2021 MEASURING TFP: THE ROLE OF PROFITS, ADJUSTMENT COSTS, AND CAPACITY UTILIZATION
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MEASURING TFP: THE ROLE OF
PROFITS, ADJUSTMENT COSTS,
2021
AND CAPACITY UTILIZATION
Documentos de Trabajo
N.º 2143
Diego Comin, Javier Quintana, Tom Schmitz
and Antonella TrigariMEASURING TFP: THE ROLE OF PROFITS, ADJUSTMENT COSTS, AND CAPACITY UTILIZATION
MEASURING TFP: THE ROLE OF PROFITS, ADJUSTMENT COSTS, AND CAPACITY UTILIZATION (*) Diego Comin DARTMOUTH AND CEPR Javier Quintana BANCO DE ESPAÑA Tom Schmitz BOCCONI UNIVERSITY Antonella Trigari BOCCONI UNIVERSITY (*) Lorenzo Arcà, Gabriele Romano, Saverio Spinella and Sviatoslav Tiupin provided outstanding research assistance. We are grateful to Klaas de Vries, Robert Inklaar and Robert Stehrer for their help with EU KLEMS, and to Kimberly Bayard, Aaron Flaaen, Norman Morin and Justin Pierce for their help with US capacity utilization data. We also thank John Earle, Simon Goerlach, Basile Grassi, Christoph Hedtrich, Robert Inklaar, Pete Klenow, Kenneth Judd, Thomas Le Barbanchon, Nicola Pavoni, Pau Roldan-Blanco, Luca Sala, Fabiano Schivardi and seminar participants at Bocconi, Groningen, Pavia, Konstanz, ZEW Mannheim and the 2021 European and North American Summer Meetings of the Econometric Society for useful comments. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 72073. Documentos de Trabajo. N.º 2143 December 2021
The Working Paper Series seeks to disseminate original research in economics and finance. All papers have been anonymously refereed. By publishing these papers, the Banco de España aims to contribute to economic analysis and, in particular, to knowledge of the Spanish economy and its international environment. The opinions and analyses in the Working Paper Series are the responsibility of the authors and, therefore, do not necessarily coincide with those of the Banco de España or the Eurosystem. The Banco de España disseminates its main reports and most of its publications via the Internet at the following website: http://www.bde.es. Reproduction for educational and non-commercial purposes is permitted provided that the source is acknowledged. © BANCO DE ESPAÑA, Madrid, 2021 ISSN: 1579-8666 (on line)
Abstract We develop a new method for estimating industry-level and aggregate total factor productivity (TFP) growth. Our method accounts for profits and adjustment costs, and uses firm surveys to proxy for changes in factor utilization. Using it to compute TFP growth rates in the United States and in five European countries since the early 1990s, we obtain results that substantially differ from the ones obtained with standard methods (i.e., Solow growth accounting and the utilization-adjusted method of Basu, Fernald, and Kimball, 2006). In every European country, our TFP series is less volatile and less cyclical than the standard ones, with striking differences during the Great Recession and Eurozone crisis. In the United States, our method indicates higher TFP growth overall and a more gradual productivity slowdown. Keywords: productivity, business cycle, capacity utilization, profit margins, adjustment costs. JEL classification: E01, E30, O30, O40.
Resumen Desarrollamos un nuevo método de estimación del crecimiento de la productividad total de los factores (PTF) a nivel tanto sectorial como agregado. Nuestro método tiene en cuenta tanto beneficios como costes de ajuste y usa encuestas empresariales para aproximar cambios en la intensidad de la utilización de factores. Usándolo para estimar tasas de crecimiento de la TFP para Estados Unidos y cinco economías europeas desde principios de la década de los noventa, obtenemos resultados sustancialmente distintos de los producidos por metodologías estándar [es decir, la contabilidad de crecimiento de Solow y el método con ajustes de utilización de Basu, Fernald y Kimball (2006)]. Para cada economía europea, nuestras series de PTF son menos volátiles y menos cíclicas que las series estándar, con notables diferencias durante la Gran Recesión y la crisis de la zona del euro. En los Estados Unidos, nuestro método señala, en general, un mayor crecimiento de la PTF y una ralentización más gradual de la productividad. Palabras clave: productividad, ciclo económico, utilización de factores, márgenes empresariales, costes de ajustes. Códigos JEL: E01, E30, O30, O40.
1 Introduction
According to Robert Solow’s famous definition, Total Factor Productivity (TFP) growth
is the part of output growth that cannot be explained by growth in inputs (Solow, 1957).
It therefore measures how efficiently a firm, an industry or an entire country use their
resources. Over the last 65 years, TFP growth has been one of the most important statistics
in macroeconomics, playing a key role for the analysis of short and long-run phenomena.
In his seminal paper, Solow did not only introduce TFP growth as a concept, but also
proposed a simple method to measure it. He noted that under perfect competition, the
elasticity of output with respect to a given input must be equal to the sales share of that
input (i.e., to the ratio of input spending to sales). Therefore, TFP growth can be computed
as the difference between output growth and a sales-share-weighted average of input
growth rates. Such “Solow residuals” are still the most common measure of TFP growth.
They have allowed researchers to repeatedly confirm Solow’s main finding, namely that
TFP growth - most often attributed to technological progress - is the main driver of long-run
economic growth (Jones, 2016).
However, Solow residuals from standard datasets (e.g., the BLS multifactor productivity
database in the United States or EU KLEMS in Europe) are problematic for short-run
analysis. The main problem is due to changes in capacity utilization, that is, changes in the
intensity with which firms use their inputs. For instance, in a recession, workers typically
perform less tasks per hour of work. As this fall in labour input is not recorded in standard
datasets, their Solow residuals spuriously decrease during recessions. The state-of-the-art
approach to dealing with this issue is due to a series of influential papers by Basu, Fernald
and Kimball (Basu and Fernald, 2001; Basu, Fernald and Kimball, 2006). Basu, Fernald and
Kimball (henceforth, BFK) show that under some assumptions, fluctuations in hours per
worker are one-to-one related to fluctuations in capacity utilization, and can therefore be
used to proxy the latter. This method underlies the widely used series for capacity-adjusted
quarterly TFP growth in the United States introduced by Fernald (2014a). It effectively
decomposes the Solow residual into a first part capturing changes in utilization, and a
second part capturing “true” TFP growth.
The Solow and BFK methods have greatly enhanced our understanding of TFP dynamics
and set standards in the literature. However, they also rely on strong assumptions. Our
paper points out some limitations of these assumptions and proposes alternative ways to
address the underlying measurement issues. In particular, we revisit the measurement
of capacity utilization and the related question of factor adjustment costs, two important
business cycle issues. We also relax the zero-profit assumption of the standard methods,
which conflicts with the rising empirical evidence for positive profits.
