# 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 Trigari

MEASURING 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 cycle

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 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 Great

Finally, 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) from

KLEMS (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 productionαVL 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 apositive 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 , Kt 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 TFPgrowth 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 andthe 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 ΓF HtF 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 − t 1 V V V

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