Foreign Demand for Domestic Currency and the Optimal Rate of Inflation

Transcription

1 STEPHANIE SCHMITT-GROHÉ MARTÍN URIBE Foreign Demand for Domesic Currency and he Opimal Rae of Inflaion We characerize he Ramsey opimal rae of inflaion in a model wih a foreign demand for domesic currency. In he absence of such demand, he model implies ha he Friedman rule deflaion a he real rae of ineres is opimal. We show analyically ha in he presence of a foreign demand for domesic currency, his resul breaks down. Calibraed versions of he model deliver opimal annual raes of inflaion beween 2% and 10%. The domesically benevolen governmen imposes an inflaion ax o exrac resources from he res of he world in he form of seignorage revenue. Keywords: JEL codes: E5, E41 foreign demand for money, opimal rae of inflaion, Ramsey equilibrium, Friedman rule. MORE THAN HALF OF U.S. currency circulaes abroad. Porer and Judson (1996), for insance, esimae ha a he end of 1995, $200 $250 billion of he $375 billion of U.S. currency in circulaion ouside of banks was held abroad. The foreign demand for U.S. currency has remained srong across ime. The 2006 Treasury, Federal Reserve, and Secre Service repor on he use of U.S. currency abroad esimaes ha as of December 2005, abou $450 billion of he $760 billion of circulaed U.S. banknoes are held in oher counries. Goldberg (2010) updaes his figure and concludes ha foreign holdings of U.S. currency had reached $580 billion in March The esimaed size of he foreign demand for U.S. currency suggess ha he majoriy of he seignorage income of he Unied Saes is generaed ouside of is borders. A second currency enjoying a srong foreign demand is he euro. A repor by he European Cenral Bank (ECB, 2011) esimaes ha in 2010 abou 25% of euro We would like o hank an anonymous referee for commens. STEPHANIESCHMITT-GROHÉ is a Professor of Economics, Deparmen of Economics, Columbia Universiy, CEPR, and NBER ( ss3501@columbia.edu). MARTÍN URIBE is a Professor of Economics, Deparmen of Economics, Columbia Universiy and NBER ( marin.uribe@columbia.edu). Received November 3, 2009; and acceped in revised form December 7, Journal of Money, Credi and Banking, Vol. 44, No. 6 (Sepember 2012) C 2012 The Ohio Sae Universiy

2 1208 : MONEY, CREDIT AND BANKING currency (or 205 billion euro) was held ouside of he euro zone. One piece of circumsanial evidence poining a he relevance ha he ECB assigns o seigniorage income generaed by foreign holdings of he euro is is decision o issue large-denominaion bills (200 and 500 euros). This decision has been inerpreed by academic researchers as an indicaion of he ECB s desire o exrac seignorage revenue from he underground economy as well as from foreign holders of euros (especially in he former Sovie block and he Middle Eas) (see, e.g., Rogoff 1998). A naural quesion is herefore wheher a counry s opimal rae of inflaion is influenced by he presence of a foreign demand for is currency. In his paper, we address his issue wihin he conex of a dynamic Ramsey problem. We show ha he mere exisence of a foreign demand for domesic money can, under plausible parameerizaions, jusify sizable deviaions from he rae of inflaion associaed wih he Friedman rule. The basic inuiion behind his finding is ha adherence o he negaive rae of inflaion associaed wih he Friedman rule would represen a welfare-decreasing ransfer of real resources by he domesic economy o he res of he world, as nominal money balances held abroad increase in real erms a he rae of deflaion. A benevolen governmen weighs his cos agains he benefi of keeping he opporuniy cos of holding money low o reduce ransacions coss for domesic agens. Our analyical resuls show ha his rade-off is resolved in favor of deviaing from he Friedman rule. Our quaniaive analysis suggess ha for plausible calibraions ha capure he range of esimaes of he size of he foreign demand for U.S. currency, he opimal rae of inflaion lies beween 2% and 10% per year. The reason why he Ramsey governmen finds i opimal o collec seignorage revenues from he res of he world is no he fac ha such revenues allow he fiscal auhoriy o lower disorionary axes. Raher, i is he fac ha he imposiion of an inflaion ax allows he domesic governmen o engineer an indirec ransfer of real resources from foreign consumers o domesic consumers. We highligh his incenive by esablishing ha in he presence of a foreign demand for domesic currency, he Friedman rule is subopimal even when he domesic governmen has access o lump-sum axaion. The res of he paper is organized as follows. Secion 1 presens a dynamic moneary model wih a foreign demand for domesic currency. Secion 2 esablishes ha he Friedman rule is opimal in he absence of a foreign demand for domesic currency and ha i fails o be opimal in he presence of such demand. Secion 3 provides he esimaes of he opimal rae of inflaion in he conex of a calibraed version of he model. Secion 4 demonsraes ha deviaions from he Friedman rule are opimal even when he domesic governmen has access o lump-sum axaion. Secion 5 concludes. 1. THE MODEL Consider an economy populaed by a large number of idenical households. Each household has preferences defined over sequences of consumpion and leisure and

