Optimal Monetary Policy under Heterogeneous Banks

Transcription

1 Opimal Moeary Policy uder Heerogeeous Baks Nao Sudo y ad Yuki Teraishi z Isiue for Moeary ad Ecoomic Sudies, Bak of Japa July 8 (Very Prelimiary) Absrac We iroduce he heerogeeous sickiesses i loa ieres rae adjusmes, which are suppored by empirical aalyses i developed couries, io he sadard New Keyesia model wih bak secor. The welfare aalysis reveals ha he ceral bak should care abou he ieres rae di erece, i.e. credi spread, bewee heerogeeous loa ieres raes ad he rs di ereces of each loa ieres rae. We show ha he ceral bak should egaively respod o he credi spread shock bewee heerogeeous loa ieres raes, i.e. idiosycraic shock i he acial marke. There he heerogeeiy i saggeredess of loa ieres raes raher ha he credi spread is quaiaively so impora o impleme he opimal moeary policy whe sickiesses of loa ieres raes are high. Fially, we show ha he ceral bak pus is prioriy o he loa ieres rae wih more sickiess raher ha a weighed average of heerogeeous loa ieres rae o achieve he opimal moeary policy whe he marke share of loas is o so di ere. Bu, as he share of he sicky loa decreases, he ceral bak should pay aeio o boh sicky ad less sicky loas. We hak Toi Brau, Simo Gilchris, Fracois Gourio, Humio Hayashi, Eric Leeper, Masao Ogaki, Wako Waaabe, ad Mike Woodford for isighful commes, suggesios ad ecouragemes. We also hak he semier paricipas a Tokyo Uiversiy for heir commes. The views expressed i his paper are hose of he auhors ad do o ecessarily re ec he o cial views of he Bak of Japa. y ao.sudou@boj.or.jp z yuuki.eraishi@boj.or.jp

2 JEL Classi caio: Keywords: E44; E5 Opimal moeary policy; Heerogeeous baks; Saggered loa coracs; Credi spread Iroducio Coveioal heory of moeary policy has bee focusig o a chael hrough which a policy shock direcly a ecs he demad side of he ecoomy (e.g. Roemberg ad Woodford (997) ad Goodfried ad Kig (997)). Rece empirical research by Barh ad Ramey (), o he oher had, prese ample evidece ha a moeary policy shock works as a cos-push shock o he rms. They argue ha whe workig capial is a esseial compoe of producio ad he rms cos is closely ied o a policy ieres rae, a moeary policy shock a ecs he oupu hrough a supply side chael as well as he radiioal demad side chael. Icorporaig his cos chael mechaism io he heoreical framework requires a cosideraio for baks, sice i mos of he couries, moeary auhoriies are o he direc loa supplier o rms. I Ravea ad Walsh (6), where exible cos chael is sudied i he New Keyesia (NK) framework, rms borrow moey from baks for he purchase of heir ipus, ad so heir margial coss are direcly a eced by he loa raes variaios. Baks are playig impora role i he model, as hey rasmi a shock of he policy ieres rae o rms cos srucure, via a bak ledig chael. How baks rasmi moeary policy shocks o heir behavior is sudied empirically i several aspecs (e.g. Kashyap ad Sei (), Haa ad Berger (99), Slovi ad Sushka (983), ad Berger ad Udel (99)). I erms of loa ieres raes, macroecoomis are aware of he wo facs o baks resposes o a policy ieres rae shock: (i) loa Kashyap ad Sei () examies he impac of he policy rae chage o ledig volume. Haa ad Berger (99) ivesigaes he impac of he policy rae chage o he deposi raes. Slovi ad Sushka (983) ad Berger ad Udel (99) ivesigae he impac of he policy rae chage o he loa raes.

3 ieres raes se by baks are sicky compared o a policy ieres rae ad (ii) resposes of he loa ieres raes o a policy ieres raes are o uiform across baks depedig o he characerisics of each baks. Slovi ad Sushka (983) esimae he ime series deermias of ieres raes o US commercial loas ad show ha commercial loa raes respod less ha oe-for-oe o chages i he marke raes i he shor ru. Berger ad Udel (99) also idicae ha bak loa ieres raes are sicky compared o a policy ieres rae usig US pael daa. Moreover, may empirical sudies for Euro Area repor ha he heerogeeiy i loa sickiess are observed across baks. For example, Gambacora (4) shows ha he shor-ru heerogeeiy i loa raes are prese amog Ialia baks. He pois ou ha well-capialized baks respod quicker o a shock. De Graeve e al (7) ad Weh () repor he aalogous resuls for Belgium ad Germay, respecively. 3 The rs coribuio of he paper is ha we icorporae his heerogeeiy io he NK model wih bak secor. We assume he moopolisic compeiio i a bak ledig marke ad iroduce heerogeeous sickiess i loa ieres rae seigs. Thus, some baks adjus he loa raes more frequely ha oher baks do so ha he resposes of he loa raes o a iovaio i he policy ieres rae di er across baks. The secod coribuio is ha we hik of he opimal moeary policy uder a secod orderly approximaed welfare fucio. Welfare aalysis reveals ha he ceral bak should respod o he ieres rae di erece, i.e. credi spread, bewee loa ieres raes ad heerogeeous rs di ereces of loa ieres raes. There he heerogeeiy i sag- Kashyap ad Sei () repors a heerogeeiy i baks resposes o a moeary policy shock, i erms of he ledig volume. They claim ha baks wih less liquid balace shee are likely o respod more o policy shocks. 3 De Graeve e al (7) obais a icomplee ad heerogeous pass-hrough i he loa marke i Belgia. They repor ha he degree of capializaio ad he size of liquidiy are resposible for he diverse respose of bakig secors o a chage i marke raes. Weh () examies Germa markes ad cocludes ha ledig raes are sickier for small baks, baks wih high savigs deposis ad baks wih a high volume of o-bak busiess. 3

4 geredess of loa ieres raes raher ha he credi spread is quaiaively so impora o impleme he opimal moeary policy whe sickiesses of loa ieres raes are high. Bu, whe sickiesses of loa ieres raes are low eough, he credi spread quaiaively becomes more crucial eleme o coduc he opimal moeary policy. Provided his heerogeeiy across baks, we derive a opimal moeary policy rule. The simulaio resuls show ha he ceral bak pus is prioriy o he loa ieres rae wih more sickiess raher ha a weighed average of heerogeeous loa ieres rae o achieve he opimal moeary policy whe he marke share of loas is o so di ere. Bu, as he share of he sicky loa decreases, he ceral bak should pay aeio o boh sicky ad less sicky loas. The hird coribuio is ha we discuss he curre issue represeed by Taylor (8) ad Cúrdia ad Woodford (8). Now, may papers ivesigae wheher he ceral baks should respod o he credi spread of ieres raes ha ecoomic ages face (e.g. Taylor (8), McCulley ad Toloui (8), ad Cúrdia ad Woodford (8)). Taylor (8) implies ha he Federal Reserve Board have egaively reaced o he credi spread i he moey marke for he las few years o simulae ecoomy eve hough such a addiioal easig eveually iduces he ecoomic boom leadig o he sub-prime morgage loa problem from fall of 7 4. I coras o Taylor (8), Cúrdia ad Woodford (8) heoreically ivesigae wheher a ceral bak should reac o he credi spread bewee saver ad borrower i cosumers. They coclude ha he ceral bak qualiaively should pay aeio o he credi spread bu he opimal moeary policy i he basic NK model wihou credi spread sill quaiaively provides a good approximaio o he opimal moeary policy i he NK model wih credi spread. Our coclusio is ha he credi 4 The respose o he credi spread is also suggesed i McCulley ad Toloui (8) ha show a oefor-oe correspodece of he policy rae o he marke credi spread i he explici ieres rae rule i US. 4

