Application of Lagrange Multipliers in the Analysis of the Stolper Samuelson Theorem

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1 Joši,H.etl.:ApplictionofLgrngeMultipliersintheAnlysisoftheStolperSmuelsonTheorem ApplictionofLgrngeMultipliersintheAnlysisoftheStolper SmuelsonTheorem HrvojeJoši FcultyofEconomicsndBusiness,UniversityofZgreb,Croti MislvJoši CrotinFinncilServicesSupervisoryAgency(HANFA),Sesvete,Croti Abstrct:StolperSmuelsontheoremisbsictheoremininterntionltrdetheorynd provides nlyticl extension of HeckscherOhlin trde theory. This pper provides theoreticlproofofstolpersmuelsontheorembyintroducinglgrngemultiplierswhilethe problemofincomeoptimiztionwithmorerestrictionsistckledwiththehelpofkrush KuhnTucker'soptimiztionconditions.Lgrngemultipliersinmodelrewgesndrents whichrepresentshdowprices.themodelnlyzestheimpositionoftriffsonnimported goodndtheimpctofchngesonfctorprices.assumingthttechniclcoefficientsof productionndfctorbundncereconstnt,optimumproductionreminsthesme.finl conclusionscnbereducedtotheclimthtnincreseinthepriceofonegoodledston boveproportionlincreseofincomeoftheproductionfctorreltivelyintensivelyusedin the production of good which price hs chnged, ccording to the StolperSmuelson theorem. Keywords:HeckscherOhlintheory,StolperSmuelsontheorem,Lgrngemultipliers,triffs 1. Introduction StolperSmuelsontheoremisoneofthemostimportnttheoremsininterntionltrde theoryndprovidesnlyticlextensionofheckscherohlintrdetheory.stolpersmuelson theoremstressesthtnincreseinthepriceofonegood,longwithotherconditions unchnged,boveproportionllyincresestherelincomeofproductionfctorintensively usedintheproductionofgoodwhosepricehsincresed.ontheotherside,theincomeof other production fctor decreses below proportionlly. Wolfgng Stolper nd Pul SmuelsonwerethefirsttoshowhowFctorproportionsmodelbestexplinsgovernment ctionsinprotectionistpolicy.ppersthtfollowedresultedsrectiontotheirmodelnd cretedcontroversiesreltedtothevlidityofmodel.theuthorsconfirmedthehypothesis thttrdeliberliztionhrmstheincomeoflbourownersifcustomrestrictionsonlbour intensive products re lowered. Such income redistribution hs its socil nd politicl connottionstkingoverthefunctionoftxesndtrnsfers. TAKE2017 TheoryndApplictionsintheKnowledgeEconomy 682

