T he St. Louis model of the early 1970s,

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1 Rober G. King is A. W. Roberson professor of economics a he Universiy of Virginia. Alexander L. Wolman is wih he Federal Reserve Bank of Richmond. The auhors hank Marianne Baxer, Michael Dosey, Marvin Goodfriend, and Peer Ireland for useful discussions during he preparaion of his aricle. We also received valuable commens during various presenaions of i, paricularly from Ben Bernanke, Michael Kiley, Preson McAfee, Edward Presco, Julio Roemberg, Chrisopher Sims, Mark Wason, and Michael Woodford. Inflaion Targeing in a S. Louis Model of he 21s Cenury Rober G. King and Alexander L. Wolman T he S. Louis model of he early 1970s, as described by Andersen and Carlson (1972), was a small-scale monearis model of economic aciviy. Is srucure was sharply a variance wih he framework of he major, larger scale macroeconomeric models used for policy analysis by he Federal Reserve, boh a he ime of he incepion of he S. Louis model and oday. The S. Louis model had four major feaures: I was sufficienly small ha one could acually undersand how i worked by looking a he model s behavioral equaions and by conducing simulaions of i. I could be used for policy analysis, specifically for sudying he effecs of moneary and fiscal policy on inflaion, oupu, and ineres raes. The monearis background of is auhors mean ha he model (1) focused on he quaniy of money as he key measure of he sance of moneary policy and (2) conained srucural linkages from money o economic aciviy ha are now widely acceped, including he cenral role of a long-run demand for money ha is a relaively sable funcion of a small number of variables. I combined shor-run non-neuraliy of changes in money wih long-run neuraliy, in line wih he perspecive of Friedman and Schwarz (1963a and b). Wih Lucas (1976) criique of policy evaluaion, he S. Louis model was largely abandoned by moneariss. The model fell vicim o he criique in a paricularly rapid and complee manner because is builders had sressed ha i conained quaniaively imporan effecs of expecaions. 1 In his aricle, we consruc a smallscale modern macroeconomic model ha can be used o sudy he effecs of alernaive moneary and fiscal policies in a manner consisen wih Lucas recommendaions. Tha is, we build a raional expecaions macroeconomic model in which he ineremporal opimizaion problems of households and firms are explicily described. We call his a S. Louis model of he 21s cenury because we believe i is he ype of small-scale macroeconomic model ha will be sysemaically employed for he purpose of policy analysis by cenral banks in he coming years. The model is monearis in five specific ways ha i shares wih he S. Louis model: Our model conains a sable demand for money ha is invarian o alernaive moneary policies. I incorporaes shor-run nonneuraliy of money wih long-run neuraliy. Fricions in he price-seing process yield a shor-run non-neuraliy of money ha is quaniaively and economically imporan. 1 Andersen and Carlson (1972) discuss he cenral role of expecaions in he S. Louis model. We furher discuss heir modeling of expecaions below.

2 Once one akes ino accoun anicipaed price change, hese fricions lead o no quaniaively imporan long-run rade-off beween inflaion and real aciviy. Our model places a cenral emphasis on expecaions in consumpion choices, invesmen decisions, asse valuaion, and price deerminaion in he radiion of Irving Fisher and Milon Friedman. We use his model o sudy an imporan quesion in moneary economics ha is currenly policy-relevan a cenral banks around he world: Wha are he consequences of adoping an inflaionargeing rule for moneary policy? We provide deailed informaion on wo versions of his quesion. Firs, since mos cenral banks seek o arge raes of inflaion ha are low by he hisorical sandards of heir counries, we ask, Wha are he benefis (or coss) of having a low rae of inflaion? Second, since criics of inflaion arges are frequenly concerned ha hese may inerfere wih sabilizaion policy, we ask, In a seing wih sicky prices and demand-deermined oupu, how do he responses of he economy o various shocks differ from hose which would occur if prices were flexible? Our model s answers o hese quesions are very monearis. Firs, here are major benefis o seing low raes of inflaion. Tha is, Friedman s prescripion for long-run moneary policy (seing he nominal ineres rae o zero) approximaely holds in he long run of our model economy even hough we have some imperfec marke fricions of he form sressed by Tobin (1972a and b). Even if one does no go all he way o a zero nominal ineres rae, here are quaniaively imporan long-run welfare gains from moving from an inflaion rae of 5 percen o price sabiliy (0 percen inflaion). Second, under a regime of perfec inflaion argeing (in which he pah of he price level is specified exacly by he moneary auhoriy), we find ha he responses of our sicky price model are essenially idenical o hose of a flexible price economy. Our analysis hus suggess ha here are major benefis o implemening an inflaion-argeing regime wih a low arge rae of inflaion. To furher explore issues ha arise in he implemenaion of his regime, we consider (1) he dynamic behavior of he economy if a arge pah for he price level is implemened wih an ineres rae rule and (2) he dynamic behavior of real aciviy if a credible, low inflaion pah is implemened immediaely. We find ha i is indeed feasible o implemen a policy of argeing he pah of he price level wih an ineres rae rule. We also find ha here is zero cos of immediaely inroducing a credible low inflaion program. THE ST. LOUIS MODEL CIRCA 1972 The S. Louis model of macroeconomic aciviy was a small macroeconomeric model conaining five srucural equaions, as spelled ou in Andersen and Carlson (1972): a oal spending equaion, a price equaion, a long-erm ineres rae equaion, a definiion of anicipaed price changes based on he long-erm ineres rae equaion, and an Okun s law (1962) link beween an oupu gap and unemploymen. In addiion, in he background, here were mehods of measuring he rend level of real economic aciviy and he normal level of unemploymen. Money Demand and Expendiure The cenerpiece of he S. Louis model was a oal spending equaion, pu forward in Andersen and Jordan (1968), ha linked he change in nominal gross naional produc o four-quarer disribued lags of changes in he nominal money sock and of high-employmen federal governmen expendiures. Given ha he effecs of high-employmen expendiures were small, his specificaion 84

