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Federal Reserve Bank of Chicago
Evaluating the Calvo Model of Sticky
Prices
Martin Eichenbaum and Jonas D.M. Fisher
WP 2003-23
Evaluating the Calvo Model of Sticky Prices
Martin Eichenbaum∗ and Jonas D.M. Fisher†
November 2003
Abstract
This paper studies the empirical performance of a widely used model of nominal
rigidities: the Calvo model of sticky goods prices. We describe an extended version of
this model with variable elasticity of demand of the differentiated goods and imperfect
capital mobility. We find little evidence against standard versions of the model without
the extensions, but the estimated frequency of price adjustment is implausible. With
the extended model the estimates are more reasonable. This is especially so if the
sample is split to take into account a possible change in monetary regime around 1980.
∗
Northwestern University, NBER and Federal Reserve of Chicago.
Federal Reserve Bank of Chicago
†
1. Introduction
In this paper we analyze the aggregate implications of the widely used model of sticky prices
due to Calvo (1983).1 This model makes the simplifying assumption that the number of
firms adjusting prices in any given period is exogenous. We address the question of whether
models with this assumption make sense empirically.2
To do this, we study a generalized version of the Calvo model. Standard versions of
the Calvo model assume that monopolistically competitive firms face a constant elasticity
of demand. Following Kimball (1995), we allow for the possibility that the elasticity of
demand is increasing in the firm’s price. Another standard assumption is that capital can be
instantaneously reallocated after after a shock. Following Sbordone (2002), we also consider
the possibility that capital is fixed in place and is not reallocated in response to shocks.
These two extensions to the Calvo model in principle enable it to account for the dynamics
of inflation with lower degrees of price rigidities.
The parameters of the extended Calvo model are not separately identified using aggregate
time series data. In particular, one cannot separately identify the probability that a firm
reoptimizes its price, the nature of demand elasticities, and the degree of capital mobility.
Still, we can identify the frequency of reoptimization if we have information or priors about
demand elasticities and the degree of capital mobility.
Our main findings are as follows. We find strong evidence against the standard Calvo
model, that is the version of the model with a constant demand elasticity, mobile capital, and
no lag between the time that firms reoptimize and the time that they implement their new
plans. This is true regardless of whether or not we allow for a structural break in monetary
policy in the early 1980s. When we allow for a one period implementation lag, the model
is no longer rejected. Interestingly, this is the specification adopted in Chari, Kehoe and
McGrattan (2001) and Christiano, Eichenbaum and Evans (2001) among others.
Evidently, allowing for a one quarter delay in the implementation on new prices renders
1
See for example, Clarida, Gali and Gertler (2001), Christiano, Eichenbaum and Evans (2001), Erceg,
Henderson and Levon (2000), Rotemberg and Woodford (1997, 2003), Woodford (2003), and Yun (1996).
2
We do not consider models which endogenize the number of firms changing prices, such as in the ones
studied by Dotsey, King and Wolman (1999) and Goloslov and Lucas (2003). While models of this kind
seem very promising they are more difficult to work with than Calvo-style models. There is some theoretical
evidence that endogenizing the number of firms setting prices may not effect policy analysis for the inflation
rates typically seen for the US. For example. For example, Burstein (2002) shows that for moderate changes
in the growth rate of money (less than or equal to 5% on a quarterly basis), traditional Calvo models are a
good approximation to a model where the number of firms adjusting prices is endogenous.
2
the standard Calvo model consistent with the aggregate data in a statistical sense. But that
does not mean the estimated model makes economic sense. Here the model does less well.
Specifically, according to our point estimates, firms reoptimize prices on average roughly once
every two and a half years. This seems implausible on a priori grounds. More importantly, it
is inconsistent with results based on microeconomic data (see for example Bills and Klenow
(2002)).
In the extended Calvo model the estimated frequency of reptimization may be more
reasonable. Based on the full sample results, the model with immobile capital and nonconstant elasticity of demand is consistent with the view that firms reoptimize prices roughly
once every year, using the relatively low upper bound for the demand elasticity as suggested
by Bergin and Feenstra (2000). Using the higher benchmark elasticity of Chari, Kehoe and
McGrattan (2000), the model with mobile capital and non-constant elasticity of demand is
consistent with the view that firms reoptimize prices roughly once every three quarters.
Even more favorable results emerge if we take as given that there is split in the sample
period owing to a change in monetary policy. Averaging across the two subsamples, we find
that the model with mobile capital and the non-constant elasticity of demand is consistent
with the view that firms reoptimizing prices every three quarters, if we are willing to use
the higher benchmark elasticity. The model with immobile capital and the non-constant
elasticity of demand is consistent with the view that firms reoptimize prices more often than
once every three quarters, regardless of the which benchmark elasticity we assume.
The rest of the paper is organized as follows. First we describe the extended Calvo model
and our empirical methodology. Then we discuss the empirical results. In the penultimate
section we interpreting the parameters of the estimated Calvo model. In the final section we
summarize our results.
2. The Calvo Model of Sticky Prices
In this section we display an extended version of the Calvo model. In the first subsection
we consider a version of the model in which intermediate good firms face a non constant
elasticity of demand for their output. In addition, we allow for a finite lag between the
time firms reoptimize prices and when they implement new plans. In the next subsection we
incorporate into our analysis the assumption that firms’ capital is fixed in place and is not
reallocated in response to shocks.
3
2.1. The Calvo Model with Non Constant Elasticity of Demand
At time t, a final good, Yt , is produced by a perfectly competitive firm. The firm does
so combining a continuum of intermediate goods, indexed by i ∈ [0, 1] using the following
technology suggested by Kimball (1995):
Z
0
1
(1)
G(Yit /Yt )di = 1.
Here G is increasing, strictly concave and G(1) = 1. In addition, Yit is the input of intermediate good i. The standard Dixit-Stiglitz specification corresponds to the special case:
G(Yit /Yt ) = (Yit /Yt )(µ−1)/µ , µ > 1.
(2)
For convenience we refer to the general version of G(·) that we work with as the non Dixit
-Stiglitz aggregator.
The final goods firm chooses Yt and Yit to maximize profits
Pt Yt −
Z
1
0
Pit Yit di
subject to (1). Here Pt and Pit denote the time t price of the final and intermediate good i,
respectively. The first order conditions to the firm’s problem imply
0−1
Yit = Yt G
µ
¶
Pit Yt
.
λt
(3)
Here λt ,the time t Lagrange multiplier on constraint (1), is given by:
λt = R
Pt Yt
.
0
G (Yit /Yt ) · (Yit /Yt )di
(4)
Throughout the symbol ‘0 ’ denotes the derivative operator and G0−1 (·) denotes the inverse
function of G0 (·). Our assumptions on G(·) imply that the firm’s demand for input Yit is a
decreasing in its relative price.3
0
3
Here we have used the fact that, given our assumptions on G, if x = G −1 (z),then dG0−1 (z)/dz =
00
1/G (x).
4
Intermediate good i ∈ [0, 1] is produced by a monopolist who uses the following technol-
ogy:
Yit = At Kitα L1−α
it
(5)
where 0 < α < 1. Here, Lit and Kit denote time t labor and capital services used to
produce intermediate good i, respectively. Intermediate good firms rent capital and labor
in economy wide perfectly competitive factor markets. The variable At denotes possible
stochastic disturbances to technology.
Profits are distributed to the firms’ owners at the end of each time period. Let st denote
the representative firm’s real marginal cost. Given our assumptions on factor markets, all
firms have identical marginal costs. Consequently, we do not index st by i. Marginal cost
depends on the parameter α and factor prices which the firm takes as given. The firm’s time
t profits are:
·
¸
Pit
− st Pt Yit ,
Pt
where Pit is the price of intermediate good i.
