Strategic Complementarities and Optimal Monetary Policy

Transcription

1 Sraegic Complemenariies and Opimal Moneary Policy Andrew T. Levin, J. David Lopez-Salido, and Tack Yun Board of Governors of he Federal Reserve Sysem Augus 2006 In his paper, we show ha sraegic complemenariies such as rm-speci c facors or quasikinked demand have crucial implicaions for he design of moneary policy and for he welfare coss of oupu and in aion variabiliy. Recen research has mainly used log-linear approximaions o analyze he role of hese mechanisms in amplifying he real e ecs of moneary shocks. In conras, our analysis explicily considers he nonlinear properies of hese mechanisms ha are relevan for characerizing he deerminisic seady sae as well as he second-order approximaion of social welfare in he sochasic economy. We demonsrae ha rm-speci c facors and quasi-kinked demand curves yield markedly di eren implicaions for he welfare coss of seady-sae in aion and in aion volailiy, and we show ha hese consideraions have dramaic consequences in assessing he relaive price disorions associaed wih he Grea In aion of JEL Classi caion Sysem: E3, E32, E52 Keywords: rm-speci c facors, quasi-kinked demand, welfare analysis Acknowledgemens: We appreciae commens and suggesions from Susano Basu, Larry Chrisiano, John Fernald, Jinill Kim, John Williams, Michael Woodford, and paricipans in seminars a Columbia Universiy, he NBER-SI, he San Francisco Fed, and he Bank of Finland. The opinions expressed here are solely hose of he auhors and do no necessarily re ec he views of he Board of Governors of he Federal Reserve Sysem or of anyone else associaed wih he Federal Reserve Sysem. Division of Moneary A airs, Mail Sop 7, Federal Reserve Board, Washingon, DC 2055 USA phone ; fax andrew.levin@frb.gov, david.lopez-salido@frb.gov, ack.yun@frb.gov

2 Inroducion The New Keynesian lieraure has emphasized he role of sraegic complemenariies also referred o as real rigidiies in reducing he sensiiviy of prices wih respec o marginal cos and hereby amplifying he real e ecs of moneary disurbances. Several forms of sraegic complemenariy including rm-speci c facors, inermediae inpus, and quasikinked demand have observaionally equivalen implicaions for he rs-order dynamics of aggregae in aion. However, here has been relaively lile analysis of he nonlinear characerisics of hese mechanisms ha may be relevan for deermining he seady-sae properies of he economy and for assessing he welfare coss of sochasic ucuaions. In his paper, we show ha he speci c formulaion of sraegic complemenariy has crucial implicaions for he design of moneary policy and for he welfare coss of oupu and in aion variabiliy. In conducing his analysis, we formulae a dynamic general equilibrium model ha incorporaes boh quasi-kinked demand and rm-speci c facors. We follow Kimball (995) in specifying a generalized aggregaor funcion ha allows for a non-consan elasiciy of demand while nesing he Dixi-Sigliz aggregaor as a special case. In addiion, our speci caion of he producion funcion encompasses a general degree of rm-speci ciy of boh capial and labor, ha is, he proporion of variable vs. xed inpus of each facor used by each individual rm. In calibraing he overall degree of real rigidiy, we consider several disinc combinaions of he srucural parameers ha yield he same slope of he New Keynesian Phillips Curve (NKPC) and hen proceed o deermine he exen o which hese alernaive calibraions in uence he nonlinear properies of he model. Our seady-sae analysis shows ha quasi-kinked demand and rm-speci c inpus have markedly di eren implicaions for he coss of deerminisic in aion and for he degree o which he opimal seady-sae in aion rae under he Ramsey policy di ers from ha of he Friedman rule. 2 In doing so we derive a non-linear expression for he evoluion Following Kimball (995), many auhors have analyzed he implicaions of sraegic complemenariies for equilibrium in aion dynamics, such as Woodford (2003, 2005), Alig, Chrisiano, Eichenbaum and Linde (2005), and Dosey and King (2005a,b). Mos of hese mechanisms are reminiscen from he lieraure on nominal and real rigidiies originaed wih he seminal work of Ball and Romer (990) and surveyed by Blanchard (990) and Blanchard and Fisher (989). 2 We follow Khan, King and Wolman (2003), and more recenly Schmi-Grohe and Uribe (2005 a,b),

3 of he relaive price disorion and average markup under each source of sraegic complemenariies. The di eren naure of he sraegic linkage among rm s incenive o changes price is a he core of he asymmeric resuls ha we emphasize in his paper. Thus, if he source of real rigidiy is coming from he presence of quasi-kinked demand, he e ecs of negaive in aion ends o dramaically shrink he pro s of non-adjusing rms by moving consumers demand away from is producs o oher. If, on he conrary, here is a fracion of xed facors, hen he higher he seady sae in aion he higher are he oupu coss associaed wih he presence of price dispersion. To characerize he welfare implicaions of real rigidiies in he sochasic economy, we follow he linear-quadraic approach of Woodford (2003) in deriving he second-order approximaion of condiional expeced household welfare. 3 For any given combinaion of nominal and real rigidiies, we nd ha he welfare coss of in aion variabiliy are an order of magniude smaller when he real rigidiy arises from quasi-kinked demand raher han rm-speci c facors. Thus, he characerisics of opimal moneary policy also depend crucially on he paricular form of real rigidiy. The nal sage of our analysis gauges he welfare coss of he Grea In aion by using he observed ime series for U.S. in aion o consruc he corresponding sequence of relaive price disorions under alernaive assumpions abou he form of sraegic complemenariy. Given a moderae degree of nominal rigidiy (namely, an average duraion of 2-/2 quarers beween price changes), we deermine he degree of facor speci ciy or quasi-kinked demand needed o mach he esimaed slope of he NKPC. In he case of quasi-kinked demand, he high and volaile in aion of only generaes a modes degree of ine ciency arising from relaive price dispersion. In conras, he case of rm-speci c facors yields dramaically higher welfare coss: in his case, he Grea In aion generaes relaive price disorions ha reduce he level of aggregae oupu by 0 percen or more. Levin and Lopez-Salido (2004), and Levin e al. (2005) in using Lagrangian mehods o obain he rsorder condiions of he underlying Ramsey problem o compue opimal long-run policy under commimen in disored economies. 3 Woodford (2003, 2005) uses second-order approximaions o characerize he welfare implicaions of rm-speci c inpus bu does no consider he case of quasi-kinked demand. 2