2 Joši,H.etl.:ApplictionofLgrngeMultipliersintheAnlysisoftheStolperSmuelsonTheorem PurposeofthispperistotestStolperSmuelsontheoremwiththepplictionofLgrnge multipliers.theoreticlprtofthepperreliesonthefctorproportionsmodel,lsoknown stheheckscherohlinmodel.aslongsthessumptionsofmodelrestisfiedtheltter yieldslogicllyvlidconclusions.onthequntittivesidekrushkuhntuckeroptimiztion conditionsreusedinordertosolvemultiplerestrictionsproblem.thetermshdowprices inthispperrefertothepricesofinputslsoreferredtosthelgrngemultipliers. Moreover,onthetheoreticlsidesomebsicssumptionsofFctorproportionsmodel (HeckscherOhlintrdetheory)relidoutinordertocrryoutlogicllyvlidconclusions.In ourpperweusedmodelpproch,i.e.expliningthesubjectthroughmodelssumptions, theoremdefinitionndinitilconditions.lernerpercedigrmservessnnlyticltool tointerpretshdowprices,whilethetheorysubstntillyuseslinerlgebr,lgrnge functionndkrushkuhntuckeroptimiztionconditions. ExpectedresultsmnifestinnlyticlbrekdownofStolperSmuelsontheoremthrough rigorousmthemticllogic.theresultoftriffimpositioninsmllcountryintheoryre Lgrngemultipliersorshdowpricesndreinterpretedccordingtothedefinitionofthe Theorem.IftheresultsofthenlysisindictefindingsinlinewiththedefinitionofStolper Smuelsontheorem,Lgrngemultiplierscnbecceptedssuitbleinstrumentfor nlysisofcustomsimpositiononredistributionofincomeholders,nmelyworkersnd cpitlowners.thepperdoesnotcontinempiriclnlysisorreldt.vluedded resultingfromthisreserchreflectindeeperunderstndingofhowtriffimpositioninsmll countrychngesinputprices,mostnotblywgesndrentspricesoflbourndcpitl. Although,thereremnyppersinthisfield,dditionlreserchboutconnectionbetween StolperSmuelsontheoremndLgrngemultipliersrelcking.Implictionsofthispper re theoreticl, nevertheless they serve s tool for policy mkers in the nlysis of interntionltrdebrrierssuchstriffs,quotsndothersimilrtrdeinstruments. 2. TheoreticlbckgroundoftheStolperSmuelsontheorem TheoreticlbckgroundoftheStolperSmuelsontheoremiscloselyreltedtotheHeckscher Ohlin theory of interntionl trde. HeckscherOhlin theory of interntionl trde is formultedbyswedisheconomistseliheckscherndbertilohlininerly1920's(heckscher 1919,Ohlin1933).ItlensonDvidRicrdo'stheoryofcomprtivedvntgesndsttes thtcpitlbundntcountrywillexportcpitlintensiveproductswhilethelbour bundntcountrywillexportlbourintensiveproducts.accordingtodvidricrdothe source of comprtive dvntges lies in different reltive lbour productivity while comprtivedvntgesbsedonheckscherohlintrdetheoryrisessolelyfromreltive fctorbundnce.furthermore,heckscherohlinmodelssumestwofctorsofproduction, (lbourndcpitl)ndthtisthemindifferencebetweenthesetwoclssicltheoriesof comprtivedvntges.thessumptionsoftheheckscherohlin'stheoryndstolper SmuelsontheoremrederivedfromthebseHeckscherOhlintheoryndreccordingto Heckscher1919ndOhlin1933sfollows: Therereonlytwocountries(countryAndcountryB),twogoodsorproducts(nd Y)ndtwofctorsofproduction(lbour(L)ndcpitl(K)). TAKE2017 TheoryndApplictionsintheKnowledgeEconomy 683

3 Joši,H.etl.:ApplictionofLgrngeMultipliersintheAnlysisoftheStolperSmuelsonTheorem GoodislbourintensivewhilegoodYiscpitlintensive. Bothcountriesusethesmetechnology. Productionfunctionsrehomogeneouswithconstntreturnstoscle. Differentitioninproductionisincompleteinbothcountries. Consumerpreferencesreidenticlinbothcountries. Goodsndfctormrketsinbothntionsreperfectlycompetitive. Thereexistsperfectmobilityoffctorsofproductionwithinechcountrywhilecross borderfctormobilityisnotpossible. Thererenotrnsporttioncosts,triffsorothertrdebrriers. AllproductioninputsreusedPretooptimllyinbothcountries. Importsndexportsbetweentwocountriesreblnced. Aslongsthessumptionsofthemodelremettheconclusionsresultingfromthemodel willbeccurtendconsistent.thecoreofheckscherohlintheoryisoftenexplinedwith thefollowingfigure(figure1). Figure1:HeckscherOhlintheorem AlthoughcountrieshveidenticltechnologyndsmepreferencesinpointC,miniml necessryconditionforbilterltrdeisreltivedifferencebetweenbundnceofproduction fctorsbetweencountriesandb.stolpersmuelsontheoremisoneofthekeytheoremsin interntionltrdetheory,thetheoremththsttrctedthettentionofeconomists, politicinsndscientistscrosstheglobeforthelst70yers.alongwithheckscherohlin Theorem,FctorpriceequliztiontheoremndRybczynskitheoremitrepresentsthecoreof HeckscherOhlintrdetheory.StolperSmuelsontheoremisfundmentlmicroeconomic bsisfornlysistheimpctofinterntionltrdeonincomefromlbour(wge)ndcpitl (rent).thestrtingpointofstolpersmuelsontheoremisinheckscherohlintheory,nmely the2x2x2modelthtencompssestwocountries,twogoodsndtwoproductionfctors. TAKE2017 TheoryndApplictionsintheKnowledgeEconomy 684

