George Alogoskoufis, Dynamic Macroeconomics, 2016 Chaper 15 A Model wih Periodic Wage Conracs In his chaper we analyze an alernaive model of aggregae flucuaions, which is based on periodic nominal wage conracs. The model is based on Gray (1976), Fischer (1977) and Taylor (1979), who emphasized he periodic adjusmen of nominal wages raher han he periodic adjusmen of prices. 1 In he Gray-Fischer-Taylor model, nominal wage conracs are assumed o be negoiaed a he beginning of every period, or a he beginning of alernae periods. In addiion, nominal wages are assumed o remain fixed for he duraion of he conrac. Nominal wages depend on prior expecaions abou he evoluion of he price level, produciviy and all oher shocks. If inflaion urns ou o be higher han expeced, hen real wages fall, firms demand more labor, and employmen rises. The opposie happens when inflaion urns ou o be lower han expeced. The model builds on one of he key insighs of he General Theory, namely he shor run rigidiy of nominal wages, as envisaged by labor marke conracs. In all oher respecs i is based on ineremporal opimizaion on he par of boh households and firms. The specific model inroduced in his chaper is a dynamic, sochasic general equilibrium model, in which non indexed nominal wage conracs are negoiaed periodically by insiders in he labor marke. There are wo disorions in he model compared o he new classical model wihou capial ha we analyzed in Chaper 13. The firs is a real disorion, arising from he fac ha ousiders are disenfranchised from he labor marke. As a resul, wage conracs do no seek o mainain full employmen and here is a posiive naural rae of unemploymen. The 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, before he realizaion of curren nominal and real shocks. Thus, nominal wages are se on he basis of prior raional expecaions abou he various unobserved shocks. 2 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. Thus, nominal shocks and, by exension, moneary policy, are able o affec flucuaions in boh inflaion and real variables such as oupu, employmen, unemploymen, real wages and he real ineres rae. 1 The Fischer (1977) and Taylor (1979) models are models wih saggered wage conracs. See Blanchard and Summers (1986), Lindbeck and Snower (1986) and Gofries (1992) for insider-ousider models of 2 he labor marke.
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, which affec he relaion beween real wages and produciviy. 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 ineremporal uiliy funcion. Thus, boh he consumpion funcion and he money demand funcion are derived from ineremporal microeconomic foundaions. The model is also characerized by exogenous shocks o produciviy and preferences for consumpion and money demand. Thus, he model is in essence a dynamic sochasic model ha incorporaes many of he feaures of he AS-AD version of he Keynesian model ha we presened in chaper 12. We analyze aggregae flucuaions in his model under wo alernaive moneary regimes. The firs is an exogenous process for he rae of growh of he money supply and he second is a feedback ineres rae rule, according o which he nominal ineres rae responds o deviaions of inflaion from he arge of he cenral bank, and deviaions of oupu from is naural level. Conrary o he new classical model, moneary shocks affec real variables in his model, causing emporary deviaions of oupu, employmen, unemploymen, real wages and he real ineres rae from heir naural levels. The exac variance of such deviaions depends on he moneary rule. Under an exogenous process for he rae of growh of he money supply, all shocks affec aggregae flucuaions. Under a feedback ineres rae rule, only produciviy and nominal ineres rae shocks affec aggregae flucuaions. We hus demonsrae he dependence of aggregae flucuaions no only on exogenous shocks, bu on he form of he moneary policy rule followed by he cenral bank. We also exend he model o accoun for persisence in deviaions of unemploymen and oupu from heir naural levels. The exension is based on a dynamic model of he Phillips Curve, in which unanicipaed shocks o inflaion and produciviy have persisen effecs on unemploymen, and hese persisen effecs are compaible wih full ineremporal opimizaion on he par of labor marke insiders. The propagaion mechanism ha causes unanicipaed nominal and real shocks o produce persisen deviaions of unemploymen and oupu from heir naural rae is he parial adjusmen of labor marke insiders o employmen shocks. We demonsrae ha under a Taylor rule, he only shocks ha canno be compleely neuralized by moneary policy are produciviy shocks and, of course, moneary policy shocks. Flucuaions of deviaions of unemploymen and oupu from heir naural raes display persisence and are driven by hese wo ypes of shocks. Because of he endogenous persisence of deviaions of unemploymen from is naural rae, he equilibrium inflaion rae also displays persisence around he inflaion arge of he cenral bank. 15.1 Insiders, Wage Seing and he Phillips Curve The wage seing model inroduced in his chaper combines and exends wo srands of he lieraure.!2
