On the Taylor Rule and Optimal Monetary Policy in a Natural Rate Model. George Alogoskoufis*

Transcription

1 On he Taylor Rule and Opimal Moneary Policy in a Naural Rae Model by George Alogoskoufis* June 015. Revised May 016 Absrac This paper invesigaes he sabilizing role of moneary policy in a dynamic, sochasic general equilibrium model of he naural rae, in which non indexed nominal wages are periodically se by labor marke insiders. This nominal disorion allows for nominal shocks o have emporary real effecs, and hus, for moneary policy o be able o affec shor run flucuaions in boh inflaion and real oupu. We derive and analyze opimal moneary policy in he presence of real and nominal shocks, and highligh he properies of he opimal moneary policy rule. The opimal moneary policy rule is second bes, as i canno compleely neuralize produciviy shocks, and is associaed wih a radeoff beween he sabilizaion of inflaion and oupu. In he presence of moneary policy shocks, hese also affec oupu and inflaion and canno be fully neuralized. The opimal policy can be replicaed by a se of appropriaely paramerized Taylor rules, according o which deviaions of he curren nominal ineres rae from is naural rae, depend on deviaions of inflaion from arge and oupu from is naural level. This se of opimal Taylor rules is no unique. Provided ha he moneary auhoriies aach a sufficienly low weigh o deviaions of oupu from is naural level, he opimal policy could also be replicaed hrough a unique, appropriaely paramerized Wicksell rule, according o which deviaions of he nominal ineres rae from is naural rae depend only on deviaions of inflaion from arge. Keywords: moneary policy, aggregae flucuaions, Taylor rule, insiders and ousiders, naural rae JEL Classificaion: E3, E4, E5 * Deparmen of Economics, Ahens Universiy of Economics and Business, 76, Paission sree, GR-10434, Ahens, Greece. The auhor would like o acknowledge financial suppor from a deparmenal research gran of he Ahens Universiy of Economics and Business. alogoskoufis@me.com Web Page: alogoskoufisg.com.

2 The design of policies ha cenral banks should follow in order o sabilize he economy is one of he mos imporan concerns of moneary economics and pracical policymaking. There is wide agreemen abou he key objecives of moneary policy, which are none oher han price sabiliy and a sable pah for oupu and employmen, bu a number of oher disagreemens remain regarding he naure of appropriae policies. The long sanding argumens abou rules versus discreion have been resolved in favor of rules. Ye, here is sill a very lively debae abou he appropriae design of such rules. 1 Anoher long sanding debae has concerned he appropriae insrumen of moneary policy. In recen years he debae has iled in favor of ineres rae raher han money supply rules, in view of he difficulies in conrolling, or even defining, he appropriae measure of he money supply. The lieraure on ineres rae rules in he pas weny years has focused on he analysis of one paricular such rule, he Taylor (1993, 1999) rule. This rule is an exension of he classic Wicksell (1898) rule, for an economy in which here is a shor run radeoff beween inflaion and unemploymen. The Taylor rule has been shown o describe he moneary policy of he USA and oher advanced economies relaively well, since a leas he early 1980s. The Taylor rule has been analyzed exensively, and a number of proposals have been pu forward for improvemens in design and he choice of parameers of his paricular rule. 3 This paper conribues o he discussion of he pros and cons of he Taylor rule, by analyzing opimal moneary policy in he conex of a naural rae model. We use a dynamic, sochasic general equilibrium model economy, in which nominal wages are se periodically by insiders in he labor marke, and are non indexed. There are wo disorions in he model economy of his paper. One is a real disorion, arising from he fac ha ousiders are disenfranchised from he labor marke, and he second is a nominal disorion, arising from he fac ha nominal wage conracs are no indexed, and can only be reopened a he beginning of each period, bu no during he period. The srucure of he labor marke, as well as he assumpion of compeiive produc markes, consiue he mos imporan differences of his model from he sandard new keynesian model, as exposed for example in Woodford (003) and Gali (008). On he oher hand, he presen model is more parsimonious ha he sandard new keynesian model, and is characerized by simpler 1 Among he firs o discuss he dilemma of rules versus discreion was Simons (1936). The dilemma was analyzed more rigorously by Kydland and Presco (1977) in he conex of models wih raional expecaions. Fischer (1990) surveys he relevan lieraure. The classic analysis of he appropriae choice of moneary insrumens was Poole (1970). In an imporan paper, Sargen and Wallace (1975) demonsraed ha under raional expecaions, a non-coningen ineres rae arge leads o price level indeerminacy and insabiliy. However, i is now acceped ha his problem does no arise in he case of coningen ineres rae rules ha make he nominal ineres rae depend on he price level (McCallum 1981), or inflaion. See Woodford (003) Ch.1 for he relevan argumens. In addiion, cenral banks, have in he las hiry years been consisenly using ineres raes as heir main moneary policy insrumen. As noed by Bernanke (006), In pracice, he difficuly has been ha, deregulaion, financial innovaion, and oher facors have led o recurren insabiliy in he relaionships beween various moneary aggregaes and oher nominal variables.. See Clarida, Gali and Gerler (1999), Gali (008, 011a,b), Taylor and Williams (011) and Woodford (003, 011) for 3 surveys of he findings of his lieraure. Woodford (003) and Orphanides (008) survey he analysis of ineres rae rules from Wicksell (1898) o Taylor (1993, 1999).!

