Technical Note: A Monetary Model of the Exchange Rate with Informational Frictions

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1 Technical Noe: A Moneary Model of he Exchange Rae wih Informaional ricions Enrique Marinez-Garcia ederal Reserve Bank of Dallas Curren Draf: Augus Absrac This echnical noe is developed as a mahemaical companion o he paper A Moneary Model of he Exchange Rae wih Informaional ricions. I conains hree basic calculaions. irs I include a succinc discussion of he opimizaion problem for households firms and financial inermediaries. I also derive he equilibrium condiions of he model. Second I compue he zero-inflaion zero-curren accoun (deerminisic) seady sae. Third I describe he approach of log-linearizing he equilibrium condiions around he seady sae. Simulaneously I explain he sysem of equaions ha consiues he basis for he paper. Commenary is provided whenever necessary bu i remains limied given he scope of his noe. The algebra is also abbreviaed whenever possible as longasidoesnoobscurehemainderivaions. JEL Classificaion: KEY ORDS: Asymmeric Informaion Raional Expecaions Equilibrium Bilaeral Exchange Raes ia Money inancial Inermediaion This is he companion echnical noe for he paper A Moneary Model of he Exchange Rae wih Informaional ricions. The exended version of he paper is available as Dallas ed GMPI P #2. All remaining errors are mine alone. Please repor any misakes ypos and inconsisencies you find. The views expressed in his noe do no necessarily reflec hose of he ederal Reserve Bank of Dallas or he ederal Reserve Sysem. Enrique Marinez-Garcia ederal Reserve Bank of Dallas. Correspondence: 2200 N. Pearl Sree Dallas TX Phone: + (24) enrique.marinez-garcia@dal.frb.org. ebpage: hp://dallasfed.org/research/bios/marinezgarcia.hml.

2 A Inroducion ere I presen he derivaion of he (symmeric) equilibrium condiions and he deerminisic seady sae characerizing he wo-counry economy in my paper A Moneary Model of he Exchange Rae wih Informaional ricions. I also sudy he linearizaion of he resuling sysem of equaions. Since he model is buil around wo (almos-)symmeric counries my analysis is subsanially iled owards he problem of he home counry unless oherwise noed. B The Equilibrium Condiions The equilibrium condiions of he model pin down he policy rules used by raional households o deermine heir consumpion-savings opimal pah by firms o decide heir opimal pricing policy and by financial inermediaries o solve heir opimal porfolio allocaion problem. These equaions are complemened wih a pair of sable money demand and labor supply funcions per each household and a resource consrain ha summarizes he budgeary consrains of all he economic agens in he economy. Asses inancial Inermediaries ome ouseholds oreign ouseholds Labor Labor ome Goods oreign Goods igure. The Srucure of he Economy B. The Represenaive ousehold s Problem The Opimizaion in he ome Counry. he home counry is expressed as ( + X X βτ E E exp ξ +τ " (C j+τ bc +τ ) + The maximizaion problem of he domesic household j in µ χ M d j+τ P +τ h nλ j+τ B j+τ + Mj+τ d + +τ L s j+τ + Π j+τ + TR j+τ P +τ C j+τ #) κ +ϕ L s +ϕ j+τ + B j+τ exp(i +τ ) M d j+τ io

3 where λ j refers o he Lagrange muliplier and he expecaions operaor E { } E { } is condiional on a given informaion se. Individual consumpion C j money demand Mj d demand for domesic bonds B j and labor supply L s j are effecively chosen a ime by he household iself. The habi componen is observable a ime because all informaion up o ime is public bu is reaed as an exernaliy. The curren domesic preference shock ξ is also observable. ouseholds selec heir opimal sraegy a ime prior o he opening of he markes. Even hough consumpion prices P wages and ineres raes i are no observable households undersand ha he informaion in allows hem o generae imperfec predicions of hem. Naurally his could affec heir choices. In fac if second-order effecs are negligible (as i will be assumed whenever log-linearizing he equilibrium condiions) and informaion is common and perfec (ha is he realizaion of all shocks is observable by all agens up o ime prior o making a decision) hen hey may be able o anicipae correcly he prices ha will prevail in each marke before i opens up (perfec foresigh). or racabiliy I assume ha he households pre-commi o a pair of sable money demand and labor supply rules i.e. χ Ã! Mj d = exp (i ) () P (C j bc ) exp (i ) P = κ (C j bc ) L s ϕ j. (2) These rules correspond o he firs-order condiions of he domesic households problem for M d j and Ls j in he sandard model. This means ha households pre-commi o operae in he money and labor markes as price-akers. Relaxing his characerizaion would be concepually sraighforward bu adds more complexiy o he algebraic derivaion of he model wihou significaively alering is qualiaive resuls. In his sense I can re-inerpre he households problem as an opimizaion program wih random bond prices and focus on he effec ha informaion has on he more relevan consumpion-savings margin. Taking () and (2) as given he firs-order necessary condiions for he home counry are reduced o C j : exp(ξ )(C j bc ) λ j P 0 =0if C j > 0 B j : λ j E exp (i ) + βe λ j+ 0 =0if B j > 0. Afer some simple algebra I derive he following equilibrium condiion for an inerior soluion " B j : E exp (i ) = βe exp µ # C j+ bc P ξ + (3) C j bc P + which resaes he sandard firs-order condiion replacing he price of he domesic bond wih is expeced value. The budge consrain is saisfied wih equaliy ex pos. The Opimizaion in he oreign Counry. The maximizaion problem of he foreign household j can be characerized in a similar fashion. Analogously I sar wih a pre-commimen o M d : χ Ã P Mj d C j bc! = E exp (i ) (4) L s : P = κ Cj bc L s ϕ j (5) In fac he household akes as given he prices P (h) for all varieies h [0n] and P (f) for all varieies f (n ]. 2

4 and I obain he following firs-order condiion for an inerior soluion B : E exp (i ) = βe exp ξ + Ã Cj+! bc P Cj bc The budge consrain is saisfied wih equaliy ex pos. The Demand Curves. are aggregaed by means of a CES index as C j P +. (6) Thehomeandforeignconsumpion bundles of he household C j C j C j and C j = C j = " µ Z # θ " θ n C j (h) θ θ µ Z # θ θ dh Cj θ = C (f) θ θ θ df n 0 n n " µ Z θ n Cj (h) θ θ dh n 0 # θ θ C j = " µ Z θ C (f) θ θ df n n # θ θ while aggregae consumpion C j and Cj isdefined wih anoher CES index as C j = C j = n σ C σ σ σ j +(n) n σ C σ σ j +(n) σ σ C σ σ σ j C j σ σ σ σ. The domesic and foreign households are also pre-commied o a sandard allocaion across varieies and bundles. Given he srucure of preferences he soluion o he sub-uiliy maximizaion problem implies ha he households demands for each variey are given by C j (h) = n C j (f) = µ P (h) n P µ P (f) θ C j C j (h) = n P µ P (h) P θ C j C j (f) = n θ C j if h [0n] (7) µ P (f) P while he demands for he bundles of home and foreign goods are simply equal o µ P Cj σ = n C j Cj = n P µ P Cj = ( n) P σ C j C j =( n) θ C j if f (n ] (8) µ P σ P Cj (9) µ P σ P Cj. (0) Because he problem of all domesic households is idenical (symmeric) I drop he subscrip j from now on. Similarly for he foreign households. 3

