Key ideas: trading financial assets, complete security markets, incomplete security markets, implicit state claims prices. Arrow-Debreu equilibrium 3

Transcription

1 Eentil Microeconomic -- EQUILBRIUM IN FINANCIAL MARKETS 8.2 SECURITY MARKET EQUILIBRIUM Key ide: trding finncil et, complete ecurity mrket, incomplete ecurity mrket, implicit tte clim price Arrow-Debreu equilibrium 3 Security mrket equilibrium 4 Equivlence wit pnning 6 Multiple Period, Stte, Commoditie nd Production 3 Replcing A-D Stte-Contingent Mrket by Trde in Security Mrket 7 Incomplete mrket 22 Jon Riley November 5, 202

2 Eentil Microeconomic -2- In te previou ection we exmined equilibrium wen conumer trde directly in tte- contingent mrket. Here we exmine equilibrium wen conumer trde finncil et. Initilly we conider one-good excnge economy in wic tere re S tte. Conumer H = {,..., H} tte-contingent endowment ω = ( ω,..., ω S ). Let x = ( x,..., x S ) be i conumption vector. We ume tt conumer trictly increing expected utility function S U ( x ) = π u ( x ). = Jon Riley November 5, 202

3 Eentil Microeconomic -3- Arrow-Debreu equilibrium We introduce mrket for ec of te S commoditie. Te lloction { x X } H i n A-D equilibrium lloction if for ome p >> 0, (i) p x p ω conumption lloction re in budget et (ii) U ( xˆ ) > U ( x ) p xˆ > p x no trictly preferred lloction i in conumer budget et (iii) H H F f x = ω + = = f = y mrket cler. Jon Riley November 5, 202

4 Eentil Microeconomic -4- Security mrket (SM) equilibrium Suppoe tt individul do not trde tte-contingent commoditie but inted trde finncil et. Aet, A = {,..., A} price P nd tte-contingent dividend d = ( d,..., ds ). Note: Te AD equilibrium i equivlent to n SM equilibrium in wic tere re S ecuritie. Aet, wit price p dividend of p in tte nd dividend of zero in every oter tte. Conumer cn trnfer welt cro tte by buying nd elling et. Aet olding: Conumer portfolio of et olding ξ = ( ξ,..., ξ A ). Conumer begin wit no finncil et o te mrket vlue of ti portfolio cnnot be poitive. Given our umption tt utility i trictly increing te mrket vlue of te portfolio cnnot be negtive. Ten te portfolio contrint i A P ξ = 0. Given uc portfolio, conumer finl conumption of x = ω + d ξ. A Jon Riley November 5, 202

5 Eentil Microeconomic -5- Becue we ve plced no retriction on te ign of te dividend, n et price my be poitive or negtive. Definition: SM equilibrium S S An lloction { x X, ξ } were H x = ω + dξ i (Wlrin) ecurity mrket equilibrium lloction if for ome P= ( P,..., P A ), A (i) P ξ = 0 A =,..., H conumption lloction re in budget et (ii) ( ˆ ) ( ) ˆ U ω + d ξ > U x P ξ > 0 A no trictly preferred lloction i in conumer budget et (iii) x = ω, nd ξ = 0. H H H mrket cler. Jon Riley November 5, 202

6 Eentil Microeconomic -6- Spnning Suppoe firt tt tere re t let mny et tte nd tt te dividend vector { d } pn A S. Witout lo of generlity, we my relbel te et o tt te dividend vector of te firt S et re linerly independent. Ten, ny lloction in S cn be expreed liner combintion S ξ d of te firt S et. It follow tt for every tere i ome portfolio = ξ = 0, > ) uc tt ξ (wit x ω = A d ξ. (8.2-) Jon Riley November 5, 202

