working Price-Level and Interest-Rate Targeting in a Model with Sticky Prices by Charles T. Carlstrom and Timothy S.

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1 working p a p e r Price-Level and Ineres-Rae Targeing in a Model wih Sicky Prices by Charles T. Carlsrom and Timohy S. Fuers FEDERAL RESERVE BANK OF CLEVELAND

2 Working Paper 9819 Price-Level and Ineres-Rae Targeing in a Model wih Sicky Prices by Charles T. Carlsrom and Timohy S. Fuers Charles T. Carlsrom is an Economis a he Federal Reserve Bank of Cleveland. Timohy S. Fuers is Associae Professor of Economics a Bowling Green Sae Universiy, Bowling Green, Ohio, and a visiing consulan wih he Federal Reserve Bank of Cleveland. Working papers of he Federal Reserve Bank of Cleveland are preliminary maerials circulaed o simulae discussion and criical commen on research in progress. They may no have been subjec o he formal ediorial review accorded official Federal Reserve Bank of Cleveland publicaions. The views saed herein are hose of he auhors and are no necessarily hose of he Federal Reserve Bank of Cleveland or of he Board of Governors of he Federal Reserve Sysem Working papers are now available elecronically hrough he Cleveland Fed's home page on he World Wide Web: hp:// Augus 1998

3 Price-Level and Ineres-Rae Targeing in a Model wih Sicky Prices by Charles T. Carlsrom and Timohy S. Fuers This paper examines a sandard sicky price moneary model. The equilibrium condiions of he model are perurbed relaive o he canonical real business cycle model by wo varying disorions: marginal cos and he nominal rae of ineres. The paper explores he implicaions of wo moneary policies ha are frequenly advocaed: (1) an inflaion arge and (2) an ineres rae arge. Under an inflaion rae arge, marginal cos is sabilized while he nominal rae is variable. In conras, under an ineres rae arge, he nominal rae is sabilized bu marginal cos is (in general) variable. Boh policies are subjec o sunspo flucuaions arising from he endogenous movemen of he money sock. These flucuaions can be avoided by eliminaing he conemporaneous response of he money sock o innovaions in he environmen.

4 Price-Level and Ineres-Rae Targeing in a Model wih Sicky Prices Charles T. Carlsrom a Timohy S. Fuers b a Federal Reserve Bank of Cleveland, Cleveland, Ohio, 44101, USA. b Deparmen of Economics, Bowling Green Sae Universiy, Bowling Green, Ohio, 43403; and Consulan, Federal Reserve Bank of Cleveland. Augus 1998 ABSTRACT: This paper examines a sandard sicky price moneary model. The equilibrium condiions of he model are perurbed relaive o he canonical real business cycle model by wo varying disorions: marginal cos and he nominal rae of ineres. The paper explores he implicaions of wo moneary policies ha are frequenly advocaed: (1) an inflaion arge and (2) an ineres rae arge. Under an inflaion rae arge, marginal cos is sabilized while he nominal rae is variable. In conras, under an ineres rae arge, he nominal rae is sabilized bu marginal cos is (in general) variable. Boh policies are subjec o sunspo flucuaions arising from he endogenous movemen of he money sock. These flucuaions can be avoided by eliminaing he conemporaneous response of he money sock o innovaions in he environmen. Key Words: General Equilibrium; Money and Ineres Raes; Moneary Policy. JEL Classificaion: D51; E42; E52. The views saed herein are hose of he auhors and no necessarily hose of he Federal Reserve Bank of Cleveland or hose of he Board of Governors of he Federal Reserve Sysem.

5 1. Inroducion This paper examines wo radiional moneary policies in an economy in which money is non-neural because some nominal goods prices mus be se in advance. The assumed nominal rigidiy leads o a real rigidiy: in he absence of cenral bank acion, he marke economy responds sluggishly o real shocks o he environmen. A benevolen cenral banker will aemp o eliminae his real rigidiy by varying money growh in response o he real shocks. However, informaional resricions make i unlikely ha he cenral bank can vary he money supply in response o fundamenal shocks. Insead, he cenral bank mus rely on a more readily observable arge. This paper considers wo frequenly advocaed arges: a price level arge, and an ineres rae arge. The model s equilibrium condiions are perurbed relaive o he canonical real business cycle model by wo varying disorions: marginal cos and he nominal rae of ineres. The firs disorion arises because of sicky prices, while he laer arises from he cash-in-advance consrain. The wo arges price level and ineres raes--have opposing effecs on hese wo disorions. Under a price level arge, marginal cos is sabilized bu he nominal rae is variable. Under an ineres rae arge, he nominal rae is sabilized bu marginal cos is (in general) variable. Thus choosing he arge is isomorphic o choosing which disorion is variable and which is sabilized. Boh policies are subjec o sunspo flucuaions arising from he endogenous movemen of he money sock. Tha is, here is no a unique money reacion funcion ha suppors eiher a price level or an ineres rae arge, and each of hese reacion funcions leads o differen real oucomes. These sunspo flucuaions can be avoided 1

