Banking, Inside Money and Outside Money


 Marshall Powers
 3 years ago
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1 Banking, Inide Mone and Ouide Mone Hongfei Sun Deparmen of Economic Univeri of Torono (Job Marke Paper) Abrac Thi paper preen an inegraed heor of mone and banking. I addre he following queion: when boh individual and bank have privae informaion, wha i he opimal wa o ele deb? I develop a dnamic model wih microfounded role for bank and a medium of exchange. I eablih wo main reul: r, marke can improve upon he opimal dnamic conrac a he preence of privae informaion. Marke price full reveal he aggregae ae and help olve he incenive problem of he bank. Secondl, i i opimal for he bank o require loan be eled wih horerm inide mone, i.e. bank mone ha expire immediael afer he elemen of deb. Shorerm inide mone dominae ouide mone becaue he former make i le col o induce ruhful revelaion and achieve more e cien rik haring. Ke word: banking, inide mone, ouide mone JEL clai caion: E4, G2 I am graeful o Shouong Shi for guidance and inpiraion. I hank David Mill for inighful commen and uggeion. I have alo bene ed from converaion wih eminar paricipan a he Univeri of Torono, he 26 Midwe Macroeconomic Meeing, he 26 Annual Meeing of he Canadian Economic Aociaion, he 26 Cleveland Fed Summer Workhop on Mone, Banking and Pamen and he 26 Reearch on Mone and Marke Workhop. Thi reearch i parl uppored b Shouong Shi Bank of Canada Fellowhip. However, he opinion expreed here i m own and doe no re ec he view of he Bank of Canada. All error are m own. Mailing addre: 5 S. George S., Univeri of Torono, Torono, Onario M5S 3G7, Canada. addre:
2 Inroducion The main goal of hi paper i o inegrae he banking heor wih he monear heor. I addre he following queion: given ha boh individual and bank have privae informaion, wha i he opimal wa o ele deb? Thi i a fundamenal queion concerning an modern econom, where boh ouide mone ( a mone) and inide mone (creaed b bank and pamen em) are ued o faciliae rade. How o ele deb e  cienl i criical for he performance of he banking em a a major ource of lending. There are everal apec o hi iue. For example, wh hould deb be eled wih mone? Which i a beer inrumen for elemen, inide mone or ouide mone? To anwer hee queion, I develop a dnamic model wih microfounded role for bank and a medium of exchange. There are wo pe of fricion in he econom. The r one i lack of ineremporal double coincidence of wan. Thi, along wih paial eparaion and limied communicaion, give rie o a role of mone a he medium of exchange. The econd fricion i wolaered privae informaion. On one hand, agen have privae informaion abou heir random endowmen. Hence banking ha a role in providing rikharing. In paricular, banker can o er dnamic conrac o help agen mooh conumpion over ime. However, he conrac mu be incenive compaible for individual o ruhfull make pamen. On he oher hand, banker have privae informaion abou he uncerain aggregae endowmen becaue he can ler ou he idioncraic hock b aggregaing he repor of individual agen. Thi creae a role for marke o help olve he incenive problem on he bank ide. Indeed, marke a he elemen age generae informaionrevealing price uch ha banker canno lie abou he aggregae ae. In he model, a banking ecor arie endogenoul a he beginning of ime and provide dnamic conrac o agen. According o he conrac, banker lend mone o agen a he beginning of a period and agen ele he curren deb wih banker a he receive endowmen a he end of he period. Each period, he amoun of he loan enilemen of
3 an agen depend on he individual hior of pa elemen (i.e. hi hior of repored endowmen) and he equence of price a elemen age. I eablih wo main reul in hi paper. Fir, marke can improve upon he opimal dnamic conrac in he preence of privae informaion on he bank ide. Marke of good for mone a he elemen age generae price ha full reveal he aggregae ae. Thi colel olve he incenive problem of banker. However, if deb are required o be eled wih real good, no marke will arie a he elemen age. Therefore, deb elemen mu involve mone in order o e cienl dicipline banker. Second, he opimal inrumen for elemen i he kind of inide mone ha expire immediael afer each elemen. I call i oneperiod inide mone. Inducion of ruhful revelaion i le col wih oneperiod inide mone han wih ouide mone or inide mone of an longer duraion, which leave agen beer inured again idioncraic rik. Agen canno bene from holding oneperiod inide mone acro period becaue i expire righ afer a elemen (which happen a he end of a period). In hi cae, he onl pro able wa for one o defaul i o ave and conume one own endowmen, which i no ver deirable. In conra, when ouide mone i valued, an agen nd i more pro able o defaul b carring ouide mone acro period han aving endowmen. The reaon i ha he agen can ue he hidden ouide mone o bu hi preferred conumpion good. Thu he gain of defaul i higher wih ouide mone han wih oneperiod inide mone. The ame argumen applie o inide mone of longer duraion. Longerermed inide mone funcion imilarl o ouide mone and involve higher incenive o mirepreen in period when he curren iue of mone doe no expire. Therefore, oneperiod inide mone help he opimal dnamic conrac implemen beer allocaion. In equilibrium, more e cien rikharing i achieved and welfare i improved. The ke o he above reul i he iming of he expiraion of inide mone, which i exacl when each elemen of deb i done. Once an agen obain uch inide mone for he elemen, making he pamen o he bank i nohing bu giving up ome worhle 2
4 objec. However, hi i no rue if ouide mone i required for elemen. Ouide mone will ill be valuable o he agen afer he elemen age. Hence he incenive o defaul are much ronger wih ouide mone. No urpriingl, in aion of ouide mone can be ued o correc incenive. Wih ouide mone geing le valuable a ime goe on, inducion of ruhful revelaion end o ge le col. The model of hi paper i buil upon Andolfao and Noal (23) and Sun (JME, forhcoming). Andolfao and Noal (23) conruc a model wih paial eparaion, limied communicaion fricion and limied informaion fricion. The explain wh mone creaion i picall aociaed wih banking. Sun (JME, forhcoming) addree he problem of monioring bank wih undiveri able rik and how ha here i no need o monior a bank if i require loan o be repaid parl wih mone. A marke arie a he repamen age and generae informaionrevealing price ha perfecl dicipline he bank. Thi reul i renghened in he curren paper of mine, which feaure an enduring relaionhip beween banker and he conraced agen. In conra o he aic conrac udied in Sun (JME, forhcoming), here I how ha even he more ophiicaed conrac form, dnamic conrac, can ue he help of marke o deal wih he incenive problem of banker. M work i complemenar o he lieraure ha examine he funcioning of inide mone and ouide mone, e.g. Cavalcani and Wallace (999), Williamon (24), He, Huang and Wrigh (25, 26) and Sun (JME, forhcoming). Cavalcani and Wallace (999) ud a random maching model of mone and prove ha inide mone ha he advanage of faciliaing rade beween banker and nonbanker becaue wih inide mone banker are no conrained b rading hiorie. One of he iue addreed b Williamon (24) i he implicaion of privae mone iue for he role of ouide mone. Inide mone ha he advanage of being exible and i repond o unanicipaed hock beer han ouide mone. He, Huang and Wrigh (25, 26) ud mone and banking in a mone earch model. Bank liabiliie are ideni ed a a afer inrumen han cah while cah i 3
