Advanced Microeconomic Theory · Outline • Partial Equilibrium Analysis • General Equilibrium...
Transcript 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.
• Acompetitiveequilibriumallocationwillemergeatapricethatmakesconsumers’purchasingplanstocoincidewiththefirms’productiondecision.
AdvancedMicroeconomicTheory 3
PartialEquilibriumAnalysis
• Firm:– Giventhepricevector𝑝∗,firm𝑗’sequilibriumoutputlevel𝑞%∗ mustsolve
max)*+,
𝑝∗𝑞% − 𝑐%(𝑞%)
whichyieldsthenecessaryandsufficientcondition
𝑝∗ ≤ 𝑐%3(𝑞%∗),withequalityif𝑞%∗ > 0– Thatis,everyfirm𝑗 producesuntilthepointinwhichitsmarginalcost,𝑐%3(𝑞%∗),coincideswiththecurrentmarketprice.
AdvancedMicroeconomicTheory 4
PartialEquilibriumAnalysis
• Consumers:– Consideraquasilinearutilityfunction
𝑢7 𝑚7, 𝑥7 = 𝑚7 + 𝜙7(𝑥7)
where𝑚7 denotesthenumeraire,and𝜙73 𝑥7 > 0,𝜙733 𝑥7 < 0 forall𝑥7 > 0.
– Normalizing,𝜙7 0 = 0.Recallthatwithquasilinearutilityfunctions,thewealtheffectsforallnon-numeraire commoditiesarezero.
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PartialEquilibriumAnalysis
– Consumer𝑖’sUMPismax
@A∈ℝD,EA∈ℝD𝑚7 + 𝜙7(𝑥7)
s. t. 𝑚7 + 𝑝∗𝑥7IJKLMNOPNQR.
≤ 𝑤@A + ∑ 𝜃7%(𝑝∗𝑞%∗ − 𝑐%(𝑞%∗)VWJXYKZ
)[%\]
IJKLMWNZJ^W_NZ(NQRJ`aNQKbPWJXYKZ)
– Thebudgetconstraintmustholdwithequality(byWalras’law).Hence,
𝑚7 = −𝑝∗𝑥7 + 𝑤@A + ∑ 𝜃7% 𝑝∗𝑞%∗ − 𝑐%(𝑞%∗)[%\]
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PartialEquilibriumAnalysis
– Substitutingthebudgetconstraintintotheobjectivefunction,
maxEA∈ℝD
𝜙7 𝑥7 − 𝑝∗𝑥7 +
𝑤@A + ∑ 𝜃7% 𝑝∗𝑞%∗ − 𝑐%(𝑞%∗)[%\]
– FOCswrt 𝑥7 yields𝜙73 𝑥7∗ ≤ 𝑝∗,withequalityif𝑥7∗ > 0
– Thatis,consumerincreasestheamounthebuysofgood𝑥 untilthepointinwhichthemarginalutilityheobtainsexactlycoincideswiththemarketpricehehastopayforit.
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PartialEquilibriumAnalysis
– Hence,anallocation(𝑥]∗, 𝑥c∗, … , 𝑥e∗, 𝑞]∗, 𝑞c∗, … , 𝑞[∗)andapricevector𝑝∗ ∈ ℝf constituteaCEif:
𝑝∗ ≤ 𝑐%3(𝑞%∗),withequalityif𝑞%∗ > 0𝜙73 𝑥7∗ ≤ 𝑝∗,withequalityif𝑥7∗ > 0
∑ 𝑥7∗e7\] = ∑ 𝑞%∗
[%\]
– Notethatthetheseconditionsdonotdependupontheconsumer’sinitialendowment.
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PartialEquilibriumAnalysis
• Theindividualdemandcurve,where𝜙73 𝑥7∗ ≤ 𝑝∗
AdvancedMicroeconomicTheory 9
PartialEquilibriumAnalysis• Horizontallysummingindividualdemandcurvesyieldstheaggregatedemandcurve.
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PartialEquilibriumAnalysis
• Theindividualsupplycurve,where𝑝∗ ≤ 𝑐%3(𝑞%∗)
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PartialEquilibriumAnalysis
AdvancedMicroeconomicTheory 12
• Horizontallysummingindividualsupplycurvesyieldstheaggregatesupplycurve.
PartialEquilibriumAnalysis• Superimposingaggregatedemandandaggregatesupplycurves,weobtaintheCEallocationofgood𝑥.
• ToguaranteethataCEexists,theequilibriumprice𝑝∗ mustsatisfy
max7𝜙73 0 ≥ 𝑝∗
≥ min%𝑐%3 0
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PartialEquilibriumAnalysis
• Also,since𝜙73 𝑥7 isdownwardslopingin𝑥7,and𝑐%3(𝑞7) isupwardslopingin𝑞7,thenaggregatedemandandsupplycrossatauniquepoint.– Hence,theCEallocationisunique.
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PartialEquilibriumAnalysis
• Ifwehavemax7𝜙73 0 < min
%𝑐%3 0 ,
thenthereisnopositiveproductionorconsumptionofgood𝑥.
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PartialEquilibriumAnalysis
• Example6.1:– Assumea perfectlycompetitiveindustryconsistingoftwotypesoffirms:100firmsoftypeAand 30firmsoftypeB.
– Short-runsupplycurveoftypeAfirmis𝑠k 𝑝 = 2𝑝
– Short-runsupplycurveoftypeBfirmis𝑠m 𝑝 = 10𝑝
– TheWalrasianmarketdemandcurveis𝑥 𝑝 = 5000 − 500𝑝
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PartialEquilibriumAnalysis
• Example6.1 (continued):– Summingtheindividualsupplycurvesofthe100type-A firmsandthe30type-B firms,𝑆 𝑝 = 100 q 2𝑝 + 30 q 10𝑝 = 500𝑝
– Theshort-runequilibriumoccursatthepriceatwhichquantitydemandedequalsquantitysupplied,
5000 − 500𝑝 = 500𝑝,or𝑝 = 5– Eachtype-A firmsupplies:𝑠k 𝑝 = 2 q 5 = 10– Eachtype-B firmsupplies:𝑠m 𝑝 = 10 q 5 = 50
AdvancedMicroeconomicTheory 17
ComparativeStatics
AdvancedMicroeconomicTheory 18
ComparativeStatics• Letusassumethattheconsumer’spreferencesareaffected
byavectorofparameters 𝛼 ∈ ℝt,where𝑀 ≤ 𝐿.– Then,consumer𝑖’sutilityfromgood𝑥 is𝜙7(𝑥7, 𝛼).
• Similarly,firms’technologyisaffectedbyavectorofparameters𝛽 ∈ ℝx,where 𝑆 ≤ 𝐿.– Then,firm𝑗’scostfunctionis𝑐%(𝑞%, 𝛽).
• Notation:– �̂�7(𝑝, 𝑡) istheeffectivepricepaidbytheconsumer– �̂�%(𝑝, 𝑡) istheeffectivepricereceivedbythefirm– Perunittax:�̂�7 𝑝, 𝑡 = 𝑝 + 𝑡.
• Example:𝑡 = $2,regardlessoftheprice𝑝– Advaloremtax(salestax):�̂�7 𝑝, 𝑡 = 𝑝 + 𝑝𝑡 = 𝑝 1 + 𝑡
• Example:𝑡 = 0.1 (10%).
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ComparativeStatics• IfconsumptionandproductionarestrictlypositiveintheCE,then
𝜙73 𝑥7∗, 𝛼 = �̂�7 𝑝∗, 𝑡 foreveryconsumer𝑖𝑐%3(𝑞%∗, 𝛽)=�̂�%(𝑝∗, 𝑡) foreveryfirm𝑗
∑ 𝑥7∗e7\] = ∑ 𝑞%∗
[%\]
• Thenwehave𝐼 + 𝐽 + 1 equations,whichdependonparametervalues𝛼,𝛽 and𝑡.
• Inordertounderstandhow𝑥7∗ or𝑞%∗ dependsonparameters𝛼 and𝛽,wecanusetheImplicitFunctionTheorem.– Theabovefunctionshavetobedifferentiable.
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ComparativeStatics
• ImplicitFunctionTheorem:– Let𝑢(𝑥, 𝑦) beautilityfunction,where𝑥 and𝑦 areamountsoftwogoods.
– If��(E̅,��)�E
≠ 0 whenevaluatedat(�̅�, 𝑦�),then��(E̅,��)�E
𝑑𝑥 + ��(E̅,��)��
𝑑𝑦 = 0whichyields
𝑑𝑦(�̅�)𝑑𝑥 = −
𝜕𝑢(�̅�, 𝑦�)𝜕𝑥
𝜕𝑢(�̅�, 𝑦�)𝜕𝑦
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ComparativeStatics
– Similarly,if��(E̅,��)��
≠ 0 whenevaluatedat(�̅�, 𝑦�),then
𝑑𝑥(𝑦�)𝑑𝑦 = −
𝜕𝑢(�̅�, 𝑦�)𝜕𝑦
𝜕𝑢(�̅�, 𝑦�)𝜕𝑥
forall(�̅�, 𝑦�).
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ComparativeStatics
– Similarly,if𝑢(𝑥, 𝛼) describestheconsumptionofasinglegood𝑥,where𝛼 determinestheconsumer’spreferencefor𝑥,and��(E,�)
��≠ 0,
then
𝑑𝑥(𝛼)𝑑𝛼 = −
𝜕𝑢(𝑥, 𝛼)𝜕𝛼
𝜕𝑢(𝑥, 𝛼)𝜕𝑥
– Theleft-handsideisunknown– Theright-handsideis,however,easiertofind.
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ComparativeStatics
• Salestax (Example6.2):– Theexpressionoftheaggregatedemandisnow𝑥(𝑝 + 𝑡),becausetheeffectivepricethattheconsumerpaysisactually𝑝 + 𝑡.
– Inequilibrium,themarketpriceafterimposingthetax,𝑝∗(𝑡),musthencesatisfy
𝑥 𝑝∗ 𝑡 + 𝑡 = 𝑞(𝑝∗(𝑡))– ifthesalestaxismarginallyincreased,andfunctionsaredifferentiableat 𝑝 = 𝑝∗ 𝑡 ,𝑥′ 𝑝∗ 𝑡 + 𝑡 q (𝑝∗3 𝑡 + 1) = 𝑞′(𝑝∗(𝑡)) q 𝑝∗3 𝑡
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ComparativeStatics
– Rearranging,weobtain𝑝∗3 𝑡 q 𝑥3 𝑝∗ 𝑡 + 𝑡 − 𝑞3 𝑝∗ 𝑡
= −𝑥3 𝑝∗ 𝑡 + 𝑡– Hence,
𝑝∗3 𝑡 = − E� �∗ � b�E� �∗ � b� �)� �∗ �
– Since𝑥(𝑝) isdecreasinginprices,𝑥3 𝑝∗ 𝑡 + 𝑡 < 0,and𝑞(𝑝) isincreasinginprices,𝑞3 𝑝∗ 𝑡 > 0,
𝑝∗3 𝑡 = − E� �∗ � b�E� �∗ � b�
��)� �∗ �
D
= −��= −
AdvancedMicroeconomicTheory 25
ComparativeStatics
– Hence,𝑝∗3 𝑡 < 0.–Moreover,𝑝∗3 𝑡 ∈ (−1,0].– Therefore,𝑝∗ 𝑡 decreasesin𝑡.
§ Thatis,thepricereceivedbyproducersfallsinthetax,butlessthanproportionally.
– Additionally,since𝑝∗ 𝑡 + 𝑡 isthepricepaidbyconsumers,then𝑝∗3 𝑡 + 1 isthemarginalincreaseinthepricepaidbyconsumerswhenthetaxmarginallyincreases.§ Since𝑝∗3 𝑡 ≥ 1,then𝑝∗3 𝑡 + 1 ≥ 0,andconsumers’
costoftheproductalsoraiseslessthanproportionally.
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ComparativeStatics
• Notax:– CEoccursat𝑝∗ 0and𝑥∗ 0
• Tax:– 𝑥∗ decreasesfrom𝑥∗ 0 to𝑥∗ 𝑡
– Consumersnowpay𝑝∗ 𝑡 + 𝑡
– Producersnowreceive𝑝∗ 𝑡 forthe𝑥∗ 𝑡 unitstheysell.
AdvancedMicroeconomicTheory 27
ComparativeStatics• SalesTax(ExtremeCases):
a) Thesupplyisveryresponsivetopricechanges,i.e.,𝑞3 𝑝∗ 𝑡 islarge.
𝑝∗3 𝑡 = − E� �∗ � b�E� �∗ � b� �)� �∗ �
→ 0
– Therefore,𝑝∗3 𝑡 → 0,andthepricereceivedbyproducersdoesnotfall.
– However,consumersstillhavetopay𝑝∗ 𝑡 + 𝑡.§ Amarginalincreaseintaxesthereforeprovidesan
increaseintheconsumer’spriceof𝑝∗3 𝑡 + 1 = 0 + 1 = 1
§ Thetaxissolelybornebyconsumers.
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ComparativeStatics
– Thepricereceivedbyproducersalmostdoesnotfall.
– But,thepricepaidbyconsumersincreasesbyexactlytheamountofthetax.
AdvancedMicroeconomicTheory 29
• Averyelasticsupplycurve
ComparativeStaticsb) Thesupplyisnotresponsivetopricechanges,i.e.,
𝑞3 𝑝∗ 𝑡 isclosetozero.
𝑝∗3 𝑡 = − E� �∗ � b�E� �∗ � b� �)� �∗ �
= −1
– Therefore,𝑝∗3 𝑡 = −1,andthepricereceivedbyproducersfallsby$1foreveryextradollarintaxes.§ Producersbearmostofthetaxburden
– Incontrast,consumerspay𝑝∗ 𝑡 + 𝑡§ Amarginalincreaseintaxesproducesanincrease
intheconsumer’spriceof𝑝∗3 𝑡 + 1 = −1 + 1 = 0
§ Consumersdonotbeartaxburdenatall.
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ComparativeStatics
• Inelasticsupplycurve
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ComparativeStatics
• Example6.3:– Consideracompetitivemarketinwhichthegovernmentwillbeimposinganadvaloremtaxof𝑡.
– Aggregatedemandcurveis𝑥 𝑝 = 𝐴𝑝�,where𝐴 > 0 and𝜀 < 0,andaggregatesupplycurveis𝑞 𝑝 = 𝑎𝑝�,where𝑎 > 0 and𝛾 > 0.
– Letusevaluatehowtheequilibriumpriceisaffectedbyamarginalincreaseinthetax.
