Week 7: Dynamic Price Setting

Transcription

1 00 Week 7: Dynamic Price Seing Week 7: Dynamic Price Seing Rmer begins Chaer 7 n dynamic new Keynesian mdels wih a general framewrk fr dynamic rice seing In ur analysis f menu css and real/nminal rigidiy f rices, we have cnsidered nly a saic framewrk: a ne-ime decisin wih given iniial rices Acual rice decisins ccur in a dynamic seing in which he rices se day are likely ersis in ne erid. The key quesin f his secin is: Wha is he dynamic-imal rice se if he rice is likely be in effec fr mulile erids? The inuiive answer is an average f he eeced saic-imal rice fr each fuure erid, weighed by he rbabiliy ha he rice is in effec fr ha erid. Mdeling dynamic rice seing Mdel f Rmer s Secin 7. Msly similar ur revius mdels Ms f he cmleiy in his mdel is eiher n ineresing r is ulimaely assumed away by Rmer in his arimain ward he end f he secin We will fcus n he basics f he mdel ha cmes u raher han he deails f he derivain (Sme maerial n he derivain are in he Cursebk if yu wan hem.) Saic-imal rice Based n cndiins a ime (if we knw hem) here is a rice ha is imal fr he firm in ha erid. This is jus he same ha we have had in he mdels f Chaer 6 b y The nly hing new here is he b, which is inrduced fr cnvenience ge rid f he ln(/(-)) erm, since we will assume ha hey cancel Le and b ln 0 by assumin, s Frm Rmer s (7.6), ln m m This is ur saic-imal rice fr erid Dynamic-imal rice Fllwing ur inuiin, suse ha q is he rbabiliy ha a rice se a ime 0 is sill in effec a ime Fied-rice cnracs wuld have q = fr durain f cnrac, and 0 aferward

2 Predeermined Prices: Fischer Mdel 0 Dynamic-imal rice is weighed average f saic-imal rices wih weighs given by rbabiliies q : qe0, where we he 0 q denminar is he sum f he weighs, which we need make sure ha he weighs add u ne q Le, where he denminar can be inerreed as he eeced q 0 number f erids ha a rice will be in effec. 0 Dynamic-imal 0 E m 0 0 Fr fied-rice cnrac f lengh n, fr he n erids f he n cnrac and zer aferward Fr Calv mdel wih rbabiliy f rice re-se f er erid, q and / 0 T deermine he imal rice and he resuling dynamics f he ecnmy, we need describe he aern f rice re-seing ver ime Fischer mdel f redeermined rices : Tw-erid verlaing cnracs wih differen rice fr firs and secnd erids Taylr mdel f fied-rice cnracs: Tw-erid verlaing cnracs wih same rice fr bh erids Calv mdel: Fied rbabiliy ha rice will be re-se in any erid Predeermined Prices: Fischer Mdel Adaain f Fischer s riginal mdel based n wage cnracs, where firm and wrkers can se differen wages fr he differen erids f he cnrac Tw equal-sized ses f firms: Gru A ses rices a beginning f dd-numbered erids Gru B ses rices a beginning f even-numbered erids Nain: is rice se fr erid fr firs erid f cnrac Se a beginning f is rice se fr erid fr secnd erid f cnrac Se a beginning f Ne ha in general

3 0 Predeermined Prices: Fischer Mdel Gru A Gru B Duble lines are imes a which rices are se (new cnracs) is aggregain f rice level in erid A end f, gru ses E and E Frm imal rice-seing mdel abve, Puing hese geher, E m E E m E m E E m E m Taking he eecain as f f he equain (which is needed in he bm ne), E E E m Em E (wih las equaliy cming frm eressin abve) S E m E m E m And E m E m E m E m E m Em Em Em Em S E m E m E m. SRAS curve is hriznal a his redeermined rice level and uu is deermined by AD

