TIME CONSISTENT COLLECTION OF OPTIMAL SEIGNIORAGE: A UNIFYING FRAMEWORK

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1 TIME CONSISTENT COLLECTION OF OPTIMAL SEIGNIORAGE: A UNIFYING FRAMEWORK Berhold Herrendorf Universiy of Warwick Absrac. This paper reviews he exising lieraure on he ime consisency problem of seigniorage collecion when moneary policy is deermined by opimal axaion consideraions. I develops a unifying accouning framework and suggess a general measure for seigniorage, which encompasses he sandard measures employed in he lieraure. In addiion, he ex ane opimal soluion o he opimal axaion problem is derived and inerpreed in relaion o he Ramsey principle. We show ha he differen recommendaions of he public finance lieraure, i.e. he Friedman rule of opimal deflaion, [moderaely] posiive inflaion, and seigniorage maximizing inflaion, are specific soluions o he opimal axaion problem. The paper coninues wih a formal illusraion of he ime consisency problem of he ex ane opimal policy and he characerizaion of he ime consisen soluion under discreion. As possible soluions o he ime consisency problem, we consider repuaional forces, insiuional reforms ha esablish cenral bank independence, and specific ways of asse and deb managemen. In paricular, i is formally shown ha a modified version of he asse and deb managemen scheme suggesed by Persson, Persson, and Svensson is no only necessary bu also sufficien for opimaliy, alhough his did no hold in heir model. Keywords. Asse and deb managemen; inflaion ax; opimal axaion; seigniorage; ime consisency. 1. Inroducion Seigniorage, or he public secor s revenues from he creaion of money, has araced a grea deal of aenion in moneary economics. The recen European ineres in he issue has cerainly been relaed o he forhcoming esablishmen of a European cenral bank during he process of European moneary inegraion; see, for insance, Dornbusch (1988), Drazen (1989), Grilli (1989), Canzoneri and Rogers (1990), V egh and Guidoi (1990), or Alesina and Grilli (1991). The main problem is ha he revenues from he creaion of money played very differen budgeary roles across he member counries of he European union: in conras o he counries in Norhern Europe, he Souhern European members of he European Union have used seigniorage o a relaively large exen, in order o finance governmen expendiure. Drazen (1989), for example, esimaed he share of seigniorage in ax revenues for he period as 5.9 percen in Spain, /97/ JOURNAL OF ECONOMIC SURVEYS Vol. 11, No. 1 Blackwell Publishers Ld. 1997, 108 Cowley Rd., Oxford OX4 1JF, UK and 350 Main S, Malden, MA 02148, USA.

2 2 HERRENDORF 6.2 percen in Ialy, 9.1 percen in Greece, and 11.9 percen in Porugal. Oher calculaions of seigniorage for Souhern European counries include Bruni, Penai and Pora (1989), Grilli (1989), Repullo (1991), and Gros (1993). Seigniorage in hese sudies is repored as a share of GDP or GNP and varies beween 2 and 4 percen. 1 The budgeary problems of he Easern European saes afer he srucural break in 1989 have also drawn aenion o he imporance of seigniorage as a revenue insrumen; see, for example, Hochreier, Rovelli and Winckler (1996). As is he case for many developing counries, mos of he emerging democracies are highly dependen on seigniorage, because hey have poorly developed excise and income axaion sysems, ogeher wih significan black marke aciviies, raher underdeveloped domesic capial markes and only limied access o inernaional capial markes. The resuls of Oblah and Valeninyi (1994) indicae quaniaively how imporan seigniorage is in Easern Europe. For Hungary, which is among he more sable Easern European counries, hey value he average share of he revenues from money creaion a 3.5 percen of GDP over he period of , wih a peak of 4.8 percen in Measured as he average share of Hungarian ax revenues during his period, he seigniorage ransferred o he governmen amouned o around 10 percen. The scale of hese figures does no appear o be unrealisic in comparison o esimaes for Souhern European counries, which appear o have more ferile alernaive revenue sources. The heoreical lieraure on seigniorage has sared from he wo rail-blazing, formal conribuions of Bailey (1956) and Cagan (1956). 2 However, i was Phelps (1973) who provided he heoreical framework used in he modern lieraure on he opimal collecion of seigniorage from a public finance viewpoin. Essenially, Phelps inegraed he revenues from he creaion of money ino a public finance framework of opimal axaion, which calls for an opimal combinaion of all possible revenue sources. In his conex, he erm opimaliy is used in a normaive sense, and means ha a combinaion of he differen revenue insrumens raises he required funds a he lowes possible welfare cos. The lieraure deparing from Phelps work has shown ha opimal inflaion raes may be eiher posiive or negaive, depending on he microfoundaions of money and on he efficiency of alernaive revenue insrumens, such as oupu axes. The only consensus in moneary heory seems o be ha, in general, he opimal inflaion rae is no he seigniorage-maximizing one. Anoher srand of he lieraure has focussed on he inheren ime consisency problem, which has been well recognized since Calvo (1978a,b) applied he classic ideas exposed in Kydland and Presco (1977) o moneary economics. The quesion of ime consisency arises here because, afer individuals have deermined heir inflaion expecaions, he policy maker may choose o creae unexpeced inflaion in order o reduce he real value of nominal, public liabiliies vis-a-vis he privae secor, ha is, nominal balances or non-indexed governmen bonds. This is welfare-improving from he poin of view of he policy maker because unexpeced inflaion does no disor individuals pas decisions and, hus, causes lower social cos han expeced inflaion. Moreover, he revenue per uni of