3 dynamics, hus allowing us o derive analyical soluions ha help sharpen our undersanding of he relevan policy issues. The real disorion in our model makes he naural rae of unemploymen inefficienly high, while he nominal disorion allows for nominal shocks o have emporary real effecs, and hus, for moneary policy o be able o affec flucuaions in boh inflaion and real oupu and employmen. Obviously, moneary policy canno correc he real disorion bu only he effecs of he nominal disorion on aggregae flucuaions. The model is characerized by an expecaions augmened Phillips curve, in which deviaions of oupu and employmen from heir naural level depend on unanicipaed curren inflaion and unanicipaed produciviy shocks. Aggregae demand is deermined by he opimal behavior of a represenaive household, wih access o a compeiive financial marke, choosing he pah of consumpion and real money balances in order o maximize is iner-emporal uiliy funcion. The model is also characerized by exogenous shocks o produciviy, and preferences for consumpion and money demand. We firs characerize opimal moneary policy in he presence of hese real and nominal shocks. We demonsrae ha under he opimal moneary policy rule, he only shocks ha canno be compleely neuralized by moneary policy are produciviy shocks, and shocks o moneary policy iself. Since produciviy shocks are supply shocks, whose real effecs can only be offse hrough unanicipaed inflaion, hey cause a radeoff beween deviaions of inflaion from arge, and oupu from is naural level, even under he opimal policy. This is no he case for demand shocks, which, wih he excepion of moneary policy errors, can be fully neuralized by moneary policy, hrough appropriae changes in nominal ineres raes. I is for his reason ha, in he presence of produciviy shocks, he opimal policy is second bes even in he absence of moneary policy errors. 4 The opimal moneary policy can in principle be implemened eiher using he nominal ineres rae, or he money supply as he moneary insrumen. To conform wih he pracice of cenral banks in recen decades, we concenrae on ineres rae rules. We proceed o invesigae he properies of he Wicksell (1898) and Taylor (1993, 1999) rules, and compare heir effecs o he opimal policy. The Wicksell rule requires he cenral bank o se he ineres rae a a level differen from is naural rae, only in response o deviaions of inflaion from arge. The Taylor rule is more general, and requires he cenral bank o se he ineres rae a a level differen from is naural rae in response o wo variables, deviaions of inflaion from is arge, and deviaions of oupu from is naural level. We esablish four main proposiions regarding hese rules: Firs, an appropriaely paramerized Taylor rule can replicae he opimal policy. I requires he cenral bank nominal ineres rae o reac one o one o flucuaions in he naural real ineres rae, o overreac o deviaions of inflaion from he cenral bank arge, and o reac posiively o deviaions of real oupu from is naural rae. Second, he opimal Taylor rule is no unique. An opimal Taylor rule can be defined for more Produciviy shocks disrup he divine coincidence characerizing he ypical imperfecly compeiive new 4 keynesian model. The divine coincidence, noed by Blanchard and Gali (007) for he sandard new keynesian model, means ha sabilizing deviaions of inflaion from arge also sabilizes deviaions of oupu from is naural level. Blanchard and Gali (007) inroduced exogenous real wage rigidiy o he new keynesian model o break his link.!3

4 han one pair of parameers deermining he response of nominal ineres raes o deviaions of inflaion from arge, and deviaions of oupu from is naural level, alhough here exiss a unique lineal relaion beween hese wo parameers. Third, an appropriaely paramerized Wicksell rule, is equivalen o an opimal Taylor rule, provided ha he weigh aached o oupu relaive o inflaion in he preferences of he cenral bank is sufficienly low. Oherwise, he opimal Wicksell rule may no saisfy he Taylor principle, which guaranees he sabiliy of he inflaionary process. The opimal Wicksell rule, if i exiss and is sable, is unique. Fourhly, we demonsrae ha a policy of absolue inflaion argeing is only opimal if he cenral bank is only concerned wih deviaions of inflaion from arge, and no deviaions of oupu from is naural level. In a separae analysis, delegaed o an appendix, we examine a simplified, bu widely used, version of he Taylor rule, he rule wih a consan inercep. This is equivalen o assuming ha he cenral bank does no reac o flucuaions in he naural real ineres rae. This rule is shown o be subopimal in he conex of our model, as i generaes persisen, policy induced, flucuaions of inflaion and high variabiliy of deviaions of oupu from is naural level. Given he consan inercep in he rule, persisen flucuaions of inflaion are he only way for he real ineres rae o equilibrae he produc marke in he face of persisen real shocks o he naural rae of ineres. In addiion, because all shocks affec inflaion under his rule, deviaions of oupu from is naural level depend on innovaions in all real shocks, and no only produciviy shocks, as under he opimal rule. Thus, a consan inercep Taylor rule could resul in poenially significan coss for he moneary auhoriies, in erms of persisen flucuaions of inflaion, and higher variabiliy of deviaions of oupu from is naural level han he opimal rule. The res of he paper is as follows: In secion 1 we presen a wage seing model, based on insiders and ousiders in he labor marke, and derive a Phillips Curve ype posiive relaion beween unanicipaed inflaion and deviaions of oupu and employmen from heir naural levels. The wage seing model inroduced in his paper combines and exends wo srands of he lieraure. Firs, he Gray (1976)-Fischer (1977) model of predeermined nominal wages, according o which nominal wages are se a he beginning of each period, and remain fixed for one period. Because shocks o inflaion are no known when nominal wage conracs are negoiaed, unanicipaed inflaion reduces real wages and causes employmen o increase along a downward sloping labor demand curve. Thus, he model produces a Phillips Curve ype of radeoff beween unanicipaed inflaion and oupu and employmen. The second srand of he lieraure is he insider-ousider heory of wage deerminaion of Lindbeck and Snower (1986), Blanchard and Summers (1986) and Gofries (199). According o his approach, here is an asymmery in he wage seing process beween insiders, who already have jobs, and ousiders who are seeking employmen. Ousiders are disenfranchised from he labor marke, and wages are se by insiders, who seek o maximize he real wage consisen wih heir own employmen, and no wih he employmen of he full labor force. This causes he naural rae of unemploymen o be inefficienly high. The oal number of insiders in he economy is assumed o be subjec o sochasic shocks, resuling in exogenous flucuaions in he naural rae of unemploymen and he naural level of employmen and oupu.!4

