The Optimal Inflation Tax*

Transcription

1 Review of Economic Dynamics 2, Ž Aricle ID redy , available online a hp: on The Opimal Inflaion Tax* Isabel Correia and Pedro Teles Banco de Porugal and Uni ersidade Caolica Poruguesa, Lisbon, Porugal Received Augus 18, 1997 We deermine he second bes rule for he inflaion ax in moneary general equilibrium models where money is dominaed in rae of reurn. The resuls in he lieraure are ambiguous and inconsisen across differen moneary environmens. We derive and compare he opimal inflaion ax soluions across he differen environmens and find ha Friedman s policy recommendaion of a zero nominal ineres rae is he righ one. Journal of Economic Lieraure Classificaion Numbers: E31, E41, E58, E Academic Press Key Words: Friedman rule; inflaion ax 1. INTRODUCTION This paper addresses he issue of he opimal inflaion ax in moneary general equilibrium models where money is dominaed in rae of reurn. Friedman Ž addresses his issue in a firs bes environmen, where lump-sum axes are available. He proposes a moneary policy rule ha generaes a nominal ineres rae equal o zero, corresponding o a zero inflaion ax and o a negaive rae of inflaion. The inuiion is simple: since he marginal cos of supplying money is negligible, he marginal benefi should equal he marginal cos, and so he nominal ineres rae should be se equal o zero. We are ineresed in he more relevan second bes resuls, i.e., when he governmen mus finance governmen expendiures wihou having access o lump-sum axaion. Here, he lieraure is inconsisen, paricularly across differen moneary environmens. The key inconsisency, which will be our main focus here, is ha while in models ha explicily specify ransacion echnologies he Friedman rule is a general resul, in models * We wan o hank V. V. Chari, Mike Dosey, Rody Manuelli, and Harald Uhlig for very valuable commens. Remaining errors are ours. Financial suppor by PRAXIS XXI is graefully acknowledged $30.00 Copyrigh 1999 by Academic Press All righs of reproducion in any form reserved.

2 326 CORREIA AND TELES wih money in he uiliy funcion he resuls are ambiguous. As poined ou by Woodford Ž 1990., eiher he Phelps or he ani-phelps resul is possible, depending upon deails of specificaion. In any case, he convenional wisdom is sill he inuiion in Phelps Ž 1973., ha in a second bes environmen liquidiy is a good ha should be axed, jus as any oher good. We clarify he issues involved and find ha he Friedman rule is he opimal policy. The reason for he generaliy of his resul is he fac ha money is a free good. We show ha since he cos of producing he good is zero, he opimal uni ax is also zero, under general condiions, ranslaing ino a robus opimal rule of a zero nominal ineres rae. This resul is imporan in ha i ranslaes ino a very clean policy recommendaion, independen of he parameerizaion of he economy. The class of general equilibrium models ha incorporae he feaure of dominance in rae of reurn, and in which we perform he welfare analysis, are designed in a somehow ad hoc fashion. 1 Where his is more clearly so is in models where he preferences depend on he real quaniy of money, as proposed by Sidrauski Ž and Brock Ž The fac ha he use of money for ransacions is no explici in hese models led Clower Ž o propose a cash-in-advance resricion. Lucas Ž and Lucas and Sokey Ž used his approach in a general equilibrium framework. A more complee ransacions echnology, where i is assumed ha ime is subsiuable for he use of money, was addressed by McCallum Ž 1983., Kimbrough Ž 1986., and McCallum and Goodfriend Ž Two major second bes axaion ses of rules in he public finance lieraure have been used o jusify he opimal inflaion ax resuls: he Diamond and Mirrlees Ž opimal axaion rules of inermediae goods and Ramsey s Ž axaion rules of final goods, furher developed by Akinson and Sigliz Ž The Diamond and Mirrlees Ž opimal axaion rules, derived for he case of consan reurns o scale producion funcions, are he basis for he resuls in he lieraure of moneary models 2 wih ransacions echnologies. In Correia and Teles Ž we show ha he Friedman rule is he opimal soluion in hese moneary models for all homogeneous ransacions coss funcions. We also show ha he inerpreaion of his resul is no a direc exension of he heorem of Diamond and Mirrlees bu is relaed o he free good characerisic of money and o he special srucure of producion and axaion implied in his class of models. 1 In conras, models where he purpose is o generae an equilibrium posiive price for fia money are more fundamenally specified. The seminal papers are Samuelson Ž 1958., Grandmon and Younes Ž 1973., Bewley Ž 1980., Townsend Ž 1980., and Kiyoaki and Wrigh Ž In hese models, he perfec subsiuabiliy beween money and bonds implies a zero nominal ineres rae, and he policy issue is he deerminaion of he real ineres rae. 2 See Kimbrough Ž 1986., Guidoi and Vegh Ž 1993., and Chari e al. Ž

3 THE OPTIMAL INFLATION TAX 327 Akinson and Sigliz Ž esablished ha i is opimal no o disor he relaive prices beween consumpion of differen goods when he preferences are separable in leisure and homoheic in he consumpion goods. These rules were applied o cash credi goods economies by Lucas and Sokey Ž and Chari e al. Ž There, he inflaion ax ranslaes ino discriminaed effecive axes on credi and cash goods. The Friedman rule is opimal under he Akinson and Sigliz Ž condiions for uniform axaion. The Ramsey Ž rules were also used o explain he resuls in models wih money in he uiliy funcion. Phelps Ž uses his srucure, wih exogenous facor prices, and concludes ha he opimal inflaion ax is posiive. Chamley Ž aims a generalizing Phelps Ž resuls o a general equilibrium model and concludes ha he Friedman rule is he opimum only in he firs bes case. Siegel Ž sresses he cosless naure of liquidiy services bu concludes ha his characerisic does no affec he resul of a sricly posiive ax on hose services. Drazen Ž saes ha he disincion beween he cosliness and coslessness of he producion of goods is imporan in he deerminaion of he second bes soluion. Neverheless, he concludes ha i appears difficul o say even wheher he opimal inflaion rae will be posiive or negaive. These resuls are disurbing because hey are no consisen wih he general opimaliy resul of he Friedman rule in ransacions echnology models. One oher reason for he apparen inconsisency in he opimal inflaion ax resuls is ha he approach o he second bes problem is no uniform. Some auhors impose condiions of saionariy on he second bes problem. In his hird bes soluion he Friedman rule is never he opimal soluion. The main conribuion of his paper is o show ha he Friedman rule is indeed a general resul in he se-up where liquidiy is modeled as a final good. I urns ou ha he generalized use of he Ramsey Ž rules o jusify he Phelps resul is misleading and explains he ambiguiy and he apparen inconsisency in he resuls in he differen moneary environmens. The rules on opimal axaion of final goods apply o ad valorem axes on cosly goods. In general, he opimal ad valorem consumpion axes are sricly posiive. Since he goods are cosly, he corresponding uni axes are also sricly posiive. Bu money is assumed o have a negligible producion cos. If his is he case, hen he only ax ha can generae posiive revenue is a uni ax, and in any case he nominal ineres rae is by consrucion a uni ax. The general resul ha he ad valorem ax rae on real balances is sricly posiive can ranslae in he limi, when he coss of producing money are made arbirarily small, ino an opimal zero nominal ineres rae. This is he bes inuiion for why he Friedman rule is a general resul.

