IMES DISCUSSION PAPER SERIES

Optimal Trend Inflation

Klaus Adam and Henning Weber
Discussion Paper No. 2018-E-7

INSTITUTE FOR MONETARY AND ECONOMIC STUDIESBANK OF JAPAN
2-1-1 NIHONBASHI-HONGOKUCHOCHUO-KU, TOKYO 103-8660
JAPAN

You can download this and other papers at the IMES Web site:

https://www.imes.boj.or.jp
Do not reprint or reproduce without permission.

NOTE: IMES Discussion Paper Series is circulated inorder to stimulate discussion and comments. Views
expressed in Discussion Paper Series are those of
authors and do not necessarily reflect those of
the Bank of Japan or the Institute for Monetaryand Economic Studies.

IMES Discussion Paper Series 2018-E-7
July 2018

Optimal Trend Inflation
Klaus Adam* and Henning Weber**
Abstract
Sticky price models featuring heterogeneous firms and systematic firm-level productivity trends deliver radically different predictions for the optimal inflation rate
than their popular homogenous-firm counterparts: (1) the optimal steady-state inflation rate generically differs from zero and (2) inflation optimally responds to
productivity disturbances. We show this by aggregating a heterogenous-firm modelwith sticky prices in closed form. Using firm-level data from the U.S. Census Bureau, we estimate the historically optimal inflation path for the U.S. economy. In
the year 1977, the optimal inflation rate stood at 1.5%, but subsequently declined
to around 1.0% in the year 2015. Inflation rates up to twice these numbers can berationalized if one considers product demand elasticities more in line with the trade
literature or if one considers firms that (partially) index prices to lagged inflation
rates.
Keywords: optimal inflation rate; sticky prices; firm heterogeneity
JEL classification: E52, E31, E32*University of Mannheim & CEPR (E-mail: adam@uni-mannheim.de)**Deutsche Bundesbank (E-mail: henning.weber@bundesbank.de)
We would like to thank Matthias Kehrig for supporting us through access to the LBD databaseduring early stages of this project. We also thank Pierpaolo Benigno, Saki Bigio, V.V. Chari,
Argia Sbordone, Stephanie Schmitt-Grohé, Michael Woodford, seminar participants at the2016 CSEF-CIL-UCL conference in Alghero, the 12th Dynare conference in Rome, 2016, the2016 EUI Workshop on Economic Policy Challenges, Florence, the 2016 German Economic
Association (VfS) meeting, the 2017 SNB Research Conference, the 2017 ECB conference on“Understanding Inflation”, and the 2017 EABCN meeting at the Banque de France forcomments and suggestions. Special thanks go to the editor John Leahy and anoynmous referees
for very valuable suggestions on how to improve the paper. The views expressed in this paper
are those of the authors and do not necessarily reflect the official views of the Bank of Japan,
the Deutsche Bundesbank, or the Eurosystem.

1

Introduction

This paper introduces heterogeneous …rms and empirically plausible …rm-level productivity trends into an otherwise standard sticky-price economy. It shows that some of thefundamental implications of canonical sticky-price models with homogeneous …rms fail tosurvive within such a generalized setup. The optimal steady-state in‡ation rate genericallydi¤ers from zero and in‡ation optimally responds to productivity disturbances, unlike insettings with homogeneous …rms. Moreover, the paper documents that the predictions of
the homogeneous …rm model turn out to be non-robust in the sense that they are discontinously a¤ected by the presence of …rm heterogeneity. We thus present an example inwhich microeconomic heterogeneity matters for macroeconomic policy prescriptions, anissue that has attracted renewed interest (Ahn et al. (2017), Kaplan and Violante (2014)).
Due to the technical di¢ culties associated with aggregating heterogeneous-…rm models, it is standard in the sticky-price literature to abstract from all …rm-level heterogeneity
beyond that generated by price adjustment frictions themselves. As is well known, priceadjustment frictions then tightly anchor the optimal steady-state in‡ation rate at zero,see Woodford (2003).1 As we show, this rather robust but somewhat puzzling implication of standard sticky-price models arises precisely because of the homogeneity assumption. Homogeneity implies that the productivity of price-adjusting …rms equals that of
non-adjusting …rms. With economic e¢ ciency requiring relative prices to re‡ect relative
productivities, it calls for price-adjusting …rms to charge the same price as charged on
average by non-adjusting …rms, i.e., it calls for zero in‡ation.2
The present paper extends the basic sticky-price setup by introducing …rm heterogeneity and systematic …rm-level productivity trends. Such …rm-level trends are clearlypresent in micro data, but are routinely abstracted from in the sticky-price literature.
New …rms, for example, tend to be initially small, i.e., tend to be initially unproductive
when compared to existing …rms.3 Some of the young …rms become more productive overtime and grow, others become unproductive and exit the economy. We show how suchlife-cycle related productivity dynamics cause the average productivity of price-adjusting…rms to generally di¤er from the average productive of non-adjusting …rms. Economice¢ ciency then requires that adjusting …rms set on average di¤erent prices than existing…rms, which causes in‡ation or de‡ation to be optimal in steady state. We show this byaggregating the non-linear sticky price model with heterogeneous …rms in closed form andby deriving analytical expressions for the optimal in‡ation rate.
1

Section 2 discusses a range of extensions of the basic framework considered in the literature and theirimplications for the optimal in‡ation rate.
2
Yun (2005) shows, using a setting with homogeneous …rms, that if initial prices do not re‡ect initial
productivities, the optimal in‡ation rate can display deterministic transitory deviations from zero.
3
This does not rule out that new …rms are in age-adjusted terms more productive than old …rms. Our
setup will allow for this possibility.

1The heterogeneous …rm model that we present is formulated in abstract terms andallows for a variety of economic interpretations through which …rm heterogeneity arises.
One interpretation is - as alluded to above - that heterogeneity arises from …rm entry
and exit and the associated life-cycle dynamics of …rm productivity. This is also theinterpretation that we shall consider in our empirical analysis. Yet, as explained in themain text, the model can equally be interpreted as one in which heterogeneity arises fromproduct substitution or product quality improvements.
To …x ideas, consider a sticky-price model with Calvo type or menu-cost type priceadjustment frictions in which a measure

0 of randomly chosen …rms becomes unpro-

ductive and exits the economy each period. Exiting …rms are replaced by a measureofyoung new …rms. Our setup then features three systematic productivity trends, each ofwhich has di¤erent implications for the optimal in‡ation rate. First, there is a common
trend in total factor productivity (TFP), which a¤ects all …rms equally. The common
TFP trend captures general-purpose innovations that are adopted by all …rms simultaneously. As in a standard homogeneous-…rm model, it does not a¤ect the optimal in‡ation
rate. Second, there is an experience trend in …rm-level TFP, which determines how …rms
accumulate experience with age. The experience trend may capture productivity gains
from learning-by-doing or other forms of experience accumulation. As we show, this productivity trend generates a force towards positive in‡ation rates. Third, there is a cohort
productivity trend, which determines the productivity level of newly entering …rms. This
trend captures the fact that new …rms tend to bring new technologies into the economy
that are not (yet) used by other …rms.4 The cohort trend will be a force towards making
de‡ation optimal.
Taken together, the optimal steady-state in‡ation rate in our setting depends on thestrength of the experience trend relative to the strength of the cohort trend, whenever
there is some positive …rm turnover ( > 0). The optimal steady-state in‡ation rate isitself independent of the …rm turnover rate, as long as> 0. Yet, in the absence of…rm turnover ( = 0), the optimal steady-state in‡ation rate collapses to zero, i.e., to the
optimal in‡ation rate of a homogeneous …rm model. It is in this sense, that the in‡ation
predictions of the homogeneous …rm model turn out to be non-robust.
To obtain economic intuition for these …ndings, consider two polar settings. The …rst
setting abstracts from the presence of a cohort trend and considers a setting where the only
trend is that …rms accumulate experience over time.5 If an old …rm becomes unproductive
and exits the economy, the new …rm that replaces it will not have accumulated any
experience yet. The new …rm will thus be less productive than the remaining set of
4

Newly entering …rms are endowed with the cohort productivity level, in addition to the common TFP
component, and then gradually accumulate experience over time.
5
As mentioned before, we can abstract from the common TFP trend, as it does not a¤ect the optimalin‡ation rate.

2

old …rms.6 From a welfare standpoint, the optimal price of new …rms should therefore
exceed the average price of existing …rms, so as to accurately re‡ect relative productivities.
Achieving this requires either that new …rms choose higher prices or that old …rms reduce
prices, or a combination thereof.
In the presence of sticky prices, price reductions by old …rms are costly in welfare terms.
In time-dependent price adjustment models, they lead to ine¢ cient price dispersion due
to asynchronous price adjustment; in state-dependent pricing models, they require …rms
to pay adjustment costs. Therefore, it is optimal to implement the e¢ cient relative priceexclusively by having new …rms charge higher prices, while all other …rms hold theirprices steady. Clearly, this implies that the aggregate in‡ation rate must be positive inthe steady state.
Now consider the second polar setting, in which there is no experience e¤ect andthe only trend is a positive cohort trend. New …rms are then more productive than theexisting set of old …rms, thus optimally charge lower prices than existing …rms. This
makes negative rates of in‡ation optimal.7
We also determine in closed form the optimal dynamic response of the in‡ation rate
following shocks to experience and cohort productivity. We show that such shocks havefairly persistent e¤ects on the optimal in‡ation rate, especially in settings in whichispositive but close to zero.
To estimate the optimal in‡ation rate for the United States, we devise a modelconsistent estimation approach, which is based on …rm-level information from the Business
Dynamics Statistics (BDS) of the U.S. Census Bureau. The BDS is based on the Longitudinal Business Database (LBD) and covers all private sector establishments in the UnitedStates from the year 1977 onwards. We rely on …rm-level information because aggregate
information fails to identify the in‡ation-relevant cohort and experience trends. This isso because in‡ation-neutral TFP trends mask the underlying in‡ation-relevant …rm-level
trends at the aggregate level. The …rm-level information allows us to estimate the historically optimal in‡ation path for the U.S. economy in a model-consistent way and fora setting where the actual in‡ation rates implemented by the Federal Reserve may havebeen suboptimal. Our estimation shows that the optimal U.S. in‡ation rate was strictly
positive throughout the years 1977-2015. In our benchmark estimation, it ranges between
1.5% in 1977 and a temporary low of around 0.85% during the Great Recession; the mostrecent estimate for the year 2015 is 1%.
In‡ation rates up to twice these numbers can be rationalized if one is willing to assume
lower demand elasticities, in line with estimates in the trade and industrial organization
literatures, or if one consider settings in which sticky-price …rms index partially prices to
6

The new …rm will be in age-adjusted terms more productive than all old …rms, once one allows for apositive cohort trend.
7
Due to price-setting frictions, it is again not optimal that old …rms adjust prices.

3

past in‡ation rates. Optimal in‡ation rates above 1% are large by the standards of the
sticky-price literature, e.g., Diercks (2017).
The remainder of the paper is structured as follows. Section 2 discusses the related
literature. Section 3 presents our heterogeneous-…rm model with sticky prices and section
4 presents a special case that illustrates our main result in rigorous but simple terms.
Section 5 analytically aggregates the general model, and section 6 shows that the ‡exibleprice equilibrium in this model is …rst best when a Pigouvian output subsidy corrects…rms’monopoly power. The main optimal in‡ation result is presented in closed-form in
section 7. Section 8 discusses the optimal steady-state in‡ation rate. It shows how the
optimal in‡ation rate jumps discontinuously when moving from a standard sticky-priceeconomy ( = 0) to one including …rm turnover ( > 0). Section 9 determines the utility
costs of implementing suboptimal in‡ation. Section 10 presents our estimation approachand our empirical results. Section 11 discusses the robustness of our …ndings towards
various extensions. A conclusion brie‡y summarizes. Proofs and technical material are
relegated to a series of appendices.

2

Related Literature

Few papers discuss the relationship between the optimal in‡ation rate and productivity trends. All of them focus on aggregate or sectoral productivity trends and …nd thatthe optimal in‡ation rate is (slightly) negative. Amano et al. (2009) consider an economy with aggregate productivity growth and sticky wages and prices. They show howmonetary policy a¤ects wage and price mark-ups and that this can make it optimal to implement de‡ation, so as to reduce wage mark-ups. Wolman (2011) considers a two-sector
sticky-price economy with sectoral productivity trends. Despite the absence of monetary
frictions, the optimal in‡ation rate is either negative or close to zero in his setting.
Golosov and Lucas (2007) and Nakamura and Steinsson (2010) consider sticky-pricesetups with heterogeneous …rms and study monetary non-neutrality within these setups.
They do not consider the issue of the optimal in‡ation rate. Firms in their settings are
subject to random idiosyncratic productivity shocks. This di¤ers from the present setup
which features idiosyncratic shocks that give rise to systematic productivity adjustments(as implied by the cohort and experience trends). The idiosyncratic nature of productivity
shocks in Golosov and Lucas (2007) and Nakamura and Steinsson (2010) causes …rms
with very positive or very negative idiosyncratic productivity shocks to adjust prices.
The productivity of price-adjusting …rms is thus on average similar to the productivity of
non-adjusting …rms, suggesting zero in‡ation to be optimal.
The present paper is also related to a large literature studying the determinants of
optimal in‡ation, most of which …nds that the optimal in‡ation rate is either negative or4

close to zero. None of these papers makes a connection between the optimal in‡ation rateand …rm-level productivity dynamics.
In classic work, Kahn, King and Wolman (2003) consider a homogeneous-…rm modeland explore the trade-o¤ between price adjustment frictions, which call for price stability,
and monetary frictions, which call for a Friedman-type de‡ation. They demonstrate how aslight rate of de‡ation is optimal in such frameworks. In a comprehensive survey, SchmittGrohé and Uribe (2010) document the robustness of these …ndings to a large number ofnatural extensions. They show that taxation motives, including the presence of untaxed
income, foreign demand for domestic currency (Schmitt-Grohé and Uribe (2012a)), as well
as a potential quality bias in measured in‡ation rates (Schmitt-Grohé and Uribe (2012b)),
are all unable to rationalize signi…cantly positive rates of in‡ation.
Adam and Billi (2006, 2007) and Coibion, Gorodnichenko and Wieland (2012) explicitly incorporate a lower bound on nominal interest rates into sticky-price economies.
They …nd that fully optimal monetary policy is consistent with close to zero average
rates of in‡ation. While zero lower bound episodes make it optimal to promise in‡ationin the future, these promises should only be made conditionally on being at the lower
bound, which happens rather infrequently; see Eggertsson and Woodford (2003) for an
early exposition.
A number of papers …nd positive average rates of in‡ation to be optimal in the presence of downward nominal wage rigidities. Kim and Ruge-Murcia (2009) argue that such
rigidities generate optimal in‡ation rates of approximately 0.35% in a model featuring
aggregate shocks only. Looking at a setting with idiosyncratic shocks, Benigno and Ricci(2011) also …nd a positive steady-state in‡ation rate to be optimal.8 Carlsson and Westermark (2016) consider a setting with nominal wage rigidities and search and matching
frictions in the labor market. They show how a standard U.S. calibration of the model
implies failure of the Hosios condition and justi…es an annual in‡ation rate of about 1.2%.
Schmitt-Grohé and Uribe (2013) analyze the case for temporarily elevated in‡ation in theEuro Area due to the presence of downward rigidity of nominal wages.
Brunnermeier and Sannikov (2016) show that the optimal in‡ation rate can also be
positive in a model without nominal rigidities. They present a model with undiversi…able
idiosyncratic capital income risk in which the optimal in‡ation rate increases with theamount of idiosyncratic risk.
There is also a literature studying endogenous …rm entry decisions in homogeneous …rm
economies, focusing on the e¤ect of in‡ation on …rm entry (Bergin and Corsetti (2008),
Bilbiie et al. (2008)). Bilbiie et al. (2014) document that the welfare optimal in‡ation
rate is positive whenever the bene…t of additional varieties to consumers falls short of8

Since positive in‡ation has no welfare costs in their setup, they do not quantify the optimal in‡ation
rate.

5

the market incentives for creating these varieties. In‡ation then reduces the value ofcreating varieties and brings …rm entry closer to its e¢ cient (lower) level. The present
paper abstracts from endogenous …rm entry decisions and thus from the implication ofmonetary policy for the entry margin. Instead, it considers a setting with heterogeneous…rms in which entry and exit is driven by exogenous productivity dynamics.
Part of the sticky price literature incorporates trend in‡ation via exogenous in‡ation
trends (Ascari and Sbordone (2014), Cogley and Sbordone (2008)). Trend in‡ation inthese setups results from a central bank pursuing an exogenous and potentially timevarying in‡ation target. The present paper is concerned with determining the optimalin‡ation rate and how it relates to microeconomic fundamentals.

3

Economic Model

We consider a cashless economy with nominal rigidities and monopolistically competitive…rms. The model is entirely standard, except for the more detailed modeling of …rmlevel productivity and price adjustment dynamics. Speci…cally, we augment the standard
sticky-price setup by idiosyncratic …rm-level productivity adjustments that arrive in conjunction with a price adjustment opportunity. This gives rise to a setting with heterogeneous …rm-level productivities in which the productivity of price-adjusting …rms is notnecessarily equal to that of non-adjusting …rms.
For simplicity, we derive our results within a time-dependent price adjustment modelà la Calvo (1983). As we argue in section 11.1, our main result in proposition 2 below
remains unaltered if we look instead at a setting where price adjustment frictions take theform of menu costs. The next section introduces our generalized …rm setup in abstract
terms. Section 3.2 provides alternative economic interpretations of the setup.

