Optimal Central Bank Lending

Transcription

1 Opimal Cenral Bank Lending Andreas Schaber 1 Universiy of Cologne This version: January 9, 215 Absrac We analyze opimal moneary policy in a sicky price model wih open marke operaions. The cenral bank ses he policy rae and can, addiionally, conrol he amoun of money by raioning money supplied agains eligible securiies. Opimal policy under money raioning is shown o enhance welfare in he long-run and in he shor-run compared o a convenional opimal policy regime where money supply is no raioned and saiaes money demand. Specifically, his propery is shown o apply when privaely issued deb is eligible in open marke operaions, which allows he cenral bank o separaely aler coss of borrowing and he size of ransacions for which money is required. JEL classificaion: E4; E5; E32. Keywords: Opimal moneary policy, cenral bank insrumens, collaeralized lending, money raioning 1 Universiy of Cologne, Cener for Macroeconomic Research, Alberus-Magnus-Plaz, 5923 Cologne, Germany, Phone: , schaber@wiso.uni-koeln.de.

2 1 Inroducion In he macroeconomic lieraure of he las wo decades, he implemenaion of moneary policy has focussed on how a risk-free shor-erm nominal ineres rae should be se. 2 Money supply is hen passively adjused by he cenral bank o saiae money demand, which means ha money is supplied unil he privae agens marginal valuaion of money accords o he marginal coss of holding money in equilibrium. Ye, cenral banks ypically i.e. in non-crisis imes) refrain from fully accommodaing money demand, as, for example, he US Federal Reserve "has creaed wha is called a srucural deficiency. Tha is, i has creaed permanen addiions o he supply of reserve balances ha are somewha less han he oal need" such ha he open marke "desk is in a posiion o add balances emporarily o ge o he desired level". 3 Likewise, he European Cenral Bank has in general no fully accommodaed liquidiy demand of counerparies by applying "allomen raes" less han one for is main refinancing operaions. 4 This indicaes ha cenral banks can in fac conrol boh, he nominal ineres rae or, policy rae) as he price of money and he amoun of money, by raioning he supply of reserves. 5 This paper shows ha a cenral bank can enhance welfare by raioning he amoun of money supplied in open marke operaions. Accouning for he fac ha reserves are supplied agains eligible asses, we consider a collaeral consrain for open marke operaions, where he concep of collaeral is used here like by cenral banks in a broader sense and refers o he propery of repurchase agreemens being a form of collaeralized lending. When he cenral bank supplies money a a price below he marginal valuaion of money by privae agens, he laer are willing o acquire money agains eligible asses unil he collaeral consrain becomes binding. The policy rae is hen decoupled from he marginal rae of ineremporal subsiuion, which frees up insrumens ha can change privae agens access o money and ineres raes on eligible deb securiies. Under money raioning, he cenral bank can hus conrol he amoun of money as well as ineres raes, which allows o separaely induce changes in he size of ransacions for which money is required and in borrowing coss. Compared o a moneary policy regime ha fully accommodaes money demand, he cenral bank can herefore reduce disorionary effecs on he allocaion more effecively under money raioning simply by having addiional insrumens a is disposal. This is demonsraed by applying a sylized macroeconomic model, where money raioning is shown o enhance welfare in he long-run and in he shor-run. 2 See, e.g, several chapers in 12]. Excepions are analyses of unconvenional moneary policies ha are applied in crisis imes, like 1], 13], or 14]. 3 See Fedpoin "Open Marke Operaions" a hp:// 4 Deails on he European Cenral Bank s allomen rae decisions can be found in 11]. 5 Responses of US Federal Reserve and he European Cenral Bank o he recen financial crisis, i.e. seing ineres raes close o zero and acively expanding he supply of reserves via lending faciliies and direc asse purchases, also indicae ha cenral banks can simulaneously conrol ineres raes and he quaniy of money. 1

3 We examine opimal moneary policy in a framework wih fricions ha are sandard in he lieraure see 17], 2], 6], or 23]). Specifically, we allow for goods prices o be se in an imperfecly flexible way, for ransacion fricions i.e. cash consrains), and for ime varying mark-ups. In his framework, we explicily consider ha money is supplied by he cenral bank only in exchange for eligible asses. Since he laer serve as imperfec) subsiues for money, he ineres raes on eligible deb securiies relae o he price of money in open marke operaions and hus o he policy rae. For asses ha are non-eligible, invesors hen demand ineres raes ha are higher due o an illiquidiy premium. 6 These ineres raes relae as usual o he nominal marginal rae of ineremporal subsiuion, which reflecs he opporuniy coss of money holdings and, herefore, deermines he privae agens marginal valuaion of money. When he policy rae is se a a lower level, privae agens are willing o hold money up o he maximum amoun supplied by he cenral bank agains eligible asses, such ha money supply is effecively raioned. Money raioning becomes a relevan opion for a welfare maximizing cenral bank if i canno implemen he firs bes allocaion under a convenional single insrumen regime. If, for example, only ransacion fricions are presen, saiaing money demand a zero nominal ineres raes implemens firs bes, such ha money raioning would be undesirable in his case. If, however, effi ciency requires a posiive nominal marginal rae of ineremporal subsiuion, e.g. when nominal rigidiies call for price sabiliy, raioning money supply is advanageous: Suppose ha he cenral bank ses like in a convenional regime he policy rae equal o he nominal marginal rae of ineremporal subsiuion, such ha money supply is no raioned and he cenral bank has only one insrumen a is disposal. Unless ransacion fricions are no presen and he special case of "divine coincidence" applies, 7 firs bes will hen no be implemenable. Now suppose ha he cenral bank ses he policy rae below he desired nominal marginal rae of ineremporal subsiuion and consrains access o money by a se of eligible asses ha are no abundanly available. 8 Then, money supply is raioned, while privae agens are no worse off han under a convenional regime. The cenral bank can in his case deal wih muliple disorions, i.e. i can simulaneously sabilize prices and vary he coss of borrowing o reduce supply side disorions), by separaely adjusing he supply of money o manipulae aggregae demand and alering he ineres raes on eligible deb securiies. Hence, here are gains from money raioning when here exis fricions ha, on he one hand, call for non-zero nominal ineres raes and, on he oher hand, canno be neuralized jus by seering aggregae demand wih a single insrumen. 6 This is consisen wih empirical evidence by 18] on he yield spread beween corporae bonds and reasuries. 7 The erm divince coincidence has been used by 3] o describe he propery of a sandard New Keynesian model which does no feaure a conflic beween he goals of an opimizing cenral bank. 8 If money were insead supplied in an unconsrained way e.g. agains abundanly available collaeral), he nominal marginal rae of ineremporal subsiuion would hen fall o he level of he policy rae, implying ineffi cien levels of inflaion and real aciviy. 2

