CONCEPTSOFOPTIMALITYANDTHEIRUSESNobelMemorialLecture,December11,1975byTJ ALLI NGC.KOOPMANSYaleUniversity,NewHaven,Connecticut,USAAccordingtoafrequentlyciteddefinition,economicsisthestudyofbestuseofscarceresources.Thedefinitionisincomplete.Secondbestuseofre-sources,andoutrightwastefuluses,haveequalclaimtoattention.Theyaretheothersideofthecoin.Forourpresentpurposethephrasebestuseofscarceresourceswillsuffice.However, each of the two nouns and two adjectives in this phrase needs furtherdefinition.Thesedefinitionsinturnneedtobevariedandadjustedtofitthespecificcircumstancesinwhichthevariouskindsofoptimizingeconomicdecisionsaretobetaken.Iwillassumethatthemaininterestofthisgatheringisintherangeofapplications of the idea of best use of scarce resources, and in the ways in whichthemaincategoriesofapplicationsdifferfromeachother.Ishallthereforedescribemathematicalideasandtechniquesonlytoanextenthelpfulfortheexplorationofthatrangeofapplications.A good place to start is with the productionprogramsoftheindividualplantorenterpriseforashortperiodahead.Theresourcesthenincludethecapacitiesofthevariousavailablepiecesofequipment.Inacentrallydirectedeconomytheymayalsoincludetheallotmentsofnationallyallocatedprimaryinputssuchasfuels,rawmaterials,laborservices.Inamarketeconomywithsomecapitalrationingonesingleallotmentofworkingcapitalavailableforthepurchaseofprimaryinputsatgivenmarketpriceswouldtaketheplaceofmostoftheprimaryinputallotments.Ineitherinstitutionalframework,anespeciallysimpleprototypeproblemisobtainedifonefixesthequantityofrequiredoutputoftheproductorproductsmadebytheenterprise,whilepricesfortheprimaryproductsaregiven.Thenthetermuseofresourcesstandsforachoiceofatechnicalprocessoracombinationofprocessesthatmeetsthatrequirementwithinthegivenconstraints. Bestuseisa,orifunique,thatchoicethatmeetstherequirementatminimumcostofprimaryinputs.Economistshavedifferedastowhetherthisproblembelongsineconomics.InthetwentiestheBritisheconomistA.C.Pigoustated. . . itisnotthebusinessofeconomiststoteachwoolenmanufacturershowtomakeandsellwool,orbrewershowtomakeandsellbeer...ThiswasnottheattitudeofeconomistsinseveralotherEuropeancountries.Inparticular,therewasinthethirtiesalivelydiscussionamongScandinavianandGermaneconomistsconcerningmodelsofproductionpossibilitiesandtheiruseinachievingefficiencywithintheenterprise.TheNordiskTidsskriftforTekniskkonomiprovidedanimportantmediumforthesediscussions.240 EconomicSciences1975Significantcontributions1weremadebyCarlson,Frisch,GloerfeltTarp,Schmidt,Schneider,Stackelberg,andZeuthen.Thusthesituationattheendofthethirtieswasoneinwhichimportantpracticalproblemsinthebestuseofresourceswithintheenterprisehadbeenneglectedbyeconomistsinseveralcountries,andhadbeentakenupbyonlyahandfulofeconomistsinafewothercountries.Inaddition,theproblemswereofakindinwhichspecialknowledgepossessedbyotherprofessions,mathe-maticians,engineers,managers,waspertinent.Onecouldthereforehaveexpectedimportantnewcontributionstocomefromtheseneighboringpro-fessions.Thisispreciselywhathappened,andseveraltimesover.ChronologicallyfirstwasthepublicationbythemathematicianLeonidV.Kantorovich(1939,inRussia)ofa68-pagebookletentitled,intranslation(1960),Mathe-maticalMethodsofOrganizingandPlanningofProduction.Theimportanceofthispublicationisduetothesimultaneouspresenceofseveralideasorele-ments,someofwhichhadalsobeenpresentinearlierwritingsindifferentpartsofeconomicsormathematics.Ienumeratetheelements.(1) Amodelofproduction intermsofafinitenumberofdistinctproductionprocesses, eachcharacterizedbyconstantratiosbetweentheinputsandoutputsspecifictotheprocess.Thiselementhasalonghistoryineconomics.ItisfoundinWalras(1874,LeGon41;1954,Lesson20),Cassel(1919,Ch.IV),themathematicianvonNeumann(1936),Leontief(1936,1941), alldealingwithmodelsofthepro-ductive system as a whole. However, the feature most important for our purposewaspresentonlyintheclassicalwritersinthetheoryofinternationaltrade2andinthemodelsofvonNeumannandofKantorovich.Thisfeatureisthattheoutputofone-and-the-samerequiredcommoditycaningeneralbeachievedbymorethanoneprocess.Thesamespecifiedvectorofoutputsofallrequiredcommoditiescanthereforeingeneralbeobtainedastheoutcomeofmanydifferentcombinationsofprocesses.Twosuchcombinationsmaydifferinthelist