1
Following the tradition of the growth accounting literature, our approach is founded on
a simple dynamic model in which firms minimize costs and take input prices as given. This
framework shows the potential limitations of the BFK proxy method. Indeed, both shocks to
the relative cost of hours per worker and changes in the composition of the labour force blur
the relationship
7
BANCO DE ESPAÑA between hours per worker and unobserved utilization. These limitations
DOCUMENTO DE TRABAJO N.º 2143
are empirically relevant, especially in Europe. Therefore, we propose an alternative proxy:
capacity utilization rates from firm surveys. Utilization surveys - a common business cycleFollowing the tradition of the growth accounting literature, our approach is founded on
a simple dynamic model in which firms minimize costs and take input prices as given. This
framework shows the potential limitations of the BFK proxy method. Indeed, both shocks to
the relative cost of hours per worker and changes in the composition of the labour force blur
the relationship between hours per worker and unobserved utilization. These limitations
are empirically relevant, especially in Europe. Therefore, we propose an alternative proxy:
capacity utilization rates from firm surveys. Utilization surveys - a common business cycle
indicator
which in many
conflicts withcountries
the rising - ask firms toevidence
empirical report the forratio between
positive profits.actual and full capacity
using
output. cost
Following shares
In ourthe and
model, including
this measure
tradition adjustment
of the growth is unaffected costs)
accounting on changes
by composition
literature, our in the
effects capacity
approach and relative utilization
is founded factor
on
survey. 3 The residual from this regression is our measure of industry-level TFP growth. This
prices, 1
a simpleand proportional
dynamic model to changes
in which in actual
firms minimize unobserved
costs andutilization.
take input prices as given. This
approach
Our focus
framework is similar
shows to
on capacity BFK,
the potential who regress
utilization
limitations the
leadsofus standard
the toBFK
alsoproxySolow
consider residual
the closely
method. on changes
Indeed, related in
both shocks hours
issue of
to
per
the worker.
adjustment
relative costs.However,
cost ofAdjustment our dependent
hours per worker costs are and variable
anchanges
important accounts
in the for
conceptual profits
composition and
explanation adjustment
for fluctuations
of the labour costs,
force blur
and
in we
thecapacity use
relationship a different
utilization. 2
betweenTheyutilization
hours alsoperproxy.
matter
worker forandTFPunobserved
growth, as utilization.
they create aThese wedge between
limitations
are Finally,
effectivewe
the empirically andshow thehow
relevant, measuredto use these
especially growth results
in Europe.rate of to compute
capital and
Therefore, aggregate
we labour inputs.
propose TFP growth, which
Nevertheless,
an alternative proxy: is
probably
Solow
capacity and the
BFK most
utilization relevant
assumerates from
frommacroeconomic
thefirmoutset that statistic.
surveys. adjustment
Utilization In thecostspresence
surveys of non-zero
are- negligible.
a common In profits,
contrast,
business we
we
cycle
can no longer
estimate
indicator the
in many relycountries
parameterson standard - askaggregation
of adjustment firms to costs results.
reportfunctions Instead,
the ratio forbetween we use
capital and the
actual recent
labour
and by insights
full using
capacity of
our
Baqaee and
model’s
output. Euler
In ourFarhi (2019)
equations,
model, tomeasure
consistently
thisfollowing aismethod aggregate
unaffected byindustry-level
introduced by Hall (2004).
composition TFP growth
effects rates. factor
and relative
We and
We
prices, implement
also engageour
proportional withmethod
the by estimating
recent
to changes debate
in actual aboutindustry-level and aggregate
the role ofutilization.
unobserved profits 1
(Gutierrez TFPand growth rates
Philippon,
for the
2017; Our United
Basu,
focus 2019;States
on (between
Karabarbounis
capacity 1989
utilization andand Neiman,
leads 2018) and
us to2019;also the five largest
Barkai,
consider 2020; European
De
the closely Loecker,
related economies
Eeckhout
issue of
(between the
and Unger, costs.
adjustment early 1990s
2020).Adjustment and
The Solowcosts 2015).
and BFKare anDoing
methods so,
important we
both obtain TFP
assume that
conceptual series that
profits are
explanation are substantially
forzero. In light
fluctuations
different
of the recent
in capacity from the ones
evidence,
utilization. obtained
2 They
we doalso by standard
notmatter
want to methods.
forimpose
TFP growth, These
this assumption differences
as they create are
a priori. mainly
a wedge Instead, driven
between we
by our
show
the treatment
that
effective if firms
and the of profits
make and growth
positive
measured ournegative)
(or new rate utilization
capitalproxy,
ofprofits, factor while adjustment
elasticities
and labour inputs. costs
are Nevertheless,
equal and
to cost
aggregation
shares
Solow rather
and BFK choices
than
assume play
sales a more
shares.
from the To modest
convert
outset role.
that sales to costcosts
adjustment shares,arewe estimateInindustry-level
negligible. contrast, we
In
estimate Europe,
profits. This our
the requires
parameters most striking
us toofcompute
adjustmentfinding
a rental is that
costsrate TFP was
of capital,
functions essentially
following
for capital flat
andthe during
seminal
labour by the Great
approach
using our
Recession
of Hall and
model’s and
Euler Euro
Jorgenson crisis,
equations, while
(1967).
following the Solow
In most
a method and
countries BFK methods
and industries,
introduced suggest
by Hall (2004). a substantial
we find positive profits. decrease.
ThisWe
Thus, result
wealso is partly
obtain
engage higher due
with torecent
output
the profits.
elasticities
debatePositive
for labour
about profits
the roleandlower our (Gutierrez
materials
of profits estimate
than standard for
andthe output
methods,
Philippon,
elasticity
as costofshare
the Basu,
2017; capital,
2019; and capital
ofKarabarbounis
these inputs fell andless
exceeds Neiman,than
their other inputs
sales share.
2019; Barkai,At during
the same
2020; theLoecker,
De crisis.
time, we Thus,obtain
Eeckhout our
method
lower
and attributes
output
Unger, more
elasticities
2020). The Solowofforthecapital.
fall in
and BFK output
This istoimportant
methods a fall
bothinassume
inputs and
that less
for productivity profitsto measurement,
TFP. This effect
are zero. In light is
as
particularly
capital
of behaves
the recent strong
evidence,in Southern
differently wefrom do not Europe,
other
want where
inputs both
to impose profits theare
inthis highand
short
assumption and inathe
the crisis was severe.
long Instead,
priori. run. we
Our
show new utilization
Combining
that if firms themakeproxy
new positivealso plays
elements(or a crucial
discussed
negative) role.
soprofits,In many
far, we factor countries,
obtainelasticities BFK-style
industry-level utilization
TFP growth
are equal to cost
adjustment
by running
shares rather regressions
anthaninstrumental have
sales shares. a weak
variable first stage
regression
To convert and
sales to an
of cost insignificant
a modified
shares, Solow second
we estimate stage,
residual while
(computed
industry-level our
survey
using measure delivers much stronger results. Accordingly, in allinfive countries, the survey
1 Thecost
profits. Thisshares
requires andusincluding
to compute adjustment
a rental costs)
rate on changes
of capital, following
fact that our proxy is unaffected by shocks to relative factor prices is not only an advantage over
thethe capacity
seminalutilization
approach
proxy
survey. delivers
3and a TFP series
TheJorgenson
residual thatregression
is
Inless volatile and less industries,
cyclical thanwe the one obtained with
of Hall
hours per worker, but alsofrom (1967).
over this
other most
proxies is
haveourbeen
thatcountries measure
and
suggested of in
industry-level
the literature find TFP growth.
positive
(e.g., This
profits.
electricity use).
the hours
2 For
approach
Thus, per
instance,
we obtain worker
BFK
is similar write
higher proxy.
that
to BFK,output For instance,
“internal
who adjustment
regress the
elasticities the standard
costs
forstandardare
labour and deviation
required
Solow to model of
residual
materials our
why
than series
industries
onstandard for
vary aggregate
utilization
changesmethods,
in hours
in response to idiosyncratic changes in technology or demand” (Basu et al., 2006, P. 1422).
