How square is the policy frontier?

Transcription

1 How square is he policy fronier? SGB Henry, M Sachi and D Vines Firs Version December 000 This version January 00 Cenre for Inernaional Macroeconomics Deparmen of Economics, Universiy of Oxford Absrac: This paper assesses he implicaions of discouning on a resul derived by Bean (998): ha in a model of moneary policy where policy acs wih a lag, he oucomes of moneary policy are very similar for a wide range of weighings of he (non-discouning) moneary auhoriy s objecive funcion, wih respec o inflaion sabiliy versus oupu sabiliy. We show ha when he auhoriy discouns he fuure, oucomes become more sensiive o preferences, and ha i is imporan o ake he discoun rae ino accoun when examining he quesion of how he auhoriy s remi should be specified. Keywords: JEL Nos: Moneary Policy, Policy Fronier, Discouning E53, E58, E6 We are graeful o Sephen Hall, Adrian Pagan and John Muellbauer for helpful commens and suggesions. All errors remain he responsibiliy of he auhors. We are graeful o he Leverhume Trus for financial suppor under gran no. F/08 59A, EMU and European Macroeconomic Policy in a Global Conex.

2 . Inroducion Here we address an imporan finding derived by Bean (998). This finding, when applied o he new UK moneary arrangemens, implies ha i does no maer ha he inflaion remi does no specify he relaive weigh ha he Bank of England should place on oupu volailiy vis-à-vis inflaion volailiy. The resul is obained under he assumpion ha he auhoriy does no discoun he fuure. In pracice, however, some discouning is likely o occur and planning horizons migh be relaively shor; for he U.K., members of he Moneary Policy Commiee are appoined for a erm of hree years, and so he average remaining erm of exising members is one and a half years. Given ha a single ime period in Bean (998) is inerpreed as corresponding o around one year, he assumpion of no discouning by he cenral bank should no be aken for graned. We examine how he above finding alers when discouning is inroduced. Recen research in moneary policy has incorporaed models wih a more realisic reamen of he lag srucure wih which policy akes effec. Svennson (997), Ball (997) and Bean (998) are recen examples. In hese models, opimal policy ypically akes he form of a Taylor rule. Bean (998) demonsraes in his seing, ha he weighing of he loss funcion of he moneary auhoriy wih respec o oupu volailiy vis-à-vis inflaion volailiy does no affec he opimal policy rule in such a way as o grealy aler he oucome of sabilisaion policy. This can be seen in he shape of he Bean policy fronier (he se of efficien oucomes) which is relaively recangular. Figure (), which will be described furher below, shows he policy fronier and he opimal oucomes for wo differen weighings of he loss funcion. These are close, boh in erms of heir absolue disance relaive o he magniude of he variance of supply and demand shocks, and also in erms of he raio of oupu volailiy o inflaion volailiy obained in each case. We find ha his resul is very robus o variaions in he esimaes of he parameers of he model. This holds o such an exen ha i is almos independen of he empirical analysis underaken by Bean himself in he original paper we find ha he propery of recangulariy holds for any se of plausible esimaes. His analysis is herefore likely o apply o cases oher han he U.K. I can also be shown ha excep in he unrealisic case where policy affecs inflaion conemporaneously, he recangulariy resul is also robus o simple variaions in he lag srucure used. Our main finding is ha he recangulariy resul, however, is dependen on he planning horizon of he policymaker. We argue ha when discouning is inroduced, he weighing of he loss funcion becomes increasingly imporan in affecing he rade-off. In a model wihou lags, he ineres rae is ypically a linear funcion of curren shocks. When policy acs wih a lag, he role of policy is o conrol he effecs of pas shocks which remain in he sysem via persisence. Policy

