世界銀行 -糧食商品產(chǎn)量增長模式 Yield Growth Patterns of Food Commodities Insights and Challenges

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PolicyResearchWorkingPaper10990

YieldGrowthPatternsofFoodCommoditiesInsightsandChallenges

JohnBaffesXiaoliEtienne

WORLDBANKGROUP

DevelopmentEconomicsVicePresidencyOfficeoftheChiefEconomist

December2024

PolicyResearchWorkingPaper10990

Abstract

Understandingglobalfoodproductionandproductivitypatternsiscrucialforpolicyandin-vestmentdecisionsaimedataddressingpoverty,foodinsecurity,andclimatechange.Thispa-perdevelopscomprehensivecalorific-basedproductionandyieldindexesfor144crops,covering98percentofglobalagriculturallandandfoodoutput.Theseindexesprovidestandardizedmeasuresacrossvariouscropsandvarieties,facilitatingcomparisonofagriculturalproductivityandconsolidatingcountryandregionalcon-tributionstoglobalfoodproduction.UtilizingaBox-Coxtransformation,theanalysisfindsthatalinearmodelbestapproximatesyieldgrowth.Thefindingsrevealthat,atanaggregatelevel,therehasbeennodiscernableslowdowninglobalyieldgrowthoverthepastsixdecades.Thistranslates

intoanaverageannualyieldincreaseequivalenttonearly33kilogramsofwheatperhectare.Theseresultssuggestthatanyobserveddecelerationinspecificcommodities,regions,orcountrieshasbeenoffsetbygainsinothers.Whilethesefindingsarereassuringfromaglobalfoodsupplyperspective,cautioniswarrantedaboutthesustainabilityofproductionandtheaffordabilityoffood.Theseconcernsareparticularlyrelevantasglobalfooddemandincreasesduetopopula-tionandincomegrowth,andasthepressuresfromclimatechangeintensify.Thestudyunderscorestheimportanceofadoptingstrategicandsustainableagriculturalpracticestoensurecontinuedfoodsecurityinthefaceofevolvingglobalchallenges.

ThispaperisaproductoftheOfficeoftheChiefEconomist,DevelopmentEconomicsVicePresidency.ItispartofalargereffortbytheWorldBanktoprovideopenaccesstoitsresearchandmakeacontributiontodevelopmentpolicydiscussionsaroundtheworld.PolicyResearchWorkingPapersarealsopostedontheWebat

/prwp

.Theauthorsmaybecontactedatjbaffes@.

ThePolicyResearchWorkingPaperSeriesdisseminatesthefindingsofworkinprogresstoencouragetheexchangeofideasaboutdevelopmentissues.Anobjectiveoftheseriesistogetthefindingsoutquickly,evenifthepresentationsarelessthanfullypolished.Thepaperscarrythenamesoftheauthorsandshouldbecitedaccordingly.Thefindings,interpretations,andconclusionsexpressedinthispaperareentirelythoseoftheauthors.TheydonotnecessarilyrepresenttheviewsoftheInternationalBankforReconstructionandDevelopment/WorldBankanditsaffiliatedorganizations,orthoseoftheExecutiveDirectorsoftheWorldBankorthegovernmentstheyrepresent.

ProducedbytheResearchSupportTeam

YieldGrowthPatternsofFoodCommodities:InsightsandChallenges

JohnBaffes

TheWorldBank

XiaoliEtienne

UniversityofIdaho

JELCLASSIFICATION:Q10,Q18,Q11

KEYWORDS:yieldgrowth,calorie-basedapproach,foodcommodities,foodinsecurity

AnearlierversionofthepaperappearedinPloSONE.TheprojecthasbeenpartlysupportedbytheWorldBank’sProspectsGroup,theIntramuralResearchProgramoftheU.S.DepartmentofAgriculture,NationalInstituteofFoodandAgriculture,ResearchCapacityFund(AccessionNo.7005605),andtheIdahoWheatCommission.Viewsandfindingsarethoseoftheauthors,nottheaffiliatedinstitutions.

