The Evolution of Cash Transactions: Some Implications for Monetary Policy

Transcription

1 The Evoluion of Cash Transacions: Some mplicaions for Moneary Policy STACEY L. SCHREFT AND BRUCE D. SMTH * November 997 Las Revised: February 999 Absrac This paper considers he implicaions for moneary policy of a decreasing demand for ouside money. finds ha even perpeual declines in he demand for base money pose no hrea o he radiional mehods employed for conducing moneary policy. The effecs of such reducions in he demand for cenral bank liabiliies, however, do depend on how moneary policy is conduced. Four moneary policy regimes are analyzed. Wih a policy of nominal-ineres-rae argeing, a secular decline in he volume of cash ransacions unambiguously leads o acceleraing inflaion. A policy of mainaining a fixed composiion of governmen liabiliies leads o acceleraing (deceleraing) inflaion if agens have sufficienly high (low) levels of risk aversion. nflaion argeing produces falling nominal and real ineres raes, while a policy of fixing he rae of money growh can easily lead o indeerminacy and endogenous oscillaion in ineres raes. is argued ha a policy of fixing he composiion of governmen liabiliies has several advanages if i is known ha agens are no oo risk averse and ha he asympoic demand for base money is small. f his informaion is no known, hen ineres-rae or inflaion argeing have an advanage because heir consequences are no sensiive o such environmenal feaures. * Schref, Research Deparmen, Federal Reserve Bank of Kansas Ciy, Kansas Ciy, MO 6498, , sschref@frbkc.org; Smih, Deparmen of Economics, Universiy of Texas, Ausin, TX 7872, , bsmih@mundo.eco.uexas.edu. The views expressed in his paper are no necessarily hose of he Federal Reserve Bank of Kansas Ciy or he Federal Reserve Sysem.

2 has long been hough ha moneary policy affecs he macroeconomy by inducing variaions in he supply and demand of ouside money. Recenly, however, he radiional sources of he demand for ouside money have been in pronounced decline. For example, coninuous echnological improvemen in elecronics and communicaion sysems has made possible he developmen of several new, noncash means of paymen. As he use of hese new paymen insrumens has grown, here has been a pronounced shif away from he use of cash in ransacions. According o The Nilson Repor (997), cash accouned for 20 percen of he dollar volume of U.S. paymens made by consumers in 990 and 8 percen in 996, and is projeced o accoun for only 6 percen in 2000 and 2 percen in A he same ime, many counries have eliminaed reserve requiremens on mos or all inermediaries, while oher counries ha have no done so (like he Unied Saes) have permied financial innovaions ha render reserve requiremens virually inconsequenial. 3 As a resul, i seems enirely possible ha he demand for base money may virually or enirely vanish in he no-oo-disan fuure. These developmens necessiae he reopening of an age-old quesion in moneary economics: Wha does a declining demand for base money and, perhaps, a demand ha is declining wihou limi imply for moneary policy? 4 Does i imply ha cerain mehods of conducing moneary policy may become infeasible? Does i mean ha some mehods may lead o unbounded inflaion or indeerminacy? And, mus a cenral bank know wheher he use of ouside money will disappear alogeher or jus become minimal in order o deermine he bes course of acion? This paper proposes a simple framework for hinking abou hese issues. begins by presening a pure-exchange, overlapping-generaions model wih only wo primary asses governmen-issued fia currency and governmen bonds. To generae a demand for cash ransacions, he model incorporaes spaial separaion and limied communicaion along he lines For example, Lindsey and Wallich (989, p. 23) asser ha variaions in he supply of reserves relaive o he demand for hem, wih associaed impacs on he cos of reserves, oher ineres raes, and he sock of money, are he iniial channels hrough which mos cenral banks of developed capialis counries use heir policy insrumens o affec he macroeconomy. 2 Cash is used, and will coninue o be used in he foreseeable fuure, for mos small consumer ransacions. Such ransacions consiue he majoriy of all ransacions, bu heir oal dollar value is relaively small (see Nilson 997). 3 See Sellon and Weiner (996). 4 Wicksell (898) appears o be he firs o consider his quesion.

3 2 of Townsend (987). To generae a role for banks, he model includes shocks o liquidiy needs along he lines of Diamond and Dybvig (983). Togeher hese feaures imply a derived demand for base money ha depends on he need for currency in paymens and he demand by banks for cash reserves. 5 The volume of cash ransacions is hen assumed o evolve over ime in a way ha affecs he oal demand for base money. 6 Four convenional mehods for conducing moneary policy are considered. The cenral bank can conduc moneary policy eiher by seing he mix of governmen bonds and money ousanding or by seing he money-supply growh rae. Alernaively, i can se eiher a nominal-ineres-rae or an inflaion-rae arge. To minimize complicaions, for each policy he cenral bank is assumed o se only a single once-and-for-all arge value for he variable i conrols. This differs from Woodford (998), which allows for policies wih feedback rules. The four major findings can be summarized as follows. Firs, all of hese sandard mehods for conducing policy are feasible, even in an asympoically cashless economy, in he following sense. A each dae, well-defined condiions deermine all real and nominal quaniies, and he cenral bank can achieve is arge wih convenional open marke operaions. The implicaion is ha convenional mehods for conducing moneary policy can coninue o work as expeced in he face of a declining demand for base money. Second, hree of he policies considered inflaion argeing, nominal-ineres-rae argeing, and fixing he bond-o-money raio imply no indeerminacies, even if he demand for base money asympoically goes o zero. A policy of fixing he money growh rae implies no indeerminacies only if agens are no oo risk averse. Oherwise, i does lead o indeerminacies and may also lead o endogenous flucuaions ha migh no disappear asympoically. ndeed, moneary flucuaions due o self-fulfilling prophecies can persis indefiniely, even if he demand for base money is declining o zero. Third, he rae of inflaion will necessarily remain bounded under hree of he policies inflaion-argeing, ineres-rae argeing, and a consan rae of money creaion even if he 5 The model is in fac a pure-exchange version of ha in Champ, Smih, and Williamson (996) and Schref and Smih (997, 998). 6 A relaed paper is Dow (995), which sudies he effec on he price level of an exogenous decline in he need for cash in a cash-in-advance model wihou banks. Moneary policy in his model consiss of adjusmens in he quaniy of money accomplished hrough lump-sum ransfers.

