Et−1 π ∗ y ∗ y yˆt −1 0 0 ˆt 0 1 1 Et−1 ˆt+1 0 0 1 ud,t Et−1 um,t ρm 0 0 um,t−1 em,t us,t = 0 ρs 0 us,t−1 + es,t . ud,t 0 0 ρd ud,t−1 ed,t The model is purely forward-looking, so the MSV solution will only contain the exogenous processes. Formally, equation (A.9) simplifies to ¯wt−1 + ket yt = c and the PLM will take the same form.
C.1
Determinacy
Using the solution methodology outlined in Evans and Honkapohja (2001, p. 252) we can find the conditions for determinacy in a model with lagged expectations. Under rational expectations, determinacy requires that the eigenvalues of matrix −1 1 −φπ −φy 0 0 0 1 −κ 0 β 0 0 1 0 1 0 1 1
5
lie inside the unit circle. It is immediate to see that one eigenvalue is equal to zero. As discussed in LaSalle (1986, p. 28), the remaining eigenvalues are inside the unit circle if and only if the next two conditions are met: 1−β φy > 1, κ φy + κφπ > β − 1,
φπ +
(D1) (D2)
provided that φy + κφπ > −1. The last inequalities correspond to equations (7) and (8) in the main text.
C.2
E-stability
¯ = 0) The fact that the MSV solution does not depend on lagged endogenous variables (i.e., b substantially facilitates the investigation of the stability properties of the model under learning, as we can focus on the single Jacobian matrix DTc = ϕ0 ⊗ β1 + I ⊗ β0 .
(A.15)
The following proposition shows that transparency allows a central bank to better anchor inflation expectations with respect to opacity.1 Proposition 1. The rational expectations equilibrium (REE) of the model is E-stable when the following conditions are satisfied. (i) Under transparency iff φπ + φy
1 − βρ (1 − ρ) (1 − βρ) >ρ− , κ κ φy > ρ (1 + β) − 2.
(TR1) (TR2)
(ii) Under opacity iff (TR1) holds and φy >
1 (Ψ(β, κ, ρ) + κφπ ) , (2 − ρ)
(OP)
where Ψ(β, κ, ρ) = [ρ (1 + β) − 2] [(2 − βρ) (2 − ρ) − κρ] . Proof. Under transparency, agents use their PLM only to predict output and inflation. In fact, the central bank reveals the parameters of the Taylor rule and agents can form policy-consistent forecasts of the interest rate, given their predictions of output and inflation. In practice, we can 1
Our conditions are slightly different from the ones in Bullard and Mitra (2002) because we assume lagged expectations and omit the constant term in the PLM. Lagged expectations imply, in turn, that our conditions involve the persistence parameter of the exogenous processes. In what follows, we will have conditions that hold for every exogenous process, so we should write ρi for i = {d, m, s}. For the sake of brevity we just write ρ without subindex.
6
directly substitute the Taylor rule inside the Euler equation and obtain the system " # π ˆt yˆt
" =
0 −φπ
# um,t " # um,t−1 # " #" # " #" ∗ ∗ 0 κ 0 0 0 0 ˆt+1 β 0 Et−1 π Et−1 π ˆt κ + + us,t + us,t−1 . ∗ ∗ 0 0 1 −ρ 0 0 1 1 Et−1 yˆt+1 −φy Et−1 yˆt ud,t ud,t−1
Note that the Jacobian matrix of the T -map will still be given by (A.15), and will be a block diagonal matrix whose blocks are identical and given by "
ρβ
#
κ
ρ − φπ ρ − φy
.
To check that all eigenvalues have real part lower than z, we apply the Routh-Hurwitz criterion to the characteristic polynomial of matrix "
ρβ − z
#
κ
,
ρ − φπ ρ − φy − z
(A.16)
that is λ2 + a1 λ + a2 = 0 with a1 = 2z − ρ + φy − βρ, a2 = (z − βρ) (z − ρ + φy ) − κ (ρ − φπ ) . The Routh-Hurwitz criterion ensures that all eigenvalues have real parts below z if and only if a1 , a2 > 0. E-stability requires that all eigenvalues are less than 1 in real part. Therefore, we rewrite the Routh-Hurwitz conditions for z = 1 and obtain φπ + φy
1 − βρ (1 − ρ) (1 − βρ) >ρ− , κ κ φy > ρ (1 + β) − 2.