4 Joši,H.etl.:ApplictionofLgrngeMultipliersintheAnlysisoftheStolperSmuelsonTheorem Firstreserchinthefieldoftrdepolicynditsinfluenceonpricesofproductionfctorsis ssocitedwiththeworkofwolfgngstolperndnobelluretepula.smuelsonintheir seminl pper "Protection nd Rel Wges" from The forementioned uthors empiriclly tested the existence of cuslity between the trde policy decisions nd consequentilimpctonfctors prices.theirnlysisconfirmedthergumentthttrde liberliztionwouldbedmgingtoincomeownersiftheyreducethecustomsrestrictionson lbourintensiveproducts.thisredistributionofincomehsitssocilndthuspoliticl connottions,becuseittkesoverthefunctionoftxndtrnsfers.wemywonderwht theimportnceofthistheoremis.economictheorysuggeststhtintroductionofbrriersto freetrdehsitscostsndthtthefirstbestsolutionisfreetrdewithoutnyobstclesto trde.secondbestsolutionconcernsthesitutionwhereoptimlsolutioncnnotbestisfied. Trdeliberliztionshould,intheory,incresecountry'snetsocilwellbeing.However,if prtilinterestsretobeconsidered,domesticproducersofcountry sprotectedimported good,whichisthesubjectofliberliztion,willhvelessndlessprofits.thequestionwhich riseiswhotkestheburdenoftrdeliberliztionndwhorepsbenefitsfromit.the nswerisgivenbythestolpersmuelsontheorem. Figure2:StolperSmuelsontheorem StolperSmuelsontheorememphsizesthttheincreseinthepriceofonegood,longwith other conditions unchnged, increses the rel income of production fctor which is intensivelyusedintheproductionofgoodwhosepricehsincresed.infigure2grphicl representtion of the StolperSmuelson theorem is given. According to Stolper nd Smuelson theorem the rise in price of lbourintensive product, other conditions unchnged, it will boveproportionlly increse the price (income) of lbour which is intensivelyusedinproductionofproductwhiletheincomeofcpitlwillfllbelow proportionlly. TAKE2017 TheoryndApplictionsintheKnowledgeEconomy 685

5 Joši,H.etl.:ApplictionofLgrngeMultipliersintheAnlysisoftheStolperSmuelsonTheorem 3. TheproofofStolperSmuelsontheoremndtheuseofLgrngemultipliers ThelogicofStolperSmuelsontheoremisexplined,prtfromdescriptivemethods,bystrict mthemticllogic,whichimpliesknowledgeofdifferentilndmtrixclculus.thefollowing mthemticlpostultesreshortenedndpurifiedversionofthemthemticlnlysisof StolperSmuelsontheorem(seeSilberberg1990ndSurnovic2007). Assumethereretwocountriesinperfectlycompetitivemrket(countryAndcountryB). Bothcountriesproducetwogoods(ndY)withtwofctorsofproduction(lbour(L)nd cpitl(k)).countryaislbourbundntwhilecountrybiscpitlbundnt. K L A K L B (1) GoodislbourintensivewhilegoodYiscpitlintensive. K L K (2) L Y Productionfunctionisthefunctionofbothfctorsofproduction: f L K (3) Y f L K (4) whereli ndki rethequntitiesofproductionfctorsthtreusedintheproductionof goods ndy. Assumethtproductionfunctionsexhibitpositivebutdecresingreturnstoechfctor i i f f (, L K i scle(crs). i f nd L i i f, K i i )ndrechrcterizedbyconstntreturnsto Modelisconstrinedwithfixedendowmentoflbourndcpitlwhichressumedtobe homogeneous nd freely mobile between industries nd four technicl coefficients of production; L,,K nd. L K Y L (5) Y K (6) Tking into ccount model constrints quntities of production cn be expressed s: L K L K, ndy, Y. (7) L K Theobjectiveistomximizentionloutput(z): z p p Y (8) Y TAKE2017 TheoryndApplictionsintheKnowledgeEconomy 686