TFP
perthe
as growth
worker. in
cost share Eurozone
However,of these countries
our inputs
dependent is only
exceeds half
variable as large
their accounts
sales share.as the one
for profits of
At the same the BFK measure,
and adjustment
time, we obtain and
costs,
its
andcorrelation
lower weoutput with realutilization
use a different
elasticities value added
for capital. growth
proxy.This is is 0.14 (against
important 0.52 for themeasurement,
for productivity BFK measure).as
In
capital the
Finally, United
behaves we show States,
how we
differently to findother
use
from that inputs
these aggregate
results2both TFP
to compute
in the increased
aggregate
short andonin average
TFP byrun.
the growth,
long 1.02%
whichper is
year
probablybetween
Combiningthe most 1989 and
therelevant 2018,
new elements around
macroeconomic 0.05
discussed percentage
statistic.
so far, In wethe points more
presence
obtain than suggested
of non-zeroTFP
industry-level by
profits,
growth the
we
BFK
by and
canrunning Solow
no longer an relymethods.
on standard
instrumental As in Europe,
aggregation
variable profits
regression play
results. an important
of aInstead,
modified role:
weSolow
use the positive profits
recent(computed
residual lower
insights of
our
Baqaee estimate
and Farhi (2019) to consistently aggregate industry-level TFP growth rates. other
for the output elasticity of capital, and as capital has grown faster than
1 The fact that our proxy is unaffected by shocks to relative factor prices is not only an advantage over
inputs over the period, we attribute less of output growth to capital and TFPmoregrowthto TFP. We
hoursWe perimplement
worker, but also our over
method other by estimating
proxies that haveindustry-level
been suggested and in theaggregate
literature (e.g., electricityrates
use).
also note
2 For a particularly
instance, BFK write strong
that upward
“internal adjustment
adjustment costs are
for the United States (between 1989 and 2018) and the five largest European economies of TFP
required growth
to model between
why 2005
industries vary and 2009,
utilization
in response to idiosyncratic changes in technology or demand” (Basu et al., 2006, P. 1422).
(between
3 We use the earlyoil,
monetary, 1990s andand
financial 2015). Doingshocks
uncertainty so, we
as obtain TFPfor
instruments series thatutilization.
capacity are substantially
different from the ones obtained by standard methods. These differences are mainly driven
by our8 treatment of profits and our new utilization
BANCO DE ESPAÑA DOCUMENTO DE TRABAJO N.º 2143
2 proxy, while adjustment costs and
aggregation choices play a more modest role.3
In Europe, our most striking finding is that TFP was essentially flat during the GreatFinally, we show how to use these results to compute aggregate TFP growth, which is
probably the most relevant macroeconomic statistic. In the presence of non-zero profits, we
can no longer rely on standard aggregation results. Instead, we use the recent insights of
Baqaee and Farhi (2019) to consistently aggregate industry-level TFP growth rates.
We implement our method by estimating industry-level and aggregate TFP growth rates
for the United States (between 1989 and 2018) and the five largest European economies
(between the early 1990s and 2015). Doing so, we obtain TFP series that are substantially
different from the ones obtained by standard methods. These differences are mainly driven
by our treatment of profits and our new utilization proxy, while adjustment costs and
aggregation choices play a more modest role.
In Europe, our most striking finding is that TFP was essentially flat during the Great
Recession and Euro crisis, while the Solow and BFK methods suggest a substantial decrease.
driven
This both is
result bypartly
our treatment of profitsPositive
due to profits. and by our utilization
profits lower ourproxy. Thus, while
estimate for thetheoutput
Solow
and BFK methods
elasticity of capital,suggest an abrupt
and capital slowdown
fell less than other around the during
inputs year 2005 (Fernald,
the crisis. 2014b;
Thus, our
Gordon, 2016), we find that TFP growth was still 0.7% per year between
method attributes more of the fall in output to a fall in inputs and less to TFP. This effect is 2005 and 2009,
before dropping
particularly to 0.3%
strong betweenEurope,
in Southern 2009 and 2018.
where This suggests
profits are highthatandthere mightwas
the crisis have been
severe.
a further
Our drop in productivity
new utilization proxy alsogrowth
plays a after therole.
crucial Great In Recession.
many countries, BFK-style utilization
adjustment regressions have a weak first stage and an insignificant second stage, while our
Relatedmeasure
survey literature
deliversFollowing Solow (1957),
much stronger results. many researchers
Accordingly, in allhave assembledthe
five countries, extensive
survey
industry-level
proxy delivers agrowth accounting
TFP series datasets.
that is less volatileLeading
and lessexamples for the
cyclical than thisoneapproach
obtainedarewith
EU
KLEMS
the hours(O’Mahony
per worker and Timmer,
proxy. 2009) orthe
For instance, thestandard
BLS multifactor
deviation productivity
of our series database. These
for aggregate
high-quality
TFP growth in datasets
Eurozone are countries
the basis isforonly
our half
empirical
as large work.
as theHowever,
one of thetheir
BFK Solow residuals
measure, and
do not consider profits, adjustment costs, or changes in utilization. 4
its correlation with real value added growth is 0.14 (against 0.52 for the BFK measure).
There
In the is a large
United literature
States, on each
we find that of these aspects.
aggregate The needon
TFP increased to average
adjust TFP by growth
1.02% per for
changes in capacity utilization has long been recognized. 5 Costello (1993) and Burnside,
year between 1989 and 2018, around 0.05 percentage points more than suggested by the
Eichenbaum
BFK and Solow and Rebelo (1995)
methods. propose
As in Europe, electricity
profits play anconsumption
important role: (in positive
the latter case,lower
profits joint
with hours per
our estimate forworker)
the output as elasticity
a proxy for capital services,
of capital, and as capitalwhilehas Field
grown(2012) relies
faster thanon the
other
unemployment
inputs rate. Imbs
over the period, (1999) less
we attribute develops an alternative
of output growth to capitalmodel-based
and more methodology.
to TFP. We
Currently, the BFK method is the leading approach on this
also note a particularly strong upward adjustment of TFP growth between 2005 and issue. Its application has2009,
been
largely
driven limited
both to US
by our data, with
treatment only two
of profits andexceptions that we proxy.
by our utilization are awareThus, of.while
Inklaarthe(2007)
Solow
3 We use monetary, oil, financial and uncertainty shocks as instruments for capacity utilization.
uses
and BFKthe BFK method
methods for European
suggest an abruptcountries
slowdown and finds that
around the resulting
the year TFP measures
2005 (Fernald, 2014b;
remain strongly
Gordon, 2016), procyclical.
we find thatHe TFPconcludes
growth was that still
hours per per
0.7% worker
yearmay not be2005
between an appropriate
and 2009,
utilization proxytoin0.3%
Europe, but 2009
does and
not propose ansuggests
alternative. 6 More recently, Huo,
before dropping between 2018.
