The Real Exchange Rate, the Current Account, and Foreign Debt in a Stochastic Optimizing Model with Imperfect Capital Flows. Jafar A.

Transcription

1 The Real Echange Rae, he Curren Accoun, and Foreign Deb in a Sochasic Opimizing Model wih Imperfec Capial Flows Jafar A. Khondaker Deparmen of Economics Carleon Universiy, Oawa

2 The Real Echange Rae, he Curren Accoun, and Foreign Deb in a Sochasic Opimizing Model wih Imperfec Capial Flows Jafar A. Khondaker Absrac This paper develops a hree-good sochasic ineremporal opimizing model of a small open economy o analyse he dynamics of he real echange rae, he curren accoun and deb accumulaion wih consrained access o he world capial marke. The model is solved and simulaed, and analysed using Canadian daa for he period wih ools borrowed from he real business cycle modelers. The model s equilibrium sochasic processes indicae ha real shocks (o echnology and erms of rade) can eplain he behaviour of he real echange rae in erms of volailiy and persisence. Thus, for eample, he model s impulse response funcions show ha a shock o echnology in he epor-good secor ha raises produciviy one percen above is seady-sae level leads immediaely o an appreciaion o be followed by a depreciaing pah for more han fory years afer he shock. We also find ha some degree of sensiiviy of he real ineres rae o he sock of foreign deb gives he model sabiliy, and allows he model o beer fi he daa in erms of producing a mach wih he sandard deviaions in he daa.. Inroducion The las wo decades have winessed a large volume of work direced a modeling he dynamics of he real echange rae, he curren accoun and asse accumulaion, and long-run eernal indebedness in response o various demand and supply shocks in an ineremporal opimizing seing. A parial lis of his work includes Buier (987), Brock (988), Frenkel and Razin (987), Masuyama (987), Obsfeld (982, 989), Gavin (992), Sen and Turnovsky (989), Devereu and Shi (99), Murphy (989), Karayalcin (994), and Serven (995). While he disurbances sudied in hese papers are differen, all of hese papers share a common mehodology: hey derive he underlying behavioural relaionships from ineremporal opimizaion by a represenaive agen. These models also assume he eisence of a perfec world capial marke for deb so ha a counry can borrow or lend as much as desired a he given world ineres rae. 2 This implies ha a seady sae will be aained if and only if he rae of ime preference becomes equal o he paramerically given world ineres rae. If he wo raes are iniially differen, he sysem has no equilibrium unless one or boh raes are endogenous. Some auhors simply assume ha he wo raes are always equal. 3 Frenkel and Razin (987), Buier (987), Brock (988), Obsfeld (989), Murphy (989), Karayalcin (994), Serven (995), Devereu and Shi (99) eamine various fiscal shocks; Gavin considers moneary shocks; Masuyama (987) considers a shock in he form of a permanen change in he price of impored inermediae goods, and Sen and Turnovsky (989) sudy he effecs of ariffs. Devereu and Shi and Karayalcin also eamine he effecs of produciviy shocks. 2 There is a class of models ha assumes an imperfec capial marke for debs in analyzing growh and deb problems of highly indebed counries. See, for eample, Bardhan (967) for an early conribuion in his area. A more recen eample is Bhandari, Haque, and Turnovsky (99). These auhors, however, do no consider he dynamic ineracions of he real echange rae, he curren accoun and foreign deb ha is he main focus of he presen work. 3 See, for eample, Froo and Rogoff (99), Turnovsky (993), Murphy (989) and Serven (995).

