Optimal Interest Rate Policy ina SmallOpen Economy

Transcription

1 Optimal Iterest Rate Policy ia SmallOpe Ecoomy Eric Parrado y New York Uiversity Adrés Velasco z Harvard Uiversity December 2000 Prelimiary Draft Abstract Usig a optimizig model we derive the optimal moetary ad exchage rate policy for a small stochastic ope ecoomy with imperfect competitio ad short ru price rigidity. The optimal moetary policy has a exact closed-form solutio ad is obtaied usig as welfare criterio the utility fuctio ofthe represetative aget. The paper cocludes that the optimal policy depeds cruciallyo thesourceof stochastic disturbaces affectig the ecoomy, much as i the stadard literature pioeered by Poole (1970). Uder the optimal policy the exchage rate oats, but there is a positive correlatio betwee domestic ad rest-of-the-world iterest rates. That meas that the omial exchage rate does ot carry all the burde of adjustmet, ad may therefore move relatively little i equilibrium. This may provide a theoretical ratioale for the fear of oatig recetly documeted empirically for ope emergig ecoomies. JEL Classi catio: E52, F41. Keywords: Small Ope Ecoomy, Optimal Moetary Policy, Exchage Rates. We are thakful to Adrés Escobar for useful suggestios. y ecp200@yu.edu z adres_velasco@harvard.edu

2 1. Itroductio After the exchage rate crises of the last decade may small ope ecoomies, both rich ad poor, have adopted exible exchage rates i combiatio with some kid of moetary or iterest rate rule. Exactly what such a rule should look like, however, remais very much a ope questio. I closed ecoomies, i atio-targetig or Taylor-type rules are commo, eve though the optimality of such rules is yet to be established aalytically. I the ope ecoomy, host of additioal tricky issues tur up. Should moetary policy respod systematically to the omial or real exchage rate? Equivaletly, shouldthe oat be completely clea or ot? If dirty, should moetary policy act through direct itervetio i the exchage market or throughchages i target iterest rates? If the exchage rate is targeted, however loosely, how to avoid the credibility problems that are edemic to this kid of policy? These crucial questios are oly owbegiig to receive systematic attetio i the cotext of state-of-the-art models. 1 There is work o optimal moetary policy that cosiders di eret model structures with price stickiess. However, previous research has bee based o closed or two-coutry models, or has lacked of a adequate welfare criterio for ope ecoomies. It seems importat to have for the small ope ecoomy a simple represetatio of the optimal moetary policy, derived explicitly from the utility fuctio of the represetative aget. The aim of this paper is to develop such a optimizig model ad its correspodig policy rule. We cosider a small stochastic ope ecoomy withimperfect competitio ad short-ruprice rigidity (oe-periodomial cotracts for prices.) The utility fuctio of the represetative local aget has a exact closed-form solutio, ad both the rst ad the secod momets of variables matter for utility. Usig this welfare fuctioal we ca compute the optimal iterest rate policy, which also as a exact, closed-form solutio. 2 I particular, we study howmoetary policy should respod to shocks to productivity ad iteratioal iterest rates. Implicitly, this sheds light o the questio ofwhat the optimal exchage rate system should be. We also compare, i terms of welfare, this optimal moetary policy with alterative rule-of-thumb regimes. Our results appear to be i accordace to covetioal wisdom ad empirical 1 I the cotext of a earlier geeratio of models, some of these issues wer discussed, for istace, by Fischer (1977) ad Flood (1979). 2 We cosider oly iterest rate rules, but give moey demad oe could easily determie the quatity movemets that could implemet or support the chose iterest rules. 2

3 evidece. We d that the optimal moetary policy of this small ope ecoomy requires a positive correlatio betwee domestic ad the rest of the world s iterest rates. Ideed, the parameter associated to the foreig iterest rate shock i the reactio fuctio of the moetary authority is positive. It is also less tha oe, which implies that the optimal policy does ot require a full o set by the domestic iterest rate. The ituitio is as follows. A uaticipated shock of the foreig iterest rate ca cause movemets i the level of domestic cosumptio ad the real exchage rate. Iparticular, a positive foreigiterest rate iovatio, other thigs equal, wouldleadto a depreciatio of the exchage rate, a ectig domestic cosumptio via both the level ad the volatility of the relative price of home cosumptio. Some smoothig of the movemets i the real exchage rate is therefore called for. The moetary authority of our small ope ecoomy model, whe faced with such a exteral shock (a foreig iterest rate shock,) should respod icreasig its iterest rate util it compesates, to some extet, for this exteral rate hike. Hece, uder the optimal policy the exchage rate oats, but give the required co-movemet betwee domestic ad rest-of-the-world iterest rates, the omial exchage rate does ot carry all the burde of adjustmet, ad may therefore move relatively little i equilibrium. This may provide a theoretical ratioale for the fear of oatig recetly documeted empirically for ope emergig ecoomies. 3 We also study the optimal iterest rate reactio to a temporary productivity shock, adshow that iterest rate policy should ot react whefacedwith such a iovatio. The ituitio i this case is give by the fact that domestic output is demad-determied ithe short-ru; therefore, a temporary positive productivity shock does ot produce ay e ect o cosumptio or the real exchage rate; the productivity icrease is simply absorbed by a fall i domestic labor supply. Sice price exceeds margial cost (because of moopolistic competitio), domestic output remais uchaged. How is our cotributio i this paper related to the earlier literature? Recetly a wave of cotributios has attempted to explai macroecoomic cosequeces of moetary policy usig two di eret approaches. First, i dyamic eo-keyesia models, which cosider optimizig agets, omial rigidities, ad a key role for moetary policy, the solutio is determied umerically. These models compare alterative moetary policies ad exchage rate regimes. 4 O the other had, 3 See Hausma et al, (1999) ad Calvo ad Reihart (2000). 4 See Clarida, Galí ad Gertler (1999) for a survey of this approach. For ope ecoomy appli- 3