3 TIME CONSISTENT COLLECTION OF OPTIMAL SEIGNIORAGE 3 unexpeced inflaion is generally larger han ha per uni of expeced inflaion, due o he fac ha individuals can only ake expeced inflaion ino accoun when making heir decisions. For his reason, creaing unexpeced inflaion has ofen been viewed as a capial levy. Irrespecive of is imporance, however, he ime consisency problem is ofen assumed away in he public finance lieraure on opimal seigniorage collecion by resricing aenion o siuaions in which he policy maker is ommied. Exising reviews also end o be focused eiher on he opimal axaion problem or on he ime consisency problem of opimal moneary policy making. 3 In conras, he presen paper discusses he collecion of seigniorage boh in he conexs of opimal axaion and of ime consisency. I may, herefore, be viewed as a supplemen o exising discussions, raher han as a subsiue for hem. The paper is organized as follows. Secion 2 conains an exension of he model suggesed by Barro (1983). Emphasis is pu on he derivaion of a general measure of seigniorage from he consolidaed public secor flow budge consrain and on he discussion of he social loss from inflaion. The resuling framework encompasses he mos commonly used measures of seigniorage. Secion 3 conains he formal derivaion of he ex ane opimal soluion o he opimal axaion problem and is inerpreaion in relaion o he Ramsey (1927) principle of opimal axaion. In paricular, he differen recommendaions of he public finance lieraure on opimal inflaion, noably he Friedman (1969) rule, [moderaely] posiive inflaion, or even seigniorage-maximizing inflaion raes, are shown o be soluions o he opimal axaion problem in specific cases. In secion 4, he ime inconsisency of he ex ane opimal soluion is proved and he ime consisen soluion is characerized and discussed. The nex secion discusses hree possible soluions o he ime consisency problem, namely, repuaional effecs [in subsecion 5.1], insiuional reforms ha esablish cenral bank independence [in subsecion 5.2] and specific ways of managing public asses and deb as proposed by Persson, Persson and Svensson (1987) [in subsecion 5.3]. I is shown ha a modified version of heir scheme is no only necessary bu also sufficien o ensure he ime consisency of he ex ane opimal policy under discreion, hough his did no hold rue in heir more general model [Calvo and Obsfeld (1990)]. Finally, he implicaions of he resuls are discussed in secion 6 and a lis of used symbols is given a he end of he paper. 2. The opimal axaion problem In his secion, a unifying framework is developed by exending Barro (1983) s seigniorage model. 4 In subsecion 2.1, a simple endowmen economy wih money is oulined, in which prices and inflaion can be deermined in a meaningful way. The consolidaed flow budge consrain of he public secor is hen derived [in subsecion 2.2] and a deailed discussion of seigniorage is provided [in subsecion 2.3]. Subsecion 2.4 conains he moivaion and specificaion of he social loss from he differen revenue insrumens. The secion ends wih a formal saemen of he opimal axaion problem in subsecion 2.5.

4 4 HERRENDORF 2.1 A model economy Consider a deerminisic, closed economy in which he realizaions of all exogenous variables oher han he policy insrumens are known wih cerainy. To concenrae on he issues relaed o seigniorage, assume ha he sream y 0, y 1, y 2, of he homogeneous real oupu or endowmen good is exogenously given. For convenience, real and nominal variables will be denoed by lower and upper case leers respecively. For insance, if P is he price level in period, his convenion implies ha nominal oupu can be expressed as Y = y P. Prices in he endowmen economy are supposed o be perfecly flexible. In order o deermine he price level, i is assumed ha individuals wan o hold nominal balances wih a real value m during period, implying ha nominal money demand equals m P. 5 Denoing he nominal money supply by M, he price level follows from equilibrium in he money marke, i.e. from equaliy beween nominal money supply and money demand, M =m P P = M X. (1) m The inflaion rae beween he periods 1 and is defined as π = P P 1. (2) This backward-looking definiion of inflaion is differen from he more common, forward-looking measure (P P 1 ) P 1. However, specificaion (2) is appropriae in he presen conex because he inflaion rae will be inerpreed as a ax rae on real money holdings. This ax rae ough o converge o one if he price level a goes o infiniy, provided ha he price level a 1 is finie; see Obsfeld (1991a,b) for a similar definiion. 6 Bonds may be issued eiher by privae individuals or by he governmen. All bonds are assumed o pay he same ex ane nominal ineres rae i in any period, ha is, for simpliciy, privae and governmen bonds are aken o be perfec subsiues. Since raional individuals require ineres paymens ha fully compensae for expeced inflaion, i is deermined by he Fisher relaion 1 +i = P e (1 +r )= 1 +r Xi P 1 e = r +π e, (3) 1 π 1 π e e e where P and π are he period price level and inflaion rae, as expeced in period 1. To avoid confusion, we noe here ha his version of he Fisher relaion differs from he sandard version 1 + i =(1+r )(1+ π e ), due o he differen definiion of inflaion chosen. However, his difference is no crucial for small inflaion raes; aking logarihms, boh specificaions imply he same familiar approximaion i r + π e. In order o deermine he demand for real balances, we posulae he exisence of a sable aggregae money demand funcion. As is sandard in he lieraure, he P

5 TIME CONSISTENT COLLECTION OF OPTIMAL SEIGNIORAGE 5 demand m for real balances in period is aken o depend negaively and sricly convexly on he real opporuniy cos ρ of holding money insead of ineres bearing bonds. ρ is given by he expeced presen discouned, real value of nex period s ineres paymens 1 P ρ = i e +1 = ( ) i +1 i +1 =. (4) (1 +r +1 ) P +1 e 1 π +1 1+r +1 1+i +1 Using (3), one may also wrie ρ as ρ =(r +1 + π e +1) (1 + r +1 ), which shows ha ρ π e +1 1 = 0. (5) 1+r +1 Inequaliy (5) implies ha real balances depend negaively and sricly convexly on π e +1. Assuming ha he real ineres rae is exogenously given, one may herefore express real balances as a funcion of he expeced inflaion rae, m =m (π e +1 ), m (π e m (π +1 ) +1 ) 0, m (π π e e +1 ) 0. (6) +1 The index on m indicaes ha he demand for real balances depends on he real ineres rae and, possibly, on addiional ime-varying facors such as changes in financial echnology. I should be menioned ha his specificaion of he demand for real balances can be derived from individual uiliy maximizaion when money is modelled as an argumen of a separable uiliy funcion; see Obsfeld (1991b). Moreover, Cagan (1956) money demand funcions of he ype α 0 exp( α 1 π e +1) [α 0, α 1 being consans], which are ofen used in moneary economics, are a special case of (6). Evenually, by subsiuing (6) ino (1) and changing from levels ino growh raes, he realized inflaion rae follows as a funcion of he raes of change of supplied base money and demanded real balances: = π M M e e m (π +1 ). (7) e m (π +1 ) Since in equilibrium he auhoriy knows he raional individual inflaion expecaion, one can ake he realized inflaion rae, insead of he nominal money supply, as he policy insrumen under he conrol of he auhoriy. This is is analyically more convenien The consolidaed budge consrain of he public secor We assume for simpliciy ha real governmen expendiures, g, are exogenously given. 8 In order o finance he nominal value of hese expendiures, G, he governmen may collec oupu axes, T, use he profi of he cenral bank, X, or o raise funds hrough increasing he oal sock B of governmen bonds. The

