T he stated long-term goal of monetary

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1 JANUARY/FEBRUARY 999 Rober Dimar is a mahemaician and William T. Gavin is vice presiden and research coordinaor for he Federal Reserve Bank of S. Louis. Finn E. Kydland is a professor of economics a Carnegie Mellon Universiy and visiing scholar of he Federal Reserve Bank of S. Louis. Daniel R. Seiner provided research assisance. The Inflaion- Oupu Variabiliy Tradeoff and Price-Level Targes Rober Dimar, William T. Gavin, and Finn E. Kydland T he saed long-erm goal of moneary policy in he Unied Saes and around he world is price sabiliy. Eigh counries in he world now have explici arges for inflaion. Many more, including he Unied Saes, appear o operae as if hey have implici arges for inflaion. There is an ongoing debae abou how sricly one should ry o arge inflaion. The idea is ha if one ried o keep inflaion oo close o a arge, here would be a significan increase in he variabiliy of oupu and ineres raes. There is a suble bu imporan disincion o be made abou he difference beween argeing inflaion in he shor run (say, every monh) and argeing a paricular average inflaion rae over many monhs. By argeing a long-moving average of zero inflaion, or a horizonal price-level pah, he cenral bank would have an operaional arge for price sabiliy, bu would no be required o keep inflaion on an exac pah each monh or quarer. Objecions o price-level argeing usually assume ha any economic disurbance ha caused he price level o deviae from he arge would require he cenral bank o reac immediaely, and harshly, o ge he price level back on rack. Bu, here is no reason for his. Wheher argeing inflaion more closely in he long-run would lead o more or less shor-run variabiliy of inflaion and oupu depends on how he economy works and how he cenral bank runs moneary policy. By price-level argeing we mean ha he cenral bank announces a pah for he price level. I may be fla or i may be changing a a rae of x percen per year. For x 0, he price level pah will be horizonal. In any case, he noion of a price-level arge means ha he cenral bank will arge a long-run average inflaion rae, seing objecives ha correc for pas deviaions from he arge. Technically, we define a price-level-argeing regime as one in which he logarihm of he price level has a deerminisic rend. An inflaion-argeing regime is one in which he logarihm of he price level has a uni roo and follows a sochasic rend. Resuls in his paper apply o a price-level arge wheher he average inflaion rae is zero or no. Taylor (979) inroduced he idea of using he inflaion/oupu variabiliy radeoff o examine alernaive moneary policy rules. Using a raional expecaions model wih saggered wage conracs, he explained why he choice facing policymakers in a dynamic seing involves he radeoff beween oupu variabiliy and inflaion variabiliy. In his raional expecaions framework here is no long-run radeoff beween levels of oupu and inflaion. Policymakers can, however, choose alernaive poins along an inflaion/ oupu variabiliy fronier by varying he relaive weigh hey pu on inflaion versus oupu sabilizaion. Using a simplified version of Taylor s framework, Svensson (997b) shows ha, for a given level of oupu variabiliy, he shor-run variabiliy of inflaion depends on he amoun of persisence in he oupu gap and on wheher he cenral bank arges an inflaion rae or a pah for a price index. He shows ha if he oupu gap is persisen enough, he cenral bank should arge a For a more deailed descripion of he inuiion underlying he inflaion/oupu variabiliy radeoff, see Taylor (994). 3