4 For he analysis of opimal moneary policy, we apply a model ha is suffi cienly simple o faciliae he derivaion of closed form resuls, while i accouns for some main characerisics of cenral bank pracice. Specifically, we consider money being supplied ourigh agains reasury securiies and under repurchase agreemens repos), which lead o privae secor money holdings as well as posiive money injecions. In addiion o reasury securiies, we also consider corporae loans, which are issued by firms who rely on working capial, as eligible asses for money supply operaions. 9 Under money raioning, he cenral bank hen alers he disorive loan rae via policy rae adjusmen, while i can separaely conrol he amoun of money supplied agains collaeral in open marke operaions. Given ha supply side disorions exis implying ha here is no divine coincidence), money raioning enables he cenral bank o enhance welfare by influencing firms borrowing coss and by simulaneously implemening a desired level of aggregae demand wih disinc insrumens. 1 We compare opimal policy under commimen, which is shown o be associaed wih money raioning, wih a convenional opimal policy regime where money supply is no raioned. The laer regime is characerized by he ideniy of he policy rae and he nominal marginal rae of ineremporal subsiuion. Decoupling hese raes under money raioning endows he cenral bank wih more han one insrumen and allows o conrol he long-run inflaion rae independenly from he policy rae. The cenral bank can herefore reduce welfare coss of price sickiness by sabilizing he price level via money supply adjusmens and i can simulaneously se he policy rae o reduce he coss of borrowing money; his sraegy being impossible under a convenional regime where he policy rae is linked o inflaion in he long-run by he Fisher equaion. Under a non-raioned money supply, borrowing coss herefore end o be higher and long-run welfare losses compared o firs bes) are a muliple of he long-run welfare losses under an opimal policy regime wih money raioning. We furher show ha effecs of cos-push shocks, which are ypically considered in he New Keynesian lieraure and found o subsanially conribue o business cycle flucuaions see 24]), can be neuralized under money raioning. More precisely, he cenral bank can neuralize effecs on firms marginal coss by lowering he policy rae or increasing he fracion of eligible loans, and can simulaneously adjus he oal amoun of acceped collaeral o induce a level of aggregae demand ha closes he oupu gap. For a special case, where he disorion beween cash goods and credi goods is eliminaed, i can be shown ha money raioning even 9 The US Federal Reserve mainly acceped "Treasuries only" in pre-28 open marke operaions. However, corporae deb securiies like commercial papers ha relae o inraperiod loans in he model have also been considered as subsiues for reasury deb in case of "large budge surpluses and he associaed seep reducions in Treasury deb" see 4]). 1 We furher examine an alernaive model version, where we accoun for heerogenous households who borrow/lend for consumpion purposes and we consider consumpion loans as eligible asses see Appendix F). For his alernaive version, we also find ha moneary policy can enhance welfare via money raioning, hough, he welfare gains are less pronounced han for he benchmark model. 3

5 enables implemening he firs bes allocaion. 11 The paper relaes o sudies on opimal moneary policy under commimen, in paricular, wih sicky prices and ransacion fricions, like 17], 2], and 6]. These sudies show ha he cenral bank should predominanly sabilize prices and deviae from he Friedman rule see 23], for an overview). Opimal policy is also mainly characerized by price sabiliy when prices are sicky and axes are disorionary, even hough inflaion serves as a subsiue for axaion see 21] and 2]). For he case where suffi cienly many ax insrumens are available, 9] show how he effecs of price sickiness can be neuralized, while 1] show ha he cenral bank can off-se effecs of ransacion fricions and of pre-se prices when i simulaneously conrols he policy rae and he money growh rae. The paper furher relaes o sudies where a cenral bank is assumed o direcly lend o he privae secor, raher han o use privae deb as collaeral as assumed in his paper). Analyzing moneary policy in a model wih sicky prices and imperfecions in privae financial inermediaion, 1] show ha direc cenral bank lending is associaed wih coss ha differ from privae coss of inermediaion and can be beneficial a imes of unusual financial disress. Similarly, 13] develop a model where privae inermediaries face balance shees consrains, while he cenral bank can inelasically raise funds a fixed coss per uni len o he privae secor. Like 14], who augmen 13] by inroducing idiosyncraic invesmen risks, hey show ha direc lending is beneficial in crises siuaions when privae inermediaries are financially consrained. In conras o hese sudies, we do no consider credi originaion by he cenral bank and absrac from coss of cenral bank operaions. The paper is organized as follows. Secion 2 presens he benchmark model. In Secion 3, we describe he role of money raioning. In Secion 4, we describe he policy problem under commimen and presen some closed form resuls for a special case. Secion 5 provides comparisons o a convenional opimal policy regime wihou money raioning. In Secion 6, we discuss limis o money raioning and differences o direc cenral bank lending. Secion 7 concludes. 2 The model In his Secion, we presen he model, which feaures fricions ha are sandard in he New Keynesian lieraure on moneary policy, namely, sicky prices, poenially ime varying mark-ups, and ransacion fricions. The laer are modelled by liquidiy consrains for households, who rely on money for purchases of a cash good in conras o purchases of a credi good), and firms, who rely on working capial. The main difference o sandard models is ha money is supplied only in exchange for asses in open marke operaions. There, he cenral bank conrols he price of money 11 Noably, opimal moneary policy under money raioning is no limied o policy rae adjusmens, which is mos apparen a he zero lower bound where privae secor behavior can sill be affeced via money supply insrumens, as shown by 16] in a companion paper. 4

