Optimal Monetary Policy When LumpSum Taxes Are Unavailable: A Reconsideration of the Outcomes Under Commitment and Discretion*


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1 Opimal Moneary Policy When LumpSum Taxes Are Unavailable: A Reconsideraion of he Oucomes Under Commimen and Discreion* Marin Ellison Dep of Economics Universiy of Warwick Covenry CV4 7AL UK Neil Rankin Dep of Economics Universiy of Warwick Covenry CV4 7AL UK Augus 2005 * We acknowledge helpful conversaions wih many people: Berhold Herrendorf, Henrik Jensen, Jonahan Thomas and Ted To, o name jus a few. Thanks also o wo anonymous referees, and o paricipans in numerous seminars. All errors and opinions are of course our own. Marin Ellison acknowledges suppor from an ESRC Research Fellowship, Improving Moneary Policy for Macroeconomic Sabiliy in he 2s Cenury (RES ).
2 2 Absrac We reexamine opimal moneary policy when lumpsum axes are unavailable. Under commimen, we show ha, wih alernaive uiliy funcions o ha considered in Nicolini s relaed analysis, he direcion of he incenive o chea may depend on he iniial level of governmen deb, wih low deb creaing an incenive owards surprise deflaion, bu high deb he reverse. Under discreion, we show ha he economy will no necessarily end o he Friedman Rule, as Obsfeld found. Insead i may end o he criical deb level a which here is no cheaing incenive under commimen, and inflaion and could well be posiive here. Keywords ime consisency, opimal inflaionax smoohing, discreion, commimen, Friedman Rule JEL Classificaion E52, E6
3 . Inroducion As is well known, opimal, welfarebased moneary policy, even in a flexprice, efficienly funcioning economy, is subjec o a ime consisency problem if he governmen does no have lumpsum axes or ransfers a is disposal. This is because surprise inflaion may be a subsiue source of nondisoring revenue, alleviaing he problem of being unable o reach he firsbes, Friedman Rule oucome of a zero nominal ineres rae. In his world, moneary policy mus be implemened by openmarke swaps of money and governmen deb, and he opimal secondbes policy is o use deb o smooh ineremporally he disorions caused by inflaion. Surprise inflaion can help reduce hese disorions. Alhough his problem is familiar, significan puzzles abou i remain. Firs is ha he opimal moneary policy is likely o be degenerae, in ha he bes surprise rae of inflaion o pick is infiniy, because his maximises he lump sum of revenue appropriaed by he governmen. Aenion was drawn o his feaure by Lucas and Sokey (983). However, a welldefined opimum can be obained if money eners he economy in such a way ha here is a welfare cos of curren inflaion, since his cos mus hen be balanced agains he benefis. This feaure was inroduced in an ineresing conribuion by Nicolini (998). The presence of he cos, on he oher hand, leads o a second puzzle: i may now be he case ha he ime consisency problem akes he form of an incenive o creae surprise deflaion (i.e. inflaion lower han expeced). This invers all our usual ideas abou ime inconsisency in moneary policy. A hird puzzle arises if we assume policy is conduced under discreion, raher han (as implici in he discussion so far) under commimen. In his case lack of rus in he governmen s projeced policy migh be conjecured o lead o higher expeced and acual inflaion; bu in fac i has been argued ha discreion will lead in he long run o lower inflaion in paricular o convergence o he Friedman Rule where inflaion is negaive. Such an argumen is presened in wo papers by Obsfeld (99, 997). Here we refer o he case of Obsfeld s analysis where he auhoriies objecive funcion is as close as possible o privae uiliy funcions.
4 2 In his paper we reconsider his opimal moneary policy problem. We do so using Nicolini s model of a simple cashinadvance economy, hereby avoiding he firs of he abovemenioned puzzles. We exend his analysis of he case of commimen a lile, bu our main conribuion is o he case of discreion, which he did no sudy. The model enables us o conduc a pure welfarebased analysis of opimal policy under discreion, avoiding Obsfeld s need o posulae an ad hoc objecive funcion for he policymaker. Our analysis shows ha he second and hird of he abovemenioned puzzles are relaed. Specifically, we find ha under discreion i is no necessarily rue ha in he long run he economy will converge on he Friedman Rule. Depending on consumer preferences, i may converge on a differen seady sae where inflaion is above he FriedmanRule level, and quie possibly posiive. We call his he imeconsisen seady sae. The force which makes such a seady sae possible urns ou o be he incenive which exiss owards surprise deflaion under commimen. This laer couneracs he more familiar incenive owards surprise inflaion, and makes i possible ha, under commimen, a criical level of inheried governmen deb exiss such ha he wo incenives exacly cancel ou, leading o no empaion o behave in a imeinconsisen manner. I ranspires ha his criical deb level is he same as he one associaed wih he ime consisen seady sae o which he economy may converge under discreion. The paper is organised as follows. Secion 2 describes he srucure of he economy. We examine opimal moneary policy under commimen in Secion 3. In Secion 4 we sudy opimal moneary policy under discreion, providing boh analyical resuls and numerical compuaions. Secion 5 concludes. 2. The Srucure of he Economy The economy consiss of many idenical households who consume he single ype of oupu good, supply a single ype of labour, and hold money  moivaed by a cashinadvance consrain  and bonds. Markes are perfecly compeiive and all prices are flexible. The governmen issues money and bonds. Bonds are aken o be real, or indexed, as in he
5 3 papers already cied. 2 A key consrain on he governmen is ha i does no have access o lumpsum axes and ransfers: he money supply mus be changed hrough open marke operaions, i.e. purchases or sales of bonds in exchange for money. Since our focus is purely on moneary policy, we shall ignore convenional disoring axes, and rea governmen spending on goods and services as exogenous. Technology akes he form y = n, where y is oupu and n is labour inpu. Hence compeiive firms will se a price equal o he wage, and make zero profis. For he represenaive household, he opimisaion problem is: maximise β [ Uc αn ] Σ= 0 ( ) (U > 0, U < 0) s.. pc M, M + bp( + R) + pn = pc + M + b p, for = 0,...,, wih M 0, b 0 given. () M,b are, respecively, he quaniies of money and real bonds held a he sar of period ; R is he nominal ineres rae beween  and ; c is consumpion; n is labour supply; and p is he price level. The key feaure of his problem is ha M 0 and b 0 are given. In his way he model incorporaes a welfare cos of curren inflaion : a rise in p 0 (and equal proporional rise in all p, > 0) depresses he real value of he household s iniial cash holdings, so ha (provided he cashinadvance consrain is binding) curren consumpion is squeezed. This liquidiy squeeze occurs because he household is assumed o be unable o visi he asse marke a he sar of he period: in any period, goods markes open before asse markes. Such a iming assumpion was inroduced by Svensson (985), and is he reverse of he more common iming assumpion in cashinadvance models associaed wih Lucas (e.g. Lucas 2 Alhough i would be more realisic o assume nominal bonds, his would obviously make i easier o generae a surprise inflaion resul, so o assume real bonds canno be said o faciliae our main conclusion. An ineresing relaed analysis which does assume nominal bonds is by DiazGimenez e al. (2002).
