Real exchange rate variability in a two-country business cycle model

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1 Real exchange rae variabiliy in a wo-counry business cycle model Håkon Trevoll, November 15, 211 Absrac Real exchange rae flucuaions have imporan implicaions for our undersanding of he sources and ransmission of inernaional business cycles, and he degree of inernaional risk sharing. In mos wo-counry business cycle models, he real exchange rae is deermined by differences in consumpion and leisure across counries, and is sandard deviaion is significanly smaller han we see in he daa. Wih recursive preferences he marginal uiliy of consumpion oday also depends on innovaions in agens fuure uiliies. Real exchange rae movemens are herefore parially disconneced from curren quaniies. Wih permanen coinegraed echnology shocks and home bias, he model generaes realisic variabiliy in he real exchange rae. The mechanism is similar o relaed work on asse pricing, and he model produces highly volaile sochasic discoun facors wih more realisic asse pricing implicaions. In addiion, here is modes improvemen in he cross-counry correlaions of quaniies, as invesmen and employmen are posiively correlaed across counries. JEL Classificaion Codes: E32; F31; F32; F44 Keywords: Inernaional business cycles, recursive preferences, real exchange rae dynamics, home bias. I would like o hank David Backus, Fernando Leibovici, Tom Sargen, Ma Smih, Chrisopher Tonei and San Zin for helpful commens and useful discussions. I would also like o hank paricipans a he Macroeconomics Lunch Seminar and he Inernaional Reading group a NYU Sern for useful feedback. All remaining errors are my own. I welcome commens, including references o relaed papers I inadverenly overlooked. New York Universiy;

2 1 Inroducion A cenral puzzle in inernaional macroeconomics and finance is he high volailiy of he real exchange rae. Sandard inernaional business cycle models ha focus on quaniies canno mach his feaure of inernaional asse prices wih convenional parameer values. The mechanism proposed here o resolve his ension beween explaining movemens in quaniies and prices, is o model agens wih recursive preferences of he form discussed in Epsein and Zin 1989) in he inernaional business cycle framework. The real exchange rae is he relaive price of consumpion across counries. In he sandard model, where agens have power uiliy over a composie of consumpion and leisure, only he shor-run differences in hese quaniies across counries affec he curren currency value. Wih recursive preferences, he exchange rae adjuss o reflec boh curren and fuure differences in he allocaions. Agens wih recursive preferences wan o smooh uiliy across fuure saes as well as smoohing consumpion and leisure. They are herefore sensiive o he long-run prospecs of he economy, and news abou fuure consumpion growh affecs heir marginal uiliy of consumpion oday. If he news implies fuure differences in consumpion growh across counries, he effec on marginal uiliies differ and his is refleced in he real exchange rae. In he model presened here, wo counries specialize in he producion of inermediae goods. These goods are raded inernaionally, and used as inpus in he producion of final goods ha are allocaed o consumpion and invesmen. There is home bias in final goods producion, so flucuaions in he relaive price of inermediae inpus affec he cos of consumpion differenly across counries. Home bias herefore gives rise o variabiliy in he real exchange rae. Uncerainy arises from flucuaions in produciviy growh. Wih a uni roo in he produciviy process hese shocks have permanen effecs, bu for he wo-counry model o be consisen wih balanced growh, produciviies are also assumed o be coinegraed. The difference in produciviy across counries is herefore saionary, bu persisen, which leads o persisen flucuaions in he relaive price of inermediae goods. Togeher he hree key componens of he model deliver a highly variable real exchange rae: A coinegraed produciviy process delivers persisen differences in echnology levels and hence, persisen differences in he relaive price of inpus o final goods producion. Home bias implies ha hese relaive price differences are refleced in he relaive cos of consumpion. Produciviy shocks provide news abou persisen differences in consumpion growh across counries and when agens have recursive preferences he corresponding response in he real exchange rae is amplified. In he equilibrium of he model, he amoun of rade is deermined joinly by he degree of home bias and he elasiciy of subsiuion beween inermediae goods. These parameers

3 are chosen o ensure a realisic level of rade in he seady sae. However, as a high degree of home bias is imporan o generae real exchange rae volailiy, a low elasiciy of subsiuion is necessary o mach he level of impors. Using a low value of his elasiciy is common in he inernaional macroeconomics lieraure, and in our model, complemenariy beween inermediae goods implies ha employmen is posiively correlaed across counries. This holds for boh addiive and recursive preferences, bu wih recursive preferences he cross-counry correlaion in invesmen is also posiive. The larger relaive price movemens following produciviy shocks in ha model and he corresponding increases in he wage and renal rae of capial, induce households o increase boh labor supply and invesmen. When agens have recursive preferences, solving for he equilibrium wih rade inroduces addiional complexiy. Since he model feaures complee financial markes, a social planner s problem can be solved for equilibrium allocaions, bu o obain a recursive formulaion of his problem an addiional sae variable is required. One inerpreaion of he addiional sae is ha he Pareo weighs on he wo counries evolve over ime. Wih his inerpreaion, he dynamics of he Pareo weighs are driven by he heerogeneous responses of households uiliies o produciviy shocks, ha arise from home bias. The agens desire o smooh heir coninuaion uiliies across saes carries over o he social planner. Due o home bias however, higher produciviy for one of he inermediae goods, makes i easier for he planner o provide uiliy o he households who consume he final good produced wih a bias owards he more abundan inermediae inpu. Therefore he planner will no equalize agens coninuaion uiliies. Insead he planner parly smoohs uiliies by increasing he share of inermediae goods allocaed o he counry hi wih a relaively lower produciviy shock. The increase in he share of available goods is refleced in an increase in he Pareo weigh on he counry wih he worse shock. These dynamics in he Pareo weighs in he soluion o he planner s problem reflec he same forces ha drive he variabiliy in he real exchange rae in he decenralized equilibrium. Since Tallarini 2) i is well known ha he alernaive preference specificaion applied in his paper enables a closed producion economy o joinly mach key momens of boh quaniies and asse prices. This feaure also applies in he inernaional seing. The model is hus able o joinly mach he volailiy of he real exchange rae relaive o oupu and produce sochasic discoun facors consisen wih Hansen-Jagannahan bounds. The high volailiy of sochasic discoun facors necessary o mach hese bounds arise from agens sensiiviy o news abou fuure consumpion growh. This shows up as addiional componens of agens sochasic discoun facors, and wih permanen produciviy shocks, hese long-run componens increase he volailiy of hose facors. Some sandard puzzles in inernaional economics remain. In he opimal risk sharing arrangemen consumpion is more correlaed across counries in he model han in he daa, and i is more correlaed across counries han oupu is. In addiion, he larger movemens in he real exchange rae in he model wih recursive preferences, sill correspond wih relaively larger increases in consumpion in he counry hi wih he posiive shock o 2