5 le expenive o hold. In equilibrium, agen ma nd i opimal o hold a mix of boh. Sun (JME, forhcoming) eablihe ha wih muliple bank, inide mone help achieve beer oucome han ouide mone doe. The reaon i ha he compeiion of privae monie drive up he equilibrium reurn of mone and improve welfare. A prohibiion on inide mone iue no onl eliminae mone compeiion bu alo rigger freerider problem among banker, which decreae welfare. All he above paper focu on he role of inide mone and ouide mone a alernaive inrumen o faciliae rade. In conra, hi paper of mine ake a new e no le imporan perpecive, which i he e cienc of alernaive monear inrumen for eling deb. Thi paper develop an inegraed heor of mone, banking and dnamic conrac, which i b far a rare e or in he lieraure. A relaed previou work i b Aiagari and Williamon (2). The ud mone, credi and dnamic conrac. In heir model, nancial inermediarie wrie longerm conrac wih conumer. Mone i eenial becaue of limied paricipaion in he nancial marke. There are incenive problem due o privae informaion and limied commimen. Wih limied commimen, in aion ha a large impac on he diribuion of welfare and conumpion. In conra, here incenive problem are caued b privae informaion and aggregae uncerain. I i eenial o have conrac ha require elemen be made wih mone, in order o cope wih he incenive problem of banker. Boh inide mone and ouide mone are examined o derive he mo e cien pamen em for inducion of ruhful revelaion. The remainder of he paper i organized a follow. Secion 2 decribe he environmen of he model. Secion 3 udie banking wih ouide mone. Secion 4 examine banking wih inide mone. Secion 5 explore banking wih cocirculaion of inide mone and ouide mone. Secion 6 udie he exience and uniquene of he banking equilibrium. Secion 7 conclude he paper. 4
6 2 The environmen Time i dicree and ha in nie horizon, = ; ; :::;. Each period coni of hree ubperiod, indexed b = ; 2; 3. There are hree iland indexed b i = a; b; c. Each iland i populaed b a coninuum of agen who have uni ma, live forever and dicoun acro ime wih facor 2 (; ). A an poin in ime, here are onl wo iland in communicaion, from which agen can freel vii each oher. The equence of communicaion a an dae i he following: iland a and b a =, iland b and c a = 2, and iland c and a a = 3. Traveling agen reurn o heir naive iland a he end of he ubperiod. Agen on iland i receive endowmen of pe i good. Tpe b good are endowed a = of all, pe c good a = 2 of all and pe a good a = 3 of all. For individual pe b and pe c agen, he endowmen i deerminiic a for all, where < <. However, he endowmen of a pe a agen i ochaic: =, where and are boh random variable and E ( ) =. Here i an aggregae hock, which i common o all pe a agen. I i i.i.d. acro ime according o he probabili deni funcion f () and he cumulaive diribuion funcion F (). The variable i an idioncraic hock. I i i.i.d. over ime and drawn in uch a wa ha he law of large number applie acro pe a agen, according o PDF g () and CDF G (). Boh f () and g () have uppor [; ]. Le h () and H () denoe he PDF and CDF of, repecivel. B Rohagi wellknown reul, h () = R f () g d. The realizaion of, no or peci call, i privae informaion of he agen. All agen know abou f () and g (). The aggregae endowmen of pe a good i no publicl obervable. Endowmen are received prior o he arrival of an raveling agen a he ar of each. All good are perihable. In paricular, pe b and pe c good can la for onl one ubperiod and canno be ored acro ubperiod. Tpe a good, however, can la for wo ubperiod. Tha i, he endowmen of pe a good a = 3 of become inconumable For he diribuion of he produc of wo coninuou random variable, ee Rohagi (976). 5
7 aring = 2 of +. Agen preference are a follow: X U a = E u C;b a + "C a ;a = U b = E X = U c = E X = C;c b + C;b b C;a c + C;c c where he funcion u : R +! R i wice coninuoul di ereniable wih u > and u <, and C i ;j denoe a pe i = a; b; c agen conumpion of dae pe j = a; b; c good. Tha i, he upercrip characerize he agen and he ubcrip decribe he conumpion good. I i given ha C a ;a =. Noe ha agen can eiher conume heir own endowmen or anoher paricular pe of good. In conra o pe i = b; c agen, pe a agen onl conume heir own endowmen a one ubperiod over. 2 The preference parameer " i a ver mall poiive number, i.e. < ". Tha i, pe a agen rongl prefer pe b good o heir own endowmen. There i lack of ineremporal double coincidence of wan among variou pe of agen. In paricular, pe a agen would like o rade endowmen for pe b good. However, pe b agen do no value pe a good. Tpe b agen can conume pe c good, bu pe c agen do no value pe b good. Similarl for pe c and pe a agen. Thi lack of double coincidence of wan, ogeher wih he limied communicaion fricion, generae a role for mone. A he beginning of ime, each pe a agen i endowed wih M uni of orable a objec called ouide mone. Agen can rade mone for good oher han heir own endowmen (ee Figure ). Wih random endowmen, pe a agen mone income will alo be random. Banking ha a role in providing rikharing o a o 2 Thi aumpion, along wih he aumpion ha pe a good can la for wo ubperiod, i inended o implif anali bu i no criical for he main reul. A a reul of hee aumpion, a pe a agen currenperiod deciion of ruhfull eling deb i independen of hi conumpion of pe b good earlier hi period. 6
8 e cienl inure pe a agen again he idioncraic rik. [Iner Figure ] 3 The banking arrangemen A banking ecor arie endogenoul a he beginning of =. Each pe a agen chooe o be a banker or a nonbanker. Banker o er longerm conrac o nonbanker, o help hem mooh conumpion over ime. end up o ering he ame equilibrium conrac. Banking i compeiive and he banker Becaue of he free enr o banking, he equilibrium conrac i uch ha individual banker and nonbanker earn he ame expeced lifeime uili. Wihou lo of generali, i i convenien o hink of banker work ogeher a one inermediar, i.e. he bank. Boh he bank and nonbanker commi o he conrac. All erm of he conrac are public informaion. Marke rade are compeiive. The bank aim o inure pe a agen again he idioncraic endowmen hock. Perhap he mo raighforward banking arrangemen i a follow. A each =, he bank o er mone in exchange for he endowmen of pe b agen and hen allocae pe b good e cienl among pe a agen. Then a each = 3, he bank collec pe a endowmen, give he endowmen o pe c agen in exchange for mone, and hen allocae he re of he pe a good (if an) e cienl among pe a agen. There are woided incenive problem aociaed wih a banking arrangemen a decribed above. On one hand, incenive problem arie due o privae informaion a he individual level. For pe a agen, none of he individual endowmen, conumpion and mone holding i obervable. I focu on incenive compaible allocaion. Tha i, an banking arrangemen mu be uch ha individual pe a agen (boh banker and nonbanker) will ruhfull reveal heir endowmen hroughou ime. On he oher hand, he bank ha he incenive o lie abou he aggregae ae. Noe ha he bank collec pe 7
