Exchange Rates, Oil Price Shocks, and Monetary Policy in an Economy with Traded and Non-Traded Goods.

Transcription

1 CAEPR Working Paper # Exchange Raes, Oil Price Shocks, and Moneary Policy in an Economy wih Traded and Non-Traded Goods. Michael Plane Indiana Universiy Augus 14, 2009 This paper can be downloaded wihou charge from he Social Science Research Nework elecronic library a: hp://ssrn.com/absrac= The Cener for Applied Economics and Policy Research resides in he Deparmen of Economics a Indiana Universiy Bloomingon. CAEPR can be found on he Inerne a: hp:// CAEPR can be reached via a caepr@indiana.edu or via phone a by NAME. All righs reserved. Shor secions of ex, no o exceed wo paragraphs, may be quoed wihou explici permission provided ha full credi, including noice, is given o he source.

2 Exchange Raes, Oil Price Shocks, and Moneary Policy in an Economy wih Traded and Non-Traded Goods. Michael Plane Augus, 2009 Absrac This paper examines moneary policy responses o oil price shocks in a small open economy ha produces raded and non-raded goods. When only labor and oil are used in producion and prices are sicky in he non-raded secor he behavior of inaion, he nominal exchange rae, and he relaive price of he non-raded good depends crucially upon wheher he raio of he cos share of oil o he cos share of labor is higher for he raded or non-raded secor. If he raio is smaller (higher) for he raded secor hen a policy ha fully sabilizes non-raded inaion causes he nominal exchange rae o appreciae (depreciae) and he relaive price of he non-raded good o rise (fall) when here is a surprise rise in he price of oil. Similar resuls can hold for a policy ha sabilizes CPI inaion. Under a policy ha xes he nominal exchange rae, non-raded inaion rises (falls) if he raio is smaller (larger) for he raded secor. Analyical resuls show ha a policy of xing he exchange rae always produces a unique soluion and ha a policy of sabilizing non-raded inaion produces a unique soluion so long as he nominal ineres rae is raised more han one-for-one wih rises in non-raded inaion. A policy ha sabilizes CPI inaion, however, produces muliple equilibria for a wide range of calibraions of he policy rule. Keywords: oil prices, moneary policy, inaion, exchange raes JEL Classicaions: F41, E52, Q43 Dae: Augus, Any commens would be grealy appreciaed. I would like o hank my advisor Ed Bue for many helpful commens abou his paper. As always, I assume full responsibiliy for any and all errors. Deparmen of Economics, Ball Sae Universiy, Whiinger Business Building, 2000 W. Universiy Ave., Muncie, IN 47304, or by a mdplane@bsu.edu 1

3 1 Inroducion A search of he recen lieraure urns up a number of papers wrien on how policy makers should respond o oil price shocks, including bu no limied o Leduc and Sill (2004), Dhawan and Jeske (2007), Plane (2009a), and Bodensein, Erceg, and Guerrieri (2008). Wih almos no excepions his work makes use of models suied for a large developed counry, such as he Unied Saes. For policy makers from small open economies who may be concerned wih nominal exchange raes, real exchange raes, and rade balances, here is a dearh of resuls available. This is rue despie he fac ha many small open economies are heavy users of oil producs and hence are vulnerable o oil price shocks in he same way ha large economies are. In his paper I address his issue by developing a coninuous ime, small open economy model ha can be used o invesigae he performance of dieren specicaions of moneary policy when he price of oil changes. The economy is small, fully inegraed ino world capial markes, and uses labor and oil o produce boh raded and non-raded goods. Prices are sicky in he non-raded secor so ha moneary policy may have imporan real eecs on he economy. To my knowledge his is he rs paper in his area of work ha uses a dynamic general equilibrium model ha conains raded and non-raded goods. Using his model I consider he performance of hree dieren specicaions of moneary policy. The rs is an inaion argeing policy ha sabilizes he inaion rae of prices in he non-raded secor (non-raded inaion). The second policy considered is an inaion argeing policy ha sabilizes CPI inaion. The hird policy xes he nominal exchange rae hrough inervenion in he foreign exchange marke. I presen boh analyical and numerical resuls regarding he behavior of he model's variables under each policy and analyical resuls which show he condiions under which each policy generaes a unique soluion. The main resuls are he following. Firs, I show ha wheher he raio of he cos share of oil o labor is higher in he raded or non-raded secors is crucial o pinning down he 1

4 behavior of many of he model's variables. For example, under a policy ha successfully sabilizes non-raded inaion around is arge level i is possible for he nominal exchange rae o appreciae or depreciae and he relaive price of he non-raded good o rise or fall depending upon he raios. If he raio is higher in he non-raded secor hen a rise in he price of oil causes he nominal exchange rae o appreciae and he relaive price of he non-raded good o rise. If he raio is lower for he non-raded secor hen he opposie occurs. For cerain calibraions of he moneary policy rule, a similar resul holds for he policy ha sabilizes CPI inaion. Under a policy of xed exchange raes, he response of non-raded inaion is pinned down by he raio and rises (falls) when he raio is higher (lower) in he non-raded secor. The inuiion behind hese resuls is ha he oil price shock may have asymmeric eecs on coss (and hence inaion) in he raded and non-raded secors. A policy of sabilizing non-raded inaion works by oseing any changes in cos in he non-raded secor by properly adjusing he nominal exchange rae over ime. Bu since coss may rise or fall in he non-raded secor (holding he exchange rae xed) his policy may bring abou nominal exchange rae appreciaion or depreciaion. Similar inuiion explains he resuls for he xed exchange rae policy. Under his policy he nominal exchange rae is no used o change coss in he non-raded secor and consequenly inaion rises (falls) when coss rise (fall) in he non-raded secor. I also derive analyical resuls ha show under wha condiions hese policies produce a unique soluion. The resuls show ha a policy ha xes he nominal exchange rae always produces a unique soluion while a policy which sabilizes non-raded inaion produces a unique soluion so long as he nominal ineres rae is raised by more han one-for-one wih movemens in non-raded inaion. Under a policy ha sabilizes CPI inaion, however, muliple equilibria occur for a wide range of calibraions for he policy rule. The analyical resuls show ha he weigh of he non-raded good in he CPI is very imporan in deermining wha areas of he parameer space produce indeerminacy. The paper proceeds as follows. Secion 2 inroduces he deails of he model and he soluion 2