6 6 HERRENDORF sylized governmen flow budge consrain in nominal erms, which requires equaliy beween nominal expendiures and nominal revenues a he end of any period, is herefore given by G +(1+i )B o o 1 = B + T + X. (8) Nominal ax revenues in (8) equal he produc of he average ax rae τ and nominal oupu Y, T = τ Y. (9) Furhermore, nominal cenral bank profi, X, is deermined by he ineres receips on he previous period s cenral bank porfolio: X = i (B cb 1 + C 1 ). (10) In equaion (10), B cb 1 and C 1 denoe he sock of governmen bonds and privaely issued bonds in he cenral bank porfolio respecively. To wrie X in his form, wo simplifying assumpions mus be made. Firsly, he nominal ineres rae needed o induce he cenral bank o hold privaely issued asses mus be he same as he one he privae secor requires o hold governmen bonds. This excludes he possibiliy of implici cenral bank subsidies o he privae secor hrough acceping ineres raes on C 1 ha are below he marke rae; compare Klein and Neumann (1990) for furher discussion. Secondly, he operaing cos of he cenral bank have been se equal o zero. This is no o say ha hese cos can be negleced in pracice. For Germany, for insance, Klein and Neumann (1990) esimaed hem a around 10% of he oal seigniorage in he 1980s. Given he presen ineres, however, he operaing cos are assumed o be exogenous and can hus be suppressed. Throughou mos of he paper, he cenral bank is modelled as oally dependen on he governmen; he eniy comprising boh of hem is synonymously referred o as he auhoriy, he policy maker or he public secor. 9 Subsiuing (10) ino (8) and using he fac ha changes in base money mus resul from open marke operaions, i.e. M = (B cb + C ), we obain he consolidaed, nominal flow budge consrain of he public secor, G +(1+i )(B 1 C 1 )=(B C )+T + M. (11) where B denoes he sock of governmen bonds no held by he cenral bank, i.e. B B o B cb. Governmen bonds held by he cenral bank cancel when consolidaing he budge consrain, because hey are an asse of he cenral bank and a liabiliy of he governmen. Since i will be needed below when he Persson, Persson and Svensson (1987) soluion is discussed, we allow for he possibiliy ha a share (1 ω ) of governmen bonds is indexed o realized inflaion, ω [0, 1]. 10 While boh ypes of bonds pay he same ex ane nominal ineres rae, he ex pos nominal ineres rae facor on indexed bonds is (1 + r ) (1 π ), as opposed o (1 + r ) (1 π e ) on non-indexed bonds. 11 Dividing equaion (11) by he price level and using he fac ha P 1 P =1 π, an expression for he consolidaed public flow

7 TIME CONSISTENT COLLECTION OF OPTIMAL SEIGNIORAGE 7 consrain in real erms is obained: g + (1 +r ) (1 ω 1 ) b π (1 +r 1 π e ) (ω 1 b 1 c 1 ) =(b c )+τ y + m +π m 1. (12) Adding (1 + r )(ω 1 b 1 c 1 ) on boh sides of (12) and rearranging gives he following version of he consolidaed flow consrain in real erms: (b c )=[g +r (b 1 c 1 )] [τ y + m + π m 1 +(π π e )(1+ i )(ω 1 b 1 c 1 )]. (13) Equaion (13) highlighs he fac ha he public secor mus increase he real value of is ne ineres-bearing liabiliies [lef hand side] when is oal real expendiures [firs sum in brackes on he righ hand side] exceed is oal real revenues [second sum in brackes on he righ hand side]. 2.3 The revenues from he creaion of money Toal seigniorage In his subsecion, an expression for oal [real] seigniorage, s, is derived and inerpreed. Defining seigniorage in he broades possible sense as he sum of all public secor revenues resuling from he monopoly power o issue money, s comprises all erms in (13) ha would be absen in a non-moneary barer economy. In a barer economy, here would be no cenral bank asses c and no real balances m, i.e. c = m = 0. Moreover, all governmen bonds would be like indexed bonds, implying ω = 0. Hence, he flow budge consrain of he public secor in a barer economy would read as simply b =(g +r b 1 ) τ y. (14) One may, herefore, wrie he consolidaed real flow consrain in a moneary economy as b =(g +r b 1 ) (τ y +s ). (15) Subracing (13) from (15) gives a formal expression for seigniorage, s = [m (π e +1) c ]+π m 1 (π e )+r c 1 +(π π e )(1+ i )(ω 1 b 1 c 1 ), (16) which comprises he following componens: 1. The real revenue from hose increases in desired real balances ha are no used o purchase addiional cenral bank asses, i.e. (m c ). 2. The reducion of he real value of nominal money holdings due o realized inflaion, i.e. π m 1. This erm is usually referred o as he inflaion ax, aking he realized inflaion rae π as he ax rae and real balances m 1 as he ax base.

