Topics in Macroeconomics

Transcription

1 Topics in Macroeconomics Volume 3, Issue Aricle 3 To Furnish an Elasic Currency : Banking, Aggregae Risk, and Welfare Michael B. Loewy Universiy of Souh Florida Topics in Macroeconomics is one of The B.E. Journals in Macroeconomics, produced by The Berkeley Elecronic Press (bepress). hp:// Copyrigh c 2003 by he auhors. All righs reserved. No par of his publicaion may be reproduced, sored in a rerieval sysem, or ransmied, in any form or by any means, elecronic, mechanical, phoocopying, recording, or oherwise, wihou he prior wrien permission of he publisher bepress.

2 To Furnish an Elasic Currency : Banking, Aggregae Risk, and Welfare Absrac The Federal Reserve Ac calls upon he newly creaed Banks... o furnish an elasic currency... since such acion was hough o be welfare improving during imes of high currency demand. This paper considers he welfare implicaions of an elasic currency regime wihin he conex of an overlapping generaions model of fia money ha includes banks ha face aggregae risk. Alhough in he economy s saionary equilibrium he bank chooses o hold boh fia currency and illiquid capial in is porfolio, i would prefer o hold addiional amouns of currency (were hey available) during periods of high wihdrawal demand. To remedy his problem, a cenral bank is inroduced ha offers zero ineres, inraperiod loans of currency via a discoun window. When he cenral bank opimally chooses he size of he loan, i is shown ha he resuling saionary equilibrium suppors he economy s golden rule allocaion.

3 Loewy: Elasic Currency and Welfare 1 1. Inroducion In he official ile of he ac esablishing he Federal Reserve Sysem, he newly creaed Federal Reserve Banks are called upon... o furnish an elasic currency.... As Friedman and Schwarz (1963) poin ou, his language was likely included in he ile since he framers of he ac were very much ineresed in finding ways o increase liquidiy, and hence o lower ineres raes, during periods of high currency demand such as harves ime. Having he cenral bank provide addiional liquidiy during such periods is consisen wih he real-bills docrine which recommends ha he money supply vary wih is demand. 1 The goal of his paper is o invesigae he welfare implicaions of an elasic currency regime wihin he conex of a dynamic general equilibrium model of banking in which privae banks are subjec o random variaions in he size of heir wihdrawal demand. I is shown ha during periods of high wihdrawal demand, a emporary increase in he supply of fia currency on he par of he cenral bank is welfare improving. As he injecion of currency is boh finie and is removed from circulaion a he end of he period, his policy is shown o cause a emporary fall in he rae of ineres and o do so wihou causing unbounded money growh. Hence, he model is consisen wih Friedman and Schwarz s (1963) observaion ha ineres raes during harves ime were lower afer he creaion of he Federal Reserve Sysem han before and shows ha a common criicism of real-bills policies, namely Wicksellian indeerminacy, need no arise. 2 The modeling framework uilized here begins wih a moneary version of Diamond and Dybvig s (1983) banking model wih he bank subjec o aggregae risk regarding he magniude of is wihdrawal demand. In order o provide a relaively simple way o obain valued fia currency in equilibrium, he bank is placed in a hree-period-lived overlapping generaions framework. Similar modeling approaches are found Smih (1987), Loewy (1991, 1998), Freeman (1996, 1999), Zhou (2000), and Kahn and Roberds (2001). In my model, he bank acquires fia currency, reserves, and an illiquid real asse, capial, o suppor he deposis of young agens. Since he bank is assumed o be unable o observe ex ane he measure of middle-aged agens seeking o wihdraw and currency dominaes capial in rae of reurn over one period, he bank finds i opimal o finance he wihdrawals of early-arriving middle-aged agens solely wih currency. As i is also he case ha capial dominaes currency in rae of reurn over wo periods, i is also opimal for he bank o exhaus is reserves in financing hese paricular wihdrawals. Thus, any lae-arriving middle-aged agens and all members of he same generaion who wish o wihdraw when old receive only capial. While having he laer receive only capial is desirable, having he former do so is no since hese agens, who receive less consumpion han do hose who arrive earlier, are made o bear all of he risk associaed wih he variaion in wihdrawal demand. 3 Those middle-aged agens who do receive currency hen exchange i for goods in he spo marke for currency. This serves o ransfer he currency back o he bank where i now serves as an asse supporing he deposis of he nex generaion. The shor-run rae of reurn dominance of fia currency implies ha if middle-aged wihdrawal demand proves o be high, hen he bank would prefer o finance (in par or in oal) he wihdrawals of laearriving agens using currency were i available. One mechanism for is provision is a cenral bank ha operaes a ype of discoun window. Here, his policy akes he form of a zero-ineres, inraperiod loan of fia currency in exchange for (claims on) immaure capial. 4 Given ha he bank akes raes of reurn on money as given, i follows ha he bank will accep however much currency he cenral bank is willing o offer. From he vanage poin of he cenral bank however, a finie, opimal upper bound for he loan does 1 See Sargen and Wallace (1982), Freeman (1996), and Sproul (2000a, b). 2 Such criicisms appear in Mins (1945), Friedman and Schwarz (1963), Laidler (1984), and Selgin (1989), among ohers. 3 Tha he opimal response o high wihdrawal demand is a parial suspension of wihdrawals is also seen in Wallace (1988, 1990), Loewy (1998), and Green and Lin (2000). 4 Freeman (1996) shows ha he opimal ineres rae on such loans is zero when defaul risk is absen as i is here. Produced by The Berkeley Elecronic Press, 2003

