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Relative Price Shocks and Inflation
WP 22-07
Francisco Ruge-Murcia
McGill University
Alexander L. Wolman
Federal Reserve Bank of Richmond
Relative Price Shocks and In‡ation
Francisco Ruge-Murciay and Alexander L. Wolmanz
May 2022
Abstract
In‡ation is determined by interaction between real factors and monetary policy. Among the
most important real factors are shocks to the supply and demand for di¤erent components of the
consumption basket. We use an estimated multi-sector New Keynesian model to decompose the
behavior of U.S. in‡ation into contributions from sectoral (or “relative price”) shocks, monetary
policy shocks, and aggregate real shocks. The model is estimated by maximum likelihood with
U.S. data for the post-1994 period in which in‡ation and the monetary policy regime appeared
to be stable. In addition to providing a broad decomposition of in‡ation behavior, we enlist
the model to help us understand the in‡ation shortfall from 2012 to 2019, and the dramatic
in‡ation movements during the COVID pandemic.
JEL classi…cation: E31, E52, E58
Key Words: Monetary policy, sectoral shocks, in‡ation shortfall, COVID-19.
This research received …nancial support from the Social Sciences and Humanities Research Council of Canada
and the Fonds de recherche du Québec. This paper represents the views of the authors and not of the Federal Reserve
Bank of Richmond, the Federal Reserve Board, or the Federal Reserve System. The authors would like to thank
Stephanie Schmitt-Grohé for helpful conversations, and Elaine Wissuchek and Zachary Edwards for outstanding
research assistance.
y
Department of Economics, McGill University. Email: francisco.ruge-murcia@mcgill.ca
z
Federal Reserve Bank of Richmond. Email: alexander.wolman@rich.frb.org
1.
Introduction
How do factors that drive relative price changes across consumption categories a¤ect in‡ation? In
theory, monetary policy can o¤set these factors and perfectly stabilize in‡ation. This does not
happen in practice for two broad reasons: …rst, monetary policy may be unable to control in‡ation
perfectly; second, monetary policy may choose not to control in‡ation perfectly, instead following a
policy that results in equilibrium ‡uctuations in in‡ation. Following the latter branch, we ask what
is the contribution of “relative price” shocks to the behavior of in‡ation. We focus on in‡ation
determination in a stable monetary policy regime, namely the post-1994 United States. The paper
makes three contributions, all based on an estimated multi-sector New Keynesian model. First,
we provide summary information on the importance of relative price shocks for the behavior of
in‡ation since 1994. Second, we examine the in‡ation shortfall from target that occurred between
2012 and 2019 in the U.S., decomposing that shortfall into policy and real shocks. Third, we apply
the model to studying the dramatic change in in‡ation during the COVID pandemic, providing an
out-of-sample version of that same decomposition.
The relative in‡uence of monetary policy and real factors on in‡ation is one of the most important and long-studied questions in monetary economics. Any such analysis immediately confronts
the issue of monetary policy regimes. At one extreme, if there is a stable underlying monetary
policy regime, then the tools of rational expectations and local approximation of DSGE models
may be appropriate. At another extreme, clearly identi…ed breaks in regime can be the source of
facts against which theories are evaluated, but the researcher must decide how to model information sets of private agents and the policymaker. The fundamental assumption of this paper is that
from January 1995 to December 2019 the United States was in a stable, well-understood monetary
policy regime and that it is appropriate to use a locally approximated DSGE model with rational
expectations to study that period.
The …rst contribution involves basic analysis of in‡ation and relative price shocks. The model
has 15 consumption categories, shocks to productivity whose volatility and dynamics di¤er across
categories, and Rotemberg costs of price adjustment that are allowed to vary across categories.
We estimate the model on monthly data from 1995-2019 by maximum likelihood. The observable
variables are rates of price change for the 15 consumption categories, and the level of the nominal
interest rate. We decompose in‡ation into the contribution of monetary policy shocks and shocks
to productivity in each of the consumption categories. Over the sample, in‡ation was low and
stable, but nonetheless exhibited substantial month-to-month volatility. Our estimates explain
that volatility primarily with sectoral shocks, which we label relative price shocks. One way we
1
evaluate the model …t is through its ability to match the historical relationship between the monthly
in‡ation rate and the distribution of relative price changes. As discussed in Hornstein, Ruge-Murcia
and Wolman (2022), from 1995 to 2019 there was a tight negative relationship between the monthly
in‡ation rate and the share of consumption expenditures exhibiting relative price inreases. The
model reproduces and explains that relationship.
The second contribution is to decompose the shortfall of in‡ation from target in the 20122019 period. According to this decomposition, which uses the smoothed estimates of the model’s
structural shocks, the largest contributors to the cumulative shortfall of the price level from the
2% trend implied by the Fed’s target were shocks to gasoline and energy goods and to health care.
Note that we are intentionally describing causality; absent a model, when one analyzes in‡ation and
category price changes, it is only possible to describe the extent to which a particular category’s
price change accounts for in‡ation. The model allows us to provide estimates of the extent to which
particular shocks caused the in‡ation shortfall. While nontrivial, the monetary policy contribution
is relatively small (about 17%).
A leading theoretical explanation for the in‡ation shortfall is that it resulted from interaction
between the lower bound on nominal interest rates and the Fed’s implicit policy rule. See for
example Bianchi, Melosi and Rottner (2021). Our current results cannot speak directly to that
view because we do not impose a lower bound on nominal interest rates. However, during much
of the period when the shortfall occurred, the interest rate was at its lower bound, and we do
use interest rate data in estimation. The 17% contribution from monetary policy shocks can be
interpreted as a lower bound for the e¤ects of the zero bound.
The third contribution is an out-of-sample analysis of the high in‡ation episode that began
in March 2021 during the COVID-19 pandemic, and is ongoing as of this writing. As with the
in‡ation shortfall, we decompose in‡ation into contributions of the sectoral shocks and monetary
policy. Here, however, the smoothed estimates of the model’s shocks use out-of-sample data, as our
estimation sample ends in December 2019. In conducting local analysis around the model’s (trending) steady state, we assume that the same policy rule remains in place during this high in‡ation
episode. But has in‡ation in fact remained anchored? Our approach describes the contributions to
in‡ation of policy shocks and real shocks under the assumption that in‡ation remains anchored. If
one has outside information about the shocks, then that can be used together with our estimates to
evaluate the anchoring question. Summarizing what we …nd: of the 2:7% in‡ation overshoot from
March 2021 through November 2021, the largest contributor is shocks to the motor vehicle sector,
at 31%. According to the model, monetary policy shocks explain 22% of the in‡ation overshoot.
There is a large and varied literature on relative prices and in‡ation. One branch focuses on the
2
role of oil shocks in driving short-run in‡ation ‡uctuations; see for example Kilian and Zhou (2021)
and the references therein. Our paper embodies a generalization of that idea: oil shocks are the
most volatile relative price shocks, but in any given period some other category might experience
an unusually large shock that accounts for a sizable movement in in‡ation— for example, motor
vehicles during the COVID period. Some other branches of the literature were concerned mainly
with periods when in‡ation was not stable— in particular a set of papers from the 1970s and ’80s
that studied causality in the opposite direction, from in‡ation to the variability of relative prices. A
key reference is Parks (1978). In principle, our model can have in‡ation (i.e., policy) a¤ect relative
price variability but in practice that channel is weak, as one would expect in a stable policy regime.
The papers closest to ours are Ball and Mankiw (1995), Balke and Wynne (2000), Reis and
Watson (2010), and Smets, Tielens, and Van Hove (2019). In Ball and Mankiw (1995), menu costs
mean that in‡ation tends to move with the price change of sectors hit by the largest shocks to
their desired price; prices don’t change in sectors hit by small shocks to their desired price. In our
model, Rotemberg costs mean that any …rm hit by a shock will adjust its price; in equilibrium,
the interaction of policy and shocks delivers a relationship between in‡ation and the distribution
of actual relative prices consistent with the data. Balke and Wynne (2000) conduct an analysis
quite similar to ours, but their model has ‡exible prices and monetary policy is characterized
by a constant money growth rule. Reis and Watson’s (2010) empirical analysis using a factor
model leads them to conclude that “most of the variation in standard aggregate in‡ation indices
is associated with relative-price movements.” Our estimated DSGE model leads us to the same
conclusion. Like Smets, Tielens, and Van Hove (2019) we use the Kalman …lter to decompose the
behavior of in‡ation into aggregate and sectoral shocks. They study network interactions absent
from our model, and estimate over a longer sample in which in‡ation was not stable. They do not
address the in‡ation-shortfall or COVID episodes.