4 3 demand for base money vanishes in he limi. Under he fourh policy a fixed bond-o-money raio he price level necessarily sabilizes in finie ime if cash asympoically goes ou of use and if agens are no oo risk averse. When hese condiions do no hold, his policy is he only one ha leads o unbounded inflaion in an aysmpoically cashless economy. Finally, as he las observaion indicaes, o predic he consequences of mainaining a fixed bond-o-money raio, i is necessary o know a good deal abou he preferences of bank deposiors and he asympoic magniude of cash use. The same is rue under a policy of fixing he money growh rae, bu no wih inflaion or ineres-rae argeing. Since deposior preferences and he long-run demand for cash are difficul o deermine wih precision, his finding highlighs an advanage of inflaion or ineres-rae argeing. The quesion of he welfare consequences of he differen policies is more vexing. will ypically no have an unambiguous answer wihou specifying how he uiliy of differen generaions should be weighed. The analysis does, however, yield one conclusion regarding welfare. Of he policies ha migh be observed in pracice (ha is, ruling ou policies ha generae susained deflaion or a zero nominal rae of ineres), here is only one ha can eliminae he disorions associaed wih he fricions ha generae moneary exchange. is he policy of fixing he bond-o-money raio, and i evenually eliminaes such disorions only if agens are no oo risk averse. Bu ineresingly, hese disorions will be eliminaed in finie ime even if cash never goes ou of use, so long as moneary policy is no oo igh and long-run cash use is no oo large. The remainder of he paper proceeds as follows. Secion describes he environmen, and secion 2 presens he model s equilibrium condiions. Secions 3 hrough 6 analyze he various moneary policies considered, while secion 7 compares he properies of he differen policy regimes. Finally, Secion 8 concludes by discussing he role ha some of he assumpions play in generaing he resuls obained.. Environmen Consider an infinie-horizon economy, wih =,2,... indexing ime. The economy consiss of wo idenical islands, each inhabied by an infinie sequence of wo-period-lived

5 4 overlapping generaions. Each island has, a he sar of each dae, a coninuum of ex ane idenical young agens of measure one. A he iniial dae, each region also has an idenical old generaion. Agens are endowed wih ω > 0 unis of he economy s single nonsorable consumpion good when young. They have no endowmen when old. n addiion, hey derive uiliy from consumpion only when old, denoed by c. The uiliy funcion common o all agens is u(c)= c ρ ( ρ), wih ρ > 0. Because agens care only abou consumpion when old, hey save heir enire endowmen when young. They hold all heir savings, eiher direcly or indirecly, in he economy s primary asses. Two primary asses are available o agens in he economy: money (fia currency) and one-period, defaul-free governmen bonds. 7 M denoes he per capia value of he moneary base on each island, and B denoes he nominal per capia supply of governmen bonds. Each bond issued a is a claim o unis of currency a +. Thus, is he gross nominal ineres rae a. The ime price level is p, which is common across islands. Thus, in real erms, he per capia supplies of money and bonds are m M /p and b B /p, respecively. The governmen, hrough is cenral bank, has a variey of moneary policy opions. can fix eiher he bond-o-money raio or he rae of money creaion. Alernaively, i could arge eiher he nominal ineres rae or he inflaion rae. Subsequen secions of his aricle consider each of hese policy opions. Regardless of wha opion he governmen chooses, policy mus be conduced so ha he governmen budge consrain is saisfied: R b = ( M M ) p + b, () where R p p is he gross real ineres rae. M 0 > 0 and B 0 0 are given as iniial condiions. 8 The division of he economy ino islands inroduces he economically imporan feaure of spaial separaion ino ransacions. There is no communicaion across islands while ransacions are being conduced. Limied communicaion and spaial separaion imply ha agens canno exchange privaely issued claims across islands. n addiion, money is assumed o 7 Thus far he environmen has much in common wih Woodford (998), who also considers a pure exchange economy wih bonds and money as he only primary asses. 8 This modeling of he governmen essenially consolidaes he Treasury and cenral bank. One can hus hink of he cenral bank here as being able o issue deb if necessary o ensure ha i can conduc moneary policy.

