The-Fundamentals-of-Land-Prices-and-Urban-Growth

JOURNALOFURBANECONOMICSx,295-306(1989)TheFundamentalsofLandPricesandUrbanGrowth*DENNISR.CAPOZZAANDROBERTW.HELSLEYFacultyofCommerceandBusinessAdministration,UniversityofBritishColumbia,2053MainMall,Vancouver,BritishColumbia,CanadaV6TIY8ReceivedOctober7,1986;revisedMarch24,1987Inaverysimplemodelinwhichcapitalisdurableandlandownershaveperfectforesight,thepriceofurbanlandhasfouradditivecomponents:thevalueofagriculturallandrent,thecostofconversion,thevalueofaccessibility,andthevalueofexpectedfuturerentincreases,agrowthpremium.Inrapidlygrowingcities,thegrowthpremiummayeasilyaccountforhalfoftheaveragepriceoflandandmaycreatealargegapbetweenthepriceoflandattheboundary(minusconversioncost)andthevalueofagriculturallandrent.Q1989AcademicPI~SS,I~C.I.INTRODUCTIONDuringthelast15years,thetheoryofintraurbanlocationandlandusehasbeenextendedtoaddressthedynamicsofurbangrowthanddecline.Thisextensionhasfundamentalimplications.First,thedeterminantsofurbanspatialstructurearedifferentinstaticanddynamicmodels.Forexample,inastaticmonocentricmodel[15],lotsizesincreaseanddensitiesdecreasewithdistancefromemploymentsinceequilibriumlandrentsdeclinetooffsettherisingcostofcommuting.However,inadynamicmodelwithdurablehousingandmyopiclandowners[l,91,urbandevelop-mentisanincrementalprocess,wheredensitiesdependsolelyoneconomicconditionsatthetimeofdevelopment.Densitiesmayneverthelessdeclinewithdistance,becauseeconomicconditionschangeovertimeinparticularways(incomesincreaseortransportationcostsfall),butlandrentsandpricesandpopulationdensitiesmayalsorisewithcommutingdistance11,16,81.Second,basiccharacteristicsofcitiesthatseemanomalousinastaticcontextareeasilyexplainedwhentheintertemporalnatureofurbandevel-opmentisacknowledgedInparticular,dynamicmodelsofurbanspatialdevelopmentanalyzehousingabandonment[l],theagingandreplacementofstructures[4,171,amixtureoflandusesatasinglelocation[7],anddiscontinuousspatialdevelopmentpatterns[14,5,16,12,71.Thislast*WearegratefultoRichardAmott,EdwinMills,andWilliamWheatonforhelpfulcomments.ThefinancialsupportoftheRealEstateCouncilofBritishColumbiaisgratefullyacknowledged.2950094-1190/89$3.00Copyright01989byAcademicPress.Inc.Allrightsofreproductioninanyformresewed.296CAPOZZAANDHELSLEYapplication,towhichthepejorativeterm“urbansprawl”isoftenattached,isespeciallyinteresting,sincediscontinuousdevelopmentcanbeParetoefficientinadynamiccontext.Third,urbangrowthhasimplicationsforthepriceofland.Inastaticcontext,landpriceisproportionaltolandrent,andthepriceoflandattheboundaryofanurbanareaequalsthevalueofagriculturallandrent.Inadynamiccontext,neitherpropertyholds.Usingaverysimplemodelinwhichcapitalisdurableandlandownershaveperfectforesight,weshowthatlandpricehasfouradditivecomponents:thevalueofagriculturallandrent,thecostofconversion,thevalueofaccessibility,andthevalueofexpectedfuturerentincreases.Wealsoshowthat,inadynamiccontext,anefficientlandmarketnaturallyproducesagapbetweenthepriceoflandattheboundary(minusconversioncost)andthevalueofagriculturallandrent.Thepaperhassixparts.PartIIdescribesthemodel,andPartIIIpresentsthesolutionforlandrentsandprices.InPartIVweaddtheassumptionthatpopulationgrowthisexponentialandderiveanexpressionfortheaveragepriceoflandintheurbanarea.InPartVweusethemodeltoanalyzeintermetropolitanlandpricedifferentialsandthegapbetweenthepriceoflandandthevalueofagriculturallandrentsattheboundaryofanurbanarea.PartVIsummarizesthepaper.II.THEMODELOurmodeliscloselyrelatedtoearlierurbangrowthmodels,especiallythedurablecapital,perfectforesightmodelsofAmott[2]andWheaton[16].Infact,themodelanalyzedinthispaperisessentiallyaspecialcaseoftheworkofthese(andother)authors,sinceweassumethatlotsizesaref&d.AssumptionsAnurbanareaislocatedonahomogeneousplain2+radiansofwhicharesuitablefordevelopment.AttimetE[0,cc)thereareN(t)identicalhouseholdsintheurbanareaandtheyderiveutilityfromacompositenumerairegoodXandlandL.TheutilityfunctionU(X,L)ishomoge-neousofdegree1,continuousandincreasinginXandL.Consumptionoflandisfixedatzunitsperhousehold.Sincetheutilityfunctionishomogeneousofdegree1wemaywriteu(x,r;)=Lu(x/z,1)=24(x/i;),whereu(X/z)isacontinuousandincreasingfunctionoftheratioofothergoodstoland.LANDPRICESANDURBANGROWTH297Employmentisconcentratedatapointcentralbusinessdistrict(CBD)towhichallhouseholdscommutedaily,andlocationsareindexedbytheirdistancezfromtheCBD.ThecostofcommutingaunitofdistanceisapositiveconstantT.Theboundaryoftheurbanarea,Z(f),expandsinannularringsof2+radiansasthepopulationgrows.LandRentAteachpointintimelandrentRsatisfiesthebudgetconstraint,y=X+Rz+Tz,(2)whereyishouseholdincome.Ifthetimepathofutilityisu(t)thentherentfunctionisR(t,z)=(l/z)(y-Tz)-u-+(t)].ThederivativesofR(t,z)withrespecttodistanceandtimeareR,(t,z)=-T/z,(3)(4andR,(t,z)=-u-“[~(t)]u’(t).(5)Equation(4)isthefamiliarspatialequilibriumconditionthatrentfallswithdistancetooffsetrisingtransportationcosts.Equation(5)showsthatifincomesareconstant,thenlowerutilitylevelsaccompanyrisingrentsastheurbanareagrows.ThePriceofLandIflandownershaveperfectforesightandthelandmarketisc