4 Before proceeding furher i is useful o brie y examine he NKPC under he assumpion of Calvo-syle saggered price seing, = E f + g + p mc ; () where is he in aion rae and mc is he logarihmic deviaion of real marginal cos from is seady-sae value. Noice ha he slope of he NKPC is expressed as he produc of wo coe ciens: p re ecs he degree of nominal rigidiy, and re ecs he degree of sraegic complemenariy in price-seing behavior. When he value of p is calibraed using microeconomic evidence suggesing relaively frequen price adjusmen, hen a small value of (corresponding o a high degree of real rigidiy) is needed o accoun for he low esimaed slope of he NKPC. In a nushell, our analysis indicaes ha alernaive forms of sraegic complemenariy may yield he same value of bu have markedly di eren implicaions for moneary policy and welfare. 4 The remainder of his paper is organized as follows. Secions 2 and 3 describe our speci caions for quasi-kinked demand and rm-speci c inpus, respecively, elaboraing on he nonlinear characerisics as well as he implicaions for he degree of real rigidiy in price-seing behavior. In Secion 4 describes how do we calibrae he degree of real rigidiies. Secion 5 evaluaes he coss of seady-sae in aion associaed wih hese forms of sraegic complemenariy. Secion 6 uses linear-quadraic mehods o characerize he social welfare funcion and he properies of opimal moneary policy in he sochasic economy. Secion 7 considers he exen o which hese sraegic complemenariies have markedly di eren implicaions regarding he coss of he Grea In aion. Secion 8 concludes. Finally, in he appendix A we presen he deails on how o calibrae he curvaure of he demand curve and we relae i wih he preceding lieraure; in Appendix B we presen he key derivaions of he paper. 4 Some papers have emphasized how o damp ucuaions in marginal coss hrough elasic supply mechanisms. Among hose are he possibiliy ha he rms can adjus is capaciy uilizaion, he exisence of an elasic labor supply (Dosey and King (2005b)). Alernaively, allowing for sicky price and sicky nominal wages also end o generae persisen responses in real marginal coss in response o nominal shocks (see, e.g., Chrisiano, Eichenbaum and Evans (2005)). We do no consider hese mechanisms in his paper. 3

5 2 Quasi-kinked Demand In his secion, we describe an economy where he producion of nal goods requires a coninuum of di ereniaed goods, indexed by a uni inerval, and a single monopolisic compeior produces each ype of hese di ereniaed goods. In order o generae sraegic complemenariies in price-seing, we begin wih a Kimball-ype of household preference for di ereniaed goods and hen move ono a producion funcion emphasizing he role played by rm-speci c xed-capial Demand srucure The represenaive household seeks o maximize E 0 P =0 discoun facor. The household s uiliy in period has he form U = C N ( M P ) 0 U, where 2 (0; ) is he where C is an aggregaor of he quaniies of he di eren goods consumed by households ha i will be de ned laer, N denoes hours worked, and M P (2) denoes is real balances, and he parameer > 0 capures risk aversion aiudes; 0 > 0, and 0 is he inverse of he Frisch labor supply elasiciy; and nally, 0 0, and > 0 is relaed o he semielasiciy of real balances o (gross) nominal ineres raes. Laer on will become clear why do we allow for money balances o direcly in uence household uiliy. 6 We assume ha he economy is populaed by a coninuum of monopolisically compeiive rms producing di ereniaed inermediae goods. These goods are hen used as inpus by a (perfecly compeiive) rm producing a single nal (consumpion) good. Following Kimball (995) we assume ha each rm faces an endogenous demand elasiciy ha dampens is incenive o raise is price in response o an increase in is marginal cos of producion. Formally, he nal good is produced by a represenaive, perfecly compeiive, rm wih he following general echnology R G(e Y 0 (j)) dj = ; where Y e (j) = Y(j), and Y (j) is 5 In he Appendix we also describe how he model works once we allow for he exisence of inermediae (maerials) inpus in he producion of di ereniaed goods. 6 The cashless economy corresponds o he limiing case in which 0 becomes arbirarily small. In he Appendix we describe he fairly sandard rs order condiions associaed o his problem. 4 Y

6 Figure : Quasi-Kinked Demand he quaniy of inermediae good j used as an inpu. The funcion G sais es ha G 0 > 0, G 00 < 0, and G() =. The nal good rm chooses inpu demands Y (j) o maximize pro s, subjec o he previous echnological consrain. 7 While hese general assumpions are su cien for obaining a rs order approximaion, our analysis requires a speci c choice of funcional form for he aggregaor, G. Thus, following Dosey and King (2004), we consider he following aggregaor: G( e Y ) = + h ( + ) e Y i + (3) where he composie parameer = (( + ))=(( + ) >. ), and he elasiciy parameer The parameer deermines he degree of curvaure of he rm s demand curve. When = 0, he demand curve exhibis consan elasiciy, as in he Dixi-Sigliz formulaion. When < 0, each rm faces a quasi-kinked demand curve; in e ec, consumers have a saiaion level of demand for each good, so ha a drop in is relaive price only simulaes a small increase in demand, while a rise in is relaive price generaes a large drop in demand. In his paper, we will focus on non-negaive values of his parameer; however, i is ineresing o noe ha when > 0, consumers have a subsisence level of demand for each good, implying ha pricing decisions exhibi sraegic subsiuabiliy. Given expression (3) he soluion of he rm problem yields he se of demand schedules given by where e P (j) = P(j) P ey (j) = + h ep (j) (+ ) (+ ) +, P is he aggregae price level and P (j) corresponds o he inerme- R e P 0 (j) (+ ) (+ ) dj. Afer diae goods price; and he Lagrange muliplier = imposing a zero pro condiion, hen he aggregae price index can be wrien as follows: 7 See for deails Woodford (2003), Dosey and King (2005a), Eichenbaum and Fisher (2004), and Klenow and Willis (2005). i (4) 5

