Chapter 14 A Model of Imperfect Competition and Staggered Pricing

Transcription

1 George Alogoskoufis, Dynamic Macroeconomic Theory, 205 Chaper 4 A Model of Imperfec Compeiion and Saggered Pricing In his chaper we presen he srucure of an alernaive new Keynesian model of aggregae flucuaions. The model is a dynamic sochasic general equilibrium model based on monopolisic compeiion in produc markes, and we analyze i assuming boh full adjusmen of wages and prices and saggered pricing. The model wih full adjusmen of wages and prices is comparable o he new classical model wihou capial, presened in Chaper, and he model wih saggered pricing is comparable o he new keynesian model wih periodic nominal wage conracs of Chaper 3. The imperfecly compeiive new Keynesian model has wo imporan differences from he ypical perfecly compeiive new classical model. Firs, insead of fully compeiive markes for goods and services, i assumes ha markes are characerized by condiions of monopolisic compeiion. Firms do no ake prices as given, bu deermine prices so as o maximize profis. Because of monopolisic compeiion, employmen, real oupu, consumpion and real wages are deermined a a lower level han in he corresponding compeiive model, even when here is complee flexibiliy in prices and wages. However, by iself his difference does no resul in major differences from he new classical compeiive model regarding he naure of macroeconomic flucuaions. Second, in he imperfecly compeiive new Keynesian model i is usually assumed ha firms adjus prices only gradually. Two observaionally equivalen versions of gradual price adjusmen have dominaed he lieraure. The one is he Roemberg (982 a,b) model of monopolisic price adjusmen, and he second is he Calvo (983) model of saggered pricing. In he Roemberg model, firms balance he coss of adjusing prices agains he coss of deviaing from he profi maximizing opimal price. They end up gradually adjusing prices, so as o gradually approach he opimal price. In he Calvo model, i is assumed ha only a fixed proporion of firms have he freedom o adjus prices in any given period. This resuls in he remaining firms no being able o adjus prices. Alhough opimal pricing akes his resricion ino accoun in advance, he aggregae price level adjuss only gradually. These wo alernaive assumpions lead o models wih price level sickiness, which differ significanly from he new classical models, and share many of he properies of he new keynesian model wih periodic nominal wage conracs analyzed in Chaper 3. In he new keynesian model wih periodic nominal wage conracs i is only deviaions of curren inflaion from prior expecaions of curren inflaion ha resul in deviaions of oupu and unemploymen from heir naural raes. In he saggered pricing model, i is deviaions of inflaion from expeced fuure inflaion ha are associaed wih such deviaions. This resuls in a differen ype of Phillips curve, called he new keynesian Phillips curve, which differs from he radiional

2 George Alogoskoufis, Dynamic Macroeconomic Theory, 205 Chaper 4 expecaions augmened Phillips curve, as curren inflaion depends on curren expecaions of fuure inflaion, and no prior expecaions of curren inflaion. The imperfecly compeiive new Keynesian model has he following srucure: Deviaions of inflaion from he arge of he cenral bank are deermined by he new Keynesian Phillips curve, and depend of expeced fuure inflaion and deviaions of real oupu from is naural level, as he laer cause an increase in nominal marginal coss and hence prices. The deviaions of aggregae demand from he naural level of real oupu depend on he new Keynesian IS curve, which, as in he periodic nominal wage conracs model of Chaper 3, depend on deviaions of he curren real ineres rae from is naural level. The nominal ineres rae is deermined by he cenral bank, which follows a Taylor ineres rae rule. According o he Taylor rule, he nominal ineres rae reacs posiively o deviaions of curren inflaion from he cenral bank arge, as well as deviaions of real oupu from is naural level. Afer presening he properies of his model, we analyze he effecs of moneary and real shocks on flucuaions in real oupu and he price level (inflaion). The imperfecly compeiive new keynesian model wih saggered prices can, unlike he classical model, explain moneary cycles, i.e aggregae flucuaions caused by moneary shocks. These shocks are ransmied o real variables, and, o he exen ha hey persis over ime, have persisen real effecs. However, due o he absence of labor marke disorions, his model canno accoun for involunary unemploymen. A more saisfacory new keynesian model mus combine he labor marke disorions of he periodic nominal wage conracs model of Chaper 3, wih he produc marke disorions of he model of his chaper. 4. An Imperfecly Compeiive New Keynesian Model In his secion we examine in deail he srucure of an imperfecly compeiive new Keynesian model. The basic model ha we analyze has much in common wih he new classical model. I is a dynamic sochasic general equilibrium model wih wo imporan differences from he new classical model. Firs, insead of perfecly compeiive markes for goods and services we assume ha markes are characerized by condiions of imperfec (monopolisic) compeiion. Firms do no ake prices as given, bu have he power o deermine prices ha maximize profis. Because of imperfec compeiion, in equilibrium, employmen, real oupu, consumpion and real wages are deermined a a lower level han in he corresponding compeiive model, even when here is complee flexibiliy in prices and wages. However, by iself his difference does no resul in maerial differences from he compeiive classical model wih respec o he naure of aggregae flucuaions. If his was he only difference, we could well alk abou an imperfecly compeiive new classical model. See Gali (2008) for a fuller presenaion and analysis of his model.!2

