What is Measured in National Accounts?

Board of Governors of the Federal Reserve System International Finance Discussion Papers ISSN 1073-2500 (Print) ISSN 2767-4509 (Online) Number 1375 May 2023 What is Measured in National Accounts? Franc¸ois de Soyres, Alexandre Gaillard, and Henry Young Please cite this paper as: de Soyres, Franc¸ois, Alexandre Gaillard, and Henry Young (2023). “What is Measured in National Accounts?,” International Finance Discussion Papers 1375. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/IFDP.2023.1375. NOTE: International Finance Discussion Papers (IFDPs) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the International Finance Discussion Papers Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers. Recent IFDPs are available on the Web at www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at www.ssrn.com.

What is Measured in National Accounts? * François de Soyres† Alexandre Gaillard‡ Henry Young§ FederalReserveBoard PrincetonUniversity UniversityofMichigan May2023 Abstract Most statistical agencies construct sectoral real GDP using double deflation and base periodprices. Whenthebaseperiodpriceusedforintermediateinputsisnotequaltotheir marginalrevenueproduct,suchaswhenfirmsapplyamarkup,realGDPfluctuationsbecomemechanicallylinkedtovariationsinintermediateinputs.Thisisbecausetheseinputs generate profits that are incorporated into real value added. Taking this channel into account,wedemonstratethatrealGDPreportedinnationalaccountssubstantiallydiverges from a theory-consistent "physical" value added. This, in turn, has implications for the measurementofproductivity. Between1999and2021,"physical"productivitycumulative growthintheFinancesectorwas15pplowercomparedtotheSolowResidual,whileitwas 15pphigherintheManufacturingsector. Keywords: EconomicMeasurement,NationalAccounts,Markups,Productivity. JELClassification: E1,E3,O4,O51 *Theviewsinthispaperaresolelytheresponsibilityoftheauthorsandshouldnotnecessarilybeinterpretedas reflectingtheviewsoftheBoardofGovernorsoftheFederalReserveSystemorofanyotherpersonassociatedwith theFederalReserveSystem.Allerrorsareourown. †Address: BoardofGovernorsoftheFederalReserveSystem,2051ConstitutionAvenueNW,Washington,DC. Email:francois.m.desoyres@frb.gov;Correspondingauthor. ‡Address:PrincetonUniversity–SchoolofPublicandInternationalAffairs,237JRRabinowitzBuilding,Princeton,j.Email:ag6575@princeton.edu. §Address: University of Michigan- Department of Economics, 611 Tappan Ave, Ann Arbor, MI. Email: hlryoung@umich.edu.

1 Introduction Since the publication of their 2021 “Blue Book”, the UK’s Office for National Statistics started tomeasurerealGDPinthenationalaccountsusingdoubledeflation. Thismethodologicalupdate follows the premise that “double deflation is internationally accepted as the best approach to producing volume estimates of industry Gross Value Added”. Indeed, double-deflation, taking the difference between gross output and intermediate inputs, has been the international standard since the System of National Accounts (SNA 1993) was adopted by the international communitytofacilitateinternationalcomparisonsofnationaleconomicstatistics. IntheUnitedStates, this accounting practice has been used since 1992, while it has been the standard practice in the Euro Area since the creation of Eurostat. Although double deflation is the state-of-the-art methodforrealvalue-addedmeasurement,itsaccuracyreliesonseveralassumptionsthatare not met in practice. As a result, our understanding of relative growth rates across sectors is biased. These limitations are important for researchers that match their quantitative models withthedataproducedbystatisticalagencies. Tounderstandaccountingproceduresinnationalaccounts, itisusefultocomebacktothe notion of Real value added (RVA). In its original concept described by Fabricant (1940), Sims (1969)andArrow(1974),RVAisanidealindexofanindustry’snetphysicaloutputthatcanbe implicitlyderivedfromtheproductionfunctionifitseparatesprimaryfactorsandintermediate inputs. Using Arrow (1974)’s words: we can "imagine capital and labor cooperating to produce an intermediate good called real value added, which in turn cooperates with materials to produce the final product.”1 Concretely, consider an economy in which gross output is produced through the combinationYoflaborandcapitalinputs,aswellasintermediateinputsX. AccordingtoSims (1969), the definition of real value added corresponds only to the quantity Y, which can be referred to as Physical Value-Added (PVA). Changes in the quantity of intermediate inputs X onlyimpactPVAwhenitaffectsthequantityY producedintheeconomy. In practice, statistical agencies do not observe Physical Value-Added. Instead, national accountants construct a measure of "statistical value added" using double deflation, a method that consists of taking the difference between gross output and intermediate inputs, both valued using base period prices. Using base period prices to measure real GDP is a standard procedure, as merely counting the quantity of goods produced in a country would be meaningless: what would it mean, for example, to compute real value-added as in the automotive sector by taking the difference between the number of cars produced and the number of its 1ThisquoteistakenfromArrow(1974),pp4-5. 1