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ComparativeStatics
• Example6.3 (continued):– Thechangeinthepricereceivedbyproducersat𝑡 =0 is
𝑝∗3 0 = −𝑥3 𝑝∗
𝑥3 𝑝∗ − 𝑞3 𝑝∗
= −𝐴𝜀𝑝∗��]
𝐴𝜀𝑝∗��] − 𝑎𝛾𝑝∗��] = −𝐴𝜀𝑝∗�
𝐴𝜀𝑝∗� − 𝑎𝛾𝑝∗�
= −𝜀𝑥(𝑝∗)
𝜀𝑥(𝑝∗) − 𝛾𝑞(𝑝∗) = −𝜀
𝜀 − 𝛾– Thechangeinthepricepaidbyconsumersat𝑡 = 0 is
𝑝∗3 0 + 1 = −𝜀
𝜀 − 𝛾 + 1 = −𝛾
𝜀 − 𝛾AdvancedMicroeconomicTheory 33
ComparativeStatics
• Example 6.3(continued):–When𝛾 = 0 (i.e.,supplyisperfectlyinelastic),thepricepaidbyconsumersinunchanged,andthepricereceivedbyproducersdecreasesbetheamountofthetax.§ Thatis,producersbearthefulleffectofthetax.
–When𝜀 = 0 (i.e.,demandisperfectlyinelastic),thepricereceivedbyproducersisunchangedandthepricepaidbyconsumersincreasesbytheamountofthetax.§ Thatis,consumersbearthefullburdenofthetax.
AdvancedMicroeconomicTheory 34
ComparativeStatics
• Example 6.3(continued):–When𝜀 → −∞ (i.e.,demandisperfectlyelastic),thepricepaidbyconsumersisunchanged,andthepricereceivedbyproducersdecreasesbytheamountofthetax.
–When𝛾 → +∞(i.e.,supplyisperfectlyelastic),thepricereceivedbyproducersisunchangedandthepricepaidbyconsumersincreasesbytheamountofthetax.
AdvancedMicroeconomicTheory 35
WelfareAnalysis
AdvancedMicroeconomicTheory 36
WelfareAnalysis• Letusnowmeasurethechangesintheaggregatesocialwelfareduetoachangeinthecompetitiveequilibriumallocation.
• Considertheaggregatesurplus𝑆 = ∑ 𝜙7(𝑥7)e
7\] − ∑ 𝑐%(𝑞%)[%\]
• Takeadifferentialchangeinthequantityofgood𝑘 thatindividualsconsumeandthatfirmsproducesuchthat∑ 𝑑𝑥7e
7\] = ∑ 𝑑𝑞%[%\] .
• Thechangeintheaggregatesurplusis𝑑𝑆 = ∑ 𝜙73(𝑥7)𝑑𝑥7e
7\] − ∑ 𝑐%3(𝑞%)𝑑𝑞%[%\]
AdvancedMicroeconomicTheory 37
WelfareAnalysis
• Since– 𝜙73 𝑥7 = 𝑃(𝑥) forallconsumers;and• Thatis,everyindividualconsumesuntilMB=p.
– 𝑐%3(𝑞%) = 𝐶3(𝑞) forallfirms• Thatis,everyfirm’sMCcoincideswithaggregateMC)
thenthechangeinsurpluscanberewrittenas𝑑𝑆 = ∑ 𝑃(𝑥)𝑑𝑥7e
7\] − ∑ 𝐶3 𝑞 𝑑𝑞%[%\]
= 𝑃(𝑥)∑ 𝑑𝑥7e7\] − 𝐶3(𝑞)∑ 𝑑𝑞%
[%\]
AdvancedMicroeconomicTheory 38
WelfareAnalysis
• Butsince∑ 𝑑𝑥7e7\] = ∑ 𝑑𝑞%
[%\] = 𝑑𝑥,and𝑥 = 𝑞
bymarketfeasibility,then𝑑𝑆 = 𝑃 𝑥 − 𝐶3 𝑞 𝑑𝑥
• Intuition:– Thechangeinsurplusofamarginalincreaseinconsumption(andproduction)reflectsthedifferencebetweentheconsumers’additionalutilityandfirms’additionalcostofproduction.
AdvancedMicroeconomicTheory 39
WelfareAnalysis• Differentialchangeinsurplus
AdvancedMicroeconomicTheory 40
p
x,q
p(x)
c'(x)
dx
x(.), p(.)
c’(.), q(.)
x0 x1
WelfareAnalysis
• Wecanalsointegratetheaboveexpression,eliminatingthedifferentials,inordertoobtainthetotalsurplusforanaggregateconsumptionlevelof𝑥:
𝑆 𝑥 = 𝑆, + ∫ 𝑃 𝑠 − 𝐶3 𝑠 𝑑𝑠E,
where𝑆, = 𝑆(0) istheconstantofintegration,andrepresentstheaggregatesurpluswhenaggregateconsumptioniszero,𝑥 = 0.– 𝑆, = 0 iftheinterceptofthemarginalcostfunctionsatisfies𝑐%3 0 = 0 forall𝐽 firms.
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WelfareAnalysis• Surplusataggregateconsumption𝑥
AdvancedMicroeconomicTheory 42
WelfareAnalysis
• Forwhichconsumptionlevelisaggregatesurplus𝑆 𝑥 maximized?– Differentiating𝑆 𝑥 withrespectto𝑥,
𝑆3 𝑥 = 𝑃 𝑥∗ − 𝐶3 𝑥∗ ≤ 0or,𝑃 𝑥∗ ≤ 𝐶3 𝑥∗
– Thesecondorder(sufficient)conditionis𝑆33 𝑥 = 𝑃3 𝑥∗
�− 𝐶33 𝑥∗
b< 0
• Hence,𝑆 𝑥∗ isconcave.• Then,when𝑥∗ > 0,aggregatesurplus𝑆 𝑥 ismaximizedat𝑃 𝑥∗ = 𝐶3 𝑥∗ .
AdvancedMicroeconomicTheory 43
WelfareAnalysis
• Therefore,theCEallocationmaximizesaggregatesurplus.
• ThisistheFirstWelfareTheorem:– EveryCEisParetooptimal(PO).
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WelfareAnalysis
• Example6.4:– Consideranaggregatedemand𝑥 𝑝 = 𝑎 − 𝑏𝑞andaggregatesupply𝑦 𝑝 = 𝐽 q �
c,where𝐽 isthe
numberoffirmsintheindustry.– TheCEpricesolves
𝑎 − 𝑏𝑝 = 𝐽 q �c
or𝑝 = c�c�b[
– Intuitively,asdemandincreases(numberoffirms)increases(decreases)theequilibriumpriceincreases(decreases,respectively).
AdvancedMicroeconomicTheory 45
WelfareAnalysis• Example 6.4(continued):
– Therefore,equilibriumoutputis
𝑥∗ = 𝑎 − 𝑏2𝑎
2𝑏 + 𝐽 =𝑎𝐽
2𝑏 + 𝐽– Surplusis
𝑆 𝑥∗ = � 𝑝 𝑥 − 𝐶3 𝑥 𝑑𝑥E∗
,
where𝑝 𝑥 = ��E�
and𝐶3 𝑥 = cE[.
– Thus,
𝑆 𝑥∗ = �𝑎 − 𝑥𝑏 −
2𝑥𝐽 𝑑𝑥
E∗
,=
𝑎c𝐽4𝑏c + 2𝑏𝐽
which is increasing in the number of firms 𝐽.
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GeneralEquilibrium
AdvancedMicroeconomicTheory 47
GeneralEquilibrium
• Sofar,weexploredequilibriumconditionsinasinglemarketwithasingletypeofconsumer.
• Nowweexaminesettingswithmarketsfordifferentgoodsandmultipleconsumers.
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GeneralEquilibrium:NoProduction
• Consideraneconomywithtwogoodsandtwoconsumers,𝑖 = {1,2}.
• Eachconsumerisinitiallyendowedwith𝐞7 ≡(𝑒]7 , 𝑒c7 ) unitsofgood1and2.
• Anyotherallocationsaredenotedby𝐱7 ≡(𝑥]7 , 𝑥c7 ).
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GeneralEquilibrium:NoProduction
• Edgeworthbox:
AdvancedMicroeconomicTheory 50
GeneralEquilibrium:NoProduction
• Theshadedarearepresentsthesetofbundles(𝑥]7 , 𝑥c7 ) forconsumer𝑖 satisfying𝑢](𝑥]], 𝑥c]) ≥ 𝑢](𝑒]], 𝑒c])𝑢c(𝑥]c, 𝑥cc) ≥ 𝑢c(𝑒]c, 𝑒cc)
• Bundle𝐴 cannotbeabarterequilibrium:– Consumer1doesnotexchange𝐞 for𝐴.
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• 𝐼𝐶7 istheindifferencecurveofconsumer𝑖,whichpassesthroughhisendowmentpoint𝐞7.
GeneralEquilibrium:NoProduction
• Notallpointsinthelens-shapedareaisabarterequilibrium!
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• Bundle𝐵 liesinsidethelens-shapedarea– Thus,ityieldsahigherutility
levelthantheinitialendowment𝐞 forbothconsumers.
• Bundle𝐷,however,makesbothconsumersbetteroffthanbundle𝐵.– Itlieson“contractcurve,”in
whichtheindifferencecurvesaretangenttooneanother.
– Itisanequilibrium,sinceParetoimprovementsarenolongerpossible
GeneralEquilibrium:NoProduction
• Feasibleallocation:– Anallocation𝐱 ≡ (𝐱], 𝐱c, … , 𝐱e) isfeasible ifitsatisfies
∑ 𝐱7e7\] ≤ ∑ 𝐞7e
7\]
– Thatis,theaggregateamountofgoodsinallocation𝐱 doesnotexceedtheaggregateinitialendowment𝐞 ≡ ∑ 𝐞7e
7\] .
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GeneralEquilibrium:NoProduction
• Pareto-efficientallocations:– Afeasibleallocation𝐱 isParetoefficientifthereisnootherfeasibleallocation𝐲 whichisweaklypreferredbyallconsumers,i.e.,𝐲7 ≿ 𝐱7 forall𝑖 ∈𝐼,andatleaststrictlypreferredbyoneconsumer,𝐲7 ≻ 𝐱7.
– Thatis,allocation𝐱 isParetoefficientifthereisnootherfeasibleallocation𝐲makingallindividualsatleastaswelloffasunder𝐱 andmakingoneindividualstrictlybetteroff.
AdvancedMicroeconomicTheory 54
GeneralEquilibrium:NoProduction
• Pareto-efficientallocations:– ThesetofParetoefficientallocations(𝐱], … , 𝐱e)solves
max𝐱¬,…,𝐱+,
𝑢](𝐱])
s. t. 𝑢%(𝐱%) ≥ 𝑢�% for𝑗 ≠ 1,and∑ 𝐱7e7\] ≤ ∑ 𝐞7e
7\] (feasibility)where𝐱7 = (𝑥]7 , 𝑥c7 ).
– Thatis,allocations(𝐱], … , 𝐱e) areParetoefficientiftheymaximizesindividual1’sutilitywithoutreducingtheutilityofallotherindividualsbelowagivenlevel𝑢�%,andsatisfyingfeasibility.
AdvancedMicroeconomicTheory 55
GeneralEquilibrium:NoProduction
– TheLagrangianis𝐿 𝐱], … , 𝐱e; 𝜆c, … , 𝜆e, 𝜇 =
𝑢] 𝐱] + 𝜆c 𝑢c 𝐱c − 𝑢�c +⋯+𝜆e 𝑢e 𝐱e − 𝑢�e + 𝜇 ∑ 𝐞7e
7\] − ∑ 𝐱7e7\]
– FOCwrt 𝐱] = (𝑥]], 𝑥c]) yields�f�E²
¬ =��¬(𝐱¬)�E²
¬ − 𝜇 ≤ 0foreverygood 𝑘 ofconsumer1.
– Foranyindividual𝑗 ≠ 1,theFOCsbecome�f
�E²* =
��*(𝐱*)�E²
¬ − 𝜇 ≤ 0AdvancedMicroeconomicTheory 56
GeneralEquilibrium:NoProduction
– FOCswrt 𝜆% and𝜇 yield𝑢%(𝐱%) ≥ 𝑢�% and∑ 𝐱7e7\] ≤
∑ 𝐞7e7\] ,respectively.
– Inthecaseofinteriorsolutions,acompactconditionforParetoefficiencyis
³´¬(𝐱¬)³µ²¬
³´¬(𝐱¬)³µ¶¬
=³´*(𝐱*)³µ²¬
³´*(𝐱*)
³µ¶*
or𝑀𝑅𝑆],c] = 𝑀𝑅𝑆],c%
foreveryconsumer𝑗 ≠ 1.– Graphically,consumers’indifferencecurvesbecometangenttooneanotherattheParetoefficientallocations.
AdvancedMicroeconomicTheory 57
GeneralEquilibrium:NoProduction
• Example6.5 (Paretoefficiency):– Considerabartereconomywithtwogoods,1and2,andtwoconsumers,𝐴 and𝐵,eachwiththeinitialendowmentsof𝐞k = (100,350) and𝐞m =(100,50),respectively.
– Bothconsumers’utilityfunctionisaCobb-Douglastypegivenby𝑢7 𝑥]7 , 𝑥c7 = 𝑥]7𝑥c7 forallindividual𝑖 = {𝐴, 𝐵}.
– LetusfindthesetofParetoefficientallocations.
AdvancedMicroeconomicTheory 58
GeneralEquilibrium:NoProduction
• Example (continued):– Paretoefficientallocationsarereachedatpointswherethe𝑀𝑅𝑆k = 𝑀𝑅𝑆m.Hence,
𝑀𝑅𝑆k = 𝑀𝑅𝑆m ⟹ E¶¹
E¬¹= E¶º
E¬ºor𝑥ck𝑥]m = 𝑥cm𝑥]k
– Usingthefeasibilityconstraintsforgood1and2,i.e.,𝑒]k + 𝑒]m = 𝑥]k + 𝑥]m𝑒ck + 𝑒cm = 𝑥ck + 𝑥cm
weobtain𝑥]m = 𝑒]k + 𝑒]m − 𝑥]k𝑥cm = 𝑒ck + 𝑒cm − 𝑥ck
AdvancedMicroeconomicTheory 59
GeneralEquilibrium:NoProduction
• Example (continued):– Combiningthetangencyconditionandfeasibilityconstraintsyields
𝑥ck(𝑒]k + 𝑒]m − 𝑥]k)E¬º
= (𝑒ck + 𝑒cm − 𝑥ck)E¶º
𝑥]k
whichcanbere-writtenas
𝑥ck =»¶¹b»¶º
»¬¹b»¬º𝑥]k =
¼½,b½,],,b],,
𝑥]k = 2𝑥]k
forall𝑥]k ∈ [0,200].