5 spread is quaiaively o so impora o coduc he opimal moeary policy, which suppor Cúrdia ad Woodford (8). Bu, we also show ha he sadard opimal moeary policy ca o be a good approximaio i heerogeeous acial marke. This meas ha he heerogeeiy i saggeredess of loa ieres raes is very crucial o achieve he opimal moeary policy. The paper is orgaized as follows. Secio shows he empirical evidece. Secio 3 describes our model wih saggered loa raes. Secio 4 aalyzes he welfare implicaio i our model. Secio 5 ivesigaes he properies of opimal moeary policy. Secio 6 gives discussio. Secio 7 cocludes. Facs The heerogeous sickiess of loa raes are observed i a large lieraure sice here are variey of ways ha pricig decisios ca di er across baks. A coexisece of muliple bakig sysems uder a sigle currecy is oe source of he heerogeeiy. Iroducio of euro provides is clear example. I he Euro area, while a moeary policy is coduced by oe isiuio, he pricig behaviors of baks are di ere across coureies, maybe re ecig he di ereces i he degree of compeiio (Sorese ad Werer (6)). Sorese ad Werer (6) esimae he sickiess of bak loa raes for e couries i he euro area. They repor ha eve corollig for he loa ypes (e.g., Morgage loas, Cosumer loas), he speed of loa rae adjusme across cories is di ere i saisically sigi ca maer. Their resuls for he sickiess of log-erm loas rae o eerprises caegory imply ha upo he iovaio i he policy rae by basis, loa raes i Germay is revered by 45% while hose i Belgium is revered oly by % i he ex period. As we will discuss below, wih hese observed heerogeeiy i he loa ieres rae dyamics across couries i he euro area, our model provides a fair 5

6 descripio ha ECB faces afer he uiozaio. 5 The sickiess of loa raes may di er wihi a coury re ecig he variey of bak ypes. The respose of loa raes o a chage i he marke rae di er, depedig o he size of baks, liabiliy of baks. For each coury of Euro, exisig empirical sudies have provided he ample evidece for he heerogeeiy. Gambacora (4) ivesigaes he Ialia baks o d ou he baks liabiliy srucures are he impora deermias of heerogeous resposes of loa raes across baks. Weh () repors for he Germa baks ha large baks ad baks wih few savig deposis adjus heir ledig raes more quickely. Similarly for Japa, accordig o he repor released from Bak of Japa (BOJ (7, 8)), he loa raes se by larger baks (Ciy Baks) ed o respod quicker ha hose se by smaller baks (Regioal Baks, Regioal Baks II) do. 6 Thus he cosideraio for he heerogeeiy i loa rae sickiess is also impora for he implemeaio of moeary policy for oe coury. Heerogeeiy of loa rae sickiess arises from he di ereces associaed wih he loa ype, oo. I Sorese ad Werer (6), observed loa sickiess se by baks may vary depedig o he erms of he loa, such as he ype of borrowers or he legh of he loa 5 Io ad Ueda (98) shows ha he prime loa ieres rae adjusme speeds are very di ere bewee U.S. ad Japa. They coclude ha he prime loa rae adjusme i U.S. is much faser ha ha i Japa. 6 BOJ (7) repors ha Ciy Bak eeds abou hree o four quarers ad Regioal Baks ad Regioal Baks II eed abou ve o six quarers o adjus he loa ieres raes i he rs wo quarers of 6. Bak of Japa releases he loa rae series for several groups of baks. Ciy Baks are he baks ha have aio-wide bruches, whose mai busiess aciviies are basig o large ciies. Regioal Baks ad Regioal Baks II are comparaively smaller size of baks ad mos of heir bruches are limied i speci c prefecures. The able below repors he summary saisics o illusrae he characerisics of each bak group. Number of Baks Share i Deposi Share i Loas/Discous Ciy Baks 6 3.7% 33.6% Regioal Baks % 6.% Regioal Baks II 4 6.5% 7.6% 6

7 corac. The sudy for Belgia baks by De Graeve e al. (7) show ha shor-erm pass-hrough for ives loa rae is larger ha cosumer credi or morgage. Berger ad Udel (99), usig he pael daa of he U.S. baks, show ha he sickiess of he loa rae is larger for secured loas ad xed-rae loas, ha usecured loas ad oaig-rae loas. 3 Model We iroduce he heerogeeous saggered omial loa ieres rae coracs bewee privae baks ad rms io a model based o a sadard NK framework buil by Woodford (3). The model cosiss of four ages: cosumers, rms, a ceral bak, ad privae baks. 3. Cos Miimizaio I his model, we have wo cos miimizaio problems. The rs deermies he opimal allocaio of di ereiaed goods for he cosumer. The secod deermies he opimal allocaio of labor services, give he loa raes ad wages for he rm s preside. For he cosumer, we assume ha he cosumer s uiliy from cosumpio is icreasig ad cocave i he cosumpio idex, which is de ed as a Dixi-Sigliz aggregaor as i Dixi ad Sigliz (977), of budles of di ereiaed goods f [; ] produced by rm s projec groups as Z C c (f) df ; where C is aggregae cosumpio, c (f) is a paricular di ereiaed good alog a coiuum produced by he rm s projec group f, ad > is he elasiciy of subsiuio 7

8 across goods produced by projec groups. For he cosumpio aggregaor, he appropriae cosumpio-based price idex is give by Z P p (f) df ; where P is aggregae price ad p (f) is he price o a paricular di ereiaed good c (f). As i oher applicaios of he Dixi-Sigliz aggregaor, he cosumer s allocaio across di ereiaed goods a each ime mus solve a cos miimizaio problem. This meas ha he relaive expediures o a paricular good is decided accordig o: p (f) c (f) = C : () P A advaage of his cosumpio disribuio rule is o imply ha he cosumer s oal expediure o cosumpio goods is give by P C. di ereiaed goods i he rm secor. We use his demad fucio for Firms opimally hire di ereiaed labor as price akers. This opimal labor allocaio is carried ou hrough wo-sep cos miimizaio problems. Firm f hires all ypes of labor. There, each rm has o use wo ypes of loa, sicky loas ad less sicky loas. Privae baks rese loa ieres raes wih loger ierval i sicky loa ad hey rese wih shorer ierval i less sicky loa. To replicae his siuaio, we assume ha o ace a labor cos for labor ype h [; ); he rm has o use sicky loa, ad o ace he cos for labor ype h [; ]; i has o use less sicky loa. We ca hik of his seig as a rm uses sicky loa o some projec which is characerized by labor ype h, bu uses less sicky loa o some projec which is characerized by labor ype h. (more ad more) Whe hirig a labor from h [; ), porio of he labor cos associaed wih labor ype h; which we deoe as ; is aced by borrowig from he bak h. The, he rs-sep cos 8

