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1 TitleTHE MODIGLIANI-MILLER THEOREM IN A Author() TAKEKUMA, HIN-ICHI Citatio Iue Date Type Techical Report Text Verio publiher URL Right Hitotubahi Uiverity Repoitory

2 Graduate chool of Ecoomic Hitotubahi Uiverity Dicuio Paper erie# THE MODIGLIANI-MILLER THEOREM IN A DYNAMIC ECONOMY by hi-ichi Takekuma April 200

3 THE MODIGLIANI-MILLER THEOREM IN A DYNAMIC ECONOMY HIN-ICHI TAKEKUMA Graduate chool of Ecoomic, Hitotubahi Uiverity, Kuitachi, Tokyo , apa takekuma.eco.hit-u@com.home.e.p April 200 Abtract A dyamic ecoomy with market of equitie ad bod i coidered. The ratioal expectatio equilibrium i defied i a aet pricig model ad a coditio uder which the Modigliai-Miller theorem hold i how. I a aggregate model the exitece of a ratioal expectatio equilibrium i proved. Keyword: The Modigliai-Miller theorem; ratioal expectatio; aet pricig. EL claificatio: C02, C6, D80, D90, O4 I. Itroductio I thi paper, a geeral model of a dyamic ecoomy i preeted ad the equilibrium of ratioal expectatio for the ecoomy i defied. I the model, we re-examie the Modigliai-Miller theorem, which aert that the value of a firm i idepedet of it debt-equity ratio. I the cotext of a dyamic geeral equilibrium model, we will how that the M-M theorem hold i a much more geeral framework. The validity of the theorem deped heavily o the ratioality of coumer expectatio. I the proof of the M-M theorem, which origiate i the paper by Modigliai-Miller (958), it i uually aumed that the gro retur of a firm deped oly o the tate of the ecoomy, ice the theorem i baed o tatic equilibrium rather tha dyamic aalyi. I a

4 dyamic ecoomy, the profit of firm are determied depedig o the behavior of all ecoomic aget, epecially their expectatio. The purpoe of thi paper i to how that M-M reult are till valid i a dyamic equilibrium of ratioal expectatio. Alo, i a aggregate model of the ecoomy where all coumer are idetical, we prove the exitece of ratioal expectatio equilibria. Our model i a geeralizatio of the aet pricig model preeted by Luca (978). Aet pricig model have become etablihed tool ad have bee applied to variou aalye by may author, e.g., Cotatiide ad Duffie (996), Brav et al. (2002), ad Kocherlakota ad Pitaferri (2009). The M-M theorem wa origially proved i a tatic framework ad wa exteded to the geeral equilibrium model of tiglitz (969). Alo, the theorem wa coidered by Diamod (967) ad DeMarzo (988) i dyamic ecoomie i which expectatio are ot icorporated. I thi paper, the theorem will be recoidered ad proved i a geeral model with ratioal expectatio. Thi paper i formulated i the followig fahio. I ectio II, a geeral model of a dyamic ecoomy i preeted ad i ectio III, the equilibrium for the ecoomy i defied. I ectio IV, a coditio for the equilibrium uder which the Modigliai-Miller theorem hold i how. I ectio V, a aggregate model i which all coumer are idetical i preeted. I ectio VI, the exitece of equilibria for a aggregate ecoomy i proved ad the M-M theorem i how to hold i the equilibrium. II. A Geeral Model I thi ectio, we coider a geeral dyamic ecoomy i which there are ifiitely may coumer ad fiitely may firm. The et of coumer i deoted by a atomle meaure pace (A, A, ν), where A i the et of all coumer, A i a σ-field coitig of ome ubet of A, ad ν i a meaure defied o A o that ν(a)=. O the other had, we aume that there are fiitely may firm ad the umber of firm i. I a ecoomy, there are kid of commoditie ad the commodity pace i deoted by -dimeioal Euclidia pace R. We aume that all commoditie ca be ued a coumptio good a well a capital good. The coumptio et of each coumer i the o-egative orthat of pace R, which i deoted by. The et of poible utility fuctio of coumer i deoted by U. The utility fuctio of each coumer i ucertai but i a elemet of U. We aume that U i a et of ome real-valued cotiuou fuctio defied o ad i edowed with the topology of uiform covergece. The family of poible productio et of firm i deoted by Y. 2