6 4 (982)). Is value in he presen conex is ha i capures he idea ha curren inflaion has a cos because agens canno insananeously adjus heir porfolios. 3 From he firsorder condiions for he household s problem we can readily derive: U'( c ) = α[ + R ], =,...,, (2) + p [ + R + ] =, = 0,...,, (3) p β (in which i has been assumed ha R > 0, so ha he cashinadvance consrain binds). (3) indicaes ha he real ineres rae is a consan equal o he inverse subjecive discoun rae. The nominal ineres rae hus moves oneoone wih he expeced inflaion rae. (2) indicaes ha consumpion is uniquely and negaively relaed (since U < 0) o he nominal ineres rae, or equivalenly o he expeced inflaion rae. This reflecs he disoring effec of he inflaion ax: he earnings from an increase in curren labour supply canno be spen immediaely, owing o he cashinadvance consrain, and heir fuure value is hence eroded by expeced inflaion, which herefore provides a disincenive o labour supply and so o consumpion. Wriing m M /p, we noe ha m = c provided ha he cashinadvance consrain binds, which reminds us ha expeced inflaion equivalenly reduces he demand for real balances. Equilibrium in he privae secor of he economy may now be deermined. A his poin, suppose an arbirary moneary policy defined by a sequence of moneary growh raes, { µ } 0 (where µ [M + M ]/M ). Then equilibrium consumpion mus saisfy: β c = U ( c ) c + + α + µ, (4) which has been obained from (2), (3) and p + /p = (+µ )c /c + (he laer being implied by m = c ). Under perfec foresigh (4) deermines c in a pure forwardlooking manner, as a 3 As Nicolini (998) shows, i can be inerpreed as a more compac version of he limied paricipaion model of Grossman and Weiss (983) and Roemberg (984). There, agens are divided ino wo groups who are only allowed o visi he bank in alernae periods.
7 5 funcion of curren and fuure moneary growh raes { µ + s } s = 0. Wih consumpion ied down by (4), oupu is deermined via he goods marke clearing condiion: y = c + g (5) where g is he level of governmen spending. A key role is played in wha follows by he governmen s budge consrain. The singleperiod consrain may be wrien: bp( + R) + pg = M M + b p. (6) Noe he absence of lumpsum axes or ransfers. Under our iming assumpions, (b,m ) are predeermined in period, and he only policy acion open o he governmen (since we rea g as exogenous) is an openmarke sale or purchase of bonds, which raises or lowers b +, lowering or raising M +, respecively. We may inegrae (6) forwards, incorporaing (3) (and appealing o a No Ponzi Game condiion), o obain: b0( + R0) = Σ= 0βµ m g. (7) β This is he governmen s ineremporal budge consrain expressed in erms of cashflow seigniorage, i.e. µ m. We may also rewrie i as: b0( + R0) = m0 + Σ= β Rm g. (8) β (8) gives he same consrain in erms of opporuniycos seigniorage, i.e. R m. 4 Noice ha m 0 eners (8) differenly from m,m 2, ec. Algebraically speaking, his is he source of ime inconsisency, as we discuss furher below. 4 See Herrendorf (997) for a useful discussion of hese wo definiions. To obain (8) from (7), noe ha [M +  M ]/p can be rewrien as m + β[+r + ]  m.
8 6 3. Opimal Moneary Policy Under Commimen In his secion we assume ha here is some commimen mechanism which obliges he governmen o adhere o he moneary policy which i chooses in period 0, { µ } 0. Is opimisaion problem is hen: maximise w.r.. { µ } 0 Σ 0 β [ Uc ( ) αc αg] = ( ) ( ) β α s.. + R0 b0 + g = c0 +Σ= β U c c, β c = U ( c ) c + + α + µ for = 0,...,, wih (+R 0 )b 0, g given. (9) We have subsiued ou R using (2), m using m = c, and n using he producion funcion and (5). I is clear ha, raher han rea he µ s as he conrol variables, we can equivalenly rea he c s as he conrol variables, leaving he µ s o be deermined residually by he difference equaion consrains. This reduces he problem o: maximise w.r.. { c } 0 Σ= 0 β [ Uc ( ) αc αg] ( + R ) b g = c Σ β U ( c ) c β α. (0) s = We easily see ha his problem has an unconsrained maximum where: U ( c ) = α for all. () This is he firsbes, Friedman Rule soluion, where R = 0. The governmen s ineremporal budge consrain in general prevens he aainmen of his oucome unless iniial governmen deb is sufficienly negaive. Thus, in a secondbes siuaion, c will be less han is FriedmanRule level, and an opimal policy mus rade off he deviaion of c from his wih he deviaion of any oher c s.