7 = + implies Y = C. R e P 0 (j) (+ ) (+ ) R dj + + P e 0 (j) dj. Finally, he marke-clearing condiion In Figure we plo he log of relaive demand for alernaive values of ; for his purpose, we calibrae he demand elasiciy parameer = 7, which yields a markup of 6 percen in he seady sae wih zero in aion. 8 when = 0, corresponding o he Dixi-Sigliz formulaion. Of course, he demand curve is log-linear The value of = 2 falls in he lower end of he range considered by Eichenbaum and Fisher (2004); in his case, he demand curve exhibis quie srong curvaure. Finally, = 8 implies a very high degree of curvaure approaching ha of a ruly kinked demand curve. Thus, he presence of quasi-kinked demand implies ha a drop in he rms relaive price only simulaes a small increase in demand, while a rise in is relaive price generaes a large drop in demand. Tha is, consumers will cosesly move away from relaive expensive goods bu do no run ino inexpensive ones. The producion funcion for a ypical inermediae goods rm j is given by: Y (j) = A K (j) N (j) (5) where A represens an exogenous oal facor produciviy shifer, K (j) and N (j) represen he capial and labor services hired by rm j, and he parameer represens he shor run elasiciy of oupu o capial. In his secion, we implicily assume ha boh inpus can be perfecly reallocaed across rms, so he model corresponds o he one considered by Erceg, Henderson and Levin (2000) and Chrisiano, Eichenbaum and Evans (2005). In he nex secion we exend furher he model in such a way ha he capial sock is xed a he rm level. We now urn o he comparison of he demand curve speci ed in (4) wih he sandard Dixi-Sigliz ype of preferences. The prooypical demand curves under he Dixi-Sigliz ype can be derived by seing = 0, which in urn implies ha he muliplier = in equaion (4), so ha he elasiciy of demand is consan across rms, and i is deermined by he elasiciy of subsiuion among di ereniaed goods. Under he quasi-kinked de- 8 As noed in he Appendix, he calibraion of he curvaure of demand depends crucially on he assumpion abou he seady sae markup. 6

8 mand expression (4), he demand elasiciies of di ereniaed goods vary wih heir relaive demands. Formally, i can be easily shown ha he elasiciy of demand for good j, denoed by ( e Y j ), can be wrien as follows: ( e Y j ) = + e Y j : (6) In he absence of a producion subsidy, he desired markup of individual rms is given as ( e Y j ) ( e Y j ) ( e Y j ), and depends on he rm s relaive demand, e Y j. In general, in a non-zero seady sae in aion, and under < 0, he elasiciy ( e Y ) is decreasing in he relaive demand. Hence, a increase in nominal demand ha increases marginal coss will end o reduce rm s desired markup so reducing he incenives o increase prices in response o he changes in demand. Noice ha for = 0, he previous expression corresponds o he sandard Dixi- Sigliz demand funcion, () = =, where he desired markup is consan and a funcion of he parameer. Nowihsanding, once we deparure from he consan elasiciy of demand, by calibraing he seady sae markup is no enough o pin down he degree of curvaure of he demand funcion,. 9 We will urn o his issue in secion The Firm s Price-Seing Decision Because of he presence of marke power, inermediae rms are assumed o se nominal prices in a saggered fashion, according o he sochasic ime dependen rule proposed by Calvo (983). Each rm reses is price wih probabiliy each period, independenly of he ime elapsed since he las adjusmen. Thus, each period a measure of producers rese heir prices, while a fracion keep heir prices unchanged. 0 Given he assumpion of perfec facor mobiliy across rms, all rms have he same real marginal cos, which is given by he raio of he real wage o he marginal produc of 9 The reason is ha he calibraion of only involves second order derivaives of G, while higher order derivaives will be necessary o undersand he implicaions of he curvaure of demand for price adjusmen. See below for deails. 0 We do no assume an indexaion clause for hose rms ha can no reopimize is price. This is in line wih recen wih micro evidence for he U.S. (Bils and Klenow, 2004) and various European counries (Alvarez e al. (2006)). 7

9 Table : Quasi-Kinked Demand and Price-Seing Behavior ep = Z2 + p Z + (+ ) Z3 Z ep +(+ ) Z = E f (+ ) + Z + g + Y (+ ) Z 2 = E f (+ ) + Z 2+ g + Y Z 3 = E f + Z 3+ g + Y (+ ) MC labor, i.e. MC = w N =( )Y. In he nex secion we analyze he e ecs of relaxing his assumpion. Table indicaes he rs-order condiions for each rm ha reses is price conrac in a given period. The opimal price is denoed by P, and he relaive price P e = P P. The rm s opimal price depends on he aggregae gross in aion rae = P =P, aggregae real marginal cos MC, and aggregae demand Y. The sochasic variables Z, Z 2, and Z 3 are described by recursive expressions in he able. Noe ha he erm (+ p ) represens a producion ax when p > 0 or a subsidy when p < Relaive Price Disorions The labor inpus of individual rms are linearly aggregaed o obain a measure of he aggregae labor, i.e. N = R 0 N (j)dj. Subsiuing individual gross producion funcion ino he de niion of he aggregae labor, we have a producion relaion beween he aggregae oupu and labor: Y = ( A )K N where he measure of relaive price disorion, denoed by, can be wrien as follows: = + Z 0 ( (+ ) ep (j) (+ ) + )dj: (7) The aggregae variables are he sum of homogeneous capial and labor, i.e. K = R 0 K (j)dj, N = R 0 N (j)dj. Goods marke equilibrium requires ha Y (j) = C (j), for all j 2 [0; ], and Y = R 0 Y (j)dj. 8

