Banking, Inside Money and Outside Money

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Banking, Inside Money and Outside Money"

Transcription

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 hor-erm inide mone, i.e. bank mone ha expire immediael afer he elemen of deb. Shor-erm 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 micro-founded 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 wo-laered privae informaion. On one hand, agen have privae informaion abou heir random endowmen. Hence banking ha a role in providing rik-haring. 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 informaion-revealing 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 one-period inide mone. Inducion of ruhful revelaion i le col wih one-period 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 one-period 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 one-period inide mone. The ame argumen applie o inide mone of longer duraion. Longer-ermed 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, one-period inide mone help he opimal dnamic conrac implemen beer allocaion. In equilibrium, more e cien rik-haring 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 informaion-revealing 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 non-banker 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 free-rider 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 long-erm 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 co-circulaion 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 ub-period, 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 ub-period. 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 well-known 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 ub-period. Tpe a good, however, can la for wo ub-period. 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 ub-period 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 rik-haring o a o 2 Thi aumpion, along wih he aumpion ha pe a good can la for wo ub-period, i inended o implif anali bu i no criical for he main reul. A a reul of hee aumpion, a pe a agen curren-period 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 non-banker. Banker o er long-erm conrac o non-banker, 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 non-banker earn he ame expeced life-ime 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 non-banker 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 wo-ided 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 non-banker 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 non-banker m 2 R + uni of ouide mone o nance hi dae- conumpion of pe b good; (ii) a = 3 of each, he non-banker 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 non-banker dae- conumpion of pe b good i nanced b hi endowmen of M uni of ouide mone. Afer mone pamen o non-banker, 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 non-banker. 4. Timing of even Timing of even i illuraed b Figure 2. In an, a he beginning of =, he bank allocae mone among non-banker 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 non-banker expeced life-ime uili precribed b he conrac. Correpondingl, W i a banker expeced life-ime uili. Le 2 [; ] be he equilibrium meaure of banker (i.e. he ize of he bank) and hence he equilibrium meaure of non-banker. De niion A banking equilibrium coni of a conrac wih he iniial promied value v o a repreenaive non-banker 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 life-ime 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 beginning-of- 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 life-ime 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 beginning-of- 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 non-banker 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 non-banker = 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 non-banker 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 non-banker 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, non-banker 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 non-banker have repored. For an, each non-banker 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 nex-period 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 end-of-period- 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 r-be conrac ha nance conumpion of uni of pe b good ever period. Conrain (3) and (4) are he incenive compaibili conrain for a non-banker 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 non-banker 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 nex-period 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 righ-hand ide of (3) i he pao if he non-banker 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 nex-period conumpion or be raded for mone o bu pe b good. The non-banker chooe, he fracion of endowmen o be ored, o maximize hi gain of defaul. The r erm in he maximizaion problem (on he righ-hand 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 righ-hand 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 promie-keeping conrain. All he value promied o non-banker 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 non-banker 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 non-banker. 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. One-period 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, one-period inide mone (OPIM). Namel, i i iued a he beginning of each and expire a he end of afer he curren-period 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 non-banker 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 non-banker 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 non-banker 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 non-banker and conrain (5) for a repreenaive banker. The righ-hand 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 non-banker 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 nex-period 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 one-period 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 one-period 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 one-period 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 []). One-period 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 nex-period 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 rik-haring 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 one-period inide mone. The overall welfare of he econom i alo improved becaue he expeced life-ime 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 one-period 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 r-be 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 rik-haring, no even when " =. 8

20 5.2 Inide Mone wih Longer Duraion The previou ecion udie a pecial kind of inide mone, one-period inide mone. Welfare i improved wih one-period 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 non-banker 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 one-period 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 nex-period 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 one-period 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 one-period inide mone. No 2

22 urpriingl, incenive compaibili i ill more col wih -period inide mone han wih one-period inide mone. Hence Propoiion 4. 6 Co-circulaion of inide mone and ouide mone In hi ecion I ud co-circulaion of inide mone and ouide mone. Previoul, i ha been eablihed ha one-period 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 co-circulaion of ouide mone and one-period 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 non-banker mu ell ( ) uni of endowmen for curren-period 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 one-period 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 beginningof-dae- 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 co-circulaion of inide mone and ouide mone coni of a conrac wih he iniial promied value v o a repreenaive non-banker 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 nex-period 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 co-circulaion 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, co-circulaion of one-period 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 lump-um 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 non-banker 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 nex-period 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 one-period 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. econd-be) 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 uili-maximizing 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 econd-be if he equilibrium meaure of banker i no negligible relaive o ha of non-banker ( > ). However, a eablihed b Propoiion -3, he main reul of hi paper i robu o an banking conrac: one-period inide mone can help he banking conrac achieve beer allocaion for an. Tha i o a, if he econd-be allocaion can be achieved in he banking equilibrium, i can onl be achieved if he conrac require pamen be made wih one-period 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 micro-founded 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 wo-ided 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 informaion-revealing 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 one-period inide mone, fewer reource are needed o reward ruhful revelaion and agen are beer inured again idioncraic rik. Agen canno bene from holding one-period 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 one-period inide mone. The ame argumen applie o inide mone of longer duraion. Longer-ermed 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 one-period inide mone, which help he opimal dnamic conrac implemen beer allocaion. In equilibrium, more e cien rik-haring 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 r-order 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 one-period 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 life-ime 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 one-period 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 righ-hand 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 r-be 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 co-circulaion 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 one-period 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 life-ime 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 non-banker 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 r-be 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