6 Joši,H.etl.:ApplictionofLgrngeMultipliersintheAnlysisoftheStolperSmuelsonTheorem subjectto: L K Y L (9) Y K (10) f L K (11) f (12) where p LK reexogenousvriblesndzisendogenousvrible. p Y ThisproblemcnbesolvedusingLgrngefunctionndLgrngemultipliers.Lgrnge function(lgrnge1811)isgivenbyexpression(13): MxL p f p Y wl f L K Y L Y rk K Y (13) wherew,r, nd rethelgrngemultipliers.wgererentsreoftenreferredtos theshdowpricesoflbourndcpitl. Sincethisisproblemofoptimiztionwithmultipleconstrints,itcnbegenerlizedbyusing the KrushKuhnTucker optimiztion conditions (see Nerli 2008). KrushKuhnTuckeroptimumconditionswereoriginllynmedfterHroldW.Kuhnnd AlbertW.Tucker(KuhnndTucker1951).ItwslterdiscoveredthtWilimKrushfirst sttedtheseconditionsinhis1939mster'sthesis(krush1939). ThestrtingpointofStolperSmuelsontheoremistheequilibriumrequirementofthe compny'szeroprofitwherepriceisequltomrginlcost: Differentiting(14)withrespectto w r p (14) L K Y w r p (15) p yields: w L r K L w K r (16) p p p p Rerrngingtheboveequtionitcnbewritten: L w p K r p p L K (17) w r p Itispossibletoprovethtw givingexpression(19). p L r p K givingeqution(18)nddifferencing(15) TAKE2017 TheoryndApplictionsintheKnowledgeEconomy 687

7 Joši,H.etl.:ApplictionofLgrngeMultipliersintheAnlysisoftheStolperSmuelsonTheorem w r L K (18) p p w r (19) p p Theterms(18)nd(19)cnbesolvedssystemnddisplyedinmtrixform L K w p r p (20) Expression(20)cnbesolvedusingCrmer'sRuletoget: w p L K (21) r p L K (22) wheretheterminnomintor isdeterminntofthefirst(left)mtrix. L K Finlconclusionscnbereducedtotheclimthtnincreseinthepriceofonegoodleds tonboveproportionlincreseofincomeoftheproductionfctorreltivelyintensively usedintheproductionofgoodwhichpricehschnged,ccordingtothestolpersmuelson theorem. 4. ApplictionofLgrngemultipliersinthenlysisoftheStolperSmuelsontheorem InthischpterisgivennexmpleofuseofLgrngemultipliersinthesitutionoftriff impositiononprotectedimportedgoodyincountryutopi.supposethtutopihs100 unitsoflbour(l)nd80unitsofcpitl(k).itissmllcountrythtexportsgoodtworld priceof3unitsndimportsgoodytthesmeprice.thessumptionofthesmepricefor bothgoodsistkenduetofreemrketconditionswherepricesofbothgoodsreequlnd mount1.inthissitutionbothcountriescnchievelrgerwelfrefromtrde. px px (1) py py Technicl(inputoutput)coefficientsofproductionregiveninTble1: A B TAKE2017 TheoryndApplictionsintheKnowledgeEconomy 688

