The infinite-dimensional sticky-information Phillips curve is cast as a finite-dimensional timevarying system of difference equations in order to directly assess determinacy in the model with demand given by the forward-looking IS equation and monetary policy by an interest rate rule. An equivalence to the model without lagged expectations holds (albeit tenuously) for the particular specification and a common truncation method produces spurious determinacy.Determinacy, Taylor rule, Sticky Information, Time-Varying Difference Equations

Alexander Meyer-Gohde

English

Research Papers in Economics

 
 
 
 
 
 
 
 
 
 
 
SFB 649 Discussion Paper 2011-006 
Sticky Information  
and Determinacy 
 
Alexander Meyer-Gohde*  
 
 
* Humboldt-Universität zu Berlin, Germany 
 
 
 
 
This research was supported by the Deutsche 
Forschungsgemeinschaft through the SFB 649 "Economic Risk". 
 
http://sfb649.wiwi.hu-berlin.de 
ISSN 1860-5664 
 
SFB 649, Humboldt-Universität zu Berlin 
Spandauer Straße 1, D-10178 Berlin 
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N
 Sticky Information and Determinacy
Alexander Meyer-Gohde∗†
Abstract
The inﬁnite-dimensional sticky-information Phillips curve is cast as a ﬁnite-dimensional time-
varying system of difference equations in order to directly assess determinacy in the model with
demand given by the forward-looking IS equation and monetary policy by an interest rate rule.
An equivalence to the model without lagged expectations holds (albeit tenuously) for the partic-
ular speciﬁcation and a common truncation method produces spurious determinacy.
JEL classiﬁcation: C62; E31; E43; E52
Keywords: Determinacy; Taylor rule; Sticky Information; Time-Varying Difference Equations
1 Introduction
The sticky-information model of Mankiw and Reis (2002),1 with an inﬁnite regress of lagged
expectations, cannot be brought into the canonical form of Blanchard and Kahn (1980) to assess
determinacy (existence of a unique, bounded equilibrium). I analytically derive the determinacy
properties for a standard New Keynesian model with sticky information by recasting the system
as a time-varying system of difference equations. I show that for standard dynamic IS demand
and the interest rate rule examined here, the parameter restriction to ensure determinacy is the
same as would be obtained by examining the model without lagged expectations. Such an equiv-
alence need not hold in general, however, as the non-singularity constraints and ﬁnite variational
bounds satisﬁed by the particular model analyzed here need not be satisﬁed by other models with
lagged expectations. With analytical results in hand, I conclude by demonstrating that a standard
truncation method produces spurious determinacy.
∗Alexander Meyer-Gohde; Humboldt-Universit¨ atzu Berlin; Institute for Economic Theory II; Spandauer Straße
1; 10178 Berlin; Germany. Tel.:+49-30-2093-5720. E-mail: alexander.meyer-gohde@wiwi.hu-berlin.de
†I am grateful to Michael Burda and Frank Heinemann, as well as participants of the 2009 Midwest Macroeco-
nomic Meetings and the Verein f¨ ur Socialpolitik 2008 Annual Meeting and of research seminars at the HU Berlin
and the FU Berlin for useful comments, suggestions, and discussions. This research was supported by the Deutsche
Forschungsgemeinschaftthrough the SFB 649 “Economic Risk”.
1Forarecentoverviewofapplicationsandextensionsofstickyinformation,seeMankiwandReis (Forthcoming).
12 A Sticky-Information Model
A basic sticky-information New Keynesian model can be written as2
yt = Et [yt+1]−a1Rt +a1Et [pt+1] (1)
pt =
1−l
l
xyt +(1−l)
¥
å
i=0
liEt−i−1[pt +x(yt −yt−1)] (2)
whereyt istheoutputgap, pt inﬂation, andRt thenominalinterestrate. Equation(1)isadynamic
IS-curve and (2) is Mankiw and Reis’s (2002) sticky-information Phillips curve. Here, a1 and
x are assumed positive,3 and 0 < 1−l < 1 is a ﬁrm’s probability of receiving an information
update.
Monetary policy will be described by the following rule for the interest rate to close the
model
Rt = fRRt−1+fp[(1−yp)Et [pt+1]+yppt]+fyyt (3)
where 0 ≤ fR < 1 describes the degree of interest-rate smoothing, 0 ≤ fp < ¥ of inﬂation target-
ing, and 0 ≤ fy < ¥ of output-gap targeting. The coefﬁcient 0 ≤ yp ≤ 1 nests contemporaneous
inﬂation targeting (yp = 1) and inﬂation forecast targeting (yp = 0) into the rule.
3 Endogenous Fluctuations and Determinacy
Without loss of generality, I abstract from exogenous driving forces.4 By examining the inﬁ-
nite moving average representation of the model in response to endogenous ﬂuctuations (i.e.,
to sunspot shocks), the system of difference equations originating from the model of sticky in-
formation will yield a non-autonomous or time-varying system of homogenous non-stochastic
difference equations.
Consider a sunspot shock that occurs at time 0 and denote with xt the response of the variable
x in period t to the sunspot. The response of the model, deﬁned by (1), (2), and (3), is given by
2See, e.g., Trabandt (2007) for a ﬁrst-principles derivation analogous to Woodford (2003, Ch. 4).
3See, e.g., Woodford (2003, pp. 160–164 & 243–245)
4With bounded exogenous driving forces, the boundedness of the particular solution will rest on that of the
homogenous solution. See Taylor (1986), Woodford (2003, pp. 252, & 636) and Pituk (2002).
2the system of deterministic time-varying difference equations
yt = yt+1−a1Rt +a1pt+1 (4)
lt+1pt =
 