3 was widely viewed as resuling from a monearis specificaion of he demand for money one wih an income elasiciy of uniy and an ineres rae elasiciy of zero. In he macroeconomeric model, his specificaion was he economeric embodimen of he Friedman and Schwarz (1963a and b) mehod of aking nominal income as proximaely exogenous. I could be employed o discuss he effec of moneary changes on nominal income, as in Andersen and Jordan (1968), wihou discussing he division of nominal income ino price level and real oupu. Price Adjusmen As elaboraed in Andersen and Carlson (1972), he S. Louis model employed a price equaion o describe he division of nominal income ino prices and oupu. Tha specificaion relaed he change in he price level o wo facors. Firs, anicipaed changes in prices enered wih a coefficien of 0.86, so ha he price equaion was aken o be imporanly influenced by he beliefs of price seers. Second, a four-quarer disribued lag of changes in nominal income was included as a measure of demand pressure in he macroeconomy. The builders of he S. Louis model conrased heir approach wih he approach used o consruc radiional price equaions, such as hose presened by Ecksein and Fromm (1968). Those who buil he S. Louis model emphasized ha heir racable specificaion included imporan expecaions effecs and permied he simulaneous deerminaion of prices and oupu. This laer feaure was sufficienly novel, relaive o he sandard economeric pracice of relaing price changes o real demand pressure measures, ha i warraned exensive discussion by Tobin (1972a and b) in his summary of he conference ha led o he Ecksein (1972) volume. Since he coefficiens on he nominal demand variables summed o 0.10, he model also implied a Phillips curve defined by Andersen and Carlson (1972) o be he join ime pahs of unemploymen and inflaion arising from heir model, which involved only a very small effec of susained inflaion on real aciviy. This small rade-off is conrased wih oher conemporary sudies such as de Menil and Enzler (1972), Klein (1972), Bodkin (1972), and Ecksein and Wyss (1972). Expecaions An unusual feaure of he S. Louis model was is modeling of expecaions. Andersen and Carlson (1972) sressed ha expeced prices played a key role in he model and ha heir explici specificaion of an expecaions mechanism allowed hem o explore he rade-off beween inflaion and unemploymen a various horizons. They used he long-erm ineres rae as an imporan empirical indicaor of expeced inflaion and, for his reason, also included a srucural equaion deermining he long-erm ineres rae. The deails of heir expecaions scheme were sharply criicized by Gordon (1972) on heoreical and empirical grounds, alhough many pracical macroeconomiss now rouinely use he long-erm nominal ineres rae as a guide o long-erm inflaion expecaions. A ST. LOUIS MODEL OF THE 21ST CENTURY Many of he srucural equaions of our model economy are obained by sudying he opimizaion problems of households and firms. In addiion, he srucural equaions include resource consrains and he policy rule of he cenral bank. Moneary Services and he Demand for Money In our model economy, here is a welldefined demand for money ha one can deermine from a problem of cos minimizaion. Tha is, one can define a demand for money condiional on a paern 85

4 2 Our iming convenion for money demand embodies a view of he average cash balance holdings as a durable good (as in Friedman, 1969) and is bes exposied as follows. Suppose ha he evoluion of bonds is 1 B +1 +M +P c = 1+ R Y +B +M 1, where Y is some measure of nominal income and B is he quaniy of bonds mauring a. Accordingly, he dae cos of holding a uni of money from o +1(M ) is he discouned value of he ineres foregone, R, 1+ R as in he ex. of expendiure deermined as par of a more general uiliy maximizaion problem. The demand for money sems from a ransacions echnology a shopping ime echnology in he erminology of McCallum and Goodfriend (1987) which governs he amoun of ime ha mus be spen o underake real consumpion aciviy wihin he period of our discree ime seup. This shopping ime funcion specifies ha ime devoed o ransacions aciviy depends negaively on he raio of he amoun of real money balances (m ) o he amoun of real consumpion expendiure (c ): (1) h = h m c. Throughou, we use noaion in which lowercase leers refer o real variables and uppercase leers refer o nominal variables so ha m is he quaniy of real balances and is equal o M /P, where M is he sock of money held during he period and P is he price level. The opporuniy cos of a uni of ime spen shopping is given by he real wage rae. The opporuniy cos of choosing o hold a uni of money during period is he discouned value of ineres forgone nex period, so ha he renal price of a uni of cash balances is R 1+ R 2. Minimizing he flow real cos R w h + m, 1+ R hrough selecion of he quaniy of real balances accordingly requires ha (2) w Dh ( m c ) 1 ( = R, c 1+ R ) which implicily defines he demand for money. In his expression, Dh ( m c ), is he marginal ime saving ha arises from holding an addiional uni of cash balances per uni of consumpion expendiure. (We use Dh o denoe he derivaive of he h funcion wih respec o he raio m/c, viewing i as a funcion of his single variable.) In our heoreical analysis, we assume ha his ime saving is diminishing in he raio (m/c). Our empirical work jusifies his assumpion. Noice ha if he real wage and real consumpion grow a he same rae and he nominal ineres rae is consan over ime, hen our specificaion implies ha here is consan consumpion velociy. Tha is, here is no rend in he raio m/c. In our empirical analysis below, we specify ha he funcion Dh ( m c ), akes a specific parameric form Dh ( m c ) [( = m c ) / ] 1_v, which hen implies ha (3) ( m c ) = [ + 1+ R R ( c w )] v This funcional form for real money demand per uni of expendiure is close o he consan elasiciy srucure ha is frequenly sudied in he lieraure. Noably, if =0, hen here is a loglinear money demand funcion, log (m ) = log (. ) + (1 v) log (c ) vlog + vlog (w ). R 1+R 86

