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Working Paper Series
Notes on the Inflation Dynamics of the
New Keynesian Phillips Curve
WP 07-04
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
Andreas Hornstein
Federal Reserve Bank of Richmond
Notes on the Inflation Dynamics of the
New Keynesian Phillips Curve∗
Andreas Hornstein†
Federal Reserve Bank of Richmond Working Paper 2007-04
Abstract
These notes contain the derivations for results stated without proof in Hornstein
(2007). First, I derive the log-linear approximation of the inflation dynamics in the
Calvo-model with elements of backward-looking pricing when the approximation takes
place around a positive average inflation rate. I derive a version of the “hybrid” New
Keynesian Phillips Curve (NKPC) that can be estimated using standard GMM techniques. Second, I characterize the inflation dynamics implied by the NKPC when
marginal cost follows an AR(1) process. For this purpose I derive the autocorrelation
and crosscorrelation structure of inflation.
JEL Classification: C63, E31, E32
Keywords: Inflation, Nominal Rigidities, New Keynesian Phillips Curve
∗
Any opinions expressed in this paper are those of the author and do not necessarily reflect those of the
Federal Reserve Bank of Richmond or the Federal Reserve System.
†
Federal Reserve Bank of Richmond, andreas.hornstein@rich.frb.org
1. The NKPC at a steady state with positive inflation
We study a standard Calvo (1983)-type model with monopolistically competitive firms and
nominal rigidities. There is a continuum of firms that produce differentiated products, that is,
they face downward sloping demand curves. Firms set the nominal prices of their products,
but they are limited in their ability to adjust their prices. In particular, whether a firm can
optimally adjust its price is random, and the probability of price adjustment is constant over
time. There is exogenous price indexation for firms that do not reoptimize their price. If
a firm cannot readjust its price, the price increases in proportion to last period’s aggregate
inflation rate. This indexation scheme has been used by Christiano, Eichenbaum and Evans
(2005). We derive a log-linear approximation of the equilibrium around a positive steady
state inflation rate. The derivation follows Ascari (2004) and Cogley and Sbordone (2005,
2006). Finally, we show how the equilibrium conditions have to be modified when the price
indexation scheme is replaced with “rule-of-thumb” price adjusters; that is, some firms never
set their prices optimally. Rather they index their prices to the optimal price adjustment of
the previous period, Galí and Gertler (1999).
1.1. The environment
Aggregate output is a Dixit-Stiglitz (1977) aggregator of a continuum of differentiated products on the unit interval
∙Z
¸
θ/(θ−1)
1
yt =
(θ−1)/θ
yt (i)
di
,
(1.1)
0
where the substitution elasticity is greater than one, θ > 1. Each differentiated good, yt (i),
is produced by a monopolistically competitive firm that sets the nominal price for its own
product, Pt (i). Assume that production of the final good is competitive, then the price
index of the final good, Pt , is simply its unit cost given the prices for differentiated goods,
Pt ≡
∙Z
1
Pt (j)1−θ dj
0
1
¸ 1−θ
.
(1.2)
The demand function for a firm’s differentiated product is declining in its relative price,
pt (i) ≡ Pt (i) /Pt ,
yt (i) = yt pt (i)−θ .
(1.3)
Production of the firm’s differentiated product is assumed to be such that it yields a convex
cost function
st yt−γ
· yt (i)1+γ ,
(1.4)
ct [yt (i)] =
1+γ
1
with constant own output elasticity, 1 + γ. The firm’s own cost also depends on aggregate
demand, and st will denote the aggregate marginal cost index in terms of the final good.
1.2. The price index
The evolution of the aggregate price index is determined by the optimal price setting of
the monopolistically competitive firms and the limits on nominal price adjustment. In any
period a firm has the opportunity to optimally reset its nominal price with probability 1 − α.
All firms that can adjust their price will choose the same price Xt since they are identical.