3 STEPHANIE SCHMITT-GROHÉ AND MARTÍN URIBE : 1209 described by he uiliy funcion β U(c, h ), (1) =0 where c denoes he consumpion, h denoes he labor effor, and β (0, 1) denoes he subjecive discoun facor. The single-period uiliy funcion U is assumed o be increasing in consumpion, decreasing in effor, and sricly concave. A domesic demand for real balances is inroduced ino he model by assuming ha nominal money holdings, denoed by M d, faciliae consumpion purchases. Specifically, consumpion purchases are subjec o a proporional ransacion cos s(v ) ha is decreasing in he household s money-o-consumpion raio, or consumpion-based money velociy, v = P c M d, (2) where P denoes he nominal price of he consumpion good in period. The ransacion cos funcion, s(v), saisfies he following assumpions: (i) s(v) is nonnegaive and wice coninuously differeniable; (ii) here exiss a level of velociy v > 0, o which we refer as he saiaion level of money, such ha s(v) = s (v) = 0; (iii) (v v)s (v) > 0forv v; and (iv) 2s (v) + vs (v) > 0 for all v v. Assumpion (ii) ensures ha he Friedman rule, ha is, a zero nominal ineres rae, need no be associaed wih an infinie demand for money. I also implies ha boh he ransacion cos and he disorion i inroduces vanish when he nominal ineres rae is zero. Assumpion (iii) guaranees ha in equilibrium money, velociy is always greaer han or equal o he saiaion level. Assumpion (iv) ensures ha he demand for money is a decreasing funcion of he nominal ineres rae. Households are assumed o have access o one-period nominal bonds, denoed as B, which carry a gross nominal ineres rae of R when held from period o period + 1. Households supply labor services o compeiive labor markes a he real wage rae w. In addiion, households receive profi income in he amoun from he ownership of firms, and pay income axes a he fla rae τ. The flow budge consrain of he household in period is hen given by P c [1 + s(v )] + M d + B = M d 1 + R 1 B 1 + P (1 τ )(w h + ). (3) In addiion, i is assumed ha he household is subjec o he following borrowing limi ha prevens i from engaging in Ponzi-ype schemes: M+ d j lim + R + j B + j j j s=0 R +s 0. (4)

4 1210 : MONEY, CREDIT AND BANKING This resricion saes ha in he long run, he household s ne nominal liabiliies mus grow a a rae smaller han he nominal ineres rae. I rules ou, for example, schemes in which households roll over heir ne debs forever. The household chooses sequences {c, h,v, M d, B } =0 o maximize (1) subjec o (2) (4), aking as given he sequences {P,τ, R,w, } =0 and he iniial condiion M 1 d + R 1 B 1. The firs-order condiions associaed wih he household s maximizaion problem are (2), (3), and (4) holding wih equaliy, and v 2 s (v ) = R 1 R, (5) U h(c, h ) U c (c, h ) = (1 τ )w 1 + s(v ) + v s (v ), (6) U c (c, h ) 1 + s(v ) + v s (v ) = β R P P +1 U c (c +1, h +1 ) [1 + s(v +1 ) + v +1 s (v +1 )]. (7) Opimaliy condiion (5) can be inerpreed as a domesic demand for money or a domesic liquidiy preference funcion. Given our mainained assumpions abou he ransacions echnology s(v ), he implied domesic money demand funcion is decreasing in he gross nominal ineres rae R. Furher, our assumpions imply ha as he ineres rae vanishes, or R approaches uniy, he domesic demand for money reaches a finie maximum level given by c /v. A his level of money demand, households are able o perform ransacions coslessly, as he ransacions cos, s(v ), becomes nil. Opimaliy condiion (6) shows ha a level of money velociy above he saiaion level v, or, equivalenly, an ineres rae greaer han zero, inroduces a wedge, given by 1 + s(v ) + v s (v ), beween he marginal rae of subsiuion of consumpion for leisure and he real wage rae. In addiion, he labor supply disorion has a ax componen given by 1 τ, making he oal wedge beween he marginal rae of subsiuion of leisure for consumpion and he real wage rae equal o (1 τ )/[1 + s(v ) + v s (v )]. This wedge induces households o move o an inefficien allocaion feauring oo much leisure and oo lile consumpion. The wedge is increasing in he nominal ineres rae and in he income ax rae, implying ha he larger is he nominal ineres rae or he income ax rae, he more disored is he consumpion leisure choice. Opimaliy condiion (7) is a Fisher equaion, saing ha he nominal ineres rae mus be equal o he sum of he expeced rae of inflaion and he real rae of ineres. I is clear from he Fisher equaion ha ineremporal movemens in he nominal ineres rae creae a disorion in he real ineres rae perceived by households. Final goods are produced by compeiive firms using he echnology F(h ) ha akes labor as he only facor inpu. The producion funcion F is assumed o be increasing and concave. Firms choose labor inpu o maximize profis, which are

