Closing small open economy models

Transcription

1 Journal of Inernaional Economics 61 (003) locae/ econbase Closing small open economy models a b, Sephanie Schmi-Grohe, Marın Uribe * a Deparmen of Economics, Rugers Universiy, New Brunswick, NJ 08901, USA b Deparmen of Economics, Universiy of Pennsylvania, Philadelphia, PA 19104, USA Received 14 November 001; received in revised form 13 February 00; acceped 6 March 00 Absrac The small open economy model wih incomplee asse markes feaures a seady-sae ha depends on iniial condiions and equilibrium dynamics ha possess a random walk componen. A number of modificaions o he sandard model have been proposed o induce saionariy. This paper presens a quaniaive comparison of hese alernaive approaches. Five differen specificaions are considered: (1) A model wih an endogenous discoun facor (Uzawa-ype preferences); () a model wih a deb-elasic ineres-rae premium; (3) a model wih convex porfolio adjusmen coss; (4) a model wih complee asse markes; and (5) a model wihou saionariy-inducing feaures. The main finding of he paper is ha all models deliver virually idenical dynamics a business-cycle frequencies, as measured by uncondiional second momens and impulse response funcions. The only noiceable difference among he alernaive specificaions is ha he complee-asse-marke model induces smooher consumpion dynamics. 00 Elsevier B.V. All righs reserved. Keywords: Small open economy; Saionariy; Complee and incomplee asse markes JEL classificaion: F41 1. Inroducion Compuing business-cycle dynamics in he sandard small open economy model is problemaic. In his model, domesic residens have only access o a risk-free *Corresponding auhor. Tel.: ; fax: addresses: uribe@econ.upenn.edu (M. Uribe), grohe@econ.rugers.edu (M. Uribe) / 0/ $ see fron maer 00 Elsevier B.V. All righs reserved. doi: /s (0)

2 164 S. Schmi-Grohe, M. Uribe / Journal of Inernaional Economics 61 (003) bond whose rae of reurn is exogenously deermined abroad. As a consequence, he seady-sae of he model depends on iniial condiions. In paricular, i depends 1 upon he counry s iniial ne foreign asse posiion. Pu differenly, ransien shocks have long-run effecs on he sae of he economy. Tha is, he equilibrium dynamics posses a random walk componen. The random walk propery of he dynamics implies ha he uncondiional variance of variables such as asse holdings and consumpion is infinie. Thus, endogenous variables in general wonder around an infiniely large region in response o bounded shocks. This inroduces serious compuaional difficulies because all available echniques are valid locally around a given saionary pah. To resolve his problem, researchers resor o a number of modificaions o he sandard model ha have no oher purpose han o induce saionariy of he equilibrium dynamics. Obviously, because hese modificaions basically remove he buil-in random walk propery of he canonical model, hey all necessarily aler he low-frequency properies of he model. The focus of he presen sudy is o assess he exen o which hese saionariy-inducing echniques affec he equilibrium dynamics a business-cycle frequencies. We compare he business-cycle properies of five variaions of he small open economy. In Secion we consider a model wih an endogenous discoun facor (Uzawa, 1968 ype preferences). Recen papers using his ype of preferences include Obsfeld (1990), Mendoza (1991), Schmi-Grohe (1998), and Uribe (1997). In his model, he subjecive discoun facor, ypically denoed by b, is assumed o be decreasing in consumpion. Agens become more impaien he more hey consume. The reason why his modificaion makes he seady-sae independen of iniial condiions becomes clear from inspecion of he Euler equaion l5 b(c )(1 1 r)l 11. Here, l denoes he marginal uiliy of wealh, and r denoes he world ineres rae. In he seady-sae, his equaion reduces o b(c)(1 1 r) 5 1, which pins down he seady-sae level of consumpion solely as a funcion of r and he parameers defining he funcion b(? ). Kim and Kose (001) compare he business-cycle implicaions of his model o hose implied by a model wih a consan discoun facor. They find ha boh models feaure similar comovemens of macroeconomic aggregaes. We also consider a simplified specificaion of Uzawa preferences where he discoun facor is assumed o be a funcion of aggregae per capia consumpion raher han individual consumpion. This specificaion is arguably no more arbirary han he original Uzawa specificaion and has a number of advanages. Firs, i also induces saionariy since he above Euler equaion sill holds. Second, he modified Uzawa preferences resul in a model ha is compuaionally much simpler han he sandard Uzawa model, for i conains one less Euler equaion and one less Lagrange muliplier. 1 If he real rae of reurn on he foreign bond exceeds (is less han) he subjecive rae of discoun, he model displays perpeual posiive (negaive) growh. I is sandard o eliminae his source of dynamics by assuming ha he subjecive discoun rae equals he (average) real ineres rae.