3 George Alogoskoufis, Dynamic Macroeconomic Theory, 205 Chaper 4 Second, we assume ha here is saggered price adjusmen, i.e. ha firms do no have he abiliy o change heir prices a all imes. This assumpion is wha makes he model new keynesian, as i leads o a model in which he price level adjuss gradually owards he equilibrium price level. As a resul of gradual price adjusmen, real variables deviae from heir naural raes, and moneary shocks can have real effecs, as was he case wih he model wih predeermined wages examined in Chaper 3. Gradual adjusmen of nominal wages and prices is hus he key elemen ha differeniaes he new keynesian from he new classical approach o macroeconomic flucuaions. Wih full and immediae adjusmen of wages and prices, nominal disurbances do no affec real variables, even if here is imperfec compeiion. 4.. The Represenaive Household The problem of he represenaive household under monopolisic compeiion has one difference from he corresponding problem under perfec compeiion. The difference is ha because of monopolisic compeiion, he household consumes differeniaed producs. The represenaive household maximizes,! E 0 (4.) + ρ u(c, N ) =0 where C is consumpion, N is labor supply, and ρ is he pure rae of ime preference. Consumpion consiss of all produced goods, which are defined on he basis of a consan index j in he inerval [0,]. Aggregae consumpion is hus given by, ε ε ε ε! C = C ( j) dj (4.2) j=0 where ε is also a parameer of he preferences of he represenaive household, and more precicely, he elasiciy of subsiuion beween goods. We assume ha ε>. The sequence of budge consrains under which he household maximizes iner-emporal uiliy is given by, P ( j)c ( j)dj + B B + W N T j=0 + i (4.3) The household mus also saisfy he ransversaliy condiion,! lim E B 0 T (4.4) T where P(j) is he price of good j, W he nominal wage, i he nominal ineres rae, B a nominal one period bond, and T an exogenous ransfer of nominal income o he household (dividends, governmen ransfers or axes).!3

4 George Alogoskoufis, Dynamic Macroeconomic Theory, 205 Chaper 4 Apar from he decision abou aggregae consumpion and labor supply, which we have analyzed in he relevan secion of Chaper, he household mus now decide on he disribuion of is consumpion expendiure among he various goods. This requires he maximizaion of he consumpion bundle (4.2) for any level of expendiure. One can easily deduce ha his implies,! C ( j) = P ( j) (4.5) C() P ε for any good j in he inerval [0,], where P is he average price level, defined as, ε ( j=0 )! P = P ( j) ε dj (4.6) In addiion, when he household follows his opimal allocaion policy, we also have ha,! P ( j)c ( j)dj = P C (4.7) j=0 (4.7) suggess ha oal consumpion expendiure can be wrien as he produc of he aggregae consumpion index and he aggregae price index. Subsiuing (4.7) in he sequence of budge consrains (4.3), we ge,! P C + B B + W N T (4.8) + i This sequence of budge consrains is he same as he sequence of budge consrains of he represenaive household in he compeiive classical model. As a resul, he firs order condiions for consumpion and labor supply are analogous o he ones of he new classical model we analyzed in Chaper.! u N = W (4.9) u C P! = (4.0) + i + ρ E u C+ P u C P + We assume, as in he new classical model wihou capial of Chaper, ha he uiliy funcion is defined by, θ +λ! U(C, N ) = C (4.) θ N + λ!4

5 George Alogoskoufis, Dynamic Macroeconomic Theory, 205 Chaper 4 where /θ is he iner-emporal elasiciy of subsiuion in consumpion, and /λ he iner-emporal elasiciy of subsiuion of labor supply. Assuming ha preferences ake he form of (4.), he firs order condiions (4.9) and (4.0) can be wrien in log-linear form as,! w p = θc + λn (4.2) ( )! c = E (c + ) (4.3) θ i E (π ) ρ + where lower case leers denoe he logarihms of he corresponding variables. π is he rae of inflaion. (4.2) and (4.3) are analogous o he ones in he new classical model wihou capial in Chaper The Represenaive Firm and Opimal Pricing We assume ha oupu is produced by a se of firms denoed by a coninuous index j defined in he inerval [0,]. Each firm produces a differeniaed produc under condiions of monopolisic compeiion. All firms have access o he same producion echnology, denoed by he producion funcion,! Y ( j) = A L ( j) α (4.4) where Α>0 and 0<α< are exogenous echnological parameers, common o all firms. L(j) is employmen of labor by firm j. The parameer α is consan, while A is assumed o follow an exogenous sochasic process. The opimal price of each firm, if i can choose is price in every period, is given by he maximizaion of is profis, under he consrain of he producion funcion (4.4) and he demand funcion for is produc (4.5). Each firm akes he average price P, he average wage W and he level of oal demand C as given. The per period profis of firm j are given by,! P ( j)y ( j) W L ( j) (4.5) From he firs order condiions for a maximum of (4.5), under he consrains (4.4) and (4.5), he opimal price is deermined as,! P ( j) = ε W (4.6 ) ε ( α )A L ( j) α The opimal price is a fixed muliple of he firm s marginal cos, which equals he expression in brackes. The muliple depends on he elasiciy of subsiuion beween goods in he preferences of consumers, which deermines he price elasiciy of demand of heir produc, and herefore he profi margin of he firm. In he case of perfec compeiion ha we examined in Chaper, he!5