3.1

Technology, Prices and Price Adjustment Opportunities

Each period t = 0; 1; : : : there is a unit mass of monopolistically competitive …rms indexed
by j 2 [0; 1]. Each …rm j produces output Yjt , which enters as an input into the productionof an aggregate consumption and investment good Yt according toYt =where 1 <

R1

1

dj
Y
0 jt

1

;

< 1 denotes the price elasticity of product demand. Let Pjt denote the pricecharged by …rm j in period t. Firms can adjust prices with probability 1(0(1)

each period< 1). The arrival of a Calvo price adjustment opportunity is thereby idiosyncratic

and independent of all other exogenous random variables in the economy.
We augment this standard setting by a second price adjustment opportunity that
arrives with probability

0 each period. This second adjustment opportunity is idio6

syncratic across …rms, but arrives in conjunction with a …rm-level productivity change,
as described in detail below. In particular, let

jt 2 f0; 1g denote the idiosyncratic i.i.d.

random variable governing this second price and productivity adjustment and let

jt = 1indicate the arrival of such an adjustment event for …rm j in period t (Pr( jt = 1) = ).
We shall informally refer to the event jt = 1 as the occurrence of a -shock. We introduce
such -shocks in abstract form below and discuss alternative economic interpretations in
section 3.2.
Letting Kjt and Ljt denote the amount of capital and labor used by …rm j, respectively,…rm output Yjt is given by1

Yjt = At Zjt Kjt

1

1

Ljt

(2)

Ft ;

where At captures common productivity, Zjt …rm-speci…c productivity, and Ft
potential presence of …xed costs for operating the …rm. We have

0 the

1 and to be consistent

with balanced growth, we assume
1

Ft = f ( et )1for some f

0, where(3)

et captures the growth trend in the balanced growth path, as
9de…ned in equation (27) below. Common productivity evolves according to
A t = a t At 1 ;…rm-speci…c productivity according to(Zjt =gt Zjt 1

if

jt = 0Qt

if

jt = 1;(4)

where Qt is given by
Qt = qt Qt 1 :We also assume that at = a at , qt = q qt , and gt = g gt with

(5)
a q g
t ; t ; t > 0 being stationary

shocks with an arbitrary contemporaneous and intertemporal covariance structure, satisfying E[ at ] = E[ qt ] = E[ gt ] = 1. To obtain a well-de…ned steady state and to insure that
relative prices in the ‡exible-price economy remain bounded, we assume throughout the
paper
(1

) (g=q)

1

< 1:

(6)

Productivity dynamics in the previous setting feature three trends: (1) the common
growth trend at ; (2) the experience growth trend gt , which applies in the absence of shocks; and (3) the productivity growth trend qt , which determines the e¤ects of -shocks
9

In the absence of aggregate technology growth, the formulation of …xed costs in equation (3) corresponds to that used in Melitz (2003).

7

on technology. Each of these three growth trends has a di¤erent implication for the
optimal in‡ation rate.
To understand the productivity dynamics implied by the previous setup, consider …rst
the special case with = 0. In the absence of idiosyncratic -shocks to …rm technology, all…rms experience the same productivity growth rate at gt . Such a setting with homogeneous
productivity growth across all …rms is the one routinely considered in the sticky-priceliterature.10
Next, consider the case

> 0 and let sjt denote the number of periods that haveelapsed since …rm j last experienced a -shock (i.e., j;t sjt = 1 and j;et = 0 for et =sjt + 1; :::; t). Firm-speci…c productivity Zjt in equation (4) can then be written ast

Zjt = Gjt Qt sjt ;
where
Gjt =

(

for sjt = 0

1

gt Gjt 1 otherwise,

and where Qt follows equation (5). This alternative formulation illustrates that all …rms
hit by a -shock in t upgrade idiosyncratic productivity to Zjt = Qt , so that Qt canbe interpreted as capturing a "cohort e¤ect" of productivity dynamics, where cohorts
are determined by the arrival time of the last -shock. Following a -shock, the …rm
experiences productivity gains, as described by the process Gjt , as long as no further-shocks arrive. Since the productivity gains Gjt are lost with the arrival of the next shock, one can interpret the process Gjt as capturing "experience" or "learning-by-doinge¤ects" associated with the cohort production technology Qt sjt . Following a -shock inperiod t, our speci…cation thereby implies that …rm productivity increases (temporarily
decreases) if Qt has been growing faster (slower) than Gjt since the time of arrival of
the last -shock prior to period t. Note, however, that as long as Qt displays a positive
growth trend (qt > 0), …rms always become more productive over time in experienceadjusted terms, even if Qt grows slower than Gjt . Indeed, in a setting with> 0, thelong-term growth rate of …rms’ productivity is determined by the process at qt , as theexperience growth rate gt generates - due to the occasional reset - only temporary levele¤ects for …rm productivity.
As usual, we de…ne the aggregate price level as
Pt

Z 1

1
1

Pjt1 dj

:

(7)

0

10

For the case = 0, our setting still allows for a non-degenerate initial distribution of …rm productivities. Typically, this initial distribution is also assumed to be degenerate in the sticky-price literature. As
we show below, the additional assumption of a degenerate initial distribution is not key for the conclusionthat zero in‡ation is optimal, as long as initial prices re‡ect initial productivities, see Yun (2005) for a
discussion of this and related issues in a homogeneous …rm setting.

8

Cost minimization in the production of …nal output Yt implies
Yjt = (Pjt =Pt )

Yt :

(8)

which shows that the price level can be expressed as an expenditure-weighted average ofthe prices in the di¤erent expenditure categories, in line with the practice at statisticalagencies:
Pt =

Z 1
0

Yjt
Yt

Pjt dj,(9)Note that the price index contains in any period t all the goods available during thisperiod. This is clearly an idealized notion of how price indices are actually computed
by statistical agencies. We discuss in section 11.3 how results are altered if the productbasket underlying the price index is biased towards ‘older’products.
Using the price index in equation (9), we de…ne the gross in‡ation rate ast

To the extent that

Pt =Pt 1 :-shocks capture product substitutions, this assumes that productsthat become unavailable are replaced by a new products in the subsequent product basket,
so that the in‡ation rate is computed using the price level associated with the old productbasket in the earlier period and the price level associated with the new basket in thesubsequent period. This assumption is in line with the sampling procedures typicallypursued by the BLS and other statistical o¢ ces. The section "Item replacement andquality adjustment" in chapter 17 of the BLS Handbook of Methods (BLS (2015)), forinstance, describes how the changeover of discontinued product versions is handled. If adata collector cannot …nd anymore a product version that was previously contained in thebasket, the collector replaces it with a new version. The price of the old version enters
the previous price index and the price of the new version enters the current price index.11

3.2

Alternative Interpretations of the Firm Setup

The previous section de…ned -shocks ( jt = 1) as an idiosyncratic change in …rm-levelproductivity that is associated with a price adjustment opportunity. This section presents
three alternative economic interpretations of -shocks that highlight alternative economic
sources of …rm heterogeneity and that explain why productivity changes may plausibly
be associated with price ‡exibility at the …rm level.
11

The BLS also seeks to adjust for quality di¤erences across versions. Armknecht et al. (1996) showsthat about 3% of products are discontinued each month. Their table 9.2 shows that more than 50% of
the replacement versions fall into the category "direct comparisons", for which no quality adjustment ismade; for the remaining replacements there is either a direct quality adjustment or quality adjustment is
imputed via di¤erent methods. As will become clear in section 3.2, our setup is consistent with statisticalagencies making such quality adjustments.

9Figure 1: Productivity dynamics in a setting with …rm entry and exit

Firm entry and exit. It is possible to interpret -shocks as a …rm exit and entry
event. Indeed, this is the interpretation that we adopt in our empirical application of
the model in section 10. Speci…cally, the event

jt = 1 can be interpreted as an eventin which …rm j becomes permanently unproductive and thus exits the economy. Eachexiting …rm is then replaced by a newly entering …rm to which we assign for simplicity
the same …rm index j. The variable Qt then captures the productivity level of the cohortof …rms that enters in period t, and Gjt captures the experience accumulated over the
lifetime of …rm j. The assumption that …rms’prices are ‡exible following a -shock should
then be interpreted as newly entering …rms being able to freely choose the price of theirproduct. It is worth noting that …rm entry and exit rates are high in the United States,
see …gure 3 in Decker et al. (2014).
Figure 1 illustrates the …rm-level productivity dynamics for the empirically plausible
setting in which the cohort trend is positive (q > 0), but less strong than the experience
trend (g > q). To simplify the exposition, the …gure depicts the deterministic dynamics
and abstracts from the common TFP trend a, which does not a¤ect the distributionof relative productivities across …rms. The line labeled log(Qt ) in the …gure indicatesthe cohort trend and captures the productivity of newly entering …rms at each point intime. The lines starting at the cohort trend line capture the productivity dynamics of the10

entering cohorts over time. Since g > q, the productivity of existing …rms grows fasterthan the productivity of new entrants, so that existing …rms are initially more productive
and thus larger than newly entering …rms. In experience-adjusted terms, however, newly
entering …rms are the most productive …rms in the economy. The downward-pointing
dashed arrows indicate the productivity losses of exiting …rms that have been hit by a-shock. For simplicity, the …gure assumes that their productivity permanently drops tozero. As should be clear from the …gure, the entry and exit dynamics imply an exponential
distribution for …rm age. Coad (2010) shows that such an age distribution is empiricallyplausible and how it generates, together with (productivity) growth shocks, a Pareto
distribution for …rm size, in line with the observed …rm size distribution.
Product substitution. The eventjt= 1 can also be interpreted as an event inwhich the product previously produced by …rm j is no longer demanded by consumers.
Firm j reacts to this by introducing a new product, which - for simplicity - is assigned thesame product index j. The variable Qt then captures the productivity level associatedwith products that are newly introduced in t and Gjt captures experience accumulationin producing the new product. Product substitutions, e.g., in the form of new productversions or models, take place rather frequently in the data and are also prevalent in the
CPI baskets of statistical agencies (see section III.C in Nakamura and Steinsson (2008)
for evidence on the rate of product substitution in the U.S. CPI). Evidence provided in
Moulton and Moses (1997), Bils (2009) and Melser and Syed (2016) furthermore showsthat the prices of new products are typically higher than those that they replace, evenafter accounting for quality improvements.12 It thus appears reasonable to assume price‡exibility for new products (see also Nakamura and Steinsson (2012)).
Quality improvements. Let Qjt denote the quality of the product produced by …rm
j in period t. De…ning Qjt = Qt sjt , the event

jt = 1 captures the situation in which…rm j upgrades the quality of its product from level Qt 1 sj;t 1 to level Qt . Let aggregate
output produced with intermediate inputs of di¤erent quality be given by
Yt =

Z 1
0

1

Qjt Yejt1

dj;and let …rm j’s output of quality level Qjt be given by1
Yejt = At Gjt Kjt

1

1

Ljt

Ft ;

where Gjt now captures experience e¤ects associated with producing quality Qjt . Finally,
let Pejt denote the price of a unit of good j of quality level Qjt . Assuming that statistical
12

Evidence provided in Bils (2009) shows that in‡ation for durables ex computers over the period
1988-2006 averaged 2.5% per year, but when including only matched items, the in‡ation rate was -3.7%
per year.

11

agencies perfectly adjust the price level for quality changes over time, we have0

Pt = @

Z 1

Pejt
Qjt

0

!1

111

djA

:

As is easily veri…ed, this setup with quality improvements is mathematically identical tothe one with productivity changes spelled out in the previous section.13 Again, it appears
natural to assume that …rms can ‡exibly price goods with improved quality features.

3.3

Optimal Price Setting

Firms choose prices, capital and hours worked to maximize pro…ts. While price adjustment is subject to adjustment frictions, factor inputs can be chosen ‡exibly. Letting Wt
denote the nominal wage and rt the real rental rate of capital, …rm j chooses the factor
input mix so as to minimize production costs Kjt Pt rt + Ljt Wt subject to the constraints
imposed by the production function (2). Let
Ijt

Ft + Yjt =(At Qt sjt Gjt )11denote the units of factor inputs (Kjt

1

Ljt ) required to produce Yjt units of output. As

appendix A.1 shows, cost minimization implies that the marginal costs of Ijt are given by
M Ct =

1

Wt
1=

Pt rt
1 1=

1

1

(10)

:

Now consider a …rm that either experienced a -shock or a Calvo shock in period t andthat can freely choose its price. Let
keep its previous price (0denote the Calvo probability that the …rm has to< 1), the …rm will not be able to reoptimize its price withprobability (1) at any future date, i.e., whenever it undergoes neither a - shock nor

14

The price-setting problem of a …rm that can optimize its price in perioda Calvo shock.

t is thus given by
max Et
Pjt

1
X

( (1

i=0

))i

t;t+i

Pt+i

[(1 + )Pjt+i Yjt+i

M Ct+i Ijt+i ](11)

s:t: Ijt+i = Ft+i + Yjt+i =At+i Qt sjt Gjt+i ;Yjt+i = (Pjt+i =Pt+i )Pjt+i+1 =Yt+i ;

t+i;t+i+1 Pjt+i :The quality-adjusted price Pejt =Qjt and the quality-adjusted quantity Yejt Qjt then correspond to theprice Pjt and quantity Yjt , respectively, in the previous section.
14
In any period, the …rm can adjust its price with probability due to the occurrence of a -shock andwith probability (1)(1) due to the occurrence of a Calvo price adjustment shock.
13

12

wheredenotes a sales tax/subsidy andt;t+idenotes the representative household’s

discount factor between periods t and t + i. The …rst constraint captures the …rm’s
technology, the second constraint captures the demand function faced by the …rm, as
implied by equation (1), and the last constraint captures how the …rm’s price is indexed
over time (if at all) in periods in which prices are not reset optimally. We consider generalprice indexation schemes and allow
15

to period t + i.

t+i;t+i+1 to be a function of aggregate variables up

In the absence of indexation, we havet+i;t+i+1 = 1 for all i?
Appendix A.2 shows that the optimal price Pjt can be expressed as

Pjt?
Pt

Qt sjt Gjt
Qt

=

1
11+

Nt
;
Dt

0.

(12)

where the variables Nt and Dt are independent of the …rm index j and evolve recursively
according to
M Ct
Nt =
+ (1
Pt At Qt
Dt = 1 + (1

)Et

)Et
"

"

Yt+1
( t;t+1 )
t;t+1
Yt

Yt+1
( t;t+1 )1
t;t+1
Yt

Pt+1
Pt

Pt+1
Pt
1

qt+1
gt+1
#

Nt+1

#

(13)
(14)

Dt+1 :

Equation (12) shows that the optimal reset price of a …rm depends only on how its own
productivity (At Qt sjt Gjt ) relates to the productivity of a …rm hit by a -shock in periodt (At Qt ), as well as on the aggregate variables (Nt ; Dt ). It is precisely this feature which
permits aggregation of the model in closed form. Equation (12) furthermore shows that
more productive …rms optimally choose lower prices. With homogeneous …rms, relative
productivity is always equal to one (Qt sjt Gjt =Qt = 1) and equations (12)-(14) thenreduce to the ones capturing the optimal price in a standard homogeneous-…rm modelwith sticky prices.

3.4

Household Problem

There is a representative household with balanced growth consistent preferences given by
!
1
1
X
[C
V
(L
)]
1
t
t
t
E0
;
(15)
t
1
t=0
where Ct denotes private consumption of the aggregate good, Lt labor supply,
ence shock with E[ t ] = 1 and

2 (0; 1) the discount factor. We assume

15

t a prefer-> 0 and thatWe only require that price indexation is such that the price-setting problem remains well de…ned,
that price indexation does not give rise to multiplicities of the optimal in‡ation rate and that t;t+1 = 1
in a steady state without in‡ation. For instance, when indexing occurs with respect to lagged in‡ationaccording to t;t+1 = ( t ) with
0, we rule out > 1 to avoid non-existence of optimal plans andrule out = 1 to avoid multiplicities of the steady-state in‡ation rate.

13V ( ) is such that period utility is strictly concave in (Ct ; Lt ) and that Inada conditions
are satis…ed. The household faces the ‡ow budget constraint
Z 1
Bt
Wt
Bt 1
jt
Ct + Kt+1 += (rt + 1 d)Kt +
Lt +
dj +
(1 + it 1 )
Pt
Pt
Pt
0 Pt

Tt ;

where Kt+1 denotes the capital stock, Bt nominal government bond holdings, it 1 thenominal interest rate, Wt the nominal wage rate, rt the real rental rate of capital, d thedepreciation rate of capital,

jt nominal pro…ts from ownership of …rm j, and Tt lumpsum taxes. Household borrowing is subject to a no-Ponzi scheme constraint. The …rstorder conditions characterizing optimal household behavior are entirely standard and arederived in appendix A.3. To insure existence of a well-de…ned balanced growth path, we
assume throughout the paper that
< (aq) :3.5

Government

To close the model, we consider a government which faces the budget constraint
Z 1
Bt 1
Pjt
Bt
=
(1 + it 1 ) +Yjt dj Tt ;
Pt
Pt
Pt
0
where

denotes a sales subsidy, which will be used to correct for the monopolistic distor-

tions in product markets. The government levies lump sum taxes Tt , so as to implement abounded state-contingent path for government debt Bt =Pt .16 Since we consider a cashless
limit economy, there are no seigniorage revenues, even though the central bank controls thenominal interest rate. We furthermore assume that monetary policy is not constrained bya lower bound on nominal interest rates. The equilibrium concept is standard and de…ned
in appendix A.5.

