Advanced Microeconomic Theory · Outline • Partial Equilibrium Analysis • General Equilibrium Analysis • Comparative Statics • Welfare Analysis Advanced Microeconomic Theory

AdvancedMicroeconomicTheory

Chapter6:PartialandGeneralEquilibrium

Outline

• PartialEquilibrium Analysis• GeneralEquilibrium Analysis• ComparativeStatics• WelfareAnalysis

AdvancedMicroeconomicTheory 2

PartialEquilibriumAnalysis

• Inacompetitiveequilibrium(CE),allagentsmustselectanoptimalallocationgiventheirresources:– Firmschooseprofit-maximizingproductionplansgiventheirtechnology;

– Consumerschooseutility-maximizingbundlesgiventheirbudgetconstraint.

• Acompetitiveequilibriumallocationwillemergeatapricethatmakesconsumers’purchasingplanstocoincidewiththefirms’productiondecision.



• Firm:– Giventhepricevector𝑝∗,firm𝑗’sequilibriumoutputlevel𝑞%∗ mustsolve

max)*+,

𝑝∗𝑞% − 𝑐%(𝑞%)

whichyieldsthenecessaryandsufficientcondition

𝑝∗ ≤ 𝑐%3(𝑞%∗),withequalityif𝑞%∗ > 0– Thatis,everyfirm𝑗 producesuntilthepointinwhichitsmarginalcost,𝑐%3(𝑞%∗),coincideswiththecurrentmarketprice.



• 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.



– 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%∗)[%\]



– Substitutingthebudgetconstraintintotheobjectivefunction,

maxEA∈ℝD

𝜙7 𝑥7 − 𝑝∗𝑥7 +

𝑤@A + ∑ 𝜃7% 𝑝∗𝑞%∗ − 𝑐%(𝑞%∗)[%\]

– FOCswrt 𝑥7 yields𝜙73 𝑥7∗ ≤ 𝑝∗,withequalityif𝑥7∗ > 0

– Thatis,consumerincreasestheamounthebuysofgood𝑥 untilthepointinwhichthemarginalutilityheobtainsexactlycoincideswiththemarketpricehehastopayforit.



– 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.



• Theindividualdemandcurve,where𝜙73 𝑥7∗ ≤ 𝑝∗


PartialEquilibriumAnalysis• Horizontallysummingindividualdemandcurvesyieldstheaggregatedemandcurve.



• Theindividualsupplycurve,where𝑝∗ ≤ 𝑐%3(𝑞%∗)




• Horizontallysummingindividualsupplycurvesyieldstheaggregatesupplycurve.

PartialEquilibriumAnalysis• Superimposingaggregatedemandandaggregatesupplycurves,weobtaintheCEallocationofgood𝑥.

• ToguaranteethataCEexists,theequilibriumprice𝑝∗ mustsatisfy

max7𝜙73 0 ≥ 𝑝∗

≥ min%𝑐%3 0



• Also,since𝜙73 𝑥7 isdownwardslopingin𝑥7,and𝑐%3(𝑞7) isupwardslopingin𝑞7,thenaggregatedemandandsupplycrossatauniquepoint.– Hence,theCEallocationisunique.



• Ifwehavemax7𝜙73 0 < min

%𝑐%3 0 ,

thenthereisnopositiveproductionorconsumptionofgood𝑥.



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

– Short-runsupplycurveoftypeAfirmis𝑠k 𝑝 = 2𝑝

– Short-runsupplycurveoftypeBfirmis𝑠m 𝑝 = 10𝑝

– TheWalrasianmarketdemandcurveis𝑥 𝑝 = 5000 − 500𝑝



• 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


ComparativeStatics


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%).


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.


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


ComparativeStatics

– Similarly,if��(E̅,��)��

≠ 0 whenevaluatedat(�̅�, 𝑦�),then

𝑑𝑥(𝑦�)𝑑𝑦 = −

𝜕𝑢(�̅�, 𝑦�)𝜕𝑦

𝜕𝑢(�̅�, 𝑦�)𝜕𝑥

forall(�̅�, 𝑦�).


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.


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


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

= −��= −


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.


ComparativeStatics

• Notax:– CEoccursat𝑝∗ 0and𝑥∗ 0

• Tax:– 𝑥∗ decreasesfrom𝑥∗ 0 to𝑥∗ 𝑡

– Consumersnowpay𝑝∗ 𝑡 + 𝑡

– Producersnowreceive𝑝∗ 𝑡 forthe𝑥∗ 𝑡 unitstheysell.


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.


ComparativeStatics

– Thepricereceivedbyproducersalmostdoesnotfall.

– But,thepricepaidbyconsumersincreasesbyexactlytheamountofthetax.


• 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.


ComparativeStatics

• Inelasticsupplycurve


ComparativeStatics

• Example6.3:– Consideracompetitivemarketinwhichthegovernmentwillbeimposinganadvaloremtaxof𝑡.