5 Employmen and oupu are deermined by compeiive firms, which se employmen in each period a he level which equaes he real wage o he marginal produc of labor. The marginal produc of labor is subjec o persisen produciviy shocks, which affec boh labor demand, and he oupu produced for given employmen. In secion, we analyze he demand side. Consumpion and money demand are deermined by a represenaive household, able o borrow and lend freely in a compeiive financial marke, a he marke ineres rae. Money eners he uiliy funcion of he represenaive household, and he demand for real money balances is proporional o consumpion, and inversely relaed o he nominal ineres rae. The Euler equaion for consumpion deermines he evoluion of privae consumpion. The preferences of he represenaive household for consumpion and real money balances are subjec o persisen sochasic shocks, which shif boh he Euler equaion for consumpion and he demand for money funcion. In secion 3 we analyze produc and money marke equilibrium. Produc marke equilibrium is achieved hrough adjusmen of he real ineres rae, which equaes aggregae demand and aggregae supply. The money marke equilibrium condiion depends on wheher he cenral bank follows a money supply rule, or an ineres rae rule. The demand for real money balances is proporional o real oupu and inversely relaed o he nominal ineres rae. If he cenral bank follows an ineres rae rule, he money supply adjuss endogenously o equilibrae he money marke, while if he cenral bank follows a money supply rule, nominal ineres raes are endogenously deermined by he equilibrium condiion in he money marke. In secion 4 we characerize opimal moneary policy. Moneary policy is decided by a cenral bank, which minimizes an iner-emporal quadraic loss funcion ha depends on deviaions of inflaion from a fixed arge, and oupu from is naural level. Since he cenral bank does no seek o achieve full employmen, bu seeks o sabilize oupu around is lower naural level, he opimal policy does no resul in inflaion bias relaive o he cenral bank arge. We derive he opimal policy, and show ha, in he presence of supply shocks, i is second bes, in ha he cenral bank canno perfecly sabilize eiher deviaions of inflaion from is arge, or oupu from is naural level. However, such deviaions are non persisen under he opimal policy, and only depend on produciviy shocks and moneary policy shocks. Oher aggregae demand shocks are fully neuralized hrough appropriae flucuaions in nominal and real ineres raes. In secion 5 we analyze he properies of he Wicksell (1898) and Taylor (1993, 1999) rules. We show how an appropriaely paramerized Taylor rule can replicae he opimal policy. We also show ha he opimal Taylor rule is no unique, as i can replicae he opimal policy under alernaive combinaions of parameers defining he response of nominal ineres raes o deviaions of inflaion from arge, and deviaions of oupu from is naural level. We also demonsrae ha an appropriaely paramerized Wicksell rule, wih a sufficienly high opimal response of he nominal ineres rae o deviaions of inflaion from arge, is equally effecive wih an opimal Taylor rule and ha, if i exiss, i is unique. Finally, we prove ha a policy of absolue inflaion argeing is only opimal if he cenral bank is concerned only wih deviaions of inflaion from arge, and no deviaions of oupu from is naural level. The sub-opimal simple Taylor rule, wih a consan inercep, is analyzed in Appendix. In he las secion we sum up our conclusions.!5

6 1. Insiders, Wage Seing and he Phillips Curve Consider an economy consising of compeiive firms, indexed by i, where i [0,1]. Labor is he only variable facor of producion, and firms deermine employmen by equaing he marginal produc of labor o he real wage. 1.1 Oupu, Employmen and Labor Demand The producion funcion of firm i is given by, 1 α! Y (i) = A L(i) (1) where Y(i) is oupu, A is exogenous produciviy, and L(i) is employmen. is a ime index, where =0,1,. 0<1-α<1 is he elasiciy of oupu wih respec o employmen. Employmen is deermined by firms, which maximize profis by equaing he marginal produc of labor o he real wage. Thus, employmen is deermined by he condiion ha,! (1 α )A L(i) α = W (i) () P where W(i) is he nominal wage of firm i, and P is he price for he produc of firm i. Since he produc marke is assumed o be compeiive, all firms face he same price, and P(i)=P for all firms. In log-linear form, (1) and () can be wrien as,! y(i) = a + (1 α )l(i) (3)! l(i) = l _ 1 (4) α (w(i) p a ) where! l _ = ln(1 α ) α Lowercase leers denoe he logarihms of he corresponding uppercase variables. (3) deermines oupu as a posiive funcion of employmen, and (4) deermines employmen as a negaive funcion of deviaions of real wages from produciviy. 1. Wage Seing and Employmen in an Insider Ousider Model Nominal wages are se by insiders in each firm, a he beginning of each period, before variables, such as curren produciviy and he curren price level are known. Thus, nominal wages are se on he basis of he raional expecaions of insiders abou hese shocks. Nominal wages remain consan for one period, and hey are rese a he beginning of he following period. Thus, his model is characerized by he real disorions emphasized by Lindbeck and Snower (1976), leading o an inefficienly high naural rae of unemploymen, and by he nominal wage sickiness of he Gray (1976), Fischer (1977), Gofries (199) models. Employmen is deermined ex pos by firms, given!6

7 he conrac wage, afer he curren price level and produciviy have been revealed. This se up leads o nominal shocks and moneary policy having emporary real effecs. The number of insiders, who a he beginning of each period deermine he conrac wage, is assumed exogenous. The key objecive of insiders is o se a nominal wage which, given heir raional expecaions abou he price level and produciviy, will minimize deviaions of expeced employmen from he pool of insiders. The expecaions on he basis of which wages are se depend on informaion available unil he end of period -1, bu no on informaion abou prices and produciviy in period. On he basis of he above, we assume ha he objecive of wage seers is o make expeced employmen saisfy a pah ha minimizes he following quadraic iner-emporal loss funcion, min E 1 n _ β s s=0 1 l(i) +s n_ (i) +s " is he logarihm of he number of insiders. β=1/(1+ρ)<1 is he discoun facor, wih ρ being he pure rae of ime preference. (5) is minimized subjec o he labor demand equaion (4). We assume ha he oal number of insiders in he economy is always sricly smaller han he labor force. We hus assume ha, 1! n _ (i) di = n _ < n,! (6) i=0 where n is he log of he labor force. From he firs order condiions for a minimum of (5), wages are se so ha expeced employmen for each firm saisfies, (5)! E 1 l(i) = n _ (i) (7) Inegraing over i, expeced aggregae employmen mus hen saisfy,! E 1 l = n _ (8) (8) is he same as (7) wihou he i index. (8) deermines he naural level of employmen, solely on he basis of he number of insiders in he wage seing process. Since he number of insiders is assumed o always be smaller han he labor force, he "naural" level of employmen is inefficienly low. Acual employmen is deermined by firms, afer he nominal wage has been se, and afer informaion abou curren prices, produciviy and oher shocks has been revealed. Inegraing he labour demand funcion over he number of firms i, aggregae employmen is given by,!7

8 ! l = l _ 1 (9) α (w p a ) From (8) and (9), he conrac wage saisfies,! w = E 1 p + E 1 a α(n _ l _ ) (10) The wage is se so as o make expeced employmen equal o he number of insiders, and is based on one period ahead expecaions abou he price level and produciviy. 1.3 An Expecaions Augmened Phillips Curve Subsiuing (10) in (9), acual employmen evolves according o, ( ) = n _ + 1 ( α π E 1π + a E 1 a )! l = n _ + 1 (11) α p E p + a E a 1 1 where,! π = p p 1 is he rae of inflaion. From (11), employmen deviaes from is naural level o he exen ha here are unanicipaed shocks o inflaion and produciviy. Unanicipaed increases in inflaion cause a reducion in real wages and increase labour demand and employmen, while, unanicipaed increases in produciviy increase produciviy relaive o real wages, and hus increase labour demand and employmen. We can define he unemploymen rae as,! u! n l (1) We can define he naural rae of unemploymen as, 5! u _! n n _ (13) From (11), (1) and (13), i follows ha, ( )! u = u _ 1 (14) α π E 1π + a E 1 a The unemploymen rae deviaes from is naural rae as a resul of unexpeced shocks o inflaion and produciviy, because boh reduce real wages relaive o produciviy, compared wih he prior expecaions of wage seers. (14) has he form of an expecaions augmened Phillips curve, which The concep of he naural rae of unemploymen is due o Friedman (1968). To quoe, The "naural rae of 5 unemploymen, is he level ha would be ground ou by he Walrasian sysem of general equilibrium equaions, provided here is imbedded in hem he acual srucural characerisics of he labor and commodiy markes, including marke imperfecions, sochasic variabiliy in demands and supplies, he cos of gahering informaion abou job vacancies and labor availabiliies, he coss of mobiliy, and so on. (p. 8). I is worh noing ha Friedman defined he naural rae of unemploymen in an analogous way o he naural rae of ineres of Wicksell (1898). He made a direc reference o Wicksell, and was fully aware of he analogy beween he wo conceps.!8

