Advanced Microeconomic Theory .Outline • Partial Equilibrium Analysis • General Equilibrium Analysis

  • View
    214

  • Download
    0

Embed Size (px)

Text of Advanced Microeconomic Theory .Outline • Partial Equilibrium Analysis • General Equilibrium...

  • AdvancedMicroeconomicTheory

    Chapter6:PartialandGeneralEquilibrium

  • Outline

    PartialEquilibrium Analysis GeneralEquilibrium Analysis ComparativeStatics WelfareAnalysis

    AdvancedMicroeconomicTheory 2

  • PartialEquilibriumAnalysis

    Inacompetitiveequilibrium(CE),allagentsmustselectanoptimalallocationgiventheirresources: Firmschooseprofit-maximizingproductionplansgiventheirtechnology;

    Consumerschooseutility-maximizingbundlesgiventheirbudgetconstraint.

    Acompetitiveequilibriumallocationwillemergeatapricethatmakesconsumerspurchasingplanstocoincidewiththefirmsproductiondecision.

    AdvancedMicroeconomicTheory 3

  • PartialEquilibriumAnalysis

    Firm: Giventhepricevector,firmsequilibriumoutputlevel% mustsolve

    max)*+,

    % %(%)

    whichyieldsthenecessaryandsufficientcondition

    %3(%),withequalityif% > 0 Thatis,everyfirm producesuntilthepointinwhichitsmarginalcost,%3(%),coincideswiththecurrentmarketprice.

    AdvancedMicroeconomicTheory 4

  • PartialEquilibriumAnalysis

    Consumers: Consideraquasilinearutilityfunction

    7 7, 7 = 7 + 7(7)

    where7 denotesthenumeraire,and73 7 > 0,733 7 < 0 forall7 > 0.

    Normalizing,7 0 = 0.Recallthatwithquasilinearutilityfunctions,thewealtheffectsforallnon-numeraire commoditiesarezero.

    AdvancedMicroeconomicTheory 5

  • PartialEquilibriumAnalysis

    ConsumersUMPismax

    @AD,EAD7 + 7(7)

    s. t. 7 + 7IJKLMNOPNQR.

    @A + 7%(% %(%)

    VWJXYKZ

    )[%\]

    IJKLMWNZJ^W_NZ(NQRJ`aNQKbPWJXYKZ)

    Thebudgetconstraintmustholdwithequality(byWalraslaw).Hence,

    7 = 7 + @A + 7% % %(%)

    [%\]

    AdvancedMicroeconomicTheory 6

  • PartialEquilibriumAnalysis

    Substitutingthebudgetconstraintintotheobjectivefunction,

    maxEAD

    7 7 7 +

    @A + 7% % %(%)

    [%\]

    FOCswrt 7 yields73 7 ,withequalityif7 > 0

    Thatis,consumerincreasestheamounthebuysofgood untilthepointinwhichthemarginalutilityheobtainsexactlycoincideswiththemarketpricehehastopayforit.

    AdvancedMicroeconomicTheory 7

  • PartialEquilibriumAnalysis

    Hence,anallocation(], c, , e, ], c, , [)andapricevector f constituteaCEif:

    %3(%),withequalityif% > 073 7 ,withequalityif7 > 0

    7e7\] = %[%\]

    Notethatthetheseconditionsdonotdependupontheconsumersinitialendowment.

    AdvancedMicroeconomicTheory 8

  • PartialEquilibriumAnalysis

    Theindividualdemandcurve,where73 7

    AdvancedMicroeconomicTheory 9

  • PartialEquilibriumAnalysis Horizontallysummingindividualdemandcurvesyieldstheaggregatedemandcurve.

    AdvancedMicroeconomicTheory 10

  • PartialEquilibriumAnalysis

    Theindividualsupplycurve,where %3(%)

    AdvancedMicroeconomicTheory 11

  • PartialEquilibriumAnalysis

    AdvancedMicroeconomicTheory 12

    Horizontallysummingindividualsupplycurvesyieldstheaggregatesupplycurve.