3 The essence of his argumen goes as follows: here as is ypical in models of moneary policy, he relaive conrol of oupu or inflaion deviaions depends on he exen o which supply shocks are conrolled a igh conrol of supply shocks corresponds o igh conrol of inflaion and lax conrol of oupu, and vice versa. However, he variances of boh fuure oupu and inflaion depend posiively on he amoun of inflaionary pressure already wihin he sysem. A myopic auhoriy akes his as given and policy is sensiive o is preferences over oupu and inflaion sabiliy. However, a far-sighed auhoriy, regardless of is preferences, has an incenive o conrol inflaion now since his will lower he fuure variance of boh oupu and inflaion. This resuls in policy being less sensiive o preferences. Dynamic moneary policy models generally rely on some form of persisence in eiher oupu or inflaion. For insance Lockwood, Miller and Zhang (998) discuss persisence in employmen while Mash (000) discusses persisence in inflaion. Boh of hese models are characerised by inflaion bias, which resuls from he auhoriy argeing a level of oupu higher han he naural rae. The level of he discoun rae is hen shown o have implicaions for he consequen issues of ime inconsisency and delegaion; for insance Mash (000) shows ha an auhoriy wih long planning horizons offses he problem of ime inconsisency. Persisence can also be explored in he case where he auhoriy arges he naural rae of oupu and hence here is no inflaion bias. Clark, Goodhar and Huang (999) provide a discussion of he difference beween policy made under commimen and discreion in he case where here is no inflaion bias bu here is persisence of inflaion. Bean also adops a framework wih persisence in inflaion and wihou inflaion bias (i.e. in which he auhoriy arges he naural rae of oupu). In comparison wih he framework pu forward by Clark e al., Bean places a greaer emphasis on he lag srucure and a lesser emphasis on he reamen of expecaions. 3 Wha we show here is ha in he Bean framework, planning horizons can have a significan impac on he recangulariy argumen i.e. ha if planning horizons are shor, he relaive weighing of inflaion and oupu variabiliy in he loss funcion does have a much greaer impac on he oucome of opimal policy. hen acs by conrolling persisence, and so he opimal ineres rae is hen ofen a funcion of he persising variables raher han he shocks. Wih simple lag srucures, his can ake he form of a Taylor rule. Bean argues ha independence of he cenral bank is likely o achieve his siuaion, since i is hen immune o he poliical pressures which cause oupu o be argeed a a level higher han he naural rae. He also suggess ha lenghening he erm of MPC members would furher insulae he MPC from shor-erm poliical pressures. Our analysis broadly suppors his measure since i could also be expeced o reduce discouning. 3 This is of course enirely naural given he differen issues ha he models analyse. For example, Clark e al. assume policy affecs boh oupu and inflaion conemporaneously. However, aking ino accoun ha he policy insrumen is he nominal rae, hey allow a role for inflaionary expecaions in he evoluion of oupu unlike he models of Ball (997), Bean (998) and Svensson (997).

4 Secion provides an ouline of he heoreical analysis presened by Bean (998) and Ball (997), and secion 3 provides he corresponding analysis in he case of discouning. Svensson (997) also solves his problem in he case of discouning; neverheless, we presen a soluion here which sylisically is more in keeping wih ha of Ball (997) and Bean (998) since i allows a clearer discussion of recangulariy argumen. 4 Secion 4 discusses he implicaions of discouning on recangulariy and secion 5 concludes.. An Ouline of he Bean Model Bean (998) and Ball (997) use he following equaions for aggregae demand and supply. Aggregae demand is given by y µ r λ η wih η ~ N (0,σ η ) (D) = y where r, he real ineres rae, is he policy insrumen. Aggregae supply is given by an acceleraionis Philips curve: 5 π = π αy ε wih ε ~ N (0,σ ε ) (S) So real ineres raes affec oupu wih a lag of one period, oupu affecs inflaion wih a furher lag, and one period is inerpreed as corresponding o a ime span of around one year. The loss funcion of he non-discouning moneary auhoriy is given by: L = Var( π ) βvar( y) (L) Bean provides a clear derivaion of he opimal rule. The key observaion is ha he rule mus be of he form: ( π αy) Ey = ρ Eπ = ρ () The consan ρ is referred o as he feedback parameer. Equaion () implies ha oupu moves in such a way ha as far as possible pas demand shocks, which increase he volailiy of boh oupu and inflaion, are removed from he sysem. The level of ρ deermines o wha exen supply shocks, which have opposing effecs on oupu and inflaion volailiy, are conrolled a high ρ corresponds o a igh conrol of supply shocks and so inflaion. In effec, by imposing () we are choosing an efficien sequencing marix described below in 4 In paricular of he feedback parameer, defined below. Svennson s soluion, in many ways more direc, is ideal for his consideraion of he inflaion forecas as an inermediae arge. 5 ζ This he reduced form of he equaion π = ( ω) Επ ωπ ζy ε where α = ω. 3

