WORKING PAPER SERIES Inflaion-Targeing, Price-Pah Targeing and Indeerminacy Rober D. Dimar and William T. Gavin Working Paper 2004-007B hp://research.slouisfed.org/wp/2004/2004-007.pdf March 2004 Revised December 2004 FEDERAL RESERVE BANK OF ST. LOUIS Research Division 4 Locus Sree S. Louis, MO 6302 The views expressed are hose of he individual auhors and do no necessarily reflec official posiions of he Federal Reserve Bank of S. Louis, he Federal Reserve Sysem, or he Board of Governors. Federal Reserve Bank of S. Louis Working Papers are preliminary maerials circulaed o simulae discussion and criical commen. References in publicaions o Federal Reserve Bank of S. Louis Working Papers (oher han an acknowledgmen ha he wrier has had access o unpublished maerial) should be cleared wih he auhor or auhors. Phoo couresy of The Gaeway Arch, S. Louis, MO. www.gaewayarch.com
Inflaion-Targeing, Price-Pah Targeing and Indeerminacy By Rober D. Dimar and William T. Gavin Absrac In his paper, we examine he areas of indeerminacy in a flexible price RBC model wih shopping ime role for money and a cenral bank ha uses an ineres rae rule o arge inflaion and/or he price level. We presen analyical resuls showing ha, alhough inflaion argeing ofen resuls in real indeerminacy, a price level arge generally delivers a unique equilibrium for a relevan range of policy parameers. Keywords: Inflaion argeing, price-pah argeing, indeerminacy JEL Classificaion: C62, E52 Original Dae: Sepember 26, 2003, Revised December 2, 2004 Rober D. Dimar William T. Gavin Mahemaician Vice Presiden and Economis Research Deparmen Research Deparmen Federal Reserve Bank of S. Louis Federal Reserve Bank of S. Louis P.O. Box 442 P.O. Box 442 S. Louis, MO 6366-0442 S. Louis, MO 6366-0442 dimar@sls.frb.org gavin@sls.frb.org The views expressed here are hose of he individual auhors and do no necessarily reflec official posiions of he Federal Reserve Bank of S. Louis or of he Federal Reserve Sysem. We hank Jim Bullard and Chuck Carlsrom for helpful commens on an earlier draf.
. Inroducion Economiss working in heoreical models have long undersood ha finding soluions o models wih ineres rae rules and forward-looking agens can be problemaic because a relevan range of parameer values ofen leads o muliple soluions (or, equivalenly, areas of indeerminacy). Sargen and Wallace (975) showed ha ineres raes rules could lead o price level indeerminacy in models wih forwardlooking agens. McCallum (98, 986) showed ha modifying he rule o include he lagged money supply was one of many possible ways o eliminae indeerminacy in he price level. He has argued ha many cases of indeerminacy are he resul of inappropriae specificaion choices. However, he issue abou indeerminacy in moneary models is sill ineresing because i urns ou ha he condiions for real deerminacy in dynamic general equilibrium models depend criically on model specificaions of policy rules ha are hough o correspond o feaures in modern economies. Benhabib and Farmer (999) summarize lieraure in which researchers have used models wih indeerminacy o explain sicky prices and real effecs of moneary policy. For he policy economis, he issue is wheher inappropriaely designed policy insiuions increase he probabiliy of asse pricing bubbles and oher self-fulfilling prophecies ha can be generaed in models wih muliple equilibria. If so, hen policymakers wan o design insiuions ha eliminae his poenial source of insabiliy. For a discussion of indeerminacy in he conex of policy design see McCallum (2003), Woodford (2003a,b) and Benhabib, Schmi-Grohe and Uribe (2003).