1−lt+1 
xyt −xl
 
1−lt 
yt−1 (5)
Rt = fRRR
t−1+fp[(1−yp)pt+1+yppt]+fyyt (6)
with R−1 = 0, where (4) and (6) correspond straightforwardly to (1) and (3), and where (5)
follows from (2) after noting that both the response of variables and expectations dated before 0
are zero.5
Equation (5) gives the time-varying difference equation described by the sticky-information
Phillips curve (2). As t → ¥, the foregoing converges to the “unrestricted” perfect-foresight
version of the model, given by yt = lyt−1 as all outdated information sets are updated. The
lagged expectations serve to transition the Phillips curve from having a positive trade-off at time
0, given by lp0 = (1−l))xy0, to being vertical with no trade-off in the limit. This contrasts
with the sticky-price Phillips curve, which always posits the same dynamic trade-off between
inﬂation and output: pt −bpt+1 = kyt under perfect foresight.6 Although the model itself is time
invariant, the response of a variable under sticky information is time varying: the equilibrium
relationships between the responses of endogenous variables to a shock change as the shock
becomes more outdated. The model will be determinate (sunspots can be ruled out), if the only
sequence of impulse responses to a sunspot shock that remains bounded is the trivial sequence of
zeros for all variables at all horizons; i.e., if the only bounded response of endogenous variables
to sunspots is no response at all.
Lagging (5) forward and noting the additional initial condition yields the following system


lt+2 −x
 
1−lt+2 
0
a1 1 −a1
−fp(1−yp) 0 1




pt+1
yt+1
Rt

 =


0−xl
 
1−lt+1 
0
0 1 0
fpyp fy fR




pt
yt
Rt−1

 (7)
for i = 0,1,2,..., with R−1 = 0 and lp0 = (1−l)xy0.
The foregoing system has two initial conditions but three variables. As the system is ho-
mogenous, one solution is pt,yt,Rt = 0, t = 0,1,..., but it may not be the only bounded solu-
tion. Different potential solutions can be indexed by different values for the “missing” initial
condition—i.e., a value for y0 or p0; if the system (7) is stable, then it will remain bounded for
5See, likewise, Mankiw and Reis’s (2002) Appendix.
6The notation follows Woodford (2003).
3any such ﬁnite initial condition and, thus, the sunspots cannot be ruled out. If the system, how-
ever, has a one-dimensional unstable manifold that can be associated with this condition, then
the boundedness requirement will providethe missinginitial condition and sunspotscan be ruled
out.
Proposition 3.1. The model given by (1), (2), and (3) is determinate iff |
fR+fpyp
1−fp(1−yp)| > 1.7
Proof. The system of difference equations in (7) can be inverted to yield
 
pt+1 yt+1 Rt
 ′ = (C+D(i))
 
pt yt Rt−1
 ′ (8)
so long as fp(1−yp)  = 1+ lt+2
(1−lt+2)xa1
, ∀t ≥ 0. Where
C =



fpyp
1−fp(1−yp)
a1fy+1−l
a1(1−fp(1−yp))
fR
1−fp(1−yp)
0 l 0
fpyp
1−fp(1−yp)
a1fy+(1−l)fp−fpyp
a1(1−fp(1−yp))
fR
1−fp(1−yp)