10 Table 2: Quasi-Kinked Demand and Relaive Price Disorions = (+ ) + (+ ) + (+ ) 2; ; + + 2; + + 3; ; = ( )( e P ) (+ ) + (+ ) ; 2; = ( )( e P ) (+ ) + (+ ) 2; 3; = ( ) e P + 3; As emphasized by Goodfriend and King (997), he relaive price disorion resuls in a missallocaion of aggregae oupu across alernaive uses of goods, so ha i appears as a echnological shifer ha reduces aggregae oupu. In Table 2 we describe he main componens of he previous de niion of price dispersion. In paricular, as described in he rs row of he able, he previous expression for can be wrien as a non-linear funcion of wo di eren weighed average measures of price dispersions, ; R 0 e P (j) (+ ) dj and 2; R 0 e P (j) (+ ) dj. Noice also ha he Lagrange muliplier can be expressed as a funcion of he 2; measure of dispersion as (+ ) follows: = 2;. The second row of he able corresponds o he relaionship beween wo measures of dispersion ha comes from he de niion of aggregae prices, where we inroduce a new measure of price dispersion, 3; R 0 e P (j) dj. Finally, he las hree rows show ha, following Yun (996), he Calvo-ype of saggered price seing allows us o wrie he hree measures of relaive price disorion in a recursive form. Noice ha he case of Dixi-Sigliz consan elasiciy of demand corresponds o = 0, Hence, 2; = =, = ;, and expression (7) corresponds o he sandard equaion linking aggregae in aion and he relaive price of he newly se prices (see e.g., Schmi-Grohé and Uribe (2005a, b)). 9

11 2.4 Implicaions for Real Rigidiies As shown in he Appendix a log linear approximaion o he price equaion of his model corresponds o (). In paricular, he frequency of price adjusmen and he exogenous discoun facor deermine he degree of nominal rigidiy: p = ( )( ) Furhermore, he degree of real rigidiy can be expressed as follows: = where is he seady-sae markup a zero in aion. Noice ha < 0 implies ha he parameer <, and he magniude of declines wih he absolue value of. 3 Firm-Speci c Marginal Coss We now consider he implicaions of assuming ha each rm has a xed allocaion of capial raher han being able o obain any desired amoun on an aggregae renal marke. In his case, he rm s real marginal cos (de aed by he aggregae price index) may di er from he average real marginal cos, and we denoe he raio as g MC (j) = MC (j)=mc. For ease of presenaion his secion assumes a Dixi-Sigliz demand srucure ( = 0), bu we will subsequenly consider he general model wih boh quasi-kinked demand and rm-speci c capial, and he equaions for he general case may be found in he Appendix. 3. The Deerminaion of Marginal Coss We exend he producion funcion considered in he previous secion so ha, for any rm j, i can be wrien as follows 2 Y (j) = A K fk K (j) vk N fl N (j) vl (8) where fk > 0, vk > 0, fl > 0, vl > 0, and fk + vk + fl + vl =. Noice ha f = fl + fk represens he oal fracion of inpu facors (capial sock and labor) ha 2 The case of inermediae inpus as an addiional source of sraegic complemenariy is analyzed in he Appendix. 0

12 remains xed a he rm level, K and N. In paricular, if fl = fk = 0, he producion funcion (8) corresponds o he one considered in he previous secion (expression (5)). Absen he consideraion of maerial inpus, and relaive o he assumpion of common facor markes, he presence of a xed facor of producion (capial) a he rm level will generae shor run decreasing reurn in oher facors (i.e. labor). This will imply ha equilibrium wages will vary across rms, and so marginal coss. In he absence of perfec reallocaion of facors across rms; he rm s marginal is increasing in is own oupu, where he elasiciy of marginal cos o oupu (given he real wages) depend upon he exisence of shor run reurns o scale in he variable facors. Formally, he deviaions of rm s marginal coss from he (average) norm will become an increasing funcion of he deviaion of he rm s oupu relaive o he average, i.e. gmc (j) = Y e f (j) f (9) wih decreasing reurns o labor, rms ha mainain a high relaive oupu will face a lower relaive marginal cos han he average. Thus, he exisence of xed facors because of he exisence of local labor and capial markes implies ha price adjusers rying o undercu ohers o boos is own demand would also raise own s marginal coss. This is he naure of he real rigidiy ha induces he adjusers o have less incenive o price up. 3.2 The Firm s Price-Seing Decision Limiing he possibiliy of reallocaion of capial across rms change he represenaion of he opimal price conrac of he rms ha are allowed o changes is price a ime. In paricular, he pro maximizaion condiion ( rs row of Table ) has he following form: + f ep f = ( + p )( ) Z2 Z (0) I is ineresing o noe ha his expression di ers o he one presened in he rs row of Table in wo respecs. Firs, he variables Z and Z 2 corresponds o he ones of he previous model under = 0. Second, he exisence of rm speci c facors maers for he lef hand side of he previous expression. Hence, i is possible o explicily solve for he