4 328 CORREIA AND TELES In clarifying he puzzles in his lieraure, we ake ino accoun mainly Ž. i he free good characerisic of real balances, Ž ii. he fac ha models wih money in he uiliy funcion are reduced forms of more explici moneary models, and ha Ž iii. he Ramsey problem is unresriced. We exend he resuls for money in he uiliy funcion models obained by Chari e al. Ž and esablish he links beween he resuls obained in he differen ypes of moneary models. In paricular, we relae he resuls from a model wih money in he uiliy funcion o hose from ransacions echnology models obained by Correia and Teles Ž The opimal rules in he money in he uiliy funcion model are derived in Secion 2. We also show, in his secion, ha he Friedman rule is a general resul by esablishing an equivalence beween he money in he uiliy funcion models and he underlying ransacions echnologies models. In Secion 3 we provide he main inuiion for he resuls. In Secion 4 we compare he resuls o hose from models wih credi goods. In Secion 5 we discuss he robusness of he resuls o alernaive specificaions of he available axes and alernaive iming srucures. Secion 6 conains he conclusions. 2. MONEY IN THE UTILITY FUNCTION We use he general equilibrium model of a moneary economy developed by Sidrauski Ž and laer used by Brock Ž 1975., Woodford Ž 1990., and Chari e al. Ž o discuss he opimaliy of he Friedman rule in firs bes and second bes environmens. The economy is populaed by a large number of idenical infiniely lived households, wih preferences given by ž / M Ý 0 P V c,, h, Ž 2.1. where c, M, P, and h represen, respecively, consumpion in period, money balances held from period o period 1, he price of he consumpion good in unis of money in period, and leisure in period. The uiliy funcion shares he usual assumpions of concaviy and differeniabiliy; i is increasing in M P as long as M P m* Ž c, h. and nonincreasing for M P m* Ž c, h.. m* Ž c, h. is he saiaion funcion ha represens he saiaion level, ha is, he poin where cash balances... are held o saiey, so ha he real reurn from an exra dollar is zero Ž Friedman, The characerizaion of he funcion m* Ž c, h. is crucial for he deerminaion of he opimum quaniy of money, as will be shown in he nex secion. Alhough Friedman Ž does no fully characerize his poin, he examples he presens imply ha he poin of

5 THE OPTIMAL INFLATION TAX 329 saiaion is finie. Phelps Ž also assumes ha here is full liquidiy or liquidiy saiaion when he nominal ineres rae is no sricly posiive and he demand for real balances is finie. The discussion in Brock Ž on he opimum quaniy of money is also for a finie saiaion level. The echnology of producion of he privae good and he public consumpion good is linear wih uniary coefficiens. The represenaive household Žha implicily solves he problem of he firm. chooses a sequence c 4, h, M, B 0, given a sequence of prices and income axes, P, i, 4, and iniial condiions for W M Ž i. B, o saisfy a sequence of budge consrains: 1 1 Pc M 1 B 1 Ž 1. PŽ 1 h. M Ž 1 i. B, 0 M0 B0 W 0, Ž 2.2. ogeher wih a no-ponzi games condiion. B is he number of bonds held from period o period 1, and i is he nominal reurn on hese bonds. 1 h is he labor supply. The se of budge consrains can be wrien as a unique ineremporal budge consrain: Ý Ý Ý QPc QiM QPŽ 1.Ž 1 h. W, Ž where W M Ž 1 i. B and Q 1 ŽŽ 1 i.... Ž 1 i In he compeiive equilibrium he following marginal condiions mus hold: VcŽ. P Ž 1 i 1., 0 Ž 2.4. V Ž 1. P c 1 Ž 1. V V, 0 Ž 2.5. c h V iv, 0, Ž 2.6. m c and he resources consrains are c g 1 h, 0, Ž 2.7. where g is he level of public spending in period. We assume ha g is consan, g g. We proceed o characerize he opimal policy in his environmen.

6 330 CORREIA AND TELES 2.1. The Opimal Policy The firs bes policy in his economy is given by he maximizaion of Ž 2.1. subjec o he resources consrain Ž In his soluion Vc Vh and Vm 0, 0. If he governmen could collec lump-sum axes, i would be possible o decenralize his soluion by seing a consan nominal ineres rae equal o zero. I is clear ha he Friedman rule is opimal simply because real balances are a free good, in he sense ha hey do no require resources o be produced. Since he social marginal cos of money is equal o zero, hen he level of money balances ha characerizes he firs bes is m*, i.e., he level for which marginal uiliy is zero. The soluion can be decenralized by seing he privae marginal cos of holding real balances idenical o zero, i.e., a zero nominal ineres rae. The opimal policy problem is more ineresing when he governmen canno levy lump-sum axes. To deermine he second bes Ž Ramsey. soluion, we consruc he implemenabiliy consrain, subsiuing Ž 2.4., Ž 2.5., and Ž 2.6. ino he ineremporal budge consrain Ž 2.3.: W 0 Ý Ž c h. Ý m Ž c m P 0 Vc V Ž 1 h. V m V V. Ž 2.8. The soluion of he maximizaion of Ž 2.1. subjec o he implemenabiliy consrain Ž 2.8. and he resources consrains Ž 2.7., and given he iniial nominal wealh, W 0, is he following: P0 is se a an arbirarily large 4 number, and he firs-order condiions for c, h, m are as follows: Ž. 0 Vc Vc Vc c c Vh c 1 h Vm c m, 0 Ž. Vh Vh Vc h c Vh h 1 h Vm h m, 0 Ž 2.9. Ž V V V c V Ž 1 h. V m 0, 0, Ž m m cm hm mm where is he shadow price of he implemenabiliy consrain, i.e., i measures he marginal excess burden of governmen deficis in his second bes world. measures he shadow price of resources. This second bes allocaion can be decenralized using he insrumens i and, 0. Given Ž 2.6., he discussion of wheher he Friedman rule is opimal in his environmen is equivalen o he discussion of wheher V m is zero, for 0. Condiions Ž 2.9. Ž 2.11., ogeher wih he resources consrain Ž 2.7. and he implemenabiliy consrain Ž 2.8., define he sa-