5 STEPHANIE SCHMITT-GROHÉ AND MARTÍN URIBE : 1211 given by = F(h ) w h. The firs-order condiion associaed wih he firm s profi maximizaion problem gives rise o he following demand for labor F (h ) = w. (8) The res of he world demands domesic nominal money balances in he amoun M f. We assume ha he demand by foreigners for real domesic currency is a funcion of he level of foreign aggregae absorpion, denoed as c f, and he domesic nominal ineres rae: M f P = c f L(R, x ), (9) where he funcion L is assumed o be decreasing in is firs argumen. The variable x is a vecor of shifers assumed o be exogenous. 1 Le he velociy of foreign-held domesic currency, denoed by v f, be given by v f = P c f M f. The governmen prins money, issues nominal one-period bonds, and levies axes o finance an exogenous sream of public consumpion, denoed as g, and ineres obligaions on he ousanding public deb. Accordingly, he governmen s sequenial budge consrain is given by (10) M d + M f + B = M d 1 + M f 1 + R 1 B 1 + P g P τ F(h ). (11) Implici in he sequenial budge consrain of he governmen is he assumpion ha he governmen s consumpion ransacions are no subjec o a moneary fricion like he one imposed on privae purchases of goods. Combining he household s and he governmen s sequenial budge consrains yields he following aggregae resource consrain: [1 + s(v )]c + g = F(h ) + M f M f 1 P. (12) I is clear from his resource consrain ha he domesic economy collecs seignorage revenue from foreigners whenever nominal money balances held by foreigners increase. 1. In an online appendix, available on he websie of he Journal of Money, Credi and Banking and on ha of he auhors, we provide some microfoundaions for his specificaion of he foreign demand for domesic currency.

6 1212 : MONEY, CREDIT AND BANKING A compeiive equilibrium is a se of sequences {v, w, v f, c, h, M d, M f, B, P } =0 saisfying (2) and (4) holding wih equaliy, (5) (12), and R 1, (13) given policies {R,τ } =0, he exogenous sequences {g, c f } =0, and he iniial condiions M 1 d + R 1 B 1 > 0 and M f 1. Equilibrium condiion (13) imposes a zero lower bound on he nominal ineres rae. Such a bound is required o preven he possibiliy of unbounded arbirage profis creaed by aking shor posiions in nominal bonds and long posiions in nominal fia money, which would resul in ill-defined demands for consumpion goods by households. Our primary goal is o characerize he Ramsey opimal rae of inflaion. To his end, we begin by deriving he primal form of he compeiive equilibrium. Then, we sae he Ramsey problem. And finally, we characerize opimal fiscal and moneary policy. 1.1 The Primal Form of he Compeiive Equilibrium Following a long-sanding radiion in public finance, we sudy opimal policy using he primal-form represenaion of he compeiive equilibrium. Finding he primal form involves he eliminaion of all prices and ax raes from he equilibrium condiions so ha he resuling reduced form involves only real variables. In our economy, he real variables ha appear in he primal form are consumpion, hours, and domesic and foreign money velociy. The primal form of he equilibrium condiions consiss of wo equaions. One equaion is a feasibiliy consrain, given by he resource consrain (12), which mus hold a every dae. The oher equaion is a single, presen-value consrain known as he implemenabiliy consrain. The implemenabiliy consrain guaranees ha a he prices and quaniies associaed wih every possible compeiive equilibrium, he presen discouned value of consolidaed governmen surpluses equals he governmen s oal iniial liabiliies. Combining equaions (5), (9), and (10), one can express foreign money velociy as a funcion of domesic money velociy alone: v f = χ(v ), (14) where we have omied he exogenous argumen x, which will be regarded as consan hroughou he analysis. The following proposiion provides he primal form of he compeiive equilibrium. PROPOSITION 1. (Primal Form of he Compeiive Equilibrium). Given he iniial condiions (R 1 B 1 + M d 1 ) and M f 1 and he iniial price level P 0, sequences

7 STEPHANIE SCHMITT-GROHÉ AND MARTÍN URIBE : 1213 {c, h,v } =0 saisfy he feasibiliy condiion [1 + s(v 0 )]c 0 + g 0 = F(h 0 ) + c f 0 χ(v 0 ) M f 1 (15) P 0 in period 0 and [1 + s(v )]c + g = F(h ) + c f χ(v ) c f 1 ( 1 v 2 χ(v 1 ) 1 s (v 1 ) ) U c(c 1, h 1 ) γ (v 1 ) for all > 0, he implemenabiliy consrain γ (v ) βu c (c, h ), (16) and β U c (c 0, h 0 ) R 1 B 1 + M 1 d {U c (c, h )c + U h (c, h )h } =, 1 + s(v =0 0 ) + v 0 s (v 0 ) P 0 (17) v v and v 2 s (v ) < 1, if and only if hey also saisfy he se of equilibrium condiions (2) and (4) holding wih equaliy, and (5) (13). PROOF: See he Appendix. 2. THE RAMSEY EQUILIBRIUM The governmen is assumed o be benevolen oward domesic residens. This means ha he welfare funcion of he governmen coincides wih he lifeime uiliy of he domesic represenaive agen, and ha i is independen of he level of uiliy of foreign residens. The Ramsey problem consiss in choosing a se of sricly posiive sequences {c, h,v } =0 o maximize he uiliy funcion (1) subjec o (15), (16), and (17), v v, and v 2s (v ) < 1, given R 1 B 1 + M 1, M f 1, and P 0.Wefixhe iniial price level arbirarily o keep he Ramsey planner from engineering a large unexpeced iniial inflaion aimed a reducing he real value of predeermined nominal governmen liabiliies. This assumpion is regularly mainained in he lieraure on opimal moneary and fiscal policy. Wrie he feasibiliy consrain (16) as H(c, c 1, h, h 1,v,v 1 ) = 0 and he implemenabiliy consrain (17) as =0 β K (c, h ) = A(c 0, h 0,v 0 ). Le he Lagrange muliplier on he feasibiliy consrain (16) be denoed by ψ, he Lagrange muliplier on he implemenabiliy consrain (17) be denoed by λ, and he Lagrange muliplier on he consrain v v be denoed by μ. Then, for any > 0, he

