Theoreical Economics Leers, 203, 3, 82-87 hp://dxdoiorg/04236/el20333030 Published Online June 203 (hp://wwwscirporg/journal/el) Asse Prices, Nominal Rigidiies, and Moneary Policy: Role of Price Indexaion Kengo Nuahara,2 Deparmen of Economics, Senshu Universiy, Kanagawa, Japan 2 The Canon Insiue for Global Sudies, Tokyo, Japan Email: nui@iscsenshu-uacjp Received April 7, 203; revised May 7, 203; acceped June 0, 203 Copyrigh 203 Kengo Nuahara This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied ABSTRACT A recen paper by Carlsrom and Fuers [ Asse Prices, Nominal Rigidiies, and Moneary Policy, Review of Economic Dynamics, Vol 0, 2007, pp 256-275] finds ha moneary policy response o share prices is a source of euilibrium indeerminacy because an increase in inflaion implies a high real marginal cos and low share prices in a sicky-price economy We find ha if he New Keynesian Phillips curve has a lagged inflaion erm caused by price indexaion, his effec is weakened Moreover, euilibrium indeerminacy caused by he moneary policy response o share prices never arises if all he firms ha canno re-opimize heir prices follow price indexaion Keywords: Asse Prices; Moneary Policy; Euilibrium Deerminacy; Price Indexaion Inroducion A paper by Carlsrom and Fuers [] shows ha euilibrium indeerminacy arises if moneary policy responds o share prices in a sandard sicky-price economy An increase in inflaion reduces firm s profis, and he share prices decline, since hey reflec he firm s profis Then, he moneary policy response o share prices implicily weakens overall reacions o inflaion This is a source of euilibrium indeerminacy In his paper, we exend he model of [] by inroducing price indexaion and show ha euilibrium deerminacy is likely o arise Under price indexaion, he New Keynesian Philips curve is hybrid and has a lagged inflaion erm I is shown ha he effec of an increase in inflaion on real marginal coss is weakened hrough he hybrid Phillips curve Moreover, euilibrium indeerminacy caused by he moneary policy response o share prices never arises if all he firms ha canno re-opimize heir prices follow price indexaion An increase in inflaion increases he real marginal cos under he sicky-price seing wihou price indexaion, since a fracion of firms canno change heir prices This increase in he real marginal cos implies low share prices Then, he moneary policy response o share prices implicily weakens overall reacions o inflaion Conrary o his, firms following price indexaion can keep heir real marginal cos consan in he long run since he pas inflaion reflecs his increase in inflaion [2] emphasize he inflaion persisence by empirical analyses, and [3] develop a model wih he hybrid New Keynesian Phillips curve Many sae-of-he-ar DSGE models a la [4-6] employ price indexaion Therefore, i is imporan o consider he ype of New Keynesian Phillips curve used when we invesigae he relaionship beween moneary policy and share prices The res of his paper is organized as follows Secion 2 inroduces our model Secion 3 presens he main resuls and heir inerpreaion Finally, Secion 4 presens our concluding remarks 2 The Model We employ a sandard sicky-price model wih shares, like [] The difference beween our model and heirs is ha we inroduce price indexaion in sicky prices 2 Households The household begins period wih M cash balances, B one-period nominal bonds ha pay R gross riskfree ineres rae, S shares of sock ha sell a price Q The uiliy funcion is M C H M UC, H, V, () P P Copyrigh 203 SciRes
K NUTAHARA 83 where 0, 0, 0, V is increasing in and concave, C denoes consumpion, H denoes labor supply, P denoes aggregae price level, and M P denoes real cash balances a he end of period The budge consrain of household is PC M B PQ S PW H M R B P Q D S X where W denoes wage rae, D denoes dividends of share, and X denoes moneary injecion The firs order condiions of households are C H W, C C R QC C Q D, where P P denoes gross inflaion The firs euaion is he inraemporal opimizaion condiion, he second is he Euler euaion for consumpion, and he las is he Euler euaion for share The las euaion can be rewrien as familiar asse prices euaions: 22 Firms R, (2) Q Q D (3) There are compeiive final-goods firms and monopolisically compeiive inermediae-goods firms and The producion echnology of final-goods firms is d 0 Y Y i i, (4) where denoes he elasiciy of subsiuion and Y i denoes oupus of inermediae-goods indexed by i The profi imizaion of final-goods firms implies he demand curve for Y i as Y i P i P Y, (5) where P i denoes he price level of inermediaegoods indexed by i Combining Euaions (4) and (5) yields he following price index for inermediae goods: d, (6) 0 P P i i The inermediae-goods firms are monopolisically compeiive, and hey produce inermediae-goods Y i employing labor H i from households The producion funcion of inermediae-goods firm is Y i H i, (7) The cos minimizaion problem implies W Z, (8) where Z denoes he Lagrange muliplier