Essential Microeconomics EXISTENCE OF EQUILIBRIUM Core ideas: continuity of excess demand functions, Fixed point theorems

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1 Essetial Microecoomics EXISTENCE OF EQUILIBRIUM Core ideas: cotiuity of excess demad fuctios, Fixed oit teorems Two commodity excage ecoomy 2 Excage ecoomy wit may commodities 5 Discotiuous demad 9 A ecoomy wit bouded roductio sets 0 Demad ad suly corresodeces Ubouded roductio sets 4 Jo Riley October 9, 202

2 Essetial Microecoomics -2- Walrasia equilibrium i a Excage ecoomy Cosider a excage ecoomy wit strictly icreasig, strictly covex refereces. Cosumer, =,..., H wit edowmet ω > 0 as a cosumtio set 2 X =. By te Teorem of te maximum, te coice of cosumer, x (, ω ) is a cotiuous fuctio. Tus te aggregate excess demad fuctio H z( ) = ( x (, ω ) ω ) is cotiuous. = For te two commodity case it is easy to see tat tere must be a Walrasia equilibrium. We ormalize te two rices so tat tey sum to. (Te set of o-egative vectors summig to is called te uit simlex). By Walras Law z( ) z ( ) = Cosider te followig rice adustmet rocess. F z ( ) ( ) = were z ( ) = Max{ z ( ),0}. z ( ) Jo Riley October 9, 202

3 Essetial Microecoomics -3- Suose tat tere is excess demad for commodity (ad ece excess suly of commodity 2) so tat z ( ) = z( ). Te te ew rices are F( ) z ( ) F ( ) = ( ). 2 = ad 2 z( ) z Note F( ) ad F ( ) 0 ad 2 F( ) F2( ) =. Tus F mas te uit simlex ito te uit simlex. Sice utility is strictly icreasig, demad for commodity icreases witout boud as 0. Tus z ( ) z ( ) lim F ( ) = lim = lim = 0 0 z ( ) 0 z( ) Similarly, demad for commodity 2 icreases witout boud as 2 0. Terefore z ( ) lim F ( ) = lim == lim = 0 z ( ) z2( ). Note fially tat te cotiuity of z( ) imlies tat F( ) is cotiuous. Jo Riley October 9, 202

4 Essetial Microecoomics -4- Sice 2 = we ca write te adustmet maig as F( ). Te maig for F( ) is deicted. We ave sow tat F( ) > 0 for sufficietly close to zero ad F( ) 45 lie F( ) < 0 for sufficietly close to. Te, give te cotiuity of F () tere must be At least oe rice (0,) at wic F( ) = 0. z ( ) Hece = z ( ) Terefore z( ) = 0. ad so z ( ) = 0. O Figure : Existece of a WE Jo Riley October 9, 202

5 Essetial Microecoomics -5- A excage ecoomy wit may commodities Proositio 5.3-: Existece of a WE i a excage ecoomy Suose U ( x ) =,..., H is strictly icreasig ad strictly quasi-cocave over X = ad cosumer as edowmet ω > 0 X. Te tere exists a WE. Arguig as i te two commodity case tere exists a cotiuous excess demad fuctio z( ). Wit more ta 2 commodities we aeal to te followig dee matematical teorem. Brouwer s Fixed Poit Teorem Let S be closed ad bouded ad suose tat te cotiuous maig = mas eac elemet of S ito a oit i S. Te for some x S, gx ( ) = x. gx ( ) ( g( x),..., g( x)) Proof of Proositio 5.3-: As i te two-commodity case, let be te uit simlex, tat is { 0, = }. = Jo Riley October 9, 202