8 8 HERRENDORF 3. The real ineres paymens r c 1 on he previous period s sock of privaely issued asses held by he cenral bank. 4. The revenue from unexpeced devaluaions of ineres bearing, nonindexed, ne public liabiliies, i.e. (π π e )(1+ i )(ω 1 b 1 c 1 ). Noice ha governmen deb eners he expression for seigniorage differenly from privaely issued asses held by he cenral bank. All changes of he sock of privaely issued asses in he cenral bank porfolio as well as he real value of ineres paymens on hose asses affec real seigniorage. However, only he changes of he real value of governmen bonds resuling from unexpeced inflaion are par of seigniorage. This comes from he fac ha all oher budgeary effecs of governmen bonds would also occur in a barer economy, and can hus no be relaed o he revenues from he governmen s monopoly o issue money Seigniorage from expeced inflaion In order o undersand he componens of seigniorage more deeply, we disinguish beween [real] seigniorage from expeced and from unexpeced inflaion. Consider firs a siuaion in which all inflaion is fully anicipaed and denoe he seigniorage from expeced inflaion by s e e. Seing π = π in (16), one obains s e = [m (π e +1) c ]+π e m 1 (π e )+r c 1. (17) e I should be poined ou ha s may well comprise componens ha were no expeced in period 1. An example would be he effec of an unexpeced change in financial echnology on s e. However, since, in general, such an unexpeced change is no caused by he creaion of surprise inflaion, i is correcly included in s e, raher han in he revenues from unexpeced inflaion o be defined below. Two exreme cases may shed some addiional ligh on he expression for s e : The cenral bank implemens moneary policy solely hrough openmarke operaions in governmen bonds: In his case, cenral bank credi is exended exclusively o he governmen, and c = 0 in every period. Thus, he seigniorage from expeced inflaion follows from (2) and (17): s e = m + π e m 1 = M. (18) P e Since s e is now given by he real value of changes in nominal base money, (18) has a imes been called he cash-flow or he moneary measure of seigniorage; see Repullo (1991) and Klein and Neumann (1990) respecively. 2. The cenral bank implemens moneary policy solely hrough open marke operaions in privaely issued bonds: In his case, cenral bank credi is exclusively exended o he privae

9 secor, and m = c. Using (3) and (4), expression (17) hen becomes: s e = (π e i +r )m 1 =(1 +r ) m 1 =(1 +r )ρ 1 m 1. (19) 1 +i Hence, he seigniorage from expeced inflaion equals he value of he nominal ineres paymens ha he privae secor foregoes when holding non-ineres-bearing money insead of ineres-bearing asses. Noice firs ha since he real opporuniy cos of holding money was defined in erms of period 1 oupu [compare (4)], i needs o be muliplied by he facor 1 + r in order o obain a period value. Secondly, for small raes of real ineres and expeced inflaion, one has he approximaion s e i m 1. This is he opporuniy cos measure of seigniorage commonly employed in he lieraure; compare, for insance, Johnson (1969a,b), Barro (1982) and Gros (1993). I has ofen been dispued wheher or when i is appropriae o use he cash-flow measure or he opporuniy cos measure. Since mos real world cenral banks exend credi o boh he governmen and he privae secor, neiher measure is appropriae in general for calculaing he seigniorage from expeced inflaion accruing in any single period. Insead, (17) ough o be used. However, in sylized infinie horizon models such as he presen one, his is no a problem because he presen discouned value s e of he sum of all fuure period-by-period seigniorage from expeced inflaion is unaffeced by alernaive ways of implemening moneary policy. In order o see his, recall he above assumpions ha he real ineres rae is exogenous, ha he Fisher relaion is valid and ha he cenral bank does no subsidize he privae secor. In addiion, we assume he validiy of he sandard ransversaliy condiion for he evoluion of c, lim R c = 0, (20) 2 where R denoes he marke discoun facor, R 1 = 1 +r 0, R 0 = 1, R = i=1 1. (21) 1 +r i The ransversaliy condiion (20) ludes cases in which he cenral bank holds seigniorage back by hoarding asses ad infinium. Providing he above condiions hold, s e can be shown o equal s e e R s = R =0 =0 i +1 1+i +1 m +(1 +r 0 )c 1 (1 π m 0 ) 1, where c 1 and m 1 are he given iniial values of privaely issued bonds held by he cenral bank and of real balances respecively. (22) implies ha he presen discouned value of seigniorage from expeced inflaion is invarian wih respec o he differen ways of implemening moneary TIME CONSISTENT COLLECTION OF OPTIMAL SEIGNIORAGE 9 (22)

10 10 HERRENDORF policy. Inuiively, his resul arises because an addiional purchase of privaely issued asses decreases curren seigniorage by he same amoun as i increases he presen value of fuure seigniorage [hrough he addiional revenues from ineres paymens and sales of hese asses]. The invariance of s e wih respec o he implemenaion of moneary policy is imporan for wha follows. I means ha resricions on he cenral bank s open marke operaions do no affec he ineremporal budgeary posiion of he public secor. We will come back o his when considering specific ways of asse managemen ha impose such resricions in order o cure he ime consisency problem of moneary policy. Before moving on o a discussion of he real seigniorage from surprise inflaion, i should be menioned ha he chosen measure of real seigniorage maers crucially in empirical sudies, which ineviably work wih a finie horizon. This is rue, because, over a finie horizon, addiional cenral bank purchases of privaely issued asses may well affec he presen value of real seigniorage from expeced inflaion; see Klein and Neumann (1990). Alhough rarely apiaed, he empirically correc measure [in he absence of unexpeced inflaion] is (17), possibly wih some sligh modificaions reflecing insiuional peculiariies of he counry under consideraion. This was sressed in a series of empirical sudies by Klein and Neumann (1990) and Neumann (1992, 1995). In conras, Barro (1982) or Fischer (1982) are examples of empirical work ha crudely use he opporuniy cos measure (19), or he cash flow measure (18) respecively. The resuling differences beween he figures for real seigniorage can be remarkably large. For example, over he period from 1960 o 1973, Klein and Neumann (1990) esimaed for Germany ha he governmen received on average only around 13% of he seigniorage ha Fischer (1982) found by using he cash-flow measure Seigniorage from surprise inflaion Expression (16) for oal seigniorage shows ha unexpeced inflaion yields seigniorage oo. Subracing (17) from (16), he seigniorage from surprise inflaion amouns o s s =(π π e )[m 1 (π e ) + (1 + i )(ω 1 b 1 c 1 )]. (23) Surprise inflaion yields revenues for wo reasons: firs, nominal balances carried over from he previous o he curren period are deermined on he basis of he expeced inflaion rae π e ; compare (6). The public secor can hus reduce he real value of nominal balances hrough inflaing more han was expeced by he privae secor. Second, he ineres bearing ne liabiliies of he public secor ha are no indexed are only proeced agains expeced inflaion. Surprise inflaion, herefore, reduces he real value of public ne liabiliies as long as he non-indexed par is posiive, i.e. ω 1 b 1 c 1 > The social losses from axaion and inflaion The previous discussion has shown ha he public secor may finance is expendiures hrough he revenues from oupu axaion and seigniorage, where