5 The Price Indexes. Under sandard resuls on funcional separabiliy I infer ha he price indexes corresponding o he specified srucure of preference are P = P = P = h n P Z n n 0 n σ +( n) P σ i σ () P (h) θ θ dh (2) Z n P (f) θ θ df. (3) ome and foreign households have idenical ases and herefore heir respecive price indexes are symmeric i.e. P = P = P = h n P σ +(n) P σ i σ (4) Z n P (h) θ θ dh (5) n 0 n Z B.2 The Represenaive irm s Problem n P (f) θ θ df. (6) The problem of he firm becomes dynamic in he presence of nominal price rigidiies à la Calvo (983). ollowing G. Benigno (2004) I allow for differences in he degree of price sickiness across producer locaions and beween domesic and foreign markes (i.e. α 6= α 6= α 6= α ). The Opimizaion in he ome Counry. henever echnologies are linear in labor he opimizaion problem in he domesic and foreign markes can be easily separaed for any given firm. This fac urns ou o be very convenien o compue he opimal pricing rules. Le me define he ineremporal marginal rae of subsiuion (IMRS) as β τ Ξ +τ β τ exp µ C +τ bc +τ P ξ +τ ξ C bc P +τ for any τ 0. Then he maximizaion problem of firm h in he domesic marke discouned on he basis of he domesic household IMRS is expressed as X βα τ E ½Ξ +τ ey d +τ (h) ep (h) ¾ +τ exp (a +τ ) while in he foreign marke is given by X βα τ E ½Ξ +τ e Y d +τ (h) S +τ e P (h) ¾ +τ exp (a +τ ) where e Y d +τ (h) and e Y d +τ (h) refer o he demand funcions for a fixed ime price and he expecaions operaor E { } E { } is condiional on a given informaion se. The prices of he good in each marke e P (h) and e P (h) are effecively chosen a ime by he firm iself. The curren domesic produciviy shock a is observable and all informaion up o ime is public. irms selec heir opimal pricing sraegy a ime prior o he opening of he markes. Even hough 4

6 consumpion prices 2 P wages and ineres raes i are no observable firms undersand ha he informaion in allows hem o generae imperfec predicions of hem. Naurally in his conex heir forecas of wages has implicaions for heir opimal choices and he dynamics of prices. or racabiliy I assume ha firms pre-commi o saisfy he consumer s demand derived in (7)(0) a he fixed prices unil hey can re-opimize again i.e. ey d +τ (h) = ey d +τ (h) = Ã! θ ep µp σ (h) +τ C +τ P +τ P +τ Ã! θ ep µp (h) P +τ +τ P +τ σ C +τ for any τ 0. The firs-order necessary condiions for he problem of a monopolisic compeior can be expressed as follows ep (h) : ep (h) : X X " βα τ E nξ +τ e Y d +τ (h) o θe (Ξ +τ e Y d +τ (h) Ã P(h) =0if P e (h) > 0 " Ã o βα τ E nξ +τ Y e d +τ (h) S +τ θe (Ξ +τ Y e d +τ (h) =0if e P (h) > 0. +τ exp(a +τ ) P (h)!)# 0 S +τ P (h) +τ exp(a +τ ) P (h)!)# 0 Afer some algebra his would give me he following equilibrium condiions for an inerior soluion X n o ep (h) : P e (h) = θ βα τ E Ξ +τ Y e d +τ (h) +τ exp(a +τ ) X θ n o (7) (βα ) τ E Ξ +τ ey +τ d (h) X n o ep (h) : ep (h) = θ βα τ E Ξ +τ Y e d +τ (h) +τ exp(a +τ ) X θ n o (8) (βα ) τ E Ξ +τ Y e d +τ (h) S +τ which correspond o he sandard firs-order condiions for he problem of firm h. The consumer s demand is fully saisfied ex pos in every period a he chosen prices. The Opimizaion in he oreign Counry. The maximizaion problem of he foreign firm f in he domesic and foreign markes can be characerized in a similar fashion. Analogously I sar wih a precommimen o ey d +τ (f) = ey d +τ (f) = Ã! θ ep µp σ (f) +τ C +τ P +τ P +τ Ã! θ ep µp (f) P +τ +τ P +τ σ C +τ 2 In realiy each firm knows wha he average firm no allowed o re-se prices does because all informaion up o is public. If he firm was allowed o re-se prices i would also know he opimal price of is variey. In a symmeric equilibrium his implies ha all producers in each counry have enough informaion o infer he price of he bundle of all heir varieies in he domesic and foreign marke. 5

7 and I obain he following firs-order condiions for an inerior soluion P ½ ep (f) : P e (f) = θ βα τ E Ξ +τ Y e d θ P n (βα ) τ E Ξ e +τ Y d ep (f) : e P (f) = θ θ P βα τ E nξ +τ e Y d +τ (f) µ +τ exp(a +τ) ¾ +τ (f) )o +τ (f) +τ exp(a +τ P (βα ) τ E n Ξ +τ e Y d +τ (f) S +τ o (9) o (20) where he expecaions operaor E { } E { } is condiional on a given informaion se andhe foreignimrsis β τ Ξ +j β τ exp ξ +j ξ µ C+j bc +j P C bc P+j. The demand in each marke is fully saisfied ex pos every period a he chosen prices. B.3 The Represenaive inancial Inermediary s Problem The maximizaion problem of he financial inermediary z is expressed as ½ E X z+ exp (i ) X z λ ¾ S Bz 2 2 where λ z refers o he Lagrange muliplier and he expecaions operaor E { } E { I } is condiional on a given informaion se I. Noice ha he ne posiion in foreign asses is effecively chosen a ime by he inermediary iself. Inermediaries observe he money supply shock in boh counries m and m and all informaion up o ime is public. Inermediaries selec heir opimal sraegy a ime prior o he opening of he markes. Even hough consumpion prices P and P wages and ineres raes i and i and nominal exchange raes S are no observable inermediaries undersand ha he informaion in I allows hem o generae imperfec predicions of hem. Naurally heir forecas of he nominal ineres rae spread and he depreciaion of he nominal exchange rae will affec heir porfolio allocaion as well as he insrumened levels of inernaional borrowing and lending (beween he home and foreign counry). or racabiliy I assume ha inermediaries pre-commi o saisfy he ineremporal budge consrain wih equaliy i.e. X z+ =exp(i ) X z + S Bz S+ exp (i i ) S where X z denoes he wealh of inermediary z a ime andbz is he ne foreign asse posiion of inermediary z a ime. Thefirs-order necessary condiion of he inermediary can be expressed as follows ½ Bz : E λs Bz + S ¾ + exp (i i ) 0 =0if Bz > 0. S Afer some re-arranging his would give me he following equilibrium condiion for an inerior soluion ½ ¾ Bz S+ : E exp (i i ) λs Bz =0 (2) S which is similar o he uncovered ineres pariy equaion wih he addiion of a risk premium erm linked o he ne foreign asse posiion and he replacemen of he ineres rae differenial by is expeced value. The ineremporal budge consrain is saisfied wih equaliy ex pos. Because he problem of all financial inermediaries is idenical I drop he subscrip z from now on. 6

