Does External Habit Formation Help Increase Real Exchange Rate

Transcription

1 Does Exernal Habi Formaion Help Increase Real Exchange Rae Persisence? An Analyical Characerizaion Yu-Ning Hwang a, Deparmen of Economics Naional Chengchi Universiy March 2007 Absrac The objecive of his paper is o invesigae he effec of exernal habi formaion on real exchange rae persisence in a wo-counry dynamic sochasic general equilibrium (DSGE) model. The purchasing power pariy (PPP) puzzle saes ha high persisence and volailiy of he real exchange rae found in empirical sudies can no be capured in a single model wih convenional specificaions. Par of he puzzle is resolved hrough a DSGE model wih nominal rigidiy: he model generaes enough real exchange rae volailiy bu no enough persisence. Habi formaion was suggesed as an addiional facor conribuing o real exchange rae persisence. In his paper, we focus on he consumpion exernaliies caused by exernal habi formaion o undersand how habi socks affec real exchange raes. The analyical soluions show ha exernal habi formaion decreases he persisence of real exchange raes. The main reason is ha he exernaliy caused by exernal habi formaion has an opposie effec on he real exchange rae o ha of real shocks. Alernaive specificaions of he habi-relaed parameers do no reverse he negaive effec of exernal habi formaion on real exchange rae persisence. Keywords: Habi Formaion, Real Exchange Rae, Purchasing Power Pariy Puzzle JEL Classificaions: F3; F4; E32 a Correspondence: Deparmen of Economics, Naional Chengchi Universiy, 64, Sec. 2, Tz-nan Rd., Wenshan, Taipei 6, Taiwan. Tel: ex yuning@nccu.edu.w. This is he firs chaper of my disseraion. I hank Professor Sephen Turnovsky for his supervision of my Ph.D. disseraion. I also hank Professor Yu-Chin Chen, Richard Harman, Phillip Brock for helpful commens and suggesions. Responsibiliy of any error is mine.

2 . Inroducion One of he primary puzzles in inernaional macroeconomics is he purchasing power pariy puzzle. The exensive proposiion of purchasing power pariy addresses ha he relaive inflaion raes beween he wo counries shall be refleced in nominal exchange rae movemens. Based on his asserion, he real exchange rae shall be consan. However, empirical sudies show ha he real exchange rae exhibis high persisence and high volailiy. The difficuly in reconciling hese wo properies of he real exchange rae in a heoreical model is called he purchasing power pariy puzzle. Earlier empirical findings have been surveyed by Rogoff (996), and more recen effors devoed o his issue wih alernaive approaches include Chen and Rogoff (2003) and Murray and Papell (2005). Despie he many aemps in he lieraure o explain his puzzle, such as Bergin and Feensra (2002), i remains unresolved. The mos well-known paper explaining real exchange rae persisence and volailiy is Chari, Kehoe and McGraan (2002). They use a dynamic sochasic general equilibrium (DSGE) model and successfully generae boh he volailiy and persisence of he real exchange rae. The auhors calibraion resuls show ha he real exchange rae in he DSGE model is approximaely four o five imes more volaile han GDP, as observed from he OECD daa. However, he persisence of he real exchange rae ranges from 0.63 o 0.68 per quarer, smaller han he 0.83 found empirically. Long persisence, close o he empirical finding, can be obained only when a nominal rigidiy of 2 o 3 years is assumed. Requiring such a high degree of nominal rigidiy is unreasonable; ha he model fails o generae enough persisence in he real exchange rae remains a persisence puzzle. Habi formaion was suggesed as one of he possible soluions o he persisence puzzle. Because, in a DSGE model he real exchange rae is deermined by he consumpion raio across counries when here are complee asse markes of radable securiies, facors ha can

3 2 increase persisence in consumpion may help reinforce he real exchange rae persisence. Of hese facors, habi formaion has been considered as he primary one. The imporance of ime non-separable uiliy funcions has been recognized in various empirical sudies. In he lieraure, here are wo ypes of habi formaion specificaions: inernal and exernal. The specificaion of inernal habi formaion is he habi persisence commonly indicaed in he lieraure. The noion assers ha consumers ake he accumulaion of heir own pas consumpions as he reference sock and do no wan heir curren consumpion o deviae oo much from pas consumpion habis. The persisence of consumpion and mos of he real variables are increased accordingly. In addiion o is wide applicaions o growh and macroeconomics relaed lieraure, habi formaion has imporan implicaions in he evaluaion of moneary policies, as noed in Fuhrer (2000). While moneary policy evaluaion is based on welfare, namely, expeced uiliy, he specificaion of uiliy funcion maers because i deermines how he variables in he economy reac o moneary shocks. Fuhrer (2000) finds ha a model wih habi persisence can generae more humpshaped responses of oupu o moneary shocks, closer o he ones observed in he daa. A subsequen sudy, Amao and Laubach (2004), also suppors his finding. On he oher hand, he specificaion of exernal habi formaion saes ha consumers wan o cach up wih ohers and hus ake he accumulaion of pas average consumpion in he economy as he reference sock. This is known as caching up wih he Joneses in he lieraure. As a resul, he higher consumpion of each consumer imposes an exernal effec on ohers. The imporance of consumpion exernaliy has been recognized in economics, as noed in Liu and Turnovsky (2005). The consumpion exernaliy is mos commonly used in he asse pricing lieraure, such as Campbell and Cochrane (999) and Abel (999). Consumers wih habi formaion preferences do no like drasic changes in consumpion, hus hey require a

4 3 higher risk premium o hold risky asses ha may resul in a sharp decline in fuure consumpion. The objecive of his paper is o invesigae he effec of habi formaion on real exchange rae persisence in a DSGE model wih nominal rigidiy. The inroducion of habi formaion o DSGE models is no new. While DSGE models have been widely used in inernaional macroeconomics, many recen sudies adop habi formaion in he uiliy funcion as a more general specificaion of he uiliy funcion, in an aemp o increase he persisence of real variables. However, Jusiniano and Preson (2004), among ohers, find ha habi formaion does no increase real exchange rae persisence, even hough habi formaion does improve he fiing of he model o he daa. Alhough his finding is pervasive, no one has ye provided an explanaion. In his paper, we adop an analyically racable model o characerize he role of habi formaion in inernaional macroeconomics, which is implici in a generalized DSGE model where he derivaion of analyical soluions is no feasible. Our model is similar o a sandard DSGE model in general, bu differs in some imporan respecs. Because we do no inend o solve he purchasing power pariy puzzle, bu insead focus on habi formaion s effec on he real exchange rae, we make several assumpions o faciliae he derivaion of analyical soluions. Based on he analyical soluions, we can shed ligh on he role of exernaliies caused by he habi socks in he economy, how he exernal effecs are passed across counries and how hey affec he real exchange rae and is persisence. To do so, firs, we ake exernal habi formaion, raher han inernal habi persisence. Second, insead of saggered pricing, he nominal rigidiy is on wages, which are prese one period ahead. In his model, he deviaion from he purchasing power pariy is no from he nominal rigidiy, bu from he nonradable goods. Wih his specificaion, we can simplify boh