4 produciviy. In he benchmark calibraion he model can herefore no accoun for he disconnec beween real exchange raes and relaive consumpions idenified by Backus and Smih 1993). 1.1 Relaed lieraure There are large lieraures on boh inernaional business cycles and real exchange rae flucuaions. One closely relaed paper is Rabanal, Rubio-Ramirez, and Tuesa 211) who also sudy he issue of real exchange rae variabiliy in an inernaional business cycle model. As we do here, hey build on he framework inroduced by Backus e al. 1992, 1994) and furher exended by Heahcoe and Perri 22). Rabanal e al. find ha a model where rade in financial asses is resriced o a non-coningen bond can generae a high volailiy of he real exchange rae relaive o oupu when produciviy shocks are coinegraed. Here he focus is on a differen mechanism ha arises due o he alernaive specificaion of preferences. The model presened here can herefore address a broader se of quesions in inernaional asse pricing. Our resuls complemen hose of Rabanal e al. and show ha high real exchange rae volailiy is no necessarily due o incomplee financial markes. Oher examples are Raffo 29) and Mandelman e al. 211). These papers consider models wih invesmen specific echnology shocks in addiion o shocks o produciviy. The former argues ha an appropriaely calibraed process for invesmen specific shocks enables he model o explain several puzzles in inernaional business cycles, including highly variable real exchange raes. The laer however, argues ha when a carefully esimaed process for he shocks o invesmen is fed ino he model, is explanaory power is grealy diminished. Colacio and Croce 211a) sudy he implicaions for inernaional asse pricing of inroducing recursive preferences in a wo-counry endowmen economy. They follow Bansal and Yaron 24) and inroduce consumpion processes feauring small, bu persisen long-run componens. In heir model here is perfec home bias and no rade in goods, and heir resuls require ha he long-run componens are perfecly correlaed across counries. They argue ha growh and diffusion of echnology can jusify heir assumpions, and in his paper a coinegraed produciviy process endogenously implies long-run dynamics similar o hose ha hey impose. In ha respec our paper feaures a similar mechanism o Kalenbrunner and Lochsoer 21) who show how long-run risk arises endogenously in a producion economy where agens have Epsein-Zin preferences. This paper is also relaed o he lieraure on recursive preferences wih muliple agens. Lucas and Sokey 1984), Kan 1995) and Anderson 25) discuss he formulaion and soluion of a recursive planner s problem in such environmens. Anderson focuses on he dynamics of Pareo weighs in a one good economy wih heerogeneiy in preference parameers. He finds ha his leads o a degenerae saionary disribuion. Colacio and Croce 211c) on he oher hand focus on Pareo weigh dynamics in an endowmen economy 3

5 wih wo goods where agens derive uiliy from an aggregae consumpion good wih he wo inermediae goods as inpus. They prove ha here is a non-degenerae saionary disribuion when heerogeneiy beween agens arises from home bias in he aggregaion. Colacio and Croce 211b) applies his framework in an asse pricing model feauring rare disasers. The economy presened here feaures a similar seup o Colacio and Croce, bu we consider a business cycle model wih producion and endogenous labor supply. In addiion, we do no resric he ineremporal and inraemporal elasiciies of subsiuion o be uniy. A low elasiciy of subsiuion beween inermediae goods enables he model o joinly generae realisic levels of goods rade and volaile real exchange raes. 1.2 Organizaion This paper is organized as follows. Secion 2 describes an inernaional business cycle model where agens have recursive preferences and derives he equilibrium condiions for ha economy. Secion 3 discusses a recursive formulaion of he social planner s problem which allows us o solve for equilibrium allocaions. In secion 4 we calibrae he model using sandard parameer choices from he inernaional macroeconomics lieraure and discuss our main resuls. Secion 5 considers some alernaive parameerizaions o highligh he main forces ha drive he increased relaive volailiy of he real exchange rae in he model wih recursive preferences. Secion 6 concludes. 2 The model The model presened here is similar o he inernaional real business cycle models sudied recenly in Rabanal, Rubio-Ramirez, and Tuesa 211) and earlier in Heahcoe and Perri 22) and Backus, Kehoe, and Kydland 1992, 1994). Our innovaion is o model households wih recursive preferences in his framework. There are wo counries referred o as home h, and foreign f. In each counry an inermediae good is produced which is raded inernaionally, and a final good is produced using he wo inermediae goods as inpus. For convenience we will occasionally refer o he final good produced in counry i {h, f} as final good i, and he inermediae good produced in counry i as inermediae good i. Inernaional financial asse markes are complee. 2.1 Uncerainy Time is discree. Each period here is a realizaion of a random even z, and we le z = z,..., z ) denoe he hisory of evens up o and including ime. The random evens are shocks o produciviy in he wo counries, and in general allocaions a ime are funcions of he hisory of hese shocks z. 4