6 George Alogoskoufis, Dynamic Macroeconomic Theory, 205 Chaper 4 elasiciy of subsiuion ends o infiniy, and he price ends o marginal cos. In he case of monopolisic compeiion wih ε >, as we have assumed, he opimal price is higher han he marginal cos of labor. As all firms have he same producion funcion and face he same nominal wage and he same demand funcion for heir produc, hey will all choose he same price. Consequenly, he price level is defined as,! P = ε W (4.6) α ε ( α )A L Taking he logarihm of he producion funcion (4.4) for he represenaive firm, and equaion (4.6) for he opimal price, we ge,! y = a + ( α )l (4.7)! w p = a αl µ (4.8) where, ε! a = ln A,! µ = ln. ε ln( α ) a is he logarihm of he exogenous produciviy shock, and he consan µ is he logarihm of he markup on marginal cos, minus he logarihm of he coefficien of decreasing reurns o labor Equilibrium wih Full Price Flexibiliy Solving he model under he assumpion of full flexibiliy of prices, one can show ha flucuaions in employmen, oupu, consumpion and real wages are a funcion only of he exogenous shocks o produciviy, while flucuaions in he real ineres rae are a funcion of he expeced change in produciviy, jus as in he classical model wih he assumpion of perfec compeiion. In he basic form of his model we shall assume ha here are is no invesmen or public consumpion. Thus, in equilibrium, labor supply would be equal o labor demand by firms, and consumpion will be equal o oupu.! n = l (4.9)! y = c (4.20) The model consiss of equaions (4.2), (4.3), (4.7) and (4.8) and he equilibrium condiions (4.9) and (4.20). The model deermines employmen, oupu, consumpion, real wages and he real ineres rae as funcions of he exogenous shock o produciviy a. The real ineres rae is deermined by he Fisher equaion as,!6

7 George Alogoskoufis, Dynamic Macroeconomic Theory, 205 Chaper 4! r = i E (π + ) (4.2) Solving he model for he five endogenous variables, we ge,! l N = n N = φa + n _ (4.22) θ where,! φ = and,! n _ µ =. θ( α ) + α + λ θ( α ) + α + λ! y N = c N =ψ a + y _ (4.23) + λ where,! ψ = + ( α )φ = and,! y _ = ( α )n _. θ( α ) + α + λ ( ) N = χa + ω _! w p (4.24) θ + λ where,! χ = αφ = and,! ω _ = ( θ( α ) + λ)n _. θ( α ) + α + λ! r N = ρ +θψ E (Δa + ) (4.25) (4.22), (4.23), (4.24) and (4.25), along wih he equilibrium condiions (4.9) and (4.20), deermine he five endogenous real variables as funcions of he exogenous produciviy shock. Superscrip N (naural) denoes he equilibrium values of he relevan variables, which, according o he Friedman definiion are heir naural raes. Oupu, consumpion and real wages are posiive funcions of he produciviy shock a, while employmen is a posiive funcion of he produciviy shock only if θ<, i.e. only if he elasiciy of iner-emporal subsiuion is greaer han one. If θ> employmen is a negaive funcion of produciviy, while if θ= employmen is independen of produciviy. This applies because if θ< he iner-emporal subsiuion effec dominaes on he income effec, following a change in produciviy and real wages. If θ> he income effec dominaes on he iner-emporal subsiuion effec, which in he case where θ= he wo effecs cancel each oher ou, and employmen is no affeced. No oher facor affecs flucuaions in real variables. We see ha, as in he compeiive real business cycle model, moneary facors such as he money supply and nominal ineres raes have no effec on he evoluion of real variables. However, in his model here is a significan difference from he compeiive model of Chaper. Because of monopolisic compeiion, which implies a posiive margin of prices over marginal coss of firms, boh employmen and oupu, as well as consumpion and real wages, are deermined a a lower level han in he case of perfec compeiion. Monopolisic compeiion implies a disorion!7

8 !! George Alogoskoufis, Dynamic Macroeconomic Theory, 205 Chaper 4 in he marke of goods and services, which leads o lower equilibrium employmen and oupu and o lower real wages han wih perfec compeiion. 2 If he produciviy shock follows a saionary sochasic process wih mean zero, hen, from (4.22), he log of he seady sae employmen level will be equal o, n _ µ ln( α ) ln(ε / (ε )) = = θ( α ) + α + λ θ( α ) + α + λ If ε>, he seady sae employmen level will be lower han in he case of perfec compeiion. Under perfec compeiion, goods are perfec subsiues in he preferences of consumers. Thus, seady sae employmen would be equal o, lim n _ ln( α ) = ε θ( α ) + α + λ Thus, because of imperfec compeiion, his model implies under employmen relaive o a fully compeiive model, even when here is full flexibiliy of prices and wages. Through (4.23) and (4.24), his under employmen implies ha seady sae oupu and seady sae real wages will also be lower compared o perfec compeiion. In all oher respecs, his model resembles he new classical compeiive real business cycle model analyzed in Chaper Saggered Price Adjusmen In conras o he new classical model, in new keynesian models one assumes gradual and no full adjusmen of wages and prices owards heir equilibrium values. In Chaper 3 we analyzed a model of predeermined nominal wages, which were se a he beginning of each period, and assumed fully flexible prices. Here we shall assume gradual adjusmen of prices and fully flexible wages in a compeiive labor marke. A number of alernaive new keynesian models of gradual price adjusmen under monopolisic compeiion have been developed in he lieraure. We shall concenrae on one of hem, he Calvo (983) model, which is based on saggered pricing. 3 Following Calvo (983), we shall assume ha firms canno freely adjus heir prices in every period. For each firm, he probabiliy of adjusing prices in any period is equal o -γ, which is consan and independen of he lengh of ime ha has elapsed since he las price adjusmen by he firm. Thus, in each period, a proporion -γ of all firms adjus heir prices, and he remaining proporion γ do no adjus heir prices. As wih predeermined nominal wage conracs in he model 2 See Akerlof and Yellen (985), Mankiw (985), Blanchard and Kiyoaki (987) and Ball and Romer (990) for he firs generaion of new keynesian models ha relied on monopolisic compeiion. 3 An observaionally equivalen model, he Roemberg (982 a,b) model of quadraic coss of adjusing prices, is analyzed in he Annex o his Chaper.!8