intermediate inputs? One needs a common unit of account, which is why base period prices arenecessaryfortheconstructionofsectoralrealvalueadded,aswellasaggregaterealGDP. The main contribution of this paper is to demonstrate a potential bias that may arise from the use of double deflation in comparing theory-based measures of real GDP and productivity with their statistical counterparts. We show that double deflation implicitly assumes that the base period price used to deflate inputs reflects both the marginal cost of inputs and the marginal revenue that their usage can generate. If this assumption holds, then real GDP is indeedanaccuratemeasureofPVA. The crux of the issue is that while RVA cannot be constructed without prices, the presence ofdistortedpricescreatesmeasurementissues. Inthepresenceofdistortions(markups,taxes, orotherfactors),usingpricestoconstructRVAcreatesawedgebetweenthestatisticalmeasure andthetheory-consistentPVAmeasure. Itisimportanttonotethatpricedistortionsneednot varyovertimetocreateabiasinRVAmeasurement. Evenwithconstantmarkups,intermediateinputsgeneratemorerevenuesthantheircost, meaningusingmoreinputsresultsinmore statisticalvalue-added,evenifdomesticfactors(laborandcapital)andtechnologyremainunchanged. This happens because a sector’s net operating surplus is included in the statistical agencies’measureofRVA.Consequently,asectorcancreatestatisticalvalue-addedbyincreasingitsprofits,ceterisparibus. Our analysis shows that real GDP in the national accounts and PVA differ mainly for sectors with three characteristics: (i) markups are large, (ii) the intermediate input share is large, and (iii) variations of PVA and intermediate inputs usage are not collinear. This last conditionisinterestingbecauseitillustrateswhyameasurementbiascanariseevenwhenmarkups are constant. To gain clarity, consider first a situation with constant markups and where intermediate inputs are proportional to PVA. In this case, gross output (which is a combination of intermediate inputs and PVA) is also proportional to PVA, and so are profits. This, in turn, implies that a one percent change in PVA is associated with the same proportional change in profits,andhenceprofitsdonotbiasPVA’smeasuredgrowthrate. ThingsaredifferentwhenintermediateinputsandPVAarenotcollinearthough,becauseinsuchacasePVAandprofitsdo nothavethesamegrowthrate. Thiswedgethencreatesadisconnectbetweenthevalue-added measuredbystatisticalagenciesanditstheoreticalcounterpart. UsingdatafromtheBureauofEconomicAnalysis(BEA),wethenevaluatehowprofitsand distortedpricesingeneralhavebiasedourmeasureofsectoraloutputandsectoralproductivity intheUS.WeusetheBEA’sdataonnetoperatingsurplustobuildameasureofmarkupsatthe 2

sector level, as well as standard input-output analysis, to construct a new version of national account statistics.2 We then compute a measure of PVA for each sector and compare it to the statisticalmeasureofrealGDP. Our results highlight the potential biases in widely used data and how they can affect our understandingofeconomicactivity. Wecomparedtheevolutionofstatisticalrealvalue-added (orsectoralrealGDP)toourmeasureofPVA,takingfluctuationsinnetoperatingsurplusinto account. OuranalysisoftheUSeconomyshowsthatbetween1999and2021,thereisa1.24pp differenceincumulativegrowthbetweenourPVAmeasurementandrealGDPasmeasuredby the BEA. While differences between statistical and physical objects are not significant for the USeconomyasawhole,individualsectorsrevealinterestingdiscrepancies. Wefocusedontwosectors,FinanceandManufacturing,whichillustratedifferentbiasesin nationalstatistics. Bothsectorsheavilyrelyonintermediateinput,withinputcostsaccounting for an average of 55% of total sales in Finance and 70% in Manufacturing between 1999 and 2021. While year-to-year growth rates are highly correlated, the cumulative bias over time leads to significant differences by the end of our sample. Normalizing both measures in 1999, wefindthatPVA’scumulativegrowthin2021ismorethan13pphigherthanrealGDPgrowth in the Manufacturing Sector. In Finance, PVA’s cumulative growth is 20pp lower than real GDPgrowth. Inotherwords,nationalaccountsseverelyunderestimatedthegrowthofPVAin Manufacturing,whileitoverestimateditinFinance. Equipped with our measure of PVA, we constructed a measure of “Physical Productivity” asthefluctuationsinPVAthatarenotduetoobservedmovementsinlaborandcapital,which wecanthencomparetothemorestandardSolowResidual. Resultsshowedthatbytheendof oursamplein2021,cumulativegrowthofphysicalproductivityisabout15ppsmallerthanthat oftheSolowresidualinFinance,whileitis15pplargerthanSolowResidualinManufacturing. Once again, national accounts’ data overestimated productivity growth in Finance, while it underestimatedproductivitygrowthinManufacturing. Relation to the literature The importance of using base period prices in real GDP measurementhasbeenanalyzedinseveralcontexts,mostnotablyKehoeandRuhl(2008)andBurstein andCravino(2015). Theroleofmarkupsingeneratingalinkbetweenintermediateinputand measuredproductivityhasbeendiscussedinseveralpaperssuchasHall(1988)andBasuand Fernald(2002),andmorerecentlyinGopinathandNeiman(2014). Themismatchbetweenthe 2TheBEAdefinesitsmeasureofnetoperatingsurplusas: “aprofits-likemeasurethatshowsbusinessincomeafter subtractingthecostsofcompensationofemployees(received),taxesonproductionandimportslesssubsidies,andconsumption offixedcapitalfromvalueadded,butbeforesubtractingfinancingcostsandbusinesstransferpayments.” 3

theory-consistent measure of real value added (PVA, defined from the production function) andstatisticalrealGDPmeasuredinthenationalaccountsissignificantinmanycontexts. For instance,usingdoubledeflationtomeasurerealGDPinamacromodelhasasubstantialimpact on the statistical properties of real GDP in the model’s simulations. In de Soyres and Gaillard (2021), we argue that double deflation is critical to understanding cross-country real GDP comovement. It is a way to reconcile data and model-based simulations and can help solve the "TradeComovementPuzzle"inanInternationalBusinessCyclemodelwithmarkups. Overall, our findings underscore the importance of paying close attention to the measurementofeconomicvariablesusedinpolicymakingdecisionsandacademicresearch,especially regardingtheriseofmarketpowerandmarkupintheUS(DeLoeckeretal.,2020). Finally,itisimportanttonotethattheanalysisaimstoquantifyhowthepresenceofmarkups createsameasurementissueinnationalaccountsdata,whichisconceptuallydistinctfromthe topic of resource misallocation. Recent papers, such as Baqaee and Farhi (2020), highlighted how markup heterogeneity across firms implies that the allocative efficiency of the US economy has changed over time. As production factors are reallocated to high-markup firms, this reallocation process accounts for approximately half of aggregate TFP growth from 1997 to 2015. While both misallocation and measurement are significant issues, they highlight differentaspectsofhowmarkupsimpacttheeconomy. 2 An Accounting Framework with Input-Output Linkages Consideraneconomywith J sectorsindexedby j. Wepresentasimpleaccountingframework and show how the presence of a wedge between total sales and total cost at the sectoral level gives rise to a disconnect between Real Value Added (RVA or real GDP) (as measured by statistical agencies) and what we call "Physical Value Added". For any variable A, we define the proportionalchangeas A(cid:98)t = A ∆ t A − t 1 . 2.1 Production Gross output (GO ) in sector j is produced using labor L , physical capital K , productivity jt jt jt Z ,andintermediateinputs X ,suchthat: jt jt GO = (cid:16) Z L αjK 1−αj (cid:17)γj ·X 1−γj, (1) jt jt jt jt jt 4