6 and he amoun of money supplied agains eligible asses, which consis of shor-erm reasuries and corporae loans. By deciding on how much money is supplied agains hese asses, he cenral bank influences heir prices compared o non-eligible asses, since agens inernalize he propery of asses o serve as imperfec) subsiues for money. Following common cenral bank pracice, we assume ha money is supplied ourigh as well as under repurchase agreemens repos), which ensures posiive money injecions in each period and ha cenral bank ransfers consis of oal ineres earnings. 2.1 Timing of evens Households, indexed wih i, 1], ener a period wih money Mi, 1 H, one-period governmen bonds B i, 1, and coningen claims D i,. Afer aggregae shocks are realized a he beginning of he period, he cenral bank ses is insrumens, i.e. i announces he maximum amoun of money as a fracion of eligible asses held by he counerpary κ B and κ, see below), and ses he policy rae R m. The remainder of he period unfolds as follows. Firs, he labor marke opens, where a perfecly compeiive inermediae goods producing firm j hires workers n j,. We assume ha i has o pay wages before he goods are sold. Since i does no hold any financial wealh, firm j borrows working capial o finance is wage bill L j, /R L j, P w n j,, 1) where w denoes he aggregae real wage rae and P he final goods price. Firm j borrows he amoun L j, /Rj, L and repays he amoun L j, a he end of he period, such ha Rj, L is he ineres rae on he inraperiod loan. Second, open marke operaions are conduced, where he cenral bank supplies money ourigh or under repos agains eligible asses a he price R m. Household i receives new money injecions) from he cenral bank I i, agains eligible asses, i.e. corporae loan conracs and governmen bonds. Specifically, he cenral bank supplies money agains fracions of randomly seleced reasuries κ B and loan conracs κ, such ha I i, is consrained by he following condiion, which we summarize as he collaeral consrain : 12 I i, κ B B i, 1 /R m ) + κ L i, /R m ). 2) Afer receiving I i,, household i delivers L i, /R L o firms according o he loan conracs. I hen holds money, bonds, and loans o he amoun M H i, 1 +I i, L i, /R L ), B i, 1 B c i,, and L i, L R i,, where B c i, are reasuries received by he cenral bank and LR i, are loans under repos, such ha 12 Though, he erm collaeral only applies o repos and no o ourigh purchases, i is like by cenral banks used in a broader sense hroughou he paper, for convenience. 5

7 I i, = Bi, c /Rm ) + L R i, /Rm ). Third, wages are paid, and inermediae as well as final goods are produced. Then, he goods marke opens, where cash goods c in conras o credi goods c can only be purchased wih money. Hence, household i faces he cash-in-advance consrain P c i, I i, + Mi, 1 H L i, /R L ) + P w i, n i,, 3) where w i, denoes he individual wage rae. Household i s sock of money hen equals M i, = Mi, 1 H + I i, L i, /R L ) + P w i, n i, P c i, and is sock of reasuries equals B i, = B i, 1 Bi, c. Fourh, before household i eners he asse marke, i receives governmen ransfers P τ i,, and dividends of firms and reailers, which sum up o P δ i,. Repurchase agreemens are hen seled, i.e. household i buys back loans L R i, = M i, L Rm and reasuries Bi, R = M i, RRm from he cenral bank, where Mi, L and M i, R denoe money supplied emporarily agains loans and reasuries. In he asse marke, reurns from mauring asses are paid, loans are repaid, and reasuries are issued a he price 1/R ) as well as coningen claims. Household i s asse marke consrain is hus given by B i, /R ) + E q,+1 D i,+1 ] + M H i, 4) B i, + B R i, + M i, M L i, + M R i,)r m + L i, + D i, + P c i, + P τ i, + P δ i,, where q,+1 denoes a sochasic discoun facor see Secion 2.3). The cenral bank ransfers seigniorage o he reasury and reinvess payoffs from mauring bonds in newly issued bonds and leaves aggregae money supply unchanged a his sage, 1 M H i, di = 1 M H i, 1 +I i, M L i, M R i, )di. 2.2 Firms There is a coninuum of idenical inermediae goods producing firms indexed wih j, 1]. They exis for one period, are perfecly compeiive, and are owned by he households. A firm j disribues profis o he owners and hires he aggregae labor inpu n j, a a common wage rae w. We assume ha wages have o be paid before goods are sold. For his, firm j borrows cash L j, from households a he price 1/R L j, see 1) and repays he loan a he end of he period.13 I hen produces he inermediae good according o IO j, = a n α j,, where α, 1) and a is sochasic wih an uncondiional mean equal o one, and sells i o reailers. Following relaed sudies see e.g. 6] and 23]), we consider a consan subsidy τ p o eliminae long-run disorions, such ha 13 In Appendix D, we show ha he possibiliy of reained earnings does no o affec our main resuls. 6

8 he problem of a profi-maximizing firm j is given by max 1 + τ p )P J, a n α j, P w n j, L j, Rj, L 1)/Rj,, L s.. 1), 5) {n j,,l j, } where P J, denoes he price for he inermediae good. The firs order condiions are given by 1 + τ p ) P J, /P ) αn 1 α j, = w + ς j, w, R L 1 = ς j,, and ς j, l j, /R L ) w n j, ] =, where l j, = L j, /P and ς j, is he muliplier on 1). Defining τ n = τ p /1 + τ p ) as he producion or wage) subsidy rae, labor demand and loans saisfy: P J, /P ) a αn α 1 j, = 1 τ n ) w R L j,, 6) l j, /R L j, = w n j,, 7) if R L > 1, while l j, /R L ) w n j, holds insead of 7) if R L = 1. Firms ransfer profis o he owners in a lump-sum way. Condiion 6) shows ha he working capial consrain 1) can disor labor demand hrough he coss of borrowing R L. Since inermediae goods producing firms are ex-ane idenical, hey can only exhibi differen labor demands if hey face differen coss of borrowing 6). This would, for example, be he case, if lenders perceive loans of differen firms as imperfec subsiues. We exclude his possibiliy by assuming ha he cenral bank reas loans of all firms in an ex-ane idenical way see Secion 6 for a discussion). To inroduce sicky prices, we assume ha here are monopolisically compeiive reailers who re-package inermediae goods IO = 1 IO j,dj. A reailer k, 1] produces one uni of a disinc good y k, wih one uni of he inermediae good purchased a he common price P J, ) and sells i a he price P k, o perfecly compeiive bundlers. They bundle he disinc goods y k, o a final good y ha can be used for consumpion as a cash good or as a credi good. Boh goods are hus produced according o he same echnology y = 1 y ε 1 ε k, dk) ε ε 1 and are sold a he same price P. The cos minimizing demand for y k, is hen given by y k, = P k, /P ) ε y. Following 5], we assume ha each period a measure 1 φ of randomly seleced reailers may rese heir prices independenly of he ime elapsed since he las price seing, while a fracion φ, 1) of reailers do no adjus heir prices. A fracion 1 φ of reailers ses heir price o maximize he expeced sum of discouned fuure profis. For φ >, he firs order condiion for heir price P is given by P = ε ε 1 E s= φβ)s q,+s y +s P+smc ε +s E s= φβ)s q,+s y +s P+s ε 1, 8) where mc = P J, /P denoes reailers real marginal cos. Wih perfecly compeiive bundlers, he price index P for he final good saisfies P 1 ε = 1 P 1 ε k, dk. Using ha 1 P 1 ε k, dk = 1 φ) s= φs 1 ε 1 ε 1 ε P s holds, and aking differences, leads o P = 1 φ) P + φp 1 1 ε. 7