4 2 Topics in Macroeconomics Vol. 3 [2003], No. 1, Aricle 3 exis since alhough more currency increases he consumpion of lae-arriving middle-aged agens, ceeris paribus, i also reduces he rae of reurn on money which in urn reduces he consumpion of all currencyholding agens. In his case, I show ha he resuling saionary moneary equilibrium no only Pareo dominaes he one ha he bank can achieve in he absence of cenral bank credi, i also suppors he golden rule allocaion, namely he allocaion ha maximizes seady-sae ex ane uiliy subjec o he economy s feasibiliy consrains. The opimal cenral bank loan achieves his allocaion by being of exacly he righ size so as o eliminae he parial suspension of wihdrawals ha oherwise arises when wihdrawal demand is high and so spreads he risk associaed wih variable wihdrawal demand across all middle-aged agens. While he cenral bank can uilize is discoun window o achieve a Pareo-improving allocaion, i should be noed ha his resul ough o be viewed as a posiive, raher han a normaive, implicaion of he model. Such a viewpoin is needed here because he paper akes he exisence and liquidiy-provision role of he cenral bank as given wihou jusifying why he cenral bank exiss or engages in his aciviy. In addiion, he paper makes no claim ha he privae secor is unable o achieve he same Pareo-improving allocaion on is own. Inasmuch as his abiliy would preclude he exisence of a cenral bank, he model includes a pair of legal resricions ha preven he privae secor from engaging in aciviies ha serve o duplicae wha he paper akes as he given funcion of he cenral bank. Alhough Freeman (1996, 1999) and Zhou (2000) consider moneary banking models similar o ha used here, heir approaches differ inasmuch as each considers environmens in which here exiss inrageneraional borrowing and lending. They hen focus on how a cenral bank can overcome he liquidiy problems ha arise in he clearing of nominal deb hrough he use of eiher open marke operaions or a discoun window. While deb per se does no exis in my model, liquidiy problems neverheless arise and for reasons similar o ha of Freeman and Zhou a mismach beween he number of agens who wish o consume and he amoun of funds available o finance his consumpion. In conras, Champ e al. (1996) and Williamson (1998) provide a differen general equilibrium moneary model of banking, one in which he need for currency arises from deposiors demanding liquidiy in order o move from one locaion o anoher raher han from uncerainy over he iming of consumpion as is he case here. The remainder of his paper is organized as follows. The nex secion oulines he economy and he sequence of evens wihin each period. Secion 3 considers he commercial bank s problem while he exisence of a saionary moneary equilibrium is he subjec of Secion 4. Secion 5 describes he bank s problem in he presence of cenral bank credi and he corresponding problem of he cenral bank. I hen esablishes he exisence a saionary moneary equilibrium ha is he soluion o he cenral bank s problem and herefore is also he equilibrium ha will arise when he commercial bank is provided wih he opimal currency injecion. The secion finishes by defining a golden rule allocaion and showing ha he saionary moneary equilibrium wih cenral bank credi suppors such an allocaion. Secion 6 concludes. 2. The Environmen 2.1 Fundamenals Consider an economy of hree-period-lived overlapping generaions where a each dae $ 1 a coninuum of ex ane idenical agens (aken o be of measure 1) is born. The generaion o which each agen belongs is public informaion. Each agen born a ime is endowed wih one uni of he economy s lone consumpion good during ime and no unis a any oher dae. Following Bhaacharya and Gale (1987), Freeman (1988), Loewy (1991, 1998), Qi (1994) and ohers, I assume ha each generaion consiss of wo ypes of agens, hose who consume during heir second period of life (early consumers) and hose who consume during heir hird period of life (lae consumers). An agen s ype is assumed o be privae informaion wih he laer no becoming revealed o he agen unil he beginning of his second period of life. Besides facing he risk of ype, agens and he bank also face a form of aggregae risk. Specifically, he measure of early consumers in generaion $ 1 is assumed o be a random variable, λ. To keep maers hp://

5 Loewy: Elasic Currency and Welfare 3 simple, λ is assumed o have a wo-poin suppor where prob { = λ 1 } = 1 - π, prob { = λ 2 λ λ } = π, 0 < π < 1 and 0 < λ 1 < λ 2 < 1. Le e and l be he ime +1 (early) and +2 (lae) consumpion of an agen born a ime $ 1. Since an agen s ype deermines when he consumes, i also deermines his uiliy. Thus, le each agen s uiliy funcion be given by U(e, l, θ) = θ ln(e ) + (1 ) θ)ln(l ), where θ = 0 if he agen is a lae consumer and θ = 1 if he agen is an early consumer. 5 The economy s single echnology, capial, may be accessed eiher direcly or indirecly hrough an inermediary. Per capia holdings of capial, k, earn a fixed gross rae of reurn of x when held for one period and of X when held for wo periods. I is assumed ha 0 < x < 1 < X. Thus, inerruping capial premaurely is cosly. Noe ha he inabiliy o verify an agen s ype implies ha IOUs backed by such goods will never be raded. A = 1 here also exis he agens of generaion 0 who are aken o be of measure 1. To simplify he discussion of equilibria below, I assume ha all members of generaion 0 are early consumers and hence seek o maximize heir consumpion, e 0. These agens are assumed o hold in aggregae M > 0 unis of fia currency. In wha follows, I make he simplifying assumpion ha he supply of fia currency is fixed save for he presence of any loans ha he cenral bank may exend should he commercial bank reques hem. Las, le p be he ime goods price of currency and m be he ime per capia sock of currency. For nonnegaive values of p and m, le q = p m define per capia real balances. If p > 0 for all $ 1, hen he one-period rae of reurn on real balances, R = p +1 /p is well defined. The presence of aggregae risk implies ha in equilibrium R is condiional on he known sequence 6 λ s > &1 s'1 as well as on he sochasic process 6 λ s > 4 s'. However, because in wha follows i is he realizaion of λ ha is mos relevan o he analysis, le R i, i = 1, 2, represen he ime rae of reurn on real balances when λ ' λ i, i = 1, Aciviy wihin each period A = 1, he young agens of generaion 1 join he middle-aged agens of generaion 0 a a locaion where a his and a all oher daes here also exis he economy s commercial bank, is cenral bank, and is spo marke for currency. The commercial bank hen offers is opimal conrac o he members of generaion 1 who deposi heir enire endowmen ino he bank. 6 Assuming, as will be he case in equilibrium, ha he bank s conrac implies ha reserves be par of is porfolio, i eners he ime 1 currency spo marke on behalf of he members of generaion 1 by supplying ime 1 good in exchange for currency ha is inelasically supplied by members of generaion 0. Following rade, generaion 0 agens consume. Once his concludes, all members of generaion 1 remain a he locaion unil he sar of period 2 while all members of generaion 0 exi he economy. Consider nex he sequence of evens ha occur during each dae $ 2. The period begins wih he arrival of generaion a he locaion and hese agens deposiing heir endowmen wih he bank in exchange for he bank s opimal conrac. Coinciding wih his aciviy, he members of generaion -1 learn heir ype while he members of generaion!2 (which exis for $ 3) remain inacive. Second, middle-aged agens who have learned ha hey are early consumers randomly and uniformly conac he bank in order o wihdraw while hose who have learned ha hey are lae consumers remain inacive unil he following period when hey will wihdraw. Third, he early consumers of generaion -1 and he lae consumers of generaion!2 wihdraw. Given ha early consumers randomly and uniformly arrive a he bank and ha in equilibrium lae consumers will self-selec o wihdraw in he nex period, i 5 Diamond and Dybvig (1983), Wallace (1988), Jacklin (1993), Temzelides (1997), and Adão and Temzelides (1998) assume insead ha he rae of reurn is sufficienly high o induce lae consumers o do jus ha raher han consuming early. In conras, Jacklin (1987), Jacklin and Bhaacharya (1988), Chari and Jagannahan (1988), and Alonso (1996) assume ha he preferences of boh ypes are such ha hey consume in boh he second and hird period of life. 6 This follows from he risk-sharing abiliies of he bank coupled wih he inabiliy of agens o issue capial-backed claims. Produced by The Berkeley Elecronic Press, 2003