3 This that there might have been
Levchenko
a further drop andinPandalai-Nayar
productivity growth(2020)after
use the
theBFK
Great method to calculate utilization-adjusted
Recession.
TFP series for a large panel of countries. Their baseline estimates impose that the relation
between hours per worker
Related literature and Solow
Following utilization is the
(1957), same
many in all countries.
researchers Our results
have assembled instead
extensive
suggest heterogeneity
industry-level across countries
growth accounting and Leading
datasets. problemsexampleswith thefor hours
this per workerare
approach proxy
EU
in Europe. In general, our main contribution to this literature
KLEMS (O’Mahony and Timmer, 2009) or the BLS multifactor productivity database. These is the use of capacity
utilization
high-quality surveys as aare
datasets new theproxy.
basis We
for show that this proxy
our empirical work. does not require
However, assumptions
their Solow residualson
relative factor prices,
do not consider profits,is adjustment
robust to changes
costs, orin changes
employment composition
in utilization. 4 and labour market
There
4 TFP is a largeobviously
measurement literature onmany
faces eachother
of these aspects.
challenges that The
we doneed to adjust
not consider TFP
here. For growth
instance, for
we
ignore
changes measurement
in capacity issues relating tohas
utilization quality
long improvements
been recognized. 5
and new products
Costello(Boskin,
(1993) Dulberger, Gordon,
and Burnside,
Griliches and Jorgenson, 1996; Aghion, Bergeaud, Boppart, Klenow and Li, 2017). We also do not attempt to
Eichenbaum
measure intangible andcapital
Rebelo (1995)
(Corrado, propose
Haskel, electricity
Jona-Lasinio and consumption (in the
Iommi, 2012; Crouzet andlatter case,
Eberly, joint
2021).
5 Solow himself was aware of the issue, and proposed a correction dealing specifically with capital
with hours per worker) as a proxy for capital services, while Field (2012) relies on the
utilization: “Lacking any reliable year-by-year measure of the utilization of capital I have simply reduced [the
unemployment
capital stock] by therate.
BANCO DE ESPAÑA
fractionImbs
9 DOCUMENTO DE TRABAJO N.º 2143
of the(1999) develops
labor force unemployedaninalternative
each year [..].model-based methodology.
This is undoubtedly wrong, but
probably getsthe
Currently, closer
BFK to the truth than
method making
is the no correction
leading approachat all”
on(Solow, 1957, Its
this issue. P. 314).
application has been
6 Planas, Roeger and Rossi (2013) propose a statistical filtering method to extract trend TFP growth for
largely
Europeanlimited
countriesto(alsoUS relying
data, with only two
on capacity exceptions
utilization surveys).that weapproach
Their are aware of.from
differs Inklaar
BFK and(2007)
fromKLEMS (O’Mahony and Timmer, 2009) or the BLS multifactor productivity database. These
high-quality datasets are the basis for our empirical work. However, their Solow residuals
do not consider profits, adjustment costs, or changes in utilization.4
There is a large literature on each of these aspects. The need to adjust TFP growth for
changes in capacity utilization has long been recognized.5 Costello (1993) and Burnside,
driven both by our treatment of profits and by our utilization proxy. Thus, while the Solow
Eichenbaum and Rebelo (1995) propose electricity consumption (in the latter case, joint
and BFK methods suggest an abrupt slowdown around the year 2005 (Fernald, 2014b;
with hours per worker) as a proxy for capital services, while Field (2012) relies on the
Gordon, 2016), we find that TFP growth was still 0.7% per year between 2005 and 2009,
unemployment
institutions, andto rate.
is 0.3% Imbs (1999)
empirically relevant develops
in all an alternative
countries considered. model-based
7 methodology.
before dropping between 2009 and 2018. This suggests that there might have been
Currently,
Adjustmentthe BFK method is the leading approach oninthistheissue. Its application
literaturehas been
a further drop incosts have also
productivity received
growth some
after the attention
Great Recession. productivity (Berndt
largely
and Fuss, limited
1986;toBrynjolfsson,
US data, with onlyand
Rock twoSyverson,
exceptions that we
2018). Forare aware Basu,
instance, of. Inklaar
Fernald (2007)
and
uses the
Shapiro (2001)BFK method for
have Following European
computedSolow countries
a TFP (1957),
series formany and finds
the United that the
Stateshave resulting
thatassembled TFP
accounts for measures
capital
Related literature researchers extensive
remain
adjustment strongly
costs. procyclical. theyHe
Whileaccounting concludes
calibrate that hours
a capital per worker
adjustment function mayusing
not be an appropriate
external evidence
industry-level growth datasets. Leading examples for this approach are EU
utilization proxy in Europe, but does not propose an alternative. 6 More recently, Huo,
and assume that there are no adjustment costs for
KLEMS (O’Mahony and Timmer, 2009) or the BLS multifactor productivity database. These labour, we estimate adjustment costs by
Levchenko
using our model’sand Pandalai-Nayar
Euler (2020) use several the BFKrecent method to calculate utilization-adjusted
high-quality datasets areequations.
the basis Finally,
for our empirical work.papersHowever,have explored
their Solow the residuals
effects of
TFP series
positive for
profits ona large panel
TFP measurement of countries. Their
(Karabarbounis baseline estimates impose Meier andrelation
that the
do not consider profits, adjustment costs, or changesand Neiman, 2019;
in utilization. 4 Reinelt,
between
2020; hoursand
Crouzet per worker2021; and utilization is the same in allthe countries. Our results instead
There is a largeEberly,
literature on Piton, each of2021). these We examine
aspects. The need implications
to adjust TFP of profits
growthforfora
suggest
broad setheterogeneity across countries and to theproblems with the hoursour perpaper
worker proxy
changes inofcapacity
countries. More
utilization importantly,
has long been best of our
recognized. 5knowledge,
Costello (1993) and is the first
Burnside,
in Europe.
to jointly In general,
account for profits, our adjustment
main contribution andto this literature is the use ofaggregate capacity
Eichenbaum and Rebelo (1995) proposecosts electricity utilization,
consumption and to (inconsistently
the latter case, joint
utilization
the resulting surveys as a new proxy.
industry-level We show that this proxy does not require assumptions on
with hours per worker) as TFP a proxy series.for capital services, while Field (2012) relies on the
relative factor
The remainder prices, is
of this robust
paper to changes
is structured in as employment
follows. composition
Section 2 lays out andthelabour
dynamic market
cost
unemployment rate. Imbs (1999) develops an alternative model-based methodology.
institutions, and is empirically relevant in all countries considered. 7
minimization
Currently, the model
4 TFP measurement that disciplines
BFK obviously
method is themany
faces our
otheranalysis.
leading challengesSection
approach on this
that we do 3issue.
describes our
Its application
not consider here.TFPFor estimation
has been
instance, we
method
ignoreAdjustment
and
measurement costs
compares have
issues it alsothe
to
relating received
to standard
quality some ones.
improvements attention andin
Section newthe
4 productivity
discusses
products the
(Boskin, literature
data.