Our paper also relates to the literature that discusses how the Phillips curve can appear to
‡atten when policy is conducted optimally so as to stabilize in‡ation. McLeay and Tenreyro (2020)
is a leading example. While we do not study optimal policy, in practice the Federal Reserve did
stabilize in‡ation from 1995-2019, and that is embodied in our estimated model. We emphasize that
the remaining small ‡uctuations in in‡ation were associated with variation in relative prices across
categories. Borio, Disyatat, Xia, and Zakrajšek (2021) also emphasize that in a stable monetary
policy regime, sector-speci…c factors can play a major role in driving in‡ation.
The paper proceeds as follows. Section 2 describes the model and its balanced growth path.
Section 3 describes the empirical analysis and presents the parameter estimates. Section 4 presents
3
the basic analysis of in‡ation and relative price shocks. Sections 5 and 6 cover the in‡ation shortfall
and COVID in‡ation, respectively. Section 7 concludes.
2.
The Model
The economy consists of an in…nitely-lived representative household, continua of …rms in a …nite
number of sectors, and a monetary authority. Firms are monopolistic-competitors that each produce
a di¤erentiated good using labor as the sole input of production.
2.1
Households: Preferences, Constraints, and Optimality
The household maximizes
E
1
X
t
ln (Ct ) +
1
t=
where E denotes the conditional expectation as of time ,
is consumption, Nt is hours worked, and
and
!
Nt )1
(1
;
(1)
2 (0; 1) is the discount factor, Ct
are positive preference parameters. The time
endowment is normalized to be one. Consumption is an aggregate of goods produced in di¤erent
sectors s = 1; 2; : : : ; S,
Ct =
S
Y
s
( s)
s
(cs;t ) ;
(2)
s=1
where
s
2 (0; 1) are aggregation weights that satisfy
S
X
s
= 1 and cs;t is consumption of goods
s=1
produced in sector s. Under the Cobb-Douglas speci…cation in (2), the elasticity of substitution
between goods produced in di¤erent sectors is one.
Within each sector, there is a continuum of monopolistically-competitive …rms that each produce
a di¤erentiated good. The household’s preferences for these goods are represented by the CES
aggregator
cs;t =
Z
(
1)=
=(
ci;s;t
1)
;
(3)
where ci;s;t is consumption of the good produced by …rm i in sector s and
> 1 is the elasticity
of substitution between goods produced in the same sector. Since this elasticity is larger than
one, goods produced in the same sector (e.g., apples and pears) are closer substitutes than goods
produced in di¤erent sectors (e.g., apples and nails).
In every period, the household faces the budget constraint
Pt Ct + Bt
Pt wt Nt + (1 + Rt
4
1 ) Bt 1
+ Dt ;
(4)
where Pt is the aggregate price level, Bt is nominal bonds, wt is the real wage, Rt is the net nominal
interest rate, and Dt are pro…ts from …rms, which are transferred to the household in the form of
dividends. The price index satis…es
Pt =
S
Y
(Ps;t ) s .
(5)
s=1
The solution to the household’s maximization problem shows that the optimal labor supply
satis…es
(1
Nt )
1=Ct
= wt ;
(6)
meaning that the marginal rate of substitution between leisure and consumption equals the real
wage; the optimal consumption of good i produced in sector s is
ci;s;t =
Pi;s;t
Ps;t
s
where
Ps;t =
Z
Ps;t
Pt
1=(1
1
Ct ;
(7)
)
1
Pi;s;t
dl
(8)
is a sector-speci…c price index; and the optimal (total) consumption satis…es the intertemporal
Euler equation
1
=
Pt Ct
2.2
(1 + Rt ) Et
1
Pt+1 Ct+1
:
(9)
Firms: Technology, Market Structure, and Optimality
Firm i in sector s produces output using the technology
yi;s;t = ezt ezs;t ni;s;t ;
(10)
where yi;s;t is output, ezt and ezs;t are, respectively, the aggregate and sectoral levels of productivity,
and ni;s;t is labor input. Labor is completely mobile across …rms and sectors. Sectoral productivity
follows the trend-stationary process,
zs;t =
as;t =
where t is a time index,
s
s
st
+ as;t ;
s as;t 1
(11)
+ "s;t ;
(12)
is a constant, as;t represents stochastic deviations from the time trend,
2 ( 1; 1) is a parameter, and "s;t is an independent and identically distributed (i.i.d.) innovation
drawn from a normal distribution with mean zero and constant conditional variance
2.
s
The
deviation from the time trend has di¤erent persistence and variance across sectors. The trend itself
5
is also di¤erent across sectors, which is important for our model to account for the trends in relative
prices observed in the data. Aggregate productivity follows the process,
zt = zt
where
1
+ "t ;
(13)
2 ( 1; 1) is a parameter and "t is an i.i.d. innovation drawn from a normal distribution
with mean zero and constant conditional variance
2 .1
Each …rm must incur a cost when it changes its nominal price. The cost is speci…ed in units
of the aggregate consumption good, is convex in the size of the adjustment, and is proportional to
the quantity produced by the …rm. The per-unit cost,
i;s;t
where
s
0,
trend, and
=
0 and
and
s
(Pi;s;t ; Pi;s;t
1)
=
i;s;t ,
1
s;t s
2
for …rm i in sector s in period t is
e
+
s s
Pi;s;t
Pi;s;t 1
0 are parameters that satisfy 0
s
+
s
2
1
;
(14)
1,
s;t
is a deterministic
are, respectively, the steady-state in‡ation rate and the steady-state sectoral
rate of price change. The parameter
s
controls the degree of price rigidity and takes value zero in
the special case where prices are completely ‡exible. Since
s
is indexed by s (but not by i), price
rigidity di¤ers across sectors, but is constant across …rms within a sector. The deterministic trend
s;t
is speci…ed as
s;t
S
Y
= (1=e s )t
(e k )
k=1
k
!t
We require that price-adjustment costs grow at the trend rate
for sector s grows at rate e
w
.
s;t
because the labor part of costs
=e s ; if the price-adjustment part of costs were to grow at a di¤erent
rate, then one or the other components of costs would become negligible in the long run. This
S
Y
means that there would be no balanced growth path. We will see below that e w =
(e s ) s , so
s=1
that the trend can be written as
s;t
= e(
w
s )t
.
Finally, the factors in the denominator of (14) determine the degree and the form of indexation
as follows. If
=
adjustment. If 0 <
s
= 0, then there is no indexation and …rms incur a cost for any nominal price
+
s
< 1, then there is partial indexation. If
= 1 and
s
= 0, then there is
complete indexation to the aggregate in‡ation rate. Thus, price increases equal to the steady-state
rate of aggregate price change are costless. If
= 0 and
s
= 1, then there is complete indexation
to the sectoral in‡ation rate and price increases at the steady-state rate of sectoral price change
1
Results would be unchanged if we were also to assume a trend in aggregate productivity, for instance, equal to the
growth rate of aggregate output and with sectoral trends adjusted accordingly because what matters in this model
are relative productivity trends rather than absolute ones.
6
are costless. And, if
+
s
= 1 with
< 1 and
s
< 1, then there is complete indexation to an
average of the aggregate and sectoral price changes.