3 S. Schmi-Grohe, M. Uribe / Journal of Inernaional Economics 61 (003) Finally, he quaniaive predicions of he modified Uzawa model are no significanly differen from hose of he original model. In Secion 3 we sudy a model wih a deb-elasic ineres-rae premium. This saionariy inducing echnique has been used, among ohers, in recen papers by Senhadji (1994), Mendoza and Uribe (000), and Schmi-Grohe and Uribe (001). In his model, domesic agens are assumed o face an ineres rae ha is increasing in he counry s ne foreign deb. To see why his device induces saionariy, le p(d ) denoe he premium over he world ineres rae paid by domesic residens, and d he sock of foreign deb. Then in he seady-sae he Euler equaion implies ha b[1 1 r 1 p(d)] 5 1. This expression deermines he seady-sae ne foreign asse posiion as a funcion of r and he parameers ha define he premium funcion p(? ) only. Secion 4 feaures a model wih convex porfolio adjusmen coss. This way of ensuring saionariy has recenly been used by Neumeyer and Perri (001). In his model, he cos of increasing asse holdings by one uni is greaer han one because i includes he marginal cos of adjusing he size of he porfolio. The Euler equaion hus becomes l [1 1 c9(d )] 5 b(1 1 r)l 11, where c(? ) is he porfolio adjusmen cos. In he seady-sae, his expression simplifies o 1 1 c9(d) 5 b(1 1 r), which implies a seady-sae level of foreign deb ha depends only on parameers of he model. The models discussed hus far all feaure incomplee asse markes. Secion 5 presens a model of a small open economy wih complee asse markes. Under complee asse markes, he marginal uiliy of consumpion is proporional across counries. So one equilibrium condiion saes ha U c(c ) 5 au *(c * ), where U denoes he period uiliy funcion and sars are used o denoe foreign variables. Because he domesic economy is small, c* is deermined exogenously. Thus, saionariy of c* implies saionariy of c. For he purpose of comparison, in Secion 6 we also sudy he dynamics of he sandard small open economy model wihou any ype of saionariy-inducing feaures, such as he economy analyzed in Correia e al. (1995). In his economy, he equilibrium levels of consumpion and ne foreign asses display a uni roo. As a resul uncondiional second momens are no well defined. For his reason, we limi he numerical characerizaion of his model o impulse response funcions. All models are calibraed in such a way ha hey predic idenical seady-saes. The funcional forms of preferences and echnologies are also idenical across models. The basic calibraion and parameerizaion is aken from Mendoza (1991). The business-cycle implicaions of he alernaive models are measured by second momens and impulse responses. The cenral resul of he paper is ha all models wih incomplee asse markes deliver virually idenical dynamics a businesscycle frequencies. The complee-asse-marke model induces smooher consumpion dynamics bu similar implicaions for hours and invesmen. Secion 8 presens a sensiiviy analysis. I shows ha he main resuls of he paper are robus o alernaive preference specificaions. In addiion, i explores he

4 166 S. Schmi-Grohe, M. Uribe / Journal of Inernaional Economics 61 (003) relaionship beween he magniude of he parameers deermining saionariy and he speed of convergence o he long-run equilibrium.. Model 1: Endogenous discoun facor Consider a small open economy populaed by a large number of idenical households wih preferences described by he following uiliy funcion: ` E O u U(c, h ), (1) 0 50 u 5 1, () 0 u 5 b(c, h )u $ 0, (3) 11 where b c, 0, b h. 0. This preference specificaion allows he model o be saionary, in he sense ha he non-sochasic seady-sae is independen of iniial condiions (namely, he iniial level of financial wealh, physical capial, and oal facor produciviy). The evoluion of foreign deb, d, is given by d 5 (1 1 r )d y 1 c 1 i 1 F(k k ), (4) where r denoes he ineres rae a which domesic residens can borrow in inernaional markes in period, y denoes domesic oupu, c denoes consump- ion, i denoes gross invesmen, and k denoes physical capial. The funcion F(? ) is mean o capure capial adjusmen coss and is assumed o saisfy F(0)5 F 9(0)5 0. Small open economy models ypically include capial adjusmen coss o avoid excessive invesmen volailiy in response o variaions in he domesic foreign ineres rae differenial. The resricions imposed on F ensure ha in he non-sochasic seady-sae adjusmen coss are zero and he domesic ineres rae equals he marginal produc of capial ne of depreciaion. Oupu is produced by means of a linearly homogeneous producion funcion ha akes capial and labor services as inpus, y 5 AF(k, h ), (5) where A is an exogenous sochasic produciviy shock. The sock of capial evolves according o k 5 i 1 (1 d )k, (6) 11 where d [ (0, 1) denoes he rae of depreciaion of physical capial. ` Households choose processes hc, h, y, i, k 11, d, u11j50 so as o maximize he uiliy funcion (1) subjec o Eqs. () (6) and a no-ponzi consrain of he form

5 S. Schmi-Grohe, M. Uribe / Journal of Inernaional Economics 61 (003) d 1j lim E ]]]] j # 0. (7) j ` P (1 1 r s) s51 Leing u, h and l denoe he Lagrange mulipliers on Eqs. (3) and (4), he firs-order condiions of he household s maximizaion problem are Eqs. (3) (7) holding wih equaliy and: l 5 b(c, h )(1 1 r )E l (8) 11 l 5 U (c, h ) h b(c, h ) (9) c c h 5EU(c, h ) 1 E h b(c, h ) (10) U (c, h ) 1h b(c, h ) 5 l AF(k, h ) (11) h h h l [1 1 F 9(k11 k )] 5 b(c, h )El11fA11F k(k 11, h 11) 1 1 d 1 F 9(k k ) g (1) 1 These firs-order condiions are fairly sandard, excep for he fac ha he marginal uiliy of consumpion is no given simply by U c(c, h ) bu raher by U (c, h ) b (c, h )h. The second erm in his expression reflecs he fac ha an c c increase in curren consumpion lowers he discoun facor (b, 0). In urn, a uni c decline in he discoun facor reduces uiliy in period by h. Inuiively, h equals he presen discouned value of uiliy from period 1 1 onward. To see his, ierae he firs-order condiion (10) forward o obain: ` u 1j S]] u D 1j 1j j51 11 h 5E O U(c, h ). Similarly, he marginal disuiliy of labor is no simply U h(c, h ) bu insead U h(c, h ) b h(c, h )h. In his model, he ineres rae faced by domesic agens in world financial markes is assumed o be consan and given by r 5 r. (13) The law of moion of he produciviy shock is given by: ln A 5 r ln A 1 e ; e NIID(0, s ); $ 0. (14) e A compeiive equilibrium is a se of processes hd, c, h, y, i, k 11, h, lj saisfying Eqs. (4) (1) all holding wih equaliy, given Eqs. (13), (14), A 0, d 1, and k 0. We parameerize he model following Mendoza (1991), who uses he following funcional forms for preferences and echnology: 11