9 7 I is sraighforward o derive he following se of firsorder condiions for he problem (0): U ( c ) α = 0 U ( c ) α + [ σ ( c ) ] U ( c )/ α for =,...,. (2) Here, σ(c ) is he relaive risk aversion measure of curvaure of U(c ), i.e. cu ( c ) / U ( c ). I is immediae from his ha c is he same for all =,...,. I is furher apparen ha his common c ( c, say) is in general differen from c 0. Hence an opimal ime pah of c ypically akes he form of eiher a sep up or a sep down (see Figure ). Inspecion of (2) furher shows ha if σ < hen we obain c 0 < c (a sep up ), and if σ > hen we obain c 0 > c (a sep down ). In he special case where σ =, he opimal ime pah is fla. 5 The above resuls are as in Nicolini (998). Nicolini however resrics aenion o uiliy funcions of he consan relaive risk aversion (CRRA) class, so ha σ is an exogenous parameer. Here we will explore he consequences of more general uiliy funcions. A visual aid o doing so is provided by Figure 2. The diagram explois he fac ha, under an opimal policy, c = c for =,...,, in order o represen he problem in reduced form. The indifference curves over (c 0,c) are given by he lifeime uiliy funcion wrien as Uc c g+ Uc c g, and he governmen s ineremporal budge ( 0) α 0 α β[ β] [ ( ) α α ] consrain conracs o: c β 0 () ( 0) 0 U c c c = + β α R b β g. (3) Figure depics his as a backwardbending curve. This reflecs he seigniorage Laffer curve : as fuure inflaion raes (and hus nominal ineres raes) are raised, c falls, bu neverheless fuure seigniorage revenue a firs rises, permiing less revenue o be raised hrough curren seigniorage, and hus permiing higher c 0. However (depending on he shape of U(c)) he Laffer curve may have a peak, such ha revenue sars o fall as fuure inflaion coninues o be increased and c coninues o be reduced. Iniial governmen deb, (+R 0 )b 0, 5 Corresponding o hese ime pahs of consumpion, i is easy o show ha he ime pahs of deb are a sep down, a sep up, and fla, respecively.
10 8 acs as a parameer which fixes he posiion of he budge consrain: higher iniial deb shifs i o he lef. By varying iniial deb we hus race ou a locus of poins of angency wih he indifference curves  he expansion pah. The equaion of he expansion pah is jus he firsorder condiion (2), wih c subsiued by c. Consider now he shape of he expansion pah. Firs, i clearly passes hrough he poin ( c, c ), where c is he Friedman Rule consumpion level given by (). Second, from he resuls presened earlier, he expansion pah mus lie above (below) he 45 o line a levels of c such ha σ(c) < (>). I follows ha if U(c) is such ha here exiss a value c c a which σ(c) =, hen he expansion pah mus cross he 45 o line a his criical value of c. (For his o be of ineres, we obviously need c c < c ). Third, we would inuiively expec he expansion pah o be upwardsloping (i.e. under an opimal policy lower iniial governmen deb would be used o raise boh c 0 and c) 6. Hence, fourh, if c c exiss hen he quesion arises as o wheher he inersecion wih he 45 o line occurs from above or from below. Differeniaing (2) and evaluaing where c 0 = c = c c, we obain: dc α =. (4) dc0 c α + cσ [ U α] c This is clearly less han or greaer han one as σ (c c ) is (respecively) posiive or negaive. Tha is, he expansion pah cus he 45 o line from above if he relaive risk aversion parameer is increasing in consumpion a he inersecion poin, and from below if i is decreasing. 7 Figure 2 illusraes he case where an inersecion of he expansion pah and he 45 o line exiss, and where i occurs from above. In his case here mus also exis a criical value of (+R 0 )b 0 such ha, if iniial deb happened o ake his value, he opimal policy would be o choose c 0 = c = c c. The associaed budge consrain is he one which passes hrough he poin C. If iniial deb were slighly higher han his level, he budge consrain would lie slighly farher o he lef, and he opimum would be a a poin like A, where c 0 < c. Conversely if 6 The condiion for his is ha he RHS of (2) be decreasing in c. I urns ou ha he same condiion is necessary for he opimisaion problem s secondorder condiions o be saisfied, so we need his condiion o hold. I holds provided ha σ(c) is increasing, consan, or no oo srongly decreasing, in c. 7 (4) is no he correc expression for he slope a he FriedmanRule poin. The slope here is insead given simply by σ, and so is greaer or less han one as σ is greaer or less han one.
11 9 iniial deb were slighly lower, he opimum would be a a poin like B, where c 0 > c. In he case in Figure 2, hen, he shape of he opimal ime pah of consumpion depends on he iniial level of deb: high deb gives rise o a sep up shape, and low deb o a sep down shape. Wih he CRRA uiliy funcion used by Nicolini (998), on he oher hand, he expansion pah lies eiher always above, or always below, he 45 o line, and he magniude of iniial governmen deb is herefore irrelevan, qualiaively speaking, o he shape of he opimum ime pah of consumpion. In order o assess wheher here is a poenial ime consisency problem wih he opimal policy, i mus be compared wih he opimal policy if he governmen were permied o reopimise in period. This reopimisaion is only hypoheical, since we have assumed ha here is a commimen mechanism o preven he governmen from acually deviaing from is period0 plan. Le us denoe he opimal choice of c from he perspecive of period s (s ) as c s. We are paricularly ineresed in wheher c is smaller or greaer han c 0. Since c = M /p and M is predeermined, c < c 0 indicaes ha he governmen in period would wish o se p higher han was planned in period 0, i.e. ha here is an incenive o surprise inflaion. Conversely, if c > c 0, he incenive is o surprise deflaion. Following Nicolini s mehod we may readily show ha, if he period0 opimal ime pah akes he form of a sep up, hen in period here is an incenive o surprise inflaion; while if i akes he form of a sep down, hen in period here is an incenive o surprise deflaion. A summary skech of he wo possible relaionships beween he period0 and period opimal ime pahs for consumpion is hus as in Figure. Togeher wih he earlier resuls, his hen implies ha if here exiss a criical consumpion level (sricly less han he Friedman Rule level) a which he coefficien of relaive risk aversion in consumpion equals one, hen associaed wih his is a criical level of governmen deb such ha, if iniial deb happens o ake his value, here is no ime inconsisency. Moreover, if relaive risk aversion is increasing (decreasing) in consumpion a he criical level, hen iniial deb above he criical deb level will be associaed wih a empaion o creae surprise inflaion (deflaion); and iniial deb below, wih a empaion o creae surprise deflaion (inflaion). This is an exension of Nicolini s (998) main finding. I