The …rm chooses output, labor input, and the price of its good to maximize pro…ts, where costs
comprise labor costs and adjustment costs. The maximization is subject to the demand function
(7), the demand associated with other …rms’ adjustment costs,2 and the technology (10). The
solution to this problem delivers the following optimality condition for …rm i in sector s:
1
Pi;s;t
Pi;s;t
wt
= zt t as;t + s;t s
1
+
s sP
Pt
e e se
2 e
i;s;t 1
Pi;s;t
Pi;s;t
s
1
s;t
+
+
s
s
s sP
Pi;s;t 1
e
e
i;s;t 1
1
Pt+1
s
+ s;t+1
Et (1 + Rt ) 1
e + s s
Pt
Pi;s;t+1
e + s s Pi;s;t
1
Pi;s;t+1
Pi;s;t
1
Ps;t
Ps;t+1
2
(15)
ys;t+1
:
ys;t
This equation relates the optimal price selected by …rm i to its marginal cost, including that
associated with current and future expected price adjustments.
2.3
Monetary Policy
The monetary authority sets the nominal interest rate following the rule,
1 + Rt = (1 + Rt
where
2 ( 1; 1),
1)
+ (1
, and
)(1= ) exp( +
y
+
(
t
)+
y
(ln(Yt )
ln(Y )) + ut );
(16)
are parameters representing responsiveness to the lagged interest
rate, in‡ation, and output, respectively,
rate of output (see below),
y
t
is the gross in‡ation rate (= Pt =Pt
1 ),
y
is the growth
is a policy parameter denoting the in‡ation target, Yt is aggregate
output, Y is output in the balanced-growth steady state, and ut is a disturbance that represents
movements in the interest rate beyond the control of the central bank.3 This disturbance follows
the process
ut = ut
where
1
+ & t;
(17)
2 ( 1; 1) is a parameter and & t is an i.i.d. innovation drawn from a normal distribution
with mean zero and constant conditional variance
2.
&
2
Note that since price adjustment costs require each …rm to purchase the …nal good, this implies that each …rm
demands output of all other …rms in the economy.
3
We also estimated the model under a policy rule where the monetary authority responds to average quarterly
in‡ation. Parameter estimates are very similar to those reported here and support the same conclusions. These
results are available from the authors upon request.
7
2.4
Equilibrium and Balanced Growth Path
The equilibrium of the model is symmetric within sectors but asymmetric across sectors. That
is, in equilibrium, all …rms within a sector are identical and make exactly the same choices (labor
input, price, and output), but …rms in di¤erent sectors make di¤erent choices. In particular, …rms
in di¤erent sectors choose di¤erent prices and, hence, rates of price change will di¤er across sectors.
In a symmetric equilibrium we can simplify the optimality conditions by dropping the i subscripts,
and we impose market clearing in the di¤erent markets. Since we will approximate the model
solution around a steady state with long-term growth, it is also helpful to derive some relationships
between the steady state growth rates of di¤erent variables.
2.4.1
Sectoral Phillips Curve
The fact that …rms in the same sector are identical in equilibrium means that the price charged by
each …rm is equal to the sectoral price index,
Pi;s;t = Ps;t ; i 2 (0; 1); s = 1; 2; : : : ; S.
The …rm’s optimality condition can then be written as the sectoral Phillips curve,
1
+
where
s;t
s;t+1
Ps;t
wt
= zt t as;t +
Pt
e e se
1
s
Et
+
s s
e
= Ps;t =Ps;t
1.
s;t
s;t s
e +
1
1 + Rt
1
s s
t+1 s;t+1
1
1
2
e
e
s;t+1
+ s
s
s;t
+ s
1
s
ys;t+1
ys;t
1
2
;
As in previous literature that derives sectoral Phillips curves in multi-
sector economies (e.g., Imbs, Jondeau and Pelgrin, 2011, and Rubbo, 2022), the slope of the
linearized Phillips curve is heterogenous across sectors, in our case as a result of heterogeneity in
price rigidity.
2.4.2
Goods Market Clearing
In each sector, there is a goods market clearing condition,
ys;t = cs;t + fs;t ,
where fs;t denotes the output from sector s used for adjustment costs. Note that this is di¤erent
from
s;t ,
the adjustment costs incurred by sector s, and is given by
fs;t =
s
Ps;t
Pt
8
1
Ft .
Aggregate adjustment costs are
Ft =
S
X
s=1
2.4.3
s;t
+ s
s
s;t
2
e
2
1
j
ys;t :
Steady-state Growth Rates
The model has an exogenous growth rate for the overall price index in the balanced growth path.
That growth rate, , is determined by the in‡ation target in the policy rule. The model also has
S exogenous trend growth rates for sectoral productivity,
s.
Each sector’s output growth rate
is equal to its productivity growth rate. Productivity growth rates determine the growth rates of
consumption and the real wage. Then, the in‡ation target, along with the productivity growth
rates, determine the growth rates of sectoral prices as follows:
c
=
w
=
S
X
s s,
s=1
s
=
+
w
s
=
+
S
X
k
(
k
s) .
k=1
The trend growth (or decline) in sectoral relative prices is determined entirely by a real factor,
namely sectoral relative productivity growth,
s
=
s+
S
X
k k.
k=1
This point bears stressing: while sectoral relative prices can move around temporarily because of
monetary policy shocks or the interaction between real shocks and the policy rule, the trend in relative prices is invariant to monetary policy and to parameters representing price stickiness. Finally,
note that because we incorporate the exogenous trend
output growth (
y)
s;t
in the adjustment cost speci…cation,
is equal to consumption growth and adjustment cost growth on the balanced
growth path, both in the aggregate and by sector.
The model is solved using a …rst-order perturbation with the rational-expectation solution of
the linearized system found using the approach in Klein (2000).
3.
Empirical Analysis
In this section, we describe the data used for estimation and the estimation method, and we discuss
the parameter estimates.
9
3.1
Data
The data used to estimate the model are monthly observations of the nominal interest rate and rates
of price change for …fteen consumption expenditure categories of the U.S. economy from 1995M1 to
2020M1. The sample starts around the time that the Federal Reserve Board settled on implicitly
targeting an in‡ation rate of 2% per year (see the transcripts of the meeting of the Federal Open
Market Committee on January 31). The in‡ation target was o¢ cially adopted in January 2012 and
applies to the personal consumption expenditures (PCE) price index.
The nominal interest rate is the e¤ective federal funds rate and was taken from the FRED
website of the Federal Reserve Bank of St. Louis (fred.stlouisfed.org). The original interest rate
series quoted as a net annual rate is transformed into a monthly rate and quadratically detrended
to account for its secular decline over the sample period. Since in‡ation has been stable over the
sample, this secular decline is attributable to the persistent decrease in the natural real rate. This
decrease has been widely documented and studied in previous literature (see, for example, Laubach
and Williams, 2003, King and Low, 2014, and Rachel and Summers, 2019). Because our model
does not contain elements that can capture the decrease in the natural real rate, we follow an
econometric strategy that accounts for this decrease by means of a nonlinear time trend, and we
focus instead on the cyclical component of the nominal interest rate.
The …fteen categories comprise the entirety of PCE. The sectors are motor vehicles and parts,
furnishings and household durables, recreational goods, other durable goods, food consumed at
home, clothing and footwear, gasoline and other energy goods, other nondurable goods, housing and
utilities, health care, transportation services, recreation services, food services and accommodations,
…nancial services and insurance, and other services. The raw data used to construct the sectoral
price changes are seasonally adjusted price indices available from the website of the Bureau of
Economic Analysis (www.bea.gov). To be consistent with the model solution, which describes the
dynamics of the variables in deviation from their steady state values, all data series are demeaned
prior to the structural estimation of the model.
3.2
Estimation
The model is estimated by the method of maximum likelihood (ML) using the Kalman …lter to
evaluate the likelihood function. The state equation of the state-space representation of the model
solution is the joint process of exogenous and predetermined variables,
Xt+1 = HXt + vt+1 ,
where Xt = (zt ; a1;t ; : : : ; aS;t ; ut ; Rt
1 ; P1;t 1 ; : : : ; PS;t 1 )
10
0
and vt = ("t ; "1;t ; : : : ; "S;t ; & t ; 0; 0; : : : ; 0)0
are (2S + 3)
1 vectors, and H is a (2S + 3)
(2S + 3) matrix whose elements are the parameters of
the exogenous shock processes (in the …rst S + 2 elements of Xt ) and the coe¢ cients of the decision
rules of the predetermined variables in the last S + 1 elements of Xt .