of processes included - andinthelevelsofactivityassignedtotheprocessestheybothuse.Itisduetothiselementofchoicebetweenalternativewaysofachievingthesameendresultthatagenuineoptimizationproblemarises.ItistruethatWalrasalsooptimized(1954,Lesson36)onthechoiceofpro-cesses,butfromaninfinitecollectiondefinedbyadifferentiableproductionfunction.Itispreciselythischoiceofamoregeneralcollectionofprocessesthatdelayedtherecognitionbyeconomistsoftheapplicationsthatareourpresent topic:(2)Theperceptionofawiderangeofpracticalapplications ofthemodeltoindustriesthatthemselvesaresourcesforthedatarequiredbytheseapplications.1ForreferencestotheseauthorsandtothePigouquotation,seeKoopmans(1957),p.185.2SeethereferencestoTorrens(1815),Ricardo(1817),Mill(1852),Graham(1923),andothersinthesurveyarticlebyChipman(1965).T.C.Koopmans 241Theseincludedthetransportationproblemtobediscussedbelow,anagri-culturalproblem,andvariousindustrialapplications.Thedefinitionandcollectionofavailabledataofadifferent,moreaggregative,kindwasalsoanimportantelementinLeontiefsinput-outputanalysis.(3)Thedemonstrationthatwithanoptimalsolutionofthegivenproblem,whetherofcostminimizationoroutputmaximization,onecanassociatewhatinWesternliteraturehasbeencalledshadowprices, oneforeachresource,intermediatecommodityorend-product.Kantorovichstermin1939wasresolvingmultipliers,whichhechangedtoobjectivelydeterminedvaluations"3inhisbookof1959.Ingeneral,thesevaluations are equal to the first derivatives, of the negative of the cost minimum,withrespecttothespecifiedavailabilitiesofthegoodsinquestion.Inmathe-maticalterminologythesevaluationshavealsobeencalleddualvariables,incontrastwiththeactivitylevelsassignedtotheprocesses,whicharethencalledprimalvariables.AnalogousdualvariablesoccuralsoinvonNeu-mannsmodelofproportionalgrowth,withaninterpretationaspricesincompetitivemarkets.(4) The identification of a separation theorem for convex sets due to Minkowski asamathematicalbasisfortheexistenceofthedualvariables4.(5) The computationof optimal values of the primal and associated dual variablesforillustrativeexamples,andsomeindicationstowardcalculatingsuchsolutionsinmorecomplexcases.Finally,briefbutpreciseexplanationsof(6)Theinterpretationofthedualvariablesasdefiningequivalenceratios(ratesofsubstitution)betweendifferentprimaryinputsand/ ordifferentrequiredoutputs,and(7) the additional interpretation of the dual variables asguides for thecoordina-tion of allocativedecisionsmade in different departments or organizations.Ishallreturnto(7)below.Kantorovichsworkof1939didnotbecomeknownintheWestuntilthelatefiftiesorearlysixties.MeanwhilethetransportationmodelwasredevelopedintheWestwithoutknowledgeoftheworkonthistopicbyKantorovich(1942,reprinted1958)andKantorovichandGavurin(1940,1949).TheWesterncontributionsweremadebyHitchcock(1941),Koopmans(memodated1942,publ i shed1970;arti cl esof1949and1951(wi thRei ter),Dantzi g(Ch.XXI I I inKoopmans,ed.,1951).ThegenerallinearmodelwasrediscoveredanddevelopedbyGeorgeB.Dantzigandothersassociatedwithhim,undertheinitialstimulusoftheschedulingproblemsoftheUnitedStatesAirForce.Thetermlinearpro-gramming cameintouseforthemathematicalanalysisandcomputationalproceduresassociatedwiththismodel.Acompactearlypublicationofthis3Obektivnoobuslovlennyeotsenki.4ForthispurposevonNeumannhadusedtheheaviertoolofatopologicalfixed-pointtheorem. The dispensibility of this for his purpose was shown later by Gale (1956) and byKoopmansand Bausch (1959, Topic 5).242 EconomicSciences1975workcanbefoundinavolumeentitledActivityAnalysisofProductionandAllocation, editedbyKoopmans(1951).SubstantialfurtherdevelopmentsappearedinsuchJournalsasEconometrica,ManagementScience,OperationsResearch,andwerebroughttogetherinDantzigsLinearProgrammingandExtensions(1963),abookthatwasmanyyearsinthemaking.Thesedevelopments,inwhichmanymathematiciansandeconomiststookpart,wentsubstantiallybeyondtheearlierworkofKantorovich,inseveraldirec-tions.Inoteonlyafewoftheextensionstotheelementslistedabove.