9 ! George Alogoskoufis, Dynamic Macroeconomic Theory, 205 Chaper 4 of Chaper 3, his assumpion has criical implicaions for he properies of he model, he naure of aggregae flucuaions and he effecs of moneary shocks and moneary policy. 4 Under his assumpion, in period, he expeced fuure duraion of any price conrac is given by, ( γ ) sγ s = γ s=0 γ From he definiion of he price level, and he fac ha all firms ha rese heir prices in period se he same price, i follows ha,! P = γ ( P ) ε + ( γ ) P _ ε ε (4.26) where! is he price se by he firms ha rese heir prices in he curren period. P _ From (4.26) one can show ha he dynamic adjusmen of he price level is given by, ε _ P! P (4.27) P = γ + ( γ ) P In he seady sae wih zero inflaion we have ha, _ ε! P = P = P (4.28) A linear logarihmic approximaion of (4.27) around he zero inflaion seady sae yields, _! p p! ( γ ) p p (4.29) From (4.29) i follows ha inflaion is posiive if firms ha se prices in he curren period se hem a a higher level han he average price of he previous period Opimal Pricing wih Saggered Price Adjusmen In order o analyze he adjusmen of inflaion, one hus has o examine how firms decide on heir opimal price, aking ino accoun he fac ha for a period in he fuure hey may no be able o readjus heir prices, while heir compeiors have he opion of readjusing heir own prices. The problem of he firm ha decides on he price i is abou o se in period is o se he price ha maximizes he expeced presen value of is profis, given ha he probabiliy of readjusing is price in any fuure period is equal o -γ. Thus, all firms ha readjus heir prices in period maximize, 4 See Yun (996) for he firs analysis of he new keynesian dynamic sochasic general equilibrium model under he assumpion ha prices are se as posulaed by Calvo (983).!9

10 ! George Alogoskoufis, Dynamic Macroeconomic Theory, 205 Chaper 4! γ s s E (4.30) + i +z P _ Y z=0 +s W +s L s=0 +s under he consrains of he producion funcion,! L +s = Y α +s (4.3a) A +s and he demand funcion, ε _ P! Y +s = Y +s (4.3b) P +s where,! and! is he volume of oupu and employmen in period +s, of he firm ha has se Y +s L +s is prices in period. The higher he relaive price of he firm in any period, he lower he demand for is produc and hus he volume of is oupu and employmen. From he firs order condiions for a maximum i follows ha, α +ε γ s s P _ ε ε P _ α W E z=0 + i +z (ε ) Y +s +s Y +s α P +s ( α ) P +s P +s A +s = 0 s=0 (4.32) (4.32) implies ha he expeced presen value of revenues from he opimal price is equal o he expeced presen value of he marginal cos of producion, augmened by he profi margin ε/(ε-) of he firm. I is worh noing ha, as we have already shown (equaion (4.6)), if he firm could deermine is prices in every period, he price of he produc in each period would be equal o he marginal cos of producion plus he same profi margin. However, if he firm canno adjus prices in every period, as is assumed in he Calvo (983) model, pricing follows he rule (4.32). Assuming ha in he seady sae inflaion is equal o zero, (4.32) can be ransformed in logarihmic deviaions from he seady sae equilibrium, using a log linear Taylor approximaion. Thus, in logarihms we shall have ha,! p _! ( βγ ) ( βγ ) s E p (4.33) s=0 +s + ω µ + w +s p +s + ( α α y a +s +s ) where,!0

11 ! George Alogoskoufis, Dynamic Macroeconomic Theory, 205 Chaper 4! β = and! ω = α. + ρ α + ε < Consequenly, firms ha rese heir prices in period will choose a price which corresponds o a weighed average of he curren and expeced fuure price levels, plus a margin µ on a weighed average of he curren and expeced fuure level of real marginal coss. The discoun facor of a fuure period +s depends on he probabiliy ha he firm will no be able o rese is price in he fuure period +s, which equals γ s,, imes he discoun rae β s. Furhermore, he par of pricing which depends on he expeced marginal cos of he firm depends negaively on he elasiciy of demand for he produc of he firm, hrough he parameer ω. Using he fuure mahemaical expecaions operaor F, (4.33) can be wrien as,! p _ βγ! (4.34) βγ F p + ω βγ µ + βγ F w p + ( α α y a ) Subsiuing (4.34) in he equaion for he adjusmen of he average price level (4.29) we ge ha, βγ! p = γ p + ( γ ) (4.35) βγ F p + ω βγ µ + βγ F w p + ( α α y a ) Muliplying boh sides of (.35) by -βγf, afer some rearrangemens, we ge ha, ( γ )( βγ )! (+ β)p p βe p + = ω µ + w p + ( (4.36) γ α α y a ) (4.36) is he equaion of adjusmen of he price level owards he seady sae price level, which is a consan markup on he marginal cos of producion. In order o examine he shor run behavior of he model, we mus inroduce he equilibrium condiions in he markes for goods and services, labor and money Equilibrium in he Marke for Goods and Services and he New Keynesian IS Curve Equilibrium in he marke for good j implies ha, Y ( j) = C ( j) As a resul, equilibrium in he marke for all goods requires ha,! Y = C (4.37) where Υ is oal oupu, defined in he same way as oal consumpion C in equaion (4.2).!