8 1214 : MONEY, CREDIT AND BANKING firs-order condiions of he Ramsey problem are U c (c, h ) + λk c (c, h ) + ψ H 1 (c, c 1, h, h 1,v,v 1 ) + βψ +1 H 2 (c +1, c, h +1, h,v +1,v ) = 0, (18) U h (c, h ) + λk h (c, h ) + ψ H 3 (c, c 1, h, h 1,v,v 1 ) + βψ +1 H 4 (c +1, c, h +1, h,v +1,v ) = 0, (19) ψ H 5 (c, c 1, h, h 1,v,v 1 ) + βψ +1 H 6 (c +1, c, h +1, h,v +1,v ) + μ = 0, (20) (v v)μ = 0; μ 0; v v. (21) We do no include he consrain v 2 s (v ) < 1 in he Lagrangian. Therefore, we mus check ha he soluion o he above sysem saisfies his consrain. 2.1 Opimaliy of he Friedman Rule in he Absence of a Foreign Demand for Money When he foreign demand for domesic currency is nil, M f = 0, any policy oher han he Friedman rule fails o be Ramsey opimal. To see his, noe ha when M f = 0 (or, equivalenly, when c f /χ(v ) = 0), he firs-order condiion of he Ramsey problem wih respec o v, equaion (20), becomes ψ s (v )c + μ = 0. Consider any level of velociy v greaer han v, he level called for by he Friedman rule. Our assumpions regarding he ransacions cos echnology imply ha s (v) > 0 for any v >v. Also, he fac ha he period uiliy funcion U is sricly increasing implies ha ψ > 0. I hen follows from he above expression ha μ is sricly posiive when v >v. This resul and he fac ha, by assumpion, v >v,imply ha opimaliy condiion (21) is violaed. We conclude ha in he case of no foreign demand for domesic currency, if a Ramsey equilibrium exiss, hen i mus be characerized by a zero nominal ineres rae for all > 0. This is a sandard resul in he heory of opimal moneary and fiscal policy. 2.2 Failure of he Friedman Rule in he Presence of a Foreign Demand for Money We now presen he main resul of his paper, namely, ha he Friedman rule ceases o be Ramsey opimal in he presence of a foreign demand for domesic currency. To faciliae he exposiion, we resric aenion o he seady sae of he Ramsey equilibrium. Tha is, we resric aenion o soluions o (16) and (18) (21) in which he endogenous variables c, h, v, ψ, and μ are consan given consan levels for

9 STEPHANIE SCHMITT-GROHÉ AND MARTÍN URIBE : 1215 he exogenous variables g and c f. To esablish he failure of he Friedman rule when M f > 0, we show ha a Ramsey equilibrium in which v equals v is impossible. In he seady sae, he opimaliy condiion (20) when evaluaed a v = v becomes: c f χ(v) s (v)v (1 1β + v ) + μ = 0. Under our mainained assumpions regarding he ransacions cos echnology, s (v) is posiive. Also, under reasonable calibraions, he consan 1/β 1, which equals he seady-sae real ineres rae, is smaller han he velociy level v. Then, he firs erm in he above expression is posiive. This implies ha he muliplier μ mus be negaive, which violaes opimaliy condiion (21). 2 We conclude ha in he presence of a foreign demand for domesic currency, if a Ramsey equilibrium exiss, i involves a deviaion from he Friedman rule. The inuiion behind his resul is ha he presence of a foreign demand for domesic currency inroduces an incenive for he fiscal auhoriy o inflae in order o exrac resources, in he form of seignorage, from he res of he world (whose welfare does no ener he domesic planner s objecive funcion). Indeed, a any negaive inflaion rae (and, mos so a he level of inflaion consisen wih he Friedman rule), he domesic counry acually derives negaive seignorage income from he res of he world, because foreign money holdings increase in real value as he price level falls. On he oher hand, levying an inflaion ax on foreign money holdings comes a he cos of axing domesic money holdings as well. In urn, he domesic inflaion ax enails a welfare loss, because domesic households mus pay elevaed ransacion coss as hey are forced o economize on real balances. Thus, he Ramsey planner faces a rade-off beween axing foreign money holdings and disoring he domesic real allocaion. We have demonsraed analyically ha he resoluion of his rade-off leads o an inflaion rae above he one called for by Friedman s rule. We now urn o he quesion of how large he opimal deviaion from he Friedman rule is under a plausible calibraion of our model. 3. QUANTIFYING DEVIATIONS FROM THE FRIEDMAN RULE To gauge he quaniaive implicaions of a foreign demand for money for he opimal rae of inflaion, we parameerize he model and solve numerically for he seady sae of he Ramsey equilibrium. We adop he following funcional forms for he period uiliy funcion, he ransacions cos echnology, and he foreign demand 2. One may argue ha he assumpion 2s (v) + vs (v) > 0 for all v v, which implies ha he nominal ineres rae is a sricly increasing funcion of v for all v v and, in paricular, ha he elasiciy of he liquidiy preference funcion a a zero nominal ineres rae is finie is oo resricive. Suppose, insead, ha he assumpion in quesion is relaxed by assuming ha i mus hold only for v>vbu no a v = v. In his case, a poenial soluion o he firs-order condiion of he Ramsey problem wih respec o v is v = v, provided ha s (v) = 0.