– Aggregatedemandcurveis𝑥 𝑝 = 𝐴𝑝�,where𝐴 > 0 and𝜀 < 0,andaggregatesupplycurveis𝑞 𝑝 = 𝑎𝑝�,where𝑎 > 0 and𝛾 > 0.

– Letusevaluatehowtheequilibriumpriceisaffectedbyamarginalincreaseinthetax.


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.


ComparativeStatics

• Example 6.3(continued):–When𝜀 → −∞ (i.e.,demandisperfectlyelastic),thepricepaidbyconsumersisunchanged,andthepricereceivedbyproducersdecreasesbytheamountofthetax.

–When𝛾 → +∞(i.e.,supplyisperfectlyelastic),thepricereceivedbyproducersisunchangedandthepricepaidbyconsumersincreasesbytheamountofthetax.


WelfareAnalysis


WelfareAnalysis• Letusnowmeasurethechangesintheaggregatesocialwelfareduetoachangeinthecompetitiveequilibriumallocation.

• Considertheaggregatesurplus𝑆 = ∑ 𝜙7(𝑥7)e

7\] − ∑ 𝑐%(𝑞%)[%\]

• Takeadifferentialchangeinthequantityofgood𝑘 thatindividualsconsumeandthatfirmsproducesuchthat∑ 𝑑𝑥7e

7\] = ∑ 𝑑𝑞%[%\] .

• Thechangeintheaggregatesurplusis𝑑𝑆 = ∑ 𝜙73(𝑥7)𝑑𝑥7e

7\] − ∑ 𝑐%3(𝑞%)𝑑𝑞%[%\]


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%

[%\]


WelfareAnalysis

• Butsince∑ 𝑑𝑥7e7\] = ∑ 𝑑𝑞%

[%\] = 𝑑𝑥,and𝑥 = 𝑞

bymarketfeasibility,then𝑑𝑆 = 𝑃 𝑥 − 𝐶3 𝑞 𝑑𝑥

• Intuition:– Thechangeinsurplusofamarginalincreaseinconsumption(andproduction)reflectsthedifferencebetweentheconsumers’additionalutilityandfirms’additionalcostofproduction.


WelfareAnalysis• Differentialchangeinsurplus


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.


WelfareAnalysis• Surplusataggregateconsumption𝑥


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∗ .


WelfareAnalysis

• Therefore,theCEallocationmaximizesaggregatesurplus.

• ThisistheFirstWelfareTheorem:– EveryCEisParetooptimal(PO).


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).


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.


GeneralEquilibrium


GeneralEquilibrium

• Sofar,weexploredequilibriumconditionsinasinglemarketwithasingletypeofconsumer.

• Nowweexaminesettingswithmarketsfordifferentgoodsandmultipleconsumers.


GeneralEquilibrium:NoProduction

• Consideraneconomywithtwogoodsandtwoconsumers,𝑖 = {1,2}.

• Eachconsumerisinitiallyendowedwith𝐞7 ≡(𝑒]7 , 𝑒c7 ) unitsofgood1and2.

• Anyotherallocationsaredenotedby𝐱7 ≡(𝑥]7 , 𝑥c7 ).



• Edgeworthbox:



• Theshadedarearepresentsthesetofbundles(𝑥]7 , 𝑥c7 ) forconsumer𝑖 satisfying𝑢](𝑥]], 𝑥c]) ≥ 𝑢](𝑒]], 𝑒c])𝑢c(𝑥]c, 𝑥cc) ≥ 𝑢c(𝑒]c, 𝑒cc)

• Bundle𝐴 cannotbeabarterequilibrium:– Consumer1doesnotexchange𝐞 for𝐴.


• 𝐼𝐶7 istheindifferencecurveofconsumer𝑖,whichpassesthroughhisendowmentpoint𝐞7.


• Notallpointsinthelens-shapedareaisabarterequilibrium!


• Bundle𝐵 liesinsidethelens-shapedarea– Thus,ityieldsahigherutility

levelthantheinitialendowment𝐞 forbothconsumers.

• Bundle𝐷,however,makesbothconsumersbetteroffthanbundle𝐵.– Itlieson“contractcurve,”in

whichtheindifferencecurvesaretangenttooneanother.

– Itisanequilibrium,sinceParetoimprovementsarenolongerpossible


• Feasibleallocation:– Anallocation𝐱 ≡ (𝐱], 𝐱c, … , 𝐱e) isfeasible ifitsatisfies

∑ 𝐱7e7\] ≤ ∑ 𝐞7e

7\]

– Thatis,theaggregateamountofgoodsinallocation𝐱 doesnotexceedtheaggregateinitialendowment𝐞 ≡ ∑ 𝐞7e

7\] .



• Pareto-efficientallocations:– Afeasibleallocation𝐱 isParetoefficientifthereisnootherfeasibleallocation𝐲 whichisweaklypreferredbyallconsumers,i.e.,𝐲7 ≿ 𝐱7 forall𝑖 ∈𝐼,andatleaststrictlypreferredbyoneconsumer,𝐲7 ≻ 𝐱7.