6 168 S. Schmi-Grohe, M. Uribe / Journal of Inernaional Economics 61 (003) v 1g fc v h g 1 U(c, h) 5]]]]]] 1 g 1 v c b(c, h) 5f1 1 c v h g 1 a F(k, h) 5 k h 1a f F(x) 5] x ; f. 0. As is well known, he funcional forms of he period uiliy funcion and he discoun facor imply ha he marginal rae of subsiuion beween consumpion and leisure depends only on labor. In effec, combining Eqs. (9) and (11) yields v 1 h 5 AF(k, h ) (15) h The righ-hand side of his expression is he marginal produc of labor, which in equilibrium equals he real wage rae. The lef-hand side is he marginal rae of subsiuion of leisure for consumpion. The above expression hus saes ha he labor supply depends only upon he wage rae and in paricular ha i is independen of he level of wealh. We also follow Mendoza (1991) in assigning values o he srucural parameers of he model. Mendoza calibraes he model o he Canadian economy. The ime uni is mean o be a year. The parameer values are shown in Table 1. All parameer values are sandard in he real-business-cycle lieraure. I is of ineres o review he calibraion of he parameer c defining he elasiciy of he discoun 1 v facor wih respec o he composie c h /v. Given he focus of our paper, his parameer is imporan because i deermines he saionariy of he model and he speed of convergence o he seady-sae. The value assigned o c is se so as o 1 mach he average Canadian rade-balance-o-gdp raio. To see how in seadysae his raio is linked o he value of c, use Eq. (1) in seady-sae o ge 1 5S D h r1 d ] ]] k a 1/1a I follows from his expression ha he seady-sae capial labor raio is independen of he parameer c 1. Given he capial labor raio, equilibrium condiion (15) implies ha he seady-sae value of hours is also independen of c and given by 1 Table 1 Model 1: Calibraion g v c1 a f r d r se

7 S. Schmi-Grohe, M. Uribe / Journal of Inernaional Economics 61 (003) a h 5F(1 a)s]] D G r 1 d a /1a 1/v 1 Given he seady-sae values of hours and he capial labor raio, we can find direcly he seady-sae values of capial, invesmen (i 5 dk), and oupu a 1a (k h ), independenly of c 1. Now noe ha in he seady-sae he rade-balanceo-gdp raio, b/f(k, h), is given by 1 (c 1 i)/f(k, h). Then, equilibrium condiion (8) implies he following seady-sae condiion relaing he radebalance-o-gdp raio o c 1: b(f(k, h) b i, h)(1 1 r) 5 1. Using he specific funcional form for he discoun facor, his expression can be wrien as: 1/c 1 h b i (1 1 r) 1] 1 ]] ]] ]]]]]]] v 5 1 F(k, h) F(k, h) F(k, h) F This expression can be solved for c1 given b/f(k, h), a, r, d, and v. All oher hings consan, he larger is he rade-balance-o-oupu raio he larger is he required value of c Model 1a: Endogenous discoun facor wihou inernalizaion Consider an alernaive formulaion of he endogenous discoun facor model where domesic agens do no inernalize he fac ha heir discoun facor depends on heir own levels of consumpion and effor. Alernaively, suppose ha he discoun facor depends no upon he agen s own consumpion and effor, bu raher on he average per capia levels of hese variables. Formally, preferences are described by Eqs. (1), (), and u 5 b(c, h )u $ 0, (16) 11 where c and h denoe average per capial consumpion and hours, which he individual household akes as given. The firs-order condiions of he household s maximizaion problem are Eqs. (), (4) (7), (16) holding wih equaliy and: l 5 b(c, h )(1 1 r )E l (17) 11 l 5 U (c, h ) (18) c U (c, h ) 5 l AF(k, h ) (19) h h l [1 1 F 9(k k )] 5 b(c, h )E l fa F (k, h ) 1 1 d k v G 1 F 9(k k ) g (0) 1 In equilibrium, individual and average per capia variables are idenical. Tha is, 11

8 170 S. Schmi-Grohe, M. Uribe / Journal of Inernaional Economics 61 (003) c 5 c h 5 h (1) () A compeiive equilibrium is a se of processes hd, c, h, c, h, y, i, k 11, lj saisfying Eqs. (4) (7), (17) () all holding wih equaliy, given Eqs. (13), (14), A 0, d 1, and k 0. Noe ha he equilibrium condiions include one Euler equaion less, Eq. (10), and one sae variable less, h, han he sandard endogenous- discoun-facor model. This feaure faciliaes he compuaion of he equilibrium dynamics. We evaluae he model using he same funcional forms and parameer values as in Model Model : Deb elasic ineres rae In Model, saionariy is induced by assuming ha he ineres rae faced by domesic agens, r, is increasing in he aggregae level of foreign deb, which we denoe by d. Specifically, r is given by r 5 r 1 p(d ), (3) where r denoes he world ineres rae and p(? ) is a counry-specific ineres rae premium. The funcion p(? ) is assumed o be sricly increasing. Preferences are given by Eq. (1). Unlike in he previous model, preferences are assumed o display a consan subjecive rae of discoun. Formally, u 5 b, where b [ (0, 1) is a consan parameer. The represenaive agen s firs-order condiions are Eqs. (4) (7) holding wih equaliy and l 5 b(1 1 r )E l (4) 11 U (c, h ) 5 l, (5) c U (c, h ) 5 l AF(k, h ). (6) h h l [1 1 F 9(k11 k )] 5 bel11fa11f k(k 11, h 11) 1 1 d 1 F 9(k k ) g (7) 1 Because agens are assumed o be idenical, in equilibrium aggregae per capia deb equals individual deb, ha is, 11