2 JANUARY/FEBRUARY 999 Gavin and Sockman (99) explain why a sociey ha cares abou inflaion (no price level) sabiliy may sill prefer a price level arge if he source of inflaion shocks is unobservable o he public. 3 An appendix in Svensson (997b) shows ha inroducing money wih a conrol error in he inflaion equaion would no change his resuls. pah for he price level. Svensson also explains why a price-level arge can be used as a commimen mechanism o eliminae he inflaion bias ha resuls when a cenral bank ries o arge an unrealisically high level of oupu. In his paper, we explain how he inflaion-oupu variabiliy radeoff changes if he cenral bank chooses o arge a predeermined pah for he price level raher han an inflaion rae. Our analysis is more ransparen han Svensson s because we do no ry o disinguish beween cases of commimen and discreion, nor do we consider he case where he cenral bank ries o achieve an unrealisic objecive for oupu. We assume ha he cenral bank canno commi credibly o more han one period a a ime. Since he cenral bank does no ry o achieve unrealisically high levels of oupu, he seady sae inflaion raes are he same for boh inflaion and price-level argeing regimes. INFLATION VERSUS PRICE-LEVEL TARGETING IN A SIMPLE PHILLIPS CURVE MODEL The basic model described here is from Svensson (997a, 997b). The model is consisen wih a wide range of sicky-price models in which moneary policy can have imporan real effecs. The model has hree main elemens: a muliperiod objecive funcion for he cenral bank, an aggregae supply equaion, and a raional expecaions assumpion. The cenral bank minimizes an ineremporal quadraic loss funcion: () L = β λ y + π π * ( ), = 0 ( ) where y is he deviaion of oupu from he arge level (which we assume is he underlying rend in real oupu) and (π π * ) is he deviaion of inflaion from he cenral bank s inflaion arge. The cenral bank discouns fuure variabiliy in he oupu gap and inflaion by he facor. The parameer, λ, relaes he cenral bank s preference for oupu sabiliy o is preference for inflaion sabiliy. The economy is modeled as a shorrun aggregae supply curve wih persisence in he oupu gap: () The inroducion of a lagged oupu gap in his equaion is imporan for comparing inflaion and price-level argeing. Concepually, he lag will be inroduced any ime fricion prevens insananeous and complee adjusmen of oupu o unexpeced changes in he price level. This fricion could be induced by wage conracs, menu coss, ransacion coss, incomplee markes, capial adjusmen coss, ec. The slope of he shor-run Phillips Curve is given by which deermines he response of he oupu gap o unexpeced inflaion (π π e ). Wih his aggregae supply curve and raional expecaions, ha is, π e =E π, he cenral bank s opimizaion problem implies a radeoff beween oupu and inflaion variabiliy. Minimizing his loss funcion subjec o he aggregae supply curve leads o a rule for inflaion ha is coningen on he size of he oupu gap: (3) y e = ρ y + α ( π π ) + ε. A A * αλρ π = p p = π y βρ αλ βρ α λ ε, + where he superscrip A indicaes ha he variable is deermined by he inflaion-argeing rule and p is he logarihm of he price level. The inflaion rae se in each period is equal o he inflaion arge wih counercyclical adjusmens proporional o he lagged oupu gap and he curren echnology shock. Following Svensson, we assume he cenral bank can conrol inflaion direcly. 3 Deails of he soluion procedure are presened in he appendix. If he cenral bank cares abou deviaions of he price level raher han he 4

3 J ANUARY/FEBRUARY 999 inflaion rae, he naural logarihm of he price level will replace he inflaion rae in he loss funcion. We reformulae he objecive funcion as below: (4) where he arge pah for he price level may be consan or may be rising a a consan rae. The cenral bank s rule for achieving he arge pah is given by: (5) β * ( ) = 0 L = λ y + ( p p ), B * αλρ p = p y βρ αλ, βρ + α λ ε implying he following rule for he inflaion rae: expecaions, he model s Phillips Curve implies ha oupu is given by: βρ (7) y = ρ y +. βρ + α λ ε As he relaive weigh on oupu variabiliy, λ, ges large, he coefficien on he error erm ends o zero as does he variance of he oupu gap. If he variance of ε is σ, hen he above decision rule for y implies ha he uncondiional variance of he oupu gap is: ( βρ ) (8) σ y = σ ε. ( ρ )( βρ + α λ) Afer noing ha ε is uncorrelaed wih y, we can use he decision rule for π o calculae he uncondiional variance of inflaion as: (6) π B B = p p = π αλρ ( y y ) βρ αλ ( βρ α λ ε ε + ), * (9) αλρ σ π = σ βρ αλ + ( βρ + α λ) y σ ε, where we have used he assumpion ha he price-level arge, p *, is given by p * =π * +p *. The superscrip B indicaes ha he variable is deermined by he price-level argeing rule. Wih he pricelevel arge, he cenral bank s reacion funcion, Equaion 6, has hree elemens on he righ-hand side. The firs is he seadysae inflaion embodied in he arge pah for he price level. The second and hird are proporional, counercyclical adjusmens o he change in he oupu gap from period o period and he change in he echnology shock from period o period, respecively. The radeoff beween inflaion and oupu is qualiaively differen under he wo differen regimes, inflaion argeing and price-level argeing. In an inflaionargeing regime, he bank ses inflaion, π A, as shown in Equaion 3. Wih raional which can be simplified o yield an expression only involving σ, namely: α λ (0) σ π = σ ε. ( ρ )( βρ + α λ) In a price-level-argeing regime, he cenral bank ses he inflaion rae, π B, as in Equaion 5. Once again assuming raional expecaions, p e = E p, he following ime series process for he oupu gap is derived from he model s Phillips Curve, βρ () y = ρy +. βρ + α λ ε Noe ha his process for he oupu gap looks idenical o Equaion 7, which was derived in he inflaion-argeing regime. The parameer λ, however, has a differen F EDERAL RESERVE BANK OF ST. LOUIS 5