6 2.2 Households In each counry he represenaive, infiniely lived, agen maximizes a recursive uiliy funcion over an aggregae of he final consumpion good and leisure. We will refer o he represenaive consumer in counry i as agen i. As in Epsein and Zin 1989) and Weil 199), preferences are modeled using he formulaion from Kreps and Poreus 1978). The uiliy of agen i {h, f} is defined by 1 U i, = [1 β) Ci,1 τ n i, ) 1 τ ) ρ + β E Ui,+1) α ) ] ρ/α 1/ρ 1) Here τ is he share of consumpion in he consumpion-leisure bundle, β is he discoun facor, he ineremporal elasiciy of subsiuion IES) over he consumpion-leisure aggregae is 1/1 ρ), and risk aversion owards saic gambles over he aggregae is measured by 1 α. When households have a preference for leisure he measure of risk aversion over consumpion depends on boh he household s consumpion and labor margins, and is given by τ1 α). See Swanson 29) for a deailed discussion of measures of risk aversion wih leisure in he uiliy funcion. The household maximizes 1) subjec o a budge consrain, a capial evoluion equaion, and a given iniial value of he capial sock K i,. The budge consrain in counry h is given by C h, z ) + I h, z ) + q z z )A h, z z )dz = z +1 z W h, z )n h, z ) + r h, z )K h, z 1 ) + A h, z ) 2) Here C h, z ) is unis of consumpion of he final good in counry h, I h, z ) is invesmen, n h, z ) [, 1] is he fracion of ime spen working, and K h, z 1 ) is he capial sock in counry h a he beginning of period. W h, z ) is he wage in counry h, and r h, z ) is he renal rae of capial. The prices of boh facor inpus are measured in unis of he final good in counry h. The household has access o a complee se of sae coningen securiies ha deliver one uni of he final good from counry h. A h, z ) is he number of such claims owned by he represenaive agen in counry h, and q z +1 z ) is he price of one such claim in ime + 1 afer hisory z measured in unis of final good h. Households issue sae coningen securiies subjec o sae-by-sae deb limis ha ensure ha i is always feasible for he household o repay is sae coningen deb. 2 1 The special cases where ρ = or α = are easily handled by aking he appropriae limis. For breviy hese limiing expressions are no presened here. 2 These limis are ofen referred o as naural deb limis, and in he equilibrium of his model, hey are never binding. 5

7 The budge consrain in counry f is similar, bu is wrien in erms of he final good in ha counry. So he prices of he facor inpus W f, z ) and r f, z ) are measured in erms of he final good in f. The budge consrain is hus C f, z ) + I f, z ) + 1 e z q z z )A f, z z )dz = ) z +1 z W f, z )n f, z ) + r f, z )K f, z 1 ) + A f,z ) e z ) 3) where e z ) is he real exchange rae defined as he relaive price of he final good in counry f relaive o he final good in counry h. Finally, he capial evoluion equaion in counry i {h, f} is K i,+1 z ) = 1 δ)k i, z 1 ) + I i, z ) 4) where δ is he rae of depreciaion which is he same across counries. 2.3 Firms In each counry here are producers of inermediae and final goods. In his secion we describe hese firms problems and he process for produciviy in he wo counries Final goods producion A represenaive firm in he final goods secor produces using a CES producion funcion where he inpus are he inermediae goods produced in boh counries. In each counry here is a bias in he final goods producion owards he inermediae good produced in ha counry. The demand for inermediae good i in counry j is denoed by X ij, for i, j {h, f}. The producion funcions of he final goods in he wo counries are given by G h Xhh, z ), X hf, z ) ) [ = G f Xfh, z ), X ff, z ) ) = ηx hh, z )) σ 1 σ [ 1 η)x fh, z )) σ 1 σ ] + 1 η)x hf, z )) σ 1 σ σ 1 σ + ηx ff, z )) σ 1 σ ] σ σ 1 The parameer σ is he elasiciy of subsiuion beween he wo inermediae goods 3, and home bias in final goods producion is given by each counry puing weigh η > 1/2 on he inermediae good produced in ha counry. 3 For σ = 1 he final goods producion funcion is Cobb-Douglas. 5) 6

8 The represenaive final goods producers ake he prices of he inermediae inpus as given and solve he following problem: max G X ih, z ),X if, z i Xih, z ), X if, z ) ) q ih, z )X ih, z ) q if, z )X if, z ) 6) ) where q ij, z ) is he price in counry i of inermediae good j in unis of final good i Inermediae goods producion In each counry, a represenaive firm in he inermediae goods secor produces using capial and labor wih a sandard consan reurns o scale producion funcion. These firms ake prices of goods and facor inpus as given and maximize profis by solving max q ii,z )K i, z 1 ) θ Z i, z )n i, z ) ) 1 θ Wi, z )n i, z ) r i, z )K i, z 1 ) n i, z ),K i, z 1 ) where Z i, z ) is he labor-augmening produciviy in counry i, and θ is he share of capial in producion which is he same across counries. Gross domesic produc in counry i in unis of final good i is denoed Y i, z ) and is given by 7) Y i, z ) = q ii, z )K i, z 1 ) θ Z i, z )n i, z ) ) 1 θ 8) Produciviy Following Rabanal e al. 211) we le log Z h, and log Z f, be coinegraed of order C1, 1). The law of moion for he log-firs differences of produciviy in he home and he foreign counry is specified by he following VECM: log Zh, log Z f, ) = φh φ f ) φ1h + [log Z h, 1 κ log Z f, 1 ] φ 1f ) εh, + ε f, ) 9) where 1, κ) is he coinegraing vecor, ε h,, ε f, ) T N, Σ), and is he firs-difference operaor. 7

9 2.4 Inernaional rade and prices Inernaional rade in he inermediae inpus is cosless, and he law of one price holds. As he prices in he respecive counries are denoed in erms of he local final goods, he following equaion holds for he price of inermediae good i {h, f}: q hi, z ) = e z )q fi, z ) 1) Therefore he real exchange rae can be expressed in erms of he prices of eiher of he inermediae goods in he wo counries. e z ) = q hi,z ) q fi, z, i = h, f 11) ) In addiion, he erms of rade is defined as he price of impors in counry h relaive o expors from counry h i.e. he price of inermediae good f relaive o inermediae good h): p z ) = q hf,z ) q hh, z 12) ) Oher variables of ineres relaed o inernaional rade are he impor raio and ne expors. For counry i we denoe he impor raio ir i, z ) and, following Heahcoe and Perri 22), define i as he raio of impors o non-raded domesic inermediae good producion: ir i, z ) = X ij,z ) X ii, z ) j i 13) Ne expors as a fracion of GDP in counry i is denoed nx i, z ) and is given by nx i, z ) = q ii,z )X ji, z ) q ij, z )X ij, z ) Y i, z ) j i 14) 2.5 Marke clearing The model is closed wih marke clearing condiions in he markes for he final and inermediae goods in each counry, as well as in he inernaional financial asse marke. The sae-coningen claims are in zero ne supply, so afer every hisory s we have for every possible sae s +1 in he subsequen period ha A h, z +1 z ) + A f, z +1 z ) = 15) 8