Under opacity, the Jacobian matrix (A.15) is a block diagonal matrix with identical blocks
0
φπ φy
0 βρ −1 ρ
κ . ρ
Again, to check that all the eigenvalues are below a certain threshold z we apply the RouthHurwitz criterion to the characteristic polynomial of matrix −z φπ 0 βρ − z −1
ρ
7
φy
κ . ρ−z
(A.17)
The characteristic polynomial is λ3 + a1 λ2 + a2 λ + a3 = 0, with a1 = 3z − ρ − βρ, a2 = φy + z (z − βρ) − κρ − (βρ − 2z) (z − ρ) , a3 = κφπ − (φy − κρ) (βρ − 2z) − zφy + z (z − ρ) (z − βρ) + κρ (z − βρ) . The Routh-Hurwitz criterion ensures that all eigenvalues have real parts below z if and only if a1 > 0, a3 > 0, a1 a2 > a3 . The necessary and sufficient conditions for E-stability can be obtained by setting z = 1. After some algebra, we get 3 − βρ − ρ > 0, φπ + φy
(OP1)
1 − βρ (1 − ρ)(1 − βρ) >ρ− , κ κ 1 φy > (Ψ(β, κ, ρ) + κφπ ) . (2 − ρ)
(OP2) (OP3)
Note that (OP1) is always true, (OP2) coincides with (TR1), and (OP3) is relabelled as (OP).
Figure A.1 illustrates how the two determinacy conditions (D1)-(D2) and the three Estability conditions define the relevant regions for determinacy and E-stability under transparency and opacity in the space (φπ , φy ). The figure corresponds to the left upper panel of Figure 1 in the main text. To grasp the main results, it is instructive to confine the analysis to the positive quadrant for (φπ , φy ). To facilitate the reader, we highlight the main implications of the propositions by using a numbered list of corollaries. Corollary 1. Assume φπ , φy ≥ 0. If the REE is determinate (i.e., (D1) holds), then it is always E-stable under transparency, while this is not true under opacity. This follows immediately from the fact that for φy ≥ 0 the long-run Taylor principle (D1) implies (TR1) and (TR2) is always satisfied. Note that, if the determinacy conditions hold, then the equilibrium is E-stable under transparency, but the contrary is not true, in contrast to Bullard and Mitra (2002): E-stability does not imply determinacy.2 Moreover, determinacy does not imply E-stability under opacity. The learnable region of the parameter space is thus smaller when the central bank is opaque. This retraces a similar result in Preston (2006) that shows that under opacity the Taylor principle is not sufficient for E-stability in the infinite horizon approach. We have shown that the standard Euler-equation approach delivers the same message: under opacity the E-stability conditions are more stringent than the determinacy conditions. Corollary 2. Assume φπ , φy ≥ 0. If the REE is determinate (i.e., (D1) holds), then it is E-stable under opacity iff φy >
1 (2−ρ)
(Ψ(β, κ, ρ) + κφπ ) .
As Eusepi and Preston (2010) in the infinite horizon case, we were able to obtain an analytical expression for the E-stability condition under opacity. Condition (OP) implies a lower bound 2
This is because we do not use the constant term in the PLM.
8
D1
φy >
φY
κ (1 − φπ ) 1− β OP
φy > Indeterminacy and E-stability under TR and OP
1 (Ψ (β, ρ ) + κφπ ) 2−ρ
Determinacy and E-stability under TR e OP
TR1 1 - βρ 1 - βρ φπ + φy > ρ − (1 − ρ ) κ κ
Determinacy and E-stability under TR E-unstable under OP 1
φΠ TR2 φy > ρ + ρβ − 2
β −1
D2 φy > β − 1 − κφπ
Figure A.1: Determinacy and E-stability regions under transparency and opacity from Proposition 1. on φy that increases with φπ . When the central bank is opaque, an aggressive response to inflation can destabilize expectations, unless it is counteracted by an increase in the response to output. Moreover, condition (OP) implies that transparency is an essential part of the inflation targeting framework, because an interest rate rule that responds only to inflation would lead either to E-instability or to indeterminacy under opacity, depending on whether the Taylor principle is satisfied or not. Finally, note there is a region of the policy parameter space (φπ , φy ) where despite the Taylor principle not being satisfied, and thus the REE being indeterminate, the latter is E-stable. In general: Corollary 3. The Taylor principle is not a necessary condition for E-stability neither under transparency nor under opacity. To see that κ is the key parameter that affects the E-stability conditions, assume β = ρ = 1.3 Then, the determinacy and E-stability conditions under transparency collapse to the standard Taylor principle, because (D1) and (TR1) become φπ > 1, while (D2) and (TR2) are always satisfied in the positive quadrant. The additional E-stability condition under opacity (OP) simply reduces to φy > κφπ , and κ determines the lower bound of the relative weight φy /φπ necessary to ensure E-stability. As evident from Figure A.2, it follows that the higher the slope of the Phillips curve, the greater the difference between the E-stability regions under transparency and opacity. Hence: 3
This is just for the sake of exposition without loss of generality. The same argument applies for β, ρ ∈ (0, 1).