9 2.3 Households There is a coninuum of infiniely lived households indexed wih i, 1] and wih idenical asse endowmens and preferences. Insananeous uiliy decreases wih working ime and increases wih consumpion, which consiss of cash goods c i, and credi goods c i,. Following 19], we assume ha cash and credi goods are disinc from he households perspecive. 14 Household i maximizes he expeced sum of a discouned sream of insananeous uiliies E β uc i,, c i,, n i, ), 9) = where E is he expecaion operaor condiional on he informaion se in he iniial period, β, 1) is he subjecive discoun facor, and he uiliy funcion is given by uc i,, c i,, n i, ) = c 1 σ i, 1) + γ c 1 σ i, 1)] 1 σ) 1 χn 1+η i, 1 + η) 1, where σ >, χ >, η, and γ. Households are assumed o monopolisically supply differeniaed labor services n i, ha are ransformed ino aggregae labor inpu n employed for he producion of inermediae goods. The ransformaion is conduced via he aggregaor n 1 1/ζ = 1 n1 1/ζ i, di, where we follow 24] and assume ha he elasiciy of subsiuion ζ varies exogenously over ime according o a saionary process. Cos minimizaion leads o he following demand for differeniaed labor services n i,, n i, = w i, /w ) ζ n, wih w 1 ζ = 1 w 1 ζ i, di. 1) A household i is iniially endowed wih money M H i, 1 >, governmen bonds B i, 1 >, and coningen claims D i,. Before household i eners he goods marke, where i relies on cash for goods purchases see 3), i migh lend money o firms. I can hen acquire money in open marke operaions up o fracions κ and κ B of randomly seleced loans and reasuries see 2). In he goods marke, household i can use wages, money holdings ne of loans, and addiional cash from curren period open marke operaions for is consumpion expendiures see 3). In he asse marke, household i receives payoffs from mauring asses, buys reasuries, and borrows/lends using a full se of nominally sae coningen claims. Dividing he period price of one uni of nominal wealh in a paricular sae of period + 1 by he period probabiliy of ha sae gives he sochasic discoun facor q,+1. The period price of a random payoff D i, from invesmens in sae coningen claims in period + 1 is hen given by E q,+1 D i,+1 ]. Subsiuing ou he socks of bonds and money held before he asse marke opens, Bi, and M i,, in 4), he asse 14 Credi goods are sold on rade-credi, which is neiher explicily specified in his paper nor considered as a pledgeable asse. In Appendix F, we analyze consumpion loans in an alernaive model version and allow for he possibiliy ha hey are eligible in open marke operaions. 8

10 marke consrain of household i can be wrien as M H i, 1 + B i, 1 + L i, 1 1/R L ) + P w i, n i, + D i, + P δ i, + P τ i, 11) M H i, + B i, /R ) + E q,+1 D i,+1 ] + I i, R m 1) + P c i, + P c i,, while is borrowing is resriced by he no-ponzi game condiion lim s E q,+s D i,+s+1 as well as by M H i, and B i,. The erm I i, R m 1) in 11) measures he coss of money acquired in open marke operaions. Maximizing 9) subjec o he collaeral consrain 2), he cash-in-advance consrain 3), labor demand 1), he asse marke consrain 11), and he borrowing consrains, for given iniial values M H i, 1, B i, 1, and D i,, leads o he following firs order condiions for boh ypes of consumpion goods, working ime, injecions, and loans c σ i, = λ i, + ψ i,, 12) γ c σ i, = λ i,, 13) µ χn η i, = w λi, + ψ i, ), 14) ψ i, = R m 1) λ i, + R m η i,, 15) R m λi, + η i, ) = R L λi, + η i, κ ), 16) where µ = ζ /ζ 1) denoes a sochasic wage mark-up, λ i, he muliplier on he asse marke consrain 11), η i, he muliplier on he collaeral consrain 2), and ψ i, he muliplier on he cash-in-advance consrain 3). Noably, he laer affecs he consumpion and leisure choices 12) and 14), implying c σ i, = µ χn η i, /w. Furher, he following firs order condiions for holdings of bonds, money, and coningen claims, and complemenary slackness condiions, λ i, = βr E λ i,+1 + κ B +1 η i,+1 π +1, 17) λ i,+1 + ψ i,+1 λ i, = βe, π +1 18) q,+1 = β λ i,+1, π +1 λ i, 19) = ψ i, I i, + Mi, 1 H L i, /R L ) + P w n i, P c i, ], 2) = η i, κ B B i, 1 + κ L i, R m I i, ], 21) as well as 2), 3), and 11) wih equaliy since λ i, >, see 13) and he ransversaliy condiions hold. Combining 15), 17), and 18) o R E λ i,+1 + κ B +1 η i,+1) /π+1 ] = E R m +1 λ i,+1 + η i,+1 )/π +1 ], shows ha household i is indifferen beween invesing in money or invesing in reasuries and convering a fracion κ B +1 of hem ino cash in he nex period a he rae Rm +1. 9