7 THE OPTIMAL INFLATION TAX 331 ionary soluion for c, h, m,, and, for 0, since, from he compeiive equilibrium condiion Vm iv c, he soluion for he nominal ineres rae is saionary. If he Friedman rule holds, i holds for every period. So he issue is wheher, for 0, Vm Vm 0 is a soluion of he sysem of equaions. Because money is a free good, as is clear from he resources consrain Ž 2.7., he muliplier of he resources consrain does no show up in condiion Ž In his second bes soluion he social marginal benefi of using money, for he households, is equal o he marginal excess burden, i.e., he marginal cos due o he fac ha a change in m affecs he budge 3 consrain of he governmen. This can be seen by rewriing Ž as V V V c V Ž 1 h. V m, m m cm hm mm given ha a he opimum, 0, he relevan issue is he deerminaion of he sign of he erm in parenheses, i.e., he impac on governmen revenue of an increase in m, holding he quaniies of he oher goods consan. The following is an example of he general properies ypically discussed in he lieraure. An increase in m corresponds o a decrease in he nominal ineres rae and has a negaive impac on he revenue from seigniorage, Vmm.SoVm Vmmm 0. The sign of he expression depends hen on he cross-derivaives. Suppose Vcm 0 and Vhm 0. In his case, he expression would be negaive, meaning ha an increase in m has a negaive effec on oal governmen revenue. This means ha he marginal excess burden would be sricly posiive. The implicaion is ha he Friedman rule would no be opimal. The Friedman rule is opimal if he marginal excess burden of real balances is zero a he opimum. In Proposiion 1 we idenify condiions in which his is he case. These are local condiions a he poin of saiaion in real balances. We make he assumpion, ha we jusify fully in Secion 2.2, ha he saiaion poin in real balances does no depend on leisure, bu on he level of ransacions only. This assumpion is used o show he main proposiion in he paper. Assumpion 1. The saiaion poin in real balances is a funcion of consumpion only, m* Ž c.. 3 The implemenabiliy consrain was consruced using he budge consrain of he households, bu an equivalen consrain can be obained using he governmen budge consrain. The marginal effec on he implemenabiliy condiion corresponds o a symmeric marginal effec on he condiion expressed in erms of he governmen budge consrain.

8 332 CORREIA AND TELES PROPOSITION 1. In models wih money in he uiliy funcion, he Friedman rule is he opimal policy when he saiaion poin in real balances is such ha m* or m* kc, where k is a posii e consan. Proof. We verify wheher Eq. Ž is saisfied when Vm 0. m* was assumed o be a funcion of c only, herefore a he saiaion poin, Vhm 0. When m*, we mus have V Ž c, m*. 0 and V Ž c, m*. mm mc 0. We assume ha, when Vm 0, and herefore he nominal ineres is zero, he inflaion ax revenue, Vm m, is zero. Any reasonable specificaion of a model wih money in he uiliy funcion mus have his propery. Therefore lim Ž V m. m 0, or lim Ž V V m. m m m m mm 0. Then m* verifies he equaion. When he saiaion poin in real balances is finie, we find from V Ž m*, c. 0 ha m V c, m* dm*. V c, m* dc cmž. mmž. Ž. So, expression 2.11 evaluaed a he saiaion poin in real balances can be wrien as dm* c VmŽ c, m*. 1 VmmŽ c, m*. m* 1, dc m* where V Ž c, m*. m 0. When m* kc, we have ha Ž dm* dc.ž c m*. 1, and so V Ž c, m*. cm c V Ž c, m*. mm m* 0. Therefore m kc also saisfies he Ramsey firs- order condiion, Ž Noice ha when V is separable in leisure and homoheic in consumpion and real balances, hen he raio of real balances and consumpion is a funcion of he ineres rae alone, and herefore he relevan elasiciy is uniary. From Proposiion 1, he Friedman rule is opimal. These are he sufficien condiions for he opimaliy of he Friedman rule idenified by Chari e al. Ž In summary, we have shown in his secion ha he firs bes and he second bes opimal inflaion ax rules coincide, when he saiaion poin in real balances is infinie or when i is characerized by a uniary elasiciy wih respec o consumpion. The reason for his is ha in hose cases he increase in real balances has a zero effec on governmen revenues a he saiaion poin, and consequenly he marginal coss are equal o zero in boh problems. In he nex secion we go beyond he reduced form of