5 4 analyical and algebraic complicaions. However, wih he complee asse markes, he Cobb- Douglas uiliy funcion and he sicky wage, he real exchange rae is forced o be one if he iniial deb is zero. To assure he movemens of he real exchange rae, we need o impose he condiion ha he iniial securiy is nonzero. Third, he logarihmic form of he money demand uiliy funcion leads o consan nominal ineres raes and isolaes he nominal exchange rae movemens from habi formaion changes. This simplified case provides us wih he inuiion concerning habi formaion s effec on he real exchange rae hrough relaive wage. Las, we assume ha labor is he only producion inpu. Technology shocks are auocorrelaed, bu he nominal shocks are no. Therefore, real shocks are he primary source of persisence, in addiion o habi formaion, in his model. The analysis is cenered on he radeoff beween he effec of habi formaion and produciviy shocks on relaive wage. The resuls show ha exernal habi formaion decreases real exchange rae persisence, while increasing he persisence of real variables. The negaive effec is mainly caused by he adverse effec of habi formaion on he level of real exchange rae o ha of pas produciviy shocks. Alhough he resuls may srongly depend on he assumpions we make, he inuiion can be carried wihou losing generaliy. Wihou saggered pricing, exernal habi formaion does no influence real exchange rae volailiy. In addiion o he discussion based on analyical soluions, we also conduc calibraions o examine, numerically, he effecs of habi formaion on he AR() coefficien of he real exchange rae. The resuls show ha he real exchange rae persisence declines wih he increase in he imporance of habi formaion and he adjusmen speed of he habi sock accumulaion. In spie of differen reamens of he habi-relaed parameers, habi formaion s negaive effec on he real exchange rae canno be reversed.

6 5 The srucure of his paper is as follows. In Secion 2 and Secion 3, respecively, we presen he specificaion of he basic model and he analyical soluions derived o demonsrae he effecs of he produciviy shocks and exernal habi formaion on wages. In Secion 3, we provide he inuiion behind he resuls. Secion 4 hen presens a sensiiviy analysis based on calibraion, while Secion 5 conducs alernaive specificaions of he parameers. Finally, in Secion 6, we presen our conclusions and discuss areas for fuure research. 2. The model 2. Producion There are wo counries, home and foreign. Each counry produces boh radable and nonradable goods. Y H and YN denoe raded and nonraded goods j produced in home Y F Y N counry, while and denoe raded and nonraded goods in foreign counry. There is a coninuum of varieies for each ype of goods. Home goods are indexed from j (0,) while foreign goods are indexed from j (, 2). Labor is he only inpu in he producion process. Traded and nonraded goods j are produced according o he producion funcions: ( φ )/ φ YH,() j A LH,(,) j i di 0 φ/( φ ) = (a) ( φ )/ φ YN,() j A LN,(,) j i di 0 φ/( φ ) = (b)

7 6 L( ji, ) is he demand for labor ype i in he producion of good j. φ is he elasiciy of subsiuion among labors in he producion. produciviy shock. A represens he auocorrelaed counry specific As described, he labor marke is a monopolisic compeiion marke. Wage sickiness is he primary source of nominal rigidiy and each worker has a cerain marke power o prese he wage. The composie wage, of produc: W, is derived from he cos minimizaion of producing one uni φ W W() i di 0 /( φ ) = (2) Thus, he labor demand for labor i in he producion of good j can be obained: φ Wi () Y( j) L (,) j i = W A (3) Due o wage sickiness, labor demand deermines he labor employmen in he labor marke. Produciviy shocks have direc effecs on he labor employmen, because he same amoun of goods can be produced by fewer workers when a posiive produciviy shock occurs. Thus, he marginal cos for he home firm is W/ A, from he producion funcion in Eq. (). The logarihm of home produciviy shock, indicaed by lower-case a, evolves according o he following AR() process: 2 ( ρ ) ρ ε, ε σ a = a+ a + N(0, ) (4a) a a a, a, a Foreign producion funcions and wages are in he parallel way and indicaed by aserisks. Foreign produciviy shock behaves analogously: 2 a = ρa a + ρaa + εa,, εa, N(0, σ a ) (4b)

8 And he foreign marginal cos is W A. / Consumpion Consumers in each counry consume boh home and foreign produced radable goods and domesic nonradable goods. For each individual i in he home counry, he composie consumpion index is consruced in he Cobb-Douglas form: γ CC T C = γ ( γ) γ N γ (5a) Where C and C are he indices of radable and nonradable consumpions. The radable T N consumpion index is composed of equal shares of home and foreign radable goods, C F : C H and C = 2C C (5b) /2 /2 T H F The composie consumpion index for each ype of goods is in he CES form: θ/( θ ) () / d θ θ CH = CT j j, 0 θ/( θ ) 2 () / d θ θ CF = CT j j, θ/( θ ) () / d θ θ CN = CN j j (5c) 0 θ is he price elasiciy among goods. The aggregae price index for he composie consumpion is /2 /2 P= P P, where P = P P (6a) γ γ T N T H F where Wih his specificaion of he composie consumpion index, he elasiciy of subsiuion beween each ype of good is equal o one. As noed in Obsfeld and Rogoff (998), he curren accoun will be equal o zero when he iniial deb is assumed zero, even hough here are asse markes o share risks across counries. This poin will be discussed in more deail when we alk abou he marke clearing condiions.