12 0 says ha if relaive risk aversion is no a consan, hen he direcion of ime inconsisency may depend on he level of iniial governmen deb. An inuiively plausible oucome, we would argue, arises in he case of increasing relaive risk aversion. Here, while he raher unorhodox incenive o creae surprise deflaion dominaes for low levels of deb, he more convenional incenive o creae surprise inflaion dominaes for high levels. This oucome occurs if he expansion pah has he shape shown in Figure 2. 8 A naural followup quesion concerns how likely i is o find a uiliy funcion possessing he properies jus highlighed. Alhough he CRRA funcion is a common uiliy funcion wih some convenien feaures, here is of course no shorage of alernaives. Consider, for example, quadraic uiliy, Uc () = cc [ ˆ c/2]. (Wih his, ĉ α is needed o ensure a nonnegaive FriedmanRule consumpion level.) σ(c) = c/[ cˆ c] for his funcion, so a value of c such ha σ = clearly exiss, namely a c = ĉ /2. Moreover, σ is clearly everywhere increasing in c. The associaed expansion pah hen looks exacly like ha in Figure 2, provided ha ĉ > 2α. 9 Anoher common funcional form, he consan absolue risk aversion (CARA) funcion, can similarly yield an expansion pah like ha in Figure 2. We hence conclude ha he preferences required in order for he above resul o apply are no especially unusual. I is rue ha CRRA preferences are widely used in macroeconomics, parly because hey have he convenien propery of being consisen wih balanced growh. However, quadraic preferences are also widely used in consequence of some oher convenien properies: for example, in he presence of uncerainy hey generae he exac consumpionasarandomwalk resul, and hey underpin meanvariance porfolio analysis. In he presen paper our aim is o explore he implicaions of some familiar uiliy funcions as a heoreical exercise, raher han o claim any one funcion fis he facs well. Clearly 8 A referee poins ou ha, in an exension o his main analysis, Nicolini (998) also noes ha here may be a criical level of iniial deb such ha here is no empaion o ime inconsisency. This is where σ > and deb is nominal. Nominal deb generaes an incenive owards surprise inflaion which could exacly offse he incenive owards surprise deflaion. However Nicolini does no develop he analysis of his case. 9 This condiion ensures ha ĉ /2 is less han he FriedmanRule consumpion level, ĉ α. If i is violaed, he expansion pah lies everywhere above he 45 o line. Even hough quadraic uiliy by iself does no herefore guaranee an expansion pah like ha in Figure 2, we can sill say ha, wih quadraic uiliy, if an incenive o surprise deflaion is ever o exis (i.e. if par of he expansion pah is ever o lie below he 45 o line), hen a criical consumpion level as defined in he main ex mus also exis.
13 much more elaboraion of our barebones model would be needed before i could be confroned wih he daa. A furher naural quesion concerns why he direcion of ime inconsisency should depend on he size of σ. As noed, in a CIA model wih Lucas s iming assumpion, he empaion o chea in moneary policy is always owards surprise inflaion. In he presen framework, he incenive o raise all curren and fuure prices equiproporionally in order o appropriae revenue in a nondisoring way is counerbalanced by he incenive no o generae a large welfare cos of curren inflaion hrough he liquidiysqueeze mechanism explained earlier. The direcion of he empaion o chea depends on how hese incenives change, relaive o he incenives for he seing of subsequen periods inflaion, as ime advances. Firs, observe ha a empaion o reduce curren inflaion relaive o wha was planned would arise if an increase in curren inflaion (increase in p 0 /p , from he perspecive of period 0) were o provide a smaller gain in he PV of seigniorage and a larger loss of lifeime uiliy 0 han a projeced increase in fuure inflaion (increase in p /p 0, from he perspecive of period 0). This is because, when he fuure arrives, he seigniorage benefi would be smaller han i was from he vanage poin of he period before and he uiliy cos would be larger, so he governmen would perceive an incenive o deviae downwards from wha i had planned. Such changes in incenive occur when σ > because fuure consumpion demand is hen inelasic wih respec o fuure inflaion (or, equivalenly, +R ), as can be seen from (2); while curren consumpion demand is always unielasic wih respec o curren inflaion, as follows from he cashinadvance consrain. Hence a uni increase in p 0 /p  causes a larger loss of lifeime uiliy han a (/β)uni increase in p /p 0. I also causes a smaller gain in he PV of seigniorage, because he inelasic fuure consumpion demand means a weaker dampening effec on revenue of he shrinkage in he ax base as he inflaion rae is raised, so ha seigniorage increases more srongly wih fuure han wih curren inflaion. 0 Through is direc effec on uiliy, ignoring he indirec effec via is budgeary impac
14 2 4. Opimal Moneary Policy Under Discreion µ s s 0 If he governmen is unable o commi o a given policy plan made in period, { + }, hen i is no raional for households forecass of fuure µ = +s s, as of ime (denoe e hese by µ + s ), o equal hose in he plan. Insead, forecass should be based only on variables observable a ime. The governmen s inheried sock of deb is one obvious observable on which o condiion forecass, since i is clear from he previous secion ha iniial deb is a key deerminan of he governmen s opimal policy choices. Oher observables could also be used, such as curren and pas values of µ. However, his would inroduce an elemen of repuaionbuilding behaviour ino policy. Since we wish o focus on purely discreionary behaviour, we use only deb as he basis for privae forecass. Formally, he concep of equilibrium employed will be ha of Markovperfec equilibrium. Firs, i is useful, as Obsfeld does, o define he concep of governmen commimens k b( + R ) + Σ β g. (5) s s= 0 + s Commimens is jus he spending, in presenvalue erms, over which he governmen does no have discreion. Formally, we will use k raher han deb as he sae variable in wha follows, alhough, given our assumpion ha g is imeinvarian, k is simply deb plus he consan, g/(β). Noe ha he governmen s budge consrain in erms of k is (cf. (6)): k = µ m + β k +. (6) We now suppose ha, in period, households forecas (µ +,k +2 ) using he rules: µ = ˆ( φ k ), (7) e + + k = ψˆ ( k ). (8) e This is he same basic idea as in Obsfeld (99, 997). His 99 paper sudies a small open economy version of he opimal inflaion ax smoohing problem, while his 997 paper does he same for a closed economy. The key difference from he presen analysis, as noed in he Inroducion, is ha Obsfeld s model does no include a welfare cos of curren inflaion. This obliges him o use a governmen objecive funcion which differs from he privae uiliy funcion, being obained by adding on an ad hoc cos of curren inflaion erm o he laer.