5 More generally, he parameer allows here o be a finie saiaion level of real cash balances, m = [ ] v c, which occurs as a limiing case when he nominal ineres rae is zero. We choose his parameric form for he Dh(m /c ) funcion because we wan o generae a demand for money ha is relaively convenional, while leaving open he issue of wheher here is a finie saiaion level of cash balances. Consumpion Demand and Labor Supply Ineremporal uiliy maximizaion leads o demand for consumpion (c ) and supply of labor (n ), given he allocaion of ime o shopping (h ) discussed above. Expeced dae lifeime uiliy is given by E U E j=0 j u(c +j,l +j ), where he momenary uiliy funcion implies ha boh consumpion and leisure (l ) are goods. Laer in he analysis we adop a parameric specificaion of his funcion ha has been much used in he real business cycle lieraure. Individuals choose coningency plans for consumpion, labor supply, real balances, shopping ime, and leisure o maximize expeced uiliy subjec o a presen value budge consrain ha links income and expendiure, E j=0, j P +j c +j E j=0, j P +j [ w R j jn j m 1 + R j +j] + oher wealh, as well as a ime consrain ha resrics work, shopping ime, and leisure, (4) n +j + l +j + h +j =1. In addiion, as discussed above, here is a echnology linking shopping ime o curren expendiure, c and real cash balances, m. Since hese specificaions are sandard, we will review hem relaively briefly. Firs, in he presen value budge consrain, we are discouning nominal cash flows a dae +j by he discoun facor,j. 3 Second, also in his expression, we have real prices of work and cash balance holding: w is he real wage (W /P where W is he nominal wage) and he renal price of a uni of real cash is R /(1 + R ), as discussed above. Third, he ime consrain includes ime spen shopping, as well as a working and a leisure: Marke work is a residual given deerminaion of leisure demand from uiliy maximizaion and ime spen shopping from cos minimizaion. Maximizaion of uiliy subjec o hese consrains leads o he following efficiency condiions for consumpion, leisure, and money balances. These efficiency condiions are srucural equaions of our small-scale macroeconomic model. Since hey conain expecaions, hey mus be evaluaed as par of a complee raional expecaions soluion, bu presenaion of hem provides us wih he opporuniy o describe he main implicaions ha hey have, aking expecaions as exogenous. Firs, he efficien selecion of consumpion requires (5) E j D 1 u (c +j,l +j ) =Λ E, j P +j [ 1 w + j Dh( m + j c ) ( m j + j c )], 2 +j for j = 0, 1,... Second, he efficien selecion of labor requires (6) E j D 2 u (c +j,l +j ) =Λ E,j [P +j w +j], for j = 0, 1,... Third, he efficien paern for cash balance holdings requires ha w +j Dh ( m c + j )( + j c 1 +j ) = R+j 1+ R for j = 0, 1,... In hese expressions Λ is he Lagrange muliplier on he wealh consrain and hus represens he lifeime uiliy gain from an addiional dollar of wealh a. We use he noaion D i u o de- +j, 3 In he ineres of making he model seup as simple as possible noaionally, we are being a lile cavalier abou he role of uncerainy. For cerain cash flows, our discoun facor is linked o one period ineres raes by,j=[(1+r )...(1+R +j 1 )] 1, where R is he nominal ineres rae from o 1. 87

6 noe he parial derivaive of he uiliy funcion wih respec o is ih argumen. In he firs of hese efficiency condiions, noice ha he shopping ime echnology means ha he Beckerian full price of a uni of consumpion a dae +j involves a ime cos of shopping,, j P +j [ w +j Dh( m + j c )( m + j + j c )], 2 + j as well as he sandard marke cos componen, j P + j. In he second, noice ha here is he sandard equaing of he coss and benefis of forgoing a uni of leisure and supplying i o he markeplace. Togeher wih he lifeime budge consrain, hese wo equaions implicily deermine consumpion and leisure demand a all daes in line wih a permanen income approach o hese wo acions. The hird condiion is he money demand cos minimizaion condiion discussed above. Labor Demand, Invesmen, and Marginal Cos As in Roemberg (1987) and Blanchard and Kiyoaki (1987), we assume ha firms have some marke power, behaving as monopolisic compeiors. (We conras his o a perfec compeiion siuaion in some places below.) In his subsecion, we focus on he srucural equaions arising from he represenaive firm s decision problem ha concern producion, labor, and invesmen demand. In he nex subsecion, we focus on pricing implicaions. I is accordingly useful o hink abou breaking he firm ino hree separae pars for planning purposes. Firs, he producion uni akes as given he oupu level of he firm and he renal price of capial (a ransfer price from he invesmen uni): I deermines labor demand and capial demand so as o minimize cos. Second, he invesmen uni akes as given he marke prices of invesmen goods and he renal price of capial: I deermines invesmen so as o maximize he value of he firm. The aciviies of hese wo unis are explored in his secion. Third, he pricing uni deermines he price of he firm s oupu, aking ino accoun demand condiions and coss. Throughou, he firm is aken o maximize he expeced presen value of is profis, E V = E j=0,j [P +j y +j W +j n +j P +j i +j], subjec o he producion echnology for is final produc, (7) y +j = a +j f (n +j,k +j), and he evoluion equaion for is capial, (8) k +j +1 k +j = (i +j /k +j)k +j k +j, where (i/k) is a posiive, increasing and concave funcion ha embodies coss of adjusmen for he capial sock. To describe decisions of he producion uni, we employ sandard microeconomic condiions for cos minimizaion. For his purpose, we assume ha he oupu level is given a any arbirary level y +j and ha capial can be rened a price Z. Then, he saic condiions for cosminimizaion are: (9) Ψ +j a +j D 1 f ( n +j,k +j ) =W +j, and (10) Ψ +j a +j D 2 f ( n +j,k +j ) =Z +j, where Ψ +j is he Lagrange muliplier on he consrain y +j = y +j and Ψ +j accordingly is inerpreable as nominal marginal cos. The convenional soluion o his problem implies ha marginal cos is a funcion of he level of oupu; he produciviy shifer, a +j ; and he facor prices, W +j and Z +j. I also depends on he form of he producion funcion: We assume ha he producion funcion (equaion 7) is consan reurns-o-scale, so ha marginal cos is independen of oupu and hus wrie Ψ +j =Ψ( a +j,w +j,z +j ). In erms of he decisions of he invesmen uni, we assume ha he firm chooses he opimal invesmen paern o maximize is presen discouned value 88