4 JANUARY/FEBRUARY 999 Figure The Oupu-Inflaion Variabiliy Tradeoff Variabiliy of he Oupu Gap Inflaion Targe inerpreaion here, as he bank s preferences are differen. The uncondiional variance of he oupu gap as a funcion of his parameer is given by he same expression as noed in Equaion 8. The uncondiional variance of he inflaion rae, however, is given by he following expression: () Price-Level Targe β = 0.99 α = 0.50 ρ = Variabiliy of Inflaion α λ σπ = σ ε. ( + ρ)( βρ + α λ) Regardless of wheher he cenral bank is argeing inflaion or he price level, a small weigh on he oupu gap leads he bank o srive for keeping inflaion or he price level close o is arge. A he exreme, where he cenral bank places no weigh on deviaions of he oupu gap, he variance of he gap is deermined by persisence in he oupu gap, ρ, and he variance of echnology shocks. Here, he bank opimizes by fixing inflaion, or he price level, a is arge in every period. There is no inflaion variabiliy, no inflaion uncerainy, and a simple auoregressive process for he oupu gap. Conversely, a large weigh on he deviaion of he oupu gap from he arge would lead he bank o use he Phillips Curve consrain o closely conrol he oupu gap by leing inflaion vary more. We graphically display he difference beween he inflaion/oupu variabiliy radeoffsin he wo regimes by firs expressing he oupu gap variance and he inflaion variance as funcions of he preference parameer λ while holding he parameers of he Phillips Curve consan. For a given λ, he bank s decision rules can be used o calculae an uncondiional variance for boh inflaion and he oupu gap (a single poin in Figure ). Varying he bank s preferences by varying λ will deermine he locaion of he curve represening he rade off beween σ π and σ y. A sample pair of variance radeoff curves are displayed in Figure. For he chosen se of parameer values, he variance radeoff under he price-level-argeing regime lies everywhere below ha for he inflaion-argeing regime. Thus, given his paricular se of parameers, sociey would prefer he price-level-argeing regime. More can be said abou he relaive posiionof hese radeoff curves. If we examine he expressions for he uncondiional variances of he oupu gap and inflaion derived above, we can fully describe he posiion of hese curves in erms of he auoregressive parameer, ρ, in he Phillips Curve equaion. Noe ha in eiher regime, if he bank places no weigh on deviaions of he oupu gap from arge, hen he bank simply ses he inflaion rae, or he price level, equal o is arge in every period. Thus, in he limi as he parameer λ approaches 0, he uncondiional variance of inflaion approaches 0, while he uncondiional variance of oupu approaches ha of he simple firs-order auoregressive process y = y +ε. Thus, he wo radeoff curves inersec he σ y -axis a he same poin. If he cenral bank s weigh on deviaions of he oupu gap from arge becomes large, hen he cenral bank ses he oupu gap equal o is arge and manipulaes he inflaion rae o reach his goal. Thus, as he parameer λapproaches infiniy, he variance of oupu approaches 0. Examining he expressions for he uncondiional variance of inflaion shows ha as λapproaches infiniy, he variance of inflaion under an inflaionargeing regime approaches (α ( ρ )), and he variance of inflaion under a pricelevel-argeing regime approaches 6