6 5 be he only asse ha can be carried across islands and hus is he only asse ha can be used in inerlocaion exchange. This gives money an advanage over bonds in erms of liquidiy, which permis money o be dominaed in rae of reurn. 9 Each period, afer porfolio-allocaion decisions are made, a fracion π (0,) of he young agens from each island discover ha hey have o relocae o he oher island before he period ends. Those agens facing relocaion have o hold all heir wealh in he form of currency when hey move or hey will be unable o ransac on heir new island. To capure he evoluion of cash ransacions in he economy, π is assumed o saisfy π π = µ ( π π), (2) where π is he long-run value of π and µ (0,] is a known consan. Thus, π is known a he sar of period, alhough he ideniies of he specific agens facing relocaion are no known. Equaion (2) capures a number of possibiliies regarding he evolving role of currency in exchange. When µ =, π is consan and he economy is saionary. When µ < and π > π, he volume of ransacions ha require currency shrinks over ime. ndeed, when π = 0, cash may, and wih posiive nominal ineres raes will, evenually go ou of use alogeher. While π is defined as he fracion of agens who are relocaed, a secular reducion in π can proxy for a number of scenarios in which improvemens in communicaion and recordkeeping reduce he need for cash in ransacing. n one such scenario, all young agens are relocaed beween periods, wih fracion π of hem relocaed o a par of heir desinaion island where communicaion is cosly enough o preclude he use of checks or oher insrumens in exchange. A declining value of π hus corresponds o susained improvemens in communicaion ha increase he use of noncash insrumens in inerlocaion exchange. n any even, he model s assumpions on inerlocaion exchange imply ha agens who learn hey will be relocaed will wan o conver all heir asses ino currency. Random relocaions hus play he same role here ha liquidiy-preference shocks play in Diamond and Dybvig (983). And as in Diamond-Dybvig, agens will wan o insure agains premaure asse liquidaion. This insurance can be provided efficienly (see Greenwood and Smih 997) hrough a bank ha announces deposi-reurn schedules ha depend on deposior-wihdrawal daes (i.e., 9 The noion ha bonds are no useful in inerlocaion exchange could be moivaed by he realisic assumpion ha hey mus be issued in relaively large denominaions.

7 6 relocaion saus), akes deposis, and chooses how o allocae he deposied funds beween money (ha is, cash reserves) and governmen bonds. There is free enry ino banking, so compeiion ensures ha bank profis are zero in equilibrium. As in Diamond-Dybvig, banks of his ype inermediae all savings. Now consider he deerminaion of deposi-reurn schedules. Le r m (π ) (r n (π )) denoe he sae-coningen gross real reurn on deposis offered by a ypical bank o agens who are (are no) relocaed a. Banks announce hese reurns, aking he reurns offered by oher banks as given. A Nash equilibrium is a deposi-reurn schedule (r m (π ), r n (π )) for each bank such ha, given his reurn schedule, no oher bank has an incenive o aler is se of announced reurn schedules. Compeiion among banks for deposiors implies ha, in equilibrium, banks choose deposi-reurn schedules o maximize he expeced uiliy of a represenaive deposior subjec o a se of resource consrains, which are described below. Given his behavior by banks, young agens choose o deposi heir enire endowmen ω, implying ha all savings are inermediaed. 0 Nex, consider he represenaive bank s porfolio-allocaion decision. Le m denoe he reserves (i.e., real balances) a represenaive bank chooses o hold per deposior a, and le b denoe he real value of bonds ha he bank holds per deposior a. Then m and b mus saisfy m + b ω, 0. By he law of large numbers, a fracion π of a bank s deposiors mus relocae a. Thus, he represenaive bank mus pay π r m (π )ω o hose agens a. Relocaed agens mus be given currency, so he bank s paymens o agens who move are consrained by is holdings of reserves: π rm ( π ) ω m p p +, 0. (3) Because agens who move a carry ino + he currency hey receive upon wihdrawing heir deposis, he promised reurn on deposis in (3) includes he gross real reurn on money, p p +. To he fracion π of a bank s deposiors who do no relocae a, he bank mus pay ( π )r n (π )ω upon wihdrawal. Assuming ha >, money is dominaed in rae of reurn, and he bank does no carry cash balances beween periods. Paymens o nonmovers, hen, are 0 f >, all young agens sricly prefer inermediaed o uninermediaed savings a. f =, young agens are indifferen beween holding bank deposis and holding currency. is easy o verify ha (a) some savings mus be inermediaed in his case so long as B > 0, and (b) here is no loss of generaliy in assuming ha all savings are inermediaed.