5 (3). To see he mahemaical basis for equaion () - see Svensson (997) - we simply recognise ha Ey can be considered as he conrol variable for his dynamic problem. Maximising he loss funcion (which due o he linear-quadraic srucure of he model is a quadraic funcion of expeced inflaion deviaions) subjec o he consrain (S) gives a firs order condiion ha is a rule of he form (). Equaion () hen deermines he form of he opimal ineres rae rule, which is a Taylor rule, {[ λ αρ] ρπ }/ µ r = y () The curren values of inflaion and oupu in erms of las period s values are hen given by he following marix, which we shall refer o as he sequencing marix: y αρ = α ρ y π π η ε (3) I is imporan o noe ha he sequencing marix is of rank in his sysem, his is a consequence of following an efficien policy given by (). As will be described below, his rank condiion can be seen as he essenial cause of recangulariy in he case when here is no discouning. Equaion (3) is used o calculae he uncondiional variances: 6 Var( y) = Var( π ) σ η [ ] [ ] αρ αρ (/ αρ) αρ σ ρ / α ε (4) These expressions for he variances are subsiued ino he objecive funcion (L) and his allows us o calculae he value of he feedback parameer ρ ha minimises he loss. This is he posiive roo of he equaion: βρ αρ = 0 (5) The issue of he recangulariy of he policy fronier is easily described using (L). Once he opimal rule has been implemened, for a wide range of values of β, here is a small difference in he final values of Var(y) and Var(π) ha are obained. Figure () shows he shape of he policy fronier which, replicaing ha in Bean, is shown in he space of he 6 Bean uses he resul ha if X=BX - E where B is a conformable marix and E is a vecor noise process, hen Vec[Var(X)] = [ (B B)] Vec[Var(X)]. 4

6 sandard deviaions of oupu and inflaion. The oucomes for wo differen values of β, β= and β=3, are shown and can be seen o be close as described in he inroducion.. The Effec of Parameers Esimaes and Lag Srucure on Recangulariy In figure, we use he se of parameer esimaes obained by Bean (998) in his original empirical invesigaion. These are σ η =.55 and σ ε =.5 for he demand and supply shocks respecively and α = We find ha he recangulariy propery is robus o realisic aleraions in he parameer values and lag srucure of he model (accordingly, we use Bean s parameer esimaes in all subsequen figures 7 ). To examine he effec of lag srucure, we consider he following alernaive equaions for demand and supply respecively: y µ r λy η (D) = π = π αy ε (S) In (D) he lag in ineres raes has been removed so policy affecs oupu conemporaneously, while in (S) he lag in oupu has been removed so oupu affecs inflaion conemporaneously. If we consider he wo economies (S) & (D) and (S) & (D), we find ha he propery of recangulariy remains. In boh of hese economies, policy affecs inflaion wih a one period lag: his is enough o ensure ha he policy fronier becomes verical as he preferences of he auhoriy end o pure inflaion argeing, since policy canno miigae o any exen he conemporaneous supply shock. Only in he unrealisic model (S) & (D), where policy acs wihou lags and he fac ha policy can conrol he conemporaneous supply shock implies ha he policy fronier does no become verical in he limi, is he shape of he policy fronier noiceably less recangular. 8 Figure () shows he shapes of he policy fronier for all hese various models ogeher wih he oucomes for β= and β=3. 3. The Opimal Rule wih Discouning We will now use a loss funcion wih discoun facor δ: L s E δ ( π s βy ) (L) s s= 0 = 7 A recangular fronier is obained from wide ranges of parameer values. Deails available from he auhors. 8 In he model wihou lags, and for small values of β, decreasing persisence in inflaion in he model furher reduces recangulariy. This is very much no he case in he model wih lags, see foonoes and 9. 5