9
Corollary 4. The higher the slope of the Phillips curve (κ), the smaller the E-stability region under opacity.
D1 and TR1
φπ >1
φy Indeterminate and E-unstable under TR and OP
OP
Determinate and E-stable under TR e OP
φy > κφπ
Determinate and E-stable under TR E-unstable under OP
1
φπ
Figure A.2: The importance of the slope of the Phillips curve, κ.
C.3
Speed of convergence under transparency
The following proposition characterises the slope of T -map under transparency. Proposition 2. Under zero trend inflation and transparency, the largest eigenvalue of the Jacobian matrix of the T -map is smaller than z in real part iff φπ + φy
(z − ρ) (z − βρ) z − βρ >ρ− , κ κ φy > ρ (1 + β) − 2z.
(A.18) (A.19)
Proof. The conditions can be found by simply evaluating the Routh-Hurwitz conditions in matrix (A.17) for a generic threshold z. For a given z, these conditions correspond to the two lines that form the iso-eigenvalue curves in the plane (φπ , φy ), which we showed in the upper-left panel of Figure 2a in the main text. The first line represents the upper frontier that rotates clockwise as z decreases. Condition (A.19) represents a horizontal line that shifts upwards as z decreases. 10
Corollary 5 (Root-t convergence under transparency). Under zero trend inflation and transparency, the conditions for root-t convergence can be obtained from (A.18) and (A.19) by setting z = 0.5. Namely, φπ + φy
1/2 − βρ (1/2 − ρ) (1/2 − βρ) >ρ− , κ κ φy > ρ (1 + β) − 1.
(A.20) (A.21)
Condition (A.21) analytically supports the idea that, for a common calibration such that ρ(1 + β) > 1, root-t convergence is infeasible when the central bank responds only to inflation (φy = 0). Corollary 6. Assume φy , φπ > 0. If ρ = 0, the real part of the largest eigenvalue is always negative, so there is root-t convergence. More generally, assuming β ≈ 1, there exists a value ρ∗ such that root-t convergence obtains for ρ < ρ∗ . The last corollary follows from the two conditions above and highlights the role of ρ. The persistence of the shock process affects the speed of convergence to the REE: the higher is the persistence of the exogenous processes, the slower is generally the convergence. Intuitively, ρ increases the unconditional variance of the exogenous processes, so that agents find more difficult to learn the systematic component of the dynamics of the economy. Note that there is always a sufficiently low value of ρ that guarantees root-t convergence for any possible parameter combination in the determinacy and E-stability region. The following proposition demonstrates how the policy parameters and the degree of price stickiness affect the slope of the T -map, thus influencing the speed of convergence. Proposition 3 (Transparency, convergence and policy design). Let λ be the largest (in real part) eigenvalue of the Jacobian matrix of the T -map. Under zero trend inflation and transparency, λ is real iff ∆ = [ρ(β − 1) + φy ]2 + 4κ(ρ − φπ ) > 0, and complex otherwise. Moreover, the following points hold. (i) Effect of φπ : • if λ is real, it is strictly decreasing with φπ ; • if λ is complex, its real part does not depend on φπ . (ii) Effect of φy : • if λ is real, ∂λ ≶ 0 ⇐⇒ φπ ≶ ρ; ∂φy • if λ is complex, its real part is strictly decreasing with φy . (iii) Effect of κ: • if λ is real, ∂λ ≶ 0 ⇐⇒ φπ ≷ ρ; ∂κ 11
• if λ is complex, its real part does not depend on κ. Proof. Under transparency, the slope of the T -map corresponds to the largest real part of the eigenvalues of matrix (A.16) evaluated at z = 0. The largest eigenvalue is therefore