13 variable P e bu i appears raised o he power + f f, which absen he resricion of facor mobiliy, f = 0, corresponds o he baseline model usually considered in he lieraure. 3.3 Relaive Price Disorions For convenience, we normalize rm-speci c capial K =, rm-speci c labor N =, and he aggregae sock of variable capial, R K (j)dj =. Thus, he relaion beween aggregae oupu and he variable labor inpu can be expressed as follows: Y = ( A ) N vl where is a new measure of ime relaive price disorion which follows he following low of moion f = ( )( P e (j)) f + f f () Noice ha he previous expression becomes he prooypical model considered in several papers under he assumpion of f = Implicaions for Real Rigidiies Under he assumpion of xed facors a he rm level, he rs-order aggregae price dynamics coninue o be described by he NKPC given in equaion. The nominal rigidiy coe cien p is he same as de ned in Secion 2.4, bu he real rigidiy coe cien is now expressed as follows: = + : f f I is worh noing ha he higher he elasiciy of demand,, and he higher is he elasiciy of oupu o he fracion of xed inpus, i.e. f, he lower is he pass-hrough coe cien from marginal coss o prices. In paricular, under he assumpion ha f = 3, and = 7, hen = 0:22, so ha he presence of a xed facor implies ha prices will respond around a 0:22 per cen o an per cen increase in he marginal coss. While, assuming a higher elasiciy, =, and a lower elasiciy o he xed facor, say f = 2, he value of is reduced o 0:08. 2

14 4 Calibraion Table 3 describes he baseline parameer values ha we use o calibrae he model. Much of hese values closely follow hose recenly esimaed (see, for insance, Levin e al. (2005)) and hey are also in line wih mos of business cycle lieraure. We calibrae he model so ha each period corresponds o a quarer, hus we se he discoun facor = 0:99. We allow for a moderae amoun of nominal sickiness, i.e. he probabiliy of changing prices and is se equal o 0:6, which implies ha prices are xed slighly longer han wo consecuive quarers (see e.g. Bils and Klenow (2004)). When we allow for boh quasi-kinked demand and rm-speci c inpus, he slope coe cien of he NKPC akes he following form: = + f f where < 0. In he baseline Calvo model sudied in Woodford (2003) and many ohers, he parameer =, so ha assuming = 0:6 i implies a value for he slope of he NKPC of 0:27, which is higher han he esimaes in he lieraure (see, e.g. Gali,Gerler and Lopez-Salido (200), Sbordone (2002), and more recenly Eichenbaum and Fisher (2004)). In general, hose auhors nd ha he esimaes for he slope coe cien range beween 0:03 o 0:05 (see e.g. Woodford (2005) for a recen discussion on hese values.) Hence, in order o mach he aggregae esimaes wih he micro evidence on price sickiness, i is necessary a low value for he parameer. We will now urn o see how he di eren sraegic complemenariies can be se as o a cerain amoun of real rigidiies. In his paper we assume ha he slope coe cien is equal o 0:025, hen he required amoun of pass-hrough from marginal coss o prices, is around 0:09. We also assume ha he seady sae markup is around 6% (in paricular we se = 7 which implies ha = :6). Under hese assumpions, he required curvaure parameer,, o obain such a value for he pass-hrough coe cien is = 8. 3 If he only source of sraegic 3 In he Appendix we show how o relae he curvaure parameer wih he exising lieraure on quasikinked demand. 3

15 Table 3: Calibraed Parameer Values Parameer Descripion Value Discoun Facor 0.99 Risk Aversion (Frisch) Labor Supply Elasiciy Oupu elasiciy o capial 0.33 Price Elasiciy of Demand 7 Probabiliy of Changing Prices 0.60 Inverse of Money Demand Elasiciy 2 Figure 2: Quasi-Kinked Demand and Seady-Sae In aion complemenariy comes from he exisence of xed facors a he rm level, hen in order o mach he value of we need a value for f = 0:58. If we combine boh fricions, hen reducing f o 0:5 implies ha we only required much smaller amoun of curvaure for he demand coe cien, i.e. = 2, o ge he required degree of pass-hrough. 5 The Coss of Seady-Sae In aion In his secion, we solve for he non-linear seady sae of he models o compare he implicaions of he wo ypes of sraegic complemenariies for he coss of seady sae in aion hrough is e ecs on average markup and he relaive price disorion. We will use he Dixi-Sigliz model as a reference model ha helps in clarifying he disinc implicaions of alernaive sraegic complemenariies on boh he average markup and he relaive price disorions. As noed before, he calibraion we use o compare he non-linear implicaions of he models is such ha all of hem generae he same slope of he NKPC, hence hey are observaionally equivalen in erms of he rs order approximaion of in aion dynamics. In Figure 2 we plo he average markup and he relaive price disorion as a funcion of he seady sae in aion in he model wih quasi-kinked demand funcions calibraed for he wo values of, 2 and 8, discussed in he previous secion. The rs ineresing feaure of his deviaion from he sandard Dixi-Sigliz preferences is he srong asymmery 4