6 Essetial Microecoomics -6- We coose a maig g( ) tat raises te rice of every commodity for wic tere is excess demad ad lowers every oter rice. Cosider te followig maig. z ( ) g( ) = z ( ) = were z ( ) = Max{ z ( ),0}. Note tat ˆ = z( ) is strictly larger ta for eac market i wic tere is excess demad ad is equal to i a market for wic tere is excess suly. Note also tat ˆ = z ( ). = = Tus g( ) is i te uit simlex. At oits i te iterior of te uit simlex (so tat >> 0) te excess demad fuctio is cotiuous. However te maig is ot defied at boudary oits sice excess demad is ifiite. Tus we caot aeal directly to te fixed oit teorem. To deal wit tis roblem we costrai eac cosumer s otimizatio roblem to lie i a comact set. We defie Ω= { x 0 x kω, k > } ad cosider te followig modified maximizatio roblem for cosumer =,..., H. Max{ U ( x ) x ω, x Ω }. x Jo Riley October 9, 202

7 Essetial Microecoomics -7- Sice te feasible set is comact for all, a maximum for tis modified roblem exists. Give strict quasi-cocavity, tis maximum is uique. Tus for eac tere is a uique excess demad z ( ) By te Teorem of te Maximum, tis maig is cotiuous. Hece te maig g( ) is cotiuous. Aealig to Brouwer s Fixed Poit Teorem, tere is some suc tat z ( ) g( ) = z ( ) = (5.3.) We ow argue tat >> 0. Suose istead tat = 0. At tis rice te uer boud for cosumtio of commodity is kω. Sice utility is strictly icreasig x ( ) = kω. Tus for eac cosumer tere is excess demad ad so z ( ) > 0. But te z ( ) g ( ) = > 0=. z ( ) = Tis violates (5.3.2). Jo Riley October 9, 202

8 Essetial Microecoomics -8- Let I be te idex set of commodities for wic excess demad is ositive. By Walras Law, z( ) = 0 so tis caot be true of all markets. Terefore i <. (5.3.2) i I From (5.3.2) i i i = z ( ) ( z ( )) =, for all i I. Summig over i I, i = i i I i I = z ( ) ( )( z ( )). But z ( ) = 0 for I. Terefore i =, cotradictig (5.3.3). i I Terefore tere is o market i wic tere is excess demad. It follows immediately from Walras Law tat tere ca be o market i wic tere is excess suly. Tus H z( ) = x ( ) ω = 0. It = follows tat x ( ) ω, =,..., H. Te for eac cosumer te costrait x kω is ot bidig ad so te rice vector remais a Walrasia equilibrium rice vector for ay cosumtio set cotaiig Ω. I articular, tis olds if te cosumtio set is. Q.E.D. Jo Riley October 9, 202

9 Essetial Microecoomics -9- Discotiuous Demad If demad is ot cotiuous, it is easy to see wy tere may be o Walrasia equilibrium. Figure 5.3- illustrates te roblem. Suose tat eac cosumer as te same edowmet vector ad refereces. Tus, te Walrasia equilibrium allocatio must be te o trade allocatio. Note tat te budget costrait ˆ x= ˆ ω touces te idifferece curve at A ad A tus tere are two equilibrium demads at te rice vector. At ay iger rice ratio / 2tere is excess demad for commodity 2 ad for ay lower rice ratio tere is excess demad for commodity. Tus tere is o equilibrium. A ˆ x= ˆ ω ω 2 ˆ A ω Figure 2: No Walrasia equilibrium ω x Jo Riley October 9, 202

10 Essetial Microecoomics -0- A ecoomy wit roductio Wile te roductio vector of a firm may iclude ositive ad egative elemets (oututs ad iuts) we will cotiue to assume tat te cosumtio set. Tis may seem restrictive because a X cosumer may suly labor services. However suc services are readily icororated by givig te cosumer a iitial edowmet. Te te cosumer sells some of is labor edowmet but is cosumtio remais ositive. Te roof of te ext roositio follows closely te existece roof for te excage ecoomy. Proositio 5.3-2: Existece of WE wit bouded roductio Suose U ( x ) =,..., H is strictly quasi-cocave over X ad cosumer as edowmet X f ω. Suose also tat te roductio sets f =,..., Y F are closed, strictly covex ad bouded from above ad 0 Y f. Te tere exists a Walrasia Equilibrium. Jo Riley October 9, 202