AdvancedMicroeconomicTheory 60
GeneralEquilibrium:NoProduction
• Example (continued):– ThelinerepresentingthesetofParetoefficientallocations
AdvancedMicroeconomicTheory 61
GeneralEquilibrium:NoProduction
• Blockingcoalitions:Let𝑆 ⊂ 𝐼 denoteacoalitionofconsumers.Wesaythat𝑆 blocks thefeasibleallocation𝐱 ifthereisanallocation𝐲 suchthat:1) Allocationisfeasiblefor𝑆. Theaggregateamountof
goodsthatindividualsin𝑆 enjoyinallocation𝐲coincideswiththeiraggregateinitialendowment,i.e.,∑ 𝐲7�
7∈x = ∑ 𝐞7�7∈x ;and
2) Preferable.Allocation𝐲makesallindividualsinthecoalitionweaklybetteroffthanunder𝐱,i.e.,𝐲7 ≿ 𝐱7where𝑖 ∈ 𝑆,butmakesatleastoneindividualstrictlybetteroff,i.e.,𝐲7 ≻ 𝐱7.
AdvancedMicroeconomicTheory 62
GeneralEquilibrium:NoProduction
• Equilibriuminabartereconomy:Afeasibleallocation𝐱 isanequilibriumintheexchangeeconomywithinitialendowment𝐞 if𝐱 isnotblocked byanycoalitionofconsumers.
• Core: Thecoreofanexchangeeconomywithendowment𝐞,denoted𝐶(𝐞),isthesetofallunblocked feasibleallocations𝐱.– Suchallocations:
a) mutuallybeneficialforallindividuals(i.e.,theylieinthelens-shapedarea)
b) donotallowforfurtherParetoimprovements (i.e.,theylieinthecontractcurve)
AdvancedMicroeconomicTheory 63
GeneralEquilibrium:NoProduction
AdvancedMicroeconomicTheory 64
GeneralEquilibrium:CompetitiveMarkets
• Bartereconomydidnotrequirepricesforanequilibriumtoarise.
• Nowweexploretheequilibriumineconomieswhereweallowpricestoemerge.
• Orderofanalysis:– consumers’preferences– theexcessdemandfunction– theequilibriumallocationsincompetitivemarkets(i.e.,Walrasianequilibriumallocations)
AdvancedMicroeconomicTheory 65
GeneralEquilibrium:CompetitiveMarkets
• Consumers:– Considerconsumers’utilityfunctiontobecontinuous,strictlyincreasing,andstrictlyquasiconcaveinℝbÁ .
– HencetheUMPofeveryconsumer𝑖,whenfacingabudgetconstraint
𝐩 q 𝐱7 ≤ 𝐩 q 𝐞7 forallpricevector𝐩 ≫ 𝟎yieldsauniquesolution,whichistheWalrasiandemand𝐱(𝐩, 𝐩 q 𝐞7).
– 𝐱(𝐩, 𝐩 q 𝐞7) iscontinuousinthepricevector 𝐩.
AdvancedMicroeconomicTheory 66
GeneralEquilibrium:CompetitiveMarkets
– Intuitively,individual𝑖’sincomecomesfromsellinghisendowment𝐞7 atmarketprices𝐩,producing 𝐩 q 𝐞7 = 𝑝]𝑒]7 + ⋯+ 𝑝Å𝑒Å7 dollarstobeusedinthepurchaseofallocation𝐱7.
AdvancedMicroeconomicTheory 67
GeneralEquilibrium:CompetitiveMarkets
• Excessdemand:– SummingtheWalrasiandemand𝐱(𝐩, 𝐩 q 𝐞7) forgood𝑘 ofeveryindividualintheeconomy,weobtaintheaggregatedemandforgood𝑘.
– Thedifferencebetweentheaggregatedemandandtheaggregateendowmentofgood 𝑘 yieldstheexcessdemandofgood𝑘:
𝑧Å 𝐩 = ∑ 𝑥Å7 (𝐩, 𝐩 q 𝐞7)e7\] − ∑ 𝑒Å7e
7\]where𝑧Å 𝐩 ∈ ℝ.
AdvancedMicroeconomicTheory 68
GeneralEquilibrium:CompetitiveMarkets
–When𝑧Å 𝐩 > 0,theaggregatedemandforgood𝑘 exceedsitsaggregateendowment.§ Excessdemandofgood𝑘
–When𝑧Å 𝐩 < 0,theaggregatedemandforgood𝑘 fallsshortofitsaggregateendowment.§ Excesssupplyofgood 𝑘
AdvancedMicroeconomicTheory 69
GeneralEquilibrium:CompetitiveMarkets
• Differenceindemandandsupply,andexcessdemand
AdvancedMicroeconomicTheory 70
Σi=1
Ieki
Σi=1
Ixk(p,pei)i
zk(p)
pk
xk
pk
xk0
pk*
Equilibriumprice,wherezk(p)=0
GeneralEquilibrium:CompetitiveMarkets
• Theexcessdemandfunction𝐳 𝐩 ≡𝑧Å 𝐩 , 𝑧Å 𝐩 , … , 𝑧Å 𝐩 satisfiesfollowingproperties:
1) Walras’law:𝐩 q 𝐳 𝐩 = 0.– Sinceeveryconsumer𝑖 ∈ 𝐼 exhaustsallhisincome,∑ 𝑝Å q 𝑥Å7 (𝐩, 𝐩 q 𝐞7)ÁÅ\] = ∑ 𝑝Å𝑒Å7Á
Å\] ⇔∑ 𝑝Å 𝑥Å7 𝐩, 𝐩 q 𝐞7 − 𝑒Å7ÁÅ\] = 0
AdvancedMicroeconomicTheory 71
GeneralEquilibrium:CompetitiveMarkets
– Summingoverallindividuals,∑ ∑ 𝑝Å 𝑥Å7 𝐩, 𝐩 q 𝐞7 − 𝑒Å7Á
Å\]e7\] = 0
–Wecanre-writetheaboveexpressionas∑ ∑ 𝑝Å 𝑥Å7 𝐩, 𝐩 q 𝐞7 − 𝑒Å7e
7\]ÁÅ\] = 0
whichisequivalentto∑ 𝑝Å ∑ 𝑥Å7 𝐩, 𝐩 q 𝐞7e
7\] − ∑ 𝑒Å7e7\]
ɲ 𝐩
ÁÅ\] = 0
– Hence,∑ 𝑝Å q 𝑧Å 𝐩ÁÅ\] = 𝐩 q 𝐳 𝐩 = 0
AdvancedMicroeconomicTheory 72
GeneralEquilibrium:CompetitiveMarkets
– Inatwo-goodeconomy,Walras’lawimplies𝑝] q 𝑧] 𝐩 = −𝑝c q 𝑧c 𝐩
– Intuition:ifthereisexcessdemandinmarket1,𝑧] 𝐩 > 0,theremustbeexcesssupplyinmarket2,𝑧c 𝐩 < 0.
– Hence,ifmarket1isinequilibrium,𝑧] 𝐩 = 0,thensoismarket2,𝑧c 𝐩 = 0.
–Moregenerally,ifthemarketsof 𝑛 − 1 goodsareinequilibrium,thensoisthe𝑛th market.
AdvancedMicroeconomicTheory 73
GeneralEquilibrium:CompetitiveMarkets
2) Continuity:𝐳 𝐩 iscontinuousat𝐩.– ThisfollowsfromindividualWalrasiandemandsbeingcontinuousinprices.
3) Homegeneity:𝐳 𝜆𝐩 = 𝐳 𝐩 forall𝜆 > 0.– ThisfollowsfromWalrasiandemandsbeinghomogeneousofdegreezeroinprices.
• WenowuseexcessdemandtodefineaWalrasianequilibriumallocation.
AdvancedMicroeconomicTheory 74
GeneralEquilibrium:CompetitiveMarkets
• Walrasianequilibrium:– Apricevector𝐩∗ ≫ 0 isaWalrasianequilibriumifaggregateexcessdemandiszeroatthatpricevector, 𝐳(𝐩∗) = 0.§ Inwords,pricevector𝐩∗ clearsallmarkets.
– Alternatively,𝐩∗ ≫ 0 isaWalrasianequilibriumif:1) EachconsumersolveshisUMP,and2) Aggregatedemandequalsaggregatesupply
∑ 𝑥7 𝐩, 𝐩 q 𝐞7e7\] = ∑ 𝐞7e
7\]
AdvancedMicroeconomicTheory 75
GeneralEquilibrium:CompetitiveMarkets
• ExistenceofaWalrasianequilibrium:– AWalrasianequilibriumpricevector𝐩∗ ∈ ℝbbÁ ,i.e., 𝐳(𝐩∗) = 0,existsiftheexcessdemandfunction 𝐳(𝐩) satisfiescontinuity andWalras’law(Varian,1992).
AdvancedMicroeconomicTheory 76
GeneralEquilibrium:CompetitiveMarkets
• Uniqueness ofequilibriumprices:
AdvancedMicroeconomicTheory 77
satisfied violated
GeneralEquilibrium:CompetitiveMarkets
• Example6.6 (Walrasianequilibriumallocation):– Inexample6.1,wedeterminedthat
𝑀𝑅𝑆k = 𝑀𝑅𝑆m = �¬�¶
E¶¹
E¬¹= E¶º
E¬º= �¬
�¶
– LetusdeterminetheWalrasiandemandsofeachgoodforeachconsumer.
– Rearrangingthesecondequationabove,weget𝑝]𝑥]k = 𝑝c𝑥ck
AdvancedMicroeconomicTheory 78
GeneralEquilibrium:CompetitiveMarkets
• Example6.6 (continued):– Substitutingthisintoconsumer𝐴’sbudgetconstraintyields
𝑝]𝑥]k + 𝑝]𝑥]k = 𝑝] q 100 + 𝑝c q 350or𝑥]k = 50 + 175 �¶
�¬whichisconsumer𝐴’sWalrasiandemandforgood1.
– Pluggingthisvaluebackinto𝑝]𝑥]k = 𝑝c𝑥ck yields
𝑝] 50 + 175 �¶�¬
= 𝑝c𝑥ck
or𝑥ck = 175 + 50 �¬�¶
whichisconsumer𝐴’sWalrasian demandforgood2.AdvancedMicroeconomicTheory 79
GeneralEquilibrium:CompetitiveMarkets
• Example6.6 (continued):– Wecanobtainconsumer𝐵‘sdemandinananalogousway.Inparticular,substituting𝑝]𝑥]m = 𝑝c𝑥cm intoconsumer𝐵‘sbudgetconstraintyields
𝑝]𝑥]m + 𝑝]𝑥]m = 𝑝] q 100 + 𝑝c q 50or𝑥]m = 50 + 25 �¶
�¬whichisconsumer𝐵’sWalrasian demandforgood1.
– Pluggingthisvaluebackinto𝑝]𝑥]m = 𝑝c𝑥cm yields𝑝] 50 + 25 �¶
�¬= 𝑝c𝑥cm
or𝑥cm = 25 + 50 �¬�¶
whichisconsumer𝐵’sWalrasian demandforgood2.AdvancedMicroeconomicTheory 80
GeneralEquilibrium:CompetitiveMarkets
• Example6.6 (continued):– Forgood1,thefeasibilityconstraintis
𝑥]k + 𝑥]m = 100 + 100
50 + 175 �¶�¬
+ 50 + 25�¶�¬
= 200�¶�¬= ]
c– PluggingtherelativepricesintotheWalrasiandemandsyieldsWalrasianequilibrium:
𝑥]k,∗, 𝑥c
k,∗, 𝑥]m,∗, 𝑥c
m,∗; �¬�¶
= (137.5,275,62.5,125; 2)
AdvancedMicroeconomicTheory 81
x1A
x2B
x1Bx2A
0A
0B
100
100
200 200
ICAICB
x2=2x1
InitialEndowment
AA
WEA
CoreAllocations
GeneralEquilibrium:CompetitiveMarkets
• Example6.6 (continued):– Initialallocation,– Coreallocation,and–Walrasianequilibriumallocations(WEA).
AdvancedMicroeconomicTheory 82
GeneralEquilibrium:CompetitiveMarkets
• EquilibriumallocationsmustbeintheCore:– Ifeachconsumer’sutilityfunctionisstrictlyincreasing,
theneveryWEAisintheCore,i.e.,𝑊(𝐞) ⊂ 𝐶(𝐞).
• Proof (bycontradiction):– TakeaWEA𝐱(𝐩∗) withequilibriumprice𝐩∗,but𝐱(𝐩∗) ∉ 𝐶(𝐞).
– Since𝐱(𝐩∗) isaWEA,itmustbefeasible.– If𝐱(𝐩∗) ∉ 𝐶(𝐞),wecanfindacoalitionofindividuals𝑆 andanotherallocation 𝐲 suchthat
𝑢7(𝐲7) ≥ 𝑢7(𝐱7(𝐩∗, 𝐩∗ ⋅ 𝐞7)) forall𝑖 ∈ 𝑆AdvancedMicroeconomicTheory 83
GeneralEquilibrium:CompetitiveMarkets
• Proof(continued):– Theaboveexpression:
§ holdswithstrictinequalityforatleastoneindividualinthecoalition
§ isfeasibleforthecoalition,i.e.,∑ 𝐲7�7∈x = ∑ 𝐞7�
7∈x .
– Multiplyingbothsidesofthefeasibilityconditionby𝐩∗ yields
𝐩∗ ⋅ ∑ 𝐲7�7∈x = 𝐩∗ ⋅ ∑ 𝐞7�
7∈x– However,themostpreferablevector𝐲7 mustbemorecostlythan𝐱7(𝐩∗, 𝐩∗ ⋅ 𝐞7):
𝐩∗𝐲7 ≥ 𝐩∗𝐱7 𝐩∗, 𝐩∗ ⋅ 𝐞7 = 𝐩∗ ⋅ 𝐞7
withstrictinequalityforatleastoneindividual.AdvancedMicroeconomicTheory 84
GeneralEquilibrium:CompetitiveMarkets
• Proof(continued):– Hence,summingoverallconsumersinthecoalition 𝑆,weobtain
𝐩∗ q ∑ 𝐲7�7∈x > 𝐩∗ q ∑ 𝐱7 𝐩∗, 𝐩∗ ⋅ 𝐞7�
7∈x = 𝐩∗ ⋅ ∑ 𝐞7�7∈x
whichcontradicts𝐩∗ q ∑ 𝐲7�7∈x = 𝐩∗ ⋅ ∑ 𝐞7�
7∈x .