4

The Optimal In‡ation Rate for a Special Case

We now illustrate the paper’s main result using a special setting without economic disturbances, without capital in production ( = 1), and without price-indexation ( t;t+1 = 1
for all t). Additionally, we consider the special household preferences17
E0

1
X

t

(log Ct

Lt )

t=0

16
17

The household’s transversality condition will then automatically be satis…ed in equilibrium.
These are obtained from equation (15) for V (L) = e L and for the limiting case ! 1.

14

and a strictly positive rate of -shocks ( > 0). The household’s …rst order conditions for
consumption and hours worked imply
for all t:

Wt = Ct Pt(16)

In a balanced growth path with constant hours worked, aggregate consumption grows at
rate18

Ct+1= aq
for all t;(17)
Ct
where a is the productivity growth rate common to all …rms and q the cohort productivity
growth rate. The experience growth rate (g) does not contribute to long-run growth
because accumulated experience is lost once …rms are hit by a -shock.
Next, consider the …rm side of the economy. A necessary condition for optimality
in the …rm sector is - as in any standard sticky-price model without …rm level trends that …rms do not want to change prices over time.19 For each cohort of …rms, production
technology progresses at the common TFP growth rate (a) and at the experience trend
growth rate (g). For constant prices to be optimal at the …rm level, nominal wages mustgrow at the rate ag, so as to o¤set the e¢ ciency gains on the real side. This fact together
with equation (16) implies
Ct+1 Pt+1
= ag
Ct Pt

for all t;

which together with equation (17) shows that the optimal in‡ation rate must satisfy
Pt+1
g
= :
Pt
q

(18)

For this in‡ation rate to be su¢ cient for achieving optimality, one only has to insure thatthe level of prices that …rms charge in the …rst period following a -shock is the correctone, in the sense that they do not involve a mark-up over production costs. Wages arethen equal to the marginal product of labor. As in the standard New Keynesian modelwith homogeneous …rms, this can be achieved with the help of a Pigouvian output subsidy
that corrects …rms’monopoly power. The optimal in‡ation rate (18) then achieves fulle¢ ciency for the economy: product prices are equal to marginal production costs and thewage equals the marginal product of labor.20
The special case presented above highlights a number of aspects that will generalize
to the fully ‡edged model: (1) The Calvo assumption does not drive the optimal in‡ation
18

All …rms experience the TFP growth rate a. In addition, all …rms will eventually be hit by a -shockand thus experience the latest cohort productivity level Qt , where Qt latter grows at the rate q. Aggregate
labor productivity thus grows at the rate aq.
19
Otherwise, the presence of asynchronous price adjustment opportunities leads to price di¤erences
between …rms that - on technological grounds - should charge identical prices.
20
In the absence of the optimal output subsidy, the in‡ation rate in equation (18) still achieves productive e¢ ciency.

15

result: since …rm-level prices never need to change under the optimal in‡ation rate, thesame in‡ation rate would be optimal in a setting with menu cost frictions. (2) The
experience growth rate (g) and the cohort growth rate (q) have opposite e¤ects on the
optimal in‡ation rate because g a¤ects the growth rate of technology in a cohort of …rms,
but not the aggregate growth rate, while the reverse is true for q. (3) The ‡exibility ofprices following a -shock is key for achieving e¢ ciency.
The remaining part of the paper shows that once one appropriately adjusts for thepresence of price indexing, the steady state result (18) survives in more general environments. We furthermore characterize the optimal in‡ation policy outside of the steadystate, where the relationship between marginal costs and consumption is considerably
more complex. Achieving e¢ ciency then requires that the optimal in‡ation rate ‡uctuates around its steady state value.

5

Analytical Aggregation with Heterogeneous Firms

This section outlines the main steps that allow us to aggregate the model in closed form. In
a …rst step, we derive a recursive representation describing the evolution of the aggregate
price level Pt over time. In a second step, we derive a closed-form expression for theaggregate production function. In a last step, we show how to appropriately detrend
aggregate variables, so as to render them stationary.
Evolution of the aggregate price level. Let Pt? s;t k denote the optimal price of a…rm that last experienced a -shock in t
(s
Qk

j=1

s and that has last reset its price in t0). In period t, this …rm’s price is equal tok?
t k;t Pt s;t k ,where

t k+j 1;t k+j captures the cumulative e¤ect of price indexation (withthe absence of price indexation). Lett (s) = (1

t k;t

t k;t

=

1 in

t (s) denote the weighted average price in period tof the cohort of …rms that last experienced a -shock in period t
raised to the power of 1ks, where all prices are, i.e.,
)

s 1
X

k

( t k;t Pt? s;t k )1+s

( t s;t Pt? s;t s )1 :

(19)

k=0

There ares…rms that have not had a chance to optimally reset prices since receiving

the -shock and (1) k …rms that have last adjusted k < s periods ago. From equation(7) it follows that one can use the cohort average prices
price level as
Pt1

=

1
X

(1

)s

t (s) to express the aggregatet (s);

(20)

s=0

where

is the mass of …rms that experience a -shock each period and (1share of those …rms that have not undergone another -shock for s periods.
16)s is theTo express the evolution of Pt in a recursive form, consider the optimal price Pt? s;t ofa …rm that sustained a -shock s > 0 periods ago, but can adjust the price in t due to?
the occurrence of a Calvo shock. Also, consider the price Pt;tof a …rm where a -shock

occurs in period t. The optimal price setting equation (12) then implies
gt
qt

?
Pt;t
= Pt? s;t

gt s+1
qt s+1(21):

The previous equation shows that a stronger cohort productivity trend (higher values forq) causes the …rm that experiences a -shock in period t to choose lower prices relative to…rms that experienced -shocks further in the past, as a stronger cohort trend makes this…rm relatively more productive. Conversely, a stronger experience e¤ect (higher values
for g) increases the optimal relative price of the …rm that underwent a -shock in t. The
net e¤ect depends on the relative strength of the cohort versus the experience e¤ect.
Appendix A.4 shows how to combine equations (19), (20), and (21) to obtain a recursive representation for the evolution of the aggregate price level given by
Pt1? 1= (Pt;t
)

+ (1

)(1

)

(pet )
1

1
? 1
(Pt;t
)

+ (1

)( t 1;t Pt 1 )1 ;(22)

where pet summarizes the history of shocks to cohort and experience productivity andevolves recursively according to(pet )

1) pet 1 gt =qt= + (1

1(23):

The last term on the r.h.s. of equation (22) captures the price-level e¤ects from the share(1) of …rms that experienced neither a Calvo shock nor a -shock. These …rms

keep their old price (Pt 1 on average), adjusted for possible e¤ects of price indexation,
as captured by the indexation term
captures the price e¤ects of the masst 1;t .

The …rst term on the r.h.s. of equation (22)of …rms that experienced a -shock in period t;?
these …rms optimally charge price Pt;t
. The second term captures the average price of…rms that experienced a Calvo shock in period t; their share is (1)(1) and they set

a price that on average di¤ers from the price charged by …rms hit by a -shock, dependingon the value of pet . This latter aspect in equation (22) is the key di¤erence relative to thestandard model without …rm heterogeneity in productivity. A stronger experience trend(a higher value for gt ), for instance, increases (pet )1, and - ceteris paribus - causes …rms

hit by a Calvo shock to choose a lower value for the optimal reset price. A stronger cohort
trend (a higher value for qt ) has the opposite e¤ect. Overall, the interesting new featureis that price dynamics now depend on the productivity trends.
In a setting where all …rms have identical productivity trends, e.g., where the cohorte¤ect is as strong as the experience e¤ect (qt = gt for all t), equation (23) implies that pet
converges to one, causing the price level to eventually evolve according toPt1= [ + (1)(1

? 1)] (Pt;t
)

17

+ (1

)( t 1;t Pt 1 )1 ;

which is independent of productivity developments at the …rm level. If in addition there
are no -shocks ( = 0), the previous equation simpli…es further to
Pt1? 1)(Pt;t)= (1+ ( t 1;t Pt 1 )1 ;which describes the evolution of the aggregate price level in the standard Calvo modelwith homogeneous …rms.
Aggregate production function. In appendix A.6 we show that aggregate output Yt
can be written as

At Qt

Yt =

t

1

1

Kt

1

Lt

(24)

Ft ;

where Kt denotes the aggregate capital stock, Lt aggregate hours worked and
t =

Z 1
0

Qt
Gjt Qt sjt

Pjt
Pt

dj

?
Pt;t
Pt

+ (1

)

(25)

evolves recursively according to
t =

+ (1

)(1

)

(pet )
1

1

qt
gt

t

t 1:

(26)

t 1;t

TFP in the aggregate production function (24) is a function of the TFP of the latest
cohort hit by the -shock, At Qt , and of the adjustment factort.

The latter is de…nedin equation (25) and captures a …rm’s productivity relative to that of the latest cohort,
Qt = Qt sjt Gjt , and weights this relative productivity with the …rm’s production share(Pjt =Pt ) . Equations (24) and (25) thus show how relative price distortions may lead toaggregate output losses by negatively a¤ecting aggregate technology, e.g., by allocating
more demand to relatively ine¢ cient …rms. The evolution of the adjustment factor overtime is described by equation (26) and depends on …rm-level productivity trends - amongst
other ways - through the variable pet . In the limit with homogeneous …rm trends (i.e.,
qt = gt ), pet converges to one and the evolution oft becomes independent of productivity

realizations. If - in addition - there are no -shocks ( = 0), then equation (26) simpli…es
further tot = (1

)

?
Pt;t
Pt

t

+

t 1;

t 1;t

which is the equation capturing the distortions from price dispersion within a standard
homogeneous-…rm model.
Balanced Growth Path. One can obtain stationary aggregate variables by rescaling
them by the aggregate growth trendet = (At Qt =

18

e
t) ;

(27)

e
t denotes the e¢ cient adjustment factor chosen by the planner, de…ned in equation(31) below. Speci…cally, the rescaled output yt = Yt = et and the rescaled capital stock
kt = Kt = et are now stationary and the aggregate production function (24) can be written

where

as

e
t

yt =

1

1

kt

t

1

Lt

f

(28)

:

In the deterministic balanced growth path, the (gross) trend growth rate

e
t =

e
e
t= t 1is constant and equal to (aq) and hours worked are constant whenever monetary policy
implements a constant in‡ation rate. Appendices A.7 and A.8 write all model equations
using stationary variables only and appendix A.9 determines the resulting deterministic
steady state.

6E¢ ciency of the Flexible-Price Equilibrium

This section derives the e¢ cient allocation and the conditions under which the ‡exibleprice equilibrium is e¢ cient. Appendix B shows that the e¢ cient consumption, hours andcapital allocation fCt ; Lt ; Kt+1 g1
t=0 solves
max

fCt ;Lt ;Kt+1 g1
t=0

E0

1
X

t

t

t=0

s:t: Ct + Kt+1 = (1
where
e
t

1

At Qt

(Kt )1d)Kt +Z 1
0!

[Ct V (Lt )]1
1
e
t

Qt
Gjt Qt sjt

1

1

!11

dj

(29)
(Lt )

1

Ft ;

(30)

(31)

;

which evolves according to
(

e 1
t)

= + (1)1et 1 qt =gt

:

(32)

Constraint (30) is the economy’s resource constraint, when expressing aggregate output using the aggregate production function (24). The e¢ cient productivity adjustment
factoret showing up in the planner’s production function is de…ned in equation (31); itsrecursive evolution is described by equation (32). The …rst-order conditions of problem(29)-(30) shown in appendix B are necessary and su¢ cient conditions characterizing thee¢ cient allocation.
Decentralizing the e¢ cient allocation requires that …rms’prices, which entert andthus in the aggregate production function (24), satisfy certain conditions. In particular,
equations (25) and (31) imply thatt =et is achieved if prices satisfy

1
Qt
Pjt
= e
:
Pt
t Gjt Qt sjt
19

(33)

The previous equation requires relative prices to accurately re‡ect relative productivities. Furthermore, as in models without …rm heterogeneity, one has to eliminate …rms’monopoly power by a Pigouvian subsidy to obtain e¢ ciency of the market allocation. We
thus impose the following condition:
Condition 1 The sales subsidy corrects …rms’market power, i.e.,

1
1 1+

= 1.

Appendix C then proves the following result:
Proposition 1 The ‡exible-price equilibrium ( = 0) is e¢ cient if condition 1 holds.
The proof of the proposition shows that condition (33) holds under ‡exible prices,
so that one achievest =et and thereby productive e¢ ciency.

In the presence of theassumed sales subsidy, consumer decisions are also undistorted, which means that thevalues of consumption, hours worked and capital in the ‡exible-price equilibrium are
identical to the values that these variables assume in the e¢ cient allocation.

7

Optimal In‡ation with Sticky Prices: Main Result

This section determines the optimal in‡ation rate for an economy with sticky prices(> 0). It derives the optimal rate of in‡ation for the nonlinear stochastic economy

with heterogeneous …rms in closed form and shows how in‡ation optimally depends on
the productivity growth rates at ; qt and gt . As it turns out, the optimal in‡ation rate
implements the e¢ cient allocation (the ‡exible-price benchmark) and is independent ofat .
To establish our main result in the most straightforward manner, we impose an assumption on initial conditions, in particular on how …rms’initial prices and initial productivities are related. Similar conditions are imposed in sticky-price models with homogeneous …rms, where it is routinely assumed that there is either no or a su¢ ciently
small dispersion of initial prices, so that relative prices re‡ect relative productivities. We
impose:Condition 2 Initial prices in t =Pj; 1 /

1 re‡ect …rms’relative productivities, i.e.,
1Q 1 sj; 1 Gj; 1

for all j 2 [0; 1]:

We discuss the e¤ects of relaxing this condition below. The following proposition
states our main result:

20

Proposition 2 Suppose conditions 1 and 2 hold. The equilibrium allocation in the stickyprice economy is e¢ cient if monetary policy implements the gross in‡ation rate
?
t =

?
t 1;t

1

(

1

e
t)

1

! 11

for all t

0;

(34)

?
?
1 in the absencet 1;t captures price indexation between periods t 1 and t ( t 1;te
of indexation) and t is de…ned in equation (31) and evolves according to equation (32).

where

The proof of proposition 2, which is contained in appendix D, establishes that withthe optimal in‡ation rate, …rms choose relative prices as in the ‡exible-price equilibrium.
This result is established by showing that (1) …rms hit by a -shock choose the same
optimal relative price as in the ‡exible price economy, and that (2) …rms hit by a Calvoshock optimally choose not to adjust their price, which avoids the emergence of pricedispersion between otherwise identical …rms. This, together with the fact that (3) initial
prices re‡ect initial productivities, ensures that all relative prices are identical to thosein the ‡exible-price equilibrium. Under the assumed output subsidy, it then follows thathousehold allocations are also identical to the ‡exible-price equilibrium, which has beenshown to be e¢ cient (proposition 1).
It may be surprising that a single policy instrument, namely the aggregate in‡ation
rate, can insure that the following two relative prices are optimal: the relative pricebetween …rms hit by -shocks and sticky-price …rms, and the relative price between …rms
with a Calvo adjustment opportunity and sticky-price …rms. To understand this result,note that the optimal price setting equation (12), which holds independently of the reason
for why …rms i and j can adjust prices, implies
Pjt?
At Qt sit Git
=
:
?
Pit
At Qt sjt Gjt
This shows that all price-adjusting …rms set prices such that relative prices amongst themselves re‡ect inversely relative productivities. Moreover, this holds true independently ofthe value of the in‡ation rate.21 Therefore, the in‡ation rate can be used to insure that
relative prices between the set of price-adjusting …rms and the set of non-adjusting …rms
are optimal, thereby achieving e¢ cient relative prices amongst all …rms.
In the absence of price indexation ( ?t 1;tfunction of the variable1), the optimal in‡ation rate?
t is only aet , which summarizes the distribution of relative productivities

across …rms, see equation (31). Since these relative productivities are independent of thecommon TFP growth rate at , it follows that the optimal in‡ation rate does not depend
21

The fact that adjusting …rms choose optimal relative prices amongst themselves is also a feature ofhomogeneous …rm models. What we show here is that this generalizes to our setting with heterogeneous
…rms.