10 In he markes for he final and inermediae goods he marke clearing condiions are given by G i Xih, z ), X if, z ) ) = C i, z ) + I i, z ) i {h, f} 16) K i, z 1 ) θ Z i, z )n i, z ) ) 1 θ = Xhi, z ) + X fi, z ) i {h, f} 17) 2.6 Definiion of equilibrium Given he law of moion for produciviy defined by equaion 9), an equilibrium in his economy is a se of allocaions for households in boh counries, C i, z ), n i, z ), I i, z ), K i, z ), and A i, z ) for i {h, f}; allocaions for inermediae good producers in boh counries X ij, z ) for i, j {h, f}; allocaions for final good producers in boh counries G i X ih, z ), X if, z )) for i {h, f}; inermediae goods prices in boh counries q ij, z ) for i, j {h, f}; facor prices in boh counries W i, z ) and r i, z ), and he prices of saeconingen asses q z +1 z ) such ha: i) given prices, he households allocaions solve he households problem; ii) given prices, he inermediae goods producers allocaions solve he inermediae goods producers problem; iii) given prices, he final goods producers allocaions solve he final goods producers problem; iv) markes clear. 2.7 Equilibrium condiions The equilibrium condiions include he firs-order condiions of he households, and he inermediae and final goods producing firms in boh counries. In addiion, here are he laws of moion for produciviy and capial, he producion funcions, and he marke clearing condiions Households The households firs-order condiions for consumpion and labor supply are 1 β)τu i, z ) 1 ρ C i, z ) τρ 1 1 n i, z )) 1 τ)ρ = λ i, z ) 18) 1 τ τ C i, z ) 1 n i, z ) = W i,z ) 19) where λ i, z ) is he lagrange muliplier on he household s budge consrain. Wih he preferences specified in equaion 1), he households sochasic discoun facors m i,+1 z +1 z ) are given by 9

11 m i,+1 z +1 z ) = Ci,+1 z +1 z ) τρ 1 ) 1 ni,+1 z +1 z ) 1 τ)ρ ) Ui,+1 z +1 z ) α ρ ) β C i, z ) 1 n i, z 2) ) ζ U i,+1 ) 1/α where ζ U i,+1 ) = E Ui,+1) α is he cerainy equivalen of fuure uiliy, and E is he condiional expecaion operaor condiioning on informaion available a ime. The firs-order condiion wih respec o capial delivers he sandard equaion for pricing reurns 1 = E mi,+1 r i, δ) z ) 21) The remaining firs-order condiion for he households wih respec o he choice of saeconingen asses, deliver he following expressions for he asse prices: q z +1 z ) = πz +1 z )m h,+1 z +1 z ) = πz +1 z )m f,+1 z +1 z e z ) ) e +1 z +1 z ) 22) This delivers he relaionship beween he sochasic discoun facors across counries and real exchange rae dynamics discussed in Backus, Foresi, and Telmer 21) Firms The final goods firms firs-order condiions for demand of inermediae inpus from boh counries deermine he prices of he inermediae goods in each counry in erms of he final good in ha counry. G i Xih, z ), X if, z ) ) X ij, z = q ij, z ) 23) ) The inermediae goods firms firs-order condiions imply ha labor and capial are paid heir marginal produc: W i, z ) = 1 θ)q ii, z )K i, z 1 ) θ Z i, z ) 1 θ n i, z ) θ 24) r i, z ) = θq ii, z )K i, z 1 ) θ 1 Z i, z )n i, z ) ) 1 θ 25) 1

12 2.8 Terms of rade and real exchange rae The expressions for inermediae goods prices in equaion 23) and he definiion of he erms of rade in equaion 12) gives he following equilibrium expressions for he erms of rade in erms of inermediae goods allocaions in eiher counry h or f: p z ) = 1 η η Xhh, z ) 1 ) σ η = X hf, z ) 1 η Xfh, z ) 1 ) σ X ff, z ) 26) The equaions for inermediae goods prices, he definiion of he real exchange rae in equaion 11), and equaion 26) for he erms of rade imply he following expression which shows he equilibrium relaionship beween he real exchange rae and he erms of rade: 1 η) e z σ + η σ p z )) 1 σ ) = η σ + 1 η) σ p z )) 1 σ ) 1 1 σ 27) Remaining condiions The remaining condiions necessary o solve for equilibrium are he laws of moion of capial 4); he law of moion of produciviy 9); he relaionships beween inermediae goods prices across counries 1); and he marke clearing condiions 15), 16), and 17). 3 The planner s problem Since inernaional financial markes are assumed o be complee, we can solve he social planner s problem for he compeiive equilibrium allocaions in his economy, and hen use he allocaions o solve for prices. When households have recursive preferences as specified in equaion 1) as opposed o sandard addiive preferences, an addiional sae variable is necessary o obain a recursive value funcion for he planner. The formulaion used here is ha he planner maximizes he uiliy of agen h subjec o a consrain on he uiliy of agen f. The addiional sae variable is hen he promised uiliy o agen f, Ūf,. 4 In he recursive formulaion of he problem allocaions a ime are only a funcion of he vecor of sae variables a ime raher han he enire hisory. This vecor is labeled s and is given by s = K h,, K f,, Ūf,, Z h,, Z f, ) 28) 4 For an alernaive formulaion of a recursive planner s problem see Colacio and Croce 211b,c). They solve a similar problem as proposed by Lucas and Sokey 1984) and Kan 1995). 11