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1.Introduction

Astheglobalpopulationapproaches10billionbymid-century,agriculturalproductivitywillbecomeincreasinglycriticalinfeedingtheworld(Rayetal.2013).Overthepastsixdecades,productivityimprovementshaveaccountedformuchofthegrowthinfoodpro-duction.However,therateofyieldgrowthforfoodcommodities,akeymeasureofproductivity,hasbeenperceivedtohavestagnatedinrecentdecades(Alstonetal.2009;FAO2009;Grassinietal.2013;Rayetal.2012).Thisassessmenthasledtoconcernsre-gardingfoodavailability,especiallyinlow-andmiddle-incomecountrieswherepopula-tiongrowthratesarethehighest,includingSub-SaharanAfrica(VanIttersumetal.2016).Theslowdownhasalsobeencitedasacauseofcommoditypricespikesandpricevola-tility(HelblingandRoache2011).

Traditionally,yieldgrowthpatternshavebeenanalyzedusingsingle-commodity,weight-baseddata(e.g.,kilogramsormetrictonsperunitofland)atcountry,regional,andgloballevels.Whilethebulkoftheempiricalstudiesfindswidespreadyieldgrowthdecelerationorstagnation(Grassinietal.2013;Lietal.2016;Rayetal.2013),closerex-aminationrevealsconsiderableheterogeneityacrosscropsandregions.Notably,high-incomecountriesthatexperiencedlargeyieldgainsinresponsetotheGreenRevolutionearlierinthe20thcenturyappearedtohaveexperiencedstagnationorevendeceleration.Certainlow-incomecountries,especiallyinSub-SaharanAfrica,alsofacedsimilarprob-lemsduetolimitedaccesstohigh-yieldingvarietiesandproductioninputs.However,contrarytothestagnationanddecelerationnarrative,yieldgrowthaccelerationhasbeenobservedforvariouscommoditiesandregions(Finger2010).

Thisheterogeneityinyieldgrowthpatternsacrosscropsandregionsraisestwocrucialresearchquestions:Firstly,hasaggregateglobalcropyieldexperiencedaslow-downorstagnation?Secondly,howsimilarordissimilarareyieldgrowthpatternsacrosscommodities,countries,orregions?Althoughsingle-commoditymodelsareusefulforidentifyingsupply-drivenissuessuchastheeffectsofweatherpatterns,climatechange,ortechnologicalimprovementsforaspecificcrop,theydonotprovideacomprehensiveassessmentofaggregateyielddynamicsacrossdiversecropsandregions.Indeed,ana-lyzingsustainabilityissuesandaddressingfoodsecurityconcernsrequiresamodelingframeworkthatallowsustoevaluatetheaggregategrowthpatternsnotonlyacrosstheentirefoodcropspectrum,butalsoaccountsforchangingcroppatternsthatmaybedrivenbyinputcosts,domesticandtradepolicies,aswellasdemand-sideconsiderations,includingchangesintastesandpreferences.

Againstthisbackground,thepresentpaperaddressesthetworesearchquestions

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byintroducingacalorific-basedapproachtotacklethelimitationsoftheexistinglitera-ture.Insteadoftraditionalweight-basedmeasuressuchasmetrictonsorkilograms,weconvertannualcropproductionintocalorificcontentandsubsequentlyaggregatetheproductionandyieldofallfoodcommoditiesintosinglemetrics.Thisaggregationac-countsforheterogenousyieldpatternsduetochangesinproductioncomposition,thetransitionfromlow-yieldtohigh-yieldcropsandvarietiesaswellasshiftsincountryandregionalsignificance.Additionally,thisapproachprovidesaneasy-to-implement,yetstandardizedanduniversalframeworkforanalyzingandcomparingyieldgrowthatanylevel,fromindividualcropstocommoditygroupstoglobalaggregates.