5 procedure. Secion 3 presens he resuls. Finally, he paper concludes in secion 4. 2 The Model The model is a coninuous ime, perfec foresigh model of a small, open economy. The economy is small in ha i is a price-aker wih regards o he raded good and oil producs. Economic aciviy on he household side is conrolled by a represenaive agen who derives uiliy from raded and non-raded consumpion goods and dis-uiliy from providing labor o he raded and non-raded secors. The agen has access o a domesic currency, a domesic nominal bond, and radable real bond. Money is moivaed by assuming ha holdings of real money balances provides uiliy o he agen. On he producion side, he raded and non-raded goods are produced using oil and labor. The raded good is produced by a represenaive rm operaing under perfec compeiion while he non-raded good is an aggregae of diereniaed producs produced by a coninuum of rms operaing under monopolisic compeiion. As far as noaion is concerned, dx is he dierenial of he variable X, Ẋ is he ime derivaive of X, X o is he seady sae value of X, and ˆX is he log-dierenial of X, i.e. ˆX = dx. X 2.1 Prices The raded good is he numeraire and sells domesically a a price of ep, where e is he nominal exchange rae and P is he world price of he raded good in dollars. The nominal exchange rae measures he number of unis of domesic currency per 1 dollar and he rae of depreciaion is χ = ė e. For convenience I assume ha P is consan and equal o 1 so nominal variables are deaed by e. In general, nominal variables are denoed wih a ilde while heir real versions are no. 3

6 The nominal price of he non-raded good is given by P n, wih an inaion rae of π n = P n P n. The relaive price of he non-raded good is denoed as P n and is conrolled by he dierenial equaion P n = P n (π n χ). (1) Real wages are denoed as W n and W for he non-raded and raded secors, respecively. The dollar price of oil in world markes is denoed as P o. Two ocial measures of prices are also used here. The rs is he Consumer Price Index CP (CPI), P I, which in his model is also equivalen o he core CPI as households do no direcly purchase oil producs. The oher is he GDP deaor, P GDP. The formulas used o calculae hese are given laer. 2.2 The Agen's Opimizaion Problem The agen's insananeous uiliy funcion is of he form C 1 1 τ 1 1 τ + κ 2 ( ) 1 1 m τ P CP I 1 1 τ (L n + L ) 1+ 1 µ κ , µ where C ( C, C n) = [C σc 1 σc ] σc + κ 1 C n σc 1 σc 1 σc, aggregaes non-raded consumpion goods C n and raded consumpion goods C o produce an aggregae measure of consumpion, C. The parameers τ, σ c, and µ are he ineremporal elasiciy of subsiuion, he elasiciy of subsiuion beween raded and non-raded consumpion goods, and he wage elasiciy of labor supply, respecively. Toal labor supply is L = L n + L where L is he amoun of labor provided o he raded secor and L n = 1 0 Ln (i)di is aggregae labor supplied o rms in he non-raded secor. The erm 4

7 m = M e exchange rae. is real holdings of domesic currency while P CP I is he CPI deaed by he nominal I simplify he uiliy funcion by replacing C wih is indirec uiliy version. This is done by solving for he Marshallian demand funcions of C and C n and subsiuing hem back ino C, i.e. maximizing C subjec o he consrain ha C + P n C n = E, where E is real aggregae consumpion expendiure. Afer his is done he erm represening uiliy from consumpion is ( ) 1 1 E τ P CP I 1 1 τ The funcional form for he uiliy funcion implies ha he exac equaion for P CP I is. P CP I = ( ) κ σc 1 P n1 σc 1 σc. (2) CPI inaion is herefore π CP I P n = χ + γ n P, (3) n where γ n is he share of non-raded consumpion ou of aggregae consumpion spending. Real wealh is given by A = m + b + b, (4) where b is holdings of he radable real bond and b is real holdings of a nominal bond raded domesically. The real ow consrain for he agen is given by A = W n L n + W L + r b + (i χ) b + T + Π n E χm. (5) In he budge consrain T is lump sum ransfers from he governmen, Π n is rebaed pros from he non-raded secor, r is he real ineres rae paid on he radable bond, and i is he nominal ineres rae paid on he domesic bond. 5

8 The agen's problem is o maximize ( ) 1 1 E τ P CP I τ + κ 2 ( ) 1 1 m τ P CP I 1 1 τ (L n + L ) 1+ 1 µ κ µ e ρ d, (6) subjec o equaions (4) and (5). Dening λ 1 as he muliplier on he ow consrain, he rs order condiions for he household's problem can be wrien as E 1 τ P CP I 1 1 τ ) 1 τ κ 2 ( m E = λ 1, (7) = i, (8) i = r + χ, (9) κ 3 L 1 µ = λ 1 W n, (10) κ 3 L 1 µ = λ 1 W, (11) λ 1 λ 1 = ρ r. (12) The rs order condiions for E and m are sandard. The rs order condiion for b provides a no-arbirage condiion ha ses he domesic nominal ineres rae i equal o he world real ineres rae plus he rae of depreciaion. Equaions (10) and (11) se he foregone uiliy los from working more in each secor equal o he respecive bene of working more in ha secor. As labor is inersecorally mobile and wages are exible, combining hese equaions gives he condiion ha W = W n. Since he wo are equal a all imes I simply refer o he real wage as W. The remaining equaion hen reads κ 3 L 1 µ = λ1 W. This equaion pins down he aggregae labor supply decision of he agen as a funcion of he marginal uiliy of income and he real wage. Finally, he rs order condiion for A 6