21

on the realizations of at . In contrast, the cohort productivity growth rate qt and theet , see equation (32), albeit in opposite directions: aexperience growth rate gt do a¤ect

stronger cohort productivity growth rate qt decreases the optimal in‡ation rate, while astronger experience growth rate gt increases the optimal in‡ation rate.
For the special case in which all …rms have identical productivity trends ( = 0 oretgt = qt for all t) or even identical productivities (= 1), the optimal gross in‡ation

rate is equal to one in the absence of price indexation, as in a standard homogeneous-…rm
model. Perfect price stability is then optimal at all times.
To understand the economic logic underlying equation (34), we use equation (32) toobtain the following alternative expression for the optimal in‡ation rate:
?
t =

?
t 1;t

at gt At 1eQt 1
t 1

At Qt

(35)

:

e
t

The denominator in the previous expression (At Qt =et ) is the average productivity ofall …rms in the e¢ cient allocation, see equation (24). The numerator is the average
productivity of all …rms except those that received a -shock in period t: their average
productivity was At 1 Qt 1 =e
t 1 in period t

1 and has grown since by the aggregate

TFP growth rate at and the experience growth rate gt . The productivity of all …rms isin turn a weighted sum of the productivity of …rms with a -shock and …rms without-shocks:
At Qt
e
t

=

(At Qt )1+ (1) at gt1

At 1 Qt 1
e
t 1

! 11

;

where At Qt denotes the productivity of …rms with a -shock. The optimal in‡ation rate
can thus be expressed as
?
t =

1

?
t 1;t

;

1

(rpt )

1

+ (1

)

(36)

1

where rpt denotes the relative relative productivity between …rms with and without a-shock and is given by
rpt

At Qt
at gt At 1eQt 1:

t 1

In the absence of price indexation (

?
t 1;t

1), we thus havet = 1 whenever this measureof relative productivity equals one, which is the case in a homogeneous …rm model. If-shock …rms are less productive (rpt < 1), we haveare more productive (rpt > 1), we havet > 1.

Conversely, if -shock …rms

t < 1.

Price indexation by non-adjusting …rms ( ?t 1;t 6= 1), say because of indexation to

the lagged in‡ation rate, introduces additional components into the optimal aggregate

in‡ation rate. In particular, it requires that price-adjusting …rms, i.e., …rms hit by either22

a -shock or a Calvo shock, also adjust their price by the indexation component. Thisway prices continue to accurately re‡ect relative productivities at all times. This explains
why indexation a¤ects the optimal in‡ation rate one-for-one in equations (34)-(36).
Although proposition 2 assumes that …rms’initial prices accurately re‡ect the initial
relative productivities, the initial productivity distribution itself is unrestricted. We conjecture that for a setting where condition 2 fails to hold, one would obtain additionaltransitory and deterministic components to the optimal in‡ation rate, as in the homogeneous …rm setting studied by Yun (2005). The in‡ation rate stated in proposition 2
would then become optimal only asymptotically.
Interestingly, it follows from the proof of proposition 2 that the in‡ation rate (34)continues to ensure productive e¢ ciency (but not full e¢ ciency) in settings where condition 1 fails to hold. From the theory of optimal taxation it then follows that it remains
optimal to implement the in‡ation rate (34), as it is suboptimal to distort intermediate
production as long as (distortionary) taxes on …nal goods are available.

8

The Optimal Steady-State In‡ation Rate

This section discusses the optimal steady-state in‡ation rate implied by the model. To
simplify the discussion, we initially abstract from price indexation.
Proposition 2 makes it clear that in the case in which the productivity of all …rms grows
at the same rate ( = 0), which includes as a special case the setting with homogeneous…rms, we obtain?
t = 1. For= 0, the optimal (gross) steady-state in‡ation rate is thus

trivially equal to one, independently of the values assumed by (a; g; q).
For the casefrom> 0, the optimal steady-state in‡ation rate jumps discontinuously away?
t = 1, with the optimal steady-state in‡ation rate being itself independent of the22value of> 0.

The following lemma summarizes this result:Lemma 1 Suppose conditions 1 and 2 hold, there are no economic disturbances, there is
no price indexation ( ?t 1;t1) and> 0. The optimal in‡ation rate then satis…es
lim

t!1

?
t = g=q:

Proof. From equations (6) and (32) it follows that limt!1 (It then follows from proposition 2 that limt!1

(37)
e
t)

1

= [1 (1) (g=q)1]= .

?
t = g=q.

Since we allow for arbitrary initial productivity distributions, the absence of shocks
does not necessarily imply that the optimal in‡ation rate is constant from the beginning.
22

Note that the e¢ cient allocation also discontinuously jumps when moving from = 0 to > 0, as in
the former case e¢ cient aggregate growth is equal to (ag) and in the latter case it is equal to (aq) in
steady state. Appendix E shows that the discontinuity of the optimal steady-state in‡ation rate is notdue to the discontinuity of the associated aggregate allocation.

23

This only happens asymptotically, once the productivity distribution converges to its
stationary distribution (in detrended terms).23 The lemma provides the in‡ation ratethat is asymptotically optimal as this stationary distribution is reached.24
Surprisingly, the optimal long-run in‡ation rate is completely independent of the intensity of -shocks. To understand the source of this invariance, consider a setting where-shocks capture …rm exit and entry events and where g > q, so that newly entering …rms
are less productive than the set of continuing old …rms. A higher value for

implies that

more young and relatively unproductive …rms are amongst the set of price-setting …rms.
This calls - ceteris paribus - for a higher in‡ation rate. Yet, the productivity distributionof continuing old …rms is not invariant to changes in : a higher

also implies more …rmturnover and thus less experience accumulation. Continuing old …rms thus tend to be lessproductive relative to new entrants, which calls for lower in‡ation rates. In net terms,
these two e¤ects exactly cancel each other.
On empirical grounds, it appears plausible to assume g > q so that the optimal steadystate in‡ation rate from lemma 1 is positive. For the case where -shocks capture …rmturnover, one obtains g > q from the fact that young …rms are small relative to old…rms, see also section 10 below. Likewise, interpreting -shocks as representing productsubstitution shocks, g > q implies that new products are more expensive than the average
product and that their relative price is falling over the product life cycle. Both of thesefacts are in line with evidence provided by Melser and Syed (2016). Thus, while thetheoretical setup allows the optimal steady-state in‡ation rate to be potentially negative,these empirical considerations suggest positive in‡ation to be optimal in steady state.
Interestingly, aggregate productivity dynamics turn out not to be informative about
the optimal in‡ation rate. The aggregate steady-state growth is equal to (aq) and isdriven by a factor that a¤ects the optimal in‡ation rate (q) and a factor that does nota¤ect it (a). Moreover, the experience e¤ect (g) has no aggregate growth rate implications,but a¤ects the optimal in‡ation rate. Determining the optimal in‡ation rate thus requires
either studying the …rm-level productivity trends g and q or the relative productivities ofold versus new …rms, see equations (35) and (36). We shall come back to this issue in theempirical section 10.
Finally, we discuss the e¤ects of price indexation on the optimal steady state in‡ation
rate. For

> 0 the optimal long-run in‡ation rate is then given bycase where prices are indexed to lagged in‡ation according to
2 [0; 1), we thus obtain

lim

t!1

?
1
t = (g=q)231?
t 1;t =?

(g=q). For the?
t 1

for some:

When = 0, the initial distribution remains unchanged (in detrended terms).
The transitional dynamics can easily be derived from proposition 2 using the initial productivity
distribution and equation (32).
24

24

Standard forms of price indexation thus amplify the divergence of the optimal gross in‡ation rate from one.

9

The Welfare Costs of Strict Price Stability

This section shows that suboptimally implementing strict price stability, as suggested bysticky-price models with homogeneous …rms, gives rise to strictly positive welfare costs
whenever g 6= q. We derive this fact …rst analytically for a special case, as this allows
considering the limit! 0. In a second step, we illustrate the source of the welfare losses

and their magnitude numerically.

The following proposition shows that - as long as g 6= q - there is a strictly positive

welfare loss that is bounded away from zero when implementing strict price stability; thisholds true even for the limit
F.

! 0. The proof of the proposition is contained in appendix

Proposition 3 Suppose conditions 1 and 2 hold, there are no economic disturbances,> 0, …xed costs of production are zero (f = 0), there is no price indexation ( ?t 1;t

1),

and the disutility of work is given by
V (L) = 1with

> 1 andL ;> 0. Assume g=q > (1), so that a well-de…ned steady state withstrict price stability exists.
Consider the limit ( e )1

! 1 and a policy implementing the optimal in‡ation rate?
t from proposition 2, which satis…es limt!1

?

?
t =

= g=q. Let c(?

) and L(?

) denotethe limit outcomes for t ! 1 for consumption and hours worked, respectively, under thispolicy. Similarly, let c(1) and L(1) denote the limit outcomes under the alternative policyof implementing strict price stability. Then,
L(1) = L(

?

)

and
c(1)
=
c( ? )

1(1
1)(g=q)
(1
)

1

1

1
1

(1
(1

) (g=q) 1)(g=q) 1

For g 6= q the previous inequality is strict and lim !0 c(1)=c(

?

!

1:

(38)

) < 1:

We now illustrate the nature of the relative price distortions that are generated by asuboptimal rate of in‡ation and how they give rise to welfare losses. Panel A in …gure 2reports the mean cohort price (relative to the price of all …rms), depicted on the y-axis,
as a function of the cohort age in quarters (x-axis). It does so once for a setting wheremonetary policy implements the optimal in‡ation rate, which for illustration is assumed
25

A. Relative Cohort Pricecohort m ean for different
inflation rates

1.2

B. Relative Cohort Pricem ean and +/- 2 std. dev.
0% inflation rate

1.2

0% Inflation Rate
Optim al Inflation Rate

1.18

1.15

1.16
1.14

1.1
1.12
1.1
1.05
1.08
1.06

1

1.04
1.02

0

5

10

15

20

0.95

25

0

5

10

15

20

25

tim e since last -shock

tim e since last -shock

Figure 2: Relative prices and in‡ation

to be 2% per year (in net terms), and once where monetary policy pursues strict pricestability.25 Panel A shows that young cohorts charge a higher (relative) price and that thisprice decreases over the lifetime of the cohort. Under the optimal in‡ation rate, the decline
happens at a constant rate.26 Under strict price stability, however, …rms anticipate that
their relative prices will not necessarily fall, due to Calvo price stickiness. This causes
them to initially "front load" prices, i.e., in an environment with strict price stability
young cohorts charge initially lower prices than under the optimal in‡ation rate. Overtime, some …rms in the cohort will get the opportunity to lower their prices in response to
Calvo shocks, but the average relative price of the cohort will eventually be slightly higher
than under the optimal in‡ation rate. Beyond these distortions in average cohort prices,
the suboptimal in‡ation rate also generates price distortions within a cohort of …rms.
This is illustrated in panel B of …gure 2. Panel B reports the mean cohort price and thetwo standard deviation bands of the cross-sectional price distribution within the cohort,assuming monetary policy pursues strict price stability. It shows that suboptimal in‡ation
not only gives rise to distortions in the mean price but also to substantial amounts of pricedispersion within the cohort. In contrast, price dispersion at the cohort level is zero under
the optimal in‡ation rate.
Figure 3 reports the steady-state value for the ratio
25

e
t=

t

(y-axis) as a function

Figure 2 is computed using g = 1:020:25 ; q = 1; = 0:75; = 0:035; = 3:8 and assumes that theinitial productivity distribution is equal to the stationary distribution (in detrended terms).
26
The …gure assumes that no shocks hit the economy.

26

1.01

Aggregate productivity distortion

e

/

1.005

1

0.995

0.99

0.985

0.98
-10

-8

-6

-4

-2

0

2

4

6

8

Steady state inflation rate (annualized, in %)Figure 3: Aggregate productivity for di¤erent in‡ation rates (optimal rate is 2%)of the implemented steady-state in‡ation rate (x-axis), when the optimal in‡ation rate
is 2% per year.27 The aggregate production function (24) shows that one can interpretet=t as a measure of the aggregate productivity distortion that is implied by the relative
28

price distortions associated with suboptimal in‡ation rates.

The …gure shows that a 10

percentage point shortfall of the in‡ation rate below its optimal value of 2% is associatedwith an aggregate productivity loss equal to about 1%. In the process, the productivity
losses arise rather nonlinearly: a shortfall of in‡ation of 2 percentage points below its
optimal value is associated with an aggregate productivity loss of just 0.05%. Furthermore,in‡ation losses are asymmetric, with above-optimal in‡ation leading to relatively larger
losses. For instance, increasing in‡ation 8 percentage points above its optimal value
generates a productivity loss of 0.94%, while decreasing in‡ation by the same amount
below its optimal value leads to a productivity losses of only 0.37%.

10

The Optimal In‡ation Rate for the United States

Using the theory developed in the preceding sections, we now quantify the optimal in‡ation rate for the United States. In doing so, we interpret -shocks as events in which
production establishments are closed down and replaced by new establishments, in line
27
28

The …gure is based on the same parameterization as …gure 2.
Appendix F shows that c(1)=c( ? ) = ( e = ) :27

with the "…rm entry and exit" interpretation spelled out in section 3.2. This interpretation has the advantage that one can use readily available establishment-level information
to estimate the optimal in‡ation rate.
An alternative approach for estimating the optimal in‡ation rate could make use ofeven more disaggregate product-level information, which amounts to interpreting -shocks
as events in which old products (old quality levels) are replaced by new products (newquality levels). Our interpretation of -shocks as establishment entry and exit shocks
does not preclude that on top of the considered establishment dynamics, additional product substitution dynamics are present at the establishment level. For instance, it would
be relatively straightforward to enrich our framework by adding a Poisson shock, whichcauses establishments to occasionally replace their current product by a new product.
These additional product substitutions at the establishment level could also be associatedwith additional idiosyncratic adjustments to establishment productivity (or product quality).29 Provided the establishment can again freely choose the prices of newly introduced
products, all our theoretical results about the optimal in‡ation rate remain unchanged.
The empirical strategy for estimating the optimal in‡ation rate presented below would
equally remain una¤ected. In light of this, we feel comfortable with our interpretation of-shocks as establishment entry and exit events.
c

Let Lt denote the average employment per establishment in period t and let Lt denotethe average employment of continuing establishments, i.e., all establishments except theones that newly entered in period t. The following proposition shows how the optimal?
t can be inferred from these employment measures:in‡ation rate

Proposition 4 Suppose conditions 1 and 2 hold and that there are no …xed costs in production (f = 0). Suppose monetary policy implements the potentially suboptimal in‡ationratet and …rms’prices are indexed according tot 1;t ( t 1;tindexation). The optimal in‡ation rate net of price indexation

1 in the absence of price?
?
?
t = t 1;t ( t 1;t

1 in

the absence of price indexation) then satis…es
1
t
e
t

1

(1

)(

t = t 1;t )

1

1

(1

)

?
?
t = t 1;t

1

!

c

1

=

1 (1
)Lt =Lt?
(1)( t = ?t 1;t )

1

1

for t

0;
(39)

29

For example, equation (2) could be generalized to1

ejt K
Yjt = At Zjt Z
jt

1

1

Ljt

Ft ;

ejt is the productivity component that is speci…c to the product currently produced by estabwhere Z
ejt = Zejt 1 whenever the product is not substituted and Z
ejt being an iid draw fromlishment j, with Z
ejt ] = 1, whenever there is a Poisson event indicating a productsome stationary distribution with E[1=Z
substitution.

28

wheret=et evolves recursively according to

t
=
e
t

h

1

+ (1
with

1=

e

(1
)

)
(

1

?
?
t = t 1;tt = t 1;t )?
?
t = t 1;t

!

i

1

(1

)(

t = t 1;t )

1

1

(1

)

?
?
t = t 1;t

1

!

1

t 1
;
e
t 1

(40)

1 = 1.

The proof of proposition 4 can be found in appendix G. Proposition 4 shows how one
can determine the optimal in‡ation path?
t for a sticky-price economy that is subject tostochastic disturbances and in which monetary policy implements potentially suboptimalin‡ation rates. In particular, equations (39) and (40) allow to infer?
?
t = t 1;t for all t

0,

given values for the parameters ( ; ; ), the price indexation rule , the observed actualin‡ation ratect and the employment ratio Lt =Lt .

An instructive special case of proposition 4 arises when monetary policy is optimal(

t = t 1;t =?
?
t = t 1;t for all t). We then havet =yields
c

?
t =

?
t 1;t

Lt
Lt

e
t for all t and equation (39) directly

! 11

(41)

:

For this special case, the optimal in‡ation rate is then only a function of price indexc

ation ( ?t 1;t ), the ratio of average employment of continuing establishments (Lt ) over
the average employment of all establishments (Lt ), and the demand elasticity parameter. This is closely related to our theoretical result in equation (35), which shows thatthe optimal in‡ation rate depends on price indexation ( ?t 1;t ) and the ratio of average
productivity of continuing …rms (at gt At 1 Qt 1 =…rms (At Qt =et 1 ) over the average productivity of allet ). The special case with optimal monetary policy thus illustrates how ourestimation approach uses establishment employment information together with information about the demand elasticity

to determine establishment productivity.30 In fact, theemployment ratio in equation (41) raised to the power 1=(

1) is equal to the produc-

tivity ratio showing up in equation (35). This holds true because productivity di¤erencestranslate into price di¤erences, which in turn translate into employment di¤erences, as
determined by the elasticity of product demand .
30

It may seem desirable to directly estimate …rm-level productivities. It is, however, di¢ cult to measurephysical productivity at the …rm or establishment level because output prices are typically not observedat this level of observation. As is explained in Foster, Haltiwanger and Syverson (2008), the productivity
literature usually measures revenue productivity instead of physical productivity at the …rm level, whichde‡ates …rm-level output with some industry-level price index. In our setting, …rms’revenue productivity
is completely unrelated to their physical productivity in the absence of …xed costs of production. For the
few industries for which physical and revenue productivities can both be observed, the two productivity
measures can be rather di¤erent; see Foster, Haltiwanger and Syverson (2008).