11 TIME CONSISTENT COLLECTION OF OPTIMAL SEIGNIORAGE 11 oal seigniorage is composed of he seigniorage from expeced and surprise inflaion, s = s e + s s. In his subsecion, we discuss he social loss or, equivalenly, he reducion in social welfare, arising from he use of he differen revenue insrumens. The presen value l of he sum of curren and fuure period-by-period social loss, l, discouned a he marke real ineres rae is given as 15 l =0 R l. (24) The period social loss is assumed o ake he addiively separable form l l 1 (τ )+l 2 (i )+l 3 (π ). (25) which reflecs he following sources of social loss: The social loss from he use of he endowmen ax is expressed by he erm l 1 (τ ). Since oupu is exogenously given, here are no social cos due o ax disorions in he presen model. 16 Insead, he social loss from oupu axaion is assumed o arise from ax collecion or enforcemen cos, as in Barro (1979, 1983) and Mankiw (1987). These collecion cos do no show up in he public secor budge consrain, because i is implicily assumed ha privae agens have o pay hem in addiion o heir ax paymens. One may moivae ax collecion cos by he assumpion ha individuals ry o avoid axes because axes reduce heir consumpion. If uiliy has he sandard convexiy properies, hese aciviies will be inensified more han proporionally wih increasing average axes. Hence, we assume ha he collecion cos are monoonically and sricly convexly increasing in he average ax rae. 2. The erm l 2 (i ) capures he social loss due o expeced inflaion. Expeced inflaion leads o a posiive opporuniy cos of holding money and consequenly o lower holdings of real balances han are socially opimal. Avoidable banking and shopping aciviies hen become necessary, giving rise o higher ransacion cos. These well known shoeleaher cos increase in he expeced inflaion rae and are minimized when he nominal ineres rae is zero and he saiaion level of real balances is held. Given he sandard convexiy properies of uiliy, hese cos are also sricly convex in expeced inflaion and he larger hey are, he more sensiively real balances reac o expeced inflaion [i.e. he higher is he ineres elasiciy of real balances]. 3. In conras o expeced inflaion, acual inflaion does no cause social loss from subopimally low holdings of real balances. This holds rue because privae agens have already deermined heir desired holdings of nominal balances when he auhoriy decides wheher or no o creae more inflaion han is expeced. The social cos of acual inflaion, which are represened by he erm l 3 (π ), come insead from menu cos, in ha

12 12 HERRENDORF cosly adjusmens of all non-indexed nominal conracs are required. These cos are also assumed o increase sricly convexly in he acual inflaion rae. I should be menioned ha here is an addiional possibiliy by which acual inflaion may cause social loss. If acual inflaion exceeds expeced inflaion and if he populaion is heerogeneous, hen he resuling unexpeced inflaion may give rise o an unwaned redisribuion among privae agens. For example, if a posiive share of privaely issued bonds is no indexed [which appears o be a realisic supposiion], hen unexpeced inflaion ransfers real resources from crediors o debors. The resuling social cos could be capured by including a erm of he form l 4 (π π e ) in (25), which has been suggesed by Grossman (1990). However, his would no change he resuls of he subsequen analysis in an imporan way, as long as he loss erm l 3 (π ) is preserved. 17 Afer having moivaed he differen erms in (25), we need o impose some addiional srucure on he social loss funcion. So far, specificaion (25) only requires separabiliy beween he social losses from differen revenue insrumens, which is a sandard, simplifying, bu no innocuous assumpion. For example, i ludes he possibiliy ha realized inflaion affecs he social loss from oupu axaion. Higher inflaion could, for example, increase he collecion cos of he oupu ax when here are significan collecion lags. The real value of ax revenues would hen be reduced unil hey were colleced; see Tanzi (1977) for he classic argumen on collecion cos due o inflaion, and Dixi (1991) or Mourmouras and Tijerina (1994) for reconsideraions. While his may have ineresing implicaions, i is no of primary ineres here. In addiion o separabiliy, he previous discussion of he differen componens of l suggesed ha social loss mus saisfy he following sandard requiremens: (i) Social loss depends posiively and sricly convexly on each of he differen policy insrumens. (ii) No using an insrumen implies ha he minimum [marginal] social loss of zero. e (iii) The social loss, as well as he marginal loss, go o infiniy if τ, π or π approach one. [This is o say ha infinie social loss and infinie marginal social loss are caused by ax raes of one hundred percen.] In order o ensure ha all of he above properies are saisfied, he following funcional form of social loss, similar o he one suggesed by Barro (1983), is used from now on: l=κ 1 exp 1 + κ 1 τ 2 2 [exp( i 2 ) 1] + κ 3 exp 1. (26) 1 π 2 A lile hough reveals ha (26) saisfies he properies (i), (ii) and (iii). τ 2 π 2

13 2.5 The saemen of he opimal axaion problem We now sae formally he opimal axaion problem of finding a sequence of ax raes and inflaion raes ha minimizes social loss while saisfying he consolidaed flow budge consrain in every period. To his end, he Lagrangian is defined as he sum of he presen discouned value of he period-by-period social loss, and he produc of a Lagrange muliplier λ and he consolidaed real flow consrain (13), =0 TIME CONSISTENT COLLECTION OF OPTIMAL SEIGNIORAGE 13 R {[l 1 (τ ) +l 2 (i ) +l 3 (τ )] λ [ b g r b 1 +τ y +s e +s s ]}, (27) where l i, s e and s s are given by (26), (17) and (23) respecively. The opimal axaion problem is o minimize he Lagrangian wih respec o b, τ and π ( =0,1,2 ). In order o ensure ha he opimal axaion problem is well defined, some addiional assumpions need o be made. To begin wih, he moneary auhoriy is supposed o implemen base money changes by a policy ha uniquely deermines c for any chosen inflaion rae. Possible examples of such a policy have been described in subsecion 2.3. Moreover, ω is reaed as exogenously given for he momen. Below, i will be shown ha one can impose resricions on c and ω such ha he ime consisency problem is solved. Furhermore, in addiion o he ransversaliy condiion (20) on c, he fulfilmen of a ransversaliy condiion on he evoluion of b is required, lim 2 R b = 0. (28) This is he well known solvency or no Ponzi-game condiion ha he growh rae of governmen deb be smaller han he real rae of ineres. Wihou his condiion he opimal axaion problem would no be well defined because he public secor could cover is expendiure by issuing deb a an acceleraing rae. 18 We herefore assume from now on ha he iniial deb level and he exogenously given sream of governmen expendiure are such ha he public secor is solven, ha is, ha he flow consrain can be me wihou violaing (28). 19 Finally, he auhoriy is implicily assumed o honour is deb, implying ha deb repudiaion is excluded as a possibiliy of reducing governmen liabiliies. The convenional jusificaion for his assumpion is ha a deb defaul desroys he auhoriy s repuaion for being a reliable debor. This leads o fuure cos ha presumably ou-weigh he benefi of defauling. Such cos, for insance, may arise from subsequen borrowing difficulies. However, Chari and Kehoe (1993a,b) have recenly challenged his argumen by showing for a closed economy ha even he mos severe repuaional consequences of deb repudiaion may no be sufficien o preven an opimizing governmen from defauling. Hence, here is sill a conroversial heoreical quesion of why mos real world governmens do honour heir deb [a leas in normal imes ]. Despie his fac, we have only limied space and resric aenion o a governmen ha does no consider defauling on is deb as a feasible policy opion.

