Homework3

Inthisexercise,youwillanalyzethesupply-demandequilibriumofacityundersomespecialsimplifying assumptions about land use. The assumptions are:

1. alldwellingsmustcontainexactly1,500squarefeetoffloorspace,regardlessof

location,and

1. (ii)apartmentcomplexesmustcontainexactly15,000squarefeetoffloorspaceper

squareblockofland area.

Theseland-userestrictions,whichareimposedbyazoningauthority,meanthatdwellingsizesand building heights do not vary with distance to the central business district. Distance is measured in blocks.

Supposethatincomeperhouseholdequals$25,000peryear.Itisconvenienttomeasure moneyamountsinthousandsofdollars,sothismeansthaty=25,whereyisincome.

Nextsupposethatthecommutingcostparametertequals0.01.Thismeansthataperson livingtenblocksfromtheCBDwillspend0.0110=0.1peryear(inotherwords,$100) gettingtowork.

Theconsumer'sbudgetconstraintisc+pq=y-tx,whichreducestoc+1,500p=25-

0.01xundertheaboveassumptions.

Sincehousingconsumptionisfixedat1,500,theonlywaythatutilitiescanbeequalforallurbanresidentsisforbreadconsumptionctobethesameatalllocations.Theconsumptionbundle(thebread,housingcombination)willthenbethesameatalllocations,yieldingequalutilities.

Forctobeconstantacrosslocations,thepricepersquarefootofhousingmustvarywithxinawaythatallowstheconsumertoaffordafixedamountofbreadafterpayinghis rentandhiscommutingcost.Letc*denotethisconstantlevelofbreadconsumptionforeachurbanresident.Forthemoment,c*istakenasgiven.We'llseebelow,however,thatc*musttakeonjusttherightvalueorelsethecitywillnotbeinequilibrium.

1. Substitutingc*inplaceofcinthebudgetconstraintc+1,500p=25-0.01x,solve forpintermsofc*andx.Thesolutiontellswhatthepricepersquarefootmustbeata givenlocationinorderforthehouseholdtobeabletoaffordexactlyc*worthofbread.How doespvarywithlocation?

Recallthatthezoninglawsaysthateachdevelopedblockmustcontain15,000square feetoffloorspace.Supposethatannualizedcostofthebuildingmaterialsneededtoconstructthismuchhousingis90(thatis,$90,000).

1. Profitpersquareblockforthehousingdeveloperisequalto15,000p-90-r, whererislandrentpersquareblock.Inequilibrium,landrentadjustssothatthisprofitis identicallyzero.Setprofitequaltozero,andsolveforlandrentintermsofp.Then substituteyourpsolutionfrom(a)intheresultingequation.Theresultgiveslandrentrasa functionofxandc*.Howdoeslandrentvarywithlocation?

Sinceeachsquareblockcontains15,000squarefeetofhousingandeachapartmenthas1,500squarefeet,eachsquareblockofthecityhas10householdslivingonit.Asaresult,acity witharadiusofblocks canaccommodate10²households(²istheareaofthecity insquareblocks).

1. Supposethecityhasapopulationof200,000households.Howbig

mustitsradiusbeinordertofitthispopulation?Roundofftothenearestblock.

1. Inorderforthecitytobeinequilibrium,housingdevelopersmustbidawayenoughlandfromfarmerstohousethepopulation.Supposethatc*=15.5,whichmeansthateachhouseholdinthecityconsumes$15,500worthofbread.Supposealsothatfarmersofferayearlyrentof$2,000persquareblockofland,sothat= 2. Substitutec*=15.5intothe

landrentfunctionfrom(b),andcomputetheimpliedboundaryofthe city. Usingyour

answerto(c),decidewhetherthecityisbigenoughtohouseitspopulation.Ifnot,adjust c*untilyoufindavaluethatleadsthecitytohavejusttherightradius.

1. Usingtheequilibriumc*from(d)andtheresultsof(a)and(b),writedownthe equationfortheequilibriumlandrentfunction.WhatistherentpersquareblockattheCBD (x=0)andattheedgeofthecity?Plotthelandrentfunction.Howmuchdoesahousehold livingattheedgeofthecityspendoncommuting?
2. Supposethatthepopulationofthecitygrowsto255,000.Repeat(c),(d),and(e)for thiscase(butdon'trepeatthecalculationinvolvingc*=15.5).Explainyourfindings.How doespopulationgrowthaffecttheutilitylevelofpeopleinthecity?Theanswercomesfromlookingatthechangeinc*(sincehousingconsumptionisfixedat1,500squarefeet,the utilitychangecanbeinferredbysimplylookingatthechangeinbreadconsumption).Notethatbecausetheyarefixed,housingconsumptiondoesn'tfallandbuildingheightsdon't riseaspopulationincreases,ashappenedinthemodelinchapter2.Aretheeffectsonr andthesame?

1. Nowsupposethatpopulationisbackat200,000(asin(c))butthatrisesto3(thatis, farmersnowoffer$3,000rentpersquareblock).Notethat,unlikeinthechapter,thevaluecan'tchangeas rises(whatisthereason?).Repeat(d),(e)forthiscase. Compareyouranswerswiththosein(f).

1. Nowsupposethatinsteadofbeinglocatedonaflatfeaturelessplain,theCBDis locatedontheocean(wherethecoastisperfectlystraight).Thismeansthatonlyahalf- circleoflandaroundtheCBDisavailableforhousing.Howlargemustbetheradiusofthis half-circlebetofitthepopulationof200,000residents?Usingyouranswer,repeat(d)and (e),assumingthatallparametersarebackattheiroriginalvalues.Arepeopleinthiscoastalcitybetterorworseoffthanpeopleintheinlandcityof(c)and(d)?(Assumeunrealisticallythatpeople don't value the beach!) Can you give an intuitive explanation for your answer?

1. Finally, focus again on the inland city, and suppose that the zoning authority imposes a building height restriction. This restriction limits housing square footage per block to 7,500, half the previous amount. The cost of building materials per square block falls from 90 to 43 (note that the cost is less thanhalfasmuchbecauseofdiminishingreturns).Findthenewvalueof(comparetheanswerin(h)) and repeat (d) and (e). How does the height restriction affect the utility of urban residents? Explain intuitively why this effect emerge. Does the restriction seem to be a good policy?