7 given he renal price Z +j. This leads o a pair of efficiency condiions, (11) P =Q D (i /k ), and (12) Q =E { D ( k i,1z +1 +[ + 1 ) + 1 ( k i + 1 ) + 1 i k +1 ], 1Q +1} In accord wih he invesmen echnology ha Hayashi (1982) and ohers have used o raionalize a Tobin s q-heory of invesmen, he firs of hese specificaions indicaes ha he invesmen rae i /k is deermined by he raio of he (shadow) price of insalled capial (Q ) o he price of replacemen capial (P ), ha is, i is a funcion of he raio q = Q/P. The evoluion of Q over ime akes ino accoun he discouned value of renals accruing in he fuure period, as well as he effec of capial accumulaion on nex period s capial sock and adjusmen coss. 4 Price Seing The price-seing srucure ha we employ follows Calvo (1983), in ways ha are recommended by Roemberg (1987) and are similar o he recen work of Yun (1996). Firms are assumed o be able o change heir prices only in specific saes of naure and mus oherwise saisfy all demand a he quoed price. As we will see, his laer requiremen ha we rea as one of he insiuions of he markeplace has imporan consequences for opimal price seing. The price-adjusmen even occurs wih probabiliy 1 so ha wih probabiliy η he firm is suck wih a predeermined nominal price. Accordingly, he expeced ime of price fixiy is (1 )1+ (1 ) n 1 (1 ) +..., which is equal o (1 ) 1. We assume ha he average firm adjuss is price every four quarers (once per year) so ha =.75, bu we also experimen wih higher values. The sochasic price-seing specificaion also implies a (saionary) disribuion of firms in erms of he ime ha hey las adjused heir price. The fracion of firms ha las adjused price j periods ago, j, is given by j = (1 ) j. This price-seing specificaion capures wo key feaures of price seing ha have been much emphasized by macroeconomiss working on price adjusmen: The iming and magniude of price adjusmens vary widely across firms in ways ha appear sochasic o an ouside observer. However, given ha he probabiliy of price adjusmen η is exogenous in he Calvo seing, he frequency of price adjusmen canno adjus o variaions in he sae of he economy: I canno change eiher wih he average rae of inflaion or wih he sage of he business cycle. 5 We assume ha firms are monopolisic compeiors and ha each faces a dae demand schedule of he form (13) y d j,= d (P j /P ), if i las adjused is price j periods ago and seleced he price P j. 6 In his expression, d is a demand shifer ha depends on he sae of he economy and P is he aggregae price level ha evolves according o (14) P =[ j=0 j( P j) (1 )] ( 1 ) 1. Using he saionary fracions given above, he dynamics of he aggregae price level P can be reduced o (15) P = [(1 )(P ) (1 ) + (P 1) (1 ) ] ( ), where (P 1) (1 ) represens he influence of he prices ch a rged by he fracion of firm s ha do no currenly change prices in period and (1 )(P ) (1 ε) re p re sens he influence of he firms ha do change prices in period. In his ime-dependen price-adjusmen framework, a firm ha is raionally The inroducion of invesmen adjusmen coss is no necessary o any of he main conclusions ha follow from in our analysis below. However, as a consequence of he evidence summarized in Chirinko (1993), here is a good reason for incorporaing his srucural aspec ino a modern macroeconomic model. 5 Neverheless, he Calvo seup is a naural saring poin for our analysis: In Dosey, King, and Wolman (1996), we argue ha a somewha more general ime-dependen specificaion is formally a firs-order approximaion o a richer sae-dependen price-adjusmen model. 6 This specificaion requires ha we view he consumpion and invesmen goods as he same consan elasiciy of subsiuion aggregaor of differeniaed producs. Blanchard and Kiyoaki (1987) derive such individual produc demand funcions for a model wih jus a consumpion good. 89

8 seing is price in period will choose o equae marginal revenue and marginal cos in a discouned, expeced value sense. Dynamic marginal revenue semming from a change in he price is given by E j =0,j n j [(1 )y d j, +j ]. The form of his expression reflecs he fac ha he price a j will remain a he chosen level P wih probabiliy n j. Similarly, he dynamic marginal cos associaed wih a price change is E j=0, j j [( ε)ψ +j (P ) 1 y d j, +j ]. We can simplify hese expressions as in he sandard saic case and hen equae dynamic marginal cos and revenue o obain a price ha i is opimal o charge: E j, j Ψ + j d + j P + j ( 16) P * j =0 =. 1 j, j d + j P+ j Three implicaions of his expression are worh elucidaing. Firs, if marginal cos were consan over ime a he level Ψ, hen we obain so ha he raio E j =0 P = 1 Ψ, = 1, is inerpreable as he markup jus as in he saic case. Second, in he dynamic seing, price seing is influenced by he scale of oupu (demand) unless he enire sequence of oupus in he numeraor and denominaor expressions are scaled by a common facor. Third, opimal price seing in he Calvo environmen involves forecasing boh demand and coss. Taking he price-seing rule (equaion 16) and he price index (equaion 15) ogeher, i is clear ha here is long-run homogeneiy of nominal prices in erms of nominal coss: W P=P = Ψ=. 1 1 ad1 f (n,k) Accordingly, as all of he oher behavioral equaions of our model are wrien in real erms, i follows ha our model will display long-run neuraliy of money: Permanen changes in he quaniy of M will ulimaely affec prices and no oupu, even wih shor-run rigidiy of prices. However, here will be a deparure from superneuraliy, in ha he same fricions ha lead o (1) he demand for money and (2) he shor-run rigidiy of prices will mean ha here will be real effecs of susained inflaion. In he nex secion we quanify hese real effecs. The Marginal and Average Markup Given ha firms are charging differen prices in our seing, i is clear ha here will be differen values of markups across firms. We define he marginal markup as ha earned by firms which are currenly adjusing price, ha is, =P / Ψ. We define he average markup as he raio of he aggregae price level o marginal cos, ha is, =P / Ψ. In a seady sae wih zero inflaion, boh of hese consrucions are equal o he saic markup = 1, bu in seady saes wih inflaion or deflaion, here will no longer be his equaliy. Similarly, in response o business flucuaions, he fac ha par of he price level is predeermined will lead o differen imeseries variaion in he marginal and average markup. 90