(2)Extensionoftherangeofapplicationsto animalfeedingproblems,inventoryandwarehousingproblems,oilrefineryoperations,electricpowerinvest-ments5andmanyotherproblems.(3, 4) Further clarification of the mathematical relations between primal and dualvariablesandtheirextensiontononl i nearprogrammi ngby6 Tucker,Gale,Kuhnandothers.ThisworkalsotracedadditionalmathematicaloriginsorprecursorsforthedualitytheoryoflinearprogrammingintheworkongametheorybyvonNeumann(1928and,withMorgenstern,1944)andbyVille(1938),andinworkonlinearinequalitiesbyGordan(1873),Farkas(1902),Stiemke(1915),Motzkin(1936)andothers.7(5) The development by Dantzig of thesimplexmethodformaximizingalinearfunctionunderlinearconstraints(includinginequalities)andthefurtherimprovementstothismethodbyDantzigandothers.Thesimplexmethodhasbecometheprincipalstartingpointforafamilyofalgorithmsdealingwithlinearandconvexnonlinearallocationproblems.Thesemethodscanbesetupsoastocomputeoptimalvaluesofbothprimalanddualvariables.Mostimportanttoeconomictheoryaswellasapplicationwasafurtherextensionof(7)into(7) analysis of the roleoruseofprices towardbestallocationofresources,eitherthroughtheoperationofcompetitivemarkets,orasaninstrumentofnationalplanning.Theseideas,again,havealonghistoryineconomics.Inregardtocompetitivemarkets,theygobackatleastasfarasAdamSmith(1776),andwereeloquent-lyrestatedanddevelopedbyHayek(1945).ImportantwritersontheuseofpricesinsocialistplanningwereBarone(1908),Lange(1936),andLerner(1937,1938).ThenewelementintheworkbyKoopmans(1949,1951)andSamuelson(1949,1966)wastheuseofthelinearmodeland,inmyowncase,theattempttodevelopwhatmaybecalledapre- institutionaltheoryofalloca-tionofresources.ItwasalreadyforeshadowedintheworkofLangeandLernerthathypotheticalperfectcompetitionandhypotheticalperfectplanningboth imply efficient allocation of resources - although neitheroccursinreality.* See, for instance, Mass and Gibrat (1957).6SeeGale,KuhnandTucker(1951),KuhnandTucker(1950),and,forasummary,Tucker(1957).7ForreferencesseeDantzig(1963),Ch.2-3.T. C. Koopmans 243Itthereforeseemedusefultoturntheproblemaround,andjustpostulateallocativeefficiencyasamodelforabstract,pre-institutionalstudy.Thereafter,onecangoontoexplorealternativeinstitutionalarrangementsforapproxi-matingthatmodel.Ibelievethatthelinearmodeloffersagoodfootholdforthispurpose.First,itmakesarigorousdiscussioneasier. Secondly,themostchallengingnon-linearity-thatconnectedwithincreasingreturnsto scale - in fact underminescompetition. Italsogreatlyescalatesthemathematicalandcomputationalrequirementsforgoodplanning.Thelinearmodel,therefore,makesanaturalfirstchapterinthetheoryofbestallocationofresources.Initssimplestformitleadstothefollowingsymmetricrelationshipsbetweenactivitylevelsoftheprocessesandthe(shadow)pricesoftheresourcesandgoodsproduced:(7)(a)Everyprocessinusemakesazeroprofit,(b)Noprocessinthetechnologymakesapositiveprofit,(c)Everygoodusedbelowthelimitofitsavailabilityhasazeroprice,(d)Nogoodhasanegativeprice.ThesesamerelationshipsarearecurrentthemeinthefirsttwochaptersofKantorovichs(1959)book,whichalsowasmanyyearsinthemakingpriortopublication.I twassubsequentlytranslatedintoFrenchandEnglish,thelatterunderthetitleTheBestUseofEconomicResources.Thegistofthebooksrecommendationsisthatsocialistplanningcanachievebestattainmentofthegoalsetbytheplanningbodythroughcalculationsthatensurethefulfillmentoftheseorsimilarconditionsforoptimality.Kantorovichdidnotgomuchbeyondhisearlierremarksonthequestionsconcerningpossibleuseofapricesystemfordecentralizationofdecisions.Thisbecameamajortheme,however,intheabstractworkofKoopmans(1951,Sec.5.12),intheworkontwo-levelplanningbyKornaiandLiptk(1962,1963,1965)inrelationtoplanninginHungary,andinthatbyMalinvaud(1967)stimulatedbyexperienceswithplanninginFrance.TheprincipalcomputationalcounterpartofthisworkwasdevelopedbyDantzigandWolfe(1960,1961) underthenamethedecompositionprinciple.ThethirdchapterofKantorovichsbookdealswiththeproblemofinvest-mentplanningtoenlargetheproductionbase.Theprincipalemphasisisonthe concept of the normal effectiveness of capital investment. This is a discount