7 Noe ha when he moneary auhoriy minimises (L) he long run mean of boh variables is zero, and so his is he same as minimising (L) in he limi as δ. Our analysis in his secion proceeds in hree sages. Firs we idenify wo mechanisms by which he inroducion of discouning can affec opimal policy. Secondly we show ha as an auhoriy becomes more paien, boh of hese mechanisms cause he auhoriy o be more concerned wih conrolling inflaion regardless of is iniial preferences. Thirdly we describe he effec of his on he sensiiviy of policy o he weighing of he objecive funcion. The sensiiviy of policy o he weighing of he objecive funcion is now primarily wha we refer o when we discuss propery of recangulariy, raher han he shape of he fronier; he shape of he policy fronier in erms of he long-run variances of oupu and inflaion will no change when discouning is inroduced. However, he shape of policy fronier in he space of shor-run variances of oupu and inflaion is much less recangular, as seen below. Wih discouning, he opimal rule mus sill ake he same form () as above, bu he feedback parameer ρ will now be a funcion of boh β and δ. Allowing discouning alers opimal policy in wo ways. In his sysem, ineres raes affec oupu wih a one period lag and inflaion wih a wo period lag. Since policy now affecs oupu and inflaion wih differen lags, changing he discoun facor acs o change he relaive weigh on oupu in he loss funcion: for a given β, as he auhoriy becomes more myopic, i becomes more concerned wih oupu sabilisaion since i can only influence inflaion a a longer horizon. We will call his mechanism he weighing mechanism. Discouning alers policy by anoher mechanism described below which we shall refer o as he invesmen mechanism. To separae he effecs of he wo mechanisms, we make he following ~ reparamerisaion β = βδ. Varying δ while holding ~ β consan hen allows us o isolae he effec of he invesmen mechanism, since he relaive weighing in he loss funcion on oupu and inflaion deviaions a he horizon a which hey are affeced by curren policy does no hen change. Since he opimal rule under discouning remains of he same form (), he sequencing marix also akes he same form (3) wih respec o he feedback parameer. Because of he lag srucure, a ime he ineres rae decision affecs E[y ] and E[π ]. From (3) we have, ] = [ αy π ] E[ y ρ C (6) ] = ( ρα) [ αy π ] E[ π C (7) where he consans rewrie (6) and (7) as C = σ and η C = α σ η σ. Noing ha α y π = E π ] we can ε [ 6

8 ] = E[ π ] E[ y ρ C (8) ] = ( ρα) E[ π ] E[ π C (9) Equaions (8) and (9) provide he key o undersanding he oher effec of discouning on recangulariy (he invesmen mechanism ). If we ake E[π ] as given, he relaive deviaions of oupu and inflaion will be deermined by he feedback parameer: a high feedback parameer conrols inflaion bu leaves oupu far from is arge, whereas a low feedback parameer does he opposie. This is he rade-off ha a myopic (δ=0) auhoriy faces. However, he rade-off appears differen for a long-sighed auhoriy. In he fuure, oupu and inflaion deviaions will depend no only on he value of he feedback parameer, bu boh depend on he expeced deviaion of inflaion in preceding periods. By geing inflaion o arge now, he auhoriy reduces he expeced deviaions of boh inflaion and oupu in he fuure. Reducing inflaion now can be seen as an invesmen which creaes long-erm sabiliy of boh inflaion and oupu. So as he auhoriy looks o he fuure, regardless of wheher i is primarily concerned wih oupu or inflaion sabiliy, i becomes more concerned wih conrolling inflaion in he presen. Hence he policies of auhoriies wih differen preferences converge as hey become more far-sighed. 9 Noe ha in general, if he sequencing marix is of rank, here will be a paricular linear combinaion of (expeced) oupu and inflaion on which boh inflaion and oupu depend. As an auhoriy becomes more far-sighed, is policy will become more concerned wih conrolling his. Here, a far-sighed auhoriy will conrol inflaion more ighly han a myopic one and have a higher feedback parameer. We now verify his. I is shown in he appendix ha using he above reparamerisaion he feedback parameer is given by he equaion: ~ ~ [ β ( )( δ ) α] = 0 βδρ α ρ (0) For he opimal policy rule we ake he posiive roo of (0), and i can be seen ha his lies in he range [ 0, α ]. Noe ha for he case δ = equaion (0) gives he feedback parameer given by (5) above. Differeniaing (0) wih respec o δ while holding β ~ consan: 9 This invesmen argumen depends on here being some persisence in inflaion. One migh hen imagine ha removing persisence from he sysem would reduce recangulariy. This is no so: while reducing persisence in inflaion decreases he srengh of he invesmen mechanism, in a model wih lags i also reduces he rade-off beween oupu and inflaion. Wihou any persisence, he policy fronier is upward sloping and policy compleely insensiive o preferences. To see his, make he following observaion. If a policy maker cares only abou oupu sabiliy she jus uses ineres raes o conrol demand. If she cares abou inflaion as well, she uses oupu o reduce he persisence in inflaion (see foonoe ) hereby increasing he volailiy of oupu. This is he rade-off. If here is no persisence in inflaion however, she has no need o do his; minimising oupu volailiy hen minimises inflaion volailiy. Persisence causes he rade-off, bu also inroduces a mechanism by which discouning alers he opimal policy oucome. 7