3 However, for any given se of iniial condiions deermining he srengh of ime preference and he world ineres rae, his oucome would be an improbable acciden. Devereu and Shi (99), and Karayalcin (994) used recursive preferences o circumven his problem by allowing he implied rae of ime preference o be endogenous. This requires he rae of ime preference o increase wih he level of uiliy, and herefore, wih he level of consumpion. Such an assumpion is no paricularly appealing and has been subjec o criicism. 4 A second way of bringing he wo raes ino equaliy is o follow Yaari (965) and Blanchard (985) and assume a finie probabiliy of deah. 5 Given he eensive empirical evidence relaing o he violaions of real ineres rae pariy (see, for eample, Mishkin (984), Marson (995, 997)), we adop a relaively new and more realisic approach in his paper.we develop an opimizing model of a small open economy o analyse he dynamics of he real echange rae, he curren accoun and deb accumulaion wih consrained access o he world capial marke. 6 In his srucure, he real rae of ineres paid on foreign deb is increasing in he sock of foreign deb. 7 Our objecive is o eamine he impac of echnology and erms-of-rade shocks on he real echange rae and he curren accoun. In his model, ineremporal efficiency condiions place resricions on he rae of growh of consumpion and he consumpion rae of ineres. Given ha he rae of ineres on inernaional borrowing is increasing in he sock of foreign deb, hese efficiency condiions govern he rae of change of savings and invesmen, and, herefore, he size of he curren accoun. To compue he equilibrium sochasic processes of a hree-secor ineremporal model, we borrow numerical soluion mehods from real business cycle heory. The model developed, however, is disinc from a ypical real business cycle model in a number of imporan respecs. Charles Plosser (989) describes real business cycle models as asking he quesion: how do raional maimizing individuals respond over ime o changes in he economic environmen and wha implicaions do hose responses have for he equilibrium oucomes of aggregae variables? The aggregae variables are hen viewed as being he oucomes of decisions made by many individual agens maimizing uiliy subjec o heir producion and resource consrains. For more han a decade and a half now, paricularly since he publicaions of Kydland and Presco (982) and Long and Plosser (983), his school of hough has been he focus of aenion of a significan porion of he research on dynamic macroeconomics. More recenly, however, Gregory Mankiw (989), while 2 4 See, for eample, Turnovsky (997, p. 39). 5 Cardia (99) and Macklem (993) adop his approach. 6 Pichford (995) develops a simple one-secor open economy model of ineremporal choice assuming ha real ineres rae is an increasing funcion of oal borrowing. Harjes (997) applies a one-secor open economy real business cycle model o German daa, and assumes ha he rae of reurn on foreign bonds is decreasing in he sock of foreign bonds in Germany. Alhough an essenial analyical ool of aggregae economics, he one-secor framework is of limied use in an inernaional environmen where eogenous shocks affec differen secors of he economy differenly. 7 This assumpion may no a all seem unrealisic given he evidence repored by Bayoumi e al. (995) from risky U.S. saes facing he non-linear deb supply curve. Hermalin and Rose (999) provide an ehausive analysis of how risks presen in he domesic capial markes end o be eacerbaed in he inernaional cone. In heir applicaion of he Bank s Quarerly Projecion Model (QPM), economiss a he Bank of Canada (see Macklem e al. 995) have recenly incorporaed he impac of he deb and deficis on risk premiums by calibraing an increase in he risk premium by.7 basis poins for one percenage poin increase in he deb-o-gdp raio, when he deb-o-gdp raio is above 5 percen. This calibraion is based on he evidence repored by Alesina e al. (993) for 2 OECD counries for he period 979 o 989. Alhough Macklem e al. analysis applies o governmen deb, hey also show ha Canada s foreign deb is largely driven by he need o finance he governmen deb. Indeed, he auhors show ha, if here were no governmen deb, Canada would be a ne foreign credior.