rateappliedtofuturereturnsandtocontemplatedinvestmentsandotherfuturecosts,intheevaluationandselectionofinvestmentprojects.ThisideahadbeenproposedearlierbyNovozhilov(1939).ThepointemphasizedbyKantorovichisthatthepricestobeusedincalculatingreturnsandcostsshouldbetheobjectivelydeterminedvaluationsdeterminedbyhismethods,fortheselectiontohaveanoptimalresult.TheseproposalswereatthetimenewtothepracticeofSovieteconomicplanning.Ibelievethattheprincipleofthenormaleffec-tivenessofcapitalinvestmenthasgainedincreasingacceptanceinSoviettheoryandpracticesincethattime.Thereisanobviousformalanalogywiththeprofitabilitycriterionforinvestmentplanningusedbythefirminamarketeconomy,usinganticipatedmarketpricesandtheappropriatemarketrateofinterest.Theinstitutionalframeworkcontemplatedis,ofcourse,fundamentally244 EconomicSciences1975different.Ibelievethattheunderlyingpre-institutionaloptimizingtheoryisthesame.Summingup,IseetwoprincipalmeritsinthedevelopmentsIhavereviewedsofar.Oneistheirinitiallypre-institutionalcharacter.Technologyandhumanneedsareuniversal.Tostartwithjusttheseelementshasfacilitatedandintensifiedprofessionalcontactsandinteractionsbetweeneconomistsfrommarketandsocialistcountries.Theothermeritisthecombinationandmergingofeconomictheory,mathematicalmodeling,datacollection,andcomputationalmethodsandalgorithmsmadepossiblebythemoderncom-puter.Agenuineamalgamofdifferentprofessionalcontributions!Thelinearmodel,followedbytheconvexnonlinearmodel,haveprovidedtheprovinggroundforthesedevelopments-and-theirmostconspicuouslimitation:Thenonconvex nonlinearitiesassociatedwithincreasingreturnsto scale - i.e., with thegreaterproductivityoflarge-scaleproductioninmanyindustries - require quite differentmethodsofanalysis,andalsoraisedifferentproblemsofinstitutionalframeworksconducivetobestallocation.Inowproceedtoaratherdifferentclassofapplicationsoftheideaofbestallocationofscarceresources.Thisfieldisusuallyreferredtoasthetheoryofoptimal economic growth. In most studies of this kind made in the countries withmarketeconomiesthereisnotanidentifiableclienttowhomthefindingsaresubmittedaspolicyrecommendations. Noristhereanobviouschoiceofobjectivefunction,suchascostminimizationorprofitmaximizationinthestudiesaddressedtoindividualenterprises.Thefieldhasmoreofaspeculativecharacter.Themodelsstudiedusuallycontainonlyafewhighlyaggregatedvariables.Oneconsidersalternativeobjectivefunctionsthatincorporateoremphasizevariousstrandsofethical,political,orsocialthought.Theseobjectivesarethentriedouttoseewhatfuturepathsoftheeconomytheyimplyunderequallysimplifiedassumptionsoftechnologyorresourceavail-ability.Theprincipalcustomersaimedforareothereconomistsormembersofotherprofessions,whoaresomewhatclosertothemakingofpolicyrecom-mendations.Thesemaybethoseengagedinmakingmoredisaggregatedopti-mizingmodelsofgrowththatincorporatenumericalestimatesoftechnologicalorbehavioralparameters.(Ishallreturntothisfieldofdevelopmentpro-gramming below.)Orthehoped-forcustomersmaybepolicyeconomistswhomayfinditusefultohavethemoreabstractideasofthisfieldinthebackoftheirmindwhencopingwiththeday-to-daypressuresforoutcomesratherthancriteria.Thequestionoftheclientleisevenmorebafflingwhentheproblemcon-cernsgrowthpathsfortimespanscoveringseveralgenerations.Whatcanatbestberecommendedinthatcaseisthesignalthepresentgenerationgives,thetraditionitseekstostrengthenorestablish,forsucceedinggenerationstotakeofffrom.The classic in the optimal growth field is a paper published in 1928 by FrankRamsey,knownalsoastheauthorofequallyfundamentalpapersonthefoundationsofmathematicsandonsubjectiveprobability.Hisdefinitionofbestinvolvesthemaximizationofasum(orintegral)ofutilityflowstobeT.C.Koopmans 245derivedfromfutureconsumption.Usingacontinuoustimevariable,RamseyschoiceofobjectivefunctionisalimitingcaseofabroaderclassoffunctionswhichIshallconsiderfirst,Hereitdenotestheaggregateconsumptionflowasoftimet,andu(c)isautilityflowservingasanevaluatingscorefortheconsumptionflowc.Oneduchoosesthefunctionu(c)soastoincreasewithcbutatadecreasingrate-dCas c