For a firm that cannot reoptimize the nominal price, which happens with probability α, its
nominal price is partially indexed to lagged general inflation, π t−1 = Pt−1 /Pt−2 , that is, its
price will increase in proportion to last period’s aggregate inflation rate
Pt (i) = πρt−1 Pt−1 (i) ,
(1.5)
with ρ ∈ [0, 1] being the indexation factor.
Substituting for the firms that adjust their prices and the firms whose prices are indexed
to past inflation in the price index equation (1.2) we get
Pt
1
½
¾ 1−θ
Z 1
¤1−θ
£ ρ
1−θ
=
(1 − α) Xt + α
dj
πt−1 Pt−1 (j)
0
1
∙
¸ 1−θ
Z 1
ρ(1−θ)
1−θ
1−θ
= (1 − α) Xt + απ t−1
Pt−1 (j) dj
0
i 1
h
ρ(1−θ) 1−θ 1−θ
1−θ
= (1 − α) Xt + απ t−1 Pt−1
.
Dividing through by the aggregate price index we get an expression that relates current
inflation to the optimal current relative price, xt ≡ Xt /Pt , and current and past inflation,
1 = (1 − α) x1−θ
+ α(π ρt−1 π −1
)1−θ .
t
| {z t }
(1.6)
≡ψ t
1.3. Optimal price setting
A firm chooses its nominal price to maximize the expected present value of future profits in
terms of the final good for the duration that it cannot reoptimize its price:
Et
∞
X
τ =0
ατ qt,τ {pt+τ (i) yt+τ (i) − ct+τ [yt+τ (i)]} ,
2
(1.7)
where qt,τ is the discount factor for period t + τ relative to period t. Given price indexation
(1.5), the firm’s relative price evolves according to
ρ
pt+τ (i) = π −1
t+τ π t+τ −1 pt+τ −1 (i) = ψ t+τ pt+τ −1 (i)
(1.8)
during the time the firm cannot reoptimize its nominal price. Repeated substitution yields
the τ -period ahead relative price as a function of the optimally chosen relative price and
subsequent inflation
pt+τ (i) = Πτj=1 ψ τ +j xt (i) = Ψt,τ xt (i) for τ > 0,
(1.9)
and Ψt,0 ≡ 1. Thus for a firm that sets its relative price in period t and does not have the
opportunity to reset its price optimally in future periods, future demand can be written as
a function of the optimally chosen relative price and future inflation
yt+τ (i) = yt+τ pt+τ (i)−θ = yt+τ [Ψt,τ xt (i)]−θ .
(1.10)
The FOC for the optimal relative price xt (i) is
0 = Et
"∞
X
ατ qt,τ {Ψt,τ yt+τ (i)
τ =0
oi
¤
£
−θ−1
x
(i)
−θ Ψt,τ xt (i) − c0t+τ [yt+τ (i)] yt+τ Ψ−θ
t
t,τ
"∞
½
¾#
X
θ
= Et
xt (i) −
st+τ · [Ψt,τ xt (i)]−θγ Ψ−1
ατ qt,τ yt+τ Ψ1−θ
t,τ
t,τ
θ
−
1
τ =0
"∞
½
¾#
X
θ
−(1+θγ)
xt (i)1+γθ −
.
st+τ Ψt,τ
ατ qt,τ yt+τ Ψ1−θ
= Et
t,τ
θ
−
1
τ =0
(1.11)
We can solve expression (1.11) for the optimal relative price
θ Ct
with
θ − 1 Dt
"∞
#
X
−(1+γ)θ
= Et
ατ qt,τ gt,τ Ψt,τ
st+τ ,
xt (i)1+γθ =
Ct
Dt = Et
=0
" τ∞
X
#
,
ατ qt,τ gt,τ Ψ1−θ
t,τ
τ =0
(1.12)
(1.13)
(1.14)
where gt,τ = yt+τ /yt denotes the growth rate of aggregate demand from period t to period
3
t + τ . Recursive definitions of C and D are
h
i
−(1+γ)θ
Ct = st + αEt qt,1 gt,1 ψ t+1
Ct+1
£
¤
Dt = 1 + αEt qt,1 gt,1 ψ 1−θ
t+1 Dt+1 .