4 Predeermined Prices: Fischer Mdel 03 Frm he AD curve: y m m E m E m E m This equain eresses y as a funcin f w AD surrises m E m E m E m is he curren erid surrise AD is las erid s surrise AD The frmer is like he Lucas mdel s erm (wihu he b cefficien), shwing ha curren AD shcks will cause uu flucuains The laer is a lagged uu shck indicaing ha AD shcks affec uu fr w erids: he lengh f he lnges cnrac ha was fied befre he shck was knwn Inuiin Suse a shck ccurs AD ha is learned in erid,, and are all se befre he cnrac is signed Alhugh is se afer he shck is revealed, rice seers will have cmee agains which is already se lw/high because he infrmain was n knwn T he een ha here is real rigidiy, firms will wan kee clse avid geing far away frm her gru s rice Because he aggregae rice des n fully adjus in erid, uu in will be affeced by he shck ccurring in erid Oimal mneary licy Suse ha m f v, where f is he Fed s mneary licy acin and v is a randm (velciy?) shck AD Assume ha v fllws a randm walk, s v v wih being a whie nise shck ( is uncrrelaed wih anyhing ha haened befre ) The Fed ses f based n infrmain knwn a he beginning f erid, s i has n infrmain advanage ver rivae rice-seers Suse ha he Fed ries sabilize uu wih a licy feedback rule f he frm f, where is he Fed s resnse las erid s shck. The Fed des n knw his erid s shck s his is he bes i can d Aggregae demand is m f v v v m v If everyne knws he Fed s licy rule and arameer, hen

5 04 Predeermined Prices: Fischer Mdel because E 0, and E m v m E m v v E m v because E E 0, and E m E m v v Then, using he eressin fr y: y m E m E m E m Wha is he imal chice f minimize uu flucuains arund imal value f 0? Fed can d anyhing abu Seing = eliminaes he effec f lagged shck Fed s imal licy is ffse he ne-erid lagged shck make he reviusly se rice he crrec rice This was he firs f he new Keynesian mdels rvide an elicily imal licy rule wih a siive sabilizing effec f mneary licy

6 Fied-Price Cnracs: Taylr Mdel 05 Fied-Price Cnracs: Taylr Mdel Firms ha face rice-adjusmen css may n find i easy se a differen rice fr he w erids. Which is mre imran, decisin-making css r acual rice-adjusmen css? Culd be eiher because bh kinds f css are imran Fischer mdel wuld reduce decisin-making css because decisins wuld nly need be made every w erids (csly labr negiains, fr eamle), bu rices wuld have adjus every erid Taylr mdel assumes ha firms se same rice fr bh erids f heir cnrac. We als adjus he infrmain assumin allw hem knw m befre hey se Gru A Gru B Our rbabiliies ha he currenly se rice is sill in frce erids laer are q0, q and q 0,. This means ha and 0,. S imal E Subsiuing he saic-imal ricing frmula: m Em E Le s assume ha m fllws a randm walk, s m m u and Em m because E u 0 Wih, we have m m E m 4 E Slving fr, m E This is a secnd-rder difference equain in, cmlicaed by he resence f he eecain erm Rmer des he sluin by he mehd f undeermined cefficiens We wn wrry abu he mahemaics f he sluin, fcus n inuiin.

7 06 Calv Mdel f Prbabilisic Price Adjusmen Ne frm he diagram ha mus cmee agains (which is knwn) and + (which is n) T he een ha here is real rigidiy, firms will wan se in a way ha is n far frm hese as and fuure rices. This leads a lng-ailed effec f an AD shck n y Suse ha here is a big AD shck u in erid Firms seing rice in will n fully adjus because hey wan/need cmee wih firms wh se las erid This is jus like in he Fischer mdel where firms firs-erid-fcnrac rice had cmee wih he revius cnrac s rice Bu here, he same rice hlds ver in + because hey d n ge se differen rices fr he w erids Firms seing rice in + will n fully adjus shck because hey have cmee wih, which was n fully adjused Likewise, + will be adjus a lile mre, bu n fully because i mus cmee agains + Full rice adjusmen he shck will ccur nly gradually (asymically) ver ime During adjusmen rcess, y = m will be nn-zer as he change in will nly gradually mach he change in m Fischer mdel: real effecs f AD shcks las as lng as he lnges cnrac Taylr mdel: real effecs f AD shcks die away gradually desie -erid cnracs Frmal sluin:, y y u wih 0 < < 0 < < Smaller mre real rigidiy and larger = full adjusmen m Calv Mdel f Prbabilisic Price Adjusmen This is he mdel ms cmmnly used in mdern sicky-rice mdels In each erid, a randmly seleced fracin f firms change heir rices, wih he res keeing rice he same is in he revius erid The new rice se by firms changing rice in is as in he Taylr mdel Because share f firms kee same rice as las erid, Fr his analysis, Rmer re-inrduces he discun facr ( ) because he lags beween rice changes can be quie lng. We discun fuure erids a rae because fuure rfis are less valuable us