5 JANUARY/FEBRUARY 999 (α (+ρ)). Therefore, assuming ha he radeoff curves are convex for all parameer values, he radeoff curves under pricelevel-argeing regimes will lie below hose for inflaion-argeing regimes as long as (3) ( α ( + ρ)) < ( α ( ρ )), or equivalenly, ρ /. 4 Noe ha he relaive posiion of he radeoff curves does no depend on α, he slope of he shorrun Phillips Curve, or on β, he cenral bank s discoun facor. We can gain some insigh for he relaive placemen of he curves under he above condiion by considering wha happens as he auo-regressive parameer, ρ, approaches. As his happens, he oupu gap sars o behave more and more like a random walk. Under he inflaion-argeing regime, he bank ses he inflaion rae proporional o he oupu gap. Consequenly, if he oupu gap behaves like a random walk, so will he inflaion rae. Under he price-levelargeing regime, however, he bank ses he inflaion rae proporional o he change in he oupu gap. Thus, even if he oupu gap becomes non-saionary as ρapproaches, he ime pah of he inflaion rae remains saionary under such a regime. EMPIRICAL EVIDENCE The simple Phillips Curve model represens popular wisdom abou he radeoff beween inflaion and oupu variabiliy. I is insrucive o examine esimaes of he persisence in he oupu gap. We use U.S. gross domesic produc (GDP) daa where we calculae hree differen measures of he oupu gap from hree differen measures of poenial GDP: Congressional Budge Office (CBO) esimaes. A quadraic ime (QT) rend calculaed using he logarihm of real GDP. A Hodrick-Presco (HP) rend also calculaed using logarihm real GDP. Table Oupu Gap Under Alernae Definiions of Trend GDP CBO.7% Quadraic Hodrick- CBO Time Presco Quadraic Time % Hodrick-Presco % Values on he diagonal are he sandard deviaions of he oupu gap variously measured. Values on off-diagonals are he correlaion coefficiens beween he respecive measures of he oupu gap. Daa are quarerly U.S. GDP from 949:Q o 998:Q. The quadraic ime gap is calculaed as he residual in he following regression: where yis he logarihm of GDP ande^is he esimaed residual. The Hodrick-Presco gap is he deviaion from rend calculaed using he filer described in Presco (986). Table y = consan + β Time + β Time + e Esimaes of Persisence in he Oupu Gap Using U.S. GDP Daa y = c + ρ y + ω y + e i i Definiion of Trend Esimae of ρ In each case, we calculae he oupu gap as he difference beween he logarihm of real GDP and he alernae esimaes of he rend. Table shows he sample sandard deviaions and correlaions beween he differen measures of he oupu gap. The esimae based on he quadraic ime rend is he mos variable and he mos highly correlaed wih he CBO esimae. We assume he CBO esimae is closes o he daa ha he policymakers acually use. Table shows he esimaes of he auoregressive parameer calculaed for each measure of he oupu gap. The Sandard Error CBO Quadraic Time Hodrick-Presco Daa are quarerly U.S. GDP from 949:Q o 998:Q. 4 Svensson (997b) derived a similar resul for he discreion case. 7

6 JANUARY/FEBRUARY 999 Table 3 Esimaes of Persisence in he Oupu Gap (Using indusrial Producion o Measure Oupu) y = c + ρ y + ω y + e i i Counry Hodrick Presco Filer Quadraic Time Trend Filer Esimae of ρ Sandard Error Esimae of ρ Sandard Error Belgium Canada France Germany Ialy Japan Neherlands Sweden Unied Kingdom Unied Saes Daa for he G-0 are quarerly averages of monhly indusrial producion from 957: o 997: published by he Inernaional Moneary Fund. equaion used o esimae ρis shown a he op of Table. The properies of he disribuion for his esimae were discussed in Dickey and Fuller (98). By consrucion, he oupu gap is saionary so here is no prior reason o expec esimaes of ρ o be close o uniy. We find surprisingly high esimaes of ρ using boh he QT gap (0.9) and he CBO gap (0.9), however. The HP rend follows he acual series more closely han he oher wo series. The sandard deviaion is much smaller and he esimae of ρis only 0.7. Even in his case, however, he esimae is sill more han four sandard deviaions larger han This confirms Svensson s resul ha if one believes in his oupu/inflaion variabiliy radeoff, hen seing a price-level arge would mos likely resul in a more efficien se of opions for he Fed han would an inflaion arge. We also have esimaed he persisence of he oupu gap in he G-0 counries. There is a lack of hisorical quarerly GDP daa for he G-0, so we measured he persisence of he oupu gap in hese counries by aking quarerly averages of indusrial producion and using boh he HP and QT filers (see Table 3) o consruc he oupu gap. Using he QT filer, ρ is esimaed o be greaer han 0.50 and highly significan in all he counries. Using he HP filer, he resuls are mixed. Only in one case is he poin esimae below 0.50, bu in over half of he cases, he esimae is wihin one sandard deviaion of CONCLUSION In his paper we describe a popular model of moneary policy in which he cenral bank minimizes a discouned, muliperiod loss funcion ha includes deviaions of infla- 8