6 4 Topics in Macroeconomics Vol. 3 [2003], No. 1, Aricle 3 λ &1 follows ha he realizaion of becomes public informaion as soon as he bank observes he arrival (or no) of he marginal early consumer beyond measure λ 1. 7 Fourh, hose generaion -2 and -1 agens who wihdraw during ime receive a paymen of currency and/or capial ha is consisen wih he bank s opimal conrac. Assuming, as will be he case in equilibrium, ha some of hese agens will receive some or all of heir paymen in he form of currency, hen if λ &1 ' λ 1 hese agens proceed o he spo marke where hey exchange heir currency for ime good which is supplied by he bank using he deposis of generaion. 8 Afer he spo marke clears, hose agens whose wihdrawal consised in par or in whole of currency now consume as do any oher members of heir generaion who may have received only capial. On he oher hand, if λ &1 ' λ 2, hen once he bank deermines ha wihdrawal demand is high, i has an incenive o borrow currency o help finance he wihdrawals of lae-arriving agens. Alhough earlyarriving middle-aged agens represen a possible source of such currency and he bank may be able o idenify hese agens since hey have already visied he bank, i is assumed ha here exiss a legal resricion prohibiing such loans. This implies ha he bank mus urn o he cenral bank for such a loan (should i be available). By he same oken, early and lae-arriving middle-aged agens may be able o idenify each oher (since hey are of he same ype and in he same locaion) and wish o engage in side rades prior o he opening of he spo marke. Here, oo, i is assumed ha here exiss a legal resricion prohibiing such rades. Hence, hose middle-aged agens holding currency may only exchange i for goods in he spo marke and once hey do so, hey, along wih any oher members of generaions -1 and -2 holding goods, consume. Las, afer consumpion is complee, aciviy during ime concludes wih generaion -2 exiing he economy while generaions -1 and remain a he locaion. If he bank had received a loan, i is repaid a his ime. Table 1 illusraes he sequence in which he differen evens comprising ime occur. 3. The Bank s Problem A key feaure of he environmen of his paper is he presence of fia currency and is use as a reserve asse on he par of banks. No only does he presence of currency add a second asse o he bank s porfolio, bu i also necessiaes saisfying a se of marke clearing condiions as par of he equilibrium. I hen follows ha he analysis of boh he bank s problem and he equilibrium possess a degree of complexiy no seen when fia currency is absen. Therefore, in order o ry o keep he analysis as racable as possible, I make he common, simplifying assumpion ha he bank arranges is porfolio on a generaion-by-generaion basis. Under his muual-like srucure, he ime +1 wihdrawals made by he members of generaion (or -1) are compleely financed using only he asses ha he bank holds on behalf of ha specific generaion. By design, his assumpion simplifies maers by precluding he bank from uilizing any direc form of inergeneraional ransfer (alhough such ransfers do arise via he spo marke for currency). 9 7 As Diamond and Dybvig (1983, Prop. 1) show, i is his inabiliy o observe λ &1 ex ane ha precludes he bank from providing agens wih he opimal risk-sharing allocaion. Green and Lin (2000) argue ha Diamond and Dybvig s resul may be an arifac of he limied se of conracs ha he bank is permied o offer. Assuming a finie number of agens and hence he presence of aggregae risk, Green and Lin show ha when he bank opimally adjuss is paymens on an agen-by-agen basis condiional upon he paymens i has made previously and he paymens i expecs o make o any remaining agens implies ha he equilibrium yields he ex ane opimal risk-sharing allocaion. Hence, bank runs are precluded. Peck and Shell (2003) show ha his need no be he case, however, should he preferences of early and lae consumers differ and deposiors no know heir place in line. 8 Since he bank holds his currency agains generaion s deposis and also uses i o finance heir wihdrawals in subsequen periods, currency serves as a reserve asse. By he same oken, since capial is also par of he bank s porfolio, i can usefully be hough of as consiuing illiquid loans. 9 In conras, Qi (1994) assumes ha ime +1 deposis are available o finance ime +1 wihdrawals. By he same oken, his effecively leads him o view ime invesmens in he wo echnologies as anoher ype of wihdrawal. hp://

7 Loewy: Elasic Currency and Welfare 5 Table 1 Sequence of Evens wihin Period Sage Young Agens (generaion ) Middle-aged Agens (generaion -1) Old Agens (generaion -2) Commercial Bank 1 Ener economy; deposi endowmen wih commercial bank Presen a locaion; learn ype Presen a locaion Offers opimal conrac o young agens 2 Early consumers randomly and uniformly conac bank 3 Early consumers wihdraw; λ &1 publically observed Lae consumers wihdraw Paymens made; may receive loan from cenral bank if λ &1 ' λ 2 4 If hold currency, ener spo marke; afer rade, all θ = 1 agens consume If hold currency, ener spo marke; afer rade, all θ = 0 agens consume Eners spo marke; demands currency on behalf of young agens 5 Remain a common locaion Remain a common locaion Exi he economy Any loan repaid o cenral bank In order o make he disincion beween reserves and capial meaningful, he only wihdrawals ha are admissible are hose ha can be financed using anywhere from 0 o 100% of he wo asses ha he bank has on hand a he ime ha wihdrawals are made. 10 This implies ha he bank mus choose he share of each asse ha i will use o finance wihdrawals a each dae. Furhermore, hese shares mus sum o one. Le χ 1 and κ 1 be he shares of reserves and capial held on behalf of he members of generaion ha he bank uses o finance he ime +1 wihdrawals of he firs measure λ 1 generaion early consumers. Should a marginal early consumer beyond measure λ 1 arrive a he bank, le χ 2 and κ 2 be he shares of his generaion s reserves and capial offered o he nex measure λ 2 ) λ 1 early consumers who arrive a he bank. Beyond measure λ 2 early consumers, he bank is assumed o fully suspend wihdrawals. This in urn implies ha generaion lae consumers will self-selec o pospone heir wihdrawals. Hence, he observaion (or no) of his marginal early consumer is equivalen o observing he realizaion of λ. In addiion o he reserves and capial ha he bank holds a ime +1 on behalf of generaion, a his dae i also holds he reserves and capial of generaion -1 which remain afer having financed ha generaion s ime wihdrawals. Since he quaniy of he laer depended upon he realizaion of λ &1, so oo does he amoun of reserves and capial available a ime +1 o finance generaion -1 s wihdrawals. In paricular, if λ = λ 1 &1, hen a ime +1 he bank has he share 1 & χ 1 &1 of generaion -1 s reserves and 1 & κ 1 &1 10 In oher words, he bank canno offer agens wihdrawals ha require ha i issue more han he quaniy of currency or capial i has on hand by ransforming one asse ino he oher. Produced by The Berkeley Elecronic Press, 2003