8 Calv Mdel f Prbabilisic Price Adjusmen 07 Thus, he weigh we aach any fuure erid s saic-imal rice is rrinal he discuned rbabiliy ha he rice is sill in effec q raher han jus he rbabiliy q q This mdifies he definiin f slighly: q Subsiuing in q 0, he denminar sum is q, s 0 0 As usual, he rice ha he firm ses is 0 0 E E Frm his equain, we can derive he new Keynesian Phillis (r SRAS) curve by subsiuin Fllwing Rmer s analysis n , we searae u he = 0 erm frm he summain ge E E Nw ne ha E 0 E E E because i is knwn a, s E This says ha his erid s dynamic-imal rice is a weighed average f he curren saic-imal rice and ur curren eecain f ne erid s dynamic-imal rice, wih weighs and The mre ha we discun he fuure (eiher because is smaller r because is larger making i less likely ha curren rices will sill be in effec), he greaer he weigh we u n curren saic-imal rice and he smaller he weigh we u n he fuure The inflain rae is Inflain = share f firms changing rice ercenage by which hse changing rice change i

9 08 Sae-Deenden Pricing We can u he equain in erms as fllws: E Frm he saic-imal ricing equain, E E, s y, and y E y E y E > 0, s his can be hugh f as a Phillis curve r an SRAS curve relaing inflain eeced inflain and uu Ne ha if =, s we ignre discuning, hen eeced fuure inflain has a uniary imac n curren inflain This lks very much symmeric he Lucas suly curve and he Friedman-Phels versin f he Phillis curve because y > 0 when > E() Hwever, he eecain here is E, n E, which leads sme cunerfacual imlicains abu inflain dynamics Sae-Deenden Pricing A majr criicism f he mdels we have sudied s far is ha he frequency f rice changes is egenus and ime-deenden Alernaive is sae-deenden ricing, where decisin change rice deends n hw far he rice deviaes frm imal rice, regardless f when rice was las se Analgy invenry behavir fr reail sre: Firms migh chse ime-deenden rdering licies: rder new gds weekly wih quaniy deending n eising invenries Firms migh chse sae-deenden rdering licies: rder new gds when invenries ge belw a aricular level In invenry mdels and sae-deenden rice-seing mdels, he imal adjusmen rule has he frm f an Ss-rule When falls sme (negaive, if > 0) hreshld level s, i reses rice s ha i i i i is sme (siive, if > 0) arge level S Rmer s Secin 7.5 cvers w sae-deenden mdels, bu we wn d he deails Calin-Sulber mdel In nging inflain, firms lg-rices will zigzag abve and belw linear rend line (wih cnsan inflain) A any mmen in ime, firms lg-rices will be evenly disribued alng he inerval s, S, wih s < 0. Each ime a firm s rice drs s, i increases rice S