9 arises because nominal wages are se for one period and before curren inflaion and produciviy are known. We can also express his expecaions augmened Phillips curve in erms of oupu. From he loglinear version of he firm producion funcion in (3), aggregaing over firms, we ge an aggregae producion funcion in log-linear form, as,! y = a + (1 α )l (15) Subsiuing (11) in he log-linear version of he producion funcion (15), oupu supply evolves according o,! y = y _ + 1 α ( (16) α π E 1π + a E 1 a ) where,! y _ = (1 α )n _ + a (17) is he naural level of oupu. Unexpeced shocks o inflaion and produciviy cause oupu o be higher han is naural level, as hey cause employmen o be higher han is own naural level. (16) can be seen as he oupu version of he expecaions augmened Phillips curve, or as a shor run oupu supply funcion. 1.4 The Naural Rae of Unemploymen and he Naural Level of Oupu I is worh disinguishing beween he naural level of oupu and he full employmen level oupu. Full employmen oupu is given by, f! y = (1 α )n + a (18) Full employmen oupu is always higher han naural oupu in his model. The reason is ha equilibrium employmen is lower han full employmen, since he pool of insiders, who are ones who deermine equilibrium employmen hrough heir wage seing power, is smaller han he labor force. Thus, because of his real disorion in he labor marke, he naural level of oupu is inefficienly low, and he naural rae of unemploymen is inefficienly high. From (17) and (18), he relaion beween he naural rae of unemploymen and deviaions of oupu from full employmen oupu is given by, f! y y _ = (1 α )(n n _ ) = (1 α )u _ (19) This is he real disorion in his model, a disorion ha canno be addressed by moneary policy. The effecs of he nominal disorion, i.e he real effecs of unanicipaed changes in prices, can of course be parly offse by moneary policy.!9

10 . The Deerminaion of Aggregae Consumpion and Money Demand We nex urn o he deerminaion of aggregae demand. We assume ha he economy consiss of a large number of idenical households j, where j [0,1]. Each household member supplies one uni of labor, and unemploymen impacs all households in he same manner. The proporion of insiders is he same for all households. In addiion, he proporion of he unemployed is also assumed o be he same for all households. The represenaive household chooses (aggregae) consumpion and real money balances o maximize, s 1 θ 1 1! E (0) s=0 1+ ρ 1 θ V C +sc 1 θ M M +s + V +s P +s subjec o he sequence of expeced budge consrains,! E F +s+1 (1+ i +s ) F +s i +s M +s + P +s ( Y +s C +s T +s ) (1) 1+ i +s = 0 where! F = B + M. ρ denoes he pure rae of ime preference, θ is he inverse of he elasiciy of iner-emporal subsiuion, i he nominal ineres rae, F he curren value of household financial asses (one period nominal bonds B and money M), Y real non ineres income and T real axes ne of ransfers. V C and V M denoe exogenous sochasic shocks in he uiliy from consumpion and real money balances respecively. From he firs order condiions for a maximum,! V C C θ = λ (1+ i )P () θ M M! V (3) P = λ i P 1+ ρ! E λ +1 = E (4) 1+ i +1 λ where λ is he Lagrange muliplier in period. ()-(4) have he sandard inerpreaions. () suggess ha a he opimum he household equaes he marginal uiliy of consumpion o he value of savings. (3) suggess ha he household equaes he marginal uiliy of real money balances o he opporuniy cos of money. Finally, (4) suggess ha a he opimum, he real ineres rae, adjused for he expeced increase in he marginal uiliy of consumpion, is equal o he pure rae of ime preference.!10

11 From (), (3) and (4), eliminaing λ, 1 θ C M V! (5) P = C i M V 1+ i ( ) θ V C! E +1 C +1 = 1+ ρ V C C (6) P i P (5) is he money demand funcion, which is proporional o consumpion and a negaive funcion of he nominal ineres rae, and (6) is he familiar Euler equaion for consumpion. Log-linearizing (5) and (6), ( ) θ! m p = c 1 (7) θ ln i 1+ i + 1 θ v M C ( v ) ( ) + 1 θ (v C E v +1! c = E c +1 1 (8) θ i E π +1 ρ C ) where lowercase leers denoe naural logarihms, and,! π = p p 1 is he rae of inflaion. We hen urn o he deerminaion of equilibrium in he produc and money markes. 3. Equilibrium in he Produc and Money Markes Since here is no capial and invesmen in his model, produc marke equilibrium implies ha oupu is equal o consumpion.! Y = C (9) Subsiuing (9) in (7) and (8), we ge he money and produc marke equilibrium condiions,! m p = y 1 (30) θ ln i 1+ i + 1 θ v M C ( v ) ( ) + 1 θ (v C E v +1! y = E y +1 1 (31) θ i E π +1 ρ C ) (30) is he money marke equilibrium condiion, he equivalen of he LM Curve, and (31) is he produc marke equilibrium condiion, he equivalen of he IS Curve. Since oupu demand depends on deviaions of he real ineres from he pure rae of ime preference, he real ineres rae is he relaive price ha adjuss o equilibrae oupu demand wih oupu supply. No oher relaive price can play his role, as he real wage is deermined in order o make expeced labor demand equal o he number of insiders in he labor marke.!11