12 George Alogoskoufis, Dynamic Macroeconomic Theory, 205 Chaper 4 Subsiuing he Euler equaion for consumpion (4.3) in he equilibrium condiion (4.37), he logarihm of real oupu is deermined by, ( )! y = E (y + ) (4.38) θ i E π + ρ As in he predeermined nominal wage conracs model of Chaper 3, (4.38) is ofen referred o as he new keynesian IS curve, as i is derived from he equilibrium condiion for he marke for goods and services. Compared o he convenional IS curve, (4.38) conains he raional expecaion abou he fuure volume of oupu and depends on he real and no jus he nominal ineres rae. Is advanage over he convenional IS curve is ha i has been derived from firm microeconomic foundaions, and ha is parameers depend on deep srucural parameers, such as he pure rae of ime preference of he represenaive household ρ, and he iner-emporal elasiciy of subsiuion in consumpion /θ Labor Marke Equilibrium and he New Keynesian Phillips Curve We nex urn o he equilibrium condiion in he labor marke. We assume ha in conras o produc prices ha adjus gradually, nominal wages adjus immediaely in order o equae he demand and supply of labor in each period. This assumpion, is he exac opposie of he assumpion made in he periodic wage conracs model of Chaper 3, and is made for reasons of analyical simpliciy. Thus, he only sickiness which is analyzed in his version of he new Keynesian model is he gradual adjusmen of prices raher han wages. This means ha flucuaions in employmen are he resul of iner-emporal subsiuion by households and ha no involunary unemploymen exiss. Gali (20) and ohers have analyzed his model wih he addiional assumpion of rigidiy no only in prices bu also in nominal wages. In his case here are flucuaions in he unemploymen rae due o he fac ha wages are no equae he demand wih he supply of labor in each period, as is assumed here. Due o he gradual adjusmen of prices, firms produce so as o saisfy aggregae demand a he given prices in each period. Aggregae oupu is deermined a he level which is deermined by aggregae demand, and differs from is naural level, which is he level ha would prevail if here was immediae price adjusmen by all firms. As a resul, aggregae oupu, employmen, consumpion, real wages and he real ineres rae, differ from heir naural levels and display flucuaions which depend on nominal as well as real disurbances. From he price adjusmen equaion (4.36), we can deduce an equaion for flucuaions in inflaion. Expressing (4.36) as an inflaion equaion we have ha, ( γ )( βγ )! π = βe π + + ω µ + w p + ( (4.39) γ α α y a ) where π is he rae of inflaion, defined as,! π = p p.!2

13 George Alogoskoufis, Dynamic Macroeconomic Theory, 205 Chaper 4 (4.39) implies ha curren inflaion is greaer han discouned fuure inflaion, if he curren marginal cos of labor, plus he margin µ is higher han he curren price level p. The reason is ha firms seing prices in he curren period pos larger price increases han (discouned) expeced fuure inflaion, in order o offse he higher curren marginal cos of labor. The assumpion of equilibrium in he labor marke means ha we can subsiue he real wage in (4.39) from he firs order condiion (4.2) for he represenaive household. Using (4.2), he condiion for equilibrium in he marke for goods and services c=y, and he producion funcion (4.7), (4.39) can be rewrien as, ( ) N! π = βe π + +κ y y (4.40) where y N is he naural rae of real oupu, i.e. he oupu ha would be produced if here was full flexibiliy of prices, and is given by (4.23). The parameer κ is defined as, ( γ )( βγ ) θ( α ) + λ + α! κ = > 0. γ α + ε (4.40) is referred o as he new keynesian Phillips curve, and consiues he second imporan behavioral equaion of he imperfecly compeiive new keynesian dynamic sochasic general equilibrium model. The reason ha deviaions of oupu from is naural rae cause inflaion o increase relaive o expeced fuure inflaion is ha higher oupu implies a higher real marginal cos of labor, and hus calls for price increases by firms ha have he opporuniy o change heir curren price which exceed discouned expeced fuure inflaion. Like he new keynesian IS curve, he new keynesian Phillips curve has been derived from explici microeconomic foundaions, and is parameers are funcions of deep srucural parameers describing he preferences of households, he echnology of producion and he price seing echnology The Srucure of he New Keynesian Model wih Saggered Pricing Equaions (4.38) and (4.40), along wih equaions (4.23) and (4.25) for he naural level of real oupu and he real ineres rae consiue he basic srucure of he imperfecly compeiive new keynesian model. Deviaions of inflaion from discouned expeced fuure inflaion are deermined by he new keynesian Phillips curve (4.40), as a funcion of deviaions of real aggregae demand and oupu from he naural level of oupu. Deviaions of real oupu from is naural level are deermined by he new keynesian IS curve, which depends on deviaions of he real ineres rae from is naural level. The new keynesian IS curve can be expressed as,!3

14 George Alogoskoufis, Dynamic Macroeconomic Theory, 205 Chaper 4 ( ) θ i E N ( π + r )! y y N N = E y + y + (4.4) where he naural levels of oupu and he real ineres rae y N, r N are deermined by (4.23) and (4.25). In order o close he model we mus consider he deerminaion of he nominal ineres rae. In conras o he new classical model, due o saggered price adjusmen, flucuaions in real variables canno be deermined wihou reference o moneary facors. Moneary facors and moneary policy deermine no only he price level and inflaion, as in he new classical model, bu also flucuaions in real variables such as real oupu, consumpion and employmen, real wages and he real ineres rae The Taylor Rule for he Nominal Ineres Rae We shall analyze he model under he assumpion ha he cenral bank follows a Taylor (993) rule of he form,! i = ρ + η π π + η y (y y N ) + v (4.42) where ηπ and ηy are posiive coefficiens, and v is an exogenous sochasic disurbance in he nominal ineres rae. I is worh noing ha because he consan in his rule is equal o ρ, his rule is consisen wih zero seady sae inflaion. 5 As we already menioned in Chaper 3, his rule implies a counercyclical moneary policy. When inflaion is posiive, he cenral bank increases nominal ineres raes in order o reduce i. When employmen is low, i.e. when oupu is lower han is naural level, he cenral bank reduces nominal ineres raes in order o increase employmen and nudge oupu owards is naural level. As we have shown already in Chaper 3, his feedback ineres rae rule does no resul in inflaion and price level indeerminacy if he Taylor principle is saisfied, i.e. if he reacion of nominal ineres raes o inflaion is sufficienly srong. 6 Having now fully deermined he new keynesian model wih saggered pricing, we can analyze how nominal and real disurbances produce aggregae flucuaions. 4.2 Sochasic Shocks and Aggregae Flucuaions Subsiuing (4.42) for he nominal ineres rae in he new keynesian IS curve, and using he new keynesian Phillips curve, he model can be wrien in marix form, as, 5 Noe ha he Taylor rule assumed is simpler han he Taylor rule we posulaed in he model of Chaper 3. The nominal ineres rae does o reac o shocks ha change he naural rae of ineres, such as produciviy shocks, and hus produciviy shocks urn ou o affec deviaions of oupu from is naural rae and inflaion. Obviously, one can also analyze he imperfecly compeiive new keynesian model under he assumpion ha moneary policy follows a rule for he money supply and no nominal ineres raes. See Gali (2008). 6 See Chapers 0, and 3 for discussions of he properies of ineres rae rules, as well as Woodford (2003), for a more exensive and complee analysis.!4