6 by specifying he money reacion funcion. One naural way of doing his is o limi he conemporaneous response of he money sock o innovaions in he environmen. The disadvanage of his approach is ha i hinders he economy s response o real shocks. There is a voluminous lieraure on opimal moneary argeing. A seleced review of papers close in spiri o he curren analysis include Carlsrom and Fuers (1995), Ireland (1996), and Goodfriend and King (1997). Carlsrom and Fuers (1995) examine ineres-rae argeing in a limied paricipaion environmen and demonsrae ha one advanage of an ineres rae peg is ha i enhances he privae economy's abiliy o efficienly respond o shocks. 1 Ireland (1996) demonsraes ha in a sicky price model he firs-bes moneary policy implies pegging he nominal rae o zero. 2 Finally, Goodfriend and King (1997) use a sicky price framework o demonsrae he advanages of price-level argeing. An ineresing implicaion of all hese analyses is ha by overcoming he source of moneary non-neuraliy, a welfare-enhancing moneary policy exacerbaes oupu flucuaions arising from echnology shocks. The paper proceeds as follows. The nex Secion oulines he basic model. Secions 3 and 4 examine he effecs of a price-level arge and ineres-rae arge, respecively. Secion 5 considers hese arges under he addiional consrain ha he 1 See Lucas (1990), Fuers (1992), and Chrisiano and Eichenbaum (1992), for analyses of limied paricipaion models. 2 The phrase firs-bes is no quie accurae. The disorion creaed by he assumpion of imperfec compeiion canno be undone by any moneary policy. Ireland (1996) demonsraes ha he opimal moneary policy in his disored world is sill characerized by pegging he nominal rae o zero. Carlsrom and Fuers (1998) demonsrae ha here are some concerns wih his firs-bes approach o policy. The foremos problem is ha he combinaion of indeerminae real cash balances and demanddeermined oupu opens up he possibiliy of sunspo equilibria ha would no arise if he nominal rae were pegged above zero. The reason hese equilibria do no arise in he case of a posiive nominal rae is ha a binding cash consrain places an addiional resricion on equilibrium behavior. 2

7 money supply is predeermined. Secion 6 concludes. 2. A Model wih Sicky Prices We follow Chari, Kehoe, and McGraan (1996), and uilize a model of imperfec compeiion in he inermediae goods marke. 3 Since he model has become fairly sandard, we will be quie economical in our presenaion. Households are infiniely lived, wih preferences given by E 0 β U ( c,1 L ) = 0 where E 0 denoes he expecaion operaor condiional on ime-0 informaion, β (0,1) is he personal discoun facor, c is ime- consumpion, L is ime- labor, and he leisure endowmen is normalized o uniy. To purchase consumpion goods, households are subjec o he following cash-in-advance consrain: P c M + M ( G 1) B s where P is he price level, M denoes beginning-of-period cash balances, M s denoes he per household money supply, and G denoes he (gross) money supply growh rae, G M s +1/ M s. Noe ha moneary injecions are carried ou as lump sum ransfers a he beginning of each period. B denoes he household's choice of one-period nominal bonds, each promising o pay R dollars a he end of ime. The exisence of hese bonds has no effec on he equilibrium of his represenaive agen economy, bu simply allows us o deermine he equilibrium nominal rae of ineres. The household's 3 Chari, Kehoe, and McGraan (1996), are in urn building on he imperfec compeiion model of 3

8 ineremporal budge consrain is given by: M M M ( G 1) + P ( w L + r K ) P c P [ K 1 (1 δ ) K ] +Π + B ( R s + 1= + + where w and r are he wage and renal raes, K is he capial sock, δ is he rae of capial depreciaion, and Π denoes firm profis. Noe ha we are assuming ha capial accumulaion occurs a he household level (and ha invesmen is a credi good). Final goods producion in his economy is carried ou in a perfecly compeiive indusry ha uilizes inermediae goods in producion. The CES producion funcion is given by Y = { [ y ( i) ] di} 1 0 ( η 1)/ η η/( η 1) where Y denoes he final good, and y (i) denoes he coninuum of inermediae goods, each indexed by i [0,1]. The implied demand for he inermediae good is hus given by y (i) = Y [P (i)/ P ] -η 1) where P (i) is he dollar price of good i, and P is he final goods price. Inermediae goods firm i is a monopolis producer of inermediae good i. Fracion ν of hese firms se heir prices flexibly wihin each period, while he remainder (1-ν) mus se heir price one period in advance. The variable ν is hus a measure of price flexibiliy. Below we will refer o hree naural varians of he model: (1) wih ν = 1, his is a flexible price model, (2) wih ν beween zero and one, his is a sicky price model, and (3) wih ν = 0, his is a model wih rigid prices. Oher han he difference in he iming of pricing, he firms are all symmeric, so we will henceforh Blanchard and Kiyoaki (1987). 4

9 drop he firm-specific noaion. Le f P denoe he flexible price, while s P will denoe he pre-deermined (or sicky) price. The final goods price (or aggregae price level) is given by he appropriae average of hese wo prices: s(1 { η ) f (1 ν + ν η ) 1 (1 ) P P } η P =. (1) The flexible price is given by a consan mark-up over he marginal cos (z ) of producion: 1 f η P = P z. (2) η 1 The erm in brackes will appear frequenly below, so we define z (η-1)/η < 1. In a model wih flexible prices, equaion (2) and he assumpion of symmery implies ha z = z. Combining (1)-(2), we have s P = h z P whereh z 1 ν ( ), ( ) z 1 ν z 1 η 1/( 1 η) (3) The funcion h is increasing so ha innovaions in marginal cos correspond wih changes in he price level. As for he sicky price, i is given by he soluion o he following maximizaion problem: P = arg max E s 1 µ + 1 P PY P s η P P s z where E -1 is he expecaion condiional on ime -1 informaion. The inermediae goods firm is owned by he household, and pays is profis ou o he household a he end of each period. Because of he cash-in-advance consrain on household 5