9 miimizaio problem for he allocaio of di ereiaed labor from h [; ) is give by: Z mi l (h;f) [ + r (h)] w (h) l (h; f)dh; subjec o he aggregae domesic labor supply o rm f: " Z L (f) l (h; f) dh where r (h) is sicky loa ieres rae applied o he employme of a paricular labor ype h, l (h; f) is he di ereiaed labor ipu wih respec o h ha is supplied o rm f, ad is a preferece parameer o di ereiaed labors. The sicky loa bak h has some moopoly power over seig loa ieres raes. Thus, we assume he moopolisic compeiio o he loa marke where he rasacio bewee baks ad rms ake place. The relaive demad o di ereiaed labor is give as follows: [ + r (h)] w (h) ; () # ; l (h; f) = L where As a resul, we ca derive: Z Z f[ + r (h)] w (h)g dh : (3) [ + r (h)] w (h) l (h; f) dh = L (f) : Through a similar cos miimizaio problem, we ca derive he relaive demad for each ype of di ereiaed labor from h [; ] as: l h; f ( = + r L h w h ) ; (4) where Z + r h w h dh ; (5) 9

10 ad where r aced by bak h. The: h is he less sicky loa ieres rae, ad is a porio of he labor cos Z + r h w h l h; f dh = L (f) : Accordig o he above wo opimaliy codiios, he rms opimally choose he allocaio of di ereiaed workers bewee he wo groups. Because rms have producio fucio ha hires workers from h [; ) ad ( ) workers from h [; ], he secod-sep cos miimizaio problem describig he allocaio of di ereiaed labor bewee hese wo groups is give by: subjec o he labor idex: mi L (f) + L (f) ; L ;L el (f) [L (f)] L (f) ( ) : (6) The, he relaive demad fucios for each di ereiaed ype of labor are derived as follows: L (f) = e L (f) e ; (7) ad L (f) = ( ) L e (f) e : e ; (8) Therefore, we ca obai he followig equaios: L (f) + L (f) = e L e (f) ; [ + r (h)] w (h) l (h; f) = el (f) ; (9) e

11 ad l h; f ( + r h w h ) = L e (f) ; () e from equaios (), (4), (7), ad (8). We ca ow clearly see ha he demad for each di ereiaed worker depeds o wages ad loa ieres raes, give he oal demad for labor. Fially, from he assumpio ha he rms ace par of he labor coss by loas, we ca derive: q (h; f) = w (h) l (h; f) [ + r (h)] w (h) = w (h) el (f) ; e ad q h; f = w h l h; f = w h ( + r h w h ) L e (f) : e These codiios demosrae ha he demads for each di ereiaed loa also deped o he wages ad loa ieres raes, give he oal labor demad. For aggregae labor demad codiios, we ca obai followig expressio: 3. Cosumer el = Z el (f) df: We cosider he represeaive cosumer who derives uiliy from cosumpio ad disuiliy from a supply of work. The cosumer maximizes he followig uiliy fucio: ( X UT = E U(C T T ) T = Z V (l T (h))dh Z V (l T (h))dh ) ;

12 where E is a expecaio codiioal o he sae of aure a daa. The fucio U is icreasig ad cocave i he cosumpio idex as show i he las subsecio. The budge cosrai of he cosumer is give by P C + E [X ;+ B + ] + D B + ( + i )D + + Z Z w (h)l (h)dh w (h)l (h)dh + B + F ; () where B is a risky asse, D is he amou of bak deposi, i is he omial deposi rae se by he ceral bak from o +, w (h) is he omial wage for labor supply, l (h), o he rm s busiess ui of ype h, B = R B (h)dh is he omial divided semmig from he owership of baks, F = R F (f)df is he omial divided from he owership of he rms, ad X ;+ is he sochasic discou facor. We assume a complee acial marke for risky asses. Thus, we ca hold a uique discou facor ad ca characerize he relaioship bewee he deposi rae ad he sochasic discou facor: + i = E [X ;+ ] : () Give he opimal allocaio of cosumpio expediure across he di ereiaed goods, he cosumer mus choose he oal amou of cosumpio, he opimal amou of risky asses o hold, ad a opimal amou o deposi i each period. Necessary ad su cie codiios are give by U C (C ; ) = ( + i )E U C (C + ; + ) P ; (3) P + U C (C ; ) U C (C + ; + ) = X ;+ P P + :

13 Togeher wih equaio (), we ca d ha he codiio give by equaio (3) expresses he ieremporal opimal allocaio o aggregae cosumpio. Assumig ha he marke clears, so ha he supply of each di ereiaed good equals is demad, c (f) = y (f) ad C = Y, we ally obai he sadard New Keyesia IS curve by log-liearizig equaio (3): x = E x + (bi E + br ); (4) where we ame x he oupu gap ha is de ed i he ex secio, + i aio, ad br he aural rae of ieres. br will be a exogeous shock. Each variable is de ed as he log deviaio from is seady saes (excep x ad. Also, he log-liearized versio of variable m is expressed by bm = l(m =m), where m is seady sae value of m.). We de e U Y U Y Y Y >. I his model, he cosumer provides di ereiaed ypes of labor o he rm ad so holds he power o decide he wage of each ype of labor as i Erceg, Hederso ad Levi (). We assume ha each projec group hires all ypes of workers i he same proporio. The cosumer ses each wage w (h) for ay h i every period o maximize is uiliy subjec o he budge cosrai give by equaio () ad he demad fucio of labor give by equaio (). 7 The we have he followig relaio ad w (h) = V l [l (h)] P U C (C ) ; (5) w h P = V l l h U C (C ) : (6) I his paper, we assume ha he cosumer supplies is labors oly for he rm, o for he privae bak. We use he relaios give by equaios (5) ad (6) i he rm side. 7 We assume a exible wage seig i a sese ha he cosumer ca chage wage i every period. 3

14 3.3 Firms There exiss a coiuum of rms populaed over ui mass [; ]. Each rm plays wo roles. Firs, each rm decides he amou of di ereiaed labor o be employed from boh h [; ) ad h [; ], hrough he wo-sep cos miimizaio problem o he producio cos. Par of he coss of labor mus be aced by exeral loas from baks. For example, o ace he coss of hirig workers from h [; ), he rm mus borrow from baks ha provide sicky loa. However, o ace he coss of hirig workers from h [; ], he rm mus borrow from baks ha provide less sicky loa. Here, we assume ha rms mus use all ypes of labor ad herefore borrow boh sicky ad less sicky loa by he xed proporio. 8 Secod, i a moopolisically compeiive goods marke, where idividual demad curves o di ereiaed cosumpio goods are o ered by cosumers, each rm ses a di ereiaed goods price o maximize is pro. Prices are se i a saggered maer as i he Calvo (983) - Yu (99) framework. As is sadard i he New Keyesia model followig he Calvo (983) - Yu (99) framework, each rm f reses is price wih probabiliy ( discoued value of pro, which is give by: E X T = ) ad maximizes he prese i T X ;T hp (f) c ;T (f) T e LT e (f) ; (7) where we assume he producio fucio as y (f) = F ( e L T (f)). The producio fucio is a icreasig ad cocave. Here, he rm ses p (f) uder he Calvo (983) - Yu (99) framework. The prese discoued value of he pro give by equaio (7) is furher rasformed io: X E T = T p (f) X ;T (p (f) C T T e LT e (f)) : P T 8 The same srucure is assumed for employme i Woodford (3). 4