The firs srand of he lieraure is he insider-ousider heory of wage deerminaion of Lindbeck and Snower (1986), Blanchard and Summers (1986) and Gofries (1992). 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. Second, he model incorporaes he Gray (1976)-Fischer (1977) model of predeermined nominal wages, according o which nominal wage conracs are negoiaed a he beginning of each period, and wages remain fixed for one period. Because curren shocks, including curren 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 posiive shor run relaion beween unanicipaed inflaion and oupu and employmen, i.e an expecaions augmened Phillips curve. 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. 15.1.1 Oupu, Employmen and Labor Demand 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. The producion funcion of firm i is given by, 1! Y (i) = A L(i) (15.1) where Y(i) is oupu, A is exogenous produciviy, and L(i) is employmen. is a ime index, where =0,1,. 0<1-<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 )A L(i) = W (i) (15.2) 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, (15.1) and (15.2) can be wrien as,! y(i) = a + (1 )l(i) (15.3)!3
! l(i) = l _ 1 (15.4) (w(i) p a ) where! l _ = ln(1 ) 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. 15.1.2 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 (1992) models. Employmen is deermined ex pos by firms, given he conrac wage, afer he curren price level and produciviy have been revealed. This se up leads o emporary real effecs of nominal shocks and moneary policy. 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 an employmen arge deermined by insiders in each firm. 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 in each firm is o make expeced employmen saisfy a pah ha minimizes he following quadraic ineremporal loss funcion, min E 1 β s s=0 1 2 l(i) +s n_ (i) +s 2 (15.5) " is he logarihm of he number of insiders in each firm. 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 (15.4). n _ 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,! (15.6) i=0!4
where n is he log of he labor force. From he firs order condiions for a minimum of (15.5), wages are se so ha expeced employmen for each firm saisfies,! E 1 l(i) = n _ (i) (15.7) Inegraing over i, expeced aggregae employmen mus hen saisfy,! E 1 l = n _ (15.8) (15.8) is he same as (15.7) wihou he i index. (15.8) 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,! l = l _ 1 (15.9) (w p a ) From (15.8) and (15.9), he conrac wage saisfies,! w = E 1 p + E 1 a (n _ l _ ) (15.10) 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. 15.1.3 An Expecaions Augmened Phillips Curve Subsiuing (15.10) in (15.9), acual employmen evolves according o, ( ) = n _ + 1 ( π E 1π + a E 1 a )! l = n _ + 1 (15.11) p E p + a E a 1 1 where,! π = p p 1 is he rae of inflaion. From (15.11), 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 also increase labour demand and employmen. We can define he unemploymen rae as,!5
! u! n l (15.12) We can define he naural rae of unemploymen as, 3! u N! n n _ (15.13) From (11), (12) and (13), i follows ha,! u = u N 1 (15.14) 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. (15.14) has he form of an expecaions augmened Phillips curve, which 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 (15.3), aggregaing over firms, we ge an aggregae producion funcion in log-linear form, as,! y = a + (1 )l (15.15) Subsiuing (15.11) in he log-linear version of he producion funcion (15.15), oupu supply evolves according o,! y = y N + 1 ( (15.16) π E 1π + a E 1 a ) where, ( )! y N = (1 )n _ + a (15.17) 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. (15.16) can be seen as he oupu version of he expecaions augmened Phillips curve, or as a shor run oupu supply funcion. I is worh noing ha, if wage seing did no ake place in advance bu during he acual period, here would be no real effecs from unexpeced inflaion and no employmen effecs from shocks o produciviy. Essenially, we would have a quasi classical model, wih a posiive naural rae of unemploymen and no Phillips curve. Thus, he assumpion ha nominal wages are se in 3 The concep of he naural rae of unemploymen is due o Friedman (1968) and was analyzed in he conex of an expecaions augmened Phillips curve in Chaper 12.!6
advance, and before curren shocks o inflaion and produciviy become known o wage seers, is very imporan for he properies of his model. 15.1.4 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,! y F = (1 )n + a (15.18) Full employmen oupu is always higher han he naural level of 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 behavior, 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 (15.17) and (15.18), he relaion beween he naural rae of unemploymen and deviaions of oupu from full employmen oupu is given by,! y F y N = (1 )(n n _ N ) = (1 )u (15.19) This is he real disorion in his model. Because of he inefficiency in he labor marke, due o he marke power of insiders, he equilibrium level ( naural level ) of employmen is lower han full employmen, he naural level of oupu is lower han full employmen oupu, and he naural rae of unemploymen is posiive. Furhermore, equilibrium unemploymen is involunary, as he unemployed ousiders would be prepared o work a he prevailing real wage. 15.2. 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. Thus, if H is he number of households and N is he aggregae labor force, each household has N/H members. Of hose, some are insiders in he labor marke, and he res are ousiders. 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 in order o maximize, s 1 θ 1 1! E (15.20) 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,!7
! E F +s+1 (1+ i +s ) F +s i +s M +s + P +s ( Y +s C +s T +s ) (15.21) 1+ i +s = 0 where! F = B + M. ρ denoes he pure rae of ime preference, θ is he inverse of he elasiciy of ineremporal subsiuion, i he nominal ineres rae, F he curren value of he financial asses of he household (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 (15.22) θ M M! V (15.23) P = λ i P 1+ ρ! E λ +1 = E (15.24) 1+ i +1 λ where λ is he Lagrange muliplier in period. (15.22)-(15.24) have he sandard inerpreaions. (15.22) suggess ha a he opimum he household equaes he marginal uiliy of consumpion o he value of savings. (15.23) suggess ha he household equaes he marginal uiliy of real money balances o he opporuniy cos of money. Finally, (15.24) 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. From (15.22), (15.23) and (15.24), eliminaing λ, 1 θ C M V! (15.25) P = C i M V 1+ i ( ) θ ( ) θ V C! E +1 C +1 = 1+ ρ V C C (15.26) P +1 1+ i P (15.25) is he money demand funcion, which is proporional o consumpion and a negaive funcion of he nominal ineres rae, and (15.26) is he familiar Euler equaion for consumpion. Log-linearizing (15.25) and (15.26),! m p = c 1 (15.27) θ ln i 1+ i + 1 θ v M C ( v )!8
( ) + 1 θ (v C E v +1! c = E c +1 1 (15.28) θ i E π +1 ρ C ) where lowercase leers denoe naural logarihms, and, π = p p 1 is he rae of inflaion. 4 We hen urn o he deerminaion of equilibrium in he produc and money markes. 15.3 Equilibrium in he Produc and Money Markes Since here is no capial and invesmen in his model, and no governmen expendiure, produc marke equilibrium implies ha oupu is equal o consumpion.! Y = C (15.29) This produc marke equilibrium condiion allows us o subsiue oupu for consumpion in he money demand funcion and he Euler equaion for consumpion, and derive he new keynesian LM and IS curves. 15.3.1 The New Keynesian IS and LM Curves Subsiuing (15.29) in (15.27) and (15.28), we ge he money and produc marke equilibrium condiions,! m p = y 1 (15.30) θ ln i 1+ i + 1 θ v M C ( v ) ( ) + 1 θ (v C E v +1! y = E y +1 1 (15.31) θ i E π +1 ρ C ) (15.30) is he money marke equilibrium condiion, he equivalen of he LM Curve in he radiional Keynesian model, and (15.31) is he produc marke equilibrium condiion, he equivalen of he IS Curve. (15.30) and (15.31) are ofen referred o as he new keynesian LM curve and he new keynesian IS curve respecively. 15.3.2 The Naural and he Curren Real Ineres Rae 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. 4 Technically, since he logarihm of he expecaion of a produc (or raio) of wo random variables is no equal o he sum (or difference) of he expecaions of he logarihms of he relevan random variables, (15.28) mus also conain second order erms, depending on he covariance marix of consumpion, inflaion and shocks o preferences for consumpion and inflaion. Assuming ha all exogenous shocks are independen saionary sochasic processes, hese second order erms are consan and can be ignored.!9
" George Alogoskoufis, Dynamic Macroeconomics, 2016 Chaper 15 The real ineres rae is defined by he Fisher (1896) equaion, 5 r = i E π +1 (15.32) The naural real ineres rae is deermined by he produc marke equilibrium condiion, when oupu is a is naural level. From (15.17) and (15.31), he naural real ineres rae is deermined by, ( )! r N = ρ θ (1 ) n _ E n _ +1 (15.33) + 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. 6 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 (15.31) wih he oupu supply funcion (15.16). I is hus deermined as, ( )! r = r N θ 1 ( π E 1 π + a E 1 a ) (15.34) 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 Wicksellian mechanism in his model. We shall reurn o his mechanism when we discuss alernaive ineres rae rules. 15.3.3 Equilibrium Flucuaions wih Exogenous Real Shocks In wha follows, we shall assume ha he logarihms of he exogenous shocks o preferences and produciviy follow saionary AR(1) processes.! v C = η C v C C 1 + ε (15.35) 5 See he quoaion from Fisher (1896) in Chaper 6, on he disincion beween nominal and real ineres raes. The concep of a naural rae of ineres was inroduced by Wicksell (1898). To quoe, There is a cerain rae of 6 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. 102). I is worh noing ha Friedman (1968) defined he naural rae of unemploymen in direc analogy o he definiion of he naural rae of ineres by Wicksell (1898). He made a direc reference o Wicksell, and was fully aware of he analogy beween he wo conceps.!10
!! v M = η M v M 1 + ε (15.36) A a = η A a 1 + ε (15.37) 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 and he number of insiders are fixed. We hus assume ha, is fixed a n, and ha he exogenous number of insiders also follows a saionary AR(1) process, of he form,! 0 < n _ < n (15.38) From (15.38), and he definiion of he naural rae of unemploymen in (15.13), we also have ha,! u N = u N = n n _! (15.39) Thus, he naural rae of unemploymen is assumed consan. Wih hese assumpions, curren employmen, unemploymen, oupu, real wages and he real ineres rae, as funcions of he exogenous shocks and shocks o inflaion, evolve according o, ( )! l = n _ + 1 (15.40) π E A 1π + ε where! n _ is given by (15.38). ( )! u = u N 1 (15.41) π E A 1π + ε where! u _ is given by (15.39).! y = y N + 1 (15.42) π E A ( 1π + ε ) where,! y N = (1 )n _ + a. ( )! w p = (w p) N π E 1 π (15.43) where,!(w p) N = η A a 1 (n _ l _ ). ( )! r = r N θ 1 A ( π E 1 π + ε ) (15.44) where! r N C = ρ θ(1 η A )a + (1 η C )v!11