– Therefor,allWEAsmustbepartoftheCore,i.e.,𝐱(𝐩∗) ∈ 𝐶(𝐞)
AdvancedMicroeconomicTheory 85
GeneralEquilibrium:CompetitiveMarkets
• Remarks:1) TheCore𝐶(𝐞) containstheWEA(orWEAs)
§ Thatis,theCoreisnonempty.
2) SinceallcoreallocationsareParetoefficient(i.e.,wecannotincreasethewelfareofoneconsumerwithoutdecreasingthatofotherconsumers),thenallWEAs(whicharepartoftheCore)arealsoParetoefficient.
AdvancedMicroeconomicTheory 86
GeneralEquilibrium:CompetitiveMarkets
• FirstWelfareTheorem:EveryWEAisParetoefficient.
– TheWEAliesonthecore(thesegmentofthecontractcurvewithinthelens-shapedarea),
– ThecoreisasubsetofallParetoefficientallocations.
AdvancedMicroeconomicTheory 87
x1A
x2B
x1Bx2A
0A
0B
ICA
ICB
CoreAllocations
e
WEA
Contractcurve(Paretoefficientallocations)
ConsumerA
ConsumerB
GeneralEquilibrium:CompetitiveMarkets
• FirstWelfareTheorem
AdvancedMicroeconomicTheory 88
GeneralEquilibrium:CompetitiveMarkets
• SecondWelfareTheorem:– Supposethat𝐱� isaPareto-efficientallocation(i.e.,itliesonthecontractcurve),andthatendowmentsareredistributedsothatthenewendowmentvector𝐞∗7 liesonthebudgetline,thussatisfying
𝐩∗ q 𝐞∗7 = 𝐩∗ q 𝐱�7 foreveryconsumer𝑖– Then,thePareto-efficientallocation𝐱� isaWEAgiventhenewendowmentvector𝐞∗.
AdvancedMicroeconomicTheory 89
Consumer 1,
Consumer 2
Budget line.Same slope in both cases, i.e.,
Contract curve
C
C
GeneralEquilibrium:CompetitiveMarkets
• Secondwelfaretheorem
AdvancedMicroeconomicTheory 90
GeneralEquilibrium:CompetitiveMarkets
• Example6.7(WEAandSecondwelfaretheorem):
– Consideraneconomywithutilityfunctions𝑢k =𝑥]k𝑥ck forconsumer𝐴 and𝑢m = {𝑥]m, 𝑥cm} forconsumer𝐵.
– Theinitialendowmentsare𝐞k = (3,1) and𝐞m =(1,3).
– Good2isthenumeraire, i.e.,𝑝c = 1.
AdvancedMicroeconomicTheory 91
GeneralEquilibrium:CompetitiveMarkets
• Example6.7(continued):1) ParetoEfficientAllocations:– Consumer𝐵’spreferencesareperfectcomplements.Hence,heconsumesatthekinkofhisindifferencecurves,i.e.,
𝑥]m = 𝑥cm
– Givenfeasibilityconstraints𝑥]k + 𝑥]m = 4𝑥ck + 𝑥cm = 4
substitute𝑥cm for𝑥]m inthefirstconstrainttoget𝑥cm = 4 − 𝑥]kAdvancedMicroeconomicTheory 92
GeneralEquilibrium:CompetitiveMarkets
• Example6.7(continued):1) ParetoEfficientAllocations:– Substitutingtheaboveexpressioninthesecondconstraintyields
𝑥ck + (4 − 𝑥]k)E¶º
= 4 ⟺ 𝑥ck = 𝑥]k
– Thisdefinesthecontractcurve,i.e.,thesetofParetoefficientallocations.
AdvancedMicroeconomicTheory 93
GeneralEquilibrium:CompetitiveMarkets
• Example6.7(continued):2) WEA:– Consumer𝐴’smaximizationproblemis
maxE¬¹,E¶¹
𝑥]k𝑥ck
s. t. 𝑝]𝑥]k + 𝑥ck ≤ 𝑝] q 3 + 1– FOCs:
𝑥ck − 𝜆𝑝] = 0𝑥]k − 𝜆 = 0
𝑝]𝑥]k + 𝑥ck = 3𝑝] + 1where𝜆 isthelagrangemultiplier.
AdvancedMicroeconomicTheory 94
GeneralEquilibrium:CompetitiveMarkets
• Example6.7(continued):2) WEA:– Combiningthefirsttwoequations,
𝜆 = E¶¹
�¬= 𝑥]k or𝑝] =
E¶¹
E¬¹
– FromParetoefficiency,weknowthat𝑥ck = 𝑥]k.Hence,
𝑝] =E¶¹
E¬¹= 1
AdvancedMicroeconomicTheory 95
GeneralEquilibrium:CompetitiveMarkets
• Example6.7(continued):– SubstitutingboththepriceandParetoefficientallocationrequirementintothebudgetconstraint,
1 q 𝑥]k + 𝑥]k = 1 q 3 + 1or𝑥]k∗ = 𝑥ck∗ = 2
– Usingthefeasibilityconstraint,2⏟E¬¹+ 𝑥]m = 4 or𝑥]m∗ = 𝑥cm∗ = 2
– Thus,theWEAis
𝑥]k∗, 𝑥ck∗; 𝑥]m∗, 𝑥cm∗;�¬�¶
= (2,2; 2,2; 1)
AdvancedMicroeconomicTheory 96
GeneralEquilibrium:Production
• Letusnowextendourpreviousresultstosettingwherefirmsarealsoactive.
• Assume𝐽 firmsintheeconomy,eachwithproductionset 𝑌%,whichsatisfies:– Inactionispossible,i.e.,𝟎 ∈ 𝑌%.– 𝑌% isclosedandbounded,sopointsontheproductionfrontierarepartoftheproductionsetandthusfeasible.
– 𝑌% is strictlyconvex,solinearcombinationsoftwoproductionplansalsobelongtotheproductionset.
AdvancedMicroeconomicTheory 97
Production set of firm j
GeneralEquilibrium:Production
• Productionset 𝑌% forarepresentativefirm
AdvancedMicroeconomicTheory 98
GeneralEquilibrium:Production
• Everyfirm𝑗 facingafixedpricevector𝐩 ≫ 0independentlyandsimultaneouslysolves
max�*∈Ó*
𝐩 q 𝑦%
• Aprofit-maximizingproductionplan𝑦%(𝐩) existsforeveryfirm𝑗,anditisunique.
• Bythetheoremofthemaximum,boththeargmax,𝑦%(𝐩),andthevaluefunction, 𝜋%(𝐩) ≡𝐩 q 𝑦%(𝐩), arecontinuousin𝑝.
AdvancedMicroeconomicTheory 99
Isoprofit where profits are maximal
Isoprofit lines for profit levelwhere
profit maximizing production plan
GeneralEquilibrium:Production
• 𝑦%(𝑝) existsandisunique
AdvancedMicroeconomicTheory 100
GeneralEquilibrium:Production
• Aggregateproductionset:– Theaggregateproductionsetisthesumofallfirms’productionplans(eitherprofitmaximizingornot):
𝑌 = 𝐲|𝐲 = ∑ 𝑦%[%\] where𝑦% ∈ 𝑌%
– A joint-profitmaximizingproductionplan𝐲(𝐩) isthesumofeachfirm’sprofit-maximizingplan,i.e.,
𝐲 𝐩 = 𝑦](𝐩)+𝑦c(𝐩)+⋯+ 𝑦[(𝐩)
AdvancedMicroeconomicTheory 101
GeneralEquilibrium:Production
• Inaneconomywith𝐽 firms,eachofthemearning𝜋%(𝐩) profitsinequilibrium,howareprofitsdistributed?– Assumethateachindividual𝑖 ownsashare𝜃7% offirm𝑗’sprofits,where0 ≤ 𝜃7% ≤ 1 and∑ 𝜃7%e
7\] =1.
– Thisallowsformultiplesharingprofiles:§ 𝜃7% = 1:individual𝑖 ownsallsharesoffirm𝑗§ 𝜃7% = 1/𝐼:everyindividual’sshareoffirm𝑗 coincides
AdvancedMicroeconomicTheory 102
GeneralEquilibrium:Production
– Consumer’sbudgetconstraintbecomes𝐩 q 𝐱7 ≤ 𝐩 q 𝐞7 + ∑ 𝜃7%[
%\] 𝜋%(𝐩)where∑ 𝜃7%[
%\] 𝜋%(𝐩) isnewrelativetothestandardbudgetconstraint.
– Letusexpressthebudgetconstraintsas𝐩 q 𝐱7 ≤ 𝐩 q 𝐞7 + ∑ 𝜃7%[
%\] 𝜋% 𝐩@A 𝐩
⟹ 𝐩 q 𝐱7 ≤ 𝑚7(𝐩)where𝑚7 𝐩 > 0 (givenassumptionson𝑌%).
AdvancedMicroeconomicTheory 103
GeneralEquilibrium:Production
• Equilibriumwithproduction:–Westartdefiningexcessdemandfunctionsandusesuchadefinitiontoidentifythesetofequilibriumallocations.
– Excessdemand:Theexcessdemandfunctionforgood𝑘 is𝑧Å 𝐩 ≡ ∑ 𝑥Å7 (𝐩,𝑚7 𝐩 )e
7\] − ∑ 𝑒Å7e7\] − ∑ 𝑦Å
%(𝐩)[%\]
where∑ 𝑦Å%(𝐩)[
%\] isanewtermrelativetotheanalysisofgeneralequilibriumwithoutproduction.
AdvancedMicroeconomicTheory 104
GeneralEquilibrium:Production
– Hence,theaggregateexcessdemandvectoris𝐳 𝐩 = (𝑧] 𝐩 ,𝑧c 𝐩 , … , 𝑧Á 𝐩 )
–WEAwithproduction:Ifthepricevectorisstrictlypositiveinallofitscomponents,𝐩∗ ≫ 0,apairofconsumptionandproductionbundles(𝐱 𝐩∗ , 𝐲 𝐩∗ ) isaWEAif:1) Eachconsumer𝑖 solveshisUMP,whichbecomes
the𝑖th entryof𝐱 𝐩∗ ,i.e., 𝐱7(𝐩∗,𝑚7 𝐩∗ );2) Eachfirm𝑗 solvesitsPMP,whichbecomesthe
𝑗th entryof𝐲 𝐩∗ , i.e.,𝐲%(𝐩∗);AdvancedMicroeconomicTheory 105
GeneralEquilibrium:Production
3) Demandequalssupply∑ 𝐱7(𝐩∗,𝑚7 𝐩∗ )e7\] = ∑ 𝐞7e
7\] + ∑ 𝐲%(𝐩∗)[%\]
which is the market clearing condition.– Existence:Assumethat
§ consumers’utilityfunctionsarecontinuous,strictlyincreasingandstrictlyquasiconcave;
§ everyfirm𝑗’sproductionset𝑌% isclosedandbounded,strictlyconvex,andsatisfiesinactionbeingpossible;
§ everyconsumerisinitiallyendowedwithpositiveunitsofatleastonegood,sothesum∑ 𝐞7e
7\] ≫ 0.Then,thereisapricevector𝐩∗ ≫ 0 suchthataWEAexisits,i.e.,𝑧 𝐩∗ = 0.
AdvancedMicroeconomicTheory 106
GeneralEquilibrium:Production
• Example6.8(Equilibriumwithproduction):– Consideratwo-consumer,two-goodeconomywhereconsumer𝑖 = {𝐴, 𝐵} hasutilityfunction𝑢7 = 𝑥]7𝑥c7 .
– Therearetwofirmsinthiseconomy,andeachofthemusecapital(𝐾)andlabor(𝐿)toproduceoneoftheconsumptiongoodseach.
– Firm1producesgood1accordingto𝑦] = 𝐾],.ܽ𝐿],.c½.– Firm2producesgood2accordingto𝑦c = 𝐾c,.c½𝐿c,.ܽ.– Consumer𝐴 isendowedwith(𝐾k, 𝐿k) = (1,1),whileconsumer𝐵 isendowedwith(𝐾m, 𝐿m) = (2,1).
– LetusfindaWEAinthiseconomywithproduction.AdvancedMicroeconomicTheory 107
GeneralEquilibrium:Production
• Example6.8(continued):– UMPs:Consumer𝑖’smaximizationproblemis
maxE¬A ,E¶A
𝑥]7𝑥c7
s. t. 𝑝]𝑥]7 + 𝑥c7 = 𝑟𝐾7 + 𝑤𝐿7
where𝑟 and𝑤 arepricesforcapitalandlabor,respectively.
– FOC:�¬�¶= 𝑀𝑅𝑆],c7 ⟹ �¬
�¶= E¶A
E¬A⟹𝑝]𝑥]7 = 𝑝c𝑥c7
for𝑖 = {𝐴, 𝐵}.AdvancedMicroeconomicTheory 108
GeneralEquilibrium:Production
• Example6.8(continued):– Takingtheaboveequationforconsumers𝐴 and𝐵,andaddingthemtogetheryields
𝑝](𝑥]k + 𝑥]m) = 𝑝c(𝑥ck + 𝑥cm)
where𝑥]k + 𝑥]m istheleftsideofthefeasibilitycondition𝑥]k + 𝑥]m = 𝑦] = 𝐾],.ܽ𝐿],.c½.
– Substitutingbothfeasibilityconditionsintotheaboveexpression,andre-arranging,yields
�¬�¶= Þ¶ß.¶àf¶ß.áà
Þ¬ß.áàf¬ß.¶à
AdvancedMicroeconomicTheory 109
GeneralEquilibrium:Production
• Example6.8(continued):– PMPs:Firm1’smaximizationproblemis
maxÞ¬,f¬
𝑝]𝐾],.ܽ𝐿],.c½ − 𝑟𝐾] − 𝑤𝐿]– FOCs:
𝑟 = 0.75𝑝]𝐾]�,.c½𝐿],.c½𝑤 = 0.25𝑝]𝐾],.ܽ𝐿]�,.ܽ
– Combiningtheseconditionsgivesthetangencyconditionforprofitmaximization
âã= 𝑀𝑅𝑇𝑆f,Þ] ⟹ â
ã= 3 f¬
Þ¬
AdvancedMicroeconomicTheory 110
GeneralEquilibrium:Production
• Example6.8(continued):– Likewise,firm2’sPMPgivesthefollowingFOCs:
𝑟 = 0.25𝑝c𝐾c�,.ܽ𝐿c,.ܽ𝑤 = 0.75𝑝c𝐾c,.c½𝐿c�,.c½
– Combiningtheseconditionsgivesthetangencyconditionforprofitmaximization
âã= 𝑀𝑅𝑇𝑆f,Þc ⟹ â
ã= ]
¼f¶Þ¶
AdvancedMicroeconomicTheory 111
GeneralEquilibrium:Production
• Example6.8(continued):– Combiningboth𝑀𝑅𝑇𝑆 yields,
3 f¬Þ¬= ]
¼f¶Þ¶⟹Þ¬
f¬= 9 Þ¶
f¶§ Intuition:firm1ismorecapitalintensivethanfirm2,i.e.,itscapitaltolaborratioishigher.