9 θ PH = P i di i= 0 2 θ θ θ T(), PF = PT() i di, N N() i= θ θ P = P i di (6b) i= 0 According o he composie consumpion and price indices, he home demand for each commodiy is: 8 Ph T () () CT h = P H θ θ PT CH, ( f ) CT f = CF P, () P () N h CN h = F P N θ C N (7a) Home demand funcions for home and foreign radable goods are C H P H = 2 P T C T, C F P F = 2 P T C T (7b) and he radable and nonradable good demands are: C PT T = γ P C, C N PN = ( γ) P C (7c) Wih his composie consumpion index, oal spending of radable goods akes he share of γ in he oal expendiure, PC. The consumpion and price indices in he foreign counry are in he analogous form wih aserisks. 2.3 Opimizaion Consumers face he ineremporal maximizaion problem as follows: max ρ i i C ( s ) M ( s) κ i υ E β χln ( L ( s + α )) (8a) = 0 s ρ ( ) P H s s υ +

10 s.. i i i + + ( + ) ( + ) Ps C s M s Bs sd s s + i i i (, ) [ (, ) (, )] d Ps Wis L s + M s + D s + js + js j+ Ps Ts H N 0 (8b) 9 An individual i obains he uiliy from he consumpion of composie good,, real money holding and he disuiliy from labor supply. β measures he subjecive discoun facor, and ρ > 0 denoes he curvaure of he uiliy. We assume ha money demand eners he uiliy funcion in a logarihmic form for analyical simpliciy. This is a special case of he general CRRA uiliy funcion form, under which closed-form soluions of money demand and he i C nominal exchange rae arise. 2 This hus permis us o absrac he discussion of he real exchange rae from he nominal exchange rae bu o focus on relaive price. 3 υ indicaes he inverse of labor supply elasiciy. Tha υ assures a posiive elasiciy of labor supply. χ and κ are posiive consans deermining he uiliy of real money holdings and he disuiliy caused by labor supplied. The primary variaion of his model is habi formaion on consumpion, H, in he uiliy funcion, which urns he uiliy funcion ino a ime non-separable form. Here, we adop exernal habi formaion, which is known as caching up wih he Joneses in lieraure. Wih exernal habi formaion, consumers ake he accumulaed pas average consumpion of he enire economy as he reference habi sock. The main propery of exernal habi formaion is he consumpion exernaliy of each individual s consumpion on he consumpions of ohers. An increase in he consumpion of one consumer in he previous period raises he exernal habi sock in he economy and increases he marginal uiliy of consumpion for all oher consumers in his period. Thus, all consumers have he incenive o increase curren 2 See Obsfeld and Rogoff (998), Devereux and Engel (998) and Appendix.A. 3 This is he special case in Devereux and Engel (998) as noed below.

11 0 consumpion o cach up wih he aggregae consumpion level when ρ > and α > 0. However, from he individual perspecive, he impac of one s consumpion on he aggregae consumpion is negligible, hus individual consumers do no ake ino accoun his exernal effec in he opimizaion. As shown below, he exernal effecs are spilled over o anoher counry hrough inernaional rades in he wo-counry model. An alernaive characerizaion of consumpion exernaliy, as in Gali (994), is called keeping up wih he Joneses. Curren average consumpion, insead of he pas, is aken as he reference sock; ha is, H = C and consumpion behaviors may vary wih he sign of α. When α is posiive, consumers are jealous and wan o keep up wih he Joneses, and hus greaer aggregae consumpion induces higher consumpion. On he oher hand, when α is negaive, he higher aggregae consumpion decreases boh he marginal uiliy and consumpion, in which case consumers end o run away from he Joneses. This occurs because people admire he consumpion of ohers. The following analysis is primarily based on he caching wih he Joneses case wih posiive α, as mos sudies are. The exensive analysis of negaive α is conduced by calibraion. The consumpion uiliy is characerized as he raio of curren consumpion o he habi sock, wih α as he imporance of habi sock on he uiliy of consumpion, which lies beween 0 and. 4 When α = 0, he habi sock is irrelevan o he decision making, and hus he uiliy funcion collapses o he convenional specificaion. Moreover, when ρ =, even hough habi formaion remains in he uiliy funcion, i does no affec he marginal uiliy of 4 We follow he specificaion in Abel (990) of aking he raio of consumpion o habi sock. This specificaion has a drawback in ha he uiliy funcion is no guaraneed o be concave; hus he opimal consumpion derived from he firs-order condiion is no guaraneed opimal. Oher sudies, such as Campbell and Cochrane (999) and Ljungqvis and Uhlig (2000), ake he difference beween curren consumpion and habi sock o avoid his problem. Moreover, he alernaive specificaion does no require ρ > o assure he posiive effec of habi sock on he marginal uiliy of an addiional uni of consumpion.

12 consumpion. To ensure he posiive effec from he habi sock on he marginal uiliy of consumpion, we assume ρ >. 5 The habi sock, he accumulaion of pas consumpions, adjuss wih ime. Le he exernal habi sock H evolve wih pas aggregae consumpion in he following process, similar o he one specified in Alvarez-Cuadrado, Monerio and Turnovsky (2004): H H C λ = H (9) C represens he average consumpion of he economy a he period. λ is he adjusmen speed of habi sock accumulaion, which lies beween 0 and. Higher λ implies ha he adjusmen of habi sock accumulaion falls more heavily on he laes consumpion. If λ =, H = and he habi sock in he economy depends on he aggregae consumpion from he C previous period only. We assume ha asse markes are complee; ha is, here is a complee se of sae- 6 coningen securiies D( s + ) wih he price B ( s s) +. The household, as a worker, receives wages from her labor supplied and, as a firm, collecs profis from producion. he per capia lump-sum ransfer from he governmen, financed by he creaion of money. The governmen s budge consrain is as follows: PT = M M PT is Foreign consumers face he same opimizaion problem: 5 In mos sudies, such as Gali (994) and Ljungqvis and Uhlig (2000), hey do no require ρ >. Here, we follow Amao and Laubach (2004) o assure he posiive effec of habi sock on marginal uiliy. 6 Because all he exogenous shocks are lognormal, here will be a coninuum of saes. The specificaion of discree saes in he budge consrain can be exended o coninuous saes direcly.