10 Mdels wih Inflain Ineria 09 As ime asses, each firm s rice falls in relain, bu hse a he very bm mve he kee average = In Calin-Sulber mdel, a change in m is neural because i simly causes mre r fewer firms a he bm mve he, leaving average m unchanged Danziger-Glsv-Lucas mdel This is a mre realisic, bu analyically mre cmle, mdel f sae-deenden ricing There are bh relaive and aggregae rice shcks (as in Lucas mdel) and he aggregae shcks may have nn-zer eeced value s ha he rend inflain rae is siive (r negaive) Firms resnd heir relaive rice being far n eiher side f heir arge level by raising r lwering rice In his mdel, an uneeced change in AD will affec firms asymmerically s ha mre firms n ne end will change rices bu hse n he her end will n have ffseing change Mdels wih Inflain Ineria Evidence f inflain ineria If he nly surce f rigidiy is rice sickiness, hen inflain shuld be rivially easy s: Suse ha inflain has been nging a 5% frever and ha suddenly he grwh rae f m drs 0% (assuming y has n rend) Nw i 0, s everyne chses kee rice cnsan Inflain ss dead Ouu des n fall In his mdel, rice sickiness leads rice ineria, bu here is n inflain sickiness r inflain ineria There is subsanial evidence ha inflain has ineria: Inflain ineria is cnsisen wih backward-lking Phillis curve Backward-lking Phillis curve: E y y High uu is assciaed wih high inflain relaive earlier erids This means ha lwering inflain ends lead recessins Frward-lking new Keynesian Phillis curve: E y y High uu is assciaed wih high inflain relaive fuure eecain

11 0 Mdels wih Inflain Ineria This means ha inflain is eeced fall when uu is high Ball s analysis f disinflain finds ha reducins in inflain are alms always accmanied by recessins in uu Ecnmeric evidence is mied Chrisian, Eichenbaum, and Evans CEE adjus he Calv mdel s ha i is n changing nminal rices ha haens infrequenly, bu changing real rices Beween re-ricing inervals, firms rices g u a revius erid s rae f inflain raher han saying fied Yu can hink f his as full indeain f he defaul nminal rices lagged inflain (similar wage indeain wih = in rblem se) This kind f adjusmen behavir is cnsisen wih css f deciding n a ricing sraegy (decisin css) raher han css f elici rice changes (menu css) This alers basic ricing equain Calv: CEE: Rmer des he algebra n ge E y, wih This equain has bh Lagged effecs hrugh he indeain by he firms ha d n adjus heir sraegies and Frward-lking effecs hrugh aniciary rice-seing by he fracin f firms ha d adjus sraegies The CEE mdel can elain inflain ineria, bu why d firms adjus ricing sraegies nly eridically and inde lagged inflain in beween? Mankiw-Reis sicky infrmain mdel Mankiw and Reis build a mdel in which he reasn why firms rese ricing sraegies infrequenly is he cs f acquiring he necessary infrmain In he Mankiw-Reis mdel, each firm can se a rice ah fr curren and fuure daes based n is available infrmain They mdel he arrival f infrmain in a simle way: In each erid, a fracin f firms receives curren infrmain abu he ecnmy and reses is rice ah be imal given ha infrmain The fracin f firms ha d n ge new infrmain cninue n he rice ahs ha hey se in he revius erid The sluin, which we will n sudy in deail, is

12 Mdels wih Inflain Ineria a E m E m i i i i 0 y a E m E m i i i i 0 The effec a ime f a new nugge f rice infrmain arriving a ime i is divided beween uu and rice (as i mus be if m = y + ) wih fracin a i f he effec being n rice and ( a i) being n uu i a i i Rmer shws ha, wih being he elasiciy f imal rice wih resec aggregae demand m Mankiw and Reis shw dynamics f hree mdels in heir aer: backward-lking Phillis curve, sicky rices, and sicky infrmain Three shcks: One-ime fall in m Uneeced reducin in grwh rae f m Aniciaed reducin in grwh rae f m

13 New Keynesian DSGE Mdels New Keynesian DSGE Mdels The wrld f macrecnmics in he 000s has been dminaed by schasic simulains f dynamic new Keynesian mdels These mdels are usually buil arund New Keynesian IS curve, erhas wih mdels f invesmen and cnsumin grafed n New Keynesian Phillis curve, erhas wih inflain ineria buil in hrugh CEE r Mankiw-Reis Mneary-licy funcin fr seing ineres raes like he MP curve There are many, many variains n his verall framewrk and a majriy f macrecnmic aers in he las en years uses ne Yu culd, if yu waned, elre simulains f hese mdels using Dynare and sme f he mdels ha are available ublicly