8 6 Topics in Macroeconomics Vol. 3 [2003], No. 1, Aricle 3 of heir capial available o finance heir wihdrawals. On he oher hand, if λ &1 ' λ 2, hen he available shares of hese asses fall o 1 & χ 1 &1 & χ2 &1 and 1 & κ 1 &1 & κ2 &1. Since he consumpion of a generaion early consumer can poenially vary wih he realizaion of λ (as i can affec boh he shares of reserves and capial paid ou as well as he rae of reurn on real balances), le e 2h be he ime +1 consumpion of a generaion early consumer when λ = λ 2 and he agen is among he firs (h = f) measure λ 1 agens o wihdraw or among he second (h = s) measure λ 2 ) λ 1 agens o wihdraw. Analogously, le e 1f be he ime +1 consumpion of a generaion early consumer when λ = λ 1 (in which case h = f mus necessarily hold). Similarly, he consumpion of a generaion lae consumer depends upon he realizaions of boh λ and λ %1 since he former deermines boh he shares of reserves and capial ha are available and R i while he laer deermines (among oher hings) R i %1. To his end, le l ij be he ime +2 consumpion of a generaion lae consumer when λ ' λ i and λ %1 ' λ j, i, j = 1, 2. The bank is aken o be a zero-profi, zero-cos insiuion. Given is muual-like srucure, is objecive is o maximize he ex ane expeced uiliy of is deposiors a each dae. Hence, for each dae $ 1, he objecive of he bank U B, can be wrien as U B ' (1 & π) λ 1 u(e 1f ) % (1 & λ 1 )7(1 & π)u(l 11 ) % πu(l 12 )? % π λ 1 u(e 2f ) % (λ 2 & λ 1 )u(e 2s ) % (1 & λ 2 )7(1 & π)u(l 21 ) % πu(l 22 )?. (1) Given he iming of evens porrayed in Table 1, he muual-like srucure of he bank, and he feasibiliy consrains discussed above, a each dae $ 1 he bank solves he following problem: Problem B1 Choose 6e 1f o maximize U B subjec o, e 2f, e 2s, l ij, χ i, κi,(i, j ' 1, 2), q, k > λ 1 e 1f ' χ 1 R 1 q % κ 1 xk, (2) λ 1 e 2f ' χ 1 R 2 q % κ 1 xk, (3) (λ 2 & λ 1 )e 2s ' χ 2 R 2 q % κ 2 xk, (4) (1 & λ 1 )l 1j ' (1 & χ 1 ) R 1 R j %1 q % (1 & κ 1 ) Xk, j ' 1, 2, (5) (1 & λ 2 )l 2j ' (1 & χ 1 & χ 2 )R 2 R j %1 q % (1 & κ 1 & κ 2 )Xk, j ' 1, 2, (6) q % k ' 1, (7) 0 # χ i, κi # 1, i ' 1, 2; 0 # χ 1 % χ 2 # 1; 0 # κ 1 % κ 2 # 1, (8) Generaion s Self&Selecion Consrain, (9) aking raes of reurn on real balances as given. Eqs. (2) and (3) illusrae how he bank finances he wihdrawals of he firs measure λ 1 generaion early consumers. In paricular, because he bank canno observe λ ex ane, he same shares of reserves and hp://

9 Loewy: Elasic Currency and Welfare 7 capial mus be offered o each regardless of is evenual realizaion. Eq. (4) shows how he wihdrawals of he remaining measure λ 2 - λ 1 early consumers are financed should λ = λ 2. Eqs. (5) and (6) describe how he bank finances he wihdrawals of generaion lae consumers. In each case, he bank mus use he socks of reserves and capial ha are sill available afer generaion s wihdrawals in he previous period have been compleed. The consumpion of lae consumers is also dependen upon he realizaion of λ %1 which deermines (in par) R j %1. Eqs. (7) and (8) describe how he bank allocaes is ime deposi plus he admissibiliy resricions on he shares. Consider nex generaion s self-selecion consrain. Since he early consumers of generaion never delay heir wihdrawal, only ha generaion s lae consumers require such a resricion. Specifically, lae consumers will self-selec o wihdraw in he period in which hey consume when heir expeced uiliy from doing so under he bank s conrac exceeds heir uiliy from wihdrawing premaurely and using he proceeds o finance consumpion in he nex period. However, as in Diamond and Dybvig (1983), risk sharing plus he presence of a oal suspension of wihdrawals implies ha he consrain does no bind ex ane. In ligh of his, a derivaion of he consrain is available from he auhor upon reques. 4. Equilibrium Given he saionariy inheren in he economy s srucure, i is naural o consider he exisence of a saionary equilibrium and, in paricular, of a saionary moneary equilibrium. 11 In order o define and hen prove he exisence and uniqueness of such an equilibrium, i is firs necessary o consider formally he marke for real balances. A each dae he rae of reurn on real balances mus be such ha he demand for real balances on behalf of he members of generaion mus equal he supply of real balances on he par of he early consumers of generaion -1 and (for $ 3) he lae consumers of generaion However, because he measure of early consumers can vary, marke clearing varies wih he realizaions of λ &1 (for $ 2) and λ &2 (for $ 3). Given ha he members of generaion 0 supply M > 0 unis of currency in aggregae a = 1, marke clearing a ha dae requires ha q 1 ' p 1 M. (10) Since here may be eiher measure λ 1 or λ 2 early consumers in generaion 1, he economy s wo legal resricions imply ha a = 2 one of he following wo marke clearing condiions mus be saisfied, q 2 ' χ 1 1 R 1 1 q 1, (11) Similarly, one of he following four expressions mus hold for all $ 3, q 2 ' (χ 1 1 % χ2 1 )R 2 1 q 1. (12) 11 Admiedly, here may also exis non-saionary moneary equilibria. In any such equilibrium, he rae of reurn on real balances falls over ime o is lower bound of x. As i does so, he bank subsiues capial for real balances and he economy approaches a nonmoneary equilibrium in which all saving akes he form of capial. Given he addiional levels of complexiy such equilibria enail, hey are no considered here. 12 This assumes, as is he case, ha he self-selecion consrain is non-binding in equilibrium. Produced by The Berkeley Elecronic Press, 2003

10 8 Topics in Macroeconomics Vol. 3 [2003], No. 1, Aricle 3 q ' χ 1 &1 R 1 &1 q &1 % (1 & χ1 &2 )R 1 &2 R 1 &1 q &2, (13) q ' χ 1 &1 R 1 &1 q &1 % (1 & χ1 &2 & χ2 &2 )R 2 &2 R 1 &1 q &2, (14) q ' (χ 1 &1 % χ2 &1 )R 2 &1 q &1 % (1 & χ1 &2 )R 1 &2 R 2 &1 q &2, (15) q ' (χ 1 &1 % χ2 &1 )R 2 &1 q &1 % (1 & χ1 &2 &χ2 &2 )R 2 &2 R 2 &1 q &2. (16) Eq. (11) provides he = 2 marke clearing condiion when λ 1 ' λ 1 while eq. (12) provides he expression when λ 1 ' λ 2. Likewise, for $ 3, eqs. (13) - (16) provide he relevan marke clearing condiions as ( λ, ) = (λ 1, λ 1 ), (λ 2, λ 1 ), (λ 1, λ 2 ), or (λ 2, λ 2 &2 λ &1 ) respecively. In each case, he supply of real balances varies o reflec he amouns of reserves ha he bank issues o generaion -1 early consumers and o generaion -2 lae consumers. Naurally, hese quaniies reflec he size of middle-aged wihdrawal demand during boh he curren and he previous period. Wih eqs. (10) - (16) in hand, a saionary moneary equilibrium (SME) may now be defined. Definiion 1 Given M > 0 and he sochasic process 6 λ > 4 '1, a SME consiss of scalars which are sricly posiive, (p 1, R 1, R 2 ); nonnegaive, (e 1f, e 2f, e 2s, l 11, l 12, l 21, l 22 ); and bounded beween zero and one, (q, k, χ 1, χ 2, κ 1, κ 2 ); such ha boh Problem B1 is solved and he currency marke clears for every $ 1. I can now sae and prove he paper s firs resul, namely ha given logarihmic uiliy and he muual-like srucure of he bank, here exiss a unique saionary moneary equilibrium if and only if a mild regulariy condiion on he parameers is saisfied. Proposiion 1 There exiss a unique SME if and only if 1 $ π/x + (1 - π)/x. (The proof of his and all subsequen proposiions appear in he appendix.) To esablish Proposiion 1, I propose a se of equilibrium values for he model s endogenous variables and hen show ha if and only if he condiion given in he proposiion is saisfied, hen hese are values are indeed he unique SME. In order o formulae he proposed values, i is firs necessary o observe ha in order for he currency marke o clear in any SME, i mus be ha case ha R 1 = R 2 = 1 and χ 2 = 0 (where saionariy permis he ime argumen o be dropped). Second, since in equilibrium he rae of reurn on money exceeds x, i is opimal for he bank o pay currency o as many early consumers as possible and, by he same oken, o pay immaure capial o as few of hem as possible. The former implies ha χ 1 + χ 2 = 1 while he laer, along wih he random arrival of deposiors, implies ha κ 1 = 0. Furhermore, because marke clearing requires ha χ 2 = 0, we mus also have ha χ 1 = 1. Wih all he currency opimally paid ou o earlyarriving middle-aged agens, i follows ha lae-arriving middle-aged agens receive only immaure capial so ha κ 2 > 0. Since he uiliy funcion is assumed o be logarihmic, solving for he equilibrium values of q and κ 2 under hese condiions is sraighforward. The condiion of he proposiion is hen invoked o guaranee ha χ 2 is indeed equal o zero and herefore ha χ 1 is indeed equal o one. As for he necessiy of he condiion, since any saionary moneary equilibrium mus have R 1 = R 2 = 1, χ 1 = 1, and χ 2 = 0, log uiliy implies ha when he firs-order condiions for χ 1 and χ 2 are evaluaed a he proposed values, hen he condiion given in he proposiion mus hold. As for he condiion given in he proposiion iself, noe ha i can be rearranged o read x(x - 1)/(X - x) $ π. Therefore, his necessary and sufficien condiion for exisence imposes an upper bound on he hp://

11 Loewy: Elasic Currency and Welfare 9 probabiliy of middle-aged wihdrawal demand being high. Consider, hen, he implicaions of an increase in π. In his case, he expeced marginal uiliy of lae-arriving middle-aged agens rises reflecing he increased chance ha such agens will seek o wihdraw. Since marke clearing implies ha currency dominaes immaure capial in rae of reurn, he increase in π increases he pressure on he bank o finance a leas par of he wihdrawals of hese lae arrivers using currency. When π has increased o he poin where he above condiion binds, his pressure is sufficienly srong ha i is now opimal for he bank o rade off an increase in χ 2 agains an equal decrease in χ 1. Once his occurs, he SME is no longer susainable. 13 Finally, alhough Proposiion 1 is predicaed upon logarihmic uiliy, i can be generalized somewha. Le u(c) ' (c 1&γ & 1)/(1 & γ) wih γ 0 N (1, g). By coninuiy, here also exiss a unique SME for any such γ. These equilibria will again exhibi R 1 = R 2 = 1, χ 1 = 1, and χ 2 = κ 1 = 0 and so will exis if and only if π, x, and γ are no oo large and/or X is no oo small Cenral Bank Lending and Opimaliy As shown in he previous secion, in a SME he bank finds i opimal o use all of is reserves o finance he wihdrawals of early-arriving middle-aged agens. In equilibrium, his resul mus hold despie ha fac ha he ex ane uiliy of deposiors is sricly increasing in boh χ 1 and χ 2. Consequenly, were i possible o use currency o finance he wihdrawals of lae-arriving middle-aged agens wih currency, hen he ex ane uiliy would increase. One mechanism o provide for such paymen is o assume ha he cenral bank operaes a ype of discoun window. 15 In he presen conex, his is aken o mean ha he cenral bank sands ready o lend fia currency a zero ineres up o a predeermined cap of is own choosing in exchange for some of he commercial bank s immaure capial (or claims hereon) which serves o provide parial collaeralizaion of he loan. 16 The bank hen uses he loan, in he amoun of M say, and, poenially, some of is curren sock of currency, M, and ne-of-collaeral immaure capial o finance he wihdrawals of lae-arriving early consumers. The laer, and he early-arriving early consumers hen supply M + M unis of currency o he spo marke. Following he clearing of he spo marke, a which poin all ousanding unis of currency have reurned o he commercial bank, he laer repay he loan in exchange for is collaeral. 17 Since he addiional currency is removed from circulaion by he end of he period, here is no measured increase in he money supply from he beginning of he period o he end. By avoiding unbounded money growh, a feaure also found in Freeman (1996, 1999) and Zhou (2000), Wicksellian indeerminancy does no occur. Le χ 3 be he size of he loan (measured as a percenage of he exising money supply) ha he bank receives o help finance he wihdrawals of he lae-arriving early consumers of generaion and le χ 3 be he maximum loan permied by he cenral bank (again measured as a percenage of he exising money 13 Similarly, his condiion will bind if he value of x is sufficienly high or he value of X is sufficienly low. 14 As γ increases, opimal risk sharing increases. This again pus pressure on he bank o raise e 2s relaive o e 2f by rading off an increase in χ 2 agains an equal-sized decrease in χ Recall ha loans of currency from he early-arriving middle-aged o he bank, which consiue anoher mechanism, are prohibied. 16 This policy is a hybrid of wha is currenly used by he cenral banks of he EMU and he U.S. The former charge no nominal ineres on such loans bu require full collaeralizaion. The laer imposes a cap on he amoun of he loan and charges ineres on ha par which exceeds 10% of a bank s risk-based capial. In pracice, such charges are quie small or zero. See Zhou (2000). 17 The bank will always do so since i needs he collaeral o finance he wihdrawals of lae consumers nex period. Produced by The Berkeley Elecronic Press, 2003

