Mathematical Investigations in the Theory of Value and Prices - Irving Fisher - Google ブックス

CHAPTERII.MECHANISM.

1.

Scarcely a writer on economics omitstomakesomecomparisonbetweeneconomicsandmechanics.Onespeaksofa"roughcorrespondence"betweentheplayof"economicforces"andmechanicalequilibrium.Anothercomparesuniformityofpricetothelevel-seekingofwater.Another(Jevons)compareshislawofexchangetothatofthelever.Another(Edgeworth)figureshiseconomic"system"asthatofconnectedlakesofvariouslevels.Anothercomparessocietytoaplasticmasssuchthata"pressure"inoneregionisdissipatedinall"directions."Infacttheeconomistborrowsmuchofhisvocabularyfrommechanics.Instancesare:Equilibrium,stability,elasticity,expansion,inflation,contraction,flow,efflux,force,pressure,resistance,reaction,distribution(price),levels,movement,friction.Thestudentofeconomicsthinksintermsofmechanicsfarmorethangeometry,andamechanicalillustrationcorrespondsmorefullytohisantecedentnotionsthanagraphicalone.YetsofarasIknow,noonehasundertakenasystematicrepresentationintermsofmechanicalinteractionofthatbeautifulandintricateequilibriumwhichmanifestsitselfonthe"exchanges"ofagreatcitybutofwhichthecausesandeffectsliefaroutside.

inthetheoryofvalueandprices.25§2-Inordertosimplifyourdiscussionthefollowingpreliminarysuppositions*aremade:(1)Asingleisolatedmarketlargeenoughtopreventoneman'sconsctouslyinfluencingprices.(2)Agivenperiod,sayayear.(3)Duringthisperiodtherateofproductionandconsumptionareequalandsuchthatstocksleftoverfromlastyearandstocksheldoverfornextmayhaveaninfluencewhichisunvaryingorwhichisnotafunctionofquantitiesproducedandconsumedduringtheyear.Theirinfluenceisaccountedforintheformofthecurvestobeemployedjustasistheinfluenceofclimate,population,politicalconditions,etc.(4)Eachindividualinthemarketknowsallprices,actsfreelyandindependentlyandpreservesthesamecharacteristicsduringtheperiod,sothattheformsofhisutilitycurvesdonotchange.(5)Allarticlesconsideredareinfinitelydivisibleandeachmanfreetostopproducingandconsumingatanypoint.(6)Themarginalutilityofconsumingeachcommoditydecreasesastheamountconsumedincreases,andthemarginaldisutilityofproducingeachcommodityincreasesastheamountproducedincreases.(7)AsstatedinChapterI,§4,theutilityofeachcommodityisindependentofthequantitiesofothercommoditiesandlikewisefordisutility.§3.Infig.2letthecurveMNbedrawnwithaxesOEandOA.Thiscurveissuchthattheshadedarearepresentsanyamountofthegivencommodityconsumedbythegivenindividualinthegivenperiodoftime,andtheordinate(drawndownward)fromOtoRrepresentsitsmarginalutility.Thefigureevidentlyinterpretsthefactthatasthequantityofcommodityincreasesitsmarginalutilitydecreasesandviceversa.\OAindicateswhatthemarginalutilitywouldbeifonlyaninfinitesimalquantityofthecommoditywereconsumed.Furthermoreletaglasscistern(fig.2)beformedhavingthefigureOAMNforitsfront*Theseare(essentially)thoseofAuspitzundLieben.fForthefurtherpropertiesofthecurveMNanditsrelationtothecurvesofJevons,AuspitzundLiebenandFleemingJenkin,seeAppendixI,DivisionII.