10 1216 : MONEY, CREDIT AND BANKING for domesic money: U(c, h) = ln(c) + θ ln(1 h); θ>0, s(v) = Av + B/v 2 AB, and χ(v) = v. The assumed ransacions cos funcion implies ha he saiaion level of velociy is v = B/A and a demand for money of he form M d P = c. B A + 1 R 1 A R The assumed form for he funcion χ implies idenical relaionships beween he nominal ineres rae and domesic-money velociy in he domesic and he foreign economies. We follow Schmi-Grohé and Uribe (2004) and se β = 1/1.04, θ = 2.90, B = , and g = 0.04 for all, which implies a share of governmen spending of abou 20%. 3 We se c f = 0.06 and A = o mach he empirical regulariies ha abou 50% of he U.S. currency (or abou 26% of M1) is held ouside of he Unied Saes and ha he M1-o-consumpion raio is abou 29%. 4 Finally, we se he level of deb in he Ramsey seady sae o 20% of GDP. 5 We develop a numerical algorihm ha delivers he exac soluion o he seady sae of he Ramsey equilibrium. The mechanics of he algorihm are as follows: (i) Pick a posiive value of λ. (ii) Given his value of λ, solve he nonlinear sysem (16) and (18) (21) for c, h, v, ψ, and μ. (iii) Calculae w from (8), τ from (6), R from (5), π from (7), v f from (14), M d/p from (2), and M f /P from (10). (iv) Calculae he seady-sae deb-o-oupu raio, which we denoe by s d B /(P y ), from (11), aking ino accoun ha y = h. (v)ifs d is larger han he calibraed value of 0.2, 3. To idenify he parameer B, in Schmi-Grohé and Uribe (2004), we esimae he implied money demand equaion v 2 = B/A + 1/A(R 1)/R on quarerly U.S. poswar daa. We measure v as he raio of nondurable consumpion and services expendiures o M 1,andR as he 3-monh T-bill rae. We esimae he money demand equaion using OLS and IV echniques. 4. For an esimae of he amoun of U.S. currency circulaing abroad, see he join press release of he Board of Governors of he Federal Reserve Sysem and he Deparmen of he Treasury of Ocober 25, 2006, available online a 5. This deb level implies ha he pre-ramsey reform deb-o-oupu raio in he economy wihou a foreign demand for domesic currency and wih a prereform inflaion rae of 4.2% is abou 44%. The reason why he Ramsey seady-sae level of deb is much lower han he pre-ramsey-reform level is ha he reform induces a drop in expeced inflaion of abou 8%, which causes a large asse subsiuion away from governmen bonds and oward real money balances. The overall level of governmen liabiliies (money plus bonds) is relaively unaffeced by he Ramsey reform.

11 STEPHANIE SCHMITT-GROHÉ AND MARTÍN URIBE : 1217 TABLE 1 RAMSEYPOLICY WITH FOREIGNDEMAND FORDOMESTICCURRENCY M f M f +M d M f +M d Pc π R τ No foreign demand: c f = Baseline calibraion: c f = Higher foreign demand: c f = Low domesic demand: A = High ineres elasiciy: B = B High deb-o-oupu raio: = Py Lump-sum axes Lump-sum axes and g = NOTE: The baseline calibraion is: A = , B = , B Py = 0.2, and c f = The ineres rae, R, and he inflaion rae, π, are expressed in percen per annum, and he income ax rae, τ, is expressed in percen. lower λ. If, insead, s d is smaller han he calibraed value of 0.2, hen increase he value of λ. (vi) Repea seps (i) (v) unil s d has converged o is calibraed value. Table 1 presens our numerical resuls. The firs row of he able shows ha when foreign demand for domesic currency is nil, which we capure by seing c f = 0, hen as we have shown analyically above he Friedman rule is Ramsey opimal; ha is, he nominal ineres rae is zero in he seady sae of he Ramsey equilibrium. The inflaion rae is 3.85% and he income ax rae is abou 18%. In his case, because he foreign demand for domesic currency is nil, he domesic governmen has no incenives o levy an inflaion ax, as i would generae no revenues from he res of he world. The second row of he able considers he case ha he foreign demand for domesic currency is posiive. In paricular, we se c f = 0.06 and obain ha in he Ramsey seady sae, he raio of foreign currency o oal money is 22% and ha oal money holdings are 26% of consumpion. Boh figures are broadly in line wih observaion in he U.S. economy. The able shows, again in line wih he analyical resuls obained above, ha he Ramsey opimal rae of ineres is posiive; ha is, he Friedman rule is no longer opimal. Of greaer ineres, however, is he size of he deviaion from he Friedman rule. The able shows ha he Ramsey opimal inflaion rae is 2.10% per year abou 6 percenage poins higher han he value ha obains in he absence of a foreign demand for domesic currency. The opimal rae of ineres now is 6.2%. When we increase foreign demand for domesic currency by assuming a larger value of foreign demand, c f = 0.1, hen he share of foreign holdings of domesic currency in oal money increases by 10 percenage poins o 0.32 and he Ramsey opimal inflaion rae is more han 10% per year. In his calibraion, he benefi from collecing an inflaion ax from foreign holdings of currency appears o srongly dominae he coss ha such a high inflaion ax represens for domesic agens in erms of a more disored consumpion leisure choice and elevaed ransacion coss. The larger inflaion ax revenue of he governmen relaxes he budge consrain of he governmen allowing for a decline in he Ramsey opimal ax rae of abou 1.5 percenage poins.