8 7 financed solely wih he bank s holdings of bonds; ha is, ( π ) r ( π ) ω R b, 0. (4) n Le γ 0 denoe he reserve-deposi raio. Then equaions (3) and (4) can be wrien as rm ( π ) γ ( p p +) / π, 0, (5) r ( π ) R ( γ ) / ( π ), 0. (6) n n a Nash equilibrium, r m (π ), r n (π ), and γ are chosen o maximize m ρ ρ π ( r ( π ) ω ) ( π )( r ( π ) ω ) + ( ρ) m n subjec o (5) and (6). The opimal reserve-deposi raio for his problem is given by 2 γ = + F HG KJ π ( ρ)/ ρ π r γ (, π ). (7) f > a, i is easy o verify ha r m (π ) < r n (π ). n his case, agens receive less han complee insurance agains he even of being relocaed. The reason is ha banks mus hold reserves in order o provide insurance coverage, and wih >, holding reserves involves an opporuniy cos ha causes agens no o desire complee insurance. f, however, =, hen i is easy o verify ha r m (π ) = r n (π ) = p /p = p /p. The aainmen of a zero nominal ineres rae hus eliminaes a disorion, much as i does in cash-in-advance models. Some properies of he funcion γ (, π ) will be useful for fuure reference. Differeniaion of (7) esablishes ha he ineres elasiciy of reserve demand is γ (, π ) γ (, π ) = (( ρ ) ρ)[ γ (, π )]. Clearly, γ (, π ) ( > ) 0 as ρ (>). The ambiguiy in he sign of γ (, π ) derives from convenional income and subsiuion effecs. A higher value of, ceeris paribus, increases he opporuniy cos of holding reserves. The subsiuion effec causes banks, acing on behalf of deposiors, o subsiue away from low-yielding asses. A higher value of, however, also increases he income a bank earns on is bond holdings. Sandard income effecs cause he bank o wan o raise he consumpion of relocaed agens, which i can do only by holding more reserves. The subsiuion (income) effec dominaes if ρ < (>). f he nominal ineres rae is zero, he bank may carry cash reserves beween periods. also is indifferen regarding he composiion of is asses. Noneheless, m and b can be deermined as par of a general equilibrium. For he sake of breviy, his case is no analyzed here. 2 Again, his expression applies only if >.

9 8 n addiion, he elasiciy of he demand for reserves wih respec o he volume of cash ransacions is π γ 2(, π ) γ (, π ) = [ γ (, π )] ( π ) > 0. Thus, higher relocaion probabiliies (i.e., a larger volume of ransacions ha require cash) induce banks o hold higher levels of reserves, oher hings equal. Finally, if ρ (>), hen γ (, π ) [0,π] ( [π,]) for all >. nuiively, his propery reflecs he role ha banks play in providing insurance o agens who have o move. When agens are relaively risk averse (ρ > ), banks reserve-deposi raios exceed he share of deposis ha will be wihdrawn early (i.e., γ (, π ) [π,]). This allows banks o pay movers a rae of reurn on deposis greaer han he rae of reurn on currency, and hey pay nonmovers a rae of reurn less han he rae of reurn on bonds. immediaely follows from he relaionship beween ρ and γ (, π ) ha π γ (, π ) γ (, π ) ( < ) if ρ (>). n oher words, he elasiciy of reserve demand wih 2 respec o π is no less han (less han) one if agens are no more (more) risk averse han hey would be wih logarihmic uiliy. 3 nuiively, a change in π represens a change in he risk of relocaion. As deposiors become relaively more risk averse, banks reserve holdings respond more srongly o such a change. 2. General Equilibrium An equilibrium for he economy described above saisfies hree condiions. Firs, he money marke mus clear. Given ha all beginning-of-period demand for base money derives from banks, if > his requiremen implies ha γ (, π ) ω = M p m,. (8) Second, he governmen-bond marke mus clear: [ γ (, π )] ω = B p b,, (9) 3 Suppose ha he environmen described above is replaced wih he simpler assumpion ha a cash-in-advance consrain applies o a fracion π of each agen s purchases, wih π evolving as in he ex. Then he elasiciy of he demand for base money wih respec o π will be one. As will be apparen from he subsequen discussion, his will subsanially reduce he se of possible equilibrium oucomes. See Dow (995) and Woodford (998) for formulaions of his ype.

10 9 if >. Le β denoe he raio of bonds o money ousanding (b / m ). Then (8) and (9) imply ha whenever >, and β mus saisfy he condiion β = [ γ (, π )] γ (, π ),. (0) Finally, he governmen budge consrain, (), mus be saisfied a all daes. Using he definiion of β, () can be wrien as m ( + β ) = m [ R β + ( p p )], 2. () Subsiuing (8) ino () and using he relaionship R p / p yields he equivalen equilibrium condiion 4 ( + β ) γ (, π ) = ( p p ) γ (, π )( + β ), 2, (2) which obains whenever >. For a given specificaion of governmen policy, an equilibrium is a sequence {, p, β } ha saisfies (0), (), and (2) a all daes. The remaining secions of his aricle characerize he properies of equilibria under alernaive assumpions abou he conduc of moneary policy. 3. An Exogenous Bond-o-Money Raio The governmen could choose o conduc moneary policy by fixing once and for all a value of β, he bond-o-money raio. Differen choices of β correspond o differen open marke sances: high (low) values of β are associaed wih igh (loose) moneary policy, as convenionally conceived. For such a policy regime, he following resul obains: PROPOSTON : When he demand for cash for ransacions is decreasing over ime, a policy ha fixes he bond-o-money raio leads o secularly rising (falling) ineres raes and inflaion depending on wheher ρ > (<). f he asympoic use of currency is (is no) small enough, hen he nominal ineres rae and he inflaion rae approach zero (remain posiive). Wih β fixed, equaions () and (0) indicae ha he nominal ineres rae mus adjus o keep he raio of reserve demand o bond demand equal o he raio of reserve supply o bond 4 The appropriae version of his condiion for he iniial dae, =, is ( + β ) γ (, π ) = M p, wih M 0 0 > 0 exogenously given.