  • PartialEquilibriumAnalysis Superimposingaggregatedemandandaggregatesupplycurves,weobtaintheCEallocationofgood.

    ToguaranteethataCEexists,theequilibriumprice mustsatisfy

    max773 0

    min%%3 0

    AdvancedMicroeconomicTheory 13

  • PartialEquilibriumAnalysis

    Also,since73 7 isdownwardslopingin7,and%3(7) isupwardslopingin7,thenaggregatedemandandsupplycrossatauniquepoint. Hence,theCEallocationisunique.

    AdvancedMicroeconomicTheory 14

  • PartialEquilibriumAnalysis

    Ifwehavemax773 0 < min% %

    3 0 ,

    thenthereisnopositiveproductionorconsumptionofgood.

    AdvancedMicroeconomicTheory 15

  • PartialEquilibriumAnalysis

    Example6.1: Assumea perfectlycompetitiveindustryconsistingoftwotypesoffirms:100firmsoftypeAand 30firmsoftypeB.

    Short-runsupplycurveoftypeAfirmisk = 2

    Short-runsupplycurveoftypeBfirmism = 10

    TheWalrasianmarketdemandcurveis = 5000 500

    AdvancedMicroeconomicTheory 16

  • PartialEquilibriumAnalysis

    Example6.1 (continued): Summingtheindividualsupplycurvesofthe100type-A firmsandthe30type-B firms, = 100 q 2 + 30 q 10 = 500

    Theshort-runequilibriumoccursatthepriceatwhichquantitydemandedequalsquantitysupplied,

    5000 500 = 500,or = 5 Eachtype-A firmsupplies:k = 2 q 5 = 10 Eachtype-B firmsupplies:m = 10 q 5 = 50

    AdvancedMicroeconomicTheory 17

  • ComparativeStatics

    AdvancedMicroeconomicTheory 18

  • ComparativeStatics Letusassumethattheconsumerspreferencesareaffected

    byavectorofparameters t,where . Then,consumersutilityfromgood is7(7, ).

    Similarly,firmstechnologyisaffectedbyavectorofparameters x,where . Then,firmscostfunctionis%(%, ).

    Notation: 7(, ) istheeffectivepricepaidbytheconsumer %(, ) istheeffectivepricereceivedbythefirm Perunittax:7 , = + .

    Example: = $2,regardlessoftheprice Advaloremtax(salestax):7 , = + = 1 +

    Example: = 0.1 (10%).

    AdvancedMicroeconomicTheory 19

  • ComparativeStatics IfconsumptionandproductionarestrictlypositiveintheCE,then

    73 7, = 7 , foreveryconsumer%3(%, )=%(, ) foreveryfirm

    7e7\] = %[%\]

    Thenwehave + + 1 equations,whichdependonparametervalues, and.

    Inordertounderstandhow7 or% dependsonparameters and,wecanusetheImplicitFunctionTheorem. Theabovefunctionshavetobedifferentiable.

    AdvancedMicroeconomicTheory 20

  • ComparativeStatics

    ImplicitFunctionTheorem: Let(, ) beautilityfunction,where and areamountsoftwogoods.

    If(E,)E

    0 whenevaluatedat(, ),then(E,)E

    + (E,)

    = 0whichyields

    () =

    (, )

    (, )

    AdvancedMicroeconomicTheory 21

  • ComparativeStatics

    Similarly,if(E,)

    0 whenevaluatedat(, ),then

    () =

    (, )

    (, )

    forall(, ).

    AdvancedMicroeconomicTheory 22

  • ComparativeStatics

    Similarly,if(, ) describestheconsumptionofasinglegood,where determinestheconsumerspreferencefor,and(E,)

    0,

    then

    () =

    (, )

    (, )

    Theleft-handsideisunknown Theright-handsideis,however,easiertofind.

    AdvancedMicroeconomicTheory 23

  • ComparativeStatics

    Salestax (Example6.2): Theexpressionoftheaggregatedemandisnow( + ),becausetheeffectivepricethattheconsumerpaysisactually + .