10 consumpion, he firm discouns is profis using µ +1 βu c (+1)/P +1, he marginal uiliy of $1 in ime +1. The firm's opimal prese price is given by: s η E 1 P = η 1 E Using (3), his can be wrien as η+ 1 { µ } + 1P zy η { } 1 µ + 1P Y η η+ 1 [ z Y [ h( z )] ] = E [ z Y [ h( z ] 1 µ µ + 1 )] E. (4). As for producion, he inermediae firm rens capial and hires labor from households and uilizes a CRS producion funcion, α y = θ f(k,h ) θ K H 1 α where θ is a measure of aggregae produciviy, K denoes capial, and H denoes labor. Wih z as marginal cos, we hen have r = z θ f K (K,H ) and w = z θ f H (K,H ). As for he echnology variable, we assume ha i evolves in he sandard fashion: θ = exp(ζ ), where ζ = ρ θ ζ -1 + ε θ, E -1 (ε θ ) = 0, where ρ θ is he auocorrelaion coefficien, and ε θ is he innovaion in echnology. 4 Moneary policy is given by a process for P or R. In eiher case, G is endogenous wih he cenral bank varying he money supply growh rae passively o achieve he argeed price level or nominal rae. The exreme versions of hese wo policies are an inflaion rae peg P +1 = πp wih π > β, and an ineres rae peg, R = R > 1. There are four markes in his economy, he labor marke, he money marke, he bond marke, and he goods marke, and he respecive marke-clearing condiions are 4 Alhough we resric he analysis o echnology shocks, he principle heoreical conclusions of he paper (eg., Proposiions 1 and 2) are valid for a wider se of shocks including fiscal behavior. 6

11 given by L = H, M = M s, B = 0, and c + K +1 = θ f(k, L ) + (1-δ) K. (5) A recursive compeiive equilibrium is given by saionary decision ha saisfy (3), (4), (5), and he following: U 1 ()/U c () = z θ f L (K, L )/R (6) U ( ) c U = β E R c ( + 1)[ z θ f K R ( K , L + 1 ) + (1 δ )] (7) U c ( ) P = β R U c ( + 1) E P 1 + (8) P c = M G (9) Before proceeding, a few observaions abou he sicky price equilibrium are in order: Firs, (6)-(7) are remarkably similar o he sandard real business cycle (RBC) Euler equaions for labor choice and capial accumulaion. They differ only in ha boh are disored from he Pareo opimum by he nominal rae of ineres (R ) and he marginal producion cos (z ). As neiher disorion eners ino he resource consrain (5), R and z can be inerpreed as disorionary axes ha are rebaed o he represenaive household. Using his public finance inerpreaion, τ c = (1-1/R ) can be 7

12 viewed as a consumpion ax, and τ w = (1-z ) as he ax rae on wage and renal income. Sandard argumens from public finance sugges ha consan ax raes would be preferred o variable ones. Hence, one naural goal of moneary policy is o sabilize R and z. Second, a log-linear approximaion o he sysem implies ha we can use (4) o se z +1 = z in he capial accumulaion equaion (7). Tha is, flucuaions in he implici ax on facor income are no persisen bu arise only in he period of a shock. Thus flucuaions in he marginal cos disorion have an effec on curren labor inpu, bu (o a log-linear approximaion) have no effec on capial accumulaion. We will henceforh refer o hese flucuaions as movemens in he wage ax. This behavior is in conras o he nominal ineres rae disorion where flucuaions are likely o be persisen. For example, under an inflaion rae arge a shock ha leads o persisen movemens in he real rae will lead o persisen movemens in he nominal rae. Third, recall ha if all prices are flexible hen marginal cos is consan, z = z. Hence, he sicky price assumpion affecs real behavior via movemens in he marginal cos disorion on facor income. Conversely, for a given ineres rae pah, a moneary policy ha sabilizes z is a policy ha causes he sicky price model and he corresponding flexible price model (wih he same ineres rae pah) o be observaionally equivalen in real variables. In he nex wo secions, he economy s behavior under price level argeing (Secion 3) and ineres rae argeing (Secion 4) are discussed. There are wo cenral hemes. Firs, eiher arge is poenially consisen wih he sicky price economy responding o shocks as would a flexible price economy. Second, boh arges are 8

13 suscepible o real indeerminacy and sunspo flucuaions. To avoid hese bad equilibria, Secion 5 analyzes he wo arges under a pre-deermined money supply. Such a policy avoids he sunspo equilibria, bu also hinders he economy s response o real shocks. 3. Equilibrium wih Price Level Targeing. In a model in which some firms mus pre-se prices, here are clear advanages o he cenral bank eliminaing uncerainy abou he fuure price level. In paricular, we have he following: Proposiion 1: Consider a moneary policy given by a pre-deermined price level policy p = p(k -1,θ -1 ), where p P / M. Consider wo economies, one wih flexible prices and one wih sicky prices. Assume ha moneary policy is conduced according o he same pre-deermined price level policy in he wo economies. Then he sicky price economy and he flexible price economy are observaionally equivalen. This equivalence includes he behavior of he nominal rae of ineres and he money growh process. Proof: Wih P pre-deermined, (3) implies ha z is predeermined. From (4) we hen have ha z = z. Thus, real behavior in he sicky price and flexible price economies are idenical. The supporing money growh process is hen given by he cash consrain (9). The implied nominal rae is given by he Fisher equaion (8). Remark: The equivalence in Proposiion 1 does no necessarily hold in he case of 9