12 " 3.1 The Naural Real Ineres Rae and he Curren Real Ineres Rae The real ineres rae is defined by he Fisher (1896) equaion, 6 r = i E π +1 (3) The naural real ineres rae is deermined by he produc marke equilibrium condiion, when oupu is a is naural rae. From (17) and (31), he naural real ineres rae is given by, ( )! r _ = ρ θ (1 α ) n _ E n _ +1 (33) + a E a ( +1) + v C C E v +1 The naural real ineres rae is equal o he pure rae of ime preference, bu also depends posiively on deviaions of curren shocks o consumpion from anicipaed fuure shocks, and negaively on deviaions of curren produciviy shocks from anicipaed fuure shocks, as well as deviaions of he curren naural level of employmen from is anicipaed fuure level. Thus, real shocks ha cause a emporary increase in he naural level of oupu reduce he naural real rae of ineres, in order o bring abou an corresponding reducion in consumpion and mainain produc marke equilibrium. On he oher hand, real shocks ha cause a emporary increase in consumpion, require an increase in he naural real rae of ineres, in order o induce lower consumpion, and mainain produc marke equilibrium. 7 Because of he nominal rigidiy of wages for one period, he curren equilibrium real ineres deviaes from is naural rae. The curren real ineres rae is deermined by he equaion of he oupu demand funcion (31) wih he oupu supply funcion (16). I is hus deermined by, ( )! r = r _ θ 1 α ( π E 1 π + a E 1 a ) (34) α Unanicipaed shocks o inflaion or produciviy, which cause a emporary rise in curren oupu relaive o is naural level, also reduce he curren real ineres rae relaive o is naural rae. This is he well known Wicksellian mechanism, emphasized for he firs ime by Wicksell (1898). We shall reurn o his mechanism when we discuss alernaive ineres rae rules in secions 4 and Equilibrium Flucuaions wih Exogenous Preference, Produciviy and Labor Marke Shocks We shall assume in wha follows ha he logarihms of he exogenous shocks o preferences and produciviy follow saionary AR(1) processes. 6 To quoe from Fisher (1896), When prices are rising or falling, money is depreciaing or appreciaing relaive o commodiies. Our heory would herefore require high or low ineres according as prices are rising or falling, provided we assume ha he rae of ineres in he commodiy sandard should no vary. (p. 58). The rae of ineres in he commodiy sandard is he real ineres rae, and rising or falling prices are expeced inflaion.the Fisher equaion was furher elaboraed in Fisher (1930), where i was made even clearer ha Fisher referred o expeced inflaion. The concep of a naural rae of ineres was inroduced by Wicksell (1898). To quoe, There is a cerain rae of 7 ineres on loans which is neural in respec o commodiy prices, and ends neiher o raise nor o lower hem. This is necessarily he same as he rae of ineres which would be deermined by supply and demand if no use were made of money and all lending were effeced in he form of real capial goods. I comes o much he same hing o describe i as he curren value of he naural rae of ineres (p. 10).!1

13 !!! v C = η C v C 1 + ε (35) v M = η M v M 1 + ε (36) A a = η A a 1 + ε (37) where he auoregressive parameers saisfy, processes. 0 < η, and ε C, ε M, ε A C,η M,η A < 1, are whie noise We shall furher assume ha he (log of he) labor force is fixed a n, and ha he exogenous number of insiders is also fixed. We hus assume ha,! n > n _ > 0 (38) Thus he naural rae of unemploymen is assumed consan a,! u _ = n n _ > 0 (39) This does no imply any loss of generaliy, as we shall concenrae on flucuaions of unemploymen around is naural rae. Wih hese assumpions, curren employmen, unemploymen, oupu, real wages and he real ineres rae, as funcions he exogenous shocks and shocks o inflaion, evolve according o, ( )! l = n _ + 1 (40) α π E A 1π + ε where! is given by (38). n _ ( )! u = u _ 1 (41) α π E A 1π + ε where! is given by (39). u _! y = y _ + 1 α (4) α π E A ( 1π + ε ) where,! y _ = (1 α )n _ + a. ( ) _! w p = (w p) π E 1 π (43) _ where,!(w p) = η A a 1 α(n _ l _ ). ( )! r = r _ θ 1 α A ( π E 1 π + ε ) (44) α where! r _ C = ρ θ(1 η A )a + (1 η C )v!13

14 The naural raes (or levels) of real variables evolve as funcions of he exogenous real shocks. However, unanicipaed inflaion, and innovaions in produciviy, by reducing real wages relaive o produciviy, cause a emporary increase in employmen and oupu above heir naural level, and a emporary reducion in unemploymen and he real ineres rae below heir naural raes. We nex urn o moneary policy and he deerminaion of he inflaion rae. 4. Opimal Moneary Policy We approach he deerminaion of he inflaion rae, and oher nominal variables, assuming ha he moneary auhoriies deermine moneary policy in order o minimize deviaions of inflaion from a consan inflaion arge π* and deviaions of oupu from is naural level. We hus assume, ha, hrough he appropriae policy insrumen, he moneary auhoriies choose he inflaion rae in order o minimize, 1 1! Λ = E (45) s=0 1+ ρ ( π π * +s ) + ψ y +s y_ +s where ψ is he relaive weigh of oupu relaive o inflaion in he preferences of he moneary auhoriies. In conras o wage seers, i is assumed ha he moneary auhoriies have full informaion abou curren shocks when hey deermine moneary policy. 4.1 Opimal Moneary Policy s We shall examine opimal moneary policy in he form of an inflaion rae ha minimizes (45), subjec o he oupu supply supply funcion (4). From he firs order condiions for a minimum,! E π = π * ψ (1 α ) E (y y _ (46) α ) where! E π is he opimal inflaion objecive of he cenral bank a ime. From (46), under he opimal policy, he curren inflaion objecive of he cenral bank deviaes from he consan seady sae inflaion arge π*, o he exen ha oupu deviaes from is naural rae. By reducing inflaion relaive is consan inflaion arge in he case of a posiive deviaion of oupu from is naural rae, he cenral bank reduces he oupu deviaion. The opposie happens in he case of a negaive deviaion of oupu from is naural rae. By increasing inflaion relaive is consan inflaion arge in he case of a negaive deviaion, he cenral bank again reduces he deviaion of oupu. We shall assume ha he conduc of opimal moneary policy is also subjec o a moneary policy error, which could be due eiher o an inaccurae esimae of he naural rae of oupu, or o an implemenaion error in he conduc of moneary policy. Thus, from (46), curren inflaion under he opimal policy is given by,! π = E π + ε N = π * ψ (1 α ) (y y _ (47) α ) + ε N!14