Dulberger, (Berndt
Section
Gordon, 5
largely limited to US data, with only two exceptions that we are aware of. Inklaar (2007)
Griliches
and and
Fuss,our Jorgenson, 1996;
1986; Brynjolfsson, Aghion, Rock Bergeaud, Boppart,
and Syverson, Klenow and
2018).costs Li, 2017).
For instance, We also do not attempt
Basu, adjustments,
Fernald and to
presents
uses
measure BFKestimates
theintangible method
capital for for European
output
(Corrado, elasticities,
Haskel, countries
Jona-Lasinio adjustment
andandfinds
Iommi, that2012; and
the utilization
resulting
Crouzet TFP measures
and Eberly, 2021).
Shapiro
and5 Solow(2001)
Section 6
himself have
analyses
was awarecomputed
our theaestimates
offinal TFP series
issue, for
and proposed forTFPthegrowth
aUnited States
rates.
correction thatspecifically
Section
dealing accounts
7 for capital
concludes.
with capital
remain strongly procyclical. He concludes that hours per worker may not be an appropriate
utilization: “Lacking any reliable year-by-year measure
adjustment costs. While they calibrate a capital adjustment function using of the utilization of capital I have simply reduced
external evidence [the
utilization 6
capital stock] proxy in Europe,
by the fraction of the butlabordoes not propose
force unemployed an alternative.
in each year [..]. This is More recently,
undoubtedly wrong, Huo,
but
and assume
probably thattothere
gets closer are than
the truth no adjustment
making no correctioncosts for labour,
at all” (Solow,we1957,
estimateP. 314).adjustment costs by
Levchenko and Pandalai-Nayar (2020) use the BFK method to calculate utilization-adjusted
2 6Planas,
using Aourworkhorse
Roeger and
model’s Rossi
Euler model
(2013)
equations. propose
Finally,a statistical
several filtering method have
recent papers to extract trend TFP
explored the growth
effects for
of
TFP series
European for a large
countries panel on
(also relying of capacity
countries. Their surveys).
utilization baselineTheirestimates
approach impose
differs that
from theBFK relation
and from
positive
ours by theprofits
fact thatonit TFP ameasurement model(Karabarbounis and Neiman, 2019; Meier and Reinelt,
between hours peruses
worker statistical
and utilization insteadisofthe thesame
economic structure
in all imposed
countries. by cost
Our minimization.
results instead
2.1 Crouzet
2020; Production and Eberly,technology2021; Piton, 2021). We examine the implications of profits for a
suggest heterogeneity across countries and problems with the hours per worker proxy
broad set of countries. More importantly, to the 4 best of our knowledge, our paper is the first
in Europe.
Inputs We In general,
assume that our main contribution
the economy is composed to ofthisI industries.
literature is In the
eachuse of capacity
industry i and
to jointly account for profits, adjustment costs and utilization, and to consistently aggregate
utilization
time period surveys as a new proxy.firm
t, a representative We produces
show that output this proxy Yi,tdoes not require
by using capital, assumptions
two types on of
the resulting industry-level TFP series.
relative
labour, and factor prices, isPrecisely,
materials. robust tooutput changes in employment
is given by composition and labour market
The remainder of this paper is structured as follows. Section 2 lays out the dynamic cost
4 TFP measurement obviously faces many ourotheranalysis.
challengesSection
that NweF do
not considerour here.
For estimation
instance, we
minimization model that disciplines Ki,t 3 describes
i,t productsV(Boskin,
TFP
ignore measurement F F F V V Dulberger, Gordon,
Yi,t = Zi,t Fiissues Ki,trelating
Φi to quality ; E improvements
i,t Ni,t Ψ
i,t Hones. i
and new ; Ei,t Hi,t Ni,t ; Mi,t , (1)
method and compares
Griliches and Jorgenson, 1996; Aghion, it to Kthe standard
i,t−Bergeaud,
1 Section
Boppart, Klenowi,tandN F 4 discusses
Li, 2017). We the data.
also do not Section
attempt 5
to
−1
measure
presents intangible capital (Corrado,
our estimates for output Haskel, Jona-Lasinio
elasticities, and Iommi,
adjustment 2012;
costs Crouzet
and and Eberly,
utilization 2021).
adjustments,
5 Solow himself was aware of the issue, and proposed a correction dealing specifically with capital
where Zi,t is 6industry
and Section analyses TFP ourandfinal is a neoclassical
Fi estimates for of TFP production function.
utilization: “Lacking any reliable year-by-year measure thegrowth
utilization rates. Section
of capital I have7simply
concludes.
reduced [the
Asstock]
capital shown by in
theequation
fraction of (1),the laborthe force
capital input isinthe
unemployed product
each year [..].ofThistheiscapital stockwrong,
undoubtedly Ki,t andbut
probably gets closer to the truth than making no correction at all” (Solow, 1957, P. 314).
an6internal adjustment cost factor Φi that depends on the growth rate of the capital stock.
2 Planas,
Next, Athere
European
Roeger and Rossi (2013) propose a statistical filtering method to extract trend TFP growth for
workhorse
are two
countries (also types
relying model
of labour
on capacity inputs:
utilizationquasi-fixed
surveys).labour (denoted
Their approach by the
differs fromsuperscript
BFK and from F
ours by the fact that it uses a statistical model instead of the economic structure imposed by cost minimization.
7 As we discuss in Section 3.2, capacity utilization surveys are obviously not perfect (Shapiro, 1989, 1996).
2.1
However, Production
our results suggesttechnology
that they contain valuable information and behave in line with their theoretical
counterparts. This is consistent with the recent results 4 of Boehm and Pandalai-Nayar (2020), who also find
Inputs
that Wemeasures
empirical assume of that the economy
capacity is composed
utilization behave of Itheoretical
in line with industries. In each industry i and
priors.
time period t, a representative firm produces output Yi,t by using capital, two types of
labour,10and materials. Precisely, output is given
BANCO DE ESPAÑA DOCUMENTO DE TRABAJO N.º 2143
5 by
F
Ni,t
Ki,t F F F V V V
Yi,t = Zi,t Fi Ki,t Φi ; Ei,t Hi,t Ni,t Ψi F
; Ei,t Hi,t Ni,t ; Mi,t , (1)The remainder of this paper is structured as follows. Section 2 lays out the dynamic cost
minimization model that disciplines our analysis. Section 3 describes our TFP estimation
method and compares it to the standard ones. Section 4 discusses the data. Section 5
presents our estimates for output elasticities, adjustment costs and utilization adjustments,
and Section 6 analyses our final estimates for TFP growth rates. Section 7 concludes.
2 A workhorse model
2.1 Production technology
and subject
Inputs We to adjustment
assume that thecosts)
economyand variable
is composedlabour of (denoted
I industries. by the superscript
In each industryVi andand
not subject to adjustment costs). For
time period t, a representative firm produces output each type , N stands for the number of
i,t Yi,t by using capital, two types of workers of
this type,
labour, and
for the number of hours per worker, and E for the number of tasks a worker
Hi,t
materials. Precisely, output is given by i,t
undertakes in one hour (“worker effort”). Adjustment costs for quasi-fixed labour are
N F
captured by the function Ψi , which Ki,t depends F on
F F the growth i,t rate of quasi-fixed
V V V employment.