11 Likewise, 16) shows ha he loan rae R L depends on he fracion of loans eligible as collaeral in open marke operaions, κ. Using 12), 15), and 18), condiion 16) can be wrien as 1/R L = 1 κ ) c σ i,βe c σ i,+1 π 1 +1 ] + κ /R m, 22) which shows ha he inverse of he loan rae is a convex combinaion of he inverse of he nominal marginal rae of ineremporal subsiuion of consumpion of he cash good c σ i, βe c σ i,+1 π 1 +1 ] and he inverse of he policy rae 1/R m. When loans are no fully eligible, κ < 1, here can be a spread beween he policy rae and he loan rae, i.e. an illiquidiy premium, while for κ = 1 boh raes are idenical, R L 2.4 Public secor The cenral bank ransfers seigniorage revenues P τ m = R m. Oherwise, κ =, he loan rae equals c σ i, /βe c σ i,+1 π 1 +1 ]). o he reasury, which issues one-period bonds and pays a subsidy a a consan rae. The supply of shor-erm governmen bonds is specified in a simple way. Specifically, we assume ha he oal amoun of shor-erm reasuries B T, which are eiher held by households or he cenral bank, grows a some exogenously deermined rae Γ > β, B T = ΓB T 1, 23) given B 1 T >. Noe ha we do no aim o measure oal public deb by he sock of shor-erm bonds B T, which can be inerpreed as -bills. To absrac from furher effecs of fiscal policy, we assume ha he governmen has access o lump-sum ransfers P τ, which balance he budge. Is budge consrain is hus given by B T /R ) + P τ m = B T 1 + P τ + P τ p mc a 1 nα j, dj. In open marke operaions, he cenral bank supplies money ourigh and emporarily in repos agains reasuries, M H = 1 M i, Hdi and M R = 1 M i, R di, and agains corporae loans under repos, M L = 1 M i, L di. A he beginning of each period, is sock of reasuries equals Bc 1 and he sock of ousanding money equals M 1 H. I hen receives an amoun Bc of reasuries and loans L R in exchange for money a he amoun B c /R m ) + L R /R m ). Before he asse marke opens, where he cenral bank rolls over mauring asses, repos are seled. Is budge consrain hus reads B c /R ) B c 1 + P τ m = R m M H M 1 H ) + R m 1) M L + M R ). 24) Accouning for common cenral bank pracice, we assume ha he cenral bank ransfers is earnings from holding asses and from open marke operaions o he reasury, P τ m = 1 1/R ) B c + R m 1) M H M 1 H + M L + M R ). Subsiuing ou ransfers in 24) shows ha cenral bank asse holdings evolve according o B c B c 1 = M H M 1 H. Assuming ha iniial values for cenral bank s asses and liabiliies saisfy B 1 c = M 1 H, delivers he cenral bank s balance shee B c = M H. 25) 1

12 The cenral bank has hree main insrumens. I ses he policy rae R m 1 and can decide how much money o supply as fracions of eligible asses, for which i can adjus he wo addiional insrumens, κ and κ B, in a sae coningen way. We assume ha he cenral bank ses κ B beween zero and one, κ B, 1] and κ larger or equal o zero, κ. To accoun for he case where 1) is no binding if R L = 1), we resric κ o be smaller han he wages-o-loans raio κ κ, where κ = R L 1 w n j,dj/ 1 l j,dj, such ha κ, min {1, κ }]. By saisfying κ κ, he cenral bank only acceps loans up o he amoun ha is issued for curren producion expendiures. 15 If 1) is binding, i.e. when R L > 1, κ κ implies κ 1. Finally, he cenral bank can decide wheher money is supplied in exchange for reasuries via repos or ourigh while loans are only raded under repos). Specifically, i conrols he raio of reasury repos o ourigh purchases of bonds Ω > : M R = Ω M H, where a suffi cienly large value for Ω ensures ha injecions are always posiive, I i, >. The raio Ω can furher be adjused in he long-run o implemen he desired long-run inflaion arge see Proposiion 7). 2.5 Equilibrium We resric our aenion o symmeric equilibria, where all households and inermediae goods producing firms behave in an idenical way. There will be no arbirage opporuniies and markes clear, n = 1 n j,dj = 1 n i,di, y = 1 c i,di + 1 c i,di = c + c, and 1 L i,di = 1 L j,dj = L. Aggregae socks of asses saisfy 1 D i,+1di =, 1 M i, Hdi = M H, 1 M i, Rdi = M R, 1 M i, L di = M L, 1 B i,di = B, 1 I i,di = I = M H M 1 H + M R + M L, and B T = B + B c. Using he laer and 25), imply household bond holdings o saisfy b = b T m H, where b, b T, and m H denoe real values of governmen liabiliies b = B /P, b T = B T /P, and m H = M H /P. Since inermediae goods producing firms behave in an idenical way, heir aggregae oupu saisfies IO = a n α. Reailers can differ wih regard o heir prices, which migh lead o dispersed reail prices. Marke clearing for he inermediae goods marke, IO = 1 y k,dk hen implies for aggregae oupu a n α = 1 P k,/p ) ε y dk y = a n α /s, where s is a measure of price dispersion, s = 1 P k,/p ) ε dk. Following 25] and 22], we resric our aenion o he case where here is no iniial price dispersion, s 1 = 1. Taking differen cohors of price adjusing firms ino accoun, s can be wrien as s = 1 φ)z /P ) ε + 1 φ)φz 1 /P ) ε + 1 φ)φ 2 Z 2 /P ) ε +... and herefore as s = 1 φ) l= φl Z ε l Πl s=1 πε +1 s. Taking differences, ε leads o s = 1 φ) Z + φs 1 π ε. Defining Z = P /P and rewriing he denominaor and numeraor in a recursive way, 8) can be wrien as Z = 19) Z 1, = γ c σ y mc + φβe π ε +1 Z 1,+1 and Z 2, = γ c σ reail prices furher gives 1 = 1 φ) Z 1 ε ε ε 1 Z 1,/Z 2,, where using 13) and y + φβe π ε 1 +1 Z 2,+1. Aggregaion over + φπ ε 1. A definiion of he compeiive equilibrium is 15 In conras, loans ha are no issued by firms o finance he wage bill, which migh be he case when 1) is no binding, are no eligible. 11

13 given in Appendix A. 3 The role of money raioning To see how money raioning can affec he equilibrium allocaion, consider firs he cash-in-advance consrain 3), which implies P c M H +M R +M L in equilibrium. I gives rise o non-neuraliy of moneary policy if i is binding. To see when his is he case, use he condiions 12) and 18), which imply c σ c = βe σ +1 π +1 + ψ and ha he muliplier on he cash-in-advance consrain ψ saisfies in equilibrium ψ = c σ 1 1/R )], 26) where R is defined as he nominal marginal rae of ineremporal subsiuion of cash good consumpion, 1/R = βe σ +1 /P +1 c. Households are indifferen beween 1/R c σ /P unis of he means of paymen in period, which is required for consumpion purchases, and one uni in period + 1. They are herefore willing o give up R 1 in order o ransform one uni of accoun ino one uni of money oday. Thus, a posiive value R 1 > reflecs like in sandard models a sricly posiive valuaion for money and implies ha households will no hold more money han needed for consumpion expendiures. Then, ψ > see 26) and he cash-in-advance consrain is binding see 2), while i is no binding, ψ =, if R = 1. Now consider he collaeral consrain 2), which in equilibrium reads Noably, he insrumens κ B M H M H 1 + M L + M R κ B B 1 /R m ) + κ L /R m ). 27) and κ ener he se of equilibrium condiions see Definiion 1 in Appendix A) only via he collaeral consrain 27) and he asse pricing condiions 16) and 17), and joinly wih he muliplier η. Hence, if 27) is no binding, η =, he insrumens κ B and κ will no affec he equilibrium allocaion and prices, and he model reduces o a sandard model see Definiion 2 in Appendix A). To see when 27) is binding, use ha 12), 15), and 18) imply c σ = R m c λ + η ) and λ = βe σ +1 π +1 in equilibrium, and eliminae λ, which leads o he following condiion for he muliplier on he collaeral consrain η Condiion 28) shows ha, when he policy rae R m η = c σ 1/R m ) 1/R )]. 28) is smaller han R, he muliplier η is sricly posiive and he collaeral consrain is binding see 21), such ha access o money depends on he insrumens κ B) and on holdings of eligible asses. In his case, he cash-in-advance consrain is binding as well, ψ > see 26), given ha R m 1. Households can hen ge money in exchange for eligible asses a he price, R m 1, which is below heir marginal valuaion of money, R 1. Hence, hey use eligible asses as much as possible o ge money in open marke operaions, 12