6.003 Homework #4 Solutions 6.3 Homewk #4 Soluion Problem. Laplace Tranfm Deermine he Laplace ranfm (including he region of convergence) of each of he following ignal: a. x () = e 2(3) u( 3) X = e 3 2 ROC: Re() > 2 X () = x ()e d

More information

How Much Can Taxes Help Selfish Routing?

How Much Can Taxes Help Selfish Routing? How Much Can Taxe Help Selfih Rouing? Tim Roughgarden (Cornell) Join wih Richard Cole (NYU) and Yevgeniy Dodi (NYU) Selfih Rouing a direced graph G = (V,E) a ource and a deinaion one uni of raffic from

More information

A Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM)

A Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM) A Brief Inroducion o he Consumpion Based Asse Pricing Model (CCAPM We have seen ha CAPM idenifies he risk of any securiy as he covariance beween he securiy's rae of reurn and he rae of reurn on he marke

More information

Topic: Applications of Network Flow Date: 9/14/2007

Topic: Applications of Network Flow Date: 9/14/2007 CS787: Advanced Algorihm Scribe: Daniel Wong and Priyananda Shenoy Lecurer: Shuchi Chawla Topic: Applicaion of Nework Flow Dae: 9/4/2007 5. Inroducion and Recap In he la lecure, we analyzed he problem

More information

Chapter 13. Network Flow III Applications. 13.1 Edge disjoint paths. 13.1.1 Edge-disjoint paths in a directed graphs

Chapter 13. Network Flow III Applications. 13.1 Edge disjoint paths. 13.1.1 Edge-disjoint paths in a directed graphs Chaper 13 Nework Flow III Applicaion CS 573: Algorihm, Fall 014 Ocober 9, 014 13.1 Edge dijoin pah 13.1.1 Edge-dijoin pah in a direced graph 13.1.1.1 Edge dijoin pah queiong: graph (dir/undir)., : verice.

More information

2.4 Network flows. Many direct and indirect applications telecommunication transportation (public, freight, railway, air, ) logistics

2.4 Network flows. Many direct and indirect applications telecommunication transportation (public, freight, railway, air, ) logistics .4 Nework flow Problem involving he diribuion of a given produc (e.g., waer, ga, daa, ) from a e of producion locaion o a e of uer o a o opimize a given objecive funcion (e.g., amoun of produc, co,...).

More information

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes Even and Odd Funcions 23.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl

More information

How has globalisation affected inflation dynamics in the United Kingdom?

How has globalisation affected inflation dynamics in the United Kingdom? 292 Quarerly Bullein 2008 Q3 How ha globaliaion affeced inflaion dynamic in he Unied Kingdom? By Jennifer Greenlade and Sephen Millard of he Bank Srucural Economic Analyi Diviion and Chri Peacock of he

More information

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes Even and Odd Funcions 3.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl

More information

What is a swap? A swap is a contract between two counter-parties who agree to exchange a stream of payments over an agreed period of several years.

What is a swap? A swap is a contract between two counter-parties who agree to exchange a stream of payments over an agreed period of several years. Currency swaps Wha is a swap? A swap is a conrac beween wo couner-paries who agree o exchange a sream of paymens over an agreed period of several years. Types of swap equiy swaps (or equiy-index-linked

More information

SAMPLE LESSON PLAN with Commentary from ReadingQuest.org

SAMPLE LESSON PLAN with Commentary from ReadingQuest.org Lesson Plan: Energy Resources ubject: Earth cience Grade: 9 Purpose: students will learn about the energy resources, explore the differences between renewable and nonrenewable resources, evaluate the environmental

More information

On the Connection Between Multiple-Unicast Network Coding and Single-Source Single-Sink Network Error Correction

On the Connection Between Multiple-Unicast Network Coding and Single-Source Single-Sink Network Error Correction On he Connecion Beween Muliple-Unica ework Coding and Single-Source Single-Sink ework Error Correcion Jörg Kliewer JIT Join work wih Wenao Huang and Michael Langberg ework Error Correcion Problem: Adverary

More information

YTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment.

YTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment. . Two quesions for oday. A. Why do bonds wih he same ime o mauriy have differen YTM s? B. Why do bonds wih differen imes o mauriy have differen YTM s? 2. To answer he firs quesion les look a he risk srucure

More information

Niche Market or Mass Market?

Niche Market or Mass Market? Niche Marke or Mass Marke? Maxim Ivanov y McMaser Universiy July 2009 Absrac The de niion of a niche or a mass marke is based on he ranking of wo variables: he monopoly price and he produc mean value.

More information

INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS

INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS Ilona Tregub, Olga Filina, Irina Kondakova Financial Universiy under he Governmen of he Russian Federaion 1. Phillips curve In economics,

More information

The Twin Agency Problems in Corporate Finance - On the basis of Stulz s theory -

The Twin Agency Problems in Corporate Finance - On the basis of Stulz s theory - The Twin Agency Problem in Corporae Finance - On he bai of Sulz heory - Von der Fakulä für Machinenbau, Elekroechnik und Wirchafingenieurween der Brandenburgichen Technichen Univeriä Cobu zur Erlangung

More information

Chapter 7. Response of First-Order RL and RC Circuits

Chapter 7. Response of First-Order RL and RC Circuits Chaper 7. esponse of Firs-Order L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural

More information

Efficient Risk Sharing with Limited Commitment and Hidden Storage

Efficient Risk Sharing with Limited Commitment and Hidden Storage Efficien Risk Sharing wih Limied Commimen and Hidden Sorage Árpád Ábrahám and Sarola Laczó March 30, 2012 Absrac We exend he model of risk sharing wih limied commimen e.g. Kocherlakoa, 1996) by inroducing

More information

Infrastructure and Evolution in Division of Labour

Infrastructure and Evolution in Division of Labour Infrarucure and Evoluion in Diviion of Labour Mei Wen Monah Univery (Thi paper ha been publihed in RDE. (), 9-06) April 997 Abrac Thi paper udie he relaionhip beween infrarucure ependure and endogenou

More information

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees

More information

Robust Bandwidth Allocation Strategies

Robust Bandwidth Allocation Strategies Robu Bandwidh Allocaion Sraegie Oliver Heckmann, Jen Schmi, Ralf Seinmez Mulimedia Communicaion Lab (KOM), Darmad Univeriy of Technology Merckr. 25 D-64283 Darmad Germany {Heckmann, Schmi, Seinmez}@kom.u-darmad.de

More information

WHAT ARE OPTION CONTRACTS?

WHAT ARE OPTION CONTRACTS? WHAT ARE OTION CONTRACTS? By rof. Ashok anekar An oion conrac is a derivaive which gives he righ o he holder of he conrac o do 'Somehing' bu wihou he obligaion o do ha 'Somehing'. The 'Somehing' can be

More information

Subsistence Consumption and Rising Saving Rate

Subsistence Consumption and Rising Saving Rate Subience Conumpion and Riing Saving Rae Kenneh S. Lin a, Hiu-Yun Lee b * a Deparmen of Economic, Naional Taiwan Univeriy, Taipei, 00, Taiwan. b Deparmen of Economic, Naional Chung Cheng Univeriy, Chia-Yi,

More information

Mathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)

Mathematics in Pharmacokinetics What and Why (A second attempt to make it clearer) Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions

More information

11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements

11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements Inroducion Chaper 14: Dynamic D-S dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuing-edge

More information

Representing Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1

Representing Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1 Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals

More information

Graphing the Von Bertalanffy Growth Equation

Graphing the Von Bertalanffy Growth Equation file: d:\b173-2013\von_beralanffy.wpd dae: Sepember 23, 2013 Inroducion Graphing he Von Beralanffy Growh Equaion Previously, we calculaed regressions of TL on SL for fish size daa and ploed he daa and

More information

Formulating Cyber-Security as Convex Optimization Problems

Formulating Cyber-Security as Convex Optimization Problems Formulaing Cyber-Securiy a Convex Opimizaion Problem Kyriako G. Vamvoudaki, João P. Hepanha, Richard A. Kemmerer, and Giovanni Vigna Univeriy of California, Sana Barbara Abrac. Miion-cenric cyber-ecuriy

More information

Curbing Emissions through a Carbon Liabilities Market: A note from a climate skeptic s perspective

Curbing Emissions through a Carbon Liabilities Market: A note from a climate skeptic s perspective 2014-20 Curbing Emiion hrough a Carbon Liabiliie Marke: A noe from a climae kepic perpecive Eienne Billee de Villemeur, Juin Leroux Série Scienifique Scienific Serie Monréal Février 2014/February 2014