15 I should be oed ha price seig is idepede of he loa ieres rae seig of privae baks. The opimal price seig of p (f) uder he siuaio i which maagers ca rese heir prices wih probabiliy ( E X T = ) is give by: () T U C (C T ) y ;T (f) P T where we subsiue equaio (). " p (f) equaio (8), equaio (8) ca be ow rewrie as: X E () T p U C (C T ) y ;T (f) (f) P where Z ;T (f) = T = 8 < Z + L r (h) : T L e # T (f) = ; ;T (f) By furher subsiuig equaios (5) ad (6) io ( P P T V l [l T (h)] U C (C e L ;T ;T (f) Z ;T (f) = ; (9) ) = dh9 8 < Z + S r : h ( Vl lt e ) L ;T (f) = dh9 U C (C ;T (f) ; By log-liearizig equaio (9), we derive: " b X TX # ep (f) = E () T H; + RL;T b + RS;T b + cmc ;T (f) ; () T = =+ where L (+R L) + L R L ad ( ) S (+R S) + S R S are posiive parameers, ad we de e he real margial cos as: ; : where cmc ;T (f) Z cmc ;T (h; f) dh + Z cmc ;T mc ;T (h; f) V l [l T L e ;T (f) U Y (C T ;T (f) ; h; f dh; 5

16 ad We also de e: ad mc ;T h; f V l lt L e ;T (f) U Y (C T ;T (f) : R L; R S; Z Z ep (f) p (f) P ; r (h) dh; () P P : r h dh; () The, equaio () ca be rasformed io: " b X ep (f) = E () T ( + ) cmc T + RL;T b + RS;T b + T = # TX ; (3) =+ where we make use of he relaioship: cmc ;T (f) = cmc T " b ep (f) # TX ; =+ where is he elasiciy e L ;T ;T (f) margial cos as: where ad cmc T Z wih respec o y. We furher deoe he average real cmc T (h) dh + Z mc T (h) V l [l T (h)] U Y (C T ) mc T lt cmc L e T H;T h V l L e T : U Y (C T H;T h dh; 6

17 The poi is ha ui margial cos is he same for all rms i he siuaio where each rm uses all ypes of labor ad loas wih he same proporio. Thus, all rms se he same price if hey have a chace o rese heir prices a ime. I he Calvo (983) - Yu (99) seig, he evoluio of he aggregae price idex P is described by he followig law of moio: Z p (f) df = Z p (f) df + ( ) Z p (f) df; where ad =) P = P + ( ) (p ) ; (4) P p Z Z p (f) p (f) The curre aggregae price is give by he weighed average of chaged ad uchaged prices. Because he chaces of reseig prices are radomly assiged o each rm wih equal probabiliy, a aggregae price chage a ime should be evaluaed by a average of price chages by all rms. By log-liearizig equaio (4), ogeher wih equaio (3), df; df: we ca derive he followig New Keyesia Phillips curve: = cmc + RL; b + RS; b + E + ; (5) where he slope coe cie ( )( ) (+ ) is a posiive parameer. This is quie similar o he sadard New Keyesia Phillips curve, bu coais loa ieres raes as cos compoes. Here, accordig o he discussio i Woodford (3), we de e he aural rae of oupu Y from equaio (9) as 7

18 = + L R + S R 8 < Z ( : 8 < Z ( Vl l e ) L (f) = dh9 : U C (Y (f) ; V l [l (h)] U C (C ) L e ) (f) = dh9 (f) where, uder he aural rae of oupu, we assume a exible price seig, p (f) = P, ad assume o impac of moeary policy, r (h) = r (h) = R, ad so hold y (f) = Y. l (h), l (h), e L (f), ad e L (f) are he amou of labor uder Y, respecively. The, we have cmc = (! + )( b Y b Y ); where b Y l(y =Y ), ad b Y l(y =Y ), ad! is a sum of he elasiciy of he margial disuiliy of work wih respec o oupu icrease ad he elasiciy of wih respec F (F (y)) o oupu icrease. 9 The, by de ig x Y b Y b, we ally have where (! + ). 3.4 Privae baks = x + RL; b + RS; b + E + ; (6) There exiss a coiuum of privae baks populaed over [; ]. There are wo ypes of baks; baks ha provide sicky loas populae over [; ) ad baks ha provide he less sicky loas populae over [; ]. Each privae bak plays wo roles: () o collec he deposis from cosumers, ad () uder he moopolisically compeiive loa marke, o se di ereiaed omial loa ieres raes accordig o heir idividual loa demad 9! +! w, where! w is he elasiciy of margial disuiliy of work wih respec o oupu icrease i V l(l (h); ) U Y (Y ; ). We ca see more deailed derivaio i Woodford (Ch. 3, 3). 8

19 curves, give he amou of heir deposis. We assume ha each bak ses he di ereiaed omial loa ieres rae accordig o he ypes of labor force as examied i Teraishi (7). Saggered loa coracs bewee rms ad privae baks produce a siuaio i which he privae baks x he loa ieres raes for a cerai period. A sicky loa bak oly provides a loa o rms whe hey hire labor from h [; ). However, a less sicky loa bak leds oly o rms whe hey hire labor from h [; ]. Firs, we describe he opimizaio problem of a bak ha provide sicky loa. This bak ca rese loa ieres raes wih probabiliy ' L followig he Calvo (983) - Yu (99) framework. Uder he segmeed evirome semmig from di ereces i labor supply, privae baks ca se di ere loa ieres raes depedig o he ypes of labor. As a cosequece, he privae bak holds some moopoly power over he loa ieres rae o rms. Therefore, he bak h chooses he loa ieres rae r (h) o maximize he prese discoued value of pro : E X T = ' L T X;T q ;T (h; f) f[ + r (h)] ( + i T )g : The opimal loa codiio is ow give by: E X T = ' L T P P T U C (C T ) U C (C ) q ;T (h) + L r (h) L f[ + r (h)] ( + i T )g = : (7) Because he sicky loa baks ha have he opporuiy o rese heir loa ieres raes will se he same loa ieres rae, he soluio of r (h) i equaio (7) is expressed oly wih r. I his case, we have he followig evoluio of he aggregae loa ieres rae idex: + R L; = ' L ( + R L; ) + ' L ( + r ) : (8) By log-liearizig equaios (7) ad (8), we ca deermie he relaioship bewee he 9

20 loa ad deposi ieres rae as follows: where L 'L +(' L ), L br L; = L E b RL;+ + L b R L; + L 3 b i ; (9) 'L, ad +(' L ) L 3 'L +(' L ) ( ' )(+i) are posiive parame- +R L ers. This equaio describes he sicky loa ieres rae (supply) curve by he baks. Similarly, from he opimizaio problem of bak h ha provide less sicky loa, we ca obai he relaioship bewee loa ieres rae ad deposi ieres raes as follows: br S; = S E b RS;+ + S b R S; + S 3 b i ; (3) where ' is he Calvo parameer for bak h. We assume ' L > ' s. S 'S ' S, ad +(' S ) S 3 'S ( ' S )(+i) are posiive parameers. +(' S ) +R S The marke loa clearig codiios are expressed as: +(' S ), S q ;T (h) = Z q ;T (h; f)df; ad q ;T h = Z Z Z q ;T h; f df; q ;T (h)dh = D T ; q ;T h dh = ( ) D T : 4 Opimal Moeary Policy Aalysis 4. Approximaed Welfare Fucio We derive a secod order approximaio o he welfare fucio followig Woodford (3). The deail of derivaio is show i Appedix A.