Intermediate good firms set prices according to a variant of the mechanism spelled out
in Calvo (1983). In each period, a firm faces a constant probability, 1 − θ, of being able
to reoptimize its nominal price. So on average a firm reoptimizes its price every (1 − θ)−1
periods. Following the literature, we assume that the firm’s ability to reoptimize its price is
independent across firms and time. For now we leave open the issue of what information set
the firm has when it resets its price.
We consider two scenarios for what happens if a firm does not reoptimize its price. In
the first scenario, the firm adopts what we call the static indexing scheme, i.e. it updates its
price according to the rule:
Pit = π̄Pit−1 .
(6)
Here π̄ is the long run average gross rate of inflation.4 In the second scenario, the firm adopts
what we call the dynamic indexing scheme, i.e. it sets its price according to5
Pit = π t−1 Pit−1 .
4
5
(7)
Other authors who make this assumption include Erceg, Henderson and Levin (2000) and Yun (1996).
See Christiano, Eichenbaum and Evans (2001) for a discussion of this form of indexation.
5
Let P̃t denote the value of Pit set by a firm that can reoptimize its price. In addition,
let Ỹt denote the time t output of this firm. Our notation does not allow P̃t or Ỹt to depend
on i because all firms who can reoptimize their price at time t choose the same price (see
Woodford, 1996 and Yun, 1996). In what follows we focus, for convenience, on specification
(6).The firm chooses P̃t to maximize
Et−τ
∞
X
l=0
h
i
(βθ)l vt+l P̃t π̄ l − st+l Pt+l Ỹt+l
(8)
subject to (3). Here, Et−τ denotes the conditional expectations operator and the firm’s t − τ
information set which includes the realization of all model variables dated t − τ and earlier.
In addition, vt+l is the time-varying portion of the firm’s discount factor. The intermediate
good firm views st , Pt , vt and λt as exogenous stochastic processes beyond its control.
Let p̃t = P̃t /Pt . Log linearizing the first order condition of the firm around the relevant
steady state values we obtain:
where
pb̃t = Et−τ
∞
X
"
l
(βθ) π̂ t+l + AEt−τ ŝt +
l=1
A=
∞
X
l=1
l
(βθ) (ŝt+l − ŝt+l−1 )
#
1 + G00 (1)/G0 (1)
.
2 + G000 (1)/G00 (1)
(9)
(10)
Throughout x̂t denotes the percent deviation of a variable xt from its steady steady state
value.
Several features of (9) are worth emphasizing. First, if inflation is expected to be at
its steady state level and real marginal cost is expected to remain constant after time t,
then the firm sets pb̃t = AEt−τ ŝt . Second, suppose the firm expects real marginal costs to be
higher in the future than at time t. Anticipating those future marginal costs, the firm sets
pb̃t higher than AEt−τ ŝt . It does so because it understands that it may not be able to raise
its price when those higher marginal costs materialize. Third, suppose firms expect inflation
in the future to exceed its steady state level. To avoid a decline in its relative price, the firm
incorporates expected future changes in the inflation rate into pb̃t .
The degree to which pb̃t responds to current and future values of ŝt is increasing in A
which in turns depends on the properties of G(·). One way to interpret A is that it governs
the degree of pass through from a rise in marginal cost to prices. For example, according
6
to (9), a highly persistent 1% increase in time t marginal cost from its steady state value,
induces the firm to initially raise its relative price by approximately A percent.
A different way to interpret A involves the elasticity of demand for a given intermediate
good, η(x),
η(x) = −
G0 (x)
xG00 (x)
(11)
where x = Ỹ /Y. In the Appendix we show that
A=
1
,
(ζ − 1)² + 1
(12)
P̃ ∂η(1)
,
η(1) ∂ P̃
(13)
where
²=
which is the percent change in the elasticity of demand due to a one percent change in the
the relative price of the good, evaluated in steady state. The variable ζ denotes the firm’s
steady state markup
η(1)
.
η(1) − 1
ζ=
(14)
In the standard Dixit Stiglitz case, ² is equal to zero and A is equal to one.
Relation (12) and (9) imply that the larger is ², the lower is A and the less responsive is
pb̃t to current and future values of ŝt . Other things equal, a rise in marginal cost induces a
firm to increase its price. A higher value of ² means that, for any given rise in its price, the
more elastic is the demand curve for the firm’s good. So, relative to the case where ² = 0,
the firm will raise its by price by less.
Zero profits in the final goods sector and our assumption about the distribution of θ
across firms and time imply,
Pt Yt =
Z
1
0
0−1
Pit Yit = (1 − θ)P̃t Yt+l G
Ã
P̃t Yt
λt
!
0−1
+ θπ̄Pt−1 Yt G
µ
¶
π̄Pt−1 Yt
.
λt
Linearizing this relationship about steady state yields
pb̃t =
θ
π̂ t
1−θ
7
(15)
Combining (9) and (15) we obtain
π̂ t = βEt−τ π̂ t+1 +
(1 − βθ) (1 − θ)
AEt−τ ŝt .
θ
(16)
When τ = 0 and A = 1 (the Dixit Stiglitz case), (16) reduces to the standard relationship
between inflation and marginal costs studied in the literature.6
Iterating forward on (16) yields
π̂ t =
∞
X
(1 − βθ) (1 − θ)
AEt−τ
β j ŝt+j
θ
j=0
(17)
Relation (17) makes clear a central prediction of the model: deviations of inflation from
its steady state value depend only on firms’ expectations of current and future deviations
of real marginal cost from its steady state value. The lower is A, i.e. the more sensitive is
the elasticity of demand for intermediate goods to price changes, the less responsive is π̂ t to
changes in expected values of ŝt+j . Similarly, the higher is θ, the smaller will be the response
of π̂t to expected changes in marginal cost. So the version of Calvo model considered in this
subsection has two distinct mechanisms which can account for a small response of inflation
to movements in marginal cost.
In the case where firms adopt the dynamic indexing rule, (7), the linearized first order
condition is
(
pb̃t = Et−τ ŝt +
∞
X
l=1
l
(βθ) (ŝt+l − ŝt+l−1 ) +
∞
X
l=1
l
(βθ) (π̂ t+l − π̂ t+l−1 )
)
(18)
and (16) takes the form
∆π̂ t = βEt−τ ∆π̂ t+1 +
(1 − βθ) (1 − θ)
AEt−τ ŝt .
θ
6
(19)
We derived (16) by linearizing around steady state inflation π̄. Various authors assume that firms which
do not reoptimize prices, leave their price unchanged, i.e. pit = pit−1 . The model is then linearized around
π̄ = 1. Since π̂t is defined as the percentage deviation from steady state, (16) does not depend on the assumed
value of π̄.
8
In addition (17) is replaced by
∞
X
(1 − βθ) (1 − θ)
β j ŝt+j
∆π̂ t =
AEt−τ
θ
j=0
(20)
Here it is the first difference of π̂ t that is a weighted average of the the conditional expected
value of current and future values of the deviation of real marginal cost from its steady state
value.
2.2. Allowing for Fixed Firm Capital
Standard variants of the Calvo model assume that capital is perfectly mobile and that firms
instantly reallocate their capital after a shock. In conjunction with the assumption that
labor is perfectly mobile, this implies that firms’ have the same marginal cost. Sbordone
(2002) considers a variant of the Calvo model in which A = 1, where firm capital is fixed in
place and is not reallocated in response to shocks. This implies that intermediate good firms
do not have the same marginal cost. The fixed capital assumption enables the Calvo model
to account for the time series behavior of inflation with lower degrees of price rigidities, i.e.
lower values of θ.