8 B.4 The Resource Consrain In his model he conrol variables for households are money demand labor supply and bond-holdings (savings). Given a choice of conrols consumpion is deermined from he budge consrain of he household. Consumpion herefore adjuss ex pos o ensure ha demand and supply equae in all markes. The conrol variable for he financial inermediaries is heir ne foreign asse posiion while wealh adjuss ex pos o guaranee ha he ineremporal budge consrain is always saisfied. The conrol variables for firms are he prices hey charge for heir own variey in he domesic and foreign markes for as long as he noreopimizaion spell lass while ex pos oupu supply adjuss o guaranee ha ex pos he consumer s demand in every marke is saisfied a he chosen prices. This discussion is needed here because in a model wih incomplee and asymmeric informaion I need o disinguish beween ex ane decisions and ex pos realizaions. Agens eiher choose heir conrols opimally or pre-commi o a rule before he markes open. Their decisions depend on how hey forecas he price sysem for he economy and he error involved in ha. Then markes open and prices clear hem. ence consumpion for households wealh for financial inermediaries and oupu for firms should adjus ex pos o ensure ha individual and aggregae consrains are always enforced. In oher words ex pos he budge consrainshaveobesaisfied in each sae of he world ha he economy reached. Aggregaing he ex pos budge consrain (under equaliy) for every household in he home counry I ge ha B exp (i ) + M d = B + M d + L s + Π + TR P C. Using he domesic money marke clearing condiion i.e. M d =exp(m ) and he domesic governmen budge consrain i.e. TR =exp(m ) exp (m ) i follows ha B exp (i ) = B + L s + Π P C. The profis of he firm disribued o shareholders are Π = n Z n 0 [P (h) nc (h)+s P (h)( n) C (h)] dh n Z L d (h) dh. R The labor marke and goods marke clearing condiions require ha L s = n L d (h) dh and Y s (h) = nc (h)+( n) C (h) respecively. In a symmeric equilibrium L s = L d (h). Then if I combine his wih he demand curves described in (7) and (9) I can wrie ha B exp (i ) Z n = B + [P (h) nc (h)+s P (h)( n) C (h)] dh P C n 0 " Z n µ θ µ P (h) P = B + dh# n 0 P P σ nc + P " Z n µ P θ µ + (h) P dh# S P σ n 0 P ( n) C P C P where B is he per capia domesic household s bond-holdings. Using he price sub-indexes in (2) and (5) i is easy o prove ha n Z n 0 µ P (h) P θ dh = n Z n 0 µ P θ (h) P dh = 7

9 so I can express he resource consrain as follows B exp (i ) = B + P µ P P = B + P " µp P σ nc + S P µ P P σ nc + RS µ P P σ ( n) C P C σ # ( n) C P C where he real exchange rae is defined as RS S P P. This holds rue in per capia erms independenly of wheher he prices are flexible or sicky. Le me denoe B he ne domesic asse posiion of he financial inermediaries i.e. B = X + S + B. I assume ha bonds are in zero-ne supply herefore B = B. Moreover i mus hold ha he ne value of invesmen flows of he financial inermediaries ino he foreign bond marke (expressed in unis of he domesic currency) mus be he reciprocal of he ne value of invesmen flowscominginohe domesic bond marke i.e. exp (i ) B B = S exp (i ) B B. ence I obain he following resource consrain " µp exp (i ) BR = S P B R + S P P σ µ P σ # nc + RS P ( n) C C (22) where B R SB P is he real value of he ne foreign asse posiion of a represenaive inermediary. Equaion (22) is he so-called curren accoun equaion. C The Deerminisic Seady Sae By consrucion he uncondiional mean of he shocks driving he model is zero 3 i.e. Eθ =(Em Em Eξ Eξ Ea Ea ) T = 0 T. I require ha all shocks be evaluaed a heir uncondiional mean in seady sae. I also conjecure he exisence of a (symmeric) deerminisic seady sae in which prices consumpion and he nominal exchange rae are consan i.e. P + = P = P P+ = P = P C + = C = C C+ = C = C S + = S = S. Then I characerize he zero-inflaion seady sae as follows: Given my conjecure and he Euler equaions in (3) and (6) i follows ha he seady sae ineres rae in boh counries is idenical and equal o he inverse of he rae of ime-preference β =exp i =exp i. ere I used he assumpion ha Eξ = Eξ =0. rom he firm s firs-order condiions in (7) (20) I 3 Noice ha I no only require ha Eξ = Eξ =0. I also impose ha Eξ = Eξ =0. 8

10 obain ha he law of one-price (LOOP) holds in seady sae i.e. ep (h) = P e (h) S = θ (23) θ ep (f) = P e (f) = θ S θ. (24) ere I used he assumpion ha Ea = Ea =0. Noice ha he pricing policy corresponds o he soluion of he monopolisic compeior s problem under flexible prices. In oher words prices are se by he Dixi- Sigliz monopolisic price-seing rule in seady sae and firms charge a mark-up over marginal coss. Moreover profis per uni of oupu in seady sae are proporional o seady sae wages i.e. θ and θ respecively. I wrie he seady sae price sub-indexes in equaions (2) (3) and (5) (6) as follows P = e P (h) P = e P (h) (25) P = e P (f) P = e P (f) (26) and P = P S P = P S. Using he price indexes in equaions () and (4) evaluaed a heir seady sae I also infer ha he real exchange rae should be equal o one P = SP RS SP P =. rom he porfolio allocaion rule obained in (2) I know ha he (nominal and real) ne foreign asse posiion of he financial inermediaries in seady sae is zero i.e. B = B R =0ifλ6= 0. I follows from he curren accoun in equaion (22) ha in seady sae C = = ÃP! σ ÃP! σ nc + ( n) C P ÃP P P! σ h nc +( n) C i. rom he seady sae price index implied by () i is possible o re-wrie he curren accoun as C = ÃP P! σ h nc +( n) C i so I infer ha he raio of he consumpion level across counries can be expressed as a funcion of he raio of domesic good prices relaive o foreign good prices in seady sae C C = ÃP! σ. (27) P 9

11 The labor marke and he goods marke clearing condiions imply ha in seady sae L s = L d (h) =Y s (h) Y s (h) = nc (h)+( n) C (h) L s = L d (h) =Y s (f) Y s (f) = nc (f)+( n) C (f). ere I employed he assumpions ha echnologies are linear in labor and Ea = Ea =0.Usinghedemand equaions in (7) (0) I argue ha for any variey h [0n] and f (n ] i mus hold rue in seady sae ha L s = ÃP nc (h)+(n) C (h) = P L s = ÃP nc (f)+(n) C (f) = P! σ h nc +( n) C i (28)! σ h nc +( n) C i. (29) The labor supply equaions in (2) and (5) evaluaed a heir seady sae values give me ha P = κ ( b) C L sϕ P = κ ( b) C L sϕ. ence using (23) (29) I obain ha real wages saisfy hese pair of condiions θ P = θ P P = κ ( b) C ÃP! σ ϕ h nc +(n) C i (30) P θ P = = κ ( b) C ÃP! σ ϕ h nc +(n) C i. (3) θ P P P Afer some re-arranging i immediaely follows ha P P = If I use (27) now o replace he raio C C Igeha µ C ÃP C P! ϕσ. P P = ÃP! (σ)ϕσ. P Therefore as long as ( σ) ϕσ 6= his equaliy can only hold if P = P. As a resul I immediaely 0