Giannoni (2000) invesigaes he opimaliy of alernaive ineres rae rules in a New Keynesian economy. Using a sicky-price model, he shows ha while indeerminacy is presen for some range of parameers in he inflaion argeing regimes, hey disappear when he cenral bank arges a price pah. In his paper we show ha a similar resul also holds in a flexible-price model. 2. The Economic Model The model we use here is a slighed modified version of he moneary business cycle model developed in Dimar, Gavin and Kydland (2004). In each period, infinielylived consumers decide how o allocae ime beween work, leisure, and ransacionrelaed aciviies such as rips o he bank, shopping, and so on. Larger money balances carried in from he previous period make he shopping aciviy less ime consuming, leaving more ime for work and leisure. New money eners he economy as a governmen ransfer bu does no reduce shopping ime unil he nex period. 2 The governmen ses a arge for he nominal ineres rae on bonds and ransfers whaever amoun of money is necessary o achieve i. Many idenical households inhabi he model economy. Each household maximizes E β u( c, ), () 0 = 0 where 0 < β < is a discoun facor, c is consumpion expendiure, and l is leisure ime. The curren-period uiliy funcion is linear in leisure bu oherwise general. 2 Noe ha his is iming convenion used in Kydland (989). 2
The household s sock of capial, k, is governed by he law of moion, k = ( δ ) k + + i, (2) where 0 < δ <, δ is he depreciaion rae, and i is invesmen. Household ime spen on ransacions-relaed aciviies in period is given by m ω Ω P ω 0, (3) m is he nominal sock of money, and P is he price of physical goods relaive o ha of money. The parameer, Ω, represens he level of shopping ime echnology. By resricing Ω and ω o have he same sign and ω <, he amoun of ime saved increases as a funcion of real money holdings, bu a a decreasing rae. Noe ha his shopping ime funcion differs from Dimar, Gavin and Kydland (2003) because i does no include consumpion expendiures. Wih leisure linear in uiliy, his assumpion allows us o reduce our dynamic sysem o only wo equaions. Leisure in period is ω m = T n ω0 +Ω, (4) P where T is he oal ime available and n is ime spen in marke producion. Aggregae oupu, Y, is produced using labor and capial inpus: Y = C + I = z N K, (5) θ θ where I is he oal of invesmen expendiures and z is he level of echnology. We assume a Cobb-Douglas producion funcion wih labor share θ : 0 < θ <. The nominal budge consrain for he ypical individual is 3
Pc + Pk + b + m = Pzn k + P δ k + Rb + m + v (6) θ θ + + ( ), where b + are nominal bonds carried ino period, and v is a nominal lump-sum ransfer from he governmen. R is he gross nominal ineres rae earned on bonds which are assumed o be in zero ne supply in equilibrium. The governmen ransfers money balances direcly o households according o is policy rule. I produces money and conducs policy a zero cos and does no have any expendiures or revenues. A law of moion analogous o ha for individual capial describes he aggregae quaniy of capial. The disincion beween individual and aggregae variables is represened here by lowercase and uppercase leers, respecively. Compeiive facor markes will imply ha in equilibrium each facor receives is marginal produc. The cenral bank manipulaes he moneary ransfer o implemen an ineres rae rule of he ype: ln R = κ + ν ln P + ν (ln P ln P ), (7) + + 2 + where ln R + is he period + nominal ineres rae arge chosen by he cenral bank. The ineres rae is se as a funcion of he price level and he inflaion rae. The Dynamic Sysem. The agen s choice of any four variables say leisure, labor, capial and bonds will deermine he ohers via his budge and ime consrains. Here, we assume ha uiliy is linear in leisure: u l ( ) i ε (8) where ε is an arbirary consan. We define he labor-capial raio as x = n / k and compue he firs order condiions for labor, capial, and bonds in erms of his raio. The firs order condiion for labor is given as: 4
u c ε i (9) zθ x θ ( ), which equaes he marginal uiliy of consumpion o he marginal uiliy of labor relaive o he marginal produc of labor. The firs order condiion for capial is given as: u c + θ ( ) + E z( ) x + ( ) β β θ + δ + u i i = 0, (0) c which equaes he ineremporal rae of subsiuion of consumpion in uiliy o he ineremporal rae of ransformaion of oupu in producion. The firs order condiion for bonds is given as: u u ( i) = E R+ β ( i + ), () P c P+ c which relaes he nominal ineres rae o he inflaion rae and he ineremporal rae of subsiuion in consumpion. Subsiuing he firs order condiion for labor (9) ino he firs order condiions for capial (0) and bonds (), we ge: { ( θ) δ } x θ = βe x θ z x θ + + +, and (2) P θ R + x = β E x θ +. (3) P+ In he seady sae, ( ) + P = ln P / β = z θ x θ + δ and R = / β. Afer defining x = ln x and, we log linearize (2) and (3) o ge wo firs order difference equaions in he log of he price level and he log of he labor-capial raio: βθ( δ ) x = E x +, and (4) θ 5