9 THE OPTIMAL INFLATION TAX 333 models wih money in he uiliy funcion o inquire, firs, how hese local properies can be jusified, and second, wheher i can be argued ha he case of he uniary elasiciy of real balances a he saiaion poin is he relevan case Equi alence wih a Transacions Technology Model In his secion we show ha if he models wih money in he uiliy funcion are seen as reduced forms of ransacions echnology moneary models, hen he preference specificaions mus be resriced. These resricions correspond o he condiions under which he Friedman rule is opimal. Feensra Ž shows ha i is possible o esablish an equivalence beween a moneary model wih a ransacions echnology, where he preferences depend only on consumpion ne of ransacion coss, and a model wih money in he uiliy funcion. He esablishes a correspondence beween he se of assumpions characerizing he ransacions echnology and he assumpions on he uiliy funcion expressed as a funcion of real balances. Here we exend Feensra s Ž resuls o a world where he original preferences are defined over consumpion and leisure, Ý Uc, Ž h u. 0, and he ransacion coss are measured in unis of ime. The ransacions coss funcion is represened by s lm, Ž c., where s is he ime spen in ransacions. This is he ype of shopping ime specificaion of McCallum Ž The maximizaion problem is as follows: u 4 Problem 2. Choose c, M, B, h, s o maximize 0 subjec o u Ý 0 Ž. U c, h, Pc M 1 B 1 Ž 1. Pn M Ž 1 i. B, 0 s lž c, m. 1 h u n s and M B W The ransacions echnology is characerized by he following assumpion. 4 Ž. Ž. McCallum 1990 consrucs foonoe 7 he indirec uiliy funcion associaed wih his ype of shopping ime coss. However, he does no derive he properies of his uiliy funcion.

10 334 CORREIA AND TELES Assumpion 2. The ransacions coss funcion s lm, Ž c. has he following properies: Ž. a s 0, lm,0 Ž. 0. Ž b. lc 0. Ž. c lcc 0, lmm 0. Ž d. l Ž m, c. 0 defines m mc. Ž. l Ž m, c. m m 0 when m m. Ž. e l is resriced o ensure ha Problem 2 is a concave problem. The following are wo firs-order condiions of Problem 2: UcŽ. 1 lcž., 0 U už. 1 h 1 lmž. i, 0. 1 The opimal choice of m is such ha he privae agens choose he poin where he privae value of using money, l Ž 1. m, is equal o is opporuniy cos, i. The implemenaion of he Friedman rule, i 0, implies ha lm 0. We call he privae problem defined in he las secion Problem 1. The following proposiion saes he equivalence beween he wo problems. PROPOSITION 2. Gi en a moneary model wih an explici ransacions coss funcion Ž Problem 2,i. is possible o consruc an equi alen model wih Ž. u money in he uiliy funcion Problem 1, where h h lc, Ž m. and Vc, Ž m, h. Uc, Ž h lc, Ž m... If Assumpion 2 is saisfied in Problem 2, hen VŽ c, m, h. is characerized by he following condiions: Ž. a V is conca e and so Vcc 0, Vmm 0. Ž b. V Ž c, m, h. m 0. Ž. c V Ž c, m, h. 0 and V Ž c, m, h. mh mh 0 a he saiaion poin Vm 0. Ž d. m m* are idenical funcions of c only. Proof. The equivalence is for a given pair of funcions Ž U, l.. The soluion of Problem 2 is he vecor Ž ˆˆ c, m, h u.. Then here is a funcion V such ha Žˆˆ c, m, ˆh h u lc, Ž ˆˆ m.. solves Problem 1. Condiions Ž.Ž. a d are saisfied since Ž. i From he concaviy of Problem 2, V is concave, Vcc Ucc 2l U u l 2 U u u l U u 0, and V U ul l 2 U u c ch c h h cc h mm h mm m h hu 0.

11 THE OPTIMAL INFLATION TAX 335 Ž ii. V U u m h lm 0. Therefore Vm 0 if and only if lm 0. Ž iii. V U u ul U ul U u ul 0 and Ž ii. mh h h m h mh h h m boh imply ha Vmh 0, a he poin Vm 0. Ž iv. V 0 defines m* Ž c. and l 0 defines mc. Ž. So, from Ž ii. m m, m m*. As we discussed in he las secion, he deerminaion of wheher he Friedman rule is opimal in a model wih money in he uiliy funcion depends crucially on he funcional form of he funcion m* Ž c, h., i.e., he funcion defined by seing he marginal uiliy of money equal o zero. By he equivalence resul we verify ha he properies of his funcion are idenical o he properies of he funcion defined by l Ž c, m. m 0. From his, i resuls ha Ž. 1 We can jusify he hypohesis made in Secion 2.1 ha leisure is no an argumen of he funcion m*. Ž. 2 When he ransacions coss funcion is homogeneous of degree q, lm is homogeneous of degree q 1 and can be represened by lm LŽ m c. c q 1, Ž and, a he poin l 0, Lm c Ž. 0 defines m m* Ž c. m kc. So, homogeneous ransacions coss funcions correspond o he case described in Secion 2.1, where he funcion m* has uniary elasiciy and he Friedman rule is always opimal. In fac, as was shown by Correia and Teles Ž 1996., in moneary models wih explici ransacions echnologies, he Friedman rule is he Ramsey soluion for homogeneous ransacions echnologies. Ž. 3 When he ransacions coss funcion, s lc, Ž m., is associaed wih a ransacions producion funcion c fž s, m. which verifies Inada condiions, lm 0 is equivalen o m*. In his case a he poin lm 0, we have lmm lmc 0. Again he marginal condiion of he second bes problem is saisfied for lm 0, and so he Friedman rule is opimal. This corresponds o he case of m*, described in Secion 2.1. Ž. 5 The case of elasiciy of m* lower or greaer han one corresponds o he case of nonhomogeneous ransacions coss funcions. We do no know of any work where i is argued on heoreical or empirical grounds ha he ransacions coss funcion ough o be nonhomogeneous. A he heoreical level he microfoundaions of his funcion are obained from Baumol Ž and Tobin Ž 1956., from he generalizaion of Barro Ž 1976., from Guidoi Ž or from Jovanovic Ž All of hese forms are homogeneous of degree zero. In Marshall Ž he proposed and esimaed ransacions coss funcion is homogeneous of degree one.