9 gives he condiion ha λ 1 = 0 for all as ρ = r. 2.3 Producion Producion in he raded secor is done by a represenaive rm operaing under perfec compeiion. Technology in his secor is [ Q = A L σ 1 σ ] σ + b 1 O σ 1 σ σ 1, (13) where σ is he elasiciy of subsiuion beween labor and oil and b 1 is a disribuion parameer. The real uni cos funcion for his echnology is φ = 1 [ ] 1 W 1 σ + b σ A 1 P o1 σ 1 σ. (14) The facor demands for labor and oil are given by L = A σ 1 W σ Q, (15) and O = A σ 1 b σ 1 P o σ Q, (16) respecively. Producion in he non-raded secor is done by a coninuum of rms operaing under monopolisic compeiion. Each rm produces a diereniaed good, Q n (i), which i sells for p(i). Technology for rm i is [ Q n (i) = A n L n (i) σn 1 σn ] σn + b 2 O n (i) σn 1 σn 1 σn, (17) where σ n is he elasiciy of subsiuion beween oil and labor and b 2 is a disribuion 7

10 parameer. Coss for he non-raded rm, denominaed in dollars, is φ n = 1 [ ] 1 W 1 σ n + b σn A n 2 P o1 σn 1 σn. (18) Noe ha as φ n is coss denominaed in dollars. Coss denominaed in erms of he nonraded good are herefore given by φn P n. The facor demands for L n (i) and O n (i) are respecively L n (i) = φ nσn A nσn 1 W σn Q n (i), (19) O n (i) = φ nσn A nσn 1 P o σn b σn 2 Q n (i). (20) I assume ha he diereniaed goods are aggregaed ino a nal non-raded good, Q n, by a represenaive rm operaing under perfec compeiion. This rm uses he echnology [ 1 Q n = 0 Q n (i) θn 1 θ n ] θ n θ n 1, where θ n is he elasiciy of subsiuion beween he various goods. The zero pro condiion for he represenaive rm implies ha he demand for good i is Q n (i) = ( p(i) P n ) θ n Q n, (21) and ha he price index is [ 1 P n = 0 ] 1 p(i) 1 θn di 1 θ n. (22) Prices for he individual rms, p(i), are sicky according o seup similar o Calvo (1983). Each rm i reses is price when i receives a sochasic signal o do so. The signal follows a Poisson process wih an average waiing ime of 1, so ha ω can be calibraed o conrol ω he degree of price sickiness. I also assume ha prices grow a rae equal o he seady sae level of inaion. This ensures ha he markup in seady sae is only aeced by he degree of subsiuabiliy beween he goods, θ n, and no he seady sae rae of inaion 8

11 or oher facors. Readers ineresed in more deails or derivaions regarding his are referred o Plane (2009b). Given ha a rm receives a signal o rese is price, he rs order condiion for p(i) is found by solving he pro maximizaion problem, max p(i) ωe (s )(ρ+ω) λ 1s λ 1 ( p(i) e πn o (s ) ) 1 θ n P s n Q n s φn s Q n s P n s ( p(i) e πn o (s ) ) θ n ds. (23) P n s The rs order condiion is (θ n 1) ( ωe (s )(ρ+ω) λ 1s p(i) e πn o (s ) λ 1 P s n ) 1 θ n Q ns = θn ( ) ωe (s )(ρ+ω) λ θ 1s φ n s Qn s p(i) e πn o (s ) n λ 1 Ps n ds. P s n (24) I impose a symmeric equilibrium so ha each rm ha has he opporuniy o choose is price chooses he same price, p. Afer imposing he symmeric equilibrium and performing a good deal of algebra, linearizing he rs order condiion around a seady sae produces an equaion ha governs he evoluion of π n, π n = ω(ρ + ω) ( dφ n φ n o dp n ) + ρdπ n. (25) Po n This equaion is jus a coninuous ime analogue of he usual dierence equaion ha occurs in discree ime models and holds for all values of π n o. Noe ha inaion in he non-raded secor is a funcion of coss denominaed in erms of he non-raded good, φ n P n, no φ n. This occurs because from he poin of view of he non-raded rms wha maers is he price of non-raded good, no he price of he raded good. 1 Aggregaion is done linearly across rms. Equaion (21) aggregaed gives 1 0 Q n (i)di = n Q n, 1 The fac ha I have chosen he raded good as he numeraire does no change he resuls. I only aecs he way in which he variables are measured. 9

12 where n is a sae variable measuring price dispersion in he non-raded secor, n = 1 0 ( p(i) P N ) θ n di. (26) Aggregaion of equaions (19) and (20) gives he aggregae demand for labor and oil, L n = φ nσn A nσn 1 W σn Q n n, (27) O n = φ nσn A nσn 1 P o σn b σn 2 Q N n. (28) I can be shown ha he law of moion for n is n = θ n n (π n π n o ) + ωp θn ω n, (29) where p = p P n. A rs order approximaion of his equaion gives n = ωd n, which shows ha n is irrelevan for he pahs of he variables o a rs order. 2.4 Gross Domesic Produc The use of inermediae inpus means ha GDP is no equal o he sum of gross oupu in he raded and non-raded secors. I calculae real GDP, Q g, as Q g = Q + P n Q n P o (O + O n ). (30) The GDP deaor, P GDP is calculaed as he raio of nominal GDP o real GDP, and GDP deaor inaion is π GDP = P GDP P GDP. 10

13 2.5 The Public Secor The public secor consiss of he cenral bank and he scal auhoriies, hereafer simply he governmen. The balance shee of he cenral bank is M = M d + eb g, where M d is he domesic componen of he nominal money supply and b g is ocial holdings of he radable bond. The growh of he money supply is given by Ṁ = Ṁ d + eḃ g. The consolidaed governmen budge consrain is ṁ = T + ḃ g r b g χm. (31) For all cases considered here I assume ha T passively adjuss o clear he governmen budge consrain. 2.6 Marke Clearing in he Non-Traded Secor Toal oupu mus equal oal spending on he non-raded consumpion good. The marke clearing condiion in he non-radable secor is herefore Q n = C n. (32) Noe ha oupu in he non-raded secor is demand deermined so i adjuss o mee demand for he consumpion good. 11