(1.15)
(1.16)
1.4. Log-linear approximation
We now derive a log-linear approximation of the price index equation (1.6) and the FOC
for optimal price setting, (1.12), (1.15), and (1.16), at a steady state associated with some
inflation rate π̄, marginal cost s̄, discount factor q̄, and aggregate demand growth rate, ḡ.
The steady state expressions for equations (1.6), (1.12), (1.15), and (1.16) are
1 = (1 − α) x̄1−θ + απ̄ (ρ−1)(1−θ) ,
θ C̄
,
x̄1+θγ =
θ − 1 D̄
s
,
C̄ =
1 − αq̄ḡπ̄ (1−ρ)(1+γ)θ
1
.
D̄ =
1 − αq̄ḡπ̄ (1−ρ)(θ−1)
(1.17)
(1.18)
(1.19)
(1.20)
Note that the steady state values depend on the inflation rate.
The log-linear approximation of the price index equation (1.6) is
0 = (1 − α) x̄1−θ x̂t + απ̄ (ρ−1)(1−θ) ψ̂ t .
Using the steady state condition (1.17) this can be rewritten as
απ̄ (ρ−1)(1−θ)
απ̄ (ρ−1)(1−θ)
x̂t = −
ψ̂ = −
ψ̂ .
(ρ−1)(1−θ) t
(1 − α) x̄1−θ t
|1 − απ̄{z
}
(1.21)
ϕ0
The log-linear approximations of the equations characterizing optimal prices setting are
ĉt − dˆt
,
1 + γθ
£
¤
ĉt = 1 − αq̄ḡπ̄ (1−ρ)(1+γ)θ ŝt
|
{z
}
ϕ3
i
£
¤ h
+ αq̄ḡπ̄ (1−ρ)(1+γ)θ Et ẑt+1 − (1 + γ) θψ̂ t+1 + ĉt+1 ,
|
{z
}
ϕ2
i
£
¤ h
dˆt = αq̄ḡπ (1−ρ)(θ−1) Et ẑt+1 − (θ − 1) ψ̂ t+1 + dˆt+1 ,
|
{z
}
x̂t =
ϕ1
4
(1.22)
(1.23)
(1.24)
with ẑt+1 ≡ q̂t,1 + ĝt,1 . Collecting terms, we can rewrite expressions (1.23) and (1.24) as
h¡
i
¢
1 − ϕ2 L−1 ĉt + ϕ2 (1 + γ) θψ̂ t+1 − ϕ2 ẑt+1 = ϕ3 ŝt ,
h¡
i
¢
Et 1 − ϕ1 L−1 dˆt + ϕ1 (θ − 1) ψ̂ t+1 − ϕ1 ẑt+1 = 0,
Et
(1.25)
(1.26)
where L denotes the lag operators, Lj xt = xt−j for all integers j.
Combining equations (1.21), (1.22), (1.25), and (1.26) defines the following difference
equation in the change of the relative price of a firm that cannot reoptimize its nominal
price
h¡
¢¡
¢ i
−ϕ0 (1 + θγ) Et 1 − ϕ1 L−1 1 − ϕ2 L−1 ψ̂ t
h¡
oi
¢n
−1
= Et 1 − ϕ1 L
ϕ3 ŝt − θ (1 + γ) ϕ2 ψ̂ t+1
h
i
¡
¢
+Et (θ − 1) ϕ1 1 − ϕ2 L−1 ψ̂ t+1 + (ϕ2 − ϕ1 ) ẑt+1 .