7 J ANUARY/FEBRUARY 999 ion and oupu from arge levels. This minimizaion is consrained bya shor-run radeoffbeween inflaion and real oupu. This simple model suggess ha he choice beween an inflaion arge and a pricelevel arge depends on characerisics of real oupu. If he oupu gap is relaively persisen, hen argeing he price level resuls in a beer se of policy opions for he cenral bank. We presen evidence from he G-0 counries showing ha convenionally measured oupu gaps are highly persisen. The policy implicaion of assuming raional expecaions and his Phillips Curve model is ha cenral banks should se objecives for a price level, no an inflaion rae. REFERENCES Dickey, David A., and Wayne A. Fuller. Likelihood Raio Saisics for Auoregressive Time Series wih a Uni Roo, Economerica 49 (June 98), pp Gavin, William T., and Alan Sockman. Why a Rule for Sable Prices May Dominae a Rule for Zero Inflaion, Economic Review, Federal Reserve Bank of Cleveland, (99 Quarer I), pp. -8. Hodrick, Rober J., and Edward C. Presco. Poswar U.S. Business Cycles: An Empirical Invesigaion, Journal of Money, Credi, and Banking, 9 (February 997), pp. -6. Presco, Edward C. Theory Ahead of Business Cycle Measuremen, Carnegie-Rocheser Conference Series on Public Policy. New York: Norh-Holland, (Auumn986). Svensson, Lars E.O. Opimal Inflaion Targes, Conservaive Cenral Banks, and Linear Inflaion Conracs, American Economic Review 87 (March 997a), pp Price Level Targeing vs. Inflaion Targeing: A Free Lunch? Insiue for Inernaional Economic Sudies, Sockholm Universiy, Augus 997b. An earlier version was published in Augus 996 as NBER Working Paper 579. Taylor, John B. The Inflaion/Oupu Variabiliy Tradeoff Revisied, in Goals, Guidelines, and Consrains Facing Moneary Policymakers, Jeffrey C. Fuhrer, ed., The Federal Reserve Bank of Boson Conference Series 38 (994), pp Esimaion and Conrol of a Macroeconomic Model wih Raional Expecaions, Economerica 47 (Sepember 979), pp

8 JANUARY/FEBRUARY 999 Appendix APPENDIX: SOLUTION OF THE CENTRAL BANK S OPTIMIZATION PROBLEM Since he cenral bank s objecive under eiher he inflaion argeing or price-level-argeing regime is quadraic and is consrains are linear, i is possible o guess ha linear-decision rules solve he bank s opimizaion problem. We show ha subsiuing he conjecured linear rules ino he firs-order condiions for he bank s opimizaion problem and equaing coefficiens will yield he decision rules described in he ex. We rea inflaion expecaions as equilibrium variables are reaed in a dynamic general equilibrium model. Tha is, we suppose ha he bank bases is decisions a ime solely on he sae variables y and ε while inflaion expecaions are lef o be deermined by a raional expecaions condiion. Consider firs he inflaion-argeing regime. We form he bank s Lagrangian as: (A) E 0 ( ) * λy + π π β y ρy = 0 µ, e απ ( π ) ε where he µ s are a sequence of random mulipliers. The bank s firs-order condiions ake he form: (A) λ y µ + βρ E µ = 0, + when aken wih respec o he sequence of y s, and he form: * (A3) ( π π )+ αµ = 0, when aken wih respec o he sequence of π s. Eliminaing he mulipliers from hese expressions gives he following Euler equaion: (A4) We now posi a linear decision rule for inflaion of he form: (A5) π =A +A y - +A 3 ε. If expecaions formed a ime - are raional hen: (A6) π e =A +A y -. Hence, he consrain imposed by he aggregae supply relaion (Equaion in he aricle) yields a decision rule for y direcly of he form: (A7) y =ρy - +(αa 3 +)ε. Noe ha decision rules are invarian so ha π + can be deermined by ieraing on he rule for π o yield he following expression: (A8) λy ( ) α π π * + βρ * E( π+ π )= 0. α π+ = A + Ay + A3ε = A + A ( ρy + ( αa + ) ε)+ A ε Subsiuing Equaions A5, A7, and A8 ino he Euler Equaion A4 above, aking expecaions, and equaing consan erms and coefficiens on he saes yields values for A, A, and A 3 in erms of parameers of he model. Deermining he bank s decision rules in he case of a price-level-argeing regime proceeds in a similar fashion. The only difference is he bank s price-level arge changes over ime, and hence p * mus ener as a sae variable in he bank s decision rules. Since his arge evolves in a deerminisic manner as p * =p * +π *, however, i is sill. 30

9 J ANUARY/FEBRUARY 999 possible o posulae a decision rule of he form p = A + A p * + A 3 y + A 4 ε, and ierae on i o calculae p + in erms of ime saes. Afer subsiuing he resulan linear rules ino he bank s Euler equaion and equaing coefficiens, we obain he decision rule for price-level argeing given in he ex. 3

10 J ANUARY/FEBRUARY 999 3