13 max E i ( H ( s )) ρ C s M s i ( s ) κ υ i β + χ ln ( L ( s )) (0a) α = 0 s ρ P υ + 2 s.. i i i + + ς ( + ) ( + ) P s C s M s s B s s D s s + H, N, 0 i i i (, ) [ () ()]d P s W i s L s + M s + D s + j + j j+ P s T s (0b) Where s ς is he nominal exchange rae in he sae s a period, defined as he home currency price of one foreign currency. The foreign habi sock is accumulaed in an analogous process: λ H H H C = () And he foreign governmen budge consrain is balanced in a similar form: PT = M M Boh he home and foreign money supplies are random walks and heir growh raes are lognormally disribued as follows: 7 2 M = μ M, η = η+ ε, ε (0, σ ) (2a) η, η, N η M μ, M = η η ε ε 2 N σ η, = +, (0, ) (2b) where η = log μ. Moneary auhoriies in hese wo counries ac independenly and neiher reacs o shocks. η and η are he means of η and η. The disribuions of he money supply growh raes are assumed as no auocorrelaed and iid... η 7 As noed in Devereux and Engel (998), lognormally disribued money supplies are no necessary for he derivaion of close-form soluions. Close-form soluion can be obained by assuming normal disribuion of money supplies.

14 2.3. Firs-Order Condiions 3 In each period, consumers choose he consumpion, money holding and sae-coningen securiies o maximize he uiliy. The firs-order condiions of each individual i are as follows: χ C H C H M P P i ρ α( ρ ) i ρ α( ρ ) + + = E i β + (3a) ( ) = βπ ( ) B s s s s ρ α ( ρ ) + + i ρ α ( ρ ) P( s + ) i C s H s P s + + C s H s (3b) Eq. (3a) is he condiion wih respec o he money held ineremporally ha consumers equae wih he benefi and cos from he holding of one more currency. Rewriing his equaion, we can obain he Euler equaion: where / ( i) ( + = B s+ s ) s+ M i C = (4) P i i i ρ α( ρ ) + H χ i. The derivaion is in Appendix I.A. From his equaion, we know ha he nominal ineres rae is deermined by he change in consumpions across periods, insead of he level. Thus i will be smooher han in he convenional model. Eq. (3b) describes he condiion for consumers o deermine he holdings of saeconingen securiies. π ( s+ s) is defined as π( s ) / + π( s ), indicaing he probabiliy of he occurrence of sae given he curren sae s. When α = 0 or ρ =, habi formaion s + has no effec on he consumpion and hese wo condiions collapse o he convenional ones. Foreign consumers solve he same problem and parallel firs-order condiions arise: i ρ α ( ρ ) i ρ α ( ρ ) C H C+ H+ = E i β + χ M P P (5a)

15 ( + ) = βπ( + ) B s s s s i α ( ρ ) C s H s ς ( s) P s ρ α ( ρ C s H s ) ς ( s+ ) P s+ ρ ( + ) ( + ) i (5b) Money Demand Funcion As poined ou by Obsfeld and Rogoff (998) and Devereux and Engel (998), he logarihmic uiliy funcion yields he closed-form soluion of he money demand funcion from he money marke equilibrium condiion, Eq. (3a). In Appendix I.A, we show ha consumpion can be wrien as M χμ = μ β C H ρ α( ρ) 2 σ /2 P e η (6a) The main difference beween his equaion and hose in he papers menioned above is he dependence of he money demand on he habi sock, no jus on curren consumpion. When α = 0 or ρ =, he money marke equilibrium condiion is insulaed from habi formaion. 8, 9 From his equaion, given he money supply and price, he habi sock has a posiive effec on consumpion. Moreover, he habi sock does no influence consumpion s responses o shocks. 8 The resul comes from he assumpion of he logarihmic form of he money demand uiliy funcion. Obsfeld and Rogoff (998) use a general CRRA form of uiliy funcion: M ε ε P where /ε is he curvaure. When ε =, he form collapses o he one we adop in his paper. In he more general form case, we are no able o obain closed-form soluions, bu he approximaion around nonsochasic seady-sae variables is needed. The logarihmic uiliy funcion and he complee asse markes, equaion (), lead o he simplified behavior of he nominal exchange raes as shown below. 9 In Bouakez (2003), by aking a general specificaion of money demand uiliy funcion, shows ha real exchange rae persisence decreases wih he value of he parameer ε. The highes value of he real exchange rae persisence is reached when ε =.

16 5 Because individuals are symmeric in each counry, C i = C = C. i will be dropped hroughou he paper and C in habi sock accumulaion will be replaced by C. While his sudy focuses on he persisence, his equaion provides us wih he informaion on how he economy reacs o shocks. Combining Eq. (6a) wih Eq. (4), we can see ha he nominal ineres rae shall remain consan. In he face of a posiive nominal shock, Eq. (6a) shows ha boh curren consumpion and price levels increase. (Noe however ha he adjusmens of he prices o he moneary expansion are primarily on he nominal exchange rae movemens since wages are predeermined). While he real ineres rae mus drop wih he higher curren consumpion, he higher curren price level increases he expeced fuure price level. Wih he logarihmic money demand uiliy funcion, he movemens of he real and nominal ineres raes offse wih each oher, as poined ou by Devereux and Engel (998). The impac of a posiive produciviy shock will be a bi differen. While a echnological innovaion occurs on he producion side, i affecs he good demand by lowering he prices. As shown below in he opimal pricing, a posiive real shock leads o lower prices and induces greaer consumpions. The lower price from he curren posiive produciviy shock leads o lower expecaion o fuure price level, while he expeced consumpion growh shall accompany lower real ineres rae. Again, hese wo effecs offse wih each oher and hus he nominal ineres rae is consan. The consan nominal ineres rae remains in he presence of he habi sock. The effec of he habi sock on he real ineres rae will be offse by is effec on he price movemens, mainly on wages. The foreign money demand funcion follows analogously: M P χμ = 2 σ /2 e η μ β C H ρ α( ρ) (6b)

17 2.3.3 The Risk-Sharing Condiion and he Nominal Exchange Rae 6 Wih he complee asse markes, from Eq. (4b) and (5b), we are able o derive he risksharing condiion as follows: Q ρ C ρ α( ρ) C C H α ς P H = = = ( ) P ρ α ρ ρ C H C C α H (7) Q is defined as he level of he real exchange rae. Again, when α = 0 or ρ =, his risksharing condiion collapses o he sandard one wihou habi formaion. Moreover, in he case of λ =, H = C. The far righ erm in Eq. (7) shows ha he real exchange rae deermines boh he level and growh rae of consumpion across counries. This risk-sharing condiion shows ha he raio of he marginal uiliy of consumpion moves along wih he real exchange rae. The marginal uiliies of consumpion across counries will be equalized if he purchasing power pariy holds and hus consumpions across counries shall be idenical every period. As a resul, even hough habi formaion is presen, consumpions are idenical from he iniial sae. However, he purchasing power pariy in his model does no hold in general. The deviaion of he real exchange rae from one is primarily driven by he presence of nonradable goods, given ha he iniial deb is no zero as discussed below. Throughou his paper, we assume ha all he parameers are idenical beween counries, excep for being asymmeric in α and λ in he furher analyses by calibraion. Wih he idenical coefficiens and by subsiuing Eq. (6a) and (6b) ino he risk-sharing condiion, we can obain he nominal exchange rae as he raio of money supply:

18 7 M ς = (8) M From his equaion, we know ha he nominal exchange rae does no overshoo in reacing o nominal shocks. In spie of exernal habi formaion, his resul is same as he logarihmic case in he sandard uiliy funcion specificaion in Devereux and Engel (998). As discussed, he impac of a posiive nominal shock will be exacly refleced in he nominal exchange rae movemens, causing he permanen depreciaion of home currency. This simple form of he nominal exchange rae implies ha he effecs of habi formaion on he real exchange rae resul, in heir enirey, from heir effecs on relaive price, or relaive wage in his model. 2.4 Prese Wage Each household faces he labor demand and deermines he wage one period ahead o opimize her expeced uiliy. Wage is deermined o equalize he expeced reurn from he uiliy of consumpion because of higher wage and he expeced loss measured by he disuiliy from greaer labor supply induced by higher wage. 0 Prese wage is expressed as W κφ E υ { L } α( ρ) = H φ L ρ E ( C) P (9) ( φ/ φ ) is he consan markup from consan elasiciy of labor demand. The wage obained here is almos he same as he one in Obsfeld and Rogoff (2000) bu differs in he 0 The effecs of sicky wages as nominal rigidiy on he economy may no change he persisence in he economy qualiaively, bu can affec i quaniaively. Anderson (998) saes ha sicky wage would produce higher oupu persisence. However, in his paper, we are rying o discuss he effec of habi formaion on he real exchange rae qualiaively. This issue may no be relevan in a qualiaive sense. Wih sicky wage, marginal cos is sensiive o curren produciviy shocks and hus produciviy shock would have a greaer effec on he level of prices and real exchange rae han he sicky price case. Bu he qualiaive effec from habi formaion on he real exchange rae would be negaive in boh cases.

19 8 way ha exernal habi formaion eners he equaion because of is effec on he expeced marginal uiliy of consumpion. Since habi sock is known a ime, i lies ouside he expecaion operaor. Foreign wage is in he analogous form. 2.5 Opimal Pricing Since he price elasiciy of he demand for each produc is consan a θ, he markup over he marginal cos of a monopolisic compeiion firm is also consan and he opimal pricing for each ype of good is as follows: P θ W = P =, θ A N, H, θ W P = P N, F, = θ A (20) P H, θ W = P = ς θ A ς H,, W θ F, = ς = ς PF, θ A P In his model, firms are allowed o charge differen prices for he goods sold in differen markes, which is known as pricing o marke (PTM) in he lieraure. However, if θ = θ, P = ς P, hus he law of one price holds. Even hough he nominal exchange rae variaion H H has complee pass-hrough ono he prices for radable goods sold ouside he producers counries, he real exchange rae deviaes from uniy due o nonradable goods. Subsiuing opimal pricing in erms of wages in Eq. (6a) and (20), he real exchange rae wrien as: Q can be γ ςp ςw A Q P W A = = (2)

20 9 This equaion discloses he reason ha we use he sicky wage wih nonradable goods, insead of sicky price, o examine real exchange rae movemens. In he model, he purchasing power pariy holds in a model wih radable goods only, while he law of one price holds. In he lieraure, he facor ha drives he deviaion of real exchange rae from one is primarily he local-currency price (LCP), where he prese prices for goods sold in home and foreign counries are denominaed in consumers currency respecively and may differ from each oher. The deailed discussions abou he alernaive nominal rigidiy are in Hwang (2006). Using Eq. (2) and he risk-sharing condiion, Eq. (7), he labor employmen can be significanly simplified. Similarly, he erms of rade can be wrien as: ς P τ = = F, ςw A PH, W A (22) 2.6 Marke Clearing Condiions In equilibrium, good markes have o clear: CH, + CH, = YH,, C = N, Y (23) N, Y N where Y H, and, are he producions of home radable and nonradable goods, respecively. By subsiuing he demands for radable goods in Eq. (7) ino he commodiy marke equilibrium condiion in expendiure, we can obain he following condiion: Wih he LCP, we have o conduc he sudy wih four prices; ha is, prices for home and foreign radable goods in boh counries. This will significanly increase algebraic loads. Devereux and Engel (2000) and Obsfeld (2004) obain he closed-form soluions under LCP by simplifying he model wih perfecly elasic labor supplies, namely, υ =. This is no feasible in a sudy on habi formaion, because he effec of habi formaion disappears when υ =, as shown below.

21 P Y = P Y = P C + ς P C 2 2 H, H, H, H, T, T, T, T, 20 Thus, home and foreign radable incomes are equal. As Obsfeld and Rogoff (998, 2000) poin ou, consumers wih Cobb-Douglas preferences and isoelasic uiliy funcion for he composie consumpion good have no incenive o conduc securiy rades, if he iniial deb is zero. As a resul, he inernaional asse marke is redundan. However, from Eq. (5a), we know ha he expendiure of nonradable goods akes a fixed share, γ, in he oal expendiure. Because he expendiure and income will be idenical for he nonradable goods in each counry, he following condiion should hold: PN, CN, = PN, YN, = ( γ) PC If he curren accoun is zero, he naional income ideniy should follow: PC = P Y + P Y (24) H, H, N, N, Thus, he home raded income shall also be a fixed proporion of he oal expendiure: PY H H = γpc When his holds, he marke clearing condiion, Eq. (23), implies ha he real exchange rae will deermine relaive consumpion in he global economy: ξ P C = (25) P C Combined wih he risk-sharing condiion, Eq. (7), for complee asse markes in a model wihou habi formaion, consumpions have o be idenical across counries. This implies ha he purchasing power pariy mus hold. 2 Tha is, he uniy real exchange rae is required for boh goods marke equilibrium and zero asse ransacion in he asse marke equilibrium, even 2 In Obsfeld and Rogoff (2000), here is no asse marke. Relaive consumpion is solely deermined by he real exchange rae, as in Eq. (24), bu does no necessarily equalize across counries.