12 infer ha in a symmeric seady sae prices and consumpion saisfy hese equaliies P = P = P P = P = P ToT = ToT = T = T = S = C (h) = C (f) =C C (h) =C (f) =C C = C. This means ha perfec inernaional risk-sharing holds in seady sae and hence consumpion is fully equalized across counries. Using he fac ha P = P combined wih (30) allows me o derive he following characerizaion of seady sae consumpion κ ( b) µ +ϕ. (32) C = C θ = θ θ Noice how consumpion in seady sae is relaed o he mark-up charged by firms i.e. θ he habi formaion parameer i.e. b and he preference weigh on labor disuiliy i.e. κ. This funcion is decreasing and convex in he mark-up. Noice also ha he seady sae oupu and labor equilibrium levels are idenical o he righ-hand side in (32) because i follows ha L s = L s = Y s (h) =Y s (f) =C = C. I evaluae he money demand equaions in () and (4) a heir seady sae values and obain ha P C = P C = b µ β χ. (33) ere I employed he fac ha money markes clear and Em = Em =0. This equaion implies ha a version of he quaniy heory of money holds in seady sae. The lef-hand side appears expressed in erms of nominal consumpion. owever in seady sae consumpion equals oupu in each counry and his is consisen wih he quaniy heory. The righ-hand side should include he velociy of money imes he money supply. By consrucion money supply is equal o one (since Em = Em =0) in seady sae. Therefore he righ-hand side of (33) corresponds o he seady sae money velociy by analogy. Money velociy is relaed o he rae of ime preference i.e. β which is also he inverse gross nominal ineres rae in seady sae o he habi formaion parameer i.e. b and o he preference weigh on uiliy from real balances i.e. χ. Subsiuing (32) in (33) I infer ha he seady sae consumpion price index is equal o µ P = P = κ +ϕ χ ϕ ( b) +ϕ ( β) θ +ϕ. (34) θ urhermore combining (23) (26) wih (34) and P = P = P = P I also obain ha he seady sae wages are equal o µ ϕ = +ϕ = κ χ ϕ ( b) +ϕ ( β) θ +ϕ. (35) θ This pins down he enire seady sae of he model and saisfies my iniial conjecure. Equaions (34) and (35) bohdependonheraeofimepreference i.e. β he habi formaion parameer i.e. b hemark-up θ charged by firms i.e. θ and he preference weigh on labor disuiliy and on uiliy from real balances i.e. κ and χ respecively. Consumpion prices are clearly increasing in he mark-up while wages are decreasing whenever + ϕ>. Noice also ha he seady sae ineres rae β onlyaffecs prices and wages bu no consumpion or oupu in seady sae. Le me assume for he sake of argumen ha his parameer is an insrumen ha can be chosen by a cenral bank or alernaively ha he uncondiional mean of money supply can be se o be differen han

13 one. Boh specificaions produce similar resuls. My calculaions sugges ha he seady sae oucome is consisen wih he noion of long-run neuraliy of money. Moneary variables have only effecs on he price and wage levels bu no on producion. owever in he shor-run he presence of nominal rigidiies (as well as oher fricions) in he model means ha moneary variables can have real effecs ha have o be accouned for. D The Linearizaion of he Equilibrium Condiions I linearize he equilibrium condiions around he deerminisic zero-inflaion zero-curren accoun seady sae(seealsokinge al. 988). I approximae all variables in logs excep he real ne foreign asse posiion of financial inermediaries (in levels) (e.g. P. Benigno 200 and Thoenissen 2003). I denoe bx ln X ln X he deviaion of a variable in logs from is seady sae and X b X X he deviaion of C a variable in levels from is seady sae relaive o seady sae consumpion (or oupu). The approximaion echnique of each equilibrium condiion is applied as follows: (i) Le me assume he equilibrium condiion can be expressed as X i = i (X... X q )andx q mus be approximaed in levels. irs I re-express he equaion in logs ln X i =ln i e ln X... e ln Xq X q. Second I ake a firs-order approximaion in he log and level variables around he seady sae i.e. ln X i ln X i + X q i e ln X... e ln Xq X q e j= i e ln X... e ln X q Xq ln X j ln X j ln X j + X j i e ln X... e ln Xq X q + Xq X q i e ln X... e ln X q Xq X q or simply bx i X q i X... X q X j bx j + i j= X j i X... X q ln G k e ln X... e ln X q + X q j= G k e ln X... e ln X q Xq X... X q X q C i X... X q b Xq assuming ha X i = i X... X q 6=0. In his sense each log variable on he righ-hand side of he previous equaion is simply weighed by is seady sae elasiciy. (ii) Le me assume he equilibrium condiion can be expressed as G k (X... X q )= k (X... X q )and X q mus be approximaed in levels. irs I re-express boh equaions in logs ln G k e ln X... e ln Xq X q = ln k e ln X... e ln Xq X q. Second I ake a firs-order approximaion in he log and level variables around he seady sae i.e. G k e ln X... e ln Xq X q G k e ln X... e ln X q X q + Xq X q G k e ln X... e ln X q Xq X q ln k e ln X ln... e Xq + X q j= k e ln X... e ln X q Xq k e ln X... e ln Xq X q + Xq X q k e ln X... e ln X q Xq X q X j e ln X j ln X j ln X j + k e ln X... e ln X q X q X j e ln Xj ln X j ln X j + 2

14 or simply X q j= X q j= G k X... X q X j k X... X q X j X j G k X... X q bx j + G k X j k X... X q bx j + k X...X q C Xq b X q G k X...X q X... X q C Xq b k X... X q X q assuming ha G k X... X q = k X... X q 6=0. (iii) Le me assume he equilibrium condiion can be expressed as G k (X... X q )= k (X...X q ) X q mus be approximaed in levels and G k X...X q = k X... X q =0. Then he equilibrium condiion can sill be approximaed as follows X q G k X... X q X j bx j + G k X... X q CX j= X j X b q q X q k X... X q X j bx j + k X... X q CX j= X b q. q X j This formula is paricularly useful o log-linearize he resource consrain. (iv) If he equilibrium condiion conains leads and lags of cerain variables I approximae he equaion around hem oo. If he equaion conains expecaions hen I use he fac ha he expecaion of he log is approximaely equal o he log of he expecaion. More deails on he log-linearizaion are provided as needed in he res of his echnical noe. D. The CPI he RS and he RP Equaions Le me sar by describing he log-linearizaion of he domesic and foreign consumpion-based price indexes defined in () and (4). In seady sae i mus hold ha P = P and P = P. ence aking a firs-order approximaion of he price indexes I derive he CPI equaions as bp nbp +( n) bp (36) bp nbp +( n) bp. (37) Idefine he domesic and foreign inflaion raes in deviaions as bπ bp bp and bπ bp bp respecively. Idefine he real exchange rae as RS SP P and he relaive prices in he home and foreign counry respecively as T P P approximaely equal o and (T ) P And he RP equaions ake he form of P. Therefore i immediaely follows ha he RS equaion is brs = bs + bp bp (38) bs + nbp +(n) bp nbp +(n) bp. b = bp bp (39) b = bp bp. (40) I denoe world relaive prices b and he discrepancy on relaive prices across counries b R asfollows b nb ( n) b b R b + b. 3

15 ence he RP equaions can also be expressed in erms of b and b R as b = b nb R b = b +(n) b R. S P P Idefine domesic and foreign erms of rade respecively as ToT P P P. Then I can infer ha P S P P and ToT ToT = co bp bs bp + b co = co = bs + bp bp + b whichinurnallowsmeore-wrieherealexchangeraeas brs nb ( n) b bp bs bp = b co. Relaive prices and erms of rade represen differen measures of he value expressed in unis of he local currency of impored goods in erms of he domesic good. The relaive prices indicae he cos of replacing one uni of impors bough in he local marke wih one uni of he domesically-produced good. The erms of rade raio in urn expresses he cos of replacing one uni of impors wih one uni of expors o he foreign marke. These calculaions show ha movemens in he real exchange rae can be hough as he resul of differences beween world relaive prices and domesic erms of rade. These formulas and definiions are very useful in he derivaion of he linearized equilibrium condiions of he model. D.2 The Demand-Side of he Economy D.2. The Invesmen-Savings Equaions: IS and IS To obain he IS and IS equaions I need o log-linearize he Euler equaions in (3) and (6) around he seady sae i.e. E exp µ C+ bc P ξ + exp (i ) C bc P + β {z } {z } = 0 + G + E exp ξ µ C+ bc P + C bc P+ {z } + exp (i ) β {z } G + = 0. In seady sae holds ha f = g and f = g hen he log-linearizaion around he seady sae can be expressed as h i E bf+ bg + 0 h i E bf + bg + 0 4