12 336 CORREIA AND TELES Braun Ž esimaes he degree of homogeneiy of he ransacion cos funcion o be In summary we can conclude ha once he equivalence beween models wih money in he uiliy funcion and explici ransacions echnologies models is esablished, he local properies used in Secion 2.1 o obain he opimaliy of he Friedman rule are associaed wih global properies of he ransacions coss funcions. Besides, hese global properies are no resricive, since hey include all homogeneous ransacions coss funcions. 5 In Secion 2.3 we provide furher argumens in favor of he robusness of he Friedman rule. Even if he elasiciy condiions for he opimaliy of he Friedman rule are no me, i.e., he implici ransacions echnology is no homogeneous, he calibraed resuls are sill very close o ha prescripion How Far Can he Opimal Policy De iae from he Friedman Rule? In he previous secions we have shown ha he local condiions for he opimaliy of he Friedman rule in a model wih money in he uiliy funcion are general, once an equivalence is esablished beween he model wih money in he uiliy funcion and a ransacions echnology model. In any case, if hose condiions are no me, i is imporan o know he magniude of he opimal inflaion ax. In his secion we compue he opimal policy for calibraed examples where he relevan elasiciy is differen from one. When he elasiciy is higher han one, V Ž c, m*. c V Ž c, m*. cm mm m* 0. If we assume ha he marginal benefi of real balances is always nonnegaive, hen for all of he preference specificaions ha we have used, he opimal allocaion is Vm 0, corresponding o he Friedman rule. We have performed a numerical analysis for a class of preferences specificaion where he elasiciy of he saiaion funcion is sricly lower han one, specifically for he limi case, where he elasiciy of m* is equal o zero. The resuls are depiced in Figs The calibraion is made, using as borderline cases he examples of Calvo and Guidoi 7 and Lucas 26. The insananeous uiliy funcions are addiively separable: U c HŽ h. i Ž m., where HŽ h. h Ž E 2.Ž h. 2 and i 1, 2. We consider 5 The inroducion of capial will no aler ha resul. Suppose ha he ransacions coss funcion is defined as before. The inroducion of capial as an inpu in he producion of he consumpion good has no consequence in he Ramsey soluion ha defines he opimal choice of m. The main difference is ha now he marginal uiliy of labor is no consan and depends on he level of he sock of capial. The Ramsey soluion will have a ransiional period, bu lm 0, and consequenly Vm 0, also characerize ha ransiion. If he ransacions coss funcion is modified in a way ha capial is also an inpu in he producion of ransacions, he resuls are mainained once we impose ha he ransacion coss funcion is homogeneous in consumpion, real balances, and capial.

13 THE OPTIMAL INFLATION TAX 337 FIG. 1. Welfare cos of he inflaion ax and money income raio, for he uiliy funcion calibraed as in Calvo and Guidoi 7. The curve U.S. is he fied schedule of he M1 NNP raio on long-erm ineres raes, for U.S. daa. i wo possible funcions: 1 Ž m. m ŽB D lnž m.. and 2 Ž m. 2 A 1 m m k. A, B, D, E, and k are parameers, and k represens he consan saiaion level in real balances. Governmen expendiures are se a g The firs uiliy funcion, 1, is iniially calibraed wih he numbers provided by Calvo and Guidoi Ž 1993., B 0.65, D 0.5, E 1. Figure 1 shows he resuling welfare cos of he inflaion ax, in unis of consumpion, as well as he corresponding equilibrium schedule for he raio of real balances o income, as a funcion of he nominal ineres rae. The U.S. line is he log-linear M1oNNP schedule esimaed by Lucas Ž for U.S. daa Ž elasiciy is The opimal nominal ineres rae is large, around 10%, bu noice ha he calibraed money income raio and ineres rae schedule are no consisen wih he U.S. daa. Figure 2 shows he same curves for a differen calibraion ha beer fis he U.S. money demand schedule: B 0.046, D The semielasiciy is now 7. The opimal nominal ineres rae is considerably smaller. Figures 3 and 4 sill represen he same wo curves, for he FIG. 2. Welfare cos of he inflaion ax and money income raio, for he uiliy funcion in Calvo and Guidoi 7 calibraed o fi U.S. daa. The curve U.S. is he fied schedule of he M1 NNP raio on long-erm ineres raes, for U.S. daa.

14 338 CORREIA AND TELES FIG. 3. Welfare cos of he inflaion ax and money income raio, when he uiliy from 2Ž. 2 real balances is m A 1 m m k, calibraed o fi U.S. daa. k 1 is he saiaion level in real balances. The curve U.S. is he fied schedule of he M1 NNP raio on long-erm ineres raes, for U.S. daa. second preferences specificaion, 2. Wih his uiliy funcion, in he limi, for an arbirarily large k, he real balances-o-consumpion raio, as a funcion of he nominal ineres rae, is a log-linear schedule. For k 1, he opimal nominal ineres rae is smaller han 0.1% Ž Fig. 3.. Figure 4 shows he opimal nominal ineres rae when k 0.4, generaing levels of velociy a he saiaion poin ha have been observed for considerably higher nominal ineres raes. The opimal nominal ineres rae is less han 1%. The conclusion is ha for reasonable levels of k, he Friedman rule is a very good approximaion o he opimum. FIG. 4. Welfare cos of he inflaion ax and money income raio, when he uiliy from 2Ž. 2 real balances is m A 1 m m k. k 0.4 is he saiaion level in real balances. The curve U.S. is he fied schedule of he M1 NNP raio on long-erm ineres raes, for U.S. daa.

15 THE OPTIMAL INFLATION TAX MONEY IS A FREE GOOD In Secion 2, we have shown ha he Friedman rule is opimal when here is no effec on governmen revenues of changing real balances from he full liquidiy level. The argumens for he second bes axaion rules of final goods in he public finance lieraure are differen from hese. There, he opimal axes on differen goods depend on he comparison of he respecive marginal effecs on governmen revenues. In paricular, for i o be opimal no o ax final goods, when he alernaive choice is an income ax, he marginal effec on governmen ne revenues of a change of one uni of labor used o produce any of he goods should be equal. Akinson and Sigliz Ž derived condiions under which his is he opimal rule. Our resuls show ha he condiions on preferences o obain he opimaliy of he Friedman rule are more general han he ones derived in Akinson and Sigliz Ž and herefore exend he resul of Chari e al. Ž ha idenified hose condiions as sufficien condiions for he opimaliy of he Friedman rule. 6 The homoheiciy and separabiliy condiions of Akinson and Sigliz Ž correspond o uiliy funcions where he marginal rae of subsiuion beween consumpion and real balances depends only on he raio of hese wo variables. This is one example of he condiions in Proposiion 1. The condiions in Proposiion 1 are much less resricive hough, since hey mus hold only in he neighborhood of i 0. We now show how he very appealing argumen of he disribuion of disorions among differen goods in he economy can be reconciled wih he zero inflaion ax resul derived for moneary economies. Wha disinguishes money from any oher consumpion good is he fac ha an addiional uni of real money does no require relevan marginal resources. We hink ha his is he righ way of describing fia money. In he following exercise we analyze how he axaion rules are affeced when he cos of producing a good m becomes arbirarily small. Consider a saionary real economy corresponding o he moneary economy we have sudied, bu for he fac ha m is now a consumpion good ha is produced wih unis of ime, which implies ha he price of m in unis of he oher consumpion good c is. The ad valorem ax on m is m. The budge consrain of he households is wrien as c Ž 1 m. m Ž 1.Ž 1 h.. Noice ha he equivalen uni ax would be T he correspond- Ž.Ž.. ing equaion in he moneary economy is c im 1 1 h. m m Ž 6 As hey poin ou, he condiions are no necessary.