4 there are also more tractable models, i which the solutio is obtaied aalytically. These models are mostly ot dyamic, but they permit closed-form solutios ad exact utility calculatios. 5 Three features set our paper apart from the literature cotaied by these two approaches. First, i cotrast to a large umber of papers that have assumed a closed ecoomy or two equal-size coutry models, we have assumed a small ope ecoomy. I particular, the e ects of variability of the exchage rate play a key role ithe reactio of the ecoomy to speci c shocks. Secod, our approach derives the optimal moetary policy from a welfare fuctio based o the utility of the cosumer, ad ot arbitrary loss fuctios preset i most of the literature. Fially, it is also di eret from previous aalyses i that we cosider the iterest rate as the istrumet of moetary policy rather tha moey supply, as it is foud i may models based oobstfeld ad Rogo (1995.) Aexceptio is the paper by Galí ad Moacelli (2000), which cosiders a small ope ecoomy versio of a dyamic New Keyesia model with staggered price-settig à la Calvo. The paper is orgaized as follows. Sectio 2 cotais a descriptio of the theoretical model. Sectio 3 presets the aalysis of the optimal moetary policy. Sectio 4 presets a closed form solutio of the welfare fuctio based o the utility fuctio of the represetative aget. The al sectio summarizes the results ad their implicatios for models of moetary policy. 2. A Sticky-Price Model The model is a stochastic small ope ecoomy versio of Obstfeld ad Rogo (1998) ad Corsetti ad Peseti (2000) i which the curret accout is always zero i equilibrium. I the small ecoomy, Home agets cosume a variety of Home ad Foreig goods. Every aget is both cosumer ad producer who maufactures moopolistically a sigle tradable good. They ca hold two types of assets, moey ad bods, ad supplies labor implicitly. As Obstfeld ad Rogo (1996) stress, moopoly plays a key role i the aalysis because it helps to justify the Keyesia catios, see Moacelli (1998), Parrado (2000), ad Svesso (2000), amog others. Rotemberg ad Woodford (1998, 1999) deal with closed ecoomy models. 5 A recet survey is i Lae (2000). See Corsetti ad Peseti (2000) for a determiistic twocoutry model. For stochastic two-coutry equally size models see Obstfeld ad Rogo (1998, 2000). 4

5 assumptio that output is demad determied i the short ru whe prices are xed. The key assumptio of the model is that agets set ext period s prices before productio ad cosumptio is realized. Havig described the geeral setup of the model, we proceed i three steps. First, we explai the coutry ad ecoomic size aspects of the model. Secod, we outlie the mai buildig blocks of the model ad its micro-foudatios. Fially, i sectio 3 we embed these relatioships i a otherwise covetioal model Coutry Size I this model, a small ope ecoomy (Home) ad the rest of the world (Foreig) compose our ecoomy. Home agets are idexed by the iterval [0;], while Foreig agets reside o the iterval (;1]. We assume that, as i Galí ad Moacelli (1999), the rest of the world is a limitig case of a ope ecoomy, i which the cosumptio of the small ope ecoomy goods is isigi cat. I other words, we cosider the rest ofthe world as a closed ecoomy where its aggregate cosumer price idex (CPI) is exactly equal to the price idex of goods produced withi this ecoomy. Similarly, every idividual has a moopoly i producig a sigle tradable good, also idexed byi the iterval[0;1]. Thus,idicates both the populatio size ad the ecoomic size of the Home coutry. As we demostrated, the model requires that the weight i the utility fuctio (ecoomic size) is the same as the coutry size Idividual Prefereces ad Techology The small ope ecoomy has a cotiuum of cosumers-producers idexed by i 2[0; ], where a home represetative idividual maximizes the expected value of ( Ut i =E X 1 µ 1 s t Ã(Cs i)1 ½ t 1+± 1 ½ ~ k s 2 (Yi s!); )2 (2.1) s=t where ± is the rate of time preferece ad 1=½ is the elasticity of itertemporal substitutio. The otatio E t [x t+j ] represets the expectatio of variable x t+j coditioal o iformatio available at t. The utility is separable i two argumets: 6 See appedix 1. 5

6 1. Cosumptio (C i ): Prefereces over goods over time (utility is cocave i cosumptio). 2. Idividual Output (Y i ): Idividuals produce output with labor iput. This term captures the disutility the idividual experieces from havig to produce more output. The stochastic parameter ~ k represets a iverse productivity shock. The aggregate real cosumptio idexc t for ay persoi is give by C t = C H;t C1 F;t (1 ) 1 ; (2.2) wherec H;t adc F;t are the cosumptio quatities that Home agets cosume of domestic ad foreig goods, respectively. Ad is the share of the total cosumptio of the small ope ecoomy goods cosumed by both regios. The two cosumptio subidexes are symmetric ad are de ed, as i Dixit ad Stiglitz (1977), by C H;t = " µ1 1 µ Z 0 C(j) µ 1 µ dj # µ µ 1 ; C F;t = " µ 1 Z # µ 1 µ 1 µ 1 C(j) µ 1 µ dj : 1 (2.3) Aalogously to Obstfeld ad Rogo (1998) ad Corsetti ad Peseti (2000), the elasticity of substitutio across goods produced withi a coutry is µ > 1, while the elasticity of substitutio betwee idices of the small ope ecoomy ad the rest of the world is1. Rest-of-the-worldagets have idetical prefereces. Foreig values ofthe correspodig domestic variables will be deoted by a asterisk ( ) ad may di er from home variables. Prefereces over cosumptio goods are symmetric across regios because it is assumed that the elasticity of substitutio (1=½) ad the rate of time preferece (±) are the same across coutries. With the speci catioallowig for a cotiuum of di eretiated goods iboth coutries, we are explicitly assumig that goods markets are imperfectly competitive while labor markets are perfectly competitive. Implicitly, techology is liear i labor across coutries, which is competitively supplied. Speci cally, labor is homogeeous with respect to labor of other agets; hece, idividuals do ot have ay degree of moopolistic power. Agets simply take the real wage as give ad 6