of he cos minimizaion problem, and i can be inerpreed as he real marginal cos Inermediae goods firms se heir prices subjec o Calvo-ype price saggeredness wih price indexaion The price can be re-opimized a period only wih probabiliy Among firms ha canno reopimize heir prices, a fracion firms index heir prices o he pas inflaion As in [5,6], under his seing, we obain he hybrid New Keynesian Phillips curve, π π π z, (9) where π and z denoe he log deviaions from a seady sae of inflaion and he real marginal cos, respecively 23 Moneary Policy We assume ha moneary auhoriy follows a Taylor rule: r π, (0) where r and denoe he log-deviaions from a seady sae of R and Q, respecively If 0, a cenral bank responds o asse price flucuaions 24 Euilibrium The marke clearing condiions are H H d, i i () B 0, (2) S (3) The resource consrain is C Y, (4) and he aggregae producion funcion is Y H, (5) where is a measure of resource cos of price dispersion: P i d i (6) 0 P In his paper, we ignore effecs from he price dispersion for simpliciy Copyrigh 203 SciRes
84 K NUTAHARA We focus on a euilibrium where all monopolisic compeiive firms are symmeric in his paper The firm s profis are paid ou as dividends o he shareholders For simpliciy, we assume ha he measure of firms is eual o he measure of households The dividend of inermediae-goods firms is given by By Euaion (8), he dividend is wrien by 25 Linearized Sysem D Y WH (7) D Z Y (8) The linearized sysem is given as follows: c w, c c r π, (9) (20) d π r, (2) z d c z (22) z w z π π π z (23) (24) r π (25) where he lower leers denoe he log-deviaions from a seady sae As shown by [], he dividend is given by where A d Az z z (26) We employ an assumpion on A following [] Assumpion A 0 Under his assumpion, an increase in he real marginal cos decreases he dividend The euilibrium sysem is reduced o he following marix form: 0 0π 0 0 0 π 0 A z 0 0 0 π 0 π 0 0 z 0 0 0 where 0 The firs euaion is he consumpion Euler Euaion (20); he second, he New Keynesian Phillips curve (24); and he hird, he Euler euaion for share (2) In his paper, we impose he following resricion Assumpion 2 2 We make his assumpion o easily prove Proposiion However, according o our numerical robusness check, he resul in his paper is robus even if 2 3 Main Resuls 3 Resuls The main resuls of his paper are as follows Proposiion Under Assumpions and 2 a necessary and sufficien condiion for euilibrium deerminacy is A If, here is euilibrium indeerminacy or no saionary euilibrium Proof See Appendix QED A a limi of 0, he hreshold is he same as he hreshold in Proposiion of Carlsrom and Fuers (2007) The hreshold depends on he fracion of price indexaion firms, Proposiion 2 is increasing in Proof Since, we obain A QED Then, in he case where he fracion of price indexaion is large, euilibrium deerminacy is likely o arise even if moneary policy responds o share prices Especially, in he case where, euilibrium indeerminacy never arises even if moneary policy responds o share prices Proposiion 3 Under Assumpions and 2, if all he firms follow price indexaion,, hen euilibrium deerminacy arises Proof QED lim 32 A Taylor Principle Inerpreaion The Taylor principle esablishes ha a permanen increase in he inflaion rae leads o a more-han-proporionae increase in he nominal ineres rae Following [] and [7], we inerpre our resuls according o his principle Copyrigh 203 SciRes
K NUTAHARA 85 A one percenage poin permanen increase in he inflaion rae causes he marginal cos o increase by percenage poin hrough he New Keynesian Phillips curve By he definiion of, his can be rewrien as (27) This decreases dividends and share prices by A The oal effec on he nominal rae is given by A (28) If his oal response is greaer han uniy, he rule saisfies he Taylor principle If 0, he coefficien of is sricly posiive Th us, moneary policy response o share prices weakens he oal response o inflaion and is a source of euilibrium indeerminacy However, his effec is decreasing in since his coefficien of is decreasing in This is because he effec of an increase in inflaion on he real marginal cos, hrough he hybrid Phillips curve, is weakened In paricular, if, hen h is coefficien is zero and he oal effec on he nominal ineres rae of inflaion is Therefore, moneary policy response o share prices is no a source of euilibrium indeerminacy in his case Under he sicky-price seing wihou price indexaion, a fracion of firms canno change heir prices in every period Then, a permanen increase in inflaion implies a low real marginal cos Under he sicky-price seing wih price indexaion, a fracion of firms ha canno re-opimize heir prices indexes heir prices o he pas inflaion In he long run, firms following price indexaion can keep heir real marginal cos consan since he pas inflaion reflecs his increase in inflaion Therefore, a permanen increase in inflaion does no change he real marginal cos if all he firms ha canno re-opimize heir prices follow price indexaion 33 A