7 Eentil Microeconomic -7- Note: Portfolio ξ tifie x ω d ξ. (8.2-2) = A Propoition 8.2-: An A-D equilibrium i SM equilibrium if te et dividend pn Proof: Step : Portfolio contrint tified S Let { x X } H be n A-D equilibrium wit equilibrium price p. It follow from (8.2-) tt p x p ω = p d ξ. A Define P = p d, =,..., A. Ten p x p ω = Pξ. A An A-D lloction mut lie in individul budget et. Terefore Pξ = p x p ω 0. Tu te portfolio contrint re tified. A Jon Riley November 5, 202

8 Eentil Microeconomic -8- Note: Portfolio ξ tifie x ω d ξ. (8.2-3) Step 2: Aet mrket cler Summing (8.2-) over, = A x ω = d ξ = d ξ = D ξ H H A A H H were D i mtrix of te S column vector of dividend. In te A-D equilibrium mrket cler, tu D H ξ = x ω = 0. H Becue te column of D re linerly independent it follow tt D i invertible nd o ξ = 0. Tu et mrket cler. H Jon Riley November 5, 202

9 Eentil Microeconomic -9- Step 3: Any trictly preferred lloction i not feible. Finlly let xˆ = d ˆ ξ + ω be trictly preferred to A x. Becue x i A-D equilibrium wit ttecontingent price vector p, it follow tt xˆ mut cot trictly more. Tt i ˆ ˆ ˆ p x p ω = p d ξ = Pξ > 0. A A It follow tt ny trictly preferred portfolio violte te portfolio budget contrint. Ten te lloction { x, ξ } H = i n SM equilibrium lloction wit et price P = p d, =,..., A. Q.E.D. Jon Riley November 5, 202

10 Eentil Microeconomic -0- We now prove te convere. Te proof i imilr. We begin wit SM equilibrium in wic te dividend of te A et pn te complete tte pce S. Given completene we cn define implicit price for ec tte. We ten ow tt ti implicit price vector i n A-D equilibrium. Jon Riley November 5, 202

11 Eentil Microeconomic -- Propoition 8.2-2: An SM equilibrium i n A-D equilibrium if te et dividend pn Proof: Let { x X S, ξ S } H = were x A ω dξ = S = + be n SM equilibrium lloction nd let P= ( P,..., P A ) be te equilibrium vector of ecurity price. Relbel te et o tt te dividend vector of te firt S et re linerly independent. A in te proof of te previou propoition define D to be te mtrix of column of dividend vector. Becue tee column re independent, tere exit unique tte-contingent p uc tt p D = P. (8.2-4) Alo, for ec, tere exit ome portfolio ξ = ( ξ,..., ξ ) uc tt Dξ =. Ti portfolio mut S ve trictly poitive price, oterwie conumption in tte i unbounded. Tu P ξ > 0, =,..., S. Rewriting tee reult in mtrix form, D[ ξ,..., ξ S ] = I nd P [ ξ,..., ξ S ] > 0. Subtituting from (8.2-2), p D[ ξ,..., ξ ] = p I = p >> 0. S Tu te vector p i trictly poitive. Jon Riley November 5, 202

12 Eentil Microeconomic -2- Te ret of te proof prllel tt of te previou propoition. Suppoe tt individul trictly prefer x ˆ over te SM equilibrium lloction ome portfolio ˆ ξ uc tt x. Becue te et mrket re complete tere i xˆ ω =D ˆ ξ. From te definition of n SM equilibrium, ti cnnot tify te individul portfolio contrint. Terefore ˆ P ξ > 0 nd o ( ) ˆ ˆ p x ω = p D ξ = P ξ > 0. Tu no trictly preferred conumption bundle i feible if te price vector i p. It follow tt te lloction i n A-D equilibrium t te price vector p > 0. Q.E.D. Jon Riley November 5, 202