D(i) = a(i)D
a(i) =
li+2
(1−li+2)a1x(1−fp(1−yp))+li+2
D =
 
−1
1−fp(1−yp) a1
−fp(1−yp)
1−fp(1−yp)
 ′ 
fpyp 1−l+a1[fy−x(1−l)(1−fp(1−yp))] fr
 
Using the ratio test, å
¥
i=0|a(i)| < ¥, and thus
¥
å
i=0
||D(i)|| ≤ ||D||
¥
å
i=0
|a(i)| < ¥ (9)
Noting (9) and following Ludyk’s (1985) Theorem 3-29, the system in (8) is stable if C is
stable and, from Ludyk’s (1985) Theorem 3-12, will remain bounded for any bounded initial
conditions.
Examining the eigenvalues of C, z1 = 0, z2 = l, z3 =
fR+fpyp
1−fp(1−yp), the ﬁrst two of which are
necessarily inside the unit circle. If |z3| < 1, then (8) is stable for any set of bounded initial
conditions. In this case, the boundedness condition will be insufﬁcient to pin down the missing
initial condition and one cannot rule out sunspot equilibria (i.e., the model is indeterminate).
Should |z3| > 1, then z3 is a simple dominant eigenvalue. Noting (9) and following Pituk’s
(2002) Theorem 1, solutions of (8) are related asymptotically to solutions of the system xt+1 =
Cxt via limt→¥
 
z−t
3
 
pt yt Rt−1
 ′ 
= gX, where g is a constant and X is the eigenvector of C
corresponding to z3. The eigenvector is
 
1 0 1
 ′ and as |z3| > 1, pt and Rt will be unbounded
7The analysis will abstract from cases with eigenvalues on the unit circle, following Woodford (2003, p. 254).
4unless g = 0, which, following Pituk’s (2002) Theorem 3, would require
 
p0 y0 R−1
 ′ =
 
0 0 0
 ′. Hence the requirement of boundedness provides the system with an additional re-
striction that rules out the sunspot equilibria (i.e., the model is determinate).
Should fp(1−yp) = 1+ lt+2
(1−lt+2)xa1
for some t (say t), then (7) cannot be brought into the
form of (8) for all t. The singularity of the coefﬁcient matrix at t provides one linear restriction,
which, when combined with the two original restrictions, implies that Rt−1 = yt = pt = 0, t ≤ t.
The recursion then delivers two new initial conditions,
 
1−lt+2 
xyt+1 = lt+2
fp Rt and pt+1 =
x(1−l)
lt+2(1−xa1)+la1xytt+1+
a1x(l−lt+2)
lt+2(1−xa1)+la1xRtt, which result in a non-singular recursion for i = it+
1,it+2,..., with the same stability characteristics as in the recursion without the singularity.
4 Equivalence and Specious Determinacy
It is conspicuous that the parameter bound for determinacy is independent of the parameters out-
side of the interest rate rule. This independence is related to the equivalence of the determinacy
bounds to those in a frictionless version of the model. To see this, note that in the absence of
lagged expectations, (5) reduces to yt = lyt−1, which is necessarily stable. Thus, determinacy of
the system without lagged expectations can be ascertained by means of the following system
Rt = Et [pt+1]
Rt = fRRR
t−1+fp[(1−yp)Et [pt+1]+yppt]
Following Blanchard and Kahn (1980), |
fR+fpyp
1−fp(1−yp)| > 1 is required for determinacy. This is, of
course, the same bound as in (3.1). This equivalence is, however, more tenuous than one might
infer from Wang and Wen (2006). As can be seen in the proof of (3.1), both singular coefﬁcient
matrices and inﬁnite variation—if (9) does not hold—can cause this equivalence to break down:
neither of which can be a priori ruled out.
Generally,8 the sticky-information model needs to be truncated when a particular solution is
sought. The truncation used by Trabandt (2007) and Andr´ es, L´ opez-Salido, and Nelson (2005),
which eliminates the tail of the distribution of lagged expectations, leads to a specious determi-
nacy region for an otherwise indeterminate monetary policy rule.
8See Meyer-Gohde (2010) for an overview.
5Equation (2) is truncated at some I < ¥ as
pt =
1−l
l
xyt +(1−l)
I−1
å
i=0
liEt−i−1[pt +x(yt −yt−1)]
To simplify the calculations, consider pure inﬂation forecast targeting by the monetary au-
thority: the special case of yp = fR = fy = 0 in (3). The system can now be written in matrix
form as
0 =
I
å
i=0
AiEt−i[Yt+1]+
I
å
i=0
BiEt−i[Yt]+
I
å
i=0
CiEt−i[Yt−1]
where Yt =
 