where α ∈ (0,1) and γ ∈ (0,1). We consider the presence of a markup wedge (µ ) which j j jt measuresthedifferencebetweentotalsalesandtotalcost(TC ): jt P GO jt jt µ = . (2) jt TC jt Thesituationwhereµ > 1canarisebecausefirmsaremakingprofitsorforotherreasons, jt for example the presence of sales taxes. The definition of µ and the production structure jt provides a relationship between the intermediate input share of total cost, (1−γ ), and the j shareoftotalsales, P j X t−1 Xjt−1 ,suchthat: Pjt−1 GOjt−1 P j X t−1 X jt−1 1−γ j = . (3) P GO µ jt−1 jt−1 jt−1 2.2 FromsectoralGDPtoaggregateGDP Atthesectorallevel,wecandefinethechangeinsectoralrealGDP(denotedsRGDP forsector jt j)betweent−1andtbykeepingpricesattheirbaseperiodvalueandusingthechangeingross outputsandinputssothat: sR (cid:92) GDP = P jt−1 ∆GO jt −P j X t−1 ∆X jt = P jt−1 GO jt−1 (cid:34) ∆GO jt − P t X −1 ∆X jt (cid:35) , (4) jt sRGDP sRGDP GO P GO jt−1 jt−1 jt−1 jt−1 jt−1 (cid:124) (cid:123)(cid:122) (cid:125) = ω jt−1 where ω is the ratio of total sales to value added and is therefore larger or equal to one. jt−1 Whentotalsalesequaltotalcost,ω issimplyequaltotheinverseofthevalueaddedsharein jt−1 grossoutput, ω = Pjt−1 GOjt−1 = 1/γ . However,inthepresenceofawedgebetween jt−1 wjt−1 Ljt−1 +rjt−1 Kjt−1 j salesandcost,wecanusethedefinitionofµ andequation(3)toobtain: jt−1 P jt−1 GO jt−1 1 µ jt−1 ω = = = . (5) jt−1 P jt−1 GO jt−1 −P j X t−1 X jt−1 1− P j X t−1 Xjt−1 γ j +µ jt−1 −1 Pjt−1 GOjt−1 Note that when the markup wedge µ > 1, the ratio of total sales to value added ω is jt−1 jt−1 smallerthan1/γ . j Atthenationallevel,thechangeinaggregaterealGDPofthecountrycanbedefinedasthe 5

sumofthevalueaddedineachsector,givenby: R (cid:92) GDP = ∑ j J =1 P jt−1 ∆GO jt −P j X t−1 ∆X jt = ∑ J P jt−1 GO jt−1 (cid:34) ∆GO jt − P t X −1 ∆X jt (cid:35) , (6) t RGDP RGDP GO P GO t−1 j=1 t−1 jt−1 jt−1 jt−1 (cid:124) (cid:123)(cid:122) (cid:125) = d jt−1 where d defines the Domar weights which is the ratio of sector j’s sales to aggregate value jt−1 added. TheDomarweightsd combineindustry-levelgrowthintoaggregategrowthbyweightjt−1 ing each sector’s output share by its value added share. By construction, since the numerator is gross output (sales) and the denominator is value added, the weights sum up to more than one by construction. To better understand the Domar weights, we decompose them into a sales-to-value-addedratioatthesectorallevelandasectorweight,resultingin: P GO sRGDP sRGDP jt−1 jt−1 jt−1 jt−1 d = · = ω · . (7) jt−1 jt−1 sRGDP RGDP RGDP jt−1 t−1 t−1 Using(7)togetherwith(4)and(6),wecanreconcileindustryandcountry-levelRGDP: (cid:92) ∑ J sRGDP jt−1 (cid:92) RGDP = ·sRGDP . (8) t jt RGDP j=1 t−1 Equation (8) indicates that the growth rate of country-level RGDP is the weighted sum of industry-level RGDP growth rates, with weights proportional to each industry’s share of aggregatevalueadded. 2.3 ANewRealGDPDecomposition PhysicalValueAdded(PVA) InordertoprovideamorecompletedecompositionofrealGDP changes, we follow the approach of Fabricant (1940) and Arrow (1974) and define Physical Value Added (PVA) as the "quantity" of the value-added bundle used in gross output production. Specifically, we define PVA as the sum of labor and capital inputs weighted by their respectivesharesintotalfactorincome,plusatermfortotalfactorproductivity: (cid:91) PVA jt = Z(cid:98)jt +α j(cid:98)L jt +(1−α j )K(cid:98)jt . (9) 6