7 the decide how much to work. O the other had, rms hire labor ad produce goods that are imperfect substitutes of each other. As a cosequece, each rm ejoys some moopoly power ithe goods market. Cosequetly, omial rigidities are represeted oly by prices i the goods market. Therefore, the degree of moopolistic competitio is measured by the elasticity of goods substitutio (µ=µ ) Prices ad Demad Curve Facig Each Moopolist Home prices idexes for the two precedig cosumptio baskets, deoted byp H;t adp F;t are de ed as P H;t = 1 Z 0 1 P T (j) 1 µ 1 µ 1 dj ; PF;t = 1 Z 1 1 P T (j) 1 µ 1 µ dj ; (2.4) where the domestic currecy price idex for overall real cosumptioc t is give by P t =P H;t P1 F;t : (2.5) The Lawof Oe Price (LOP) holds across all idividual goods sice agets of the small ope ecoomy ad the rest of the world have idetical prefereces, so thatp t (j)=s t P t (j); 8j 2[0;1], wherep t(j) adp t (j) are the prices of good j i the small ecoomy ad the rest of the world, respectively, ads t represets the omial exchage rate. Equivaletly,P F;t =S t P H;t adp H;t=S t P F;t. De e the real exchage rate asq t = StP t. This relative price will play a key role i Pt what follows. P t =(P H;t ) S t P H;t 1 =(PH;t ) (S t P t )1 (2.6) The rest of the world behaves, i the limit, as a closed ecoomy. This fact etails that foreigers expediture share i home goods is egligible, i.e.,p t = P H;t. Hece, we ca usep H;t orp t i the previous equatios. The small ecoomy commodity demad fuctios (home good demad: j=h ad foreig good demad: j = f) resultig from cost miimizatio are: C t (h)= 1 Pt (h) P H;t µ C H;t; C t (f)= 1 1 µ Pt (f) C F;t: (2.7) P F;t 7

8 Usig the de itio of total cosumptio (equatio (2.2)), we ca derive the demad fuctios for home ad foreiggoods C H;t = PH;t P t 1 C t ; C F;t=(1 ) PF;t P t 1 C t : (2.8) Thus, pluggig equatio (2.8) ito equatio (2.7), it follows that the Home demad for Home ad Foreig goods is give by Pt (h) C t (h)= P H;t µ PH;t P t 1 Pt (f) C t ; C t (f)= P F;t µ PF;t P t 1 C t : (2.9) 2.4. Asset Markets ad Idividual ad Govermet Budget Costraits Home Agets ow a equal share of all domestic rms. They ca also hold two assets: a domestic bod deomiated i terms of home currecy (B t ) ad a foreig bod (B t ) deomiated i terms of foreig currecy. Thus, the idividual household costrait, expressedi uits oftradable goods, is give by P t B t +S t B t +P tc t =(1+~{ t )P t B t 1 +(1+~{ t )S tb t 1 +Pi H;t Yi H;t ; (2.10) whereb t adb t deote ed of periodt 1 bods holdigs. The variableyh;t i is output produced by each aget, whileph;t i is its domestic currecy price ad~{ t ad~{ t are omial iterest rates at home ad abroad, respectively Market Clearig i the Small Ope Ecoomy Regardless of whether prices are sticky or exible, tradable goods market requires that output equals demad, From which we get [P t C t +(1 )P t C t ] = P H;tY; (1 )[P t C t +(1 )P t C t ] = (1 )P F;tY : (2.11) Y t P F;t=Y t P H;t : (2.12) This coditio takes the same form as Corsetti ad Peseti (2000) ad Obstfeld ad Rogo (1998). Expressio (2.12), together with the assumptio that the iitial et iteratioal asset holdigs,b, is0, implies that the curret accout is 8

9 always zero. Furthermore, this equatio shows that coutries has costat shares of world icome at all times idepedetly of ay shock. Therefore, coutries always cosume all their real icome, because of the assumptio of isoelastic prefereces over total cosumptio ad the costat real icome share, i.e., the curret accout is always0despite of the shocks. This implies that 7 3. Equilibrium Y t = P t P H;t C t ; Y t = P t P H;t C t : (2.13) Whe computig the equilibrium we assume that the Cetral Bak uses a iterest rate rule, ad as we stress before, prices are set oe period i advace of the realizatio of the shocks First Order Coditios The represetative cosumer chooses his optimal holdigs of bods, cosumptio ad wage level to maximize his expected utility (equatio (2.1)) subject to the usual budget costrait (equatio (2.10)). We wage-settig problem i the ext sectio, ad focus o the other choices faced by the cosumer here. If t is the Lagrage multiplier associated withbudget costrait 2.10, the FONCs with respect to cosumptio, domestic bods, ad foreig bods respectively are µ 1 t=(1+~{ t ) 1+± t=(1+~{ t ) µ 1 1+± t= C i t ½ (3.1) E t ½ E t ½ ¾ P t t+1 P t+1 t+1 P t P t+1 S t+1 S t ¾ (3.2) (3.3) Combiig the rst two we have ½ C i C i ½(1+~{t+1 t ½= Et t+1 ) P ¾ t ; (3.4) P t+1 7 It is importat to ote that the key assumptio to get this result is to assume that ecoomic size is the same as populatio size. See appedix 1. 9

10 which is the traditioal itertemporal Euler equatio for total real cosumptio. I tur, combiig coditios (3.2) ad (3.3) we have ½ ¾ P E t t t+1 (1+~{ t ) (1+~{ t P )S t+1 =0; (3.5) t+1 S t which is the stadard arbitrage coditiobetwee domestic ad foreig assets. Assumig that the atural logarithms of the exogeous variables are joitly ormally distributed, 8 we ca express the equilibrium coditios i logs. For the sake of clarity, we de e the atural logarithm of ay variable X by x, ad the datet 1 ucoditioal variace ofx t,var t 1 [x t ], by¾ 2 x. Takig logs of (3.4) we ca express the cosumptio Euler equatio as a fuctio of edogeous variaces: ½(E t c t+1 c t ) = ±+ fi t+1 (E t p t+1 p t )g (3.6) + 1 Ã µ 1 2 µ! 1 ½ 2 ¾ 2 c 2 + ¾ 2 q ¾ +2½ cq ; wherei=log(1+~{). We express the variace of the price i terms of the variace of the real exchage rate. Sicep t =p H;t +(1 )p F;t,p F;t =s t +p H;t, ad both p H;t adp H;t are predetermied, the variace ofp t depeds oly o the variace of the omial exchage rate, i.e.,¾ 2 p =(1 )2 ¾ 2 s. Furthermore, the variace of the real exchage rate,¾ 2 q, is composed by the variace of the omial exchage rate, the variace of the price level, ad the covariace betwee them. However, after some simpli catios, we get that¾ 2 q =2 ¾ 2 s. Thus,¾2 p = 1 2¾ 2 q. Similarly, for the same reasos, istead ofthe covariace betweecosumptio ad price, we iclude a expressio for the covariace betwee cosumptio ad the real exchage rate, i.e.,¾ cp = 1 ¾cq. 8 A variable X is log ormally distributed if x = l(x)» N(¹ x ;¾ 2 x). Thus, if l(x) = x the X = e x. I this case E[X] = E[e x ] = m(x), where m(x) is the momet geeratig fuctio for x ad is give by Z " 1 M(x) = e x 1 p e (x ¹ 2# X ) 2¾ 2 X dx 2¼¾ 2 X Therefore, 1 E[X] = e ¹ x+ 1 2 ¾2 x 10