12 1218 : MONEY, CREDIT AND BANKING Row 4 of Table 1 considers a calibraion ha implies a weaker demand for money boh domesically and abroad. Specifically, we lower he coefficien A in he ransacions cos funcion by a facor of 4. Because he demand for money is proporional o he square roo of A, his parameer change implies ha he raio of money o consumpion falls by a facor of 2. In he Ramsey seady sae, he money-o-consumpion raio falls from 26% o 13%. The relaive imporance of foreign demand for money is unchanged. I coninues o accoun for 22% of oal money demand. The opimal rae of inflaion is virually he same as in he baseline case. The reason why he inflaion ax is virually unchanged in his case is ha he reducion in A induces proporional declines in boh he domesic and he foreign demands for domesic currency. The decline in foreign money demand is equivalen o a decline in c f and herefore induces he Ramsey planner o lower he rae of inflaion. A he same ime, he decline in he domesic demand for money reduces he cos of inflaion for domesic agens, inducing he Ramsey planner o inflae more. In our parameerizaion, hese wo opposing effecs happen o offse each oher almos exacly. Row 5 of Table 1 analyzes he sensiiviy of our resuls o raising he ineres elasiciy of money demand. Under a higher ineres elasiciy, he Ramsey opimal rae of ineres and inflaion are lower han in he baseline case. The nominal ineres rae falls from 6% o 3% and he inflaion rae falls from abou 2% o 1%. In his case, while he Ramsey policy deviaes from he Friedman rule, he deviaion is no large enough o render posiive inflaion Ramsey opimal. The las row of he able shows ha our resuls are very lile changed when we increase he seadysae deb level. We conclude from he resuls presened in Table 1 ha he rade-off beween collecing seignorage from foreign holders of domesic currency and keeping he opporuniy cos of holding money low for domesic agens is overwhelmingly resolved in favor of collecing seignorage income from foreign holdings of domesic currency. The numerical resuls of his secion sugges ha an inflaion arge of abou 2% per annum may be raionalized on he basis of an incenive o ax foreign holdings of domesic currency. This argumen could, in principle, be raised o explain average raes of inflaion in counries such as he Unied Saes, which we used as a poin of reference in our calibraion, or in he euro area, whose currency is held widely in easern Europe, Russia, and cerain pars of Asia minor. However, he fac ha a number of developed counries whose currency is no used ouside of heir geographic borders, such as Ausralia, Canada, and New Zealand, also mainain inflaion arges of abou 2% per year, indicae ha he reason why inflaion arges in he developed world are as high as observed may no exclusively originae from he desire o exrac seignorage revenue from foreigners. Our invesigaion calls aenion o he fac ha, all oher hings equal, counries whose currencies are demanded ouside of heir borders should have greaer incenives o deviae from he Friedman rule han counries whose currencies are locally circumscribed. We also view our finding as a challenge for fuure research o ascerain why counries whose monies are widely held abroad do no appear o exploi his margin fully.

13 STEPHANIE SCHMITT-GROHÉ AND MARTÍN URIBE : LUMP-SUM TAXATION The reason why he benevolen governmen finds i desirable o deviae from he Friedman rule in he presence of a foreign demand for currency is no enirely o finance is budge wih seignorage revenue exraced from foreign residens. Raher, he governmen imposes an inflaion ax on foreign residens o increase he oal amoun of resources available o domesic residens for consumpion. To show ha his is indeed he correc inerpreaion of our resuls, we now consider a variaion of he model in which he governmen can levy lump-sum axes on domesic residens. Specifically, we assume ha he labor income ax rae τ is zero a all imes, and ha he governmen ses lump-sum axes o ensure fiscal solvency. A compeiive equilibrium is hen given by sequences {v, v f, c, h, M d, M f, P } =0 saisfying (2), (5), (7), (9), (10), (12), (13), and U h(c, h ) U c (c, h ) = s(v ) + v s (v ), (22) given an ineres rae sequence {R } f =0, and he exogenous sequences {c, g } =0. One can show ha, given he iniial condiion M f 1 and he iniial price level P 0, sequences {c, h,v } =0 saisfy he feasibiliy condiions (15) and (16), he labor supply equaion (22), and v v and v 2 s (v ) < 1, if and only if hey also saisfy he se of equilibrium condiions (2), (5), (7), (9), (10), (12), (13), and (22). This primal form is essenially he same as he one associaed wih he economy wih disorionary axes and governmen spending excep ha he implemenabiliy consrain is replaced by equaion (22), which saes ha in equilibrium, labor demand mus equal labor supply. Noing ha equaion (22) appears in boh he sandard and he primal forms of he compeiive equilibrium, i follows ha he proof of he above saemen is a simplified version of he one presened in he Appendix. The Ramsey problem hen consiss in maximizing he uiliy funcion (1) subjec o he feasibiliy consrains (15) and (16) and he resricions v v and v 2 s (v ) < 1, given P 0 and M f 1. Row 7 of Table 1 presens he seady sae of he Ramsey equilibrium in he economy wih lump-sum axes. All parameers of he model are calibraed as in he economy wih disorionary axes. The able shows ha he opimal rae of inflaion equals 0.85%. This means ha he presence of a foreign demand for money gives rise o an opimal inflaion bias of abou 5 percenage poins above he level of inflaion called for by he Friedman rule. This inflaion bias emerges even hough he governmen can resor o lump-sum axes o finance is budge. The opimal inflaion bias is smaller han in he case wih disorionary axes. This is because disorionary axes, hrough heir depressing effec on employmen and oupu, make