Notethat,unlikethetraditionaldefinitionofvalueadded,whichsubtractsintermediateinputs fromgrossoutput,PVAmeasuresthephysicalquantityofvalueaddedusedinproduction. Real Value Added (RVA) In Equation (8), real GDP change in sector j can be expressed as a weighted sum of changes in labor, productivity, physical capital, and intermediate inputs usagefromequation(1)usingstandardloglinearization. UsingthePVAdefinition(9)and(3), realGDPfluctuationscanfinallybedecomposedinto(i)fluctuationsinPhysicalValueAdded (PVA)and(ii)fluctuationsinprofitsderivedfromintermediateinputusage: (cid:34) (cid:35) (cid:92) ∑ J (cid:16) (cid:17) P j X t−1 X jt−1 RGDP t = d jt−1 γ j Z(cid:98)jt +α j(cid:98)L jt +(1−α j )K(cid:98)jt +(1−γ j )X(cid:98)jt − X(cid:98)jt P GO j=1 jt−1 jt−1 (cid:34) (cid:35) ∑ J (cid:16)(cid:91) (cid:17) (cid:16) µ jt−1 −1 (cid:17) = d jt−1 γ j PVA jt +(1−γ j ) ·X(cid:98)jt . (10) µ j=1 jt−1 (cid:124) (cid:123)(cid:122) (cid:125) MarkupEffect Thepresenceofmarkups(µ > 1)createsamismatchbetweenRGDPmovementsandflucjt−1 tuationsinthe"quantityofvalueadded"asmeasuredbyPVA.Thisdisconnectappearsin(10) in two ways: (i) the "markup effect" term shows that any change in intermediate input usage affects statistical value added, and (ii) markups affect the Domar weight d in (5) such that jt−1 theratioofnominalvalueaddedtosales(ω )isnotequalto1/γ . jt−1 j 2.4 SolowResidual(SR)andPhysicalProductivity(Z) The real GDP decomposition presented in (10) also bears important implications for the measureofproductivity. WedefinetheSolowResidual(SR)atthesectorallevelas: (cid:92) S(cid:99)R jt = sRGDP jt −α j(cid:98)L jt −(1−α j )K(cid:98)jt (11) Whichcanberewrittenas: (cid:16) (cid:17) (cid:0) (cid:1) S(cid:99)R jt = γ j ω jt−1 Z(cid:98)jt + γ j ω jt−1 −1 α j(cid:98)L jt +(1−α j )K(cid:98)jt (12) (cid:16)µ −1 (cid:17) jt−1 + ω jt−1 (1−γ j ) X(cid:98)jt µ jt−1 Expression(12)highlightshowthepresenceofmarkupsaffectstheestimationofphysicalproductivity(Z)andemphasizesthedistinctionbetweenSRand Z. S(cid:99)R jt capturesthefluctuations inRGDPthatgobeyondthephysicalchangesofdomesticfactorsupply(i.e., beyondchanges 7

in LαK1−α). We can use the definition (11) and expression (12) to derive physical productivity jt jt Z(cid:98)jt asafunctionofobservablevariables: (cid:92) sRGDP (1−γ )(µ −1) jt j jt−1 Z(cid:98)jt = −α j(cid:98)L jt −(1−α j )K(cid:98)jt − X(cid:98)jt . (13) γ ω γ µ j jt−1 j jt−1 2.5 TakingtheAccountingMeasurestotheData To fully grasp the extent of the measurement bias discussed earlier, we proceed to quantify it. This will provide a thorough and precise understanding of the discrepancies between the statisticallymeasuredrealvalueaddedandproductivitymeasures,andtheirtheoreticalcounterpartsusedinmacroeconomicmodels. 2.5.1 Data We rely on the Bureau of Economic Analysis (BEA) National Accounts data to conduct our analysis. Table1presentsthespecificdatafromtheBEAthatweuse,andTable4inAppendix provides a mapping between our model variables and the corresponding BEA data. It is important to note that our computations are not affected by the level of "real" variables, which allowsustouseBEAvariablesexpressedinindexedquantities. Table.1. DescriptionofBEAdatausedthroughouttheanalysis. Object Indicatora Unit GOnominal NominalGrossOutput MillionsofDollars,Annual. jt VAnominal NominalValueAdded MillionsofDollars,Annual. jt IInominal NominalIntermediateInputs MillionsofDollars,Annual. jt NOS NominalNetOperatingSurplus MillionsofDollars,Annual. jt COMP EmployeeCompensationbyIndustry MillionsofDollars,Annual. jt GO_Q GrossOutput Index2012=100,Quantities,Annual. jt VA_Q ValueAdded Index2012=100,Quantities,Annual. jt CAP_QI CapitalStock Index2012=100,Quantities,Annual. jt II_QI IntermediateInputs Index2012=100,Quantities,Annual. jt EMP Full-TimeandPart-TimeEmployees ThousandsofPeople,Annual. jt aNotes:https://www.bea.gov/resources/guide-interactive-gdp-industry-accounts-tables 2.5.2 ConstructionofVariables The procedure involves several steps to obtain the necessary variables for the mapping of the model. First,themarkupratio µ iscomputedusingdataonNetOperatingSurplus(NOS). jt−1 8

The markup ratio measures the wedge between total cost and total sales at the industry level, anditcanberecoveredusingdataonnominalgrossoutput(=totalsales)andNOSusing: GOnominal jt µ = (14) jt GOnominal −NOS jt jt Next, the data on total sales and intermediate input payment is used to back out γ as j follows: PX X IInominal jt−1 jt−1 jt−1 1−γ = µ · = µ · (15) j jt−1 P jt−1 GO jt−1 jt−1 GOn jt o − m 1 inal where IInominal is the BEA variable recording nominal spending on intermediate inputs. In jt−1 practice, we take the average over time for each sector. In practice, the average over time is takenforeachsector. Inthedata,paymentstocapitalarecomputedasthedifferencebetweenmeasuredsectoral value added and payments to labor, which means payment to capital cannot be used in the calibration. Using the definition of µ and the fact that the labor share of total cost is equal jt−1 toγ α ,wecanobtaintheexpressionthecapitalshareα using: j j j w L γ α µ w L µ COMP jt−1 jt−1 j j jt−1 jt−1 jt−1 jt−1 jt−1 = ⇒ α = = (16) P jt−1 GO jt−1 µ jt−1 j γ j P jt−1 GO jt−1 γ j GOn jt o − m 1 inal Theaverageovertimeisagaintakenforeachsector. Finally, the Domar weights ω and d are computed directly from the data using jt−1 jt−1 current-yearpricesofgrossoutputandvalueadded,suchthat: P GO GOnominal VA_Q jt−1 jt−1 jt−1 jt ω = = , d = ω · (17) jt−1 SectoralRGDP jt−1 VAn jt o − m 1 inal jt−1 jt−1 VA_Q t Table4inAppendixsummarizestheconstructionofallvariables. 3 Results: Quantifying the Measurement Bias Inthissection,wefocusontheUnitedStatesandexaminetheconsequencesofmeasuringreal value-added inaccurately at both the aggregate and sectoral levels. To achieve this, we follow a three-step process. Firstly, we construct all variables as outlined in Table 4 in Appendix (cid:91) and estimate Z(cid:98) using (13). Secondly, we use our computed Z(cid:98) to calculate PVA using (9), and 9