More information

Present Value Methodology

Present Value Methodology Presen Value Mehodology Econ 422 Invesmen, Capial & Finance Universiy of Washingon Eric Zivo Las updaed: April 11, 2010 Presen Value Concep Wealh in Fisher Model: W = Y 0 + Y 1 /(1+r) The consumer/producer

More information

Mortality Variance of the Present Value (PV) of Future Annuity Payments

Mortality Variance of the Present Value (PV) of Future Annuity Payments Morali Variance of he Presen Value (PV) of Fuure Annui Pamens Frank Y. Kang, Ph.D. Research Anals a Frank Russell Compan Absrac The variance of he presen value of fuure annui pamens plas an imporan role

More information

Economics 140A Hypothesis Testing in Regression Models

Economics 140A Hypothesis Testing in Regression Models Economics 140A Hypohesis Tesing in Regression Models While i is algebraically simple o work wih a populaion model wih a single varying regressor, mos populaion models have muliple varying regressors 1

More information

State Machines: Brief Introduction to Sequencers Prof. Andrew J. Mason, Michigan State University

State Machines: Brief Introduction to Sequencers Prof. Andrew J. Mason, Michigan State University Inroducion ae Machines: Brief Inroducion o equencers Prof. Andrew J. Mason, Michigan ae Universiy A sae machine models behavior defined by a finie number of saes (unique configuraions), ransiions beween

More information

Inductance and Transient Circuits

Inductance and Transient Circuits Chaper H Inducance and Transien Circuis Blinn College - Physics 2426 - Terry Honan As a consequence of Faraday's law a changing curren hrough one coil induces an EMF in anoher coil; his is known as muual

More information

The Chase Problem (Part 2) David C. Arney

The Chase Problem (Part 2) David C. Arney The Chae Problem Par David C. Arne Inroducion In he previou ecion, eniled The Chae Problem Par, we dicued a dicree model for a chaing cenario where one hing chae anoher. Some of he applicaion of hi kind

More information

4.8 Exponential Growth and Decay; Newton s Law; Logistic Growth and Decay

4.8 Exponential Growth and Decay; Newton s Law; Logistic Growth and Decay 324 CHAPTER 4 Exponenial and Logarihmic Funcions 4.8 Exponenial Growh and Decay; Newon s Law; Logisic Growh and Decay OBJECTIVES 1 Find Equaions of Populaions Tha Obey he Law of Uninhibied Growh 2 Find

More information

Fortified financial forecasting models: non-linear searching approaches

Fortified financial forecasting models: non-linear searching approaches 0 Inernaional Conference on Economic and inance Reearch IPEDR vol.4 (0 (0 IACSIT Pre, Singapore orified financial forecaing model: non-linear earching approache Mohammad R. Hamidizadeh, Ph.D. Profeor,

More information

New Evidence on Mutual Fund Performance: A Comparison of Alternative Bootstrap Methods. David Blake* Tristan Caulfield** Christos Ioannidis*** and

New Evidence on Mutual Fund Performance: A Comparison of Alternative Bootstrap Methods. David Blake* Tristan Caulfield** Christos Ioannidis*** and New Evidence on Muual Fund Performance: A Comparion of Alernaive Boorap Mehod David Blake* Trian Caulfield** Chrio Ioannidi*** and Ian Tonk**** June 2014 Abrac Thi paper compare he wo boorap mehod of Koowki

More information

Economics Honors Exam 2008 Solutions Question 5

Economics Honors Exam 2008 Solutions Question 5 Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I

More information

Formulating Cyber-Security as Convex Optimization Problems Æ

Formulating Cyber-Security as Convex Optimization Problems Æ Formulaing Cyber-Securiy a Convex Opimizaion Problem Æ Kyriako G. Vamvoudaki,João P. Hepanha, Richard A. Kemmerer 2, and Giovanni Vigna 2 Cener for Conrol, Dynamical-yem and Compuaion (CCDC), Univeriy

More information

The Transport Equation

The Transport Equation The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be

More information

Calculation of variable annuity market sensitivities using a pathwise methodology

Calculation of variable annuity market sensitivities using a pathwise methodology cuing edge Variable annuiie Calculaion of variable annuiy marke eniiviie uing a pahwie mehodology Under radiional finie difference mehod, he calculaion of variable annuiy eniiviie can involve muliple Mone

More information

A Re-examination of the Joint Mortality Functions

A Re-examination of the Joint Mortality Functions Norh merican cuarial Journal Volume 6, Number 1, p.166-170 (2002) Re-eaminaion of he Join Morali Funcions bsrac. Heekung Youn, rkad Shemakin, Edwin Herman Universi of S. Thomas, Sain Paul, MN, US Morali

More information

Appendix A: Area. 1 Find the radius of a circle that has circumference 12 inches.