13 In each period he planner chooses allocaions and a promised uiliy o agen f for each possible realizaion of he exogenous sae variables nex period. The planner s problem is hen: Given s, choose C i,, n i, [, 1], K i,+1, I i,, X ij,, for i, j {h, f}, and he funcion {Ūf,+1Z h,+1, Z f,+1 )} for all possible realizaions of Z h,+1, Z f,+1 ), in order o solve Qs ) = max [1 β) C τ h, 1 n h,) 1 τ ) ρ + β EQs+1 ) α s )) ρ/α] 1/ρ subjec o [ 1 β) C τ f, 1 n f,) 1 τ ) ρ + β E Ū f,+1 Z h,+1, Z f,+1 ) α s ) ) ρ/α ] 1/ρ Ū f, 29) + he laws of moion for capial and produciviy in equaions 4) and 9), and he marke clearing condiions for final and inermediae goods in equaions 16) and 17). For noaional convenience we capure he dependence of all he choice variables on he sae s by wriing variable Xs ) as X in he formulaion above. Noe ha since preferences exhibi non-saiaion we can assume ha consrain 29) holds wih equaliy a he opimum. The uiliy obained by agen f is hen given by U f s ) = Ūf,,. 3.1 Solving he planner s problem The raio of he firs-order condiions wih respec o he consumpions of he wo agens implies he following condiion for he planner s allocaions of consumpion across he wo counries: λ s ) Cf s ) C h s ) ) τρ 1 ) 1 nf s ) 1 τ)ρ = es ) 3) 1 n h s ) The erm λ is he raio of Pareo weighs ha deermines how he planner allocaes goods across counries, and equaion 3) shows ha if he Pareo weigh raio is varying over ime, hen he real exchange rae is no only deermined by curren differences in consumpion and leisure. To derive he expression for λ s ), le λ U s ) be he lagrange muliplier on he planner s promise-keeping consrain 29), and define λ s ) as ) λ Uf s ) 1 ρ s ) = λ U s ) 31) U h s ) The Pareo weigh on agen h is hen given by µ s ) where µ s ) = λ s ) 32) 12

14 and he Pareo weigh on agen f is 1 µ s ). Equaion 31) implies ha here is a oneo-one mapping beween he coninuaion uiliy sae variable Ūf, and he Pareo weigh raio λ s ). We herefore consider he opimal allocaions funcions of λ and discuss he dynamics of he model in erms of he dynamics of his raio. For a given realizaion of he produciviy shocks nex period he firs-order condiion wih respec o Ūf s +1 ), he envelope condiion wih respec o Ūf,, and he definiion of λ s ) imply λ s +1 ) λ s ) = ) Uh s +1 ) ρ α ) Uf s +1 ) α ρ 33) ζ U h,+1 ) ζ U f,+1 ) The sandard model where agens have addiive expeced uiliy preferences is he case wih α = ρ, and equaion 33) shows ha in ha case he Pareo weigh raio is consan. I is hen common o se he Pareo weigh on each counry o 1/2 so ha λ s ) = 1. Therefore i is no crucial o emphasize how allocaions depend on his raio, and his dependence can be lef implici. In he model wih recursive preferences, we also assume ha he iniial Pareo weigh on each counry is 1/2. However, we consider he Pareo weighs as varying over ime in he following sense: consider he soluion o a planner s problem solved a ime wih an iniial Pareo weigh raio λ s ) = 1. In a fuure ime period wih sae s, he Pareo weigh raio ha deermines allocaions is given by λ s ). The iniial ime- Pareo weigh has no changed, bu if anoher planner were o solve he planner s problem a, he raio of ime- Pareo weighs would have o be λ s ) for he allocaions from he ime- soluion o be opimal for he ime- problem. The variaion in he Pareo weigh raio governed by equaion 33) capures he hisory dependence in he opimal allocaions. Equaion 33) shows ha changes in he Pareo weigh raio are relaed o innovaions in fuure uiliy. Tha is, o he realizaions of coninuaion uiliy relaive o he cerainy equivalen. In he decenralized problem discussed in secion 2, he innovaions o fuure uiliy are addiional componens of he households sochasic discoun facors in equaion 2). I is common in he lieraure on recursive preferences o decompose he sochasic discoun facors ino a shor-run componen SRC) and a long-run componen LRC) see for example Kalenbrunner and Lochsoer 21)), as follows: ) Ci s +1 ) τρ 1 ) 1 ni s +1 ) 1 τ)ρ ) Ui s +1 ) α ρ m i s +1, s ) = β C i s ) 1 n i s ) ζ U i,+1 ) } {{ } } {{ } SRC i s +1,s ) LRC i s +1,s ) 34) The dynamics in he Pareo weigh raio herefore correspond o heerogeneous responses in he long-run componens of he sochasic discoun facors o produciviy shocks. These heerogeneous responses reflec how agens marginal uiliies of consumpion are affeced differenly by shocks o echnology hrough heir impac on fuure uiliy. Equaion 3) 13