12 10 Topics in Macroeconomics Vol. 3 [2003], No. 1, Aricle 3 supply). 18 Wih his addiional currency now available o i, a each dae $ 1 he bank s problem becomes: Problem B2 Choose 6e 1f o maximize U B, eq. (1), subjec, e 2f, e 2s, l ij, χ i, κi,(i, j ' 1, 2), χ3, q, k > o eqs. (2), (3), (5) - (9), and (λ 2 & λ 1 )e 2s ' [χ 2 % χ 3 ]R 2 q % κ 2 xk, (17) 0 # χ 3 # χ 3. (18) Since he bank akes raes of reurn as given, inspecion of eqs. (17) and (18) implies ha he bank will choose he maximum ransfer ha he cenral bank will allow. Assuming ha χ 3 is no so large as o drive currency ou of circulaion (which implies ha R 2 > x; see fn. 18 below), hen he opimal saionary bank conrac again ses χ 1 ' 1 and χ 2 ' κ 1 ' 0. These resuls obain for he same reasons as in he previous secion, namely ha under saionariy marke clearing implies ha R 1 = 1 (see eq. [13]), so ha rae of reurn dominance and he inabiliy of he bank o deermine he measure of early consumers implies ha i pay reserves o as many middle-aged agens as possible and pay immaure capial o as few such agens as possible. Alhough i is no longer necessary for he bank o choose a sricly posiive value of κ 2 because he wihdrawals of lae-arriving early consumers may now be financed using currency, as shown below he bank will neverheless again find i opimal o do so. As for he remaining pars of he bank s opimal conrac, hey depend upon he value of R 2 which in urn depends upon he value of χ 3. To deermine his value requires consideraion of he cenral bank s problem and he associaed saionary equilibrium. The cenral bank is assumed o solve he same problem as does he commercial bank wih he excepion ha he cenral bank chooses χ 3 raher han χ 3 (so ha eq. [18] no longer applies) and i inernalizes he effecs ha is choice of χ 3 has on raes of reurn hrough he economy s marke clearing condiions. I follows ha he cenral bank s problem may be wrien as Problem CB Choose 6e 1f o maximize U B, eq. (1), subjec, e 2f, e 2s, l ij, χ i, κi 3,(i, j ' 1, 2), χ, q, k > o eqs. (2), (3), (5) - (9), (λ 2 & λ 1 )e 2s ' [χ 2 % χ 3 ]R 2 q % κ 2 xk (19) and marke clearing condiions in he presence of cenral bank lending, eq. (10) for = 1; eq. (11) and for = 2; and for $ 3 eqs. (13), (14), q 2 ' [χ 1 2 % χ2 2 % χ 3 2 ]R 2 1 q 1, (20) q ' (χ 1 &1 % χ2 &1 % χ 3 &1 )R 2 &1 q &1 % (1 & χ1 &2 )R 1 &2 R 2 &1 q &2, (21) q ' (χ 1 &1 % χ2 &1 % χ 3 &1 )R 2 &1 q &1 % (1 & χ1 &2 & χ2 &2 )R 2 &2 R 2 &1 q &2 (22) where he assumpion ha he addiional currency revers o he cenral bank a he end of he period implies 18 The ime +1 loan in real erms equals p +1 M. Muliplying and dividing by p M implies ha p %1 M ' R 2 q M/M. I hen follows ha χ 3 ' M/M. hp://

13 Loewy: Elasic Currency and Welfare 11 ha i does no appear in he erms represening he supply of real balances of old agens. In order o simplify he analysis and o be consisen wih wha appears above, I henceforh resric maers o saionary moneary equilibria ha are soluions o Problem CB, and, because he commercial bank is a price aker and so ses χ 3 ' χ 3, are also soluions o Problem B2. Denoing any such equilibrium as a SME CB, he definiion of same follows: Definiion 2 Given M > 0 and he sochasic process 6 λ, a SME CB > 4 '1 consiss of scalars which are sricly posiive, (p 1, R 1, R 2 ); nonnegaive, (e 1f, e 2f, e 2s, l 11, l 12, l 21, l 22, χ 3 ); and bounded beween zero and one, (q, k, χ 1, χ 2, κ 1, κ 2 ); such ha Problem CB is solved for every $ 1. Assuming logarihmic uiliy, I now show ha under cerain condiions, here exiss a SME CB. Proposiion 2 If x $ X -1, π is sufficienly close o zero, and λ 2 is sufficienly close o one, hen here exiss a unique SME CB. As in a SME, he marke clearing condiions in a SME CB uniquely deermine he saionary equilibrium values of he raes of reurn and wo of he four share parameers. In paricular, he saionary version of eq. (13) implies ha R 1 = 1 while he saionary version of eq. (14) implies ha χ 2 = 0 and herefore ha χ 1 = 1. The saionary version of eiher eq. (21) or (22) hen implies ha R 2 = (1 + χ 3 ) Furhermore, i is again he case ha he uniform arrival of deposiors and rae of reurn dominance imply ha κ 1 = 0. Given hese resuls, he remaining firs-order condiions deermine a sysem in q, κ 2, and χ 3 which is shown o have a unique soluion when λ 2 is sufficienly large. The remaining wo resricions guaranee ha χ 1 = 1 and χ 2 = 0 are indeed opimal for he cenral bank. Through is elasic currency provision, a SME CB relaxes he consrain on issuing currency o laearriving middle-aged agens. When he loan is of he opimal size, i implies ha e 2f = e 2s so ha risk sharing is now borne by all middle-aged agens when λ ' λ 2 raher han by jus hose who arrive lae. Consequenly, a SME CB suppors an allocaion ha Pareo dominaes he one ha a SME suppors. Moreover, since he soluion o Problem B2 ses χ 3 = χ 3 and he relevan marke clearing condiions are now given by eqs. (13), (14), (21), and (22), i follows ha in equilibrium he soluion o Problem B2 is equivalen o ha of Problem CB and hence yields he same Pareo-improving allocaion. I remains o deermine wheher he SME CB suppors an allocaion ha is also he soluion o some form of planning problem. If so, hen no furher welfare gain is possible beyond ha aribuable o having an elasic currency. Given he economy s overlapping generaions srucure and he focus on saionary equilibria, a naural planning problem o consider is one ha chooses he saionary allocaion ha maximizes he seady-sae ex ane uiliy of all agens born a daes $ 1 subjec o he economy s seady-sae feasibiliy consrains. I denoe he soluion o his problem as he golden rule allocaion since i represens his economy s analog o wha Freeman (1996) refers o as golden rule allocaions in his model. Unlike he cenral bank, he planner is assumed o be able o observe he realizaion of λ. While his suggess ha he planner can make he porfolio of generaion coningen upon his value, in fac his is no he case. If λ is no observed unil he beginning of ime +1, hen his is rivially rue. If λ is observed a he beginning of ime, hen o make agens porfolios coningen upon his value he planner mus have access o a one-period echnology wih a rae of reurn equal o a leas one. This is so because he planner can achieve a reurn of one using an inergeneraional ransfer whereby he par of he endowmen of generaion young agens is ransferred o middle-aged agens who wish o consume in period. 20 However, because no such echnology exiss, he planner has no opion oher han o uilize he inergeneraional 19 This expression implies ha χ 3 canno exceed 1/x! 1 since a his poin R 2 = x and any furher increase will drive currency ou of circulaion should wihdrawal demand be large. 20 I is no opimal o use he ransfer o finance he consumpion of old agens since hey always do beer o receive maure capial. Produced by The Berkeley Elecronic Press, 2003