9 S. Schmi-Grohe, M. Uribe / Journal of Inernaional Economics 61 (003) Table Model : Calibraion of parameers no shared wih Model 1 b d c d 5 d. (8) A compeiive equilibrium is a se of processes hd, d 11, c, h, y, i, k 11, r, ` lj50 saisfying Eqs. (4) (7), and Eqs. (3) (8) all holding wih equaliy, given Eq. (14), A 0, d 1, and k 0. We adop he same forms for he funcions U, F, and F as in Model 1. We use he following funcional form for he risk premium: s dd p(d) 5 c e 1, d where c and d are consan parameers. We calibrae he parameers g, v, a, f, r, d, r, and se using he values shown in Table 1. We se he subjecive discoun facor equal o he world ineres rae; ha is, 1 b 5 ]]. 1 1 r The parameer d equals he seady-sae level of foreign deb. To see his, noe ha in seady-sae, he equilibrium condiions (3) and (4) ogeher wih he assumed d / d form of he ineres-rae premium imply ha 1 5 bf1 1 r 1 cse 1dg. The fac ha b(1 1 r) 5 1 hen implies ha d 5 d. If follows ha in he seady-sae he ineres rae premium is nil. We se d so ha he seady-sae level of foreign deb equals he one implied by Model 1. Finally, we se he parameer c so as o ensure ha his model and Model 1 generae he same volailiy in he currenaccoun-o-gdp raio. The resuling values of b, d, and c are given in Table. 4. Model 3: Porfolio adjusmen coss In his model, saionariy is induced by assuming ha agens face convex coss of holding asses in quaniies differen from some long-run level. Preferences and echnology are as in Model. In conras o wha is assumed in Model, here he ineres rae a which domesic households can borrow from he res of he world is consan and equal o he world ineres, ha is, Eq. (13) holds. The sequenial budge consrain of he household is given by c ] d 5 (1 1 r )d y 1 c 1 i 1 F(k k ) 1 (d d ), (9)

10 17 S. Schmi-Grohe, M. Uribe / Journal of Inernaional Economics 61 (003) where c and d 3 are consan parameers defining he porfolio adjusmen cos funcion. The firs-order condiions associaed wih he household s maximizaion problem are Eqs. (5) (7), (5) (7), (9) holding wih equaliy and l [1 c (d d )] 5 b(1 1 r )E l (30) 3 11 This opimaliy condiion saes ha if he household chooses o borrow an addiional uni, hen curren consumpion increases by one uni minus he marginal porfolio adjusmen cos c (d d 3 ). The value of his increase in consumpion in erms of uiliy is given by he lef-hand side of he above equaion. Nex period, he household mus repay he addiional uni of deb plus ineres. The value of his repaymen in erms of oday s uiliy is given by he righ-hand side. A he opimum, he marginal benefi of a uni deb increase mus equal is marginal cos. ` A compeiive equilibrium is a se of processes hd, c, h, y, i, k 11, r, lj50 saisfying Eqs. (5) (7), (13), (5) (7), (9), and (30) all holding wih equaliy, given Eq. (14), A 0, d 1, and k 0. Preferences and echnology are parameerized as in Model. The parameers g, v, a, f, r, d, r, and se ake he values displayed in Table 1. As in Model, he subjecive discoun facor is assumed o saisfy b(11 r) 5 1. This assumpion and Eq. (30) imply ha he parameer d deermines he seady-sae level of foreign deb (d 5 d ). We calibrae d so ha he seady-sae level of foreign deb equals he one implied by Models 1, 1a, and (see Table ). Finally, we assign he value o c 3, which ensures ha his model and Model 1 generae he same volailiy in he curren-accoun-o-gdp raio. This parameer value is almos idenical o ha assigned o c in Model. This is because he log-linearized versions of Models and 3 are almos idenical. Indeed, he models share all equilibrium condiions bu he resource consrain (Eqs. (4) and (9)), he Euler equaions associaed wih he opimal choice of foreign bonds (Eqs. (4) and (30)), and he ineres rae faced by domesic households (Eqs. (13) and (3)). The log-linearized versions of he resource consrains are he same in bohmodels. The log-linear approximaion o he domesic ineres rae is given by 1 1 r 5 1 cd(1 1 r) d in Model and by 1 1 r5 0 in Model 3. In urn, he log-linearized ˆ 1 versions of he Euler equaion for deb are l 5 c d(1 1 r) dˆ 1 E ˆ l11 in Model and ˆ l 5 c ddˆ 1 E ˆ 3 l11 in Model 3. I follows ha for small values of c and c3 saisfying c 5 (11 r)c Models and 3 will imply similar dynamics Model 4: Complee asse markes All model economies considered hus far feaure incomplee asse markes. In hose models agens have access o a single financial asse ha pays a risk-free real rae of reurn. In he model sudied in his secion, agens have access o a