λ=
ρ(1 + β) − φy +
q [ρ(β − 1) + φy ]2 + 4κ(ρ − φπ ) 2
=
ρ(1 + β) − φy + ∆1/2 . 2
To evaluate the effects of policies, first notice that ∆ = ∆(φπ , φy , κ). (i) Effect of φπ : If the maximum eigenvalue is real (∆ ≥ 0) then largest real part is given by λ itself. It follows that increasing φπ leads to a decrease in λ and, in turn, to a faster rate of convergence. In fact, ∂λ κ = − 1/2 < 0. ∂φπ ∆ If the eigenvalues are complex conjugates (∆ < 0), the real part equals [ρ(1 + β) − φy ]/2, which does not depend on φπ . (ii) Effect of φy : If the maximum eigenvalue is real, then ρ(β − 1) + φy ∂λ 1 = − 1/2 ∂φy 2 2∆ 1 1 − = r 2 4κ(ρ−φπ ) 2 1 + [ρ(β−1)+φ 2 ] y
Hence
In other words, if φπ > ρ then
∂λ ≶ 0 ⇐⇒ φπ ≶ ρ. ∂φy ∂λ ∂φy
> 0, and vice versa. It follows that a higher φy increases
the maximum eigenvalue and then decreases the speed of convergence, if φπ > ρ (and vice versa). If the eigenvalues are complex conjugates, the real part equals [ρ(1 + β) − φy ]/2 which is decreasing in φy . (iii) Effect of κ: If the maximum eigenvalue is real, then ρ − φπ ∂λ = . ∂κ ∆1/2 Hence
∂λ ≶ 0 ⇐⇒ φπ ≷ ρ. ∂k
12
If the eigenvalues are complex conjugates, the real part equals [ρ(1 + β) − φy ]/2, which does not depend on κ.
Before hitting the locus ∆ = 0, a higher response of monetary policy to inflation (i.e., higher φπ ) decreases the largest eigenvalue, and hence tends to increase the speed of convergence. When it hits the locus ∆ = 0, further increases in φπ have no effects on the real part of the eigenvalue (point (i) in Proposition 3). Therefore: Corollary 7. Under transparency the speed of convergence is non-decreasing in φπ . The effect of a higher response to output (higher φy ) is instead ambiguous: a higher φy decreases the real part of the eigenvalue below the locus ∆ = 0, because in our calibration φπ > ρ, while it increases the real part of the eigenvalue above the locus ∆ = 0 (point (ii) in Proposition 3). To be on the locus ∆ = 0 and increase the speed of convergence, policy has to move towards the centre of the determinacy/E-stability region, thus it has to increase both φπ and φy both in the case of opacity and transparency. Hence, responding to the output gap is important to increase the speed of convergence.
References Ascari, G. and T. Ropele (2009): “Trend Inflation, Taylor Principle and Indeterminacy,” Journal of Money, Credit and Banking, 41, 1557–1584. ´tivier (1990): Adaptive Algorithms and StochasBenveniste, A., P. Priouret, and M. Me tic Approximations, Berlin: Springer-Verlag. Bullard, J. and K. Mitra (2002): “Learning about monetary policy rules,” Journal of Monetary Economics, 49, 1105–1129. Eusepi, S. and B. Preston (2010): “Central Bank Communication and Expectations Stabilization,” American Economic Journal: Macroeconomics, 2, 235–71. Evans, G. W. and S. Honkapohja (2001): Learning and Expectations in Macroeconomics, Princeton, New Jersey: Princeton University Press. LaSalle, J. P. (1986): The stability and control of discrete processes, New York, NY: SpringerVerlag New York, Inc. Marcet, A. and T. J. Sargent (1995): “Speed of Convergence of Recursive Least Squares: Learning with Autoregressive Moving-Average Perceptions,” in Learning and Rationality in Economics, ed. by A. Kirman and M. Salmon, Oxford, UK: Basil Blackwell, 179–215. Preston, B. (2006): “Adaptive learning, forecast-based instrument rules and monetary policy,” Journal of Monetary Economics, 53, 507–535.