The observation equation is
Qt = GXt ,
where Qt = (Rt ;
1;t ; : : : ;
S;t )
0
is (S + 1)
1 vector and G is a (S + 1)
(S + 1) matrix whose
elements are the coe¢ cients of the decision rules of the interest rate and sectoral price changes in
Qt . The coe¢ cients of the decision rules are nonlinear functions of the structural parameters. As
it is well known, this estimation approach is equivalent to using a Bayesian estimation strategy
with di¤use priors. Hansen and Sargent (2013, ch. 8) shows that the ML estimator obtained by
applying the Kalman …lter to the state-space representation of dynamic linear models is consistent
and asymptotically e¢ cient. Standard errors are estimated by the square root of the diagonal
elements of (T I)
1
where T is the sample size and I is the information matrix, which is computed
using the outer product of the scores at the maximum.
A di¢ culty that we face in estimating this model is that the steady state has to be computed
numerically in every iteration of the algorithm that maximizes the likelihood function. Given the
relatively large size of our model, this computation is time-consuming. In order to address this
challenge, we …rst …x the parameters that determine the steady state and, with these parameters
set, we estimate the parameters that determine the dynamics of the model by ML. The parameters
that determine the steady state are …xed as follows. The weight of leisure in the utility function
( ) is set to 1:8, such that households work 1=3 of the time in steady state. The parameter that
determines the elasticity of substitution between goods from the same sector ( ) is …xed to 10.
This value is standard in the literature and implies a mark-up of approximately 10%. We assume
complete indexation, meaning that
+
s
= 1. Provided that the latter condition is satis…ed, the
steady state is independent of the relative magnitude of
and
s,
and we assume
=
s
= 1=2.
The discount rate ( ) is …xed to 0:998.
The consumption weights are computed using the consumption expenditure shares in each
sector. Recall that the optimal consumption of good i produced in sector s is
ci;s;t =
s
Pi;s;t
Ps;t
Ps;t
Pt
1
Ct :
Using the fact that the equilibrium is symmetric within sectors, which implies Pi;s;t = Ps;t , and
solving for
s
delivers
s
=
Ps;t cs;t
,
P t Ct
11
where Ps;t cs;t are expenditures in sector s and Pt Ct are total consumption expenditures. Estimates
of
s
for each sector are reported in Column 1 of Table 1.
Finally, the model suggests a natural strategy to estimate the trends in sectoral productivity
based on the result that sectoral price changes in steady state are
s
for s = 1; 2 : : : ; S.
Solving for
s,
=
so that
+
s
=
s;
c
+
c
(18)
s,
and using the fact that the model
implies that the steady state values of aggregate in‡ation, aggregate consumption growth, and
sectoral price changes are equal to their respective unconditional means, delivers estimates of
s
for each sector. These estimates are reported in the Column 2 of Table 1.
3.3
Parameter Estimates
ML estimates of the price rigidity parameter in each sector and the parameters of all shock processes
are reported in Table 2. There is substantial heterogeneity in price rigidity across sectors and the
null hypothesis that rigidity is the same in all sectors is strongly rejected by the data (see the p-value
of the Wald test reported in the last row of Table 2). Heterogeneity in price rigidity across product
categories has been documented by previous literature using highly disaggregated components of
the consumer price index (CPI) (see, among others, Bils and Klenow, 2004, Klenow and Kryvtsov,
2008, and Nakamura and Steinsson, 2008) and estimated multi-sector dynamic equilibrium models
(see, e.g., Bouakez, Cardia, and Ruge-Murcia, 2014). In line with that literature, we …nd that
service prices are generally the most rigid in the U.S. economy. The only exceptions in our sample
are transportation services and …nancial services and insurance, for which the null hypothesis that
prices are ‡exible (i.e.,
= 0) cannot be rejected at the 5% level. Quantitatively, the most rigid
prices are those of food services and accommodations, and housing and utilities. The prices of some
categories of durables and nondurable goods are also rigid, meaning that the hypothesis
= 0 can
be rejected at standard levels (e.g., food at home and motor vehicles). There are, however, other
categories like furnishing and clothing for which the hypothesis cannot be rejected. In terms of the
expenditure shares, 70% of consumption by U.S. household may be considered to have rigid prices.
Sectoral productivity shocks are persistent with autoregressive coe¢ cients ranging from 0:455
to 0:998. The hypothesis that the coe¢ cients are statistically the same in all sectors can be rejected
at standard levels and the p-value of the Wald test is less than 0:001. The former estimate (0:455)
corresponds to food services and accommodations and all other estimates are concentrated in the
interval between 0:962 and 0:998. We explore the possibility that the above rejection is driven
by food services and accommodations by performing the Wald test excluding this sector. Since
12
the p-value of this test is also less than 0:001, we conclude that the …nding that the persistence
of productivity shocks is heterogenous across sectors is not only due to the moderate estimate for
food services and accommodations.
There is large heterogeneity in the standard deviation of productivity innovations, and the
hypothesis that they are the same in all sectors is strongly rejected by the data. The largest standard
deviation is that of gasoline and energy goods, followed by food services and accommodations,
which are one order of magnitude larger than that of other sectors. The …nding that the standard
deviation of productivity shocks to gasoline and energy goods is large helps explain why even though
their price rigidity parameter is of moderate magnitude and statistically signi…cant, the observed
frequency of price adjustments for these goods is high. For instance, Bils and Klenow (2004, Table
A1 in p. 983) report that the estimated average monthly frequency of price changes for gasoline
over 1995–1997 ranges between 0:762 and 0:789 depending on its grade.
The estimate of the standard deviation of the aggregate productivity innovation is small and
not statistically signi…cant. This has two implications. First, the estimate of the autoregressive
coe¢ cient of the productivity shock is poorly identi…ed (see the large standard error in Table
2). Second, as we will see below, aggregate productivity plays a limited role in explaining the
aggregate and sectoral in‡ation dynamics. For this reason, the conclusions in this paper are robust
to considering a restricted version of the model without an aggregate productivity shock.
Table 3 reports the ML estimates of the monetary policy rule under the heading “Benchmark.” The smoothing parameter is large (0:733) and in line with values reported elsewhere in
the literature. The coe¢ cients of in‡ation and output are positive, as expected, but imprecisely
estimated. It is interesting to compare these estimates, based on the full model, with reduced-form
estimates obtained under minimal assumptions and also reported in Table 3 under the heading
“Unrestricted.” The latter are estimates of the …rst-order linearization of (16) computed by ML
but without imposing the restrictions of the model. Estimates are quantitatively similar suggesting
that the restrictions of the model do not limit its ability to correctly represent the dynamics of the
monetary policy rule. (The last two columns of Table 3 will be discussed later).
3.4
Predicted Moments
Table 4 reports the standard deviations and autocorrelations of the nominal interest rate, aggregate
in‡ation, and sectoral price changes predicted by the model and compares them with U.S. data.
In contrast to method of moments estimators (e.g., GMM or SMM), our ML estimation procedure
does not explicitly target these moments. The table shows that the model quantitatively captures
the high persistence of the interest rate, the low autocorrelation of month-to-month in‡ation and
13
sectoral price changes, and the volatility of all series, notably the large standard deviation of price
changes in gasoline and other energy goods. The correlation between the two sets of moments is
high: 0:994. We, therefore, conclude that our model does a good job reproducing the key moments
of the data.
4.
Relative Price Shocks and In‡ation over the Entire Sample
With parameter estimates in hand, we can now turn to the model’s implications regarding the role
of relative price shocks in driving in‡ation dynamics. We begin with impulse response analysis and
variance decompositions. Then we focus on a summary statistic for the distribution of relative price
changes that has not— to our knowledge— previously been examined in the literature. We conclude
this section with a discussion of the (non)importance of sticky prices for some of our qualitative
…ndings.
4.1
Impulse Responses
Figure 1 reports the responses of relative prices (Ps;t =Pt ) to a negative productivity shock in each
sector. For example, panel A reports the response of the relative price of motor vehicles (thick line)
and other relative prices (thin lines) to a negative productivity shock in the motor vehicles sector.