The basic intuition for this claim can be described as follows. When capital is mobile,
firm i takes marginal cost as given. As shown in the Appendix, when capital is immobile, firm
i’s marginal cost depends partly on economy wide factors but is also an increasing function
of its own level of output. Consider a shock that raises the economy-wide component of
marginal costs, say a rise in the real wage rate. Other things equal firm i will respond by
raising its price. But this reduces its output and leads to a countervailing fall in the firm
specific component of marginal cost. On net, this leads the firm to raises its price by less
than it would if capital were perfectly mobile.
We now describe a version of the Calvo model with immobile capital and with intermediate good firms that do not necessarily face a constant elasticity of demand for their goods.
We refer the reader to the appendix for details. Consistent with Galí et. al. (2001) and
Sbordone (2002) we assume that the capital stock of firm i is proportional to the aggregate
capital stock.
With static indexation scheme, the analog to (16) in this model is given by
9
π̂ t = βEt−τ π̂ t+1 +
(1 − βθ) (1 − θ)
· A · D · Et−τ ŝt
θ
(21)
where A is defined in (12) and D is given by
D=
1
ζ
α
1 + A 1−α
(ζ−1)
.
(22)
When A = 1, (21) reduces to the case considered Sbordone (2002). Under dynamic indexation we obtain
∆π̂ t = ∆βEt−τ π̂ t+1 +
(1 − βθ) (1 − θ)
· A · D · Et−τ ŝt
θ
(23)
As long as α is between zero and one, and the steady state value of the markup ζ exceeds
one, then D ≤ 1. So for any given value of θ, fixed firm capital, like a non constant elasticity
of demand, reduces the response of π̂ t to movements in ŝt .
3. Assessing the Empirical Plausibility of the Model
This section describes our empirical methodology and the data used in the analysis.
3.1. Methodology
In principle there are a variety of ways to evaluate our model empirically. For example, one
could embed it in a fully specified general equilibrium model of the economy. This would
involve, among other things, modeling household labor and consumption decisions, credit
markets, fiscal policy and monetary policy. If in addition, one specified the nature of all the
shocks impacting on the economy, one could estimate and test the model using a variety
of statistical methods like maximum likelihood.7 Another strategy would be to assess the
model’s predictions for a particular shock, such as a disturbance to monetary policy or a
shock to technology.8
Here we apply the econometric strategy pioneered by Hansen (1982) and Hansen and
Singleton (1982) and applied recently to the Calvo model by Galí and Gertler (1999) and
7
For examples of this approach see for example Ireland (2002) and Cho and Moreno (2002) and Moreno
(2003).
8
See for example Christiano, Eichenbaum and Evans (2001) and Altig, Christiano, Eichenbaum and Linde
(2003), respectively.
10
Galí, Gertler and López-Salido (2001). The idea is to exploit the fact that in any model
incorporating Calvo pricing, certain restrictions must hold. One can analyze these restrictions, without making assumptions about other aspects of the economy. Of course, in the
end, we need a fully specified model of the economy within which to assess the consequences
of alternative policies. The approach that we discuss here has the advantage of focusing on
the empirical plausibility of one key building block that could be an element of many models.
To derive the testable implications of the model, it is convenient to focus on the model
with static indexation and define the random variable
ψ t+1 = π̂ t − β π̂ t+1 −
(1 − βθ)(1 − θ)
· A · D · ŝt .
θ
(24)
The method below provides a procedure for simultaneously estimating the structural parameters of interest and testing this implication. Throughout we set β equal to 0.96 on an
annual basis.
Since π̂ t is in agents’ time t − τ information set, (16) can be written as:
Et−τ ψ t+1 (σ) = 0.
(25)
where σ denotes the structural parameters of the model. It follows that
Eψ t+1 (σ)Xt−τ = 0
(26)
for any k dimensional vector Xt−τ in agents’ time t − τ information set. We exploit (26) to
estimate the true value of σ, σ 0 , and test the overidentifying restrictions of the model using
Hansen’s (1982) Generalized Method of Moments procedure.9 Our estimate of σ is
σ̂ = arg min
JT (σ),
σ
(27)
JT (σ) = gT (σ)0 WT gT (σ),
(28)
where
9
We require that {π̂t , ŝt , Xt } is a stationary and ergodic process.
11
and
T h
i
1 X
gT (σ) = ( )
ψ t+1 (σ)Xt−τ .
T t=1
(29)
Here T denotes the size of our sample and WT is a symmetric positive definite matrix that can
depend on sample information. The choice of WT that minimizes the asymptotic covariance
matrix of σ̂ is a consistent estimate of the spectral density matrix of {ψ t+1 (σ 0 )Xt−τ } at
frequency zero. Our theory implies that ψ t+1 (σ)Xt−τ has a moving average representation
of order τ . So we choose WT−1 to be a consistent estimate of
τ
X
E[ψ t+1+k (σ)Xt+k−τ ][ψ t+1+k (σ)Xt+k−τ ]0
(30)
k=−τ
The minimized value of the GMM criterion function, JT , is asymptotically distributed as
a chi-squared random variable with degrees of freedom equal to the difference between the
number of unconditional moment restrictions imposed (k) and the number of parameters
being estimated.10
One does not have to impose the restriction that ψ t+1 (σ)Xt−τ has an MA(τ ) representation constructing an estimate of WT−1 . Specifically, one could allow for higher order serial
correlation in the error term than the theory implies. However, as we describe below, whether
one does so or not has an important impact, in practice, on inference.
It is evident from (24) and (27) - (28) that θ, A and D are not separately identified. All
that can be identified given the assumptions made so far is the parameter
c=A·D·
(1 − β θ̂)(1 − θ̂)
.
θ̂
But given given any value of c and assumed values for A and D, one can deduce the implied
estimate of θ, θ̂. When capital is mobile, we have D = 1 and θ̂ can be derived from the
relation
A=
θ̂c
(1 − β θ̂)(1 − θ̂)
10
.
According to relation (17), π̂ t is predetermined at time t − τ . If we were only interested in assessing the
hypothesis that inflation is predetermined at time t − τ, we could test whether any variable dated between
time t − τ and t has explanatory power for time t inflation.
12
When firm capital is fixed, one can deduce θ̂ using (22),
A=
θ̂c
D(1 − β θ̂)(1 − θ̂)
and values for α and ζ. These expressions imply that with information or priors about the
nature of demand for goods and the degree of capital mobility we can identify the frequency
of reoptimization which is required to render the extended Calvo model consistent with the
data.
3.2. Data
Our benchmark sample period is 1959:1 - 2001:4. However, numerous observers have argued
that there was an important change in the nature of monetary policy with the advent of
the Volker disinflation in the early 1980s. It is also often argued that the Fed’s operating
procedures were different in the early 1980s than in the post-1982 period. Accordingly we
reestimated the model over two distinct subsamples: 1959:1-1979:2 and 1982:3- 2001:4. We
report results for two measures of inflation: the GDP deflator and the price deflator for
personal consumption expenditures.11 We measure π̂ t as the difference between actual time
t inflation and the sample average of inflation.
With mobile capital, real marginal costs are equal to the real product wage divided by
the marginal product of labor. Given the production function (5), this implies that real
marginal cost is proportional to labor’s share in national income, Wt Lt /(Pt Yt ), where Wt
is the nominal wage. In practice we measure Wt Lt as nominal labor compensation in the
non-farm business sector and we measure Pt Yt as nominal output of the non-farm business
sector. The variable ŝt is then measured as the difference between the log of our measure of
labor’s share and its mean. This is a standard measure of ŝt which has been used by Galí
and Gertler (1999), Galí et. al. (2001) and Sbordone (2001). As it turns out, this is the
correct measure of ŝt even when capital is not mobile (see the Appendix).