14 compleely rigid prices (ν = 0). Wih rigid prices, he relaionship beween marginal cos and he price level (3) is broken, so ha a predeermined price level does no predeermine marginal cos. Thus here exis equilibria in which marginal cos responds o innovaions (real or sunspo). There is of course an equilibrium in which marginal cos is consan and Proposiion 1 is saisfied. The rigid-price case is an ineresing counerexample o he equivalence resul, bu is likely empirically irrelevan, so henceforh we will ignore i by assuming ν > 0. The inuiion for Proposiion 1 is quie clear. In a flexible price economy a predeermined price level policy implies ha firms would be willing o se heir prices one period in advance. Hence, such a moneary policy causes he flexible price economy o ac as if i were a sicky price economy. Or, in oher words, in he sicky price economy operaing under such a policy he "sicky" firms would no wan o aler heir prices even if hey could do so coslessly. This moneary policy eliminaes he sicky price rigidiy by making prices rigid. Proposiion 1 of course holds for a price level peg. This is he policy advocaed by Goodfriend and King (1998) and King and Wolman (1997). These papers demonsrae ha in a Calvo-syle (1983) price-seing environmen, he observaional equivalence of Proposiion 1 occurs only wih a price level peg. 5 A price level peg is required for observaional equivalence in a Calvo seing because some firms wai an arbirarily long amoun of ime before having he opion of adjusing heir prices. 5 The counerexample in he Remark (where no firms are conemporaneously flexible) also holds in a Calvo-syle economy. 10

15 Proposiion 1 is a naural exension of his resul o an economy in which prices are fixed for a finie number of periods. Despie he simpliciy of a price-level peg here are problems wih such a policy. Carlsrom and Fuers (1998) demonsrae ha in a flexible price economy an inflaion peg causes real indeerminacy. This implies an immediae corollary o Proposiion 1: in a sicky price economy, a policy of argeing inflaion causes real indeerminacy. 6 This arises because under such a policy he nominal rae moves oo closely wih he real rae. Since he nominal rae acs as a disorionary consumpion ax, his policy implies raising consumpion axes when consumpion is low. This policy creaes self-fulfilling behavior: by lowering curren consumpion, agens increase he real rae, which (under an inflaion rae peg) implies ha he implied consumpion ax rises, hus validaing he iniial consumpion reducion. Under an inflaion rae arge, marginal cos is consan, bu here is nohing o pin down he real and hence nominal rae of ineres. While his indeerminacy resul under inflaion argeing is robus o a wide range of parameer values, i is paricularly easy o see in a special case. Suppose ha labor is inelasic and he shocks are iid. Under a price level peg, we can use he Fisher equaion (8) o express he capial equaion (7) as c { U ( + 2)[ zθ f ( K,1) + (1 )]} U ( 1) = β E 1 + c + 1 K + δ Log-linearizing his equaion yields ~ E Q K ~, K ~, K ~, θ ~, θ ( ) = 0 6 Goodfriend and King (1997) ignore hese problems by ignoring he effec of he nominal rae of ineres on he work effor decision (6) and he capial accumulaion decision (7). 11

16 where he ~'s denoe percen deviaions from he seady sae. Noe ha neiher curren capial nor curren produciviy ener direcly ino his equaion. For deerminacy, we need wo explosive roos. If we do have wo explosive roos, hen he only equilibrium pah is for capial o immediaely jump o he seady sae and say here forever. This equilibrium is no only peculiar, i is also unlikely. For all plausible parameer values here will be only one explosive roo. In his case, he equilibrium (scrolled back o ime-) is given by ~ K ~ ~ + 1 = λk + γθ. In his case, λ is given by he sable roo of he following equaion: λ 2 1 β f U = 1 + λ + β U" 1. β The larger roo exceeds one, so we are ineresed in he smaller roo. For ease of discussion, le us assume ha his roo is posiive. 7 The corresponding characerisic equaion for he RBC model is given by 2 λ = 1 β f U λ. β U" β The sable roo of he RBC equaion will (of course) no solve he characerisic equaion of he inflaion peg economy. Since his RBC roo is posiive and less han one, casual inspecion reveals ha he roo of he inflaion peg is smaller han ha of he RBC model. Tha is, in comparison o he opimal pah, convergence o he seady-sae is faser under an inflaion peg. The inuiion is as follows. Suppose ha capial sars 7 For all plausible calibraions, his smaller roo is posiive. If i is negaive, here are oscillaory dynamics. 12

17 below he seady sae. Along he pah he real rae will be falling. Under an inflaion peg his implies ha he nominal rae or implici consumpion ax is falling along he pah. The raional agen hus pospones consumpion and increases capial accumulaion hereby speeding he movemen o he seady sae. As for he response o he innovaion, γ is free--he equilibrium is consisen wih any response of invesmen o he produciviy shock! (And we can, of course, add an arbirary sunspo random variable o his soluion.) The inuiion is similar o he previous paragraph. Suppose ha we consider he case of γ = 0. In his case, invesmen is consan and consumpion moves sharply wih he echnology shock. Under an inflaion peg, an increase in curren consumpion implies ha he real and nominal raes fall. Referring back o he capial equaion (7), his emporarily low nominal rae simulaes consumpion and deers invesmen o such an exen ha capial does no respond o he shock. To eliminae his real indeerminacy under inflaion rae argeing, we need anoher real variable o be pre-deermined. We ake up his issue in Secion Equilibrium wih Ineres Rae Targeing. The previous discussion suggess ha o avoid indeerminacy he cenral bank should dampen he nominal rae s response o he real rae. This suggess an ineres rae peg will eliminae his form of indeerminacy. However, Carlsrom and Fuers (1998) demonsrae ha a a zero nominal rae here is real indeerminacy in a sicky price model. The nex proposiion exends his resul o he case where nominal raes are posiive. In an exac reversal of he indeerminacy under inflaion rae argeing, his 13