15 !! George Alogoskoufis, Dynamic Macroeconomic Theory, 205 Chaper 4 y ~ θ βη π E y ~ + = π θ + η y +κη π θκ κ + β(θ + η y ) + r ρ v θ + η E ( π + ) y +κη π κ (4.43) where,! y ~ N = y y is he percenage deviaion beween curren real oupu and is naural level. The percenage deviaion beween curren real oupu and is naural level is ofen referred o as excess oupu. Is opposie, is referred o as he oupu gap. When excess oupu is posiive, he economy produces more han is naural level, while when i is negaive (he oupu gap is posiive) i produces less han is naural level. We can see from (4.43) ha he flucuaions of excess oupu and inflaion depend on boh ypes of shocks. Real shocks which affec r N -ρ, and he nominal ineres rae shock v. The parameers deermining aggregae flucuaions in he imperfecly compeiive new keynesian model depend on he preferences of he represenaive household, (θ, λ, ε and ρ), he echnology of producion (α), marke srucure (ε), he price adjusmen mechanism (γ) and he parameers of he moneary policy rule (ηπ and ηy). Given ha boh excess oupu and inflaion are non predeermined variables, he soluion will be unique only if he marix of coefficiens of fuure expecaions has boh eigenvalues inside he uni circle. 7 Under he assumpion ha he coefficiens of he Taylor rule ηπ and ηy are posiive, one can show ha a necessary and sufficien condiion for a unique soluion is, 8 ( ) + ( β)η y > 0! κ η π (4.44) In wha follows we shall assume ha (4.44) is saisfied. (4.44) requires a sufficienly pronounced reacion of nominal ineres raes o inflaion, as, solving for ηπ,(4.44) can be expressed as, η π > ( β) η y κ For example, if he reacion of he nominal ineres rae o excess oupu is zero (ηy =0), hen he reacion of nominal ineres raes o inflaion mus exceed uniy Implicaions of a Nominal Ineres Rae Shock In order o invesigae how purely moneary shocks can cause flucuaions in real variables, le us assume ha he shock o he nominal ineres rae v in he Taylor rule follows a firs order auoregressive process of he form, 7 See Mahemaical Annex 5 or Blanchard and Kahn (980). 8 See Bullard and Mira (2002).!5

16 George Alogoskoufis, Dynamic Macroeconomic Theory, 205 Chaper 4 v! v = ρ v v + ε (4.45) where,! 0 < ρ v <, and! ε v N(0,σ 2 v ). A posiive value for ε v is a conracionary moneary shock, which leads o a rise in nominal ineres raes for given inflaion and excess oupu. A negaive value is an expansionary moneary shock, leading o a fall in nominal ineres raes for given inflaion and excess oupu. Solving he model in (4.43), or alernaively he hree equaions (4.40)-(4.42), under he assumpion ha here are only moneary and no real shocks, flucuaions of excess oupu and inflaion are deermined by,! y ~ = ( βρ v )Λ v v = ρ v y ~ ( βρ )Λ v v vε (4.46) v! π = κλ v v = ρ v π κλ v ε (4.47) where,! Λ v = (4.48) ( βρ v ) θ( ρ v ) + η y ( ) +κ (η π ρ v ) > 0 One can easily show ha Λv is posiive o he exen ha (4.44) is saisfied. As a resul, an exogenous increase o he nominal ineres rae leads o an fall in boh excess oupu and inflaion, while an exogenous reducion in he nominal ineres rae leads o a rise in excess oupu and inflaion. Furhermore, o he exen ha he shock is persisen, he effecs on oupu and inflaion persis, due o he exisence of saggered pricing. If he shock in non-persisen, hen he effecs on excess oupu and inflaion are non persisen as well. Noe ha in he new keynesian model wih predeermined periodic wage conracs, analyzed in Chaper 3, even persisen moneary shocks have non persisen effecs, unless here is endogenous persisence in employmen and oupu. Obviously, moneary shocks are no he only ones causing aggregae flucuaions in he new keynesian model wih saggered pricing. Real shocks cause aggregae flucuaions oo Implicaions of a Real Produciviy Shock We have already demonsraed ha real shocks, such as shocks o produciviy a, cause flucuaions of he naural rae of real variables. However, due o saggered pricing, he evoluion of real variables deviaes from he evoluion of heir naural raes. As a resul, given he moneary policy rule (4.42), real shocks cause flucuaions in excess oupu and inflaion. In order o invesigae such flucuaions, le us assume ha he shock o produciviy follows an firs order auoregressive process of he form, a! a = ρ a a + ε (4.49)!6