Rotemberg and Woodford (1999) discuss possible corrections to this measure that are
appropriate for different assumptions about technology. These include corrections to take
11
All data sources are listed in the Appendix. We also considered the price deflator for the non farm
business sector and the consumer price index (CPI) and found that our key results are insensitive to these
alternative measures.
13
into account a non-constant elasticity of factor substitution between capital and labor, and
the presence of overhead costs and labor adjustment costs. We redid our analysis for these
alternative measures of marginal costs and found that they do not affect the qualitative
nature of our results.
Consider next the instrument vector Xt−τ . Let Zt denote the four dimensional vector
consisting of the time t value of real marginal cost, quadratically detrended real GDP, inflation, and the growth rate of nominal wages in the non farm business sector. Our specification
of Xt−τ is given by12
Xt−τ = {1, Zt−τ , ψ t−τ }0 .
4. Empirical Results
In this section we present our empirical results. To facilitate comparisons with the literature,
we report point estimates of θ corresponding to the identifying assumption that capital is
mobile and G(·) in (1) is of the Dixit-Stiglitz form. The first subsection reports results for
the case in which there are no delays in implementing new optimal price decisions (τ = 0).
When A = 1 and capital is mobile, this corresponds to the standard Calvo model. In the
second subsection, we discuss the impact of allowing for a delay in implementing new optimal
price decisions. In the third subsection, we modify the model to allow for variants of the
rule of thumb firms considered by Galí and Gertler (1999).
4.1. The Standard Calvo Model
We begin by analyzing results for the standard Calvo model (τ = 0) in the case where firms
adopt the static indexing scheme. The top panel of Table 1 summarizes results obtained using
the full sample. We report our estimate of the parameter θ (standard error in parenthesis)
and the JT statistic (p-value in square brackets). The label L refers to the maximal degree of
12
Gali and Gertler (1999) use an instrument list consisting of a constant and lagged values of Zt where
the latter is augmented to included an index of commodity prices and the spread between the annual
interest rate on the ten year Treasury Bond and three month bill. We redid our basic analysis setting Xt
to {1, Zt−j , j = 0, 1, 2, 3}0 and {1, Zt−j , j = 1, 2, 3, 4}. Gali et. al (2001) adopt the same specification as we
do but set Xt = {1, Zt−j , j = 1, 2, 3, 4}. It turns out that the point estimates are similar across different
specifications of Xt , including the specification of Xt used in this paper. However, using a larger set of
instruments leads to misleading inference about the plausibility of the overidentifying restrictions implied by
the model. Specifically, often we cannot reject the model with a larger set of instruments on the basis of the
JT statistic but we can do so with the smaller set of instruments.
14
serial correlation that we allow for when estimating the weighting matrix WT . We consider
two values for L : (1) L = 0, which corresponds to the degree of serial correlation in ψ t+1
implied by this version of the model, and (2) L = 12, the value used by Galí and Gertler
(1999). Both values of L are admissible. But, by setting L to zero we are imposing all of
the restrictions implied by the model. This may lead to greater efficiency of our estimator
and more power in our test of the model’s overidentifying restrictions.
Recall that Table 1 presents our estimates of the model’s parameters, under the assumption that A and D equal one. Notice that θ is estimated with relatively small standard
errors. In addition the point estimate itself is reasonably robust across the different inflation measures and the two values of L. The point estimates range from a low of 0.87 to a
high of 0.91. This implies that on average firms wait between 7.5 and 11 quarters before
reoptimizing their prices.
We hesitate to attribute much importance to these point estimates. When L = 12 the
model cannot be rejected at the 5% significance level, although it can be rejected at the
1% significance level. However, when we set L = 0 the model is strongly rejected for both
inflation measures. Evidently, imposing all of the relevant restrictions implied by the model
on the weighting matrix has an important impact on inference.
The middle and bottom panels of Table 1 report our sub sample results. Note that when
L = 12, there is virtually no evidence against the model for either measure of inflation,
regardless of which subsample we consider. In the first sample period, when L = 0, the
model is rejected at the 5% significance level for both measures of inflation. Interestingly,
when L = 0, the model is decisively rejected using data from the second subsample when we
measure inflation using the GDP deflator. There is considerably less evidence against the
model in this case when we use the PCE deflator based measure of inflation. Comparing
the point estimates in the three panels, we see that inference about θ is reasonably robust
to allowing for a split in the sample. As above, we are hesitant to attach much importance
to this result in light of the overall statistical evidence against the standard Calvo model.
Table 2 reports results when we allow for dynamic indexation. Our estimation strategy
is the same as the one described above except that the random variable ψ t+1 is defined as
ψ t+1
"
#
(1 − βθ)(1 − θ)
= ∆π̂ t − β∆π̂ t+1 −
Aŝt .
θ
15
(31)
The key thing to note is that this version of the model is also rejected when we set L = 0.
4.2. Alternative Timing Assumptions
Table 3 reports the results of estimating the model when τ = 1 and we assume that firms
adopt the static indexation scheme (6). In the previous subsection we showed that imposing
the degree of serial correlation in ψ t+1 implied by the model on the estimator of the weighting
matrix WT improves the power of our statistical tests. So for the remainder of the analysis
we report results only for the case where these restrictions are imposed. In the case of τ = 1,
this means setting L = 1. The instrument used are
Xt−1 = {1, Zt−1 , ψ t−1 }0
(32)
Two key results from Table 3 are worth reporting. First, regardless of which sample
period we consider or which measure of inflation we use, there is virtually no statistical
evidence against the model. Second, θ is estimated with reasonable precision with the point
estimates ranging from a low of 0.83 to 0.91. This corresponds to firms changing prices on
average from between 6 quarters and 11 quarters. This frequency seems high relative to
evidence based on microeconomic data (see for example Bils and Klenow 2003).
Table 4 reports the results of estimating the model when τ = 1 and we assume that
firms adopt the dynamic indexation scheme (7). As with the static indexing scheme, there
is virtually no statistical evidence against the model. Moreover the point estimates of the
parameters θ are quite similar, now ranging from a low 0.83 to a high of 0.89.
We conclude that allowing for a one period lag (τ = 1) in the implementation of new
pricing plans is sufficient to overturn our statistical evidence against the standard Calvo
model with either indexing scheme. But it is not sufficient to generate economically plausible
parameter estimates of the degree of price stickiness. This is true for both the static and
dynamic indexing schemes. Of course this conclusion is conditional on the assumption that
intermediate goods are combined via a Dixit -Stiglitz technology to produce final goods
(A = 1) and that capital is mobile across firms (D = 1). Before exploring the quantitative
trade-off between the parameter A, capital immobility and the degree of price stickiness,
we investigate the claim that the standard Calvo model must be modified to allow for the
presence of non-optimizing firms.
16
4.3. ‘Rule of Thumb’ Firms
Galí and Gertler (1999) have argued it is necessary to allow for backward looking ‘rule of
thumb’ firms to render the Calvo model consistent with the data. In the previous section
we argued that there was little evidence against the Calvo model, amended to allow for a
one quarter delay between when firms reoptimize their price plans and when they actually
implement the new plan. That argument was based on the JT statistic that formed the basis
of a test of the model’s over identifying restrictions. Galí and Gertler (1999) argue that this
test may have low power against specific alternatives. In this section, we accomplish two
tasks. We begin by confirming Galí and Gertler’s result that there is evidence of backward
looking firms under the static indexation scheme. We then show that this evidence disappears under the dynamic indexing scheme. For simplicity we derive the model under the
assumption that capital is mobile and A = 1.