11 O the other had, the rest of the world log Euler equatio takes a closedecoomy form because of our assumptio of egligible cosumptio of small ecoomy goods by foreigers. I that case, the log versio of the foreig Euler equatio becomes ½ E t c t+1 c t = ±+ i t+1 E t p ª t+1 p t (3.7) ½ 2 ¾ 2 c +¾2 p +2½¾ c p +¾2 Â ; wherei =(1+~{ ). Next we log-liearize arbitrage coditio 3.5: µ 3 i t =i t +E ts t+1 s t + 2 ¾ 2 s ; (3.8) I real terms it becomes i t (E t p t+1 p t )=i t µ 3 1 E t p t+1 t p +(Et q t+1 q t )+ 2 2¾2 q : (3.9) The equatio characterizes the real ucovered iterest parity coditio (UIP C), i which movemets i the real exchage rate are give by ex-ate real iterest rate di eretials Aggregate Demad ad Output Determiatio Market clearig for our small ope ecoomy require that domestic output be equal to demad. Previous equatios have see that domestic expediture i home goods is a fractio of al expeditures. Speci cally, pluggig equatio (2.13) ito equatio (2.8), oe obtais this obvious coditio for home cosumptio of home goods i terms of domestic productio C H;t =Y t : (3.10) Similarly, usig the same set of equatios, we ca get a expressio for home cosumptio of foreig goods C F;t =(1 ) P H;t S t P H;t Y t : (3.11) 11

12 Combiig these two coditios, we ca express overall cosumptio i terms of domestic output adterms of trade Ã! 1 P C t = H;t S t P H;t Y t : (3.12) The log versio ofthis equatioiterms of domestic output ad the real exchage rate will be µ 1 c t =y t q t : (3.13) Note that i derivig equatio (3.13) we used the assumptio that the foreig coutry behaves as a closed ecoomy, i.e.,p t =P H;t ad thatq t=(s t +p t p H;t). 9 This equatio establishes a simple relatioship likig domestic cosumptio with output ad a proportioal factor of the real exchage rate Summary of Equilibrium I previous subsectios, oliear stochastic equatios appear quite complex. However, with the assumptio of logormal disturbaces, they lead to a rather simple closed form solutio. Speci cally, we assume that the atural logarithms of the exogeous variables the productivity shock (k t ) ad the foreig iterest rate shock (" t ) are all joitly ormally distributed. Thus, we ca fully describe the equilibrium dyamics of the small ope ecoomy with eqs. (3.6), (3.9), (3.13), ad a moetary rule for the domestic iterest rate. For the sake of clarity, we rewrite the system i this sectio, reorgaizig some equatios ad replacig the expected value of all the variables with their steady state values. The reaso is that we are dealig with trasitory uaticipated shocks, ad hece, after the period of the shock all variables adjust freely to the pre-shock steady state level. Therefore, iaalyzig the e ects of moetary policy i presece of price stickiess it is eough to focus what happes i the period of the shock. ½(c t ¹c) = ± fi t+1 (E t p t+1 p t )g (3.14) Ã 1 µ 1 2 µ! 1 ½ 2 ¾ 2 c 2 + ¾ 2 q ¾ +2½ cq ; 9 See Appedix 2. 12

13 µ 1 (c t ¹c)=(y t ¹y) (q t ¹q); (3.15) i t (E t p t+1 p t )=i t E t p t+1 p t (qt ¹q)+ 1 2 ¾2 q : (3.16) We have three equatios ad four ukows: cosumptio,c t ; output,y t ; the real exchage rate, q t ; ad the expected real iterest rate. The fourth equatio that will close the system is the domestic cetral bak s policy rule. The equilibrium dyamics for the rest of the world take the equivalet form, but for a closed ecoomy wherep t =P H;t. ½(c t ¹c ) = ± i t+1 E t p t+1 p t ª 1 ½ 2 ¾ 2 c 2 +¾2 p +2½¾ c p ; (3.17) (c t ¹c )=(y t ¹y ): (3.18) Agai i the rest of the world case, we have fewer equatios tha ukows: two of the former ad three of the latter: cosumptio, c t ; output, y t ; ad the expected real iterest rate. The last equatio, as i the small ope ecoomy, is the policy fuctio. 4. Price-Settig Home agets set prices for period t based o period t 1 iformatio ad must satisfy all the demad at the quoted prices. It follows that the problem of home agetii periodt 1 is to choose its price,ph;t i, to maximize its objective fuctio (equatio(2.1)), but with the expected value coditioal o date t 1 iformatio, i.e., ( µ " 1 (Ct i E )1 ½ t 1 1+± 1 ½ ~ k t 2 (Yi t #): )2 (4.1) The maximizatio of equatio (4.1) is subject to the demad for idividual goods (equatio (2.9)) ad the idividual s itertemporal budget costrait (equatio (2.10)).Thus, the FONC is 13