also obtain the time series for the level of these variables from their growth rates (the "hat (cid:92) variables"). Finally, we compare the properties of statistical variables, such as RGDP and S(cid:99)R, (cid:91) withtheir"physical"counterparts, PVAand Z(cid:98). 3.1 AFirstGlanceintotheMainStatistics (cid:92) In general, statistical variables such as RGDP and S(cid:99)R may be either more or less volatile than (cid:91) theirtheoreticalcounterparts,PVAandZ(cid:98). Thedifferencesarisefromthepresenceofprofits,as measuredbyNetOperatingSurplus,andintermediateinputs. Dependingonthesector,profits can be positively or negatively correlated with PVA and/or Z, leading to divergent behavior betweenthestatisticalandphysicalvariables. We present a comprehensive overview of our findings in Table 2, which shows selected momentsofstandardvariableslikerealGDPandSolowResidual,aswellasourownphysical variables(PVAand Z). Table5inAppendixalsodisplaysallothersectorsoftheUSeconomy. Table.2. Selectedpropertiesof"statistical"and"Physical"variables.a Sector Average Relativevolatility(σ) Correlation(ρ) (cid:91) (cid:92) (cid:91) σ(PVA) σ(Z(cid:98)) (cid:91) (cid:92) RGDP PVA ρ(PVA,RGDP) ρ(Z(cid:98),S(cid:99)R) (cid:92) σ(RGDP) σ(S(cid:99)R) USAggregate 2.20% 2.21% 0.92 0.95 1.00 0.97 Finance 2.77% 2.41% 1.19 1.18 0.99 0.99 Manufacturing 2.00% 2.54% 1.07 1.12 0.99 0.99 RetailTrade 1.68% 1.52% 1.02 1.06 0.99 0.98 ComputerSystems 8.90% 8.91% 1.01 1.00 1.00 1.00 Agriculture 2.12% 3.32% 1.76 1.71 0.98 0.98 aVariablesconstructedusingdatafromtheBEA. The study’s findings indicate that there are significant differences between the measured real gross domestic product (RGDP) and the theoretical production value added (PVA) in the finance, manufacturing, and agricultural sectors. Furthermore, the theoretical measures are generallymorevolatilethantheactualmeasuresatthesectorlevel,especiallyintheagricultural sector. Finally, there is a significant correlation between the theoretical and actual measures. Surprisingly,wefindthatthemeasurementbiascanbepositiveornegative. Asaresult,when weaggregatethepositiveandnegativebiases,theytendtocanceleachotherout. Figure 1 illustrates that while the average ratio of measured aggregate GDP to true GDP is close to unity, there is a significant and noteworthy dispersion within sectors, even after aggregating growth rates across all periods. Specifically, the mean difference between ∏ (1+ t 10

(cid:92) (cid:91) RGDP )and∏ (1+PVA )is6ppacrosssectors,andthestandarddeviationis0.33,indicating t t t considerablemeasurementbiasdispersion.3 Therefore,itisimportanttocarefullyconsiderthe impactofthisdispersionontheinterpretationofthedata. Figure1. DispersionofstatisticalRGDPvsPVA. 300 200 100 0 −2 0 2 4 ratio RGDP to PVA ytisneD Across sectors, all periods 15 10 5 0 0.0 0.5 1.0 1.5 2.0 ratio RGDP to PVA ytisneD Across sectors, cumulative 3.2 TracingtheEvolutionofValueAddedandProductivity Wenowgobeyondpresentingtablesanddelvedeeperintotheresultsofourstudy. Usingthe estimated growth rates of Physical Value Added and Physical Productivity, we construct the time series for these variables and compare them with their standard statistical counterparts. ThiscomparisonispresentedinFigures2fortheUSaggregateandforkeysectors. Ourfindings,whichareconsistentwiththosepresentedinTable2,revealsignificantbiases in three key sectors of the economy: Finance and Insurance, Manufacturing, and Agriculture, Fishing, Forestry. At almost all periods of time, when these biases are aggregated across sectors, their impact on the overall economy is relatively minor, a fact that is again somewhat surprising. Specifically, we find that the measured real gross domestic product (RGDP) in 2020 exhibited a bias of 17 percentage points higher in the Finance and Insurance sector, while it was 21 percentage points lower in the Manufacturing sector. In addition, we observed more pronounced biases in technology of 15 percentage points in the Finance sector and 15 percentage pointsintheManufacturingsector. 3Itisworthnotingthattheaggregatemeasurementbiasisweightedbysectorimportance. Ifwetakethisinto account,theweightedmeanwouldbearoundabout0. 11

Figure2. StatisticalRGDPvsPVA(toppanel)andSRvsTechnology(bottompanel)(index100=1997). Physical Value Added (PVA) Real GDP as measured by the BEA Aggregate Economy Finance and Insurance Manufacturing 180 160 160 160 140 140 140 120 120 120 100 100 100 2000 2005 2010 2015 2020 2000 2005 2010 2015 2020 2000 2005 2010 2015 2020 Retail Trade Computer Systems Design Agriculture, Fishing, Forestry 140 600 200 130 400 160 120 110 200 120 100 2000 2005 2010 2015 2020 2000 2005 2010 2015 2020 2000 2005 2010 2015 2020 Solow Residual Technology (Z) Aggregate Economy Finance and Insurance Manufacturing 210 130 130 120 180 120 110 150 110 100 120 100 90 2000 2005 2010 2015 2020 2000 2005 2010 2015 2020 2000 2005 2010 2015 2020 Retail Trade Computer Systems Design Agriculture, Fishing, Forestry 250 175 100 200 150 95 150 125 100 90 100 2000 2005 2010 2015 2020 2000 2005 2010 2015 2020 2000 2005 2010 2015 2020 12