29

For the more general case with suboptimal monetary policy, the expression on theleft-hand side of equation (39) starts to generally di¤er from one. In fact, as a result ofine¢ cient price dispersion, aggregate productivity (At Qt =it assumes in the e¢ cient allocation (At Qt =the productivity wedge

10.1

t=

e
t ), so that 1=t ) will fall short of the value
1= et . The evolution oftet under suboptimal policies is captured by equation (40).

Baseline Estimation

We estimate the optimal in‡ation rate using the result derived in proposition 4 and thebaseline parameters reported in table 1. The parameters refer to a sticky-price model atcannual frequency because the employment ratio Lt =Lt is observed at annual frequency.
(0:55)4 , where 0:55 is the baseline value chosen

The price stickiness parameter satis…es

in the quarterly sticky-price model of Coibion, Gorodnichenko and Wieland (2012).31 The
probability for -shocks is set to 11.5% per year, which is the midpoint between the average
establishment birth rate (12.4%) and the average establishment exit rate (10.7%) overthe period 1977 to 2015 reported in the Business Dynamics Statistics (BDS) of the U.S.
Census Bureau. We set the elasticity of product demand equal to 7, which is the baseline
value considered in Coibion, Gorodnichenko and Wieland (2012) and Gorodnichenko andWeber (2016). It implies a steady state mark-up of around 17% over production costs. In
general, we consider price indexation rules that index prices to lagged in‡ation accordingtot 1;t = (for some
indexation.

t 1)

;

2 [0; 1). For our baseline estimation, we consider a setting without priceParameter

Assigned value

Price stickiness0.0915-shock probability

11.5%

Demand elasticity

7

Price indexation

0

Table 1: Baseline parameters (annual model)c

We determine the employment ratio Lt =Lt using information from the BDS. The BDS
is based on the Longitudinal Business Database (LBD) and provides information on the
31

The implied price duration is neverthesless larger than in the quarterly model of Coibion, Gorodnichenko and Wieland (2012). Section 10.2 shows that our quantitative results are robust to making
prices less sticky.

30

1.7
1.6

Percentage Points

1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
1980

1985

1990

1995

2000

2005

2010

2015

Years

Figure 4: Optimal U.S. in‡ation?
t , baseline estimation

number of establishments and establishment employment by establishment age. The
database covers the universe of private establishments in the United States for the years
1977-2015. We compute Lt by dividing the reported "economy-wide employment" in any
year t by the "economy-wide number of establishments" of the year. Similarly, we computecthe average employment of continuing establishments Lt by …rst subtracting "jobs createdby establishment birth over the last 12 months" from "economy-wide employment" andthen dividing the result by the number of continuing establishments, which is equal tothe "economy-wide number of establishments" minus the number of "establishments born
during the last 12 month". We use the GDP de‡ator provided by the U.S. Bureau of
Economic Analysis (GDPDEF_PC1) as our measure for the actual in‡ation rate. Results
are virtually identical when using other measures of actual in‡ation.
Figure 4 depicts the optimal in‡ation rate implied by proposition 4 and our baseline
calibration. The estimated optimal in‡ation rate is positive throughout the sample period. This is in line with the empirical observation that older establishments employ (onaverage) more workers, which means that they are more productive in our framework(gt =qt > 1). The sample mean of the optimal in‡ation rate is equal to 1.1% per year,
which is a relatively large number within the sticky-price literature.32 Coibion, Gorod32

See …gure 1 in Diercks (2017), which depicts the optimal average in‡ation rate found in 100 quanti-

31

nichenko and Wieland (2012), for instance, …nd similarly high optimal average in‡ation
rate (of around 1.5%) using a model featuring a zero lower bound on nominal interest
rates and a monetary authority that follows a Taylor rule.33
As is evident from Figure 4, the optimal in‡ation rate displays a slight downward trend
over time. Especially from 1988 onwards, the optimal in‡ation rate seems to have dropped.
While the average over the period 1977-1987 was 1:3%, the average over the remaining
sample period was 1:0%. According to the model, this drop in the estimated optimalin‡ation rate implies that either the experience growth rate gt has weakened and/or thatthe cohort growth rate qt has accelerated. Both of these e¤ects cause older establishmentscto become smaller relative to new establishments (Lt =Lt drops). Independent evidence
on …rm employment growth provided in …gure 11 in Pugsley, Sedlacek and Sterk (2017)shows indeed that the employment growth rate of cohorts that entered after the year 1987
has slowed signi…cantly, which is consistent with the slight downward trend in the optimalin‡ation rate showing up in …gure 4.

10.2

Robustness of the Empirical Results

We now explore the robustness of our baseline estimation from the previous section. We
consider the e¤ects of choosing alternative parameter values for ( ; ; ; ), as well as thee¤ects of positive …xed costs in production (f0).

A …rst set of results is presented in …gure 5. Panel A in the …gure depicts the optimalin‡ation estimate when assuming ‡exible prices ( = 0) and shows that the benchmark
results remain virtually unchanged. Panel B depicts the e¤ects of assuming alternative
establishment turnover rates. As is well known, the turnover rate has signi…cantly dropped
over the considered sample period. The value of= 0:15 corresponds to the average

establishment entry and exit rate at the beginning of the sample period, while the lower
value of

= 0:095 is the one observed at the end of the sample period. The …gure showsthat alternative turnover rates within this range do not generate quantitatively signi…cante¤ects.
Panel C in …gure 5 shows the e¤ects of assuming a lower value for the demand elasticity. While macroeconomists tend to use high demand elasticities based on the observationthat pure pro…ts tend to be low, see Basu and Fernald (1997), the trade and industrial
tative optimal monetary policy studies.
33
The authors shows that for fully optimal monetary policy, as considered in the present paper, theaverage optimal in‡ation rate falls to 0.2% per year in their setting. The large drop in average in‡ationarises because Taylor rules are severely suboptimal in the vicinty of the strong non-linearities induced bya lower bound constraint. The associated welfare costs can then be reduced by increasing the intercept
term of the Taylor rule. In our setup, Taylor rules with optimal intercepts are approximately optimaland lead to a very similar average in‡ation rate as in a setting with fully optimal monetary policy. Thisis so because the welfare costs of in‡ation are relatively symmetric around the optimum, see …gure 3.

32organization literature usually estimates lower demand elasticity values.34 The value of= 3:8 considered in …gure 5 is taken from Bilbiie et al. (2012) and Bernard et al. (2003)
and is based on a calibration that …ts U.S. plant and macro trade data. The …gure showsthat the optimal in‡ation rate then approximately doubles. This increase is driven bycthe fact that with a lower demand elasticity, any given employment ratio Lt =Lt observedin the data must be associated with larger establishment-level productivity di¤erences (alarger gap between gt and qt ).
Panel D in …gure 5 considers a setting in which prices of non-optimizing …rms are indexed to lagged in‡ation. We choose= 1=2, which Coibion, Gorodnichenko and Wieland(2012) consider to be an upper bound on the range of plausible values for this parameter.
Again, this causes the optimal in‡ation rate to approximately double compared to thebaseline estimate.
Since proposition 4 assumes that there are no …xed costs in production (f = 0), we
also consider the e¤ects of allowing for positive …xed costs. Appendix H shows that thepresence of …xed costs tends to slightly increase the estimated optimal in‡ation rates, but
the e¤ect is quantitatively small.

11

Robustness of the Theoretical Results

This section considers various extensions and alternative model setups. Section 11.1 showsthat our main …nding about the optimal in‡ation rate (proposition 2) continues to apply
in a setting where price adjustment frictions take the form of menu costs. Section 11.2
discusses the e¤ects of a non-constant -shock hazard rate, while section 11.3 analyzes
the case with a price index which oversamples old products.

11.1

Menu Cost Frictions

While our results are illustrated using time-dependent price-setting frictions, our theoretical …nding from proposition 2 extends to a setting in which …rms optimally decide to pay a…xed cost to adjust their price. Since the optimal in‡ation rate in proposition 2 replicatesthe ‡exible-price allocation, …rms that do not experience a -shock have - independentlyof the nature of their price setting frictions - no incentives to adjust their prices. Sincethe ‡exible price allocation is e¢ cient, see proposition 1, monetary policy also has no
incentive to deviate from the ‡exible-price allocation. Both observations together imply
that the optimal in‡ation rate does not depend on whether price-setting frictions are stateor time dependent.35
34

These lower elasticities are consistent with pure pro…ts being low if there are …xed costs of production.
Obviously, this requires that, in a setting with menu cost frictions, -shocks lead either to these menu
cost not having to be paid at all or always having to be paid.
35

33Percentage Points

1.6

A. Flexible Prices

1.6

1.4

1.4

1.2

1.2

1

1

0.8

0.8
1980 1985 1990 1995 2000 2005 2010 2015

Percentage Points

3.5

B. Turnover Rate

C. Demand Elasticity

1980 1985 1990 1995 2000 2005 2010 2015

3.5

3

3

2.5

2.5

2

2

1.5

1.5

1

1

0.5

D. Price Indexation

0.5
1980 1985 1990 1995 2000 2005 2010 2015

Years

1980 1985 1990 1995 2000 2005 2010 2015

Years

Figure 5: Optimal in‡ation for the United States, alternative parameter assumptions

34

11.2

Non-Constant -Shock Probability

Throughout the paper we assume that …rms face a constant probability for receiving a-shock. While analytically convenient, it may on empirical grounds be attractive toconsider either increasing or decreasing hazard rate speci…cations. A decreasing hazard,
for instance, can capture the fact that young …rms might face higher exit rates. Conversely,
increasing hazards may capture the fact that products become increasingly likely to be
substituted as they age. To assess how our baseline results are a¤ected by non-constanthazard rates, we consider a setting where …rms receive a -shock with probability 1 in the…rst period after having received a -shock but with probability

subsequently. We allow

for increasing hazard rates ( 1 < ) and decreasing hazard rates ( 1 > ). The following
proposition states our main …nding:
Proposition 5 Suppose conditions 1 and 2 hold. Let

1 2 [0; 1) denote the probability ofreceiving a -shock for …rms that received a -shock in the previous period and let

2 (0; 1)denote the probability of receiving a -shock for all other …rms. The optimal in‡ation rateis then given by
?
t =

?
t 1;t

at gt At 1eQt 1
t 1

At Qt

(42)

;

e
t

e
t evolves according to

where

(
with

0

=(1

e 1
t)

=

1 +

0

0( 1

) (qt =gt )1

+ (1

)

1
e
t 1 qt =gt

;

(43)

) denoting the total mass of …rms receiving -shocks in a given

period. In the absence of economic disturbances and price indexation ( ?t 1;t1), we

have
lim

t!1

?
t = g=q:

The proof of the proposition is contained in appendix I. Equation (42) shows that the
optimal in‡ation rate is determined by the ratio of the average productivity of …rms witheet 1 ) over the average productivity of all …rms (At Qt = t ),adjusted for the possible presence of price indexation ( ?t 1;t ). This is identical to theout a -shock (at gt At 1 Qt 1 =setting with a constant hazard rate, see the discussion following our main result in proposition 2. The only di¤erence relative to the case with a constant hazard rate is thatthe recursive equation describing the evolution ofet generalizes from equation (32) toequation (43). The optimal steady state in‡ation rate remains nevertheless una¤ected by
non-constant hazard rates.

11.3

Bias Towards Older Goods in the Measured Price Index

Throughout the paper, we consider a price index capturing all products available at any
given point in time. This idealized price index di¤ers from how statistical agencies tend35

to compute price indices in practice. In particular, once products become unavailable,
the statistical agencies tend to replace them by products that have been around for sometime. This generates a bias towards older goods in the basket underlying actual priceindices.36 Since in‡ation/de‡ation is optimal in the present setup because of the entryof new products (or …rms or qualities), this raises the question whether oversampling ofolder products drives a wedge between the optimal in‡ation rate?
t based on the ideal

price index and the optimal in‡ation rate for a price index biased in favor of older goods.
To assess this question, we consider an in‡ation ratecontaining only products of "age" NNtthat is based on a price index

1 or higher. The following proposition summarizes

our main result:
Proposition 6 Suppose conditions 1 and 2 hold and the measured price index features
an age bias in the sense that it includes only products that received their last -shockat least N

1 periods ago. LetNtdenote the in‡ation rate associated with this priceindex. The equilibrium allocation in the sticky-price economy is e¢ cient if monetary
policy implements the gross in‡ation rate
N?
=
t

where

?
t 1;t
?
t N 1;t N?
t N for all tN;?
t denotes the optimal in‡ation rate for the ideal price index, as stated in propo?
?

sition 2, andcaptures price indexation (= 1 in the absence of indexation).

The proof is contained in appendix J. The proposition shows that in the absence of
price indexation ( ?t 1;t = 1 for all t), the optimal in‡ation rate for an age-biased priceindex equals the N -period lagged optimal in‡ation rate for the ideal price index. Thisis so because it takes N periods for products to enter the statistical agency’s productbasket.37 Naturally, the presence of price indexation adds some additional time shifters( ?t 1;t = ?t N 1;t N ). It follows from the proposition that the optimal steady-state in‡ation
rate is not a¤ected by the presence of an age bias. In the absence of shocks and in theabsence of price indexation, we have limt!1

12

N?
= limt!1
t?
38
t = g=q.

Conclusions

This paper shows how …rm-level productivity trends a¤ect the in‡ation rate that is optimal for the aggregate economy and that the e¤ect of …rm-level trends on the optimal
36

See the discussion on product substitutions in section III.C in Nakamura and Steinsson (2008).
?
As a result, the optimal in‡ation rate N
for the initial periods t = 0; :::; N 1 is a function of thetinitial distribution of prices prevailing at t = 1 under optimal monetary policy.
38?
?
More generally, for the case with price indexation we have limt!1 N=
t = t 1;t?
?
limt!1 t = t N 1;t N . If price indexation does not depend on the in‡ation measure targeted by the?
central bank, then we have again limt!1 N= limt!1 ?t .
t
37

36

in‡ation rate is quantitatively large by the standards of the sticky price literature. Indeed, a puzzling feature of this literature is the considerable gap between the in‡ation
rates it suggests to be optimal, which are typically close to zero or even negative, andthe signi…cantly positive in‡ation targets pursued by leading central banks around theglobe. The present paper thus contributes to bridging this gap by providing theoreticalunderpinnings for current central bank practice.
The paper also opens a range of interesting avenues for further research. In light ofthe present …nding, it appears of interest to understand …rm-level productivity trends,
including their changes over time, in greater detail and across a larger set of economies.
This would allow assessing to what extent the observed downward trend in global in‡ation
rates is fundamentally justi…ed.
It also appears interesting to incorporate …rm-level trends into richer sticky-price setups that feature characteristics that the present analysis abstracted from, for instance,
a set of di¤erent economic sectors and associated input-output linkages. Quantitatively
studying such richer frameworks would allow assessing whether or not these characteristics amplify or dampen the e¤ects of …rm-level trends on the optimal in‡ation rate.
Finally, it appears to be of interest to incorporate …rm-level trends into linear-quadratic
formulations of optimal monetary policy problems, so as to study how the optimal policy
response to cost-push disturbances is a¤ected by the presence of …rm-level trends.

References
Adam, K., and R. Billi (2007): “Discretionary Monetary Policy and the Zero Lower
Bound on Nominal Interest Rates,”Journal of Monetary Economics, 54, 728–752.
Adam, K., and R. M. Billi (2006): “Optimal Monetary Policy under Commitment
with a Zero Bound on Nominal Interest Rates,”Journal of Money, Credit and Banking,38(7), 1877–1905.
Ahn, S., G. Kaplan, B. Moll, T. Winberry, and C. Wolf (2017): “When Inequality Matters for Macro and Macro Matters for Inequality,” in NBER Macroeconomics
Annual 2017, ed. by M. Eichenbaum, and J. Parker. National Bureau of Economic
Research, Inc.
Amano, R., K. Moran, S. Murchison, and A. Rennison (2009): “Trend in‡ation,
wage and price rigidities, and productivity growth,” Journal of Monetary Economics,
56(3), 353 –364.

37

Armknecht, P. A., W. F. Lane, and K. J. Stewart (1996): “New Products and the
U.S. Consumer Price Index,”in The Economics of New Goods, ed. by T. F. Bresnahan,
and R. J. Gordon, chap. 9. University of Chicago Press.