18 indeerminacy arises because marginal cos is no pinned down. Proposiion 2: Suppose ha in an economy wih sicky prices he cenral bank pegs he nominal rae R = R. Then he economy's real behavior is indeerminae. This indeerminacy arises because he money supply is endogenous, and here is no resricion on he response of marginal cos o innovaions (including sunspos). Proof: Wih a fixed nominal rae, real behavior is given by (4)-(7). The only resricion on marginal cos is given by (4). In paricular, here is essenially no resricion on he response of z o innovaions (eiher real or sunspo). This behavior is hen suppored by he endogenous money supply process implied by (9). This real indeerminacy and poenial sunspo behavior arises because he money supply is endogenous and no uniquely deermined, and fracion (1-ν) of firms are no operaing on heir labor demand curves bu agree o supply whaever is demanded a he pre-se nominal price. These firms' hiring needs are hus deermined by he moneary auhoriy--higher money growh implies a higher nominal and, since prices are sicky, real demand for oupu. The firm absorbs he resuling flucuaions in coss in is markup: he mark-up (1/z ) varies negaively wih he rae of money growh. An immediae corollary o Proposiion 2 is ha here exiss an equilibrium in which marginal cos is consan. Tha is, here exiss a supporing money growh process ha pegs he nominal rae and marginal cos. Since marginal cos is consan, his supporing money growh process also predeermines he price level. Thus, a 14

19 combinaion policy of pegging he nominal rae wih a predeermined price level: (1) sabilizes he ineres-rae disorion, (2) sabilizes he marginal cos disorion on facor prices, and (3) is no subjec o real indeerminacy. 8 This nirvana policy is probably of lile pracical ineres since i assumes a grea deal of informaion on he par of he cenral bank. 5. Targeing wih a Pre-Deermined Money Supply. To summarize he previous wo secions, an inflaion rae peg is advanageous as i sabilizes he marginal cos disorion. Bu such a peg is clearly inconsisen wih a consan ineres rae as he nominal rae moves one-for-one wih he real rae. This laer fac implies ha here is real indeerminacy and sunspo equilibria under an inflaion rae peg. To avoid hese sunspo equilibria, he variabiliy of he nominal rae mus be dampened. An ineres rae peg can be consisen wih a consan marginal cos and, of course, a consan nominal rae. The disadvanage of such a policy is ha here are sunspo equilibria in which marginal cos flucuaes. To avoid he sunspo equilibria under he wo possible arges, we need an exra consrain on moneary policy. This Secion explores one naural possibiliy: he cenral bank pre-deermining he money sock, ie., eliminaing he money sock s response o conemporaneous innovaions. The disadvanage of his resricion is ha real cash 8 As in Goodfriend and King (1997), his policy implies ha he cenral bank varies he money supply o achieve he desired level of real balances wihou movemens in he price level. In paricular, policy responds o he innovaions in consumpion spending ha would arise in he corresponding flexible price economy. This is quie convenien as he flexible price economy is essenially a sandard RBC economy, and we have a grea deal of knowledge abou consumpion spending in ha model. Hence, a ransien (persisen) produciviy increase should be mached by a relaively small (large) increase in he money supply, while a ransien (persisen) governmen spending increase should be mached by a relaively small (large) decrease in he money supply. 15

20 balances are sicky, hus making he economy s iniial response o a shock subopimal. We begin he analysis by working ou some simple examples, and hen use he inuiion developed o discuss he impulse response funcions in a realisically calibraed business cycle model. A Labor-Only Example: Suppose ha U(c,1-L) = ln(c) L, f = (θl) 1-α. The mos ineresing difference in he economy s behavior under he wo moneary arges is in he response o an innovaion in he environmen. Hence, for simpliciy assume ha he echnology shocks are iid. As poin of comparison, recall ha RBC behavior implies ha labor is consan in his well-known example. Ineres Rae Peg Suppose firs ha he cenral bank pegs he nominal rae wih a predeermined money supply. Log preferences and predeermined money growh imply ha money growh is consan ((8)-(9)). Approximaing he funcion h around he seady-sae of z = z, we have dln(h)/dln(z ) = ν/(1-ν). Expressing he cash consrain (9) in log deviaions (hereafer denoed wih a ~) yields: ~ ν 1 z = c~ ν where we have used he fac ha wih iid shocks P s /M is a consan. Subsiuing his ino (6) and he resource consrain yields 16

21 ~ ( ν 1)(1 α) ~ L = θ (1 α αν ). + Consider he response o a one-ime shock. Conemporaneous labor inpu falls, wih he srengh of he decline deermined by he fracion of firms ha are sicky (1-ν). Labor inpu falls because marginal cos z falls (he wage ax rises). Oupu rises as long as prices are no compleely rigid (ν = 0). In he nex period, labor reurns o is seady sae as marginal cos reurns o normal. Hence, relaive o RBC behavior, he unusual effec of a shock arises only in he impac period. Inflaion Peg In conras, suppose ha he cenral bank pegs he price level. In his case, he cash consrain implies ha consumpion (and hus oupu) is predeermined, so ha labor responds sharply downward. In paricular, (6), (8), and he resource consrain yield: ~ L α ~ ~ ~ θ ( α 1) = L 1 + θ 1 Noe ha he conemporaneous decline in labor is equivalen o he behavior under he ineres rae peg only if prices are compleely rigid (ν = 0). If some firms adjus prices conemporaneously, labor responds more sharply downward under an inflaion rae peg. The irony is ha while an inflaion peg makes he sicky price economy behave as if all prices were flexible, he economy behaves wih lile flexibiliy. The conemporaneous decline in labor occurs because he nominal rae (consumpion ax) rises. Wih inflaion pegged and curren consumpion fixed, he real and hus nominal rae rises because 17