11 0 supply. Thus, whenever he nominal ineres rae is posiive, i mus evolve according o Equaion (2) hen deermines he sequence of inflaion raes: b g ρ ρ = βcπ π h /( ),. (3) p p = ( + β ) ( + β ), 2. (4) Subsiuing (3) ino (4) yields he following closed-form expression for p + / p : p + p = + /( ρ) ρ/ ρ β ( π ( π )) b g,. (5) ( + β) Furhermore, he real rae of ineres, R, evolves according o R = p / p + = (+β) /(+β ). The real ineres rae hus inheris he behavior of he nominal ineres rae. The evoluion of and p + / p depends on he magniudes of π and ρ. There are wo cases. Case : r < From (3) and (4), when ρ < a declining role for currency in he paymens sysem mus lead o a nominal ineres rae and inflaion rae ha are falling over ime. The quesion remains of how far hey will fall. f βπ / ( π) <, so ha he asympoic use of currency in paymens is sufficienly small, hen (3) implies ha here exiss a finie dae, denoed by T, such ha = for all T. Once he nominal ineres rae ceases o be posiive, he derivaion of equilibrium above is no longer valid. However, i is no hard o show ha, wih βπ ( π) <, = = p / p + mus obain for all T. Boh he nominal ineres rae and he inflaion rae herefore decline over ime unil, a some finie dae, he opporuniy cos of holding money is zero. A ha poin, agens have no incenive o economize furher on he use of currency. The price level sabilizes a is dae-t value. is noeworhy ha his resul is compleely independen of he value of β. Thus, regardless of he looseness or ighness of moneary policy, he price level ulimaely sabilizes. However, he larger β is, he longer is he period of ime over which a posiive nominal ineres rae can be observed. Also noeworhy is he fac ha mus evenually equal one even if π > 0 holds, so long as βπ b πg <. This means ha he aainmen of a zero nominal ineres rae and price sabiliy does no require ha he fracion of ransacions requiring cash goes o zero. also means ha in finie real ime banks will provide complee insurance.

12 n conras, when βπ ( π) >, so he asympoic use of currency remains sufficienly large, (3) implies ha > for all. also implies ha declines monoonically, as does he inflaion rae. Equaion (5) implies ha p + / p asympoically approaches + β /( ρ ) ρ ρ ( π / ( π)) / ( + β) > b g. Thus, whaever he asympoic use of currency, a diminishing role for currency in ransacions leads o a declining nominal ineres rae and inflaion rae. Case 2: r > When ρ >, (3) implies ha mus be rising over ime. Equaion (5) implies ha he rae of inflaion mus also be rising. Consequenly, a declining demand for currency in ransacions leads o a monoonically increasing nominal ineres rae and acceleraing inflaion. To summarize, when he cenral bank fixes he bond-o-money raio, he effecs of a diminishing role for currency in ransacions depend heavily on he magniude of ρ. When ρ < (>), he subsiuion (income) effec dominaes in he response of reserve demand o a change in he nominal ineres rae, so as π falls, he nominal rae declines (rises). And because he ineres elasiciy of reserve demand is less han (greaer han) zero when ρ < (>), a declining (rising) nominal ineres rae is associaed wih a declining (rising) inflaion rae and real rae of ineres. 4. Nominal-neres-Rae Targeing Alernaively, he cenral bank could conduc moneary policy so as o mainain a consan value > for he gross nominal ineres rae. 5 The following proposiion summarizes he effec of a declining demand for currency for ransacions under such a policy. PROPOSTON 2: A cenral bank ha pegs he nominal ineres rae in he face of a declining 5 As menioned previously, he effec of implemening he Friedman rule (a arge of = ) is no considered here. Neverheless, he resuls in his secion apply o policies ha approximae he Friedman rule arbirarily closely.

13 2 π mus coninuously increase he bond-o-money raio and mus do so a a more rapid rae han he conracion in he demand for reserves. Because he supply of base money decreases less rapidly han he demand for reserves, he ineres-rae peg commis he economy o secularly rising inflaion. mus saisfy To derive his resul, observe ha equaions () and (0), wih consan, imply ha β b g. (6) β = ( π ) π ( ρ)/ ρ Thus, as π declines, he cenral bank mus raise he bond-o-money raio o keep he nominal ineres rae consan. n effec, hen, he cenral bank mus drain base money from he economy via open marke sales. This is rue regardless of he magniude of ρ. n addiion, (0) wih = describes he evoluion of he demand for reserves. As noed previously, when he nominal ineres rae is consan, a secular decline in he value of π induces a less (more) han proporional decline in reserve demand if ρ > (<). Thus, he demand for base money may decline eiher more rapidly or more slowly han he volume of cash ransacions. Differeniaion of (6) esablishes ha (π /β )(dβ /dπ ) = ( π ). The cenral bank herefore mus increase he bond-o-money raio more rapidly han he volume of cash ransacions conracs if i is o peg he nominal ineres rae successfully. Wih respec o he rae of inflaion, (6) and (2) imply ha b b / ρ + π π p p = ( ) γ (, π ) =, ( ρ)/ ρ + π π g g 2. (7) Since γ (, π ) > 0, i follows from (7) ha he equilibrium inflaion rae mus be rising over 2 ime as π declines. nuiively, he supply of base money decreases less rapidly han he demand for reserves, driving he inflaion rae up over ime. Wih consan, i follows ha he real ineres rae declines over ime along wih π. Boh saemens hold regardless of he magniude of ρ. f π = 0, so ha asympoically cash use is zero when >, (7) implies ha lim p p = and hus ha he gross real ineres rae approaches one. However, wih >, banks provide incomplee insurance agains relocaion, even asympoically. Under nominal-

