and the Metzler ParadoxMasahiro Endoh and Koichi HamadaThe Lerner para - PDF document

and the Metzler ParadoxMasahiro Endoh and Koichi HamadaThe Lerner paradox is the possibility that a tariff on an import good might worsen a country’s terms of trade, and the Metzler paradox is the possibility that a tariff on an import good might reduce a country’s import price. In a general equilibrium framework with multiple goods, this paper shows that the combination of the invertibility of the Slutsky matrix for the world economy and its similarity across countries will preclude both of the paradoxes, and that the combination of the gross-substitutes assumption for the world demand and the substitute assumption for the demand of an import country property of goods will preclude the Lerner paradox. A modified condition for the Slutsky matrix combined with the gross substitute for the world demand will do the same for the Key Words:Lerner paradox, Metzler paradox, tariffs, terms of trade, gross substitutes,dominant diagonal matrix.JEL Codes:C20, F02, F11 oftradestructuresintermsoftheSlutskymatrix.Whentheincomee¤ectsaremoderateineachcountry,thisconditionissatis…ed.ThisseemstobeoneofthereasonsthatChipman(1990)consideredtheMetzlerparadoxalongwiththetransferparadox,becausetransfersmaygeneratedi¤erentincomee¤ectsacrosscountries.II.ModelSupposetheworldconsistsoftwocountries,AandB,andtherearen+1com-modities,j=0;1;2;:::;n.Thenon-negativevectorofdomesticpricesforcountryKisexpressedaspK=(pK0;pK1;:::;pKn)0forK=AandB,with0beingthesymbolfortransposition.Good0isthenumerairesothatpK0=1forallK.Wreferestothetotalworld.EK�pK;uK
istheexpenditurefunctionofcountryK(K=AandB),RK�pK
istherevenuefunctionofcountryK,anduKiscountryK’sutilitylevel,allde…nedonpricevectorincountryK;wherepK(pK1;pK2;:::;pKn)0;andtK(tK1;tK2;:::;tKn)0.Good0isanumeraire.LetSK�pK;uK
EK�pK;uK
�RK�pK
,theneachcountry’sbudgetconstraintandthemarketclearingconditionareasfollows:SK�p+tK;uK
�tK0(SK�p+tK;uK
=0;K=A;B;PK=A;BSK0p�p+tK;uK
=0;whereSKp=(@SK @pK1;@SK @pK2;:::;@SK @pKn).Westartfromthesituationwherenottari¤sexistforcountryAandB;andsupposeCountryBincresestBfrom0todtB.Then,pAisequaltotheinternationalpricep:pA+tB=pB(1)NotingthattA=tB=dtA=0,weobtain,h i26664duAduBdp37775=2666400�SBppdtB37775;where3 examinetheconditionsfortheabsenceoftheLerneraswellastheMetzlerparadoxwhenwestartthesituationwherenotari¤sexist.SupposegoodkisanimportofcountryB.Ifatari¤ongoodkisimposedbycountryB,thatis,tBk=dtBk�0,thenunderwhatconditionsdoesitincreasetheinternationalorexportercountry’sprice,thatis,dpAk�0[Lernerparadox]orunderwhatconditionsdoesitdecreasetheimportercountry’sprice,thatis,dpBk0[Metzlerparadox]?Onecanseebelowthatthisanalysismaybeextendedtotheworldwheretrans-portationcostsexistbutconstant,wheresubsidiesexist,andwherethetari¤stakethead-valoremtari¤s.Onecanalsoanalyzesimilarlytheworldwheretheinitiallevelsoftari¤sinbothcountiresarenotzero,exceptthatthesimilarityconditionofgrosssubstitutemartixinequation(15)shouldbeintermsofinternationalprices.De…nition:theLernerParadoxoccursif,dpAk dtBk�0;theMetzlerParadoxoccursifdpBk dtBk0.AStrongFormoftheLernerparadoxoccursiftherelativeinternationalpriceoftheimporttoanyothercommodityincreases(Lernerparadox),orthepriceoftheimportrelativetoanyothercommoditydecreases(Metzlerparadox).SincedpBk=dpAk+dtBk;anddtBk&#x]TJ/;༕ ;.9;U T; 13;&#x.299;&#x 0 