9 a endowmen and hence ge o know exacl wha he aggregae endowmen i baed on he repor of individual endowmen. In oher word, he aggregae endowmen become privae informaion of he bank. Therefore, he bank alwa ha he incenive o mirepreen he aggregae informaion unle oherwie diciplined. For example, he bank can claim an advere aggregae ae and keep he hidden good o bene i banker, inead of ranferring he good o pe c and pe a agen a i hould. The incenive problem on he bank ide i known a he problem of monioring he monior. Noe ha he bank canno be acuall moniored here becaue here i no ae veri caion echnolog in hi model. (Even if here wa, ae veri caion would be col.) One wa o induce ruhful revelaion of he bank i o deign a conrac ha make he banking pro depend on he aggregae ae announced b he bank. Tha i, o reward he bank (wih higher pro ) a i announce a high aggregae ae and o punih i (wih lower pro ) for claiming a low ae. However, hi mechanim will alo be col becaue i dior he opimal allocaion. In a nie horizon model of banking, Sun (JME, forhcoming) how ha he bank i perfecl diciplined if loan are required o be repaed wih mone. Thi reul can be readil applied here in he curren model. Inead of he bank managing all he allocaion of good, he opimal conrac require ha a lea par of he allocaion are done hrough monear pamen (from he bank o nonbanker and vice vera). A agen are obliged o make monear pamen, he mu rade endowmen for mone r. I will how laer ha marke arie accordingl on iland a and generae price ha full reveal he aggregae ae. A a reul, he incenive problem of he bank i olved colel. The bank can iue privae mone, which i alo known a inide mone. Beween inide mone and ouide mone, he bank chooe he opimal inrumen for eling deb. In wha follow, I ud di eren banking arrangemen which involve alernaive kind of mone. Then I compare he reul of he variou arrangemen and characerize he opimal banking conrac. 8
10 4 Banking wih ouide mone For now, aume ha privae mone iue i prohibied. The banking conrac require ha monear pamen be made wih onl ouide mone. The conrac peci e ha (i) a he beginning of each he bank pa he nonbanker m 2 R + uni of ouide mone o nance hi dae conumpion of pe b good; (ii) a = 3 of each, he nonbanker mu ell a fracion z of hi endowmen for ouide mone and hen conribue o he bank hi mone income p a z and he re of hi endowmen ( z), where p a i he marke price of pe a good for ouide mone. Then he bank reallocae he colleced pe a good among pe a agen. Triviall, a nonbanker dae conumpion of pe b good i nanced b hi endowmen of M uni of ouide mone. Afer mone pamen o nonbanker, he banker ue he reidual mone balance o nance heir own conumpion of pe b good. Each banker i allocaed m B 2 R + uni of ouide mone a he one of each period. A each = 3, each banker mu alo ell z uni of endowmen and conribue he income p a z and he re of hi endowmen ( z). Then banker divide he pe a good among hemelve afer he allocaion o nonbanker. 4. Timing of even Timing of even i illuraed b Figure 2. In an, a he beginning of =, he bank allocae mone among nonbanker and i banker. Then pe a agen vii iland b and rade mone for pe b good. A = 2, pe b agen rade mone for pe c good. A = 3, r pe c agen rade mone for pe a good. Then pe a agen make pamen o he bank, which i called he elemen. The bank reallocae he colleced pe a good (if an) among pe a agen. The above procedure i repeaed for all. [Iner Figure 2] 9
11 4.2 The banking equilibrium Le v be a nonbanker expeced lifeime uili precribed b he conrac. Correpondingl, W i a banker expeced lifeime uili. Le 2 [; ] be he equilibrium meaure of banker (i.e. he ize of he bank) and hence he equilibrium meaure of nonbanker. De niion A banking equilibrium coni of a conrac wih he iniial promied value v o a repreenaive nonbanker and he aociaed iniial value W o a repreenaive banker, an aggregae meaure, allocaion C;c; b C;b b ; Cc ;a; C;c c, marke price = p a ; p b ; p c uch ha: (i) given v = and, he conrac maximize W while delivering he promied v ; (ii) clear he marke of conrac, ha i, W = v ; (iii) given price and he conrac, allocaion C b ;c; C b ;b ; Cc ;a; C c ;c uiliie; (iv) price p a ; p b ; p c clear good marke for all. = maximize pe b and pe c agen Before examining he banking conrac, i i helpful o r ud he equilibrium deciion of pe b and pe c agen. Conider pe c agen be repone. Taking (p c ; p a ) a given, a repreenaive pe c agen maximize hi expeced lifeime uili: max (C c ;a ;Cc ;c ;dc +) E X = C;a c + C;c c :: p a C c ;a + d c + = d c + p c C c ;c where d c i he pe c agen beginningof mone holding. Le C c ;a; C c ;c; d c + denoe he opimal choice. Similarl, aking p b ; p c a given, a repreenaive pe b agen maximize hi expeced lifeime uili: max (C b ;c ;Cb ;b ;db +) E X = C;c b + C;b b :: p c C b ;c + d b + = d b + p b C b ;b
12 where d b i he pe b agen beginningof mone holding. Le he opimal choice. C b ;c; C b ;b ; db + denoe The equilibrium price are p b = D a = c b ;b, p c = D= b c c ;c and p a = D c =Z for all, where D i i he aggregae mone uppl o he marke b pe i agen and Z = zy i he aggregae uppl of pe a good o he marke when he aggregae endowmen i Y. I i raighforward o derive ha d b + = d c + = and C b ;b = Cc ;c = E [Z ] = ( z) for all. Neiher pe b nor pe c agen hold mone acro period becaue he receive a conan ream of endowmen. Now I proceed o ud he opimal banking conrac. Fir he bank mu decide he opimal fracion of he aggregae pe a endowmen o be raded in he marke, z. Ex ane he expeced amoun of pe a good o be aved and conumed b pe a agen ever period i ( z), which i equivalen o conuming " ( z) uni of pe b good. Suppoe inead of aving i up, he bank alo require he fracion z of he aggregae endowmen o be old o pe c agen. According o C;b b, hi will ge pe b agen o ell ( z) more uni of good o pe a agen. Since " <, i i e cien for he bank o require z =. A a reul, pe a agen mu ell all heir endowmen o pe c agen. In reurn, he aggregae conumpion of pe a agen i maximized and equal o uni of pe b good ever period. Now le c denoe a nonbanker dae conumpion nanced b he conrac. Thu, c = m where p b p b i he dae price of pe b good for ouide mone and m = M. P Wihou lo of generali, Normalize M =. The conrac precribe v = E u (c ). Correpondingl, c B = mb denoe a banker dae conumpion of pe b good and hence p b P W = E u c B. Again m B =. = Due o privae informaion of individual, pamen from he bank o a nonbanker = mu be baed on he laer repored hior of endowmen. Recall ha I focu on incenive compaible conrac. Unle oherwie aed, repored value alo repreen rue value. Denoe a nonbanker hior of repored endowmen up o period a