The size of the shock is one standard deviation of the productivity innovation,
s
(see (11)). In all
panels, the productivity shock leads to a large increase in the relative price of the good produced
in that sector. This increase is one or two orders of magnitude larger than the muted decrease
in the relative price of goods produced by the other 14 sectors.4 These 14 impulse responses are
quantitatively similar, and for this reason they sometimes appear as a single black line in Figure
1. The result that a sectoral productivity shock yields a large response in the relative price of
its own good and a mild response in other relative prices motivates our interpretation of sectoral
productivity shocks as relative price shocks.
Figure 2 reports the response of in‡ation to aggregate and sectoral shocks. The horizontal axis
is months, and the vertical axis is the in‡ation deviation from in steady state value, expressed
in percentage points at an annual rate. Panel A reports the in‡ation response to a monetary
policy shock that reduces the nominal interest by
&
(see eq. (17)). The other panels report the
in‡ation response to a negative relative price shock with size equal to one standard deviation of the
productivity innovation,
5
s.
Note that to allow comparisons across panels, the scale of the vertical
4
Recall that the aggregate price index, Pt , is a geometric average of sectoral prices (see eq. (5)) and hence the
sum of the weighted responses of relative prices must add up to zero.
5
We abstract from reporting the response to the aggregate productivity shock because its small standard deviation
(see Table 2) implies that the in‡ation response to a productivity shock of a plausible magnitude is basically zero.
14
axis is the same in all panels except for panels A, H, and O, which respectively report the responses
to the monetary policy shock and the shocks to gasoline and …nance and insurance. This …gure
shows that di¤erent shock sizes across sectors and, to some extent, di¤erent price stickiness across
sectors, imply heterogeneity in the e¤ects of relative price shocks on in‡ation. Quantitatively, the
largest e¤ects are due to relative price shocks to gasoline (panel H), …nance and insurance (panel
O), housing and utilities (panel J), other services (panel P), and food at home (panel F).
4.2
Accounting for the Variance of In‡ation
This section reports the variance decomposition for the in‡ation rate at di¤erent horizons. (Recall
that this is the proportion of the mean squared error of the forecast of in‡ation at di¤erent horizons
that is accounted for by each of the shocks.) Panel A in Figure 3 shows that the aggregate
productivity shock accounts for a negligible proportion— less than 0:001%— of the variance of the
in‡ation forecast error at all horizons. Instead, monetary policy and relative price shocks account for
basically all the variance of the in‡ation forecast error. Panels B and C show that they respectively
account for 23:7% and 76:3% of the variance of the in‡ation forecast error one-month ahead and
24:7% and 75:3% of the unconditional variance of in‡ation. The …nding that relative price shocks
account for a large proportion of the in‡ation forecasts error at all horizons is consistent with results
in Reis and Watson (2010, p. 146) who report that 76% of the movements in aggregate in‡ation
are accounted for by a relative-price index.6 A similar result is reported by Smets, Tielens, and
Van Hove (2019) who …nd that sectoral shocks, by ways of pipeline pressures, are an important
contributor to the variance and persistence of headline in‡ation.
The remaining panels in Figure 3 (D through R) report the contribution of relative price shocks
in each sector to the variance decomposition of in‡ation. Panel J shows that relative price shocks
to gasoline account for around 43% of the variance of the in‡ation forecast error one-month ahead
and for 42% of the unconditional variance of in‡ation. Panels H, L, M, P, and Q show that relative
price shocks to food at home, housing and utilities, health, food services, and …nancial services
substantially contribute to the variance of the in‡ation forecast error one-month ahead (3%, 4:7%,
4:4%, 2:8%, and 8:2%, respectively) and to its unconditional variance (2:9%, 4:6%, 4:3%, 3:1%,
and 7:8%, respectively). Research based on dynamic factor models (e.g., see Boivin, Giannoni,
and Mihov, 2009) typically …nds that aggregate factors are the main driver of aggregate in‡ation.7
6
Note, however, that the structural interpretation of their index, which is based on a model of price setting
under imperfect information, is di¤erent from our relative-price shocks because their index includes, for instance, the
unanticipated component of the rate of money growth, which is an aggregate variable.
7
Boivin, Giannoni and Mihov (2009) also emphasize di¤erential responses of sectoral prices to aggregate and
sectoral shocks. Carvalho, Lee and Park (2021) show how adding input-output linkages and labor market segmentation
can help a multi-sector New Keynesian model match these patterns.
15
Using a structural model with input-output interactions, Onatski and Ruge-Murcia (2013) show
that macroeconomics shocks can indeed be considered factors in that they nontrivially a¤ect most
variables in the model. However, principal components analysis has a hard time replicating the
macroeconomic factor space because sectoral shocks can act as aggregate shocks.
Table 5 reports the contribution of the “own” relative price shock (Column 1) and of the
monetary policy shock (Column 2) to the unconditional variance of each sectoral price change
under the benchmark model. (Columns 3 and 4 will be discussed below.) The table shows that
the “own” relative price shock accounts for most of the variance of all sectoral price changes. The
contribution of the monetary policy shock is also substantial. In contrast, the contribution of the
aggregate productivity shock and relative price shock in other sectors is negligible.8 The …nding
that the sector-speci…c shocks account for most of the variance of sectoral price changes is consistent
with results reported in Boivin et al. (2009) and Mackowiak, Moench, and Wiederholt (2009) based
on estimated dynamic factor models. Boivin et al. (p. 356) report that 85% percent of the monthly
disaggregated in‡ation ‡uctuations are attributable to sector-speci…c shocks, while Mackowiak et
al. (p. 82) report that the proportion of the variance in sectoral price changes due to sector-speci…c
shocks in their median sector is 90% with a 90% con…dence interval ranging from 79% to 95%. In
our sample, sector-speci…c shocks account for 80% of the variance of sectoral price changes in the
median sector, and 87% of our estimates (that is, 13 out of 15 sectors) fall in the interval from
62:4% to 94:8%.
It is interesting to note that there is basically no relationship between price rigidity and the
proportion of the variance that is accounted for by the monetary policy shock. The correlation
between the price rigidity parameters in Table 2 and the proportions in Column 2 is
0:287 and
not statistically signi…cant. In contrast, the correlation between the standard deviation of sectoral
productivity shocks in Table 2 and the proportions in Column 1 is 0:584 and statistically signi…cant
at the 5% level.
4.3
In‡ation and the Distribution of Relative Price Changes
Panel A in Figure 4 displays the monthly PCE in‡ation rate rate on the vertical axis against
the share of relative price increases— or equivalently, the share of price increases greater than
the in‡ation rate— from 1995 through 2019. A striking feature of this plot is the close empirical
relationship between the two variables.9 Panel B in the same …gure plots the same relationship but
8
These …gures are not reported to save space, but their sum can be computed by substracting the contributions
in Table 5 from 100% for each sector.
9
This empirical relationship is discussed in Wolman (2022) in relation to the high in‡ation observed since in March
2021, and it is examined in more detail by Hornstein, Ruge-Murcia, and Wolman (2022).
16
based on arti…cial data simulated from our estimated model. Comparing both panels shows that
the model can account for this feature of the U.S. data. This result is due to the fact that our
estimated speci…cation matches well the behavior of category price changes and, importantly, that
it does so by assigning a large role to sectoral, versus aggregate, shocks.
We now provide some intuition for Figure 4 from the perspective of the model. To begin, note
that there is a simple relationship between (i) the shares of relative price increases and decreases
and (ii) the average sizes of relative price increases and decreases. Because the average relative
price change is zero, the ratio of the share of relative price increases to the share of relative price
decreases is identical to the ratio of the average relative price decrease to the average relative price
increase,
!
r+
(1
!)
r
= 0;
which implies
!=(1
!) = ( r )=( r+ );
where ! is the share of relative price increases, and
r+ and
r
are the average relative price
increase and decrease, respectively. Note that the way we have written these equations, the average
relative price decrease is a positive number.