15 3 (Noe ha k + is observable by households in period  since hey can observe µ m and hus use (6)  so ha i is appropriae o base heir forecass upon i.) ˆ(.) φ and ψ ˆ (.) are for he momen reaed as arbirary funcions, bu hey will laer be deermined by imposing a raionaliy requiremen as par of he condiions of Markovperfec equilibrium. To generae an speriod ahead forecas, (8) may be used repeaedly in (7): µ = ˆ φ( ψˆ ( k )), (9) e s + s + n where ψˆ (.) ψˆ( ψˆ(... ψˆ(.)...)) denoes he nh ierae of he funcion ψ ˆ (.). These forecasing rules can nex be used o deermine equilibrium consumpion in period. Recall ha he equilibrium value of c is given by he privaesecor law of moion, (4), solved in a forwardlooking manner. The relevan µ s o use in his equaion are now he expeced values as given by (9), raher han he values from he governmen s policy plan, since wha couns for deermining he acual c are households expecaions. I is helpful o consider he deerminaion of c in wo sages. Firs, given k + and hus a sequence of expeced values { e µ + s } s = generaed by (9), we use (4) for periods + onwards o solve for c +. This yields households forecas, as of period, of he equilibrium value of c +. Since i e is coningen on he observed k +, we may wrie i as c ˆ( ) + = θ k+. Second, using ˆ( θ k + ) in (4) for period, we obain he equilibrium value of curren c : β c = U ( ˆ θ( k )) ˆ θ( k ) + + α + µ ek ( + ). (20) + µ Curren c hus depends on wo variables: he currenly observed µ, and he currenly observed k +, whose effec operaes via influencing expecaions abou fuure µ +s s. The funcion e(.) capures his expecaions effec. The form of e(.) derives from he form of ˆ(.) θ, which is in urn derived from he forms of ˆ(.) φ and ψ ˆ(.). The governmen now reas (20) as par of he economy s srucure, and hus as a given. Is opimisaion problem under discreion is herefore o choose { µ } 0 o maximise lifeime
16 4 uiliy of he ypical agen subjec o (20) and (6) for = 0,...,, and o given k 0. By subsiuing (20) ino (6) and ino he maximand, we can express his as: maximise w.r.. { } 0 ek ( ) ek ( ) Σ β U α αg µ µ µ + + = ek ( + ) s.. k = µ + βk + µ +, for = 0,...,, wih k 0 given. (2) (2) reveals ha he governmen s dynamic opimisaion problem under discreion  unlike ha under commimen  has a sandard recursive form. Tha is, every period, he new value of he sae (k + ) depends only on he curren value of he sae (k ) and on he curren value of he conrol (µ ), while he flow maximand also depends only on hese same wo variables (alhough i is k + which appears in he flow maximand, k + is an implici funcion of (k,µ ) via he consrain). This srucure means ha he problem can in principle be solved by dynamic programming. In urn, dynamic programming ensures ha he soluion is ime consisen. 2 The dynamic programming perspecive also makes clear ha he soluion o our problem can be expressed as a pair of feedback rules on he sae; for example µ = φ( k ), (22) k+ = ψ ( k). (23) (22)(23) define he governmen s opimal moneary policy having aken as our saring poin he public s arbirary forecasing rules, (7)(8). We noice ha (22)(23) relae he same variables as (7)(8); hence, for (7)(8) o provide raional forecass by he public, we need he funcions ˆ(.) φ and φ (.), and also ψ ˆ (.) and ψ (.), o be he same. In his case households will forecas correcly no maer wha he value of k. The discreionary, or 2 Given ha he funcion e(.) has as ye unknown properies, here is a quesion as o wheher he opimisaion problem (2) is well defined. Here we proceed as if his is he case, bu we reurn o he quesion below.
17 5 Markovperfec, equilibrium is hus a pair of forecasing rules which have he propery ha hey reproduce hemselves in he guise of opimal governmen policy rules. I is convenien, as earlier, o rewrie he opimisaion problem in order o rea he c s raher han µ s as he conrols. To do his we use (20) o subsiue ou he µ s from (2), so ha he problem is ransformed o: maximise w.r.. { c } 0 Σ 0 β [ U ( c ) αc αg] = s.. k = e( k+ ) c + β k+, for = 0,...,, wih k 0 given. (24) The firsorder condiions for his are hen easily derived: U ( c+ ) α = [ U ( c) α][ + e ( k+ )] for = 0,...,. (25) β Repeaing here he consrain from he problem (24), k = e( k ) c + βk, (26) + + we see ha (25)(26) consiue a firsorder dynamical sysem in (k,c ), which deermines he evoluion of he economy under he opimal discreionary policy. Given ha k is a predeermined variable whereas c is no, c 0 will generally be ied down in relaion o k 0 by he ransversaliy condiion. This hence deermines a paricular soluion of he sysem (25) (26) which consiues he soluion of he opimisaion problem. We denoe his soluion as: c = θ ( k ). (27) Equivalenly, (27) is he opimal feedback rule of he conrol upon he sae variable. If he opimum is such ha he economy converges on a seady sae, hen (27) is also he saddlepah soluion of (25)(26). Alhough θ (.), like e(.), is sill a presen an unknown funcion, once i has been deermined we can subsiue i back ino (26) o ge:
18 6 ek ( ) + βk = k+ θ ( k). (28) + + This implicily deermines he opimal feedback rule for k +, (23). The opimal feedback rule for µ, (22), is similarly recoverable by subsiuing (27) and (23) ino (20). The funcions e(.) and θ (.) hus play a cenral role in he Markovperfec equilibrium, since, if hey can be deermined, he oher unknown funcions φ(.) and ψ(.) follow. I is also worh noing ha in equilibrium he funcion θ (.) mus urn ou o be he same as he funcion ˆ(.) θ. This is because if households forecasing rules are always o yield correc predicions, heir forecas of c + coningen on k + mus coincide wih he governmen s opimal, and hus acual, choice of c + coningen on k +. We now aim o sudy he properies of he discreionary equilibrium. We sar wih he seady saes. Inspecion of (25) suggess wo ways in which a saionary soluion of he dynamical sysem (25)(26) may occur. Firs, (25) is clearly saisfied a he Friedman Rule, where U  α = 0 for all. We shall refer o his as he Friedman Rule seady sae (FRSS), I is inuiively clear ha if iniial governmen deb is sufficienly negaive ha he underlying secondbes problem is absen, hen ime inconsisency is removed, and a governmen which sared in his forunae posiion would have no incenive o move away from i. Obsfeld (99, 997) similarly idenifies he Friedman Rule allocaion as a seady sae of he discreionary equilibrium in his analysis. However, whereas for Obsfeld he Friedman Rule allocaion is he only seady sae, his is no necessarily rue here. A second way in which a saionary soluion of (25) could occur is if here exiss a k + a which e (k + ) = 0. More specifically, we migh conjecure ha his would be rue a he value of k + corresponding o he criical deb level as we defined i for moneary policy under commimen. The moivaion for such a conjecure is ha we know from Secion 3 ha here is no ime inconsisency if iniial deb happens o equal he criical value, and ha, under commimen, if he governmen sared wih his amoun of deb i would choose o say here. Thus i migh be hypohesised ha, wih his criical amoun of iniial deb, he opimal policy under discreion would be he same as under commimen. This second ype of seady sae we shall refer o as he ime consisen seady sae (TCSS).