4 acknowledging ha real business cycle heory has served he imporan funcion of simulaing and provoking scienific debae, predics ha i will ulimaely be discarded as an eplanaion of observed flucuaions. According o Mankiw, he wo fundamenal weaknesses of his approach are is reliance on he ineremporal subsiuion of leisure o eplain changes in employmen and is heavy reliance on echnological disurbances as he primary cause of economic flucuaions. While our model assumes ha produciviy disurbances moivae raional individuals o adjus savings and invesmen in order o smooh consumpion over ime, i does no rely on ineremporal subsiuion of leisure o eplain economic flucuaions. In our model, economic flucuaions arise from produciviy and erms of rade disurbances. McCallum (989) observes ha when business cycle models do no include a foreign secor and ignore he erms-of-rade shocks ha affec he real price of impored raw maerials, hese shocks may appear o be produciviy shocks. I is now widely believed ha large and recurren flucuaion in he erms of rade is an imporan driving force of recen economic flucuaions. The paper ha is closes o wha we are doing is ha of Mendoza (995). His analysis is perhaps he mos effecive analysis of open economy real business cycles ha encompasses evidence for he group of seven larges indusrialized counries (G-7) and 23 developing counries. While Mendoza s is no he firs real business cycle analysis of he open economy, i is more realisic in ha i does no impose purchasing power pariy and real ineres rae pariy o induce near-perfec correlaion beween consumpion and GDP. 8 We begin from he same hree-secor framework as Mendoza bu make some key deparures. Firsly, while Mendoza (995) assumes ha agens in he small open economy have access o a perfec world capial marke and uses preferences o endogenously bring he rae of ime preference ino line, we assume ha he rae of ime preference is fied and ha he real rae of ineres, which is increasing in he sock of foreign deb, comes ino line. 9 Secondly, as already menioned, we do no use ineremporal subsiuion of labor o propagae economic flucuaions hrough ime. Finally, while Mendoza assumes ha capial sock is composed enirely of impors, we assume ha invesmen is a composie commodiy wih inpus of domesic and foreign goods. This is imporan in view of he fac ha he response of invesmen o a erms-ofrade shock is likely o be proporionae o he share of impors in oal invesmen ependiure. The resuling model presens he following general approach. Raional opimizing agens smooh consumpion ineremporally and so adjus savings and invesmen decisions o boh produciviy and erms-of-rade shocks. All key variables, including he real echange rae, are hus he oucome of such opimizing decisions. To deermine wheher he model can mimic he sylized facs of an acual economy and hence become relevan for policy analysis, we parameerize, simulae and calibrae he model o Canadian daa. In he ne secion, he producion echnologies, capial accumulaion equaions, and sochasic 8 Mendoza (99) eamines he effecs of erms of rade shocks in he real business cycle model of a small open economy wih no non-raded goods assuming he eisence of a perfec inernaional capial marke. Correia e al (995) use a similar one-secor open economy model. Wihou a non-raded goods secor, hese models fail o accoun for he subsiuion effecs induced by changes in inra- and ineremporal relaive prices of non-raded goods. Macklem (993) applies a small open economy overlapping generaions model o Canadian daa o eplore he seady sae and dynamic effecs of erms-ofrade disurbances and governmen deficis. Sockman and Tesar (995) is a wo-counry real business cycle model wih nonraded goods. Backus, e al (994, 995), and, Baer and Crucini (995) use wo-counry models wihou a non-raded goods secor. 9 Noe ha in Mendoza s model, purchasing power and real ineres rae non-pariies arise from he inclusion of nonraded goods even in he presence of a perfec world capial marke. 3

5 processes are laid ou ogeher wih he preferences of an opimizing consumer-invesor. The model s inra- and ineremporal opimizing equilibrium condiions are hen derived. The model is calibraed, parameerized and solved in secion 3. In secion 4, we presen he resuls obained from he parameerized model and eamine is impulse response funcions and sensiiviies o variaions in key srucural parameers. Is simulaed momens are hen compared wih Canadian daa o see if he model can mimic he sylized facs of he Canadian economy. Secion 5 is devoed o an analysis of his paper s key conribuion. I shows how dropping he assumpion of a perfec world capial marke affecs he resuls of a ypical dynamic model of an open economy. Secion 6 presens he conclusions. 2. The Model We consider an economy wih hree goods: an impor good, an epor good, and a nonraded good. The counry has a fied labor force of size N ha is mobile beween is non-raded and epor secors. The represenaive consumer-invesor in his economy is an infiniely-lived agen who owns a (/N)h share in all firms, consumes all hree goods, and inelasically supplies her enire labor endowmen. Alhough labor is coslessly mobile beween he wo producion secors, capial is secor-specific. To accumulae capial, boh secors are assumed o need boh epors and impors. We assume ha he counry is iniially a ne-debor in he inernaional capial marke, and can borrow furher a a rae of ineres ha is increasing in is curren sock of deb. While he price of he nonraded good varies endogenously in response o local flucuaions in supply and demand, he small open economy faces eogenously deermined erms-of- rade. The source of susained growh in his economy is labour-augmening echnological change ha grows a a consan rae (o be specified shorly). I also eperiences emporary shocks o oal facor produciviy in boh producion secors Technologies and Shock Processes The echnologies used by he wo producion secors are given by: Y Z K X N h h h h Y Z K X N, () where all non-price variables are in per-capia erms. 2 Y h and Y are, respecively, he oupus of he nonraded and epor goods, X is he efficiency level of labour, subjec o labor-augmening echnological change, Z i is he sochasic parameer of he producion process, K i is per capia capial sock and N i is he fracion of he labor-force employed in secor i, (i, h). Boh producion A more complee analysis of capial formaion would include nonraded goods as inpus as well. We, however, eclude such goods here o simplify our analysis. We use he leers h,, and m as subscrips o relae a variable o he nonraded good, he epor good, and he impor good, respecively. The subscrip denoes he ime period. The absence of any such ime subscrip indicaes he seady sae level of he variable. 2 In wha follows, all variables, ecep, of course, ineres rae and he wo prices, are measured in per- capia erms, unless oherwise indicaed.