These resultssuggest thatthe currentmethods used tomeasure economicactivity inthese sectorsmaynotaccuratelycapturethetruevalueadded. Asaconsequence,theinterpretation ofthedriversofeconomicgrowthandperformanceintheseareasmayrequireabettermapping withthemetricusedineconomicmodels. 3.3 The(absenceof)RoleofMarkupFluctuations Theaboveanalysisaimstoquantifyhowthepresenceofmarkupscreatesameasurementissue in national accounts data, which is conceptually distinct from the topic of resource misallocation. Recent papers such as Baqaee and Farhi (2020) highlight how markup heterogeneity across firms implies that the allocative efficiency of the US economy changed over time, as production factors are reallocated to high-markup firms. According to their estimate, this reallocationprocessaccountsforabouthalfofaggregateTFPgrowthovertheperiod1997-2015. While both misallocation and measurement are important issues, they highlight different aspectsofhowmarkupsimpacttheeconomy. Inourcase,thesourceofthebiasisnottheheterogeneityofmarkups,northeirfluctuations overtime. Toemphasizethispoint,werecomputethegrowthrateofphysicalproductivity(Z) andPhysicalValueAdded(PVA)usingaconstantmarkupvalueof1.15–henceassumingthat all sectors share the exact same time-invariant markup of 15% (which is contrary to what our data on Net Operating Surplus suggests, but this is an illustration). The main messages hold withconstantandhomogeneousmarkups. 3.4 DecomposingtheMeasurementBias Using the sectoral version of equation (10), we can decompose sectoral real GDP (sRGDP ) jt intotwotermsthatcapturesectoralPVAchangeandsectoralmarkupeffect. (cid:92) (cid:16)(cid:91) (cid:17) ω jt−1 (1−γ j )(cid:16) (cid:17) sRGDP jt = ω jt−1 γ j PVA jt + (µ jt−1 −1)·X(cid:98)jt . (18) µ jt−1 (cid:124) (cid:123)(cid:122) (cid:125) MarkupEffect Usingω = µ /(γ +µ −1)andreorganizingterms,wecancomputetheratioofreal jt−1 jt−1 j jt−1 (cid:92) (cid:91) GDPgrowth(sRGDP )toPhysicalValueAddedgrowth(PVA ),suchthat: jt jt (cid:92) (cid:32) (cid:33) sRGDP jt X(cid:98)jt µ jt−1 γ j X(cid:98)jt = − −1 (19) (cid:91) PVA jt (cid:91) PVA jt µ jt−1 −1+γ j (cid:91) PVA jt 13

Equation (19) reveals that the direction of the wedge between real GDP growth and PVA growthisaprioriambiguous,andrealGDPgrowthcanbelargerorsmallerthanPVAgrowth depending on the level of markup (µ ), the share of intermediate inputs (γ ), and the ratio jt−1 j X(cid:98)jt . Intable3,wereporttheaveragevalueofeachoftermof(19)forselectedsectors. (cid:91) PVAjt (cid:91) (cid:92) Table.3. InvestigationofDirectionofBiasbetweenPVAandRGDP,sectoraltimeaverage. Terminequation(19) Correlation (cid:91) (cid:16) (cid:91) (cid:17) Sector (1−γ j ) µ jt−1 X(cid:98)jt /PVA jt ρ X(cid:98)jt ,PVA jt USAggregate 0.51 0.13 0.93 0.88 Finance 0.54 0.16 0.47 -0.57 Manufacturing 0.70 0.09 0.09 0.56 RetailTrade 0.39 0.10 3.67 0.22 ComputerSys. 0.34 0.03 0.08 0.16 Agriculture 0.74 0.20 -0.10 -0.63 Clearly, the size of markups and the share of intermediate goods is the highest in the Finance,ManufacturingandAgriculturesectors,threesectorsinwhichthebiaswasreportedto bethehighestinTable2. How can we understand this? From equation (19), it is clear that when markups are zero (µ = 1) or if there is no intermediate good usage (γ = 1), the two measures (sRGDP and jt PVA )wouldbeequivalent,i.e. thereisnobias. jt Supposenowthatµ > 1andγ < 1. Inthiscase,thetwomeasureswouldbesystematically biased downward or upward if PVA do not proportionally move with intermediate inputs usage. As such, the ratio X(cid:98)jt is a key element. Even with µ > 1 and γ < 1, there would (cid:91) PVAjt be no bias if gross output and physical value added were moving proportionally. Generally, thekeypropertyofthisequationisthatabiasbetweenthetheory-consistentmeasureofGDP anditsempiricalcounterpartarisesif: (i)thereisnoone-to-onemappingbetweenintermediate input fluctuations and PVA, (ii) intermediate inputs are used in the production, (iii) there are markups. Moreover,thesignofthebiasisdeterminedby Xjt −1. PVAjt It follows that sectors with a high share of intermediate inputs and high markups, such as Finance, Manufacturing, and Agriculture, are likely to exhibit the largest absolute size of the measurement bias by amplifying movements in X(cid:98)jt −1. To illustrate this point, we distin- (cid:91) PVAjt guish between two types of sectors-year couples. Those with a large intermediate input share andothers. Thefirstcategoryisdefinedassectors–yearinwhicheitherγ andµarebothlarger j thantheiraveragevalue,or(1−γ > 0.5)and(µ > 1.2). Figure3displaystherelationship j jt−1 14

betweenthemeasurementbias(theLHS)andtheratio X(cid:98)jt ontheRHS. (cid:91) PVAjt (cid:91) Figure3. TheroleofmarkupsandintermediateinputshareandrelationbetweenX(cid:98)/PVAandthebias High markups/input share Low markups/input share 1.0 0.8 0.6 0.4 0.8 1.2 1.6 2.0 Markups erahs tupni etaidemretnI 10 5 0 −5 −10 −40 0 40 80 Ratio X/PVA AVP/PDGR oitaR Eachdotcorrespondstoacouple(sector,year).Leftplot:thesolidblacklinesrepresenttheaverageµand1−γ.The (cid:92) (cid:91) horizontaldashedblacklineintherightplotreferstothecasewithoutanymeasurementbias,i.e. sRGDP= PVA, (cid:91) whiletheverticaldashedblackreferstothecasewhereX(cid:98) =PVA. It can be seen that the sensitivity of the measurement bias largely depends on the sectoral markups and intermediate shares. When both components are high, even small variations in the ratio X(cid:98)jt can result in substantial changes in the magnitude of the bias, as illustrated by (cid:91) PVAjt the large red slope in the right chart in Figure 3. Sectors in which fluctuations in intermediate inputs are large but fluctuations in technology, capital, and labor inputs are low, exhibit a strong bias if they also exhibit large markups and if intermediate inputs account for a significantproportionofproduction. To further illustrate the relationship between intermediate inputs and the measurement bias, it is helpful to examine how this sensitivity appears in our selected sectors. Figure 4 (cid:91) displays the relationship between X(cid:98)/PVA and the measurement bias. As expected, there is a clear relationship between the sectoral share of intermediate inputs, the presence of markups, andtheslopeofthebiasasintermediateinputsfluctuaterelativetothePVA. Attheaggregatelevel(inblack),weobservethatmarkupsarerelativelyhigh,withavalue close to 1.15, while the share of intermediate inputs is also high, exceeding 0.5. However, (cid:91) the variations in terms of X(cid:98)/PVA (summed over all sectors) are relatively small and centered (cid:92) (cid:91) aroundthevaluethatgeneratesexactlyameasurementbiasofzero(aratio RGDP/PVAof1), 15