15 where ε N is a whie noise nominal shock, due o an error in he implemenaion of opimal moneary policy. 8 From (47) and (4), under he opimal policy, he deviaions of real oupu from is naural level and inflaion from he consan seady sae inflaion arge are given by,! y y _ α = (48) ψ (1 α ) ξ ε A N ( + ε )! π π* = ξε A N + (1 ξ)ε (49) ψ (1 α ) where! ξ =. α +ψ (1 α ) < 1 Under he opimal inflaion policy, he deviaions of oupu from is naural level depend only on he par of he innovaion of produciviy and he nominal policy error ha is no neuralized by moneary policy. Thus, he nominal disorion arising from he fac ha nominal wages are se in advance and canno be changed, can only parially be correced hrough opimal moneary policy. There are wo reasons ha opimal moneary policy canno fully correc hese disorions. The firs reason is ha moneary policy operaes hrough he demand side. Even in he absence of moneary policy errors, moneary policy can compleely neuralize only he impac of demand side shocks, bu i can only counerac he real impac of aggregae supply (produciviy) shocks hrough unanicipaed inflaion. Thus, he opimal policy is second bes even in he absence of moneary policy errors, since he cenral bank has one insrumen, inflaion, hrough which i seeks o aain wo linearly independen arges. 9 The second reason ha opimal moneary policy canno fully correc he disorion is he assumpion ha here are moneary policy errors as well, which affec inflaion and cause oupu o deviae from is naural rae. From (49), i follows ha,! E 1 π = E π +1 = π * (50) The expeced iner-emporal welfare loss of he moneary auhoriies under he opimal policy is posiive. From (45), (48) and (49), i is given by, 8 If he error is due o an inaccurae esimae of he naural rae and he cenral bank has raional expecaions, he moneary policy error will necessarily be whie noise. I i is an implemenaion error, he whie noise assumpion means ha he cenral bank does no make sysemaic misakes. If he cenral bank only cared abou inflaion, i.e ψ=0, hen ξ=0 and he opimal policy would have been firs bes. The 9 divine coincidence, noed by Blanchard and Gali (007) for he sandard imperfecly compeiive new keynesian model wih saggered pricing, does no apply o his naural rae model. In he presence of supply shocks, inflaion sabilizaion does no resul in oupu sabilizaion, because of he innovaions in produciviy. Even under he opimal policy, here is a meaningful radeoff beween inflaion sabilizaion and oupu sabilizaion in his model.!15

16 ! Λ * = 1 1+ ρ Var(π π*) +ψvar(y y _ ρ ) = 1 1+ ρ ρ where! σ A is he variance of he innovaions in produciviy! ε A and! σ N is he variance of he moneary policy error ε Ν. 4. The Case of Inflaion Bias ξ σ A + ( 1 ξ ) ( σ N ) +ψ αξ ψ (1 α ) Noe ha if he objecive of he cenral bank depended on deviaions of oupu from full employmen oupu, raher han he naural level of oupu, hen he opimal inflaion rae would have aken he form, σ ( A +σ N ) (51) ψ (1 α )! π = π *+ u _ ξε A N + (1 ξ)ε (49 ) α In such a case, inflaion would be sysemaically higher han π* and would also depend posiively on he naural rae of unemploymen. Since he naural rae of unemploymen is posiive, here would be a posiive inflaion bias in such a case, as he inflaionary expecaions of insiders would adjus upwards, in order o neuralize he incenives of he moneary auhoriies o sysemaically ry and increase employmen and oupu above heir naural levels. 10 In wha follows, we shall confine ourselves o he case in which he objecive of he moneary auhoriies is described by (45), and he opimal policy of he moneary auhoriies does no suffer from he inflaionary bias associaed wih rying o achieve full employmen. 5. Inflaion and Oupu Flucuaions under Feedback Ineres Rae Rules We nex urn our aenion from opimal policy o feedback ineres rae rules such as he he Wicksell (1898) and Taylor (1993) rules. We confine ourselves o ineres rae rules, as, by revealed preference, he shor run nominal ineres rae is he insrumen of choice of cenral banks. This does no necessarily mean ha we ignore he implicaions of hese rules for he money supply, as, hrough he money demand funcion (7), any ineres rae rule can be ransformed ino a corresponding money supply rule. Our objecive is o compare he effecs of such rules o he opimal inflaion policy (46) and (47). 5.1 The Wicksell Inflaion Targeing Rule Wicksell (1898) was probably he earlies advocae of a sabilizing feedback ineres rae rule. He proposed ha, So long as prices remain unalered, he banks rae of ineres is o remain unalered. If prices rise, he rae of ineres is o be raised; and if prices fall, he rae of ineres is o be lowered; and he rae of ineres is henceforh o be mainained a is new level unil a furher This is he well known Kydland and Presco (1977) and Barro and Gordon (1983) inflaion bias of ime consisen 10 moneary policy. Assuming ha moneary policy is in he hands of an independen cenral banker, wih an objecive as in (45), addresses he inflaion bias problem (Rogoff 1985).!16

17 movemen of prices calls for a furher change in one direcion or he oher. (Wicksell 1898, p. 189). 11 In he conex of our model wih a non zero arge inflaion, a Wicksell rule could be wrien as, ( ) + ε i! i = r _ + π *+φ π π * (5) where r _ is he naural real rae of ineres, φ > 0, and ε i is a whie noise ineres rae policy error. Subsiuing he Wicksell rule (5) in he Fischer equaion (3), and solving for curren inflaion, inflaion is deermined by, 1! π = (53) φα +θ(1 α ) α(φ 1)π *+θ(1 α )E 1π + αe π +1 θ(1 α )ε A i αε Noe ha he inflaionary process depends on he key parameer of he Wicksell rule φ, as unanicipaed inflaion causes oupu o deviae from is naural rae. In addiion, he effecs of produciviy and moneary policy shocks on inflaion, also depend on he parameers of he Wicksell rule. Taking expecaions condiional on informaion available up o -1, we ge ha,! E 1 π = φ 1 (54) φ π *+ 1 φ E 1π +1 The difference equaion (54) is sable only if,! φ > 1 (55) Thus, he inflaion process will be sable and deerminae under a Wicksell rule of he form of (5), only o he exen ha he response of he nominal ineres rae o curren inflaion is higher han one. This condiion is he well known Taylor principle, for he sabiliy of he inflaion process under a nominal ineres rae rule. I requires ha nominal ineres raes reac more han one o one deviaions of curren inflaion from arge inflaion, in order o affec real ineres raes. This is a sufficien condiion for a sable and deerminae inflaion process. 1 If he Taylor principle is saisfied, he soluion for expeced inflaion is equal o, 11 Recall ha Wicksell was also he one who inroduced he disincion beween he naural and he curren ineres rae, and ha he was wriing a a ime when average (expeced) inflaion was zero, and before Fisher s (1896) disincion beween nominal and real ineres raes had been adoped. 1 Woodford (003), among ohers, conains a deailed discussion of he Taylor principle, and is significance for he resoluion of he price level indeerminacy problem highlighed by Sargen and Wallace (1975) for non coningen ineres rae rules.!17