16 of in aion induced on boh markup and relaive price disorions. The higher he nonlineariy in he demand funcion, he higher he asymmery of negaive and posiive in aion on boh average markup and relaive price disorion. Insead of he Dixi-Sigliz model, he exisence of non-zero seady sae in aion reduces he average markup. Secondly, he exisence of seady sae de aion ends o reduce he a sronger decrease in he average markup relaive o economies wih posiive in aion rae. As noiced by King and Wolman (999), in he Dixi-Sigliz model he average markup is minimized a zero seady sae in aion markup (which corresponds o he consan desired markup, i.e. =, given our calibraion is.6 in he Figure). Neverheless, under quasi-kinked de- mand, he presence of seady sae in aion ranslae in an asymmeric way ino he desired markup of he rms adjusing prices, so ha average markup is dramaically reduced under de aion more han i is under posiive in aion. The reason is also apparen from he righ panel, which shows how he seady-sae in aion in uences he magniude of relaive price disorions. In he quasi-kinked demand environmen, he presence of seady sae de aion induces a higher cos in erms of relaive price disorions han posiive in aion. This is he side e ecs of he asymmeric demand funcions, since he presence of de aion ends o increase he relaive price of rms adjusing prices so consumers move immediaely away from hose price seers generaion a higher oupu coss. To see his, le us consider he sandard model wih economy-wide facor markes and consan elasiciy of demand. In his case, for empirically eiher posiive or negaive in aion raes, he rms adjusing prices have srong incenives o do so o capure he demand of is compeiors. The presence of quasi-kinked demand implies ha a drop in he rms relaive price only simulaes a small increase in demand, while a rise in is relaive price generaes a large drop in demand. Tha is, consumers will cosesly ee from relaive expensive goods bu do no ock ino inexpensive ones. Suppose, for insance, ha he economy is facing a seady sae posiive in aion. On he one hand, he relaive price of he non-adjusing rm s reduces dramaically wihou generaing much gains in erms of relaive demand (i.e. relaive demand becomes relaive inelasic if relaive price is below he equilibrium). On he oher hand, given he reducion in he relaive price of he non-adjusing 5

17 Figure 3: Firm-Speci c Inpus and Seady-Sae In aion rms, here are low incenives for adjusers o change prices, so o gain some relaive demand hey have o reduce he desired markup which ends o reduce he economy-wide (average) markup. The e ecs of negaive seady sae in aion ends o dramaically shrink he pro s of non-adjuser by moving consumers demand away from is producs o oher; his ranslaes he relaive price dispersion ino higher oupu coss and generaes a fall in he desired markup of rms so making he economy more compeiive. The adjusing rms have low incenive o undercu ohers since hey will face a lower elasic demand wihou boosing is own sales, hence hey reduce heir desired markups o avoid furher reducions in heir relaive sales, which also ends o reduce he average markup (see he lef panel of Figure 2). Overall, seady sae de aion induces higher relaive price disorions and end o lower he average markup, while posiive in aion generaes, in equilibrium, less relaive price dispersion. In he previous se up, facor inpus can be coslessly reallocaed across rms so ha hey can adjus heir marginal cos in responses o seady sae in aion. Figure 3 corresponds o he same exercise in he model wih xed facors, where we plo he average markup and relaive price disorions for alernaive values of he shor run elasiciy of oupu wih respec o xed facors, i.e. f. The gure makes i clear ha, relaive o he quasi-kinked demand model, his mechanism has sharply di eren e ecs on boh he average markup and he relaive price disorion facor. I should be noed ha we se he curvaure parameer = 0 in order o isolae he pure e ecs of he presence of xed facors, so he demand side of he model is idenical o he one wih Dixi-Sigliz CES aggregaor. Laer we will discuss he implicaions of boh fricions operaing a he same ime. As can be seen from his gure, he mos noiceable feaure is he asymmery induced in boh he average markup and he relaive price disorions by he presence of posiive in aion rae. The higher is he share of xed facors in he producion funcion, he higher 6

18 Figure 4: Combining Quasi-Kinked Demand and Firm-Speci c Inpus Figure 5: Comparing he Alernaive Speci caions are he coss generaed by posiive in aion raes. As in he baseline model, he relaive price disorion is minimized a zero seady sae in aion. In a siuaion of posiive seady sae in aion, if here is a fracion of xed facors, hen price adjusers rying o undercu prices o boos heir demands would also raise heir own marginal coss. Hence, a posiive seady sae in aion booss he oupu coss of relaive price disorions, while negaive seady sae in aion leads o smaller coss, given he upward slope of he rms s marginal curve. Quaniaively, hese relaive price disorions associaed wih posiive in aion raes produce non negligible oupu cos. In paricular, a 3% seady sae in aion rae generaes an oupu loss of nearly one percen, while a 3% annual de aion is only abou half as cosly. In Figure 4, we plo he e ecs of he calibraion of he model combining boh xed facors (assuming ha f = 0:5) and quasi-kinked demand (assuming in such a = 2). In he gure we plo he model assuming = 2, which corresponds o he one discussed above wih xed facors alone. I is clear ha he join e ecs of boh fricion ips he coss of in aion on relaive price disorions. In paricular, now he presence of quasi-kinked demand and he fac ha he rms seing price can no perfecly adjus is facors in response o in aion makes de aion much more cosly han posiive in aion. In paricular, a negaive seady sae in aion of -3% (somehow closes o he one associaed wih he Friedman s rule) will generae imporan oupu coss due o he amoun of relaive price disorion ha he model imposes on he price seing rms. In addiion, he bene s of boh posiive and negaive in aion on average markup are more balanced, i.e. he average markup curve is more symmeric around he zero seady sae in aion. Finally, Figure 5 compares he four speci caions using calibraions ha imply he same magniude of real rigidiy,. 7

19 6 Opimal Policy in he Sochasic Economy In his secion, we discuss implicaions of sraegic complemenariies for he opimal policy when he economy is subjec o exogenous random shocks. In so doing, we derive he second-order approximaion o he uiliy funcion of he represenaive household, following he linear-quadraic approach of Woodford (2003). Besides, we do no include moneary disorions o creae incenive for holding a money in his secion (i.e. we se 0 = 0 in expression (2)). 6. Characerizing he Opimal Policy Problem Before proceeding, noice ha he deerminisic seady-sae equilibrium achieves he rsbes allocaion in he presence of he scal policy o eliminae he disorion associaed wih he monopolisic compeiion. Given ha he seady sae is Pareo opimal, we can characerize he rs-order approximaion of he opimal policy from opimizing he secondorder approximaion of he social welfare funcion subjec o he rs-order approximaion of equilibrium condiions. 4 Moreover, he log deviaion of he real marginal cos from is seady-sae level is proporional o he log deviaion of oupu from is rs-bes level, while heir proporionaliy is he weigh of he oupu gap in he second-approximaion o he social welfare funcion, denoed by x. As a resul, subsiuing mc = x x ino he NKPC speci ed in he inroducion (expression ()), we have an expression of he Phillips curve equaion in erms of oupu gap: = E [ + ] + ( x p )x : (2) I is noeworhy ha he parameer p (= ( )( )=) is associaed wih he average frequency of price changes under he Calvo pricing, while degree of real rigidiy is measured by = ( + f f ). Having described he consrain of he opimal policy problem, we urn o he secondorder approximaion o he social welfare funcion. I is shown in he appendix ha he 4 Woodford (2003) includes he second-order approximaion o he social welfare when he seady sae is disored, while Benigno and Woodford (2005) discuss he opimal policy when he seady sae achieves he Ramsey allocaion. 8