– Thatis,allocation𝐱 isParetoefficientifthereisnootherfeasibleallocation𝐲makingallindividualsatleastaswelloffasunder𝐱 andmakingoneindividualstrictlybetteroff.



• 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.



– 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


– 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.



• 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.



• 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



• 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].



• Example (continued):– ThelinerepresentingthesetofParetoefficientallocations



• 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.



• Equilibriuminabartereconomy:Afeasibleallocation𝐱 isanequilibriumintheexchangeeconomywithinitialendowment𝐞 if𝐱 isnotblocked byanycoalitionofconsumers.

• Core: Thecoreofanexchangeeconomywithendowment𝐞,denoted𝐶(𝐞),isthesetofallunblocked feasibleallocations𝐱.– Suchallocations:

a) mutuallybeneficialforallindividuals(i.e.,theylieinthelens-shapedarea)

b) donotallowforfurtherParetoimprovements (i.e.,theylieinthecontractcurve)




GeneralEquilibrium:CompetitiveMarkets

• Bartereconomydidnotrequirepricesforanequilibriumtoarise.

• Nowweexploretheequilibriumineconomieswhereweallowpricestoemerge.

• Orderofanalysis:– consumers’preferences– theexcessdemandfunction– theequilibriumallocationsincompetitivemarkets(i.e.,Walrasianequilibriumallocations)



• 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.



– Intuitively,individual𝑖’sincomecomesfromsellinghisendowment𝐞7 atmarketprices𝐩,producing 𝐩 q 𝐞7 = 𝑝]𝑒]7 + ⋯+ 𝑝Å𝑒Å7 dollarstobeusedinthepurchaseofallocation𝐱7.



• 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.



–When𝑧Å 𝐩 > 0,theaggregatedemandforgood𝑘 exceedsitsaggregateendowment.§ Excessdemandofgood𝑘

–When𝑧Å 𝐩 < 0,theaggregatedemandforgood𝑘 fallsshortofitsaggregateendowment.§ Excesssupplyofgood 𝑘



• Differenceindemandandsupply,andexcessdemand


Σi=1

Ieki

Σi=1

Ixk(p,pei)i

zk(p)

pk

xk

pk

xk0

pk*

Equilibriumprice,wherezk(p)=0


• 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



– 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\]

É² 𝐩

ÁÅ\] = 0

– Hence,∑ 𝑝Å q 𝑧Å 𝐩ÁÅ\] = 𝐩 q 𝐳 𝐩 = 0



– 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.



2) Continuity:𝐳 𝐩 iscontinuousat𝐩.– ThisfollowsfromindividualWalrasiandemandsbeingcontinuousinprices.

3) Homegeneity:𝐳 𝜆𝐩 = 𝐳 𝐩 forall𝜆 > 0.– ThisfollowsfromWalrasiandemandsbeinghomogeneousofdegreezeroinprices.

• WenowuseexcessdemandtodefineaWalrasianequilibriumallocation.



• 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\]



• ExistenceofaWalrasianequilibrium:– AWalrasianequilibriumpricevector𝐩∗ ∈ ℝbbÁ ,i.e., 𝐳(𝐩∗) = 0,existsiftheexcessdemandfunction 𝐳(𝐩) satisfiescontinuity andWalras’law(Varian,1992).



• Uniqueness ofequilibriumprices:


satisfied violated


• Example6.6 (Walrasianequilibriumallocation):– Inexample6.1,wedeterminedthat

𝑀𝑅𝑆k = 𝑀𝑅𝑆m = �¬�¶

E¶¹

E¬¹= E¶º

E¬º= �¬

�¶

– LetusdeterminetheWalrasiandemandsofeachgoodforeachconsumer.

– Rearrangingthesecondequationabove,weget𝑝]𝑥]k = 𝑝c𝑥ck



• 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


• 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


• 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)


x1A

x2B

x1Bx2A

0A

0B

100

100

200 200

ICAICB

x2=2x1

InitialEndowment

AA

WEA

CoreAllocations


• Example6.6 (continued):– Initialallocation,– Coreallocation,and–Walrasianequilibriumallocations(WEA).



• 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


• 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


• 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)



• Remarks:1) TheCore𝐶(𝐞) containstheWEA(orWEAs)

§ Thatis,theCoreisnonempty.

2) SinceallcoreallocationsareParetoefficient(i.e.,wecannotincreasethewelfareofoneconsumerwithoutdecreasingthatofotherconsumers),thenallWEAs(whicharepartoftheCore)arealsoParetoefficient.



• FirstWelfareTheorem:EveryWEAisParetoefficient.