§4.Thisnotionofaratioisintroducedintothefollowingmorecomplicatedmechanism(fig.8).Fig.9(anelevationoffig.8)showsthevariouscisternsofvariouscommoditiesfortheindividualI.Theordinatesrepresentmarginalutilityperunitofcommodity.Itcorrespondstofig.6,exceptthatinthelattertheutilityisperdollar'sworthofcommodity.Thetopsofthecisternsarenolongeratthesamelevel.Thecisternsarenowtofloatlikeboatsina

tank*andfreetomoveonlyvertically(beingsoconstrainedbyatelescopearrangementbeneathandnotshowninthediagram).Aglanceatfig.8orfig.10(aplanoffig.8)willshowthatanyrightandleftrowofcisternsisrelativetoasingleindividualandcorrespondstofig.6andthatanyfrontandbackrowisrelativetoasinglecommodityandcorrespondstofig.4.Thewaterinthesecisternsmustbesubjectedtotwosetsofconditions,first:thesumofallthecontentsofIA,IIA,1IIA,etc.,shallbeagivenamount(viz:thewholeofthecommodityAconsumedduringthegivenperiod)withalikegivensumfortheBrow,Crow,etc.,secondly:thesumofIA,IB,IC,etc.,eachmultipliedbyacoefficient(thepriceofA,ofB,ofC,etc.),shallbegiven(viz:thewholeincomeofIduringtheperiod)withalikegivensumforthe11row,IIIrow,etc.*Thelevelofwaterineachcisternisintendedtobethatofthelevelinthetank.Theonlyconstantcausewhichwillmakethelevelsdifferentisthedifferencebetweentheweightofthewholecisternandtheweightofthewaterdisplacedbyitswalls(partlywood)whichdifferenceisslight,maybeplusorminus,andisequaltotheweightoftheexcessordeficitofwaterinthecisternaboveorbelowtheoutsidelevel.

40Irvingfflsher—MathematicalinvestigationsTorealizethesetwosetsofconditionseachcisternisdividedintotwobyaverticalpartitionofwood.Thefrontcompartmentsareallofunitthicknessoneinch(say).Allfrontcompartmentsbelongingtothesamefront-and-backrowaremutuallyconnectedbytubes(inthetankbutnotinconnectionwiththewaterofthetank)thusfulfillingthefirstsetofconditions.Thethicknessofthebackcompartmentsisadjustablebutis(aswillsoonappear)constrainedtobealwaysequaltotheprice,thusifthepriceofAis$1,ofB$3andC$1.20,thethicknessofallcisternsintheArowwillbe1,intheBrow3andintheCrowi.2(inches).Sincethethicknessofthefrontcompartmentisunity,thecontentsofeachbackcompartmentequalsthecontentsofthefrontmultipliedbythenumberofinchesofthicknessofthebackcistern,thatisthebackcompartmentcontainsavolumeofwaterequaltotheamountofthecommoditymultipliedbyitsprice.Itcontainsthereforethemoneyvalueofthecommodity.Thedoublecisternrepresentsthedoublelightinwhiclieachcommodityiscommonlyregarded—somanypounds,yards,etc.andsomanydollar'sworth.Allbackcompartmentsofthesamerightandleftrowsaremutuallyconnectedbytubes—thatisthesumoftheircontentsisgiven—thusfulfillingthesecondsetofconditions.Thebackcompartmentscanchangetheirthicknesses,asthewallsattheright,leftandbottomareofflexibleleather;thebackplaneiskeptparalleltothewoodpartitionbytwodouble"parallelrules"notdiagramed.Thereremainstobedescribedthesystemoflevers.Thepurposeoftheseleversistokeepthecontinuousratioofmarginalutilities,thesameforallindividualsandequaltotheratioofprices.Firstthereisasystemofoblique*levers(Fl2,etc.,fig.9)connectedbyslidingpivotswiththetopsofthecisternsandhavingtheirlowerextremitieshingedtowoodenfloatsF,thehingesbeingonthelevelofthewaterofthetank.Thesefloatsarefreeonlytoshiftlaterally.ItisevidentfromthesimilartrianglesFRlandFR2infig.9thattheordinatesofthetwocisternsIAandIBareproportionaltothedistancesoftheAandBrodsRandKfromthehingeintheleftfloatF.Likewiseintherowbehind,theordinatesareproportionaltothesamedistances.Hencethefourordinatesareproportionaltoeachotherandingeneralallthe*Aconvenientangleforeachlevercanbeassuredbyacarefulselectionofcommodityunits.ThusifthemarginalutilityperpoundgivesinconvenientordinatesintheArow,reconstructthecisternsinthatrowsothattheordinatesarelengthenedtorepresentmarginalutilitypertonorshortenedfortheounce.