13 Eentil Microeconomic -3- Multiple Period, Stte, Commoditie nd Production Tu fr we ve focued on trding in ecurity mrket wit ingle commodity. We now conider n economy wit production nd L commoditie delivered t ec dte nd in ec tte. To implify f te expoition we ume tt tere re two dte, dte 0 nd dte. Let y 0 be te dte 0 production vector of firm f nd let f y be te dte production vector in tte of firm f. We write te production f f f f vector of firm f y = ( y0, y,..., y S ). Firm f production et Y f. Tt i, f f y Y. Conumer conumption vector x nd von Neumnn expected utility function ( S+ ) L S = π 0 = U ( x ) u ( x, x ). Finlly let f k be conumer reolding of firm f nd let ω be conumer endowment vector. A-D equilibrium Te A-D equilibrium of ti economy i n lloction { x } H,{ f } F = y f = nd price vector p > 0 tifying mrket clering nd individul rtionlity given ec conumer budget contrint In tt equilibrium, mrket for ll ( S + ) Lcommoditie re open t dte 0. Tu conumer cn mke ll of teir trde in te firt period. Jon Riley November 5, 202

14 Eentil Microeconomic -4- Te condition for n A-D equilibrium re (i) p yˆ > p y yˆ Y f f f f (ii) F f f p x = p ω + θ p y f = (iii) u ( xˆ ) > u ( x ) p xˆ > p x. A-D equilibrium wit pot mrket reopening in period 2 Tere i no need for ny mrket to reopen in lter period. But uppoe tt tte occur nd te L commodity mrket unexpectedly reopen. We et xˆ define become = x nd yˆ f = y f for ll, f nd ll, nd f Y to be te feible production pln in tte. Condition (i) (iii) for n A-D equilibrium (i) p yˆ > p y yˆ Y f f f f (ii) F f f = ω + θ f = p x p p y (iii) u ( xˆ ) > u ( x ) p xˆ > p x. Jon Riley November 5, 202

15 Eentil Microeconomic -5- Alo H H F x = ω + y = = f =. Condition (iii) revel tt if te tte pot price vector p i equl to te dte 0 contingent price vector p, no conumer will wi to trde gin t dte. Tu te future pot mrket ll cler in period. Of coure ny multiple of ti price vector i WE well. Ten for ny number r > te price vector p = ( + r ) p i WE future pot price vector. Equivlently p p =. + r Tu te A-D tte pot price i te preent vlue of te future pot price were te interet rte r i tte dependent. Becue it will be ueful lter, note tt becue p dollr in tte ve dte 0 vlue (i.e. preent vlue) of p p =, one dollr in tte preent vlue of + r + r. Next uppoe tt individul nticipte tt mrket will reopen. Now conumer cn purce unit of commodity l in tte in te A-D mrket t dte 0 or plce p l dollr in bnk, ern interet Jon Riley November 5, 202

16 Eentil Microeconomic -6- r nd tu ve p ( + r ) = p in tte. Tu tere i no rbitrge opportunity nd conumer re l l indifferent to weter tey trde in te A-D mrket or trde firt wit finncil intermediry nd ten in te future pot mrket. Jon Riley November 5, 202

17 Eentil Microeconomic -7- Replcing A-D Stte-Contingent Mrket by Trde in Security Mrket If it eem fr-fetced to imgine being ble to trde in tte-contingent mrket in ll commoditie, you re rigt! We now rgue tt tere i imple wy to economize on te number of mrket. Inted of trding in every commodity, ll tt i necery i for n individul to be ble to move welt cro time nd tte nd ten trde in future pot mrket. Let W be te extr welt in tte tt individul need to finnce i purce, tt i F f f = ω θ f = W p x p p y, =,..., S. (8.2-5) Summing over te conumer H H H F f f W = ( p x p ω ) θ p y = = = f = H H F f f ( p x p ω ) θ p y = = f = = H F f p x ω y = f =, becue reolding mut um to = ( ) Jon Riley November 5, 202