– TheWEAliesonthecore(thesegmentofthecontractcurvewithinthelens-shapedarea),

– ThecoreisasubsetofallParetoefficientallocations.


x1A

x2B

x1Bx2A

0A

0B

ICA

ICB

CoreAllocations

e

WEA

Contractcurve(Paretoefficientallocations)

ConsumerA

ConsumerB


• FirstWelfareTheorem



• SecondWelfareTheorem:– Supposethat𝐱� isaPareto-efficientallocation(i.e.,itliesonthecontractcurve),andthatendowmentsareredistributedsothatthenewendowmentvector𝐞∗7 liesonthebudgetline,thussatisfying

𝐩∗ q 𝐞∗7 = 𝐩∗ q 𝐱�7 foreveryconsumer𝑖– Then,thePareto-efficientallocation𝐱� isaWEAgiventhenewendowmentvector𝐞∗.


Consumer 1,

Consumer 2

Budget line.Same slope in both cases, i.e.,

Contract curve

C

C


• Secondwelfaretheorem



• Example6.7(WEAandSecondwelfaretheorem):

– Consideraneconomywithutilityfunctions𝑢k =𝑥]k𝑥ck forconsumer𝐴 and𝑢m = {𝑥]m, 𝑥cm} forconsumer𝐵.

– Theinitialendowmentsare𝐞k = (3,1) and𝐞m =(1,3).

– Good2isthenumeraire, i.e.,𝑝c = 1.



• 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


• Example6.7(continued):1) ParetoEfficientAllocations:– Substitutingtheaboveexpressioninthesecondconstraintyields

𝑥ck + (4 − 𝑥]k)E¶º

= 4 ⟺ 𝑥ck = 𝑥]k

– Thisdefinesthecontractcurve,i.e.,thesetofParetoefficientallocations.



• 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.



• 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



• 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)


GeneralEquilibrium:Production

• Letusnowextendourpreviousresultstosettingwherefirmsarealsoactive.

• Assume𝐽 firmsintheeconomy,eachwithproductionset 𝑌%,whichsatisfies:– Inactionispossible,i.e.,𝟎 ∈ 𝑌%.– 𝑌% isclosedandbounded,sopointsontheproductionfrontierarepartoftheproductionsetandthusfeasible.

– 𝑌% is strictlyconvex,solinearcombinationsoftwoproductionplansalsobelongtotheproductionset.


Production set of firm j


• Productionset 𝑌% forarepresentativefirm



• 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.


Isoprofit where profits are maximal

Isoprofit lines for profit levelwhere

profit maximizing production plan


• 𝑦%(𝑝) existsandisunique



• 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)



• 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



– 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%).



• Equilibriumwithproduction:–Westartdefiningexcessdemandfunctionsandusesuchadefinitiontoidentifythesetofequilibriumallocations.

– Excessdemand:Theexcessdemandfunctionforgood𝑘 is𝑧Å 𝐩 ≡ ∑ 𝑥Å7 (𝐩,𝑚7 𝐩 )e

7\] − ∑ 𝑒Å7e7\] − ∑ 𝑦Å

%(𝐩)[%\]

where∑ 𝑦Å%(𝐩)[

%\] isanewtermrelativetotheanalysisofgeneralequilibriumwithoutproduction.



– 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


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.



• Example6.8(Equilibriumwithproduction):– Consideratwo-consumer,two-goodeconomywhereconsumer𝑖 = {𝐴, 𝐵} hasutilityfunction𝑢7 = 𝑥]7𝑥c7 .

– Therearetwofirmsinthiseconomy,andeachofthemusecapital(𝐾)andlabor(𝐿)toproduceoneoftheconsumptiongoodseach.

– Firm1producesgood1accordingto𝑦] = 𝐾],.Ü½𝐿],.c½.– Firm2producesgood2accordingto𝑦c = 𝐾c,.c½𝐿c,.Ü½.– Consumer𝐴 isendowedwith(𝐾k, 𝐿k) = (1,1),whileconsumer𝐵 isendowedwith(𝐾m, 𝐿m) = (2,1).

– LetusfindaWEAinthiseconomywithproduction.AdvancedMicroeconomicTheory 107


• 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


• Example6.8(continued):– Takingtheaboveequationforconsumers𝐴 and𝐵,andaddingthemtogetheryields

𝑝](𝑥]k + 𝑥]m) = 𝑝c(𝑥ck + 𝑥cm)

where𝑥]k + 𝑥]m istheleftsideofthefeasibilitycondition𝑥]k + 𝑥]m = 𝑦] = 𝐾],.Ü½𝐿],.c½.