14 1220 : MONEY, CREDIT AND BANKING he pre-foreign-seignorage level of consumpion lower, raising he marginal uiliy of wealh, and as a resul providing bigger incenives for he exracion of real resources from he res of he world. The las row of Table 1 displays he seady sae of he Ramsey equilibrium in he case in which governmen consumpion equals zero a all imes (g = 0 for all ). All oher hings equal, he domesic economy has access o a larger amoun of resources han in he economy wih posiive governmen consumpion. As a resul, he governmen has fewer incenives o collec seignorage income from he res of he world. This is refleced in a smaller opimal rae of inflaion of 0.59%. I is remarkable, however, ha even in he absence of disorionary axes and governmen expendiures, he governmen finds i opimal o deviae from he Friedman rule by abou 4.5 percenage poins. This resul clearly shows ha he ulimae purpose of posiive ineres raes in he presence of a foreign demand for money is he exracion of real resources from he res of he world for privae domesic consumpion. 5. CONCLUSION In his paper, we demonsrae ha he presence of a foreign demand for domesic currency of he size observed for he U.S. dollar can inroduce incenives for he moneary auhoriy o generae posiive raes of inflaion. The inflaion rae acs as a ax on foreign holdings of domesic currency ha allows he domesic governmen o effecively exrac real resources from he res of he world. In our model, he Ramsey planner weighs his incenive agains he cos ha inflaion causes o domesic households. In he absence of a foreign demand for money, he opimal policy calls for adoping Friedman s rule, or deflaing a he real rae of ineres. We find ha for plausible calibraions of our model, he rade-off beween axing he res of he world and keeping domesic ransacions coss low is resolved in favor of axing foreign holdings of domesic currency a raes ranging from 2% o 10% per year. APPENDIX: PROOF OF PROPOSITION 1 We firs show ha plans {c, h,v } saisfying he equilibrium condiions (2) and (4) holding wih equaliy, and (5) (13) also saisfy (16) and (17), v v, and v 2s (v ) < 1. Le γ (v ) 1 + s(v ) + v s (v ). Noe ha (5), (13), and our mainained assumpions regarding s(v) ogeher imply ha v v and v 2s (v ) < 1. Le W +1 = R B + M d + M f. Use his expression o eliminae B from (11) and muliply by q 1 s=0 R 1 s o obain q ( M d + M f )( ) 1 R 1 + q+1 W +1 q W = q [P g τ P F(h )].

15 STEPHANIE SCHMITT-GROHÉ AND MARTÍN URIBE : 1221 Sum for = 0o = T o obain =0 [ q ( M d + M f )( ) 1 R 1 q (P g τ P F(h )) ] = q T +1 W T +1 + W 0. In wriing his expression, we define q 0 = 1. Solve (6) for τ and (8) for w and use F(h) = h o obain τ F(h ) = h + U h(c,h ) U c (c,h ) γ (v )h. Use his expression o eliminae τ F(h ) from he above equaion, which yields =0 { q ( M d + M f = q T +1 W T +1 + W 0. [ [ )( ) 1 R 1 q P g h + U ]]} h(c, h ) U c (c, h ) γ (v )h Use he feasibiliy consrain (12) o replace h g wih [1 + s(v )]c M f M f 1 P. { M d q P =0 M f + M f P Use (2) and (5) o replace Md P =0 ( ) 1 R 1 + [1 + s(v )]c M f 1 + U } h(c, h ) P U c (c, h ) γ (v )h = q T +1 W T +1 + W 0. (1 R 1 ) wih v s (v )c { q P v s (v )c M f + [1 + s(v )]c + M } f 1 + U h(c, h ) P R P U c (c, h ) γ (v )h = q T +1 W T +1 + W 0. Collec erms in c and replace 1 + s(v ) + v s (v ) wih γ (v ) and rearrange { q P γ (v )c + U h(c, h ) U c (c, h ) γ (v )h M f + M f P R =0 = q T +1 W T +1 + W 0. 1 P Noing ha by definiion q /R = q +1, wrie he above expression as =0 { q P γ (v )c + U } h(c, h ) U c (c, h ) γ (v )h + = q T +1 W T +1 + W 0. =0 } ( M f 1 q M f q +1 )