16 where he approximaions are as follows µ bf + b ξ + b µ bf + b ξ + b bg + bi bg + bi. µ b bc + + bc + + bc bπ + b µ b bc bπ b + Then I obain he following sysem of wo linearized Euler equaions h i b E [ bc + b bc ] E b i bπ + E h b i ξ + (4) b E bc + b bc h i h E b i bπ + E b ξ i + (42) which are convenionally denoed he IS and IS equaions respecively. The ex ane isher equaion requires ha he model-based implici real ineres rae be equal o h i br E b i bπ + h i br E b i bπ +. The growh rae of consumpion adjused by he effec of he habi is proporional o his measure of he real ineres rae and he preference-shock on consumpion. The consan of proporionaliy is a funcion of he habi parameer b andhecoefficien of relaive risk aversion. Using he linearized consumpion price indexes in (36) (37) I obain ha h i b E [ bc + b bc ] E b i nbπ + ( n) bπ + b E bc + b bc h E b i nbπ + ( n) bπ + E h b i ξ + i E h b ξ + (43) i (44) where bπ + bp + bp and bπ + bp + bp.thisisheversionofheis and IS equaions ha I use in he paper. inally using equaions (39) (40) for relaive prices I can also re-wrie he IS and IS equaions as h i b E [ bc + b bc ] E b i bπ + ( n) b + E h b i ξ + b E bc + b bc h i h E b i bπ + n b + E b ξ i +. owever in he lieraure i is common o rely on he world IS equaion IS andherelaiveis equaion IS R. These wo equaions describe he dynamics of world consumpion defined as a weighed average of domesic and foreign consumpion i.e. bc nbc +(n) bc and he dynamics of relaive consumpion i.e. bc R bc bc.theyaresufficien o describe domesic and foreign consumpion because i easily follows ha bc bc +(n) bc R bc bc nbc R. Their dynamics can be derived by compuing n (43)+( n) (44) and (43)(44) respecively. Noneheless for my purposes hey only obscure he analysis. In par because he dynamics for bc and bc R involve he decisions of wo disinc ypes of agens wih differen informaion ses. ence I sick wih he sandard IS and IS. 5

17 D.3 The Moneary-Side of he Economy D.3. The Money Marke Equaions: MM and MM To obain he MM and MM equaions I need o log-linearize he money demand equaions in () and (4) around he seady sae i.e. χ µ M d χ P (C bc ) {z } Ã P M d C bc! {z } = exp (i ) exp (i ) {z } G = exp (i ) exp (i. ) {z } G In seady sae holds ha f = g and f = g hen he log-linearizaion around he seady sae can be expressed as where he approximaions are as follows bf bg bf bg bf bm d + bp + b [bc bbc ] bf bm d + bp + bc b bbc β bg β b i bg β β b i. Then I obain a sysem of wo sable money demand funcions à la Cagan. In equilibrium he per capia money demand mus be equal o he per capia money supply i.e. bm d = bm bm d = bm. Therefore he following equaions fully characerize he money marke clearing condiions bm bp b [bc bbc ] µ β β µ bm bp bc b bbc β β bi (45) bi. (46) Real balances are coinegraed wih he nominal ineres rae wih aggregae consumpion (insead of aggregae oupu as in Cagan s money demand funcions) and wih he consumpion habi. The demand for real balances is a funcion of he habi parameer b andhecoefficien of relaive risk aversion. Using (36) (37) I obain he version of he money marke equaions ha I use in he paper µ bm nbp ( n) bp β bi (47) bm nbp ( n) bp b [bc bbc ] bc b bbc β µ β β bi (48) 6

18 which are convenionally denoed he MM and MM equaions respecively. Taking he difference beween MM and MM and using he definiion of he real exchange rae is easy o derive an expression for he nominal ineres rae spread across counries µ β bi R bm R + bc R β b bbc R + bs brs where bi R bi bi bcr bc bc and bmr bm bm. This implies ha he ineres rae differenial is inversely proporional o he money supply differenial. I also depends on consumpion consumpion habi and he differences in consumpion prices across counries measured as D.4 The Supply-Side of he Economy bp R bp bp = bs brs. Given he srucure of he (symmeric) equilibrium i suffices o explore he pricing rule of a represenaive firm wihin each counry in order o derive he AS equaions of he model. D.4. The Opimal Pricing Equaion for he Domesic irm Log-Linearizaion of he ome Marke OC. I can re-wrie he opimal pricing equaion in (7) as follows ( X " τ ep Ξ R +τ Y e d (h) +τ (h) θ µ # ) +τ P+τ =0 θ P +τ exp (a +τ ) P +τ where Ξ R +τ exp µ C +τ bc +τ P+τ ξ +τ ξ. C bc P +τ Under a symmeric equilibrium i holds rue ha L s = L d (h) = Y s (h) exp(a ). The second equaliy follows from he assumpion ha echnologies are linear in labor. I use his equilibrium condiion as well as he domesic labor supply equaion in (2) o re-wrie he pricing equaion as E X βα τ Ξ R +τ e Y d +τ (h) P (h) P +τ P +τ θκ θ (C +τ bc +τ ) ey s +τ (h)ϕ exp (( ϕ) a+τ ) P+τ P +τ where e Y s +τ (h) denoes he aggregae supply of firm h a ime + τ condiional on having heir prices fixed since period. Naurally he marke clearing condiion for variey h requires ha 4 ey +τ s (h) = ny e +τ d (h)+(n) Y e +τ d (h) Ã! θ ep µp σ Ã! θ (h) = n +τ ep µp C +τ +(n) (h) P +τ P +τ P +τ +τ P +τ σ C +τ where he second equaliy idenifies he aggregae demand based on he demand consrains of he firm. 4 Sensu srico G. Benigno s (2004) framework canno be viewed as a pure double-signal Calvo model (where each firm receives one signal o re-opimize in he domesic marke and anoher signal in he foreign marke). Benigno discouns he fac ha firms may receive a signal o re-opimize in one marke bu no he oher so he approximaes he symmeric equilibrium wih he behavior of a represenaive firm ha re-opimizes in boh markes simulaneously. or racabiliy and consisency I use he same approach here. Tha explains he characerizaion of he marke clearing condiion. =0 7

19 Le me define he following pair of funcions +τ P e (h) P+τ G θκ +τ θ (C +τ bc +τ ) ϕ Y e s +τ (h) exp (( ϕ) a+τ ) P +τ hen he firs-order condiion for price-seing becomes simply nx τ Ξ R +τ Y e d +τ (h) +τ G o +τ =0. In seady sae holds ha f = g hen he log-linearizaion around he seady sae can be expressed as nx h i o τ f b +τ bg +τ 0 where he approximaions are as follows P +τ bf +τ b ep (h) bp +τ bg +τ b (bc +τ bbc +τ )+ϕ b ey s +τ (h) ( + ϕ) ba +τ + bp +τ bp +τ s h b ey +τ (h) = n θ bep (h) bp +τ σ bp i +τ bp +τ + bc+τ + h +( n) θ bep (h) bp +τ σ bp +τ bp i +τ + bc +τ. Using equaions (39) (40) for relaive prices and equaions (36) (37) for he price indexes I can alernaively re-wrie he opimal pricing equaion as X ( + nϕθ) bep (h) bp +τ +(n) ϕθ bep (h) bp +τ τ ( n) ( + nϕσ) b +τ ( n) ϕσb +τ b (bc 0 +τ bbc +τ ) ϕbc +τ +(+ϕ) ba +τ where bp +τ bp +τ = ( n) b +τ bp +τ bp +τ = ( n) b +τ and world consumpion is bc +τ nbc +τ +(n) bc +τ. Noice ha I can re-express he price indexes bp +τ and bp +τ respecively as bp +τ = bp + X τ i= bπ +i and bp +τ = bp + X τ i= bπ +i. ence he opimal pricing equaion becomes ( + nϕθ) bep (h) bp +(n) ϕθ bep (h) bp (49) X τ X τ βα X ( + nϕθ) τ i=τ bπ +i +(n) ϕθ i=τ bπ +i + +( n) ( + nϕσ) b +τ ( n) ϕσb +τ + + b (bc +τ bbc +τ )+ϕbc +τ ( + ϕ) ba. +τ This expression shows compacly ha opimal pricing in he domesic marke depends on he opimal price se in he foreign marke. This is due o he fac ha prices a home and abroad deermine he aggregae demand he firm faces and herefore have an impac on domesic labor supply and wages. 8