13 h = ( ; ; ; ) 2 [; ] +. Since z = z B = for all, he equilibrium price i p a = Y = R g()d for all. Hence he marke price a he elemen age ( = 3) full reveal he aggregae ae, i.e. = p a E[]. In oher word, agen can infer he rue aggregae ae impl b oberving he marke price. A a reul, he bank canno mirepreen he aggregae informaion o bene i banker. Denoe he price equence of elemen age up o period a P = (p a ; p a ; ; p a ) 2 R + +. The banking conrac can be formall de ned a follow. De niion 2 A conrac i a conan and a equence of funcion f g = where : [; ] R +! R +. The conumpion ream o a nonbanker depend on hi repored hior of endowmen and he price equence of elemen age. Tha i, c = and c = (h ; P ) for all. 4.3 The conrac deign problem The conrac deign problem of he bank can be formulaed recurivel. A he end of = 3 of an, nonbanker repor curren endowmen and make he correponding pamen o he bank. Then he bank make deciion on fuure pamen and promied value according o wha nonbanker have repored. For an, each nonbanker i ideni ed wih a number v +, which i hi dicouned fuure value aring + and i wa promied o him b he bank a. The bank deliver v + b nancing a aedependen nexperiod conumpion c + and a promied value v +2 aring period + 2. Le he deni funcion + (v + ) characerize he diribuion of he promied value made b he bank o be delivered aring +. Then + i he ae variable for he bank recurive problem a he end of each period. Noe ha he = conumpion of pe b good i nanced b he agen endowmen of ouide mone. Thu c =. Since v i he lifeime expeced value promied b he conrac, i follow ha v = v u () 2
14 and 8 ><, if v = v u () (v ) = >:, oherwie : () The objecive of he recurive conrac deign problem i o maximize a repreenaive banker expeced dicouned value W + aring +, while delivering he diribuion of promied value +. Dropping ime ubcrip and leing + denoe + and +2 denoe + 2, he bank endofperiod objecive can be formulaed b he following funcional equaion: (T W + ) + = max (c B + ;c +;v +2) Z Z u c B + (; ) + W g d f () d The maximizaion problem i ubjec o he following condiion: (2) u [c + (; ; v + )] + v +2 (; ; v + ) (3) u c + (e; ; v + ) + max " ( e) + ( ) ( e) pa () + v 2[;] p b +2 (e; ; v + ) + 8 ; v + ; 8 e < u c B + (; ) u c B + (e; ) + max " B ( e) + B ( e) pa () B 2[;] p b + 8 ; 8 e < (4) Z Z fu [c + (; ; v + )] + v +2 (; ; v + )g g d f () d = v + ; 8 v + (5) +2 (w +2 ; ) = Z Z f(;v + ):w +2 =v +2 (;;v + )g g d + (v + ) dv + ; 8 (6) Z c B + (; ) g d +( ) Z Z V c + (; ; v + ) g d + (v + ) dv + = ; (7) 8 3
15 c + (; ; v + ) ; 8 ; ; v + (8) c B + (; ) ; 8 ; (9) v +2 (; ; v + ) 2 ; V ; 8 ; ; v + () where V = u () = ( ) i he value of he unconrained rbe conrac ha nance conumpion of uni of pe b good ever period. Conrain (3) and (4) are he incenive compaibili conrain for a nonbanker and a banker repecivel. Incenive compaibili require ha boh banker and nonbanker are induced o ell he enire endowmen and urn over he enire income ever period. Here c + (; ; v + ) and v +2 (; ; v + ) are a nonbanker nex period conumpion and promied value aring he period afer he nex, given ha he i currenl promied v +, hi curren endowmen i and he curren aggregae ae i. For a banker, c B + (; ) i hi nexperiod conumpion given hi curren endowmen and he curren aggregae ae. For boh parie, he pao of ruhful revelaion mu be no lower han he pao of an poible deviaion. The righhand ide of (3) i he pao if he nonbanker repor e < inead of he ruh. (Noe ha i i no feaible for an agen o claim e > becaue he would no have p a e > p a uni of mone o ubmi o he bank when hi rue endowmen i.) The mirepored endowmen can eiher be ored for nexperiod conumpion or be raded for mone o bu pe b good. The nonbanker chooe, he fracion of endowmen o be ored, o maximize hi gain of defaul. The r erm in he maximizaion problem (on he righhand ide of [3]) i he exra conumpion of ored endowmen " ( e). The econd erm i he exra conumpion of pe b good purchaed wih he mirepored mone, which i ( ) ( e) pa (). Similar logic for he righhand p b + ide of (4). Given price, an agen opimall chooe = (or B = ) if p a p b + > ". In equilibrium, p a = E() and pb + = = E()E(). Therefore, we have = B = provided 4
16 ha " < E (). Now conrain (3) and (4) can boh be impli ed: u [c + (; ; v + )] + v +2 (; ; v + ) u c + (e; ; v + ) + ( u c B + (; ) u c B + (e; ) + ( e) pa () + v p b +2 (e; ; v + )() + 8 ; v + ; 8 e < e) pa () p b + 8 ; 8 e < (2) Conrain (5) i he promiekeeping conrain. All he value promied o nonbanker mu be delivered. Conrain (6) characerize he law of moion of he ae variable, i.e. he diribuion of he promied value. Conrain (7) i he reource conrain. Conumpion of banker and nonbanker exhau uni of pe b good ever period. Conrain (8)() de ne he choice e for he choice variable. Le W () denoe he xed poin of T in (2). One can how ha W () i a ricl increaing, concave funcion from he fac ha T i a conracion mapping ha map he pace of increaing, concave funcion o ielf. The polic funcion fc + (; ; v + ) ; v +2 (; ; v + )g, ogeher wih he iniial conumpion c and he aociaed iniial promied value v, compleel characerize he lifeime conrac o a nonbanker. Hence, v = u ()+E u (c P ). Similarl, he polic funcion c B + (; ) pin down he iniial value of a repreenaive P banker, W = u () + E u c B. The equilibrium condiion v = W implie ha given, = v = u () + W ( ; ) ; (3) = where i given b (). The above condiion de ne v a a funcion of, i.e. he relaionhip beween he iniial value and he aggregae meaure ha clear he marke of conrac. The equilibrium conrac mu be he one ha o er he highe achievable v. Therefore, v = max 2[;] v (). So far I have e up he conrac deign problem and decribe he banking equilibrium for he banking conrac ha require pamen of ouide mone. The following ecion 5
17 examine he conrac ha require pamen made excluivel of inide mone. Then I compare he implicaion of he wo conrac and how ha i maer wheher inide mone or ouide mone i ued a he elemen inrumen. 5 Banking wih inide mone 5. Oneperiod inide mone Now aume privae iue of mone i permied. The bank can nance conumpion hrough allocaion of privae mone. In hi ecion, I ud he banking arrangemen where ouide mone i no valued and he bank iue a paricular kind of inide mone, oneperiod inide mone (OPIM). Namel, i i iued a he beginning of each and expire a he end of afer he currenperiod elemen are done. 3 A before, le be he equilibrium meaure of banker. The conrac peci e ha (i) a he beginning of all he bank pa he nonbanker m 2 R + uni of inide mone o nance hi dae conumpion of pe b good; (ii) a = 3 of all he nonbanker mu ell he enire endowmen for inide mone and hen conribue o he bank he mone income p a, where p a i now he marke price of pe a good for inide mone. The ame noaion are ued a in he previou ecion. In paricular, le c denoe a nonbanker dae conumpion of pe b good nanced b he conrac. Tha i, c = m p b where p b i he price of pe b good for inide mone. Denoe h a a nonbanker hior of repored endowmen up o period and P a he price equence of elemen age up o period. The conrac and he banking equilibrium are ill de ned b De niion and De niion 2, repecivel. The objecive of he recurive conrac deign problem b implemening OPIM i given 3 The expiraion of inide mone can be hough of a he objec deeriorae afer a cerain amoun of ime. Or we can inerpre i a an elecronic accoun whoe balance auomaicall become zero a he precribed poin of ime. Accordingl, a new iue of mone i impl an amoun newl ranferred ino he accoun b banker. 6