pt yt Rt
 ′. This is the canonical form of Meyer-Gohde (2010) and determinacy
can be ascertained by examining the eigenvalues (G) of the matrix pencil9
 
å
I
i=0Ci å
I
i=0Bi
0 I
 
−G
 
0 −å
I
i=0Ai
I 0
 
the determinate of which yields
  
lI+1−
 
l−lI+1 
a1x(fp−1)
 
−G
 
lI+1−
 
1−lI+1 
a1x(fp−1)
  
G3 = 0
The two “missing” eigenvalues are called “inﬁnite.” Of the remaining four eigenvalues, it is
trivial to see that three are equal to zero. Thus, determinacy will rest upon the ﬁnal eigenvalue
G =
lI+1−
 
l−lI+1 
a1x(fp−1)
lI+1−(1−lI+1)a1x(fp−1)
being outside the unit circle. This holds if
1 < fp < 1+
2lI+1
a1x(1+l−2lI+1)
This requires the interest-rate rule to satisfy the Taylor Principle and to not react “too strongly”
to expected inﬂation. Proposition 3.1, with yp = fR = fy = 0, states that the true, non-truncated
model is necessarily indeterminate. As the tail of the distribution of lagged expectations never
adjust, the truncation scheme causes the long-run Phillips curve to become non-vertical like in
the standard sticky-price model and leads to the emergence of a specious determinacy region.
9The lagged expectationsalso have to be resolvable—i.e.,Meyer-Gohde’s(2010) Equation(12) has to be invert-
ible. This held with the non-truncatedmodel and thus follows here as both models are identical up to the truncation.
6References
ANDR´ ES, J., J. D. L´ OPEZ-SALIDO, AND E. NELSON (2005): “Sticky-Price Models and the
Natural Rate Hypothesis,” Journal of Monetary Economics, 52(5), 1025–1053.
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to Replace the New Keynesian Phillips Curve,” The Quarterly Journal of Economics, 117(4),
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(Forthcoming): “Imperfect Information and Aggregate Supply,” in Handbook of Mon-
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WOODFORD, M. (2003): Interest and Prices. Princeton University Press.
7 
 
 
 
SFB 649 Discussion Paper Series 2011 
 
For a complete list of Discussion Papers published by the SFB 649, 
please visit http://sfb649.wiwi.hu-berlin.de. 
 
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Cabrera, Ostap Okhrin and Weining Wang, January 2011. 
002  "A Confidence Corridor for Sparse Longitudinal Data Curves" by 
Shuzhuan Zheng, Lijian Yang and Wolfgang Karl Härdle, January 2011. 
003  "Mean Volatility Regressions" by Lu Lin, Feng Li, Lixing Zhu and 
Wolfgang Karl Härdle, January 2011. 
004  "A Confidence Corridor for Expectile Functions" by Esra Akdeniz Duran, 
Mengmeng Guo and Wolfgang Karl Härdle, January 2011.  
005  "Local Quantile Regression" by Wolfgang Karl Härdle, Vladimir Spokoiny 
and Weining Wang, January 2011.  
006  "Sticky Information and Determinacy" by Alexander Meyer-Gohde, 
January 2011. 
 
 
SFB 649, Ziegelstraße 13a, D-10117 Berlin 
http://sfb649.wiwi.hu-berlin.de 
 
This research was supported by the Deutsche 
Forschungsgemeinschaft through the SFB 649 "Economic Risk". 

Sticky Information and Determinacy

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