1 Price Indexation and In ation Inertia

Transcription

1 Lecures on Moneary Policy, In aion and he Business Cycle Moneary Policy Design: Exensions [0/05 Preliminary and Incomplee/Do No Circulae] Jordi Galí Price Indexaion and In aion Ineria. In aion Dynamics Here we consider a variaion of he baseline sicky price model in which rms index heir prices o lagged in aion. In paricular we assume ha he price of goods sold in period + k by a rm ha las re-opimized is price in period is given by P +kj = P +k j ( +k ) () for k = ; ; ::: wih P j P and where [0; ] is he degree of indexaion. As in he baseline model rms reopimize heir prices only wih probabiliy in any given period. Wih indexaion, a rm reopimizing is price in period will maximize choose a price: max P k E Q;+k P +kj Y +kj +k(y +kj ) k=0 subjec o he indexaion rule () and he sequence of demand consrains Y +kjk = (P +kj =P +k ) C +k for k = 0; ; ; :::where Q ;+k k C+k P C P +k is he sochasic discoun facor for nominal payo s, () is he cos funcion, and Y +kjk denoes oupu in period + k for a rm ha las rese is price in period. Noice ha, as in he las secion of chaper 4, we are allowing for an exogenous, ime varying price elasiciy of demand (as in Seinsson (003)). The rs order condiion for ha problem akes he form k=0 k E Q ;+k Y +kj P ( +k jk ) P +kmc +kj = 0

2 where jk P =P k. Tha opimaliy condiion can be log-linearized around a zero in aion seady sae o yield, afer some manipulaion: p p = k=0 () k E ( )( ) + ( ) = E fp + p g + ( )( ) + ( ) fmc +k + ( ) +k fmc + ( ) () where fmc mc + and log. Noice ha in he presence of indexaion, expeced fuure in aion has a more limied impac on price seing, since rms realize hey will be able o reduce is impac on heir relaive price hrough he auomaic indexaion mechanism (albei parially and wih a lag), unil hey have a change o re-opimize again. On he oher hand, he law of moion for he price level is given by P (P ) + ( ) (P ) which can be log-linearized around he zero in aion seady sae o yield = + ( ) (p p ) (3) Combining () and (3) we obain he following second order di erence equaion describing he dynamics of in aion: = E f + g + cmc + u ( )( )( ) where and u (+( )) ( ). Using he simple relaionship beween marginal cos and he oupu gap derived in chaper 4, we can derive he following version of he new Keynesian Phillips curve in erms of quasidi erenced in aion e : e = E fe + g + ey + u (4) where ( + ').and ey is he oupu gap, undersood here as he logdeviaion of oupu from is equilibrium level under exible prices and a consan markup.

3 . Welfare Approximaion As shown in chaper 5, a second order approximaion o he represenaive consumer s uiliy losses akes he form W = E 0 ( + ') ey + var i fp (i)g As shown in Woodford (003, chap. 6)), in he presence of indexaion of he sor assumed above he following relaionship holds var i fp (i)g = ( ) Hence, we see how in he presen environmen relaive price disorions arise as a resul of deviaions in in aion from he rae a which indexed prices are increasing, no as a consequence of in aion in iself. In paricular, in he presence of full indexaion ( = ) only changes in in aion, bu no he level of in aion, have disorionary e ecs. Combining he above resuls we can derive he following approximae welfare loss funcion for an economy wih indexaion: W = E 0 ( + ') ey + e 3

4 .3 Opimal Moneary Policy wih In aion Ineria We revisi he opimal policy under discreion and commimen, in he presence of in aion ineria resuling from backward-looking indexaion by rms. For simpliciy we assume he cos-push shock u follows a whie noise process..3. The Case of Discreion The moneary auhoriy minimizes he period loss funcion y ey + e where y, subjec o he "radeo " equaion e = ey + where E fe + g + u is aken as given by he cenral bank. The opimaliy condiion for ha problem is ey = e Subsiuing his opimaliy condiion ino (4) and solving forward we obain = + + u and ey = + u Hence, and in conras wih he model wih no indexaion, he opimal policy wih discreion implies persisen deviaions of in aion form arge, even in he limiing case of whie noise disurbances. The inuiion for ha resul is sraighforward: due o backward-looking indexaion he in aionary impac of he shock remains once he shock is gone; in order o minimize relaive price disorions and he oupu gap he cenral bank should fully accommodae hose second round in aionary e ecs of he shock..3. The Case of Commimen Now he moneary auhoriy chooses a sae-coningen policy fey ; e g ha maximizes E 0 [ y ey + e ] 4