14 E t 1 ((C t i) ½ (µ 1) PH;t i µ 1 = E t 1 ( ~ k t µ " P i 1 µ 1 H;t (P H;t ) 1 µp tc ) (P H;t ) 1 µc # " µp i µ µ 1 H;t PH;t C#) : I order to simplify this expressio we ca use two facts. First,PH;t i is predetermied so we ca take out from the expected value ad secod, i symmetric equilibriumph;t i =P H;t. Hece, it follows that E t 1 (Ct ) 1 ½ (µ 1) ª µ ) =E t 1 (~ Pt C 2 t kt µ ; (4.2) or equivaletly P H;t P H;t v u t µ µ Et 1 ~kt (P t C t ) 2o P H;t = µ 1 E t 1 f(c t ) 1 ½ g : Equatio (4.2) represets the optimal preset home good price, P H;t, where µ µ is the elasticity of substitutio across goods, ad therefore, µ 1 correspods to a xed markup. I a determiistic setup, equatio (4.2) would imply that the margial utility of the real price (P H;t =P t ) is equal to a markup times the margial disutility of producig a uit of good. We ca also obtai a log versio of equatio (4.2), assumig that all exogeous shocks are logormal µ 1 2 (1+½)E t 1 c t = 2 ¾ 2 µ µ 1 +log µ P t µ q 2 1 µ 1 2 (1 ½)2 ¾ 2 c 4 E t 1 k t 1 2 ¾2 k 2 µ 1 E t 1 q t : ¾ cq (4.3) I equatio (4.3), as i Obstfeld ad Rogo (1998), the e ects of cosumptio volatility, ¾ 2 c, over expected cosumptio is uclear because it will deped o whether½ is less or higher tha3. If½<3, the the relatioship is egative. Later i the paper, whe we calculate ex-ate utility, we simplify algebraic expressios by usig log utility (½=1) preferece, ad thus, we would be withi 14

15 the rage where cosumptio variability a ect egatively expected cosumptio all the times. Similarly, both the variace of the real exchage rate,¾ 2 q, ad the productivity volatility,¾ 2 k, have a egative e ect o expected cosumptio. The higher the volatility of the real exchage rate or productivity, the lower expected cosumptio. The reaso is that i both cases, the demad for both regios goods is modi ed, which implies a chage i the ex-ate margial disutility of work. O the other had, the demad shock has a positive e ect over expected cosumptio. Aalogously, the logarithmic versio for the rest of the world is (1+½)E t c t+1 = µ2 12 (1 ½)2 ¾ 2 c (4.4) µ µ 1 +log E µ t 1 k t 1 2 ¾2 k : Notice that i the case of the rest of the world, the price settig equatio takes a closed-ecoomy structure where expected cosumptio depeds oly o foreig variables. The reaso, as we metioedbefore, is that foreigers cosumptio of Home goods is isigi cat. 5. Specifyig Moetary Policy The astute reader will have oticed that so far moey or moetary policy have ot etered the model. We could avoid itroducig moey explictly because we describe moetary policy etirely i terms of iterest rules. This meas that, whatever the shape or form of the moey demad fuctio, each cetral bak lets moey supply adjust edogeously so that a) the omial iterest rate is equal to its chose rate ad b) moey demad is satis ed. We owspecify the moetary authority reactio fuctios, which specify the settig of such chose omial iterest rates at home ad abroad Rest-of-the-World Reactio Fuctio The rest of the world follows a mechaical moetary policy. It follows from equatio (3.17) that i steady state (whec t ¹c = 0, where overbars deote steady state values) the followig real (risk-premium exclusive) iterest rate must obtai: i t+1 E t p t+1 p t =¹r =± 1 ½ 2 ¾ 2 c 2 +¾2 p +2½¾ c p : (5.1) 15

16 We assume that the actual foreig expected real iterest rate deviates stochastically from this steady state rate: where" t =log~" t has mea0ad variace¾2 ". i t+1 E t p t+1 p t =¹r +" t ; (5.2) 5.2. Small Ope Ecoomy Reactio Fuctio The moetary authority of the small ope ecoomy desigs a optimal moetary policy. Agai we caderive a expressio for the steady state iterest rate usig the log versio of the domestic Euler equatio. It follows from equatio (3.14) that the (risk-premium exclusive) steady state domestic real iterest rate that which is obtaied whec t ¹c=0 is give by i t+1 (E t p t+1 p t )=¹r=± 1 2 à ½ 2 ¾ 2 c + µ 1+ We ow postulate a reactio fuctio such that 2 µ! 1 ¾ 2 q ¾ +2½ cq : (5.3) i t+1 (E t p t+1 p t )=¹r+à k k t +à " " t (5.4) where the à k is the coe ciet associated to the productivity iovatio, while à " is the coe ciet associated with the foreig iterest rate shock. 6. A Closed-Form Solutio I this sectio we are able to characterize the optimal moetary policy usig welfare cosideratios basedo the utility fuctio of the represetative aget. The solutio requires three major steps. First, we express the price settig equatio i terms of logs ad variaces of logs of edogeous variables. Secod, we calculate how edogeous variables respod to exogeous disturbaces; the we are able to express the variaces i terms of the variaces of exogeous shocks. Fially, we optimize the welfare fuctio obtaiig the parameters values of the optimal moetary policy. 16

17 6.1. Solvig for the Real Exchage Rate ad Cosumptio I this sectio, we are iterested, give the two optimal moetary policies, i gettig the optimal solutio for the remaiig variables of the model. I particular, we eed to compute the edogeous variaces i terms of the uderlyig exogeous shocks. Usig the ucovered iterest parity coditio ad the two policy rules (eqs. (5.4) ad (5.2)), we ca compute the real exchage rate, ad cosumptioithe small ope ecoomy. Pluggig eq. (5.4) ito equatio (??) yield ½(c t ¹c)=¹r r t = à k k t à " " t : (6.1) Now, the real UIPC together with the foreig iterest rate rule turs ito µ 3 1 r t =r t (q t ¹q)+ 2 2¾2 q =¹r+à k k t+ã " " t : Combiig these two equatios, which after rearragig becomes (q t ¹q)= à k k t (à " 1)" t + µ ¾2 q (¹r ¹r ): (6.2) Equatios (6.1) ad (6.2) give c t adq t as a fuctio of the shocks ad the variace of the real exchage rate. Notice that ¹r also depeds o edogeous variaces - that is variaces that deped, i equilibrium, o what the domestic moetary authority does - ad that¹r depeds o variaces that are exogeous, ad which we catake as give. The ext step is to express utility as a fuctio of the shocks ad exogeous variaces Variaces As we oted before, we eed to express edogeous variaces i terms of exogeous variaces. With this purpose, we ca use eqs. (6.1) ad (6.2) to obtai the real exchage rate variace, the cosumptio variace, ad the covariace betwee cosumptio ad the real exchage rate, respectively µ 2 µ 2 ¾ 2 c = Ãk ¾ 2 k ½ + Ã" ¾ 2 " ; (6.3) ½ 17