11 S. Schmi-Grohe, M. Uribe / Journal of Inernaional Economics 61 (003) complee array of sae-coningen claims. This assumpion per se induces saionariy in he equilibrium dynamics. Preferences and echnology are as in Model. The period-by-period budge consrain of he household is given by Er b 5 b 1 y c i F(k k ), (31) where b11 denoes a random variable indicaing he number of asses purchased in period o be delivered in each sae of period 1 1. The variable r11 denoes he period- price of an asse ha pays one uni of good in a paricular sae of period 1 1 divided by he probabiliy of occurrence of ha sae given informaion available in period. Households are also subjec o a no-ponzi-game consrain of he form lim Eq 1jb 1j $ 0, (3) j ` a all daes and under all coningencies. The variable q represens he period-zero price of one uni of good o be delivered in a paricular sae of period divided by he probabiliy of occurrence of ha sae given informaion available a ime 0 and is given by q 5 rr 1???r, wih q ; 1. The firs-order condiions associaed wih he household s maxi- 0 mizaion problem are Eqs. (5), (6), (5) (7), (31), and (3) holding wih equaliy and lr11 5 bl 11. (33) A difference beween his expression and he Euler equaions ha arise in he models wih incomplee asse markes sudied in previous secions is ha under complee markes in each period here is one firs-order condiion for each possible sae in period 1 1, whereas under incomplee markes he above Euler equaion holds only in expecaions. In he res of he world, agens have access o he same array of financial asses as in he domesic economy. Consequenly, one firs-order condiion of he foreign household is an equaion similar o Eq. (33). Leing sarred leers denoe foreign variables or funcions, we have l* r11 5 bl 11 *. (34) Noe ha we are assuming ha domesic and foreign households share he same subjecive discoun facor. Combining he domesic and foreign Euler equaions Eqs. (33) and (34) yields l11 l* 11 ]] 5 ]]. l l*

12 174 S. Schmi-Grohe, M. Uribe / Journal of Inernaional Economics 61 (003) This expression holds a all daes and under all coningencies. This means ha he domesic marginal uiliy of consumpion is proporional o is foreign counerpar. Formally, l5 jl *, where j is a consan parameer deermining differences in wealh across counries. We assume ha he domesic economy is small. This means ha l* mus be aken as an exogenous variable. Because we are ineresed only in he effecs of domesic produciviy shocks, we assume ha l* is consan and equal o l*, where l* is a parameer. The above equilibrium condiion hen becomes l 5 c 4, (35) where c 4 ; jl* is a consan parameer. A compeiive equilibrium is a se of processes hc, h, y, i, k ` 11, lj50 saisfying Eqs. (5), (6), (5) (7), and (35), given Eq. (14), A, and k. 0 0 The funcions U, F, and F are parameerized as in he previous models. The parameers g, b, v, a, f, d, r, and s ake he values displayed in Tables 1 and. e The parameer c is se so as o ensure ha he seady-sae level of consumpion 4 is he same in his model as in Models 1 o Model 5: The non-saionary case For comparison wih he models considered hus far, in his secion we describe a version of he small open economy model ha displays no saionariy. In his model: (a) The discoun facor is consan; (b) he ineres rae a which domesic agens borrow from he res of he world is consan (and equal o he subjecive discoun facor); (c) agens face no fricions in adjusing he size of heir porfolios; and (d) markes are incomplee in he sense ha domesic households have only access o a single risk-free inernaional bond. This specificaion of he model induces a random walk componen in he equilibrium marginal uiliy of consumpion and he ne foreign asse posiion. A compeiive equilibrium in he non-saionary model is a se of processes hd, ` c, h, y, i, k 11, r, lj50 saisfying Eqs. (4) (7), (13), and (4) (7) all holding wih equaliy, given Eq. (14), A 0, d 1, and k 0. We calibrae he model using he parameer values displayed in Tables 1 and. 7. Quaniaive resuls Table 3 displays a number of uncondiional second momens of ineres observed in he daa and implied by Models 1 4. In all momens, we compue he Model 5 is non-saionary, and hus does no have well defined uncondiional second momens.

13 S. Schmi-Grohe, M. Uribe / Journal of Inernaional Economics 61 (003) Table 3 Observed and implied second momens Daa Model 1 Model 1a Model Model 3 Model 4 Volailiies: sd( y ) sd(c ) sd(i ) sd(h ) SD b sd ] y SD ca sd ] y Serial correlaions: corr( y, y 1) corr(c, c 1) corr(i, i 1) corr(h, h 1) S D b b1 corr ],] y y S 1 D ca ca1 corr ],]] y y 1 Correlaions wih oupu: corr(c, y ) corr(i, y ) corr(h, y ) S D b corr ], y y S D ca corr ], y y Noe. The firs column was aken from Mendoza (1991). Sandard deviaions are measured in percen per year. equilibrium dynamics by solving a log-linear approximaion o he se of 3 equilibrium condiions. Alhough he focus of he paper is no o assess he models abiliies o mach he daa, as a reference we include in he firs column of he able he observed second momens using Canadian daa. As poined ou by Mendoza (1991), he small open real business cycle model capures a number of feaures of business cycles in Canada. Specifically, as in he daa, all four models predic he following ranking of volailiies, in ascending order; consumpion, 3 The Malab compuer code used o compue he uncondiional second momens and impulse response funcions presened in his secion is available a uribe.