19 7 To prove ha a TCSS exiss under discreion if, in he problem under commimen, here exis criical consumpion and deb levels as defined in Secion 3, consider again he definiion of he funcion e(.) (given in (20)). Differeniaing his funcion wih respec o k, we obain: β e ( k) = [ σ( c)] U ( c) θ ( k) (29) α (in which we have equaed ˆ(.) θ wih θ (.), for he reason explained). Alhough θ ( k) is unknown, we do know from Secion 3 ha σ(c) = a he criical consumpion level c c. This is herefore sufficien o prove ha e = 0 a he corresponding criical commimens level (call his k c ). Hence a saionary soluion of (25) does indeed arise a k c. The level of inflaion a he TCSS is higher han he negaive inflaion rae (β) which prevails a he Friedman Rule. Is value depends on he uiliy funcion: from (2) and (3), he general expression for he inflaion rae is (β/α)u (c) , and a c c his could be posiive or negaive. Hence, if we can show ha here are condiions under which he discreionary equilibrium converges on he TCSS (for he general case in which he iniial k 0 k c ) hen i follows ha he longrun desinaion of he economy under discreion is no necessarily a siuaion of deflaion. We may also noe ha he level of deb a he TCSS is higher han he negaive level required o susain he Friedman Rule, and he same is rue of he level of commimens. Wheher hey are negaive or posiive in he absolue again depends on he uiliy funcion, and also on g. 3 The evoluion of he economy under discreion is governed by (23), which we saw o ake is form implicily from (28). Differeniaing (28), and evaluaing a a generic seady sae k, we have: dk + θ ( k) = ψ ( k) =. (30) dk β + e ( k) + Local convergence o eiher he FRSS or he TCSS clearly requires ha  < ψ <. Hence we shall proceed by aemping o solve for θ ( k) and e (k) a each ype of seady sae, in 3 Specifically, seing c 0 = c = c c in (3), we have k c = ( β)[(β/α)u (c c )]c c, and b c (+R c ) = k c  g/(β).
20 8 order o deermine wheher his condiion can be saisfied. An expression for e has already been obained in (29). Differeniaing (29) for a second ime yields he following expression for e, which will be useful below: 2 2 e = ( β / α)[ σ U ( θ ) + ( σ) U ( θ ) + ( σ) U θ θ ]. (3) In order o find an expression for θ, recall ha θ ( k ) is he saddlepah soluion of he dynamical sysem (25)(26). By aking a linear approximaion o his sysem abou a seady sae, we may find an expression for θ. Such a calculaion yields: + θ U α = e + ( β + e ) + β ± e + ( β + e ) + β 4 β( β + e ) 2β U U 2 2 U α 2 2. (32) We now observe ha (29), (3) and (32) consiue a sysem of hree simulaneous equaions in he four unknowns ( e, e, θ, θ ). Alhough we canno solve i as i sands, if we proceed o evaluae i a eiher he FRSS or TCSS, i urns ou ha we obain addiional resricions which are sufficien o make soluion possible. Consider firs he local dynamics of he FRSS. Seing U = α, noice ha e drops ou of (32). (29) and (32) hen consiue a sysem of wo equaions in jus wo unknowns, e and θ, from which we may hope o solve for e and θ. Appendix A presens he relevan calculaions. We show ha he sysem can be reduced o a quadraic equaion in θ as he single unknown. Once θ has been deermined, e follows from (29) and ψ from (30). Since he quadraic implies wo possible soluions for θ, here are also wo possible soluions for e and ψ. The poin of paricular ineres is wheher eiher of he soluions for ψ have absolue value less han one. The key resul demonsraed in Appendix A is ha if σ <, here is exacly one soluion for ψ wih absolue value less han one; bu if σ > here are no soluions wih absolue value less han one. Combining his wih our earlier finding, we hus conclude ha:
21 9 Proposiion The FriedmanRule consumpion level and associaed negaive deb level consiue a seady sae of he discreionary equilibrium. Moreover, if he coefficien of relaive risk aversion in consumpion is less (greaer) han one a his seady sae, hen, wihin a neighbourhood of i, a discreionary equilibrium which converges on i exiss (does no exis). In he case σ < we also find, more specifically, ha ψ lies in (0,) (ensuring ha convergence is monoonic), and ha θ and e are negaive. Thus, as he Friedman Rule is approached, deb seadily falls (or equivalenly  once deb becomes negaive  governmen asses seadily rise) and consumpion seadily rises. Inflaion also seadily falls (becomes more negaive). This is he oucome obained by Obsfeld (99, 997). Such an oucome differs from wha happens under commimen, where, if σ <, consumpion also rises along an opimal ime pah, bu only beween periods 0 and l. Deb correspondingly falls beween periods 0 and, bu remains permanenly above is Friedman Rule level. The clue as o why a governmen acing under discreion goes farher in reducing is deb over ime han a governmen acing under commimen, lies in he negaive e, as Obsfeld poined ou. e < 0 means ha lower deb (lower k + ) induces higher curren consumpion hrough affecing privae agens expecaions (recall (20)). From he governmen s perspecive, e < 0 hus increases he reurn o a marginal reducion in he deb: no only does lower deb nex period mean he fuure inflaionax revenue needed is lower, bu also, by lowering privae expecaions of fuure inflaion 4, i raises curren consumpion. By conras, under commimen, privae agens do no use he level of governmen deb as he basis for heir forecass  insead, hey rus he governmen o carry ou oday s opimal plan. Hence a change in he deb level per se does no have his added Obsfeld effec. 5 A numerical illusraion of a se of discreionary equilibria (one for each iniial value of k ) converging on he FRSS is given in Figure 3. This is calculaed for he CRRA uiliy 4 To see his, noe ha since θ is negaive, and recalling ha θ = ˆ θ, a reducion in k + raises privae agens forecass of c +. c + is negaively relaed o expeced inflaion p + /p by (2)(3). 5 Alhough, under his oucome, uiliy is higher in he long run under discreion han under commimen, lifeime uiliy from he perspecive of period 0 is lower, because consumpion in he shor run can be shown o be lower. This is as i should be, because inabiliy o commi acs as a consrain on he opimal policy.