20 Log-Linearizaion of he oreign Marke OC. I can re-wrie he opimal pricing equaion in (8) as follows ( X " τ ep Ξ R +τ Y e d +τ (h) (h) S +τ P+τ P+τ P+τ θ µ # ) +τ P+τ θ P +τ exp (a +τ ) P+τ =0 where Ξ R +τ exp µ C +τ bc +τ P+τ ξ +τ ξ. C bc P +τ Under a symmeric equilibrium i holds rue ha L s = L d (h) = Y s (h) exp(a ). The second equaliy follows from he assumpion ha echnologies are linear in labor. I use his equilibrium condiion as well as he domesic labor supply equaion in (2) o re-wrie he pricing equaion as E X βα τ Ξ R +τ e Y d +τ (h) P (h) P +τ S +τ P+τ P+τ θκ θ (C +τ bc +τ ) e Y s +τ (h) ϕ exp (( ϕ) a+τ ) P +τ where e Y s +τ (h) denoes he aggregae supply of firm h a ime + τ condiional on having heir prices fixed since period. Naurally he marke clearing condiion for variey h requires ha ey +τ s (h) = ny e +τ d (h)+(n) Y e +τ d (h) Ã! θ ep µp σ Ã! θ (h) = n +τ ep µp C +τ +(n) (h) P +τ P +τ P +τ +τ P +τ σ C +τ where he second equaliy idenifies he aggregae demand based on he demand consrains of he firm. Le me define he following pair of funcions P +τ e (h) S +τ P+τ G +τ P+τ P +τ θκ θ (C +τ bc +τ ) e Y s +τ (h) ϕ exp (( ϕ) a+τ ) P +τ hen he firs-order condiion for price-seing becomes simply nx τ Ξ R +τ Y e d +τ (h) +τ G o +τ =0. In seady sae holds ha f = g hen he log-linearizaion around he seady sae can be expressed as nx h i o τ f b +τ bg +τ 0 where he approximaions are as follows bf +τ b ep (h) bp +τ + bs +τ + bp +τ bp +τ bg +τ b ey s +τ (h) = n P +τ b (bc +τ bbc +τ )+ϕ b ey s +τ (h) ( + ϕ) ba +τ + bp +τ bp +τ h θ bep (h) bp +τ σ bp i +τ bp +τ + bc+τ + h +( n) θ bep (h) bp +τ σ bp +τ bp i +τ + bc +τ. P +τ =0 9

21 Using equaions (39) (40) for relaive prices and equaions (36) (37) for he price indexes I can alernaively re-wrie he opimal pricing equaion as nϕθ bep (h) bp X +τ +(+(n) ϕθ) bep (h) bp +τ τ ( n) ( + nϕσ) b +τ ( n) ϕσb +τ b (bc +τ bbc +τ ) ϕbc +τ +(+ϕ) ba +τ bs +τ + bp +τ bp +τ where bp +τ bp +τ = ( n) b +τ bp +τ bp +τ = ( n) b +τ and world consumpion is bc +τ nbc +τ +(n) bc +τ. Noice ha I can re-express he price indexes bp +τ and bp +τ respecively as bp +τ = bp + X τ i= bπ +i and bp +τ = bp + X τ i= bπ +i. ence he opimal pricing equaion becomes nϕθ bep (h) bp +(+(n) ϕθ) bep (h) bp (50) X τ X τ nϕθ βα X i=τ bπ +i +(+(n) ϕθ) i=τ bπ +i + τ +( n) ( + nϕσ) b +τ ( n) ϕσb +τ + + b (bc +τ bbc +τ )+ϕbc +τ ( + ϕ) ba +τ. bs +τ + bp +τ bp +τ This expression shows compacly ha opimal pricing in he foreign marke depends also on he opimal price se in he domesic marke. This is due o he fac ha prices a home and abroad deermine he aggregae demand he firm faces and herefore have an impac on domesic labor supply and wages. D.4.2 The Opimal Pricing Equaion for he oreign irm Log-Linearizaion of he ome Marke OC. follows ( X " τ Ξ R e ep +τ Y+τ d (f) P+τ (f) P+τ S +τ P+τ where I can re-wrie he opimal pricing equaion in (9) as à θ +τ θ P+τ exp a +τ Ξ R +τ exp ξ +τ ξ µ C+τ bc+τ C bc P +τ P+τ.! P +τ P +τ # ) =0 Under a symmeric equilibrium i holds rue ha L s = L d (f) = Y s (f) exp(a ). The second equaliy follows from he assumpion ha echnologies are linear in labor. I use his equilibrium condiion as well as he foreign labor supply equaion in (5) o re-wrie he pricing equaion as E X βα τ Ξ R +τ e Y d +τ (f) θκ θ C +τ bc+τ P (f) P+τ P+τ S +τ P+τ +τ (f) ey s ϕ exp ( ϕ) a P +τ +τ P +τ =0 20

22 where e Y s +τ (f) denoes he aggregae supply of firm f a ime + τ condiional on having heir prices fixed since period. Naurally he marke clearing condiion for variey f requires ha 5 ey s +τ (f) = ny e +τ d (f)+(n) Y e +τ d (f) Ã! θ P e µp σ Ã! θ (f) = n +τ e µp P C +τ +(n) (f) P +τ P +τ P +τ +τ P +τ σ C +τ where he second equaliy idenifies he aggregae demand based on he demand consrains of he firm. Le me define he following pair of funcions +τ e P (f) G +τ P +τ θκ θ P +τ S +τ P +τ C +τ bc +τ ey s +τ (f) ϕ exp ( ϕ) a P+τ +τ hen he firs-order condiion for price-seing becomes simply nx τ Ξ R e +τ Y+τ d (f) +τ G o +τ =0. In seady sae holds ha f = g hen he log-linearizaion around he seady sae can be expressed as nx h i o τ f b +τ bg +τ 0 where he approximaions are as follows bf +τ b ep (f) bp +τ bs +τ + bp +τ bp +τ bg +τ b s h b ey +τ (f) = n θ bep (f) bp +τ σ bp i +τ bp +τ + bc+τ + h +( n) θ bep (f) bp +τ σ bp +τ bp +τ P +τ bc +τ bbc +τ + ϕ b ey s +τ (f) ( + ϕ) ba +τ + bp +τ bp + bc +τ i. Using equaions (39) (40) for relaive prices and equaions (36) (37) for he price indexes I can alernaively re-wrie he opimal pricing equaion as ( + nϕθ) bep (f) bp X +τ +(n) ϕθ bep (f) bp +τ + τ +n nϕσb +τ ( + ( n) ϕσ) b +τ b bc +τ bbc +τ ϕbc +τ +(+ϕ) ba +τ 0 bs +τ + bp +τ bp +τ where bp +τ bp +τ = nb +τ bp +τ bp +τ = nb +τ 5 Sensu srico G. Benigno s (2004) framework canno be viewed as a pure double-signal Calvo model (where each firm receives one signal o re-opimize in he domesic marke and anoher signal in he foreign marke). Benigno discouns he fac ha firms may receive a signal o re-opimize in one marke bu no he oher so he approximaes he symmeric equilibrium wih he behavior of a represenaive firm ha re-opimizes in boh markes simulaneously. or racabiliy and consisency I use he same approach here. Tha explains he characerizaion of he marke clearing condiion. +τ 2