13
Table A.1. Simulated speed of convergence Trend inflation = 0%, transparency
φy
2.50 2.25 2.00 1.75 1.50 1.25 1.00 0.75 0.50 0.25 0.00
1.00
1.25
1.50
1.75
2.00
2.25
φπ 2.50
2.75
3.00
3.25
3.50
3.75
4.00
0.12 0.12 0.12 0.12 0.12 0.12 0.13 0.13 0.14 0.16 -
0.13 0.14 0.14 0.14 0.15 0.16 0.17 0.20 0.23 0.23 0.12
0.15 0.16 0.16 0.17 0.18 0.20 0.22 0.27 0.32 0.29 0.17
0.17 0.18 0.19 0.20 0.21 0.24 0.28 0.34 0.39 0.30 0.14
0.19 0.20 0.21 0.22 0.25 0.28 0.34 0.41 0.42 0.26 0.07
0.21 0.22 0.23 0.25 0.28 0.33 0.39 0.46 0.40 0.21 0.04
0.22 0.24 0.26 0.28 0.32 0.37 0.43 0.48 0.34 0.18 0.06
0.24 0.26 0.28 0.31 0.35 0.41 0.48 0.46 0.29 0.17 0.09
0.26 0.28 0.31 0.34 0.38 0.45 0.49 0.41 0.27 0.18 0.11
0.28 0.30 0.33 0.37 0.42 0.47 0.48 0.39 0.28 0.19 0.12
0.30 0.32 0.35 0.39 0.44 0.48 0.47 0.38 0.31 0.22 0.11
0.32 0.35 0.38 0.42 0.47 0.49 0.46 0.40 0.34 0.24 0.11
0.34 0.37 0.40 0.44 0.47 0.49 0.45 0.42 0.38 0.27 0.11
Trend inflation = 0%, opacity
φy
2.50 2.25 2.00 1.75 1.50 1.25 1.00 0.75 0.50 0.25 0.00
1.00
1.25
1.50
1.75
2.00
2.25
φπ 2.50
2.75
3.00
3.25
3.50
3.75
4.00
0.13 0.14 0.15 0.15 0.13 0.13 0.15 0.21 0.21 0.14 -
0.15 0.17 0.18 0.18 0.17 0.16 0.23 0.34 0.26 0.11 0.01
0.18 0.21 0.22 0.22 0.20 0.20 0.35 0.42 0.24 0.08 -
0.21 0.24 0.26 0.26 0.24 0.25 0.45 0.40 0.21 0.07 -
0.24 0.29 0.31 0.30 0.27 0.31 0.45 0.33 0.18 0.06 -
0.28 0.34 0.35 0.33 0.31 0.35 0.38 0.26 0.15 0.05 -
0.33 0.38 0.38 0.36 0.34 0.35 0.32 0.22 0.11 0.04 -
0.37 0.42 0.40 0.37 0.36 0.33 0.28 0.18 0.09 0.02 -
0.41 0.45 0.41 0.37 0.37 0.32 0.24 0.16 0.06 -
0.44 0.46 0.40 0.36 0.37 0.30 0.22 0.14 0.05 -
0.46 0.47 0.39 0.35 0.36 0.29 0.20 0.12 0.03 -
0.47 0.47 0.37 0.34 0.35 0.27 0.18 0.11 0.03 -
0.47 0.46 0.35 0.32 0.34 0.26 0.16 0.10 0.02 -
Trend inflation = 2%, transparency
φy
2.50 2.25 2.00 1.75 1.50 1.25 1.00 0.75 0.50 0.25 0.00
1.00
1.25
1.50
1.75
2.00
2.25
φπ 2.50
2.75
3.00
3.25
3.50
3.75
4.00
-
'0 '0 0.06 0.07
'0 '0 0.01 0.04 0.10 0.18 0.11
'0 '0 '0 0.03 0.07 0.09 0.18 0.23 0.30 0.19
'0 '0 0.04 0.06 0.11 0.15 0.19 0.25 0.29 0.32 0.21
0.03 0.07 0.10 0.13 0.18 0.21 0.24 0.29 0.34 0.32 0.20
