Transcription of 7.2 Application to economics: Leontief Model
1 economics: LeontiefModelWassilyLeontiefwo ntheNobel prizein economicsin a modelfortheeconomicsof a wholecountryor erent productssuch thattheinputequalstheoutputor,in otherwords, models:openmodel:someproductionconsumedi nternallyby industries,restconsumedby :Findproductionlevel if externaldemandis :entireproductionconsumedby :Findrelative priceof each ,S2,: : :,Sn. Theexchangeof productscanbedescribedby aninput-outputgraphHere,aijdenotesthenum ber of unitsproducedby industrySinecessarytoproduceoneunitby industrySjandbiis thenumber of externallydemandedunitsof :Primitive modelof theeconomy of Kansasin ed:ProductionofTotaloutput=Internalconsu mption+ExternalDemandfarmingindustry(int ons):x=0:05x+ 0:5y+8000horseindustry:y=0:01x+2000(in10 00kmhorserides).
2 In general,letx1,x2,: : :,xn, be thetotaloutputof industryS1,S2,: : :,Sn,respectively. Then8>>> <>>>:x1=a11x1+a12x1+ +a1nx1+b1x2=a21x1+a22x1+ +a2nx2+b2: : :xn=an1x1+an2x1+ +annxn+bn;sinceaijxjis thenumber of unitsproducedby industrySiandconsumedby indus-trySj. Thetotalconsumptionequalsthetotalproduct ionfortheproductof @a11: : : : : : calledtheinput-outputmatrix,Btheexternal demandvectorandXtheproductionlevel systemof linearequationsis equivalent to thematrixequationX=AX+B:In andtheproblemis cantransformthisequationas follows:InX AX=B(In A)X=BX=(In A) 1 Bif theinverseof thematrixIn Aexists.((In A) 1is thencalledtheLeontiefinverse.)For a given realisticeconomy, a ourexamplewe have:A= 0:050:50:10!; B= 8;0002;000!; X= xy!:2We obtainthereforethesolutionX=(I2 A) 1B= 1001!
3 0:050:50:10!! 1 8;0002;000!= 0:95 0:5 0:11! 1 8;0002;000!=19 10519:5! 8;0002;000!= 10;0003;000!; ,x= 10;000tonswheatandy= theexternaldemandchanges, 7;3002;500!, we get xy!0= (I2 A) 1B0=19 10519:5! 7;3002;500!= 9;5003;450!; ,onedoesn'tneedto recompute(I2 A) culty withthemodel:How to determinethematrixAfroma given knownand(aijxj)i;j=1; ::: nis thereforethematrix(aijxj)i;j=1; ::: nanddividesthej-thcolumnbyxjforj= 1,: : :,nto :Aneconomy hasthetwo industriesRandS. Thecurrent consumptionis given by thetableconsumptionRSexternalIndustryRpr oduction505020 IndustrySproduction6040100 Assumethenewexternaldemandis100unitsofRa nd100unitsofS. :Thetotalproductionis120unitsforRand200u nitsforS. We obtainX= 120200!,B= 20100!,A= 50120502006012040200!, andB0= 100100!
4 ThesolutionisX0= (I2 A) 1B0=141 96306070! 100100!= 307:3317:0!:Thenewproductionlevelsare307 :3and317:0forRandS, describedby thematrixequationX=AX; ,thereis Ais as follows:Thereis :Extendedmodelof theeconomy of Kansasin :0B@xyz1CA=0B@0:050:50:50:100:10:40:1133 118001CA0B@xyz1CA:4 IfXis a solution,alsot Xforeveryt >0is a solution.(Usually, onegetsa oneparameterfamilyof solutions.)Ifx6= 0, we canassumex= 1;000by choosingtheappropriateparametert. Oneobtainsthenthesolutionx= 1;000;y=290011 263:63;z=1800011 1636:36:Forthiscomputation,it is important to userationalnumbers( ,fractions)asmatrixentriessinceotherwise theapproximationto thematrixIn Ausuallywillbe invertibleandonlythetrivialuninteresting solutionx= 0,y= 0, andz= alsothereason,why closedeconomy, theabsoluteunitsof is therelative consumptionof a cannormalizethereforethematrixAsuch thatthesumof everyrow is a matrixeA, such Therecipe is:Dividethei-throw ofAby thei-thcomponent ofA thesumof thei-throw).
5 For ourexample,we haveA 0B@1111CA=0BB@212015223118001 CCA;leadingto thematrixeA=0BB@121102110211201272022311 802231133122311 CCA;eA 0B@1111CA=0B@1111CA:Theentriesof thematrixeA= (eai;j)i; j=1; :::; nhave thefollowingmeaning:eaijistherelative consumptionof theproductof industrySiby pricesTheconsumptionof productsis regulatedby anindustryis usedforbuyingother(ortheown)products, , (p1; : : : ; pn) thepricevector;piis therelative priceof theproductofindustrySi. We candraw the ow of moneyinto theinput-outputgraph,themoney owsin exchangefortheproducts:5 Onehas8>>> <>>>:p1=a11p1+a21p2+ +ean1pnp2=ea12p1+ea22p2+ +an2p2: : :pn=ea1np1+ean2pn+ +eannpn;sinceeaijpiis theamount paidby industrySjforproductsproducedby industrySjequalsthetotalpriceSjhasto pay to ,onecanwritethisas a matrixequation:P A=P:Thisequationcanbe transformedin thefollowingwayP In=P eAP (In eA) =(0; : : : ;0):ThematrixIn eAis (similarasIn A) notinvertible,since(In thatthisimpliesthatthereis alsoa SincewithPalsot Pfort >0 is a solution,onlytherelative pricebetweenthedi erentproductshasa well-de (continued):Assumep1= $1;000.)
6 Onegetsp2= $4000063 $ 634:92andp3= $11155500567 $ 1967:37. We cancomparetheserelative priceswiththeproductionlevelsmeasuredby theoriginalunitsandobtainthefollowingrel ativepricesper unit:p1=x=10001000= 1foronetonof wheat,p2=y 634:92263 2:4for1000kmhorseride,andp3=z 1967:371636:36 1 matrixequationforPis notof theusualformwhich we havestudiedso far,we make a nalmodi de neeeA= (eeai;j)i; j=1; :::; n;whereeeai;j=eaj;i:6 Thisgives us (justby switchingther^oleof rowsandcolumns)thepriceequationeP=eeAeP ;whereeeai;jis now therelative consumptionof industrySjby industrySi, so thatthesumof each columnis 1, thetextbook,ourmatrixeeAis againdenotedbyAandourePis denotedbyX. Thepriceequationis thereforeX=A X. However,onehasto keepin mindthatthismatrixAis di erent fromtheinput-outputmatrixAwe usedin theopenLeontiefmodel!
7 Example:LetA=0BB@1213141413141413121 CCA:Computeallwages,given thatthewagesforthe3rd productis$ 30; :LetX=0B@xyz1 CAbe thedi erent wageswithz= 30;000. We have tosolveX=AX(I3 A)X=0B@0001CA0BB@12 23 34 3423 34 34 23121 CCA0BB@xy30;0001 CCA=0BB@0001 CCA:Thissystemof linearequationsforxandyhasthesolutionx= 30;000andy=22;500. Thewagesforthe rstandsecondproductaretherefore$;30;000a nd$ 22;500.