Usingthesecalorific-basedindices,weapplystatisticalmethodstoselectthemostappropriatemodelforchartingtheyieldpathsofcommoditiesatglobal,regional,andcrop-grouplevels.Productionandcalorificcontentdatafor144majorcropscoveringthe1961-2021periodfromtheFoodandAgricultureOrganization(FAO)areincludedintheanalysis.Thesecropscombinedaccountforapproximately98percentoftheworld’sag-riculturallandarea.Theevidencesuggeststhattheaggregateglobalyieldindexhasnotbeensubjectedtogrowthdecelerationoverthepastsixdecades.Thus,slowgrowthincertaincommodities,regions,orcountries,documentedintheliteraturehasbeenoffsetbyacceleratedgrowthinothers.

Theremainderofthepaperproceedsasfollows.Thenextsectionsummarizestheliteratureonyieldgrowth.Section3discussestheaggregateyieldindex,themodelingframeworkforevaluatingyieldgrowth,andthedata.Section4presentstheresults.Thelastsectionconcludesanddiscussesavenuesforfurtherresearch.

2.ABriefReviewofLiterature

Theliteratureassessingyieldgrowthperformancecanbebroadlydelineatedintothreeprincipalstrands.Onestranddelvesintoyieldgrowthfromexperimentaldata,servingasatoolforplantscientiststodiscernperformancenuancesandfacilitatetheselectionofcropvarieties(Crow1998;Ki?retal.2009;ReissandDrinkwater2018).Thesecondstrandfocusesonevaluatingthestatisticaldistributionofyields,withtheexplicitobjectiveofassistingfarmerstomakeinformedselectionofcropvarieties,whilealsoaidingthein-suranceindustryindeterminingpremiaforpotentialcroplosses.Pioneeringcontribu-tionstothisstrandincludeanearlystudybyDay(1965),subsequentlyexpandeduponbyJustandWeninger(1999),Atwoodetal.(2003),Norwoodetal.(2004),andSherricketal.(2004).Thethirdstrand,particularlypertinentinthepresentcontextandthesubjectofthissection,examinesyieldgrowththroughthelensofsustainabilityandfoodsecurityconsiderations.

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Underthisstrandofliterature,determiningwhetheryieldgrowthhasencoun-tereddecelerationorstagnationhasbeenatopicofsubstantialdebatewithmoststudiesfocusingongrainandoilseedcrops.Atthegloballevel,Alstonetal.(2009)conductedacomprehensivestudydocumentingadiscernibleslowdowninthegrowthofgrainyields.Theirfindingsunderscoredthepotentiallyfar-reachingimplicationsforfoodpricetrendsshouldthisdecelerationpersist.Cassman(1999)contributedtothisdiscoursebyhigh-lightingthattherateofyieldincreaseforcerealcropspotentiallyfellsignificantlyshortoftheanticipatedriseinfooddemand.ExaminingthetrendsinyieldsfortheBig-4com-modities(maize,soybeans,wheat,andrice),Rayetal.(2012)concludedthat,whileyieldscontinuetoascendinmanyregions,anoteworthyproportion—rangingfrom24%to39%oftheBig-4growingareas—exhibitsaconcerningpatternofeithernegligibleimprove-ment,stagnation,oroutrightcollapse.SubsequentresearchbyRayetal.(2013)revealedthatthegrowthratesofyieldsinthesecommoditiesfallconsiderablyshortofthelevelsrequiredtomeettheprojecteddemandforfoodcommoditiesby2050.

Attheregionallevel,VanIttersumetal.(2016)assertedthatthestagnationofyieldsinSub-SaharanAfricapresentsaformidablefoodsecurityrisk,particularlyasthepopulationoftheregionisanticipatedtoreach2.1billionby2050(from1.4billionin2020).Usinggrid-leveldata,Iizumietal.(2014)foundincreasedyieldinstabilityfortheBig-4commoditiesacrossabroadregionoftheSouthernHemisphere,corroboratingpre-viousfindingsofyieldstagnationandcollapsesintheseregions.LinandHuybers(2012)examinedwheatyielddataacross47regions,findingthatapproximatelyhalfofthepro-ductionwithintheirsamplecontinuedwithalineargrowthtrend,whiletheremainderexhibitedyieldstagnation.MichelandMakowski(2013)estimatedthatwheatyieldgrowthexceeded0.06tonperhectareperyearin1961-2010invariouscountriesacrossEurope,Asia,Africa,andtheAmericas,butitexperiencedstagnationinmanyothercoun-tries.Focusingon24Africancountriesfrom1960to2012,Saitoetal.(2015)foundthat15countriesexperiencedacceleratingyieldgrowth,whiletherestsawstagnationordecline.