– Substitutingbothfeasibilityconditionsintotheaboveexpression,andre-arranging,yields

�¬�¶= Þ¶ß.¶àf¶ß.áà

Þ¬ß.áàf¬ß.¶à



• Example6.8(continued):– PMPs:Firm1’smaximizationproblemis

maxÞ¬,f¬

𝑝]𝐾],.Ü½𝐿],.c½ − 𝑟𝐾] − 𝑤𝐿]– FOCs:

𝑟 = 0.75𝑝]𝐾]�,.c½𝐿],.c½𝑤 = 0.25𝑝]𝐾],.Ü½𝐿]�,.Ü½

– Combiningtheseconditionsgivesthetangencyconditionforprofitmaximization

âã= 𝑀𝑅𝑇𝑆f,Þ] ⟹ â

ã= 3 f¬

Þ¬



• Example6.8(continued):– Likewise,firm2’sPMPgivesthefollowingFOCs:

𝑟 = 0.25𝑝c𝐾c�,.Ü½𝐿c,.Ü½𝑤 = 0.75𝑝c𝐾c,.c½𝐿c�,.c½

– Combiningtheseconditionsgivesthetangencyconditionforprofitmaximization

âã= 𝑀𝑅𝑇𝑆f,Þc ⟹ â

ã= ]

¼f¶Þ¶



• 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�,.Ü½𝐿c,.Ü½

⟹ �¬�¶= ]

¼Þ¬f¬

,.c½ Þ¶f¶

�,.Ü½



• Example6.8(continued):– Settingbothfirm’spriceoflabor,𝑤,equaltoeachotheryields0.25𝑝]𝐾],.Ü½𝐿]�,.Ü½ = 0.75𝑝c𝐾c,.c½𝐿c�,.c½

⟹ �¬�¶= 3 Þ¬

f¬

�,.Ü½ Þ¶f¶

,.c½

– Settingpriceratiofromconsumers’UMPequaltothefirstpriceratiofromfirms’PMPyields

Þ¶ß.¶àf¶ß.áà

Þ¬ß.áàf¬ß.¶à= ]

¼Þ¬f¬

,.c½ Þ¶f¶

�,.Ü½⟹ 𝐾] = 3𝐾c



• 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



• Example6.8(continued):– Settingpriceratiofromconsumers’UMPequaltothesecondpriceratiofromfirms’PMPyields

Þ¶ß.¶àf¶ß.áà

Þ¬ß.áàf¬ß.¶à= 3 Þ¬

f¬

�,.Ü½ Þ¶f¶

,.c½⟹ 𝐿] =

]¼𝐿c

– Bythefeasibilityconditions,weknowthat𝐿] +𝐿c = 𝐿k + 𝐿m = 2 or𝐿c = 2 − 𝐿].



• Example6.8(continued):– Substitutingtheaboveexpressioninto 𝐿] =

]¼𝐿c,

wefindtheprofit-maximizingdemandsforlaborusebyfirms1and2:

𝐿] =]¼2 − 𝐿] ⟹𝐿]∗ =

]c

𝐿c∗ = 3𝐿]∗ =¼c



• Example6.8(continued):– Substitutingthecapitalandlabordemandsforfirm1and2intothepriceratiofromconsumers’UMPyields

�¬�¶=

èéß.¶à è

¶ß.áà

êéß.áà ¬

¶ß.¶à = 0.82

wherenormalizingthepriceofgood2,i.e.,𝑝c =1,gives𝑝] = 0.82.



• Example6.8(continued):– Furthermore,substitutingourcalculatedvaluesintothepriceofcapitalandlaboryields

𝑟∗ = 0.75(0.82) æç

�,.c½ ]c

,.c½= 0.42

𝑤∗ = 0.25(0.82) æç

,.Ü½ ]c

�,.Ü½= 0.63



• 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


• 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



• 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.



– 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



– 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



– FOCswrt 𝐿% and𝐾%,where𝑗 = {1,2},yieldaconditionforefficiencythatweencounteredinproductiontheory

𝜕𝐹]𝜕𝐿𝜕𝐹]𝜕𝐾

=𝜕𝐹c𝜕𝐿𝜕𝐹c𝜕𝐾

– Thatis,the𝑀𝑅𝑇𝑆f,Þ betweenlaborandcapitalmustcoincideacrossfirms.

– Otherwise,welfarecouldbeincreasedbyassigningmorelabortothefirmwiththehighest𝑀𝑅𝑇𝑆f,Þ .



– Combiningtheabovetwoconditionsforefficiencyinconsumptionandproduction,weobtain

𝜕𝑈7𝜕𝑥]7

𝜕𝑈7𝜕𝑥c7

=𝜕𝐹c𝜕𝐿𝜕𝐹]𝜕𝐿

– Thatis,𝑀𝑅𝑆],c7 mustcoincidewiththerateatwhichunitsofgood1canbetransformedintounitsofgood2,i.e.,themarginalrateoftransformation𝑀𝑅𝑇],c.



– Ifwemovelaborfromfirm2tofirm1,theproductionofgood2increasesby�ú¶

�fwhilethatof

good1decreasesby�ú¬�f

.Hence,inordertoincreasethetotaloutputofgood1byoneunitweneed�ú¶

�f/ �ú¬�f

unitsofgood2.

– Intuition:foranallocationtobeefficientweneedthattherateatwhichconsumersarewillingtosubstitutegoods1and2coincideswiththerateatwhichgood1canbetransformedintogood2.



• FirstWelfareTheoremwithproduction:iftheutilityfunctionofeveryindividual𝑖,𝑢7,isstrictlyincreasing,theneveryWEAisParetoefficient.