15 shows ha hese heerogeneous responses are also refleced in movemens in he real exchange rae. Anoher way o see his is by aking logs of equaion 22): log es +1 ) = log m f s +1, s ) log m h s +1, s ) 35) The variance of he depreciaion rae log es +1 ) is hen relaed o he variances and he correlaion of he sochasic discoun facors across he wo counries: Var log es +1 )) = Var log m f s +1, s )) + Var log m h s +1, s )) 2 ρ m, Var log m f s +1, s ))Var log m h s +1, s )) 36) where ρ m, is he condiional correlaion of he sochasic discoun facors. If hey are perfecly correlaed and have equal variance, hen he real exchange rae is consan even if he variance of he sochasic discoun facors is high. However, he heerogeneous responses o produciviy shocks in he long-run componens imply ha he sochasic discoun facors are no perfecly correlaed which can lead o a variable real exchange rae. Movemens in he real exchange rae are herefore ighly linked wih he dynamics of he Pareo weigh raio. To evaluae he poenial for his mechanism o explain he high variabiliy of he real exchange rae in he daa, we solve he model numerically. 3.2 Balanced growh and resricing he coinegraing vecor Since log Z h, z ) and log Z f, z ) are boh inegraed processes we need o normalize he equilibrium condiions o obain a saionary sysem. The counry i variables ha have a rend are rescaled by dividing hrough by he level of produciviy in counry i, Z i, z ). 5 The full se of normalized equaions are presened in appendix A). This model saisfies he resricions on he producion funcion and preferences from King, Plosser, and Rebelo 1988) ha are necessary for he exisence of a balanced growh pah. Along wih a produciviy process wih a uni roo hese condiions are also sufficien in a closed economy, bu as discussed in Rabanal e al. 211), a wo-counry model requires an addiional resricion on he coinegraing vecor o ensure balanced growh. The resricion is ha he raio of produciviies Z h s )/Z f s ) mus be saionary. To see he need for his addiional resricion consider for example he normalized version of equaion 3): ) λ s ) zs ) 1 τρ cf s ) τρ 1 ) 1 nf s ) 1 τ)ρ = es ) 37) c h s ) 1 n h s ) 5 There are wo excepions o his: i) he uiliies of he households U i,z ) are divided by Z i,z ) τ because here is a rend in consumpion, bu no in leisure, and ii) he capial socks K i,z 1 ) are divided by Z i, 1z 1 ) so ha he normalized capial sock is deermined a he sar of period. 14

16 where c i s ) are rescaled consumpions, and zs ) = Z h s )/Z f s ). Since he real exchange rae and labor supply in each counry are saionary, if he raio zs ) was non-saionary hen eiher he raio of consumpions or he raio of Pareo weighs would be non-saionary. In eiher case balanced growh would no exis. A sufficien condiion for saionariy of zs ) is ha he coinegraing vecor in he law of moion for produciviy 9) is 1, 1). Tha is, we require ha κ = Soluion mehod Heahcoe and Perri 22) and Rabanal e al. 211) respecively linearize and log-linearize heir models and compue momens by averaging across simulaions. Here we follow a similar procedure, bu o adequaely capure he dynamics in his model a higher-order approximaion is necessary. The reason is ha he parameer α does no appear in a firsorder approximaion around a deerminisic seady sae. As shown in equaion 33) he difference beween α and ρ is key in deermining he dynamics of he Pareo weighs, so a good approximaion of he effec of α on he dynamics of he model is necessary. In a second-order approximaion α only shows up in he consan erms. Hence, we use a hird-order approximaion o capure he impac of α. 6,7 4 Parameerizaion and resuls The parameer values we use in our benchmark are repored in able 1. The parameer choices follow Rabanal e al. 211) who generally use values ha are sandard in he inernaional macroeconomics lieraure. 8 The discoun facor β is se o.99. The consumpion share in he consumpion-leisure composie τ, is se o.34. The ineremporal elasiciy of subsiuion 1/1 ρ) is se equal o.5, while we use wo differen values for risk aversion over he consumpion-leisure composie 1 α. This value is se eiher o 2 which implies ha agens have he addiive preferences used by Rabanal e al., or i is se o 1 which follows he lieraure using Epsein-Zin preferences in producion economies o mach asse pricing facs. 9 As is common o ha lieraure, a large value of risk aversion is necessary o generae sufficien volailiy in he sochasic discoun facors in a fricionless model wih complee markes where he only source of risk is shocks o produciviy. 1 6 See van Binsbergen e al. 28) for a discussion of he applicaion of perurbaion mehods o models wih recursive preferences. See Colacio and Croce 211c) for a discussion of he need for a hird-order approximaion o capure he dynamics in an endowmen economy wih risk-sensiive agens. 7 The coefficiens of he hird-order approximaion are easily compued using Dynare++, which can also compue approximaions of a higher order. I is herefore sraighforward o check ha our resuls are no subsanially affeced by using an approximaion of a higher order. 8 They closely follow papers such as Heahcoe and Perri 22) and Backus e al. 1992, 1994). 9 See for example van Binsbergen e al. 21), Rudebusch and Swanson 211), and Andreasen 211). 1 Barillas e al. 28) and Bidder and Smih 211) show ha such large values of risk aversion correspond o plausible fears of model misspecificaion in endowmen and producion economies respecively. 15

17 The elasiciy of subsiuion beween he wo inermediae goods σ is se o.6. Using a low value for his parameer is common in he lieraure. Examples of papers using a value below 1 are Heahcoe and Perri 22) σ =.9), Corsei, Dedola, and Leduc 28) σ =.5), Raffo 29) σ =.5), and Rabanal e al. 211) σ =.62 or =.85). The home bias parameer η ogeher wih σ deermines he impor raio in he deerminisic seady sae. The parameers for he produciviy process are based on esimaes by Rabanal e al. who use daa for he period 1973Q1 26Q4. We herefore choose η o mach he impor raio in he Unied Saes of abou 12% in ha period. Tha is, η solves he equaion ) 1 η σ.12 = 38) η The combinaion of parameers σ, η) ha fi his impor raio is key o he mechanism proposed in his paper, and in secion 5.1 we invesigae he sensiiviy of he resuls o he choice of hese values. Sandard values are also used for he oher echnology parameers. The capial share in producion θ, is se o.36, and he depreciaion rae δ is se o.25. The parameers of he VECM for produciviy are se based on he esimaes in Rabanal e al. They consruc series for labor augmening TFP for he Unied Saes and he res of he world where he laer is an aggregae consising of he Euro area plus he Unied Kingdom, Canada, Japan and Ausralia. They use daa on real GDP, hours, employmen and real capial socks. The key resul from heir esimaes are ha hey canno rejec he hypohesis ha he TFP series are coinegraed wih a coinegraing vecor equal o 1, 1). Hence, κ = 1 and he resricion required for balanced growh is saisfied. A second imporan resul is ha he coefficiens for he speed of adjusmen are significan, bu small. This implies ha he TFP processes converge slowly over ime and ha differences in produciviy across counries are persisen. Consisen wih heir esimaes we se φ 1h =.7 and φ 1f =.7. We impose a symmeric produciviy process across he wo counries and se φ h = φ f =.36. In Rabanal e al. hese esimaes differ, bu due o coinegraion he TFP processes neverheless grow a he same rae along he balanced growh pah. Their esimaes imply an annualized long-run growh rae of TFP of 1.44 percen which is also he implicaion from our choice of parameers. Finally we se he sandard deviaion of he innovaions o he produciviy processes o.15 which is he esimae from Rabanal e al. for he Unied Saes. 4.1 Deerminisic vs. sochasic seady sae I is imporan o noe ha since we solve he model wih a hird-order approximaion, cerainy equivalence does no hold. Therefore he deerminisic seady sae does no 16