18 b (2) ubjec o he ame conrain a (5)(). However, he incenive compaibili conrain are now di eren from (3) and (4): u [c + (; ; v + )] + v +2 (; ; v + ) u [c + (e; ; v + ) + " ( e)] + v +2 (e; ; v + ) (4) 8 ; v + ; 8 e < u c B + (; ) u c B + (e; ) + " ( e) (5) 8 ; 8 e < Conrain (4) i he incenive compaibili conrain for a repreenaive nonbanker and conrain (5) for a repreenaive banker. The righhand ide of he conrain are he pao of defaul. A required b he conrac, pe a agen mu ell he enire endowmen for inide mone. A a reul, ouide mone i no valued b pe b or c agen. Moreover, i i no bene cial for a nonbanker or a banker o ell an mirepored endowmen for inide mone becaue i will expire before period + come. Thu he onl pro able wa o defaul i o ave he hidden endowmen for nexperiod conumpion. Since he iniial allocaion of inide mone doe no depend on an repor of endowmen, naurall m = and c =. Again, v = u () + E u (c ). The P equilibrium conrac mu be he one ha o er he highe achievable v. Index value of banking wih ouide mone b upercrip o and value of banking wih oneperiod inide mone b upercrip I. Provided ha " < E (), we have he following propoiion: Propoiion W o () < W I () for an given. Propoiion 2 W o (v ; ) < W I (v ; ) for an given v and. Propoiion 3 v o < v I. Moreover, v I! V a "! while v o i independen of ". Proof of Propoiion 3 are provided in he Appendix. = Propoiion 2 eablih ha all ele equal banker can alwa achieve a higher uili b o ering conrac wih oneperiod inide mone han wih ouide mone. Accordingl, 7
19 he bank will chooe o implemen he former conrac. Thi reul i driven b he fac ha he incenive o defaul are weaker wih oneperiod inide mone han wih ouide mone. When ouide mone i valued, agen expec i o carr value ino he fuure. On evaluaing he opion o defaul, agen nd i more pro able o ell endowmen for ouide mone han aving hem for conumpion in he following period (given ha " < E []). Oneperiod inide mone, however, expire righ afer elemen. Thu, pe a agen canno bene from elling he hidden endowmen for inide mone. The onl bene from defaul now i o ave he endowmen for nexperiod conumpion, which i aociaed wih a much lower uili gain. Thu i i le col o induce ruhful revelaion wih OPIM. Thi allow he bank o achieve more e cien rikharing and o er higher equilibrium promied value, which i eablihed b Propoiion 3. A a reul, welfare of pe a agen i improved b he conrac ha require elemen be made wih oneperiod inide mone. The overall welfare of he econom i alo improved becaue he expeced lifeime uili of a pe b or pe c agen i rade. regardle of heir opimal deciion o Furhermore, he advanage of he OPIM conrac ge ronger a pe a agen value le of heir own endowmen. A "!, he uili gain of conuming heir own endowmen become negligibl mall. Wih oneperiod inide mone, he incenive o defaul diminih becaue neiher aving endowmen nor rading endowmen for mone i profiable. The reul approache he allocaion achieved b he unconrained rbe conrac. Tha i, c ( ; ; v ) =, c B ( ; ) = and v + ( ; ; v ) = u() for all ( ; ; v ). However, hee polic funcion obvioul do no aif conrain ()(2) of he conrac wih ouide mone. Wih ouide mone, he incenive o defaul are merel driven b he gain of holding ouide mone o he following period. Thee incenive do no go awa even if one doe no value one own endowmen. Therefore, here i no wa he conrac wih ouide mone can implemen perfec rikharing, no even when " =. 8
20 5.2 Inide Mone wih Longer Duraion The previou ecion udie a pecial kind of inide mone, oneperiod inide mone. Welfare i improved wih oneperiod inide mone han wih ouide mone. Now I urn o inide mone of more generalized form and inveigae he aociaed welfare implicaion. The bank iue inide mone ha ha a duraion of period, where i an ineger and 2 <. (Noe ha if =, inide mone never expire, which i equivalen o ouide mone in hi environmen.) Tha i, each iuance of inide mone i made a he beginning of period = ; ; 2;, and expire a he end of period = ; 2 ;. Oher han ha, he bank funcion in he ame wa a in Secion 4. De niion and De niion 2 ill appl. The objecive of he recurive conrac deign problem wih period inide mone i given b T W + + ; = max (c B + ;c +;v +2) Z Z u c B + (; ) + W+2 +2 ; + g d f () d ubjec o he ame conrain a (5)(). However, inead of conrain (3) and (4), here he incenive compaibili conrain are formulaed b he following: (6) u [c + (; ; v + )] + v +2 (; ; v + ) u [c + (e; ; v + ) + + ( ) 2 ] + v +2 (e; ; v + ) 8 ; v + ; 8 e < (7) u c B + (; ) u c B + (e; ) + + ( ) 2 (8) 8 ; 8 e < 9
21 where = 8 >< >:, if = ; 2 ;, oherwie = " ( e) 2 = max " ( e) + ( ) ( e) pa 2[;] p b + : (9) Le v and W be he iniial value of a repreenaive nonbanker and a repreenaive banker, repecivel. The banker recurive conrac deign problem now di er in period wih and wihou expiraion of mone. In period wih expiraion of mone, ha i, = ; 2 ;, he banker problem i imilar o he cae wih oneperiod inide mone. Since he curren iue of mone expire a he end of he period, he onl pro able wa for pe a agen o defaul i o ave he endowmen for nexperiod conumpion. The incenive compaibili conrain are equivalen o (4)(5). In period wihou expiraion of mone, he problem i imilar o he cae wih ouide mone. Agen would prefer o defaul b holding mone ino he nex period. Accordingl, he IC conrain are equivalen o (3)(4). Le v be he equilibrium iniial promied value wih period inide mone. Propoiion 4 v o < v < v I. The proof of Propoiion 4 i provided in he Appendix. Propoiion 4 eablihe ha welfare i he highe wih oneperiod inide mone. Banking wih period inide mone ake he econd place while he ouide mone arrangemen rank he la. Wih period inide mone, incenive o defaul in period wihou expiraion of mone are a rong a wih ouide mone. I doe provide more ringen dicipline when here i expiraion of mone a he end of a period. However, overall agen are no alwa a diciplined a wih oneperiod inide mone. No 2
22 urpriingl, incenive compaibili i ill more col wih period inide mone han wih oneperiod inide mone. Hence Propoiion 4. 6 Cocirculaion of inide mone and ouide mone In hi ecion I ud cocirculaion of inide mone and ouide mone. Previoul, i ha been eablihed ha oneperiod inide mone i he be of all kind of inide mone in ha i help he banking conrac achieve he highe welfare level. Therefore, i make ene here o focu on he cocirculaion of ouide mone and oneperiod inide mone. The conrac peci e ha (i) a he beginning of all he bank pa he nonbanker a porfolio of m I ; m o o nance hi dae conumpion of pe b good, where m I 2 R + i he amoun of curren period inide mone and m o 2 R + i he amoun of ouide mone; (ii) a = 3 of all he nonbanker mu ell ( ) uni of endowmen for currenperiod inide mone and uni of endowmen for ouide mone, where h i 2 [; ] i a conan. Then he porfolio of mone income p a;i ( ) ; p a;o mu be conribued o he bank, where p a;i i he marke price of pe a good for dae inide mone and p a;o i he marke price of pe a good for ouide mone. Triviall, m o =. Noe ha if =, he conrac reduce o one wih onl oneperiod inide mone; if =, he conrac become one wih onl ouide mone. In hi ecion I focu on 2 (; ). Now de ne P = p a;i ; p a;o ; p a;i ; p a;o ; ; p a;i ; p a;o 2 (R + R + ) + a he price equence of elemen age up o period. De niion ill applie. Le p b;o and p b;i repecivel. Then c = mi p b;i + mo p b;o be he marke price of pe b good for ouide mone and dae inide mone, n o for all. Le m B;I ; m B;o denoe a banker beginningofdae porfolio, where m B;I ; m B;o 2 R + and m B;o =. I follow ha c B = mb;i for all. p b;i + mb;o p b;o De niion 3 A banking equilibrium wih cocirculaion of inide mone and ouide mone coni of a conrac wih he iniial promied value v o a repreenaive nonbanker and 2