0.11 0.13 0.16 0.18 0.21 0.24 0.28 0.31 0.36 0.34 0.22
0.14 0.17 0.19 0.21 0.23 0.25 0.30 0.34 0.39 0.34 0.22
0.18 0.19 0.21 0.23 0.25 0.28 0.32 0.36 0.40 0.34 0.24
0.20 0.21 0.22 0.24 0.27 0.30 0.33 0.38 0.41 0.35 0.24
0.21 0.23 0.23 0.26 0.29 0.31 0.35 0.39 0.43 0.36 0.25
0.22 0.24 0.25 0.27 0.29 0.33 0.35 0.40 0.43 0.36 0.25
0.24 0.25 0.26 0.28 0.31 0.33 0.37 0.40 0.42 0.38 0.26
Trend inflation = 2%, opacity
φy
2.50 2.25 2.00 1.75 1.50 1.25 1.00 0.75 0.50 0.25 0.00
1.00
1.25
1.50
1.75
2.00
2.25
φπ 2.50
2.75
3.00
3.25
3.50
3.75
4.00
-
'0 '0 0.03 0.01
'0 '0 '0 0.06 0.12 0.13 -
'0 '0 0.01 0.05 0.08 0.11 0.18 0.26 0.23 -
0.04 0.03 0.06 0.08 0.12 0.16 0.21 0.27 0.30 0.23 -
0.08 0.10 0.12 0.15 0.18 0.22 0.27 0.32 0.31 0.22 -
0.12 0.15 0.17 0.19 0.22 0.26 0.29 0.32 0.31 0.21 -
0.16 0.18 0.20 0.22 0.25 0.28 0.32 0.35 0.30 0.18 -
0.19 0.20 0.22 0.24 0.27 0.31 0.34 0.35 0.29 0.17 -
0.21 0.22 0.24 0.27 0.29 0.33 0.37 0.35 0.26 0.15 -
0.22 0.24 0.25 0.27 0.29 0.33 0.37 0.34 0.25 -
0.23 0.24 0.26 0.29 0.32 0.35 0.37 0.33 0.23 -
0.25 0.26 0.28 0.30 0.32 0.36 0.36 0.33 0.23 -
14
Trend inflation = 4%, transparency
φy
2.50 2.25 2.00 1.75 1.50 1.25 1.00 0.75 0.50 0.25 0.00
1.00
1.25
1.50
1.75
2.00
2.25
φπ 2.50
2.75
3.00
3.25
3.50
3.75
4.00
-
-
'0
'0 '0
'0 '0
0.05 '0
'0 0.13 '0
0.04 0.19 '0
0.08 0.23 '0
0.03 0.13 0.23 '0
0.06 0.18 0.17 '0
0.08 0.22 0.11 '0
0.05 0.10 0.26 0.08 0.09
Trend inflation = 4%, opacity
φy
2.50 2.25 2.00 1.75 1.50 1.25 1.00 0.75 0.50 0.25 0.00
1.00
1.25
1.50
1.75
2.00
2.25
φπ 2.50
2.75
3.00
3.25
3.50
3.75
4.00
-
-
'0
'0 '0
'0 -
0.01 -
'0 0.07 -
0.07 0.09 -
0.16 0.09 -
0.04 0.25 0.07 -
0.08 0.33 0.08 -
0.12 0.37 0.12 -
0.05 0.16 0.33 0.12 -
Trend inflation = 6%, transparency
φy
2.50 2.25 2.00 1.75 1.50 1.25 1.00 0.75 0.50 0.25 0.00
1.00
1.25
1.50
1.75
2.00
2.25
φπ 2.50
2.75
3.00
3.25
3.50
3.75
4.00
-
-
-
-
-
-
'0
'0
'0
0.04 '0
0.06 '0
0.09 '0
0.14 '0
Trend inflation = 6%, opacity
φy
2.50 2.25 2.00 1.75 1.50 1.25 1.00 0.75 0.50 0.25 0.00
1.00
1.25
1.50
1.75
2.00
2.25
φπ 2.50
2.75
3.00
3.25
3.50
3.75
4.00
-
-
-
-
-
-
'0
'0
'0
'0 '0
0.02 '0
0.02 -
0.06 -
15