16 340 CORREIA AND TELES In he real economy wih ad valorem axaion, he opimal axaion rules of Ramsey Ž and Akinson and Sigliz Ž apply. Akinson and Sigliz Ž derive sufficien condiions for opimaliy of uniform axaion of consumpion goods. When he preferences are homoheic in c and m and separable in leisure, hen a ax on labor income and a zero ad valorem ax on boh c and m decenralize he second bes Ž Ramsey. soluion. This corresponds o m 0. If insead of he ad valorem ax, a uni ax was used, as is he case wih money, hen his uni ax is always equal o zero a he opimum. Now suppose ha he alernaive ax is a ax on c, c. The budge consrain is wrien as Ž 1 c. c Ž 1 m. m 1 h. The second bes allocaion for he choice of an inflaion ax and an income ax can be decenralized using a consumpion ax, equal o he ad valorem ax on money, m c. The equivalen uni ax T m m is posiive as long as he cos of producing he good is posiive, bu he limi is zero, when converges o zero. This same resul applies as long as he opimal ad valorem ax converges o a finie number. This holds whenever, according o he rules derived for he moneary economy, he marginal impac of m on he governmen revenue, a he saiaion poin in real balances, is equal o zero. There is a sense in which he applicaion of he condiions of Akinson and Sigliz Ž o his problem is misleading. When he choice is he opimal mix of an income ax and an inflaion ax, hen he resul ha he inflaion ax should be zero could be inerpreed as a direc applicaion of Akinson and Sigliz Ž However, ha same soluion is equivalen o a ax on consumpion and a zero uni ax on money. Akinson and Sigliz Ž condiions sill hold for he equivalen ad valorem ax on money raher han for he relevan uni ax. In any case, according o he opimal axaion rules derived for he moneary economy Ž so, for he case where goes o zero., when he impac of a marginal increase in m on governmen revenues is no zero, hen he opimal uni ax can be posiive. This corresponds o an opimal ad valorem ax ha becomes arbirarily large as he cos of producing m is made arbirarily small. 4. CREDIT GOODS In his secion we compare he resuls in he model wih money in he uiliy funcion wih he resuls in models wih credi goods. I is well known ha i is possible o esablish an equivalence beween he wo

17 THE OPTIMAL INFLATION TAX 341 models, by replacing in he model wih money in he uiliy funcion oal consumpion wih he sum of he consumpions of he wo goods and real balances wih he consumpion of he cash good. A condiion ha ensures ha real balances are smaller han oal consumpion guaranees nonnegaiviy of consumpion of he credi good. Chari e al. Ž idenify as sufficien condiions for opimaliy of he Friedman rule in he cash credi goods model he condiions of homoheiciy in he wo goods and separabiliy in leisure. The explanaion for his resul is simple. Suppose ha real money was cosly o produce and consider he ypical srucure in a cash credi goods model, where consumpion of he cash good requires ime and real money in fixed proporions. The producion funcions of he wo goods and of real money are linear. I is a direc applicaion of Akinson and Sigliz Ž ha under separabiliy and homoheiciy, he wo goods should be axed a he same rae. Under he Leonief producion srucure, a posiive ax on money would no disor he producion of he consumpion good, bu would disor he relaive consumpions of he wo goods. Therefore if he alernaive axes are a ax on income or a uniform ax on consumpion, hen i is opimal o se he ax on real balances Ž ad valorem or uni. o zero. Separabiliy in leisure and homoheiciy in he cash and credi goods imply separabiliy in leisure and homoheiciy in real balances and oal consumpion in he equivalen model wih money in he uiliy funcion. As was seen before, hese condiions imply uniary elasiciy a full liquidiy, and herefore he Friedman rule is opimal. If he uiliy funcion is no homoheic in he wo goods, hen he inflaion ax could be used as a means of achieving he opimal disorion in he wo goods. 7 In his case he relevan elasiciy is no uniary. 8 The opimal axaion issue here is very differen from he one considered before. The issue here is he deerminaion of he opimal disorion beween consumpion goods. Wih enough axaion insrumens, his issue Ž. would no even be presen, as seen in Lucas and Sokey If he ransacions echnology allowed for subsiuabiliy beween real balances and ime, hen he disorion of he consumpion of he wo goods would also imply a disorion in producion. Clearly, in his case, i would be preferable o discriminae beween he consumpion axes on he wo goods. 8 Noice ha, under his specificaion, real money balances mus be equal o he consumpion of he cash good, and herefore hey can never be made arbirarily large. So he poin of saiaion canno be infiniy, which is one of he sufficien condiions for he Friedman rule o be opimal.