– Setting bothfirm’spriceofcapital,𝑟,equaltoeachotheryields0.75𝑝]𝐾]�,.c½𝐿],.c½ = 0.25𝑝c𝐾c�,.ܽ𝐿c,.ܽ
⟹ �¬�¶= ]
¼Þ¬f¬
,.c½ Þ¶f¶
�,.ܽ
AdvancedMicroeconomicTheory 112
GeneralEquilibrium:Production
• Example6.8(continued):– Settingbothfirm’spriceoflabor,𝑤,equaltoeachotheryields0.25𝑝]𝐾],.ܽ𝐿]�,.ܽ = 0.75𝑝c𝐾c,.c½𝐿c�,.c½
⟹ �¬�¶= 3 Þ¬
f¬
�,.ܽ Þ¶f¶
,.c½
– Settingpriceratiofromconsumers’UMPequaltothefirstpriceratiofromfirms’PMPyields
Þ¶ß.¶àf¶ß.áà
Þ¬ß.áàf¬ß.¶à= ]
¼Þ¬f¬
,.c½ Þ¶f¶
�,.ܽ⟹ 𝐾] = 3𝐾c
AdvancedMicroeconomicTheory 113
GeneralEquilibrium:Production
• Example6.8(continued):– Bythefeasibilityconditions,weknowthat𝐾] +𝐾c = 𝐾k + 𝐾m = 3 or𝐾c = 3 − 𝐾].
– Substitutingtheaboveexpressioninto𝐾] = 3𝐾c,wefindtheprofit-maximizingdemandsforcapitalusebyfirms1and2:
𝐾] = 3 3 − 𝐾] ⟹𝐾]∗ =æç
𝐾c∗ =]¼𝐾]∗ =
¼ç
AdvancedMicroeconomicTheory 114
GeneralEquilibrium:Production
• Example6.8(continued):– Settingpriceratiofromconsumers’UMPequaltothesecondpriceratiofromfirms’PMPyields
Þ¶ß.¶àf¶ß.áà
Þ¬ß.áàf¬ß.¶à= 3 Þ¬
f¬
�,.ܽ Þ¶f¶
,.c½⟹ 𝐿] =
]¼𝐿c
– Bythefeasibilityconditions,weknowthat𝐿] +𝐿c = 𝐿k + 𝐿m = 2 or𝐿c = 2 − 𝐿].
AdvancedMicroeconomicTheory 115
GeneralEquilibrium:Production
• Example6.8(continued):– Substitutingtheaboveexpressioninto 𝐿] =
]¼𝐿c,
wefindtheprofit-maximizingdemandsforlaborusebyfirms1and2:
𝐿] =]¼2 − 𝐿] ⟹𝐿]∗ =
]c
𝐿c∗ = 3𝐿]∗ =¼c
AdvancedMicroeconomicTheory 116
GeneralEquilibrium:Production
• Example6.8(continued):– Substitutingthecapitalandlabordemandsforfirm1and2intothepriceratiofromconsumers’UMPyields
�¬�¶=
èéß.¶à è
¶ß.áà
êéß.áà ¬
¶ß.¶à = 0.82
wherenormalizingthepriceofgood2,i.e.,𝑝c =1,gives𝑝] = 0.82.
AdvancedMicroeconomicTheory 117
GeneralEquilibrium:Production
• Example6.8(continued):– Furthermore,substitutingourcalculatedvaluesintothepriceofcapitalandlaboryields
𝑟∗ = 0.75(0.82) æç
�,.c½ ]c
,.c½= 0.42
𝑤∗ = 0.25(0.82) æç
,.ܽ ]c
�,.ܽ= 0.63
AdvancedMicroeconomicTheory 118
GeneralEquilibrium:Production
• Example6.8(continued):– Usingconsumer𝐴’stangencycondition,weknow
𝑥ck =�¬�¶𝑥]k ⟹𝑥ck = 0.82𝑥]k
– Substitutingthisvalueintoconsumer𝐴’sbudgetconstraintyields
𝑝]𝑥]k + 𝑝c(0.82𝑥]k) = 𝑟𝐾k + 𝑤𝐿k– Plugginginourcalculatedvaluesandsolvingfor𝑥]k yields
𝑥]k,∗ = 0.64
𝑥ck,∗ = 0.82𝑥]
k,∗ = 0.53AdvancedMicroeconomicTheory 119
GeneralEquilibrium:Production
• Example6.8(continued):– Performingthesameprocesswiththetangencyconditionofconsumer 𝐵 yields
𝑥]m,∗ = 0.90𝑥cm,∗ = 0.74
– Thus,ourWEAis
𝑥]k, 𝑥ck; 𝑥]m, 𝑥cm;�¬�¶; 𝐿], 𝐿c; 𝐾], 𝐾c =
0.64,0.53; 0.90,0.74; 0.82; ]c, ¼c, æç, ¼ç
AdvancedMicroeconomicTheory 120
GeneralEquilibrium:Production
• Equilibriumwithproduction– Welfare:–WeextendtheFirstandSecondWelfareTheoremstoeconomieswithproduction,connectingWEAandParetoefficientallocations.
– Paretoefficiency:Thefeasibleallocation(𝐱, 𝐲) isParetoefficientifthereisnootherfeasibleallocation(𝐱�, 𝐲�) suchthat
𝑢7(𝐱�7) ≥ 𝑢7(𝐱7)foreveryconsumer𝑖 ∈ 𝐼,with𝑢7(𝐱�7) > 𝑢7(𝐱7)foratleastoneconsumer.
AdvancedMicroeconomicTheory 121
GeneralEquilibrium:Production
– Inaneconomywithtwogoods,twoconsumers,twofirmsandtwoinputs(laborandcapital),thesetofParetoefficientallocationssolves
maxE¬¬,E¶¬,E¬¶,E¶¶,f¬,Þ¬,f¶,Þ¶+,
𝑢](𝑥]], 𝑥c])
s. t. 𝑢c(𝑥]c, 𝑥cc) ≥ 𝑢�c
𝑥]] + 𝑥c] ≤ 𝐹](𝐿], 𝐾])𝑥]c + 𝑥cc ≤ 𝐹c(𝐿c, 𝐾c)
í tech. feasibility
𝐿] + 𝐿c ≤ 𝐿�𝐾] + 𝐾c ≤ 𝐾ó
ô inputfeasibility
AdvancedMicroeconomicTheory 122
GeneralEquilibrium:Production
– TheLagrangianisℒ= 𝑢] 𝑥]], 𝑥c] + 𝜆 𝑢c 𝑥]c, 𝑥cc − 𝑢�c+ 𝜇] 𝐹] 𝐿], 𝐾] − 𝑥]] − 𝑥c]+ 𝜇c 𝐹c 𝐿c, 𝐾c − 𝑥]c − 𝑥cc + 𝛿f 𝐿� − 𝐿] − 𝐿c+ 𝛿Þ 𝐾ó − 𝐾] − 𝐾c– Inthecaseofinteriorsolutions,thesetofFOCsyieldaconditionforefficiencyinconsumptionsimilartobartereconomics:
𝑀𝑅𝑆],c] = 𝑀𝑅𝑆],cc
AdvancedMicroeconomicTheory 123
GeneralEquilibrium:Production
– FOCswrt 𝐿% and𝐾%,where𝑗 = {1,2},yieldaconditionforefficiencythatweencounteredinproductiontheory
𝜕𝐹]𝜕𝐿𝜕𝐹]𝜕𝐾
=𝜕𝐹c𝜕𝐿𝜕𝐹c𝜕𝐾
– Thatis,the𝑀𝑅𝑇𝑆f,Þ betweenlaborandcapitalmustcoincideacrossfirms.
– Otherwise,welfarecouldbeincreasedbyassigningmorelabortothefirmwiththehighest𝑀𝑅𝑇𝑆f,Þ .
AdvancedMicroeconomicTheory 124
GeneralEquilibrium:Production
– Combiningtheabovetwoconditionsforefficiencyinconsumptionandproduction,weobtain
𝜕𝑈7𝜕𝑥]7
𝜕𝑈7𝜕𝑥c7
=𝜕𝐹c𝜕𝐿𝜕𝐹]𝜕𝐿
– Thatis,𝑀𝑅𝑆],c7 mustcoincidewiththerateatwhichunitsofgood1canbetransformedintounitsofgood2,i.e.,themarginalrateoftransformation𝑀𝑅𝑇],c.
AdvancedMicroeconomicTheory 125
GeneralEquilibrium:Production
– Ifwemovelaborfromfirm2tofirm1,theproductionofgood2increasesby�ú¶
�fwhilethatof
good1decreasesby�ú¬�f
.Hence,inordertoincreasethetotaloutputofgood1byoneunitweneed�ú¶
�f/ �ú¬�f
unitsofgood2.
– Intuition:foranallocationtobeefficientweneedthattherateatwhichconsumersarewillingtosubstitutegoods1and2coincideswiththerateatwhichgood1canbetransformedintogood2.
AdvancedMicroeconomicTheory 126
GeneralEquilibrium:Production
• FirstWelfareTheoremwithproduction:iftheutilityfunctionofeveryindividual𝑖,𝑢7,isstrictlyincreasing,theneveryWEAisParetoefficient.
• Proof (bycontradiction):– Supposethat(𝐱, 𝐲) isaWEAatprices𝐩∗,butisnot Paretoefficient.
– Since(𝐱, 𝐲) isaWEA,thenitmustbefeasible
∑ 𝐱7e7\] = ∑ 𝐞7e
7\] + ∑ 𝐲%[%\]
AdvancedMicroeconomicTheory 127
GeneralEquilibrium:Production
– Because(𝐱, 𝐲) isnot Paretoefficient,thereexistssomeotherfeasibleallocation(𝐱û, 𝐲û) suchthat
𝑢7(𝐱û7) ≥ 𝑢7(𝐱7)foreveryconsumer𝑖 ∈ 𝐼,with𝑢7 𝐱û7 > 𝑢7(𝐱7) foratleastoneconsumer.§ Thatis,thealternativeallocation(𝐱û, 𝐲û)makesatleastoneconsumerstrictlybetteroffthanWEA.
– Butthisimpliesthatbundle𝐱û7 ismorecostlythan𝐱7,𝐩∗ q 𝐱û7 ≥ 𝐩∗ q 𝐱7
foreveryindividual𝑖 (withatleastonestrictlyinequality).
AdvancedMicroeconomicTheory 128
GeneralEquilibrium:Production
– Summingoverallconsumersyields𝐩∗ q ∑ 𝐱û7e
7\] > 𝐩∗ q ∑ 𝐱7e7\]
whichcanbere-writtenas
𝐩∗ q ∑ 𝐞7e7\] + ∑ 𝐲û%[
%\] > 𝐩∗ q ∑ 𝐞7e7\] + ∑ 𝐲%[
%\]
orre-arranging
𝐩∗ q ∑ 𝐲û%[%\] > 𝐩∗ q ∑ 𝐲%[
%\]
– However,thisresultimpliesthat𝐩∗ q 𝐲û% > 𝐩∗ q 𝐲%forsomefirm𝑗.
AdvancedMicroeconomicTheory 129
GeneralEquilibrium:Production
– Thisindicatesthatproductionplan𝐲% wasnotprofit-maximizingand,asaconsequence,itcannotbepartofaWEA.
–Wethereforereachedacontradiction.
– Thisimpliesthattheoriginalstatementwastrue:ifanallocation(𝐱, 𝐲) isaWEA,itmustalsobeParetoefficient.
AdvancedMicroeconomicTheory 130
GeneralEquilibrium:Production
• Example6.9(WEAandPEwithproduction):– Considerthesettingdescribedinexample6.8.– ThesetofParetoefficientallocationsmustsatisfy
𝑀𝑅𝑆],ck = 𝑀𝑅𝑆],cm and𝑀𝑅𝑇𝑆f,Þ] = 𝑀𝑅𝑇𝑆f,Þc
–Wecanshowthat
𝑀𝑅𝑆],ck =𝑥ck
𝑥]k=0.530.64 = 0.82
𝑀𝑅𝑆],cm =𝑥cm
𝑥]m=0.740.90 = 0.82
whichimpliesthat𝑀𝑅𝑆],ck = 𝑀𝑅𝑆],cm .AdvancedMicroeconomicTheory 131
GeneralEquilibrium:Production
• Example6.9(continued):–Wecanalsoshowthat
𝑀𝑅𝑇𝑆f,Þ] = 3 f¬Þ¬= 3 ]
c/ æç= c
¼
𝑀𝑅𝑇𝑆f,Þc = 3 f¶Þ¶= 3 ¼
c/ ¼ç= c
¼
whichimpliesthat𝑀𝑅𝑇𝑆f,Þ] = 𝑀𝑅𝑇𝑆f,Þc .
– Sincebothoftheseconditionshold,ourWEAfromexample6.8isParetoefficient.
AdvancedMicroeconomicTheory 132
GeneralEquilibrium:Production
• SecondWelfareTheoremwithproduction:– Considertheassumptionsonconsumersandproducersdescribedabove.
– Then,foreveryParetoefficientallocation(𝐱û, 𝐲û)wecanfind:a) aprofileofincometransfers(𝑇], 𝑇c, … , 𝑇e)
redistributingincomeamongconsumers,i.e.,satisfying∑ 𝑇7e
7\] = 0;b) apricevector𝐩ó,suchthat:
AdvancedMicroeconomicTheory 133
GeneralEquilibrium:Production
1) Bundle𝐱û7 solvestheUMPmax𝐱A
𝑢7(𝐱7)
s. t. 𝐩ó q 𝐱7 ≤ 𝑚7 𝐩ó + 𝑇7 forevery𝑖 ∈ 𝐼whereindividual𝑖’soriginalincome𝑚7 𝐩ó isincreased(decreased)ifthetransfer𝑇7 ispositive(negative).
2) Productionplan𝐲û% solvesthePMPmax�*
𝐩ó q 𝐲%
s. t. 𝐲% ∈ 𝑌% forevery𝑗 ∈ 𝐽AdvancedMicroeconomicTheory 134
GeneralEquilibrium:Production
• Example6.10(SecondWelfareTheoremwithproduction):– ConsideranalternativeallocationinthesetofParetoefficientallocationsidentifiedinexample6.9.