T; [00;0;(7)bothparadoxesnevercoexistentatthesametime.Pleasenotethat,inthemodelofmultiplegoods,thetermsoftradeisnotsimplyde…nedasthepriceratioofasingleimportandasingleexport,butde…nedasthepriceofimportorexportintermsofthenuméraire.Thestrongde…nitioncomestocounterthisproblem.Fortunately,thegross-substituteassmuptionamongallthegoodsandthesimilarityassumptiontobede…nedbelowisinvarianttothechoiceofanuméraire.III.TheCasewithIdenticalorSimilarNetDemandStructureInthissection,weexaminethee¤ectoftari¤reductionontermsoftradeinthecasewithidenticalsubstitutestructureasapreliminaryanalysis.Aswehavestarted,weconsiderthecommoditykbeingexportedfromcountryAtocountryB,andsupposeonlythetari¤ongoodkfromcountryAtocountryBisincreasedfromzero.6 equaltotheinternationalpriceandaccordinglythetermsoftradeforcountryA,onlyalittle.ncreasesitsdomesticpriceincountryBbyalmostthesameamount.Acontrastingstorycanbetoldwhenisclosetozero.Theinternationalpricetakescareofmostoftheshock.Thischaracteristicoftheresultsforthecaseoftheidenticalexcesssupplyamongcountriesisinteresting,sincethechangeoftari¤oncommoditykdoesnota¤ectthepriceofothercommodities.PropositionI:Supposetheexcesssupplystructureofeachcountryisidentical.Then,underAssumptionIA(invertibilityofworldtradestructurematrix),neithertheMetzlerparadoxortheLernerparadoxwilloccur.Oneunitoftari¤increaseforacommoditybycountryBwillincreasethepriceofthecommodityincountryBbytherelativeweightofcountryA,,anddecreaseitspriceincountryAbytherelativeweightofcountryB,�1.OuranalysiscanbeextendedtothecasewheretradestructuresSAppandSBpparenotexactllysimilar,withemployinganapproximation.Thee¤ectoftBkonexportcountryAwithouttheassumptionofpropotionatesubstitutionmatrixabove.Fromequation(8),itcanbewrittenasdpA=�(XW)�1SBppdtB=�(XW)�1BXW+�SBpp�BXW
dtB;(12)whereB=augmin SBpp�XW andthenormofsquarematrixAij=(aij)ofordernn,andthenormofavectorpwithn-elementarede…nedinthispaperas(See,Bellman,1960,p.165)kAk=n
maxi;j(jaijj);kpk=maxj(jpjj):Thisde…nitionofthenormiseasilyshowntosatisfyregulartriangleinequalitiesattachedtothenorm2.Then,equation(12)isrewrittenas 2Inparticular,kA+BkkAk+kBkandkABkkAkkBk.8 PropositionII:UnderassumptionsIAandSimilarityConditionIwithasu¢-cientlysmall",neithercantheLernerparadoxnortheMetzlerparadoxemerge.Theresultsaboveindicatetheimportanceofthesimilarityofexcesssupplypat-ternsinassessingthepossibilityofthoseparadoxes.Ifthetradestructureissimilarenoughtogiveasmall",onecanneglectthee¤ectofatari¤changetoacommodityonthepriceofothercommodities.IV.GrossSubstitutabilityandtheAbsenceoftheMetzlerParadoxInthissection,wewilldiscussourquestionsunderAssumptionI,thatis,theassumptionofgrosssubstitutesamonggoods.Also,wewillshowthatifgoodsaregrosssubstitutes,andanothersimilarityconditionprevails,thentherewillbenoLernerparadoxnorMetzlerParadox.AssumptionIassurestheinvertibilityofXW(AssumptionIA).Theninordertoproveourmainresult,weutilizealemmaontheinverseofadominantdiagonalmatrixwhich,tothebestofourknowledge,isunseenintextbooksorintheirexercises.Fortheworldasawhole,theWarlaslawensures p0 xW=0;(15)wheretheinternationaln+1dimensionalpricevectoriswrittenas pincludingthenuméraire,thatis, p=(1;p)0andtheaugmentedsupplyvectorofn+1dimensionas xW=( x0W; x1W;:::; xnW)0; xkWbeingthenetexcesssupplyofgoodkintheworldmarketasde…nedatthebeginningofthepaper.AsHahn(1958)showedinlightofthefactthatinequilibriumtheexcesssupplyequalszero,i.e., xW=0;(16)thisimpliesintermsofthennSlutskymatrixfortheworldexcessdemandXW=[xWij],i;j=1;:::;n,p0XW0:(17)10 P=26666664p10:::00p2:::000:::pn37777775Accordingly,Thisgross-substitutabilitypropertyoftheworldexcessdemandresponsematrixassuresthestabilityofthemarket,invertibilityofXWandthefactthat(XW)�1isanegativematrix.Then,equation(20)isrewrittenastherowsumcon-ditionfortheelementofV=[vij]=[xijpj]vWii=Xj6=ivWji,fori=1;