14 14 HERRENDORF 3. The ex ane opimal policy We have now exended Barro s (1983) model o a framework rich enough o analyze he opimal axaion problem. As a benchmark case, we will firs sudy he opimal ax policy under ommimen. The erm ommimen means ha he auhoriy credibly binds iself o follow an announced policy. Many auhors, who waned o concenrae on he public finance aspecs of he opimal ax package, have resriced heir aenion o ommimen; see, for example, Drazen (1979), Kimbrough (1986a), Mankiw (1987), Grilli (1989), V egh (1989b), Yashiv (1989), Dixi (1991), or Calvo and Leiderman (1992). In secion 4, we will also characerize he soluion of he opimal axaion problem under discreion, which is relevan because mos policy makers in he real world are no ommied. Since he discreionary soluion urns ou o be welfare inferior compared o he ommimen soluion, secion 5 provides a discussion of how he ommimen oucome can be achieved. The plan for he remainder of he presen secion is as follows: in subsecion 3.1, we formally derive he soluion o he opimal axaion problem under ommimen. Subsecion 3.2 provides an inerpreaion of he opimaliy condiions in he ligh of he Ramsey (1927) principle. The secion ends wih subsecion 3.3, which is concerned wih he issue of wheher or no our resuls generalize o equilibrium models of opimal axaion. 3.1 The soluion o he opimal axaion problem under ommimen When he policy maker is ommied wihin a deerminisic environmen, unexpeced inflaion is no available as a policy insrumen and he realized inflaion rae mus equal he expeced inflaion rae. 20 Consequenly, he inflaion expecaion is one of he choice variables and he policy maker can opimize in a similar way o a Sackelberg leader. Taking accoun of he follower s reacion funcion, i.e. of he privae secor s demand for real balances, he policy maker chooses he opimal sequence of fuure socks of governmen bonds, ax raes and e inflaion raes, and ommis o sick o i. Subsiuing π = π and s s =0 (= 0, 1, 2, ) ino (27), he Lagrangian under ommimen is found o be = = = 0 =0 R {[l 1 (τ ) +l 2 (i ) +l 3 (π )] λ [ b g r b 1 +τ y +s e ]} R {[l 1 (τ ) +l 2 (i ) +l 3 (π )] λ [ (b c ) g r (b 1 c 1 )+τ y + m (π +1 )+π m 1 (π )]}. The opimal axaion problem under ommimen is o minimize (29) wih respec o b, τ and π (= 0, 1, 2 ). This minimizaion problem is well defined because he loss funcion is sricly convex and he consrains are concave e funcions in he ax rae and in he inflaion rae. Tha s is a concave funcion in e π is ensured by he assumpion ha m is convexly decreasing in expeced inflaion; compare (6). (29)

15 By he Kuhn-Tucker heorem, he following firs order condiions are necessary condiions for an inerior soluion, {b, τ, π }, o he opimal axaion problem: 21 Firs, he opimal policy has o saisfy he flow consrain (13) wih π e = π in each period. Also, he firs-order condiion for b is b = 0 X λ = λ +1. (30) We can see ha he Lagrange muliplier does no change over ime, λ λ. 22 In order o find he addiional wo firs-order condiions for τ and π, we mus firs recall ha oupu was assumed o be exogenously given and hus does no depend on he average ax rae. Moreover, since π e = π under ommimen, he Fisher relaion (3) implies ha he nominal ineres rae is affeced by he choice of π, i = r + π i X = 1 +i. (31) 1 π π 1 π Hence, for he wo addiional firs-order condiions, one obains where TIME CONSISTENT COLLECTION OF OPTIMAL SEIGNIORAGE 15 π τ = 0 X = 0 X = τ l 1 (τ ) = λ y, [ l 2 (i ) + l 3 (π )] π λ [1 ε 1 (π m )] 1 (π, ) (32a) (32b) ε 1 (π ) r + π m 1 (=0, 1, 2, ). (33) m 1 π ε 1 (π ) can be inerpreed as he [absolue-value] elasiciy of real balances in period 1, wih respec o he opporuniy cos ρ 1 of holding money. This follows from (4): ε 1 (π ) = r m +π m 1 1 ρ 1 π ρ 1 ρ = 1 m 1 m 1 ρ 1. (34) In summary, he firs-order condiions for a soluion o he opimal ax problem under ommimen are (13), (30), (32a) and (32b). By he Kuhn-Tucker Theorem, we also know ha if here exiss a sequence {b, τ, π }, saisfying hese firs-order condiions, i does indeed solve he opimal axaion problem, which ensures sufficiency. Tha such a sequence exiss can be seen from he following wo facs: on one hand, if neiher revenue insrumen is used, is marginal social loss equals zero; on he oher hand, a ax rae or inflaion rae of one resuls in an infinie marginal social loss. Because he marginal social gain of