17 George Alogoskoufis, Dynamic Macroeconomic Theory, 205 Chaper 4 where,! 0 < ρ a <, and! ε a N(0,σ 2 a ). In conras o pure moneary shocks, real shocks affec he evoluion of he naural rae of real variables, such as he real ineres rae r N. Solving he model using (4.49), and ignoring moneary shocks, we find ha flucuaions in excess oupu and inflaion are deermined by,! y ~ = θψ ( ρ a )( βρ a )Λ a a = ρ a y ~ θψ ( ρ )( βρ )Λ a a a aε (4.50) a! π = θψ ( ρ a )κλ a a = ρ a π θψ ( ρ a )κλ a ε (4.5) where,! Λ a = (4.52) ( βρ a ) θ( ρ a ) + η y ( ) +κ (η π ρ a ) > 0 From (4.50), and under he assumpion ha he auoregressive coefficien of he produciviy shocks is less ha one, a posiive produciviy shock leads o a fall in excess oupu and inflaion. This is because real oupu rises by less han is naural rae, due o he behavior of real ineres raes. To he exen ha produciviy shocks are persisen excess oupu and inflaion display persisen flucuaions in response o produciviy shocks. I is worh noing ha here is no endogenous persisence in his model. The persisence of excess oupu and inflaion arises solely because of he persisence on real and moneary shocks. However, because of saggered pricing, his persisence feeds hrough boh real and nominal variables. In he case of he periodic wage seing model of Chaper 3, i is only innovaions in nominal and real shocks ha affec flucuaions in real variables, as i is only unanicipaed inflaion ha generaes real effecs A Dynamic Simulaion of he Model I is worh simulaing he model for paricular parameer values, in order o asses is quaniaive properies. Figures 4. and 4.2 presen he dynamic effecs of boh moneary and real shocks, for values of he parameers commonly used in he lieraure. In Figure 4. we presen he dynamic effecs of a 0.25 basis poins shock ε v in nominal ineres raes. This shock leads o an auomaic increase of he nominal and he real ineres rae and reduces excess producion and inflaion. Because his shock does no affec he "naural" level of oupu, real oupu, employmen and real wages decline. The economy gradually reurns o long-run equilibrium, as he effecs of he moneary shock gradually peer ou. In Figure 4.2 we presen he dynamic effecs of a 0.25 shock ε a o produciviy. This shock leads o a prolonged rise of he naural level of oupu, a reducion of excess oupu and inflaion, an!7

18 George Alogoskoufis, Dynamic Macroeconomic Theory, 205 Chaper 4 increase in he real wage, and a decline in nominal and real ineres raes. The reducion in real ineres raes leads o an increase in acual oupu, which however is smaller han he increase of he naural level of oupu. Tha is he reason ha excess oupu falls. producion decreases. Again, he economy gradually reurns o equilibrium long as he effecs of he acual disurbance progressively peer ou. 4.3 Conclusions In his chaper we have analyzed he srucure and he properies of a new keynesian model, based on monopolisic compeiion and saggered pricing. Unlike radiional Keynesian models, in which he basic relaions are no derived explicily from microeconomic foundaions, his new Keynesian model is a dynamic sochasic general equilibrium model based on explici microeconomic foundaions, analogous o hose of new classical models. Afer we presened he properies of his model, we analyzed he effecs of moneary and real shocks o flucuaions in excess oupu and inflaion. This new keynesian model, unlike new classical models, can explain aggregae flucuaions caused by moneary and aggregae demand shocks. These shocks are ransmied o real variables and persis over ime hrough saggered pricing. Such moneary cycles canno resul from models wih immediae adjusmen of wages and prices. As in he fully compeiive new classical models, even under monopolisic compeiion, when here is full adjusmen of prices and wages, moneary shocks affec only nominal and no real variables such as oupu, consumpion, employmen, real wages and real ineres raes. If here is parial sickiness of wages and prices, as in he periodic nominal wage conracs model of Chaper 3, or he monopolisic compeiion and saggered pricing model of his Chaper, hen moneary shocks have real effecs and moneary cycles can arise. I is worh noing ha whereas he model of Chaper 3, wih real and nominal labor marke disorions could accoun for involunary unemploymen, and flucuaions in he unemploymen rae, in he model of his Chaper here is no involunary unemploymen. Oupu deviaes from is naural rae, because of saggered pricing, bu flucuaions in employmen are due o ineremporal subsiuion, and here is no involunary unemploymen, since he labor marke is assumed fully compeiive. This is a significan weakness of his paricular model. This weakness can be addressed if one were o combine he wo models, and rely on a generalized new keynesian model, wih boh saggered pricing, predeermined nominal wage conracs and real produc and labor marke disorions. In he nex chaper we delve deeper ino he labor marke disorions ha can accoun for involunary unemploymen, and presen he main elemens of an imporan class of maching models of he labor marke.!8

19 George Alogoskoufis, Dynamic Macroeconomic Theory, 205 Chaper 4 Annex o Chaper 4 The Roemberg Model of Convex Coss of Price Adjusmen An alernaive model of sluggish price adjusmen is he Roemberg (982 a,b) model of cosly price changes. For he represenaive monopolisically compeiive firm, as he one examined in secion 4..2, he opimal price is given by,! P _ = ε W (A4.) α ε ( α )A L The opimal price is a consan markup on marginal coss. Marginal coss are equal o wage coss over he marginal produciviy of labor. Noe ha because of decreasing reurns o employmen, increasing employmen and oupu implies declining marginal produciviy of labor and increasing marginal coss of producion. Using he producion funcion o subsiue ou for labor, (A4.) can also be expressed as,! P _ = ε W ( Y ) α α (A4. ) ε ( α )( A ) α An increase in oupu increases he marginal coss of producion for given wages, because of he declining marginal produciviy of labor. Hence, wih higher oupu he opimal price mus rise. In logs, (A4.) and (A4. ) imply,! p _ = µ + w a + αl = µ + w + ( (A4.2) α α y a ) where, ε! a = ln A,! µ = ln. ε ln( α ) a is he logarihm of he exogenous produciviy shock, and he consan µ is he logarihm of he markup on marginal cos, minus he logarihm of he coefficien implying decreasing reurns o labor. All firms, are assumed o be facing convex coss of adjusing prices. Roemberg (982, 983) assumes ha hey balance he coss of deviaing from heir opimal price agains he coss of adjusing prices. In he model ha follows, following Roemberg, we assume ha firms se curren prices minimizing a quadraic cos funcion which penalizes percenage deviaions of prices from he opimal price, and he adjusmen of prices from period o period. This akes he form,!9