14 such ha 27) is binding and money supply is effecively raioned by he cenral bank s collaeral requiremens. 16 To demonsrae how agens decisions are disored and how he policy insrumens can be used o address hese disorions, we will repeaedly refer o he firs bes allocaion, which is characerized in he following Proposiion. Proposiion 1 The firs bes allocaion {c, c, n } = saisfies and hus n = a 1 σ Proof. See Appendix B a n ) α = c + c, χ n ) 1+η α c ) σ = a α, and γ c ) σ = c ) σ, 29) 1 + γ 1/σ ) σ α/χ)] 1 η+1 α+σα, c = a n ) α / 1 + γ 1/σ), and c = c γ 1/σ. The model exhibis several fricions, which are sandard in he lieraure on opimal moneary policy see 6] or 23]). Firs, prices are imperfecly flexible and can be dispersed, s > 1, which leads o an ineffi cien allocaion of working ime. Second, imperfec compeiion beween labor suppliers and beween reailers lead o mark-ups over wages and over prices, i.e. µ and 1/mc, which can vary over ime due o innovaions o he subsiuion elasiciy ζ and due o sicky prices. Third, firms rely on loans o finance wages in advance, such ha labor demand is disored by he loan rae if R L > 1. Fourh, he households cash-in-advance consrain disors he choice of consuming cash goods c or credi goods c if R > 1. The effecs of hese disorions can easily be idenified by comparing 29) wih he following equilibrium condiions ha immediaely follow from 6), 12)-14), y = a n α /s, and he resource consrain: a n α = c + c ) s, 3) ] χn 1+η α c σ mc 1 = a α µ 1 τ n, 31) ) γ c σ R L = βe c σ +1 /π +1], 32) A convenional ineres rae policy ha is associaed wih a non-raioned money supply canno implemen he firs bes allocaion, which is a well-esablished resul in he lieraure on opimal moneary policy in sicky price models see 8], 17], or 6]). This resul holds rue even if no all disorions menioned above were presen, since he cenral bank has only one insrumen R m ) a is disposal. If, however, he cenral bank supplies money subjec o he collaeral consrain 27) and he laer is binding, addiional insrumens i.e. κ B and κ ) are available. 17 Then, money supply is raioned and he policy rae as well as ineres rae on eligible asses i.e. R and 16 This furher implies ha R m canno be larger han R see 28). 17 The insrumen Ω is used o ensure posiive injecions, I >, and o implemen he long-run inflaion arge see Proposiion 7), while i will no be uilized for sabilizaion purposes. 13

15 R L ) are decoupled from he marginal rae of ineremporal subsiuion, which reflecs changes in consumpion and inflaion ha can be influenced by he cenral bank s supply of money. Money raioning hus allows freeing up money supply insrumens in addiion o he policy rae and enables he cenral bank o address more han one disorion. The following Proposiion summarizes how moneary policy insrumens affec he deerminaion of he equilibrium allocaion under boh scenarios. Proposiion 2 For given sequences {s, π } =, he equilibrium sequences {c, c, n, mc, R L } = are deermined by 3)-32) and 1. under raioned money supply, i.e. under a binding collaeral consrain 27), by 1/R L = 1 κ ) βc σ E c σ +1 π 1 +1 ] + κ /R m, 33) c = κ mc / 1 τ n )]a αn α /R m ) + κ B b 1 /R m ) π 1 + m H 1π 1, 34) where {m H, b } = saisfy b + m H = Γ b 1 + m H 1) /π and m H 1 + Ω ) = m H 1 + κ B b 1 /R m ]π 1, given {a,µ } =, {Rm, κ, κ B, Ω } =, mh 1 > and b 1 >. 2. under non-raioned money supply, i.e. under a non-binding collaeral consrain 27), by given {a,µ } = and {Rm } =. Proof. See Appendix B 1/R L = βc σ E c σ +1 π 1 +1 ], Rm = R L, 35) Par 1 of Proposiion 2 refers o he case where he collaeral consrain 27) is binding, which requires he policy rae o be se according o R m < R, where R = 1/βc σ E c σ +1 π 1 +1 ]) see 28). The equilibrium condiions 33) and 34) show ha he cenral bank can influence he allocaion by changes in he hree insrumens R m, κ B, and κ. If a non-zero fracion of loans is acceped as collaeral, κ >, he policy rae direcly affecs he loan rae see 33), which alers he consumpion-labor choice according o 31). The cenral bank can furher induce changes in he demand for he cash good by ighening or expanding money supply wih all insrumens see 34), given ha he cash-in-advance consrain is hen binding as well see 26). Under money raioning he cenral bank can hus apply more han one insrumen o affec he equilibrium allocaion via he firms borrowing coss as well as via aggregae demand, and can hereby enhance welfare when he firs bes allocaion is no implemenable wih a single insrumen. If, in conras, he policy rae equals he nominal marginal rae of ineremporal subsiuion R m = R, he collaeral consrain 27) is slack and money supply is no effecively) raioned Seing R m = R is equivalen o he case where he cenral bank supplies money agains asses which are abundanly available, like coningen claims D i,, which are issued by privae agens. The policy rae can hen no be lower han he marginal coss of money R in equilibrium, since abundanly available asses can be used o acquire more money, such ha consumpion increases and R decreases unil R = R m. 14