14 3 ineres-rae argeing hen, economic disorions do no vanish wih he eliminaion of cash ransacions. 5. nflaion Targeing The cenral bank also could follow a policy of seing he inflaion rae, p / p, equal o φ for all. Assuming ha φ, so he cenral bank does no aemp o generae deflaion, he following resul obains. PROPOSTON 3: f he cenral bank arges he inflaion rae, he nominal ineres rae decreases (increases) as π decreases (increases). To see his, noice ha equaions (0) and (2) imply he saisfacion of he following condiion a each dae: = ( / φ )[ ( ) γ (, π )]. (8) f φ > and/or π > 0, hen (8) necessarily has a unique soluion wih > in every period. Now consider he effec of changes in he demand for currency in ransacions on he equilibrium evoluion of he nominal ineres rae. Differeniaion of (8) yields ( ) γ 2 (, π ) d dπ = γ (, π ) ( ) γ (, π ) ( ) γ 2( π = Fρ F, ) L [ γ (, π )] M γ (, π ) ρ HG N HG KJ Proposiion 3 follows immediaely since γ (, π ) > and > [(ρ )/ρ][( )/ ]γ (, π ). 2 0 nuiively, as he demand for cash for ransacions falls, he demand for reserves falls and he demand for bonds rises. Given he inflaion ax rae, hen, seigniorage revenue decreases, so he nominal and real ineres rae mus decline along wih π o saisfy he governmen budge consrain. Proposiion 3 has a corollary: Wih a fixed inflaion rae, as π declines, he real ineres rae necessarily falls over ime. This consequence of a diminishing role of currency in KJ O Q P.

15 4 ransacions was also observed under a nominal-ineres-rae arge. remains o consider how he bond-o-money raio mus evolve o keep he inflaion rae a is arge level. is easily esablished ha dβ / dπ < 0, so he cenral bank mus coninuously raise he raio o mainain is arge. When π = 0, (8) implies ha lim = φ. Thus, if φ >, so some inflaion occurs, he nominal ineres rae remains posiive a all daes. The gross real ineres rae asympoically approaches one. n effec, hen, he siuaion under an inflaion arge closely resembles ha under an ineres-rae arge. 6. A Consan Rae of Money Creaion nsead of argeing he inflaion rae, he cenral bank could se, once and for all, a rae of growh for he money supply. n paricular, if he moneary base evolves according o M + = σ M,, wih M 0 > 0 given and σ >, hen he effecs of a declining demand for currency for ransacions are more complicaed, as he following proposiion indicaes. PROPOSTON 4: When he cenral bank fixes he money growh rae, boh seady sae and dynamical equilibria are possible. n a seady sae he inflaion rae and nominal ineres rae are increasing in he money growh rae. The naure of dynamical equilibria depends on he value of ρ. f ρ is sufficienly small, here is a unique equilibrium ha monoonically converges o he seady sae. f i is sufficienly large, many oscillaory equilibrium pahs can approach he seady sae. The argumen underlying Proposiion 4 proceeds as follows. Wih a consan rae of money creaion, he real reurn on reserve holdings is given by p p m γ (, π ) = = 2 σm σγ(, π ),. (9) Using (9) and (0) in (2) gives he equilibrium law of moion for when > :

16 5 σ = + γ (, π ) L NM γ (, π ) γ (, π ) QP = + O F HG π π Equaions (2) and (20) govern he evoluion of he sequence {, π }. KJ / ρ, 2. (20) A. Seady-Sae Equilibria wih p > 0 n a seady sae, π = π > 0. mposing π = π - = π and = = in (20) yields he condiion ha deermines he seady-sae nominal ineres rae: F ππ σ ρ σ / b g L O. (2) HG K J F = NM H G K J QP This equaion has a unique soluion saisfying > σ >. n addiion, higher values of σ lead o higher values of he seady-sae nominal ineres rae and he rae of inflaion. The seady-sae real ineres rae rises when σ is increased iff ( ρ)σ >. For fuure reference, observe ha > σ > even for arbirarily small posiive values of π. Bu as π 0, σ. B. Dynamics wih p > 0 Simple algebraic manipulaion esablishes ha equaion (20) can be wrien as = F HG L NM L KJ N M F H G K J F HG b g π ρ σ σ π σ ( ρ)/ ρ / π = π µπ + µ π π / ρ σ µπ ( µ ) π σ π σ F KJ HG OL Q P F H G K J F N M HG K J O Q P KJ F HG K J O Q P, 2. Linearizing (22) in a neighborhood of he seady sae yields he dynamical sysem b g b g, π π = J, π π, where J is he Jacobian marix J = π, 0 µ (22) wih all parial derivaives evaluaed a he seady sae. Clearly he eigenvalues of J are µ and F =. Since /σ > in a seady sae, here are wo possibiliies regarding ρ σ = HG equilibrium dynamics. K J F H K