9 OPTIMAL INFLATION POLICY IN THE LONG RUN A naural mehod for deermining he arge rae of inflaion is o choose ha rae of inflaion which maximizes he welfare of he represenaive household. In his secion we conduc such an analysis for he long run, defined as a seady-sae siuaion in which all variables grow a consan raes in ways ha are consisen wih he model economy oulined in he previous secion. We solve he equaions of he model for he link beween inflaion and welfare, isolaing he facors ha are imporan for deermining he long-run opimal inflaion policy. In models wih perfec compeiion and coninuously adjused prices, here is a presumpion ha he opimal inflaion policy is ha which makes he privae cos of holding money equal o he social cos of producion, ha is, a nominal ineres rae of zero. Given a posiive real ineres rae, his condiion represens a prescripion for long-run deflaion policy ha was firs made by Milon Friedman (1969). Friedman s conclusion has been replicaed in a wide range of heoreical environmens. In paricular, leing he rae of inflaion be and he level of he real ineres rae be r, i follows ha he definiion (1 + R) = (1 + r)(1 + ) and he Friedman rule condiion R = 0 ogeher imply ha he opimal rae of inflaion should be f r = 1 + r, where he superscrip indicaes ha his is he Friedman rule level of he inflaion rae. Long-erm U.S. daa compiled by Ibboson and Sinquefeld (1982) indicaes ha he average real reurn on Treasury bills is beween 0 percen and 1 percen so ha his represens a recommendaion for a mos a small deflaion. Esimaing he Welfare Gains from Lower Inflaion One measure of he gains from lower inflaion may be compued as he ime savings generaed as agens increase heir cash balances, gains ha are someimes viewed as small. Following he provocaive work of Lucas (1993), here has been much recen ineres in he magniudes of hese gains. In his aricle, we use a benchmark esimae from Wolman (1996), he naure of which is displayed in Figure 1. In he firs sep, he hree-parameer money-demand funcion (equaion 3) is fi o annual U.S. ime-series daa over he period. Each annual observaion on he pair of raios, m/c and (c/w)(r/(1 + R)), eners as a do in he op panel of Figure 1. 7 The represenaive individual holds cash balances equal o 1.62 quarers of consumpion expendiure a he sample mean (denoed by a + in he op panel of Figure 1). The values of,, and are chosen o provide a bes fi in a leas squares sense, and he solid line in he op panel races ou he fied relaionship. As i urns ou, he esimaes indicae a saiaion level of real cash balances. Cash balances equal 2.7 quarers of consumpion expendiure a he esimaed saiaion level, which is somewha smaller han he maximum cash balances in he sample (abou 3.3 quarers of consumpion expendiure). In he second sep, we compue he ime savings ha arise as he inflaion rae is lowered from a benchmark level of 5 percen per year. For his purpose, we employ he esimaed parameer values obained in he firs sep and also he sample average value of c/w over Tha is, given he parameers, he marginal ime savings a any level of real balances is Dh m c = m c /ζ 1_v. Subsiuing in he formula for he real demand for money from equaion 3 and assuming ha he real ineres rae is invarian o inflaion, we can hen deermine he ime savings of moving from 7 The money sock is M1. The iners rae is a quarerly yield on commercial paper. We form m/c as he raio of nominal money o an annual nominal consumpion expendiure series (including durables). We hen divide by populaion and express he figure in erms of quarerly ime unis by muliplying by 4. Similarly, we divide nominal quarerly per capia consumpion by a nominal wage rae, measured as average hourly earnings of producion workers in manufacuring, in curren dollars. The average value of he resuling c/w series indicaes ha a represenaive worker requres 249 hours of work per quarer o purchase consumpion or abou 20 hours per week. In erms of inerpreing Figure 1, however, he reader should know ha we have rescaled c/w by dividing by average quarerly hours worked. Addiional deails on he daa (and he daa hemselves) are provided in a replicaion diskee ha you can reques from he auhors. 91

10 Figure 1a,b Time Cos of Inflaion (a) Acual and Esimaed Demand for Money (b) Time Saved In models wih monopolisic compeiion and saggered price seing, economiss have long suggesed ha he Friedman rule is no desirable. One argumen ha is someimes made for deparing from he Friedman rule is ha he markup of price over marginal cos may depend negaively on he inflaion rae. 8 Since higher levels of he markup depress economic aciviy (acing like a ax on facor supplies), he deflaion envisioned by Friedman would have social coss, as well as social benefis. Thus, here is an open issue as o he level of he opimal rae of inflaion wihin seups such as ours. The average markup of price over marginal cos ha prevails in our economy depends on wo facors, (17) = P = P P* P *. 8 Several recen papers make his poin in differing seings. Benabou (1992) suggess ha higher inflaion may raise he elasiciy of demand in a search model, hus lowering he markup. In a model ha shares he core neoclassical srucure of his aricle, bu uses a differing paern of price dynamics, Goodfriend (1995) also derives a link beween he markup and inflaion. one inflaion rae o anoher, which is he inegral of such ime savings over he relevan range. The lower panel of Figure 1 is he resul of his compuaion. We find ha a movemen from a 5 percen rae of inflaion o a 0 percen rae of inflaion would lead o a saving of abou 5 hours per quarer. Adoping he Friedman rule from an iniial posiion of 5 percen inflaion would lead o a saving of abou 7 hours per quarer. For an individual working 40 hours per week for 12 weeks in a quarer, hese ime savings represen abou 1 percen of his ime. Since he average hours worked by an average workage individual is lower, abou 20 hours per week, i follows ha he ime saving is roughly 2 percen. The Markup and Inflaion These wo facors are (1) he marginal markup (defined above as he raio of price o marginal cos for firms ha are free o adjus heir prices) and (2) he price adjusmen gap (defined as he raio of he general price level o he price charged by firms ha are free o adjus). The price adjusmen gap and inflaion. In a seady-sae siuaion, he price adjusmen gap is: 1 (1 ) P 1 (18) * P = 1 (1 ) This expression is obained by evaluaing equaion 15 in a siuaion where P and P * are growing a he rae. If here is no inflaion, hen here is no price adjusmen gap in he seady sae all firms are charging P*. A a higher rae of inflaion, i follows ha P is less han P * : A price adjusmen gap emerges. This gap reflecs he fac ha higher inflaion mechanically erodes he real value of markups and hus erodes relaive prices se by firms in pas periods. We hink of his price adjusmen gap as a feaure of sicky price models ha is 92