14 176 S. Schmi-Grohe, M. Uribe / Journal of Inernaional Economics 61 (003) oupu, and invesmen. The models also correcly predic ha he componens of aggregae demand and hours are procyclical, and ha he correlaion of he rade 4 balance wih GDP is close o zero. The models overesimae he procyclicaliy of labor. In he daa he correlaion beween hours and oupu is 0.8, whereas he models imply a perfec correlaion. This implicaion of he models is driven by he assumed preference and echnology specificaion. In effec, using he assumed Cobb Douglas form of he producion funcion we can wrie Eq. (15), which v holds for all models, as h 5 (1 a)y. Log-linearizing his expression we ge vh5 y, where a circumflex over a variables denoes is log-deviaion from he seady-sae value. I follows ha corr(h, y ) 5 1. The main resul of his paper is ha regardless of how saionariy is induced in he small open economy real business cycle model, he model s predicions regarding second momens are virually idenical. This resul is eviden from Table 3. The only noiceable difference arises in Model 4, he complee markes case, which as expeced predics less volaile consumpion. The low volailiy of consumpion in he complee markes model inroduces an addiional difference beween he predicions of his model and Models 1 3. Because consumpion is smooher in Model 4, is role in deermining he cyclicaliy of he rade balance is smaller. As a resul, Model 4 predics ha he correlaion beween oupu and he rade balance is posiive, whereas Models 1 3 imply ha i is negaive. Fig. 1 demonsraes ha Models 1 5 also imply virually idenical impulse response funcions o a echnology shock. Each panel shows he impulse response of a paricular variable in he six models. For all variables bu consumpion and he rade-balance-o-gdp raio, he impulse response funcions are so similar ha o he naked eye he graph appears o show jus a single line. Again, he only small bu noiceable difference is given by he responses of consumpion and he rade-balance-o-gdp raio in he complee markes model. In response o a posiive echnology shock, consumpion increases less when markes are complee han when markes are incomplee. This in urn, leads o a smaller decline in he rade balance in he period in which he echnology shock occurs. 8. Sensiiviy analysis 8.1. Alernaive preference specificaion Thus far, he analysis has focused on a specificaion of preferences ha implies ha he labor supply is unaffeced by variaions in household wealh. This ype of 4 Indeed, Models 1 3 predic correcly ha he rade balance is counercyclical. The correlaion is so close o zero, however, ha is sign does depend on he soluion mehod employed. Mendoza, for example, approximaes he soluion o Model 1 by discreizing he sae space and finds a small bu posiive correlaion.

15 S. Schmi-Grohe, M. Uribe / Journal of Inernaional Economics 61 (003) Fig. 1. Impulse response o a uni echnology shock in Models 1 5. Noe. Solid line: Endogenous discoun facor model; Squares: Endogenous discoun facor model wihou inernalizaion; Dashed line: Deb-elasic ineres rae model; Dash doed line: Porfolio adjusmen cos model; Doed line: complee asse markes model; Circles: Model wihou saionariy inducing elemens. preferences is commonplace in models of he small open economy. Neverheless, i is of ineres o invesigae he exen o which he resuls repored above are robus o he inroducion of preferences implying a wealh effec in labor supply. To his end, we consider a period uiliy funcion of he form f v 1v 1g c (1 h) 1 U(c, h) 5 ]]]]]]. 1 g g Under hese preferences, he marginal rae of subsiuion beween consumpion and leisure is given by U h(c, h) 1 v c ]]] 5 ]]]]. U (c, h) v 1 h c This marginal rae of subsiuion depends on he level of consumpion, whereas ha implied by he previously considered preferences does no.

16 178 S. Schmi-Grohe, M. Uribe / Journal of Inernaional Economics 61 (003) For models feauring an endogenous discoun facor (Models 1 and 1a), we now assume ha f v 1v c 1 b(c, h) c (1 h). g We se he parameer v a a value consisen wih a seady-sae value of h of 0.. This value for seady-sae hours is commonly used in he real-business-cycle lieraure and implies ha households allocae on average 0 percen of heir ime o he labor marke. As in Mendoza (1991), we assume ha he seady-sae rade-balance-o-gdp raio is percen. This long-run resricion pins down he parameer c in Models 1 and 1a, he parameer d in Models and 3, he 1 seady-sae level of consumpion in Model 4, and he iniial wealh posiion in Model 5. We se he parameers c and c3 in Models and 3 so as o capure he observed volailiy of he Canadian curren-accoun-o-gdp raio. Finally, as before, he sandard deviaion of he produciviy shock is se so as o mach he observed volailiy of Canadian GDP and he adjusmen cos parameer (f) is se so as o mach he sandard deviaion of Canadian invesmen. All oher parameer values are hose used earlier. Table 4 summarizes he calibraion of he model. Table 5 and Fig. presen a comparison of uncondiional second momens and impulse responses implied by he five models under he new preference specificaion. Overall, he mach beween he model and he daa is worse han under he baseline preference specificaion. More relevan for he purpose of his paper is he fac ha, as in he case in which labor supply is independen of wealh, Models 1 hrough 4 behave quie similarly. The only noiceable difference is again 5 inroduced by he behavior of consumpion in he complee asse marke model. 8.. Saionariy and speed of convergence The modificaions o he sandard small open economy model sudied in his paper induce saionariy. Tha is, hey eliminae he uni roo in ne foreign asses and consumpion. Bu hese saionariy-inducing mechanisms also affec he speed wih which he economy is expeced o rever o he seady-sae in response o saionary shocks. In he remainder of his secion we explore he connecion Table 4 Wealh elasic labor supply: calibraion g v c a f r d r s e 0. c c5 c The perfec correlaion beween consumpion and oupu in Model 4 would disappear if shocks o he exernal marginal uiliy of consumpion were presen.

17 S. Schmi-Grohe, M. Uribe / Journal of Inernaional Economics 61 (003) Table 5 Wealh-elasic labor supply: observed and implied second momens Daa Model 1 Model 1a Model Model 3 Model 4 Volailiies: sd( y ) sd(c ) sd(i ) sd(h ) SD b sd ] y SD ca sd ]..1 y Serial correlaions: corr( y, y 1) corr(c, c 1) corr(i, i 1) corr(h, h 1) S D b b1 corr ],] y y S 1 D ca ca1 corr ],]] y y 1 Correlaions wih oupu: corr(c, y ) corr(i, y ) corr(h, y ) S D b corr ], y y S D ca corr ], y y Noe. The firs column was aken from Mendoza (1991). Sandard deviaions are measured in percen per year. beween saionariy and mean reversion. We begin by sudying a simple economy ha whose linearized equilibrium condiions can be solved analyically. We hen perform a quaniaive analysis using he fully fledged inernaional RBC framework presened above. Consider a small, open, endowmen economy populaed by a large number of idenical households wih preferences given by ` E O u ln c 0 50 (36)