(1.27)
Collecting terms yields the following fourth-order difference equation in the inflation rate
Et
with
£¡
¢
¤
£¡
¢
¤
μ1 + μ2 L−1 + μ3 L−2 (1 − ρL) π̂ t = Et μ4 + μ5 L−1 ŝt + μ6 ẑt+1 ,
(1.28)
μ1 = (1 + θγ) ϕ0 ,
μ2 = (θ − 1) ϕ1 − (1 + θγ) ϕ0 (ϕ1 + ϕ2 ) − θ (1 + γ) ϕ2 ,
μ3 = (1 + θγ) (1 + ϕ0 ) ϕ1 ϕ2 ,
μ4 = ϕ3 ,
μ5 = −ϕ1 ϕ3 ,
μ6 = ϕ2 − ϕ1 .
We can factor the polynomial in the lead operators on the LHS of equation (1.28) as
¶
μ1 μ2 −1
−2
+ L +L
μ3 μ3
¢¡
¢
μ3 ¡
1 − λ1 L−1 1 − λ2 L−1
=
λ1 λ2
¡
¢
μ1 + μ2 L−1 + μ3 L−2 = μ3
µ
where λi are the roots of the polynomial in the lead operator L−1 . Using the polynomial
5
factorization on the LHS of equation (1.28), we get the NKPC
£¡
¢¡
¢
¤
Et 1 − λ1 L−1 1 − λ2 L−1 (1 − ρL) π̂ t = κ1 Et
∙µ
¶
¸
μ5 −1
μ6 −1
1+ L
ŝt + L ẑt ,
μ4
μ4
(1.29)
with κ1 ≡ λ1 λ2 μμ4 . Equation (1.29) can be estimated using the same GMM methods as used
3
in Galí and Gertler (1999) for the standard hybrid NKPC (1.31) below.
Conditional on the process for marginal cost, ŝt , and the effective discount factor, ẑt ,
and assuming that the roots of the polynomial in the lead operator are less than one in
absolute value, one can solve the NKPC (1.29) forward and get inflation as a function of
lagged inflation and of current and future marginal cost and the discount factor1
∙
½µ
¶
¾¸
¡
¢ ¡
¢
μ5 −1
μ6 −1
−1 −1
−1 −1
1+ L
ŝt + L ẑt . (1.30)
(1 − ρL) π̂ t = κ1 Et 1 − λ1 L
1 − λ2 L
μ4
μ4
This expression allows us to interpret the two sources of inflation persistence. First, there
is “extrinsic” persistence that inflation inherits through its dependence on the two driving
forces, marginal cost and the discount factor. Since inflation is the expected present value of
future marginal cost and discount rates, inflation will be more persistent the more persistent
are its driving forces. Second, there is “intrinsic” persistence that is inherent to inflation
through the backward-looking indexation scheme.
1.4.1. Approximation at a zero inflation rate
For an approximation around a steady state with zero inflation, π̄ = 1, the coefficients on
the lead terms in equation (1.27) are the same, ϕ1 = ϕ2 . Thus equation (1.27) simplifies to
¡
¢
−ϕ0 (1 + θγ) 1 − ϕ1 L−1 ψ̂ t = ϕ3 ŝt − ϕ1 [θ (1 + γ) − (θ − 1)] ψ̂ t+1
µ
¶
¸
∙
1
−1
L
ψ̂ t = ϕ3 ŝt .
−ϕ0 (1 + θγ) 1 − ϕ1 1 +
ϕ0
Substituting for the coefficients ϕ0 , ϕ1 , and ϕ3 yields then the standard “hybrid” NKPC.
¸
∙
¢
¡
1 − α 1 − αβ
−1
ŝt .
(1 − ρL) π̂ t =
1 − βL
α 1 + γθ
{z
}
|
(1.31)
≡κ0
1
A bounded solution for the inflation process may exist even if the roots λi are not all less than one in
absolute value. In this case the roots characterizing the process for marginal cost and the discount factor
have to be small enough, such that their product with the roots λi is less than one in absolute value.
6
1.5. Modifications with “rule-of-thumb” price adjusters
Gali and Gertler (1999) also assume that only a fraction 1 − α of all firms can adjust their
price in any period. They do not allow for price indexation when firms cannot optimally
reset their prices, ρ = 0. Instead Gali and Gertler (1999) introduce “rule-of-thumb” price
adjusters that never set their nominal price optimally: rather they set their price, X0,t ,
relative to an “average” price set in the last period, Xa,t−1 , taking into account past inflation
X0,t = Xa,t−1
Pt−1
= Xa,t−1 π t−1 .