Numerousstudiesexaminedcountry-levelcropyieldgrowthaswell.InafocusedanalysisofChina’syieldperformancespanning1980-2010,Lietal.(2016)observedyieldstagnationin50%ofrice-producingareas,54%ofmaize-producingareas,andnearly16%ofwheat-producingareas.InthecontextofanalyzingfoodsecurityinChina,Weietal.(2015)determinedthatriceyieldsfacepronouncedstagnationin53.9%oftheregionsexamined,followedby42.4%inmaizeand41.9%inwheat.Finger(2010)foundthatwhiletheyieldgrowthofmaize,barley,andryeremainedatalineartrendinSwitzerlandfrom1961to2006,othercommodities,includingwheat,experiencedyieldgrowthdecelera-tion.Examiningyieldsin29Indianstatesforthe1967-2017period,Madhukaretal.(2020)

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foundthat76%,47%,and18%oftheharvestedareasdidnotshowyieldimprovementintherecentdecadeforwheat,rice,andmaize,respectively.

Notwithstandingtheprevailing,somewhatpessimistic,outlookinthestudiesre-viewedabove,itisessentialtonotethatnotallresearchalignswithsuchaperspective.Inaninfluentialwork,Alexandratos(1999)contendedthatglobalagriculturalproductionispoisedtomeet,orpotentiallyexceed,thedemandsoffoodrequirements.Alexandratosalsoemphasizedthatthecriticalconcernliesinthepersistentchallengesofpovertyinlow-incomecountries.Similarly,aneditorialinthejournalNature(2020)offeredanu-ancedperspectiveonproductivitygrowthandfoodsecurity,notingthatdependingonresearchadvancementsinthesector,itisfeasibletomeetthe2050globalfooddemandatanacceptablecost.Ausubeletal.(2013)echoedsimilarviews,highlightingthesubstantialincreaseincropyieldsoverthepast50years,alongsideamarkeddeclineincaloricre-quirementsrelativetoGDP.

Tosummarize,theliteratureprovidesmixedevidenceonyieldgrowthtrends,withfindingsvaryingbasedonthespecificcommodities,countries,regions,andtimeperiodsanalyzed.Whilesomestudiesindicateadeclineorstagnationforcertaincropsandregions,othersreportanaccelerationorconsistencywithalineargrowthtrend.Thisraisesacriticalquestion:fromaglobalperspective,hasaggregatecropyieldexperiencedstagnationordeceleration?Moststudieshavepredominatelyemployedweight-basedyieldtoanalyzeproductivitygrowthforsinglecommodities.However,thisapproachdoesnotadequatelycapturetherateofproductivitychangeacrossallcombinedcom-modities,whichiscrucialforanalyzingfoodsecurityandsustainabilitychallenges.Inthisstudy,weproposeconstructingglobalyieldgrowthindicesbasedonthecalorificcontentofcropstoassessaggregateyieldgrowthpatternsattheglobal,regional,andcommoditylevels.Thesepatternsarediscussedinthefollowingsections.