16 1222 : MONEY, CREDIT AND BANKING Evaluae he second sum on he lef-hand side and recall ha by definiion q 0 = 1o obain =0 { q P γ (v )c + U } h(c, h ) U c (c, h ) γ (v )h + M f 1 M f T q T +1 = q T +1 W T +1 + W 0. Using he definiion of W, we can wrie he above expression as =0 { q P γ (v )c + U } h(c, h ) U c (c, h ) γ (v ( ) )h = q T +1 RT B T + MT d + R 1 B 1 + M d 1. (A1) Take limis for T. Then by (4), holding wih equaliy, he limi of he righ-hand side is well defined and equal o R 1 B 1 + M 1 d. Thus, he limi of he lef-hand side exiss. This yields: =0 { q P γ (v )c + U } h(c, h ) U c (c, h ) γ (v )h = R 1 B 1 + M 1 d. By (7), we have ha P q = β U c (c, h )/γ (v )P 0 /U c (c 0, h 0 )γ (v 0 ). Use his expression o eliminae P q from he above equaion o obain ( ) ( ) β Uc (c 0, h 0 ) R 1 B 1 + M 1 d [U c (c, h )c + U h (c, h )h ] =, γ (v 0 ) =0 which is (17). We nex show ha he compeiive equilibrium condiions imply (16). For = 0, equaion (16) follows direcly from (10) and (14). For > 0, use (10) o eliminae M f and M f 1 from (12) o obain: [1 + s(v )]c + g = F(h ) + c f v f Now, use (7) o eliminae π. This yields: [1 + s(v )]c + g = F(h ) + c f χ(v ) c f 1 1 v f. π 1 c f 1 U c (c 1, h 1 ) χ(v 1 ) R 1 γ (v 1 ) P 0 γ (v ) βu c (c, h ), Using (5) o replace R 1 yields (16). This complees he proof ha he compeiive equilibrium condiions imply he primal form condiions. We now show ha plans {c, h,v } saisfying (16), (17), v v, and v 2 s (v ) < 1 also saisfy he equilibrium condiions (2) and (4) holding wih equaliy, and (5) (13).

17 STEPHANIE SCHMITT-GROHÉ AND MARTÍN URIBE : 1223 Givenaplan{c, h,v } proceed as follows. Use (5) o consruc R and (9) o consruc v f. Noe ha under he mainained assumpions on s(v), he consrains v v and v 2s (v ) < 1 ensure ha R 1. Le w be given by (8) and τ by (6). To consruc plans for M d, M f, P +1, and B,for 0, use he following ieraive procedure: (i) se = 0; (ii) use equaion (2) o consruc M d and equaion (10) o consruc M f (recall ha P 0 is given); (iii) se B so as o saisfy equaion (11); (iv) se P +1 o saisfy (7); (v) increase by 1 and repea seps (i) (v). To show ha (12) holds use (16). Combining (10) and (14) wih (15), i is obvious ha (12) holds for = 0. To show ha i also holds for > 0, combine (10), (14), and (16) o obain: [1 + s(v )]c + g = F(h ) + M f P U c(c 1, h 1 ) γ (v 1 ) Using (5), one can wrie his expression as [1 + s(v )]c + g = F(h ) + M f P γ (v ) βu c (c, h ). M f 1 P 1 ( 1 v 2 1 s (v 1 ) ) γ (v ) βu c (c, h ). M f 1 (1/R 1 ) U c(c 1, h 1 ) P 1 γ (v 1 ) Finally, combining his expression wih (7) yields (12). I remains o be shown ha (4) holds wih equaliy. Follow he seps shown above o arrive a equaion (A1). Noice ha hese seps make use only of equilibrium condiions ha we have already shown are implied by he primal form. Now use (7) o replace P q wih β U c (c, h )/γ (v )P 0 /U c (c 0, h 0 )γ (v 0 ) o obain β ( ) ( ) [U c (c, h )c + U h (c, h )h ] = q T +1 RT B T + MT d U c (c 0, h 0 ) P =0 0 γ (v 0 ) ( ) ( ) Uc (c 0, h 0 ) R 1 B 1 + M 1 d +. γ (v 0 ) Taking limi for T, recalling he definiion of q, and using (17) yield (4) holding wih equaliy. This complees he proof. P 0 LITERATURE CITED European Cenral Bank. (2011) The Inernaional Role of he Euro. Goldberg, Linda S. (2010) Is he Inernaional Role of he Dollar Changing? Curren Issues in Economics and Finance, Federal Reserve Bank of New York, 16 (1) 1 7.

18 1224 : MONEY, CREDIT AND BANKING Porer, Richard D., and Ruh A. Judson. (1996) The Locaion of U.S. Currency: How Much Is Abroad? Federal Reserve Bullein, 82 (10), Rogoff, Kenneh. (1998) Blessing or Curse? Foreign and Underground Demand for Euro Noes, Economic Policy, 13, Schmi-Grohé, Sephanie, and Marín Uribe (2004) Opimal Fiscal and Moneary Policy under Imperfec Compeiion, Journal of Macroeconomics, 26,