21 The cosumer welfare i he home coury is give by ( X X E UT = E U(C T ) = = = = Z V (l (h))dh Z V (l (h))dh ) ; (3) The, we have a secod order approximaed loss fucio as follows: X X E UT ' E + x (x x ) + L ( R b! L; RL; b ) + S ( R b S; RS; b ) + brl; LS RS; b ; (3) where, x, L, S, ad LS are posiive parameers. I is impora o oe ha equaio (3) implies he ceral bak should care abou he heerogeeiy i he acial markes. I paricular, he ieres rae spread bewee loa ieres rae wih quick adjusme ad oe wih slow adjusme as well as he heerogeeous loa ieres rae chages are elemes i seig he policy ieres raes. This dig is o rivial because here are o such properies uder homogeeous loa ieres rae coracs, i.e. b R = b R L; = b R S; ad = L = S as: X UT ' = X + x (x x ) + ( R b R b ) : (33) = I his case, we have eiher he erm of he ieres rae spread or he erm of heerogeeous loa ieres rae chages. We oly have he erm of a homogeeous loa ieres rae chage. Clearly, he heerogeeiy i he acial marke makes he moeary policy complicaed. Whe sickiess i loa raes are he same i sicky ad less sicky loas, we sill have he erm of credi spread uder a shock i eiher of sicky or less sicky loa or di ere shocks i sicky ad less sicky loas as: + x (x x ) X UT ' = X = + L ( R b L; br L; ) + S ( R b S; + LS br b L; R S; br S; ) where R b L; ad R b S; deoe sicky ad less sicky loa ieres raes wih heerogeeous idiosycraic shocks, respecively. There heerogeeiy i loa ieres raes is i idiosycraic shocks raher ha i di erece of loa sickiess. C A ;

22 We show how he weigh i he welfare fucio chage accordig o he loa sickiess. Figure repors he value of L x ad LS x wih chagig loa ieres rae sickiess from o.7. I demosraes ha he heerogeeiy i saggeredess of loa ieres raes raher ha he credi spread is quaiaively so impora o impleme he opimal moeary policy whe sickiesses of loa ieres raes are high. Bu, whe sickiesses of loa ieres raes are low eough, he credi spread quaiaively becomes more crucial eleme o coduc he opimal moeary policy. Figure 3. Welfare weigh lamdal/lamdax lamdals/lamdax psil To small ' L, we have L x sickiess of he price adjusme. LS x. Here, we do o repor x, bu i also chages accordig o he

23 4. Opimal Moeary Policy Rule We cosider a opimal moeary policy scheme i which he ceral bak is credibly commied o a policy rule i he Timeless Perspecive. I his case, as show i Woodford (3), he ceral bak coducs moeary policy i a forward lookig way by payig aeio o fuure ecoomic variables ad by akig accou of he e ecs of moeary policy o hose fuure variables. The objecive of moeary policy is o miimize he expeced value of he loss crierio give by Eq. (56) uder he sadard New Keyesia IS curve give Eq. (4), he augmeed Phillips curve give by Eq. (6), he loa rae curve wih slow adjusme give by Eq. (3), ad he loa rae curve wih quick adjusme give by Eq. (9). The opimal moeary policy is expressed by he soluio of he opimizaio problem which is represeed by he followig Lagragia: 8 8 h i L + x + (bi + r ) x i >< X >< + L = E hx + RL; b + RS; b + + h = + 3 L R b L;+ + L R b L; + Lb i 3 i RL; b h >: >: + 4 S b R S;+ + S b R S; + Sb i 3 i RS; b 99 >= >= ; >; >; where,, 3, ad 4 are he Lagrage mulipliers associaed wih he IS curve cosrai, he Phillips curve cosrai, he loa rae curve cosrais, respecively. We differeiae he Lagragia wih respec o, x, b R L;, b R S;, ad bi o obai he rs-order codiios. The, by combiig he rs order codiios, we have a followig opimal The deailed explaaios abou he imeless perspecive are i Woodford (3). 3

24 moeary policy rule: z3 z4 ( z 5 L) ( z 6 F ) L (4R b L; E 4R b L;+ ) + LS ( b 3 R L; RS; b ) 6 x (x x ) 4 (z3z 4) ( z5l) ( z6f ) S (4R b S; E 4R b S;+ ) LS ( R b 7 L; RS; b ) 5 x (x x ) = E ( z L) ( z L) ( + x 4x ) ; (34) where z, z, z 3, z 4, z 5, z 6, z 3, z 4, z 5, ad z 6 are parameers, saisfyig z + z = + +, z z = (z >, < z < ), z 3 = z 3 ( z 6 = (z 4). L ), z L 6 = z 3 4, z 3 = S S 3, z4z 5 = ( z3 S 3 5 Opimal Moeary Policy Reacio 5. Respose o Credi Spread Shock L, z L 4 z 5 = 3 z 3 ( L 3 ), z 4 + z5 = ( z3 ), z 4 + z 5 = S ), ad S 3 We use he parameer values lised i Table 3 borrowig from Roemberg ad Woodford (997). I should be oed ha Slovi ad Sushka (983) claim ha privae baks, o average, eed a leas wo quarers ad perhaps more o adjus loa ieres raes. Thus, he average corac duraio of sicky loa ieres raes is se o be wo quarers for less sicky loa ad be hree quarers for sicky loa. We assume uexpeced oe perceage credi spread shock wih persisece.9 o u adjusme as: br L; = L E b RL;+ + L b R L; + L 3 b i + u L; ; i he loa rae curve wih more sicky 4

25 Table : Parameer values Parameers Values Explaaio.99 Discou facor 6.5 Elasiciy of oupu wih respec o real ieres rae.3 Elasiciy of i aio wih respec o oupu.66 Probabiliy of price chage '.66 Probabiliy of loa ieres rae chage i log erm loa '.5 Probabiliy of loa ieres rae chage i shor erm loa 7.66 Subsiuabiliy of di ereiaed cosumpio goods 7.66 Subsiuabiliy of di ereiaed laborers ; Raio of exeral ace o oal ace.5 Preferece for loa of log erm Figure shows he simulaio oucomes. Figure. Impulse Respose o Credi Spread Shock.5... Iflaio Oupu Gap Loa Rae Less Sicky Loa Sicky Loa 5 5 Policy Ieres Rae

26 We ca co rm ha a ceral bak decreases he policy ieres raes o miigae shock o he credi spread shock. The reaso of his is ha he ceral bak has a iceive o sabilize he loa rae ucuaio as: ( R b L; RL; b ) = L F L L h ( L) Lb io 3 i + u L; : (35) To decrease he loa rae ucuaio, he ceral bak should egaively respod o he credi spread shock u. Eveually, he oupu gaps ake posiive values haks o he moeary easig. Uder icreasig loa ieres raes, i aio raes go up due o he icrease i he producio cos. This opimal moeary policy respose o credi spread is cosise wih he reacio by he real ceral baks. For example, FRB lowers he policy ieres rae i he sub-prime morgage crisis from he fall of 7 o he curre. (sory. more ad more) 5. Weigh o Sicky or Less Sicky? (sory. more ad more) We ivesigae which of he loa raes ceral bak should cosider. We assume ha he weighs o sicky loa ad less sicky loas i producio fucio are equal, i.e. = :5, ad a symmeric shock i sicky ad less sicky loas, i.e. u L; = u S; for ay, i he baselie case o ideify he prioriy of he ceral bak. Figure 3 shows he dyamics of policy ieres rae wih various shares of he sicky loa. From he oucome i Figure, we ca co rm ha he ceral bak has o respod more aggressively o he shock i sicky loa ieres rae whe he marke share is equal i wo ypes of loa. The implicaio, however, chage accordig o he marke share of sicky loa. As he share of he sicky loa decreases, he ceral bak should more care abou he less sicky loa. 6