23 he aociaed iniial value W o a repreenaive banker, an aggregae meaure, allocaion n o C;c; b C;b b ; Cc ;a; C;c c, marke price p a;i = ; p a;o ; p b;i ; p b;o ; p c;i ; p c;o uch ha: (i) given v and, he conrac maximize W while delivering he promied v ; (ii) clear he marke of conrac, ha i, W = v ; (iii) given price and he conrac, allocaion = C b ;c ; C b ;b ; Cc ;a; C c ;c marke for all. = maximize pe b and pe c agen uiliie; (iv) price clear good h In equilibrium, p a;i R i h = = ( ) g ( ) d and p a;o R i = = g ( ) d for all. Obvioul in equilibrium, p a;o p a;i = pb;o p b;i = pc;o p c;i = ; 8: Tha i, he value of ouide mone relaive o inide mone on iland a i given b he raio of he amoun of good required o ell in repecive marke. Expecing hi, pe b and pe c agen value inide and ouide monie b he ame raio. The objecive of he recurive conrac deign problem now i given b (2) ubjec o he ame conrain a (5)(). Inead of conrain (3)(4), here he incenive compaibili conrain are formulaed b u [c + (; ; v + )] + v +2 (; ; v + ) " ( )# (2) u c + (e; ; v + ) + max " ( e) + ( ) ( e) pa;o () 2[;] p b;o + + v +2 (e; ; v + ) 8 ; v + ; 8 e < u c B + (; ) u " c B + (e; ) + max B 2[;] ( )# " B ( e) + B ( e) pa;o () p b;o + 8 ; 8 e < (2) In fac, he above conrain are equivalen o (3)(4) becaue pa;o p b;o + = = E() = pa. p b + 22
24 Similar o he cae wih excluive circulaion of ouide mone, here agen can defaul b elling endowmen for ouide mone. The exra ouide mone obained i ued o purchae more pe b good. Each uni of hidden endowmen can be convered ino pa;o p b;o + uni of nexperiod pe b good. Given price, an agen opimall chooe = (or B = ) if pa;o > ". Therefore, provided ha " < E (), we have = B = for all p b;o + equilibrium price. Thi i exacl he ame reul a in he cae wih ouide mone onl. Le v co denoe he equilibrium iniial promied value aociaed wih cocirculaion of oneperiod inide mone and ouide mone. i.e. 2 (; ). Hence he following propoiion: Propoiion 5 v co = v o. Proof of Propoiion 5 i provided in he Appendix. A a reul, cocirculaion of oneperiod inide mone and ouide mone generae he ame oucome a he ole circulaion of ouide mone. The incorporaion of inide mone ino he ouide mone em, 2 (; ), ha no impac on welfare a all. A long a ouide mone i valued, agen incenive o defaul are ju a high wih or wihou inide mone. The reaon i ha he pro abili of carring he mirepored ouide mone o he ucceeding period depend on he raio of he price of good for ouide mone, p a;o =p b;o +. Wih a conan ouide mone uppl, he price raio p a;o =p b;o + onl depend on he raio of aggregae marke upplie of good, =. The parameer, however, onl a ec he relaive value of ouide mone o inide mone. Therefore, he incenive are a rong a ever unle ouide mone i no valued, =. 6. In aion and incenive Thu far a conan mone uppl ha been aumed. Now I relax hi aumpion and explore he e ec of change in he mone uppl on incenive compaibili and welfare. According o he previou reul, he incenive o defaul are high when ouide mone i valued. Moreover, incorporaion of inide mone ino he ouide mone em doe no 23
25 help weaken he incenive in an wa. The value of carring mirepored ouide mone cruciall depend on he raio of he price of dae pe a good for ouide mone relaive o he price of dae+ pe b good for ouide mone, i.e. p a;o =p b;o +. A change in ouide mone uppl can a ec p a;o =p b;o + and hence he equilibrium oucome. In conra, an change of he ock of inide mone doe no have an impac on p a;o =p b;o +. Wihou lo of generali, he uppl of inide mone i ill aumed o be conan and normalized o one. Le M be he ouide mone uppl a dae. Aume M = ( + ) M, where i a conan. New mone i injeced a lumpum ranfer o pe a agen a he beginning of. Now 2 (; ]. Analogoul, c = mi p b;i + mo +T p b;o and c B = mb;i p b;i + mb;o +T p b;o for all, where T 2 R are he mone ranfer and aken a given b agen. Moreover, m I ; m B;I 2 R + and m o ; m B;o T. Noe ha now m o and m B;o can be negaive, which i inerpreed a pamen from a nonbanker o a banker (m o ) or reallocaion of mone among banker (m B;o ), righ afer he mone ranfer i received. The equilibrium price are p a;o = M and p b;o + = M + recall ha given price, an agen opimall chooe. Reviiing conrain (2)(2), ; B 8 >< >: =, if pa;o p b;o + 2 [; ], if pa;o p b;o + > " = " =, oherwie : (22) Le v inf denoe he equilibrium iniial promied value wih in aion of ouide mone. Hence follow propoiion: Propoiion 6 If E () =", v inf i conan in and v inf = v co = v o ; if > E () =", v inf i ricl increaing in. Alo, v inf! v I a! +. Provided ha " < E (), Propoiion 6 implie ha a high enough poiive in aion rae can correc incenive o ome exen. A a reul, ouide mone i geing le valuable 24
26 a ime goe on. If he aggregae endowmen on iland a i high, ouide mone i more col o obain. I i even more o conidering ha i will no be a valuable omorrow a i i oda. Therefore, for aggregae ae above a cerain hrehold, i.e. > E() "(+), pe a agen would chooe o ave endowmen for nexperiod conumpion if he were o defaul. Oherwie, he prefer o defaul b holding ouide mone acro period. To um up, wih a poiive in aion rae, from ime o ime pe a agen ma nd i more pro able o defaul b aving endowmen han carring ouide mone acro period. In hi cae, agen ge beer diciplined a in aion goe higher. 7 Exience and uniquene of equilibrium Now i ha been eablihed ha i i opimal for he bank o implemen he conrac wih oneperiod inide mone. Thi ecion udie he exience and uniquene of a banking equilibrium. In he banking equilibrium, he bank make allocaion of mone o nance a pe a agen conumpion according o c = and he opimal polic funcion c ( ; ; v ; ) ; v ( ; ; v ; ) ; c B ( ; ; ) for all ha olve (2) ubjec o conrain (5)() and (4)(5). The aggregae meaure clear he marke of conrac uch ha no one can o er a conrac ha achieve a higher iniial value v > v ha ai e W (v ) = v. Propoiion 7 There exi a unique equilibrium iniial value v. Propoiion 7 how ha he banking equilibrium exi and i unique. Noe ha he equilibrium oucome i no he conrained e cien (i.e. econdbe) oucome unle = in he equilibrium. When =, he ize of he bank i negligibl mall. In hi cae, he bank conrac deign problem i analogou o he e cienc problem addreed b Akeon and Luca (992) and oher, in which a planner endeavor o minimize he expeced value of he oal reource he allocae. The reaon wh he conrained e cienc 25