16 16 HERRENDORF eiher insrumen [righ hand side of he firs-order condiions] is posiive and remains finie, coninuiy implies exisence. Furhermore, since here can be a mos one soluion o a problem of minimizing a sricly convex funcion over a convex se, he uniqueness of his soluion is also ensured. Hence, he opimal axaion problem under ommimen has a unique inerior soluion. Since a firs-bes policy of financing all expendiure from a lossfree insrumen is no available when all revenue insrumens give rise o social loss, his soluion resuls in he highes achievable social welfare. For his reason, i is ofen referred o as he [bes] second-bes or he ex ane opimal policy. 3.2 The Ramsey principle of opimal axaion Inuiively, he firs-order condiions (32a) and (32b) require, for each revenue insrumen, ha he marginal social loss from an incremenal increase of ha insrumen be equal o he marginal social gain. As is sandard, he marginal gain is expressed in unis of social loss. The conversion of unis of oupu ino unis of social loss is achieved by means of muliplicaion by he shadow price, λ, of public revenues. More isely, λ is he marginal decrease of social loss ha would resul in an opimum if he governmen had one addiional uni of revenues available. 23 In order o inerpre he ex ane opimal policy furher, i is useful o express he firs-order condiions (32a) and (32b) in a differen way. Employing he definiion for he presen discouned values of seigniorage from expeced inflaion, and ha of social loss, (22) and (24), i is sraighforward o verify ha (32a) and (32b) can, equivalenly, be wrien as l π l τ =λ R y, (35a) =λ s (=0, 1, 2, ). π e (35b) The combinaion of hese wo equaions allows he eliminaion of he Lagrange muliplier: l s e l τ = R y l τ = R y l π, s e π l τ +1 R +1 y +1 π l π + 1 = (=0, 1, 2, ). π s e π +1, (36a) (36b) (36c) This version of he necessary condiions brings ou clearly ha he ex ane opimal policy mus saisfy he Ramsey (1927) principle of opimal axaion: 24 equaion (36a) is he saic firs-order condiion, requiring equaliy beween he raios of he

17 TIME CONSISTENT COLLECTION OF OPTIMAL SEIGNIORAGE 17 marginal social loss o marginal revenue for boh insrumens in any period. The wo following equaions (36b) and (36c) are he ineremporal firs-order condiions [Euler equaions]. They imply ha axes are o be smoohed over ime because, in an opimum, he raios of he marginal social loss o he marginal revenue of eiher insrumen mus be equalized beween periods. If he economy is in a seady sae, in which he marginal losses and revenues from eiher insrumen grow a he same rae, hen he firs-order condiions lead o consan oupu axes and consan inflaion raes. In his case, emporary changes in expendiure are o be financed by bond issues, which do no have real effecs in a Neo-Ricardian world. I is now shown ha he differen recommendaions of he public finance lieraure on opimal inflaion raes follow from he Ramsey principle under specific condiions. (i). The raio of he marginal social loss o marginal revenue is much larger for oupu axaion han for seigniorage I is opimal o rely mainly on seigniorage o finance public expendiure. Rearranging (36a) shows ha for very large marginal social losses from oupu axaion, i is opimal o choose he inflaion rae so as o maximize he presen value of seigniorage,- l τ X s e l π =y π l τ 0. (37) We can observe ha he seigniorage-maximizing inflaion rae is he soluion o a sandard monopoly problem, noably ha of choosing an inflaion rae such ha he marginal revenue from he producion of money equals he marginal cos, which are zero. From (32b) and (34) i emerges ha he marginal seigniorage revenue is zero if a one-percen increase in he opporuniy cos ρ 1 of holding money leads o a one-percen decrease of real balances, 1 = ε 1 (π ) = ρ 1 m 1. (38) m 1 ρ 1 This is a discree ime version of Auernheimer s (1974) uni-elasiciy rule for seigniorage-maximizing inflaion. Noice ha if aenion had been resriced o a saionary sae wih consan inflaion and consan real balances, and if he cash flow measure (18) had been used, hen he presen discouned value of [he ex ane opimal] seigniorage would have been equal o he presen discouned value of he [consan] inflaion ax, s = [(1 + r) r]π m(π ). In his case, he seigniorage-maximizing inflaion rae is characerized by Cagan s (1956) version of he uni-elasiciy rule, 1 = π m. (39) m π However, as he previous discussion has shown, using his formula is no

18 18 HERRENDORF appropriae in general, because i ignores he seigniorage due o changes in real balances; see also he discussion in Auernheimer (1974). Apar from Cagan (1956) and Auernheimer (1974), he seignioragemaximizing inflaion rae has been analysed in a variey of siuaions. Examples include Friedman (1971b), Carhcar (1974), Calvo (1978b), Siegel (1981), Auernheimer (1983), Calvo and Fernandez (1983), Brock (1984), Grossman and Van Huyck (1986), Neumann (1992), and Bordo and Redish (1993). Furhermore, Easerley, Mauro and Schmid-Hebbel (1995) provide a deailed empirical analysis for high inflaion counries. (ii). The raio of he marginal social loss o marginal revenue is much smaller for oupu axaion han for seigniorage The opimal policy implies ha a major share of he revenues is colleced hrough oupu axes and only a small share hrough seigniorage. Now, if using he oupu ax causes almos no rise in social cos, opimaliy requires ha he social losses from inflaion ough o be minimized, l τ 0 X l = 1 π y l s τ π 0. (40) Also, if he marginal social loss from a non-zero nominal ineres rae is weighed much more heavily han ha from acual inflaion [which corresponds o κ 2 κ 3 ]) hen he opporuniy cos of holding money should be driven o zero, i.e. i = 0. By he Fisher relaion (3) his is equivalen o π = r, (41) i.e. he opimal inflaion rae is he negaive of he real ineres rae. This is he Friedman (1969) rule of opimal deflaion, which has a imes also been called he Chicago rule or he opimum quaniy of money rule; see also Mary (1968) and Brock (1975). I implies a conracion of he money supply in any period, resuling in negaive seigniorage revenues, which are financed hrough oupu axaion. Friedman s resul arises under he assumpion ha oupu axaion is almos a loss-free revenue source, similar o a lump sum ax, hrough which all expendiure can be financed a zero social cos. Since money is produced wihou cos, i should hen be supplied unil he saiaion of real balances is reached. This parallels he principle ha i is welfare-opimal o supply whaever quaniy is demanded of a free good. I should be sressed ha, even if oupu axes do no give rise o significan social loss, he Friedman rule is only opimal if he cos of deflaion are negligible relaive o he social cos of expeced inflaion. (iii). For boh revenue insrumens, he raios of he marginal social loss o marginal revenue are of he same order of magniude According o he Ramsey principle, expendiures are in his case opimally financed from boh insrumens, and posiive inflaion raes are par of he opimal