27 Figure 3. Policy Rae Dyamics o Various Share of Sicky Loa.4 Policy Ieres Rae =.5 =.4 =.3 =. = This dig has may srog implicaios i paricular o he real moeary policy. The ceral bak does o eed o hik of a weighed average of loa ieres raes whe he marke share is o so di ere bewee wo ypes of loa coracs. The ceral bak should pu is prioriy o he loa ieres rae wih more sickiess o achieve a good policy. Oe reaso of his is ha he weigh o sicky loa ieres rae chage i he welfare, L, is larger ha ha o less sicky loa ieres rae chage, S. Aoher reaso is ha he same scale shock makes larger ucuaio i more sicky loa ieres rae ha i less sicky loa ieres rae. Bu, his oucome does o hold whe he share of he sicly loa is eough small. I his case, he ceral bak should pay aeio o boh siky ad less sicky loas. 7

28 6 Taylor (8) or Cúrdia ad Woodford (8) or Oher? Taylor (8) pois ou ha he FRB seems o deviae from he heoreical ieres rae pah of so called Taylor rule sice he FRB have egaively reaced o he credi spread i he moey marke for he las few years o simulae ecoomy. We ierpre his poi i our model i pariculer o wheher or o he ceral bak should respod o such a credi spread i he acial marke. To see a clearer e ec of heerogeeiy of loa coracs i erms of he moeary policy, we furher assume ' S = ad ' L. I his case, he less sicky loa baks chage heir loa ieress rae every period. The purpose of he ceral bak is give by X X UT ' + x (x x ) + L ( R b L; RL; b ) + brl; LS Sb 3 i ; = = where we use equaio (3). This implies ha a ceral bak has o pay aeio o he spread bewee he policy ieres rae ad he ieres rae which has some lags o cach up o he policy ieres rae. I his case, he erm of loa ieres rae chage i less sicky loa is zero, so he heerogeeous loa ieres rae chages is all i he erm of ha i sicky loa. Furhermore, whe we ierpre he policy rae as he riskless ieres rae ad he loa ieres rae as he risky rae wih some premium shocks, a ceral bak has o reac o he credi spread bewee risky rae ad riskless ieres rae. This implicaio srogly suppors he idea of McCulley ad Toloui (8), Taylor (8) ad Taylor ad Williams (8), which d ha he Federal Reserve Bak reaced o he credi spread. 3 3 Taylor (8) ad Taylor ad Williams (8) use he credi spread bewee hree moh LIBOR rae ad hree moh OIS rae (overigh idex swap rae as expeced overigh federal fuds raes). They also sugges ha he credi spread is o eough o explai he deviaio of Fed Fud rae from he Taylor rule. This lack of explaaio ca be compesaed by he erm of heerogeeous rs di ereces of loa ieres raes ha are iduced by heerogeeiy of loa ieres raes. (36) 8

29 Uder heerogeeous ieres raes, i is he opimal moeary policy o acively reac o heerogeeiy of loa ieres raes iducig he credi spread. Cúrdia ad Woodford (8), however, sugges ha a ceral bak eveually does o eed o respod o credi spread by showig ha he opimal moeary policy i he basic NK model wihou credi spread sill quaiaively provides a good approximaio o he opimal moeary policy i he NK model wih credi spread. We ivesigae his poi i our framework. We compare he di erece of impulse resposes i wo rules. Oe rule is he opimal moeary policy give by equaio (34) ha correcly capure he heerogeeiy of loa ieres raes. Aoher rule, he ear opimal rule A, is oe ha igores oly credi spread of loa ieres raes, i.e. LS =. We use he parameer values lised i Table 3 excep ' S = ad assume uexpeced oe perceage credi spread shock wih persisece.9. Thus, he less sicky loa ieres rae is exible. Figure 4 shows he simulaio oucomes. I his case, he pahes i wo rules are o di ere each oher. This implies ha he rule wihou aeio o he credi spread ca be a good approximaio o quaiaively achieve he opimal moeary policy, which is cosise wih Cúrdia ad Woodford (8) ha also poi ou quaiaively less imporace of aeio o he credi spread iself. Thus, he credi spread oly ha is oe appearace of he acial marke heerogeeiy as show above is o so impora. Fially, we compare he impulse resposes i he opimal moeary policy give by equaio (34) ad i he rule, called he ear opimal rule B, ha igore whole heeregeeiy of loa ieres raes i persuiig o miimize he welfare give by equaios (33). I pariculer, we assume ha a ceral bak misudersads ha here is oly less sicky loa ieres rae i he ecoomy. Thus, he ear opimal moeary policy rule is give by he sadard opimal moeary policy ha keeps a liear relaio bewee he i aio rae ad chage i he oupu gap. Figure 5 shows he simulaio oucomes. We ca co rm 9

30 ha he sadard opimal moeary policy ca o be a good approximaio o he opimal moeary policy. Therefore, we suppor he coclusios by Taylor (8) or Cúrdia ad Woodford (8), bu we isis ha whole heerogeeiies, o oly credi spread, i acial markes are very crucial o achieve he opomal moeary policy. Figure 4. Impulse Respose i Opimal Rule ad Near Opimal Rule A Iflaio Opimal Near Opimal A 5 5 Oupu Gap Opimal Near Opimal A Loa Rae Less Sicky Loa 5 (Opimal) Sicky Loa (Opimal) Less Sicky Loa (Near Opimal A) Sicky Loa (Near Opimal A) Policy Ieres Rae Opimal Near Opimal A 3

31 Figure 5. Impulse Respose i Opimal Rule ad Near Opimal Rule B Iflaio Opimal Near Opimal B 5 5 Oupu Gap Opimal Near Opimal B Loa Rae Less Sicky Loa 5 (Opimal) Sicky Loa (Opimal) Less Sicky Loa (Near Opimal B) Sicky Loa (Near Opimal B) Policy Ieres Rae Opimal Near Opimal B 7 Cocludig Remarks TBA. 3

32 Refereces [] Bak of Japa. Fiacial Sysem Repor Repors ad Research Papers, March 7. [] Bak of Japa. Fiacial Sysem Repor Repors ad Research Papers, March 8. [3] Barh M. J ad Ramey V. A. The Cos Chael of Moeary Trasmissio, NBER Workig Paper Series 7675,. [4] Berger, A. ad G. Udell. Some Evidece o he Empirical Sigi cace of Credi Raioig. Joural of Poliical Ecoomy, Vol., 99, pp [5] Calvo, G. Saggered Prices i a Uiliy Maximizig Framework. Joural of Moeary Ecoomics, Vol., 983, pp [6] Cúrdia, V. ad M. Woodford. Credi Fricio ad Opimal Moeary Policy. Mimeo, 8. [7] De Graeve, F., O. De Joghe ad V. Rudi Vader. Compeiio, Trasmissio ad Bak Pricig Policies: Evidece from Belgia Loa ad Deposi Markes, Joural of Bakig & Fiace, vol. 3, 7, pp [8] Gambacora, L How Do Baks Se Ieres Raes?, NBER Workig Paper Series 95, 8. [9] Io T. ad K. Ueda. Tes of he Equilibrium Hypohesis i Disequilibrium Ecoomerics: A Ieraioal Compariso of Credi Raioig, Ieraioal Ecoomic Review, vol., 98, pp