27 i no necearil achieved here i becaue in general he minimum reource needed o aain a given diribuion of promied value ma no exhau all he reource available. In hi model, here i no planner a he reidual claiman. A uilimaximizing privae banker can pro b reaining an poiive reidual. The compeiion in banking reache equilibrium unil he expeced value of being a banker equal he expeced value of a nonbanker. The equilibrium oucome i no he econdbe if he equilibrium meaure of banker i no negligible relaive o ha of nonbanker ( > ). However, a eablihed b Propoiion 3, he main reul of hi paper i robu o an banking conrac: oneperiod inide mone can help he banking conrac achieve beer allocaion for an. Tha i o a, if he econdbe allocaion can be achieved in he banking equilibrium, i can onl be achieved if he conrac require pamen be made wih oneperiod inide mone. 8 Concluion Thi paper ha developed an inegraed heor of mone, banking and dnamic conrac. The heor i ued o evaluae inide mone and ouide mone a alernaive inrumen for eling deb. The model ha microfounded role for boh bank and a medium of exchange. A banking ecor arie endogenoul and o er dnamic conrac o help agen mooh conumpion over ime. According o he conrac, banker lend mone o agen a he beginning of a period and agen ele he curren deb wih banker a he receive endowmen a he end of he period. Each period, he amoun of he loan enilemen of an agen depend on he individual hior of pa elemen (ha i, hi hior of repored endowmen) and he equence of price a elemen age.. The environmen i characerized b a woided incenive problem. A he individual level, agen have privae informaion abou heir random endowmen. Conrac mu be incenive compaible for individual o repor he rue endowmen. On he aggregae level, banker have privae informaion abou he uncerain aggregae endowmen. Thi incenive problem on he 26
28 bank ide give rie o a role for he marke o generae informaionrevealing price o ha he bank canno lie abou he aggregae ae. I have hown ha he opimal inrumen for elemen i he kind of inide mone ha expire immediael afer each elemen. Wih uch oneperiod inide mone, fewer reource are needed o reward ruhful revelaion and agen are beer inured again idioncraic rik. Agen canno bene from holding oneperiod inide mone acro period becaue i expire righ afer a elemen (which happen a he end of a period). A a reul, he onl pro able wa for one o defaul i o ave endowmen for one own conumpion. However, when ouide mone i valued, an agen nd i more pro able o defaul b carring ouide mone acro period han aving endowmen. Tha i, he gain of defaul i higher wih ouide mone han wih oneperiod inide mone. The ame argumen applie o inide mone of longer duraion. Longerermed inide mone funcion imilarl o ouide mone and involve higher incenive o mirepreen in period when he curren iue of mone doe no expire. Therefore, inducion of ruhful revelaion i he lea col wih oneperiod inide mone, which help he opimal dnamic conrac implemen beer allocaion. In equilibrium, more e cien rikharing i achieved and welfare i improved. The ke o he above reul i he iming of he expiraion of inide mone, which i exacl when each elemen of curren deb i done. Once an agen obain uch inide mone for he elemen, making he pamen o he bank i nohing bu giving up ome worhle objec. However, hi i no rue if ouide mone i required for elemen. Ouide mone will ill be valuable o he agen afer he elemen age. Hence he incenive o defaul are much ronger wih ouide mone. No urpriingl, in aion of ouide mone can be ued o correc incenive. Wih ouide mone geing le valuable a ime goe on, inducion of ruhful revelaion end o ge le col. 27
29 Appendix Proof of Propoiion 2. Conider he conrac problem (2) ubjec o conrain (5)() and (4)(5). Since u i ricl increaing in conumpion, conrain (5) implie ha c B + (; ) > c B + (e; ) ; 8 ; 8 e < : (23) Given +, le bc + ; ; v + ; + ; bv+2 ; ; v + ; + ; bc B + ; ; + be he opimal polic funcion for he banking conrac wih ouide mone. Tha i, he maximize he objecive of (2) ubjec o conrain (5)(2). Noe ha bc + ; bv +2 ; bc B + alo aif conrain (4)(5). Tha i, u [bc + (; ; v + )] + bv +2 (; ; v + ) u bc + (e; ; v + ) + ( e) pa () + bv p b +2 (e; ; v + ) + > u [bc + (e; ; v + ) + ( e)] + bv +2 (e; ; v + ) ; 8 ; v + ; 8 e < (24) u bc B + (; ) u bc B + (e; ) + ( e) pa () p b + > u bc B + (e; ) + ( e) ; 8 ; 8 e < (25) The above wo ric inequaliie hold becaue " < E () and 2 [; ]. Now conruc he following polic funcion uch ha (5) hold: 8 >< bc B ec B + ; ; + + ; if + ; ; 2 + =, (26) >: ; ; + ; if > 2 bc B + where and are in niel mall poiive number and aif g = g for 2 and all. Value of and exi b he ric inequali of (25). In wha follow, i will be proven ha bc + ; ; v + ; + ; bv+2 ; ; v + ; + ; ec B + ; ; + achieve a higher value of W I + + han bc+ ; ; v + ; + ; bv+2 ; ; v + ; + ; bc B + ; ; + 28
30 do. Fir noe ha given = = = = Z ec B + Z =2 Z bc B + Z bc B + Z bc B + d ; ; + g bc B Z + ; ; + + g d + ; ; + g ; ; + g ; ; + g d + d + d Z =2 Z =2 g =2 bc B + d g d ; ; + Z =2 Z =2 g g g d d d Therefore, ec B + ; ; + and bc+ ; ; v + ; + aif conrain (7). Rewrie he objecive of (2) a he following: W + + = max (c B + ;c +;v +2) Z Z u c B + (; ) g d f () d+ Z W f () d (27) Appl he rorder Talor expanion on he r erm of he above wih ec B + ; ; + : = = = Z Z Z Z Z 8 >< >: 8 >< >: 8 >< >: u ec B + ; ; + g d f () d R =2 u bc B + ; ; + + g + R =2 u bc B + ; ; + R =2 + R =2 g d d 9 >= f () d >; u bc B + ; ; + + u bc B + ; ; + g d d u bc B + ; ; + u bc B + ; ; + g R u 9 bc B + ; ; + g d + R =2 u >= bc B + ; ; + g d f () d R =2 u bc B + ; ; + g d >; 9 >= f () d >; 29
31 = > Z 8 >< >: Z Z + R =2 u bc B + u bc B + ; ; + g R u bc B + ; ; + g d ; ; + d 9 >= f () d >; u bc B + ; ; + g d f () d (28) The ric inequali hold becaue (23) implie ha u bc B + ; ; + u bc B + ; ; + for all 2 ; 2, wih an equali if and onl if =. A ha been eablihed, 2 bc+ ; bv +2 ; ec B + aif conrain (5)() and (4)(5). Noe ha W ake he ame value for bc + ; bv +2 ; ec B + and bc + ; bv +2 ; bc B + becaue of he ame polic funcion bv +2. Thu he econd erm in (27) alo ake he ame value for bc + ; bv +2 ; ec B + and bc+ ; bv +2 ; bc B +. The ric inequali in (28) how ha bc + ; bv +2 ; ec B + achieve a higher value of W I + + han bc+ ; bv +2 ; bc B + do. Therefore, bc + ; bv +2 ; bc B + canno be he opimal polic funcion for he conrac problem of banking wih oneperiod inide mone. I follow ha W I () > W o () for an given. B (3), W I > W o for an given v and. Proof of Propoiion 3. Given, conider wo opimal conrac wih aociaed iniial n value and conumpion ream of ev ; fec g = ; W f ; o n ec B and bv ; fbc g = ; W c ; bc B repecivel. Suppoe W f W c for an ev > bv. Thi mean he opimal conrac n ev ; fec g = ; W f ; o ec B achieve higher lifeime uiliie for boh banker and nonbanker han bv ; fbc g = ; W c ; o = n bc B doe. Hence he laer canno be an opimal = conrac given, which i a conradicion. Therefore, i mu be rue ha W i ricl decreaing in v given. Given, le v o and v I be he oluion o (3), repecivel for he ouide mone arrangemen and he oneperiod inide mone arrangemen. B Propoiion 2, we have v o = W o (v o ; ) < W I (v o ; ). Obvioul, v o 6= v I. Suppoe v o > v I, hen i follow = = o, ha W o (v o ; ) = v o > v I = W I v I ;. Thi i a conradicion becaue W i ricl decreaing in v. Thu, v o < v I all given. Then i mu be ha v o < v I becaue 3