5 The productio et of each firm i alo ucertai but i a elemet of Y. We aume that Y i a et of ome cloed ubet of R ad i edowed with the topology of cloed covergece. Ucertaity i the ecoomy ca be decribed by a tochatic proce. We aume that time i dicrete, ad it i deoted by the et of o-egative iteger, T={0,, 2, }. Let (Ω, F, P) be a probability pace, i.e., Ω i the et of all the tate of ature, F i the et of all poible evet ad i a σ-field coitig of ome ubet of Ω, ad P i a probability meaure. Ucertaity i coumer utility fuctio ad firm productio et i decribed by a tochatic proce {E t t T} defied o (Ω, F, P). For each t, E t i a meaurable mappig deoted by ω Ω (U, Y) U A Y, where U A i the et of all meaurable mappig from A to U ad Y i a -time product of Y, i.e., U A ={U U i a meaurable mappig deoted by a A U a U.} ad Y =Y Y. Whe tate ω of ature occur at period t, coumer utility fuctio ad firm productio et are deoted by E t (ω), ay (U, Y). The, U i a mappig, a A U a U ad value U a i the utility fuctio of coumer a A. I additio, Y i a elemet of et Y ad the -th coordiate Y of Y i the productio et of the -th firm. We aume that coumer utility fuctio ad firm productio et at each period will be kow at the begiig of the period. uppoe that a coumer ha utility u t at each period t=0,, 2,. Let δ be the dicout rate of utility, where 0<δ<. The um of dicouted utilitie that the coumer would have i =0 t δ t u t. However, ice hi utility fuctio i future are ucertai, the coumer i ot able to kow the level of utilitie that he will obtai i the future. Therefore, coumer will gue future utilitie ad behave to maximize the um of expected utilitie. O the other had, firm are able to kow their productio et at the begiig of each period ad productio take place withi oe period. Therefore, there i o ucertaity for firm ad they imply maximize their profit at each period i time. Proce {E t t T} alo decribe a traitio of ucertaity i the ecoomy. We aume that it i a Markov proce. Let =U A Y ad B() be the et of all Borel ubet of. We deote by M() the et of all meaure defied o B(), which i edowed with weak topology. Aumptio 2.: There i a cotiuou mappig from to M(), μ M(), 3

6 which ha the followig property: For each, μ i a traitio probability o, i.e., for each t T, μ (B)=Prob.{E t B E t =} for all B B(). More preciely, for each t T ad, C - μ ( B ) d( P E )( ) =P( E ( ) E ( C ) ) for all B, C B(). t t B t The exitece of uch a traitio probability mea that the ucertaity at each period doe ot deped o time, but oly o the tate at the previou period. Therefore, if =(U, Y) i realized at period t, the the ucertaity i the ecoomy after period t deped oly o =(U, Y). I additio, the traitio of ucertaity i the ame at all period, ad i thi ee, the ecoomy i tatioary. Becaue of thi tatioarity, whe we decribe the tate of the ecoomy at each period, we do ot have to how idex t of time i the argumet. III. The Defiitio of Equilibrium L Let be the et of all eetially bouded meaurable fuctio from A to. We ue a fuctio i L to deote the iitial holdig of commoditie by coumer at each period. Namely, for fuctio κ L, we deote by κ(a) the amout of commoditie held by L coumer a. Let be the et of all itegrable fuctio from A to. We ue a fuctio i L to deote the hare i firm owed by coumer at each period. Namely, for fuctio θ=(θ,, θ ) by coumer a. L, we deote by θ (a) the hare of the -th firm equity owed Let be the et of all itegrable fuctio from A to R. We ue a fuctio i L to L L deote the umber of bod owed by coumer. Namely, for fuctio β, β(a) deote the amout of bod owed by coumer a. We deote umber of bod that firm iue by vector D=(D,, D ) R, where D i the amout of bod that are iued by firm. All bod are meaured i term of moey, ad the value of a uit of bod i therefore equal to a uit of moey. 4