16 The insrumens κ and κ B are hen irrelevan for he allocaion and he loan rae equals he policy rae see par 2 of Proposiion 2). The policy rae hen affecs consumpion and expeced inflaion via he consumpion Euler equaion 35) and simulaneously via he consumpion-leisure choice see 31). Such a single insrumen regime, can suffi ce for a cenral bank o conduc opimal policy if i faces jus a single disorion. When firs bes can in fac be implemened by appropriaely seing he policy rae, money raioning can be irrelevan or even no recommendable. This is summarized for he sylized case of perfec compeiion ε and ζ ) and perfecly flexible prices φ = ) in he following Proposiion. Proposiion 3 Suppose ha compeiion is perfec, prices are perfecly flexible, and τ n =. The allocaion hen equals firs bes if R L = 1 and R = 1 for γ >, requiring R m = 1 and a nonraioned money supply. Proof. See Appendix B Under perfecly flexible prices and perfec compeiion, which also implies ha here are no cospush shocks, he privae secor behavior is only disored by he liquidiy consrains 1) and 3) ha firms and households face. These disorions are undone when holding money is cosless, which demands ha he lending rae R L equals one and, when households consume credi goods γ > ), ha he nominal marginal rae of ineremporal subsiuion R also equals one. Boh requiremens ogeher wih 33) imply ha he policy rae R m equals R, and, by 28), ha money supply is no raioned. When households do no consume credi goods, γ =, he nominal marginal rae of ineremporal subsiuion does no need o equal one, such ha firs bes can be implemened regardless wheher money supply is raioned or no. 19 For less sylized cases where prices are sicky or cos-push shocks exis, a welfare maximizing moneary policy will, however, rely on money raioning, as i allows o address more han one disorion. Specifically, policies ha aim a sabilizing prices or a neuralizing cos-push shocks and ha simulaneously srive for low borrowing coss can be conduced in a more successful way under money raioning han under a single insrumen regime. This will be shown in he subsequen Secion. 4 Opimal moneary policy In his Secion, we examine opimal moneary policy under commimen, wihou he simplifying assumpion of perfec compeiion. In he firs par, we describe some main properies of he soluion o he opimal policy problem. In he second par, we focus on he special case wihou credi goods, where money raioning enables he cenral bank o implemen firs bes. 19 For he case of a cashless economy wih sicky prices, i is well esablished see he discussion of divine coincidence in 3]) ha a single insrumen can also be suffi cien o implemen firs bes if average mark-ups are eliminaed). For his, he policy rae has o be se in a way ha induces aggregae demand o be consisen wih consan marginal coss, such ha prices are never changed. 15

17 4.1 Opimal policy under money raioning Throughou he subsequen analysis, we examine opimal moneary policy under he assumpion ha he cenral bank has access o a commimen echnology ha ensures ha policy announcemens are honored. I is well known ha opimal policy under commimen may enail a ime inconsisency when forward-looking equilibrium condiions like 32) serve as consrains o he policy problem. Following he relaed lieraure on opimal moneary policy see, e.g., 23]), we disregard he issue of ime-inconsisency and resric our aenion o ime-invarian processes of he soluion o he policy plan. 2 In general, an opimal moneary policy will seek o minimize he effecs of he disorions menioned in Secion 3. As demonsraed by Proposiion 2, he cenral bank has more insrumens available if i effecively raions money supply, i.e. if i ses he policy rae in a way ha leads o a binding collaeral consrain 27). To examine if he cenral bank acually chooses o raion money supply, we consider he collaeral consrain 27) and he cash-in-advance consrain 3) as consrains o he policy problem. When deriving he opimal policy, we resric our aenion o cases where he consrains on he cenral bank insrumens, κ B R m, 1], κ, min {1, κ }], and 1, are no binding. For example, he non-negaiviy consrain on ineres raes imposes a relevan resricion on he policy rae, R m 1, which migh be binding for large shocks or when he mean policy rae implied by he soluion o he policy problem is close o uniy. We herefore consider a producion subsidy τ n = 1 ε 1)/εµ) and sandard deviaions of shocks ha allow o implemen he opimal policy plan wihou violaing hese consrains on he insrumens in he neighborhood of he seady sae under he opimal policy. 21 Examining he policy problem, i can be shown ha he cenral bank will acually choose o raion money supply. Specifically, by seing he policy rae according o R m < R, he cenral bank can reduce he loan rae below R and offse mark-up shocks by sae coningen adjusmens of is addiional insrumens. Hence, he allocaion under he opimal policy is independen of mark-up shocks, which is summarized as follows. Proposiion 4 The opimal moneary policy under commimen is associaed wih money raioning, i.e. R m < 1/βc σ E c σ +1 π 1 +1 ]) and κ >. The policy problem is hen given by max E {c, c,n,π,s } = = β uc, c, n ), s.. 3), 32), s = φs 1 π ε +1 φ) 1 1 ε ) ε 1 φπ ε 1 ε 1, 36) for =,... and s 1 = 1, such ha he soluion o 36) is independen of cos-push shocks. Proof. See Appendix C 2 Specifically, he opimaliy condiions ha will be applied in he analysis can be inerpreed as being par of a commimen plan derived and implemened in a imeless perspecive see, e.g., 26]). 21 See Secion 6 for a discussion of he limis o money raioning. 16

18 Since he loan rae R L can be manipulaed by seing R m for κ > according o 33), wihou simulaneously alering he nominal rae of ineremporal subsiuion as in 35), he cenral bank can offse sochasic changes in he mark-up µ see 31). This is in general no possible for he case of non-raioned money supply, where 31) is in general a binding consrain o he policy problem see Appendix E). However, he remaining disorions canno compleely be removed under money raioning when households assign a posiive value o credi goods, γ >. Specifically, avoiding a welfare reducing price dispersion requires sable goods prices, whereas avoidance of he inflaion ax on he cash good calls for a deflaion according o he Friedman rule see 32). Since he choice of inflaion canno address boh disorions simulaneously, firs bes canno be achieved. 22 propery is summarized in he following Proposiion. Proposiion 5 Consider he policy problem in 36) for γ >. Under he opimal policy, prices are no sable in he long-run and he allocaion differs from he firs bes allocaion. This Proof. See Appendix C Given ha prices are sicky, he cenral bank will no fully eliminae he inflaion ax on cash goods induced by he households cash-in-advance consrain 3). However, i can subsanially reduce he disorion induced by he firms liquidiy consrain 1) under money raioning by seing he policy rae below he nominal marginal rae of ineremporal subsiuion, R m < R, and declaring a posiive fracion of loans as eligible, κ >. This is apparenly no possible under a non-raioned money supply, where he seady sae values of he policy rae and he loan rae are idenical and deermined by π/β. Thus, he possibiliy o separae he policy rae from R allows he cenral bank o reduce he long-run effecs of disorions as well as o sabilize he economy in he shorrun in a more effi cien way han under non-raioned money supply, which will be shown in Secion A special case In his Secion, we examine he special case where here exiss no cash-credi goods fricion and average borrowing coss are eliminaed, which is paricularly useful for demonsraing how money raioning improves he cenral bank s abiliy o address disorions in he shor-run. For his version of he model, we can derive a closed form soluion o he policy problem. Specifically, we assume ha households do no consume credi goods, γ =, such ha he inflaion ax does no disor he household decisions as in 32). Neverheless, he allocaion beween consumpion and working ime, which are boh associaed wih liquidiy requiremens see 1 and 3), is affeced by he loan rae see 31). Under money raioning, he cenral bank can hen se he insrumens 22 I should be noed ha he policy insrumen κ B migh be non-saionary under he opimal policy. This can be he case when he growh rae of shor-erm reasuries Γ see 23) deviaes from he long-run inflaion rae under he opimal policy see Proposiion 7). 17