11 emphasized by radiional economeric price equaions such as hose developed in Ecksein and Fromm (1968). The implici assumpion in such sudies is ha firms se P * as a fixed markup over marginal cos. Wih such an assumpion, here is a subsanial effec of a susained inflaion on he markup and on macroeconomic aciviy (displayed in Figure 2). In he op panel, we display he implicaions of equaion 18 for he average markup. To consruc his diagram, we assume ha here is a gross markup equal o 1.3 a zero inflaion (so ha he elasiciy of demand, is 4.33). We also assume ha =.75 (so ha he average duraion of price fixiy is a year). Higher raes of inflaion erode he markup, which approaches zero a an annual rae of inflaion of abou 25 percen per year. Since he markup acs like a ax on real aciviy (as discussed in Blanchard and Kiyoaki, 1987; Roemberg and Woodford, 1991; and Goodfriend, 1995), declines in is level are associaed wih he subsanial increases in oupu shown in he lower panel of Figure 2. Concreely, if he price adjusmen gap were he only srucural feaure of price dynamics, hen an increase in inflaion from 5 percen o 10 percen would raise oupu permanenly by abou 7 percen. Effecs of forward-looking price seing. In a siuaion of susained inflaion, however, our model does no imply ha price adjusing firms behave in he manner ha underlies Figure 2. In paricular, he raio of newly se prices o marginal cos he marginal markup is given by (19) P * = 1 ( 1+ ) 1 (1+ r s )(1+ ) 1- ( 1+ ) 1 (1+ s r )(1+ ) - This expression is a seady-sae version of equaion 16 in which is he real growh 1. Figure 2a,b Implicaions of he Price Adjusmen Gap (a) Markup of Price over Marginal Cos (b) Oupu Compared o Baseline of 5% Inflaion rae of he economy, r s is he real ineres rae appropriae for discouning he firm s cash flows (given by he real rae of ineres on he sock marke, abou 6.5 percen), and 1, is he markup ha would prevail in he absence of saggered price seing (ha is, wih = 0). The opimal pricing rule makes his raio depend posiively on he inflaion rae. A higher expeced rae of inflaion leads firms o se a higher price when hey are free o adjus, principally because hey know ha as long as heir nominal price remains fixed, inflaion will erode boh heir relaive price and he real value of any markup esablished oday. The former erosion means here is increasing sub- 93

12 9 Michael Woodford and Preson McAfee each forcefully poined ou his conclusion o us during presenaions of his paper. 10 This follows Baxer s (1995) analysis of ax disorions in a variable labor supply seing. siuion oward a produc as long as is price is fixed. The laer erosion means ha per-uni profis fall for as long as a price is fixed. Since firms mus fill demand a posed prices, hey aemp o couner hese wo effecs by seing higher markups in he face of higher inflaion raes. The Approximae Opimaliy of he Friedman Rule A sriking feaure of our seup is ha he Friedman rule is approximaely opimal under a wide range of assumpions abou he magniude of price adjusmen ( ), he exen of he saic markup = 1, and he specificaion of he ransacions echnology. By approximae opimaliy, we mean he following. Since he ransacions echnology implies a saiaion poin for cash balances, a he Friedman rule, here mus be a zero loss o holding a slighly smaller amoun of real cash balances. Thus, if he markup can be reduced by slighly more inflaion a he Friedman rule (as i can for he parameers ha we sudy), hen here will be an unambiguous welfare gain o increasing inflaion from he Friedman rule. 9 Alhough his poin can be esablished formally, one needs a microscopic inspecion of he welfare rade-off o find i in he experimens repored below: The maximum welfare poin occurs very close o he Friedman rule. The benchmark case. To begin, he op panel of Figure 3 shows he relaionship beween welfare in he seady sae and he rae of inflaion for a benchmark case. In his benchmark case, we assume ha =.75, = 1 = 1.3, and make oher assumpions abou he seady sae of he model presened in Table 1. (These oher parameer assumpions are convenional in he quaniaive business cycle lieraure.) The reference poin for our analysis is a siuaion of 5 percen inflaion, indicaed by * in Figure 3. In models ha absrac from variaions in labor supply, a sandard measure of welfare is he fracion of seady-sae consumpion ha an individual would give up o avoid a disorion (as in Lucas, 1990). In our seing wih variable labor supply, we use he measure of welfare depiced graphically in he lower panel of Figure 3 he amoun of addiional consumpion expendiure ha could be purchased by an individual a an iniial relaive price of consumpion and leisure. 10 Alhough we measure he change in full income along he verical axis of he lower panel of Figure 3, valuing changes in leisure and consumpion a a consan relaive price, we express he welfare effec as a fracion of measured naional income, so i is direcly comparable o oher measures of welfare losses expressed as a fracion of seady-sae naional income. The salien feaures of Figure 3 are wofold. Firs, as he rae of inflaion falls oward f = 1 percen from he reference level of = 5 percen here is a subsanial increase in welfare. Fundamenally, his increase arises because here is a decrease in shopping ime when he nominal ineres rae falls and he associaed cos of money holding falls. Recen work by Lucas (1993) has documened he magniude of hese welfare gains in economies simpler han hose sudied in his aricle, namely ones ha absrac from variable labor supply, physical capial accumulaion, and saggered price seing. He finds gains on he order of 1.5 percen of naional produc, a useful benchmark for our discussion. In our seing, he magniude of he welfare gain o pursuing he Friedman rule is 1.1 percen of naional income. The gain o moving from 5 percen inflaion o 0 percen inflaion is abou.8 percen of naional income. Second, Figure 3 also shows ha 94

13 here are quaniaively imporan welfare losses from raising inflaion from he benchmark level of 5 percen: An increase in he inflaion rae from 5 percen o 12 percen lowers welfare by 2 percen. Inerpreaion. In Figure 1, here was a gain from lowering inflaion since he sociey economized on ransacions ime. In Figure 2, here was a gain from raising inflaion inflaion eroded markups and lower markups simulaed economic aciviy. Given he magniude of he real effecs in Figures 1 and 2, one would suspec a srong case for high inflaion. Ye, when hese wo forces are combined (as in Figure 3), here is a srong case for low inflaion. The main reason for his sriking finding is ha he experimen displayed in Figure 2 leaves ou he incenives firms have o change heir price-seing behavior in a siuaion of susained inflaion. Tha is, in our model, a siuaion of posiive seady-sae inflaion involves wo offseing displacemens relaive o price sabiliy. The firs is ha inflaion opens a price adjusmen gap, lowering he average markup because P <1. * P The second is ha he marginal markup rises wih inflaion as firms seek o avoid an erosion of heir relaive price and markup. Figure 4 conrass he overall effec of inflaion on he seady-sae markup (dashed line) o he parial effec from Figure 2 ha akes ino accoun only he price adjusmen gap (solid line). There are wo disinc regimes. Over he range relevan for considering a decrease in inflaion relaive o he benchmark level of 5 percen, he average markup is relaively unaffeced by inflaion: The incenives ha firms have o raise he marginal markup are jus offse by he decline (via he price adjusmen gap) in he average markup. 11 For his reason, he welfare analysis of inflaion coincides wih Friedman s analysis. Over he range relevan for considering increases in inflaion relaive o is bench- Table 1 P a r a m e e r s Own price demand elasiciy 4.33 Probabiliy price does no change 0.75 Muliplicaive erm in h ( ) funcion Curvaure of h ( ) funcion Shif erm in h ( ) funcion s n Labor share 2 3 Quarerly depreciaion rae Invesmen adj. cos parameer 2 ((i/k) ( / )) ( 1) Quarerly real growh rae (3 percen annually) Quarerly inflaion rae (5 percen annually) Uiliy discoun facor (quarerly) Uiliy funcion: u(c,l ) = ln(c ) ln(l ) Figure 3a,b Seady Sae Welfare (a) Welfare Compared o Baseline of 5% Inflaion (b) The Welfare Measure 95