7 The equilibrium of the ecoomy i defied by pair {ψ, V}, where ψ i a correpodece from R to R R L L L R ad V i a fuctio from A R L R L L to R. Correpodece ψ: L L R L R R R i called a price correpodece ad it how coumer expectatio o equilibrium price ad iteret rate. Correpodece ψ i depicted i the followig otatio: (κ, θ, β, D; ) L R L L ψ(κ, θ, β, D; ) R R R. Elemet (κ, θ, β, D; ) i L R L L decribe a ituatio of the whole ecoomy at a period i time. Elemet (p, q, r) of et ψ(κ, θ, β, D; ) i a vector i R, where p i a vector of commodity price, q i a vector of equity price, ad r i the iteret rate of a bod. Price correpodece ψ decribe how equilibrium price ad iteret rate deped o the ituatio of the ecoomy. Fuctio V: A R R R L L R L R i called a value fuctio ad it how coumer expectatio o utilitie. Fuctio V i depicted i the followig otatio: (a, z, e, b; κ, θ, β, D; ) A R R R L R L L V a (z, e, b; κ, θ, β, D; ) R. Elemet (z, e, b) i R R R decribe the tate of coumer a. Value V a (z, e, b; κ, θ, β, D; ) i the expected value of utilitie that coumer a ca have. Value fuctio V decribe how the expected utility of each coumer deped o hi tate (z, e, b) a well a the ituatio (κ, θ, β, D; ) of the whole ecoomy. ice the meaure pace of coumer i atomle, each coumer i egligible i the whole ecoomy ad he ca therefore chooe a tate (z, e, b) idepedetly of the ituatio (κ, θ, β, D; ) of the ecoomy. I order for {ψ, V} to be a equilibrium of the ecoomy, it i required i the followig defiitio that (p, q, r) i ψ(κ, θ, β, D; ) i a vector of price ad a iteret rate which equilibrate all market ad that V a (z, e, b; κ, θ, β, D; ) i the maximum expected utility that coumer a ca have whe the ituatio of the ecoomy i (κ, θ, β, D; ). Defiitio 3.: Pair {ψ, V} of a price correpodece ad a value fuctio i a equilibrium of the ecoomy, if {ψ, V} atifie the followig: 5

8 Let (κ, θ, β, D, ) L A D R L L β dν = ad θ dν =, A ad (p, q, r) ψ(κ, θ, β, D; ) with where =(U, Y) ad =(,, ) R. The, there exit ĉ L, ( ˆ, κ ˆ, θ ˆ, β Dˆ ) L L R L, ad Y (,, ), which atify the followig coditio: () Firm maximize their profit, i.e., for each,,, ŷ p ŷ p y for all y Y. (2) Coumer maximize their expected utilitie ubect to their budget cotrait, i.e., for almot all a A, ad p ( c ˆ( a) ˆ κ ( a) )q ˆ θ ( a ) ˆ β ( a) p κ(a)q θ(a)(r)β(a) θ ( a)( p yˆ rd ) V a (κ(a), θ(a), β(a); κ, θ, β, D; )=U a ( cˆ ( a) )δ Va ( ˆ( κ a), ˆ( θ a), ˆ( β a) ; ˆ, κ ˆ, θ ˆ, β Dˆ; for all (x, z, e, b) R with U a (x)δ V ( z, e, b ; ˆ, κ ˆ, θ ˆ, β Dˆ; ) p (xz)q eb p κ(a) q θ(a)(r)β(a) θ ( a)( p yˆ rd ). (3) All market are i equilibrium, i.e., A cˆ dν κˆ dν = κ dν ˆ, =, ad = A βˆ dν ˆ θˆ dν. A A y a D A IV. The Modigliai-Miller Theorem I thi ectio, we how a coditio for value fuctio V i the defiitio of equilibrium uder which the Modigliai-Miller theorem hold. Coditio 4.: Let (κ, θ, β, D, ) L L R L with β dν = ad A A θ dν =. The, for each a A, D 6