4.3.1. Optimizing Firms with Static Indexing
As in Galí and Gertler (1999) we assume that there are two types of firms in the economy. A
fraction (1−ω) of intermediate good firms are optimizing Calvo type firms. That is, they face
a constant probability, 1 − θ, of being able to reoptimize their nominal price. As above, when
they reoptimize, they solve problem (8) subject to (3). When they do not reoptimize, they
adopt the static optimization scheme, (6). A fraction ω of intermediate good firms adopt
the rule of thumb for setting prices discussed in Galí and Gertler (1999). With probability
θ, rule of thumb firm i sets its price according to13
Pit = π̄Pit−1 .
(33)
With probability (1 − θ), the firm sets its price according to
Pt0 = π t−1 P̄t−1 .
(34)
P̄t = (1 − ω)P̃t + ωPt0 .
(35)
Here
13
This rule is precisely the same as the one considered by Gali and Gertler (1999) except that they assume
π̄ = 1.As explained above, this has no impact on the estimation equations used in the analysis.
17
and P̃t denotes the price set by firms that can reoptimize their price at time t. The aggregate
price level is given by,
·
³
Pt = (1 − θ) P̄t
´
1
1−µ
+ θ (π̄Pt−1 )
1
1−µ
¸1−µ
(36)
Log linearizing (33)-(36) and combining the resulting expressions with (??) one can show
that the analog to (16) is given by:
π̂ t =
βθ
ω
(1 − ω)(1 − βθ)(1 − θ)
Et−τ π̂ t+1 + π̂ t−1
Et−τ ŝt
φ
φ
φ
(37)
where φ = θ + ω [1 − θ(1 − β)] . When ω = 0, (37), collapses to the analog expression for π̂ t
in the standard Calvo model with static indexing.
We estimate the parameters of the model assuming τ = 1, using the methodology and
instruments described in section 3. Table 5 summarizes our results. Four key findings are
worth noting. First, using the full sample, we estimate that roughly 50% of firms behave in
a rule of thumb manner, with the exact percent depending on how we measure inflation. In
both cases, we can reject, at conventional significance levels, the null hypothesis that there
are no rule of thumb firms (ω = 0). Second, there is virtually no evidence against the overidentifying restrictions imposed by the model. Third, the point estimates of θ continue to be
implausibly large relative to evidence based on micro data. Fourth, there is little evidence of
rule of thumb firms once we allow for a split in sample if we measure inflation using the GDP
deflator. But there is still evidence that ω is greater than zero when we measure inflation
using the PCE deflator, at least in the second subsample.
Viewed overall, the results in Table 5 are consistent with Galí and Gertler’s conclusion
that to render the standard Calvo model consistent with the data, one must allow for the
presence of some firms who use backward looking rules when setting prices.
4.3.2. Optimizing Firms with Dynamic Indexing
We now modify the model considered in the previous subsection on exactly one dimension:
we assume that optimizing firms adopt the dynamic optimization scheme (7) instead of the
18
static scheme (6). With this modification, the aggregate price level is given by:
·
³
Pt = (1 − θ) P̄t
´
1
1−µ
+ θ ((1 − ω)π t−1 Pt−1 + ωπ̄Pt−1 )
1
1−µ
¸1−µ
(38)
Replacing (36) with (38) in the derivation with static indexation one can show that the
analog to (16) is now given by:
Et−1
∆π̂t −
βθ
∆π̂ t+1
φ0
−
ωθ
(1
φ0
− ωθ (1 − ω)(1 − βθ)π̂
t−1 −
φ0
− ω)∆π̂t−1
(1−ω)(1−βθ)(1−θ)
Et−τ ŝt
φ0
= 0.
(39)
where φ0 = θ(1 − ω) + ω. When ω = 0, (39), collapses to the analog expression for π̂ t in the
standard Calvo model under the dynamic indexing scheme.
We estimate the parameters of the model assuming τ = 1, using the methodology and
instruments described in section 3. Table 6 summarizes our results. Three key findings
emerge. First, our point estimates of ω are substantially smaller than those emerging under
the assumption that optimizing firms adopt the static indexation scheme. Indeed for the full
sample our point estimates are roughly equal to zero. Second, our point estimates of θ are
similar to those obtained when we estimated the model under the constraint that ω is equal
to zero (see Table 4). Perhaps most importantly, there is virtually no evidence of rule of
thumb firms. Regardless of which sample we consider, or which measure of inflation we use,
we cannot reject the null hypothesis that ω = 0. We conclude that the evidence for rule of
thumb of firms disappears once we allow for dynamic indexation.
5. Interpreting the Parameters of the Estimated Calvo Model
Our empirical results indicate that the Calvo sticky price model is consistent with the aggregate time series data if we assume that τ = 1 and optimizing firms use the dynamic
indexation scheme. Specifically, there is little evidence against the over identifying restrictions imposed by that version of the model and there is little evidence of ‘rule of thumb’
firms. However, the degree of price stickiness implied by the model is implausibly large
relative to existing microeconomic evidence.
Taken at face value, these results imply that the Calvo model can be rescued statistically,
but not in any interesting economic sense. But as stressed above, this conclusion emerges
19
under the maintained assumptions that capital is mobile across firms (D = 1) and that
A = 1. In this section we explore the quantitative nature of the trade-off between A, θ and the
assumption of capital mobility. Throughout we base our calculations on the estimated Calvo
model with τ = 1 and dynamic indexation (Table 4). Our results are displayed in Tables 7
and 8. Table 7 reports values of A and ², the percent change in the elasticity of demand due
to a one percent change in the relative price of good i,.for values for θ ∈ {0.25, 0.50, 0.60,
0.75}. Table 8 reports values of θ for two values of ², 10 and 33.
Panel A of Tables 7 and 8 reports the results of these calculations based on estimates of
c using the full sample period. As can be seen, our results are similar for the two inflation
measures (in Panel’s B and C as well). For convenience, we focus our discussion on the case
where inflation is measured using the GDP deflator. Three key features emerge from Panel
A. First, if we assume that θ ≤ 0.75, so that firms reoptimize prices at least once a year,
then A is substantially less than one. While not evident from the Table, the joint hypothesis
that θ ≤ 0.75 and A = 1 can be rejected at conventional significance levels.14 . That is,
one cannot adopt the Calvo model and simultaneously take the position that θ ≤ 0.75 and
the production technology G(·) is Dixit - Stiglitz. This is true regardless of whether or not
capital is mobile.
Second, when θ = 0.75, the value of ² in the mobile capital case is just a bit above 33,
the benchmark value considered in Chari, Kehoe and McGrattan (2000) and Kimball (1995).
Given the non linear relationship between θ and A, the implied value of ² rises very quickly
for further reductions in θ. So with this version of the model, it seems to be difficult to
rationalize values of θ substantially below 0.75. Bergin and Feenstra (2000) argue that ² is
roughly equal to 10. If we take this to be the relevant benchmark value, it is difficult for this
version of the model to rationalize values of θ much less than roughly 0.83.
Third, notice that in the immobile capital case, ² is actually negative when θ = 0.75.
That is, given our point estimate of c, if capital is immobile, firms must reoptimize prices
more frequently than once a year, for the model to be internally consistent. As it turns out,
² is zero when θ = 0.71. This is our point estimate of θ when we estimate the model assuming
capital is immobile and we impose the restriction that A = 1. Interestingly, if we assume
that ² is 10 or 33 then θ is 0.69 and 0.66, respectively. So with immobile capital, the model
14
We tested this joint hypothesis using the asymptotic distribution of A given the estimated sampling
distribution of c
20
is consistent with the view that firms reoptimize prices roughly once every three quarters.