14 2.7 The Curren Accoun In equilibrium W L = Q P o O, Π n = P n Q n P o O n W L n, and b = 0. Using hese equaions in he agen's budge consrain, and hen combining he resuling equaion wih he governmen budge consrain gives an equaion linking he curren accoun and ne foreign asse accumulaion, ḃ + ḃ g = P n Q n + Q + r (b + b g) E P o (O + O n ). (33) If one makes use of he marke clearing condiion for he non-radables secor his reduces o ḃ + ḃ g = Q + r (b + b g) C P o (O + O n ). This equaion simply saes ha he economy accumulaes foreign asses whenever he sum of income from producion of he raded good and foreign invesmen is greaer han spending on raded goods. 2.8 The Price of Oil I work under he assumpion ha shocks o he real price of oil are persisen, bu emporary, in naure. Following a surprise shock he price reurns monoonically o is seady sae level according o P o = α (P o P o o ), α > 0. (34) 2.9 Calibraion The model is calibraed o an iniial seady sae. A able a he end of he ex conains he saring values of he model's variables and he seings for he deep parameers. The following gives a brief discussion abou he calibraion choices for a few parameers which are imporan for he main resuls of he paper. The appendix conains a secion explaining 12

15 he procedure in more deail. Price Sickiness in he Non-raded Secor (ω) The average life of a price is 1. The ω empirical evidence suggess a range beween one o wo quarers using disaggregaed daa bu a longer life for more aggregaed models. I choose a value of 2 so ha on average prices are rese every half year. Elasiciy of subsiuion beween non-raded goods (θ n ) The value of θ n conrols he seady sae markup of price over marginal cos in he non-raded secor. The calibraions for his parameer are all over he map so here is no one acceped calibraion. I choose a value of 12 so ha he markup is small bu no negligible. Speed of Adjusmen of Oil Prices (α) Esimaes for his parameer exis for monhly and quarerly daa. Pre 1986 daa sugges a value beween.95 and 1 for he corresponding AR(1) coecien while pos-86 daa sugges a value around.80. This corresponds o a range for α beween.20 and 0. I choose he conservaive value of.10. Elasiciy of subsiuion beween oil and labor (σ and σ n ) Alhough empirical evidence for hese elasiciies for raded and non-raded secors is non-exisen, hese parameers can be calibraed o mach he price elasiciy of oil demand for rms. A variey of sudies using more aggregaed daa sugges ha hese elasiciies are generally small, reecing he diculies of subsiuing for oil producs in he shor run. Given he sparsiy of daa, I se σ = σ n o keep he wo secors symmeric in his regard and use a value of.50. Spending on oil by rms in he raded and non-raded secors (Po o Oo and Po o Oo n Daa on aggregae use of oil by all rms varies depending upon he price of oil, he ime frame being considered, and he counry in quesion. For he baseline calibraion I work wih Po o Oo = Po o Oo n =.03. An alernaive calibraion where Po o Oo n =.02 is also considered. As I explain laer hese wo calibraions provide cover he wo possible cases wih regards o he raio of he cos shares ha was discussed in he inroducion. 13

16 2.10 Soluion Mehod I solve he model by deriving he core dynamic sysem of dierenial equaions, he variables of which vary depending upon he policy considered. I hen perform a rs order linearizaion around he iniial seady sae and solve he dierenial equaions using sandard echniques. In all cases I check o ensure ha a soluion exiss and ha i is unique. The model is saddled wih a uni roo and o ge around his when solving he model numerically I assume ha r is a funcion of he deb-gdp raio of he economy, r = ρ + h ( b + b g P n Q n + Q P o (O + O n ) b o + b ) g,o, h < 0. (35) Po n Q n o + Q o Po o (Oo + Oo n ) This adds a small negaive eigenvalue o he dynamic sysem insead of he 0 eigenvalue. 3 Resuls Before considering specic policies I rs derive some analyical resuls ha hold regardless of he policy in place. Recall ha he real uni cos funcions for he raded and non-raded secors are, respecively, φ = 1 A [ W 1 σ + b σ 1 P o1 σ ] 1 1 σ, φ n = 1 A n [ W 1 σ n + b σn 2 P o1 σn ] 1 1 σn. Coss in he raded secor are equal o uniy and consan. By aking he oal derivaive of he equaion for φ one can herefore solve for how W changes as a funcion of he change in he real price of oil. Doing so gives Ŵ = ζ o ζ l ˆP o, (36) where ζ o and ζ l is he cos share of oil and labor, respecively, in he raded secor. Taking he oal derivaive of he equaion for φ n and using he resul for he change in W gives one 14

17 he change in φ n, where η = P o O n Q n dφ n = η ˆ P o, (37) 1 ζ o ζ l ζ n o ζ n l, (38) and he erms ζo n and ζl n are jus he cos shares for he non-raded secor. Wheher φ n rises or falls depends upon he sign of η, which is deermined enirely by wheher or no he raio of he cos share of oil o he cos share of labor is higher in he raded or non-raded secor. If he raio is higher in he raded secor hen η is negaive and φ n falls when he price of oil rises. If he raio is smaller in he raded secor hen φ n rises whenever he price of oil rises. The inuiion behind his resul is ha when η is negaive he drop in W is large enough o ose he increase in he price of oil so φ n falls. In he case when η is posiive he drop in wages is no large enough o ose he rise in he price of oil and φ n rises. Wih hese resuls in hand i is possible o begin hinking abou he behavior of inaion in he non-raded secor because he linearized dierenial equaion governing π n is a funcion of dφ n dp n. As P n is pre-deermined a any poin in ime he iniial change in P n will always be ˆP n = ê. I is hrough his mechanism ha moneary policy can aec inaion in he non-raded secor. The reason his is rue is ha changes in he nominal exchange rae aec he domesic price of oil ha he non-raded rms mus pay vis-a-vis he price of he non-raded good. If he domesic currency appreciaes (depreciaes) hen he cos of oil drops (rises) in erms of he price of he non-raded good. This does no maer for he rm in he raded secor because he price of he raded good varies direcly wih he nominal exchange rae. This will be he driving facor in many of he resuls ha follow. 15