9 V a (z, e, be ΔD; κ, θ, βθ ΔD, DΔD; )=V a (z, e, b; κ, θ, β, D; ) for all (z, e, b) R ad ΔD R. The above coditio mea that coumer expected utilitie are idepedet of ay chage ΔD of amout of bod iued by firm, a log a each coumer chage the amout of bod by e ΔD proportioally to amout e of equitie he hold. The coditio i the eece of the Modigliai-Miller theorem aertig that the value of a firm i idepedet of the amout of the firm debt. I fact, the followig propoitio how how the equilibrium price of equitie chage but the price of commoditie are uchaged. Propoitio 4.: Let {ψ, V} be a equilibrium of the ecoomy ad let u aume that value fuctio V atifie Coditio 4.. If (p, q, r) ψ(κ, θ, β, D; ), the (p, q-δd, r) ψ(κ, θ, βθ ΔD, DΔD; ) for ay ΔD. Proof: By Coditio 4., (2) of Defiitio 3. ca be rewritte i the followig fahio. For almot all a A, p ( c ˆ( a) ˆ κ ( a) )(q-δd) ˆ θ ( a ) ˆ β ( a) θˆ ( a) ΔD ad p κ(a)(q-δd) θ(a)(r)(β(a)θ(a) ΔD) V a (κ(a), θ(a), β(a)θ(a) ΔD; κ, θ, βθ ΔD, DΔD; ) =V a (κ(a), θ(a), β(a); κ, θ, β, D; ) θ ( a)( p y r( D ΔD )) =U a ( cˆ ( a) )δ Va ( ˆ( κ a), ˆ( θ a), ˆ( β a); ˆ, κ ˆ, θ ˆ, β Dˆ; U a (x)δ V ( z, e, b e ΔD; ˆ, κ ˆ, θ ˆ, β Dˆ; a =U a (x)δ Va ( z, e, b; ˆ, κ ˆ, θ ˆ β ˆ θ ΔD, Dˆ ΔD; for all (x, z, e, b) R with p (xz)(q-δd) eb p κ(a)(q-δd) θ(a)(r)(β(a)θ(a) ΔD)) θ ( a)( p y r( D ΔD)). Moreover, we obviouly have A ( ˆ β θ ΔD) dν = ( Dˆ ΔD ). Thi implie that (p, q-δd, r) ψ(κ, θ, βθ ΔD, DΔD; ). Q.E.D. 7