17 6 Case : r < When ρ <, / - = [/( ρ)](/σ) >. follows ha he seady sae is a saddle. Since is an endogenous iniial condiion, he equilibrium value mus place he economy on is sable manifold. Consequenly, here is a unique equilibrium, and ha equilibrium displays monoonic convergence o he seady sae. Case 2: r > When ρ >, / - = [/( ρ)](/σ) < 0. The seady sae is a sink (saddle) if b gb g /σ < (>) ρ. Moreover, using equaion (2), /σ < (>) ρ iff π π ( σ ) σ < ( )( ) ( ) ( ρ > ρ σ ρ ) ρ 2. Clearly, he seady sae is a sink if ρ > 2, and if eiher π is sufficienly close o zero or σ is sufficienly close o one. When /σ < ρ, any choice of sufficienly close o allows he seady sae o be approached, so dynamical equilibria are indeerminae. Moreover, equilibrium pahs approaching he seady sae will display oscillaions. n oher words, endogenously arising volailiy will be observed. Tha is, whenever ρ > 2, and whenever he asympoic use of currency is sufficienly small, a fixed money growh rae mus lead o a siuaion of indeerminacy and endogenous volailiy. Boh he nominal rae of ineres and he price level mus flucuae along any equilibrium pah. Noe ha he endogenous volailiy ha emerges migh no vanish asympoically. ndeed, for cerain values of π, and for fixed values of σ, here will exis equilibria displaying wo-period cycles. Thus, he policy of fixing a rae of money growh can easily lead o he exisence of permanen flucuaions. Such flucuaions canno occur under he oher policy regimes analyzed. C. An Asympoically Cashless Economy When π = 0, (22) reduces o b g L F N M b g O HG KJ Q P b g,. (23) F = 2 H G K J F + H G K J ρ ρ µ π ρ µ µ σ π σ σ πµπ

18 7 n a seady sae hen, π = 0 and = σ/µ. is possible o show ha here is no equilibrium wih = for sufficienly large. Tha is, he nominal ineres rae canno converge o zero, asympoically or oherwise. Thus, as in Woodford (998), he cashless-limi economy (i.e., one wih π = 0) has a seady sae nominal and real ineres rae ha is no he limiing soluion of (2) as π 0. Neverheless, he qualiaive behavior of economies wih π > 0 is similar o ha of economies wih π = 0. Wih respec o dynamics, i is similarly sraighforward o verify ha = b ρg. Thus, if ρ <, he seady sae is a saddle. The dynamical equilibrium = σ µ is unique, and monoonic convergence o he seady sae is observed. f ρ > 2, hen he seady sae is a sink. There is hen an indeerminacy of equilibrium, and pahs approaching he seady sae display damped oscillaion. Finally, i is clear ha as ρ varies, a flip bifurcaion can occur, so ha undamped oscillaion in ineres raes and inflaion is possible. These moneary flucuaions can herefore be observed even in an asympoically cashless economy. 7. Comparison of Policy Regimes Three key findings arise from he analysis of he preceding secions. Firs, a declining demand for ouside money does no imply ha any of he convenional mehods of conducing moneary policy will cease o be feasible. This is rue even if he demand for base money declines wihou limi. Second, he impac of a declining demand for currency for ransacions varies considerably across he policy regimes considered. The equilibrium ime pahs for he ineres rae, he inflaion rae, and he composiion of governmen liabiliies are quie differen across he regimes, and one mehod for conducing policy allows considerable scope for he indeerminacy of equilibrium and for endogenously generaed volailiy, while he ohers do no. Third, one mehod for conducing policy can resul in price sabiliy and zero nominal ineres raes in finie ime if π is sufficienly small, independen of he choice of policy parameer. Oher policies do no share his feaure.

19 8 n general, i is quie difficul o rank he various policies on he basis of welfare. To conduc a welfare analysis, one mus firs find a way o make he sance of moneary policy comparable across policy regimes. One way his can be done is by ensuring ha he seady sae ineres rae arising under each policy is abou he same. When his is done, he differences in welfare across regimes are found o be relaively minor and no uniform ranking of he regimes emerges. One unambiguous saemen can be made, however. When ρ < and βπ b πg <, price sabiliy and a zero nominal ineres rae are aained in finie ime under a policy ha mainains a fixed bond-money raio. A zero nominal ineres rae is aracive because i allows banks o provide complee insurance provision agains he risk of relocaion. f π = 0, hen here will be a finie dae T such ha for all T, young agens unambiguously prefer a policy of fixing he bond-money raio o a policy of argeing eiher he nominal rae of ineres or he inflaion rae. Thus, in an asympoically cashless economy, ulimaely agens will prefer he policy of fixing he bond-money raio o eiher a nominal-ineres-rae or inflaion-rae arge. Maers are quie differen, however, when ρ >. A policy of fixing he bond-o-money raio leads o an ever rising nominal ineres rae. This will no be he case under nominal-ineres-rae argeing or inflaion argeing. This and he earlier observaions sugges he following conclusion. Under he policy of mainaining a fixed bond-o-money raio or under a policy of mainaining a fixed rae of money growh he policy auhoriy mus know a good deal abou ρ and π o predic he consequences of he policy. This is no rue of ineres-rae or inflaion argeing. The lack of ambiguiy regarding economic dynamics under hose policies seems o be a poin in heir favor. 8. Concluding Remarks The preceding analysis absracs from several feaures ha are relevan o modern-day paymen sysems. For example, he focus on a pure-exchange economy makes i impossible for developmens in he echnology of paymens o affec he level of producion. Endogenizing producion levels would allow he evoluion of cash ransacions o affec real aciviy, which in