14 12 Topics in Macroeconomics Vol. 3 [2003], No. 1, Aricle 3 ransfer since o do oherwise implies a one-period reurn of x from immaure capial. Of course he same is rue even if λ is no observed unil ime +1. In eiher case, since he realizaion of λ is known by he ime agens of generaion are o consume, he planner is able o rea all middle-aged agens he same. By he same oken, he planner is also able o rea all old agens he same no maer he realizaions of λ &1 and λ. Hence, e 1f = e 1, e 2f = e 2s = e 2, l 11 = l 12 = l 1, and l 21 = l 22 = l 2. Le z be he per capia quaniy of generaion s endowmen ha is ransferred o he middle-aged agens of generaion -1. Then he golden rule allocaion is he soluion o he following problem: Problem GR Choose {e 1, e 2, l 1, l 2, κ 1, κ 2, z, k} o maximize U GR ' (1 & π) λ 1 u(e 1 ) % (1& λ 1 )u(l 1 ) % π λ 2 u(e 2 ) % (1& λ 2 )u(l 2 ) (23) subjec o λ 1 e 1 ' z % κ 1 xk, (24) λ 2 e 2 ' z % κ 2 xk, (25) (1 & λ 1 )l 1 ' (1 & κ 1 )Xk, (26) (1 & λ 2 )l 2 ' (1 & κ 2 )Xk, (27) z % k ' 1, (28) 0 # κ i # 1, i ' 1, 2 (29) where he κ i are (effecively) he same as in Problems B1, B2, and CB. I am now able o answer he quesion posed above, namely does a SME CB, and hence a SME wih elasic currency, suppor a golden rule allocaion. The following proposiion shows he answer o be yes. Proposiion 3 A SME CB suppors a golden rule allocaion. To prove he proposiion, i suffices o show ha he consrains and firs-order condiions of a SME CB are he same as hose ha solve Problem GR. Since in he equilibrium χ 1 ' 1, χ 2 ' κ 1 ' 0, and he opimal cenral bank loan serves o eliminae compleely he parial suspension of wihdrawals when wihdrawal demand is high, he proposiion follows direcly. 6. Conclusion This paper offers an example of a moneary banking model in which he bank faces aggregae risk governing is expeced wihdrawals. In he model, he bank s bes sraegy is o hold boh a fia reserve asse and an illiquid asse, capial, agains is deposis and o impose a parial suspension of wihdrawals on cerain agens once i learns ha wihdrawal demand exceeds a paricular limi. Prior o reaching his limi, he bank finances he wihdrawals of middle-aged agens enirely wih reserves. Once his limi is reached, he wihdrawals of he remaining middle-aged agens are financed enirely wih immaure capial. In conras, hp://

15 Loewy: Elasic Currency and Welfare 13 he wihdrawals of all old agens are financed enirely wih maure capial. Given ha he bank would prefer o finance (in par) he wihdrawals of lae-arriving middle-aged agens wih fia currency, a mechanism ha relaxes his consrain can be welfare improving. The one such mechanism ha his paper considers is for he cenral bank o operae a ype of discoun window whereby i lends he commercial bank addiional unis of currency ha are parially collaeralized wih (claims on) immaure capial. Assuming ha he cenral bank opimally chooses he maximum loan size, hen he resuling saionary moneary equilibrium Pareo dominaes he one wihou he loan since he bank is now beer able o spread he risk of high wihdrawal demand across early consumers. Moreover, he saionary equilibrium wih cenral bank lending suppors he golden rule allocaion and so achieves he highes level of welfare consisen wih maximizing seady-sae ex ane expeced uiliy. Colophon Some of he work on his paper, which previously circulaed under he ile Demand Deposi Banking wih a Fia Reserve Asse and Inraday Credi, was underaken while he auhor served as a Visiing Scholar a he Federal Reserve Bank of Alana. I hank he members of he Alana Fed Research Deparmen, and in paricular Will Roberds, for heir helpful discussions. The views expressed herein, however, are no necessarily hose of he Federal Reserve Bank of Alana nor of Federal Reserve Sysem. I hank as well he edior, Per Krusell, wo referees, and paricipans a he 2001 Midwes Macroeconomics Conference for heir commens and suggesions. All remaining errors are solely of my own creaion. Appendix Proof of Proposiion 1 Given logarihmic uiliy and he muual-like srucure of he bank, I claim ha {e 1f, e 2f, e 2s, l 11, l 12, l 21, l 22, q, k, χ 1, χ 2, κ 1, κ 2, p 1, R 1, R 2 } = {1, 1, x, X, X, X, X, λ 1, 1 - λ 1, 1, 0, 0, (λ 2 - λ 1 )/( 1 - λ 1 ), M/λ 1, 1, 1} is he unique saionary moneary equilibrium if and only if 1 $ π/x + (1 - π)/x. if Inspecion of eqs. (10) - (16) shows ha he conjecured equilibrium saisfies marke clearing a every dae. Likewise, inspecion of eqs. (2) - (8) shows ha he conjecured equilibrium saisfies he budge consrains of Problem B1 a every dae. Thus, in he sequel, he noaion may be simplified o read e 1f 'e 2f 'e f, l 11 and for consisency,. ' l 12 ' l 1, l 21 ' l 22 ' l 2, e 2s ' e s Consider nex Problem B1. Given R 1 = R 2 = 1, his problem may now be wrien more succincly as choosing 6e f, e s, l 1, l 2, χ 1, χ 2, κ 1, κ 2, q, k> o maximize U B ' λ 1 u(e f ) % (1& π)(1& λ 1 )u(l 1 ) % π (λ 2 & λ 1 )u(e s ) % (1& λ 2 )u(l 2 ) (A1) subjec o λ 1 e f ' χ 1 q % κ 1 x(1&q), (A2) (λ 2 & λ 1 )e s ' χ 2 q % κ 2 x(1&q), (A3) (1& λ 1 )l 1 ' (1& χ 1 )q % (1& κ 1 )X(1& q), (A4) Produced by The Berkeley Elecronic Press, 2003