23 and world consumpion is bc +τ nbc +τ +(n) bc +τ. Noice ha I can re-express he price indexes bp +τ and bp +τ respecively as bp +τ = bp + X τ i= bπ +i and bp +τ = bp + X τ i= bπ +i. ence he opimal pricing equaion becomes ( + nϕθ) bep (f) bp +(n) ϕθ bep (f) bp (5) X τ X τ ( + nϕθ) βα X i=τ bπ +i +(n) ϕθ i=τ bπ +i τ n nϕσb +τ ( + ( n) ϕσ) b +τ + + b bc +τ bbc +τ + ϕbc +τ ( + ϕ) ba +τ + +. bs +τ + bp +τ bp +τ This expression shows compacly ha opimal pricing in he domesic marke depends also on he opimal price se in he foreign marke. This is due o he fac ha prices a home and abroad deermine he aggregae demand he firm faces and herefore have an impac on foreign labor supply and wages. Log-Linearizaion of he oreign Marke OC. I can re-wrie he opimal pricing equaion in (20) as follows ( X " Ã! # ) τ ep Ξ R e +τ Y+τ d (f) (f) P+τ θ +τ P θ P+τ exp +τ a +τ P+τ =0 where Ξ R +τ exp ξ +τ ξ µ C+τ bc+τ C bc P +τ P+τ. Under a symmeric equilibrium i holds rue ha L s = L d (f) = Y s (f) exp(a ). The second equaliy follows from he assumpion ha echnologies are linear in labor. I use his equilibrium condiion as well as he foreign labor supply equaion in (5) o re-wrie he pricing equaion as E X βα τ Ξ R +τ e Y d +τ (f) θκ θ C +τ bc+τ ey s P (f) P+τ +τ (f) ϕ exp ( ϕ) a P +τ +τ where e Y s +τ (f) denoes he aggregae supply of firm f a ime + τ condiional on having heir prices fixed since period. Naurally he marke clearing condiion for variey f requires ha ey s +τ (f) = ne Y+τ d (f)+(n) Y e +τ d (f) Ã! θ ep µp σ Ã! θ (f) = n +τ ep µp C +τ +(n) (f) P +τ P +τ P +τ +τ P +τ σ C +τ where he second equaliy idenifies he aggregae demand based on he demand consrains of he firm. Le me define he following pair of funcions P +τ e (f) G +τ P+τ θκ θ C +τ bc+τ ϕ ey s +τ (f) exp ( ϕ) a P+τ +τ P +τ P +τ =0 22

24 hen he firs-order condiion becomes simply nx τ Ξ R e +τ Y+τ d (f) +τ G +τ o =0. In seady sae holds ha f = g hen he log-linearizaion around he seady sae can be expressed as nx h i o τ f b +τ bg +τ 0 where he approximaions are as follows bf +τ b ep (f) bp +τ b h θ bep (f) bp +τ σ bp i +τ bp +τ + bc+τ + h +( n) θ bep (f) bp +τ σ bp +τ bp +τ bg +τ b ey s +τ (f) = n bc +τ bbc +τ + ϕ b ey s +τ (f) ( + ϕ) ba +τ + bp +τ bp + bc +τ i. Using equaions (39) (40) for relaive prices and equaions (36) (37) for he price indexes I can alernaively re-wrie he opimal pricing equaion as X nϕθ bep (f) bp +τ +(+(n) ϕθ) bep (f) bp +τ + τ +n nϕσb +τ ( + ( n) ϕσ) b +τ bc +τ bbc +τ ϕbc 0 +τ +(+ϕ) ba +τ where b bp +τ bp +τ = nb +τ bp +τ bp +τ = nb +τ and world consumpion is bc +τ nbc +τ +(n) bc +τ. Noice ha I can re-express he price indexes bp +τ and bp +τ respecively as bp +τ = bp + X τ i= bπ +i and bp +τ = bp + X τ i= bπ +i. ence he opimal pricing equaion becomes nϕθ bep (f) bp +(+(n) ϕθ) bep (f) bp (52) X τ X τ βα X nϕθ τ i=τ bπ +i +(+(n) ϕθ) i=τ bπ +i n nϕσb +τ ( + ( n) ϕσ) b +τ + bc +τ bbc +τ + ϕbc. +τ ( + ϕ) ba +τ + b This expression shows compacly ha opimal pricing in he foreign marke depends also on he opimal price se in he domesic marke. This is due o he fac ha prices a home and abroad deermine he aggregae demand he firm faces and herefore have an impac on foreign labor supply and wages. D.4.3 The Price Index Equaions The Price Index in he ome Marke. There exiss a relaionship beween he price indexes P P and he (symmeric) opimal pricing rule ep (h) P e (h). In a symmeric equilibrium under sicky prices +τ 23

25 he aggregae domesic-good price indexes in equaions (2) and (3) can be expressed as follows 6 P = P = α P θ + α θ θ P e (h) α P θ + α θ θ P e (h). A proporion α of all domesic firms se he (symmeric) opimal price in he domesic marke ep (h) afer receiving a signal o re-opimize a ime. The remaining firms in proporion of α behave as if on average hey had kep he previous period price unchanged. The lagged erm reflecs he aggregae behavior of all firms who canno re-se prices in he domesic marke. Similarly a proporion α of all domesic firms se he (symmeric) opimal price in he foreign marke P e (h) afer receiving a signal o re-opimize a ime ; while a proporion α leaves prices unchanged from he previous period. In a zero-inflaion zero-curren accoun seady sae i mus hold ha P = ep (h) and P = ep (h). ence he log-linear approximaion becomes simply bp α bp + α b ep (h) bp α bp + α b ep (h). A sraighforward manipulaion of hese equaions ell me ha he domesic and foreign inflaion raes on he bundle of domesic goods (i.e. bπ = bp bp and bπ = bp bp ) could be approximaely compued as bπ α h i b ep (h) bp bπ α h i b ep (h) bp which implies ha he difference beween he opimal pricing rules b ep (h) and b ep (h) and he price indexes bp and bp is proporional o he inflaion rae in logs i.e. h i µ bep α (h) bp h α bπ bep i µ α (h) bp α bπ. (53) The Price Index in he oreign Marke. There exiss a relaionship beween he price indexes P P and he (symmeric) opimal pricing rule ep (f) P e (f). In a symmeric equilibrium under sicky prices he aggregae foreign-good price indexes in equaions (5) and (6) can be expressed as follows 7 P = P = α P θ + α θ θ P e (f) α P θ + α θ θ P e (f). 6 As noed before G. Benigno (2004) discouns he fac ha firms may receive a signal o re-opimize in one marke bu no he oher so he approximaes he symmeric equilibrium wih he behavior of a represenaive firm ha re-opimizes in boh markes simulaneously. I use he same approach here. Tha explains why I do no disinguish beween firms ha re-opimize in one marke only and firms ha re-opimize in boh markes. 7 As noed before G. Benigno (2004) discouns he fac ha firms may receive a signal o re-opimize in one marke bu no he oher so he approximaes he symmeric equilibrium wih he behavior of a represenaive firm ha re-opimizes in boh markes simulaneously. I use he same approach here. Tha explains why I do no disinguish beween firms ha re-opimize in one marke only and firms ha re-opimize in boh markes. 24