6 funcions are concave, wice coninuously differeniable, increasing in heir argumens, and boh display consan reurns o scale in he inpus capial and effecive labour, i.e. K and XN. In boh producion funcions, <. The sochasic parameer is assumed o be observed a he beginning of he period. I has been amply documened ha he parameer displays considerable serial correlaion wih he firs differences nearly serially uncorrelaed. To reflec his high persisence, we assume ha he sochasic echnology parameer evolves eogenously according o he following law of moion: 3 log Zi ψi log Zi + ψi log Zi + εi, (2) 2 εi ~ iid... ; σ i, 5 In he above specificaion, log(z i ) is he uncondiional mean of he process, and < ψ i <, (i, h). On he oher hand, labor augmening echnological progress akes place a a consan gross rae γ so ha X γx γ X (3) where X is he iniial level of he labor-augmening echnology. The remaining source of uncerainy in his economy is is eogenously deermined erms-ofrade. Like he shocks o produciviy, erms-of-rade shocks are also highly persisen. They are also assumed o be emporary, evolving according o he following sochasic process: log p ψ log p + ψ log p + ε, m m m m m m 2 ε ~ iid... ; σ, < ψ < m m m where p m is he erms-of-rade a ime, and p m is he seady-sae level of p m, and is an independenly and idenically disribued period-specific innovaion in p m. The erms-of-rade is defined here as he relaive price of he impor good o he epor good. 2.2 Capial Accumulaion We wan o model he capial accumulaion process in a realisic manner so ha he model can generae appropriae responses of is producion secors o eogenous shocks. The assumpion ha all facors of producion are perfecly mobile across producion secors has been fundamenal for many classic resuls in economics. 4 While he assumpion ha in he long-run all facors are effecively mobile is no unreasonable, he e-pos malleabiliy of capial is unrealisic for he shorer run. Mussa (978) eamines he adjusmen process hrough which capial can be reallocaed across 3 Noe ha his specificaion ensures ha he echnology componens converge o heir mean levels over ime. 4 Thus, many of he classic resuls from he Heckscher-Ohlin-Samuelson model rely on his assumpion. Based on similar assumpions, he Balassa-Samuelson model [Balassa (964), Samuelson (964)] ends up wih he conclusion ha demand facors have no effecs on he relaive price of he nonraded good. Wih secor-specific capial, demand facors can affec he relaive price of he nonraded good. Rogoff (992) derives such resuls wih fied socks of capial and labor in he raded and he nonraded secors. ε m

7 secors. Neary (978) noes ha reallocaion of capial rarely akes he form of physical movemen of insalled capial from one secor s use o anoher. In realiy, his is accomplished by invesmen in new capial goods in one secor and he depreciaion of old capial goods in he oher. This is a long-run process. We assume ha capial is secor-specific and boh secors accumulae capial using a homogeneous of degree one invesmen echnology ha combines epors and impors ino invesmen goods: 5 ϑ i i im,,, (4) ϑ i i I j j i h 6 where j i and j im, are, respecively, he quaniies of he epor-good and he impor-good used as inpus ino he invesmen process of he ih secor, and < ϑ i < is he share of epors in I i, he invesmen in he ih secor. Noe ha alhough he wo secors share he same echnology for combining epors and impors ino capial, we allow he share of epors in he capial of he wo secors o differ. The opimizaion problem of he represenaive consumer-invesor wih respec o capial accumulaion can now be viewed as a wo-sage process. The firs-sage deermines he opimal ime pahs for invesmens, I i ; he second sage deermines he cos-minimizing quaniies of j i and j im : j * im j * i ϑ i ϑ p i m ϑ ipm ϑ i ϑ i I i i ϑ (5) I i The associaed price q i of invesmen in he ih secor can be found as he minimized uni-cos funcion: 6 q Φ p ϑ i i m Φ ; ϑ i ϑ i ϑ ϑ + ϑ ϑ. i i i i i (6) Once capial goods are procured, addiional coss are assumed o be incurred for he insallaion of capial. Absen hese insallaion coss, invesmen would become passive, as whaever is saved could be added o he capial sock (if no used o pay off deb) a no cos, o equae capial s marginal produc o he rae of ineres ha he counry faces. Capial insallaion coss reflec he 5 Gavin (992) and Serven (995) following he analysis of Hayashi and Inoue (99) also consider impored invesmen good as an inpu in more general forms ino he invesmen process in one-secor Solow-ype models wih analysis resriced in he radiional sae-space diagrams. Lile is known abou he quaniaive implicaions of his approach. 6 The minimized uni-cos funcion in (6) is derived in Appendi A.