16 14 Topics in Macroeconomics Vol. 3 [2003], No. 1, Aricle 3 (1& λ 2 )l 2 ' (1& χ 1 & χ 2 )q % (1& κ 1 & κ 2 )X(1& q). (A5) Since he self-selecion consrain is nonbinding in equilibrium (Diamond and Dybvig, 1983), i follows ha he firs-order condiions for q, χ 1, χ 2, κ 1, and κ 2 mus saisfy (afer subsiuing ou mulipliers for marginal uiliies): u ) (e f ) ) X[(1& π)u ) (l 1 ) % πu ) (l 2 )] ' 0, (A6) 7u ) (e f ) & [(1& π)u ) (l 1 ) % πu ) (l 2 )]?q & ν 1 ' 0, ν 1 (1& χ 1 & χ 2 ) ' 0, (A7) 6π[u ) (e s )& u ) (l 2 )]>q & ν 1 % ν 2 ' 0, ν 2 χ 2 ' 0, (A8) 7xu ) (e f ) & X[(1& π)u ) (l 1 ) % πu ) (l 2 )]?(1& q) % ν 3 ' 0, ν 3 κ 1 ' 0, (A9) xu ) (e s ) & Xu ) (l 2 ) ' 0, (A10) for ν 1, ν 2, ν 3 $ 0. Le u = ln (c). Subsiuion of he conjecured equilibrium values ino eqs. (A6) - (A10) shows ha all bu (A8) hold uncondiionally. As for eq. (A8), eliminaion of ν 1 using (A7) shows ha he former may now be wrien as u ) (e f ) $ πu ) (e s ) % (1& π)u ) (l 1 ). (A11) Given he conjecured equilibrium, eq. (A11) holds if 1 $ π/x + (1 - π)/x. Uniqueness follows from he concaviy of he uiliy funcion and he convexiy of he budge se. P only if Le u = ln (c) and suppose ha he conjecured values consiue a saionary moneary equilibrium so ha hese values solve Problem B1 and marke clearing a every dae. Hence, hey saisfy, among oher hings, eqs. (A7) and (A8). Subsiuing (A7) ino (A8) o eliminae ν 1 and hen applying he conjecured equilibrium values implies ha 1 $ π/x + (1 - π)/x. P Proof of Proposiion 2 Consider firs he implicaions of marke clearing. The saionary version of eq. (13) implies ha R 1 = 1 while he saionary version of eq. (14) implies ha χ 2 = 0 and herefore ha χ 1 = 1. The saionary version of eiher eq. (21) or (22) hen implies ha R 2 = (1 + χ 3 ) -1. Consider nex opimizaion. Suppose ha R 1, R 2, χ 1 and χ 2 equal heir equilibrium values. Since hese imply ha l 11 = l 12 = l 1 and l 21 = l 22 = l 2, Problem CB consiss of choosing {e 1f, e 2f, e 2s, l 1, l 2, χ 1, χ 2, χ 3, κ 1, κ 2, q, k} o maximize U CB ' (1 & π) λ 1 u(e 1f ) % (1& λ 1 )u(l 1 ) % π λ 1 u(e 2f ) % (λ 2 & λ 1 )u(e 2s ) % (1& λ 2 )u(l 2 ) (A12) subjec o hp://

17 Loewy: Elasic Currency and Welfare 15 λ 1 e 1f ' χ 1 q % κ 1 xk, (A13) λ 1 e 2f ' χ 1 (1 % χ 3 ) &1 q % κ 1 xk, (A14) (λ 2 & λ 1 )e 2s ' (χ 2 % χ 3 )(1% χ 3 ) &1 q % κ 2 xk, (A15) and eqs. (A4) and (A5). The firs-order condiions for q, χ 1, χ 2, χ 3, κ 1 and κ 2 may hen be wrien as (1 & π)u ) (e 1f ) % πu ) (e 2s ) ' X[(1 & π)u ) (l 1 ) % πu ) (l 2 )] (A16) 6(1 & π)u ) (e 1f ) % πu ) (e 2f )@(1 % χ 3 ) &1 &[(1& π)u ) (l 1 ) % πu ) (l 2 )@(1 % χ 3 ) &1 ]@Π> q & ν 1 ' 0, ν 1 (1 & χ 1 & χ 2 ) ' 0, (A17) 6u ) (e 2s ) & u ) (l 2 )@Π>π(1 % χ 3 ) &1 q & ν 1 % ν 2 ' 0, ν 2 χ 2 ' 0, (A18) [xu ) (e 1f ) & Xu ) (l 1 )](1 & π)(1& q) % ν 3 ' 0, ν 3 κ 1 ' 0, (A19) e 2f ' e 2s, (A20) xu ) (e 2s ) ' Xu ) (l 2 ), (A21) where Π ' 1 & π % π(1 % χ 3 ) &1. Subsiuing he relevan budge consrains ino eqs. (A16), (A20), and (A21) yields a sysem in (q, κ 2, χ 3 ). Afer some algebra, his sysem reduces o (1 & λ 2 ) x(λ 2 & κ 2 ) ' (1 & π)(1 & λ1 )(1 & κ 2 ) % π(1 & λ 2 ). (A22) (1 & π)λ 1 (1 & κ 2 ) % π(λ 2 & κ 2 ) Le L(κ 2 ) and R(κ 2 ) denoe he lef- and righ-hand sides of (A22). I follows ha lim κ 2 6λ 2 L(κ2 ) ' 4 and lim where. Since and L(0) =, i follows ha if λ 2 κ 2 6 κ 2 R(κ2 ) ' 4 κ 2 ' (1 & π)λ1 % πλ 2 λ 2 < κ 2 1 & λ 2 (1 & π)λ 1 % π xλ 2 is sufficienly close o one, hen L(0) < R(0) and so here exiss a unique soluion o eq. (A22). Given his soluion, he opimal soluions for χ 3 and q are given by χ 3 ' (λ2 & κ 2 ) and. λ 1 (1 & κ 2 ) & 1 q ' x(λ 2 & κ 2 ) x(λ 2 & κ 2 ) % (1 & λ 2 ) I remains o verify ha eqs. (A17) - (A19) hold. Using eqs. (A16) and (A21), i is readily seen ha (A19) holds. As for eq. (A17), applying eq. (A16) implies ha (A17) is saisfied for ν 1 $ 0 if (1 & π)u ) (e 1f )(1 & X &1 ) % πu ) (e 2s )[(1 % χ 3 ) &1 & X &1 ] $ 0. (A23) Since (1 % χ 3 ) &1 $ x is necessary for he exisence of a moneary equilibrium, a sufficien condiion for eq. (A23) o hold is x $ X -1. Turning o eq. (A18), eliminaing ν 1 using eq. (A17) implies ha (A18) is saisfied for ν 2 $ 0 if e 1f # l 1. Eq. (A16) hen shows his condiion is saisfied if π is sufficienly close o zero. P Produced by The Berkeley Elecronic Press, 2003