– Suchas, 𝑥û]k, 𝑥ûck; 𝑥û]m, 𝑥ûcm = (0.82,1; 0.79,0.65).– Consumer𝐴’sbudgetconstraintbecomes
𝑝]𝑥û]k + 𝑝c𝑥ûck = 𝑟𝐾k + 𝑤𝐿k + 𝑇]– Recallthat
𝑝], 𝑝c; 𝐾k, 𝐿k; 𝑟, 𝑤 = 0.82,1; 1,1; 0.42,0.63remainsunchanged.AdvancedMicroeconomicTheory 135
GeneralEquilibrium:Production
– Substitutingthesevaluesintoconsumer𝐴’sbudgetconstraint
0.82𝑥û]k + 𝑥ûck = 1.05 + 𝑇]– Recallthat
�¬�¶= Eû¶¹
Eû¬¹⟹𝑥ûck = 0.82𝑥û]k
– Substituting
2 0.82 0.75Eû¬¹
= 1.05 + 𝑇] ⟹𝑇] = 0.17
AdvancedMicroeconomicTheory 136
GeneralEquilibrium:Production
– Likewiseforconsumer𝐵,hisbudgetconstraintbecomes
𝑝]𝑥û]m + 𝑝c𝑥ûcm = 𝑟𝐾m + 𝑤𝐿m + 𝑇c
– Substitutingtheunchangedvalues𝑝], 𝑝c; 𝐾k, 𝐿k; 𝑟, 𝑤 = 0.82,1; 1,1; 0.42,0.63 ,
0.82𝑥û]m + 𝑥ûcm = 1.47 + 𝑇c– Recallthat
�¬�¶= Eû¶º
Eû¬º⟹𝑥ûcm = 0.82𝑥û]m
AdvancedMicroeconomicTheory 137
GeneralEquilibrium:Production
– Substituting
2 0.82 0.79Eû¬º
= 1.47 + 𝑇] ⟹𝑇] = −0.17
– Clearly,𝑇] + 𝑇c = 0
– ThusthesetransferswillallowforthenewallocationtobeaWEA.
AdvancedMicroeconomicTheory 138
ComparativeStatics
AdvancedMicroeconomicTheory 139
ComparativeStatics• Weanalyzehowequilibriumoutcomesareaffectedbyanincreasein:– thepriceofonegood– theendowmentofoneinput
• Considerasettingwithtwogoods,eachbeingproducedbytwofactors1and2underconstantreturnstoscale(CRS).
• Anecessaryconditionforinputprices(𝑤]∗, 𝑤c∗) tobeinequilibriumis
𝑐] 𝑤], 𝑤c = 𝑝] and𝑐c 𝑤], 𝑤c = 𝑝c– Thatis,firmsproduceuntiltheirmarginalcostsequalthepriceofthegood.
AdvancedMicroeconomicTheory 140
ComparativeStatics
• Let𝑧]%(𝑤) denotefirm𝑗’sdemandforfactor1,and𝑧c%(𝑤) beitsdemandforfactor2.– Thisisequivalenttothefactordemandcorrespondences𝑧(𝑤, 𝑞) inproductiontheory.
• Theproductionofgood1isrelativelymoreintenseinfactor1thanistheproductionofgood2if
ɬ¬(ã)ɶ¬(ã)
> ɬ¶(ã)ɶ¶(ã)
whereɬ*(ã)ɶ*(ã)
representsfirm𝑗’sdemandforinput1relativetothatofinput2.
AdvancedMicroeconomicTheory 141
ComparativeStatics:PriceChange
1) Changesinthepriceofonegood,𝑝%(Stolper-Samuelsontheorem):– Consideraneconomywithtwoconsumersandtwofirmssatisfyingtheabovefactorintensityassumption.
– Ifthepriceofgood𝑗,𝑝%,increases,then:a) theequilibriumpriceofthefactormore
intensivelyusedintheproductionofgoodincreases;while
b) theequilibriumpriceoftheotherfactordecreases.
AdvancedMicroeconomicTheory 142
ComparativeStatics:PriceChange
• Proof:– Letusfirsttaketheequilibriumconditions
𝑐] 𝑤], 𝑤c = 𝑝] and𝑐c 𝑤], 𝑤c = 𝑝c– Differentiatingthemyields
�ü¬ ã¬,ã¶�ã¬
𝑑𝑤] +�ü¬ ã¬,ã¶
�ã¶𝑑𝑤c = 𝑑𝑝]
�ü¶ ã¬,ã¶�ã¬
𝑑𝑤] +�ü¶ ã¬,ã¶
�ã¶𝑑𝑤c = 𝑑𝑝c
– ApplyingShephard’s lemma,weobtain𝑧]](𝑤)𝑑𝑤] + 𝑧]c(𝑤)𝑑𝑤c = 𝑑𝑝]𝑧c](𝑤)𝑑𝑤] + 𝑧cc(𝑤)𝑑𝑤c = 𝑑𝑝c
AdvancedMicroeconomicTheory 143
ComparativeStatics:PriceChange
– Ifonlyprice𝑝] varies,then𝑑𝑝c = 0.– Hence,wecanrewritethesecondexpressionas
𝑧c](𝑤)𝑑𝑤] + 𝑧cc(𝑤)𝑑𝑤c = 0⟹ 𝑑𝑤] = − ɶ¶
ɶ¬𝑑𝑤c
– Solvingforýã¬ý�¬
inthefirstexpressionyieldsýã¬ý�¬
= ɶ¶É¬¬É¶¶�ɬ¶É¶¬
– Solving,instead,forýã¶ý�¬
yieldsýã¶ý�¬
= − ɶ¬É¬¬É¶¶�ɬ¶É¶¬AdvancedMicroeconomicTheory 144
ComparativeStatics:PriceChange
– Fromthefactorintensitycondition,ɬ¬(ã)ɶ¬(ã)
>ɬ¶(ã)ɶ¶(ã)
,weknowthat𝑧]]𝑧cc − 𝑧]c𝑧c] > 0.
– Hence,thedenominatorinbothýã¬ý�¬
andýã¶ý�¬
ispositive.
– Thenumeratorinbothýã¬ý�¬
andýã¶ý�¬
isalsopositive(theyarejustfactordemands).
– Thus,ýã¬ý�¬
> 0 andýã¶ý�¬
< 0.
AdvancedMicroeconomicTheory 145
ComparativeStatics:PriceChange
• Example6.11:– Letussolvefortheinputdemands inExample6.8:
𝑟] = 𝑝]0.75𝐾]�,.c½𝐿],.c½ ⟹ 𝑧]] = 𝐾] =¼�¬çâ
ç𝐿]
𝑤] = 𝑝]0.25𝐾],.ܽ𝐿]�,.ܽ ⟹ 𝑧c] = 𝐿] =�¬çã
éè 𝐾]
𝑟c = 𝑝c0.25𝐾c�,.ܽ𝐿c,.ܽ ⟹ 𝑧]c = 𝐾c =�¶çâ
éè 𝐿c
𝑤c = 𝑝c0.75𝐾c,.c½𝐿c�,.c½ ⟹ 𝑧cc = 𝐿c =¼�¶çã
ç𝐾c
AdvancedMicroeconomicTheory 146
ComparativeStatics:PriceChange
• Example6.11 (continued):– Sincefirm1ismorecapitalintensivethanfirm2,then𝑧]]𝑧cc − 𝑧]c𝑧c] > 0musthold,i.e.,
¼�¬çâ
ç𝐿]
¼�¶çã
ç𝐾c −
�¶çâ
éè 𝐿c
�¬çã
éè 𝐾] > 0
– Fromexample8.4,Þ¬f¬= 9 Þ¶
f¶⟹ 𝐾]𝐿c = 9𝐾c𝐿].
– Substitutingthisvalueintotheaboveexpression,
36.33 �¬�¶âã
þè − 1 > 0 ⟹�¬�¶
âã> 0.26
AdvancedMicroeconomicTheory 147
ComparativeStatics:PriceChange
• Example6.11 (continued):– Inoursolution,�¬�¶
âã= 3.08,hencethiscondition
issatisfied.– Next,observethatboth𝑧]] and𝑧cc aretriviallypositive.
– ApplyingtheStolper-Samuelsontheoremyieldsýã¬ý�¬
= ɶ¶É¬¬É¶¶�ɬ¶É¶¬
> 0ýã¶ý�¬
= − ɶ¬É¬¬É¶¶�ɬ¶É¶¬
< 0
AdvancedMicroeconomicTheory 148
ComparativeStatics:EndowmentChange
2) Changesinendowments(Rybczynskitheorem):– Consideraneconomywithtwoconsumersandtwofirmssatisfyingtheabovefactorintensityassumption.
– Iftheendowmentofafactorincreases,thena) productionofthegoodthatusesthisfactor
moreintensivelyincreases;whereasb) theproductionoftheothergooddecreases.
AdvancedMicroeconomicTheory 149
ComparativeStatics:EndowmentChange
• Proof:– Consideraneconomywithtwofactors,laborandcapital,andtwogoods,1and2.
– Let𝑧f%(𝑤) denotefirm𝑗’sfactordemandforlabor(whenproducingoneunitofoutput)
– Similarly,let𝑧Þ%(𝑤) denotefirm𝑗’sfactordemandforcapital.
– Then,factorfeasibilityrequires𝐿 = 𝑧f] 𝑤 q 𝑦] + 𝑧fc 𝑤 q 𝑦c𝐾 = 𝑧Þ] 𝑤 q 𝑦] + 𝑧Þc 𝑤 q 𝑦c
AdvancedMicroeconomicTheory 150
ComparativeStatics:EndowmentChange
– Differentiatingthefirstcondition
𝑑𝐿 = 𝑧f] q��¬�f+ 𝑧fc q
��¶�f
– Dividingbothsidesby𝐿 yieldsýff= Éÿ¬
fq ��¬�f+ Éÿ¶
fq ��¶�f
–Multiplyingthefirsttermby�¬�¬
andthesecondtermby�¶
�¶,weobtain
ýff= Éÿ¬q�¬
fq³!¬³ÿ�¬+ Éÿ¶q�¶
fq³!¶³ÿ�¶
AdvancedMicroeconomicTheory 151
ComparativeStatics:EndowmentChange
–Wecanexpress:
a) ÉÿA(ã)q�Af
≡ 𝛾f7,i.e.,theshareoflaborusedbyfirm𝑖;
b)³!A³ÿ�A≡ %∆𝑦7,i.e.,thepercentageincreasein
theproductionoffirm𝑖 broughtbytheincreaseintheendowmentoflabor;
c) ýff≡ %∆𝐿,i.e.,thepercentageincreaseinthe
endowmentoflaborintheeconomy.
AdvancedMicroeconomicTheory 152
ComparativeStatics:EndowmentChange
– Hence,theaboveexpressionbecomes%∆𝐿 = 𝛾f] q %∆𝑦] + 𝛾fc q (%∆𝑦c)
– Asimilarexpressioncanbeobtainedfortheendowmentofcapital:%∆𝐾 = 𝛾Þ] q %∆𝑦] + 𝛾Þc q (%∆𝑦c)
– Notethat𝛾f], 𝛾fc ∈ (0,1)§ Hence,%∆𝐿 isalinearcombinationof%∆𝑦] and%∆𝑦c.
– Similarargumentappliedto%∆𝐾,where𝛾Þ], 𝛾Þc ∈ (0,1).
AdvancedMicroeconomicTheory 153
ComparativeStatics:EndowmentChange
– Capitalisassumedtobemoreintensivelyusedinfirm1,i.e.,
Þ¬f¬> Þ¶
f¶or
𝛾Þ] > 𝛾f] forfirm1and𝛾Þc < 𝛾fc forfirm2
– Hence,ifcapitalbecomesrelativelymoreabundantthanlabor,i.e.,%∆𝐾 > %∆𝐿,itmustbethat%∆𝑦] > %∆𝑦c.
AdvancedMicroeconomicTheory 154
ComparativeStatics:EndowmentChange
– Thatis%∆𝐿 = 𝛾f]∧ ∧
%∆𝐾 = 𝛾Þ]
q %∆𝑦] +∥
q %∆𝑦] +
𝛾fc q (%∆𝑦c)∨ ∥𝛾Þc q (%∆𝑦c)
– Intuition:thechangeintheinputendowmentproducesamore-than-proportionalincreaseinthegoodwhoseproductionwasintensiveintheuseofthatinput.
AdvancedMicroeconomicTheory 155
ComparativeStatics:EndowmentChange
• Example6.12(Rybczynski Theorem):– ConsidertheproductiondecisionsofthetwofirmsinExample6.8,wherewefoundthat𝐾] =3𝐾c and𝐾] + 𝐾c = 𝐾ó = 3.
– Assumethattotalendowmentofcapitalincreasesto𝐾ó = 5,i.e.,𝐾c = 5 − 𝐾].
– Theprofitmaximizingdemandsforcapitalare
𝐾] = 3 5 − 𝐾] ⟹𝐾]∗ =]½ç
𝐾c =]¼𝐾]∗ =
½ç
AdvancedMicroeconomicTheory 156
ComparativeStatics:EndowmentChange
• Example6.12(continued):– Similarly,forlaborwefoundthat𝐿] =
]¼𝐿c and
𝐿] + 𝐿c = 𝐿� = 2.–Wedonotaltertheaggregateendowmentoflabor,𝐿� = 2.
– Hence,capitalusebyfirm1increasesfrom𝐾]∗ =æç
to]½ç.
– Firm1usescapitalmoreintensivelythanfirm2
does,i.e.,Þ¬f¬> Þ¶
f¶,since
ê鬶>
èéè¶.
AdvancedMicroeconomicTheory 157
ComparativeStatics:EndowmentChange
• Example6.12(continued):– Thefactordemandsforeachgoodare
𝑧Þ] =¼âã
�,.ܽand𝑧f] =
¼âã
,.c½
𝑧Þc =â¼ã
�,.ܽand𝑧fc =
â¼ã
,.c½
– Usingthevaluesfromexample6.8,wecanassignfollowingvalues:
𝛾Þ], 𝛾f], 𝛾Þc, 𝛾fc = (0.75,0.25,0.25,0.75)
AdvancedMicroeconomicTheory 158
ComparativeStatics:EndowmentChange
• Example6.12(continued):– Ourtwoequationsthenbecome
0 = 0.25 q %∆𝑦] + 0.75 q (%∆𝑦c)
0.66 = 0.75 q %∆𝑦] + 0.25 q (%∆𝑦c)– Solvingtheaboveequationssimultaneouslyyields
%∆𝑦] = 1 = 100%%∆𝑦c = −0.3333 = −33.33%
– Intuition:anincreaseintheendowmentofcapitalby½�¼¼= 0.66 = 66% entailsanincreaseingood1’s
outputby100% whilethatofgood2decreasesby33.33%. AdvancedMicroeconomicTheory 159
IntroducingTaxes
AdvancedMicroeconomicTheory 160
IntroducingTaxes:TaxonGoods• Assumethatasalestax𝑡Å isimposedongood𝑘.• Thenthepricepaidbyconsumersincreasesby𝑝Å& = (1 + 𝑡Å)𝑝Å',where𝑝Å' isthepricereceivedbyproducers.