9 ~ ~ ~ [ βδρ β ( )( δ ) α] βρ( ρ ) = 0 ρ α α () Since he erm in he square brackes is sricly posiive and ρ 0, ], i follows ha ρ is an [ α increasing funcion of δ. Unless he weighing objecive funcion is such ha i corresponds o a case of pure inflaion argeing or pure oupu sabilisaion, ρ is a sricly increasing funcion of δ. The invesmen mechanism augmens he weighing mechanism: he more paien he auhoriy he igher is he policy in conrolling inflaion. 3. Discouning in an Alernaive Framework As discussed above, he weighing mechanism relies on he fac ha policy affecs oupu and inflaion a differen horizons. Suppose we waned o consider he effec of discouning on a sysem where policy affeced boh oupu and inflaion a he same horizon. Consider for example he sysem (S) & (D) which as seen in figure () has he propery of recangulariy when here is no discouning. We refer o his as he one-lag case. The opimal rule now has he form Ey = ρπ which gives us he analogues of equaions (8) and (9): ] = π E[ y ρ C () ] = ( ρα) π E[ π C (3) where C = σ and C η = α σ η σ ε Here here is no weighing mechanism bu he invesmen mechanism remains he same: i can be shown ha he feedback parameer is given by an equaion idenical o (0) save for he reparamerisaion made above: 0 [ β ( )( δ ) α] = 0 βδρ α ρ (4) 3. Policy Equivalence Curves We can summarize as follows: he sysem of equaions for oupu and inflaion deermines how policy affecs he sequencing marix. This dicaes he form of he opimal rule in erms of a feedback parameer ha is consan over ime. Once he form of he opimal rule has been deermined, i only remains o solve for he feedback parameer o specify policy precisely. The feedback parameer is a funcion of he parameers β and δ, respecively preferences over inflaion and oupu sabiliy, and he level of discouning. 0 Deails available from he auhors. 8

10 Ignoring he weighing mechanism and considering he one-lag case for he momen, he feedback parameer is given by equaion (4). Then reaing ρ as fixed in (4) we can hen plo curves in ( β, δ ) space which are he locus of poins which yield equal values of ρ and so equivalen policy rules. We refer o hese as policy equivalence curves. Figure (3A) shows a series of such curves for equaion (4). Bearing in mind ha hese curves are conour lines represening equal values of ρ and ha conours wih lower values of ρ correspond o higher values of β, figure (3A) demonsraes graphically he resul derived above, ha for a given value of β he feedback parameer is increasing in δ i.e. ha policy becomes igher as he auhoriy becomes more far-sighed. We can also see ha as he auhoriy becomes more far-sighed and δ increases, changes in β resul in smaller changes in he feedback parameer. As argued above, when he auhoriy is far sighed, he weighing of he objecive funcion maers less in he formaion of policy. Figure (3B) shows he resuls in he original Bean framework using equaion (0) where he weighing mechanism is aken ino accoun; we can see ha boh hese feaures remain, only he effec is more dramaic. 4. The Implicaions for Recangulariy How do we assess he implicaions of hese resuls for recangulariy? As an example, we can calculae he compleely myopic feedback parameer in he one-lag case when δ=0 i.e. when he auhoriy jus looks one period ahead: α ρ = α (5) β The shor run policy fronier given by () and (3) in { E[y ], E[π ] } space depends on he curren level of inflaion, paricularly is magniude relaive o he sandard deviaion of he supply and demand shocks. I is easy o see ha his will no be recangular. Figure 4A shows how he shor-run policy fronier shifs ouwards as he curren level of inflaion increases, wih he policy oucomes marked for β= and β=3. Suppose a social planner who did no discoun he fuure waned o consider he impac of no specifying he weighing of he objecive funcion of her myopic moneary auhoriy. Would similar oucomes resul from a wide range of weighings? To address his quesion, we use he myopic feedback rule for differen values of β o calculae he long-run variances of inflaion and oupu. Noe ha in he space of long-run variances, since he form of he opimal rule is he same excep for he value of he feedback coefficien, he shape of he policy fronier will remain unchanged. The effec of myopia is shown on he disance 9