Yi,t = Zi,t Fi Ki,t Φi ; Ei,t Hi,t Ni,t Ψi F
; Ei,t Hi,t Ni,t ; Mi,t , (1)
Finally, material inputs are denoted Ki,t−1 by M i,t and are notN subject
i,t−1 to adjustment costs.
Given the focus of our analysis, it may be surprising that the production technology has
where
no roleZfori,t isa industry
utilization TFP andofFcapital.
rate i is a neoclassical
This is becauseproduction we think function.
that capital utilization is
As shown in equation (1), the capital input is the
not well modelled as a production factor per se. Instead, it is an endogenous product of the capital stock
outcome and
Ki,t that
an internal
depends on adjustment
the capital stockcost factor
and onΦall i that
otherdepends
inputs,on andthe does growth rate ofinthe
not appear capital stock.
a reduced-form
Next, there function.
production are two types of labour inputs:
8 Nevertheless, Appendix quasi-fixed
A.2 shows labourthat(denoted
modelling bycapital
the superscript
utilizationF
and subject to adjustment costs) and variable labour (denoted by the superscript V and
as 7an
As input,
we discussas itin is often3.2,
Section done in the
capacity literature,
utilization does
surveys are not affectnot
obviously
our measurement.
perfect (Shapiro, 1989, 1996).
not subject to adjustment costs). For each type
However, our results suggest that they contain valuable information , N i,t stands
and behave innumber
for the of workers
line with their of
theoretical
counterparts.
this type, Hi,t Thisforisthe
consistent
number with
ofthe recent
hours results
per of Boehm
worker, and Eand Pandalai-Nayar (2020), who also find
i,t for the number of tasks a worker
Functional
that empirical forms
measures In order toutilization
of capacity implement ourinmethod,
behave line withwe need topriors.
theoretical assume functional forms
undertakes in one hour (“worker effort”). Adjustment costs for quasi-fixed labour are
for the production function F and for the adjustment cost functions Φ and Ψ.9 We assume
captured by the function Ψi , which depends on the growth rate of quasi-fixed employment.
that the production function is Cobb-Douglas5 with constant returns to scale:
Finally, material inputs are denoted by Mi,t and are not subject to adjustment costs.
Given the focus of αK it may be surprising
our analysis,
F
that
α FL the production αVL technology
α M has
Kt F F F N t V V V
no role F (•)
for = Kt Φ
a utilization rate of capital. Nt Ψis because
Et HtThis we think Et Hthat
t Nt capitalM ,
utilization
t is
K t −1 NtF−1
not well modelled as a production factor per se. Instead, it is an endogenous outcome that
depends
where αKon + the
α FL +capital
αV stock and on all other inputs, and does not appear in a reduced-form
L + α M = 1. This is obviously a strong assumption, but it is in line with
production function. 8
the empirical evidenceNevertheless, AppendixofA.2
and the vast majority theshowsgrowth that modellingliterature.
accounting capital utilization
10
as anWeinput,
assume as itthat
is often done in thecost
the adjustment literature,
functiondoes not affect
for capital is our measurement.
2
Functional forms In orderKto t implement our
a method,
Kt we needK ∗ to assume functional forms
Φ = exp − Φ − ∗t ,
for the production functionKtF−1and for the adjustment
2 Kt−1cost Kfunctions
t −1
Φ and Ψ.9 We assume
that the production function is Cobb-Douglas with constant returns to scale:
K∗
where aΦ is a positive
parameter
αK and Kt∗−1 stands
t
for
the growth rate of
α FL capital
α Mon the
F αV
balanced path K(BGP), N L
F (•)growth a concept
EtF HtF Ntwhich wet define below.
EtV HtV NThe adjustment , cost
t F V
= Kt Φ Ψ F t Mt
K t 1 N
function for quasi-fixed employment Ψ is specifiedt−analogously,
− 1 with a parameter aΨ . It is
8 For example,F the utilization
where α K + α L + αV L + α M = 1. This is obviously a strong assumption, but it is in line with
rate of a machine depends on how often workers use it, how much electricity
it consumes, and how many material inputs it receives. The utilization rate of a restaurant building 10 depends
the
on howempirical evidence
many people work in and
thethe vast majority
restaurant, and how of thetasks
many growth accounting
(cooking, waiting) literature.
they carry out.
9 To simplify notation, we drop industry subscripts whenever this does not cause confusion.
We assume that the adjustment cost function for capital is
10 While Basu et al. (2006) allow for non-constant returns to scale, their results indicate constant returns,
and they impose these from the (Basu,
2Kimball,
11 DOCUMENTO DE TRABAJO N.º 2143
outset
in later work Fernald, Fisher and
∗
2013; Fernald, 2014a).
Moreover, Basu and Fernald (2001)
BANCO DE ESPAÑA
a
Kt argue that because Φ K K
the Cobb-Douglas
t t production function is a first-order
approximation to any production Φ = exp − − ∗ must ,
t −1 K function, deviations 2from K
this framework
t −1 K
t −1
be second-order issues.
∗ α F FL
α αVL α M
Kt αKK F F F NtF α FL V V V αV
L M α M ,
F (•) = Kt Φ Kt αK EtF HtF NtF Ψ NFtF α L Et Ht Nt αV
V V V t α
F (•) = Kt Φ KKt−t 1 EtF HtF NtF Ψ NNtF− t1 EtV HtV NtV L Mt M ,
F (•) = Kt Φ Kt−1 Et Ht Nt Ψ NtF− E t H t N t M t ,
K t −1 Nt−11
where αK + α FFL + αV L + α M = 1. This is obviously a strong assumption, but it is in line with
where αK + α FL + αV + α M = 1. This is obviously a strong assumption, but it is in 10 line with
where
the empirical + αVLL + αand
αK + α Levidence M = the This majority
1. vast is obviously of the a strong
growthassumption, accountingbut it is in 10
literature. line with
the empirical evidence and the vast majority of the growth accounting literature.10
the Weempirical
assumeevidence
that theand the vast cost
adjustment majority function of the forgrowth
capitalaccountingis literature.