14 Figure 4 The Markup and Inflaion How sensiive is he cos of inflaion o higher values of, ha is, o more ineria in prices? 11 For he parameer values we use, he markup is decreasing wih inflaion in a neighborhood of he Friedman rule. This leads o he approximae opimaliy resul discussed in he inroducion o his secion. 12 The =.8 value is an upper bound for our analysis, given our demand parameer ( or ). Wih higher values of, here is a sufficienly large cos of geing suck wih a fixed price ha i is opimal no o open a firm. This example indicaes an unforunae ension beween maching shor-run price dynamics and longer run inflaion responses in he Calvo model, one reason for he generalizaions considered in Dosey, King, and Wolman (1996). mark level, significan increases occur in he average markup. This response sems from an even larger effec on he marginal markup, ha in urn reflecs sufficien demand response o he real price declines associaed wih inflaion ha firms choose o increase he marginal markup dramaically o avoid he prospec of saisfying unprofiable demand in he fuure. Sensiiviy analysis. We now presen a brief sensiiviy analysis. Six dimensions exis along which i seems naural o explore he robusness of he relaionship beween welfare and inflaion depiced in Figure 4. How differen is he size of he cos of inflaion in our sicky price, monopolisically compeiive seing from ha in a flexible price, perfecly compeiive seing? How would his cos be overesimaed if one viewed price seers as myopic, in he same way we did in consrucing Figure 1 and as was frequenly done in radiional Keynesian macroeconomeric models? How differen would he coss of inflaion be if he economy were much less compeiive? How is he cos of inflaion affeced by eliminaing he inflaion disorion beween consumpion and leisure? How sensiive is he cos of inflaion o our assumpion ha here is a saiaion level of real cash balances? The six panels of Figure 5 display he answers o hese quesions. In each panel, we use he dashed line o re p resen he re s u l s f rom he benchmark case and a solid line o re p resen hose from he case under sudy. Variaions in compeiion, price rigidiy, and forward-looking behavior. In conducing he firs four aleraions in he model, we assumed ha he economies were each calibraed a a 5 percen inflaion rae, so ha he new and old lines all roae hrough he 5 percen inflaion poin. In panel A he effec of moving o perfec compeiion and flexible prices is shown: There is minimal difference. In panel B we provide a case similar o ha underlying Figure 2 in which price seers are assumed no o be forward-looking: I is he single case we have found of a seing in which here is a case for increased inflaion (a very srong one). In panel C we see he effec of raising he saic markup from 1.3 o 3: The effec is o slighly increase he welfare benefi from reducing inflaion. The analysis of Goodfriend (1995) can be used o explain his resul. Wih less compeiion (higher ), he wage rae is a smaller fracion of labor s marginal produc and oupu is correspondingly furher below is efficien level. Thus, when labor flows ou of ransacions aciviy and ino producive aciviy, here is a larger welfare gain. In panel D we see he effec of raising he expeced duraion of price seing from 4 quarers o 10 quarers, ha is, he effecs of an increase in from.75 o.8. There is lile effec on he benefis from lower inflaion bu a larger increase in coss of higher inflaion

15 Figure 5 a f Sensiiviy Analysis (a) Perfec Compeiion Welfare (b) Price Adjusmen Gap Only Welfare (c) Saic Markup 3 Welfare (d) Sickier Prices Welfare (e) Full Price Effec Welfare (f) No Saiaion Level of Real Balances Welfare Variaions in he money demand speci - ficaion. In panel E we show he effec of eliminaing he implicaions ha he money demand funcion has for he full price of consumpion: We se he erm w +j Dh m c + j + j m + j c 2 + j o zero in he expression (equaion 5) for opimal consumpion. We view his as he radiional neoclassical and monearis procedure of ignoring he implicaions of he cos of money holding for he full price, of consumpion i is used, for example, in mos exbooks wih srong neoclassical underpinnings, such as Bailey (1971), Barro (1990), and Abel and Bernanke (1992). However, his assumpion is he polar opposie of ha made in cash-inadvance models of money, where he ransacions echnology implicily makes variaions in he raio of consumpion o money infiniely expensive in erms of ime. Tha is, in he cash-in-advance analysis of Cooley and Hansen (1992), here is only a full price effec of inflaion, no a ime re- 97

