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

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.

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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% % %(%)

[%\]

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PartialEquilibriumAnalysis

Substitutingthebudgetconstraintintotheobjectivefunction,

maxEAD

7 7 7 +

@A + 7% % %(%)

[%\]

FOCswrt 7 yields73 7 ,withequalityif7 > 0

Thatis,consumerincreasestheamounthebuysofgood untilthepointinwhichthemarginalutilityheobtainsexactlycoincideswiththemarketpricehehastopayforit.

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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.

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PartialEquilibriumAnalysis

Theindividualsupplycurve,where %3(%)

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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.

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PartialEquilibriumAnalysis

Ifwehavemax773 0 < min% %

3 0 ,

thenthereisnopositiveproductionorconsumptionofgood.

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PartialEquilibriumAnalysis

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

Short-runsupplycurveoftypeAfirmisk = 2

Short-runsupplycurveoftypeBfirmism = 10

TheWalrasianmarketdemandcurveis = 5000 500

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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.

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ComparativeStatics

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

If(E,)E

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

+ (E,)

= 0whichyields

() =

(, )

(, )

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ComparativeStatics

Similarly,if(E,)

0 whenevaluatedat(, ),then

() =

(, )

(, )

forall(, ).

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ComparativeStatics

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

0,

then

() =

(, )

(, )

Theleft-handsideisunknown Theright-handsideis,however,easiertofind.

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ComparativeStatics

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

Inequilibrium,themarketpriceafterimposingthetax,(),musthencesatisfy

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

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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

=

=

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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.

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ComparativeStatics

Notax: CEoccursat 0and 0

Tax: decreasesfrom 0 to

Consumersnowpay +

Producersnowreceive forthe unitstheysell.

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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.

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ComparativeStatics

Thepricereceivedbyproducersalmostdoesnotfall.

But,thepricepaidbyconsumersincreasesbyexactlytheamountofthetax.

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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.

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ComparativeStatics

Inelasticsupplycurve

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ComparativeStatics

Example6.3: Consideracompetitivemarketinwhichthegovernmentwillbeimposinganadvaloremtaxof.

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

Letusevaluatehowtheequilibriumpriceisaffectedbyamarginalincreaseinthetax.

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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