10 Let {ψ, V} be a equilibrium of the ecoomy. If (p, q, r) ψ(k, θ, β, D; ), the value of firm are defied by qd. Therefore, Propoitio 4. implie that the price of firm equitie become q-δd if the amout of firm debt chage by ΔD. After D chage, the value of firm are (q-δd) (DΔD)=qD. Thu, the value of firm are uchaged ad idepedet of the amout of firm debt. I additio, price p of commoditie ad iteret rate r remai cotat. Moreover, ice V a (κ(a), θ(a), β(a)θ(a) ΔD; κ, θ, βθ ΔD, DΔD; )=V a (κ(a), θ(a), β(a); κ, θ, β, D; ) for each a A, all coumer ca attai the ame level of expected utility after D chage. Hece, Propoitio 4. implie that the equilibrium of the ecoomy i ot affected by chage of D, which i a theorem origially proved by Modigliai ad Miller (958) ad exteded to the framework of geeral equilibrium by tiglitz (989). V. A Aggregate Ecoomy I thi ectio we coider a implified ecoomy where there are may, but idetical coumer ad prove the exitece of a equilibrium for the ecoomy. I what follow, ice we aume that the coumer i the ecoomy are all idetical, we have oly to coider the behavior of a repreetative coumer. uch a aggregate model of the ecoomy i ueful particularly for macroecoomic aalye. The utility fuctio of coumer are deoted by a mappig U:A U, which i a elemet of et U A. We aume that the utility fuctio of all coumer are the ame, ad that mappig U i cotat, i.e., for ome u U, U(a)=u for all a A. Therefore, we ca regard U A a U. Thu, i thi ectio we aume that =UY, ad Aumptio 2. hold for et i thi cae. Moreover, we aume that coumer are all i the ame ituatio, ad that their holdig of commoditie, equitie, ad bod are the ame. The amout of commoditie held by coumer are decribed by fuctio κ:a which i a elemet of et. Whe coumer have the ame amout of commoditie, the fuctio κ i cotat, i.e., for ome k, κ(a)=k for all a A. Therefore, we ca regard a. et Equity holdig by coumer are deoted by a fuctio θ:a which i a elemet of L. ice the total equity of each firm i aumed to be uity, whe all coumer have the ame amout of equitie, θ(a)= for all a A. Thu, fuctio θ ca be regarded a L L 8

11 vector, ad we ca omit howig it. The umber of bod held by coumer are decribed by a fuctio β:a R, which i a elemet of et L. Whe all coumer have the ame amout of bod, the fuctio β i cotat, i.e., for ome B R, β(a)=b for all a A. Therefore, we ca regard L a R. By the above implificatio, a macro-tate (κ, θ, β, D; U, Y) of the ecoomy ca be depicted i the aggregate ecoomy by a elemet (k, B, D; u, Y) R R UY. By thi procedure, we ca defie a price correpodece ad a value fuctio for a repreetative coumer i the followig fahio. Defie price correpodece ψ by (k, B, D; ) R R R ψ(k, B, D; ) R, R R where =(u, Y). Alo, defie a value fuctio V by (z, e, b; k, B, D; ) R R R R R R V(z, e, b; k, B, D; ) R. We ca ow defie a equilibrium for the aggregate ecoomy. Defiitio 3. i reduced to the followig. Defiitio 5.: Pair {ψ, V} of a price correpodece ad a value fuctio i called a equilibrium for the aggregate ecoomy, if {ψ, V} ha the followig property: D Let (k, B, D) R R R, =(u, Y), ad (p, q, r) ψ(k, B, D; ) with B=. The, there exit xˆ R, ˆ ( kˆ, Bˆ, D) R R R, ad ŷ Y (,, ), which atify the followig coditio: () Firm maximize their profit, i.e., for each,,, p ŷ p y for all y Y. (2) Coumer maximize their expected utilitie ubect to their budget cotrait, i.e., ad p ( xˆ k ˆ )q Bˆ p kq (r)b ( p yˆ rd ), V(k,, B; k, B, D; )=u( xˆ )δ V ( kˆ,, Bˆ; kˆ, Bˆ, Dˆ; u(x)δ for all (x, z, e, b) R R R R with V ( z, e, b; kˆ, Bˆ, Dˆ; p (xz)q eb p kq (r)b ( p yˆ rd ). (3) All market are i equilibrium, i.e., 9