18 Benchmark parameers Preferences Discoun facor β =.99 Consumpion share τ =.34 Ineremporal elasiciy of subsiuion 1/1 ρ) =.5 Risk aversion 1 α = 2 or 1 α = 1 Technology E.o.s. beween inermediae goods σ =.6 Impor raio.12 η =.972) Capial share θ =.36 Depreciaion rae δ =.25 Produciviy VECM φ h = φ f =.36 φ 1h =.7, φ 1f =.7, κ = 1 Sandard deviaion of innovaions σ ε =.15 Table 1: Benchmark parameers: In he addiive preferences case we se 1 α = 2. preferences case we se 1 α = 1. In he recursive correspond o he mean of he ergodic disribuion of he variables in he model. To obain an approximaion o his mean, we sar our economy a he deerminisic seady sae and simulae he economy under a sequence of produciviy shocks se equal o zero. The economy converges o a poin which we refer o as he sochasic seady sae, where agens ake ino accoun he uncerainy in he economy. They herefore work more and inves more o mainain a higher capial sock, which lowers he reurn on capial. In our simulaions and impulse response funcions we use he sochasic seady sae as our saring poin Resuls for he benchmark parameers As he model is non-saionary we follow he approach of Rabanal e al. and compue HPfilered momens by sochasic simulaion. The model is simulaed for 125 periods saring from he sochasic seady sae, and he HP-filer is applied o he series for oupu, consumpion, invesmen, employmen and he real exchange rae. Second momens are compued from he filered series, and he averages over 1 simulaions are repored in able The firs and second rows of each panel in able 2 repor he resuls for he economy wih addiive and recursive preferences respecively. Wih addiive preferences he model discussed here is a complee markes version of he economy in Rabanal e al., and he resuls are similar o he resuls hey repor for ha case. 11 An alernaive approach is o sar simulaions a he deerminisic seady sae, bu discard a large number of periods a he sar of each simulaion as a burn-in. See for example van Binsbergen e al. 28). We find ha using his approach does no affec our resuls. 12 For some of our parameerizaions we have also compued averages over 1 simulaions and found ha his does no affec he resuls. 17

19 Exending he sandard model o allow for recursive preferences improves he model s abiliy o mach he volailiy of he real exchange rae relaive o oupu. For he benchmark parameers he sriking resul is ha he model wih recursive preferences explains more han 95% while he model wih addiive preferences only explains abou 35% of he relaive real exchange rae volailiy. As in Tallarini 2) he model also generaes a high marke price of risk when preferences are recursive and risk aversion is high. The marke price of risk in he model is abou 21% which is slighly lower han he 25% ha Tallarini finds in he daa. 13 Table 3 repors he marke price of risk for he model wih addiive and recursive preferences. In addiion i repors he sandard deviaion of he sochasic discoun facor, and he sandard deviaions of he shor-run and long-run componens of he sochasic discoun facor. The resuls show ha he model wih addiive preferences does no come close o maching he marke price of risk. Wih addiive preferences α = ρ and he sochasic discoun facor consiss of he shor-run componen only. Since he variabiliy of his componen is low, so is he marke price of risk. Wih recursive preferences on he oher hand, he variabiliy of he sochasic discoun facor is high, and his is mainly driven by he variabiliy of he long-run componen. Since here is a uni roo in he process for produciviy, echnology shocks have permanen effecs. They herefore produce large innovaions in fuure uiliy and hence, a high volailiy of he sochasic discoun facors Produciviy shocks and opimal allocaions To analyze he dynamics ha deliver he resuls discussed above, we firs consider how he opimal allocaions in his economy respond o produciviy shocks. Figures 1 and 2 plos simple impulse response funcions o a one sandard deviaion shock o produciviy in counry h. The impulse responses are ploed for he model wih addiive preferences dashed red line) and he model wih recursive preferences solid blue line). The responses in he quaniies in he wo models broadly follow he same paern. A posiive echnology shock in counry h has a permanen effec on produciviy here, while he produciviy in counry f is expeced o cach up in he long-run due o coinegraion. In he shor-run here is herefore a difference in he levels of produciviy in he wo counries. In counry h he response is ha GDP, invesmen, labor supply, and consumpion increase. Greaer supply of inermediae good h leads o an increase in he erms of rade. The increase in he relaive price of inermediae good f leads o immediae increases in GDP, labor supply and consumpion in ha counry as well. The iniial responses in counry f are smaller, bu produciviy growh is higher and brings produciviy in counry f o is new long-run growh pah. As he cross-counry difference in he level of echnology is gradually eliminaed, GDP, invesmen and consumpion increase furher. In counry h on 13 Tallarini 2) calculaes he marke price of risk from U.S. daa from 1948 : : 4. The daa used are he value-weighed NYSE porfolio and he 3-monh Treasury bill. 18

20 Resuls addiive and recursive preferences sdy) rsdc) rsdι) rsdn) rsde) ρe) Daa Addiive Recursive Correlaions y, n) y, c) y, ι) y, nx) e, c h /c f ) Daa Addiive Recursive Correlaions y h, y f ) c h, c f ) ι h, ι f ) n h, n f ) Daa Addiive Recursive Table 2: Second momens from HP-filered simulaed series: sd denoes sandard deviaion; rsd denoes sandard deviaion relaive o oupu; y, c, ι, n, e, nx) are respecively oupu, consumpion, invesmen, labor, he real exchange rae, and ne expors; ρ denoes firs-auocorrelaion. Momens are calculaed as he average over 1 simulaions of 125 periods. 19