• Proof (bycontradiction):– Supposethat(𝐱, 𝐲) isaWEAatprices𝐩∗,butisnot Paretoefficient.

– Since(𝐱, 𝐲) isaWEA,thenitmustbefeasible

∑ 𝐱7e7\] = ∑ 𝐞7e

7\] + ∑ 𝐲%[%\]



– 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).



– 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.



– Thisindicatesthatproductionplan𝐲% wasnotprofit-maximizingand,asaconsequence,itcannotbepartofaWEA.

–Wethereforereachedacontradiction.

– Thisimpliesthattheoriginalstatementwastrue:ifanallocation(𝐱, 𝐲) isaWEA,itmustalsobeParetoefficient.



• 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


• 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.



• SecondWelfareTheoremwithproduction:– Considertheassumptionsonconsumersandproducersdescribedabove.

– Then,foreveryParetoefficientallocation(𝐱û, 𝐲û)wecanfind:a) aprofileofincometransfers(𝑇], 𝑇c, … , 𝑇e)

redistributingincomeamongconsumers,i.e.,satisfying∑ 𝑇7e

7\] = 0;b) apricevector𝐩ó,suchthat:



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


• 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


– 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



– 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



– Substituting

2 0.82 0.79Eû¬º

= 1.47 + 𝑇] ⟹𝑇] = −0.17

– Clearly,𝑇] + 𝑇c = 0

– ThusthesetransferswillallowforthenewallocationtobeaWEA.


ComparativeStatics


ComparativeStatics• Weanalyzehowequilibriumoutcomesareaffectedbyanincreasein:– thepriceofonegood– theendowmentofoneinput

• Considerasettingwithtwogoods,eachbeingproducedbytwofactors1and2underconstantreturnstoscale(CRS).

• Anecessaryconditionforinputprices(𝑤]∗, 𝑤c∗) tobeinequilibriumis

𝑐] 𝑤], 𝑤c = 𝑝] and𝑐c 𝑤], 𝑤c = 𝑝c– Thatis,firmsproduceuntiltheirmarginalcostsequalthepriceofthegood.


ComparativeStatics

• Let𝑧]%(𝑤) denotefirm𝑗’sdemandforfactor1,and𝑧c%(𝑤) beitsdemandforfactor2.– Thisisequivalenttothefactordemandcorrespondences𝑧(𝑤, 𝑞) inproductiontheory.

• Theproductionofgood1isrelativelymoreintenseinfactor1thanistheproductionofgood2if

É¬¬(ã)É¶¬(ã)

> É¬¶(ã)É¶¶(ã)

whereÉ¬*(ã)É¶*(ã)

representsfirm𝑗’sdemandforinput1relativetothatofinput2.


ComparativeStatics:PriceChange

1) Changesinthepriceofonegood,𝑝%(Stolper-Samuelsontheorem):– Consideraneconomywithtwoconsumersandtwofirmssatisfyingtheabovefactorintensityassumption.

– Ifthepriceofgood𝑗,𝑝%,increases,then:a) theequilibriumpriceofthefactormore

intensivelyusedintheproductionofgoodincreases;while

b) theequilibriumpriceoftheotherfactordecreases.



• 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



– Ifonlyprice𝑝] varies,then𝑑𝑝c = 0.– Hence,wecanrewritethesecondexpressionas

𝑧c](𝑤)𝑑𝑤] + 𝑧cc(𝑤)𝑑𝑤c = 0⟹ 𝑑𝑤] = − É¶¶

É¶¬𝑑𝑤c

– Solvingforýã¬ý�¬

inthefirstexpressionyieldsýã¬ý�¬

= É¶¶É¬¬É¶¶�É¬¶É¶¬

– Solving,instead,forýã¶ý�¬

yieldsýã¶ý�¬

= − É¶¬É¬¬É¶¶�É¬¶É¶¬AdvancedMicroeconomicTheory 144


– Fromthefactorintensitycondition,É¬¬(ã)É¶¬(ã)

>É¬¶(ã)É¶¶(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.



• Example6.11:– Letussolvefortheinputdemands inExample6.8:

𝑟] = 𝑝]0.75𝐾]�,.c½𝐿],.c½ ⟹ 𝑧]] = 𝐾] =¼�¬çâ

ç𝐿]

𝑤] = 𝑝]0.25𝐾],.Ü½𝐿]�,.Ü½ ⟹ 𝑧c] = 𝐿] =�¬çã

éè 𝐾]

𝑟c = 𝑝c0.25𝐾c�,.Ü½𝐿c,.Ü½ ⟹ 𝑧]c = 𝐾c =�¶çâ

éè 𝐿c

𝑤c = 𝑝c0.75𝐾c,.c½𝐿c�,.c½ ⟹ 𝑧cc = 𝐿c =¼�¶çã

ç𝐾c



• 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



• Example6.11 (continued):– Inoursolution,�¬�¶

âã= 3.08,hencethiscondition

issatisfied.– Next,observethatboth𝑧]] and𝑧cc aretriviallypositive.