Pt−2
(1.32)
Let ω denote the fraction of these “rule-of-thumb” price setters. Then among all the firms
that can adjust their nominal price, a fraction 1 − ω of producers will adjust their price
optimally and a fraction ω will index their price to the last period’s “average” new price.
The “average” price set in the current period is then defined as
1−θ
1−θ
Xa,t
= (1 − ω) Xt1−θ + ωX0,t
(1.33)
and the current period overall price index is
Pt1−θ
= α
Z
0
1
£
¤
1−θ
1−θ
Pt−1
(i) di + (1 − α) (1 − ω) Xt1−θ + ωX0,t
1−θ
1−θ
= αPt−1
+ (1 − α) Xa,t
.
(1.34)
Dividing through equations (1.33) and (1.34) by the price level and using (1.32), we get
µ
π t−1
= (1 −
+ω
xa,t−1
πt
1 = απ θ−1
+ (1 − α) x1−θ
t
a,t ,
x1−θ
a,t
ω) x1−θ
t
¶1−θ
,
(1.35)
(1.36)
for the normalized average price adjustment xa,t ≡ Xa,t /Pt and the normalized optimal price
adjustment xt .
The steady state relations for the average price adjustment x̄a , optimal price adjustment,
x̄, and inflation, π̄, are
(1.37)
x̄a = x̄,
1 = απ̄ θ−1 + (1 − α) x̄1−θ
a .
7
(1.38)
The log-linear approximations of (1.35) and (1.36) at the steady state are
x̂a,t = (1 − ω) x̂t + ω (π̂ t−1 − π̂ t + x̂a,t−1 )
θ−1
π̂ t .
0 = (1 − α) x̄1−θ
a x̂a,t − απ̄
(1.39)
(1.40)
Using the steady state relations (1.37) and (1.38), we can combine equations (1.39) and
(1.40), eliminate the average price adjustment, and get an expression in the inflation rate
and the optimal price adjustment alone
£
¡
¢¤
£
¤
£
¤
(1 − ω) π̄1−θ − α x̂t = ωπ̄ 1−θ + (1 − ω) α π̂ t − ωπ̄1−θ π̂ t−1
(1.41)
Equation (1.41) now replaces equation (1.21) in the case of inflation indexation, and we
proceed with equations (1.22), (1.25), and (1.26) with no price indexation, ρ = 0.
2. Inflation dynamics of the NKPC
We now characterize persistence of inflation and its comovement with marginal cost when
marginal cost follows a simple AR(1) process
ŝt = δŝt−1 + εt
(2.1)
with 0 < δ < 1 and εt is an iid shock with mean zero and variance σ 2ε . For such an AR(1)
process the second moments are
σ 2ε
2
E [ŝt ŝt ] =
2 = σs
1−δ
E [ŝt ŝt−j ] = δ j σ 2s
(2.2)
and the conditional expectations of marginal cost j periods ahead are
Et [ŝt+j ] = Et [δŝt+j−1 + εt+j ] = δEt [ŝt+j−1 ] = . . . = δj ŝt .
(2.3)
2.1. Inflation dynamics of the simple NKPC
The basic NKPC approximates the inflation dynamics without indexation around a zero
steady state inflation rate
Et
£¡
¢ ¤
1 − βL−1 π t = κ0 ŝt + ut .
8
(2.4)
We obtain the basic NKPC from equation (1.31) with ρ = 0. The shock, ut , is simply
added to the NKPC, and it is assumed to be iid with mean zero and variance σ 2u , and
uncorrelated with marginal cost.2 We can solve (2.4) forward by repeatedly substituting for
future inflation, and thereby obtain the current inflation rate as a discounted present value
of future marginal cost
∞
X
π̂ t = κ0
β j Et ŝt+j + ut .