The naural raes (or levels) of real variables evolve as funcions of he exogenous real shocks. In he absence of he nominal rigidiy due o he assumpion ha nominal wages are se in advance and remain fixed for one period, he evoluion of real variables would be equal o heir naural levels. The model would in all respecs be similar o a new classical model. 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. Since inflaion is also affeced by nominal shocks, unanicipaed nominal shocks have real effecs in his model. 15.4 Aggregae Flucuaions under an Exogenous Money Supply Rule In order o close he model, we mus make assumpions abou he evoluion of nominal variables such as he money supply. We shall iniially assume ha he money supply follows a random walk wih drif, of he form, S! m = µ + m 1 + ε (15.45) where µ is a consan and ε S a whie noise shock o he money supply. (15.45) can be viewed as a sochasic consan growh rule for he money supply, followed by he cenral bank. For example, Friedman (1960) proposed such a rule for moneary policy. Wih his assumpion, he seady sae rae of growh of he money supply is equal o µ, and since growh is equal o zero in his model, seady sae inflaion is also equal o µ, and he seady sae nominal ineres rae is equal o ρ+µ. Τhe money demand funcion (15.30) can be approximaed around he seady sae nominal ineres rae ρ+µ as,! m p = y 1 (15.46) θ ln i 1+ i + 1 θ (vm v C )! m 0 + y ζ (r + E π +1 ) + 1 θ (v M v C ) where! m 0 = 1, and,! > 0. θ ln ρ + µ 1+ ρ + µ 1 1 1+ ρ + µ ζ = θ(ρ + µ)(1+ ρ + µ) ζ is he semi-elasiciy of money demand wih respec o he nominal ineres rae. Subsiuing for real oupu and he real ineres rae from (15.42) and (15.44) and solving for he price level, we ge ha,! p 1+ζ + (1+ζθ)(1 ) (1+ζθ)(1 ) (15.47) E 1 p ζ E p +1 = z!12
! George Alogoskoufis, Dynamic Macroeconomics, 2016 Chaper 15 where! z = m y N +ζ r N (1+ζθ)(1 ) ε A 1 θ (vm v C ) m 0 15.4.1 The Real Effecs of Moneary Shocks In order o solve for he price level and inflaion, we shall firs absrac from real shocks and assume ha he only shocks are moneary shocks, i.e. shocks o he money supply process ε S and shocks o money demand v M. Moneary shocks do no affec he naural level of oupu or he naural real rae of ineres, so we can normalize hem o zero. In he presence of moneary shocks, he raional expecaions soluion of (15.47) is given by, p = µ + m 1 + + (1 )(1+ζθ) ε S η M ( ) v 1 θ 1+ζ (1 η M ) From (15.48), unanicipaed inflaion is given by, (15.48)! π E 1 π = (15.49) + (1 )(1+ζθ) ε S M θ 1+ζ (1 η M ) M θ 1+ζ (1 η M ) ( ( ) + (1 )(1+ζθ)) ε ( ( ) + (1 )(1+ζθ)) ε M Subsiuing (15.49) in (15.42), we ge ha real oupu is deermined by, 1! y = y N + (15.50) + (1+ζθ)(1 ) ε 1 S M θ 1+ζ (1 η M ) ( ( ) + (1 )(1+ζθ)) ε Purely moneary shocks, such as shocks o he money supply and money demand, by causing unanicipaed changes in inflaion, cause emporary deviaions of oupu from is naural level. The reason is ha nominal wages are predeermined, based on expeced inflaion a he beginning of each period. By causing unanicipaed changes in inflaion, moneary shocks affec real wages and employmen, oupu and he real ineres rae. Thus, in his model, because of he nominal disorion of predeermined nominal wages, purely moneary shocks have emporary real effecs on oupu, employmen, unemploymen, real wages and he real ineres rae. 15.4.2 The Effecs of Real and Moneary Shocks on Prices and Oupu Real shocks would also affec inflaion, unanicipaed inflaion and flucuaions in real variables hrough z in (15.47). Real shocks in his model affec boh he naural level of oupu and he real ineres rae, and deviaions of oupu and he real ineres rae from heir naural levels, eiher hrough unanicipaed inflaion, or, in he case of produciviy shocks, direcly. We can use (15.47) o solve for he price level in erms of all he shocks. The soluion akes he following form.!13
! p = p _ + m 1 + χ A a 1 + χ C v C 1 + χ M v M 1 +ψ A ε A +ψ C ε C +ψ S ε S M +ψ M ε (15.51) where,! p _ = µ m 0 (1 )n _ +ζρ (15.52 a)! χ A = 1+ζθ(1 η ) A (15.52 b) 1+ζ (1 η A ) η A! χ C = 1+ζθ(1 η ) C η C (15.52 c) 1+ζ (1 η C ) θ 1 η! χ M = M (15.52 d) 1+ζ (1 η M ) θ ( )! ψ A = (1+ζθ)(1 ) 1+ζθ(1 η + A ) (15.52 e) (1+ζ (1 η A )) + (1+ζθ)(1 ) ( ) 1+ζθ(1 η! ψ C = C ) 1 (15.52 f) (1+ζ (1 η C )) + (1+ζθ)(1 ) θ! ψ S = (15.52 g) + (1 )(1+ζθ)! ψ M = (15.52 h) θ 1+ζ (1 η M ) ( ( ) + (1 )(1+ζθ)) From (15.51), unanicipaed inflaion is deermined by,! π E 1 π =ψ A ε A +ψ C ε C +ψ S ε S M +ψ M ε (15.53) All he relevan shocks, real and nominal, affec unanicipaed inflaion. Thus, all he relevan shocks affec oupu flucuaions as well. Subsiuing (15.53) in he oupu supply funcion we ge ha flucuaions of oupu around is naural level are given by,! y = y N + 1 (15.54) (1+ψ )ε A A +ψ C ε C +ψ S ε S M ( +ψ M ε ) Thus, innovaions in produciviy, consumpion demand, he money supply and money demand, cause deviaions of oupu from is naural level. The variance of oupu around is naural level is given by, 2! Var(y y N ) = 1 (15.55) (1+ψ A ) 2 σ 2 A +ψ C2 σ 2 C +ψ S2 σ 2 2 ( S +ψ M2 σ M )!14