– ApplyingtheStolper-Samuelsontheoremyieldsýã¬ý�¬

= É¶¶É¬¬É¶¶�É¬¶É¶¬

> 0ýã¶ý�¬

= − É¶¬É¬¬É¶¶�É¬¶É¶¬

< 0


ComparativeStatics:EndowmentChange

2) Changesinendowments(Rybczynskitheorem):– Consideraneconomywithtwoconsumersandtwofirmssatisfyingtheabovefactorintensityassumption.

– Iftheendowmentofafactorincreases,thena) productionofthegoodthatusesthisfactor

moreintensivelyincreases;whereasb) theproductionoftheothergooddecreases.



• 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



– 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�¶



–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.



– 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).



– Capitalisassumedtobemoreintensivelyusedinfirm1,i.e.,

Þ¬f¬> Þ¶

f¶or

𝛾Þ] > 𝛾f] forfirm1and𝛾Þc < 𝛾fc forfirm2

– Hence,ifcapitalbecomesrelativelymoreabundantthanlabor,i.e.,%∆𝐾 > %∆𝐿,itmustbethat%∆𝑦] > %∆𝑦c.



– Thatis%∆𝐿 = 𝛾f]∧ ∧

%∆𝐾 = 𝛾Þ]

q %∆𝑦] +∥

q %∆𝑦] +

𝛾fc q (%∆𝑦c)∨ ∥𝛾Þc q (%∆𝑦c)

– Intuition:thechangeintheinputendowmentproducesamore-than-proportionalincreaseinthegoodwhoseproductionwasintensiveintheuseofthatinput.



• 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



• 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

êé¬¶>

èéè¶.



• Example6.12(continued):– Thefactordemandsforeachgoodare

𝑧Þ] =¼âã

�,.Ü½and𝑧f] =

¼âã

,.c½

𝑧Þc =â¼ã

�,.Ü½and𝑧fc =

â¼ã

,.c½

– Usingthevaluesfromexample6.8,wecanassignfollowingvalues:

𝛾Þ], 𝛾f], 𝛾Þc, 𝛾fc = (0.75,0.25,0.25,0.75)



• 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


IntroducingTaxes:TaxonGoods• Assumethatasalestax𝑡Å isimposedongood𝑘.• Thenthepricepaidbyconsumersincreasesby𝑝Å& = (1 + 𝑡Å)𝑝Å',where𝑝Å' isthepricereceivedbyproducers.

• Ifthetaxongood1and2coincides,i.e.,𝑡] = 𝑡c,thepriceratioconsumersandproducersfaceisunaffected:

�¬(

�¶(= (]b�¬)�¬)

(]b�¶)�¶)= �¬)

�¶)

– Hence,theafter-taxallocationisstillParetoefficient.


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.


IntroducingTaxes:TaxonGoods

– Similarly,the𝑀𝑅𝑇],c stillcoincideswiththepriceratioofgoods1and2:

𝜕𝐹c𝜕𝐿𝜕𝐹]𝜕𝐿

=𝑝]'

𝑝c=𝜕𝐹c𝜕𝐾𝜕𝐹]𝜕𝐾

wherethepricereceivedbytheproducer,𝑝]',isthesamebeforeandafterintroducingthetax.


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.


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



• However,whentaxesdiffer,𝑡f ≠ 𝑡Þ,productiveefficiencynolongerholdsundersuchcondition:– Whileinputconsumerssatisfy

𝑤f&

𝑤Þ&=𝜕𝐹]𝜕𝐿𝜕𝐹]𝜕𝐾

andinputproducerssatisfy

𝑤f'

𝑤Þ'=𝜕𝐹c𝜕𝐿𝜕𝐹c𝜕𝐾



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.


AppendixA:LargeEconomiesandtheCore


LargeEconomiesandtheCore

• Weknowthatequilibriumallocations(WEAs)arepartoftheCore.

• Wenowshowthat,astheeconomybecomeslarger,theCoreshrinksuntilexactlycoincidingwiththesetofWEAs.


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.



• Whencomparingtworeplicaeconomies,thelargestwillbethathavingmoreofeverytypeofconsumer.

• Allocation𝐱7) indicatesthevectorofgoodsforthe𝑞th consumeroftype𝑖.

• Thefeasibilityconditionis∑ ∑ 𝐱7)â

)\]e7\] = 𝑟 ∑ 𝐞7e

7\]



• 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.



• 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.



– Assumethattype-1consumerweaklyprefers𝐱]]to𝐱]c,i.e.,𝐱]] ≿] 𝐱]c.§ Thisistrueforbothtype-1twins,sincetheyhavethesamepreferences.

§ Asimilarresultemergesifweinsteadassume𝐱]c ≿] 𝐱]].


In this case since both bundles lie on the same indifference curve

In this case

LargeEconomiesandtheCore• Unequaltreatmentatthecorefortype-1consumers



– 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


• Averagebundlesleadingtoablockingcoalition



– 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).