(2.5)
j=0
Substituting for the expected future marginal cost from (2.3) we get
π̂ t = κ0
∞
X
j=0
β j δ j ŝt + ut =
κ0
ŝt + ut = a0 ŝt + ut .
1 − βδ
(2.6)
Equation (2.6) is a reduced form relationship between current inflation and marginal cost.
The relationship is reduced form since it incorporates the presumed equilibrium law of motion
for marginal cost. If the law of motion for marginal cost changes, then the relation between
inflation and marginal cost will change.
The second moments of the inflation rate process are given by
E [π̂ t π̂ t−k ] = a20 E [ŝt ŝt−k ] + I[k=0] σ 2u = δ k (a0 σ s )2 + I[k=0] σ 2u
(2.7)
where I[.] denotes the indicator function, I[k=0] = 1 for k = 0 and zero otherwise. The
expected cross-product of inflation and marginal cost is
E [π̂ t ŝt+k ] = a0 E [ŝt ŝt+k ] = δ k a0 σ 2s .
(2.8)
From the second moments we obtain the autocorrelation coefficients for inflation and the
crosscorrelation coefficients of inflation and marginal cost as
a20
,
a20 + (σ u /σ s )2
a0
Corr (π̂ t , ŝt+k ) = δ k £
¤1/2 .
a20 + (σ u /σ s )2
Corr (π̂ t , π̂ t−k ) = δ k
(2.9)
(2.10)
In the simple NKPC the only source of inflation persistence is “extrinsic.” Given the assumed
law of motion for marginal cost, the inflation rate is positively correlated with marginal cost
and inherits some of the persistence properties of marginal cost. In particular, the autocorrelation coefficients of inflation are simply scaled versions of the autocorrelation coefficients of
2
When the inflation dynamics are approximated around a zero steady state inflation rate, one can interpret
the shock ut as a random disturbance to the demand elasticity parameter θ.
9
marginal cost. The scale factor depends on the importance of marginal cost for inflation, a0 ,
and the relative volatility of marginal cost shocks, σ u /σ s . The more important is marginal
cost and the bigger is the relative volatility of marginal cost, the closer is the autocorrelation
structure of inflation to that of marginal cost.
2.2. Inflation dynamics of the “hybrid” NKPC model
Consider now the “hybrid” NKPC with partial price indexation ρ
Et
£¡
¢
¤
1 − βL−1 (1 − ρL) π̂t = κ0 ŝt + ut .
(2.11)
Given the marginal cost process (2.1), the properties we have just derived for the inflation
process of the simple NKPC now apply to the transformation of the inflation process,
π̃ t = (1 − ρL) π̂ t = a0 ŝt + ut .
(2.12)
The inflation rate itself is now an infinite sum of past transformed inflation rates π̃
π̂ t = (1 − ρL)−1 π̃ t =
∞
X
ρj π̃ t−j
j=0
The second moments of the inflation rate are now defined as follows:
E [π̂ t π̂ t−k ]
XX
ρi ρj E [π̃ t−i π̃t−k−j ]
=
i≥0 j≥0
= σ 2u
XX
i≥0 j≥0
|
ρi ρj I[i=k+j] + (a0 σ s )2
{z
i≥0 j≥0
}
A(k;ρ)
For k ≥ 0, we can write the two terms A and B as
A (k; ρ) = ρ0
X
ρj I[0=k+j] + ρ1
j≥0
k
X
k+2 2
= ρ + ρ ρ + ρ ρ + ...
¢
¡
= ρk 1 + ρ2 + ρ4 + ρ6 + . . .
X
ρk
= ρk
ρ2j =
1 − ρ2
j≥0
and
10
|
ρi ρj δ |k+j−i| .
{z
B(k;ρ,δ)
ρj I[1=k+j] + ρ2
j≥0
k+1
XX
X
j≥0
}
ρj I[2=k+j] + . . .