increases.Thisexpressesthatatalltimesmoreisbetter,butlesssoifmuchisalreadybeingenjoyed(seeFigure1).Theeffectontheallocationofconsumptiongoodsbetweengenerationsissimilartotheeffectofaprogressiveincometaxonspendableincomesamongcontemporaries.cFig.1Weshallcalltheexponentiallydecayingfactore-t, Q>0,thediscountfactorforutility.Itdiminishestheweightgiventofutureutilityflowsinthesumma-tionoftheentireutilityflowoverallthefuturetoformatotalscoreU. Theweightissmallerthelargerthediscountrateforutility,Q,andthefurtheronelooksintothefuture.OnethicalgroundsRamseywouldhavenoneofthis.Ishalltaketheviewthattheimportantquestionofdiscounting utility - or forthatmatteranyotheraspectofthechoiceoftheobjectivefunction-shouldnotbesettledentirelyonapriorigrounds. Mostdecisionmakerswillfirstwanttoknowwhatagivenobjectivefunctionwillmakethemdoingivencircumstances.IshallthereforeholdQ>0forthisfirstexploration,8andturntothemathematicalmodelingofthecircumstancesintermsoftechnologicalandresourceconstraintsontheconsumptionandcapitalvariables.Oneresourceisthelaborforce.Itneednotentertheformulaebecauseitwillbeassumedtoremaininexorablyconstantovertime.Theonlyotherresourceisaninitialcapitalstockdenotedk0,historicallygivenasoftimet =0.Theuseatanytimet oflaborandofthethencapitalstockkt consists8Foranobjectivefunctionimplyingavariablediscountratethatdependsonthepathcontempl atedseeKoopmans(1960),Koopmans. Di amondandWi l l i amson(1964)andBeal sandKoopmans(1969).246EconomicSciences1975of two steps. The first and obvious step is to achieve at. all times the highest netoutputflowf (kt)thatcanbeproducedbythelaborforce,usingthecapitalstockfullyandtobesteffect.Theformgiventothefunctionf(k)summarizesandsimplifiesbroadtechnologicalexperience.Itspecifiesf(0)=0(withoutcapitalnooutput),f(k)initiallyincreasingwithkbutatadiminishingratedfiiin such a way that from some pointRofcapitalsaturationon,f(k)decreasesbecausedepreciationrisesmoresteeplythangrossoutput.(SeeFigure2.)Fig. 2kInallthistheproductflowf(k)isregardedasconsistingofasinglegood,which can be used as desired for consumption or for adding to the capital stock,Todeterminethisallocationforallt isthesecondstep.ThisisdonebestatallfuturepointsoftimeifthetotalscoreU istherebymaximized.Itmightseemasifthisconstrainedmaximizationproblemisquitedifferentinmathematicalstructurefromthosediscussedbefore.Thisisnotthecase.Themaindifferenceisthatthediscussionhasshiftedfromavectorspacetoafunctionspace,usingconventionalnotationsnotdesignedtorevealthecommonstructureofthetwoproblems. Inparticular,aslongasthecrucialconvexityassumptionsaremaintained,interpretationsintermsofshadowpricesremainvalid.TheproblemforQ>0wassolvedindependentlybyCass(1966),Koopmans(1965,1967),Malinvaud(1965,1967),thirty-fiveyearsafterRamsey.Withoutproof I indicate the nature of the solution in Figure 3. In the diagram on the left,the abscissakissetoutalongtheverticalhalf-axistheodinate-11 =f (k)alongthehori zontal half-axispointingleft.Forgiven>0,findtheuniquepointi(p)onthecurvey=f(k) inwhichtheslopefequals.Then,iftheinitialcapitalstockk,shouldhappentoequali(p),theoptimalcapitalpathremainsconstant,kt =L(p),overallthefuture. Foranyinitialstockk0lessthan01largerthanl(p),theoptimalpathshowsamonotonicandasymptoticapproachto K(p). All this is illustrated in solid lines in the top right diagram in Figure 3.ThelowerrightdiagramshowsthecorrespondingoptimalconsumptionpathQ,whichapproachestheasymptoticlevelc()=f(k()).T.C.Koopmans 247Fig.3If ptandqtweretobemarketpricesforthecapitalgoodandforitsuseinproduction,thisequalitywouldstatethatthereturnsontwoalternativedispositionsforthecapitalgoodwouldbeequal:Oneistosellthegoodnow,theotheristosellfirstonlyitsuseforashortperiod,andthereaftertosellthegooditself.Thatthisprincipleisnecessaryforoptimalityisintuitivelyplausible.Itssufficiencycanbeprovedfromtheconvexityassumptionsaboutthefunctions-u( *)and-f ( *), plus theboundaryconditionthatlimkt ==ii(@).t+=248 EconomicSciences1975WhatistheeffectofchoosingdifferentvaluesofthediscountrateQ?Figure3suggeststheanswerfortherealisticcasethattheinitialcapitalstock k, iswellbelowitsultimateleveli(e). In that case, as