11 beween oucomes for wo differen values of β; hese can be seen for β=3 and β= in figure 4B for he one-lag case. The filled circles show he oucomes for a perfecly farsighed auhoriy, whereas he hollow circles show he oucome for myopia, and as can be seen hese are appreciably furher apar. When he auhoriy is myopic oucomes are more sensiive o differences in he weighing. In his original analysis Bean provides an analysis of he excess loss ha resuls from he auhoriy choosing he wrong β in calculaing he feedback coefficien. Supposing for insance ha a social planner has a paricular value of β, say β=, he excess loss diagram shows he percenage excess loss, from he poin of view of he planner, when policy is implemened by an auhoriy wih a weighing β*. In he case wih discouning, we can augmen his analysis by supposing ha he planner and he auhoriy each have heir own discoun rae. Supposing ha he planner does no discoun he fuure (i.e. δ=), figure 5A shows he excess loss from he planner s poin of view boh for an auhoriy wih δ= and for an auhoriy wih δ=0.65 in he one-lag case ; figure 5B shows he corresponding diagram for he wo-lag case. In boh diagrams, he dashed line shows he loss funcion when he auhoriy does no discoun, and he coninuous line when i does. We can see from he policy equivalence curves in figure 4, ha when he auhoriy and he social planner have differing discoun raes, he excess loss of he social planner will be zero for a paricular weighing of he auhoriy s objecive funcion β* wih β*. This is verified in figure 5. As we can see, consideraion of he relaive discoun raes can be imporan in he excess loss raes; when he auhoriy does no discoun he fuure excess loss raes are relaively low for a wide range of values of β* whereas when he auhoriy does discoun, he excess loss can be much higher. 5. Conclusions Whils i is rue in general ha a recangular policy fronier implies ha similar policy oucomes are obained from a wide range of weighings of he objecive funcions, i should be emphasised ha an imporan implicaion of recangulariy of he policy fronier is ha i makes sense for he cenral bank o concern iself wih oupu volailiy o a leas some sricly posiive exen. As can be seen in figure, a cenral bank ha purely arges inflaion will no exploi he recangulariy of he policy fronier; i will obain an oucome a he poin A whereas i migh be considered socially desirable o be in he general viciniy of B. From I is no helpful o consider a compleely myopic auhoriy in he wo-lag case, since he auhoriy will only care abou oupu, and so we obain ρ=0 and an infinie variance of inflaion regardless of he value of β. In general, however, because of he addiional impac of he weighing mechanism, he sensiiviy of oucomes o β is furher heighened in he wo-lag case when δ is small. In he one-lag case discouning has relaively lile impac on policy when β is small bu non-zero; his is no rue in he wo-lag case because of he weighing mechanism. 0