    Inequilibrium,themarketpriceafterimposingthetax,(),musthencesatisfy

    + = (()) ifthesalestaxismarginallyincreased,andfunctionsaredifferentiableat = , + q (3 + 1) = (()) q 3

    AdvancedMicroeconomicTheory 24

  • ComparativeStatics

    Rearranging,weobtain3 q 3 + 3

    = 3 + Hence,

    3 = E b

    E b )

    Since() isdecreasinginprices,3 + < 0,and() isincreasinginprices,3 > 0,

    3 = E b

    E b

    ) D

    =

    =

    AdvancedMicroeconomicTheory 25

  • ComparativeStatics

    Hence,3 < 0.Moreover,3 (1,0]. Therefore, decreasesin.

    Thatis,thepricereceivedbyproducersfallsinthetax,butlessthanproportionally.

    Additionally,since + isthepricepaidbyconsumers,then3 + 1 isthemarginalincreaseinthepricepaidbyconsumerswhenthetaxmarginallyincreases. Since3 1,then3 + 1 0,andconsumers

    costoftheproductalsoraiseslessthanproportionally.

    AdvancedMicroeconomicTheory 26

  • ComparativeStatics

    Notax: CEoccursat 0and 0

    Tax: decreasesfrom 0 to

    Consumersnowpay +

    Producersnowreceive forthe unitstheysell.

    AdvancedMicroeconomicTheory 27

  • ComparativeStatics SalesTax(ExtremeCases):

    a) Thesupplyisveryresponsivetopricechanges,i.e.,3 islarge.

    3 = E b

    E b ) 0

    Therefore,3 0,andthepricereceivedbyproducersdoesnotfall.

    However,consumersstillhavetopay + . Amarginalincreaseintaxesthereforeprovidesan

    increaseintheconsumerspriceof3 + 1 = 0 + 1 = 1

    Thetaxissolelybornebyconsumers.

    AdvancedMicroeconomicTheory 28

  • ComparativeStatics

    Thepricereceivedbyproducersalmostdoesnotfall.

    But,thepricepaidbyconsumersincreasesbyexactlytheamountofthetax.

    AdvancedMicroeconomicTheory 29

    Averyelasticsupplycurve

  • ComparativeStaticsb) Thesupplyisnotresponsivetopricechanges,i.e.,

    3 isclosetozero.

    3 = E b

    E b ) = 1

    Therefore,3 = 1,andthepricereceivedbyproducersfallsby$1foreveryextradollarintaxes. Producersbearmostofthetaxburden

    Incontrast,consumerspay + Amarginalincreaseintaxesproducesanincrease

    intheconsumerspriceof3 + 1 = 1 + 1 = 0

    Consumersdonotbeartaxburdenatall.

    AdvancedMicroeconomicTheory 30

  • ComparativeStatics

    Inelasticsupplycurve

    AdvancedMicroeconomicTheory 31

  • ComparativeStatics

    Example6.3: Consideracompetitivemarketinwhichthegovernmentwillbeimposinganadvaloremtaxof.

    Aggregatedemandcurveis = ,where > 0 and < 0,andaggregatesupplycurveis = ,where > 0 and > 0.

    Letusevaluatehowtheequilibriumpriceisaffectedbyamarginalincreaseinthetax.

    AdvancedMicroeconomicTheory 32

  • ComparativeStatics

    Example6.3 (continued): Thechangeinthepricereceivedbyproducersat =0 is

    3 0 = 3

    3 3

    = ]

    ] ] =

    = ()

    () () =

    Thechangeinthepricepaidbyconsumersat = 0 is

    3 0 + 1 =

    + 1 =

    AdvancedMicroeconomicTheory 33

  • ComparativeStatics

    Example 6.3(continued):When = 0 (i.e.,supplyisperfectlyinelastic),thepricepaidbyconsumersinunchanged,andthepricereceivedbyproducersdecreasesbetheamountofthetax. Thatis,producersbearthefulleffectofthetax.

    When = 0 (i.e.,demandisperfectlyinelastic),thepricereceivedbyproducersisunchangedandthepricepaidbyconsumersincreasesbytheamount