20 second-order approximaion o he social welfare funcion can be wrien as follows: X =0 2 E 0 [ 2 + x 2 x ]; (3) 2 where x and are weighs for oupu gap and in aion, respecively. The weigh on oupu gap can be wrien as x = + ( + vl )= vl, hough he magniude of vl is a eced by he presence of xed labor inpus, given ha vk + vl = fl fk. However, has di eren expressions depending on sources of sraegic complemenariies: eiher = =( p ) in he case of xed facors inpus or = = p in he case of kinked demand curves. Given he second-order approximaion o he social welfare funcion and he rsorder approximaion of equilibrium condiions, he opimal policy from imeless perspecive can be wrien as = x + x in he case of xed-inpus, and = x + x in he case of kinked demand curves. We hus nd ha he opimal responses of in aion o changes in oupu gap depend on sources of sraegic complemenariies. However, since here is no mechanism ha generaes rade-o s beween in aion and oupu gap, he opimal in aion rae under imeless perspecive becomes zero regardless of sources of sraegic complemenariies. Hence, in he nex secion, we incorporae cos-push shocks ino he Phillips curve equaion o creae shor-run rade-o s beween in aion and oupu, following Clarida, Gali and Gerler (999, 200). 6.2 Fixed-Inpus a he Firm Level We begin wih he case of xed-inpus a he rm level and hen move ono he case of kinked demand curves. When sae-coningen commimen is feasible, he social planner chooses is sae-coningen plan on f, x g =0 in order o minimize X =0 E 0 [ x 2 x2 +! c ( x p x E [ + ] u )]; (4) where u represens an i.i.d. exogenous cos-push shock 5,! c represens he Lagrange muliplier for he Phillips curve in he opimizaion problem under commimen and! c = 0. 5 Alhough we do no make i explici in he previous secion, u can ake place when ax raes are subjec o exogenous variaions. 9

21 Combining he rs-order necessary condiions hen yields he opimal policy rule ha links arges: = x + x for, 0 = x 0 : Under discreion, he social planner can no make any binding commimen over is fuure policy acions, so ha i has o ake as given he public s expecaions abou he fuure. Hence, he opimizaion problem under discreion urns ou o be (5) min f ; x x 2 x2 +! d ( x p x E [ + ] u )g; (6) where! d is he Lagrange muliplier for he Phillips curve in he opimizaion problem under discreion. The rs-order condiions can be combined o yield = x : (7) I follows from (5) and (7) ha he inroducion of sraegic complemenariies hrough he xed-capial a he rm level does no a ec he opimal raio of in aion o oupu gap under boh of discreion and commimen. As a resul, we can nd ha he welfare level under he opimal policy is no a eced by he inroducion of sraegic complemenariies hrough he xed-inpus a he rm level, up o he rs-order approximaion of he opimal policy wih he second-order approximaion of he welfare. 6.3 Kinked Demand Curves Having described he opimal policies in he case of xed facors a he rm level, we solve he opimal policy problems when we allow for only kinked demand curves wihou having he xed-capial a he rm level. Noice ha he case of kinked demand curves corresponds o = =( ) and = = p. Given hese de niions of parameers, we solve he opimal policy problems similar wih hose in he previous secion. As a resul, he opimal policy under commimen can be wrien as = x + x for, 0 = x 0 : (8) The opimal policy under discreion is given by = x : (9) 20

22 I hen follows ha he inroducion of kinked demand curves reduces he opimal response of in aion o oupu, as opposed o he case of he xed facors a he rm level. The reason for his is ha he inroducion of kinked demand curves (hrough a change from he Dixi- Sigliz preference o Dosey-King ype preference) a ecs he rade-o beween in aion and oupu gap in he consrain, while i does no have any in uence on he rade-o beween in aion and oupu gap in he objecive funcion of he social planner. Furhermore, subsiuing he e ciency condiion (9) ino he social period loss funcion (3), we can nd ha he period loss funcion a he opimum under discreion urns ou o be ( + ( x p )) 2 : (20) 2 p I hus follows from (20) ha in he case of kinked demand curves, he opimal loss becomes smaller if in aion variabiliy is he same. Bu i does no mean ha he opimal loss under he same cos-push shock becomes smaller. For he sraegic complemenariies generaed by he inroducion of kinked demand curves increases he opimal response of he aggregae in aion rae o he same size of cos-push shocks. In order o see his, noice ha subsiuing (9) ino he Phillips curve speci ed in (2), solving he resuling linear di erence equaion, and hen puing he resuling soluion ino he period loss funcion yields 2 p ( + ( x p )) u2 : (2) As a resul, we can see ha he opimal loss under he same cos-push shock can be increased when he degree of real rigidiy rises wih he inroducion of kinked-demand curves. 7 Coss of he Grea In aion In his secion, we use explici funcional forms of relaive price disorion under each source of sraegic complemenariies o consruc ime-series of relaive price disorions, aking as given a se of observed ime-series of in aion raes and iniial values of relaive price disorions. Speci cally, we measure relaive price disorions ha are implied by di eren sources of sraegic complemenariies under he assumpion ha he governmen in he model achieves he same se of in aion raes observed in he U.S. economy. 2