18 ! E 1 π = E π +1 = π * (56) Subsiuing (56) in he inflaion equaion (53), inflaion under he Wicksell rule is deermined by, θ(1 α )! π = π * (57) φα +θ(1 α ) ε α A φα +θ(1 α ) ε i Comparing (60) o inflaion under he opimal moneary policy as given by (49), and assuming ha moneary policy errors are he same under he implemenaion of he opimal rule and he Wicksell rule, he opimal response of he nominal ineres rae o inflaion under he Wicksell rule is given by comparing he coefficiens of he produciviy shock εa. Comparing coefficiens we see ha he opimal φ is deermined by, θ(1 α )(1 ξ)! φ = = θ α (58) αξ ψ 1 α Thus, a Wicksell rule would be opimal, if φ is deermined as suggesed in (58). However, for he opimal φ o saisfy he Taylor principle as well, and for he inflaionary process o be sable, i mus also hold ha,! ψ < θα (59) 1 α This implies ha he opimal policy can be implemened as a Wicksell rule only in he case in which ψ, he cos of deviaions of oupu from is naural rae relaive o he cos of deviaions of inflaion from arge in he preferences of he cenral bank is sufficienly small. 13 A special case of he Wicksell rule is he absolue price level argeing rule of Fisher (1919) and Simons (1936), who wen furher han Wicksell, by suggesing a policy of complee sabilizaion of he price level. In he presen conex, his could be expressed as absolue inflaion argeing, i.e seing he nominal ineres rae so ha inflaion is always equal o he arge of he auhoriies π*. This is he limi of he Wicksell rule (53), as he response of nominal ineres raes o deviaions of inflaion from arge ends o infiniy. In his case, inflaion would end o be equal o π* a all imes. Thus, he absolue inflaion argeing rule could be expressed as he limi of (5), as φ ends o infiniy. This limiing policy would be opimal only in he case where he cenral bank is only concerned wih inflaion, and no deviaions of oupu from is naural rae, and hus he case where ψ=0. 5. The Taylor Inflaion and Oupu Targeing Rule The Taylor (1993) rule is more general han he Wicksell rule. The principle of he Taylor rule requires he moneary auhoriies o se he nominal ineres rae as a funcion of he naural real rae of ineres, plus heir inflaion arge, and adjus i on he basis of deviaions of curren inflaion For example, wih an iner-emporal elasiciy of subsiuion in consumpion 1/θ equal o uniy, and a share of labor 13 in oal oupu 1-α equal o wo hirds, his would require ha he cenral bank cares a leas wice as much for inflaion relaive o oupu.!18

19 from arge, and deviaions of oupu from is naural level. We shall assume ha his generalized Taylor rule akes he form, " i = r _ i + π *+φ 1 π π * (60) where φ 1,φ > 0. ( ) + φ (y y _ ) + ε 14 I is obvious ha he Taylor rule (60) encompasses he previous rules as special cases. The Wicksell rule (53) is a special case of he Taylor rule, wih φ=0. The Fisher-Simons rule is he limiing case of (60) as φ1, and φ=0. By subsiuing (60) in he Fisher equaion (3), afer using he real ineres equaion (44) and he oupu supply funcion (4), we ge he following process for inflaion,! π = γ 1 E 1 π + γ E π +1 + (φ 1 1)γ π * γ 1 ε A i γ ε (61) ( φ where! γ 1 = +θ )(1 α ),!. φ 1 α + ( φ +θ )(1 α ) < 1 γ = α φ 1 α + ( φ +θ )(1 α ) Noe ha he inflaionary process depends on boh parameers of he Taylor rule, as unanicipaed inflaion causes oupu o deviae from is naural rae. In addiion, he effecs of produciviy shocks on inflaion, also depend on he parameers of he Taylor rule. 15 One can show ha he process (61) is sable, if, γ1+γ<1. A sufficien condiion for his is, 16! φ 1 > 1 (6) Condiion (6) is he same as he one we derived for he sabiliy of inflaion under he Wicksell rule, and is he Taylor principle. If (6) is saisfied, hen he raional expecaions soluion of he inflaion equaion (61) is given by,! π = π * γ 1 ε A i γ ε (63) From (63), he flucuaions of inflaion around he arge of he moneary auhoriies π* only depend on he curren innovaion in produciviy and he curren moneary policy error Taylor (1993) proposed a rule is which he naural rae of ineres was consan a %. We shall erm his rule he consan inercep Taylor rule, and we analyze i in Appendix. In he body of he paper we sick o he spiri of he analysis of Taylor, and rea he naural real rae of ineres as an endogenous variable. 15 (61) being he inflaionary process from a dynamic sochasic general equilibrium, in which he policy rule of he moneary auhoriies is aken ino accoun, i does no suffer from he Lucas (1976) criique. Changing he parameers of he policy rule, would also change he parameers of he inflaionary process. 16 The raional expecaions soluion of (61) is derived in Appendix The soluion of (56) under raional expecaions is derived in Appendix 1.!19

20 The flucuaions of oupu around is naural level depend on unanicipaed inflaion and innovaions in produciviy. From (63), unanicipaed inflaion is deermined by,! π E 1 π = π π* = γ 1 ε A i γ ε (64) Subsiuing (64) in he oupu supply funcion (4), deviaions of oupu from is naural rae under he Taylor rule are deermined by,! y y _ = 1 α (65) α (1 γ )ε A i ( 1 γ ε ) As in he case of he opimal policy, under he Taylor rule (60), only innovaions in produciviy and moneary policy errors induce deviaions of inflaion from arge, and oupu from is naural level. Oher shocks, such as shocks o consumpion preferences, are fully neuralized by moneary policy, as he nominal ineres rae fully accommodaes changes in he naural rae of ineres. In wha follows we shall ignore errors in moneary policy, assuming hey have he same impac boh under he Taylor rule and he opimal policy. We shall hus concenrae on he condiions under which he reacion of he Taylor rule o produciviy shocks has he same impac as under he opimal policy. The Taylor rule (60) would be he opimal policy, if γ1 was equal o he opimal response of inflaion o innovaions in produciviy ξ derived in equaion (49). From he definiion of ξ in (49), and he definiion of γ1 in (61), his requires ha φ1 and φ mus saisfy, (φ! +θ)(1 α ) (66) φ 1 α + (φ +θ)(1 α ) = ψ (1 α ) α +ψ (1 α ) As long as he policy parameers are chosen o saisfy (66), a Taylor rule can fully replicae he opimal policy in his model. Solving (66) for he opimal φ1 we ge, α! φ 1 = (φ +θ) (67) ψ (1 α ) > 1 From (67), here is an infinie se of pairs of φ1 and φ. As long as,! φ > ψ (1 α ) θ (68) α any pair of φ1 and φ ha saisfies (67) is opimal. Assuming θ=1, 1-α=/3 and ψ=1/, φ mus be posiive. Assuming ha i is equal o 0.5, as suggesed by Taylor (1993), he opimal φ1, wih hese parameer values, is equal o 1.5, exacly as suggesed by Taylor. However, he pairs, 1.1 and 0.1, 1. and 0., 1.3 and 0.3 and so on, are also!0