5 subjec o a sequence of consrains Firs order condiions: e = E fe + g + ey + u y ey ' = 0 e + ' ' = 0 for = 0; ; ; :::and wih ' = 0. Eliminaing he mulipliers and focusing on we have ey 0 = e ey = ey e = ; ; ::: which in urn can be wrien more compacly as ey = (bp bp ) for = 0; ; ; ::: where bp p p. Subsiuing ino he in aion equaion and rearranging erms we obain ep = a ep + a E fep + g + a u where ep bp bp and a. The saionary soluion is given by: ++ p 4a a ep = ep + u (5) where (0; ). Equivalenly, in erms of deviaions of he price level from he implici arge p we have he following AR() process: o bp = ( + ) bp bp + u Noice also ha, independenly of, he oupu gap will evolve according ey = ey u Hence, and o he exen ha + >, he opimal response of in aion o a ransiory cos-push shock will display some posiive serial correlaion a shor horizons. Tha opimal ineria arises from he desire o avoid he large relaive price disorions associaed wih large deviaions of in aion from. See Fig 3. in Giannoni and Woodford (005). 5

6 Transacions Fricions In he secion we resore an explici role for money in our model by assuming ha real balances genrae uiliy. Hence, as in chaper, preferences are now represened by a discouned sum of he form E 0 X U C ; M ; N P where M denoes moneary holdings in period and P is he price index. The ow budge consrain now akes he form (once opimal allocaion of expendiures is accouned for): P C + R B + M B + M + W N T We specify he uiliy funcion o have he funcional form U C ; M ; N = C P + (M =P ) m N +' + ' As shown in Woodford (RES, 00x), in ha environmen a second order approximaion o he uiliy of he represenaive household around a zero in aion, undisored seady sae akes he form: W = E 0 + y ey + r r + :i:p: where, for simpliciy, we ignore he zero lower bound on he nominal ineres rae. Inuiion: he nominal rae acs as a ax on real balances, wih associae deadweigh loss convex in he ax rae. 6

7 . Opimal Moneary Policy wih Commimen The moneary auhoriy is assumed o choose a sae-coningen policy fey ; g ha maximizes E 0 + y ey + r r subjec o he sequence of consrains: = ey + E f + g + u ey = (r E f + g rr ) + E fey + g The Lagrangean can be se up as follows: L = E 0 ( + y ey + r r ) + ; ( ey + ) + ; (ey + r Firs order condiions: + ey + ) y ey ( ; ; + ; ) = 0 + ( ; ; ; ) = 0 r r + ; = 0 Subsiuing ; we have (for = ; ; 3; :::): y ey ; r r + r r = 0 + ( ; ; ) + r r = 0 which in urn can be combined (for = ; 3; 4; :::) y ey + + r r r r + r r = 0 7

8 Finally, rearranging erms, i yields he super-inerial Taylor rule: r = ( + ) r + r + r + y r ey (6) which is independen of he saisical properies of he disurbances ( robusly opimal ). Togeher wih he NKPC and IS, i can also be shown o have a locally unique soluion. See gure 3.3 in GW, wih impulse responses o naural real rae shock. Remarks: a policy radeo arises even in he absence of cos push shocks, resuling from he desire o avoid large ucuaions in ineres raes. in he limiing case r = 0, we recover: ey = y.. 8

9 .. A Targeing Rule Represenaion Le q r + y r ey. Then we can rewrie (6) as: ( L)( L) r = q where 0 < < <. Hence, a each period ineres rae mus be se so ha he condiion: ( L) r = is sais ed. Muliplying boh sides by k=0 k E fq +k g r ( ) we obain F () + F (ey) = y ey r r r where = y y ( ) < 0, r r( )( ) < 0, and r ( ), and F (z) ( ) P k=0 k E fz +k g which is a weighed forecas wih weighs adding up o one. Remarks: well de ned rule, even hough i does no deermine r explicily. inerpreaion: deermines he arge for he in aion forecas, as a funcion of oupu gap forecass and pas condiions. mean horizon for he relevan in aion projecion is ( ) P k=0 k k =. Under Woodford s baseline calibraion = 0:68, implying a mean forecase horizon of : quarers, much shorer han common pracice by in aion argeing cenral banks. under a welfare-heoreic inerpreaion of he loss funcion y =, hus implying = ( ), suggesing ha forecass of fuure oupu gaps should have lile e ec on he in aion forecas arge. 9