19 TIME CONSISTENT COLLECTION OF OPTIMAL SEIGNIORAGE 19 policy. In paricular, opimaliy implies ha he higher he marginal social loss from an insrumen relaive o is marginal revenue, he less revenue should be colleced from i. In summary, posiive inflaion raes have been shown o be opimal if he raio of he marginal social loss o he marginal revenue of oupu axaion is no negligible. The following feaures may even lead o he opimaliy of high inflaion: firs, ax enforcing and collecing auhoriies work inefficienly or ax evasion is significan. This is capured in our model by large marginal collecion cos for oupu axes. Second, he ineres elasiciy of real balances is small, so ha high expeced inflaion raes lead o small reducions in real balances and, hus, o small marginal social losses from expeced inflaion. This corresponds o he inuiion ha more inelasic ax bases ough o be axed more heavily. And hird, desired real balances are large for any given inflaion rae, offering a large ax base for inflaion. This would be he case if he financial secor is relaively underdeveloped and he bulk of ransacions is faciliaed in cash. Noe ha, for hese reasons, an acive black marke or shadow economy ends o increase he opimal inflaion rae, because illegal ransacions are mosly faciliaed in cash, in order o avoid axes. In ligh of hese argumens i seems worhwhile o devoe some aenion o he effecs of financial innovaions on he opimal inflaion raes and on social welfare. This issue is imporan, for insance, when he welfare effecs of he inroducion of new financial echnologies in developing counries or in Easern Europe are o be judged. For each given rae of expeced inflaion, a financial innovaion presumably reduces he desired holdings of real balances, leading o lower marginal social cos and lower marginal seigniorage from expeced inflaion. The ne effec on he opimal rae of inflaion is hus ambiguous: if he marginal social cos decrease by more [less] han he marginal revenue, higher [lower] opimal inflaion raes would resul. 25 Similarly, i is no clear how a financial innovaion affecs social welfare. Ceeris paribus, of course, i increases welfare. However, he public secor also has o increase he ax rae and he rae of inflaion when agens hold fewer real balances and seigniorage revenues decrease. These higher ax raes lower social welfare. If his effec is large enough, i may even overcompensae he iniial welfare increase, hereby resuling in a negaive ne effec on social welfare. This is more likely o happen, he more imporan seigniorage is as a source of public revenue. If individuals reduce heir desired holdings of real balances sharply, for example, because credi cards become an acceped means of paymen or an efficien chequeclearing sysem is insalled, large increases in inflaion may be ineviable o mee he revenue requiremens. 26 Auhoriies confroned wih his problem hus ough o be advised o improve he efficiency of heir ax sysem, in order o have alernaive revenue sources a hand when seigniorage decreases. 3.3 Possible generalizaions o equilibrium models of opimal axaion Having discussed he opimal policy, he quesion remains as o wheher he resuls found are a consequence of he simpliciy of our model. Needless o say,

20 20 HERRENDORF he use of an equilibrium framework would be more saisfacory. In a more complee analysis, our ax on exogenous oupu would be subsiued by a disorionary ax on labor income or on consumpion expendiure. Two basic cases can hen be disinguished. Firs, if collecing axes is cosly, as in cases 1 and 3 of he las subsecion, hen our resuls can also be obained in general equilibrium frameworks; see Aizenman (1983, 1987) and V egh (1989a). Second, in he absence of collecion cos, he only clear-cu heoreical resul is ha posiive domesic inflaion is opimal in small open economies wih currency subsiuion when he foreign nominal ineres rae is posiive; see V egh (1989b, 1995) and Guidoi and V egh (1993a). Oherwise, wheher he Friedman rule is opimal depends on he way in which money is inroduced ino he model. There are hree main alernaives: (i) money holdings ener he uiliy funcion direcly; (ii) individuals face a cash-in-advance consrain; 27 (iii) money is an inpu facor in a ransacions echnology. 28 For a long ime, here has been agreemen ha posiive inflaion is opimal even in he absence of collecion coss, if holding money yields direc uiliy or if individuals face a cash-in-advance consrain. The firs approach was used by, among ohers, Phelps (1973), Mary (1978), Chamley (1985), Persson e al. (1987), and Aizenman (1993), whereas Braun (1994) employed he second one. In conras o heir resuls, a recen conribuion by Chari, Chrisiano and Kehoe (1996) shows ha he Friedman rule remains opimal in boh cases, if he uiliy funcion is separable in leisure and if he subuiliy funcion over consumpion goods is homoheic. Noe ha hese are he condiions for he opimaliy of uniform axaion in he public finance lieraure [Akinson and Sigliz (1972)]. If he demand for money is derived from a ransacion echnology, conflicing resuls have been obained. Iniially, Drazen (1979) and Leach (1983) claimed ha a posiive inflaion rae is opimal, bu Kimbrough (1986a,b) proved hem o be wrong. Laer auhors, such as Guidoi and V egh (1993b), hough ha Kimbrough s resul depends criically on he assumpion of a ransacions echnology ha is homogeneous of degree of one. However, recenly Correia and Teles (1996) have shown ha his is incorrec and ha, indeed, he Friedman rule is opimal for ransacions echnologies ha are homogeneous of any degree. 29 In sum, i appears ha he recommendaions of equilibrium models wihou collecion cos are no enirely robus. Hence, I agree wih Guidoi and V egh (1993, p. 203) in concluding ha given ha he microfoundaions of money demand sill remain a conroversial and, o a large exen, unexplored erriory, we feel ha unil he profession reaches somehing of a consensus, no definie conclusions should be drawn. 4. The ime consisency problem In his secion, we show ha generally he ex ane opimal policy is no ime consisen when he auhoriy has discreion [subsecion 4.1]. A policy is called ime consisen when i is ex ane and ex pos opimal. Pu differenly, ime consisency means ha he planned policy deermined as opimal a he iniial dae