3.MethodsandData

Indiceshavebeenusedinseveralcontextswithincommoditymarkets,suchastheaggre-gationofvariouscommoditypricesintoasingleindexandtheconstructionofaggregateagriculturalproductivityindices.Acriticalissueintheaggregationprocessistheselec-tionofappropriateweights.Forinstance,theweightsusedintheWorldBank’scommod-itypriceindicesarebasedontheexportvaluesofemergingmarketsanddevelopingeconomies(WorldBank2024).Foragriculturalproductivitystudies,weightsaretypicallybasedonvaluesderivedfromFAOinternationalcropprices,measuredinGeary-Khamisdollarsperton,alsoknownas“internationaldollars”(PrasadaRao1993).NumerousstudieshaveemployedFAO-basedaggregation,includingAdamopoulosandRestuccia

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(2022)whostudiedcross-countryagriculturalproductivitybasedonmicro-plotleveldata;Gollinetal.(2014)whoexaminedagriculturalproductivitydifferencesacrosscoun-triesformaize,rice,andwheat;Mekonnenetal.(2015)whoinvestigatedtechnicaleffi-ciencyindevelopingcountries;andNin-Prattetal.(2010)whocomparedagriculturalproductivitygrowthinChinaandIndia.

However,usingweightsbasedonvaluesderivedfromcommoditypricesforspe-cificyearscanintroducebiasforseveralreasons.Firstly,whilethereareestablishedin-ternationalpricesequencesforkeyagriculturalcommoditiessuchastheBig-4,mostothercommodities,whicharelessfrequentlytraded,lackbenchmarkprices.Second,inmanylow-incomecountries,geographicalisolationandhightransportationcostsrendermanyagriculturalcommoditiesnon-tradable(Delgado,Minot,andTiongco2005;Delgado1995).Third,tradedistortionsoftencausedomesticpricestodivergefrominternationalbenchmarks(BaffesandGardner2003).Additionally,manycommoditypricesaresubjecttolong-termtrends,medium-termcyclicality,andshort-termvariability,implyingthatselectingdifferentsub-periodscanyieldvaryingresults(BaffesandEtienne2016;BaffesandKabundi2023).Therefore,yieldaggregationbasedonvalueterms,evenwhenad-justedforinflationorpurchasingpower,cansufferfromrapidlychangingweightsduetopricevolatility.Toaddresstheseissues,weemploycalorific-basedweightsinourcon-structionofglobalandregionalproductionandyieldindices.

Calorific-basedindiceshavebeenwidelyusedontheconsumptionsideinvariouscontexts,suchascalculatingfoodrequirementsforbalanceddietsandestimatingbudg-etaryneedstoensurehealthynutrition(Bekaert1991;SibhatuandQaim2017).However,theirapplicationontheproductionsidehasbeenlessfrequent,thoughtheyhavebeenutilizedinarangeofcontexts.Datingbackto1942,WilliamsonandWilliamson(1942)usedcalorificcontentoffoodcommoditiestodiscernpatternsinfoodconsumption.Rob-ertsandSchlenker(2009)extendedthisapproachbyconvertingtheproductionoftheBig-4commoditiestoidentifysupplyanddemandelasticitiesinthecontextofthe2007-08pricespike.Inasimilarvein,Bobenriethetal.(2013)calculatedstocks-to-useratiosformajorgrainsandanindexoftotalcaloriesfromthesegrains,servingasindicatorsofvul-nerabilitytofoodshortagesandpricespikes.D'Odoricoetal.(2014)contributedtothislineofinquirybycalculatingthewaterintensityofinternationallytradedfoodcommod-ities,employingacalorificaggregation.Additionally,Cassidyetal.(2013)leveragedthecalorificcontentoffoodcommoditiestoassessarearequirementsforhumanfoodcon-sumption.

Ourstudyextendsthisstrandofliteraturetoevaluateaggregatecropyieldgrowth.Theremainderofthissectiondetailstheconstructionofanaggregateyieldindex

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anddiscussesthekeyassumptionsunderlyingcommonyieldgrowthestimationap-proaches.Italsoexaminesthemodelingframeworkusedtoselecttheappropriatespeci-ficationforgrowthrateestimation,addressinghowtomanagestructuralbreaksandnon-linearities.Additionally,itprovidesabriefoverviewofthedatausedintheanalysis.

3.1Theaggregateyieldindex

Theaggregatecalorie-basedyieldindex,yt,iscomputedas:

wiQit/(1)

whereQitdenotesthetotaloutputofcommodityiatyeartinweightunit;wirepresentsthecalorificcontentofcommodityiperweightunit;andLitislandallocatedtocommod-ityiatyeart.Thus,ytrepresentsthenumberofcaloriesproducedperlandarea(inhec-tare).Inadditiontoagivencountry,theindexcanbeconstructedfortheentireworldoranyregionoraggregatecommoditygroupofinterest.