– 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.



– Insummary,pairofbundles 𝐱�]c, 𝐱�cc :§ makesconsumers12and22betteroffthantheoriginalallocation(𝐱]c, 𝐱cc)

§ isfeasible

– Thus,theseconsumerscanblock(𝐱]c, 𝐱cc).§ Theoriginalallocation 𝐱]c, 𝐱cc cannotbeattheCore.

– Therefore,ifanallocationisattheCoreofthereplicaeconomy,itmustgiveconsumersofthesametypetheexactsamebundle.



• 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).



• Thecoreshrinksastheeconomyenlarges.Thesequenceofcoresets𝐶], 𝐶c, … , 𝐶â isdecreasing.

• Thatis,– thecoreoftheoriginal(un-replicated)economy,𝐶],isasupersetofthatinthe2-foldreplicaeconomy,𝐶c;

– thecoreinthe2-foldreplicaeconomy,𝐶c,isasupersetofthe3-foldreplicaeconomy,𝐶¼;

– etc.


=WEAs

LargeEconomiesandtheCore• TheCoreshrinksas𝑟 increases



• 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


– Nowweneedtoshowthat,as𝑟 increases,thecoreshrinks.

–Wewilldothisbedemonstratingthatallocationsatthefrontierof𝐶] donotbelongtothecoreofthe2-foldreplicaeconomy,𝐶c.


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]



– Thelinebetween𝐱- and𝐞 includescoreallocations.§ Allpointsinthelinearepartofthecore.§ However,notallpointsinthislineareWEAs.§ Forinstance:𝐱- isnotaWEAsincethepricelinethrough𝐱- and𝐞 isnottangenttotheconsumer’sindifferencecurveat𝐱-.

– IftheCoreshrinksastheeconomyenlarges,weshouldbeabletoshowthatallocation𝐱- ∉ 𝐶c.

– Letusbuildablockingcoalitionagainst𝐱-.



– 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


– 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.



–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.



• 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).



• AlimittheoremontheCore:Ifanallocation𝐱belongstothecoreofall𝑟-foldreplicaeconomiesthensuchallocationmustbeaWEAoftheun-replicatedeconomyℰ].

• Proof (bycontradiction):– Considerthatanallocation𝐱- belongstothecoreofthe𝑟-foldreplicaeconomy𝐶â butisnot aWEA.

– Acoreallocationfortheun-replicatedeconomyℰ],𝐱- ∈ 𝐶] satisfyies𝐱- ∈ 𝐶â since𝐶] ⊃ 𝐶â.

– Allocation𝐱- mustthenbewithinthelens-shapedareaandonthecontractcurve.


LargeEconomiesandtheCore• Acoreallocation𝐱- thatisnotWEA


Consumer 1,

Consumer 2


– Considernowthelineconnecting𝐱- and𝐞.– Since𝐱- isnotaWEA,thebudgetlinecannotbetangenttobothconsumers’indifferencecurves:

�¬�¶> 𝑀𝑅𝑆 or�¬

�¶< 𝑀𝑅𝑆

– Canallocation𝐱- beattheCore𝐶â andyetnotbeaWEA?

– Letusshowthatif𝐱- isnotaWEAitcannot bepartoftheCore𝐶â either.§ Todemonstratethat𝐱- ∉ 𝐶â,letusfindablockingcoalition



– 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.



– 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.



– 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-.



– Thusif𝐱- isnotaWEA,then,𝐱- cannotbeintheCoreofthe𝑟-foldreplicaeconomyℰâ.

– Asaconsequence,if𝐱- ∈ 𝐶âforall𝑟 > 0,then𝐱-mustbeaWEA.


AppendixB:Marshall–HicksFourLawsof

DerivedDemand


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.


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),�).



• Let𝜃f ≡ 𝑤𝐿/𝑝𝑞 and𝜃Þ ≡ 𝑟𝐾/𝑝𝑞 bethecostoflaborandcapital,respectively,relativetototalsales.

• Thisimpliesthat𝜃f = 1 − 𝜃Þ .• Forcompactness,letusdefine

𝐴 ≡ 1 − (1/𝜀),�7 )𝐵 ≡ 1 + (1/𝑠f,ã7 )



• 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


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)


Marshall’sPresentation

• Assumptions:– Outputandinputsmarketsareperfectlycompetitive,𝜀),�7 = 𝑠f,ã7 = ∞,foreveryfirm𝑖

– Inputscannotbesubstitutedintheproductionprocess,𝜎 = 0

• Theexpressionfor𝑠Þ,â canbesimplifiedto

𝑠Þ,â = −𝜃Þ𝜀),�𝑠f,ã𝑠f,ã + 𝜃f𝜀),�


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.


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),�


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.


Documents

Advanced Microeconomic Theory · Outline • Partial Equilibrium Analysis • General Equilibrium Analysis • Comparative Statics • Welfare Analysis Advanced Microeconomic Theory