Appendix A: Area. 1 Find the radius of a circle that has circumference 12 inches. Appendi A: Area worked-ou s o Odd-Numbered Eercises Do no read hese worked-ou s before aemping o do he eercises ourself. Oherwise ou ma mimic he echniques shown here wihou undersanding he ideas. Bes wa

More information

The Equivalent Loan Principle and the Value of Corporate Promised Cash Flows. David C. Nachman*

The Equivalent Loan Principle and the Value of Corporate Promised Cash Flows. David C. Nachman* he Equivalen Loan Principle and he Value of Corporae Promied Cah Flow by David C. Nachman* Revied February, 2002 *J. Mack Robinon College of Buine, Georgia Sae Univeriy, 35 Broad Sree, Alana, GA 30303-3083.

More information

Morningstar Investor Return

Morningstar Investor Return Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion

More information

1 The basic circulation problem

1 The basic circulation problem 2WO08: Graphs and Algorihms Lecure 4 Dae: 26/2/2012 Insrucor: Nikhil Bansal The Circulaion Problem Scribe: Tom Slenders 1 The basic circulaion problem We will consider he max-flow problem again, bu his

More information

Relative velocity in one dimension

Relative velocity in one dimension Connexions module: m13618 1 Relaive velociy in one dimension Sunil Kumar Singh This work is produced by The Connexions Projec and licensed under he Creaive Commons Aribuion License Absrac All quaniies

More information

Trading Strategies for Sliding, Rolling-horizon, and Consol Bonds

Trading Strategies for Sliding, Rolling-horizon, and Consol Bonds Trading Sraegie for Sliding, Rolling-horizon, and Conol Bond MAREK RUTKOWSKI Iniue of Mahemaic, Poliechnika Warzawka, -661 Warzawa, Poland Abrac The ime evoluion of a liding bond i udied in dicree- and

More information

Permutations and Combinations

Permutations and Combinations Permuaions and Combinaions Combinaorics Copyrigh Sandards 006, Tes - ANSWERS Barry Mabillard. 0 www.mah0s.com 1. Deermine he middle erm in he expansion of ( a b) To ge he k-value for he middle erm, divide

More information

Sc i e n c e a n d t e a c h i n g:

Sc i e n c e a n d t e a c h i n g: Dikuionpapierreihe Working Paper Serie Sc i e n c e a n d e a c h i n g: Tw o -d i m e n i o n a l i g n a l l i n g in he academic job marke Andrea Schneider Nr./ No. 95 Augu 2009 Deparmen of Economic

More information

CHAPTER 11 NONPARAMETRIC REGRESSION WITH COMPLEX SURVEY DATA. R. L. Chambers Department of Social Statistics University of Southampton

CHAPTER 11 NONPARAMETRIC REGRESSION WITH COMPLEX SURVEY DATA. R. L. Chambers Department of Social Statistics University of Southampton CHAPTER 11 NONPARAMETRIC REGRESSION WITH COMPLEX SURVEY DATA R. L. Chamber Deparmen of Social Saiic Univeriy of Souhampon A.H. Dorfman Office of Survey Mehod Reearch Bureau of Labor Saiic M.Yu. Sverchkov

More information

Impact of scripless trading on business practices of Sub-brokers.

Impact of scripless trading on business practices of Sub-brokers. Impac of scripless rading on business pracices of Sub-brokers. For furher deails, please conac: Mr. T. Koshy Vice Presiden Naional Securiies Deposiory Ld. Tradeworld, 5 h Floor, Kamala Mills Compound,

More information

Dividend taxation, share repurchases and the equity trap

Dividend taxation, share repurchases and the equity trap Working Paper 2009:7 Deparmen of Economic Dividend axaion, hare repurchae and he equiy rap Tobia Lindhe and Jan Söderen Deparmen of Economic Working paper 2009:7 Uppala Univeriy May 2009 P.O. Box 53 ISSN

More information

Chapter 6: Business Valuation (Income Approach)

Chapter 6: Business Valuation (Income Approach) Chaper 6: Business Valuaion (Income Approach) Cash flow deerminaion is one of he mos criical elemens o a business valuaion. Everyhing may be secondary. If cash flow is high, hen he value is high; if he

More information

Circuit Types. () i( t) ( )

Circuit Types. () i( t) ( ) Circui Types DC Circuis Idenifying feaures: o Consan inpus: he volages of independen volage sources and currens of independen curren sources are all consan. o The circui does no conain any swiches. All

More information

and Decay Functions f (t) = C(1± r) t / K, for t 0, where

and Decay Functions f (t) = C(1± r) t / K, for t 0, where MATH 116 Exponenial Growh and Decay Funcions Dr. Neal, Fall 2008 A funcion ha grows or decays exponenially has he form f () = C(1± r) / K, for 0, where C is he iniial amoun a ime 0: f (0) = C r is he rae

More information

cooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)

cooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins) Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer

More information

Markit Excess Return Credit Indices Guide for price based indices

Markit Excess Return Credit Indices Guide for price based indices Marki Excess Reurn Credi Indices Guide for price based indices Sepember 2011 Marki Excess Reurn Credi Indices Guide for price based indices Conens Inroducion...3 Index Calculaion Mehodology...4 Semi-annual

More information

AP Calculus BC 2010 Scoring Guidelines

AP Calculus BC 2010 Scoring Guidelines AP Calculus BC Scoring Guidelines The College Board The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in, he College Board

More information

INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES

INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchange-raded ineres rae fuures and heir opions are described. The fuure opions include hose paying

More information

Premium indexing in lifelong health insurance

Premium indexing in lifelong health insurance Premium indexing in lifelong healh insurance W. Vercruysse 1, J. Dhaene 1, M. Denui 2, E. Piacco 3, K. Anonio 4 1 KU Leuven, Belgium 2 U.C.L., Louvain-la-Neuve, Belgium 3 Universià di Triese, Triese, Ialy

More information

Section 7.1 Angles and Their Measure

Section 7.1 Angles and Their Measure Secion 7.1 Angles and Their Measure Greek Leers Commonly Used in Trigonomery Quadran II Quadran III Quadran I Quadran IV α = alpha β = bea θ = hea δ = dela ω = omega γ = gamma DEGREES The angle formed

More information

MTH6121 Introduction to Mathematical Finance Lesson 5

MTH6121 Introduction to Mathematical Finance Lesson 5 26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random

More information

Chapter 2 Problems. 3600s = 25m / s d = s t = 25m / s 0.5s = 12.5m. Δx = x(4) x(0) =12m 0m =12m

Chapter 2 Problems. 3600s = 25m / s d = s t = 25m / s 0.5s = 12.5m. Δx = x(4) x(0) =12m 0m =12m Chaper 2 Problems 2.1 During a hard sneeze, your eyes migh shu for 0.5s. If you are driving a car a 90km/h during such a sneeze, how far does he car move during ha ime s = 90km 1000m h 1km 1h 3600s = 25m

More information

4.2 Trigonometric Functions; The Unit Circle

4.2 Trigonometric Functions; The Unit Circle 4. Trigonomeric Funcions; The Uni Circle Secion 4. Noes Page A uni circle is a circle cenered a he origin wih a radius of. Is equaion is as shown in he drawing below. Here he leer represens an angle measure.

More information

Small Menu Costs and Large Business Cycles: An Extension of Mankiw Model *

Small Menu Costs and Large Business Cycles: An Extension of Mankiw Model * Small enu Coss an Large Business Ccles: An Exension of ankiw oel * Hirana K Nah Deparmen of Economics an Inl. Business Sam Houson Sae Universi an ober Srecher Deparmen of General Business an Finance Sam

More information

Random Walk in 1-D. 3 possible paths x vs n. -5 For our random walk, we assume the probabilities p,q do not depend on time (n) - stationary

Random Walk in 1-D. 3 possible paths x vs n. -5 For our random walk, we assume the probabilities p,q do not depend on time (n) - stationary Random Walk in -D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes

More information

Answer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1

Answer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1 Answer, Key Homework 2 Daid McInyre 4123 Mar 2, 2004 1 This prin-ou should hae 1 quesions. Muliple-choice quesions may coninue on he ne column or page find all choices before making your selecion. The

More information

I. Basic Concepts (Ch. 1-4)

I. Basic Concepts (Ch. 1-4) (Ch. 1-4) A. Real vs. Financial Asses (Ch 1.2) Real asses (buildings, machinery, ec.) appear on he asse side of he balance shee. Financial asses (bonds, socks) appear on boh sides of he balance shee. Creaing

More information

Imagine a Source (S) of sound waves that emits waves having frequency f and therefore

Imagine a Source (S) of sound waves that emits waves having frequency f and therefore heoreical Noes: he oppler Eec wih ound Imagine a ource () o sound waes ha emis waes haing requency and hereore period as measured in he res rame o he ource (). his means ha any eecor () ha is no moing

More information

Option Put-Call Parity Relations When the Underlying Security Pays Dividends

Option Put-Call Parity Relations When the Underlying Security Pays Dividends Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 225-23 Opion Pu-all Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,

More information

1 HALF-LIFE EQUATIONS

1 HALF-LIFE EQUATIONS R.L. Hanna Page HALF-LIFE EQUATIONS The basic equaion ; he saring poin ; : wrien for ime: x / where fracion of original maerial and / number of half-lives, and / log / o calculae he age (# ears): age (half-life)

More information

Chapter 2 Problems. s = d t up. = 40km / hr d t down. 60km / hr. d t total. + t down. = t up. = 40km / hr + d. 60km / hr + 40km / hr