22 20 funcion wih σ = 0.5, α = 5 and β = (We have also se g = 0 for all.) A descripion of he algorihm used for he compuaions is provided in Appendix C. The FriedmanRule consumpion level in his example is c = 0.04, and he corresponding level of commimens needed o susain his is k = Panel (a) shows he e(k ) funcion for his example. This confirms he negaive effec of he sock of deb (o which commimens are here equivalen), operaing via privaesecor expecaions, on c . Panel (b) shows he θ(k ) funcion (he line labelled c ), indicaing he negaive oal effec of he sock of deb on he governmen s opimal choice of c. Noe ha he range of c values considered exends from he maximum, FriedmanRule, value down o approximaely one quarer of ha value, so ha he picure is no jus concerned wih a small neighbourhood of he FRSS. The funcion is clearly quie close o being linear. Panel (c) plos he ψ(k ) funcion and also, for reference, he 45 o line. As Proposiion predics, he slope is less han one  in fac, i is abou which confirms ha he economy does indeed converge monoonically on he FRSS for any iniial level of deb above For comparison, panel (b) also plos he wo consumpion levels which resul from he problem under commimen. The line c 0 gives consumpion in he firs period of an opimal plan as a funcion of he iniial deb k 0 ; while he line c gives consumpion in he second and all laer periods, again as a funcion of k 0. We hus see ha for he same inheried level of deb, in his example consumpion in he shor run is chosen o be lower under discreion han under commimen and, correspondingly, inflaion is chosen o be higher. When σ >, however, Proposiion indicaes ha i is no possible for he oucome obained by Obsfeld o occur in our model. The force driving he economy away from he FriedmanRule oucome in such a case is discussed below in he conex of Proposiion 2. This suggess ha in his case he desinaion may insead be he imeconsisen seady sae, where he laer exiss. We now urn o he local dynamics of he TCSS. A he TCSS iself, σ = and e = 0, as shown previously. Hence θ drops ou of (3), and he simulaneous sysem of (29), (3) and (32) can be reduced o he sysem of jus (3) and (32), in he unknowns θ and e. From his we may hope o solve for θ and e. Appendix B presens he relevan calculaions. We show
23 2 ha he sysem can be reduced o a quadraic equaion in θ as he single unknown, analogous o, bu differen from, he quadraic applying a he FRSS. Once θ has been deermined, ψ follows from (30). Since he quadraic implies wo possible soluions for θ, here are also wo possible soluions for ψ. The poin of paricular ineres is wheher eiher of he soluions for ψ have absolue value less han one. The key resul demonsraed in Appendix D is ha if σ > 0, here is exacly one soluion for ψ wih absolue value less han one; bu if σ < 0 here are no soluions wih absolue value less han one. Combining his wih our earlier findings, we hus conclude ha: Proposiion 2 If he criical consumpion and deb levels as defined for he problem under commimen exis, hen a seady sae of he discreionary equilibrium also exiss a hese consumpion and deb levels. Inflaion could be posiive or negaive a his seady sae. Moreover if, under commimen, iniial deb above he criical level was associaed wih a empaion o creae surprise inflaion (deflaion), and below, wih a empaion o creae surprise deflaion (inflaion), hen, under discreion, wihin a neighbourhood of he seady sae, an equilibrium which converges on i exiss (does no exis). As Proposiion 2 emphasises, in he neighbourhood of he criical consumpion level here is a close relaionship beween he opimal policy under commimen and ha under discreion. Moreover, in he case where under commimen a empaion o surprise inflaion is associaed wih deb above he criical level and vice versa (i.e. in he case where σ > 0), we can show ha 0 < ψ <, hus ensuring ha convergence under discreion is monoonic; and we can also show ha θ and e are negaive. If k 0 > k c, consumpion hen seadily rises over ime unil i reaches he criical level. Along his pah, inflaion and deb are falling. If, on he oher hand, k 0 < k c, hen consumpion seadily falls over ime, wih accompanying rises in deb and inflaion. By comparison, under commimen, pahs wih he same k 0 would involve a rise or a fall in consumpion (respecively) beween periods 0 and, and a corresponding fall or rise in deb during period 0, bu hey would no coninue all he way o (c c,k c ). The reason why he evoluion is carried furher under discreion is ha e < 0 a k 0 > k c and e > 0 a k 0 < k c, as follows from he fac ha e < 0 a he TCSS. e < 0 means ha here
24 22 is an Obsfeld effec, as described above, so ha he governmen perceives addiional benefis of deb decumulaion when unable o commi. e > 0, by conras, implies ha here is a reverse Obsfeld effec : now a marginal increase in endofperiod deb raises curren consumpion hrough he effec on agens expecaions, giving he governmen an incenive o accumulae deb relaive o he case in which i can commi. Inuiively, e > 0 here because his is he region of k 0 in which, under commimen, he empaion is o creae surprise deflaion. In his region, for a governmen following he opimal commied policy, afer one period he higher deb which i faces would creae an incenive for i o choose consumpion o be greaer han in he plan: under discreion, his incenive is refleced in he way deb influences consumpion hrough expecaions. 6 Figure 4 provides a numerical example of a se of discreionary equilibria (one for each iniial value of k ) converging on a TCSS. This is calculaed for he quadraic uiliy funcion wih ĉ = 0.08, α = and β = (Again, g = 0 for all.) The algorihm used is again ha described in Appendix C. The implied criical value c c, i.e. where σ(c) =, is 0.04, and correspondingly k c = The e(k ) funcion for his example is depiced in panel (a). As prediced by he heory, i shows ha he gradien is zero a k c and he shape is concave: i.e. small increases in deb here affec consumpion via he public s expecaions in opposie ways, depending on wheher deb is high or low. Panel (b) plos he oal effec of deb on he governmen s opimal choice of c. The dynamics of deb are illusraed in panel (c). To make he picure clearer, we plo k + k on he verical axis, since k + as a funcion of k urns ou o be very close o he 45 o line. I can be seen ha he TCSS a k c is indeed sable, even if convergence is very slow. Panel (c) also shows he FRSS (a k = ) and so reveals how i is unsable. For comparison, in panel (b) we also plo he wo consumpion levels which resul from he problem under commimen (cf. Figure 3). We see ha while, for deb above 6 Having esablished ha e = 0, e < 0 in he neighbourhood of he TCSS, i is sraighforward o check and confirm ha he secondorder condiions for he governmen s opimisaion problem are saisfied when e(.) has hese properies. In he neighbourhood of he FRSS we canno deermine e analyically. However our numerical experimen confirms ha a maximum exiss for he chosen parameer values, and also suggess ha under CRRA uiliy e = 0 more generally may no be a bad approximaion. I is again sraighforward o check and confirm ha he secondorder condiions are saisfied when e(.) is linear.