18 ¾ 2 q =Ã2 k ¾2 k +(Ã " 1)2 ¾ 2 " ; (6.4) 6.3. Calculatig Ex-Ate Utility ¾ cq = Ã2 k ½ ¾2 k +(Ã " 1)2 ¾ 2 " : (6.5) ½ I this sectio, we derive the ex-ate utility to get a welfare measure i a closedform. " 1 # X Max E t 1 [U t ] = MaxE t 1 tu t i 1 = U i 1 t Et 1 o wheree t 1 [Ut i]=e (C i t ) 1 ½ t 1 1 ½ kt 2 (Yi t )2. Usig the coditio of optimal price settig (equatio (4.2)) ad the fact that P t C t P H;t =Y t we get a relatio betwee the two compoets of the utility fuctio µ µ 1 µ t=0 E t 1 (C i t ) 1 ½ =E t 1 kt (Y t ) 2 : (6.6) Therefore, the expected utility i term of cosumptio will be : E t 1 U i t = 2µ (1 ½)(µ 1) 2µ E t 1 (C i t ) 1 ½ 1 ½ It is worth otig that expected utility depeds o the degree of moopolistic power, µ, eve i the logarithmic case (½ = 1). Therefore, without loss of geerality, we cacotiue withthe case of log utility of cosumptio. I that case, expected utility will be equivalet to E t U i t =Et 1 [c t ] (µ 1) ; (6.7) 2µ 18

19 where from equatio (4.3) we have that E t 1 [c t ] = 1 ( µ ½ µ µ 1 log 2 µ 2 µ 1 ¾ 2 q 2¾2 c 4 E t 1 [k t ] 1 2 ¾2 k 2 ¾ cq ) µ 1 ¾ E t 1 [q t ] (6.8) However, i this equatio we have that expected cosumptio ot oly depeds o variaces ad costats, but also o the expected real exchage rate. Combiig eqs. (2.12) ad (3.13) to express the expected real exchage rate i terms of expected cosumptio E t 1 [c t ]=E t 1 [q t ]+E t 1 [y t ]: Thus, usig the previous equatio to elimiatee t 1 [q t ] from equatio (6.8), oe obtais ( µ 1 E t 1 [c t ] = 2 ½ µ µ 1 + log µ 2 µ 1 ¾ 2 q 2¾2 c 4 ¾ cq ) E t 1 [k t ] 1 2 ¾2 k +2 µ 1 ¹y ¾ ; where¹y =E t 1 [y t ]. This would mea that expected cosumptio is uambiguously decreasig i both the variability ofcosumptioad the variability of the exchage rate. The ext step is to express expected cosumptio i terms ofexogeous variaces usig eqs. (6.3), (6.4), ad (6.5). It follows that : µ 1 2 E t 1 [c t ] = Ã 2 k ¾2 k +(Ã " 1)2 ¾ 2 " Ã 2 k ¾2 k +Ã2 " ¾2 " 2(1 ) Ã 2 k ¾2 k +(Ã " 1)2 ¾ 2 " + ½ µ µ 1 log E t 1 [k t ] 1 µ 1 2 µ 2 ¾2 k +2 ¹y ¾ : (6.9) 19

20 6.4. Optimal Moetary Policy I this sectio, we calculate the optimal values of the parameters of the moetary authority reactio fuctio. Pluggig equatio (6.9) ito equatio (6.7), the expected utility i terms of exogeous variaces ad costats would be µ E t 1 U i 1 2 t = à 2 k ¾2 k +(à " 1)2 ¾ 2 " à 2 k ¾2 k +Ã2 " ¾2 " 2(1 ) à 2 k ¾2 k +(à " 1)2 ¾ 2 " + ½ µ µ 1 log 2 µ E t 1 [k t ] 1 2 ¾2 k 2 µ 1 (6.10) ¹y ¾ µ 1 2µ : From the previous equatio we ca coclude uambiguously that the variace of the productivity ad the foreig iterest rate shocks a ect expected welfare egatively, sice0<<1. Fially, we ca choose the optimal values of the parameters associated to each shock i the cetral bak s policy fuctio to maximize the welfare fuctio. We are lookig for the policy who maximizes the expected utility. I order to do so, it is ecessary to choose the parameters of the reactio fuctio à k ad à " to maximize equatio (6.10). Therefore, à k =0; à " =1 2 : Therefore, the optimal moetary policy rule for the small ope ecoomy would take the followig form: r t =¹r+ 1 2 " t : where ¹r is the steady state level of the ex-post real iterest rate. These results appear to be i accordace to covetioal wisdom ad empirical evidece. We d that the parameter associated to the temporary productivity 20