• Ifthetaxongood1and2coincides,i.e.,𝑡] = 𝑡c,thepriceratioconsumersandproducersfaceisunaffected:
�¬(
�¶(= (]b�¬)�¬)
(]b�¶)�¶)= �¬)
�¶)
– Hence,theafter-taxallocationisstillParetoefficient.
AdvancedMicroeconomicTheory 161
IntroducingTaxes:TaxonGoods• However,ifonlygood1isaffectedbythetax,i.e.,𝑡] > 0 while𝑡c = 0 (i.e.,𝑡] ≠ 𝑡c),thentheallocationwillnotbeParetoefficient.– Inthissetting,the𝑀𝑅𝑇𝑆f,Þ isstillthesameasbeforetheintroductionofthetax:
𝜕𝐹]𝜕𝐿𝜕𝐹]𝜕𝐾
=𝑤f𝑤Þ
=𝜕𝐹c𝜕𝐿𝜕𝐹c𝜕𝐾
– Therefore,theallocationofinputsstillachievesproductiveefficiency.
AdvancedMicroeconomicTheory 162
IntroducingTaxes:TaxonGoods
– Similarly,the𝑀𝑅𝑇],c stillcoincideswiththepriceratioofgoods1and2:
𝜕𝐹c𝜕𝐿𝜕𝐹]𝜕𝐿
=𝑝]'
𝑝c=𝜕𝐹c𝜕𝐾𝜕𝐹]𝜕𝐾
wherethepricereceivedbytheproducer,𝑝]',isthesamebeforeandafterintroducingthetax.
AdvancedMicroeconomicTheory 163
IntroducingTaxes:TaxonGoods
– However,whilethe𝑀𝑅𝑆],c isequaltothepriceratio
thatconsumersface,i.e.,�¬(
�¶= (]b�¬)�¬)
�¶,itnow
becomeslargerthanthepriceratiothatproducers
face,�¬)
�¶:
𝑀𝑅𝑆],c =𝑝]&
𝑝c=(1 + 𝑡])𝑝]'
𝑝c>𝑝]'
𝑝c– Intuition:
§ Therateatwhichconsumersarewillingtosubstitutegood1for2islargerthantherateatwhichfirmscantransformgood1for2.
§ Thus,theproductionofgood1shoulddecreaseandthatofgood2increase.
AdvancedMicroeconomicTheory 164
IntroducingTaxes:TaxonInputs
• Similarargumentsextendtotheintroductionoftaxesoninputs
• Pricepaidbyproducersis𝑤@' = (1 + 𝑡@)𝑤@& forinput𝑚 = {𝐿, 𝐾}.
• Ifbothinputsaresubjecttothesametax,i.e.,𝑡f = 𝑡Þ = 𝑡,theinputpriceratioconsumersandproducersfacecoincides:
ãÿ)
ã*) =
(]b�)ãÿ(
(]b�)ã*( =
ãÿ(
ã*(
– Hence,theefficiencyconditionsisunaffected
AdvancedMicroeconomicTheory 165
IntroducingTaxes:TaxonInputs
• However,whentaxesdiffer,𝑡f ≠ 𝑡Þ,productiveefficiencynolongerholdsundersuchcondition:– Whileinputconsumerssatisfy
𝑤f&
𝑤Þ&=𝜕𝐹]𝜕𝐿𝜕𝐹]𝜕𝐾
andinputproducerssatisfy
𝑤f'
𝑤Þ'=𝜕𝐹c𝜕𝐿𝜕𝐹c𝜕𝐾
AdvancedMicroeconomicTheory 166
IntroducingTaxes:TaxonInputs
theinputpriceratiostheyfacedonotcoincide𝜕𝐹]𝜕𝐿𝜕𝐹]𝜕𝐾
=𝑤f&
𝑤Þ&≠
1 + 𝑡f 𝑤f&
1 + 𝑡Þ 𝑤Þ&=𝑤f'
𝑤Þ'=𝜕𝐹c𝜕𝐿𝜕𝐹c𝜕𝐾
– Forinstance:§ If𝑡f > 𝑡Þ,the𝑀𝑅𝑇𝑆f,Þ islargerforfirm1than2,§ Thustheallocationofinputsisinefficient,i.e.,themarginalproductivityofadditionalunitsoflabor(relativetocapital)islargerinfirm1thanin2.
AdvancedMicroeconomicTheory 167
AppendixA:LargeEconomiesandtheCore
AdvancedMicroeconomicTheory 168
LargeEconomiesandtheCore
• Weknowthatequilibriumallocations(WEAs)arepartoftheCore.
• Wenowshowthat,astheeconomybecomeslarger,theCoreshrinksuntilexactlycoincidingwiththesetofWEAs.
AdvancedMicroeconomicTheory 169
LargeEconomiesandtheCore• Letusfirstconsideraneconomywith𝐼 consumers,eachwithutilityfunction𝑢7 andendowmentvector𝐞7.
• Considerthiseconomy’sreplicabydoublingthenumberofconsumersto2𝐼,eachofthemstillwithutilityfunction𝑢7 andendowmentvector𝐞7.– Therearenowtwoconsumersofeachtype,i.e.,"twins,"havingidenticalpreferencesandendowments.
• Definean𝑟-foldreplicaeconomy ℰâ, havingconsumersofeachtype,foratotalof𝑟𝐼 consumers.– Foranyconsumertype𝑖 ∈ 𝐼,all𝑟 consumersofthattypesharethecommonutilityfunction𝑢7 andhaveidenticalendowments𝐞7 ≫ 0.
AdvancedMicroeconomicTheory 170
LargeEconomiesandtheCore
• Whencomparingtworeplicaeconomies,thelargestwillbethathavingmoreofeverytypeofconsumer.
• Allocation𝐱7) indicatesthevectorofgoodsforthe𝑞th consumeroftype𝑖.
• Thefeasibilityconditionis∑ ∑ 𝐱7)â
)\]e7\] = 𝑟 ∑ 𝐞7e
7\]
AdvancedMicroeconomicTheory 171
LargeEconomiesandtheCore
• EqualtreatmentattheCore:If𝐱 isanallocationintheCoreofthe𝑟-foldreplicaeconomyℰâ,theneveryconsumeroftype𝑖 musthavethesamebundle,i.e.,
𝐱7) = 𝐱7)�
foreverytwo“twins”𝑞 and𝑞3 oftype𝑖,𝑞 ≠ 𝑞3 ∈{1,2, … , 𝑟},andforeverytype𝑖 ∈ 𝐼.– Thatis,inthe𝑟-foldreplicaeconomy,notonlysimilar
typeofconsumersstartwiththesameendowmentvector𝐞7,buttheyalsoendupwiththesameallocationattheCore.
AdvancedMicroeconomicTheory 172
LargeEconomiesandtheCore
• Proof (bycontradiction):– Consideratwo-foldreplicaeconomyℰc
§ Theresultscanbegeneralizedto𝑟-foldreplicas).– Supposethatallocation𝐱 ≡ {𝐱]], 𝐱]c, 𝐱c], 𝐱cc} isatthecoreofℰc.
– Since𝐱 isinthecore,thenitmustbefeasible,i.e.,𝐱]] + 𝐱]c + 𝐱c] + 𝐱cc = 2𝐞] + 2𝐞c
– Assumethatallocation𝐱 doesnotassignthesameconsumptionvectorstothetwotwinsoftype-1,i.e.,𝐱]] ≠ 𝐱]c.
AdvancedMicroeconomicTheory 173
LargeEconomiesandtheCore
– Assumethattype-1consumerweaklyprefers𝐱]]to𝐱]c,i.e.,𝐱]] ≿] 𝐱]c.§ Thisistrueforbothtype-1twins,sincetheyhavethesamepreferences.
§ Asimilarresultemergesifweinsteadassume𝐱]c ≿] 𝐱]].
AdvancedMicroeconomicTheory 174
In this case since both bundles lie on the same indifference curve
In this case
LargeEconomiesandtheCore• Unequaltreatmentatthecorefortype-1consumers
AdvancedMicroeconomicTheory 175
LargeEconomiesandtheCore
– Considerthatfortype-2consumerswehave𝐱c] ≿c 𝐱cc.
– Hence,consumer12istheworstofftype-1consumerandconsumer22istheworstofftype2consumer.
– Letustakethesetwo"poorlytreated"consumersofeachtype,andcheckiftheycanformablockingcoalitiontoopposeallocation 𝐱.
– Theaveragebundlesfortype-1andtype-2consumersare
𝐱�]c = 𝐱¬¬b𝐱¬¶
cand𝐱�cc = 𝐱¶¬b𝐱¶¶
cAdvancedMicroeconomicTheory 176
In this case but we can find another bundle, , which satisfies
In this case but we can still find another bundle, , which satisfies
LargeEconomiesandtheCore
• Averagebundlesleadingtoablockingcoalition
AdvancedMicroeconomicTheory 177
LargeEconomiesandtheCore
– Desirability.Sincepreferencesarestrictlyconvex,theworstofftype-1consumerprefers 𝐱�]c ≿] 𝐱]c,§ Thatis,alinearcombinationbetween𝐱]] and𝐱]c ispreferredtotheoriginalbundle𝐱]c.
– Asimilarargumentappliestotheworstofftype-2consumer,i.e.,𝐱�cc ≿c 𝐱cc.
– Hence,(𝐱�]c, 𝐱�cc)makesbothconsumers12and22betteroffthanattheoriginalallocation(𝐱]c, 𝐱cc).
AdvancedMicroeconomicTheory 178
LargeEconomiesandtheCore
– Feasibility.Canconsumers12and22achieve(𝐱�]c, 𝐱�cc)?
– Sumtheamountofgoodsconsumers12and22needtoachieve 𝐱�]c, 𝐱�cc toobtain
𝐱�]c + 𝐱�cc = 𝐱¬¬b𝐱¬¶
c+ 𝐱¶¬b𝐱¶¶
c
= ]c𝐱]] + 𝐱]c + 𝐱c] + 𝐱cc
= ]c2𝐞] + 2𝐞c = 𝐞] + 𝐞c
– Hence,thepairofbundles 𝐱�]c, 𝐱�cc isfeasible.
AdvancedMicroeconomicTheory 179
LargeEconomiesandtheCore
– Insummary,pairofbundles 𝐱�]c, 𝐱�cc :§ makesconsumers12and22betteroffthantheoriginalallocation(𝐱]c, 𝐱cc)
§ isfeasible
– Thus,theseconsumerscanblock(𝐱]c, 𝐱cc).§ Theoriginalallocation 𝐱]c, 𝐱cc cannotbeattheCore.
– Therefore,ifanallocationisattheCoreofthereplicaeconomy,itmustgiveconsumersofthesametypetheexactsamebundle.
AdvancedMicroeconomicTheory 180
LargeEconomiesandtheCore
• If𝐱 isinthecoreofa𝑟-foldreplicaeconomyℰâ,i.e.,𝐱 ∈ 𝐶â,then(bytheequaltreatmentproperty)allocation𝐱mustbeoftheform
𝐱 = (𝐱], … , 𝐱]âKYaNZ
, 𝐱c, … , 𝐱câKYaNZ
, … , 𝐱e, … , 𝐱eâKYaNZ
)
– Allconsumersofthesametypemustreceivethesamebundle.
– Coreallocationsinℰâ are𝑟-foldcopiesofallocationsinℰ],𝐱 = (𝐱], 𝐱c, … , 𝐱e).
AdvancedMicroeconomicTheory 181
LargeEconomiesandtheCore
• Thecoreshrinksastheeconomyenlarges.Thesequenceofcoresets𝐶], 𝐶c, … , 𝐶â isdecreasing.
• Thatis,– thecoreoftheoriginal(un-replicated)economy,𝐶],isasupersetofthatinthe2-foldreplicaeconomy,𝐶c;
– thecoreinthe2-foldreplicaeconomy,𝐶c,isasupersetofthe3-foldreplicaeconomy,𝐶¼;
– etc.
AdvancedMicroeconomicTheory 182
=WEAs
LargeEconomiesandtheCore• TheCoreshrinksas𝑟 increases
AdvancedMicroeconomicTheory 183
LargeEconomiesandtheCore
• Proof:– Sinceweseektoshowthat𝐶] ⊇ 𝐶c ⊇ ⋯ ⊇ 𝐶â�] ⊇𝐶â,itsufficestoshowthat,forany𝑟 > 1,𝐶â�] ⊇ 𝐶â.
– Supposethatallocation𝐱 = (𝐱], 𝐱c, … , 𝐱e) ∈ 𝐶â.– Thereisnoblockingcoalitionto𝐱 inthe𝑟-foldreplicaeconomyℰâ.
– Wethenneedtoshowthat𝐱 cannotbeblockedbyanycoalitioninthe(𝑟 − 1)-foldreplicaeconomyeither.§ Ifwecouldfindablockingcoalitionto𝐱 inℰâ�], thenwecouldalsofindablockingcoalitioninℰâ.
§ Allmembersinℰâ�] arealsopresentinthelargereconomyℰâ andtheirendowmentshavenotchanged.AdvancedMicroeconomicTheory 184
LargeEconomiesandtheCore
– Nowweneedtoshowthat,as𝑟 increases,thecoreshrinks.
–Wewilldothisbedemonstratingthatallocationsatthefrontierof𝐶] donotbelongtothecoreofthe2-foldreplicaeconomy,𝐶c.
AdvancedMicroeconomicTheory 185
Consumer 1,
Consumer 2
Bundles in this line are core allocations
Not a WEA
In addition, yields the lowest utility for consumer 1, among all core allocations.
LargeEconomiesandtheCore• Un-replicatedeconomyℰ]
AdvancedMicroeconomicTheory 186
LargeEconomiesandtheCore
– Thelinebetween𝐱- and𝐞 includescoreallocations.§ Allpointsinthelinearepartofthecore.§ However,notallpointsinthislineareWEAs.§ Forinstance:𝐱- isnotaWEAsincethepricelinethrough𝐱- and𝐞 isnottangenttotheconsumer’sindifferencecurveat𝐱-.