12 y D xˆ k ˆ =k ˆ ad Bˆ = ˆ. From (2) i the above defiitio, ice B= D, we ca eaily ee that iteret rate r i idetermiate ad ca be ay umber. The idetermiacy of iteret rate i a peculiar pheomeo of the aggregate ecoomy, where each aget virtually borrow from himelf. I what follow, we tate the aumptio that eure the exitece of a equilibrium for the aggregate ecoomy. For et U of utility fuctio ad family Y of productio et, we aume the followig. Aumptio 5.: Let u U. The, u ha the followig propertie. () u i a cotiuou ad cocave fuctio. (2) u i a mootoe icreaig fuctio, i.e., if x x ad x x, the u ( x) > u( x ). (3) u(0)=0. (4) There exit a umber ε 0 > 0 uch that x implie u ( x) ε 0. Aumptio 5.2: Let Y=(Y,, Y ) Y. The, Y ha the followig propertie. () Y i a cloed ad covex ubet of R. (2) R ={ 0}. Y (3) There exit a umber ε > 0 uch that y Y implie y ε. Uder the above aumptio, we have the followig theorem o the exitece of a equilibrium for the aggregate ecoomy, which iclude a coditio correpodig to Coditio 4.. Theorem 5.: Uder Aumptio 5. ad 5.2, there exit a equilibrium {ψ, V} for the aggregate ecoomy that ha the followig propertie. () Value fuctio V i cotiuou ad bouded ad V(z, e, b; k, B, D; ) i mootoe o-decreaig ad cocave i (z, e, b). (2) Let (k, B, D; ) R R ad B= D. The V(z, e, be ΔD; k, B ΔD, DΔD; )=V(z, e, b; k, B, D; ) for all (z, e, b) R R R ad ΔD R. 0

13 I the above theorem, value fuctio V atifie coditio (2), which correpod to Coditio 4., ad the M-M theorem therefore hold i the aggregate model. VI. Proof of Theorem 5. I thi ectio, we will prove Theorem 5.. The proof of the theorem i a modificatio of the argumet i Takekuma (990). Let C * be the pace of all bouded cotiuou fuctio defied o R W C *, defie fuctio MW o R by MW(k, e; )=up { u( x) δ W ( z, e; x, z, xz=k e y },. For each y Y (,, ), where =(u, Y) ad Y=(Y,, Y ). The followig two lemma are the ame a Lemma 6. ad 6.2 i Takekuma (990), ad we will omit their proof. Lemma 6.: For ay W C *, MW i a fuctio that ha the followig propertie. () MW C *, i.e., MW i a cotiuou ad bouded fuctio. (2) If W(z, e; ) i mootoe o-decreaig ad cocave i (z, e), the o i MW(z, e; ) i (z, e). (3) If W(0, e; )=0 for all (e; ), the MW(0, e; )=0 for all (e; ). By () of the above lemma, we have a mappig, W C * MW C *, which i deoted by M:C * C *. Thi mappig ha the followig property. Lemma 6.2: There exit a uique fuctio W 0 C * that ha the followig propertie. () W 0 i a fixed-poit of mappig M, i.e., W 0 =MW 0. (2) For each, W 0 (z, e; ) i mootoe o-decreaig ad cocave i (z, e). (3) W 0 (0, e; )=0 for all e ad. Let k ad =(u, Y) where Y=(Y,, Y ). ice W 0 =MW 0, by Aumptio 5. ad 5.2, there exit ˆ R, ˆ R, ad x k yˆ Y (,, ) uch that

14 W 0 (k, ; )= u( xˆ) δ W0 ( kˆ, ; ad xˆ kˆ =k yˆ. (6.) D Next, let (B, D) R R with B= ad defie ubet Φ(k, B, D; ) of R R by Φ(k, B, D; )={(p, q) R R W 0 (k, ; )B- D u( x) δ ( W0 ( z, e; b e D) for all (x, z, e, b) R R R R with p (xz)q eb- D p kq up p }. (6.2) Y Lemma 6.3: Φ(k, B, D; ) φ. Proof: Defie two ubet F ad G of R F={(w, e, m) w=xz, m=b-e D, R R by u( x) δ ( W0 ( z, e; b e D) > W 0 (k, ; )B- D} y G={(w, e, m) w=k for ome y Y (,, ), e=, m=0}. By (6.) ad Aumptio 5. (2), we ca how that F φ. Alo, Aumptio 5.2 (2) implie that G φ. The covexity of F ad G follow from Lemma 6.2 (2) ad Aumptio 5. () ad 5.2 (). uppoe that F G φ. The, there exit x, z, b, ad y Y (,, ) uch that u( x ) δ ( W z b 0(, ; D) > W 0 (k, ; )B- D, x z = k y, ad b - D=0. That i, u( x ) δ W z 0(, ; >W 0 (k, ; ) ad x z =k. ice W 0 =MW 0, we have a cotradictio to the defiitio of mappig W 0. Hece, y F G = φ. By a eparatio theorem, there exit a vector (p, q) R R uch that vector (p, q, ) eparate et F ad G, that i, y p wq em p (k )q for all (w, e, m) F ad y Y (,, ). ice u i mootoe icreaig ad W 0 i mootoe o-decreaig, p>0, ad q 0. Alo, 2