Ascari, G., and A. M. Sbordone (2014): “The Macroeconomics of Trend In‡ation,”
Journal of Economic Literature, 52(3), 679–739.
Basu, S., and J. G. Fernald (1997): “Returns to Scale in U.S. Production: Estimatesand Implications,”Journal of Political Economy, 105(2), 249–283.
Benigno, P., and L. A. Ricci (2011): “The In‡ation-Output Trade-O¤ with Downward
Wage Rigidities,”American Economic Review, 101, 1436–1466.
Bergin, P. R., and G. Corsetti (2008): “The extensive margin and monetary policy,”
Journal of Monetary Economics, 55(7), 1222–1237.
Bernard, A. B., J. Eaton, J. B. Jensen, and S. Kortum (2003): “Plants andProductivity in International Trade,”American Economic Review, 93(4), 1268–1290.
Bilbiie, F. O., I. Fujiwara, and F. Ghironi (2014): “Optimal monetary policy withendogenous entry and product variety,”Journal of Monetary Economics, 64, 1–20.
Bilbiie, F. O., F. Ghironi, and M. J. Melitz (2008): “Monetary Policy and Business Cycles with Endogenous Entry and Product Variety,” in NBER Macroeconomics
Annual 2007, Volume 22, NBER Chapters, pp. 299–353. National Bureau of Economic
Research, Inc.
(2012): “Endogenous Entry, Product Variety, and Business Cycles,” Journal of
Political Economy, 120(2), 304 –345.
Bils, M. (2009): “Do Higher Prices for New Goods Re‡ect Quality Growth or In‡ation?,”Quarterly Journal of Economics, 124, 637–675.
BLS (2015): “Bureau of Labor Statistics, Handbook of Methods",”Chapter 17, Consumer
Price Indexes.
Brunnermeir, M., and Y. Sannikov (2016): “On the Optimal Rate of In‡ation,”
American Economic Review, Papers and Proceedings, 5, 484–489.
Calvo, G. A. (1983): “Staggered Contracts in a Utility-Maximizing Framework,”Journal of Monetary Economics, 12, 383–398.
Carlsson, M., and A. Westermark (2016): “Labor market frictions and optimal
steady-state in‡ation,”Journal of Monetary Economics, 78, 67 –79.

38

Coad, A. (2010): “The Exponential Age Distribution and the Pareto Firm Size Distribution,”Journal of Industry, Competition and Trade, 10(3), 389–395.
Cogley, T., and A. M. Sbordone (2008): “Trend In‡ation, Indexation and In‡ation
Persistence in the New Keynesian Phillips Curve,” American Economic Review, 98,
2101–2126.
Coibion, O., Y. Gorodnichenko, and J. Wieland (2012): “The Optimal In‡ation
Rate in New Keynesian Models: Should Central Banks Raise Their In‡ation Targets
in Light of the Zero Lower Bound?,”Review of Economic Studies, 79(4), 1371–1406.
Decker, R. A., J. Haltiwanger, R. S. Jarmin, and J. Miranda (2014): “TheSecular Decline in Business Dynamism in the U.S.,”University of Maryland mimeo.
Diercks, A. M. (2017): “The Reader’s Guide to Optimal Monetary Policy,” Available
at SSRN: https://ssrn.com/abstract=2989237.
Eggertsson, G., and M. Woodford (2003): “The Zero Interest-Rate Bound andOptimal Monetary Policy,”Brookings Papers on Economic Activity, (1), 139–211.
Foster, L., J. Haltiwanger, and C. Syverson (2008): “Reallocation, Firm
Turnover, and E¢ ciency: Selection on Productivity or Pro…tability?,” American Economic Review, 98, 394–425.
Golosov, M., and R. E. Lucas (2007): “Menu Costs and Phillips Curves,”Journal of
Political Economy, 115, 171–199.
Gorodnichenko, Y., and M. Weber (2016): “Are Sticky Prices Costly? Evidence
from the Stock Market,”American Economic Review, 106(1), 165–99.
Kaplan, G., and G. Violante (2014): “A Model of the Consumption Response to
Fiscal Stimulus Payments,”Econometrica, 82(4), 1199–1239.
Khan, A., R. G. King, and A. L. Wolman (2003): “Optimal Monetary Policy,”
Review of Economic Studies, 70(4), 825–860.
Kim, J., and F. J. Ruge-Murcia (2009): “How much in‡ation is necessary to grease
the wheels?,”Journal of Monetary Economics, 56(3), 365 –377.
Melitz, M. J. (2003): “The Impact of Trade on Intra-Industry Reallocations and Aggregate Industry Productivity,”Econometrica, 71(6), 1695–1725.
Melser, D., and I. A. Syed (2016): “Life Cycle Price Trends and Product Replacement:
Implications for the Measurement of In‡ation,”Review of Income and Wealth, 62, 509–
533.
39

Moulton, B. R., and K. E. Moses (1997): “Addressing the Quality Change Issue in
the Consumer Price Index,”Brookings Papers on Economic Activity, pp. 305–349.
Nakamura, E., and J. Steinsson (2008): “Five Fact about Prices: A Reevaluation of
Menu Cost Models,”Quarterly Journal of Economics, 123, 1415–1464.
(2010): “Monetary Non-neutrality in a Multisector Menu Cost Model,”Quarterly
Journal of Economics, 125, 961–1013.
Nakamura, E., and J. Steinsson (2012): “Lost in Transit: Product Replacement Bias½
and Pricing to Market,”American Economic Review, 102(7), 3277U3316.
Pugsley, B. W., P. Sedlacek, and V. Sterk (2017): “The Nature of Firm Growth,”
University College London mimeo.
Schmitt-Grohé, S., and M. Uribe (2010): “The Optimal Rate of In‡ation,”in Handbook of Monetary Economics, ed. by B. M. Friedman, and M. Woodford, vol. 3 ofHandbook of Monetary Economics, chap. 13, pp. 653–722. Elsevier.
(2012a): “Foreign Demand for Domestic Currency and the Optimal Rate ofIn‡ation,”Journal of Money, Credit and Banking, 44(6), 1207–1224.
(2012b): “On Quality Bias and In‡ation Targets,” Journal of Monetary Economics, 59(4), 393 –400.
(2013): “Downward Nominal Wage Rigidity and the Case for Temporary In‡ationin the Eurozone,”Journal of Economic Perspectives, 27, 193–212.
Wolman, A. L. (2011): “The Optimal Rate of In‡ation with Trending Relative Prices,”
Journal of Money, Credit and Banking, Vol. 43, 355–384.
Woodford, M. (2003): Interest and Prices. Princeton University Press, Princeton.
Yun, T. (2005): “Optimal Monetary Policy with Relative Price Distortions,” Amercian
Economic Review, 95, 89–109.

40

Appendix
A

Derivation of the Sticky-Price Economy

A.1

Cost Minimization Problem of Firms

The cost minimization problem of …rm j,
1

min Kjt rt + Ljt Wt =Pt

s:t: Yjt = At Qt sjt Gjt Kjt

Kjt ;Ljt

1

1

Ljt

Ft ;

yields the …rst-order conditions
1

0 = rt + 1
0 = Wt =Pt +
where

t

1

t At Qt sjt Gjt

Ljt
Kjt

t At Qt sjt Gjt

1

Ljt
Kjt
1

1

;

denotes the Lagrange multiplier. The …rst-order conditions imply that the

optimal capital labor ratio is the same for all j 2 [0; 1], i.e.,
Wt
Kjt
=
(
Ljt
Pt r t1):

Plugging the optimal capital labor ratio into the technology of …rm j and solving for thefactor inputs yields the factor demand functions
Ljt =

Wt
(
Pt r tKjt =Wt(
Pt r t

1

1

Ijt

(44)

Ijt :

(45)

1)
1

1)

Firm j demands these amounts of labor and capital, respectively, to combine them to Ijt ,which yields Yjt units of output. Accordingly, the …rm’s cost function to produce Ijt is
M Ct Ijt = Wt

1

Wt
(
Pt rt

1

1)

Ijt + Pt rt

Wt(
Pt r t

1

1)

(46)

Ijt ;

where M Ct denotes nominal marginal (or average) costs. This equation can be rearranged
to obtain equation (10) in the main text.

A.2

Price-Setting Problem of Firms

The price-setting problem of the …rm j, see equation (11), implies that the optimal productprice is given by
Pjt? =

1
11+

Et

P1

i=0 (

Et

(1
P1

i=0 (

))i

t;t+i Yt+i ( t;t+i =Pt+i )

(1
41

))i

M Ct+i =Pt+i
At+i Qt sjt Gjt+i

1t;t+i Yt+i ( t;t+i =Pt+i )

:

(47)

Rewriting this equation yields
Pjt?
Pt

Qt sjt Gjt
Qt

=

1
11+

Et

P1

i=0 (

))i

Yt+i
t;t+i Yt

i=0 (

))i

(1
Et

P1

(1

t;t+i Pt

Qt+i =Qt
M Ct+i
Pt+i At+i Qt+i Gjt+i =Gjt

Pt+i

t;t+i Pt

Yt+i
t;t+i Yt

1

:

(48)

Pt+i

The multi-period growth rate of the cohort e¤ect relative to the experience e¤ect corresponds to
Qt+i =Qt
qt+i=
Gjt+i =Gjt
gt+i

qt+1
;
gt+1

for i > 0, and equals unity for i = 0. Hence, this growth rate is independent of the index
j, because when going forward in time, …rms are subject to the same experience e¤ect.
Thus, we can rewrite the equation (48) according to
Pjt?
Pt

Qt sjt Gjt
Qt

1
11+

=

Nt
;
Dt

where the numerator Nt is given by
Nt = Et

1
X

( (1

i=0

))i

t;t+i

t;t+i Pt

Yt+iYt

M Ct+i
Pt+i At+i Qt+i

Pt+i

qt+i
gt+i

qt+1
gt+1

:

The numerator evolves recursively as shown by equation (13). The denominator Dt also
evolves recursively, and jointly this yields the recursive pricing equations (12)-(14).

A.3First-Order Conditions to the Household Problem

The …rst-order conditions that belong to the household problem comprise the household’sbudget constraint, a no-Ponzi scheme condition, the transversality condition, and thefollowing equations:
Wt
=
Pt

ULt
UCt

t;t+1 =

t+1 UCt+1

UCtt

1 + it

1 = Ett;t+1
t+1

1 = Et [t;t+1 (rt+1 + 1d)] :Here, we denote by U (:) the period utility function. Our assumption that U (Ct ; Lt ) =([Ct V (Lt )]11)=(1) implies
UCt = Ct V (Lt )1
ULt = Ct1

V (Lt )VLt ;where UCt = @U (Ct ; Lt )=@Ct and VLt = @V (Lt )=@Lt .
42

A.4

Recursive Evolution of the Price Level

Plugging the weighted average price of a cohort, equation (19), into the price level, equation (20), yields that
? 1)
= ( t;t Pt;t

Pt1

+

1
X

(1

)s

s=1

"

(1

)

s 1
X

k

( t k;t Pt? s;t k )1s+

( t s;t Pt? s;t s )1

k=0

Telescoping the sums yields:? 1)= ( t;t Pt;t

Pt1

+ (1

)1 (1

)( t;t Pt? 1;t )1+ ( t 1;t Pt? 1;t 1 )1

+ (1

)2 (1

)( t;t Pt? 2;t )1+ (1) ( t 1;t Pt? 2;t 1 )1

2+

( t 2;t Pt? 2;t 2 )1

+ ::: :
Collecting optimal prices that were set at the same date in square brackets yields:
Pt1

=
1
t;t

? 1
(Pt;t
)

+ [ (1

)]

+ (1

)(1

) (Pt? 1;t )1

1
t 1;t

(Pt? 1;t 1 )1+ (1)(Pt? 2;t )1+ (1)2 (Pt? 3;t )1+ :::) (Pt? 2;t 1 )1+ (1)(Pt? 3;t 1 )1

+ :::

+ (1
)(1

+ ::: :
Using equation (21) and the de…nition of pet in equation (23), we can replace the terms incurly brackets in the previous equation by pet . This yields

Pt1

? 1
= ( t;t Pt;t)

1 + (1(pet ))

1

1+ [ (1)]1 ( t 1;t Pt? 1;t 1 )1

1 + (1)+ [ (1)]2 ( t 2;t Pt? 2;t 2 )1

1 + (1)(pet 1 )1(pet 2 )1

1
1+ ::: :

Rearranging the previous equation yields
Pt1? 1= ( t;t Pt;t)

+ (1)(pet )

1

( t 1;t 1 Pt? 1;t 1 )1+ (1)( t 1;t )1+ (1)( t 2;t 1 Pt? 2;t 2 )1+ (1)(pet 1 )+ (1)(pet 2 )

1

+ :::

1

:

The term in curly brackets in the previous equation corresponds to Pt1 1 , which yields
the price level equation (22) in the main text.
43

#

:

A.5

Equilibrium De…nition

We are now in a position to de…ne the market equilibrium:
De…nition 1 An equilibrium is a state-contingent path for f(Pjt ; Ljt ; Kjt ) for j 2 [0; 1],

Wt ; rt ; it ; Ct ; Kt+1 ; Lt ; Bt ; Tt g1
t=0 such that

1. the …rms’choices fPjt ; Ljt ; Kjt g1
t=0 maximize pro…ts for all j 2 [0; 1], given the priceadjustment frictions,2. the household’s choices fCt ; Kt+1 ; Lt ; Bt g1
t=0 maximize expected household utility,
3. the government ‡ow budget constraint holds each period, and4. the markets for capital, labor, …nal and intermediate goods and government bonds
clear,
given the initial values B 1 (1 + i 1 ); K0 ; Pj; 1 , and A 1 Q 1 sj; 1 Gj; 1 , with j 2 [0; 1].

A.6

Aggregate Technology and Aggregate Productivity

To derive the aggregate technology, we combine …rms’technology to produce the di¤erentiated product in equation (2) with product demand Yjt =Yt = (Pjt =Pt )Qt =Qt sjt
Gjt

Yt
At Qt

Pjt
Pt

=

Kjt
Ljt

1

to obtain

1

Ljt

Ft :

Integrating over all …rms with j 2 [0; 1], using labor market clearing, Lt =R10

Ljt dj, andthe fact that optimizing …rms maintain the same (and hence the aggregate) capital laborratio yields
Yt
At Qt

Z 1

Qt =Qt sjt
Gjt

0

Pjt
Pt

1

dj = Kt

1

1

Lt

Ft :

Rearranging this equation and de…ning the (inverse) endogenous component of aggregate
productivity as in equation (25) in the main text yields the aggregate technology (24).
To derive the recursive representation oft shown in equation (26), we rewrite equa-tion (25) according to
t

Pt

=

Z 1

qt
gt

0

qt sjt +1
gt sjt +1(Pjt )

dj;using the processes describing the evolution of Qt and Gjt . As for the price level, we
proceed with the aggregation in two steps. First, we aggregate the optimal prices ofall …rms operating within a particular cohort. Second, we aggregate all cohorts in theeconomy. To this end, we rewrite

t =Pt
t

Pt

=

in the previous equation according to

1
X

(1

s=0

44

)s b t (s);(49)

usingb t (s) =

8
<

qt
gt

qt s+1
gt s+1(1):( P ? )t;t t;t

Ps 1

k

k=0

( t k;t Pt? s;t k )s+( t s;t Pt? s;t s )

if s

1;

if s = 0 :Substituting out for b t (s) in equation (49) yields
t

Pt

?
)
= ( t;t Pt;t

+

1
X

(1

)s

s=1

qt
gt

qt s+1
gt s+1

"

(1

)

s 1
X

k

( t k;t Pt? s;t k )+s( t s;t Pt? s;t s )k=0

We rearrange the previous equation following corresponding steps to those in appendix
A.4. This yieldstPt?
= ( t;t Pt;t
)

+ (1

)

+ [ (1)]2)(pet )

+ (1
qt
gt

1

( t 1;t Pt? 1;t 1 )qt qt 1gt gt 1+ (1( t 2;t Pt? 2;t 2 ))(pet 1 )1)(pet 2 )1

+ (1)(pet 1 )1

+ (1)(pet 2 )+ (1+ ::: :We rearrange the previous equation further to obtain thatt?
= ( t;t Pt;t
)

Pt

+ (1

)

+ (1

)

)(pet )

+ (1

1

qt
( t 1;t )(Pt? 1;t 1 )gt
qt 1( t 2;t 1 Pt? 2;t 2 )gt 1

The term in curly brackets in the previous equation is equal to
t

Pt

=

+ (1

)(pet )

1?
( t;t Pt;t
)

+ (1

)

qt
gt

1

+ ::: :

t 1 =Pt 1 , which yields( t 1;t )

t 1

Pt 1

:

Multiplying the previous equation by Pt yields equation (26) in the main text.

A.7

Consolidated Budget Constraint

Consolidating the household’s and the government’s budget constraints shown in the main
text yields
Ct + Kt+1 = (1Wtd)Kt + rt Kt +Lt +Pt

To compute aggregate …rm pro…ts denoted by

R1
0

R1

jt dj
0
Pt

R1

Pjt Yjt dj
0
Pt

!

:

(50)

jt dj, we use marginal costs in equation(46) and combine them with the factor demands for Ljt and Kjt , equations (44) and (45),
45

#

:

which yields that M Ct Ijt = Wt Ljt + Pt rt Kjt . We use this equation and product demand
Yjt =Yt = (Pjt =Pt )Z 1

to rewrite aggregate …rm pro…ts according tojt dj = (1 +)Z 1

Pjt Yjt dj= (1 + )Z 1

Pjt Yjt dj= (1 + )Pt Yt

reduces to

R10

Ljt dj and Kt =

R1
0

Z 1

(Wt Ljt + Pt rt Kjt ) dj

00with Lt =M Ct Ijt dj

0

0

0

Z 1

W t LtP t r t Kt ;Kjt dj. Thus, the consolidated budget constraint (50)Kt+1 = (1d)Kt + Yt

Dividing the previous equation by trend growthet+1 kt+1 = (1Ct :et yieldsd)kt + yt

ct ;

e
e
t = t 1 denotes the gross trend growth rate.

where

e
t =

A.8

Transformed Sticky-Price Economy?
We de…ne p?t = Pt;t
=Pt and mct = M Ct =(Pt ( et )1= ) and wt = Wt =(Pt et ) and ct = Ct = et .