22 fuure consumpion (and fuure labor inpu) rises. Thus labor responds posiively wih a lag o he shock, and hereafer he consumpion dynamics are damped oscillaions back o he seady-sae. These oscillaions are suppored by corresponding movemens in he nominal rae. Relaive o he behavior under an ineres rae rule, he perverse response o a ransien shock is remarkably persisen. An Example wih Capial and Labor: Suppose ha U(c,1-L) = ln(c)-l, f = K α (θl) 1-α, δ = 1, and he echnology shocks are iid. As poin of comparison, recall ha RBC behavior implies ha consumpion and invesmen move one-for-one wih oupu and ha labor is consan. Ineres Rae Peg In he case of an ineres rae peg, we have: ~ L = z ~ ( ν 1)(1 α) ~ = θ (1 α αν ) + c~ ~ ~ ν (1 α) ~ = K + 1 = αk + θ (1 α αν ). + The shock leads o a fall in marginal cos z (an increase in he wage ax). The exen of his decline depends upon he level of price rigidiy (ν). A he exreme of full price flexibiliy (ν=1) marginal cos and labor are consan as in he corresponding RBC model. For less han perfec flexibiliy, he emporary ax change produces a negaive labor response and hus a dampened response of consumpion and invesmen (relaive o RBC). As before, he perverse behavior of labor in response o a shock is over afer 18

23 he period of he shock (once prices adjus). Inflaion Peg In conras, under an inflaion peg, he perverse behavior is persisen. Sraighforward calculaions imply: ~ L (1 αβ z)(1 α) = (1 αβz) + (1 α) ~ K α 2 2 (1 α)(1 αβz) βz(1 αβz) + (1 α) ~ θ c~ 2 (1 α βz) ~ = (1 αβz) + (1 α) K ~ K ~ K (1 α)[(1 α) + (1 αβz)] ~ α βz(1 αβz) + (1 α) + 1 = α + θ 2. A few observaions are in order. Firs, he coefficien of capial-on-capial is exacly α, he RBC value. This is surprising given Secion 3 s demonsraion ha in an inelasic-labor model, convergence o he seady-sae is faser under an inflaion peg, ha is, he capial-on-capial coefficien is less han α. The difference arises because of he effec of labor. When capial is below he seady sae he nominal rae is relaively high and is expeced o decline. This emporarily high nominal rae leads o wo offseing effecs on capial accumulaion. Since invesmen is a credi good, here is a simulus o invesmen (he coefficien of consumpion on capial is sricly greaer han α). However, since leisure is also a credi good, here is a negaive effec on curren labor inpu and hus oupu (he coefficien of labor on capial is posiive in conras o zero for he RBC model). These wo effecs exacly cancel leading o a coefficien of capial-on-capial ha is idenical o RBC. 19

24 Second, he response of labor o an innovaion is negaive, bu no as sharply negaive as in he inflaion-peg case wihou capial. The invesmen channel is criical. Alhough consumpion is predeermined, since invesmen can respond oupu is no predeermined. Hence, oupu responds posiively o he shock. Since consumpion is predeermined movemens in oupu mus be enirely absorbed by movemens in invesmen. For example, noice ha he coefficien of invesmen on he shock is sricly greaer han he RBC value of (1-α), and ha oupu responds less srongly han in he RBC economy. Finally, in he previous model wihou capial, he labor response is always larger (less negaive) in he ineres-rae-peg economy han in he inflaion-peg economy. However, in a model wih capial, he iniial labor responses in he wo economies are idenical a he degree of price rigidiy given by 2 (1 αβ z) 1 ν ) =. (1 α β z) ( 2 Wih β =.99, α = 1/3, z = 10/11, we have a bound of (1-ν) =.54. If prices are more sicky han his, hen he iniial labor response is smaller (more negaive) in he ineresrae peg economy han in he inflaion-peg economy. Impulse Response Funcions in a Calibraed Model The previous examples illusrae he basic mechanisms a work in he numerical simulaions. A posiive echnology shock will cause he marginal cos disorion (wage ax) o rise under an ineres rae peg, and he nominal rae disorion (consumpion ax) o rise under an inflaion peg. These ax movemens will aler he responses of 20