B (k; ρ, δ) =
XX
ρi ρj δ |k+j−i|
i≥0 j≥0
=
X
i
ρ
i≥0
⎧
⎪
⎪
⎪
⎪
⎪
⎨i−k−1
X
j i−k−j
ρδ
⎪
⎪
j=0
⎪
⎪
{z
⎪
⎩|
B1i (k)
B1i (k; ρ, δ) = I[i>k] δ i−k
i−k−1
X
}
+
∞
X
j=max{0,i−k}
|
(ρ/δ)j = I[i>k] δ i−k
B2i (k; ρ, δ) = I[i≤k]
j k+j−i
ρδ
+ I[i>k]
j=0
= I[i≤k] δ k−i
{z
B2i (k)
j=0
∞
X
ρj δ k+j−i
∞
X
1 − (ρ/δ)i−k
1 − ρ/δ
ρj δ j−(i−k)
(ρδ)j + I[i>k] ρi−k
∞
X
∞
X
ρi B2i (k; ρ, δ) =
i=0
=
B (k; ρ, δ) =
=
=
=
(ρδ)j−(i−k)
j=i−k
1
1
+ I[i>k] ρi−k
= I[i≤k] δ k−i
1 − ρδ
1 − ρδ
∞
∞
i−k
X
X
1 − (ρ/δ)
ρi B1i (k; ρ, δ) =
ρi δ i−k
1 − ρ/δ
i=0
i=k+1
=
⎪
⎪
⎪
⎪
}⎪
⎭
j=i−k
∞
X
j=0
=
⎫
⎪
⎪
⎪
⎪
⎪
⎬
∞ h
i
X
1
ρk (ρδ)i−k − ρk ρ2(i−k)
1 − ρ/δ i=k+1
∙
¸
ρδ
ρ2
1
k
ρ
−
1 − ρ/δ
1 − ρδ 1 − ρ2
#
" k
∞
X
X
1
ρi δ k−i +
ρ2i−k
1 − ρδ
i=0
i=k+1
#
"
k+1
2
1
−
(ρ/δ)
ρ
1
+ ρk
δk
2
1 − ρ/δ
1 − ρ 1 − ρδ
∙
¸
2
2
2
2
k ρδ (1 − ρ ) − ρ (1 − ρδ) + ρ (1 − ρ/δ) − (ρ/δ) (1 − ρ )
ρ
(1 − ρ/δ) (1 − ρδ) (1 − ρ2 )
1
+δ k
(1 − ρ/δ) (1 − ρδ)
∙
¸
1
k
k+1 δ − 1/δ
ρ
+δ
2
1−ρ
(1 − ρ/δ) (1 − ρδ)
∙
¸
2
1δ −1
1
ρk+1
+ δk
2
δ1−ρ
(1 − ρ/δ) (1 − ρδ)
∙
¸
2
1
ρ1−δ k
δk −
ρ
2
δ1−ρ
(1 − ρ/δ) (1 − ρδ)
11
The crossproducts of inflation and marginal cost are
E [π̂ t ŝt+k ] = E [(1 − ρL) (a0 ŝt + ut ) ŝt+k ]
∞
X
=
ρj a0 E [ŝt−j ŝt+k ]
j=0
=
a0 σ 2s
∞
X
ρj δ |k+j| = a0 σ 2s C (k; ρ, δ) .
j=0
If k ≥ 0 then
C (k; ρ, δ) = δ k
∞
X
ρj δ j =
j=0
δk
,
1 − ρδ
(2.13)
and if k < 0 then
C (k; ρ, δ) =
−k−1
X
ρj δ −(k+j) +
j=0
∞
X
ρj δ(k+j)
j=−k
−k
1 − (ρ/δ)
1
+ ρ−k
1 − ρ/δ
1 − ρδ
½
2¾
1
−k
−k ρ 1 − δ
δ −ρ
.