We also use that pet = 1=et , which follows from the equations (23) and (32). This yields

46

the following equations that describe the transformed sticky-price economy.
1

1=

+ (1

t =

+ (1

)(

1
11+

p?t =
Nt =

)(

mct

Dt = 1 + (1mct =
rt kt = (etet+1 kt+1 = (1

1

+ (1

"

)

(51)

t 1;t

qt
gt

)

t
t 1

e
t 1=

= + (1ct

1 = Et

)

(52)

t 1;t

(53)
"

e
t;t+1 t+1et;t+1 t+1

yt+1
yt

t+1
t;t+1
1

yt+1
yt

t+1

qt+1
gt+1
#

Nt+1

#

Dt+1

t;t+1

(54)
(55)

1

1

(56)
(57)

1

1

Lt

d)kt + ytet = (at qt

wt =(p?t )

rt
1 1=

kt

t

e 1
t)

e 1
t)

t

1)wt Lt

yt =

(

+ (1

)Et
)Et

1

wt
1=

(p?t )1

Nt
Dt

+ (1

e
t

e 1
t)

(58)

f

(59)

ct

e
t)

(60)
1
e
t 1 qt =gt

(61)

VLt
V (Lt )

(62)
1 + it

(63)

t;t+1
t+1

1 = Et [t;t+1 =

t;t+1 (rt+1 + 1
t+1e
t+1 ct+1

t

ct

(64)

d)]
V (Lt+1 )V (Lt )

1

(65)

After adding a description of monetary policy and a price indexation rule, these seventeen
equations determine the paths of the seventeen variables it ;e ?
t ; pt ;A.9

t 1;t ; Nt ; Dt ;t ; yt ; ct ; kt ; Lt ; rt ; wt ; mct ;

e
t;

t;

t 1;t given the four exogenous shocks qt ; gt ; at ; t .

Steady State in the Transformed Sticky-Price Economy

We consider a steady state in the transformed sticky-price economy, in which g and q are
constant and the government maintains a constant in‡ation rate

, which also implies a

constant rate of price indexation .
To solve for the model variables in this steady state, we …rst solve for the ratio
as a function of model parameters and the in‡ation rate?

an expression for p as a function of=e

only. To this end, we deriveusing the equations (51) and (52). Both equations
47

can be rearranged to obtain, respectively,
(1

(1

)=

+ (1

)(

e 1

)

(p? )1(66)(1)( = ) (g=q) 1 =

+ (1

)(

e 1

(p? ) :

(67))( = ) 1)( = ) (g=q) 1

:

1

1

)( = ))

Dividing the equation (66) by the equation (67) yields
p? =

1

1

1

(1
(1

(68)

We substitute this expression for p? into the equation (67), which yields
1

=

e

e

(
(1

1

)

1

+1
)( = ) (g=q) 1e

We use equation (61) to substitute for (

1
1
1

)

)( = ) 1)( = ) (g=q) 1

(1
(1

:

on the right hand side of the previous

equation and rearrange the result to obtain
( )
e

=

1
1

)( = ) 1)(g=q) 1(1(1

where we have indicated that

1

1

(1

)(g=q)

1

!

;

(69)

( ) depends on the steady-state in‡ation rate

. For

1

) ( = ) (g=q) 1(1

later use, we de…ne the relative price distortion as
e

( )=

(70)

:

( )

Combining the pricing equations (53) to (55) yields
1
=
mc

1
11+

1
p?

1
e

1

)[ ( e )1(11] ( = ) (g=q) 1)[ ( e )1(1]( = )!

:

Using the expression for p? in equation (68) to substitute for p? in the previous equationand the solution for

( )=

e

in equation (69), we thus obtain a solution for 1=mc. Again

for later use, we denote the average markup by
( )=

1
11+

1
1

(1
(1

= 1=mc and thus obtain the solution1)( = ) 1)(g=q) 1

1

1

(1

1

(1

)[ ( e )1)[ ( e )1]( = )1] ( = ) (g=q) 1(71)

Again, we indicate here that ( ) depends on the steady-state in‡ation rate.
Now, we rewrite marginal costs in equation (56) as
1

w
mc =
(
r
and use equation (57) to obtain mc =

1)
k
L

1

r= ( ) 1 1
48

r
1 1=

r
1 1=

or

1

k
L

;

1

;

(72)

!

:

after also using= 1=mc. Analogous steps for the wage rate also imply
1

w= ( ) 1

1

1

k
L

(73)

:

Furthermore, the aggregate technology (58), the aggregate resource constraint (59) andthe household’s optimality conditions (62) to (65) imply the following four equations:
!
1 1
k
L f
y= ( )
L
VL
V (L)

w=c
1

r=

1+d

( e)

y =c+( ewhere we have used ( ) =e

1 + d)k;= ( ). To simplify these four equations further, we use theequations (72) and (73) to substitute out for w and r. Then, we express all the remaining
variables relative to hours worked, which yields the following four equations:
k
L

y
= ( )
L
c
= ( ) 1L

1

1

1+ ( )
1

k
L

1

1

k
1
k
= ( ) 1 1LLyck
= + ( e 1 + d) :
L
L
L

1

f
y

(74)
V (L)
LVL

1

1

(75)
1

1

(76)

1+d

( e)

(77)

We now show that these four equations determine the four variables y; c; L; k, given a
steady-state in‡ation rate

and assuming that the ratio of …xed costs over output, f =y,

is a calibrated parameter.
First, we solve for hours worked as a function ofby substituting the equations (74)to (76) into equation (77). This yieldsf( ) ( ) 1+ ( )
y

1

=

1

e

V (L)
LVL

+

1
( e)

1+d
1+d

!

1

1

or
V (L)
LVL

=

f
( ) ( ) 1+ ( )y= L( );
49

1

e

(

1)

1
( e)

1+d
1+d

!

;

where L( ) abbreviates the right-hand-side term, which is a function of the steady-statein‡ation rate. The previous equation provides an implicit solution for L. We obtain an
explicit solution for L, if we assume a functional form for V (L). Using that V (L) =1

L , with> 1 and> 0 yields
1
V (L)
=
LVL

L
L

and hence1=

1

L( ) =

(78)

;

L( )

+

where we have indicated that in general, steady-state hours worked L depend on thesteady-state in‡ation rate

through L( ). Recall that in order to compute L( ), theequations (69), (70) and (71) are required. The solutions for k; c, and y can be recursively
computed from the equations (74) to (76). These solutions arek( ) = ( )c( ) = ( )11y( ) = c + ( e

1

1

1

1+d

( e)
k
L

1

1

(79)

L

V (L)
VL

(80)
(81)

1 + d)k:Again, we indicate that these solutions depend on the steady-state in‡ation rate.

B

Planner Problem and Its Solution

The planner allocates resources across …rms and time by maximizing expected discounted
household utility subject to …rms’ technologies and feasibility constraints. The planner
problem can be solved in two steps. The …rst step determines the allocation of given
amounts of capital and labor between heterogenous …rms at date t. The second step
determines the allocation of aggregate capital, consumption and labor over time. Endogenous variables in the planner solution are indicated by superscript e.

B.1

Intratemporal Planner Problem

The intratemporal problem corresponds tomaxee

Ljt ;Kjt

Z 1(Yjte )

11

e 1
s:t: Yjte = At Qt sjt Gjt (Kjt
)

dj

1

(Lejt )

1

Ft ;

0

and given Let and Kte , with Let =R1

Lejt dj and Kte =
0

R1
0

e
Kjt
dj. Optimality conditions yielde
Kjt=Lejt = Kte =Let and hence that all …rms maintain the same capital labor ratio. Thus, the50e 1 1=e)(Lejt )1= := (Kjtproblem can be recast in terms of the optimal mix of input factors, Ijt
Z 1
Z 1
1
1
e
e
e
At Qt sjt Gjt Ijt Ft
max
dj
s:t: It =
Ijt
dj;
e
Ijt

0

0

with Ite = (Kte )1 1= (Let )1=

being given. Equating the …rst-order conditions to this prob-

lem for two di¤erent …rms j and k to each other yields the conditione
Zjt Zjt Ijt

1

Ft

e
= Zkt [Zkt (Ikt

Ft )]

1

;

where Zjt = Qt sjt Gjt denotes productivity of the …rm j at date t. Rearranging thisecondition yields Ijt

Ft = (Zjt =Zkt )

1

e
(Ikt

Ft ), and aggregating this equation over all

j’s yieldseIkt

Ft = R 1(Gkt Qt skt =Qt )

1

(Ite

Ft ):
(82)
(G
Q
=Q
)
jt
t
s
t
jt
0
Thus, the optimal input mix of the …rm k net of …xed costs is proportional to the optimal
1 djaggregate input mix net of …xed costs, and the factor of proportionality corresponds tothe (weighed) productivity of the …rm k relative to the (weighed) aggregate productivity
in the economy. Thus, equation (82) shows that the productivity distribution determines
the e¢ cient allocation of the optimal input mix across …rms.
To obtain the aggregate technology in the planner economy, we combine equation (82)with equation (2) and the Dixit-Stiglitz aggregator (1). This yields
0
!#
Z 1"
1
(Q
G
)
t
s
jt
jt
Yte = @At Qt sjt Gjt R 1(I e Ft )1 dj t(QG)
0
t sjt jt
0

1

1

1

djA

:

Simplifying this equation yields the aggregate technology in the planner economy,
1
1
At Qt
(Kte )1 (Let )
Yte =
Ft ;
e

(83)

t

where the e¢ cient productivity adjustment factor is de…ned as
1
Z 1
1
1
e
1= t =
Gjt Qt sjt =Qt
dj

(84)

0

and evolves recursively. To see this, rewrite equation (84) as

Assuming that the initial productivity distribution at t =1 is consistent with theassumed productivity process we have
Z 1
qt
qt sjt +1 1e
1
(1= t )=dj
gt
gt sjt +1
0
)
(
1
1
X
q
q
t
t
s+1
=
1+
(1
)s
gt
gt s+1s=1(
1
1qt
qt q t 1
2=
1 + (1)+ (1)
gt
gt gt 1= (pet )

1

:
51

+ :::

)

The last step follows from backward-iterating equation (23) and implies that the e¢ cient
productivity adjustment factor equals the relative price of …rms hit by a -shock in period
t in the economy with ‡exible prices,1=
It follows also from equation (23) thateet = pt :(85)et evolves recursively as shown in equation (32).

The intratemporal planner allocation then consists of equation (82), which determines
the e¢ cient allocation of the optimal input mix across …rms, and equations (83) and (32),which describe the aggregate consequences of the e¢ cient allocation at the …rm level.

B.2

Intertemporal Planner Problem

The intertemporal allocation maximizes expected discounted household utility subject tothe intertemporal feasibility condition,
max

e g1
fCte ;Let ;Kt+1
t=0

E0

1
X

t

e
e
t U (Ct ; Lt )

(86)

s:t:

t=0

e
Cte + Kt+1
= (1d)Kte +At Qtet

with U (:) denoting the period utility function and(Kte )1

1(Let )

1

(87)

Ft ;

e
t given by equation (32).

The …rst

order conditions to this problem comprise the feasibility constraint and
e
YLt
=

e
ULt
;
e
UCt

1 = Et(88)e
t+1 UCt+1
e
UCt
t

e
YKt+1
+1

d

(89)

;

e
e
denoting by YKt
the marginal product of capital and by YLt
the marginal product of labor.

Thus, the planner allocation for aggregate variables is characterized by the aggregate
technology, equation (83), the e¢ cient adjustment factor, equation (32), the feasibility
condition, equation (87), and the two …rst-order conditions (88) and (89).

C

Proof of Proposition 1

To show that condition (33) holds under ‡exible prices, we divide equation (22) by Pt1
and impose= 0 to …nd out that the optimal relative price p?t of …rms experiencing a-shock in period t is equal to pet . This and the equations (53) to (55) determining the
optimal relative price of …rms experiencing a -shock in t imply with
pet =

1
11+
52

mct
e
t

:

= 0 that

Combining the previous equation with the equation (85) yields
1
11+

1=

(90)

mct ;

which shows that real detrended marginal costs are constant in the economy with ‡exible
prices. From equation (12) it follows that the optimal relative price of the …rm j in the‡exible-price model satis…es
Pjt?
(Gjt Qt sjt =Qt ) =
Pt

1
11+

mct
e
t

:

Combining the previous equation with equation (90), we obtain condition (33) in the maintext. Plugging this condition into equation (25) shows that the ‡exible-price equilibrium
implements

t =

e
t.

Thus, the aggregate production function in equation (24) in the‡exible-price equilibrium is given by
Yt =with Ft = f ( et )1 1= and

e
t

At Qt
e
t

1

(Kt )1

(Lt )

1

(91)

Ft ;

e
t ) , and the resource constraint (derived in= (At Qt =Appendix A.7) is given by
Kt+1 = (1d)Kt + Yt

(92)

Ct :

The two equations (91) and (92) are the same constraints faced by the planner undere¢ cient allocation. Combined with the fact that the household decisions in the ‡exible
price economy are undistorted in the presence of the corrective sales subsidy, it follows thatthe allocation of aggregate consumption, capital, labor, and output in the ‡exible-priceequilibrium is identical to e¢ cient allocation.

D

Proof of Proposition 2

Establishing (1): First, we show that …rms hit by a -shock in period t in the stickyprice economy choose the same optimal relative price as in the ‡exible-price economy.
Let superscript e denote allocations and prices in the ‡exible-price economy, which wehave shown reproduces the e¢ cient allocation. Under ‡exible prices ( = 0) and given
condition 1, the optimal relative price implied by equation (12) for …rms with a -shockin period t is given by

? e
(Pt;t
)
M Cte
=
:
Pte
Pte At Qt> 0) and the e¢ cient allocation, combining this equation withpet =Under sticky prices (equation (13) implies
Nt= 1 + (1
pet

)Et

"

e
Yt+1
e
t;t+1
Yte

t+1
t;t+1

53

qt+1
gt+1

pet+1
pet

Nt+1
pet+1

#

:

(93)

Furthermore, equation (14) implies
Dt = 1 + (1

)Et

"

1

e
Yt+1
e
t;t+1
Yte

t+1

#

(94)

Dt+1 :

t;t+1

Firms hit by a -shock in period t in the sticky-price economy choose the same optimal
relative price as …rms receiving a -shock in period t in the ‡exible-price economy, i.e.,?
=Pt = Nt =Dt = pet or equivalently Nt =pet = Dt , if it holds that
Pt;t

pet+1
pet

qt+1
gt+1

t+1
t;t+1

(95)

= 1;

which follows from comparing the equations (93) and (94). To show that equation (95)holds under the optimal in‡ation rate stated in proposition 2, we lag this equation by one
period and rearrange it to obtaintpet = pet 1

t 1;t

gt
:
qt

Combining this equation with equation (23) implies that the optimal in‡ation rate as
de…ned in equation (34) satis…es equation (95).
Establishing (2): To show that, under the optimal in‡ation rate, …rms that are
subject to a Calvo shock in period t and hence can adjust their price do not …nd it
optimal to change their price, we need to establish thatPt? k;t =?
?
t k;t Pt k;t k ;(96)

for all k > 0. Dividing this equation by the (optimal) aggregate price level Pt? k and using?
=Pt? = pet , we obtain
the result from step (1), i.e., Pt;t

Pt? k;t=Pt? k

Pt? k;t k
Pt? k

?
t k;t

=

?
e
t k;t pt k :Using equation (21), we can rewrite the previous equation as
?
Pt;t
Pt?

qt
gt

?
Again using Pt;t
=Pt? = pet and that

pet
pet k

qt
gt

Pt?
=
Pt? k

qt k+1
gt k+1
t k;t =

qt k+1
gt k+1Qk?
et k;t pt k :j=1

t k+j 1;t k+j further delivers

?
t

?
t+1 k

?
t 1;t

?
t k;t+1 k!

= 1:

Rewriting the previous equation as?
t

qt pet?
e
t 1;t gt pt 1

qt+1 k pet+1 k?
pet k
t k;t+1 k gt+1 k
?
t+1 k

?
t 1

qt 1 pet 1?
et 2;t 1 gt 1 pt 2
54

!

=1

shows that each term in parenthesis is equal to unity under the optimal in‡ation rate,
which follows from equation (95). This establishes that …rms that can adjust their pricemaintain the indexed price as given by equation (96).
Establishing (3): We can establish the fact that the condition 2 causes initial prices
to re‡ect initial relative productivities as follows. The pricing equations (12)-(14) imply
under ‡exible prices and no markup distortion thatPjt?
Pt

Qt sjt Gjt
Qt

=

M Ct
:
Pt At Qt

For a …rm hit by a -shock in period t, this equation yields
pet =

M Ct
:
Pt At Qt

Combining both previous equations yields
Pjt?
=
Pt

Qt
Qt sjt Gjt

pet :Plugging this equation into the aggregate price level, Pt1
1=

Z 1
0

E

1

Qt
Qt sjt Gjt

Rewriting this equation and using pet = 1=

=

R1
0

Pjt1

dj, yields(pet )1 dj:et yields equation (31) for t =1.