18 342 CORREIA AND TELES 5. ALTERNATIVE TAXES AND WELFARE CRITERIA In Secion 2 we have consruced a second bes environmen assuming ha here were wo alernaive axes, an inflaion ax and a ax on labor income. For he purpose of checking he robusness of he resuls, in his secion we discuss he implicaions of considering a consumpion ax insead of he labor income ax. In addiion, we will assess he implicaions of considering alernaive iming convenions and welfare crieria in he specificaion of he second bes problem Consumpion Taxes When he level of ransacions is measured by consumpion ne of axes, he ax on consumpion, c, does no affec he ransacions coss funcion, i.e., s lc, Ž m.. In his case he indirec uiliy funcion Vc, Ž m, h. associaed wih he pair Ž U, l. is he same as described in Secion 2. As we saw in Secion 3 he second bes allocaion coincides wih he one obained when he alernaive ax is an income ax. So he condiions under which he Friedman rule is opimal are he same wheher he alernaive ax is a ax on labor income or a ax on consumpion. In summary, he irrelevance resul of he alernaive ax in money in he uiliy funcion models is exended o models of explici ransacions coss funcions, given he equivalence esablished in Secion 2. A number of auhors claim ha he inroducion of a consumpion ax should modify he ransacions coss funcion, in he sense ha he amoun of ransacions ough o be measured by consumpion gross of axes: s lc1 Ž Ž., m. c. This inroduces some changes. Adoping he same procedure as in Secion 2, i is clear ha he pair Ž U, l. corresponds o a uiliy funcion V such ha Ž Ž.. Ž. U c, h l cž 1., m V c, m, h, Ž 1.. c c Now preferences depend on he ax parameer c. Under his formulaion he Friedman rule is opimal when m*. I is also opimal when he elasiciy of m* ŽcŽ 1.. is uniary, if in addiion we impose ha, a he c saiaion poin, he underlying echnology is characerized by eiher l cž1. c 0orlcŽ1., m 0. c De Fiore and Teles Ž show ha he addiional condiions are necessary because he ransacions echnology has he undesirable propery ha i is possible o reduce he ime used for ransacions, wihou changing real consumpion and real money used o buy i, by reducing he ax on consumpion. When eiher he ransacions echnology does no

19 THE OPTIMAL INFLATION TAX 343 have ha propery or when income axes are allowed ogeher wih consumpion axes, hen hose addiional condiions are no longer necessary Alernai e Timing Con enions and Welfare Crieria I is a sandard view Žsee Woodford Ž 1990., p ha alernaive iming convenions in he decisions of he privae agens affec he resul of opimaliy of he Friedman rule. In he previous secions, he privae agens are assumed o choose financial asses, in each ime period, so ha he resuling money balances can be used for ransacions ha same period. This is he iming assumed by Lucas Ž and Lucas and Sokey Ž Alernaively, Woodford Ž assumes, in line wih Svensson Ž 1985., 10 ha he money balances ha can be used in any one period are decided he period before. The implicaion of his iming is ha here are real effecs of unanicipaed moneary shocks. In he beginning of ime, period zero, he agens canno adjus he porfolios, and herefore i is no longer opimal for he benevolen governmen o compleely deplee he real value of ousanding moneary balances. The allocaion is saionary from period one on, bu in period zero he levels of he variables are, in general, differen from he corresponding saionary levels. The saionary opimal allocaion from period one on corresponds o he Friedman rule. Woodford Ž 1990., for he sake of racabiliy, and Braun Ž and Lucas Ž propose a hird bes soluion concep: he maximizaion of welfare resriced o he soluion being saionary, i.e, he problem is resriced so ha he allocaion in period zero is he same as from period one on. When he saionariy resricion is imposed, he governmen faces a rade-off beween he low level of iniial real balances and he high seady-sae level. I is inuiive ha from his rade-off here resuls a saionary level of real balances higher han he iniial opimal level of he Ramsey soluion and lower han he high seady-sae level of he same soluion. The soluion is characerized by less han full liquidiy and a sricly posiive nominal rae of ineres. 9 Asse markes open in he beginning of period, so ha money balances used for ransacions in any period are beginning-of-period balances. Kimbrough Ž and ohers assume ha end-of-period money balances are used for ransacions ha same period. 10 A discussion of he posiive implicaions of he wo iming convenions in a cash-in-advance model is presened by Giovannini and Labadie Ž Nicolini Ž discusses ime inconsisency in a cash-in-advance model wih he wo alernaive iming convenions.

20 344 CORREIA AND TELES Ž. Lucas 1994 has compued numerically, using his crierion, he opimal policy in a ransacions echnology model. He concludes ha he opimal nominal ineres rae, alhough sricly posiive, is very close o zero. 6. CONCLUDING REMARKS In his paper we compue he second bes inflaion ax rule in models where real balances are an argumen in he uiliy funcion. We idenify local condiions ha exend he global condiions of separabiliy and homoheiciy in Chari e al. Ž as sufficien condiions for he opimaliy of he Friedman rule. Furhermore, we esablish an equivalence beween he models wih money in he uiliy funcion and more fundamenal models of ransacions echnologies and show ha he wide class of ransacions echnologies where he Friedman rule is opimal saisfy he local condiions for he opimaliy of he Friedman rule in he money in he uiliy funcion specificaion. The characerisic of real balances ha is deerminan for he general opimaliy of he Friedman rule is he fac ha money is a free good, meaning ha he producion cos of money is zero, i.e., he producion possibiliies in hese economies are no affeced by a change in he quaniy of money. For his reason, he usual inuiion of he comparison of he marginal excess burdens of alernaive axes ha give he same revenue no longer applies. The opimal decision here consiss of he following comparison: an increase in he quaniy of money generaes a benefi for he households in erms of uiliy and a cos equal o he value of he marginal effec on governmen ne revenues. A he poin of saiaion in real balances, he marginal uiliy benefi is by definiion equal o zero. We show ha under reasonable preferences specificaions, he marginal impac on governmen ne revenues is also equal o zero, a ha poin of full liquidiy. Therefore he Friedman rule is opimal. In less adequae specificaions for preferences, in erms of is microfoundaions, where he Friedman rule is no opimal, i is neverheless very close o he opimum. These are cases where he opimal implici ad valorem ax is infiniy. The main conclusion of his paper is ha he opimal axaion resuls in moneary models are much more robus han he public finance resuls derived in oher economic environmens. In paricular, he Friedman rule, i.e., a zero inflaion ax, is a general resul for moneary economic srucures wih reasonable microfoundaions. This normaive resul has no counerpar in he public finance lieraure where he opimal policies depend on he srucure of preferences and echnologies. The nea and