Chapter 2 Problems. s = d t up. = 40km / hr d t down. 60km / hr. d t total. + t down. = t up. = 40km / hr + d. 60km / hr + 40km / hr Chaper 2 Problems 2.2 A car ravels up a hill a a consan speed of 40km/h and reurns down he hill a a consan speed of 60 km/h. Calculae he average speed for he rip. This problem is a bi more suble han i

More information

BALANCE OF PAYMENTS. First quarter 2008. Balance of payments

BALANCE OF PAYMENTS. First quarter 2008. Balance of payments BALANCE OF PAYMENTS DATE: 2008-05-30 PUBLISHER: Balance of Paymens and Financial Markes (BFM) Lena Finn + 46 8 506 944 09, lena.finn@scb.se Camilla Bergeling +46 8 506 942 06, camilla.bergeling@scb.se

More information

C Fast-Dealing Property Trading Game C

C Fast-Dealing Property Trading Game C AGES 8+ C Fas-Dealing Propery Trading Game C Y Collecor s Ediion Original MONOPOLY Game Rules plus Special Rules for his Ediion. CONTENTS Game board, 6 Collecible okens, 28 Tile Deed cards, 16 Wha he Deuce?

More information

Equity Valuation Using Multiples. Jing Liu. Anderson Graduate School of Management. University of California at Los Angeles (310) 206-5861

Equity Valuation Using Multiples. Jing Liu. Anderson Graduate School of Management. University of California at Los Angeles (310) 206-5861 Equiy Valuaion Uing Muliple Jing Liu Anderon Graduae School of Managemen Univeriy of California a Lo Angele (310) 206-5861 jing.liu@anderon.ucla.edu Doron Niim Columbia Univeriy Graduae School of Buine

More information

UNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES. Nadine Gatzert

UNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES. Nadine Gatzert UNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES Nadine Gazer Conac (has changed since iniial submission): Chair for Insurance Managemen Universiy of Erlangen-Nuremberg Lange Gasse

More information

Two-Group Designs Independent samples t-test & paired samples t-test. Chapter 10

Two-Group Designs Independent samples t-test & paired samples t-test. Chapter 10 Two-Group Deign Independen ample -e & paired ample -e Chaper 0 Previou e (Ch 7 and 8) Z-e z M N -e (one-ample) M N M = andard error of he mean p. 98-9 Remember: = variance M = eimaed andard error p. -

More information

AP Calculus AB 2010 Scoring Guidelines

AP Calculus AB 2010 Scoring Guidelines AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in 1, he College

More information

Working Paper Monetary aggregates, financial intermediate and the business cycle

Working Paper Monetary aggregates, financial intermediate and the business cycle econsor www.econsor.eu Der Open-Access-Publikaionsserver der ZBW Leibniz-Informaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Hong, Hao Working

More information

DIFFERENTIAL EQUATIONS with TI-89 ABDUL HASSEN and JAY SCHIFFMAN. A. Direction Fields and Graphs of Differential Equations

DIFFERENTIAL EQUATIONS with TI-89 ABDUL HASSEN and JAY SCHIFFMAN. A. Direction Fields and Graphs of Differential Equations DIFFERENTIAL EQUATIONS wih TI-89 ABDUL HASSEN and JAY SCHIFFMAN We will assume ha he reader is familiar wih he calculaor s keyboard and he basic operaions. In paricular we have assumed ha he reader knows

More information

Pricing Fixed-Income Derivaives wih he Forward-Risk Adjused Measure Jesper Lund Deparmen of Finance he Aarhus School of Business DK-8 Aarhus V, Denmark E-mail: jel@hha.dk Homepage: www.hha.dk/~jel/ Firs

More information

Individual Health Insurance April 30, 2008 Pages 167-170

Individual Health Insurance April 30, 2008 Pages 167-170 Individual Healh Insurance April 30, 2008 Pages 167-170 We have received feedback ha his secion of he e is confusing because some of he defined noaion is inconsisen wih comparable life insurance reserve

More information

Graduate Macro Theory II: Notes on Neoclassical Growth Model

Graduate Macro Theory II: Notes on Neoclassical Growth Model Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2011 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.