Commoncalorificunitsusedincludecal(smallcalorie)andCal(largecalorie).Asmallcalorieisdefinedas“…theamountofheatrequiredtoraisethetemperatureof1gofwaterby1°Cwithatemperaturechangefrom14.5to15.5°C.ThecurrentUSDietaryReferenceIntakesdefine1calas4.186J[joules]”(Hargrove2007).Onelargecalorieisequivalentto1,000smallcalories,or1Cal=1,000cal.OftenCalisdenotedaskcal,anotationusedinfoodlabeling.Fornotationalconvenience,thepresentpaperusesKCalandMCal,whichequal1,000largecaloriesand1,000,000largecalories,respectively.Inotherwords,wedefine1KCal=1,000Cal=1,000kcal,and1MCal=1,000KCal=1,000,000Cal.

Thebenefitsofthecalorific-basedapproachovertraditionalweight-basedmeth-odscanbeillustratedwithanexampleoftwocommoditiesthatfolloweddifferentpro-ductionandyieldpaths.Milletexperiencedamodestincreaseinglobalproductionfrom25.7MMT(millionmetrictons)in1961to30.1MMTin2021.Theareaallocatedtomilletduringthisperioddeclinedbyalmostathird,leadingtoanannualyieldgrowthof6.3kg/ha.Duringthissameperiod,maizeproductionincreasedsixfoldwhiletheareadou-bled,givinganannualyieldgrowthof65.6kg/ha.Toassessthecombinedyieldgrowthperformanceforthesetwocrops,productionisconvertedfrommetrictonstocalories(3,400kcal/kgformilletand3,560kcal/kgformaize)andaggregatedintoasinglemetricbyaddingtheirglobalcalorificproduction.Theannualcalorific-basedyieldgrowthforthetwocommoditiescombinedis219KCal/ha.Thiscalorificapproachallowsustoassesstheaggregateyieldgrowthoftwocropswithdifferentgrowthpathsinawaythatthetradi-tionalweight-basedapproachcannot.

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3.2Assumptionsundertraditionalmethodsofestimatingyieldgrowthpatterns

Thegrowthratebetweenperiods1and2,denotedbyP,typicallyreportedaspercentchange,iscalculatedasP=(y1?y0)/y0,wherey1andy0denoteyieldinthecurrentandpreviousperiods.Thegrowthrateequationcanalsobewrittenasy1=(1+P)y0.Inamultiperiodcase,ifyieldgrowsatrateP,onaverage,whileineachperioditissubjectedtoastochasticshockηt,yieldattimetcanberewrittenasyt=(1+P)yt?1ηt.

Takingaone-periodlagresultsinyt?1=(1+P)yt?2ηt?1,whichuponsubstitutionandworkingbackwardtoperiod0resultsinyt=(1+P)ty0ηtηt?1...η0.Takingnatural

logarithmsofbothsidesandsettingβ0=log(y0),β1=log(1+P),andεt=∑log(ηi)

gives:log(yt)=β0+β1t+εt.Suchlogarithmictransformationisafrequentlyusedre-gressionwheretheparametersβ0andβ1areestimatedwithordinaryleastsquaresandthegrowthrateiscalculatedasP=exp(β1)?1.Thegrowthrateisoftenreportedastheestimateofβ1ratherthanP,sinceforsmallgrowthratesβ1andPareapproximatelyequal.Estimatinggrowthratesbyusinglogarithmictransformationsrestsontheassump-tionthatytgrowsatapproximatelythesamerate(inpercentterms)throughoutthesam-pleperiod,inordertorendertheerrortermwhitenoise(ηtislog-normallydistributedwithmean1).Thismodelessentiallyassumesthatyieldfollowsanexponentialgrowthpattern.