Thus, under an exogenous money supply rule, all shocks affec he variance of oupu around is naural level, as he money supply does no adjus o counerac he effecs on hese shocks on unanicipaed inflaion. 15.5 Aggregae Flucuaions under a Feedback Nominal Ineres Rae Rule In realiy, modern cenral banks do no allow he money supply o follow an exogenous process of he form of (15.45). Moneary policy usually reacs o deviaions of inflaion from arge, and/or o deviaions of oupu and unemploymen from arge. In addiion, cenral banks usually conduc moneary policy by conrolling he nominal ineres rae raher han he money supply. This is because of he difficulies in conrolling he money supply, and because he money demand funcion is subjec o shocks due o financial innovaions. 7 In wha follows we shall hus examine he behavior of he model under he assumpion ha he cenral bank follows a feedback rule for he nominal ineres rae. In paricular, we shall assume ha he cenral bank follows a feedback rule of he form,! i = r N + µ + (π µ) + φ 2 (y y _ ) + ε i (15.56) i where!,φ 2 > 0 are policy parameers, and! ε is a whie noise policy shock (error). This is a generalizaion of he Wicksell rule ha we examined in he case of he new classical model of Chaper 12 and similar o he rule we used in Chaper 15. The generalizaion is due o Taylor (1993, 1999) who showed ha in he las 35 years or so, he Federal Reserve, and oher cenral banks, follow such feedback ineres rae rules. According o his rule, he cenral bank aims for a nominal ineres rae which is equal o he naural real rae of ineres, plus a arge inflaion rae equal o µ. If acual inflaion is higher han he arge µ, hen he cenral bank raises ineres raes in order o reduce inflaion. In addiion, if oupu is higher han is naural level, and unemploymen lower han is naural rae, hen he cenral bank also raises ineres raes, in order o bring oupu back o is naural level and unemploymen back o is naural rae. Under his assumpion, our new keynesian model hus consiss of he oupu supply funcion (15.42), he Fisher equaion (15.32), he real ineres rae equaion (15.44), and he policy rule (15.51). These equaions can help deermine he price level and inflaion, oupu and he real ineres rae. Once we deermine inflaion, we can also deermine unanicipaed inflaion, and he evoluion of employmen, unemploymen and real wages, hrough equaions (15.40). (15.41) and (15.43). Subsiuing he policy rule (15.56) in he Fisher equaion (15.32), and using he real ineres rae equaion (15.44) and he oupu supply funcion (15.42), we ge he following process for inflaion.! π = γ 1 E 1 π + γ 2 E π +1 + ( 1)γ 2 µ γ 1 ε A i γ 2 ε (15.57) 7 See Bernanke (2006) for how he Federal Reserve has been conducing moneary policy.!15
(φ where,! γ 1 = 2 +θ)(1 ), and,!. + (φ 2 +θ)(1 ) < 1 γ = 2 + (φ 2 +θ)(1 ) The inflaionary process depends on he policy parameers of he Taylor rule and he oher srucural parameers of he model, such as and θ. I is driven by wo shocks. Shocks o produciviy, as hese shocks cause deviaions of oupu from is naural level, due o he fac ha nominal wages were deermined before he realizaion of hese shocks, and also shocks o he policy rule (15.51). No oher shocks affec he inflaionary process under his rule, as he nominal ineres rae adjuss o reflec changes in he naural rae of ineres, which is affeced by he oher shocks. The inflaion process (15.57) is sable if! γ 1 + γ 2 < 1. This requires ha,! > 1 (15.58) Condiion (15.58), is usually referred o as he Taylor principle, and requires ha he nominal ineres rae reacs more han one o one o deviaions of curren inflaion from is arge µ. This is a sufficien condiion for a sable and deerminae inflaion process, and we shall assume ha i is saisfied by he cenral bank. If (15.58) is saisfied, hen he raional expecaions soluion of he inflaion process akes he form,! π = µ γ 1 ε A i γ 2 ε (15.59) Inflaion deviaes from he cenral bank arge µ, only in response o curren shocks o produciviy and shocks o he seing of he nominal ineres rae. From (15.59) unanicipaed inflaion is hus given by,! π E 1 π = γ 1 ε A i γ 2 ε (15.60) Subsiuing (15.60) in he oupu supply funcion (15.42), we ge ha,! y = y N + 1 (15.61) (1 γ )ε A i ( 1 γ 2 ε ) Flucuaions of oupu around is naural level depend posiively on unanicipaed shocks o produciviy and negaively on nominal ineres rae shocks. The impac of he shocks depends on he policy parameers of he cenral bank rule, which affec γ1 and γ2. From (15.61) he variance of deviaions of oupu from is naural level is given by, 2! Var(y y N ) = 1 (15.62) (1 γ 1 ) 2 σ 2 2 ( A + γ 22 σ i ) Subsiuing (15.60) in he employmen equaion (15.40), he unemploymen equaion (15.41), he real wage equaion (15.43) and he real ineres rae equaion (15.44), we see ha unanicipaed!16
shocks o produciviy and he nominal ineres rae, also affec deviaions of hese variables from heir naural levels as well. From (15.41), he deviaion of unemploymen from is naural rae is deermined by, ( )! u u N = 1 (15.41 ) π E A 1π + ε Since unanicipaed inflaion and unanicipaed produciviy shocks are no persisen, deviaions of unemploymen from is naural rae will be non persisen eiher. For example, under a Taylor (1993) moneary policy rule, deviaions of unemploymen from is naural rae are deermined by, ( )! u u N = 1 (15.63) (1 γ )ε A i 1 + γ 2 ε From (15.61) and (15.63) we can confirm ha under a Taylor rule, only produciviy and moneary policy shocks affec flucuaions in real variables, such as oupu and unemploymen, around heir naural level. This is in conras o he exogenous rule for moneary growh, which resuls in all shocks affecing deviaions of oupu from is naural rae, and hus a higher poenial variance of oupu. Furhermore, he