We assume that the adjustment cost function for capital is
andWe
worth
assume adjustment
subject
notingto
that the adjustment
that this exponential and cost
costs) specification function
variable
for capital
islabour
similar (denoted is by
to the∗ quadratic
the superscript V and
specifications often
2
not subject to adjustment Kt For each
costs). typeaΦ , K t stands Kt∗for 2the number
N theofexponential
workers of
11
used in the literature (e.g., Φ K David
t = and exp Venkateswaran,
− aΦ Ki,tt − 2019). K∗t∗ 2However, ,
this type, Hi,tdeliversfor the number Φ K K = exp − a 2 K K − K K ,
Kt−of
t− t 1 hours per worker, andt−t 1 E for
t 1the number of istasks a worker
specification anΦ elasticity 1 =of expadjustment − 2Φ costs K t−1to− Ktt∗∗−
capital
i,t
−1 growth
, that linear in the
undertakes in one hour K t −1
(“worker effort”). 2 K
Adjustment t −1 K
costs
t−1 for quasi-fixed labour are
parameter aΦ , which will be useful for the Kt∗
estimation.
where
captured aΦby is the
a positive
functionparameter Ψi , which and depends Kt on
∗ ∗ stands
the growth for therate growth rate of capital
of quasi-fixed employment.on the
where aΦ is a positive parameter and K KKt∗−t∗ 1 stands for the growth rate of capital on the
where
Finally,astock
balanced is a positive
Φ growth
material path are
inputs parameter
(BGP), denoted and
a concept by M t−which
Kt∗i,t stands
1 and arewefor not the
define
subject growth
below. rate
Theof
to adjustment capital
costs.on
adjustment the
cost
Taking
balanced growth Using pathour functional
(BGP), a concept form −assumptions,
which
1 we define we can express
below. The TFP growth
adjustment as cost
balanced
function
Givenfor growthfocuspath
thequasi-fixed of our (BGP),
employment
analysis, a concept
it Ψmay which
is specified
be we
surprising define
analogously,
that the below. withThe
production adjustment
a parameter technology aΨ . cost
Ithas
is
function for quasi-fixed employment Ψ is specified analogously, with a parameter a Ψ . It is
function
no8role for foraquasi-fixed
utilization employment
rate of capital. Ψ is This specified
is analogously,
because
F F we think
F with
thatF acapital
parameter
utilizationaΨ . It is
For example,dZ thet = dYt −rateαKof(adK
utilization t + dΦdepends
machine t ) + α ondE howt + dHworkers
often t + dNuse t +it,dΨ howt much electricity
8 For example, the utilization rate of a machine dependsLon how
worth
not
it wellnoting
consumes, andthat
modelled how this
as
manyaexponential
production
material specification
inputs factor
it per
receives. is
se.
The
8 For example, the utilization rate of a machine depends on how often workers use it, how similar
Instead,
utilizationto
often
itthe
is
rate quadratic
workers
an of use
endogenous
a restaurantspecifications
it, how outcome
building often
much electricity that
depends(2)
restaurant, similar much electricity
worth
it
on
used how innoting
consumes,
many
the
andthat
howthis
people
literature
many
work exponential
in material
(e.g., Vthe David V specification
inputs
and
itV receives.
and how
Venkateswaran, V is
The
many utilization
tasks to the
(cooking,
2019).
ratequadratic
11 ofwaiting)
a restaurant
However, specifications
they building
carry
the out.
exponential often
depends
depends
it
onconsumes,
9how manyonand
the
people capital
how many
work +stock
in LthedE
αmaterial and ondH
t inputs
+
restaurant, all other
ittand dNinputs,
receives.
+how The
tmany M dM
and
+utilization
αtasks does , not
t rate
(cooking, of appear
a restaurant in acarry
reduced-form
building out.depends
worth
used Toin
on109how noting
simplify
the
many that
notation,
literature
people this
work exponential
we
(e.g.,
in drop
the industry
David
restaurant, specification
and subscripts
Venkateswaran,
and how is similar
whenever
many tasks thistodoes
2019).the not
(cooking, 11 waiting)
quadratic
However,
waiting)
they
specifications
cause confusion.
they the
carry out. often
exponential
specification
To simplify
production
9 While Basu delivers
notation,
function.
et al.
8an
we
(2006) elasticity
drop
Nevertheless,
allow industry
for of subscripts
adjustment
Appendix
non-constant A.2
returns costs
whenever shows
to to capital
this
scale, does
that
their not growth
causeindicate
modelling
11 results that
confusion. is
capital linear
constant in the
utilization
returns,
used
and
10 To in
specification
While
they
the
simplify
Basu
impose
literature
notation,
delivers
et al.
these (2006)
from
(e.g.,
we
an the
drop
allow David
industry
elasticity
outset for in
and
of
non-constant
later
Venkateswaran,
subscripts
adjustment
work
whenever
returns
(Basu, costs to
Fernald,
2019).
this
to
scale, does
capital
Fishertheir not
and
However,
causeindicate
growth
results
Kimball, 2013;
the
confusion.
that is exponential
linear
constant
Fernald, in the
returns,
2014a).
parameter
as
and
10an
where input,
While
they dX Basu
t ≡
impose
,these
aΦaset
ln which
ital.
Xist(2006)
−
from
will
often
ln tbe
done
Xallow
the − 1 useful
outset in
stands
for in the for
for
non-constant
later
the
literature,
the(Basu,
work
estimation.
growth
returnsdoes rate
to
Fernald, notFisher
ofaffect
scale, variable
their and our
resultsXmeasurement.
Kimball, . That
tindicate
2013; is, TFP growth
constant
Fernald, returns,
2014a).
specification
Moreover,
parameter
and they impose a delivers
Basu and
,these
whichFernald
froman
will
theelasticity
(2001)
be argue
useful
outset offor
in later adjustment
that because
the
work
the
estimation.
(Basu, costs
Fernald, toFisher
Cobb-Douglas capital growth
production
and that
2013;isFernald,
function
Kimball,function linear 2014a).
is a in the
first-order
can be
Moreover,
approximation BasuΦto
computed andany as the
Fernald
productiondifference
(2001) argue
function, between
that because
deviations the growth
the
from Cobb-Douglasrate
this framework of output
must beand
production an
second-order appropriately
is a issues.
first-order
parameter
Moreover,
approximation aΦto
Basu , which
andany Fernald will
production befunction,
(2001) useful
argue that for the
because
deviations estimation.
the Cobb-Douglas
from this framework production function is a issues.
first-order
Taking
weighted
Functional
approximationstock
average
forms Using
to any of In our
input
order
production functional
growth
to implement
function, form assumptions,
rates.
deviations ourfrom method, weweneed
this framework can mustexpressbe second-order
to assume
must TFP growth
functional
be second-order asforms
issues.
Taking stock Using our functional form assumptions, we can express TFP growth 9 as
Equation
for the production (2) conveniently
function F and summarizes
for the adjustment 6the challenges cost functions that need Φ and to be Ψ. overcome
We assume in
Taking stock Using our functional form assumptions, F we can express TFP growth as
dZt TFP dY (dKt + dΦt ) + 6inwith F F F
order
that the to production
measure growth.
= function t− α isKWhile
Cobb-Douglasgrowth L dE
6 αoutput, constant
t + the dH t + dN
capital
returns stock, dΨ
tto+scale: hours
t per worker,
dZt = dYt − αK (dKt + dΦt ) + α FL dEtF + dH F + dN F + dΨ
elasticities α, (2)
t t t
employment and materials are observable in standard F Fdatasets, F+ the output
F + dΨ the
dZ
t = dY
+α L
t V− α V ( dK
t + V dΦ t )
dEKtαK+ dHt + dNt L+ αFMtdMt L , t + V α dE + αdH
F dN t α L t α M (2)
V
parameters of the adjustment KVt cost
V functionsV F F ΦV and +NαtΨ, dM and the changes in worker effort (2) dE
F (•) = Kt Φ +α L dEt + dH EtF H t t+NdN t ΨtV F M t , EtV HtV NtV Mt ,
are not. Any TFP estimation +Kαt−1 dE
V method
V
t + dHt + dNt
therefore needs
V N
+t− αM 1 to
dMaddress
t , these three measurement
where dXt ≡ ln Xt − ln XLt−1 stands for the growth rate of variable Xt . That is, TFP growth
challenges. In line with the growth accounting tradition, we engage with these challenges
where
can be dX t ≡F ln Xas − ln X t−1 standsbetween for the growth rate rate of variable Xt .and That is,appropriately
TFP growth
where
by αcomputed
imposing K + αt VL the
+ αadditional + αdifference = 1. ThisInisparticular,
structure.