18 180 S. Schmi-Grohe, M. Uribe / Journal of Inernaional Economics 61 (003) Fig.. Wealh-elasic labor supply: impulse response o a uni echnology shock in Models 1 5. Noe. Solid line: Endogenous discoun facor model; Squares: Endogenous discoun facor model wihou inernalizaion; Dashed line: Deb-elasic ineres rae model; Dash doed line: Porfolio adjusmen cos model; Doed line: complee asse markes model; Circles: Model wihou saionariy inducing elemens. u u 5 b(c )u, 11 where u denoes he discoun facor, c denoes consumpion in period, and E0 is he mahemaical expecaion operaor given informaion a ime 0. The discoun facor is assumed o depend on he average per capia level of consumpion, c, which he represenaive household akes as given. The household faces a periodby-period budge consrain of he form d5 (1 1 r)d1 1 c y (37) where d denoes foreign deb, r denoes he world ineres rae, and y denoes he endowmen income. Income is assumed o follow a firs-order auoregressive process ( y y ) 5 r( y y ) 1 e ; r [ (1, 1), (38) 11 11

19 S. Schmi-Grohe, M. Uribe / Journal of Inernaional Economics 61 (003) where e disribues NIID(0, s ). In each period $ 0, households are subjec o he following no-ponzi-game consrain: j lim E (1 1 r) d # 0. (39) j ` 1j The household chooses c and d so as o maximize Eq. (36) subjec o he period-by-period budge consrain (37) and he no-ponzi-game condiion (39). The firs-order condiions associaed wih he household s maximizaion problem are Eqs. (37), (39) holding wih equaliy, and 1 1 ] 5 (1 1 r)b(c )E ]]. c c 11 In equilibrium we have ha c 5 c. Assume he following funcional form for he discoun facor ] c s1 1 c cd b(c ) 5 ]]]]; c $ 0, 1 1 r where ] c is some posiive consan. Then a compeiive equilibrium is a pair of sochasic processes hc, dj saisfying Eqs. (37), (39) holding wih equaliy, and 1 ] c 1 ] 5s1 1 ccd E ]], c c 11 given d1 and he exogenous sochasic process for he endowmen defined in Eq. (38). Le d 5 (y c )/r. Ifc. 0, he pair hc, dj is he non-sochasic seady-sae of he economy. Noe ha he seady-sae is independen of iniial condiions, so ha he expeced long-run levels of foreign deb and consumpion are also independen of he iniial sock of foreign deb. In his sense, he model is saionary. Linearizing he equilibrium condiions and he endowmen process around he poin hc, y, dj yields Ecˆ 5 (1 cc )cˆ 11 dˆ 5 (1 1 r)dˆ 1 cˆ yˆ 1 Eyˆ 5 ryˆ 11 where x ˆ; x x. This is a sysem of hree linear expecaional difference equaions wih wo predeermined variables, foreign deb and he endowmen. The non-explosive soluion o his sysem is ˆ ˆ 1 r d5 (1 cc )d1 ]]]]] yˆ r 1 cc 1 1 r

20 18 S. Schmi-Grohe, M. Uribe / Journal of Inernaional Economics 61 (003) r 1 cc cˆ ˆ 5(r1cc )d1 1]]]]]ˆ y r 1 cc 1 1 r If c 5 0, he equilibrium law of moion of ne foreign deb displays a uni roo. In his case he discoun facor is consan and equal o 1/(1 1 r). The expeced long-run level of foreign deb given informaion a ime 0 is lim ` Ed5 0 d1 1/(r 1 1 r)y ˆ0. This implies ha he expeced long-run level of foreign deb (and wih i consumpion) depends on he iniial level of foreign deb and he iniial realizaion of he endowmen shock. Therefore, emporary shocks have permanen effecs on he level of consumpion and foreign deb. In his sense he model is non-saionary. Local approximaion echniques are herefore invalid. For posiive values of c ha are close o 0, he coefficien on d ˆ 1 in he equaion giving he evoluion of dˆ is less han one and hus he equilibrium process for exernal deb is mean revering. Moreover, he speed of mean reversion increases wih he value of c. Also in response o innovaions in oupu he change in d is smaller he larger is c. Similarly, he speed of mean reversion in consumpion is also enhanced by higher values of c. However, he iniial impac of an oupu shock on consumpion is larger he larger is c. The inuiion behind hese resuls is simple. In his economy agens become more impaien as consumpion increases. Thus, as he elasiciy of he discoun facor increases, hey are willing o rade off a higher impac effec of an oupu shock on consumpion for a faser reurn o he seady-sae. Fig. 3 illusraes his rade-off. I depics he impulse response of deb and consumpion o a uni innovaion in oupu in period 1 for wo alernaive values of he parameer c measuring he sensiiviy of he discoun facor wih respec o consumpion. To illusrae how he parameers deermining saionariy affec he speed of mean reversion in he conex of a more realisic model wih endogenous labor supply and capial accumulaion, we plo in Fig. 4 he impulse response of Model (deb elasic ineres rae) o a produciviy shock for alernaive values of c, measuring he elasiciy of he counry premium wih respec o foreign deb. We consider hree differen values of c : (he baseline value), , and /10. I is eviden from he figure ha alering his parameer wihin a wide range around he baseline value does no affec he quaniaive predicions of he model in significan ways. We do no choose Models 1 or 1a (endogenous discoun facor) for his sensiiviy analysis because in hese models he parameer c 1, governing he saionariy of he model, also affecs he seady-sae level of consumpion and exernal deb. This problem does no arise in he simple version of Model 1a considered above in his secion. In ha model, consumpion and ne foreign deb appear o be more sensiive o wide variaions in he parameer c conrolling he saionariy of he model (see Fig. 3). Of course, alhough boh c and c deermine he saionariy of heir respecive models, changes in heir calibraed values are no direcly comparable. Performing he sensiiviy analysis wih Model 3 (porfolio adjusmen coss) delivers idenical