– IftheCoreshrinksastheeconomyenlarges,weshouldbeabletoshowthatallocation𝐱- ∉ 𝐶c.
– Letusbuildablockingcoalitionagainst𝐱-.
AdvancedMicroeconomicTheory 187
LargeEconomiesandtheCore
– Desirability.Considerthemidpointallocation𝐱�andthecoalition𝑆 = {11,12,21}.Suchamidpointinthelineconnecting𝐱- and𝐞 isstrictlypreferredbybothtypesofconsumer1.
– Ifthemidpointallocation𝐱� isofferedtobothtypesofconsumer1(11and12),andtooneoftheconsumer2types,theywillallacceptit:
𝐱�]] ≡ ]c(𝐞] + 𝐱-]]) ≻] 𝐱-]]
𝐱�]c ≡ ]c(𝐞] + 𝐱-]c) ≻] 𝐱-]c
𝐱-c] ∼c 𝐱-c]AdvancedMicroeconomicTheory 188
LargeEconomiesandtheCore
– Feasibility.Since𝐱�]] = 𝐱�]c,thenthesumofthesuggestedallocationsyields
𝐱�]] + 𝐱�]c + 𝐱-c] = 2 ]c𝐞] + 𝐱-]] + 𝐱-]c
= 𝐞] + 𝐱-]] + 𝐱-]c
– Recallthat𝐱- ispartoftheun-replicatedeconomyℰ].§ Hence,itmustbefeasible,i.e.,𝐱-] + 𝐱-c = 𝐞] + 𝐞c.§ Therefore,𝐱-]] + 𝐱-]c = 𝐞] + 𝐞c.
AdvancedMicroeconomicTheory 189
LargeEconomiesandtheCore
–Wecanthusre-writetheaboveequalityas𝐱�]] + 𝐱�]c + 𝐱-c] = 𝐞] + 𝐱-]] + 𝐱-]c
𝐞¬b𝐞¶= 𝐞] + 𝐞] + 𝐞c = 2𝐞] + 𝐞c
whichconfirmsthefeasibility.– Hence,thefrontierallocation𝐱- inthecoreoftheun-replicatedeconomydoesnotbelongtothecoreofthetwo-foldeconomy,𝐱- ∉ 𝐶c.§ Thereisablockingcoalition𝑆 = {11,12,21} andanalternativeallocation𝐱� = {𝐱�]], 𝐱�]c, 𝐱-c]} thattheywouldpreferto𝐱- andthatisfeasibleforthecoalitionmembers.
AdvancedMicroeconomicTheory 190
LargeEconomiesandtheCore
• WEAinreplicatedeconomies:– ConsideraWEAintheun-replicatedeconomyℰ],𝐱 = (𝐱], 𝐱c, … , 𝐱e).
– Anallocation𝐱 isaWEAforthe𝑟-foldreplicaeconomyℰâ iffitisoftheform𝐱 = (𝐱], … , 𝐱]
âKYaNZ, 𝐱c, … , 𝐱c
âKYaNZ, … , 𝐱e, … , 𝐱e
âKYaNZ)
– If𝐱 isaWEAforℰâ ,thenitalsobelongstothecoreofthateconomy (bythe"equaltreatmentatthecore"property).
AdvancedMicroeconomicTheory 191
LargeEconomiesandtheCore
• AlimittheoremontheCore:Ifanallocation𝐱belongstothecoreofall𝑟-foldreplicaeconomiesthensuchallocationmustbeaWEAoftheun-replicatedeconomyℰ].
• Proof (bycontradiction):– Considerthatanallocation𝐱- belongstothecoreofthe𝑟-foldreplicaeconomy𝐶â butisnot aWEA.
– Acoreallocationfortheun-replicatedeconomyℰ],𝐱- ∈ 𝐶] satisfyies𝐱- ∈ 𝐶â since𝐶] ⊃ 𝐶â.
– Allocation𝐱- mustthenbewithinthelens-shapedareaandonthecontractcurve.
AdvancedMicroeconomicTheory 192
LargeEconomiesandtheCore• Acoreallocation𝐱- thatisnotWEA
AdvancedMicroeconomicTheory 193
Consumer 1,
Consumer 2
LargeEconomiesandtheCore
– Considernowthelineconnecting𝐱- and𝐞.– Since𝐱- isnotaWEA,thebudgetlinecannotbetangenttobothconsumers’indifferencecurves:
�¬�¶> 𝑀𝑅𝑆 or�¬
�¶< 𝑀𝑅𝑆
– Canallocation𝐱- beattheCore𝐶â andyetnotbeaWEA?
– Letusshowthatif𝐱- isnotaWEAitcannot bepartoftheCore𝐶â either.§ Todemonstratethat𝐱- ∉ 𝐶â,letusfindablockingcoalition
AdvancedMicroeconomicTheory 194
LargeEconomiesandtheCore
– Bytheconvexityofpreferences,wecanfindasetofbundles(suchthosebetween𝐴 and𝐱-)thatconsumer1prefersto𝐱-:
𝐱û ≡1𝑟 𝐞
] +𝑟 − 1𝑟 𝐱-]
forsome𝑟 > 1,where]â+ â�]
â= 1.
– Consideracoalition𝑆 withall𝑟 type-1consumersand𝑟 − 1 type-2consumers.
– Letusnowshowthatallocation𝐱û satisfiesthepropertiesofacceptanceandfeasibilityfortheblockingcoalition𝑆.
AdvancedMicroeconomicTheory 195
LargeEconomiesandtheCore
– Acceptance.Ifwegiveeverytype-1consumerthebundle𝐱û],𝐱û] ≻] 𝐱-].Similarly,ifwegiveeverytype-2consumerinthecoalitionthebundle𝐱ûc =𝐱-c,then𝐱ûc ∼c 𝐱-c.
– Feasibility. Summingovertheconsumersincoalition𝑆,theiraggregateallocationis
𝑟𝐱û] + 𝑟 − 1 𝐱ûc = 𝑟 ]â𝐞] + â�]
â𝐱-] + (𝑟 − 1)𝐱-c
= 𝐞] + (𝑟 − 1)(𝐱-] + 𝐱-c)– Since𝐱- = (𝐱-], 𝐱-c) isinthecoreoftheun-replicatedeconomyℰ],thenitmustbefeasible𝐱-] + 𝐱-c = 𝐞] + 𝐞c.
AdvancedMicroeconomicTheory 196
LargeEconomiesandtheCore
– Combiningtheabovetworesults,wefindthat𝑟𝐱û] + 𝑟 − 1 𝐱ûc = 𝐞] + (𝑟 − 1)(𝐞] + 𝐞c)
𝐱-¬b𝐱-¶= 𝑟𝐞] + 𝑟 𝐞] + 𝐞c − (𝐞] + 𝐞c)
= 𝑟𝐞] + (𝑟 − 1)𝐞c
whichconfirmsfeasibility.
– Hence,𝑟 type-1consumersand𝑟 − 1 type-2consumerscangettogetherincoalitionandblockallocation𝐱-.
AdvancedMicroeconomicTheory 197
LargeEconomiesandtheCore
– Thusif𝐱- isnotaWEA,then,𝐱- cannotbeintheCoreofthe𝑟-foldreplicaeconomyℰâ.
– Asaconsequence,if𝐱- ∈ 𝐶âforall𝑟 > 0,then𝐱-mustbeaWEA.
AdvancedMicroeconomicTheory 198
AppendixB:Marshall–HicksFourLawsof
DerivedDemand
AdvancedMicroeconomicTheory 199
Marshall–HicksFourLaws
• Consideraproductionfunction𝑞 = 𝑓(𝐾, 𝐿),withpositivemarginalproducts,𝑓f, 𝑓Þ > 0.
• Assumethatthesupplyofeachinput(𝑤 𝐿 , 𝑟(𝐾)) ispositivelysloped,𝑤3 𝐿 > 0 and𝑟3 𝐾 > 0.
• Demandforoutputisgivenby𝑞 = 𝑔(𝑝),whichsatisfies𝑔3 𝑝 < 0.
• Thetotalcostis𝑤 𝐿 𝐿 + 𝑟 𝐾 𝐾.• Assumethatthecapitalmarketisperfectlycompetitive,butthelaborandoutputmarketsarenot necessarilycompetitive.
AdvancedMicroeconomicTheory 200
Marshall–HicksFourLaws• Define:
– 𝜀),� = (𝜕𝑞/𝜕𝑝)(𝑝/𝑞) asthepriceelasticityofoutput– 𝑠Þ,â = (𝜕𝐾/𝜕𝑟)(𝑟/𝐾) astheelasticityofcapitalsupplytoa
changeinitsprice– 𝑠f,â = (𝜕𝐿/𝜕𝑟)(𝑟/𝐿) astheelasticityoflaborsupplytoa
changeinthepriceofcapital– 𝑠f,ã = (𝜕𝐿/𝜕𝑤)(𝑤/𝐿) astheelasticityoflaborsupplytoa
changeinitsprice– 𝜎 astheelasticityofsubstitutionbetweeninputs
• Weusesuperscript𝑖 torefertotheelasticitythatanindividualfirmfaces(𝜀),�7 ).
• Theindustryelasticitiesdonotincludesuperscripts(𝜀),�).
AdvancedMicroeconomicTheory 201
Marshall–HicksFourLaws
• Let𝜃f ≡ 𝑤𝐿/𝑝𝑞 and𝜃Þ ≡ 𝑟𝐾/𝑝𝑞 bethecostoflaborandcapital,respectively,relativetototalsales.
• Thisimpliesthat𝜃f = 1 − 𝜃Þ .• Forcompactness,letusdefine
𝐴 ≡ 1 − (1/𝜀),�7 )𝐵 ≡ 1 + (1/𝑠f,ã7 )
AdvancedMicroeconomicTheory 202
Marshall–HicksFourLaws
• Marshall,Hicks,andAllenanalyzehowtheinputdemandofaperfectlycompetitiveinput,suchascapital,isaffectedbyamarginalchangeinthepriceofcapital:
𝑠Þ,â = −𝜃Þ𝜀),�𝐴 + 𝜎𝜀),�/𝑠f,ã 𝐴c + 𝜃f𝐴𝐵𝜎
(𝜃Þ + 𝜃f𝐵)c+𝜃Þ 𝜎/𝑠f,ã 𝐴 + 𝜃f 𝜎/𝑠f,ã 𝐴𝐵
AdvancedMicroeconomicTheory 203
Marshall–HicksFourLaws• Marshall–Hicks’sfourlawsofinputdemand(“derived
demand”)statethataninputdemandbecomesmoreelastic,whereby𝑠Þ,â decreases,in1. theelasticityofsubstitutionbetweeninputs𝜎2. theprice-elasticityofoutputdemand𝜀),�3. thecostoftheinputrelativetototalsales𝜃Þ4. theelasticityoftheotherinput’ssupplytoachangeinits
price𝑠f,ã• Weanalyzethesefourcomparativestaticsundertwo
marketstructures:1. TheMarshall’spresentation:𝜀),�7 = 𝑠f,ã7 = ∞,𝜎 = 02. TheHick’spresentation:𝜀),�7 = 𝑠f,ã7 = ∞ (noassumptionson
𝜎)
AdvancedMicroeconomicTheory 204
Marshall’sPresentation
• Assumptions:– Outputandinputsmarketsareperfectlycompetitive,𝜀),�7 = 𝑠f,ã7 = ∞,foreveryfirm𝑖
– Inputscannotbesubstitutedintheproductionprocess,𝜎 = 0
• Theexpressionfor𝑠Þ,â canbesimplifiedto
𝑠Þ,â = −𝜃Þ𝜀),�𝑠f,ã𝑠f,ã + 𝜃f𝜀),�
AdvancedMicroeconomicTheory 205
Marshall’sPresentation• Thederivativestestingthelawsare:
𝜕𝑠Þ,â𝜕𝜀),�
= −𝜃Þ(𝑠f,ã)c
(𝑠f,ã + 𝜃f𝜀),�)c𝜕𝑠Þ,â𝜕𝜃Þ
= −𝑠f,ã q 𝜀),�(𝑠f,ã + 𝜀),�)(𝑠f,ã + 𝜃f𝜀),�)c
𝜕𝑠Þ,â𝜕𝑠f,ã
= −𝜃Þ𝜃f(𝜀),�)c
(𝑠f,ã + 𝜃f𝜀),�)c• Iflaborisa“normal”input,𝑠f,ã > 0,thethreederivatives
areallnegative(thethreelawshold).• Iflaborisinferior,𝑠f,ã < 0,𝑠Þ,â isstilldecreasingin𝜀),�7
andin𝑠f,ã7 ,butnotnecessarilyin𝜃Þ.
AdvancedMicroeconomicTheory 206
Hick’sPresentation
• Assumptions:– Outputandinputsmarketsareperfectlycompetitive,𝜀),�7 = 𝑠f,ã7 = ∞,foreveryfirm𝑖
– Noconditionimposedonthesubstitutionofinputs(𝜎)• Theexpressionfor𝑠Þ,â canbesimplifiedto
𝑠Þ,â = −𝜃Þ𝜀),�𝑠f,ã − 𝜎𝜀),� − 𝜃f𝜎𝑠f,ã
𝑠f,ã + 𝜃Þ𝜎 + 𝜃f𝜀),�
AdvancedMicroeconomicTheory 207
Hick’sPresentation• Thederivativestestingthelawsare
𝜕𝑠Þ,â𝜕𝜀),�
= −𝜃Þ(𝑠f,ã + 𝜎)c
(𝑠f,ã + 𝜃Þ𝜎 + 𝜃f𝜀),�)c𝜕𝑠Þ,â𝜕𝜃Þ
= −(𝜀),� + 𝑠f,ã) +(𝑠f,ã + 𝜎)(𝜀),� − 𝜎)
(𝑠f,ã + 𝜃Þ𝜎 + 𝜃f𝜀),�)c𝜕𝑠Þ,â𝜕𝑠f,ã
= −𝜃Þ𝜃f(𝜀),� − 𝜎)c
(𝑠f,ã + 𝜃Þ𝜎 + 𝜃f𝜀),�)c𝜕𝑠Þ,â𝜕𝜎 = −
𝜃f(𝜀),� + 𝑠f,ã)c
(𝑠f,ã + 𝜃Þ𝜎 + 𝜃f𝜀),�)c• Hence,𝑠Þ,â decreasesin𝜀),�,𝑠f,ã,and𝜎 (thethreelawshold).• 𝑠Þ,â alsodecreasesin𝜃Þ iftheinputis“normal”,𝑠f,ã > 0,and
inputsarenotextremelyeasytosubstitute,𝜀),� > 𝜎.
AdvancedMicroeconomicTheory 208