21 Parameerizaion MPR sdm) sdsrc) sdlrc) Addiive Recursive Table 3: MPR denoes he marke price of risk, 1 sdm)/em); m denoes he sochasic discoun facor; SRC denoes he shor-run componen and LRC denoes he long-run componen; sd denoes he sandard deviaion. Each erm is calculaed as he average over 1 simulaions of 125 periods. he oher hand, coinegraion implies ha following he shock produciviy growh is lower han along he balanced growh pah. Thus invesmen is emporarily higher han is new long-run level and here is a hump-shaped response in consumpion and GDP. Despie he broadly similar paerns in he models wih addiive and recursive preferences, here are imporan differences beween hem. Firs, he models produce differen responses in invesmen in counry f. In boh models invesmen increases over ime in counry f as produciviy increases, bu in he model wih addiive preferences invesmen iniially drops. This leads o a negaive correlaion in invesmen across counries in ha model. In he model wih recursive preferences on he oher hand, invesmen in counry f increases in response o he shock. Tha model herefore produces he posiive cross-counry correlaion of invesmen ha we see in he daa. Second, and more imporan, are he differences in he magniudes of he responses described above. To see how hese differences arise, consider he planner s problem of allocaing resources beween he wo counries. The planner chooses how much of each inermediae good is produced and how he oal produced of each inermediae inpu is allocaed o be used for consumpion and invesmen. The share of inermediae good i {h, f} allocaed o households in counry h a ime is defined as share hi, = X hi, X hi, + X fi, 39) and he corresponding shares o households in counry f are given by share fi, = 1 share hi,. Figure 3 plos simple impulse responses for he oal producion of each inermediae good and he share of each good allocaed o each agen. The differences beween he wo models are clear from he plos in figure 3. In boh cases he planner chooses virually he same response in producion of inermediae goods, bu he allocaion of hose goods depends on agens preferences. When households have addiive preferences, he planner emporarily increases he shares of boh inermediae goods allocaed o counry h. As shown in figure 2, his allows for a relaively larger increase in invesmen and consumpion in counry h. Resources herefore move o he counry where produciviy is emporarily higher. When agens have recursive preferences on he oher hand, he planner reduces he shares of each inermediae good allocaed o agen h and increases he shares o agen f. The response in consumpion and invesmen in counry h 2

22 is herefore lower, and households work more. For households in counry f he larger shares hey receive in he model wih recursive preferences allow consumpion and invesmen o increase more and hey work less. The relaively larger share of resources allocaed o counry f in he model wih recursive preferences is refleced in he drop in he Pareo weigh on counry h, µ. When households have recursive preferences, hey dislike flucuaions in fuure uiliy as well as flucuaions in consumpion and leisure. The planner herefore aims o smooh coninuaion uiliy across saes in he fuure. Due o home bias however, he planner does no choose o equalize coninuaion uiliy across saes. When produciviy increases in counry h here is relaively more of inermediae good h being produced. Since households in h consume he final good produced using inermediae good h as he main inpu, i is easier for he planner o provide uiliy o hose households. Given he produciviy realizaion he uiliy of hose households herefore increase more. This heerogeneous response across counries is refleced in he increase in he Pareo weigh raio as shown by equaion 33). The increase in he Pareo weigh on households in counry f compensaes for he shock o produciviy ha is lower domesically han abroad, and increases he share of available inermediae goods hey receive Variabiliy in real exchange raes The dynamics in he opimal allocaions described above is suppored by corresponding price dynamics in he decenralized equilibrium. As discussed in secion 3.1, heerogeneous innovaions o fuure uiliy ha arise due o home bias and are refleced in he dynamics of he Pareo weighs, can also amplify he response of he real exchange rae. Figure 1 shows ha his is indeed he case in he baseline parameerizaion of he model. Anoher way o see ha home bias is essenial for generaing real exchange rae volailiy, is hrough equaion 27) which shows how he real exchange rae relaes o he erms of rade. Clearly, if η = 1/2 so ha here is no home bias, hen he real exchange rae is consan even when he erms of rade increase. Wih home bias on he oher hand, real exchange rae volailiy corresponds wih volailiy in he erms of rade. Figure 6 in appendix B shows impulse responses in he inermediae goods prices, he erms of rade and he real exchange rae afer a one sandard deviaion shock o produciviy in counry h. The shock leads o a persisen increase in he supply of inermediae good h relaive o inermediae good f. In he model wih recursive preferences he persisen increase leads o a larger relaive increase in he marginal uiliy of consumpion in counry f which increases he real exchange rae. This corresponds wih a larger increase in he equilibrium erms of rade which induces final goods firms in counry h o reduce heir demand of inermediae good f and o increase heir demand for inermediae good h by less han he increase in producion. Hence, he share of boh inermediae goods used in final goods producion in counry h is reduced. 21

23 .8 GDP counry h.8 GDP counry f.4 Recursive Addiive Real exchange rae Ne expors Pareo weigh µ* Tech. difference Figure 1: Impulse responses funcions afer a one sandard deviaion shock o produciviy in counry h. The impulse responses are ploed as percenage difference from he balanced growh pah for he model wih addiive preferences 1 α = 2) and wih recursive preferences 1 α = 1). 22

24 Consumpion counry h Recursive Addiive Consumpion counry f Invesmen counry h Invesmen counry f Labor counry h Labor counry f Figure 2: Impulse responses funcions afer a one sandard deviaion shock o produciviy in counry h. The impulse responses are ploed as percenage difference from he balanced growh pah for he model wih addiive preferences 1 α = 2) and wih recursive preferences 1 α = 1). 23

25 1 Producion counry h 1 Producion counry f.5 Recursive Addiive shareh,h) shareh,f) sharef,h) sharef,f) Figure 3: Impulse responses funcions afer a one sandard deviaion shock o produciviy in counry h. The impulse responses are ploed as percenage difference from he balanced growh pah for he model wih addiive preferences 1 α = 2) and wih recursive preferences 1 α = 1). The op panels show he oal producion of inermediae goods in he wo counries. sharei, j) denoes he share of inermediae good j allocaed o counry i. 24

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