18 Eentil Microeconomic -8- = 0, becue upply equl demnd in n A-D equilibrium. Ti mut be true becue tere cn be no ggregte trnfer of welt in or out of tte, even toug individul cn ift welt cro time nd tte. Similrly let W 0 be te dte 0 welt needed to finnce tee trde; tt i F f f 0 = 0 ω0 + θ f = W p p y p x. (8.2-6) Te A-D lifetime budget contrint i S S F S f f f = 0 ω0 + ω + θ = = f = = p x p x p p ( p y p y ) Tt i, te vlue of conumption equl te vlue of te endowment plu te vlue of te profit in ec period (ditributed dividend). Ti budget contrint cn be written follow. S F F f f f f p x ω k y p0 x0 ω0 θ y0 = f = f = ( ) + ( ) = 0 k p y f f Note tt we re uming tt reolder firt-period dividend of 0 0 i reponible for finncing i re of te cot of te firm dte 0 input. from firm f. Alterntively, if ti i negtive, reolder Jon Riley November 5, 202

19 Eentil Microeconomic -9- Subtituting from (8.2-3) nd (8.2-4), S = W = W 0 Tu inted of trding in te tte clim mrket, individul imply need ome wy of trnferring welt cro time nd tte. Jon Riley November 5, 202

20 Eentil Microeconomic -20- Stock Mrket Equilibrium Tu fr we ve not conidered trding in tock mrket. If te dte 0 vlue of te dte dividend of te F firm pn S, ten te previou rgument pply directly. By trding in te tock mrket nd commodity mrket, te tock mrket equilibrium replicte te A-D equilibrium. Te dte 0 mrket vlue of te firm i imply it dte 0 vlue in te A-D equilibrium, tt i P f f = p y. Jon Riley November 5, 202

21 Eentil Microeconomic -2- Perfect Foreigt At firt blu it pper tt we ve eliminted ot of mrket t no cot. Rter tn trde t dte 0 in ll future contingencie, it i equivlent to defer mny trde to lter period by moving welt cro dte vi trde in ecuritie mrket. Becue mrket re not cotle to operte, it i tempting to conclude tt trding in ecurity mrket dominte trding in tte- contingent mrket. However, tere i one very importnt qulifiction. In te A-D equilibrium, conumer nd firm oberve te price of every contingent commodity. However, in te SM equilibrium, individul trding t dte 0 do not oberve te price of commoditie in te future pot mrket. Tu wt we ve relly etblied i tt te SM equilibrium replicte te A-D equilibrium if every conumer correctly forect tee future pot price. Ti i obviouly very trong umption. However, it i not prepoterou it my eem. For mny commoditie ggregte ock tend to be igly correlted cro mrket. Tu reltive price cnge re mll. For commoditie woe reltive price do vry widely cro tte, tere i n incentive for finncil intermedirie to crete new finncil intrument. Te more preciely ti finncil engineering i focued on prticulr commodity nd tte, te more likely it i tt te tte clim price cn be inferred from te price of te finncil et. Jon Riley November 5, 202

22 Eentil Microeconomic -22- Incomplete Mrket Suppoe tt te dividend in ecurity mrket equilibrium do not pn S but inted ome ubpce of dimenion J. We firt conider pecil ce in wic te A-D equilibrium lloction i till cieved. We ten look t n exmple in wic te SM equilibrium lloction differ. Suppoe tt conumer n initil portfolio ξ of te A et. Aet (rel) dividend of d in tte. Ec conumer te me omotetic expected utility function S U( x) = π ux ( ), =,..., H. = Given te omoteticity umption te complete mrket equilibrium i te no-trde equilibrium of ingle repreenttive conumer. Te totl welt i A W = p d. Conumer, wit welt = = A ξ = W p d kwconume frction k of te totl dividend. Tt i, finl conumption i A x k d = =. It follow tt inted of trding in te A-D mrket, ec conumer cn imply purce mutul fund tt perfectly trck te mrket portfolio. Tu te A-D equilibrium lloction Jon Riley November 5, 202