( ) ( θ) P + θ x = E R + P + + x +, (5) We log-linearize he ineres-rae rule (7) o ge: Subsiuing (6) ino (5), we ge: ( ) E R ν ν E P ν P (6 ) + = + 2 + 2. ( θ) ( ν ν ) ν ( θ) P + x = E + 2 P + 2 P + x +, (7) Equaions (4) and (7) form a wo-equaion sysem wih wo roos: Where ( ) E x + x A = B E P + P, βθ δ 0 θ 0 A = and B = θ ν+ ν2 θ ν2. For a unique soluion o his sysem we need one eigenvalue inside he uni circle and he oher ouside. The eigenvalues of A B are given by ( θ ) / [ β( δ) θ ] and ( ) ( / ). The firs is beween zero and one by our model assumpions and ν 2 ν ν 2 he second is greaer han uniy for he combinaion of values shown in he shaded regions in figure. If here is no weigh on inflaion, hen he model is deermined for 0 < ν < 2. The general condiions for deerminacy are: ν ν > 0 < ν2 < 2, and ν ν < 0 < ν2 <. 2 6
Of special ineres is he case of a pure inflaion arge, i.e. v = 0. In his case, we can show ha here is indeerminacy everywhere. To see his we firs noe ha if v = 0, hen we can rewrie equaion (7) as follows: ( θ) ( ν ) ( ) ( θ) x = 2 E. P+ P + E x + (8) If we le / = P P be he inflaion rae a ime + and π = lnπ π + + + +, hen by definiion we have E π + = E P + P. Therefore equaion (8) above can be wrien as a difference equaion in E π +, and E x +, namely: x, ( ) ( ) ( ) θ x = ν E π + θ E x (9). 2 + + Combining equaions (9) and (4) as above again gives a wo variable sysem of difference equaions ha can be wrien as: where E x + x A = B, E π π + ( δ ) βθ 0 A = θ ν and θ 0 B =. θ 0 For a unique soluion, we again would need one eigenvalue of o be inside he uni circle and one eigenvalue ouside he uni circle. Here, however, one eigenvalue of is given by ( θ )/ [ β( δ) θ ] A B A B which is posiive and sricly less han by our model assumpions while he oher eigenvalue is exacly 0. The model is herefore indeerminae wih a pure inflaion argeing rule for any value ofν. 3 3 Noe ha his resul appears o be a odds wih Woodford (2003, proposiion 2.6), bu i is no. He has a cashless economy wih deerminacy condiions ha are he same as a model in which money balances a he 7
3. Concluding Remarks We find ha he indeerminacy ha can arise wih inflaion argeing disappears when he inflaion arge is replaced wih a pah for he price level. In he real world cenral banks do no se pahs for he price level, bu hey do end o arge inflaion averaged over muliple periods. Counries ofen adoped inflaion argeing in order o eliminae high and variable inflaion. As inflaion fell, hese counries seem o have seled on a single arge ha is repeaed year afer year. For example, he Bank of Canada has had he same inflaion argeing range, one o hree percen, since 995 and he Bank of England has had a single inflaion arge, 2.5 percen, since 998. Our model suggess ha doing so may be a good idea if he implemenaion of such a regime approximaes a arge pah for he price level. end of he period ener uiliy. In our model, only money balances brough ino he period ener he shopping ime funcion (and curren period uiliy). See Carlsrom and Fuers (200a) for a deailed analysis of how his iming assumpion affecs he deerminacy condiions. Carlsom and Fuers (200b) presen resuls for conemporaneous policy rules. 8
References Benhabib, J., and R.E.A. Farmer, 999, Indeerminacy and sunspos in macroeconomics, in: J.B. Taylor and M. Woodford, eds. Handbook of Macroeconomics, Vol, Elsevier Science B.V. 999, 387-448. Benhabib, J., S. Schmi-Grohé, and M. Uribe. Backward-Looking Ineres-Rae Rules, Ineres-Rae Smoohing, and Macroeconomic Insabiliy, mimeo, New York Universiy, June 2003. Carlsrom, C.T., and T.S. Fuers. "Timing and Real Indeerminacy in Moneary Models," Journal of Moneary Economics 47 (200a), 285-298., and. "Real Indeerminacy in Moneary Models wih Nominal Ineres Rae Disorions," Review of Economic Dynamics 4 (200b), 767-789. Dimar, R.D., W.T. Gavin, and F.E. Kydland. "Inflaion Persisence and Flexible Prices," Federal Reserve Bank of S. Louis Working Paper 200-00E, March 2004, forhcoming in he Inernaional Economic Review. Giannoni, M.P. Opimal Ineres-Rae Rules in a Forward-Looking Model, and Inflaion Sabilizaion versus Price-Level Sabilizaion." mimeo, Columbia Universiy, Ocober 2000. Kydland, F.E. Moneary Policy in Models wih Capial, in Dynamic Policy Games, van der Ploeg, F. and A. J. de Zeeuw eds. Elsevier Science Publishers B. V. (Norh-Holland), 989. McCallum, B.T. Muliple-Soluion Indeerminacies in Moneary Policy Analysis, Journal of Moneary Economics 50 (2003), 53-75.. Some Issues Concerning Ineres Rae Pegging, Price Level Deerminacy and he Real Bills Docrine, Journal of Moneary Economics 7 (January 986), 35-60.. Price Level Deerminacy wih an Ineres Rae Policy Rule and Raional Expecaions, Journal of Moneary Economics 8 (November 98), 39-329. Sargen, T.J., and N. Wallace. Raional Expecaions, he Opimal Moneary Insrumen and he Opimal Moneary Policy Rule, Journal of Poliical Economy 83 (975), 24-254. Woodford, M., Ineres and Prices, Princeon Universiy Press: Princeon, 2003a., Commen on: Muliple-Soluion Indeerminacies in Moneary Policy Analysis, Journal of Moneary Economics 50 (2003b), 77-88. 9
Figure : Deerminacy Condiions for Policy Parameers Weigh on he inflaion rae, ν 2 2.5 0.5 0-0.5 - -2-0 2 3 Weigh on he price level, ν 0