15 the above iequality implie that p wq em p kq up p for all (w, e, m) F. (6.3) Y uppoe that equality i (6.3) hold for ome (w, e, m) F. The, there exit x, z, e, ad b o that ad u( x )δ ( W 0( z, e ; b e D) > W 0 (k, ; )B- D p ( x z )q e b - D=p kq up p. However, by decreaig b lightly, we have a cotradictio to (6.3). Therefore, we have proved that p wq em > p kq up p for all (w, e, m) F, that i, for all (x, z, e, b) R W 0 (k, ; )B- D u(x)δ R R R with which implie that (p, q) Φ(k, B, D; ). Y Y ( W0 ( z, e; b e D) p (xz)q eb- D p k q, up p Y Q.E.D. By Lemma 6.3 we have a correpodece, (k, B, D; ) R R R Φ(k, B, D; ) R R, D where B=. Thu, we ca defie correpodece ψ: R R R R ψ(k, B, D; )=Φ(k, B, D; )R, D R R by where B=. Alo, let u defie fuctio V: R R R R R R R R by V(z, e, b; k, B, D; )=W 0 (z, e; )b-e D. The, obviouly, V i cotiuou ad bouded. Alo, by Lemma 6.2 ad the defiitio of V, we ca eaily check that fuctio V ha propertie () ad (2) i Theorem 5.. It remai to be how that {ψ, V} i a equilibrium for the aggregate ecoomy i the ee of Defiitio 5.. Lemma 6.4: If (p, q, r) ψ(k, B, D; ), the V(k,, B; k, B, D; )=u( xˆ )δ V ( kˆ,, B; kˆ, B, D; 3

16 for all (x, z, e, b) R u(x)δ R R R with V ( z, e, b; kˆ, B, D; p (xz)q eb p k q (r)b (up p Y rd ). Proof: ice (p, q) Φ(k, B, D; ), from (6.2) it follow that W 0 (k, ; )B- D u(x)δ for all (x, z, e, b) R R R R with ( W0 ( z, e; b e D) p (xz)q eb- D p k q. up p ice B= D, the defiitio of fuctio V ad (6.) ad (6.2) imply that V(k,, B; k, B, D; )=W 0 (k, ; ) B- D for all (x, z, e, b) R R R Y = u ( xˆ) δ ( W0 ( kˆ, ; B D) = u( xˆ) δ V ( kˆ,, B; kˆ, B, D; u( x) δ ( W0 ( z, e; b e D) = u( x) δ V ( z, e, b; kˆ, B, D; R with p (xz)q eb p kq (r)b (up p Y rd ). Q.E.D. Put Bˆ =B ad Dˆ =D. The, by (6.) we have p ( xˆ ẑ )q Bˆ - D=p (k )q yˆ p kq up p Y. uppoe that trict iequality hold i the above. The, by icreaig xˆ lightly, Aumptio 5. (2) immediately implie a cotradictio to Lemma 6.4. Therefore, equality hold i the above, ad we have y p ˆ = up p, Y which implie () of Defiitio 5.. Thu, Lemma 6.4 implie (2) of Defiitio 5.. (3) of Defiitio 5. alo follow from (6.). Thi complete the proof of Theorem 5.. 4

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