18 3.1 Sabilizing Non-Traded Inaion The rs policy I consider is one where he cenral bank focuses on sabilizing he inaion rae of he non-raded good, π n. To implemen he policy I assume ha he cenral bank publicly announces ha i will adjus he nominal ineres rae according o i = i o + α π (π n π n o ). (39) As he nominal ineres rae is he moneary policy arge, he nominal supply of money passively adjuss o clear he money marke and he nominal exchange rae oas. The core dynamic sysem of he model under his policy consiss of equaions (1), (12), and (25) for he jump variables P n, λ 1, and π n, and equaions (33) and (34) for he sae variables b and P o. In he rue model where r = r, or assuming ha dr is negligible and can be ignored, i is possible o derive some very useful analyical resuls for his policy. In paricular, i is possible o show he condiions under which he policy produces a deerminae soluion and o also produce some resuls abou he behavior of π n, P n, and several oher variables on he ransiion pah. This is possible because under his policy he sysem of linearized dierenial equaions for π n, P n, and P o form a subsysem ha can be solved independenly of he oher equaions in he model. The procedure for deriving he sub-sysem is as follows. The rs order condiion for b, equaion (9), gives he ineres rae pariy condiion. When linearized his equaion shows us ha di = dχ. Combining his wih a linearized version of he moneary policy rule in equaion (39) gives dχ = α π dπ n. Making use of his resul and he previously derived equaion for dφ n one can wrie he 16

19 sysem of linearized equaions for P n, π n, and P o as P n π n P o = 0 P n o (1 α π ) 0 k1 ρ k2 0 0 α dp n dπ n dp o, where k1 = k2 = ω(ρ + ω), P n o ω(ρ + ω) η. φ n o The coecien k1 is always posiive bu k2 may be greaer han, equal o, or less han 0 depending upon wheher or no η is negaive, zero, or posiive. The roos of he sub-sysem are r 1 = α, r 2 = 1 [ ] ρ 4ω(ω + ρ)(1 α π ) + ρ 2 2, r 3 = 1 [ ] ρ + 4ω(ω + ρ)(1 α π ) + ρ 2 2. Under an inaion argeing policy boh π n and P n are jump variables so if he sysem is o have a unique soluion here mus be 1 negaive eigenvalue and 2 posiive eigenvalues. The rs roo is always negaive so i remains o show under wha condiions he las wo roos are posiive. I presume rs ha he only relevan values of α π are hose greaer han 0. Under ha condiion we can see ha r 2 <=> 0 as α π <=> 1, wih he possibiliy of an imaginary roo if α π is large enough. Given his we can rule ou any value of α π 1 as a possible candidae. For r 3, close inspecion reveals ha ha roo will always be greaer han 0 for α π > 0, wih he possibiliy of an imaginary roo for large enough values of α π. In oher words, he condiion under which his policy produces a unique soluion is exacly he condiion found in he sandard closed economy New Keynesian model, namely α π > 1. 17

20 Making use of he eigenvecor associaed wih r 1 he soluion for he variables along he ransiion pah is, dp n dπ n = dp o k2(α π 1) ω(ρ+ω)(α π 1)+α(α+ρ) k2(α) ω(ρ+ω)(α π 1)+α(α+ρ) where h1 is he iniial surprise jump in he price of oil. 1 [ h 1 e α ], The soluion for π n gives us wo useful pieces of informaion. Firs, noice ha for a given calibraion ha he deviaions of π n from is arge become smaller and smaller as α π becomes larger and larger. This applies o boh he iniial jump and along he enire ransiion pah. In oher words, how much inaion varies on he ransiion pah depends direcly upon how srongly he cenral bank announces i will respond o a rise in inaion. Second, he direcion of he iniial jump in π n is enirely deermined by he sign of η, operaing hrough k2. If η is posiive hen k2 is negaive and non-raded inaion jumps up. If η is negaive hen k2 is posiive and non-raded inaion jumps down. The iniial jump in he relaive price depends upon boh he soluion jus derived and he change in he relaive price across seady saes. For he simple model considered here, however, we can ignore he laer issue because he change across seady saes is 0. 2 wih π n, he direcion of he iniial jump and he evoluion of P n on he ransiion pah is deermined solely by he sign of η. For cases when η is posiive P n iniially rises and hen declines monoonically on he ransiion pah. If η is negaive hen P n falls iniially and rises monoonically on he ransiion pah. As P n is iniially xed, his means ha he nominal exchange rae appreciaes in he former case and depreciaes in he laer case. The magniude of he iniial jump is also sensiive o he value of α π. In he limi as α π goes o inniy he coecien collapses o θ n θ n 1 η. The inuiion behind he resuls for π n and P n is simple. In order o keep π n close o is 2 Since he shock is emporary wages do no change across he seady sae and hence neiher does φ n. Using he rs order condiion for p(i) one can hen show ha P n does no change across seady sae. As 18