Discontinuity of the Optimal In‡ation Rate

This appendix compares the optimal in‡ation rate in an economy with -shocks ( > 0)
to the economy in the absence of such shocks ( = 0). We refer to the …rst economy as
the -economy and to the latter as the 0-economy. Comparing these two economies is
not as straightforward as it might initially appear: even if both economies are subject tothe same fundamental shocks (at ; qt ; gt ; t ), the e¢ cient allocation displays a discontinuity
when considering the limit ! 0. The discontinuity arises because aggregate productivity
growth in the -economy is driven by at qt , while it is driven by at gt in the 0-economy.

To properly deal with this issue, we construct a -economy whose e¢ cient aggregate
allocation (consumption, hours, capital) is identical to the e¢ cient aggregate allocation in
the 0-economy.39 We then compare the optimal in‡ation rates in these two economies andshow that the optimal in‡ation rate for the -economy di¤ers from the optimal in‡ation
rate for the 0-economy, even for the limit
39

! 0.

The two economies do of course di¤er in their underlying …rm-level dynamics.

55

Let at , qt , gt denote the productivity disturbances in the -economy and let A 1 Gj; 1 Q 1 sj; 1
for j 2 [0; 1] denote the initial distribution of …rm productivities. This, together with theprocess f jt g1
t=0 for all j 2 [0; 1], determines the entire state-contingent values for At , Qt ,Gjt , and Qt sjt for all j 2 [0; 1] and all t0.

Next, consider the 0-economy and suppose it starts with the same initial capital stockas the -economy. For the 0-economy, we normalize Q0t sjt

1 for all j 2 [0; 1] and all t

and then set the initial …rm productivity distribution in the 0-economy equal to that inthe -economy by choosing the initial conditions
A0 1 = A 1 ;G0j; 1 = Gj; 1 Q 1 sj; 1 :Finally, let the process for common TFP in the 0-economy be given by
A0t = At

Z 1
0

1

1

1

dj

Qt sjt Gjt

Z 1

G0jt

1

1
1

dj

;

0

where G0jt is generated by an arbitrary process gt0 , e.g., gt0 = gt . In this setting, it is easily
veri…ed that aggregate productivity associated with the e¢ cient allocation, de…ned as
At Qt =et = At Qt

Z 1

1

Gjt Qt sjt =Qt

1

1

dj

;

0

is the same in the -economy and the 0-economy.40 We then have the following result:
Proposition 7 Under the assumptions stated in this section, the e¢ cient allocations in
the two economies, the -economy and the 0-economy, satisfy
Ct = Ct0 ; Lt = L0t ; Kt = Kt0
for all t

0 and all possible realizations of the disturbances.

Proof.

Since At Qt =e;t

= A0t Q0t =e;0for all t, it follows from the planner’s problemt(29)-(30) and the fact that the initial capital stock is identical that both economies sharethe same e¢ cient allocation.
The following proposition shows that (generically) the optimal in‡ation rate discontinuously jumps when moving from the 0-economy to the -economy, even if both economies
are identical in terms of their e¢ cient aggregate dynamics:41
40

The fact that At Qt = et is equal to aggregate productivity in the e¢ cient allocation follows fromequations (30) and (31).
41
Recall that the optimal in‡ation rates implement the e¢ cient aggregate allocations in these
economies.

56

Lemma 2 Under the assumptions stated in this section and provided conditions 1 and 2
hold, the optimal in‡ation rate in the 0-economy is?;0
t = 1 for all t. The optimal in‡ation

rate in the -economy is given by equation (34); in particular, for gt = g and qt = q, andin the absence of price indexation, the optimal rate of in‡ation in the -economy satis…es
limt!1?;t= g=q.

Proof. The results directly follow from proposition 2 and lemma 1.
The previous result illustrates the fragility of the optimality of strict price stability
in standard sticky-price models, once non-trivial …rm-level productivity trends are taken
into account. Moreover, in combination with proposition 7, the result shows that two
economies that can be identical in terms of their aggregate e¢ cient allocations may requiredi¤erent in‡ation rates for implementing these allocations.

F

Proof of Proposition 3

Under the assumptions stated in the proposition, it is straightforward to show that the relative price distortion ( ) and the markup distortion ( ), which are de…ned in equations(69), (70) and (71), are inversely proportional to each other,( ) = 1= ( ):As a result, the solution of L determined in equation (78) in appendix A.9 simpli…es to
1(1 + )

L=1=;

because L( ) = 1 and, therefore, L no longer depends on the steady-state in‡ation rate
. This result implies that L(1) = L(?

), as stated in proposition 3.

In this case, the solutions for capital and consumption, equations (79) and (80), implyk( ) = ( )c( ) = ( )

1
1

1

1

( e

1 + d)

1

1

( e

L;
1 + d)1

V (L)
VL

;

where we explicitly indicate that steady-state capital and consumption depend on.

Comparing steady-state consumption for the policy implementing the optimal in‡ation
rate?

and the alternative policy implementing strict price stability in economies without

price indexation yields
(1)
( ?)

c(1)
=
c( ? )
57:

Equations (69) and (70) imply that the relative price distortion (?

) = 1. This yields

c(1)= (1) ;c( ? )
e

=
=

(1)
1

(1
1

1

)(g=q)
(1
)

1
1

1

(1
(1

) (g=q) 1)(g=q) 1!

;

which is the expression in proposition 3.
To show that c(1)=c(?

)

show that the inequality holds strictly, c(1)=c(of c(1)=c(?

@@(g=q)c(1)c( ? )) with respect to g=q. This yields=c(1)c( ? )?

1, note that c(1)=c(

(1
)
2
(g=q)

1

?

) = 1, if g = q and hence

if 1< 1, and g=q > (1= 1. To) < 1, for g 6= q, we take the derivative
1 (g=q)) (g=q) 1 [1(1

(1

Terms in square brackets are positive, because we have assumed that (1(see equation (6)),?

)(g=q))(g=q)

1

1]

:

<1

). Therefore, the derivative is strictly positive(g=q) > 0 and thus g=q < 1. The derivative is strictly negative if 1(g=q) < 0and thus g=q > 1. The derivative is zero if g=q = 1.

G

Proof of Proposition 4

We start by deriving equation (39) in the proposition. Average employment per …rm Lt
can be written as?

Lt = Lt + (1

c

(97)

)Lt ;

?

where Lt denotes average employment of the …rms that received a -shock in period tcand Lt average employment of the remaining …rms. Equation (2) and equation (24),respectively, imply
Yjt+ Ft = (Kjt =Ljt )1

1

Ljt
At Qt sjt Gjt
1
Yt t+ Ft = (Kt =Lt )1 Lt ;At Qt
where we used the fact that due to there being a unit mass of …rms, we have Lt = Lt .
Taking the ratio of the two previous equations and using the fact that each …rm’s capitallabor ratio is equal to the aggregate capital-labor ratio, we get
!
Ljt
1
Yjt
At Qt
At Qt
+
F
=
t
At Qt sjt Gjt Yt t
Yt t
Lt
1 + Ft AY t Qt
t

t

58

:

1

Using Ft = f ( et )1andet = (At Qt =et)from equation (3), the de…nition of detrended output yt = Yt = et ,from equation (27), the previous equation can be expressed as

Ljt
=
Lt

1+

yt

f
t=

1

Yjt
Yt t

e
t

Qt

+

Qt sjt Gjt

f
t=

yt

:

e
t

Using the product demand function (8) to substitute Yjt =Yt , we get
Ljt
=
Lt

f
1+
yt

1

e
t

e
t

f
yt

t

1

+

t

Qt
Qt sjt Gjt

t

!

Pjt
Pt

:

?
Firms that receive a -shock at date t can charge the optimal price, i.e., Pjt =Pt = Pt;t=Pt =p?t . For these …rms, the previous equation implies
?

Lt
=
Lt

1+

1

e
t

f
yt

e
t

f
yt

t

1

+

t

t

(p?t );

where we used the fact that …rms that receive a -shock are identical, so that on theleft-hand side of the previous equation, we can write average employment of these …rms
in the numerator. Using equation (97) to substitute for L?t =Lt in the previous equation
yields
1+

f
yt

e
t

1

f
yt

e
t

=1

)(

?
?
t = t 1;t )ctEquation (34) implies (

1

e
t)

e
t

)Lt =Lt

(1

(1

=

t

e
t)

[ (

t
1

1

](

e ?
t pt ):. This allows us to rewrite theprevious equation as(e ?
t pt )tet=

Using again (

e
t)

(1

)(

1

= 1 (1

(1

=

(

e
t)

?
?
t = t 1;t )1allows us to rewrite the previous equation@

From equation (51) we obtain
1

1

h c

0

t = t 1;t )

)(

1

(1

1

)

1

+ (1

e
t)

=

1

A:

1

e ? 1
t pt )

) (

as
(p?t

i1

c

f et
Lt
Lt
+
yt t
Lt
Lt
?
?
)( t = t 1;t )

1

(1

)(

t = t 1;t )

1

1

(1

)

?
?
t = t 1;t

1

!

(98)

:

1

(99)

:

Equating the right-hand sides of equation (98) and equation (99) delivers equation (39)in the proposition for the special case with f = 0.
We next derive equation (40) in the proposition. From equation (52) we have
t =

(

e
t)

1

+ (1

)

e
?
t (pt

e
t)

59

+ (1

)

qt
gt

t
t 1;t

t 1:

Equation (34) implies

(

e
t)

1

=1

Using

h

1

(1

?
t
?
t 1;t

=

t
=
e
t

h

?
?
t = t 1;t

)
e
t

gt
e
t 1

qt

1

i

(p?t

i

1

Using equation (99) to substitute (p?t

e
t)

(1

?
?
t = t 1;t )1. Substituting this into the

e
t)

+ (1

)

e
t 1
e
t

qt
gt

t

t 1
:
e
t 1

t 1;t

from equation (35) delivers?
?
t = t 1;t1)(et delivers

previous equation and dividing by
t
=
e
t

(1

)

(p?t

e
t)

+ (1

)

(

t = t 1;t )?
?
t = t 1;t

!

t 1
:
e
t 1

in the previous equation delivers equation (40)in the proposition.

H

Robustness of Results to Positive Fixed Costs

From the proof of proposition 4 in appendix G, which covers the general case with nonnegative …xed costs f

0, it follows that equation (40) continues to hold for fequations (98) and (99) it follows that equation (39) generalizes to
h c
c
! 1 0
f et
Lt
Lt
1
1
+
1
(1
)
yt t
1
(1
) ( t = t 1;t )
Lt
Lt
t
=@
e
?
?
1
1 (1
)( t = t 1;t )t
1(1) ?t = ?t 1;t0. From

1
1

i1

A:

(100)

Using equations (40) and (100), we then evaluate the sensitivity of the optimal in‡ation
estimate in steady state (yt = y) for di¤erent …xed cost, using the baseline parameterscfrom table 1. We thereby set Lt =Lt = 1:0703, which is the sample mean of this ratio in
the data andt = t 1;t = 1:031, which is equal to the sample mean of GDP de‡ator overthe considered sample period, i.e., we assume no price indexation ( t 1;tstate value ofet=t then follows from (100).

1). The steady

We consider …xed costs in a range up to

10% of total (detrended) output, f =y 2 [0; 0:1], where f =y = 0 is the case considered in
the main text. Figure 6 shows that the estimated optimal in‡ation rate is quite insensitive

to assuming alternative …xed costs values: over the considered range of …xed costs, theoptimal in‡ation rate increases, but the maximal e¤ect on the optimal in‡ation rate issmall and around 0.1%. This continues to be true for reasonably sized output ‡uctuations(yt ? y).

I

Proof to Proposition 5

We start by deriving the optimal in‡ation rate (42) and the recursive equation (43). In
the absence of price rigidities, …rms choose at all times their price such that their relative
60

2

1.8

1.6

Pe r ce n ta g e Po in ts

1.4

1.2

1

0.8

0.6

0.4

0.2

0
0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

F ix e d C o s ts f/y

Figure 6: Robustness of optimal in‡ation estimates towards positive …xed costs.

price is inversely proportional to their relative productivity. This follows from the equation(33), which determines the optimal relative price in the absence of price rigidities and isreproduced here for convenience:
1
Qt
Pjt
:
= e
Pt
t Gjt Qt sjt(101)

Condition 2 implies that the previous equation holds also for t =1.

We now show that the optimal relative price (101) can also be achieved by …rm j
in an economy with price setting frictions and non-constant -shock intensities under
the optimal in‡ation rate stated in the proposition. This is so because absent -shocks,
the optimal in‡ation rate insures that the …rm’s nominal price either remains constant(when there is no price indexation) or evolves over time in line with the price indexation
rule, while equation (101) continues to hold. Taking growth rates of equation (101) and
imposing Pjt =t 1;t Pj;t 1 , which holds in the absence of
?
t
?
t 1;t

=

e
t
e
t 1

-shocks, delivers42

gt
:
qt

The previous equation implies equation (42).
To derive equation (43), we can rewrite the de…nition of42et in equation (31) according

In the presence of -shocks, prices are ‡exible so that equation (101) can easily be achieved.

61

to43
(

e 1
t)

=

Z 1
0

=

0 + 0 (1=0 + 0 (1+ (1qt sjt +1 1
dj
gt sjt +1
1
X
qt
(1
)s 1
1)
gt
s=1

qt
gt

1

qt s+1
gt s+111 ) (qt =gt )() (qt =gt )1

0 (1

1)

1
X

(1

)s 1

1
e
t 1)

0.

s=1

where the term in parenthesis is equal to (

qt 1
gt 1

qt s
gt s

1

)

;

This delivers equation (43) inthe proposition.
In the absence of economic disturbances, equation (43) implies that
1

e

1
1

1

=

1

1+

( 1
(1

)(g=q))(g=q)

1et converges to

1
1

:

1

The steady state result in the proposition then follows from equation (42) and the assumption of no price indexation

J

?
t 1;t

1 .

Proof to Proposition 6

For simplicity, we shall refer to PtN , which contains only products of age N or higher, asNNNt = Pt =Pt 1 as the measured in‡ation rate. As before,

the measured price level and to

we let Pt denote the ideal price level (using all products) andt the ideal in‡ation rate.

The proof proceeds in two steps. In a …rst step, we derive the measured in‡ation rate?
t from proposition 2 for the ideal?
for thein‡ation rate. In a second step, we show that if monetary policy implements NtN?
in a setting where monetary policy implementstmeasured in‡ation rate, then this policy implements the same relative product prices asin the case where monetary policy implements?
t for the ideal rate.

Step 1: In analogy to equation (20), which de…nes the ideal price level, the measured
price level is de…ned as(PtN )1

=

1
X

(1

)s

t (s + N );

(102)

s=0

where the weighted average cohort pricet ( ) is de…ned in equation (19). From proposition
2 it follows that under the optimal in‡ation rate ?t , …rms with a Calvo shock do not …nd

it optimal to adjust their price, so that we have for s
Pt? s;t k =

k

0

?
?
t s;t k Pt s;t s :43

The following derivations assume that the initial productivity distribution at t =with the assumed productivity process.

62

1 is consistent

Using this result to rewrite equation (19) shows that the weighted average cohort price?
t isunder the optimal in‡ation rate?
?
1t (s) = ( t s;t Pt s;t s )

(103)

:

The previous equation implies?
t (N +s);tt (s + N ) =?
t (N +s);t N= ( ?t N;t )1!1

t N (s)t N (s):Substituting this into equation (102) yields(PtN ? )1= ( ?t N;t )1

"

1
X

)s

(1

#

t N (s)s=0;where the expression in brackets is the ideal price level de…ned in equation (20) shifted
N periods into the past. For a policy that implements the optimal in‡ation rate fromproposition 2 for the ideal in‡ation measure, we thus have
PtN ? =?
?
t N;t Pt N :(104)

From the previous equation we get that measured in‡ation is then given by
N?
=
t

?
t 1;t
?
t N 1;t N?
t N;

which is the in‡ation rate stated in the proposition.
Step 2: Using equation (103) to rearrange equation (102) delivers(PtN ? )1=

1
X)s ( ?t (s+N );t Pt? (s+N );t (s+N ) )1

(1

s=0

= ( ?t N;t Pt? N;t N )1
Dividing the previous equation by (PtN ? )1N?
?
t = t 1;t =

1

+ (1

)( ?t 1;t PtN ?1 )1 :and using equation (104) one obtains
Pt? N;t N =Pt? N
1

1

! 11

:

(105)

The previous equation shows how the relative price of …rms with a -shock (Pt? N;t N =Pt? N )N?
t .

is determined so as to be consistent with
?
t 1;t
?
t N

1;t N

?
t N,

When monetary policy targets

N?
t

=

as assumed, then equation (105) coincides with equation (34) shiftede?
?
t = Pt;t =Pt , this shows that monetary policy implements the same relative prices as a policy that implements ?t fromback by N periods. Since equation (33) implies 1=

proposition 2 for the ideal in‡ation measure.
63