11 Essetial Microecoomics -- Oe limitatio of tis roositio is tat it aeals to te strict covexity of bot refereces ad roductio sets. Wile tis is a reasoable assumtio for refereces, it excludes te ossibility of a liear tecology i wic iuts ad oututs ca be simly scaled u, at least over some rage. Haily, te assumtio of strict covexity ca be relaced by (weak) covexity. Wile te formal matematics becomes a little more comlicated, te basic ideas are similar. Cosider a two commodity world ad a sigle firm wit a roductio set as deicted below. ( 2, a b) y 2 y ( ) (, a) 2 = / b Y f /a x ( ) 2 2 y a a b y2 Fig. 3: Suly corresodece We ormalize so tat te rice of commodity is. For low outut rices, te rofit maximizig outut is zero. If te rice of commodity 2 is betwee /a ad /b te rofit maximizig outut is a. For iger rices of commodity 2 te rofit maximizig outut is a b. Jo Riley October 9, 202

12 Essetial Microecoomics -2- Note also tat, at te two critical rices of commodity, tere is a set of rofit maximizig roductio vectors. Tus we o loger ave a roductio fuctio but a set-valued roductio corresodece y2( 2). Tis corresodece is also deicted I Fig. 3. Suose tat tis is te oly firm ad tat refereces are strictly covex. Sice maximized firm rofit Π( ) is a cotiuous fuctio of te rice vector, it follows tat te aggregate demad fuctio is also cotiuous. It follows tat te suly corresodece ad demad fuctios itersect at some rice 2. For te case of strict covexity we aealed to te cotiuity of te excess demad fuctio i order to rove existece. Wit covexity we aeal to te followig smootess roerty of demad corresodeces. Defiitio: Uer emi-cotiuous corresodece Te corresodece φ is uer emi-cotiuous at 0 x if t 0 t t 0 { x } x, { y φ( x )} y imlies y φ( x ). 0 0 Cosider te suly corresodece y2( 2) deicted i Fig. 3. Note tat y2( 2) = [ aa, b]. For te t t icreasig sequece { 2} 2, y2( 2) a. Ad for te decreasig sequece t t { 2} 2, y2( 2) a b. Tus te corresodece is uer emi-cotiuous at 2. Jo Riley October 9, 202

13 Essetial Microecoomics -3- Wile we will ot derive it ere, te followig result ca be roved usig argumets arallelig tose i te roof of cotiuity for a strictly covex ecoomy. Proositio 5.3-3: Uer emi-cotiuity of te excess demad corresodece Suose U ( x ) =,..., H is quasi-cocave ad cotiuous over X ad cosumer as f edowmet ω X. Suose also tat te roductio sets Y f =,..., F are closed, strictly covex ad bouded from above ad 0 Y f. Te for te modified ecoomy i wic eac cosumtio set is bouded from above ad eac roductio set is bouded from below, te excess demad corresodece z( ) is uer emi-cotiuous ad covex. Te fial ste is to first aeal to te followig fixed oit teorem ad te argue tat te Walrasia equilibrium of te modified ecoomy is also a equilibrium of te umodified ecoomy. Kakutai s fixed oit teorem Let S be closed, bouded ad covex ad let φ : S S be a uer emi-cotiuous corresodece. If for eac x S te set φ( x) is oemty ad covex, te φ as a fixed oit. Jo Riley October 9, 202

14 Essetial Microecoomics -4- A limitatio of te roof of existece is te assumtio tat te roductio sets are bouded from above. A alterative way to guaratee existece is to assume tat demad for at least oe commodity is ubouded we a rice aroaces zero. Te followig is roved i EM. Proositio 5.3-4: Existece of Walrasia equilibrium wit ubouded excess demad Let z( ) be a cotiuous excess demad maig from te uit simlex were () z( ) is bouded from below, (2) z( ) = 0, ad (3) bdy z( ). Te tere exists suc tat z( ) = 0. A similar roof alies if z( ) is a uer emi-cotiuous corresodece. Jo Riley October 9, 202