21 arge φn P n mus remain nearly consan when he price of oil rises. As discussed earlier, he change in W is pinned down in he raded secor and his hen pins down he change in φ n, which may increase or decrease depending upon he sign of η. Bu he nominal exchange rae can adjus insananeously o adjus P n. For example, if η is posiive hen wages do no fall enough o ose he rise in he price of oil so φ n rises. In his case he nominal exchange rae mus appreciae so as o reduce he cos of purchasing oil for he non-raded rms. The relaive price of he non-raded good rises and his keeps φn P n nearly consan. The opposie resuls hold when he η is negaive. In his case he nominal exchange rae depreciaes, which raises coss in he non-raded secor by jus he righ amoun o keep non-raded inaion very close o is arge. Given he resuls for π n i is possible o derive resuls for some of he oher variables, as well. The pah of π n pins down he enire pah of χ and π CP I as dχ = α π dπ n, dπ CP I = [(1 γ n )α π + γ n ] dπ n. Under his policy boh he rae of depreciaion and CPI inaion vary direcly wih π n along he ransiion pah. I is also possible o link up he behavior of π n, i, and wha I will refer o as he non-raded real ineres rae, r n. To see where r n comes from, imagine ha he agen had access o a bond indexed o P n. This bond would have a real reurn of r n and he nominal ineres rae on his bond would be i n = r n + π n. A no-arbirage condiion would ensure ha i n = i along he ransiion pah so r n = r + (χ π n ). 19

22 A rs order approximaion of his equaion gives dr n = (α π 1)dπ n, which shows ha so long as α π > 1 hen he policy ensures ha non-raded real rae rises whenever non-raded inaion rises above is seady sae value. This is similar o he resul ha holds in closed economy New Keynesian models, excep ha in ha in ha case here is only one real ineres rae. The no-arbirage condiion also implies ha for cases where π n is kep close o is arge ha i mus be adjused o vary wih r n. Imporanly, he exac manner in which i mus be adjused depends upon η and so i is no always rue ha ineres raes need o be raised o sabilize π n. All of he resuls derived so far echnically only hold for he model where r = r while he soluion mehod used is jus an approximaion o he rue model. I remains o be shown ha numerical resuls are in line wih resuls jus derived. To show ha his is so, I solve he model numerically and plo he impulse responses for a number of variables. All plos show he percenage deviaions of he variables from heir seady sae values, wih he excepion of he inaion and ineres raes, which are in percenages, and he rade and curren accoun decis, which are in unis of GDP. The plo for aggregae consumpion expendiure, E, incorporaes relaive price changes while E(f ix) does no. The variables T B, CA, and OS are he rade balance, he curren accoun balance, and oal spending on oil by rms. Figure 1 shows he impulse responses when here is a 20 percen rise in he price of oil, wih α π calibraed o a large value. Alhough he soluion is an approximaion all of he resuls mach hose from he analyical soluions previously derived. The rise in he real price of oil causes wages o fall bu under he baseline calibraion η is posiive so he drop in wages is no big enough o ose he rise in he price of oil in he non-raded secor and φ n rises. Sabilizing π n herefore requires he currency o insananeously appreciae which causes he relaive price of he non-raded good o iniially jump up. Wih π n sabilized close o is arge, variaions in P n are due almos enirely o variaions in χ, which is above 20

23 is seady sae level along he enire ransiion pah. The nominal ineres rae and r n boh rise iniially and hen fall over ime. The behavior of he real variables basically follows wha we would expec from economic inuiion. The rise in he price of oil reduces he demand for oil in boh secors and brings abou a reducion in he amoun of oupu produced in he raded secor. Income is lower for he agen so aggregae consumpion expendiure drops, and his income eec along wih he relaive price movemen induces a sharp and subsanial drop in oupu in he non-raded secor. Real money balances decline because of he slighly higher ineres raes and he reducion in aggregae consumpion. The economy makes use of is abiliy o smooh he shock ou by borrowing signicanly from inernaional capial markes. Figure 2 makes explici he imporance of he sign of η by showing he impulse responses under he alernaive calibraion where P o o O n o is se o 0.02 insead of This alernaive calibraion makes η negaive insead of posiive so φ n falls and φn P n is kep consan only if here is an oseing depreciaion of he currency. Under he same exac specicaion of moneary policy, i is now he case ha P n, χ, r n, and i iniially fall and hen rise over ime. As he relaive price of he non-raded good is emporarily lower han ha of he raded good, oupu and consumpion of his good decline less han wha occurs under he case where η is posiive. 3.2 Sabilizing CPI Inaion In pracice cenral banks focus on sabilizing an inaion variable oher han π n, generally CPI inaion, so i is worhwhile considering he implicaions of his ype of policy. I assume ha he cenral bank implemens his policy by adjusing he nominal rae according o he rule ( ) i = i o + α π π CP I πo CP I, which is exacly he same as he rule found in equaion (39) excep ha π n is replaced wih π CP I. 21

24 As wih he policy of sabilizing π n i is possible o derive analyical resuls abou he condiions under which his policy delivers a unique soluion and analyical resuls abou he behavior of some of he variables along he ransiion pah. This is rue because under his policy i is again possible o derive a sub-sysem in P n, π n, and P o separae from he oher equaions of he model. The procedure for deriving he sub-sysem is basically he same as before. The linearized versions of he moneary policy rule, he equaion for CPI inaion, and he ineres-pariy condiion can be combined o give an equaion for χ of he form dχ = α π γ n 1 α π (1 γ n ) dπn. As before, use his equaion in he linearized equaion for P n. Once his is done one again has a se of linearized dierenial equaions for P n, π n and P o ha are funcions of hese hree variables alone. There are wo jump variables in he subsysem so here mus be wo posiive roos if here is o be a unique soluion. The main resul here is ha his policy can produce muliple equilibria in he economy for a wide range of calibraions for α π. The roos of he subsysem in his case are r 1 = α r 2 = 1 ρ 4ω(ω + ρ)(1 α π) + ρ 2 1 α π (1 γ n ) 2 r 3 = 1 ρ + 4ω(ω + ρ)(1 α π) + ρ 2 1 α π (1 γ n ) 2. From he equaion for r 2 we can immediaely see ha here will be rouble depending upon he value of α π (1 γ n ). For cases where α π < 1 1 γ n, r 2 will be posiive wih he possibiliy of a complex roo. If α π > 1 1 γ n hen r 2 will be negaive and here will be muliple equilibria. This can be seen by noing ha he rs erm in he square roo will be posiive in his case and since we are adding a posiive number o ρ 2 we are guaraneed a negaive roo. Under 22