=
1 − ρ/δ
δ 1 − ρδ
= δ −k
(2.14)
We can now calculate the autocorrelation coefficients for the inflation rate and the crosscorrelations of inflation and marginal cost as
(σ u /σ s )2 A (k; ρ) + a20 B (k; ρ, δ)
Corr (π̂ t , π̂ t−k ) =
(σ u /σ s )2 A (0; ρ) + a20 B (0; ρ, δ)
a0 C (k; ρ, δ)
Corr (π̂ t , ŝt+k ) = £
¤1/2 .
(σ u /σ s )2 A (0; ρ) + a20 B (0; ρ, δ)
(2.15)
(2.16)
In the “hybrid” NKPC there are two sources of inflation persistence, “intrinsic” and
“extrinsic” persistence. Equations (2.15) and (2.16) illustrate how the two sources of inflation
persistence ultimately affect overall inflation persistence. We can see that persistence of
marginal cost, a high δ, is relatively more important for the dynamics of inflation if marginal
cost is quantitatively important in the reduced form NKPC, that is, the coefficient a0 is
larger, and if the shocks to the NKPC are small relative to the volatility of marginal cost.
Alternatively, if marginal cost is not important or if shocks to the NKPC are relatively large,
the inflation persistence can only arise through the inherent persistence of inflation, that is,
a high ρ.
12
2.3. Inflation dynamics at positive steady state inflation
In the case of an AR(1) process for marginal cost, it is straightforward to derive the persistence properties of inflation around a positive steady state inflation rate. Adding a shock ut
to the modified NKPC and assuming that Et [ẑt+1 ] = 0 equation (1.29) simplifies to
µ
¶
£¡
¢¡
¢
¤
μ5
−1
−1
1 − λ2 L
(1 − ρL) π̂ t = κ1 1 + δ ŝt + ut .
Et 1 − λ1 L
μ4
(2.17)
Assuming that the roots of the polynomial and the persistence of marginal cost are such
that |δλi | < 1, we can divide through by the inverse lead polynomials and get
µ
¶ h
i
¡
¢ ¡
¢
μ5
−1 −1
−1 −1
Et 1 − λ1 L
ŝt + ut
1 − λ2 L
Et [(1 − ρL) π̂ t ] = κ1 1 + δ
μ4
1 + δμ5 /μ4
= κ1
ŝt + ut ,
(2.18)
(1 − λ1 δ) (1 − λ2 δ)
|
{z
}
≡a1
which is formally equivalent to equation (2.12) for the “hybrid” NKPC, but now the coefficient a1 is a function of the average inflation rate and the demand elasticity θ.
References
[1] Ascari, Guido. 2004. “Staggered Prices and Trend Inflation: Some Nuisances.” Review
of Economic Dynamics 7 (3), 642-667.
[2] Calvo, Guillermo A. 1983. “Staggered Prices in a Utility Maximizing Framework.” Journal of Monetary Economics 12 (3), 383-398.
[3] Christiano, Lawrence J., Martin Eichenbaum, and Charles L. Evans. 2005. “Nominal
Rigidities and the Dynamic Effects of a Shock to Monetary Policy.” Journal of Political
Economy 113 (1), 1-45.
[4] Cogley, Timothy and Argia M. Sbordone. 2005. “A Search for a Structural Phillips
Curve.” Federal Reserve Bank of New York Staff Report 203.
[5] Cogley, Timothy and Argia M. Sbordone. 2006. “Trend Inflation and Inflation Persistence
in the New Keynesian Phillips Curve.” Federal Reserve Bank of New York Staff Report
270.
[6] Dixit, Avinash K. and Joseph E. Stiglitz. 1977. “Monopolistic Competition and Optimum
Product Diversity.” American Economic Review 67 (3), 297-308.
13
[7] Galí, Jordi and Mark Gertler. 1999. “Inflation Dynamics: A Structural Econometric
Analysis.” Journal of Monetary Economics 44 (2), 195-222.
[8] Hornstein, Andreas. 2007. “Evolving Inflation Dynamics and the New Keynesian Phillips
Curve.” Federal Reserve Bank of Richmond Economic Quarterly, forthcoming.
14
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