16 13 Tha is, he slope of he solid line is infiniely negaive a f in Lucas s analysis raher han slighly posiive as in our analysis. 14 The impulse response funcions in Figures 6 o 10 were generaed by linearizing he model around is deerminisic seadysae growh pah, using he singular linear sysem mehods deailed in King and Wason (1995a and b). allocaion effec associaed wih cosly subsiuions in ransacions aciviies designed o avoid he inflaion ax. Panel E shows small differences in our cos of inflaion measure for small raes of inflaion, bu hese become more pronounced as he inflaion rae increases. Thus, he radiional neoclassical and monearis procedure is a good approximaion for small and moderae inflaions, bu i would likely no be for hyperinflaions. In panel F, we invesigae he effec of ruling ou saiaion in real balances (imposing he resricion = 0 when esimaing he money demand funcion). Lucas (1993) imposes such an assumpion ha has he effec of making he Friedman rule exacly raher han approximaely opimal. 13 However, in oher ways i has lile effec on our welfare compuaions. INFLATION TARGETING AND BUSINESS CYCLES Having esablished ha benefis o argeing a low rae of inflaion exis in he long run, we urn nex o evaluaing how a policy of inflaion argeing influences he dynamic response of our economy o various shocks. For his purpose, we use a perfec inflaion argeing scheme in which he cenral bank is presumed o manipulae he money supply each period so as o achieve exacly he arge rae of inflaion (an annualized 5 percen rae of inflaion in our experimens). Then we ask how real aciviy responds o produciviy and money demand shocks ha we ake o be wo major sources of disurbances. We model hese as random walk shocks because we hink hese are largely permanen in naure. The moivaion for his par of he invesigaion is o see wha gains or losses would arise if he cenral bank could arge he inflaion rae accuraely. In paricular, we are concerned wih wheher price-level argeing inroduces major gaps relaive o he real oucomes ha would arise if here were perfec price flexibiliy. The boom line is ha i does no, according o boh eyeball and formal welfare measures. In fac, compared wih a policy of money supply argeing, inflaion argeing succeeds in eliminaing hese gaps. To undersand his resul, i is necessary o undersand how real aciviy responds o shocks under money supply argeing. Our analyses of produciviy and money demand shocks hus begin wih his case. We subsequenly discuss how and why responses differ under inflaion argeing. Produciviy Shocks Figure 6 displays he impulse response funcions o a permanen produciviy shock of oupu, consumpion, invesmen, hours, he average markup, he real wage, he real ineres rae, and real balances, assuming ha he money growh rae is held consan. 14 The dashed lines describe a flexible price economy ( = 0), and he solid lines describe a sicky price economy ( =.75). To explain he marked differences beween he wo ses of curves, i is helpful o decompose he average markup as in he secion on esimaing he welfare gains from lower inflaion. We can use he decomposiion (equaion 17) a any dae o wrie he average markup as, (20) = 1 1 (P 1 /P ) (1 ) 1 1 also recognizing ha he price level evolves according o equaion 15 o describe he price adjusmen gap as a funcion of he curren and pas price level. Produciviy shock dynamics under a fixed money supply rule. Under a fixed money supply rule, we find he inuiive resul ha a permanen produciviy shock will raise oupu (Figure 6, panel A) and lower he curren price level from is rend when prices are flexible. Since capial and consumpion grow slowly oward he new seady sae, a siuaion of susained deflaion is shown in Figure 6: This is o be inerpreed as he price level rising a less han he benchmark 5 percen rae. If prices are sicky, here is also pressure for deflaion. Accordingly, he firs erm in he markup expression (equaion P *, 98

17 Figure 6 a h Money Supply Targeing, Response o a Permanen Produciviy Shock _ 0.75 (a) Oupu (b) Invesmen (c) Consumpion (d) Labor Inpu (e) Markup (f) Real Wage (g) Real Ineres Rae (h) Real Balances 99

18 15 A formal welfare comparison suppors he conclusion ha sicky price and flexible price economies are no very differen under perfec inflaion argeing. To produce his measure in an earlier draf of he aricle, we deermined he welfare effec of a shock using he dynamic equivalen of he measure described in he secion on opimal inflaion policy in he long run and illusraed in he lower panel of Figure 2. Linear approximaion mehods were used o compue hese welfare changes ha are relaed o he measure of he Hicksian wealh effecs of shocks produced in King (1993). 20) rises when he price level falls, increasing he markup: This is he flip side of he inflaion erosion effec discussed in he secion on he price adjusmen gap and inflaion. Since he produciviy shock lowers marginal cos a dae and in all fuure daes, here is a relaively small effec on he marginal markup componen P * The average markup rises in response o he produciviy shock. A rise in he markup (Figure 6, panel E) implies an increase in he wedge beween he marginal produc of labor and he real wage (Figure 6, panel F). Since he wedge will evenually reurn o is seady-sae level, here is a srong subsiuion effec ha causes labor inpu (Figure 6, panel D) o fall in he impac period. The direcion of he effec on labor conrass wih he case in which prices are flexible and markups never deviae from ε/(ε 1). In his case, labor inpu rises as agens capialize on he emporarily high real ineres rae. Thus under a money supply arge, a sicky price economy responds o shocks in a qualiaively differen manner han a flexible price economy. Produciviy shock dynamics wih a per - fec inflaion arge. The resuls differ subsanially when here is a policy of perfecly argeing he inflaion rae. Figure 7 displays comparable impulse response funcions in his case (he arge inflaion rae is 5 percen). A glance a Figure 7 reveals ha, unlike he money growh arge case, here is no qualiaive difference beween he responses under fixed and flexible prices. The average markup decomposiion is again helpful in undersanding his resul. Because inflaion argeing prevens shocks from affecing he price level, i eliminaes he price adjusmen gap channel he firs erm in he decomposiion so ha here is no endency for he markup o rise (Figure 7, panel E). Wih he markup essenially unchanged, he real wage fully inheris he increase in produciviy. Thus, a major componen of he subsiuion effec acing o. decrease labor supply disappears and, as in he flexible price model, hours rise in accord wih he opposing subsiuion effec of a emporarily high real ineres rae (Figure 7, panels F, D, and G). Thus, perfec inflaion argeing effecively makes a sicky-price economy behave like a flexible price economy: The key is ha inflaion argeing inroduces a moneary accommodaion ha avoids he necessiy for he price level o fall in response o permanen produciviy shocks. 15 Money Demand Shocks Figure 8 displays he resuls of a permanen shock o he demand for money under fixed money supply and price level pahs, wih he laer being equivalen o he oucome under flexible prices jus as i was in response o a echnology shock. Our money demand shock corresponds o a change in he cash balance efficiency condiion such ha here is a 1 percen higher demand for real balances a given levels of consumpion, he real wage rae and he nominal ineres rae. An imporan difference is eviden beween he wo ses of resuls shown in Figure 8: An increase in money demand produces a decline in oupu if here is a fixed pah of money, whereas i has no effec on oupu if policy is geared oward a sabilizing he price level. IMPLEMENTATION ISSUES We nex discuss wo ses of issues ha would be a necessary par of implemening a policy rule like ha discussed above. Firs, we ask wheher i is possible o use he nominal ineres rae as an insrumen o implemen a policy of conrolling he price level in our model economy. Second, we invesigae he consequences of immediaely ransiing o a lower inflaion rae. Implemenaion wih an Ineres Rae Insrumen Many cenral banks employ a shorerm ineres rae as he insrumen by 100