25 employmen and invesmen o produciviy shocks relaive o he RBC responses. The simulaions pu some quaniaive deail on his qualiaive frame. The model is calibraed as follows. Preferences are given by U(c,1-L) = ln(c) AL, where A is chosen o imply ha seady-sae work effor is 1/3. Assuming he ime period o be one quarer, we se β = 0.99 implying a 4% annual real rae of ineres. The capial coefficien is se a α = 0.34, and he quarerly rae of depreciaion is δ = The seady-sae level of quarerly money growh is G = 1.01 in all simulaions, implying a seady-sae annual nominal rae of 8%. The seady-sae mark-up is se a 10% implying η = 11. As for he degree of sickiness in he ineres rae peg economies, figure 1 repors resuls for ν = 0.20, ν = 0.80, and ν = 1 (he RBC economy). Figure 1 also shows he impulse response for he inflaion peg economy. The shock is a 1% produciviy shock wih ρ =.95. A key resul is ha he wo moneary policies lead o differing movemens of he implici axes. Depending on he degree of sickiness, he ax movemen under an inflaion peg can be higher or lower han he ax movemen under an ineres rae peg. Wih ν=1 he wage ax is consan. Wih ν = 0, in order o keep consumpion consan he wage ax mus increase from 9.1% o 12.5%! In conras, under an inflaion peg, o keep consumpion consan he consumpion ax need only rise from 1.98% o 2.3%. This demonsraes clearly he difference beween a consumpion ax and a wage ax in a model wih invesmen. The inermediae values of ν illusrae an imporan non-lineariy. A lile flexibiliy goes a long way. Wih ν=0 he wage ax rises by 3.4%, vs. 0.9% for ν=0.2 and 0.1% for ν=0.8. This non-lineariy also appears in he employmen response. Wih 21

26 ν = 0, employmen falls by 9.1%. A he oher exreme, wih ν = 1, employmen rises by 0.97%. Wih ν = 0.2, he employmen response is 1.6% (3/4 of he way beween 9.1% and 0.97%) while he employmen response is 0.76% (99% of he way beween 9.1% and 0.97%) wih ν=0.8. In he case of an inflaion peg, he emporary consumpion ax increase discourages work effor bu causes an invesmen surge (relaive o an RBC economy) which ends o simulae work effor. Taken ogeher, he inflaion peg s employmen response is only slighly below ha of he RBC model, while invesmen is amplified considerably relaive o RBC (6.2% vs. 5.1%). Consumpion does no respond conemporaneously, bu hen jumps up in he second period. Comparing he differen employmen responses for he ν = 0.80 case and he inflaion peg case reveals he differing disorions a work here. The shock causes he implici consumpion ax (in he inflaion peg economy) o rise by more han he implici wage ax (in he ineres rae peg economy). Bu since one is a wage ax, and he oher is a consumpion ax, he effec on employmen is much differen because of he possibiliy of alering invesmen behavior. The earlier 100% depreciaion example emphasized ha under an inflaion peg a shock leads o a persisen deviaion of he nominal rae (or implici consumpion ax) from he seady sae. In conras, under an ineres rae peg he movemen of marginal cos from seady sae ends afer one period (as soon as all prices can adjus). Wih plausible calibraion, however, his qualiaive difference in he wo moneary policies seems o be quaniaively unimporan. The deviaion of he nominal rae from seady sae is rivial in he periods afer he shock. Thus, in he period afer a shock, he wo 22

27 economies behave essenially as an RBC economy wih a consan wage ax and a consan consumpion ax. For example he capial-on-capial coefficiens are idenical while he consumpion-on-capial coefficien rises from 0.5 in he ineres rae peg economy o 0.52 in he inflaion peg economy. A Welfare Analysis We will exploi his approximae RBC-equivalence (in he period following he shock) in conducing a simple welfare experimen. Consider wo economies: (1) an economy operaing under an inflaion-peg, and (2) an economy operaing under an ineres-rae peg. Suppose boh economies begin a he non-sochasic seady-sae and are hi by a one-ime produciviy innovaion (wih ρ=0.95) in, say, period 1. The previous discussion implies ha from period 2 onwards he economies (essenially) follow RBC dynamics. To solve for he iniial response o he shock, we will use he period 1 Euler equaions and he log-linear decision rule for RBC consumpion in period 2. 9 We can hen solve his nonlinear sysem ha describes period 1 behavior wihou resoring o any furher linear approximaions. This is paricularly useful as he flucuaing axes have imporan nonlinear effecs. 10 Once we have calculaed period 1 consumpion, employmen, and invesmen, we can hen compare uiliy across he wo economies by urning period 2 capial ino uiliy unis using period 2 s RBC marginal uiliy of consumpion. Since he period 1 responses are non-linear, we perform his 9 In he case of he ineres rae peg, we also need an expression for period 2 employmen and hus add period 2 s (nonlinear) employmen Euler equaion o he sysem. 10 For example, suppose ha uiliy were log-log and we used he log-linear decision rules for period 1. Then he produciviy shock would ener he welfare comparison linearly so ha he expeced welfare 23

28 calculaion for a posiive and a negaive shock, and hen average he wo ogeher. Tha is, we are ineresed in he welfare consequences of eiher a posiive or negaive produciviy innovaion, each of which occurs wih equal probabiliy. The resuls of he welfare experimens are repored in Figures 2 and 3 (measured in uiliy differences). For very low levels of price flexibiliy (ν < 0.18), he inflaion peg dominaes he ineres rae peg. However, if he economy is more flexible han his (ν > 0.18), an ineres rae peg dominaes alhough he welfare gain is relaively small. This basic resul is largely insensiive o alernaive values for he subsiuion elasiciy (η) and money growh rae (G). These values do reveal an ineresing nonlineariy. Under an ineres rae rule, for large values of η (η = 51), sickiness is preferred o flexibiliy. Tha is, welfare is higher for 0.09 < ν < 1 han an economy wih perfec price flexibiliy (ν = 1). This implies ha a variable wage ax is preferred o a consan wage ax. This surprising resul occurs because in he former case he average value of z is larger han he non-sochasic seady-sae value of z (recall ha z = 1 is he firs-bes). This nonlineariy can bes be undersood by recalling he pricing equaion (3) and he binding cash consrain (9). Wih a predeermined money sock and sicky prices, movemens in consumpion can occur only if marginal cos (z) moves in he opposie direcion. This implici relaionship beween consumpion and marginal cos is convex. Hence, flucuaing consumpion will lead o an average level of z ha is higher han he seady-sae z. For benchmark parameers (G = 1.01, and η = 11), his effec differences would all be zero. 24