20 9 urn migh modify some conclusions abou how his evoluion affecs he behavior of he price level or he behavior of real and nominal ineres raes. And by allowing for capial accumulaion, he model would inrinsically have much richer dynamics. A second feaure from which he model absracs is he mechanism by which he demand for cash evolves. One can regard he specificaion of an exogenous law of moion for π as a reduced-form approach ha implicily akes no sand on he economic forces governing he use of cash in ransacions. Clearly, i would be more saisfying o model hese forces explicily. Likewise, i would be beer o model he choice beween cash and oher insrumens o make paymens. Modeling ha choice (for example, as done in Schref 992 and reland 994) would endogenize he evoluion of cash use and migh no aler any of he resuls derived above. Third, he model absracs from he exisence of a marke in which reserves can be borrowed and len. The presence of such a marke has in principle he poenial o subsanially affec he demand by banks for reserves. And reserve demand is a he hear of he analysis here. nroducing a marke for reserves ino he model is sraighforward. The mos naural approach is o assume ha each bank faces a sochasic demand for cash wihdrawals (ha is, a random value of π ), bu ha here is no aggregae randomness. This inroduces an addiional feaure ino a bank s decision regarding is reserve holdings: banks face uncerainy regarding wihdrawal demand. f banks mus choose heir reserve-deposi raio before observing heir wihdrawal demand, hen ex pos some banks will have more, and some will have fewer, reserves han needed o pay deposiors. This fac leads naurally o he inroducion of a marke in which banks wih a reserve surplus (defici) can lend (borrow) reserves. While he inroducion of a sochasic wihdrawal demand and a marke for reserves ha resembles oday s federal funds marke adds some noaional complexiy, i does no aler he fundamenal behavior of he model. ndeed, all he resuls repored above have close analogs when hese addiional feaures are added. Thus, absracing from a marke for reserves does no affec any qualiaive conclusions abou how an evolving demand for cash ransacions affecs he economy under alernaive mehods of conducing moneary policy. Finally, in his economy all financial ransacions are conduced hrough banks. f he model were modified o allow for a richer se of financial insiuions, agens could very well have access o financial insrumens ha help hem overcome he spaial separaion and limied

21 20 communicaion criical o heir demand for cash for ransacions. The inroducion ino he economy of such new financial insrumens likely would lead o a faser reducion in he use of cash over ime, alhough he implicaions of his observaion for he conduc of moneary policy are by no means clear. References Champ, Bruce; Smih, Bruce D., and Sephen D. Williamson, 996. Currency Elasiciy and Banking Panics: Theory and Evidence, Canadian Journal of Economics, November, 29(4), Diamond, Douglas, and Philip Dybvig, 983. Bank Runs, Deposi nsurance, and Liquidiy, Journal of Poliical Economy, June, 9(3), Dow, James P., Jr., 995. A Model of he Transiion o a World wihou Money, Mimeo, Ocober. Greenwood, Jeremy, and Bruce D. Smih, 997. Financial Markes in Developmen, and he Developmen of Financial Markes, Journal of Economic Dynamics and Conrol, January, 2(), reland, Peer N., 994. Economic Growh, Financial Evoluion, and he Long-Run Behavior of Velociy, Journal of Economic Dynamics and Conrol, May/July, 8(3/4), Lindsey, David E., and Henry C. Wallich, 989. Moneary Policy. n The New Palgrave Money, edied by John Eawell, Murray Milgae, and Peer Newman. New York: W.W. Noron. The Nilson Repor, November 997, p. 6. Schref, Sacey L., 992. Transacion Coss and he Use of Cash and Credi, Economic Theory, April, 2(2), Schref, Sacey L., and Bruce D. Smih, 997. Money, Banking, and Capial Formaion, Journal of Economic Theory, March, 73(), Schref, Sacey L., and Bruce D. Smih, 998. The Effecs of Open Marke Operaions in a Model of nermediaion and Growh, Review of Economic Sudies, 65(3), Sellon, Gordon H., Jr., and Suar E. Weiner, 996. Moneary Policy Wihou Reserve

22 2 Requiremens: Analyical ssues, Federal Reserve Bank of Kansas Ciy Economic Review, Fourh Quarer, 8(4), Townsend, Rober M., 987. Economic Organizaion wih Limied Communicaion, American Economic Review, December, 77(5), Wicksell, Knu, 898. neres and Prices. Translaed and republished 936; reprined 962. New York: Augusus M. Kelley, Chaper 6. Woodford, Michael, 998. Doing wihou Money: Conrolling nflaion in a Pos-Moneary World, Review of Economic Dynamics, January, (),