33 [] Kashyap A. K. ad J. C. Sei, Wha Do a Millio Observaios o Baks Say abou he Trasmissio of Moeary Policy?, America Ecoomic Review, America Ecoomic Associaio, vol. 9,, pp [] McCulley, P. ad Rami Toloui. Chasig he Naural Rae Dow: Fiacial Codiio, Moeary Policy, ad he Taylor Rule. Global Ceral Bak Focus, PIMCO, February, 8. [] Ravea, F. ad C. Walsh. Opimal Moeary Policy wih he Cos Chael. Joural of Moeary Ecoomics, Vol. 53, 6, pp99-6. [3] Roemberg, J. ad M. Woodford. A Opimizaio-based Ecoomeric Framework for he Evaluaio of Moeary Policy. NBER Macroecoomics Aual, Vol., 997, pp [4] Slovi, M. ad M. Sushka. A Model of he Commercial Loa Rae. Joural of Fiace, Vol. 38, 983, pp [5] Sørese, K. C. ad T. Werer. Bak Ieres Rae Pass-hrough i he Euro Area: A Cross Coury Compariso. Joural of Ieraioal Bakig ad Fiace, forhcomig. [6] Taylor, J. Moeary Policy ad he Sae of he Ecoomy. Tesimoy before he Commiee o Fiacial Services U.S. House of Represeaives, 8. [7] Taylor, J. ad J. C. Williams. A Black Swa i he Moey Marke. NBER Workig Paper Series, No. 4, 8. [8] Teraishi, Y. Opimal Moeary Policy uder Saggered Loa Coracs, Mimeo, 8. 33

34 [9] Weh, M. A., The Pass-hrough from Marke Ieres Raes o Bak Ledig Raes i Germay, Discussio Paper Series : Ecoomic Sudies,, Deusche Budesbak, Research Cere. [] Woodford, M. Ieres ad Prices: Foudaio of a Theory of Moeary Policy. Priceo Uiversiy Press, 3. [] Yu, T. Nomial Price Rigidiy, Moey Supply Edogeeiy, ad Busiess Cycles. Joural of Moeary Ecoomics, Vol. 37, 996, pp

35 Appedix A Derivaio of Loss Fucio I his subsecio, we derive a secod order approximaio o he welfare fucio (all deails of hese derivaios ad explaaios are i Appedix B). I derivaio of approximaed welfare fucio, we basically follow he way of Woodford (3). Excep x ad, log-liearized versio of variable m is expressed by bm = l(m =m), where m is seady sae value of m. 4 Uder he siuaio i which goods supply maches goods demad i every level, Y = C ad y (f) = c (f) for ay f, he welfare crierio of cosumer is give by ( ) X E UT ; = where ad UT = U(C ) Z V (l (h))dh Z V (l (h))dh; (37) Z Y y (f) df : We log-liearize equaio (37) sep by sep o derive a approximaed welfare fucio. Firsly, we log-liearize he rs erm of equaio (37). 4 You ca see Woodford (3) abou how o log-liearize a fucio.

36 UT (Y ; ) = U + U c e Y + U + U cc e Y + U c e Y + U + Order(k k 3 ) = U + Y U c ( b Y + b Y ) + U + U ccy b Y + Y U c b Y + U + Order(k k 3 ) h Y U c + Y U cc i by Y U cc g b Y + :i:p + Order(k k 3 ) = Y U cy b + = Y U c by + ( ) Y b + g Y b + :i:p + Order(k k 3 ); (38) where U U(Y ; ), e Y Y Y, :i:p meas he erms ha are idepede from moeary policy, Order(k k 3 ) expresses order erms higher ha he secod order approximaio, Y Ucc U c >, ad g Uc Y U cc. To replace e Y by b Y l(y =Y ), we use he Taylor series expasio o Y =Y i he secod lie as Y =Y = + b Y + b Y + Order(k k 3 ): Secodly, we log-liearize he secod erm of equaio (37) by a similar way. Z V (l (h); )dh = V l L(E h b l (h) + E h( b l (h)) ) + V lll E h ( b l (h)) + V l L E h b l (h) +:i:p + Order( k k 3 ) = LV l b L + ( + )b L e L b + ( + )var b hl (h) +:i:p + Order( k k 3 ) (39) where e V l V l LV ll, LV ll, h Y Lf,! p ff (f ), q ( +! )a +! e, a l A, var h b l (h) is he variace of b l (h) across all ypes of labor, ad var f bp (f) is he variace of bp (f) across all di ereiaed good prices. Here he de iio of labor sub-aggregaor is give by L " Z # l (h) dh ;

37 ad so we have b L = E h b l (h) + var b hl (h) + Order(k k 3 ) i he secod order approximaio. We use his relaio i he secod lie. Thirdly, we log-liearize he hird erm of equaio (37) by a similar way. Z V (l (h); )dh = V l L(E h b l (h) + E h (b l (h)) ) + V lll E h ( b l (h)) + V l L E h b l (h) +:i:p + Order( k k 3 ) = LV bl l + ( + )b L e b L + ( + )var b hl (h) +:i:p + Order( k k 3 ) (4) Here he de iio of labor sub-aggregaor is give by " Z L l h # dh ; ad so we have b L = E h b l (h) + var b hl (h) + Order(k k 3 ) i he secod order approximaio. We use his relaio i he secod lie. The, from equaio (39) ad equaio (4), we have Z " = LV l +:i:p + Order( k k 3 ) = LV l +:i:p + Order( k k 3 ) V (l (h); )dh + Z V (l (h); )dh b L + ( + )b L e b L + ( + )var h b l (h) +( ) b L + ( + )b L ( )e b L + ( + )var h b l (h) 4 b L e + + b L e e h b e L + ( ) + bl b L + ( + )var h b l (h) + ( + )var h b l (h) 3 5 # (4) where we use b e L = b L + ( ) b L ; 3

38 from equaio (6). The, we use he codiio ha he demad of labor is equal o he supply of labor as el = Z el (f)df = Z f ( y (f) )df; A where he producio fucio is give by y (f) = A f(l (f)), where f() is a icreasig ad cocave fucio. By akig he secod order approximaio, we have b e L = h ( b Y a ) + ( +! p h ) h ( b Y a ) + ( +! p)var f bp (f) + Order(k k 3 ); where we log-liearize he demad fucio o di ereiaed goods o derive he relaio var f l y (f) = var f l p (f), which ca be derived from he cosumer s cos miimizaio problem uder Dixi-Sigliz aggregaor, as p (f) y (f) = Y ; P h R i where he aggregae price idex is give by P p (f) df. Also, we use he relaio of h =! w ad! =! p +! w, where! w is a elasiciy of real wage uder he exible-wage labor supply wih respec o aggregae oupu. We ca rasform equaio 4