20 George Alogoskoufis, Dynamic Macroeconomic Theory, 205 Chaper 4! Λ = E β s (A4.3) s=0 2 (p +s p_ +s )2 + ξ 2 (p p +s +s )2 where p is he log of he acual price of he represenaive firm. ξ is a parameer measuring he cos of adjusmen of prices relaive o he cos of deviaions from he opimal price. From he firs order condiions for he minimizaion of (A4.3), i follows ha,! p = (A4.4) + ξ(+ β) p_ + ξ + ξ(+ β) p + ξβ + ξ(+ β) E p + The curren price, in logs, is a weighed average of he opimal price, he pas price and he expeced fuure price. The firm is forward looking, and anicipaes he fuure coss of adjusing prices, so is curren price depends no only on is pas price, bu on is expeced fuure price as well. Since his is he represenaive firm, we can ake in o be equal o he log of he price level. Expressing (A4.4) as an inflaion equaion, one ges,! π = βe π + + (A4.5) ξ p_ p where π=p-p- is he rae of inflaion. Inflaion deviaes from expeced fuure inflaion, o he exen ha he opimal price exceeds he curren price. Subsiuing for he opimal price from (A4.2), one ges, ( )! π = βe π + + (A4.6) ξ µ + w a + αl p From (A4.6), inflaion deviaes from expeced fuure inflaion, o he exen ha he marginal cos of producion plus he opimal price markup exceeds he curren price. Using he labor and produc marke equilibrium condiions o subsiue ou for he real wage and employmen, as well as he definiion of he naural rae of oupu, we can express (A4.6) as, N! π = βe π + +κ y y (A4.7) where! ( ) κ = θ( α ) + α + λ ξ( α ) > 0 (A4.7) has exacly he same form as he new keynesian Phillips curve (4.40) derived from he Calvo (983) model of saggered pricing. The only difference is in he definiion of κ which is now in erms of he parameer ξ of he Roemberg model, insead of he parameer γ of he Calvo model. Thus, he wo models of sluggish price adjusmen, he Roemberg model of coss of adjusmen of prices and he Calvo model of saggered pricing are observaionally equivalen a he aggregae level.!20

21 George Alogoskoufis, Dynamic Macroeconomic Theory, 205 Chaper 4 References Akerlof G. and Yellen J. (985), A Near-Raional Model of he Business Cycle wih Wage and Price Ineria, Quarerly Journal of Economics, 00, Supplemen, pp Ball L. and Romer D. (990), Real Rigidiies and he Non-Neuraliy of Money, Review of Economic Sudies, 57, pp Blanchard O.J. and Kahn C. (980), The Soluion of Linear Difference Equaions under Raional Expecaions, Economerica, 48, pp Blanchard O.J. and Kiyoaki N. (987), Monopolisic Compeiion and he Effecs of Aggregae Demand, American Economic Review, 77, pp Bullard J. and Mira K. (2002), Learning abou Moneary Policy Rules, Journal of Moneary Economics, 49, pp Calvo G. (983), Saggered Prices in a Uiliy Maximizing Framework, Journal of Moneary Economics, 2, pp Gali J. (2008), Moneary Policy, Inflaion and he Business Cycle, Princeon N.J., Princeon Universiy Press. Gali J. (20), Unemploymen Flucuaions and Sabilizaion Policies: A New Keynesian Perspecive, Cambridge Mass., The MIT Press. Mankiw G. (985), Small Menu Coss and Large Business Cycles: A Macroeconomic Model of Monopoly, Quarerly Journal of Economics, 00, pp Roemberg J. (982a), Monopolisic Price Adjusmen and Aggregae Oupu, Review of Economic Sudies, 44, pp Roemberg J. (982b), Sicky Prices in he Unied Saes, Journal of Poliical Economy, 90, pp Taylor J.B. (993), Discreion versus Policy Rules in Pracice, Carnegie-Rocheser Conference Series on Public Policy, 39, pp Woodford M. (2003), Ineres and Prices: Foundaions of a Theory of Moneary Policy, Princeon N.J., Princeon Universiy Press. Yun T. (996), Nominal Price Rigidiy, Money Supply Endogeneiy and Business Cycles, Journal of Moneary Economics, 37, pp !2

22 George Alogoskoufis, Dynamic Macroeconomic Theory, 205 Chaper 4 Figure 4. The Dynamic Behavior of he Saggered Pricing Model Following a Conracionary Moneary Shock Noe: The parameer values for his simulaion are: θ=, λ=, ρ=0.0, α=0.333, ε=6, γ=0.667, ηπ=.50, ηy=0.25, ρa=0.90, ρv=0.50.!22

23 George Alogoskoufis, Dynamic Macroeconomic Theory, 205 Chaper 4 Figure 4.2 The Dynamic Behavior of he Saggered Pricing Model Following a Produciviy Shock Noe: The parameer values for his simulaion are: θ=, λ=, ρ=0.0, α=0.333, ε=6, γ=0.667, ηπ=.50, ηy=0.25, ρa=0.90, ρv=0.50.!23