Consider now a point on the far right of panel A; because the share of relative price increases is
close to one, the ratio of the share of relative price increases to the share of relative price decreases
is very high, and therefore the average relative price increase is very small compared to the average
relative price decrease. In principle, this can happen in many ways— we’ll consider two. One
possibility is that in‡ation is much higher than target. This does not occur in the data or in the
model, but it’s instructive to work through what it would involve. A large share of relative price
increases and in‡ation far above target would mean that the relative price increases correspond to
nominal price increases that are much higher than target. This could happen from a combination
of a large in‡ationary aggregate shock and a small share of sectors experiencing a large increase in
productivity, so that they choose to reduce their relative price by a large amount. But according
to our estimates, large aggregate shocks are unlikely. Or, it could happen without an aggregate
shock, if most sectors experienced negative sectoral productivity shocks. But this is also unlikely,
because the sectoral shocks are uncorrelated.
A second possibility when there is a large share of relative price increases is that in‡ation is
much lower than target, which is the typical outcome in the …gure. A large share of relative price
increases and in‡ation far below target means that the relative price decreases— few in number,
correspond to nominal price decreases that are much lower than target. This could happen from
a combination of a large de‡ationary aggregate shock and a small share of sectors experiencing a
17
large (possibly additional) increase in productivity, so that they choose to reduce their relative price
by a large amount. But again, large aggregate shocks are unlikely. Or, it could happen without
an aggregate shock, if a small share of sectors experienced large positive productivity shocks, and
in response chose large nominal price decreases. While this scenario is unconditionally unlikely,
conditional on a high share of relative price increases, and conditional on the model estimates, it
is in fact likely.
The discussion thus far has not touched on the systematic behavior of monetary policy, but that
behavior is of course central to the relationship in Figure 4. The key point here is that monetary
policy responds— in the model and generally in the data— to economywide aggregates, as opposed
to sectoral price changes. When a relative price shock hits a particular sector, the desired relative
price change is accomplished mainly by a nominal price change of the same sign for that sector,
so that in‡ation moves in the same direction (Figure 2). It’s theoretically possible for monetary
policy to perfectly stabilize in‡ation (a ‡at line in Figure 4), or even to generate an upward sloping
relationship between the share of relative price increases and the in‡ation rate. But either of these
cases would require that in the face of a large relative price shock for just one sector, monetary
policy generates large nominal price changes for all other sectors in the opposite direction. This
seems implausible, and is not what we see in the data.10
We summarize as follows: most of the time, the share of relative price increases is close to
one half, and in‡ation is close to target. When the share of relative price increases is large, it
is associated with a small share of sectors experiencing large positive productivity shocks, and
choosing large nominal price decreases, which results in low in‡ation. When the share of relative
price increases is small, it is associated with a small share of sectors experiencing large negative
productivity shocks, and choosing large nominal price increases, which results in high in‡ation.
The systematic behavior of monetary policy delivers these relationships as equilibrium outcomes:
it is optimal for sectors experiencing changes in their desired relative price to move their nominal
price in the same direction, and for other sectors to respond very little.
4.4
A Flexible-Price Version of the Model
As discussed above and reported in Table 2, there is substantial heterogeneity in our estimates of
price stickiness across sectors. The price-stickiness parameter
s
is statistically signi…cant for more
than half of sectors that account for 70% of the expenditures by U.S. households. Most notably, the
10
Additionally, in New Keynesian models such as the one we employ, it is generally not optimal to stabilize the
price level in response to large relative price shocks, unless those shocks hit sectors in which nominal rigidities are
especially large. Goodfriend and King (1997) …rst made this general point; Aoki (2001) provided analytical results
in a two-sector model; and Eusepi, Hobijn and Tambalotti (2011) conducted quantitative analysis in a model similar
to ours.
18
joint hypothesis that the
s
are all equal is rejected by the data. It may be tempting to conclude
then that the relationship between relative price changes and in‡ation displayed in Figure 4 is also
closely related to price stickiness. We …nd this not to be the case.
We estimate a ‡exible-price version of the model, meaning a version of the model in section 2
subject to the constraint that the price rigidity parameters are zero in all sectors. ML estimates
of the parameters are reported in Table 6 and the last two columns of Table 3. Note in Table 6
that the estimates of the autoregressive coe¢ cients of sectoral productivity shocks are extremely
high. The reason is simply that without price rigidity, the persistence of sectoral price changes can
only be accounted for by a high autocorrelation of the sectoral shocks. Estimates of the standard
deviation of productivity innovations and Taylor rule parameters obtained under this model are
roughly consistent with those obtained under the benchmark model with sticky prices. Note that
this model is strongly rejected by the data: the likelihood ratio statistic of the test of the restriction
s
= 0 for all s (that is, that all prices are ‡exible) is 2
(25612:7
25540:8) = 143:8 which is
larger than the 5% critical value of 25. The point that we want to make here, however, is that price
rigidity is not essential for our result that the dynamics of aggregate in‡ation and sectoral price
changes are driven by relative price shocks. To see this, consider the following. First, regarding
aggregate in‡ation, monetary policy and relative price shocks respectively account for 33:7% and
66:3% of the unconditional variance of in‡ation under the ‡exible-price model, compared to the
values of 23:7% and 76:3% reported above for the benchmark model.
Second, the last two columns in Table 5 report the contribution of the “own”relative price shock
(Column 3) and of the monetary policy shock (Column 4) to the unconditional variance of each
sectoral price change under the ‡exible-price model. These estimates are quantitatively similar to
those in columns 1 and 2 obtained under the benchmark model and lead to the same conclusions.
Finally, panel C in Figure 4 shows that strong negative relationship between the monthly in‡ation
rate on the vertical axis and the share of relative price increases predicted by our model is robust
to assuming that all prices are completely ‡exible.
These results relate to work by Balke and Wynne (2000). Compared with Ball and Mankiw
(1995), whose explanation for the relationship between relative price changes and in‡ation relies on
price stickiness, Balke and Wynne consider a ‡exible-price model. There are three central elements
to their analysis. First, they point out that in a ‡exible price model, properties of the distribution of
sectoral productivity growth would be re‡ected in the distribution of relative price changes. Second,
they provide evidence that the distribution of productivity changes in fact had much in common
with the distribution of sectoral price changes. These two elements are also present in our model
in a more general setup where the data are allowed to determine the extent of price rigidity in each
19
sector (see Figure 1). Finally, they show that in a calibrated multi-sector model with ‡exible prices,
basic properties of the empirical relationship between the in‡ation and the distribution of relative
price changes are replicated when the model is driven by an estimated sectoral productivity process.
In contrast to their work, we focus on the heterogeneity in the variance of sectoral productivity
innovations and the limited role of aggregate shocks in accounting for the empirical relationship
between relative price changes and in‡ation.
5.
In‡ation Shortfall 2012-2019
We estimated the model over post-1994 U.S. data. Over that period, in‡ation averaged around 2%
and was quite stable by historical standards. Since 2012, the Federal Reserve has had a formal 2%
target for PCE in‡ation, but from 2012 to 2019, PCE in‡ation averaged only 1:4%. We focus here
on the post-2012 period and use the model to decompose the in‡ation shortfall into contributions
from the various estimated shocks.
Because mean in‡ation is 2%, if the model state variables were at their steady states and
there were no shocks, then in‡ation would be constant at 2%. The model explains an episode of
deviations from 2% in‡ation by a combination of initial state variables being away from steady
state and subsequent shocks. Figure 5 plots annual in‡ation during the undershoot period, along
with the contributions to the cumulative price level undershoot from the gasoline shock, the health
care shock and the monetary policy shock. The cumulative undershoot is measured starting from
January 2012. The contributions are based on the smoothed inferences of each shock from the
Kalman …lter with our estimated parameters. The early part of the undershoot was driven mainly
by health care shocks, and then somewhat later by gasoline shocks, whereas monetary policy became
more important starting in late 2016. Note however that the period where monetary policy began
to account for more of the undershoot coincided with the Fed’s increase in interest rates. With the
caveat that our estimation procedure does not impose the zero bound, the narrative that seems
most consistent with Figure 5 is that health care and gasoline shocks drove in‡ation below target
initially, with policy at the lower bound, then in‡ation stayed below target in part because of the
Fed’s raising the Funds rate starting late in 2015. While that rate increase was gradual by historical
standards, according to our estimated model, it represented contractionary policy shocks.
6.