12 poin A, large gains in oupu sabiliy can be obained by compromising o a small degree on inflaion volailiy, since he slope of he policy fronier becomes verical a poin A. The implicaions for recangulariy of discouning may be summarised as follows. When he auhoriy is myopic, i focuses on he shor-run policy fronier which is no recangular. The auhoriy cares less abou inflaion he more i discouns he fuure, and is acions become increasingly sensiive o he weighing of is objecive funcion. From he poin of view of a social planner who does no discoun he fuure, large excess losses can be generaed by no obviously implausible combinaions of β and δ for he auhoriy. Wha does his say abou he way in which an auhoriy s remi should be specified? Suppose sociey wans o achieve on oucome a A on he long-run policy fronier. This can be achieved easily he auhoriy should engage in pure inflaion argeing and wheher i discouns he fuure or no is irrelevan since discouning has no effec on he acions of an auhoriy ha purely arges inflaion. However, if sociey wans o exploi he recangulariy of he long-run fronier and achieve an oucome in he viciniy of B, hen i does become imporan o consider he discoun rae of he auhoriy. Given mixed argeing and no discouning, recangulariy implies ha we are fairly likely o end up a B. Of course discouning per se does no imply an oucome away from B; were he auhoriy argeing inflaion excessively a he expense of oupu sabilisaion, increased discouning could help achieve an oucome a B. Wih discouning however, policy oucomes become more sensiive o he value of β, and again many no obviously implausible combinaions of β and δ can resul in an oucome far from B. This is paricularly rue of he wo-lag case where he weighing mechanism plays an addiional role. Since he relaive weighing of a moneary auhoriy s objecive funcion is ofen no specified in is remi, as in he U.K. case, agens in he economy in general may no have a precise idea of he auhoriy s β. When i discouns he fuure less, he acions of he auhoriy become less sensiive o he value of β and ough o be more predicable, so sabiliy is likely o be improved when he auhoriy is paien. So, provided he auhoriy does no arge inflaion excessively, arrangemens ha decrease myopia 3 ough o resul in greaer long run sabiliy. Bibliography Ball L. 997 Efficien Rules for Moneary Policy NBER Working Paper W595 The reason we consider ha such a low discoun facor (or high discoun rae) migh be plausible is for he reasons menioned in he inroducion: ha members of he MPC serve a erm of hree years, so he average remaining erm is one and a half years; and ha a period in he model corresponds o around one year. 3 For insance for he U.K., by increasing he lengh of service on he MPC.

13 Bean C. 998 The new U.K. moneary arrangemens Economic Journal 08: Clark P.B., Goodhar C.A.E. and Huang H. 999 Opimal moneary policy rules in a raional expecaions mode of he Phillips curve Journal of Moneary Economics 43: Lockwood B., Miller M. and Zhang L. 997 Designing moneary policy when unemploymen persiss Economica 65: Mash R. 000 The ime inconsisency of moneary policy wih inflaion persisence mimeo Universiy of Oxford Svensson L.E.O. 997 Inflaion forecas argeing: Implemening and monioring inflaion arges European Economic Review 997 4: -46 Appendix Define V s ~ E δ ( π s βy s ). Then he loss funcion (L) can be wrien: s= 0 = (( α y π ) σ = δ V E[ π ] = δv ω) ε L (A) Since ρ only eners (A) via V, by he principle of opimaliy, ~ V = min E[ π βy δv ] ρ (A) From equaions (8) and (9) in he main ex, we can see ha V will have he form: = A0 A E[ ] V π (A3) Subsiuing equaions (8), (9) and (A) ino he righ hand side of (A) we ge: ~ ~ [( ω) ( Aδ ) βρ ) E[ π ] δa0 βc ( δa ) C ] ρα V = min (A4) ρ Using he envelope heorem, we can hen obain he firs order condiion for ρ : ~ ρα βρ = α A δ (A5) ω ( )( ) ω Comparing he coefficien of E[π ] in (A3) and (A4) we hen obain ~ A = ρα ( A δ ) βρ Subsiuing for ( ) ω ~ β ρ from (A5) gives ρα ( )( A ) ω δ A = (A6) ρ = Comparing (A5) and (A6) we can see ha ( α ) ωβ ~ ~ βδρ [ β ( )( δ ) ( α α ω) ] ρ = 0 ~ A. Subsiuing his ino (A5) gives ω (A7)

14 Figure The Bean Policy Fronier Figure Lag Srucure 3

15 Figure 3 A. Policy Equivalence Curves B Equivalence Curves wih he Weighing Effec 4

16 Figure 4 A. Shor-Run Policy Froniers B. Long-Run Policy Froniers Myopic and Far-Sighed Oucomes 5

17 Figure 5 A. Excess Losses in he One-Lag Case wih and wihou Discouning B. Excess Losses in he Two-Lag Case wih and wihou Discouning 6