23 Eentil Microeconomic -23- i cieved imply by trding in te et mrket, regrdle of te dimenion of te ubpce pnned by te dividend vector. Wt if we dd initil endowment tt re not trded in te ecurity mrket. Intuitively, long ec conumer endowment vector i pnned by te dividend vector, ti rgument will continue to old. For ten ec conumer totl endowment plu initil portfolio cn be written weigted verge of te dividend vector. We terefore conider n exmple were ti i not te ce. Jon Riley November 5, 202

24 Eentil Microeconomic -24- An exmple: Solving for n incomplete ecurity mrket equilibrium one-good, two-conumer, 3 eqully likely tte, 2 ecuritie wit dividend: d = (, 0,), d 2 = (0,,) Te two conumer ve te me logritmic expected utility function 3 3 π 3 = = U( x) = ux ( ) = ln x, =,2. Contructing n SM equilibrium It i not poible to olve nlyticlly for te olution to portfolio optimiztion problem o we contruct n equilibrium by working bckwrd. Let x = (2, 4,2) be te SM equilibrium conumption of conumer. At ti conumption bundle te mrginl rte of ubtitution or implicit price r = MRS ( x, x ) = π u ( x )/ π u ( x ) re follow: r x x x = (,, ) = (,3,). x x2 x3 Jon Riley November 5, 202

25 Eentil Microeconomic -25- Cooing n optiml portfolio Conumer, wit endowment ω nd portfolio ξ finl conumption of x = ω + ξ d He olve te following problem: 3 Mx{ π u( ω + ξ d ) P ξ 0} ξ = Lgrngin: 3 L = π u( ω + ξ d ) λp ξ, FOC: = 3 = π u ( x ) d ) λp = 0, =, 2 Eliminting te dow price, 3 π u ( x ) d = P2 = 3 P π u ( x ) d =. We cn rewrite ti in term of te implicit price. r d2 P2 = r d P Jon Riley November 5, 202

26 Eentil Microeconomic -26- Conumer : r = (,3,) (, 0,) d =, d 2 = (0,,) (,3,) (,,) 4 P2 Terefore = = (,3,) (, 0,) 2 P. Tu te finl conumption i optiml if te et price rtio i 2. Conumer 2: Note tt for n SM equilibrium r d r d P r d r d P = = 2. r d2 P2 Appeling to te rtio rule it follow lo tt = r d P were r = r2 r = (0, r2, r3). r2 + r3 r2 Terefore = + = 2 r r 3 3. Terefore te FOC old for conumer 2 if r2 = r3 =. Jon Riley November 5, 202

27 Eentil Microeconomic -27- But r =. r = (,3,). Terefore 2 (,4,2) r = MRS ( x, x ) = π u ( x )/ π u ( x ) Terefore: r x x x = (,, ) = (,4,2) x x2 x3 Tu if we cooe 2 x = 8, conumer 2 conumption vector i 2 x = (8,2,4). Conumer portfolio contrint i Ti i tified if ξ = (2, ). Ten P ξ = ξ + 2ξ = 0. x 2 ω = ξ D= (2,,) nd o ω = (0,5,) For mrket clering, ξ + ξ = 0. Terefore ξ = ( 2,) nd o ω = (0,,5). Jon Riley November 5, 202

28 Eentil Microeconomic -28- Comprion wit te AD equilibrium Becue te conumer ve te me omotetic preference, te A-D equilibrium i no-trde equilibrium for te repreenttive conumer. Te normlized A-D equilibrium price vector i x x (, p, p ) = (,, ) = (,3, ), ince x= ω = (20,6,6) x2 x3 Note tt te A-D equilibrium price vector lie between te two implicit price vector r = (,3,) nd 2 r = (,4,2) ocited wit te incomplete mrket equilibrium. p d2 65 P2 Note lo tt = < 2 =. Tu te reltive vlue of ecurity i iger wen mrket re p d 27 P complete. Terefore, if n individul or firm cn predict te effect of dding new ecurity on te reltive price of current ecuritie, tere re profitble rbitrge opportunitie. Tu in n incomplete mrket, finncil intermediry trong incentive to crete new ecuritie. Jon Riley November 5, 202