21 THE OPTIMAL INFLATION TAX 345 successful pracical recommendaion of Friedman is reinforced now ha i is shown ha is opimaliy exends o a second bes environmen. REFERENCES Akinson, A. B., and Sigliz, J. E. Ž The Srucure of Indirec Taxaion and Economic Efficiency, Journal of Public Economics 1, Barro, R. J. Ž Inegral Consrains and Aggregaion in an Invenory Model of Money Demand, Journal of Finance 31, Baumol, W. J. Ž The Transacions Demand for Cash: An Invenory Theoreic Approach, Quarerly Journal of Economics 66, Bewley, T. Ž The Opimum Quaniy of Money, in Models of Moneary Economies ŽJ. Kareken and N. Wallace, Eds.., pp Federal Reserve Bank of Minneapolis. Braun, A. Ž Fall Anoher Aemp o Quanify he Benefis of Reducing Inflaion, Federal Reser e Bank of Minneapolis Quarerly Re iew, Brock, W. Ž A Simple Perfec Foresigh Moneary Model, Journal of Moneary Economics 1, Calvo, G., and Guidoi, P. Ž On he Flexibiliy of Moneary Policy: The Case of he Opimal Inflaion Tax, Re iew of Economic Sudies 60, Chamley, C. Ž On a Simple Rule for he Opimal Inflaion Rae in Second Bes Taxaion, Journal of Public Economics 26, Chari, V. V., Chrisiano, L. J., and Kehoe, P. J. Ž Opimaliy of he Friedman Rule in Economies wih Disoring Taxes, Journal of Moneary Economics 37, Clower, R. W. Ž A Reconsideraion of he Microfoundaions of Moneary Theory, Wesern Economics Journal 6, 1 8. Correia, I., and Teles, P. Ž Is he Friedman Rule Opimal When Money is an Inermediae Good? Journal of Moneary Economics 38, De Fiore, F., and Teles, P. Ž The Opimal Mix of Taxes on Money, Consumpion and Income, mimeograph, Lisbon, Porugal: Banco de Porugal. Diamond, P. A., and Mirrlees, J. A. Ž Opimal Taxaion and Public Producion, American Economic Re iew 63, 8 27, Drazen, A. Ž The Opimal Rae of Inflaion Revisied, Journal of Moneary Economics 5, Feensra, R. C. Ž Funcional Equivalence Beween Liquidiy Coss and he Uiliy of Money, Journal Moneary Economics 17, Friedman, M. Ž The Opimum Quaniy of Money, in The Opimum Quaniy of Money and oher Essays Ž M. Friedman, Ed.., pp Chicago: Aldine. Giovannini, A., and Labadie, P. Ž Asses Prices and Ineres Raes in Cash-In-Advance Models, Journal of Poliical Economy 99, Grandmon, J.-M., and Younes, Y. Ž On he Efficiency of a Moneary Equilibrium, Re iew of Economic Sudies 40Ž. 2, Guidoi, P. E. Ž Exchange Rae Deerminaion, Ineres Raes and an Inegraive Approach o he Demand for Money, Journal of Inernaional Money and Finance 8, Guidoi, P. E., and Vegh, C. A. Ž The Opimal Inflaion Tax When Money Reduces Transacions Coss, Journal of Moneary Economics 31,

22 346 CORREIA AND TELES Jovanovic, B. Ž Inflaion and Welfare in he Seady-Sae, Journal of Poliical Economy 90, Kimbrough, K. P. Ž The Opimum Quaniy of Money Rule in he Theory of Public Finance, Journal of Moneary Economics 18, Kiyoaki, N., and Wrigh, R. Ž On Money as a Medium of Exchange, Journal of Poliical Economy 97, Lucas, R. E., Jr. Ž Equilibrium in a Pure Currency Economy, in Models of Moneary Economies Ž J. Kareken and N. Wallace, Eds.., pp Federal Reserve Bank of Minneapolis. Lucas, R. E., Jr. Ž Ineres Raes and Currency Prices in a Two-Counry World, Journal of Moneary Economics 10, Lucas, R. E. Ž The Welfare Cos of Inflaion, mimeograph, The Universiy of Chicago. Lucas, R. E., Jr., and Sokey, N. L. Ž Opimal Fiscal and Moneary Theory in an Economy Wihou Capial, Journal of Moneary Economics 12, Marshall, D. A. Ž Inflaion and Asse Reurns in a Moneary Economy, Journal of Finance 47, McCallum, B. T. Ž The Role of Overlapping Generaions Models in Moneary Economics, Carnegie-Rocheser Conference Series on Public Policy 18, McCallum, B. T. Ž Inflaion: Theory and Evidence, in Handbook of Moneary Economics Ž B. Friedman and F. Hahn, Eds.., pp New York: Norh-Holland. McCallum, B. T., and Goodfriend, M. Ž Demand for Money: Theoreical Sudies, in The New Palgra e: A Dicionary of Economics ŽJ. Eawell, M. Milgae, and P. Newman, Eds.., pp London: MacMillan. Nicolini, J. P. Ž More on he Time Inconsisency of Opimal Moneary Policy, Journal of Moneary Economics 41, Phelps, E. S. Ž Inflaion in he Theory of Public Finance, Swedish Journal of Economics 75, Ramsey, F. P. Ž A Conribuion o he Theory of Taxaion, Economic Journal 37, Samuelson, P. A. Ž An Exac Consumpion-Loan Model of Ineres Wih or Wihou he Social Conrivance of Money, Journal of Poliical Economy 6, Sidrauski, M. Ž Inflaion and Economic Growh, Journal of Poliical Economy 75, Siegel, J. Ž Noes on Opimal Taxaion and he Opimal Rae of Inflaion, Journal of Moneary Economics 4, Svensson, L. E. O. Ž Money and Asse Prices in a Cash-In-Advance Economy, Journal of Poliical Economy 93, Tobin, J. Ž The Ineres Elasiciy of he Transacions Demand for Cash, Re iew of Economics and Saisics 38, Townsend, R. M. Ž Models of Money wih Spaially Separaed Agens, in Models of Moneary Economies Ž J. Kareken and N. Wallace, Eds.., pp Federal Reserve Bank of Minneapolis. Woodford, M. Ž The Opimum Quaniy of Money, in Handbook of Moneary Economics Ž B. Friedman and F. Hahn, Eds.., pp New York: Norh-Holland.