25 he borderline case where α π = 1 1 γ n, r 2 explodes o complex negaive inniy and we also have indeerminacy. I is possible o give a clear explanaion of why his occurs by looking a he behavior of χ, i, and r n. The soluion for χ shows ha in cases where he policy produces an indeerminae soluion χ varies inversely wih π n. Subsiuing he equaion for χ ino he equaion for CPI inaion gives dπ CP I = γ n 1 α π (1 γ n ) dπn, which shows ha he change in χ is enough o force CPI inaion o also vary inversely wih π n. Subsiuing he equaion for dπ CP I ino he linearized moneary policy rule one has di = and using his in he equaion for r n one ges dr n = α π γ n 1 α π (1 γ n ) dπn, α π 1 1 α π (1 γ n ). From hese wo equaions we can see ha a rise in π n leads o a fall in boh i and r n. Bu if he inuiion carries over from oher models he drop in r n spurs furher increases in π n which, hrough he policy rule, spurs furher decreases in χ, i, and r n. When α π (1 γ n ) is greaer han 1 he policy creaes a response ha allows self-fullling expecaions o drive he economy. Noe ha when α π (1 γ n ) is less hen 1 hen hese self-fullling expecaions can no occur because a rise in π n brings abou a rise in r n and his prevens inaion from spiralling ou of conrol. In cases where here is a unique soluion, he soluions of he hree variables on he ransiion pah are dp n dπ n = dp o k2 k1 [ 1 + α+ρ ω(ρ+ω)(1 απ) 1 απ(1 γn) α(α+ρ) αk2 ω(ρ+ω)(1 απ) 1 απ(1 γn) α(α+ρ) 1 ] [ h 1 e α ], 23

26 where h1 is again he iniial surprise jump in he price of oil. So long as α π (1 γ n ) < 1, he direcion of he iniial jump in inaion is deermined by he sign of η and he magniude is a decreasing funcion of α π (1 γ n ). In paricular, in he limi as α π (1 γ n ) approaches 1 he iniial jump approaches 0. Since his can only occur if here is a very specic response from he nominal exchange rae his implies ha for his calibraion he policy will behave very closely o he one where he cenral bank fully sabilizes π n. The equaions are, unforunaely, oo complicaed o be able o derive any more analyical resuls so I now urn o numerical simulaions. I avoid he indeerminacy problem by calibraing α π o 1.99, which is near he indeerminacy boundary for calibraion used here. This also implies ha he moneary auhoriy sabilizes CPI inaion o he greaes exen possible. The impulse response funcions for he baseline calibraion are shown in gure 3. Because a calibraion of α π = 1.99 is successful a sabilizing π n close o is arge level he resuls are very similar o he ones found in gure 1. Given he similariies of he resuls for hese wo policies I refer he reader o he previous secion for a discussion abou hem. For breviy's sake, I do no display he response funcions for he alernaive calibraion, since hese will mirror closely he resuls found in gure A Crawling Peg While he number of economies ha compleely x heir exchange raes are few, many counries sill occasionally inervene in foreign exchange markes. This secion provides resuls for he polar case where he nominal exchange rae is compleely xed, as opposed o he inaion argeing schemes where he nominal exchange rae was compleely free o oa. Under a crawling peg χ is he moneary policy variable, he nominal exchange rae is pre-deermined a any poin in ime, and real money balances adjuss hrough he capial accoun. I assume for simpliciy ha χ is se equal o is seady sae level χ o. In order o defend he peg he cenral bank inervenes in he foreign exchange marke whenever 24

27 necessary using is sock of foreign exchange reserves b g. The core dynamic sysem of he model under his policy consiss of equaions (12) and (25) for he jump variables λ 1 and π n, and equaions (1), (33), and (34) for he sae variables P n, k, and P o, where k = b + b g. As wih he inaion argeing policies i is possible o derive analyically he condiions under which his policy produces a unique soluion and also some analyical resuls for several variables when r = r. Under his policy he sysem of equaions for P n, π n, and P o can be wrien as P n 0 Po n 0 dp n π n = k1 ρ k2 dπ n, P o 0 0 α dp o where k1 and k2 are he same as before. The hree eigenvalues are r 1 = α r 2 = 1 ( ) ρ 4ω(ω + ρ) + ρ 2 2 r 3 = 1 ( ) ρ + 4ω(ω + ρ) + ρ 2 2. In his case P n is a sae variables so a unique soluion requires wo negaive eigenvalues. I is obvious ha boh r 1 and r 2 will always be negaive for realisic calibraions of ω so his policy always generaes a unique soluion in he subsysem. Ineresingly enough, his policy produces a unique soluion even hough r n falls when inaion rises because he policy does no creae a feedback eec ha leads o self-fullling expecaions. Afer solving for he eigenvecors one can show ha along he ransiion pah dp n dπ n dp o = k2po n ω(ρ+ω) α(α+ρ) ρ+ 4ρ(ρ+ω)+ρ 2 2k1 αk2 1 ω(ρ+ω) α(α+ρ) 1 0 h 1e α, h 2 e r 2 25

28 where h1 is again jus he iniial jump in oil prices and 2ω(ρ + ω)k2 h2 = [ω(ρ + ω) α(α + ρ)] [ ]h1. ρ + 4ω(ρ + ω) + ρ 2 Noe ha he sign of h2 is deermined enirely by he sign of η hrough he erm k2. These soluions are more complicaed han in he previous cases and do no lead o many easily inerpreable soluions. Bu afer performing some algebra i can be shown ha he iniial jump in non-raded inaion is 2ω(ρ + ω) k2 α ρ + 4ω(ρ + ω) + ρ 2 ω(ω + ρ) α(α + ρ) h1. While i is no possible o sign he iniial jump for all possible values of he deep parameers, so long as ω is greaer han α, a likely case, he jump in inaion is pinned down by he sign of η, wih inaion rising immediaely if η is posiive and falling if η is negaive. Wih he nominal exchange rae xed variaions in φ n are no immediaely ose by movemens in P n. Therefore coss rise or fall, depending upon he sign of η, and rms ose his by raising or lowering heir prices, which leads o non-raded inaion rising or falling. Several oher soluions are available. Wih χ and r xed di = 0 a all imes. Therefore dr n = dπ n. Under a xed exchange rae policy he non-raded real ineres consequenly falls whenever non-raded inaion rises. Also, he change in CPI inaion is simply dπ CP I = γ n π n, so CPI inaion varies direcly wih non-raded inaion. As before, more comprehensive resuls can only be had by solving he model numerically. The impulse response funcions for he baseline calibraion are shown in gure 4. The rise in 26