impac of hese shocks in (15.61) and (15.62) depends on he parameers of he moneary policy rule (15.51), which can affec γ1 and γ2. Thus, in his model here is scope for moneary policy o affec he shor run flucuaions of real variables by appropriae choice of he policy parameers. We shall reurn o he issue of he appropriae role of moneary policy in Chaper 16. A final remark is in order hough. We can see from (15.59, (15.61) and (15.63) ha flucuaions of inflaion from arge, and oupu and unemploymen around heir naural levels and are he sum of wo whie noise processes, i.e whie noise processes hemselves. All deviaions las for one period and here is no persisence. This lack of persisence is a serious weakness of he model, as he persisence of aggregae flucuaions is one of he main characerisics of business cycles. 15.6 Unemploymen Persisence, Inflaion and Moneary Policy As we have presened i so far, his model can accoun for flucuaions of inflaion around he arge of he moneary auhoriies and oupu, employmen and unemploymen around heir naural raes, bu hese flucuaions are no persisen. Ye, persisence of aggregae flucuaions is one of he main characerisics of business cycles. To accoun for persisence in his model, he model mus be generalized o inroduce a propagaion mechanism for he effecs of he various shocks. 15.6.1 Nominal Wage Conracs in a Dynamic Insider Ousider Model!17
One way o inroduce unemploymen persisence in a model wih periodic wage seing has been proposed by Blanchard and Summers (1986). We shall examine a fully dynamic version of heir model. 8 Following Blanchard and Summers (1986), we assume ha he employmen objecive which deermines he nominal wage in he conrac depends on boh he exogenous number of core insiders in each firm, bu also hose who were employed in period -1. 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 insiders is o make expeced employmen saisfy a pah ha minimizes he following quadraic ineremporal loss funcion, 2 1! min E 1 β s (15.64) s=0 2 l(i) +s n_ (i) + ω ( 2 l(i) l(i) +s +s 1) 2 (15.64) is minimized subjec o he sequence of labor demand equaions (15.4), as employmen in each period is deermined ex pos by he firm. " is he logarihm of he number of core insiders. β=1/(1+ρ)<1 is he discoun facor, wih ρ being he pure rae of ime preference. ω is he weigh of recen employees relaive o core insiders in he wage seing process. As can be seen from (15.64), ousiders, i.e he unemployed of he previous period, have no influence on he wage seing process. 9 n _ We shall assume ha he oal number of core insiders in he economy is always sricly smaller han he labor force. We shall hus assume ha, 1! n _ (i)di = n _ < n,! (15.65) i=0 where n is he log of he labor force. From he firs order condiions for a minimum of (15.64), wages are se so ha expeced employmen for each firm saisfies,!( 1+ ω(1+ β) )E 1 l(i) βωe 1 l(i) +1 ωl(i) 1 = n _ (i) (15.66) Inegraing over he number of firms i, expeced aggregae employmen mus hen saisfy,!( 1+ ω(1+ β) )E 1 l βωe 1 l +1 ωl 1 = n _ (15.67) 8 The model we analyze in his secion is due o Alogoskoufis (2015). Noe ha (15.5) is a special case of (15.64) wih ω=0. Thus, in (15.5) only core insiders and no hose recenly 9 employed affec he wage seing process. An alernaive inerpreaion of (15.64) would be in erms of adjusmen coss. Insiders seek o minimize deviaions of expeced employmen from heir number, bu here is a cos o adjusing employmen from period o period. Then, ω can be inerpreed as he relaive imporance of adjusmen coss relaive o coss of deviaions from he employmen of core insiders.!18
! George Alogoskoufis, Dynamic Macroeconomics, 2016 Chaper 15 (15.67) is he same as (15.66) wihou he i index and refers o aggregae employmen. (15.67) helps explain he differences of our dynamic wage seing model from he model in he previous secion, where only core insiders, and no he employees of he previous period affeced wage conracs. In he model where only core insiders affec wage conracs, ω=0, as recen employees do no exer any separae influence in he wage seing process. Seing ω=0 in (15.67), nominal wages would be se in order o ensure ha, E 1 l = n _ which is he same as equaion (15.8). Thus, he dynamic model we are considering now, conains he saic model of he previous secions as a special case. In our more general forward looking dynamic model, from (15.67), expeced employmen is given by, 1 ω! E 1 l = (15.68) 1+ ω(1+ β) n_ + 1+ ω(1+ β) l + βω 1 1+ ω(1+ β) E l 1 +1 Thus, in our dynamic wage seing model, insiders se nominal wages in order o achieve an employmen arge which depends on core insiders, hose previously employed, bu also on expeced fuure employmen, as expeced fuure employmen will affec fuure wage seing behavior. 15.6.2 Wage Deerminaion, Unemploymen Persisence and he Phillips Curve Subracing (15.67) from he log of he labor force n, afer some rearrangemen, we ge,!( 1+ ω(1+ β) )E 1 u βωe 1 u +1 ωu 1 = u N (15.69) where,! u! n l is he unemploymen rae, and! u N! n n _ >0 is he naural unemploymen rae. The naural rae of unemploymen in his model is defined in erms of he difference beween he labor force and he number of core insiders. This is he equilibrium rae owards which he economy would converge in he absence of shocks. To solve (15.69) for expeced unemploymen, define he operaor F, as,! F s u = E 1 u +s (15.70) We can hen rewrie (15.69) as,!(( 1+ ω(1+ β) )F 0 βωf ωf 1 )u = u N (15.71) (15.71) can be rearranged as,!19