21 shock is0. I other words, the optimal moetary policy of this small ope ecoomy does ot require ay type of reactio whe faced with temporary productivity shock. The ituitio of this result is give by the fact that domestic output is demaddetermied i the short-ru; therefore, a temporary rise i productivity shock does ot produce ay e ect over cosumptio ad the real exchage rate. Thus, the moetary policy does ot eed to respod to this speci c shock, because the productivity icrease is absorbed by a fall i domestic labor. A temporary chage i productivity does ot modify the fact that price exceeds margial cost (because of moopolistic competitio), ad hece domestic output will remai uchaged. I sum, i presece of a trasitory domestic productivity shock, domestic productio is still demad-determied, ad thus it is ot ecessary a moetary respose by a moetary authority who tries to maximize agets welfare. O the other had, we also study the optimal iterest rate reactio to a temporary foreig iterest rate iovatio. We d that the optimal moetary policy of this small ope ecoomy requires a positive correlatio betwee domestic ad the rest of the world s iterest rates. Speci cally, the parameter that correspods to the foreig iterest rate iovatio i the reactio is positive but less tha oe, 0 < Ã " < 1, which implies that the cetral bak does ot require a complete reactio of domestic iterest rate, because this would provoke high cosumptio variability. I the limit, if Ã1 )Ã " Ã0, which seems plausible, would mea that the more closed the ecoomy is, the less sesitive to chages i the foreig iterest rate (or backwards, as i Galí ad Moacelli (2000), (1 ) could be iterpreted as a opeess idex. ) I this case the ituitio is as follows. A uaticipated shock of the foreig iterest rate would create movemets i the level of domestic cosumptio ad exchage rate. I particular, a positive foreig iterest rate iovatio, other thigs equal, would lead to a depreciatio of the exchage rate, a ectig egatively domestic cosumptio via both the level ad the volatility of the real exchage rate (prices are preset.) The impact of the depreciatio would be too costly i terms of sustaiig smooth cosumptio over time, which would require to partially o settig the former with some istrumet. Therefore, the moetary authority of our small ope ecoomy model, whe faced with a positive exteral shock (a foreig iterest rate shock,) should respod icreasig its policy rate util they compesate, to some extet, this exteral hike. Beyod a particular poit, a higher domestic iterest rate would lead to a o-optimal Pareto equilibrium, which meas that there is a precise domestic iterest rate respose where welfare 21

22 is maximized. I other words, this result suggests that a active role by Cetral Baks i small ope ecoomies is ecessary, but upto a certai poit; otherwise, moetary policy would begi to reduce welfare. This result would also suggest that this optimal moetary policy requires a active moetary policy ad some degree of exibility i exchage rates. The reactio of the domestic iterest rate would be less tha proportioal to the foreig iterest rate icrease, while the rest would be absorbed by the exchage rate. Therefore, a exible exchage rate or at least, a maaged exchage rate regime would be required to stabilize this small ope ecoomy. Of course, this will deped o the degree of opeess of the ecoomy. As the coutry teds to close up, the optimal regime will approach to a xed exchage rate. It is also worth otig that sice prices are preset, movemets i the omial exchage rate are fully traslated ito movemets i the real exchage rate, ad hece, depreciatios are give oly by chages i the omial exchage rate. 22

23 7. Coclusios Sice the secod half of the 1990s, a substatial umber of cetral baks have move towards moetary policies that combie exible exchage rates with some kid of moetary or iterest rate rule. I this cotext, it is essetial to discuss what type of moetary policy rule is more coveiet i terms of ecoomic stability ad coutries welfare. Research o optimal moetary policies has received limited cosideratio, especially for small ope ecoomies, ad oly recetly, there has bee a iterest to study the optimality of such policies uder ormative frameworks. Some literature o optimal moetary policy cosiders di eret model structures with price stickiess. However, previous research has bee based o closed or two-coutry equally size models, or has lacked of a adequate welfare criterio for ope ecoomies. I view ofthat, our paper exteds ObstfeldadRogo (1998, 2000) framework to a small ope ecoomy cotext i which foreig variables are ot a ected by ay small ecoomy actio. I doig so, we developeda optimizig model, i which we derived the optimal moetary ad exchage rate policy for a small stochastic ope ecoomy with imperfect competitio ad short ru price rigidity. The optimal iterest rate policy has a exact closed-form solutio ad is obtaied usig a welfare criterio derived from the utility fuctio of the represetative aget. This welfare criterio suggests that the moetary authority has to avoid large chages i both cosumptio ad the real exchage rate. I particular, we require a policy that miimizes the e ects of higher volatility i exogeous variables over cosumptio ad the real exchage rate. Hece, the welfare aalysis allows us to cosider the optimal moetary policy for a coutry, which is subject to productivity ad foreig iterest rate shocks. With such a tool, it is possible to aswer questios like: give uexpected chages i productivity ad i exteral acial coditios, what is the optimal moetary policy reactio? What determies the respose ofthe optimal moetary policy istrumet? What is the exchage rate regime associated with this optimal moetary policy? Our results appear to be i accordace to covetioal wisdom ad empirical evidece. We d that the optimal moetary policy of this small ope ecoomy requires a positive correlatio betwee domestic ad the rest of the world s iterest rates. Ideed, the parameter associated to the foreig iterest rate shock i the reactio fuctio of the moetary authority is positive. Furthermore, it should be oted that it is also less tha oe, which implies that we do ot require a complete reactio of domestic iterest rate, because this would cause high cosumptio 23

24 variability. The reactio of the domestic iterest rate would be less tha proportioal to the foreig iterest rate icrease, while the rest would be absorbed by the exchage rate. Thus, a exible exchage rate or at least, a maaged exchage rate regime wouldbe required to stabilize the small ope ecoomy. The aalysis also ds out the optimal degree of itervetio for a small ope ecoomy that udergoes a temporary productivity shock. We coclude that, the optimal moetary policy i this framework should ot react whe faced with a productivity iovatio. The reaso is that domestic output is demad-determied i the short-ru; cosequetly, a temporary icrease i productivity shock would ot produce ay e ect over cosumptio ad the real exchage rate. Therefore, moetary policy would ot eed to respod to this speci c shock, because the productivity icrease is absorbed by a fall i domestic labor. A atural ext step for further research would be to extedthe aalysis with other type of shocks, such as cost push shocks. It would be also attractive to aswer the same questios but i a model with more complex acial structure, as i Berake ad Gertler (1989) admore recetly Céspedes, Chag, ad Velasco (2000), amog others. 24