19 such ha welfare losses due o changes in mark-ups and due o price dispersion are avoided by manipulaing he loan rae and aggregae demand in a way ha eliminaes he wedges in 31) and ha implies consan goods prices. This however requires average mark-ups o be small and he average loan rae o be suffi cienly large such ha he zero ineres rae bound is no hi. For his, we follow relaed sudies on opimal moneary policy see 2] and 6]) and consider a subsidy which correcs for he deerminisic means of he mark-ups and he coss of borrowing: τ n = 1 mc/µ] /R L, 37) where µ = ζ/ζ 1) and mc = ε 1) /ε. When producion is subsidized according o 37), moneary policy under money raioning can offse changes in mark-ups by induced changes in he loan rae see 33), while money supply can be adjused o implemen a level of aggregae demand consisen wih sable prices see 34). Hence, he cenral bank can overcome he well-known radeoff beween sabilizing inflaion and closing he oupu gap by raioning money supply. This resul is summarized in he following Proposiion. Proposiion 6 If households do no consume credi goods, γ =, and he producion subsidy saisfies 37), he cenral bank can implemen he firs bes allocaion if i raions money supply. Proof. See Appendix C. As saed in Proposiion 6, he cenral bank can implemen he firs bes allocaion when i raions money supply, which endows he cenral bank wih he wo money supply insrumens κ and κ B in addiion o he policy rae R m. To see how he insrumens are adjused o implemen firs bes, inser 37) in 31) o ge χn 1+η α c σ = a α mc /mc µ /µ R L R L ]. 38) A comparison of 3) and 38) for γ = c = wih he corresponding condiions for he firs bes allocaion 29) shows ha mc /mc = µ /µ) R L /R L) and s = 1 have o be saisfied for he allocaion under he opimal policy o be idenical wih firs bes. The absence of price dispersion, s = 1, requires consan prices, π = 1, which is only consisen wih opimal price seing when marginal coss are consan, mc = mc see proof of Proposiion 7). Hence, he cenral bank can implemen he firs bes allocaion if and only if i ses is insrumens such ha mc = mc and R L /R L = µ/µ. 39) Given ha he cash-in-advance consrain and he collaeral consrain are binding see Proposiion 4), he cenral bank can adjus R m o induce he loan rae o saisfy R L /R L = µ/µ, and can se κ and κ B in a sae-coningen way ha implies a consumpion level which is consisen 18

20 wih mc = mc see Proposiion 2). In paricular, κ and κ B can be adjused according o 34) such ha consumpion equals c and by 3) working ime equals n, where c = cs, c ) and n = ns, n ). 23 Once mc = mc holds, reailers will no change heir prices, implying ha here will be no price dispersion, s = 1, and he equilibrium allocaion is idenical o he firs bes allocaion, c = c allocaion can be implemened. and n = n. The following Proposiion summarizes how he firs bes Proposiion 7 For γ = and 37), he cenral bank implemens price sabiliy and he firs bes allocaion if i ses R m 1, κ, min {1, κ }], κ B, 1], and Ω according o κ /R m = µ / µr L) 1 κ ) βc σ E c σ κ B b 1 = R m π c 1 κ s mc µ αr L /R m +1 π 1 +1 ] >, 4) ], 41) ) m H 1 π 1 R m < 1/βc σ E c σ +1 π 1 +1 ]), and lim 1+Rm Ω /κ B +1 1+Ω Γ 1+Rm Ω 1 /κ B 1+Ω 1 ) =. If loans are no eligible, κ =, he firs bes allocaion canno be implemened. Proof. See Appendix C. As described in Proposiion 7, opimal policy can implemen he firs bes allocaion by using he insrumens R m, κ, and κ B. For his, he cenral bank has o accep a leas some loans in open marke operaions, κ >, o saisfy 4). The condiions lised in Proposiion 7 furher imply ha muliple combinaions of he insrumens are consisen wih opimal policy. The cenral bank can direcly manipulae he loan rae by seing he policy rae for a given κ see 4), 24 and i can implemen he desired consumpion level c by adjusing κ B laer condiion showing ha a higher κ requires ceeris paribus a lower κ B or κ according o 41); he and vice versa. Under he opimal policy, he long-run inflaion rae equals one, which can be implemened if he cenral bank adjuss κ B or Ω in he long-run o off-se rends Γ > 1 or Γ < 1) in he supply of governmen bonds, i.e. 1+ΩRm /κ B +1 1+Ω = Γ 1+Ω 1R m /κ B 1+Ω 1 for. 25 If loans are no eligible, κ =, he loan rae R L equals R see 22) and is only affeced by he policy rae hrough is equilibrium impac on consumpion and inflaion. In his case, he remaining insrumens, i.e. R m and κ B, joinly affec he consumpion level via 41), bu hey canno simulaneously be used o off-se mark-up shocks. Pu differenly, changes in he policy rae do no direcly affec firms marginal coss if κ =, such ha all insrumens impac on he demand side and firs bes canno be implemened. This resul relaes o he well-known rade-off faced by 23 As shown in he proof of Proposiion 7, he arge values c and n are given by c = s 1+η)/η+ασ+1 α) c and n = s σ 1)/η+ασ+1 α) n, where he firs bes values c and n saisfy 29). 24 This can be done in he simples way when all loans are eligible, κ = 1, such ha 4) reduces o R m /R m = µ/µ. 25 For Γ > 1, he values of κ B have o decline over ime o implemen a saionary sequence of inflaion raes. If Γ < 1, he cenral bank has o increasingly supply money for reasuries under repos by leing he raio of repos o money supplied ourigh Ω increase over ime. 19