Numerical Example We have he fracion of firms ha follow price indexaion affecs he hreshold of he cenral bank s sance o he share prices on euilibrium indeerminacy ualiaively In his subsecion, we invesigae he uaniaive effecs of on For his exercise, we se he parameer values of he model as follows The discoun facor of households,, is 099 The relaive risk aversion,, is wo The Fr isch elasiciy of labor,, is wo The cenral bank s sance o inflaion, is 5 The seady-sae marginal cos, z, is 085, which implies ha he seady-sae markup is A recen paper by Carlsrom and Fuers [] found ha moneary policy response o share prices is a source of euilibrium indeerminacy in a sandard sicky-price model because an increase in inflaion implies a high real marginal cos and low share prices In his paper, we invesigaed a sicky-price model in which he New Keynesian Phillips curve has a lagged inflaion erm caused by price indexaion We found ha if firms follow price indexaion, he effec of an in crease in inflaion on real marginal cos is weakened and euieuilib- librium deerminacy is likely o arise Moreover, rium indeerminacy never arises if all he firms ha canno re-opimize heir prices follow price indexaion Empirical resuls suppor he significance of a backward inflaion erm in he New Keynesian Phillips curve Therefore, when we discuss he relaionship beween asse prices and moneary policy, we mus consider he ype of Phillips curve 5% These values are aken from hose employed by [] We s e he Calvo-pricing price-sickiness parameer,, is 075 following he lieraure, which implies ha firms can re-opimize heir prices abou once a year Figure shows he deerminacy and indeerminacy regions The verical axis means he cenral bank s sance o he share price The horizonal axis means he fracion of firms ha follow price indexaion The euilibrium indeerminacy arises in he upper-lef region The euilibrium deerminacy arises in he lower-righ region Then, a sronger sance of he cenral bank o he share prices induces euilibrium indeerminacy However, if he fracion of firms ha follow price indexaion is sufficienly high, euilibrium indeerminacy is no likely o arises if moneary policy responds o share prices 4 Concluding Remarks 5 Acknowledgemens I would like o hank Timohy Fuers, Masaru Inaba, and Keiichiro Kobayashi for heir helpful commens and Figure Deerminacy and indeerminacy regions Copyrigh 203 SciRes
86 K NUTAHARA suggesions Of course, he remaining errors are mine This work was funded by a Senshu Universiy research gran ( Analyses of Moneary Policy Responses o Asse Price Flucuaions ) in 20 REFERENCES [] C T Carlsrom and T S Fuers, Asse Prices, Nominal Rigidiies, and Moneary Policy, Review of Economics Dynamics, Vol 0, No 2, 2007, pp 256-275 doi:006/jred2006005 [2] J Fuhrer and G Moore, Inflaion Persisence, Quarerly Journal of Economics, Vol 0, No, 995, pp 27-59 doi:02307/2853 [3] J Gali and M Gerler, Inflaion Dynamics, Journal of Moneary Economics, Vol 44, No 2, 999, pp 95-222 doi:006/s0304-3932(99)00023-9 [4] L Chrisiano, M Eichenbaum and C Evans, Nominal Rigidiies and he Dynamic Effecs of a Shock o Moneary Policy, Journal of Poliical Economy, Vol 3, No, 2005, pp -45 doi:0086/426038 [5] F Smes and R Wouers, Shocks and Fricions in US Business Cycles: A Bayesian DSGE Approach, American Economic Review, Vol 97, No 3, 2007, pp 586-606 doi:0257/aer973586 [6] F Smes and R Wouers, An Esimaed Dynamic Sochasic General Euilibrium Model of he Euro Area, Journal of he European Economic Associaion, Vol, No 5, 2003, pp 23-75 doi:062/5424760377038345 [7] J Bullard and K Mira, Learning abou Moneary Policy Rules, Journal of Moneary Economics, Vol 49, No 6, 2002, pp 05-29 doi:006/s0304-3932(02)0044-7 Copyrigh 203 SciRes
K NUTAHARA 87 Appendix: Proof of Proposiion For euilibrium deerminacy, jus one roo should be inside he uni circle and ohers should be ouside he uni circle I is easily shown ha one roo is The hree remaining roos are he soluions of a characerisic euaion: 3 2 F x x Fx F x F 2 3, where F A F 2 0 A 0 F3 0 I is shown ha F 0 F3 0, F 0 F2 0, and F 2 A 2 0 since 2 A necessary condiion for euilibrium deerminacy is F F F F 2 3 A 0 In he case where all roos are real, i is obvious ha his condiion is also sufficien Nex, consider he case where wo roos are complex Suppose ha a bi are roos and a norm of M a b We have F x x a bix a bix r 2 2 3 2 x a r x M ar x M r where is a 0, For euilibrium deerminacy, we will show ha 2 M By he form of Fx, i is easily shown ha F x reaches a local mi nimum a r real roo in 2 2ar r 2ara 3b x x 3 Since F0 0 and F 0, i is shown ha x I suffices o show ha x a for for M a b The res of his proof, we show ha Since r x, i is obained ha If r a, we have 2 ra r ara b r 3 a a firs 2 3 4 r a r 2ara 3b ra 3b 0 and i is a con radicion Then, r a Finally, x a is shown as follows A necessary and sufficien condiion for x a is Since r a and his is easily shown 2 r 2ar a 3b a r, his condiion is reduced o 2 r ara 3 b ar 2, QED Copyrigh 203 SciRes