! George Alogoskoufis, Dynamic Macroeconomics, 2016 Chaper 15 1+ ω(1+ β)! βωf 1 F 2 F + 1 (15.72) βω β u = u N I is sraighforward o show ha if 0<β<1 and ω>0 and finie, he characerisic equaion of he quadraic in he forward shif operaor (in brackes) has wo disinc real roos, which lie on eiher side of uniy. The wo roos saisfy, 1+ ω(1+ β)! λ 1 + λ 2 = and! λ 1 λ 2 = 1 (15.73) βω β Using (15.73) we can rewrie (15.72), as,! (F λ 1 )(F λ 2 )u = 1 (15.74) βω u N Assuming λ1 is he smaller roo, we can solve (15.74) as, λ 1! E 1 u = λ 1 u 1 + (15.75) ω(1 βλ 1 ) u_ = λ 1 u 1 + (1 λ 1 )u N (15.75), which is he raional expecaions soluion of (15.69), deermines he pah of expeced unemploymen implied by he wage seing behavior of insiders. From (15.73), i is sraighforward o show ha an increase in ω, he relaive weigh of recen employees in he wage seing process, resuls in an increase in λ1, he coefficien ha deermines he persisence of expeced unemploymen. From (15.73), which defines he wo roos, i follows ha, λ 1 ω = λ 1 ω 2 > 0 Thus, he higher he weigh of recen employees relaive o core insiders in he wage seing process, he higher he persisence of unemploymen. 10 Acual unemploymen, is deermined from he employmen decisions of firms, afer informaion abou prices, produciviy and oher shocks has been revealed. Inegraing he labour demand funcion (15.4) over he number of firms i, aggregae employmen is given by,! l = l _ 1 (15.76) (w p a ) 10 For example, assuming β=0.99, wih ω=1, λ1=0.38. Wih ω=2, λ1=0.50, wih ω=10, λ1=0.73 and wih ω=100, λ1=0.91.!20
Subracing he aggregae employmen equaion (15.76) from he log of he labor force n, acual unemploymen is deermined by,! u = n l _ + 1 (15.77) (w p a ) Taking expecaions on he basis of informaion available a he end of period -1, he wage is se in order o make expeced unemploymen equal o he expression in (15.75), which defines he rae of unemploymen consisen wih he wage seing behavior of insiders. From (15.77), he wage is hus se in order o saisfy,! w = E 1 p + E 1 a + E 1 u n + l _ (15.78) where! E 1 u is deermined by (15.75). Subsiuing for he nominal wage in (15.77), using (15.78), hen he unemploymen rae evolves according o,! u = E 1 u 1 (15.79) (p E p + a E a ) 1 1 Subsiuing (15.75) in (15.79) hus gives us he soluion for he unemploymen rae.! u = λ 1 u 1 + (1 λ 1 )u N 1 (15.80) (p E p + a E a ) 1 1 From (15.80), he unemploymen rae is equal o he expeced unemploymen rae, as deermined by he behavior of insiders in he labor marke, and depends negaively on unanicipaed shocks o inflaion and produciviy. Unanicipaed shocks o inflaion reduce unemploymen by a facor which depends on he elasiciy of labor demand wih respec o he real wage, as unanicipaed inflaion reduces real wages. Unanicipaed shocks o produciviy also reduce unemploymen, as hey reduce he difference beween real wages and produciviy and increase labor demand. We can express (15.80) in erms of inflaion, by adding and subracing he lagged log of he price level in he las parenhesis. Thus, (15.80) akes he form of a dynamic, expecaions augmened Phillips Curve.! u = λ 1 u 1 + (1 λ 1 )u N 1 (15.81) (π E 1π + a E 1 a ) where π is he inflaion rae. (15.81) can be expressed in erms of deviaions of unemploymen from is naural rae, as,!21
! u u N = λ 1 (u 1 u N ) 1 (15.82) (π E 1π + a E 1 a ) From (15.82), deviaions of unemploymen from is naural level depend negaively on unanicipaed shocks o inflaion and produciviy, as hese cause a discrepancy beween real wages and produciviy, due o he fac ha nominal wages are predeermined. Unanicipaed shocks o inflaion reduce real wages and induce firms o increase labor demand and employmen beyond heir naural level. Thus, unemploymen falls relaive o is naural rae. Unanicipaed shocks o produciviy, given inflaion, cause an increase in produciviy relaive o real wages, and also cause firms o increase labor demand, employmen and oupu, beyond heir naural levels, which reduces unemploymen beyond is naural rae. I can easily be confirmed from (15.82) ha following a shock o inflaion or produciviy, unemploymen will converge gradually back o is naural rae, wih he speed of adjusmen being (1-λ1) per period. Thus, following shocks o inflaion or produciviy, deviaions of unemploymen from is naural rae will display persisence. 15.6.3 The Relaion beween Oupu and Unemploymen Persisence The persisence of employmen and unemploymen, will also be ranslaed ino persisen oupu flucuaions. Aggregaing he firm producion funcions (3), he aggregae producion funcion can be wrien as,! y = a + (1 )l (15.83) Adding and subracing!(1 )(n n _ ), he producion funcion can be wrien as,! y = y N (1 )(u u N ) (15.84) where,! y N = (1 )n _ + a (15.85) is he log of he naural level of oupu. (15.84) is an Okun (1962) ype of relaion, which suggess ha flucuaions of oupu around is naural level will be negaively relaed o flucuaions of he unemploymen rae around is own naural rae. From (15.84) and (15.82), deviaions of oupu from is naural level will be deermined by,! y y N = λ 1 (y 1 y N 1 ) + 1 (15.86) (π E 1π + a E 1 a )!22