22 2 hough here are nonradable goods. When consumpions, hus he ineremporal elasiciy of subsiuion, for every possible sae, are equal, here is no need o share risks via inernaional asse ransacions. Therefore, o examine he deviaion of he real exchange rae from he purchasing power pariy, we assume nonzero iniial deb, D 0 0. Given his condiion, shocks may induce consumers o rade sae-coningen securiies across counries and Eq. (24) does no necessarily hold in equaliy. Subsiuing he good demand funcions, Eq. (7) and he producion funcion Eq. () ino Eq. (23), he marke clearing condiion can be wrien as: γ PC γ P C PC AL = Y = + + γ 2 P 2 P P ( ) H, H, N, γ PC γ PC PC AL = Y = P 2 P P ( γ ) F, F, N, (26a) (26b) Using he opimal pricing in erms of wages in Eq. (20) and he price index in Eq. (6a), we can rewrie he equaions above as: γ/2 γ/2 γ ςw / A γ ςw / A = C+ C 2 W / A 2 W / A AL (27a) γ/2 γ/2 γ ςw / A γ ςw / A = C+ C 2 W / A 2 W / A A L (27a) From Eq. (2), we know ha he erm in he bracke is simply he erms of rade, Eq. (22). Therefore, Eq. (27) shows how he erms of rade movemens affec producions. The home producion increases when he home erms of rade worsens, because boh home and foreign demands for home goods are greaer. As we discuss laer, we will apply log-linearizaion o hese wo equaions afer subsiuing he real exchange rae from he risk-sharing condiion o derive he explici log-linear labor

23 22 employmen in erms of consumpion and habi sock. Using log-linearizaion will do no harm o his sudy, as noed below. 3. Analyical Soluions 3. Labor Employmen To derive analyical soluions and examine he effecs of habi formaion on real exchange rae persisence, we assume ha hese wo counries are exacly symmeric so ha α = α and Subsiuing consumpion from Eq. (6a) ino he home wage in Eq. (9), he predeermined home nominal wage can be wrien as: κφ E W = φ E υ { L } { LM } Since all he exogenous variables are lognormally disribued, wages can be wrien in logs: σ 2 /2 β e η κφ μ w = ln + ( υ ) E l + E m +Λ (28a) w χμ φ ( ) σ 2 /2 β e η κφ μ w = ln + ( υ ) E l + E m +Λ w (28b) χμ φ ( ) 2 2 where ( υ ) 2 ( ) 2 Λ = +, σ σ σ m 2 2 lm l υ 2 2 σ σ lm l σ m λ = λ. Λ = +. Lower-case leers indicae 2 2 he logs of hese variables. Wages are deermined by expeced labor demand, expeced money sock and he correlaion and volailiy erms, incorporaing all he momens of exogenous shocks. From his equaion, we can see ha workers end o prese higher nominal wages in reacion o uncerainies, characerized by he momen erms. The momen erms capured by

24 23 lognormal disribuion are he primary advanages of he model bu are negleced in he linear approximaion, as noed by Obsfeld and Rogoff (2000). Here, however, we pay he leas aenion o he covariance and volailiies for wo reasons. Firs, hese wo erms are irrelevan o persisence in he economy. Second, he idenical Λ w and Λ in a symmeric global w economy will be cancelled ou and absen from he real exchange rae which is deermined by relaive economic condiion across counries. From Eq. (28), we know ha he habi sock affecs wages mainly hrough is effec on he labor demand. However, when υ =, habi formaion s effecs disappear. From he marke clearing condiion in Eq. (27), we can obain he labor employmen. Since Eq. (27) is no a log-linearized equaion, o derive he closed-form soluions explicily, we subsiue he real exchange rae in erms of consumpion from he risk-sharing condiion, Eq. (7), and hen apply log-linearizaion o his equaion around he seady sae. The seady sae is defined as a non-sochasic sae where all he shocks are in heir mean values. 3 Moreover, since he curren accoun is zero, consumpions are idenical across counries, or C = C = H = H. As shown in Appendix I.B, he logarihms of labor employmen from he log-linearizaion approximaion are: 2 γ γρ+ γ γρ γ 2 + γ γ αγ ρ 2 γ αγ ρ 2 γ l = c + c + h + h a (29a) 2 γ 2 γ 2 γ 2 γ γρ γ 2 + γ γ 2 γ γρ + γ αγ ρ 2 γ αγ ρ 2 γ l = c + c + h + h a (29b) 2 γ 2 γ 2 γ 2 γ These wo equaions above are assured o be real numbers, because we had imposed he condiion ha γ in order o induce real exchange rae movemens. 3 The definiion of he seady sae is he same as he one used in Benigno (2004).

25 24 No surprisingly, his equaion shows ha consumpion demands increase domesic labor employmen. Wihou real exchange rae movemens, home consumpion demand shall also increase he labor demand in he foreign counry. However, he risk-sharing condiion saes ha greaer home consumpion, relaive o foreign consumpion, is associaed wih home real depreciaion, which lures he global demand away from foreign commodiies. Therefore, home consumpion demands end o decrease employmen abroad. These wo equaions show how shocks and exernal effecs of habi socks are shared beween counries. The following discussion of habi formaion s influence on he real exchange rae will cener on he rade-off beween he effecs of habi sock and pas produciviy shocks on relaive wage, because auocorrelaed produciviy shocks and habi socks are he primary causes of persisence in he economy and are he primary pas informaion referred o by firms in he expecaion. Their conribuions o he persisence will inerac in he wage predeerminaion. Alhough Eq. (28) and (29) show he channel ha exogenous shocks and habi socks are passed across periods, hey are no closed-form soluions. To examine heir effecs exacly, we need o obain he closed-form soluion by considering he money clearing condiion in Eq. (6). 3.2 Level of Real Exchange Rae As shown in Eq. (2), he real exchange rae is deermined by relaive wage and shock. Therefore, o sudy he real exchange rae, we may simply ake he relaive erms across counries o describe he real exchange rae. Taking he difference beween Eq. (29a) and (29b), relaive employmen is shown as:

26 2 2 + ( γ γ l l c c h h a a γ γ γρ αγ ρ 2 γ = + ) 25 Combining he expecaion of relaive labor employmen from he equaion above, he nominal exchange rae from Eq. (8), he logarihm of he money marke clearing condiion and he difference in wages from Eq. (28) yields relaive wage in erms of relaive habi sock and exogenous shocks: ( ) ( ) w w = d w h h + d w2 a a + m m (30) where α( γ)( ρ )( υ ) ( υ ) γρ( 2 γ) + ( γ) d w 2 = > 0, ρ + 2 ( ρ )( υ )( γ) ρa ( υ ) γρ( 2 γ) + ( γ ) = < 0. ρ + d w 2 2 Given Λ w =Λ w, hese erms are absen in relaive wage. Wih he direc effec of labor employmen on wages, Eq. (30) reflecs he relaive employmen. Noe ha he effecs of habi formaion on he labor employmen and on all oher variables as shown below, are governed by he imporance of habi formaion in he uiliy funcion α. When α = 0, he uiliy depends on curren consumpion solely. From his equaion, we can see ha relaive produciviy shock and habi sock engender he reverse effecs on relaive wage when α > 0. Proposiion When 0< α, relaive habi sock affecs he level of he real exchange rae in he opposie way ha relaive real shock does. Proof From Eq. (30) and he log of Eq. (2), he real exchange rae can be wrien as: h h ρ γ ε ε η η q = d + d a a + + q q2 a a, a, (3)

27 where d ρ υ ( υ )( γ) 2 ( γ) 2 ( υ ) q0 = + > 0, 26 2 α( ρ )( υ )( γ) d = < 0, q d q 0 d q2 ( γ ) υρ = > 0. Relaive habi sock, pas real shock, curren real and nominal shocks are d q 0 he major deerminans of he real exchange rae. All he shocks and consumpion exernaliies are passed across counries. When α> 0, relaive habi sock places a negaive effec on relaive wage, which resuls in a negaive effec on he real exchange rae. The effecs of relaive habi sock and real shock on he real exchange rae are in he opposie sign. This is primarily from he rade-off beween he effecs of relaive habi sock and produciviy shock on relaive wage. While higher pas home produciviy shock, relaive o foreign produciviy shock, decreases home wage, home habi sock leads o higher home wage. The lower wage from he posiive home real shock will reinforce he increase in home consumpion, bu his effec is relaively small. As a resul, relaive real shock causes he home real depreciaion and relaive habi sock acs o offse his effec. The effec of habi formaion increases wih he imporance of habi formaion, α. The reverse effecs of relaive habi sock and produciviy shock on he real exchange rae have he imporan implicaions for he persisence. The magniudes of he effecs are based on he share of radable goods in he composie consumpion γ, he ineremporal subsiuion ρ and he elasiciy of labor supply υ. Wih nonradable goods, shocks impac he home economy by γ /2, while in he foreign economy by γ /2. γ hus describes he difference beween he impacs on he home and foreign counries, and shocks fall more heavily on he domesic counry wih he presence of nonradable goods. When here is no nonradable good, γ = and he impacs on hese wo counries are idenical. As menioned above, nonradable good is he main facor ha causes

28 27 he deviaion from he purchasing power pariy in his model. Even if here is no asse marke, he purchasing power pariy shall hold in he economy wih only radable goods due o he holding of he law of one price. 4 Workers wih lower elasiciy of labor supply, higher υ, end o prese a higher wage, leading o greaer real exchange rae movemens. Since we assume ha nominal shocks are no auocorrelaed, he relaive pas money supply in relaive wage is offse by nominal exchange rae movemens. As a resul, he pas nominal shocks do no appear in he Eq. (3). As shown, boh nominal and produciviy shocks accoun for he volailiy of real exchange rae. However, he magniude of volailiy is governed only by he share of radable goods, γ, which deermines how shocks are passed across counries. Habi formaion does no influence real exchange rae volailiy due o cerain assumpions of his model, as explained below. 3.3 Persisence of Real Exchange Rae The preceding subsecion shows he effec of relaive habi sock, h h, on he level of he real exchange rae. In his subsecion, we characerize he informaion embodied in habi socks o shed ligh on how relaive exernal habi sock affecs he persisence of he real exchange rae. In conras wih Fuhrer (2000), who characerizes he change in consumpion across ime in a model wih inernal habi formaion, we characerize relaive habi formaion as he accumulaion of pas relaive shocks. Because he habi sock conains he pas 4 The resul ha he habi formaion has a posiive effec on home wages is because of he asse marke srucure and he logarihm form of he money demand funcion, which isolaes he movemen of he nominal exchange rae from any real variables in he economy. If he asse marke srucure is modified, for example, he asse marke is absen as specified in Obsfeld and Rogoff (2000), he nominal exchange rae is no deermined from he risk-sharing condiion, bu deermined by he relaive wage income across counries. In ha case, habi formaion will lead o greaer nominal exchange rae movemen. Higher home habi formaion enlarges he depreciaion of home currency and hus he world demand moves oward home goods. As a resul, he negaive effec of home habi formaion on foreign consumpion demand will dominae compared o is posiive effec on home consumpion oward foreign goods and will resul in a decrease in foreign labor demand. In his case, foreign habi formaion will have a negaive effec on he home wage insead.

29 28 informaion, i crucially influences he persisence of he variables in he economy. While habi formaion increases he persisence of real variables, as described by he habi formaion lieraure, i does no increase he real exchange rae persisence as shown below or in he calibraion. Proposiion 2 When 0< α, habi formaion reduces he persisence of he real exchange rae. Proof Rewrie he risk-sharing condiion (7) as α( ρ ) = ( ) + (32) ρ ρ c c h h q Relaive curren consumpion embodies he informaion of curren habi sock and curren real exchange rae. Le C C = due o idenical consumers and ake he logarihm of he habi accumulaion pah from Eq. () and he foreign analogy, hen he difference beween home and foreign habi socks evolves wih ( ) ( λ)( ) h h = λ c c + h h (33) Remember ha λ is he adjusmen speed of he reference sock accumulaion. From his equaion, relaive habi sock is a weighed average of pas relaive consumpion and pas habi sock. Subsiuing his equaion recursively back ino he iniial dae, relaive habi sock is simply he accumulaion of relaive pas consumpions, which are deermined by pas shocks. Therefore, when he pas shocks are accrued in he habi sock, all pas informaion is passed hrough he exernal effec on curren consumpion. Exernal habi formaion has creaed an