Qisdecreased,thatis,asthepresentvaluationofconsumptioninadistantfutureisincreased,thentheasymptoticlevelsofthecapitalstockandofconsumptionarebothincreased.However,toaccumulatetheadditionalcapitalthatmakesthisfeasible,con-sumptioninthepresentandthenearfutureisfurtherdecreased.Thus,theimpatienceexpressedbyapositivediscountratemerelydeniestouncounteddistantgenerationsapermanentlyhigherlevelofconsumptionbecausethatwouldnecessitateasubstantiallysmallerpresentconsumption.Perhapsapity,butnotasin.RamseyshowedthattheeffectofadecreaseinQgoesrightdowntobutnotbeyondthelimitingcaseofnodiscounting,Q=0.TheoptimalpathsforthatcaseareshownbydashedlinesinFigure3.Ramseyusedaningeniousmathe-maticaldevicethatgetsaroundthenonconvergenceoftheutilityintegralforQ=0,andalsoleadstoaproofsimplerthantheoneforpositiveQ.Thisnarrowescapeforvirtueisblockedoffbysomequiteplausiblemodi-ficationsofthemodeltowardgreaterrealism.Forinstance,onemayintroduceapopulation(=laborforce)Ltthatchangeswithtimeand,forinterpersonalequity,modifytheobjectivefunctiontoreadovertime.Bywayofexample,let correspondtoagrowthby3%peryear.Then,formathematicalreasonsalone,thediscountrateQhastocorrespondtoatleast3%peryearforanoptimalpathtoexist.IfonetriestokeepQ atzeroandforceexistencebyimposingafinitetimehorizonofonecentury,say,thentheoptimizationproducesanirrationalandarbitrarypile-upofconsumptiontowardtheendofthatcenturywhilethecapitalstockrunsdownto zero - or to any other prescribedlevelthatstillleavesroomforaterminalsplurge.Themodeldiscussedsofarleavesoutimportantaspectsofthemoderneconomy.Ishallcommentbrieflyontheincorporationintothemodelof(1)exhaustibleresources,(2)technologicalchange,(3)populationasapolicyvariable.With regard to exhaustibleresources,I shall only consider the extreme case ofan essentialresource. By this I mean a resource that is essential to sustaining life,isnotcapableofcompleterecycling,andhasnosubstituteeithernoworlaterwithintheremainingperiodofitsavailability.Ihaveconstructedahighlysimplifiedmodel(Koopmans,1973)ofanessen-T.C.Koopmans 249tialresourcewhichleadstoconclusionsquitedifferentfromthosereachedfortheprecedingcapitalmodel.Theonlyprocessinthetechnologyconsistsofcost-lessextractionoftheresourcecombinedwithitsimmediateanddirectcon-sumption.ThisprocessisavailableatalltimesuntilthetimeTatwhichtheresourceisexhausted.Populationisassumedgivenandconstantforthe.survivalperiod 0 t 7.AttimeTitfallstozero.Inthesecircumstances,higherpercapitaconsumptionbythoselivingearlyenoughtoshareintheavailableresourceshortensthesurvivalperiod,hencereducesthetotalnumbersharing.Thusthesurvivalperiodisnowapolicyvariable,andtherebysoispopulationinsomepartofthefuture.WenowadoptagaintheobjectivefunctionoftheRamseymodelwithdiscounting,exceptthattheintegrationextendsonlyovertheperiod of survival.Then,unlikeinthecapitalmodel,theoptimalpathnowdependsonwhereoneplacesthezeropointintheutilityscaleinwhichthefunctionU( +)isexpressed.Letussetthiszeropointsomewhatarbitrarilyattheutilityflowlevel,u(r)=0,oftheresourceconsumptionlevelr belowwhichlifecannotbesustained.Theresultingfunctionu(r) isshown(againsideways)ontheleftinFigure4.rrUFig.4Intheabsenceofdiscounting(Q=0),timenowentersonlyasascaleonwhichthefatalcut-offpointTisdetermined.Amongthosesoadmittedequalsharingisclearlyoptimal,henceoptimalconsumptionstaysataconstantlevelovertime.Figure4shows(dashedlines)howthislevel,andtherewiththesurvivalperiod,aredetermined:AllincludedclaimantsconsumeoptimallyatthatuniquepercapitalevelPthatmaximizestheutilityflowperunitofresourceflowconsumed.Notethatthisconsumptionleveliswellabovethesubsistenceminimumr.Thisresultistobeexpectedfromanobjectivefunctionthatplacesvaluenotonthenumberofincludedpeopleassuch,butonlyonthenumberofutilsenjoyedbyallalivetakentogether.Discounting(Q>0)nowintroducesunequalweights,decreasingastimegoeson,betweenpeoplewithequaltechnologicalopportunities.Theresultis(solidline)thattheconsumptionofearlierclaimantsisraisedoverthatoflaterones,withthelastclaimantsnobetteroffthanbefore.Ofnecessity,the250 EconomicSciences1975cut-offpointarrivessooner,andsomeofthoseincludedforQ =0 will not betheretopresstheirclaimifQ >0.Iholdnobrieffortherealismofthismodel.Ihavebroughtituponlyasastarkdemonstrationofthepointthattheproblemofwhetherandhowmuchtodiscountfutureutilitiescannotbeequitablyresolvedaprioriandintheabstract.Oneneedstotakeintoaccounttheopportunitiesexpectedtobeavailabletothevariousconsumersnowandlater,forthegiventechnologyandresourcebase.Doesthereexistanessentialresource,asdefinedabove,exhaustibleinlessthanastronomicalorevengeologicaltime?Ihavearguedelsewhere(Koop-mans,1973)thatthebestavailableanswertothisquestionmustcomefromthose natural scientists and engineers most able to assess what future technologymaydo,orbemadetodo,tofindsubstitutematerialsorfuelsforthosenowwithinsightofexhaustion.Ishouldaddthatgeologistsmaywelldevelopwaystopindownfurtherthebestestimatesofultimateavailability.Economistscouldbringuptherearwithmethodsforintegratingthediversepiecesofinforma-tionsoobtained.This leads to a few remarks on models that recognize technological change, andtheuncertaintybywhichitisinevitablysurrounded.AconceptualstepforwardinthisdirectionhasbeenmaderecentlybyDasguptaandHeal(1974).Theirmodelpostulatesanexhaustibleresourceforwhichasubstitutewillormaybecomeavailableatanuncertainfuturedate.Theuncertaintyisdescribedbyasubjectivelyestimatedprobabilitydistribution.Capitalaccu-mulationisalsorepresentedinthemodel.Theobjectivefunctionisthemeanvalueofthedistributionofthesumofdiscountedutilitiesoveraninfinitefutureperiod.Theoptimalpathtobefoundconsistsoftwosuccessiveseg-ments.Oneisthepathtobefollowedfromt =0uptotheasyetunknowntimet=Toftheavailabilityofthesubstitute.TheotheristhepathtobefollowedfromtimeTon,whichofcoursedependsonthesituationinwhichtheadventofthesubstitutefindstheeconomy.Inoneinterestingsub-case,theeffectofuncertaintycanberepresentedbyanequivalentadditiontothediscountrate-untilthesubstituteisavailable.Thisisthecaseinwhichthetechnologicalchangeisexpectedtobesoincisivethattheoldcapitalstockwillloseallitsvalueasaresultoftheavailabilityofthenewtechnology.Deepproblemsofchoiceofoptimalitycriterionariseinmodelsinwhichpopulation size is a decision variable, or isaffectedbydecisionvariables.Ishallcontrastonlytwodistinctalternativecriteria.One,usedinapopulationcon-textbyMeade(1955),Dasgupta(1969) andothers,isgivenbytheobjectivefunctionrepresentingindividuals alreadydiscussed.Thisfunctionmultipliestheutilityofpercapitaconsumptionbythenumberofconsumersbeforeform-ingthediscountedsum.Theothercriterion,thefunctionrepresentinggenera-tions, sumsjustthepercapitautilities.TheargumentbyArrowandKurz(1970,I.4)infavoroftheformercri-terionusestheanalogyoftwocontemporaneousislandpopulationsunderonegovernment.ItshowsconvincinglythatpercapitautilitiesforthetwoislandsT. C. Koopmans 251mustbeforetheiradditionbemultipliedbytherespectivepopulationsifonewantstoavoiddiscriminationbetweenpeopleofthetwogroups.Doesthisargumentcarryovertogenerationsthatsucceedeachotherintime?Onemightwellarguethisonthesamegroundsifpopulationincreaseisatrulyexogenousfunctionoftime,unalterablebyanypolicy,butfortunatelyonethatpermitsafeasiblepath.Onesuchcasewasalreadyconsideredabove.Butinfactpopulationcanbeinfluenced,directlybypersuasionandprovisionofinformation,orindirectlythroughothereconomicvariables.Isubmitthatthismakesadifference.Adistinctionshouldbemadeherebetweenconcernforthewelfareofthosealreadyaliveandofthoseasyetunbornwhosenumbersarestillundecided.Current social ethic urges recognition of the needs and desires of living persons,withinnations,andbetweennations-eventhoughpracticediffersfromnorm.Butthereissomethingopen-endedaboutthesameconcernforourdescendants.Howmanydescendants?Theanswertothatquestionisdifferentforthetwocriteriawearenowcomparing.OnthebasisofindicationsinarecentpaperofPitchford(1975,p,21)Iconjecture:9thatthecriterionrepresentingindividualswill,underconstraintslikethoseconsideredabove,recommendasmallerpercapitautilityforallgenerations,presentandfuture,thandoesthecriterionrepresent-inggenerations.Thisistobeexpectedbecausethecriterionrepresentingin-dividualsattachesvaluetonumbersassuch.Specificallyitwouldratethecombinationofagivenpercapitautilitylevelandpopulationsizebelowanothercombinationwitha5%lowerutility,say,andamorethan5%largerpopulation.Sotheissueappearstobeoneofquantityversusquality.Again,onewouldwanttotryoutthetwocriteria,andotherones,underavarietyofconstraintsbeforeafullyconsideredjudgmentcanbemade.Butthesimpleideaofaddingupindividualutilitiesdoesnotseemtomecompellinginitself.Thosealreadybornarecommittedtotheirexistencebytheinstinctofselfpreservation.Choosingacriterionthatlimitsbirthssoastoallowagoodexistencetoanindefinitelycontinuingsequenceofgenerationsappearssufficientasanendinitself.Whymorepeopleattheexpenseofeachofthem?Theforegoingdiscussiontacitlyassumesthatnetpopulationincreasecanbecontrolledwithoutmuchdelay,withintherangerecommendedbyoptimiza-tion.Inreality,neitherthetechniquesnortheacceptanceofbirthcontrolaresuchastosupportthatassumption.Ontheotherhanditisnotsothatpopula-tionincreasecanberegardedasentirelyexogenous.Thissuggestsrefinementofthemodelsintwodirections.9For this conjecture to he well defined, the criterion must again specify the zero point in theutilityscale.Isetitagainatthesubsistenceminimum, becauseassociatingazeroutilitywith any higher consumption level would imply a preference for the phasing out of humanlife in some adverse circumstances in which a very low consumption level permitting survivalremains feasible.252 EconomicSciences1975First,amorerealisticmodelshouldincorporateestimatesoftherelationsdescribingtheresponseofreproductivebehaviortolevelsincome,educa-tion,housing,medicalcare,andothercausativevariables.Incircumstanceswheretheresultingpathofpopulationdoesnotseriouslyreducepercapitaincomebelowwhatcouldbeachievedbyamoredirectpopulationpolicy,nofurtheractionwouldberequired.Wherethisisnotthecase,theprocesseswherebyreproductivebehaviorcanbeinfluenced,andinparticulartherela-tionbetweenresourceinputsintotheseprocessesandtheresponsestothem(promptanddelayed)needtobeincorporatedinthemodel.ConsiderationsofthiskindhavebeenintroducedintooptimalpopulationmodelsbyPitch-ford(1974).Thesecondrefinementconcernstheoptimalitycriterion.Situationsoccurinwhichwhatcanbeachievedbypopulationpoliciescanonlydiminishbutnotpreventaloweringofthepercapitaconsumptionsustainedbydomesticresources,foranextendedperiodahead.Insuchcasesarealisticviewwillrecognizeadegreeofexogeneityinthefuturepathofpopulationforsometimetocome.Onemaythenwanttoexploreoptimalitycriteriathatextendtothoseasyetunbornchildrenwhosebirthhadbetter,butcannot,bepreventedthesameconsiderationastothosealreadyborn.Wehaveconsideredtwobroadfieldsofapplicationforoptimizationmodels.Onecomprisesthedetailedanddata-orientedoptimizationofthedecisionsoftheenterpriseorpublicagency-andalsothecoordinationofsuchdecisionsthroughapricesystem,throughcentralizedplanningandmanagement,orboth.Theotheristhemorespeculativestudyofalternativeaggregatefuturegrowthpathsforanentireeconomy.InconclusionIwanttomakesomeremarksaboutthegrowingfieldofdevelopmentprogramming,in which the two strands of thought are being com-binedandmerged.Oneearlystepinthisdevelopmentwastheconstructionofamathematicalprogrammingmodelforaneconomyasofsomefutureyear,includinginvestmentsandtheflowofaidintheinterveningperiodasdecisionvariables.AnexampleisthestudyoftheeconomyofSouthernItalybyChene-ry,writingwithKretschmer(1956)andwithUzawa(1958).AnevaluativedescriptionofexperienceswithHungarianeconomicplanningalongtheselineswaswrittenbyKornai(1967).Laterstudies,suchasthatoftheMexicaneconomybyGoreux,Manneandcoauthors(1973),envisagedasequenceoffutureyears.InmostofthesestudiesdataavailabilitiesdeterminedtheuseofLeontiefsinput-outputframeworkforrepresentingtheproductionpossibili-tiesoftheeconomyasawhole.Policychoicesandoptimizationwereintro-ducedwheredatasopermitted.OneexampleisthechoicebetweendomesticproductionversusimportspaidforbyexportsintheSouthernItalystudy.OthersarethesectoraldetailintheHungarianstudies,andconcentrationontheenergyandespeciallyelectricenergysectorsintheMexicanone.Aweaknessinthetreatmentofconsumptioninoptimalgrowthmodels,notedbyChakravarty(1969),isthelackofcontinuitybetweenconsumptionlevelsinthepastandthoserecommendedbyotherwisereasonablelookingoptimalitycriteriaforthenearfuture.OneremedyproposedbyManneT.C.Koopmans 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