39 (4) as: Z = h LV l +:i:p + Order( k k 3 ) = h LV l +:i:p + Order( k k 3 ): V (l (h); )dh Z V (l (h); )dh by + ( +!)b Y!q b Y + ( ) + + ( +! p)var f l p (f) + h + h bl b L ( + )var h l l (h) ( + )var h l l (h) by + ( +!)b Y!q b Y +( ) + b R L; RS; b + ( +! p)var f l p (f) + h ( + )var h l l (h) + h ( + )var h l l (h) (4) From he secod lie o he hird lie, we use followig rasformaios: bl b L = b b = R b L; RS; b = R b L; RS; b + bl Z w (h)dh L b ; Z w (h)dh where L (+r L ) + L r L ad S (+r S ) + S r S. There we use log-liear relaios from equaio (7), equaio (8), equaio (5), ad equaio (6) ad he de iios from equaio (),equaio (3), equaio (4), equaio (5), equaio (), ad equaio (). Furhermore, we ca replace h LV l by ( Z +:i:p + Order( k k 3 ); V (l (h); )dh + Z +( ) + )Y U c as: V (l (h); ) ( ) Y b + ( +!)b Y!q Y b + ( +! 3 p)var f l p (f) 6 = Y U c 4 + h ( + )var h l l (h) + h ( + )var h l l (h) 7 R b 5 L; RS; b + (43) 5

40 Here, we use he assumpio ha disorio of he oupu level, iduced by rm s price mark up hrough 8 < Z ( : V l [l T (h)] U C (C e L ;T ;T (f) ) = dh9 ; 8 < Z ( Vl lt h : U C (C e L ;T ;T (f) ) = dh9 which would exis uder exible price ad o role of moeary policy is of order oe as i Woodford (3). 5 (44) Thus, i erms of he aural rae of oupu, we acually assume ha real margial cos fucio of rm Z() i order o supply a good f is give by ; 8 < Z Z(f) = Z(y (f); Y ; r ; ) = + L r (h) : ( V l [l T e ) L ;T (f) = dh9 U C (C ; ;T (f) ; 8 < Z + S r h ( Vl lt e ) L ;T (f) = dh9 : U C (C ; ;T (f) ; he he aural rae of oupu Y Z(Y ; Y ; R; ) = = Y ( ) is give by = + L R + S R ( ) (45) where a parameer expresses he disorio of oupu level ad is of order oe. 6 we ca combie equaio (38) ad equaio (43), 5 We assume ha he moeary policy has o impac o he level of he aural rae of oupu. 6 By assumig a proper proporioal ax o sales as Z(Y ; Y ; R; ) = we ca iduce = as i Roemberg ad Woodford (997). ( ) = + L R + S R ( ); ; The 6

41 6 U = Y U c 4 Y b ( +!) Y b + ( g +!q ) Y b var f l p (f) lvar h l l (h) lvar h l l (h) +( ) + + b R L; b RS; +:i:p + Order( k k 3 ) ( +!)(x x ) 3 + var f l p (f) = Y U 6 c 4 + l var h l l (h) + ( ) l var h l l (h) 7 +( ) + b 5 R + L; RS; b +:i:p + Order( k k 3 ); (46) where ( +! p ), l h ( + ), x b Y b Y, ad x l(y =Y ). Here Y is a soluio i equaio (45) whe =, which is called as a e cie level of oupu as de ed i Woodford (3a). I he secod lie, we use he log-liearizaio of equaio (45) as ad he relaio as by l(y =Y ) = g +!q +! ; l(y =Y ) = ( +!) + Order(k k); which is give by he relaio bewee he e cie level of oupu ad he aural rae of oupu i erms of oe by equaio (44). This expresses ha he perceage di erece bewee Y ad Y is idepede from shocks i he rs order approximaio. I agai oes ha we assume ha is of order oe. To evaluae var h b l (h) ad var h b l (h), we use he opimal codiio of labor supply ad he labor demad fucio give by equaio (9), equaio (), equaio (5), ad equaio (6). By log-liearizig hese equaios, we ally have a followig relaio var h l l (h) = var h l( + r (h)) + Order(k k 3 ); 7

42 var h l l (h) = var h l( + r (h)) + Order(k k 3 ); where ( ( +) + ) ad ( ) ( ( +) + ). The, equaio (45) is rasformed io UT = ( +!)(x x ) 3 + var f l p (f) Y U 6 c 4 + r var h l( + r (h)) + ( ) rvar h l( + r (h)) 7 +( ) + b 5 R + L; RS; b +:i:p + Order( k k 3 ); (47) where r l = h (+) ( ( +) +) ad r l = h (+)( ) ( ( +) + ). The remaiig work o derive he approximaed welfare fucio is o evaluae var h l p (f) ad var h l( + r (h)) i equaio (47). Followig Woodford (3a), we de e P E f l p (f); The we ca make followig relaios 4 var f l p (f): P P = E f l p (f) P = E f l p (f) P + ( )Ef l p (f) P = ( )E f l p (f) P ; (48) ad we also have 8

43 4 = var f l p (f) P l o = E f p (f) P (E f l p (f) P ) l o l = E f p (f) P + ( )E f p o (f) P = 4 + ( )E f l p (f) P o (P P ) (P P ) = 4 + ( )(var f (l p (f) P ) + E f l p (f) P ) (P P ) = 4 + (P P ); (49) where we use equaio (48) ad p (f) is a opimal price seig by he age f followig he Calvo (983) - Yu (99) framework. I oes ha all projec groups re-se he same price a ime whe hey are seleced o chage prices, because he ui margial cos of producio is same for all projec groups. Also, we have a followig relaio ha relaes P wih P P = l P + Order(k k ); where Order(k k ) is order erms higher ha he rs order approximaio. Here we h R i make use of he de iio of price aggregaor P p (f) df. The equaio (49) ca be rasformed as where l P P. From equaio (5), we have 4 = 4 + ; (5) ad so 4 = X s= s s; 9

44 X 4 = = ( )( ) X + :i:p + Order(k k 3 ): (5) = To evaluae var h l( + r (h)), we de e R L; ad 4 R as R L; E h l( + r (h)); The, we ca make followig relaios 4 R var h l( + r (h)): R L; R L; = E h l( + r (h)) R L; = ' L E h l( + r (h)) R L; + ( ' L ) l( + r ) R L; = ( ' L ) l( + r (h)) R L; ; (5) ad 4 R = var h l( + r (h)) R L; l( o = E h + r (h)) R L; (E h l( + r (h)) R L; ) l( = ' L o E h + r (h)) R L; + ( ' L ) l( + r ) R L; (R L; R L; ) = ' L 4 R + 'L ' L (R L; R L; ) ; (53) where we use equaio (5). Also, as i he discussio o price, we have R L; = l( + R L; ) + Order(k k ); (54)

45 where we make use of he de iio of he aggregae loa raes + R L; R q (h) Q ( + r (h))dh. The, from equaio (53) ad equaio (54), we have 4 R = ' L 4 R + ' ' ( b R L; b RL; ) ; (55) where b R L; l +R L; +r. From equaio (55), we have The, we have 4 R = ' L + 4 R + X s= ' ' s ( R ' b L;s RL;s b ) : X 4 R = = ' L ( ' L )( ' L ) X ( R b R b ) + :i:p + Order(k k 3 ): = We have a similer relaio for he less sicky loa. rasformed as: The, equaio (47) ca ally be X UT ' = X + x (x x ) + L ( R b L; RL; b ) + S ( R b S; RS; b ) + LS R b L; RS; b =! ; where Y u c, (+! ( )( ) p), x ( +!), L h (+)( L (+r L ) ) ( + + L r L ( +) ' ) L, ( ' L )( ' L ) S ( ) h ( + )( S (+r S ) ) ( ' + ) S, ad + S r S ( +) ( ' S )( ' S ) LS ( ) +. + Fially, by assumig =, we have X UT ' = where LS LS. X + x (x x ) + L ( R b L; RL; b ) + S ( R b S; RS; b ) + brl; LS RS; b =! ; (56)