26 A proporion α of all foreign firms se he (symmeric) opimal price in he domesic marke P e (f) afer receiving a signal o re-opimize a ime. The remaining firms in proporion of α behaveasifon average hey had kep he previous period price unchanged. Similarly a proporion α of all foreign firms se he (symmeric) opimal price in he foreign marke P e (f) afer receiving a signal o re-opimize a ime ; while a proporion α leaves prices unchanged from he previous period. In a zero-inflaion zero-curren accoun seady sae i mus hold ha P = P e (f) and P = P e (f). ence he log-linear approximaion becomes simply bp α bp + α b ep (f) bp α bp + α b ep (f). A sraighforward manipulaion of hese equaions ell me ha he domesic and foreign inflaion raes on he bundle of foreign goods (i.e. bπ = bp bp and bπ = bp bp ) could be approximaely compued as bπ α h i b ep (f) bp bπ α h i b ep (f) bp which implies ha he difference beween he opimal pricing rule b ep (f) and b ep (f) and he price indexes bp and bp is proporional o he inflaion rae in logs i.e. h i µ bep α (f) bp h α bπ bep i µ α (f) bp α bπ. (54) D.4.4 The Aggregae-Supply Equaions for he Domesic irm: AS and AS In order o derive he dynamics of he inflaion raes for he domesic bundle of goods i.e. bπ and bπ I need o manipulae furher he linearized firs-order condiions of he firm in (49) and (50). Thesoluion involves some algebra bu i is oherwise concepually sraighforward and can be summarized in hree seps. irs I use he resul in equaion (53) o replace he opimal prices b ep (h) and b ep (h) i.e. α α bπ α ( + nϕθ) bπ α +(n) ϕθ X τ βα X E ( + nϕθ) βα τ and βα E X i=τ bπ +i X τ +(n) ϕθ i=τ bπ +i + +( n) ( + nϕσ) b +τ ( n) ϕσb +τ + + b (bc +τ bbc +τ )+ϕbc +τ ( + ϕ) ba +τ α nϕθ bπ α α +(+(n) ϕθ) bπ α X τ X τ nϕθ βα τ i=τ bπ +i +(+(n) ϕθ) i=τ bπ +i + +( n) ( + nϕσ) b +τ ( n) ϕσb +τ + + b (bc +τ bbc +τ )+ϕbc +τ ( + ϕ) ba +τ bs +τ + bp +τ bp +τ Second I realize ha hese expressions ake he form of a discouned presen-value sysem ha can be re-wrien as he forward-looking (no-bubble) soluion of a pair of expecaional difference equaions. In. 25

27 oher words I can say ha he equaions of ineres are α ( + nϕθ) α bπ α +( n) ϕθ α bπ ½ ¾ βα ( n) ( + nϕσ) b E ( n) ϕσb + + b (bc bbc )+ϕbc ( + ϕ) ba + ( + nϕθ) bπ + + X τ +βα E X βα βα τ +( n) ϕθ bπ + + X τ i=τ bπ ++i i=τ bπ ++i + + +( n) ( + nϕσ) b ++τ ( n) ϕσb ++τ + + b (bc ++τ bbc +τ )+ϕbc ++τ ( + ϕ) ba ++τ and α ( + nϕθ) bπ α α +(n) ϕθ bπ α ½ ¾ βα ( n) ( + nϕσ) b E ( n) ϕσb + + b (bc bbc )+ϕbc ( + ϕ) ba + ( + nϕθ) bπ + +(n) ϕθbπ ++ X τ ( + nϕθ) +βα E X (βα +E ) τ X τ +( n) ϕθ (βα ) i=τ bπ ++i i=τ bπ ++i + + +( n) ( + nϕσ) b ++τ ( n) ϕσb ++τ + + b (bc ++τ bbc +τ )+ϕbc ++τ ( + ϕ) ba ++τ + where I used he law of ieraed expecaions. Therefore I obain he following ransformaion of (49) µ α ( + nϕθ) α bπ µ α +(n) ϕθ α bπ + βα ½ ( n) ( + nϕσ) b E ( n) ϕσb + b (bc bbc )+ϕbc ( + ϕ) ba ½ µ µ +βα E ( + nϕθ) α bπ + +(n) ϕθ α ¾ + bπ + ¾. Analogously I derive he following expression for he oher linearized firs-order condiion in (50) µ α nϕθ α bπ βα ½ E ½ +βα E nϕθ µ α +(+(n) ϕθ) α bπ ( n) ¾ ( + nϕσ) b ( n) ϕσb + b (bc bbc )+ϕbc ( + ϕ) ba bs + bp bp + µ ¾ bπ + +(+(n) ϕθ) bπ +. + µ α α Basedonhedefiniion of relaive prices he consumer price indexes and he real exchange rae in (36)(40) I can derive he following relaionship bs + bp bp = brs + bp bp bp bp brs +(n) bp = brs +(n) b + b bp ( n) bp bp where he discrepancy on relaive prices across counries is denoed b R b + b. 26

28 inally I noice ha he expressions I have derived so far are all funcions of he expeced inflaion rae of he domesic bundle of goods in he home as well as he foreign marke. Tha is µ α h oi ( + nϕθ) α bπ βe nbπ + βα ½ ¾ ( n) ( + nϕσ) b E ( n) ϕσb + + b (bc bbc )+ϕbc ( + ϕ) ba µ α α µ ( n) ϕθ α bπ α h n oi ( n) ϕθ α bπ βe bπ + and µ α h n oi ( + ( n) ϕθ) α bπ βe bπ + βα ½ ¾ ( n) ( + nϕσ) b E ( n) ϕσb + + b (bc bbc )+ϕbc ( + ϕ) ba βα E brs +(n) b ª + b µ α α µ nϕθ α bπ α h oi nϕθ α bπ βe nbπ +. ence in he hird sep I subsiue ou he inflaion expecaions o ensure ha each equaion is expressed as a funcion of he expecaions in one marke only. The subsiuion requires a bi of painful effor on he algebra. Afer replacing he foreign marke expecaions in he linearized pricing equaion for he domesic marke i follows ha he supply curve for he home good in he home marke is given by where o bπ βe nbπ + + kπ bπ + kπ bπ + kc E k π ½ ¾ b (bc bbc )+ϕbc ( + ϕ) ba + +k rse { brs } + k E b ª + k E b ª (55) α α α ϕθ(n) ϕθn kπ α α α (n) α α ϕθ ( n) µ µ kc (βα )(α ) (n) (βα )(α α ) ϕθ(n) α µ krs (βα )(α ) ϕθ(n) α µ µ k (βα )(α ) ((n))(+ϕσn) (βα )(α α ) ϕθ(n)(ϕσn) α ( n) µ k µ (βα )(α ) (+ϕσ(n))ϕθ(n) (βα )(α α ) ((n))ϕσ(n) α ( n). Subsiuing now he domesic marke expecaions in he linearized pricing equaion for he foreign marke i follows ha he supply curve for he home good in he foreign marke is given by n o ½ ¾ bπ βe bπ + + kπ bπ + kπ bπ + kc E b (bc bbc )+ϕbc ( + ϕ) ba + +k rs E { brs } + k E ª b + k E b ª (56) 27