;nIntermsofY=[yij]=[pixij]yWjj=Xi6=jyWijBythosepreparations,wewillshowbelowthatthediagonalelementsof�XA+XB
�1SBppareallpositivebyutilizingthefollowingcuriouslemma.StepII(Lemma)IfYW=yWijisamatrixyWii0,yWij&#x]TJ/;༕ ;.9;U T; 14;&#x.379;&#x 0 T; [00;0(i6=j),andyWii&#x]TJ/;༕ ;.9;U T; 14;&#x.379;&#x 0 T; [00;Pi6=jyWij(orPni=1yWij0),foralli,thenYWisnegativelyinvertible,i.e.(YW)�1=YW=yWij0,wheretheasteriskindicatestheelementsoftheinversematrix.Andeachnon-diagonalelementofanyrowissmallerthanitsdiagonalelement,thatis,yWii&#x]TJ/;༕ ;.9;U T; 15;&#x.035;&#x 0 T; [00;yWijfori6=j.The…rststatementonthenegativityoftheinverseisgenerallyknownastheresultunderthecolumn-sumcondition,aspecialcaseofthedominantdiagonalcondition.Thesecondconditionthatthediagonalofthematrixislargerthaneacho¤-diagonalmatirixisperhapsnew.Wecallthepropertyoftheinversematrixasthematrixof"unsurpassed"diagonal.12 assumptionofinduction.Thelastequalitycomesfromthefactthateachcolumnoftheinversematrixisorthogonaltotheelementsintheformofsummationintheprecedingvectorexceptoneitemwiththesynchronizedindex.SincealltheelementsofyWnandyWnnarenegative,yWnn�yWnjforn6=j.(Q.D.E.oftheLemma.)StepIIIWeshowthatthediagonalelementofthematrixdpA=��XA+XB
�1SBpp=�(XW)�1SBppispositiveundertheassumptionofgross-substitutesassumptionfortheworldecon-omyandsubstitutesassumptionforcountryB.De…neYW=PXWandYB=PSBpp.BecauseoftheLemma,(YW)�1hasa"unsurpassed"diagonalsuchthatyWii�yWijfori6=j.Thenweshowthattheincreaseintari¤rateincountryBproducestheincreaseindomesticpriceincountryB.LetdtB=�0;


;0;dtBk;0;


;0
0,dpB=(XW)�1SBpp
dtB=(YW)�1YB
dtB.(YW)�1=YW=yWijYB=yBijWeareinterestedinthek’thcolumnofdpB,whichisthek’thelementof�(YW)�1YBdtB=�hyWk1yWk2


yWkni26666664yB1kyB2k...yBnk37777775dtB;iswritten�yWkkyAkk�nXi=1;i6=kyWkiyAik�yWkkyAkk�yWkknXi=1;i6=kyAik014 Conventionalassumptionlikegrosssubstitutabityalonecannotprecludethepos-sibilityoftheseparadoxes,and,thoughbothparadoxiesemergefromsimilarmech-anism,theMetzlerparadoxpersistsmorerobustlythantheLernerparadox.Ourtaskhasbeentoseekexactlywhatconditionsarenecessarytoexcludetheoccurenceoftheseapparentlyanomaloussituations.Asmalllemmaonthedominantdiagonalmatrixisshowntoclarifythesituation.IftheexcessdemandstructureexpressedbytheSlutskymatirixisinvertibleandsimilareachother,gthentheseparadoxesarehardertooccur.Also,thedominanceofgrossandnetsubstitionratherthancomplementalitywillhelptheavoidanceoftheMeltzerparadox.16