Whiletheproportionalityassumptionmaybeareasonableapproximationforshortperiods(e.g.,lessthanadecade)andrelativelysmallgrowthrates,itmaybeunre-alisticwhenlongperiodsareconsidered,suchasa60-yearsampleinthecurrentcontext,duetothechangeinthebase.Analternativeand,perhaps,morerealistic,assumptioncouldbethatytgrowsbyaconstantamountineachperiod,sayμ,inwhichcasethetwo-periodyieldgrowthwouldbey1=μ+y0or,inthegeneralcase,yt=μ+yt?1.Forthemulti-periodcase,backwardsubstitutiongivesyt=μt+y0.Lettingβ0=y0andβ1=μ,andappendinganadditiveerrortermgivesyt=β0+β1t+εt.Thislineargrowthspeci-ficationimpliesthatyieldgrowsatanaveragerateofβ1units.

Thefundamentallydifferentnatureofthesespecifications,i.e.,exponentialvs.lin-eargrowthmodels,hasimportantimplicationsforassessingwhethergrowthhasdecel-eratedoraccelerated.Toillustrate,considerthatmaizeyieldgrewat2.6and1.7percentperannumduring1961-71and2011-21,respectively,thefirstandthelastdecadesofthesample.However,maizeyieldsgrewat203KCaland324KCalannuallyduringthesameperiods.Inotherwords,yieldgrowthformaizehasdroppedbymorethanone-third(basedonthelogarithmicspecificationthatassumesaconstantpercentagegrowth)buthasincreasedbymorethan50percent(basedonthelinearspecificationthatassumesaconstantamountofgrowth).

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3.3Choosingtheappropriatespecification

Thediscussionabovesuggeststhatyieldgrowthshouldbemodeledunderamoregen-eralframework.Webeginwiththefollowingspecification:

y(λ)=β0+β1t+ε,(2)

whereβ0isconstant,β1indicatestherateofyieldgrowth,εisiid(independentandidenti-callydistributed)errorterm,andλisatransformationparameter,suchthaty(λ)iseitherlog(y)(forlogarithmicspecification)ory(forlinearspecification).Inthispaperweem-ploytheBox-Coxmodeltodeterminewhichspecificationbestrepresentsthedatagener-ationprocess.Thisapproachselectsthetransformationofthedependentvariablesuchthattheresidualsapproximateanormaldistributionandexhibitreducedheteroskedas-ticity(BoxandCox1964).Sakia(1992)providesacomprehensivereviewoftheBox-Coxtransformationtechnique.NotableadvancementstotheBox-CoxtransformationincludeYeoandJohnson(2000)transformation,whichaccommodatesbothpositiveandnegativeobservations,andtheextensionbyAtkinsonetal.(2021),whichallowsfortransfor-mationsonbothsidesoftheequation.

TheBox-Coxmodelreliesonestimatingthetransformationparameterλsuchthat:

y(λ)={(yl))/λ.(3)

Allnotationsarethesameasdefinedpreviously.Equation(3)embedslinear(λ=1)andlogarithmic(λ=0)transformationofthedependentvariable.Undertheassumptionthatthereexistsaλthatmakestheerrorterminthemodelapproximatelynormal,BoxandCox(1964)derivedthelikelihoodfunctionforasetofobservations{y1,y2,…,yt}andsug-gestedusingthemaximumlikelihoodestimation(MLE)todetermineλ.FurtherdetailsontheMLEandalternativeestimationproceduresarediscussedinSpitzer(1982).

Regardlessofthevalueofλ,equation(2)assumesthattheunderlyingparameterestimateofthegrowthrateisconstantovertime.However,yieldpathsmayexhibitnon-linearities,withgrowthratesgraduallyaccelerating,decelerating,oreventakingsharpturns.Weconsidertwowaystoaccountforsuchnon-linearities.First,followingearlierstudies(e.g.,Finger2010),asquaredtimetrendisaddedtoequation(4):

y(λ)=β0+β1t+β2t2+εt,(4)

whereβ1approximatesthegrowthrateandβ2denotestherateatwhichgrowth