巴德学院利维经济研究所-聚合生产函数与索洛的“三件牛仔”（英）-2024.3.docx

aaI向Eo1.evyEconomics0InstituteofBardCollegeWorkingPaperNo.1045TheAggregateProductionFunctionandSolovv,suThreeDenials”byJesusFelipe(*)SchoolofEconomics,De1.aSalleUniversity(Manila,Philippines)andJohnMcCombie(*)UniversityofCambridge,Cambridge,UKMarch2024Authorcontactinformation:Felipe(iesus.felir)e(>dlsu.edu.uh):McCombie(jslm2(5)cam.ac.uk)The1.evyEconomicsInstituteWorkingPaperCollectionpresentsresearchinprogressby1.evyInstitutescholarsandconferenceparticipants.Thepurposeoftheseriesistodisseminateideastoandelicitcommentsfromacademicsandprofessionals.1.evyEconomicsInstituteofBardCollege,foundedin1986,isanonprofit,nonpartisan,independentlyfundedresearchorganizationdevotedtopublicservice.Throughscholarshipandeconomicresearch,itgeneratesviable,effectivepublicpolicyresponsestoimportanteconomicPrObIemSIhatPrOfbUndIyaffeciIhequalilyOfIifeinIheUnitedSlaIeSandabroad.1.evyEconomicsInstituteP.O.Box5000Annandale-on-Hudson,NY12504-5000Copyright©1.evyEconomicsInstitute2024AllrightsreservedISSN1547-366XABSTRACTThispaperoffersaretrospectiveviewofthekeypillarofSolow,sneoclassicalgrowthmodel,namelytheaggregateproductionfunction.Wereviewhowthistoolcametolifeandhowithassurviveduntiltoday,despitethreecriticismsthatundermineditsraisond,etre.TheyaretheCambridgeCapitalTheoryControversies,theAggregationProblem,andtheAccountingIdentity.Thesecriticismswereforgottenbytheprofession,notbecausetheywerewrongbutbecauseofthekeyroleplayedbyRobertSolowinthefield.Today,thesecriticismsarenotevenmentionedwhenstudentsareintroducedto(neoclassical)growththeory,whichispresentedinmosteconomicsdepartmentsandmacroeconomicstextbooksastheonlytheoryworthstudying.JE1.C1.ASSIFICATION:B22,B31,B32,B41,E13,E25KEYWORDS:AccountingIdentity,AggregationProblem,CambridgeCapitalTheoryControversies,SolowINTRODUCTIONRobertSolow,oneofthemostnotableeconomistsofthetwentiethcenturydiedDecember21,2023,attheageof99.Hewascertainlyagreatresearcher,someoneinterestedinsociety,srealproblems,andthosewhostudiedatMITduringhistenureconsiderhewasagreatinstructor.SolowarrivedatMITin1949asaprofessorofstatistics.TogetherwithPaulSamuelson,hebuiltoneofthebestregardedeconomicsdepartmentsintheworld.Botheconomistswereinstrumentalinlayingthefoundationsoftheneoclassicalsynthesis.WhilethistooksomeelementsofKeynes,GeneralTheory,itwas,insomerespects,areturntothepre-Keynesianworld.Theneoclassicalsynthesishadthreepillars:microeconomiccompetitivegeneralequilibriumtheory,thePhillipscurve,andSolow,s(1956)growthmodel.ThesynthesisassertedtheefficacyofKeynesianpolicyintheshortrun,butassertedabeliefincompetitivemarketforcesandSay,s1.awinthelongrun(inparticularinthecaseofSolow,sgrowthmodel).Asaneconomistinterestedinsociety,srealproblems,Solowwasconcerned,interalia,withthecausesofunemployment(Solow1980),andviewedthelabormarketasasocialinstitutionandnotlikethetypicalmarketforagood(Solow1990).Hetookpartinthetwomostimportantintellectualdebatesineconomicsofthetwentiethcentury:theCambridgeCapitalTheoryControversies(CCTCs)(mainlywithJoanRobinsonintheUK)andwiththenewclassicalmacroeconomists(mainlywithMiltonFriedmaninChicago).SolowwasawardedtheNobelPrizein1987forhiscontributionstothetheoryofeconomicgrowth.Solow,s(1956)growthmodelincorporatedanaggregateproductionfunction(APF).Thisconceptbecameapillarofneoclassicalmacroeconomicsandcertainlyofsubsequentneoclassicalgrowthmodels.Productionfunctionsappearedineconomicsinthenineteenthcentury.Itwasnot,however,untilCobbandDouglas(1928)thatonewasestimatedfortheAmericaneconomyusingstatisticalmethods,whichrequiredusingaggregatedata(indices)foroutput,labor,andcapital.Inthisarticle,theyfoundtheelasticitiesofoutputwithrespecttolaborandcapitaltobe0.75and0.25,respectively,closetothefactorshares.Theirworkwasinitiallyreceivedwithgreathostility,tothepointthatDouglasconsideredgivingup.Hemovedfromtheoriginalworkwithtimeseriestocross-sectionaldata,whichproducedmuchbetterstatisticalresults(McCombie1998;FelipeandMcCombie2013,chapter4).AggregateproductionfunctionscontinuedbeingusedduringthefollowingdecadesbutitwasnotuntilSolow,s(1956,1957)twoseminalpapersthattheybecameastandardconceptinmacroeconomics.Thiswasdespitesomedoubtsabouttheirtheoreticalandempiricalfoundations,asweshallseebelow.Solow,s(1956)growthmodelisstillusedtodaytointroducestudentstogrowththeory.InSolow(1957),theAPFservedasthetooltoempiricallyapportionthecontributionsoftechnicalprogress(totalfactorproductivitygrowthorSolowresidual)andthoseofcapitalandlaborgrowth,tooutputgrowth,intheUSeconomybetween1909and1949.Althoughtotalfactorproductivityhadbeencalculatedbefore,thiswasthefirsttimeithadbeenexplicitlylinkedtoproductiontheory.The1957paperlaunchedtheneoclassicalresearchprogramongrowth,inparticulargrowthaccountingexercisesandtheestimationoftotalfactorproductivitygrowth.ThepresentpaperoffersaretrospectiveviewoftheAPF,inparticularofthreefundamentalcriticismsitwassubjectedto,andofhow,despitethesecriticisms,ithassurvivedthetestoftime.ThesecritiquesaretheCambridgeCapitalTheoryControversies(CCTCs),theAggregationProblem,andtheAccountingIdentity.Itisarguedthatthesethreecriticismsunderminedtheraisond,etreoftheAPF,Solow,sgrowthmodel,andgrowthaccountingexercises.WithouttheAPF,theneoclassicalmacroeconomicsedificewouldlargelycollapse.Hence,itsputativedefenseatthetimethatthesecritiqueswereadvancedbecameamatterOfcrucialimportance.However,allthreehavelongbeenforgottenbytheprofession,andnotbecausetheywerefundamentallyflawedbutbecauseofthekeyrolethatSolowplayedinthedebates.Obviously,Solowwasnottheonlyone,buthewasthe"fther“oftheneoclassicalgrowthmodel(forwhichhereceivedtheNoblePrize)andofthisliterature.Hewasawareofthethreecriticismsandplayedamostimportantroleindismissingthem.HeclearlydidsointhecaseoftheCCTCs,ignoredtheaggregationresults,anddeniedtheveracityoftheaccountingidentityargument.Today,theseargumentsarenotevenmentionedwhenstudentsareintroducedtoneoclassicalgrowththeory,whichispresentedinmosteconomicsdepartmentsandmacroeconomicstextbooksastheonlytheoryworthstudying.Thcstandardtextbooksongrowththeoryinthe1970ssuchasWan(1971),Jones(1975)andHacche(1970),mentionedtheCCTCs.Wan(1971)andJones(1975)alsomentionedheaggregationproblems.Bythe1990s.therewasnodiscussioninthegrowthtextbooksofeitherofthese,e.g.,BarroandSala-i-Martin(1995),Jones(19982002),AghionandHowitt(1998),Wcil(2(X)5),orAcemoglu(2009).ForalternativegrowththeoriesseeSen(1970),Scott(1989),andSetterfield(2010).ItshouldbeemphasizedthatthediscussionheredoesnotintendtoundermineallthemanyimportantcontributionsthatSolowmadetoeconomics.Yet,itisimportanttosettherecordstraightontheaboveissues.Therestofthepaperisstructuredasfollows.Section2discussestheCCTCs.Section3discussestheAggregationProblem.Section4discussestheAccountingIdentitycritique.Section5concludes.TheAppendixintroducesthealgebraoftheaccountingidentityargumentandrevisitsSolow(1957).2.DENIA1.1:THECAMBRIDGECAPITA1.THEORYCONTROVERSIES“Thestudentofeconomictheoryistaughttowrite0=f(1.,C)where1.isaquantityoflabour,Caquantityofcapitaland0arateofoutputofcommodities.Heisinstructedtoassumeallworkersalike,andtomeasure1.inman-hoursoflabour;heistoldsomethingabouttheindex-numberprobleminvolvedinchoosingaunitofoutput;andthenheishurriedontothenextquestion,inthehopethathewillforgettoaskinwhatunitsCismeasured.Beforeeverhedoesask,hehasbecomeaprofessor,andsosloppyhabitsofthoughtarehandedonfromonegenerationtothenexf,(Robinson1953-54,81).