COVID In‡ation
We now use our estimated model to interpret the behavior of U.S. in‡ation in the period immediately
after our estimation sample; that period corresponds to the COVID-19 pandemic, when in‡ation
20
was at …rst volatile and then consistently far above the Fed’s 2% target.11 Because it interprets
the data through the lens of the estimated model, this analysis assumes that the U.S. economy has
remained in a rational expectations equilibrium involving local ‡uctuations around a steady state
with 2% in‡ation. One might respond sceptically that we are assuming the answer to the most
important question: has the U.S. economy remained in that equilbrium, or has the Fed’s behavior
deviated from its previous rule to such an extent that private agents no longer perceive that rule
to be in place? We do not dispute the importance of that question, and in fact see our work as
contributing to an answer: under the assumption that in‡ation has remained anchored, we provide
estimates of the contributions to observed in‡ation from monetary policy and sectoral shocks. An
evaluation of those estimates based on independent information can then help in assessing whether
in‡ation has in fact remained anchored.
As above, we decompose observed in‡ation into the contributions of the various shocks. The
data is now outside our estimation sample, but the procedure is otherwise identical. Note that this
procedure also yields a counterfactual policy analysis: in the absence of monetary policy shocks,
how would in‡ation have behaved? Figure 6 presents selected elements of the COVID in‡ation
decomposition. In this case, we plot the price level instead of in‡ation, along with the 2% trend
line and the price level paths implied by the shocks to gasoline, motor vehicles, food at home,
and monetary policy. The …gure uses data only though November 2021, so it misses some of the
high in‡ation months. (The …gure will be updated in the next revision of the paper). According
to our estimated model, expansionary monetary policy and (to a lesser extent) food at home were
signi…cant contributors to in‡ation in this period. However, their positive contributions came early,
when the Fed lowered its interest rate target, and they were compensated by a large drop in the
price of gasoline and energy goods. A few shocks (especially motor vehicles) were the main culprits
in the subsequent increase in in‡ation (along with dissipation of other shocks).
It will be interesting to see how this decomposition looks with more recent data, as Wolman
(2022) shows that at the end of 2021 and the beginning of 2022, the share of relative price increases
has been consistent with relatively low in‡ation, according to the historical relationship in Figure
4. The fact that in‡ation has in fact been quite high suggests that expansionary policy is playing
a large role.
11
Recent work that studies the macroeconomic implications of COVID-19 include Baqaee and Farhi (2022), Guerrieri, Lorenzoni, Straub, and Werning (2022), and Woodford (2022). However, contrary to our analysis, which focuses
on in‡ation, their work is concerned with the transmission of an asymmetric (COVID) shock in economies with complementarity in production (Baqaee and Farhi, and Guerrieri et al.) or a network payments structure (Woodford).
21
7.
Conclusions
We set out to evaluate the contribution of sectoral shocks— relative price shocks — to the behavior
of in‡ation, and we found their contribution to be very large indeed. For instance, we …nd that
the early part of the in‡ation undershooting after 2012 was driven mainly by shocks to health care
and to gasoline and other energy goods. We …nd that expansionary monetary policy has been a
signi…cant contributor to the high in‡ation outcomes following the COVID pandemic, but sectoral
shocks (especially motor vehicles) were the main culprits in the burst of in‡ation from May through
November 2021. To be clear, we are not claiming that in‡ation is always and everywhere a relativeprice-shock phenomenon. Rather, the lesson is simply that if monetary policy has crediblility and
behaves in a way that delivers stable in‡ation, then the in‡ation ‡uctuations that remain seem
to be driven by sectoral shocks. Our model-based analysis assumes credibility and stability of the
policy regime, and the data we use to estimate the model comes from a period when in‡ation was
stable, so the assumption is plausible.
In future work, we plan to address some of the caveats in the analysis reported in this paper.
First, we focus primarily on nominal variables but any subsquent research that also goes after the
real-nominal relationship and seeks to explain in‡ation should also be consistent with the fact that
there is a systematic relationship between the distribution of relative price changes and in‡ation.
Second, in order to simplify the analysis as much as possible, we have considered only sectoral
supply shocks. A natural and perhaps necessary extension is to add sectoral demand shocks. We
expect that the qualitative results would be unchanged, but there would be an additional factor
driving relative price changes.
22
Table 1. Sectoral Consumption Weights and Productivity Trends
Sector
Motor vehicles and parts
Furnishings and household durables
Recreational goods
Other durable goods
Food at home
Clothing and footwear
Gasoline and other energy goods
Other nondurable goods
Housing and utilities
Health care
Transportation services
Recreation services
Food services and accommodations
Financial services and insurance
Other services
23
Consumption
Weight
(1)
0:0488
0:0296
0:0311
0:0166
0:0878
0:0414
0:0308
0:0801
0:1877
0:1509
0:0343
0:0371
0:0661
0:0738
0:0840
Productivity
Trend 102
(2)
0:2605
0:3936
0:7428
0:3076
0:1710
0:3487
0:1343
0:1564
0:0891
0:0464
0:1586
0:0983
0:1003
0:0978
0:1041
Table 2. Estimates of Sectoral Parameters and
Aggregate Productivity Process
Autoregressive
Coe¢ cient
Price Rigidity
Sector
Motor vehicles and parts
Furnishings and household durables
Recreational goods
Other durable goods
Food at home
Clothing and footwear
Gasoline and other energy goods
Other nondurable goods
Housing and utilities
Health care
Transportation services
Recreation services
Food services and accommodations
Financial services and insurance
Other services
Aggregate productivity
Wald test (p-value)
Estimate
(1)
7:316
0:186
0:676
< 0:001
4:229
0:516
3:673
0:030
8:485
4:090
0:255
2:094
111:121
< 0:001
12:883
s.e.
(2)
2:362
0:625
0:660
0:023
1:572
0:584
1:159
0:345
1:752
1:591
0:527
0:994
24:342
0:348
3:966
< 0:001
Estimate
(3)
0:990
0:991
0:998
0:995
0:997
0:995
0:962
0:990
0:997
0:997
0:975
0:997
0:455
0:991
0:998
0:819
< 0:001
Note: s.e. stands for standard error. The superscripts
s.e.
(4)
0:012
0:011
0:002
0:009
0:002
0:006
0:015
0:012
0:002
0:002
0:021
0:004
0:062
0:007
0:002
216:745
Standard
Deviation
Estimate
s.e.
102
102
(5)
(6)
0:460
0:041
0:386
0:028
0:376
0:028
0:584
0:026
0:324
0:029
0:547
0:039
6:144
0:301
0:315
0:015
0:192
0:016
0:184
0:018
0:530
0:039
0:243
0:039
1:989
0:456
0:710
0:040
0:272
0:031
< 0:001
0:386
< 0:001
and y denote statistical signi…cance at the
…ve and ten percent levels, respectively. The value of the log likelihood function at the optimum
(excluding the constant term) is 25540:8.
24
Table 3. Estimates of Taylor Rule
Parameter
Smoothing parameter
In‡ation coe¢ cient 102
Output coe¢ cient 102
AR coe¢ cient
Standard deviation 102
Benchmark
Estimate
s.e.
(1)
(2)
0:733
0:146
4:312
2:725
0:310
0:227
0:673
0:176
0:049y 0:027
Note: see notes to Table 2.
25
Unrestricted
Estimate
s.e.
(3)
(4)
0:989
0:261
1:007
0:249
0:146y
0:079
0:456
0:038
0:011
< 0:001
Flexible-Price
Estimate
s.e.