M obviously the growth ajust
strong as Solow
of output
assumption, and BFK, butan it is
we in line with
assume that
where
can be dX t ≡ ln Xas
computed
L t −theln X t−1 standsbetween
difference for the growth the growth rate rate of variable
of output Xt .and That an is,appropriately
TFP growth
weighted
the empirical averageevidence of inputandare growth
theprice-takersrates.
vast majority of themarkets.growth accounting literature. 10
firms
can beminimize
computed costs
as and inthe input
growth rate ofThe nextand section lays out our
weighted
Equation average(2) ofthe
input difference
conveniently growth between
rates.
summarizes the challenges
output
that need
an appropriately
to be overcome in
We
dynamic cost assume that
minimizationthe adjustment
model.rates. cost function for capital is
weighted
Equation average(2) TFP of input growth
conveniently summarizes the challenges that need
order to measure growth. While growth the in output, the capital stock,tohours be overcome
per worker, in
order Equation
to measure (2) conveniently
TFP growth. summarizes
While
growth in output, challenges the capital that 2 need to be overcome
stock, hours per worker, in
employment and materials are
Kt observablein standard aΦ Kt datasets, Kt∗ theoutput elasticities α, the
2.2
order
employment Dynamic
to measure materials
parameters ofand
cost
TFP
the adjustment
Φ minimization
growth.
Kare
While= exp
observable growth in in output,
− standard −the ∗capital
datasets, stock,
,
the output hours per worker,
elasticities α, the
t−1cost functions Φ 2 andKtΨ, −1 andKthe t−1 changes in worker effort dE
employment
parameters and materials are observable in standard datasets, the output elasticities α, the
are
Setup not. We Anyof the estimation
TFP
assume
adjustmentmethod cost functions
that the representative therefore Φ and Ψ,to
firmneeds solves the
and the changes
address these three
cost minimization
in worker effort
measurement
problem
dE
parameters
are not. Anyof the estimation
adjustment cost functions Kt∗ Φ and Ψ,to and the changes in worker effort dE
challenges.
where a Φ isInaTFP
line
positive
with
thet growth
+∞ parameter
method
and therefore
accounting needs
tradition,
stands for
address
thewegrowth engage these
with
rate
three
these
of
measurement
capital challenges
on the
are not. AnyInTFP
challenges. lineestimation
with the method
growth therefore
1 accounting ∗
K t −1
F needs
tradition, F to address
Fwe engage V these withthree
V these Vmeasurement
challenges
by imposing
balanced growth
challenges.
min additional
E
linepath
0
Inadditional
∑ (BGP),
with
∏ 1a+concept
structure. In
rIn
particular,w
which Γ H
just
t F wet define as N Solow
t +
we below.
w Γ
and
t V The H
BFK, N
we
t adjustment
t assume that
cost
by imposing t=0 the s=growth
structure.
1 accounting
s particular, tradition,
just as Solow engage
and with these
BFK, we challenges
assume that
firms
function minimize costs and
for quasi-fixed are price-takers
employment in input markets.
is specified Thewith nextasection lays aout
our
by
firmsimposing
minimize additional
costs and
F
structure.
are F InFΨparticular,
Fprice-takers V in just
input V
analogously,
as
markets.V
Solow
V The andnext BFK, parameter
sectionwe assume
lays
Ψ . that
out
It
our
is
dynamic cost minimization +qt Λ F E model.
t H N
t t + q t Λ V E t H t N t + P M,t t M + P I
I,t t
firms
8 Forminimize
dynamic costs
cost minimization
example, the and are
utilization price-takers
model.
rate of a machine depends in input on markets.
how often workers The next usesection
it, how much lays electricity
out our(3)
it consumes, andminimization
how many material
inputs it receives. The utilization
dynamic cost model. N F rate of a restaurant building depends
Kt
on how many s.t.people work Yt =inZthe Kt Φ Kt−and
t F restaurant, ; EtF H
how
F FΨ
t Nttasks
many t
(cooking,
V V V; M
; Ewaiting)
t Ht Ntthey t , out.
carry
2.2 Dynamic
9 To simplify cost minimization
notation, we drop industry subscripts whenever this does
1
t −1
not cause confusion.
NF
2.2 Dynamic cost minimization
K t +1 =
10 While Basu et al. (2006) (1 −
allow forδK ) Kt + It , returns to scale, their results indicate constant returns,
non-constant
2.2
and theyDynamic
Setup We assume
impose cost
that
these from minimization
the
F the
representative
outset
F (Basu,
inFlater work firm
F
solves the
Fernald, cost
Fisher andminimization problem
Kimball, 2013; Fernald, 2014a).
Setup
Moreover, We
Basuassume
Nthat
t +1
and Fernald the1representative
= − δN Nt + Afirm .
t thesolves the cost minimization problem
+∞ (2001) t argue that because Cobb-Douglas
production
function
is a first-order
Setup We assume
approximation to any that the function,
production 1
representative
deviationsfirm solves theF cost Vmust
minimization problem
min E0 + ∑∞ ∏ t
1
wt Γ F Ht Nt + wt ΓV Ht aΦNt Kt issues.
Ffrom this
F framework be second-order
V V
Kt∗ 2
approximation
11 Indeed, a first-order 1 +of rour
adjustment
F costFfunction
F yields V Φ ≈ 1V VK .
min E0 t+∑∞ ∏
= 0 s t
= 1
1 +1 rs
s w Γ H
t F t t N + w Γ H − N
t V t t t−1
2
− K ∗
t −1
min E0 t∑=0 s∏ =1 F F F wtF Γ F H tF NtF + wV t ΓV Ht
V
NtV
1 + r s V 6
+tq=t 0Λ Fs =E1 t Ht Nt + qt Λ7V Et Ht Nt + PM,t Mt + PI,t It
F V V V
12
BANCO DE ESPAÑA
+qtF Λ F EtF HtF NtF + qV
DOCUMENTO DE TRABAJO N.º 2143
ΛV EtV HtV NtV + PM,t Mt + PI,t It (3)
t
+ qtF Λ F EtF HtFNtFKt+qV Λ E V HV N N V+P
F
M + P I (3)
s.t. Yt = Zt F Kt Φ Kt−1 ;t EtFVHtF Nt tF Ψ t N FttF ; EM,t t t t Mtt ,
V H VtN V ; I,t
s.t.
N
Yt = Zt F Kt Φ KKt−t 1 ; EtF HtF NtF Ψ NtF−tF1 ; EtV HtV NtV ; Mt , (3)
K = (1 − δ ) K + K t I , F F F Nt −
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