23 Figure 6: The Evoluion of he Disorion Facor (%) 7. Dixi-Sigliz Preferences Consider he sandard Dixi-Sigliz aggregaor, which leads o he following relaionship beween relaive price disorion and in aion under he Calvo pricing: = ( )( ) + : (22) This means ha if we have an observed se of he aggregae in aion rae, denoed by f g T =0, 6 i is possible o consruc a se of relaive price disorions f g T =0 using equaion (22) given an iniial value of relaive price disorion,. Besides, he iniial value of relaive price disorion is se o be = whose value is measured a he long-run average in aion rae in he periods before he in aion series begin. In order o give a concree idea abou he consrucion of he ime-series of relaive price disorion, we proceed wih he measure of relaive price disorion (22). As a benchmark choice of parameer values, values of parameers and are, respecively, se o be = 0.6 and = 7, hough various ses of parameer values will be used. Speci cally, = 0.60 means ha rms x prices on average for 2.5 quarers, while = 7 implies ha he seady sae markup, de ned as he raio of price o marginal cos, equals 7 percen, because he seady sae markup is. Furhermore, he sample in his secion covers he quarerly daa on non-farm business secor in aion rae over he period 947: :3. In addiion, a sample average of non-farm business secor in aion rae over he period 947: - 959:4 is used o compue a seady sae value of relaive price disorion: = ( + ) ( ( + ) ) : (23) The iniial value of he measure of relaive price disorion is hen se o be =. Figure 6 repors consruced series of relaive price disorion over he period 960: :3. I demonsraes ha he measure of relaive price disorion rises in early 970s, 6 In he previous secion, denoes he logarihmic di erence beween price levels a period and. In his secion, when we consruc he ime-series of relaive price disorion, we de ne as he change rae of price levels a period and, so ha = P P P. 22

24 reaches is peak around 975 and hen declines. The size of relaive price disorion a is peak is around 2 % in erms of quarerly real oupu under he se of parameer values speci ed above. Besides, relaive price disorion shows large declines afer 982, while i is sable around 990s. I is worhwhile o menion ha he measure of relaive price disorion speci ed in (22) depends mainly on he in aion rae. In addiion, when he long-run average in aion rae is se equal o zero, he seady-sae relaive price disorion disappears. The cenral bank can herefore adjus he level of relaive price disorion by conrolling he rae of in aion. This in urn implies ha he sample average of relaive price disorion can be inerpreed as represening he cos of in aion. Furhermore, radiional works on he welfare coss of in aion has focused on he size of he deadweigh loss under a money demand curve ha occurs because of in aion, as can be seen in he works of Bailey (956) and Lucas (987, 2000). The coss of relaive price disorion, however, are no associaed wih he fricions ha make households volunarily hold real money balances. The ndings explained above hus indicae ha saggered priceseing can be an independen and signi can source of he welfare coss of in aion. 7.2 Implicaions of Real Rigidiies Having discussed he oupu coss of in aion based on he sandard Dixi-Sigliz preference, we move ono he cases of kinked demand curves and xed producion inpus. In so doing, we choose values of relevan parameers in order o mach he esimaed slope coe cien of he Phillips curve. As noed earlier, his implies ha he coe cien o measure he size of xed facors inpus is f = 0.58 in he rs case, while he curvaure parameer of demand curves is se o be = -8 in he second case. Given hese parameer values, we compue ime-series of relaive price disorion aking as given he observed ime-series of aggregae in aion rae. We do his o show ha hese wo sources of sraegic complemenariies have di eren welfare implicaions, even hough hey produce he same rs-order dynamics of he aggregae in aion rae. The resuls from hese experimens can be summarized as follows. Above all, di eren 23

25 sources of sraegic complemenariies share similar ime paerns of relaive price disorion during he whole sample period. Speci cally, measures of relaive price disorion rise in early 970s and hen decline afer 982. Thus, hey show heir peaks around 975. Bu heir magniudes are dramaically di eren. In paricular, when we derive quasikinked demand curves using he Dosey and King aggregaor, relaive price disorion becomes small relaive o he case of he sandard Dixi-Sigliz preferences. In conras, allowing for xed-inpus a he rm level raises he relaive price disorion. As shown in Figure 6, he peak relaive price disorion under quasi kinked demand curves is less han 0.25 % of quarerly real oupu, whereas i becomes slighly less han 30 % in he case of xed-inpus. 8 Conclusions Many papers have sudied he role of real rigidiies o mach he aggregae response of in aion o marginal cos. Bu, in all his lieraure, he non-linear implicaions of he underlying mechanism have no been furher explored. This issue is of cenral imporance o undersand he moneary policy implicaions of micro-founded New Keynesian models. Wha are he di eren implicaions for he welfare coss of seady-sae in aion and in aion volailiy of alernaive sources of real rigidiies? Our analysis corroboraes ha alernaive real rigidiies lead o very di eren implicaions for he welfare coss of seady sae in aion and in aion variabiliy, alhough hey migh have idenical implicaions for he rs order dynamics of he in aion rae. Under he presence of quasi-kinked demand, here is a sraegic link beween rms marginal revenues and he incenive for price adjusmen. The presence of local facor markes xed facor inpus, induces a sraegic link beween he rm s marginal coss and he incenive for price adjusmen. This di eren naure of he sraegic linkage among rm s incenive o changes price is a he core of he asymmeric resuls for moneary policy ha we nd in his paper. More generally, his paper pus forward he need o carefully examine mechanisms (embedded ino New Keynesian dynamic general equilibrium models) observaionally equiv- 24

26 alen up o rs order, which may yield sharply di eren conclusions for moneary policy once he non-lineariies implied by he models are accouned for. 25

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