30 29 exra avenue o carry pas shocks and hus influence he persisence of macroeconomic variables. Subsiue c c by aking one period ahead from Eq. (32) ino Eq. (33); hus relaive habi sock becomes ( ) λ αρ h h = λ( c c ) + ( λ)( h h ) = q + ( λ) ( h h ρ + ) (34) ρ As shown, relaive habi sock embodies he lagged real exchange rae. Subsiue Eq. (34) ino he real exchange rae in Eq. (32): ( ) λ α ρ q d q ( h h ) d ρ ( a a ρ ρ = q + ( λ) + + q2 a ) ( γ)( εa, εa, lnη lnη ) + + (35) When α > 0, d <, while d 2 > 0. Because habi formaion acs in he opposie way o q 0 q pas real shocks do, he persisence of he real exchange rae caused by relaive real shocks is diminished by he exernal habi sock. Thus, relaive habi sock will have a negaive effec on he real exchange rae persisence. The ineracion beween produciviy shocks and habi socks can be summarized here. In he period, a posiive echnology shock decreases he wage. The consumpion and he habi sock are increased accordingly. These wo changes from a echnology shock will influence wages and consumpion in he nex period. While he wage for he period is decreased furher due o he expeced higher produciviy, greaer habi sock raises he wage. The lower nominal wage in period appears o be followed by lower wage in period, leading o he persisence of wage. Thus he addiional impac of he produciviy shock on he wage, hrough habi sock, dampens he direc effec, he primary cause of he wage persisence.

31 30 Noe ha we shall no hink of Eq. (30) and (3) as closed-form soluions, alhough he habi sock is reaed as an exogenous variable o he exen ha i is known one period ahead. However, he habi sock is composed of pas consumpions, which are he endogenous variables deermined from he preceding periods. Wih he closed-form soluion of consumpion given below, we are able o characerize he informaion conained in he consumpion habi more closely. 3.4 Consumpion The role of habi socks in he open economy can be more elusive from he closed-form soluion of consumpion. Subsiuing relaive wage in Eq. (30) ino he relaive opimal price and he money marke equilibrium condiion, relaive consumpion can be wrien as γ ρ c c = dc ( h h) + dc2ρa( a a ) + ( εa, εa, ) + ( η η ) (36) Where d c ( ) + ( )( 2 ) υ( γ) α ρ γ υ γ = > 0, d d q0 c2 = > 0. This equaion shows ha d q0 consumpions are deermined by exernal habi socks and exogenous shocks. Boh relaive habi sock and produciviy shock have posiive effecs on relaive consumpion. While relaive habi sock decreases he real exchange rae persisence, i reinforces he persisence of consumpion. Because relaive oupu is deermined by consumpion demand, relaive habi sock will also increase he persisence of oupu. This is consisen wih he findings in Jusiniano and Preson (2004) ha habi formaion does help increase he persisence of real variables, bu is no significan for nominal price or he real exchange rae persisence. Curren shocks also increase consumpions via lowered prices. As menioned above, he price adjusmens in reacion o moneary shocks rely on he flexible nominal exchange rae. A

32 3 posiive home nominal shock leads o home currency depreciaion, and hus benefis boh he home expor and consumpion. Subsiuing relaive consumpion for he period from Eq. (36) ino he habi sock adjusmen process in Eq. (9), relaive habi sock can be wrien as he accumulaion of all pas shocks back o he iniial sae: ( γ) λ h h = λb c c + λd ρ B a a + B ε ε + η η 2 2 ( j j 0 0 c2 a 2 j 2 j a, j a, j j j j 0 ρ = j= 0 ) where (37) B= λd c + ( λ) > 0 and λ ( c0 c0) = h h, given ha h0 h0 = 0. The c 0 derivaion is in Appendix I.C. Noe ha does no equal, as we have imposed a condiion ha he iniial deb is no zero. This equaion discloses he informaion conained in he relaive habi sock, he composiion of pas produciviy and nominal shocks. Due o he posiive effecs of shocks on consumpions, all he pas shocks are accumulaed in a posiive way. α and λ, he adjusmen speed which deermines he weigh ha more recen shocks ake in he curren habi sock, are he primary parameers ha govern he habi sock process. I is clear ha pas shocks have sronger impacs on relaive habi sock wih α, whereas he influence of λ faces he rade-off. Taking he firs-order derivaive of c 0 j λ B wih respec o λ, where j represens he periods a which he shocks are ahead of he curren period: j λb λ = + λ ( d )( j+) c The sign of his erm depends on he size of j. I is posiive for small j, while negaive for greaer j. Because a higher λ implies a higher weigh of recen shocks, he effec of recen shocks on he curren habi sock shall increase wih λ. The firs-order derivaive says

33 ( )/( λ( ) ) posiive as long as λ 32 j < d c d c. When α = 0 and λ 0, alhough he habi sock keeps accumulaing, i has no influence on he curren economy. On he oher hand, when λ = 0, regardless of he value of α, relaive habi sock says zero, as here is no such dynamic accumulaion process of he habi sock. Subsiuing Eq. (37) ino Eq. (3), he real exchange rae can be wrien as: ( γ) 2 2 j λ q = d λd ρ B ( a a ) + B ( ε ε ) + ( η η j q c2 a 2 j 2 j a, j a, j j j j= 0 ρ j= 0 ( ) ( )( ln ln ) + d ρ a a + γ ε ε + η η q2 a a, a, This equaion clearly shows ha pas shocks involved in he habi sock counerac he effec of pas produciviy shocks because of he negaive d q. In one of he special cases in Devereux and Engel (998), similar o he specificaion in his paper, hey poined ou ha money is neural in he long run, leaving no real effec one period afer he nominal shock his he economy. In our model, we can see ha money is neural in he long run if α = 0. However, money is no neural even in he very long run when α 0. Through reference socks, pas nominal shocks affec curren and fuure real variables in he economy. As noed above in he wage predeerminaion, nominal exchange rae movemens play a criical role in deermining he effecs of exernal habi sock on he real exchange rae. In his model, due o he complee asse marke srucure and he logarihmic form of he money demand uiliy funcion, he nominal exchange rae is simply he difference beween money supplies across counries. While i is insulaed from he movemen of habi formaion, i fails o offse he negaive effec of relaive habi formaion on real exchange rae persisence. ) (38)