8 Joši,H.etl.:ApplictionofLgrngeMultipliersintheAnlysisoftheStolperSmuelsonTheorem Tble1:TechniclcoefficientsoflbourndcpitlcountryUtopi i Y Li Ki Techniclcoefficientsofproductionreexpressedinccordncewiththefctorintensityof lbourndcpitlusedintheproductionoflbourintensivendcpitlintensivegood.in grphicl representtions it is often ssumed tht the reltionship between technicl coefficientsisinthertiotwotoone(2:1). First,weformultegolfunction.Thegolfunctionistomximizethentionloutput(Q) ccordingtotheexpression(2): Q p x p y (2) x Constrintfunctionsredeterminedfromthevilblequntityofproductionfctors(Lnd K)ndtechniclcoefficientsofproduction. Lx Kx y x y L (3) Ly x y K (4) Ky x y (5) TheLgrngefunctionincludesthegolfunctionndtwoconstrintfunctionsinwhich Lgrngemultipliersrewges(w)ndrent(r).Inthisexmplewgesndrentwillrepresent shdowprices. L p x p y wl x yrk x y (6) x y FirstorderconditionsreprtilderivtionsofLgrngefunctionbyx,y,wndr. Lx Ly Kx Ky L x L y px wlx rkx w r (7) py wly rky w r (8) L L Lxx Lyy x y (9) w L r K Kxx Kyy x y (10) Weclcultetheoptimumproductionquntities x iy withtheexistingconstrintsnd thepricesofproductionfctors,wgesndrents.thesolutionofsystemof4equtionswith TAKE2017 TheoryndApplictionsintheKnowledgeEconomy 689

9 Joši,H.etl.:ApplictionofLgrngeMultipliersintheAnlysisoftheStolperSmuelsonTheorem 4unknownsis:w r x y.sincethepricesofgoodsreequl,ssuming freeinterntionltrdendccordingtothefctorpriceequliztiontheorem,thefctor pricesndpricesofgoodsshouldbeequl(figure3). Figure3:Fctorpriceequliztiontheorem ThefreetrdewillequlizethepricesofgoodsndproductionfctorsincountriesAndB, despitethessumptionoftheheckscherohlinmodelthtproductionfctorsreimmobile crosscountries.weclcultentionloutputtoptimumproductionlevel: Q x y p x p y x y (11) Figure4:ProductionrestrictionsinHeckscherOhlinmodel TAKE2017 TheoryndApplictionsintheKnowledgeEconomy 690

10 Joši,H.etl.:ApplictionofLgrngeMultipliersintheAnlysisoftheStolperSmuelsonTheorem Bothproductionfctorsrefullyusedtoptimumproductionquntitiesx ndy.lgrnge multiplierswndrrepresentshdowprices(instedof i ).Iftheuseoflbourin productionincresesforsingleunitorsingleworker(fromletussy100to101),thenthe mximumcostthtmngementiswillingtopyforthtworker(themrginlcostoflbour) isoneunit.anlogously,thesmecnbeprovenwithoneunitincreseincpitluse(the opportunitycostofoneunitofcpitl). Inthecseofimpositionofimportduty( t )ongoodythedomesticpriceofthisgood incountryutopiwillbechngedfromp to t priceunits. Thegolfunctionhsbeenchnged whiletherestrictionfunctionsreminedthesme Lgrngefunctionisexpresseds: y p y Q p tx p y (12) Lx Kx x y x y L (13) Ly x y K (14) Ky x y (15) x yrk x y L p tx p y wl (16) x y ThefirstorderconditionsretheprtilderivtionsoftheLgrngefunctionbyx,y,wndr. Lx Ly Kx Ky L x L y px wlx rkx w r (17) py wly rky w r (18) L L Lxx Lyy x y (19) w L r K Kxx Kyy x y (20) Thesolutionofsystemof4equtionsnd4unknownsis:w r x y. Sincethetechniclcoefficientsofproductionhsnotchngednorthequntityofproduction fctors,the optimum vlue of theproductionofgoods ndywill remin thesme x y. TAKE2017 TheoryndApplictionsintheKnowledgeEconomy 691