suchthatthereisnotmuchhappeningintermsoftherelationshipbetweenintermediateinputs andthemeasurementbiasattheaggregatelevel. (cid:91) Figure4. TheroleofmarkupsandintermediateinputshareandrelationbetweenX(cid:98)/PVAandthebias Agriculture, forestry, fishing, and hunting Finance and insurance Manufacturing Computer systems design and related services Gross domestic product Retail trade 0.7 0.6 0.5 0.4 1.0 1.1 1.2 1.3 1.4 Markups erahs tupni etaidemretnI 4 2 0 −10 0 10 20 30 Ratio X/PVA AVP/PDGR oitaR Eachdotcorrespondstoacouple(sector,year).Leftplot:thesolidblacklinesrepresenttheaverageµand1−γ.The (cid:92) (cid:91) horizontaldashedblacklineintherightplotreferstothecasewithoutanymeasurementbias,i.e. sRGDP= PVA, (cid:91) whiletheverticaldashedblackreferstothecasewhereX(cid:98) =PVA. In the Appendix, we also present the time series of the measurement bias in the selected sector. Asalreadyshown,theFinanceandManufacturingsectorsexhibitthestrongestvolatility inthemeasurementbias. These findings underscore the importance of considering how the data are constructed when comparing them to the properties of a model. This is particularly relevant for frameworks that aim to assess the impact of markups. Therefore, a careful analysis that accounts for these issues is necessary to draw accurate conclusions about the effects of markups on the economy. 4 Conclusion This paper provides new insights into the discrepancies between statistically measured real value added and productivity measures and their theoretical counterparts used in macroeconomicmodels. Byquantifyingthemeasurementbias,weobtainacomprehensiveandaccurate understandingoftheimpactofmis-measurementofrealvalue-addedattheaggregateaswell assectorallevel. Ourresultshighlightthesubstantialandpersistentmeasurementbiasesinthe 16

currentmethodsofmeasuringrealvalueaddedandproductivity,especiallyinthefinanceand manufacturingsector. Overall,thispaperemphasizestheimportanceofaccuratemeasurement inunderstandingeconomicgrowthandfluctuations. References Arrow, K. J. (1974). The measurement of real value added. In David, P. A. and Reder, M. W., editors,NationsandHouseholdsinEconomicGrowth: EssaysinhonorofMosesAbramovitz,pages 3–19.AcademicPress. Baqaee,D.R.andFarhi,E.(2020). ProductivityandMisallocationinGeneralEquilibrium. The QuarterlyJournalofEconomics,135(1):105–163. Basu,S.andFernald,J.G.(2002). Aggregateproductivityandaggregatetechnology. European EconomicReview,46(6):963–991. Burstein,A.andCravino,J.(2015). Measuredaggregategainsfrominternationaltrade. AmericanEconomicJournal: Macroeconomics,7(2):181–218. DeLoecker,J.,Eeckhout,J.,andUnger,G.(2020). Theriseofmarketpowerandthemacroeconomicimplications. TheQuarterlyJournalofEconomics,135(2):561–644. de Soyres, F. and Gaillard, A. (2021). Value added and productivity linkages across countries. WorkingPaper. Fabricant,S.(1940).ChangesinTotalManufacturingOutput.InTheOutputofManufacturingIndustries, 1899-1937, NBERChapters, pages663–685. NationalBureau of EconomicResearch, Inc. Gopinath,G.andNeiman,B.(2014). Tradeadjustmentandproductivityinlargecrises. AmericanEconomicReview,104(3):793–831. Hall, R. E. (1988). The Relation between Price and Marginal Cost in U.S. Industry. Journal of PoliticalEconomy,96(5):921–47. Kehoe, T. J. and Ruhl, K. J. (2008). Are shocks to the terms of trade shocks to productivity? ReviewofEconomicDynamics,11(4):804–819. Sims, C. A. (1969). Theoretical Basis for a Double Deflated Index of Real Value Added. The ReviewofEconomicsandStatistics,51(4):470–471. 17

A Data Table.4. VariableconstructionusingBEAdata. ConstructionwithBEA Description GOnominal jt µ Markup jt GOnominal−NOS jt jt (cid:32) IInominal (cid:33) Intermediate input share of Gross Output. In jt γ 1− ·µ . practice, we take the time-period average for j GOnominal jt jt eachsector. (cid:32) (cid:33) COMP jt µ jt Labor Share of Value Added. In practice, we α · . j GOn jt ominal γ j taketheaverageovertimeforeachsector. II_QI jt −II_QI jt−1 Proportionalchangeinintermediateinputus- X(cid:98)jt II_QI jt−1 ageforsectorjattimet. CAP_QI jt −CAP_QI jt−1 ProportionalchangeinCapitalinputusagefor K(cid:98)jt CAP_QI jt−1 sectorjattimet. EMP jt −EMP jt−1 ProportionalchangeinLaborusage(measured (cid:98)L jt (cid:98)L jt = EMP jt−1 ashoursworked)forsectorjatt. Twopossiblemethods: µ jt−1 1. ,asin(5) γ j +µ jt−1 −1 Sectoral part of the domar weight – ratio of ω jt−1 salestovalueadded–foreachsectorjatt. GO jt−1 2. ,asin(17) VA jt−1 (cid:92) VA_QI jt −VA_QI jt−1 Proportional change in sectoral real GDP for sRGDP . jt VA_QI jt−1 eachsectorjattimet. d jt−1 ω jt−1 · V V A A _ _ Q Q I I j t t − − 1 1 Domarweightforeachsectorjattimet. 18