29 he price of oil raises φ n bu since e is xed his brings abou an immediae rise in non-raded inaion. Over ime rms adjus o he emporarily high cos of oil by raising heir prices, which causes he relaive price of he non-raded good o rise over ime. The fac ha P n is emporarily lower han wha occurs under he inaion argeing policy manifess iself in a higher level of Q n, iniially. As P n rises, however, he agen reduces consumpion of he non-raded good, which shows up as a rapid decline in Q n over he rs year. Figure 5 shows he resuls for he alernaive calibraion of he model. When η is negaive he rise in oil prices now reduces φ n which in urn causes π n o drop iniially and for P n o fall over ime, while r n iniially rises. 4 Conclusions The conribuion of his paper is o examine moneary policy responses o oil price shocks in a small, open economy wih raded and non-raded goods as opposed o he large, closed economy model ha has been he mainsay so far. This change allows one o consider issues wih exchange raes and relaive prices ha are necessarily ignored in he closed economy models. The main resul in his paper is ha because of he asymmeric eecs a rise in he price of oil has on coss in he raded and non-raded secors i is possible for he same specicaion of moneary policy o produce very dieren behavior in inaion, he nominal exchange rae, and he relaive price of he non-raded good depending upon he cos shares of oil and labor in he wo secors. Under he assumpions made in he paper he behavior of wages is pinned down in he raded goods secor. If he raio of he cos share of oil o he cos share of labor is higher (lower) in he raded secor hen coss and inaion in he non-raded secor fall (rise) unless moneary policy responds appropriaely. A successful inaion argeer who sabilizes π n does so by oseing hese changes in cos by bringing abou exacly he correc movemen needed in he nominal exchange rae. The exac movemen depends, hough, on raios and hence i is possible for his policy o cause a nominal exchange rae depreciaion 27

30 or appreciaion. A similar resul holds for an inaion argeer who sabilizes CPI inaion, for cerain calibraions of he policy rule. For similar reasons, a policy of xed nominal exchange raes, because i precludes any such adjusmen in he nominal exchange rae, can bring abou increases or decreases in non-raded inaion. The second se of imporan resuls is relaed o he abiliy of he policies o produce unique soluions. I have also shown analyically under wha condiions he hree policies produce muliple equilibria in he economy. A policy ha sabilizes non-raded inaion only produces an indeerminae soluion if he nominal ineres rae is no raised more han one-for-one wih a rise in non-raded inaion. A policy ha sabilizes CPI inaion, however, produces muliple equilibria for a wide range of calibraions. The analyical resuls show ha wheher his occurs or no is a funcion of how srongly he cenral bank pledges o respond o rises in CPI inaion and he share of he non-raded good in he CPI. If he pledged response is oo srong hen he policy response leads o self-fullling expecaions. A crawling peg, unlike he oher wo policies, always produces a unique soluion for he model considered here. While his paper has produced a number of useful and ineresing resuls, he model used is simple and highly sylized and fuure research could expand upon he model in several ways. One useful line of work would be o inroduce furher fricions in he model, such as sicky nominal wages or less hen full labor mobiliy beween he wo secors. A second line of work would be o add a second counry and make endogenous he foreign variables of he model. While his would be a signicanly more complicaed model i would allow for greaer realism and a much wider range of quesions ha could be invesigaed han wih he simple model used here. 28

31 Table 1: Calibraed Parameers and Seady Sae Values Parameer Value τ.50 σ c.50 ρ.05 ω 2 α.10 σ.50 σ n.50 µ 1 m o.08 Po n 1 Co n.50 Co.50 P o O.03 P o O n.03 29

32 Figure 1: Non-Traded Inaion, Baseline Calibraion Q m Π n r b Qn Π GDP rn W Χ Π CPI i Pn

33 Figure 2: Non-Traded Inaion, Alernaive Calibraion Q Qn W m b Χ Π n Π GDP Π CPI r rn i Pn

34 Figure 3: CPI Inaion, Baseline Calibraion Q m Π n Pn 0 r b Qn Π GDP rn W Χ Π CPI i

35 Figure 4: Fixed Raes, Baseline Calibraion Q m Π n Pn r k Qn Π GDP rn L W Χ Π CORE i

36 Figure 5: Fixed Raes, Alernaive Calibraion Q Qn W m k Χ Π n Π GDP Π CORE r rn i Pn

37 References Bodensein, M., C. Erceg, and L. Guerrieri (2008): Opimal Moneary Policy wih Disinc Core and Headline Inaion Raes, Journal of Moneary Economics, forhcoming. Calvo, G. A. (1983): Saggered Prices in a Uiliy-Maximizing Framework, Journal of Moneary Economics, 12, Dhawan, R., and K. Jeske (2007): Taylor Rules wih Headline Inaion: A Bad Idea!, Working Paper. Leduc, S., and K. Sill (2004): A Quaniaive Analysis of Oil-Price Shocks, Sysemaic Moneary Policy, and Economic Downurns, Journal of Moneary Economics, 51, Plane, M. (2009a): How Should Moneary Policy Respond o Changes in he Relaive Price of Oil?, Working Paper. (2009b): Two Calvo Pricing Models for Coninuous Time, Open Economy Models wih Traded and Non-Traded Goods, Noes. 35