32 v = max v (). 2[;] Wih ouide mone, he conrac deign problem i given b (2) ubjec o conrain (5)(2). I i obviou ha " doe no ener ino he problem a all. Therefore, v o i independen of ". Wih OPIM, " onl ener ino he incenive compaibili conrain. Conider an " < " 2. The value of he righhand ide of he IC conrain are maller wih " han wih " 2. Analogou o he proof of Propoiion, one can conruc alernaive polic funcion ha achieve a higher value for he problem wih " han he opimal polic funcion for he problem wih " 2. (Deail are omied for brevi.) Then i follow ha v I ricl increae a " decreae. When "!, he incenive o defaul diminih and he opimal conrac approache he unconrained rbe conrac. Tha i, c ( ; ; v )!, c B ( ; )! and v + ( ; ; v )! u() ( ; ; v ), which conclude he proof. for all Proof of Propoiion 4. I i raighforward ha W I + + i equivalen o W + + ; wih = for all. Given +, le c + ; ; v + ; + ; ; v +2 ; ; v + ; + ; ; c B + ; ; + ; be he opimal polic funcion for he banking conrac wih period inide mone. A he cae wih ouide mone, we have = for he problem given b (9) and hence 2 >. I follow ha when =, u [c + (; ; v + )] + v +2 (; ; v + ) u [c + (e; ; v + ) + 2 ] + v +2 (e; ; v + ) > u [c + (e; ; v + ) + ] + v +2 (e; ; v + ) 8 ; v + ; 8 e < u c B + (; ) u c B + (e; ) + 2 > u c B + (e; ) + ; 8 ; 8 e < Therefore, c + ; v +2 ; c B + aif all conrain (5)() and (4)(5). I follow ha 3
33 W I + + W + + ; for an given + ;. Analogou o he conrucion in (26), one can nd oher polic funcion ha achieve a higher value for W+ I + han c+ ; v +2 ; c B + do, which implie W+ I + > W + + ; for an given + ;. (Deail are omied for brevi.) Thi in urn implie ha W I > W for an given v and. Analogou o he proof of Propoiion 3, one can how ha v < v I. B he ame oken, W o + + i equivalen o W + + ; where = for all. Similarl, one can prove ha W o + + < W + + ; for an given + ; and hence W o W for an given v and. I follow ha v o < v. Proof of Propoiion 5. Since conrain (3)(4) are equivalen o conrain (2)(2), he conrac problem under cocirculaion of inide mone and ouide mone i exacl he ame a under excluive circulaion of ouide mone. Hence v co = v o. Proof of Propoiion 6. Plugging in he equilibrium price, (22) become ; B 8 >< >: =, if < E() " 2 [; ], if = E() " =, if > E() " : I i raighforward o how ha ; B = for an 2 [; ] if E () =". The conrac deign problem i he ame a given b (2) ubjec o he ame conrain a (5)() and (2)(2). Hence v inf i conan in and v inf = v co = v o b Propoiion 5. Provided ha > E () =", hen ; B = if > E () = [" ( + )] and ; B = if E () = [" ( + )] for given. Thu for high enough aggregae ae, i become le pro able o defaul and carr ouide mone ino he fuure. Now he recurive conrac deign problem i given b (2) ubjec o he ame conrain a (5)(). Neverhele, inead of conrain (3) and (4), he IC conrain become he following: for 32
34 > E () = [" ( + )], u [c + (; ; v + )] + v +2 (; ; v + ) u [c + (e; ; v + ) + ] + v +2 (e; ; v + ) (29) 8 ; v + ; 8 e < u c B + (; ) u c B + (e; ) + ; 8 ; 8 e < (3) and for E () = [" ( + )], u [c + (; ; v + )] + v +2 (; ; v + ) u [c + (e; ; v + ) + 2] + v +2 (e; ; v + ) (3) 8 ; v + ; 8 e < u c B + (; ) u c B + (e; ) + 2 ; 8 ; 8 e < (32) where = " ( e) and 2 = ( e) pa;o of (2) can be rewrien a: W inf + + = max (c B + ;c +;v +2) 8 >< >: R + R R p b;o + The above problem can be furher decompoed ino: R. Le = E () = [" ( + )]. Then he objecive u c B + (; ) + W inf g u c B + (; ) + W inf g d f () d d f () d 9 >= >; : W+ inf + = W inf + + ; + W inf + + ; 2 (33) where W inf + + ; = max (c B + ;c +;v +2) Z Z :: (5)() and (29)(3) u c B + (; ) + W+2 inf +2 g d f () d (34) 33
35 and W inf + + ; 2 = max (c B + ;c +;v +2) Z Z :: (5)() and (3)(32) u c B + (; ) + W+2 inf +2 g d f () d (35) Noe ha he problem in (34) i equivalen o he problem for he excluive circulaion of oneperiod inide mone, excep ha he lower bound for i now inead of zero. Similarl, he problem of (35) i equivalen o he problem for circulaion of ouide mone wih a conan mone uppl, wih he upper bound of being inead of one. I i obviou ha W+ inf +! W I + + a! and W inf + +! W o + + = W co + + a! given an +. Conider an < 2. Given +, le bc + ; ; v + ; + ; bv+2 ; ; v + ; + ; bc B + ; ; + be he opimal polic funcion for (33) given = 2. I i raighforward o ee ha bc+ ; bv +2 ; bc B + alo aif all he conrain for (33) given =. Now conruc he following polic funcion ec B + ; ; + uch ha (3) and (32) hold: 8 >< ec B + ; ; + = >: 8 >< >: bc B + ; ; + for 2 [; ] ; if 2 [; ) [ ( 2 ; ] bc B + bc B + ; ; + + for 2 ; 2 ; if 2 [ ; 2 ] ; ; + for 2 ; 2 where and are in niel mall poiive number and aif g = g for 2 ; 2 and 2 [ ; 2 ]. Value of and exi b he ric inequali of (3) given bc B + ; ; + for 2 [ ; 2 ]. Analogou o he proof of Propoiion, i can be hown ha given =, bc + ; ; v + ; + ; bv+2 ; ; v + ; + ; ec B + ; ; + achieve a higher value of W+ inf + han bc+ ; ; v + ; + ; bv+2 ; ; v + ; + ; bc B + ; ; + do. Thi implie ha W inf + + () i ricl decreaing in for an given +. Hence b he ame argumen of he proof of Propoiion 4, v inf i ricl decreaing in. Noe ha i ricl decreaing in. Thu v inf i ricl increaing in. 34
36 Proof of Propoiion 7. Conider he following polic funcion: c + ; ; v + ; + = c B + ; ; + for an given v+ and +. Triviall, v +2 ; ; v + ; + = E ; u c B + ; ; +. Given c + and v +, c B + olve he following maximizaion problem: W + = max c B + Z Z u c B + (; ) g d f () d (36) ubjec o u c B + (; ) u c B + (e; ) + " ( e) ; 8 ; 8 e < (37) Z Since =, he above problem can be rewrien a c B + (; ) g d = ; 8 (38) W + = max c B + Z Z u c B + (; ) g () d () f () d (39) ubjec o u c B + (; ) h i u c B e; + + " e ; 8 ; 8 e < (4) Z c B + (; ) g () d () = ; 8 (4) I i raighforward o how ha he oluion o hi problem c B + exi and i unique. The polic funcion c + ; v +2 ; c B + impl ha v = W (v ). Thi i rue for an 2 [; ]. B de niion, he polic funcion c B + i opimal given c + ; v +2 and hence v. Bu c+ ; v +2 ma no be he opimal polic funcion o achieve v. If he are opimal, hen i i rivial ha he banking equilibrium exi and i unique. Suppoe he are no opimal. Le fc g = be he equence of conumpion achieved b c = and polic funcion c + ; v +2 for all. Since he goal of he bank i o maximize he lifeime expeced uili of a banker, i chooe funcion c + and v +2 o minimize he expeced value of he oal reource i allocae o he nonbanker for an promied value v. Since c + ; v +2 are no opimal b aumpion, here mu be a le col equence 35
37 of allocaion oher han fc g = ha achieve v. Pu i anoher wa, here mu be allocaion ha achieve a higher value han W for a repreenaive banker while delivering he promied v. Formall, i mu be rue ha W (v ) > W and W v = W for ome v > v. Thi hold for an. Le ' (v ) = W (v ). The Theorem of he Maximum deliver ' a a coninuou funcion on v ; V, where V = u() i he value achieved b he rbe conrac. Recall from he proof of Propoiion 3 ha given, he funcion W (v ) i ricl decreaing in v. Therefore, i mu be rue ha here exi a unique v 2 v ; V ha ai e W (v ) = v for an given. The uniquene of he equilibrium value v follow becaue v = max v (). 2[;] 36
38 Figure Monear Trade 37
39 Figure 2 Timing of Even 38
6.003 Homework #4 Solutions
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