Figure5. Timeserieofthemeasurementbias 4 2 0 2000 2005 2010 2015 2020 AVP/PDGR oitaR Agriculture, forestry, fishing, and hunting Computer systems design and related services Finance and insurance Gross domestic product Manufacturing Retail trade Eachdeviationfromthehorizontalsolidblacklineat1isabias. 19

Table.5. SectoraldeviationbetweenRGDPandPVA. Industry ∏ (1+g ) µ γ t t (cid:92) (cid:91) RGDP PVA diff. (pp) t t Grossdomesticproduct 1.63 1.65 -1.86 1.15 0.51 Accommodation 1.22 1.21 1.23 1.14 0.44 Accommodationandfoodservices 1.34 1.30 3.90 1.10 0.50 Administrativeandsupportservices 2.16 2.12 3.60 1.12 0.42 Administrativeandwastemanagementservices 2.07 2.04 3.03 1.12 0.43 Agriculture,forestry,fishing,andhunting 1.54 1.83 -28.30 1.25 0.74 Airtransportation 0.98 0.97 0.93 1.03 0.53 Ambulatoryhealthcareservices 2.28 2.33 -5.81 1.12 0.39 Amusements,gambling,andrecreationindustries 1.11 1.08 3.03 1.07 0.44 Apparelandleatherandalliedproducts 0.37 0.38 -1.19 1.01 0.59 Arts,entertainment,andrecreation 1.38 1.37 1.32 1.15 0.43 Broadcastingandtelecommunications 3.08 3.34 -26.27 1.22 0.58 Chemicalproducts 1.27 1.51 -23.42 1.20 0.69 Computerandelectronicproducts 10.13 10.35 -21.95 1.03 0.39 Computersystemsdesignandrelatedservices 6.91 7.00 -9.27 1.02 0.34 Construction 0.92 0.86 6.23 1.17 0.58 Durablegoods 2.06 2.28 -21.74 1.07 0.65 Educationalservices 1.51 1.50 1.61 1.05 0.34 Electricalequipment,appliances,andcomponents 1.34 1.38 -4.16 1.12 0.62 Fabricatedmetalproducts 0.97 0.99 -2.21 1.10 0.64 Farms 1.51 1.64 -12.69 1.26 0.79 Financeandinsurance 1.78 1.61 17.38 1.19 0.54 Foodandbeverageandtobaccoproducts 1.23 1.30 -6.99 1.11 0.80 Foodservicesanddrinkingplaces 1.40 1.35 4.52 1.09 0.52 Forestry,fishing,andrelatedactivities 1.55 1.65 -9.90 1.14 0.44 Funds,trusts,andotherfinancialvehicles 1.55 0.12 143.04 1.18 0.96 Furnitureandrelatedproducts 0.76 0.76 0.25 1.08 0.65 Information 3.84 4.16 -32.70 1.20 0.53 Insurancecarriersandrelatedactivities 1.98 1.39 58.62 1.24 0.56 Legalservices 0.95 0.79 15.54 1.39 0.39 Machinery 1.11 1.14 -2.89 1.08 0.65 Manufacturing 1.54 1.75 -20.98 1.10 0.70 Mining 1.50 1.65 -15.20 1.10 0.47 Mining,exceptoilandgas 0.70 0.61 9.61 1.24 0.58 Miscellaneousmanufacturing 1.69 1.86 -16.79 1.14 0.57 20

Table.6. SectoraldeviationbetweenRGDPandPVA. Industry ∏ (1+g ) µ γ t t (cid:92) (cid:91) RGDP PVA diff. (pp) t t Motionpictureandsoundrecordingindustries 2.26 2.04 21.76 1.12 0.48 Motorvehicles,bodiesandtrailers,andparts 1.88 1.84 3.49 1.04 0.79 Nondurablegoods 1.06 1.13 -7.31 1.14 0.77 Nonmetallicmineralproducts 1.03 1.06 -2.89 1.13 0.63 Oilandgasextraction 1.62 2.28 -66.10 1.07 0.44 Otherservices,exceptgovernment 0.83 0.78 5.70 1.12 0.43 Othertransportationequipment 1.46 1.52 -6.31 1.10 0.60 Paperproducts 0.79 0.74 5.23 1.09 0.72 Performingarts,spectatorsports,museums,etc. 1.62 1.62 0.61 1.23 0.42 Petroleumandcoalproducts 0.74 0.04 70.65 1.20 0.90 Pipelinetransportation 1.93 0.79 114.49 1.23 0.45 Plasticsandrubberproducts 1.16 1.23 -6.45 1.08 0.72 Primarymetals 1.35 1.56 -21.56 1.05 0.77 Printingandrelatedsupportactivities 0.94 1.00 -5.93 1.08 0.59 Privateindustries 1.67 1.71 -3.33 1.17 0.53 Professional,scientific,andtechnicalservices 2.23 2.26 -2.06 1.17 0.41 Realestate 1.67 1.47 20.60 1.75 0.50 Realestateandrentalandleasing 1.66 1.49 16.60 1.71 0.51 Retailtrade 1.45 1.40 5.06 1.11 0.39 Securities,commoditycontracts,andinvestments 1.68 1.14 53.50 1.00 0.53 Socialassistance 1.68 1.62 6.35 1.07 0.42 Supportactivitiesformining 2.89 2.53 36.21 1.05 0.47 Textilemillsandtextileproductmills 0.57 0.57 0.49 1.02 0.70 Transitandgroundpassengertransportation 1.45 1.21 23.94 1.28 0.44 Transportationandwarehousing 1.26 1.23 3.54 1.09 0.55 Trucktransportation 1.10 1.01 8.68 1.11 0.61 Utilities 1.29 1.19 9.48 1.17 0.52 Warehousingandstorage 3.58 2.65 93.43 1.11 0.46 Wastemanagementandremediationservices 1.40 1.30 9.75 1.14 0.55 Watertransportation 1.44 1.45 -0.70 1.06 0.76 Wholesaletrade 1.57 1.41 15.48 1.19 0.42 Woodproducts 1.17 1.15 1.76 1.06 0.72 21