(15.86) shows ha deviaions of oupu from is naural level, also display persisence, because of he persisence of employmen and unemploymen. (15.86) is a dynamic oupu supply funcion. Deviaions of oupu from is naural level depend posiively on unanicipaed shocks o inflaion and produciviy, as hese cause a discrepancy beween real wages and produciviy, due o he fac ha nominal wages are predeermined. Unanicipaed shocks o inflaion reduce real wages and induce firms o increase labor demand, employmen and oupu. Unanicipaed shocks o produciviy, given inflaion, cause an increase in produciviy relaive o real wages, and also cause firms o increase labor demand, employmen and oupu, beyond heir naural levels. On he oher hand, anicipaed shocks o produciviy increase boh oupu and is naural level by he same proporion. Under unemploymen persisence, curren employmen, unemploymen, oupu, real wages and he real ineres rae, as funcions of he exogenous shocks and shocks o inflaion, evolve according o,! l = (1 λ 1 )n _ + λ 1 l 1 + 1 (15.87) π E A 1π + ε where! is he naural level of employmen. n _! u = (1 λ 1 )u N + λ 1 u 1 1 (15.88) π E A 1π + ε where! u N is he naural rae of unemploymen.! y = y N + λ 1 (y 1 y N 1 ) + 1 (15.89) π E A ( 1π + ε ) where,! y N = (1 )n _ + a.! w p = (w p) N + λ 1 (u 1 u N A ) π E 1 π + ε (15.90) where,!(w p) N = a (n _ l _ ). ( ) ( ) ( )! r = r N +θ(1 )(1 λ 1 )(u u _ ) (15.91) where! r N C = ρ θ(1 η A )a + (1 η C )v. 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 heir naural level, cause persisen increases in employmen and oupu above heir naural level, and persisen reducions in unemploymen, real wages and he real ineres rae, below heir naural raes. 15.6.4 Flucuaions of Unemploymen and Inflaion under a Taylor Rule Assume ha deviaions in he unemploymen rae persis as in (15.82), and ha he cenral bank follows a Taylor rule of he form,!23
!!!!!!! George Alogoskoufis, Dynamic Macroeconomics, 2016 Chaper 15! i = r N + µ + π π * (15.92) i where!,φ 2 > 0 and! ε is a whie noise ineres rae policy shock. We have now expressed he Taylor rule in erms of deviaions of unemploymen and no oupu from is naural rae. This does no affec he resuls, as hrough he Okun ype relaion (15.84), deviaions of unemploymen from is naural rae are a linear funcion of deviaions of oupu from is naural level. Subsiuing (15.92) in he Fisher equaion, afer using he real ineres rae equaion (15.91) and he dynamic Phillips curve (15.88), we ge he following process for inflaion,! π = γ 1 E π +1 + γ 2 E 1 π + γ 3 π 1 + γ 4 µ + γ 5 ε A + γ 6 ε i i + γ 7 ε 1 (15.93) where, γ 1 = ( ) φ 2 (u u N ) + ε i + φ 2 +θ(1 λ 1 )(1 ) + λ 1 φ γ 2 = 2 +θ(1 λ 1 )(1 ) + φ 2 +θ(1 λ 1 )(1 ) + λ 1 λ γ 3 = 1 + φ 2 +θ(1 λ 1 )(1 ) + λ 1 γ 4 = γ 5 = γ 2 γ 6 = γ 1 γ 7 = λ 1 γ 1 ( 1)(1 λ 1 ) + φ 2 +θ(1 λ 1 )(1 ) + λ 1 Noe ha, because of he persisence of unemploymen, he inflaionary process also displays persisence. I also depends on he curren expecaions abou fuure inflaion, hrough he definiion of he real ineres rae and on boh parameers of he Taylor rule, as unanicipaed inflaion causes he unemploymen rae and he real ineres rae o deviae from heir naural raes. Finally, because of he persisence in unemploymen boh curren and pas nominal ineres rae shocks affec he inflaionary process. The effecs of produciviy and nominal ineres rae shocks on inflaion also depend on he parameers of he Taylor rule. 11 In order o solve for inflaion, we firs ake expecaions of (15.93) condiional on informaion available up o he end of period -1. This yields, 1! E 1 π = E 1 π +1 + λ 1 π 1 + (φ 1)(1 λ ) 1 1 λ µ + 1 i ε 1 (15.94) + λ 1 + λ 1 + λ 1 + λ 1 (15.93) being he inflaionary process from a dynamic sochasic general equilibrium model, in which he policy rule 11 of he moneary auhoriies is aken ino accoun when agens form heir expecaions, 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.!24
!!! George Alogoskoufis, Dynamic Macroeconomics, 2016 Chaper 15 The process (15.94) has wo roos,! λ 1 and!, and will be sable if he wo roos lie on eiher side of uniy. Since! λ 1 < 1, he expeced inflaion process will be sable if,! > 1 (15.95) Condiion (15.95), is he Taylor principle. I requires ha nominal ineres raes over-reac o deviaions of curren inflaion from arge inflaion, in order o affec expeced real raes. This is a sufficien condiion for a sable and deerminae process for expeced (and acual) inflaion. 12 If (15.95) is saisfied, hen he soluion for he expeced inflaion process (15.94) is given by,! E 1 π = (1 λ 1 )µ + λ 1 π 1 + λ 1 i ε 1 (15.96) From (15.96), i follows ha,! E π +1 = (1 λ 1 )µ + λ 1 π + λ 1 i ε (15.97) Subsiuing (15.96) and (15.97) in he inflaion process (15.93), he raional expecaions soluion for inflaion is given by,! π = (1 λ 1 )µ + λ 1 π 1 ψ 1 ε A ψ 2 ε i i +ψ 3 ε 1 (15.98) where, ψ 1 = φ 2 +θ(1 λ 1 )(1 ) + φ 2 +θ(1 λ 1 )(1 ) < 1 ψ 2 = λ 1 + φ 2 +θ(1 λ 1 )(1 ) > 0 ψ 3 = λ 1 > 0 From (15.98), he flucuaions of inflaion around he arge of he moneary auhoriies µ are persisen, and depend on he curren innovaion in produciviy and curren and pas ineres rae shocks. Furhermore, he persisence of inflaion is equal o he persisence of deviaions of unemploymen and oher real variables, such as oupu, from heir naural level. The flucuaions of unemploymen and oupu around heir naural level are driven by unanicipaed inflaion and innovaions in produciviy. From (15.98), unanicipaed inflaion is deermined by, 12 Woodford (2003), among ohers, conains a deailed discussion of he Taylor principle, and is significance for he resoluion of he price level and inflaion indeerminacy problem highlighed by Sargen and Wallace (1975) for non coningen ineres rae rules.!25