Panels B and C report the analog results on the subsample periods. A number of key
results emerge here. First, as above, we can always reject, at conventional significance levels,
the joint hypothesis that θ ≤ 0.75 and A = 1. Second, we find that for both the mobile and
immobile capital cases, the model is consistent with lower values of θ once we allow for a split
in the sample. The decline in θ is particularly notable in the case of the second subsample.
For example, suppose we assume that ² = 33. Then, for the case of mobile and immobile
capital, we obtain values of θ equal to 0.67 and 0.56, respectively. If we assume that ² = 10,
we obtain values of θ equal to 0.77 and 0.60, in the two cases.
To summarize, based on the full sample results, the model with immobile capital and
the non Dixit Stiglitz aggregator is consistent with the view that firms reoptimize prizes
roughly once every three quarters. This is true even if we take as our upper bound for ² the
value of 10, suggested by Bergin and Feenstra (2000). If we take as our upper bound for ²,
the benchmark value of 33, used by Chari, Kehoe and McGrattan (2000), the model with
mobile capital and A different from 1 is consistent with the view that firms reoptimize prices
roughly once a year.
Even more favorable results emerge if we take as given that there is split in the sample
period owing to a change in monetary policy. Averaging across the two subsamples, we find
that the model with mobile capital and the non Dixit Stiglitz aggregator is consistent with
the view that firms reoptimizing prices every 3 quarters, if we are willing to assume that
² = 33. The model with immobile capital and the non Dixit Stiglitz aggregator is consistent
with the view that firms reoptimize prices more often than once every three quarters even if
we assume that ² = 10.
6. Conclusion
This paper discussed the empirical performance of the Calvo model of sticky goods prices.
We argued this model can be rescued statistically by assuming dynamic indexation and a
one quarter implementation lag of a reoptimized price. Yet, the estimated frequency of reoptimization does make much economic sense. This conclusion emerges under the maintained
assumptions that capital is mobile across firms and that output is a Dixit-Stiglitz aggregate. Finally, we explored the quantitative nature of the trade-off between non-Dixit-Stiglitz
aggregation, the frequency of reoptimization, and the assumption of capital mobility.
21
Based on the full sample results, the model with immobile capital and the non Dixit
Stiglitz aggregator is consistent with the view that firms reoptimize prizes roughly once
every three quarters, using the relatively low upper bound for the demand elasticity as
suggested by Bergin and Feenstra (2000). Using the higher elasticity of Chari, Kehoe and
McGrattan (2000), the model with mobile capital and A different from 1 is consistent with
the view that firms reoptimize prices roughly once a year.
More favorable results emerge if we take as given that there is split in the sample period
owing to a change in monetary policy. Averaging across the two subsamples, we find that
the model with mobile capital and the non Dixit Stiglitz aggregator is consistent with the
view that firms reoptimizing prices every 3 quarters, using the larger elasticity. The model
with immobile capital, the non Dixit Stiglitz aggregator and the lower elasticity is consistent
with the view that firms reoptimize prices more often than once every three quarters.
22
References
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Technology Shocks, manuscript, Northwestern University.
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[3] Calvo, Guillermo, 1983, ‘Staggered Prices and in a Utility-Maximizing Framework, Journal of Monetary Economics, 12(3): 383-98.
[4] Chari, V.V., Kehoe, Patrick and Ellen McGrattan, ‘Sticky Price Models of the Business
Cycle: Can the Contract Multiplier Solve the Persistence Problem?’, Econometrica,
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[5] Clarida, R., Mark Gertler and Jordi Gali, 2001, “Optimal Monetary Policy in Closed
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[6] Christiano, Eichenbaum and Evans (2001), Nominal Rigidities and the Dynamic Effects
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[8] Erceg, Christopher, J., Henderson, Dale, W. and Andrew T. Levin, 2000, ‘Optimal
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[10] Hansen, Lars P. and Kenneth J. Singleton (1982) ‘Generalized Instrumental Variables
Estimation of Nonlinear Rational Expectations Models’, Econometrica, Vol. 50, 12691286.
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analysis,’ Journal of Monetary Economics 44, pp. 195-222.
[13] Gali, Jordi, Mark Gertler and David Lopez-Salido, “European Inflation Dynamics,”
European Economic Review, 2001.
[14] Taylor, John ‘Aggregate Dynamics and Staggered Contracts’, Journal of Political Economy, 88, February, 1980.
23
[15] Ireland, Peter, 1997, ‘ A Small, Structural, Quarterly Model for Monetary Policy Evaluation, Carnegie Rochester Conference Series on Public Policy, 47, pages 83-108.
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[17] Mankiw, N.G. and R. Reiss, ‘Sticky Information: A Proposal to Replace the New
Keynesian Phillips Curve, Quarterly Journal of Economics, 117, Nov. 2002, 1295 1328.
[18] Moreno, A, 2003, “Reaching Inflation Stability,” manuscript Universidad de Navarra,
Spain.
[19] Rotemberg, Julio J. and Michael Woodford, 1997, ‘An Optimization-Based Econometric Framework for the Evaluation of Monetary Policy,’ National Bureau of Economic
Research Macroeconomics Annual.
[20] Rotemberg, Julio J. and Michael Woodford, 1999, ‘The Cyclical Behavior of Prices and
Costs’, Handbook of Monetary Economics, Vol. 1B, Michael Woodford and John Taylor,
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[21] Sbordone, Argia, 2001, ‘An Optimizing Model of U.S. Wage and Price Dynamics,’ Journal of Monetary Economics.
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24
Appendix
In this appendix we describe our data, describe how to interpret the parameters A and D,
and provide more detail on the version of the Calvo model with non-Dixit-Stiglitz aggregation
and immobile capital.
Data
Our data are from the Haver Analytics database. For each data series below we provide
a brief description and, in parenthesis, the Haver codes for the series used.
• Price measures: GDP deflator is the ratio of nominal GDP (GDP) and real chainweighted GDP (GDPH); personal consumption expenditures deflator (JCBM2).
• Real marginal costs: Share of labor income in nominal output for the non-farm business
sector which is proportional to the Bureau of Labor Statistics measure of nominal unit
labor costs divided by the non-farm business deflator (LXNFU/LXNFI).
• Adjusted real marginal costs: Per capita hours - hours non-farm business sector (LXNFH)
divided by over 16 population (LN16N); Capital-output ratio - annual private fixed capital (EPQ) interpolated with quarterly private fixed investment (FH) divided by GDP
(GDPH), all variables in chained 1996 dollars.
• Instruments: Quadratically detrended real GDP is the residual of a linear regression
of real GDP (GDPH) on a constant, t and t2 ; inflation is the first difference of the log
of the price measures; growth rate of nominal wages is the first difference of the log of
nominal compensation in the non-farm business sector (LXNFC).