(5)
(6)
0:721
0:095
5:122
2:481
0:037
0:111
0:621
0:130
0:052
0:018
Table 4. Empirical and Theoretical Second Moments
Variable
Nominal interest rate
Aggregate in‡ation
Sectoral price changes:
Motor vehicles and parts
Furnishings and household durables
Recreational goods
Other durable goods
Food at home
Clothing and footwear
Gasoline and other energy goods
Other nondurable goods
Housing and utilities
Health care
Transportation services
Recreation services
Food services and accommodations
Financial services and insurance
Other services
Standard
Deviation
Data Model
(1)
(2)
0:110
0:043
0:187
0:252
0:315
0:377
0:355
0:597
0:263
0:517
4:986
0:303
0:140
0:148
0:511
0:208
0:169
0:722
0:171
0:348
0:432
0:415
0:628
0:307
0:566
4:828
0:382
0:197
0:224
0:558
0:273
0:211
0:762
0:221
Autocorrelation
Data
Model
(3)
(4)
0:992
0:939
0:387
0:139
0:307
0:019
0:026
0:170
0:266
0:016
0:349
0:140
0:338
0:116
0:045
0:097
0:056
0:236
0:356
0:325
0:000
0:047
0:013
0:228
0:039
0:211
0:024
0:324
0:207
0:005
0:127
0:288
0:012
0:410
Note: The predicted moments are the sample average of the moments computed using 1000 simulations with number of observations equal to the sample size T = 301.
26
Table 5. Variance Decomposition
Variable
Sectoral price changes:
Motor vehicles and parts
Furnishings and household durables
Recreational goods
Other durable goods
Food at home
Clothing and footwear
Gasoline and other energy goods
Other nondurable goods
Housing and utilities
Health care
Transportation services
Recreation services
Food services and accommodations
Financial services and insurance
Other services
Benchmark
Own Monetary
Shock
Policy
(1)
(2)
Flexible-Price
Own Monetary
Shock
Policy
(3)
(4)
86:481
78:458
79:421
89:216
77:909
88:414
99:904
71:308
62:428
58:413
87:256
62:771
94:805
92:735
77:021
81:676
86:024
83:966
93:401
73:178
91:765
99:900
79:135
52:783
53:192
91:800
64:388
55:789
95:489
57:717
27
9:988
15:438
14:980
7:711
16:674
8:372
0:072
20:537
28:985
31:883
9:139
27:274
3:060
5:240
17:813
15:474
11:865
13:429
5:537
22:582
6:908
0:084
17:451
41:515
40:677
6:961
29:905
36:976
3:782
35:958
Table 6. Estimates of Sectoral Parameters and
Aggregate Productivity Process under Flexible-Price Model
Motor vehicles and parts
Furnishings and household durables
Recreational goods
Other durable goods
Food at home
Clothing and footwear
Gasoline and other energy goods
Other nondurable goods
Housing and utilities
Health care
Transportation services
Recreation services
Food services and accommodations
Financial services and insurance
Other services
Aggregate productivity
Autoregressive
Coe¢ cient
s.e.
Estimate
103
(1)
(2)
0:999
0:010
0:999
0:006
0:999
0:021
0:999
0:012
0:999
0:017
0:999
0:025
0:982
0:135
0:983
0:152
0:999
0:008
0:999
0:008
0:999
0:010
0:999
0:010
0:967
0:180
0:997
0:064
0:999
0:008
0:903
979:108
Standard
Deviation
Estimate
s.e.
102
102
(3)
(4)
0:316
0:012
0:369
0:015
0:354
0:013
0:583
0:025
0:247
0:009
0:517
0:017
4:948
0:115
0:305
0:007
0:128
0:005
0:134
0:006
0:504
0:018
0:202
0:009
0:175
0:007
0:716
0:010
0:162
0:007
< 0:001
0:034
Wald test (p-value)
< 0:001
< 0:001
Sector
Note: see notes to Table 2. The value of the log likelihood function at the optimum (excluding the
constant term) is 25612:7.
28
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31
Figure 1: Responses of Relative Prices to Negative Productive Shock in Own Sector
A. Motor
Vehicles
B. Furnishings
C. Recreational
Goods
0.4
0.4
0.3
0.3
D. Other
Durables
E. Food
at Home
F. Clothing
and Footwear
H. Other
Nondurables
G. Gasoline
0.6
0.3
0.6
6
0.3
0.4
0.2
0.4
4
0.2
0.2
0.1
0.2
2
0.1
0
0
0
0.4
0.3
0.2
0.2
0.1
0.2
0.1
0.1
0
0
0
-0.1
1
48
96
1
I. Housing
0.2
48
96
J. Health
1
48
96
-0.1
1
48
96
1
0.6
0.3
96
-0.1
1
M. Food
Services
K. Transportation L. Recreation
0.2
48
0
0
48
96
1
0.8
96
O. Other
Services
N. Finance
0.2
48
0.3
0.6
0.2
0.4
0.1
0.2
0.1
0.1
0.1
0.2
0
0.4
0.1
0.2
0
0
0
1
48
96
1
48
96
0
0
1
48
96
1
48
96
1
48
96
0
1
48
96
1
48
96
1
48
96
Figure 2: Inflation Responses
A. Monetary
Policy
B. Motor
Vehicles
C. Furnishings
D. Recreational
Goods
E. Other
Durables
F. Food
at Home
G. Clothing
and Footwear
1.5
H. Gasoline
2
0.6
0.6
0.6
0.6
0.6
0.6
1.5
1
0.4
0.4
0.4
0.4
0.4
0.4
1
0.5
0
1
6
12
0.2
0.2
0.2
0.2
0.2
0.2
0
0
0
0
0
0
18
1
I. Other
Nondurables
6
12
18
1
J. Housing
6
12
18
K. Health
1
6
12
18
1
6
12
18
1
6
12
18
0.5
0
1
N. Food
Services
L. Transportation M. Recreation
6
12
18
1
6
12
18
P. Other
Services
O. Finance
1
0.6
0.6
0.6
0.6
0.6
0.6
0.6
0.8
0.4
0.4
0.4
0.4
0.4
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0
0
1
6
12
18
0
1
6
12
18
0
1
6
12
18
0
1
6
12
18
0
1
6
12
18
0
0
1
6
12
18
1
6
12
18
1
6
12
18
Figure 3: Variance Decomposition of the Inflation Rate
A. Aggregate
Productivity
0.01
B. Monetary
Policy
C. All
Relative
D. Motor
Vehicles
25
E. Furnishings
0.32
F. Recreational
Goods
1.12
G. Other
Durables
H. Food
at Home
I. Clothing
and Footwear
0.38
1.34
3.05
0.7
1.1
1.32
0.37
76
24
0.31
3
0.69
1.3
1.08
0.36
2.95
0
1 12 24 36
J. Gasoline
23
1 12 24 36
75
K. Other
Nondurables
1 12 24 36
0.68
1 12 24 36
0.3
1 12 24 36
1.06
1 12 24 36
M. Health N. TransportationO. Recreation
L. Housing
43.5
1.28
1 12 24 36
1 12 24 36
1 12 24 36
P. Food
Services
Q. Finance
R. Other
Services
0.66
8.2
3.55
1.72
0.5
4.4
3
3.5
8
43
4.7
0.64
1.68
0.49
4.3
42.5
3.45
2.8
1.64
1 12 24 36
1 12 24 36
4.6
1 12 24 36
1 12 24 36
0.62
1 12 24 36
0.48
1 12 24 36
7.8
1 12 24 36
1 12 24 36
1 12 24 36
Figure 4: Inflation and the Share of Price Changes Larger than Inflation
B. Benchmark Model
C. Flexible-Price Model
1
1
0.5
0.5
0.5
0
-0.5
Monthly Inflation
1
Monthly Inflation
Monthly Inflation
A. U.S. Data
0
-0.5
-0.5
Corr = -0.55
Corr = -0.60
Corr = -0.75
-1
-1
0
0.5
Share
0
1
-1
0
0.5
Share
1
0
0.5
Share
1
Figure 5: Undershooting: Inflation and Contributions from Gasoline, Health Care, and Monetary Policy
0.01
0
-0.01
-0.02
Fitted Inflation
Gasoline
Health Care
Monetary Policy
Jan 2012
Jan 2013
Jan 2014
Jan 2015
Jan 2016
Jan 2017
Jan 2018
Jan 2019
Dec 2019
Figure 6: COVID Period: Price Level and Contribution from Selected Shocks
0.07
0.06
Actual PCE
2% Trend
Motor Vehicles
Food at Home
Gasoline
Monetary Policy
0.05
0.04
0.03
0.02
0.01
0
-0.01
Jan. 2020
Jun. 2020
Dec. 2020
Jun. 2021
Nov. 2021
@FedFRASER