More information

Return Calculation of U.S. Treasury Constant Maturity Indices

Return Calculation of U.S. Treasury Constant Maturity Indices Reurn Calculaion of US Treasur Consan Mauri Indices Morningsar Mehodolog Paper Sepeber 30 008 008 Morningsar Inc All righs reserved The inforaion in his docuen is he proper of Morningsar Inc Reproducion

More information

AP Calculus AB 2013 Scoring Guidelines

AP Calculus AB 2013 Scoring Guidelines AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a mission-driven no-for-profi organizaion ha connecs sudens o college success and opporuniy. Founded in 19, he College Board was

More information

Hedging with Forwards and Futures

Hedging with Forwards and Futures Hedging wih orwards and uures Hedging in mos cases is sraighforward. You plan o buy 10,000 barrels of oil in six monhs and you wish o eliminae he price risk. If you ake he buy-side of a forward/fuures

More information

2.6 Limits at Infinity, Horizontal Asymptotes Math 1271, TA: Amy DeCelles. 1. Overview. 2. Examples. Outline: 1. Definition of limits at infinity

2.6 Limits at Infinity, Horizontal Asymptotes Math 1271, TA: Amy DeCelles. 1. Overview. 2. Examples. Outline: 1. Definition of limits at infinity .6 Limis a Infiniy, Horizonal Asympoes Mah 7, TA: Amy DeCelles. Overview Ouline:. Definiion of is a infiniy. Definiion of horizonal asympoe 3. Theorem abou raional powers of. Infinie is a infiniy This

More information

Optimal Investment and Consumption Decision of Family with Life Insurance

Optimal Investment and Consumption Decision of Family with Life Insurance Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker

More information

Acceleration Lab Teacher s Guide

Acceleration Lab Teacher s Guide Acceleraion Lab Teacher s Guide Objecives:. Use graphs of disance vs. ime and velociy vs. ime o find acceleraion of a oy car.. Observe he relaionship beween he angle of an inclined plane and he acceleraion

More information

2.5 Life tables, force of mortality and standard life insurance products

2.5 Life tables, force of mortality and standard life insurance products Soluions 5 BS4a Acuarial Science Oford MT 212 33 2.5 Life ables, force of moraliy and sandard life insurance producs 1. (i) n m q represens he probabiliy of deah of a life currenly aged beween ages + n

More information

Chapter 8: Regression with Lagged Explanatory Variables

Chapter 8: Regression with Lagged Explanatory Variables Chaper 8: Regression wih Lagged Explanaory Variables Time series daa: Y for =1,..,T End goal: Regression model relaing a dependen variable o explanaory variables. Wih ime series new issues arise: 1. One

More information

1. y 5y + 6y = 2e t Solution: Characteristic equation is r 2 5r +6 = 0, therefore r 1 = 2, r 2 = 3, and y 1 (t) = e 2t,

1. y 5y + 6y = 2e t Solution: Characteristic equation is r 2 5r +6 = 0, therefore r 1 = 2, r 2 = 3, and y 1 (t) = e 2t, Homework6 Soluions.7 In Problem hrough 4 use he mehod of variaion of parameers o find a paricular soluion of he given differenial equaion. Then check your answer by using he mehod of undeermined coeffiens..

More information

SOLVENCY II: QIS5 FOR NORWEGIAN LIFE AND PENSION INSURANCE

SOLVENCY II: QIS5 FOR NORWEGIAN LIFE AND PENSION INSURANCE SOLVENCY II: QIS5 FOR NORWEGIAN LIFE AND PENSION INSURANCE BY KEVIN DALBY THESIS for he degree of MASTER OF SCIENCE (Modeling and Daa Anali) Facul of Mahemaic and Naural Science UNIVERSITY OF OSLO Ma 2011

More information

Diagnostic Examination

Diagnostic Examination Diagnosic Examinaion TOPIC XV: ENGINEERING ECONOMICS TIME LIMIT: 45 MINUTES 1. Approximaely how many years will i ake o double an invesmen a a 6% effecive annual rae? (A) 10 yr (B) 12 yr (C) 15 yr (D)

More information

CHARGE AND DISCHARGE OF A CAPACITOR

CHARGE AND DISCHARGE OF A CAPACITOR REFERENCES RC Circuis: Elecrical Insrumens: Mos Inroducory Physics exs (e.g. A. Halliday and Resnick, Physics ; M. Sernheim and J. Kane, General Physics.) This Laboraory Manual: Commonly Used Insrumens:

More information

Fourier series. Learning outcomes

Fourier series. Learning outcomes Fourier series 23 Conens. Periodic funcions 2. Represening ic funcions by Fourier Series 3. Even and odd funcions 4. Convergence 5. Half-range series 6. The complex form 7. Applicaion of Fourier series

More information

Entropy: From the Boltzmann equation to the Maxwell Boltzmann distribution

Entropy: From the Boltzmann equation to the Maxwell Boltzmann distribution Enropy: From he Bolzmann equaion o he Maxwell Bolzmann disribuion A formula o relae enropy o probabiliy Ofen i is a lo more useful o hink abou enropy in erms of he probabiliy wih which differen saes are

More information

Risk Modelling of Collateralised Lending

Risk Modelling of Collateralised Lending Risk Modelling of Collaeralised Lending Dae: 4-11-2008 Number: 8/18 Inroducion This noe explains how i is possible o handle collaeralised lending wihin Risk Conroller. The approach draws on he faciliies

More information