11 Joši,H.etl.:ApplictionofLgrngeMultipliersintheAnlysisoftheStolperSmuelsonTheorem Figure5:TheincreseinpriceofgoodYintheStolperSmuelsontheorem TheimpositionofimportdutyoncpitlintensivegoodYhsprotectedtheownersofcpitl sfctorthtismoreintensivelyusedintheproductionofgoodwhichpriceshschnged. Forthisresontheemployer(mngement)willbewillingtopybelowproportionllower mountoffundsforthedditionlworkerthtis priceunitsincomprisonwithoneprice unitbeforetheimportdutyimpositionwhiletheerningsfromcpitlwillrisebove proportionllyon priceunitsbecuseyisprotectedgood. Lgrngemultiplierscnbecceptedssuitbleinstrumentfornlysisofcustomsimposition on redistribution of income holders, nmely workers nd cpitl owners. Moreover, reserchersndscholrsinthereofinterntionleconomicscouldbenefitfromthepper sitdefinesthetheoreticlbsefortheempiriclreserchintrdepolicyndpublicpolicys well.limittionsofthenlysisrereltedtostrictssumptionsofstolpersmuelson theoremthtdonotpplyinreleconomy. 5. Conclusion TheimofthepperistotesttheStolperSmuelsontheoremusingLgrngemultipliers. AfterthetheoreticlbckgroundoftheStolperSmuelsontheoremtheproofofStolper Smuelson theorem nd Lgrnge multipliers is presented. The problem of income optimiztion with multiple restrictions is tckled using KrushKuhnTucker's optimum conditions.lgrngemultipliersinthemodelrewgesndrentswhichrepresentshdow prices.anexmpleoflgrngemultiplierspplictionisgiveninthesitutionoftriff TAKE2017 TheoryndApplictionsintheKnowledgeEconomy 692

12 Joši,H.etl.:ApplictionofLgrngeMultipliersintheAnlysisoftheStolperSmuelsonTheorem impositiononprotectedimportedgoodyincountryutopindconsequentlyimpcton fctorprices.sincethetechniclcoefficientsofproductionhsnotchngednorthequntity ofproductionfctors,theoptimumvlueoftheproductionofgoodsndyreminedthe sme.theimpositionofimportdutyoncpitlintensivegoodyhsprotectedtheownersof cpitlsfctorthtismoreintensivelyusedintheproductionofgoodwhosepriceshve chnged. For this reson the employer (mngement) will be willing to py below proportionllowermountoffundsforthedditionlworkerthtis priceunitsin comprisonwithonepriceunitbeforetheimportdutyimpositionwhiletheerningsfrom cpitlwillriseboveproportionllyon priceunitsbecuseyisprotectedgood.finl conclusionscnbereducedtotheclimthtnincreseinthepriceofonegoodledston boveproportionlincreseofincomeoftheproductionfctorreltivelyintensivelyusedin the production of good which price hs chnged, ccording to the StolperSmuelson theorem. ItcnbeconcludedthtLgrngemultiplierscnbecceptedssuitbleinstrumentfor nlysisofcustomsimpositiononredistributionofincomeholders,nmelyworkersnd cpitlowners.moreover,reserchersndscholrsinthereofinterntionleconomics couldbenefitfromtheppersitservessthetheoreticlbsefortheempiriclreserchin trdepolicyndpublicpolicyswell. Literture Heckscher,E.F.(1919)Theeffectofforeigntrdeonthedistributionofincome.EkonomiskTidskrift,21, ReprintedinEllis,H.S.ndL.A.Metzler,(eds.),RedingsintheTheoryofInterntionlTrde.London,Allen ndundwin. Krush, W. (1939) Minim of Functions of Severl Vribles with Inequlities s Side Constrints. M.Sc. Disserttion.Dept.ofMthemtics,Univ.ofChicgo,Chicgo,Illinois. Kuhn, H. W. nd Tucker, A. W.(1951) Nonliner progrmming.proceedings of 2nd Berkeley Symposium. Berkeley:UniversityofCliforniPress.p Lgrnge,J.L.(1811)Mécniquenlytique,sect.IV,2vols.Pris,1811. Nerli,L.(2008)Uvodumtemtikoprogrmirnje1,SecondEdition.Zgreb:Element,p.56. Ohlin,B.(1933)InterregionlndInterntionlTrde.Cmbridge,Mss.:HrvrdUniversityPress. Silberberg,E.(1990)TheStructureofEconomics:AMthemticlAnlysis".NewYork:McGrwHill. Stolper,W.F.ndSmuelson,P.A.(1941)ProtectionndRelWges,ReviewofEconomicStudies9(1):p Surnovic,S.(2007)InterntionlTrdeTheoryndPolicy.Avilblet: TAKE2017 TheoryndApplictionsintheKnowledgeEconomy 693