25 8. Appedices 8.1. Coutry ad Ecoomic Size This appedix has two purposes. First, together with the assumptio that iitial et iteratioal asset holdigs is zero, we showthe ecessary coditio of havig costat shares of per capita world real icome. Secod, for the geeral case i which both coutries cosume both home ad foreig good, we demostrate that the populatio size, deoted by, has to be equal to the ecoomic size, represeted by. The market clearig coditios require that [P t C t +(1 )P t C t ] = P HY (8.1) (1 )[P t C t +(1 )P t C t ] = (1 )P FY Y = C H +(1 )C F (8.2) (1 )Y = (1 )C H+C F Set (8.1) characterizes the equilibrium coditio i which output supplies equal demads, while set (8.2) represets the equilibrium of quatities produced ad cosumed. Dividig the two equatios of set (8.1) we ca get the followig expressio P H = Y P F Y µ 1 µ 1 (8.3) Now, pluggig equatio (8.3) ito the ratio of demads fuctios for the composite home ad foreig goods (eqs. (2.8)), we ca obtai a expressio for relative cosumptio of home ad foreig goods i terms of relative prices C H C F = The parallel foreig relatio is µ 1 PF P H (8.4) C H C F µ 1 P = F P H (8.5) 25

26 Thus, icludig eqs. (8.4) ad (8.5) ito equatio (8.3) we ca get the rst ad secod equality, respectively, of µ µ Y Y =C H 1 = C F 1 C F C H (8.6) which implies that C H =C F µ CF C H (8.7) Next use set (8.2). Dividig the two equatios of this set, we obtai the followig expressio Y 1 Y =C H+(1 )C F (1 )C H +C F (8.8) Pluggig eqs. (8.6) ad (8.7) ito equatio (8.8) ad multiplyig by C F C F C H C F = C H +(1 )C ³ F ) C H (1 )C F CF +C CH F C F = C H C F (1 ) ³ C F CH +(1 ) + Letx= C H C, the F x[(1 )x+]=x+(1 ) )x=1 )C H =C F (8.9) We also kow that C = C H C1 F ; C (1 ) = (C H) (C F) 1. Therefore, if we 1 (1 ) 1 divide these expressios ad cosider equatio (8.7) ad equatio (8.9) we get C C =C H C F = C F C H =1 )C=C (8.10) Fially, we require a balaced curret accout, i.e.,y P C=0. Factorizig PH the rst equatio of set (8.1) 26

27 Therefore, Y P C= (1 ) P µ 1 P C+ C (8.11) P H P H P H Y P C=0 iff C µ µ 1 P H C = 1 (8.12) But we already kow that C C =1, the we eed that = Real Exchage Rate ad Terms oftrade: Some Idetities This appedix presets the derivatio of some importat idetities, such as the real exchage rate adthe terms of trade. De e the real exchage rate asq t = EtP t, wheree Pt t is the omial exchage rate, while P t ad P t are the price level of the Home ad Foreig coutries, respectively. Similarly, we ca de e the terms of trade asx t = P F;t = EtP H;t, wherep PH;t PH;t F;t adp H;t are the price level of the goods produced by Foreig ad Home coutries, respectively. The geeral price level of Home ad Foreig are de es asp t =PH;t P1 F;t ad P t = P H;t 1 P, F;t respectively. Applyig atural logs to these four equatios ad itroducig oe of the key assumptios of the model, that is assumig that the share of the Home goods price is egligible i the Foreig coutry as i a closed ecoomy (i.e. set = 0.), we get the followig set of equatios q t = s t +p t p t (8.13) x t = s t +p t p H;t p t = p H;t +(1 )p F;t p t = p H;t Combiig all the four equatios we ca obtai a expressio of the real exchage rate as a fuctio of the terms of trade q t =x t (8.14) 27

28 8.3. Price Rigidity Home agets set prices for period t based o period t 1 iformatio ad must satisfy all demads at the quoted prices. It follows that the problem of home ageti i periodt 1 is to choose its price, PH;t i, to maximize its objective fuctio. It follows that the rst order coditio is " µ PC 2 # E[C 1 ½ (µ 1)]=E kµ Sice home prices are predetermiede[p H ]=P H, we ca get a expressio for home prices s µ E[kP P H = 2 C 2 ] µ 1 E[C 1 ½ (8.15) ] We also kowthat Usig eqs. (8.15) ad (8.16) µ PH P H 1 = P H P t =P H;t P1 F;t (8.16) s µ µ 1 Solvig eachcompoet (with expected value), E[kE 2(1 ) C 2 ] E[C 1 ½ ] (8.17) E ke 2(1 ) C 2 = Efexp[logk+2(1 )e+2c]g = expfe[logk]+2(1 )E[e]+2E[c]+ 1 2 ¾2 k +2(1 ) 2 ¾ 2 e +2¾2 c +2(1 )¾ ke+2¾ kc +4(1 )¾ ce Similarly, E[C 1 ½ ]=exp[(1 ½)E[c]+ 1 2 (1 ½)2 ¾ 2 c Therefore, equatio (8.17) turs to 28

29 µ PH P H 1 = µ µ 1=2 expf 1 µ 1 2 E[logk]+(1 )E[e]+1 2 (1+½)E[c] (8.18) ¾2 k +(1 )2 ¾ 2 e +[1 1 4 (1 ½)2 ]¾ 2 c +(1 )¾ ke +¾ kc +2(1 )¾ ce g Applyig logs (ad assumig that¾ ke =¾ kc, ad kowig that¾ 2 e =¾2 q, ad ¾ ce =¾ cq because prices are predetermied) we get: (1 )(p H p H E[e]) = 1 µ µ 2 log + 1 µ 1 2 E[logk]+1 2 (1+½)E[c] ¾2 k +(1 )2 ¾ 2 q +[1 1 4 (1 ½)2 ]¾ 2 c +2(1 )¾ cq But we kow that the real exchage rate is proportioal to the terms of trade i the followig way: E[q t ] = E[s t ]=(E[e]+p H p H) ) (p H p H E[e])= 1 q t (remember that prices are preset i which casee[p H;t ]=p H ade p H;t =p H ) Thus, µ 1 E[q] = 1 µ µ µ 1 2 E[logk]+1 2 (1+½)E[c] ¾2 k +(1 )2 ¾ 2 q +[1 1 4 (1 ½)2 ]¾ 2 c +2(1 )¾ cq Fially, we ca obtai a expressio for expected cosumptio (equatio (4.3)) E t 1 [c t ] = 1 ½ 2(1 ) 2 ¾ 2q 1+½ µ µ ½ ½ log µ (1 ½)2 ¾ 2 c 4(1 )¾ cq E t 1 [k] 1 µ 1 2 ¾2 2 ¾ E t 1 [q t ] ¾ 29

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