Interpreting Estimates of A
Recall that the elasticity of demand for a given intermediate good is
G0 (x)
xG00 (x)
(40)
Ỹ
Y
(41)
1 − 1/η̄
2 + G000 (1)/G00 (1)
(42)
η(x) = −
where
x=
The coefficient A can be written
A=
where
η̄ = −
G0 (1)
1 × G00 (1)
25
is the steady state elasticity of demand. Note that in steady state an intermediate good firm
sets price as a markup over marginal cost, where the markup, ζ, is
ζ=
η̄
η̄ − 1
(43)
A variety of authors have considered the value of
¯
P̃ ∂η(x) ¯¯
¯
²=
η(x) ∂ P̃ ¯x=1
This is the percent change in the elasticity of demand due to a one percent change in the
own price at the steady state. The value of ² can be derived in terms of A and η̄ (or ζ) using
(3), (40), (41), and (42)
"
#
P̃ ∂η(x) ∂x ∂ Ỹ
² =
η(x) ∂x ∂ Ỹ ∂ P̃ x=1
"
#
1 − 1/η̄
= 1 + η̄
−1
A
"
#
ζ
1
= 1+
−1
ζ − 1 ζA
Notice that under Dixit-Stiglitz, when A = 1, ² = 0. This is to be expected: under DixitStiglitz the markup is constant. We can solve this for A
A=
1
(ζ − 1)² + 1
Immobile Capital and Non-Dixit-Stiglitz Aggregation 15
Here we derive the inflation equation under static indexation. The derivation for the
dynamic indexation case follows the same basic steps.
Intermediate good firms will not have the same marginal cost if capital is fixed in place
and does not reallocate in response to shocks. The crucial feature of the derivation is
that log linearized date t + j real marginal costs for an intermediate good producer which
implements reoptimized prices at t can be written in terms of ŝt+j , which is measured in
terms of aggregates, and a firm-specific term. To achieve this result one needs to make the
auxilary assumption that firm level capital stocks are proportional to the aggregate capital
stock. With a unit measure of firms we have Kit = Kt where Kt is the aggregate stock of
capital.
15
We thank Argia Sbordone for correspondence which helped us clarify the analysis in this section.
26
Real marginal cost at date t + j firms which implement a reoptimized price plan at date
t, st+j,t is
1 Wt+j L̃t+j
st+j,t =
(44)
1 − α Pt+j Ỹt+j
where the tildes denote intermediate good firm choices. Linearizing this yields
b̃
b̃
ŝt+j,t = Ŵt+j − P̂t+j + Lt+j − Y t+j
The intermediate good firm production function implies,
Ã
Lit+j
1
= 1/(1−α)
Yit+j
At+j
Yit+j
Kt+j
(45)
!α/(1−α)
Linearizing this and substituting into (45) yields
ŝt+j,t = Ŵt+j − P̂t+j +
α b̃
α
1
K̂t+j −
Ât
Y t+j −
1−α
1−α
1−α
(46)
Recall that the demand for intermediate good i in date t + j under a reoptimized plan
implemented at t is
Ã
!
j
π̄
Y
P̃
t
t+j
Ỹt+j = Yt+j G0−1
(47)
λt+j
Linearizing this yields
b̃
Y t+j = Ŷt+j + gb̃t+j
³
´
where gb̃t+j is the linearized version of G0−1 P̃t π̄ j Yt+j /λt+j . Substituting this last expression
b̃
for Y t+j in (46) implies
α
ĝt+j
1−α
·
¸
α
α
1
+
Ŷt+j −
K̂t+j −
Ât
1−α
1−α
1−α
ŝt+j,t = Ŵt+j − P̂t+j +
(48)
The last part of the derivation involves replacing the term in square brackets with the
linearized aggregate labor-output ratio. Linearizing the aggregation technology for the final
good firm yields
Z
1
Ŷt+j =
0
Ŷit+j di
Substituting from the linearized intermediate good firm’s technology and using the definition
of aggregate labor input as the arithmetic sum of labor input across intermediate good
producers yields
Ŷt+j = At + αK̂t + (1 − α)
Z
0
1
L̂it+j di
= At + αK̂t + (1 − α)L̂t+j
27
Using this we can write
L̂t+j − Ŷt+j =
α
α
1
Ŷt+j −
K̂t+j −
Ât
1−α
1−α
1−α
(49)
Notice that the right hand side of this equation is identical to the term in square brackets
in equation (48). Using the labor share definition of st+j
b
b
ŝt+j = Ŵt+j − P̂t+j + L
t+j − Yt+j .
(50)
After substituting (49) into (48) and using (50) we achieve the desired decomposition of
ŝt+j,t :
ŝt+j,t = ŝt+j + gb̃t+j .
The objective of the intermediate firm is
max Et−τ
P̃t
∞
X
h
1/(1−α)
(βθ)j ν t+j P̃t π̄ j Ỹt+j − Wt+j A1/(1−α)
Ỹt+j
t+j
j=0
1/(1−α)
i
1/(1−α)
where Ỹt+j is given by (47) and At+j
Ỹt+j
= H̃t+j . Log linearizing the first order
condition around the relevant steady state values we obtain:
pb̃t = Et−τ
∞
X
l=1
"
l
(βθ) π̂ t+l + A · D · Et−τ ŝt +
∞
X
l=1
#
l
(βθ) (ŝt+l − ŝt+l−1 )
(51)
where A is the coefficient defined above and D is given by
D=
1
ζ
α
1 + A 1−α
(ζ−1)
Under Dixit-Stiglitz, A = 1 so the coefficient on marginal cost is
D =
1
1+
ζ
α
1−α (ζ−1)
1
=
α
1 + 1−α
µ
(1 − α)
=
1 + α(µ − 1)
which is the same coefficient displayed in Sbordone (2001) and Gali, et. al (2001).
Linearizing the zero profit condition for final good firms and combining it with the linearized first order condition yields the aggregate inflation equation under non-Dixit-Stiglitz
aggregation and immobile capital:
Et−τ
(
)
(1 − βθ) (1 − θ)
π̂ t − βπ̂ t+1 −
· A · D · ŝt = 0
θ
28
Table 1. Estimates of the Standard model with Static Indexation
Inflation
Measure
L=0
θ
JT
L = 12
θ
JT
1959:I-2001:IV
GDP Deflator 0.90
28.2
0.91
(0.05) [9e-5] (0.03)
PCE Deflator 0.87
36.9
0.88
(0.04) [2e-6] (0.02)
GDP Deflator
PCE Deflator
10.2
[0.04]
11.1
[0.03]
1959:I-1979:II
12.1
0.87
4.58
[0.02] (0.03) [0.33]
16.8
0.83
5.60
[0.02] (0.02) (0.23)
0.86
(0.05)
0.82
(0.04)
1982:III-2001:IV
GDP Deflator 0.87
15.8
0.90
6.16
(0.04) [0.003] (0.03) [0.19]
PCE Deflator
0.87
8.89
0.89
4.39
(0.04) [0.06] (0.03) [0.36]
Notes: The JT statistics are distributed as χ2 random variables with 4 degrees of freedom.
Standard errors in parenthesis. P-values in brackets. In the L = 12 cases the Newey-West
correction to the weighting matrix is used.
29
Table 2. Estimates of the Standard model with Dynamic Indexation
Inflation
Measure
θ
L=0
JT
GDP Deflator 0.81
(0.03)
PCE Deflator 0.79
(0.03)
θ
L = 12
JT
1959:I-2001:IV
35.0
0.91
[6e-7] (0.02)
47.9
0.88
[1e-9] (0.02)
10.2
[0.04]
10.8
[0.03]
1959:I-1979:II
GDP Deflator 0.76
17.5
0.87
5.59
(0.05) [0.002] (0.04) [0.23]
PCE Deflator
0.77
18.1
0.88
5.80
(0.04) [0.002] (0.03) (0.22)
1982:III-2001:IV
GDP Deflator 0.68
17.0
0.83
4.41
(0.04) [0.002] (0.08) [0.35]
PCE Deflator
0.54
14.2
0.77
4.54
(0.03) [0.007] (0.06) [0.34]
Notes: The JT statistics are distributed as χ2 random variables with 3 degrees of freedom.
Standard errors in parenthesis. P-values in brackets. In the L = 12 cases the Newey-West
correction to the weighting matrix is used.