25 23 k c, consumpion in he shor run is lower under discreion han commimen, for deb below k c he opposie is rue. Our resuls demonsrae he main claim made in he Inroducion, which is ha i is no ineviable ha under discreion he opimal policy will converge on he Friedman Rule. As we have jus argued, he force which may preven he aainmen of he Friedman Rule, or even drive he economy away from i, is he fac ha under commimen here can be an incenive o surprise deflaion raher han surprise inflaion. I may be remarked ha, alhough he condiions for local convergence o he FRSS and TCSS are no he same, i urns ou hey appear similar when viewed in erms of he expansion pah diagram. For local convergence o be possible, a eiher seady sae he expansion pah mus cu, or mee, he 45 o line from above. This is because he sabiliy condiions in Proposiions and 2 are he same as hose governing he slopes of he expansion pah a he relevan poins, as is clear from he resuls in Secion 3. Thus, if he expansion pah has he shape illusraed in Figure 2, hen, under discreionary policy, local convergence o he FRSS is no possible, bu local convergence o he TCSS is. If insead he expansion pah lies everywhere above he 45 o line  excep a he Friedman Rule poin where i mus mee i  hen he FRSS is he only seady sae under discreion, and local convergence o i is possible Conclusions Our main findings having been summarised in he Inroducion, we here commen on heir generaliy. Firs, i is no essenial o he resuls ha we have used a cashinadvance framework. A common alernaive is he moneyinheuiliy funcion approach. If his is adoped, hen, in order o capure a welfare cos of curren inflaion, an analogous modelling device o he use of Svensson s iming in cashinadvance is o use beginningofperiod nominal balances deflaed by he curren price as he real balances variable in he uiliy funcion (as is done, in a differen applicaion, by Neiss (999)). In early work we employed 7 These are he only wo possible shapes for he expansion pah under quadraic or CARA uiliy.
26 24 such an approach and obained resuls which exacly parallel hose presened here. In fac, i can be shown ha he wo approaches are mahemaically equivalen. However he inerpreaion required of he funcions is differen: wha maers in he moneyinheuiliy funcion approach is he shape of he uiliyofrealbalances funcion, raher han he shape of he uiliyofconsumpion funcion. Second, he precise condiions which we have obained in his paper for a imeconsisen seady sae of he discreionary equilibrium o exis, and for convergence o i o be possible, are ones which we would no expec o be robus o oher ways of modelling he welfare cos of curren inflaion. The way he laer is represened here is simple and convenien bu here are more sophisicaed ways in which i could be capured, such as in he limied paricipaion models referred o earlier. Neverheless we would sill expec ha a imeconsisen seady sae, as we have defined i, could occur in such models, albei wih modificaions o he exac condiions for is exisence. Hence we anicipae ha our conclusion ha opimal moneary policy under discreion does no necessarily converge on he Friedman Rule would sill apply in more general models.
27 c plan plan 0 c plan 0 plan Figure Opimum ime pahs of consumpion under commimen c c A C B expansion pah 45 o 0 c c 0 Figure 2 The governmen s opimisaion problem under commimen in reduced form
28 2 FIGURE 3 Discreionary equilibrium wih CRRA uiliy σ = 0.5, α = 5, β = 0.95; FRSS value of k = 0.04
29 3 FIGURE 4 Discreionary equilibrium wih quadraic uiliy ĉ =0.08, α = , β = 0.95; TCSS value of k = 0.04; FRSS value of k =
30 References DiazGimenez, J., Giovanei, G., Marimon, R. and Teles, P. (2002) Nominal Deb as a Burden on Moneary Policy, unpublished paper, Dep of Economics, Universidad Carlos III de Madrid Grossman, S. and Weiss, L. (983) A TransacionsBased Model of he Moneary Transmission Mechanism, American Economic Review 73, Herrendorf, B. (997) Time Consisen Collecion of Opimal Seigniorage: A Unifying Framework, Journal of Economic Surveys, 46 Lucas, R.E. (982) Ineres Raes and Currency Prices in a TwoCounry World, Journal of Moneary Economics 0, Lucas, R.E. and Sokey, N. (983) Opimal Fiscal and Moneary Policy in an Economy wihou Capial, Journal of Moneary Economics 2, Neiss, K.S. (999) Discreionary Inflaion in a General Equilibrium Model, Journal of Money, Credi and Banking 3, Nicolini, J.P. (998) More on he Time Consisency of Moneary Policy, Journal of Moneary Economics 4, Obsfeld, M. (99) A Model of Currency Depreciaion and he DebInflaion Spiral, Journal of Economic Dynamics and Conrol 5, 577 Obsfeld, M. (997) Dynamic Seigniorage Theory: An Exploraion, Macroeconomic Dynamics, Roemberg, J. (984) A Moneary Equilibrium Model wih Transacions Coss, Journal of Poliical Economy 92, Svensson, L.E.O. (985) Money and Asse Prices in a CashinAdvance Economy, Journal of Poliical Economy 93,
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