29 peaks a ν = 0.05, where he average wage ax (1-z) is 9.03% vs. a seady-sae wage ax of 9.09%. Wih greaer convexiy, his effec is magnified: wih η = 51 (a 2% markup), he average wage ax is 1.71%, while he seady-sae is 2.0%. There are a few caveas o his welfare comparison. Firs, by using he log-linear decision rule for period 2 consumpion we ignore a poenial nonlineariy. I is unclear which way his biases he resuls. However, given ha we sar he economy a he seady-sae capial sock and consider a small produciviy shock, his bias is likely o be small. Second, as already noed, he esimae assumes ha under he inflaion rae peg he nominal rae is back o he seady-sae for period 2 and onwards. This assumpion probably biases he resuls in favor of an inflaion peg. Third, and mos obviously, hese are welfare comparisons along an impulse response from a one-ime shock, and no lifeime uiliy measures in a sochasic economy. In paricular, we are ignoring all fuure nonlineariies ha would arise in an economy subjec o repeaed shocks. For example, as noed above he expeced fuure value of marginal cos is greaer han he non-sochasic seady sae. A similar condiion holds for he consumpion ax in he inflaion peg case. This discrepancy is very small: for he baseline calibraion, he average consumpion ax (1-1/R) is % vs % for he non-sochasic seady sae. These omissions end o bias he resuls in favor of he inflaion peg. 6. Conclusion This paper is a ale of disorions. The model economy is disored relaive o RBC behavior by marginal cos and he nominal rae of ineres. These disorions are 25

30 fundamenal, arising from he assumpion of sicky prices and he need o use money o faciliae ransacions. The moneary policy rule in place alers he economy s response o shocks by alering he endogenous movemen of hese disorions. A price level peg sabilizes marginal cos, bu allows he nominal rae o vary. An ineres rae peg has he exac opposie effec. As a resul, hese wo differen moneary policies have differen posiive implicaions for he real economy s response o a shock. Under an ineres rae peg, he employmen and invesmen responses are oo small (relaive o RBC). Under an inflaion peg, he invesmen response is oo large while he consumpion response is oo small. On he normaive side, hese very differen responses lead o a ranking of policies. For small degrees of price flexibiliy, he marginal cos disorion is poenially quie variable so ha an inflaion peg dominaes. If more han 20% of firms have price flexibiliy, his dominance is reversed and he ineres rae peg is preferred. However, he welfare differences of he wo policies are quie small. This near-welfare-invariance o cyclical moneary policy is similar o he resuls of Dow (1995), Carlsrom and Fuers (1995), and Ireland (1998). All of hese analyses ignore he poenial welfare cos of a moneary policy ha inroduces sunspo equilibria ino he economy. These coss could be quie large. From his perspecive, he mos useful policy suggesion is o do no harm: be somewha agnosic in he choice of cyclical moneary policy, bu make sure ha one chooses a policy rule ha does no inroduce sunspo equilibria. This laer goal is achieved by using a policy rule ha 26

31 severely resrics he immediae policy response o innovaions in he real environmen. 27

32 References Blanchard, Olivier, J., and Nobuhiro Kiyoaki, "Monopolisic Compeiion and he Effecs of Aggregae Demand," American Economic Review (77), 1987, Calvo, Guillermo A., "Saggered Prices in a Uiliy-Maximizing Framework," Journal of Moneary Economics 12 (3), Sepember 1983, Carlsrom, Charles T, and Timohy S. Fuers, "Ineres Rae Rules vs. Money Growh Rules: A Welfare Comparison in a Cash-in-Advance Economy," Journal of Moneary Economics (36), 1995, Carlsrom, Charles T, and Timohy S. Fuers, "Real Indeerminacy under Inflaion Rae Targeing" 1997 working paper. Carlsrom, Charles T, and Timohy S. Fuers, "A Commen on The Role of Counercyclical Moneary Policy," 1998, forhcoming Journal of Poliical Economy. Chari, V.V., Parick Kehoe, and Ellen McGraan, "Sicky Price Models of he Business Cycle: Can he Conrac Muliplier Solve The Persisence Problem," FRB Minneapolis working paper #217, Chrisiano, Larry, and Marin Eichenbaum, "Liquidiy Effecs and he Moneary Transmission Mechanism," American Economic Review 82, 1992, Fuers, Timohy S., "Liquidiy, Loanable Funds, and Real Aciviy," Journal of Moneary Economics 29, 1992, Goodfriend, Marvin and Rober G. King, "The New Neoclassical Synhesis and he Role of Moneary Policy," 1997 NBER Macroeconomics Annual. Ireland, Peer N., "The Role of Counercyclical Moneary Policy," Journal of Poliical Economy (104), 1996, Ireland, Peer N., A small, srucural, quarerly model for moneary policy evaluaion, (1998), forhcoming, Carnegie-Rocheser Conference Series, 47. Lucas, Rober E., Jr., "Liquidiy and Ineres Raes," Journal of Economic Theory 50, 1990,