21 S. Schmi-Grohe, M. Uribe / Journal of Inernaional Economics 61 (003) Fig. 3. Endowmen economy: impulse response o a uni oupu shock. Noe. Solid line: c 5 0.1; Dashed line: c The values assigned o he remaining srucural parameers are: r , c 5 1, and r resuls o hose presened in Fig. 4. Finally, Model 4 (complee asse markes) does no feaure a parameer governing saionariy, and hus i does no lend iself o he ype of sensiiviy analysis conduced here. 9. Conclusion In his paper, we presen five alernaive ways of making he small open economy real business cycle model saionary: wo versions of an endogenous discoun facor, a deb-coningen ineres rae premium, porfolio adjusmen coss, and complee asse markes. The main finding of he paper is ha once all five models are made o share he same calibraion, heir quaniaive predicions regarding he behavior of key macroeconomic variables, as measured by uncondiional second momens and impulse response funcions, is virually idenical. We conclude ha if he reason for modifying he canonical non-saionary small open economy model in any of he ways presened in he presen sudy is simply

22 184 S. Schmi-Grohe, M. Uribe / Journal of Inernaional Economics 61 (003) Fig. 4. Saionariy and speed of convergence: impulse responses of Model o a produciviy shock. Noe. Solid line: c5baseline value; Dashed line: c5 en imes he baseline value; Doed line: c 5 one enh he baseline value;. echnical, ha is, solely aimed a inroducing saionariy so ha he mos commonly used numerical approximaion mehods can be applied and uncondiional second momens can be compued, hen in choosing a paricular modificaion of he model he researcher should be guided by compuaional convenience. In oher words, he researcher should choose he varian of he model he finds easies o approximae numerically. In his respec Model 1, feauring an endogenous discoun facor a la Uzawa (1968) is in disadvanage vis a vis he oher models. For is equilibrium condiions conain an addiional sae variable. A second resul of our paper is ha, in line wih resuls previously obained in he conex of wo-counry real-business-cycle models by Kollmann (1996) and Baxer and Crucini (1995), wheher asse markes are complee or incomplee makes no significan quaniaive difference. This paper could be exended in several dimensions. One would be o allow for addiional sources of uncerainy, such as domesic demand shocks (i.e. governmen purchases and preference shocks) and exernal shocks (i.e. erms-of-rade and world-ineres-rae shocks). A second possible exension is o consider oher

23 S. Schmi-Grohe, M. Uribe / Journal of Inernaional Economics 61 (003) characerisics of he business cycle along which o compare he various models, such as frequency decomposiions. Finally, one could sudy addiional saionariy induces variaions of he small open economy model. For example, Cardia (1991) and Ghironi (001) among ohers bring abou saionariy by inroducing overlapping generaions wih perpeually young agens a la Blanchard (1985). Acknowledgemens We hank an anonymous referee for commens. Sephanie Schmi-Grohe hanks he Universiy of Pennsylvania for is hospialiy during he wriing of his paper. References Baxer, M., Crucini, M.J., Business cycles and he asse srucure of foreign rade. Inernaional Economic Review 36, Blanchard, O., Deb, deficis, and finie horizons. Journal of Poliical Economy 93, Cardia, E., The dynamics of a small open economy in response o moneary, fiscal, and produciviy shocks. Journal of Moneary Economics 8, Correia, I., Neves, J.C., Rebelo, S., Business cycles in a small open economy. European Economic Review 39, Ghironi, F., 001. Macroeconomic Inerdependence under Incomplee Markes, manuscrip. Boson College, Ocober. Kim, S.H., Kose, A., 001. Dynamics of open economy business cycle models: undersanding he role of he discoun facor, Tufs Universiy. Macroeconomic Dynamics, Macroeconomic Dynamics, forhcoming. Kollmann, R., Incomplee asse markes and he cross-counry consumpion correlaion puzzle. Journal of Economic Dynamics and Conrol 0, Mendoza, E., Real business cycles in a small-open economy. American Economic Review 81, Mendoza, E., Uribe, M., 000. Devaluaion risk and he business-cycle implicaions of exchange-rae managemen. Carnegie-Rocheser Conference Series on Public Policy 53, Neumeyer, P.A., Perri, F., 001. Business Cycles in Emerging Markes: The Role of Ineres Raes, manuscrip. New York Universiy, May. Obsfeld, M., Ineremporal dependence, impaience, and dynamics. Journal of Moneary Economics 6, Schmi-Grohe, S., The inernaional ransmission of economic flucuaions: effecs of U.S. business cycles on he Canadian economy. Journal of Inernaional Economics 44, Schmi-Grohe, S., Uribe, M., 001. Sabilizaion policy and he coss of dollarizaion. Journal of Money, Credi, and Banking 33, Senhadji, A.S., Adjusmen of a Small Open Economy o Exernal Shocks, Disseraion. Universiy of Pennsylvania. Uribe, M., Exchange rae based inflaion sabilizaion: he iniial real effecs of credible plans. Journal of Moneary Economics 39, Uzawa, H., Time preference, he consumpion funcion and opimum asse holdings. In: Wolfe, J.N. (Ed.), Value, Capial and Growh: Papers in Honor of Sir John Hicks. The Universiy of Edinburgh Press, Edinburgh, pp