30
Table 3: Prices Chosen One Period In Advance with Static Indexation
Inflation
Full Sample
Measure
θ
JT
GDP Deflator 0.89
6.89
(0.03) [0.14]
PCE Deflator 0.90
8.54
(0.03) [0.07]
1959:I-1979:II
θ
JT
0.84
2.96
(0.03) [0.56]
0.83
2.95
(0.05) [0.56]
1982:III-2001:IV
θ
JT
0.92
3.98
(0.04) [0.41]
0.91
3.50
[0.05] [0.48]
Notes: The JT statistics are distributed as χ2 random variables with 3 degrees of freedom.
Standard errors in parenthesis. P-values in brackets.
Table 4: Prices Chosen One Period In Advance with Dynamic Indexation
Inflation
Full Sample
Measure
θ
JT
GDP Deflator 0.88
2.65
(0.05) [0.62]
PCE Deflator 0.86
4.98
(0.05) [0.29]
1959:I-1979:II
θ
JT
0.86
0.89
(0.09) [0.93]
0.84
2.05
(0.08) [0.73]
1982:III-2001:IV
θ
JT
0.83
6.33
(0.05) [0.18]
0.83
5.61
(0.06) [0.23]
Notes: This table considers the case where firms that do not reset their price plans use the
updating scheme: Pit = π t−1 Pit−1 . The JT statistics are distributed as χ2 random variables
with 3 degrees of freedom. Standard errors in parenthesis. P-values in brackets.
Table 5: Prices Chosen One Period In Advance, Static Indexing, and Rule of Thumb Firms
Inflation
Full Sample
Measure
θ
ω
JT
GDP Deflator 0.85
0.44
0.92
(0.08) (0.17) [0.82]
PCE Deflator 0.96
0.56
1.66
(0.09) (0.22) [0.64]
1959:I-1979:II
θ
ω
JT
0.88
0.51
0.68
(0.11) (0.32) [0.97]
0.97
0.72
0.73
(0.09) (0.35) [0.87]
1982:III-2001:IV
θ
ω
JT
0.85
0.37
0.73
(0.12) (0.28) [0.87]
0.48
0.80
0.12
(0.18) (0.08) [0.99]
Notes: Notes: This table considers the case where (1 − ω) of firms that have the opportunity
to change prices do so optimally while ω are of the Gali-Gertler type, that is they set prices
according to Pt0 = π t−1 P̄t−1 . where P̄t = (1 − ω)P̃t + ωPt0 . When unable to reset the price
plan, all firms use the same updating scheme, Pit = π̄Pit−1 . The JT statistics are distributed
as χ2 random variables with 3 degrees of freedom. Standard errors in parenthesis. P-values
in brackets.
31
Table 6: Prices Chosen One Period In Advance,
Dynamic Indexing, and Rule of Thumb Firms
Inflation
Full Sample
Measure
θ
ω
JT
GDP Deflator 0.87
0.04
2.79
(0.05) (0.14) [0.43]
PCE Deflator 0.88
-0.06 4.79
(0.06) (0.15) [0.19]
1959:I-1979:II
θ
ω
JT
0.83
0.12
0.61
(0.10) (0.22) [0.89]
0.85
-0.02 1.85
(0.09) (0.18) [0.60]
1982:III-2001:IV
θ
ω
JT
0.82
0.15
6.12
(0.07) (0.26) [0.11]
0.76
0.40
5.67
(0.63) (2.73) [0.13]
Notes: This table considers the case where (1 − ω) of firms that have the opportunity
to change prices do so optimally and using dynamic indexing when they do not have the
opportunity to reset the price plan. In additin ω firms are of the exact version Gali-Gertler
considered in their paper. That is they set prices Pit0 according to Pit0 = π t−1 P̄t−1 , where
P̄t = (1−ω)P̃t +ωPt0 when they have the opportunity to reset their plan, but use Pit = π̄Pit−1 .
The JT statistics are distributed as χ2 random variables with 3 degrees of freedom. Standard
errors in parenthesis. P-values in brackets.
32
Table 7: Pass Through and Elasticity (²) with Prices
Chosen One Period In Advance and Dynamic Indexing
Panel A: Full Sample
Price
Capital
Assumed
Pass
Mobile Immobile
θ
Through
²
²
GDP Deflator
0.25
0.01 1223.9
1168.9
0.50
0.04
266.0
211.0
0.60
0.07
137.9
82.9
0.75
0.21
36.9
-18.1
PCE Deflator
0.25
0.01
954.7
899.7
0.50
0.05
205.8
150.8
0.60
0.09
105.7
50.7
0.75
0.27
26.7
-28.3
Inflation
Measure
Panel B: 1959:I-1979:II
Price
Capital
Assumed
Pass
Mobile Immobile
θ
Through
²
²
GDP Deflator
0.25
0.01
858.2
803.2
0.50
0.05
184.2
129.2
0.60
0.10
94.1
39.1
0.75
0.30
23.0
-32.0
PCE Deflator
0.25
0.01
727.7
627.7
0.50
0.06
155.0
100.0
0.60
0.11
78.4
23.4
0.75
0.36
18.0
-37.0
Inflation
Measure
Panel C: 1982:III-2001:IV
Price
Capital
Assumed
Pass
Mobile Immobile
θ
Through
²
²
GDP Deflator
0.25
0.02
618.4
563.4
0.50
0.07
130.6
75.6
0.60
0.13
55.3
10.3
0.75
0.42
13.9
-41.1
PCE Deflator
0.25
0.02
613.7
558.7
0.50
0.07
129.5
74.5
0.60
0.13
64.8
9.8
0.75
0.42
13.7
-41.2
Inflation
Measure
Note: Estimates based on with α = 1/3 and ζ = 1.1.
33
Table 8: Frequency of Reoptimization under Benchmark Demand Elasticities (²) with
Prices Chosen One Period In Advance and Dynamic Indexing
Panel A: Full Sample
Capital
Assumed
Mobile
Immobile
1
1
²
θ
θ
1−θ
1−θ
GDP Deflator
10
0.83 5.9 0.69 3.3
33
0.76 4.2 0.66 2.9
Inflation
Measure
PCE Deflator
10
33
0.81
0.73
5.3
3.7
0.66
0.63
2.9
2.7
Panel B: 1959:I-1979:II
Capital
Assumed
Mobile
Immobile
1
1
θ
²
θ
1−θ
1−θ
GDP Deflator
10
0.80 5.0 0.65 2.9
33
0.72 3.6 0.61 2.6
Inflation
Measure
PCE Deflator
10
33
0.78
0.70
4.6
3.3
0.62
0.58
2.6
2.4
Panel C: 1982:III-2001:IV
Capital
Assumed
Mobile
Immobile
1
1
²
θ
θ
1−θ
1−θ
GDP Deflator
10
0.77 4.3 0.60 2.5
33
0.67 3.0 0.56 2.3
Inflation
Measure
PCE Deflator
10
33
0.76
0.68
Note: Estimates based on α = 1/3 and ζ = 1.1.
34
4.2
3.1
0.62
0.56
2.6
2.3
Working Paper Series
A series of research studies on regional economic issues relating to the Seventh Federal
Reserve District, and on financial and economic topics.
Dynamic Monetary Equilibrium in a Random-Matching Economy
Edward J. Green and Ruilin Zhou
WP-00-1
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Marianne Baxter and Michael A. Kouparitsas
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Data Revisions and the Identification of Monetary Policy Shocks
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2
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WP-01-06
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3
Working Paper Series (continued)
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Working Paper Series (continued)
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Working Paper Series (continued)
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Banking Market Conditions and Deposit Interest Rates
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7
@FedFRASER