21 opimal. The Wicksell rule is no sable in his case, as i requires ψ<1/. In his case, he Wicksell rule would be a special case of he Taylor rule. We have hus demonsraed a number of proposiions in he conex of he model economy sudied in his paper, ha can be summarized as follows: Proposiion 1: There exiss an opimal Taylor rule of he form of (60), which resuls in a sable and deerminae inflaion process. The Taylor rule requires he cenral bank nominal ineres rae (a) o fully reflec flucuaions of he curren naural real rae of ineres, convered o a nominal rae a he cenral bank s inflaion arge, (b) o respond posiively and more han one o one o deviaions of inflaion from he cenral bank s arge, and (c), o respond posiively o deviaions of oupu from is naural rae. Proposiion : The se of opimal Taylor rule reacion parameers o inflaion and oupu is infinie, alhough a unique linear relaion of he form of (67) mus exis beween hem. Proposiion 3: As long as he weigh aached by he cenral bank o deviaions of oupu from is naural level, relaive o deviaions of inflaion from arge, is sufficienly low and saisfies (59), here exiss a sable and unique opimal Wicksell rule, which is described by (5). This is a special case of he Taylor rule (60). Proposiion 4: The Fisher-Simons absolue inflaion argeing rule, which requires he response of nominal ineres raes o deviaions of inflaion from he cenral bank arge o approach infiniy, will only be opimal if he relaive weigh aached by he cenral bank o deviaions of oupu from is naural level, relaive o deviaions of inflaion from arge, is equal o zero. We have used (67) and (68) o calculae he se of opimal Taylor rule parameers for differen values of he srucural parameers of he model and he preferences of he moneary auhoriies for oupu versus inflaion. Assuming ha θ=1 and α=1/3, which is a se of preference and echnological parameers ofen used in calibraion exercises, he pair favored by Taylor, i.e φ1=1.5, φ=0.5, suggess, in he conex of our model, a relaive weigh on oupu relaive o inflaion ψ of abou 0.5. However, he same oucome, under he same preferences of he moneary auhoriies, could be achieved by an infinie combinaion of oher parameers ha saisfy (67) and (68). On he oher hand, for a Wicksell rule o be boh opimal and sable, boh (67) and (68) mus be saisfied. Using he same values, θ=1 and α=1/3, (66) implies ha ψ mus be lower han 1/. Thus, for he Wicksell rule o be applicable, ψ mus be lower ha 1/ in he conex of our model. For example, if ψ=0.4, he Wicksell rule would be opimal wih φ1=1.5, which is uniquely deermined from (67).!1

22 One final poin is in order. In many discussions and applicaions of he Taylor rule, he curren naural real rae of ineres in (55) is replaced by he long run equilibrium real ineres rae ρ. We call his he consan inercep Taylor Rule. 18 In he conex of our model, he consan inercep Taylor rule can be shown o be sub-opimal. I is demonsraed in Appendix, ha his simpler version of he Taylor rule would resul in policy induced inflaion persisence, higher variabiliy of boh inflaion and oupu and dependence of he variance of inflaion and oupu on all real demand and supply shocks. The reason is ha under his version of he Taylor rule he cenral bank nominal ineres rae does no reac direcly o flucuaions in he naural real ineres rae, bu reacs indirecly, hrough deviaions of curren inflaion from arge, and oupu from is naural level. As a resul boh of hese deviaions display boh higher variabiliy and persisence compared o he opimal moneary policy. 6. Conclusions In his paper we have analyzed he sabilizing role of moneary policy in a dynamic, sochasic general equilibrium model of he naural rae, in which nominal wages are periodically se by labor marke insiders. There are wo disorions in his model, a real disorion, arising from he fac ha ousiders are disenfranchised from he labor marke, and a nominal disorion, arising from he fac ha nominal wage conracs are no indexed, and can only be reopened a he beginning of each period. The nominal disorion allows for nominal shocks o have emporary real effecs, and hus, for moneary policy o be able o affec shor run flucuaions in boh inflaion and employmen. We have derived and discussed opimal moneary policy in he presence of a variey of real and nominal shocks, and highlighed he properies of he opimal moneary policy rule. The opimal moneary policy rule is second bes, as i canno neuralize supply (produciviy) shocks, and he auhoriies have o accep a radeoff beween inflaion and oupu sabilizaion even under he opimal policy. We have also demonsraed ha he opimal policy can be replicaed by an appropriaely paramerized se of Taylor rules, according o which deviaions of he curren nominal ineres rae from is naural rae, depend on deviaions of inflaion from arge and oupu from is naural level. We have shown ha he se of parameers of he opimal Taylor rule is no unique in he conex of his model, alhough here is an opimal analogy beween he wo reacion coefficiens of he Taylor rule. We have also shown ha, provided he moneary auhoriies aach a relaively low weigh o deviaions of oupu from is naural rae, relaive o deviaions of inflaion from arge, he opimal policy can also be replicaed by an appropriaely paramerized unique Wicksell rule, Taylor (1993) is quie explici ha he inercep in he rule is a consan, equal o he long run equilibrium real ineres 18 rae (ρ in our model) and he inflaion arge π*. He furher specifies he sum of he wo a 4%. He also specifies φ1=1.5 and φ=0.5 on he basis of simulaions wih economeric models. Taylor and Williams (011) defend such a rule on accoun of is simpliciy. We have however adoped a broader inerpreaion of he Taylor rule in his paper, assuming ha he inercep depends on he cenral bank s esimae of he curren naural real ineres rae. Noe ha he Taylor rule (55) requires ha he auhoriies observe curren inflaion and curren real oupu, and ha hey can also deduce he curren deviaion of oupu from is naural level from (4). If hey have his informaion, hey can also deduce he naural rae of ineres in our model. Thus, he informaional requiremens of he consan inercep Taylor rule are he same as he more general rule (55).!