“InherpaperTheProductionFunctionandtheTheoryofCapitalMrs.Robinsonwasannoyedatmanyofthepracticesofacademiceconomists.Wehavereasontobegratefulforherannoyance,forsheseemstohavewrittenherarticlethewayanoystermakespearls-outofsheerirritation(Solow1955-56,101).TheCambridgeCapitalTheoryControversies(CCTCs)wereaseriesofexchangesbetweentwogroupsofeconomistsonbothsidesoftheAtlantic.Theyweretheoretical.CambridgeUKeconomists,ledinitiallybyRobinson,questionedtheverydeepfoundationsofneoclassicalmacroeconomics.CambridgeUSAeconomistsprimarilyledbySamuelsonandSolow,counterargued.Summarizingtheexchangesbetweenthetwosidesisarathercomplicatedtask.Harcourt(1969,1972)providesanexcellentsummaryanddiscussionoftheliterature.TherearemanygoodsummariesoftheCCTCs.CohenandHarcourt(2003)andFelipeandMcCombie(2013)alsoprovidesummaries.Inthefinalanalysis,thedebateswereaboutdifferentvaluetheories,theclassical(Smith,Ricardo,Marx)versustheneoclassical(Bohm-Bawerk,Jevons,Clark,Wicksteed,Wicksell,Walras),andhowthosetwotheoriesexplainpricesanddistribution.Thedebatescenteredonaseriesofissuesderivedfromandlinkedtothequestionofwhetheronecanuseanaggregatemeasureofcapitalinamacroeconomicproductionfunctionwithoutrunningintoapparentlyparadoxicalphenomena.TheAPFappearedintheCCTCsinRobinson,s(1953-54)article.Comingfromaclassicaltradition,capitalintheproductionfunctionneededtobeintheformofphysical,homogeneousmachines.Theseweretermed“leets"(Steelspeltbackwards)byRobinson.However,shearguedthatthiscouldnotbethecaseattheaggregatelevel.Atthislevel,ithadtobeavaluemeasure,withtheconsequencethatthisnotionofcapitalwasconflatingthenotionsofphysicalcapital(alistofheterogeneousmachines)andfinancialcapital(afundofresources,“money"),whichwereverydifferentinclassicaleconomics.Theneoclassicaleconomistsusedthefirstnotionofcapitalinthestudyofinterestandportfolioadjustment;capitalinphysicaltermswasusedinthetheoreticalstudyofproduction(Clark1893).Clark,however,arguedthat,overlongperiodsoftime,changesinthevalueofcapitalreflectedchangesinthestockofinvestmentgoods.Thiswasimperativeinordertoshowthat,inalaissez-faireeconomy,eachindividual(factorofproduction)whocontributedtoproductionreceivedthevalueofwhatheorsheproduced(inanormativesense,deserved).Thiswas,ofcourse,thepointofthemarginalisttheoryofproduction:thewagerateequalsthemarginalproductoflabor,andthereturnoneachdollarofcapitalequalsthemarginalproductofcapital.Hence,eachsocialclassgetswhatitcontributestoproduction(naturally,thereisaclearcontrastbetweenthisviewoftheworldandthatprovidedbythelabortheoryofvalue,inparticulartheMarxianversion).TheCambridgeCapitalControversieswereplaguedwithconfusions.Themostimportantisbetweenthelegitimacyofpostulatinganaggregateproductionfunction,andthelegitimacyofthemarginalist-neoclassical,orthesupply-demand,approachtodistribution.Theimportantquestionisthesecondone.Itisonlywhenoneacceptsthemarginalistapproachthatonemayfeellikeusingaggregateproductionfunctionswhosemarginalproductsdeterminedistribution.So,onemustask:whatkindofaggregationisneededtorthevalidityoftheneoclassicalapproachto(macro)distribution?WhatwastheproblemRobinson(1953-54)pointedoutandwhichinitiatedtheCCTCs?ShecriticizedthatthedistinctionmadebytheneoclassicaleconomistsbetweencapitalasamonetaryfundandcapitalasacollectionofdifferentcapitalgoodswascompletelylostaftertheKeynesianRevolution,inparticular,inempiricalanalyses.Infact,accordingtoRobinson,thetwoconceptionsofcapitalweremixed.Atsomepointintime,economistsbegananalyzingtheperformanceoftheeconomyintermsofanaggregateproductionfunctionwith“capital”andlaborasfactorsofproduction,andbegandiscussingtheremunerationofthesefactorsintermsoftheirmarginalproductivities.Asaconsequence,itappearedthatthedivisionoftheaggregateproductbetweenlaborandcapitalcouldbeexplainedintermsofmarginalproductivities.Robinson(1953-54)askedtheimportantquestionthattriggeredthedebate:inwhatunitis“capital“tobemeasured?Robinsonwasreferringtotheuseoftcapita,asafactorofproductioninaggregateproductionfunctions.Becausecapitalgoodsareaseriesofheterogeneouscommodities(physicalinvestmentgoods),eachhavingspecifictechnicalcharacteristics,itisimpossibletoexpressthestockofcapitalgoodsasahomogeneousphysicalentity.Robinsonclaimedthatitfollowedthatonlytheirvaluescouldbeaggregated.Suchavalueaggregate,however,isnotindependentoftherateofprofitandthusofincomedistribution.Thisheterogeneityofcapitalbecameanimportantpartofthecontroversies(atleastinitiallyandcertainlyforRobinson).TheClarkianconceptofcapital,conceivedasafairlyhomogeneousandamorphousmasswhichcouldtakedifferentforms,Robinsonargued,cannotserveinamacroeconomicproductionfunctionbecauseitisessentiallyamonetaryvalue.Sheclaimed,perhapsimplausibly,thatalthoughlaborisnotahomogeneousinput,inprincipleitcanbemeasuredinatechnicalunit,i.e.,man-hoursofwork(disregardingdifferentqualities).Thesamegoesforland(acresoflandofagivenquality).Thesearenaturalunits,sothatthemarginalproductsoflandandlaborcouldbedefinedindependentlyoftheequilibriumfactorprices.Butwhatabout(aggregate)capital?Robinsonarguedthatthestatisticsofcapitalusedinpracticehadnothingtodowiththeprevioustwonotionsofcapital.Suchstatisticsarevalues(e.g.,dollars)andnomatterhowtheyaredeflated(toconvertthemintoconstantdollars),theycontinuetobemoney-asumofvalue.Therefore,thiscannotbemadetocorrespondtoaphysicalfactorofproduction.Allthismattersbecause,itwasclaimed,itisimpossibletogetanynotionofcapitalasameasurablequantityindependentofdistributionandprices.Thatis,ifdistributionistobeexplainedbytheforcesofsupplyofanddemandforthefactorsofproduction,thenthelattermusteachhaveameasure.Thosemeasuresmustbehomogeneoussothataggregationispossible.Thiswasclaimednottobepossibleforcapitalasafactorofproduction,butonlywhenitisanamountoffinance.Thus,capitalhasnonaturalunit,akintolaborandland,whichcanapparentlybeaggregatedtogiveaquantityofaproductiveservice,thatcanthenbeusedforthedeterminationoftheirprices.Wenote,however,thatsuchaggregationisillusory,andtheaggregationproblemisnotrestrictedtocapital.WasonelaborhourbyRobinsonreallythesameasonebyQueenElizabethIIoroneofBritneySpearsintermsofproductivity?Theanswerisno.TheargumentoftheCambridge,Englandeconomistswasthatinvestmentgoods,whichmakeupthestockofrealcapitalarethemselvesproduced(i.e.,producedmeansofproduction).Thesegoodsareproducedinamarketeconomywherecapitalistsrequireprofits.Thisimpliesthatinordertoprovidea“quantityofcapita,onemustfirstknowitsprice.Infact,thepriceofanycommoditycannotbedeterminedindependentlyofthetechnicalconditionsofproductionandoftherateofprofits.Inotherwords,thepriceoftheaggregatefactorcapitalisaffectedbythedistributionofincomeamongthefactors.Thevalueofcapitalchangesastheprofitandwagerateschangesothatthesamephysicalcapitalcanrepresentadifferentvalue,whereasdifferentstocksofcapitalgoodscanhavethesamevalue(Robinson1956).Solongasthecapitalstockisheterogeneous,itsmeasurementrequiresknowledgeoftherelativevaluesofindividualcapitalgoods.Thiscanonlybeachievedifthepricevectoroftheeconomyandtherateofprofitswereknownexante.Theconsequenceisthataggregatecapital,theaggregateproductionfunction,andthemarginalproductsofthefactorscanonlybedefinedwhentherateofprofitisgiven,implyingthattheycannotbeusedtobuildatheoryoftherateofprofitordistribution.Robinson,scritiqueledtoaseriesofintertwineddebatesthatlastedtwodecadesandwhichwentfarbeyondtheoriginalquestion.Themaindebateswereaboutthefollowingfiveissues.(i) Thetheoryofdistribution,inparticulartheneoclassicalclaimthatthedistributionofincomecouldbederivedfromsometechnicalpropertiesofaneconomy,embeddedintheproductionfunction,andthatfactorsharescouldbesomehowlinkedtothemargina