8 facs ha capial socks move sluggishly and ha proraced deviaions of capial s marginal produc from he real rae of ineres are possible. We assume here, for analyical simpliciy, ha adjusmen coss are incurred only when invesmen is above or below is seady-sae level: g i φ iqi Ii Ii 2 K K i i 2 K i 7 (7) where g i is he insallaion cos. The parameer φ i > makes he insallaion cos increasing in invesmen relaive o he eising capial sock. Invesmen a ime becomes producive capial a ime +. In paricular, he capial sock in he ih secor evolves according o he law of moion: Ki + Ki + Ii δ i Ki, < δ i <, (8) where δ i is he rae of depreciaion. 2.3 Preferences The infiniely-lived represenaive consumer-invesor inelasically supplies her enire labor endowmen and owns a per-capia share in all he firms in he economy, which eniles her o a flow of profis accruing o each secor. The agen s objecive is o maimize discouned epeced lifeime uiliy given by: < β < UC E β uc φ I I F RF + PC + q I + 2 K K φ h Ih I + qh Ih + 2 K K h h h 2 K p Z K X N,, (9) where u is concave, sricly increasing, and wice coninuously differeniable in consumpion in period, C, wih,, 2,..,. Here β is he agen s discoun facor and C is a real consumpion bundle (o be specified below). Given ha consumers own all facor shares, he represenaive consumer faces a difficul problem of choosing he pahs of consumpion and invesmen in an environmen wih uncerainy abou fuure produciviy and erms-of-rade so as o maimize (9). These choices are based on epecaions over fuure prices, and are subjec o he laws of moion for capial socks in he wo secors (8) and sequences of budge consrains: 7 h h h h h 2 K Z K X N () 7 One can re-wrie he consrain in () o epress he curren accoun defici as absorpion (in erms of consumpion and invesmen) over income or oupu.

9 8 given iniial socks F, K h, and K, and he ransversaliy condiions 8 : T T lim E R q K. T lim E R F T i i () In he above specificaion, F is he represenaive agen s deb, which grows according o he discrepancy beween he value of absorpion and of income, P is he consumpion-based price inde, p h is he relaive price of he nonraded good in erms of he epor good, and R [+r(f - )] is he gross real rae of ineres on foreign deb. Noe ha wih he absence of any governmen, he consumers deb is idenical o he counry s ne foreign indebedness, and he growh of ha indebedness is he curren accoun defici of he counry. 9 The represenaive agen s conemporaneous uiliy is given by: η uc C (2) η where η > is he coefficien of relaive risk aversion. The real consumpion bundle C is given by: C C + C C θ θ h m (3) where C h, C, and C m are, respecively, he consumpion of nonradables, eporables, and imporables. 2 In he above specificaion, radables and nonradables are represened in consanelasiciy-of-subsiuion (CES) form, wih ( + ) being he elasiciy of subsiuion, and radables are epressed in Cobb-Douglas uniary-elasiciy form, wih θ being he share of eporables in radables ependiure. The ineremporal elasiciy of subsiuion in aggregae consumpion is given by η. Given any opimal oal consumpion level C obained by solving he ineremporal allocaion problem specified above, demands for he hree goods are obained by solving a saic problem of inraemporal subsiuion o minimize he consumpion cos in erms of eporables. 2 This saic opimizaion gives: 22 8 The condiions in () rule ou he possibiliy of eplosive soluions. 9 The inclusion of a governmen in he model wih governmen deb complicaes he analysis wihou adding insigh ino he impac of produciviy and erms-of-rade shocks. 2 We ignore he share parameers of raded and nonraded goods in he consumpion bundle in (3) wihou loss of generaliy. 2 See Obsfeld and Rogoff (995, p ) for derivaion of similar iner- and inra-emporally opimizing consumpion levels in a wo-good closed-economy model. 22 The saic opimizaion problem is solved in Appendi B.

10 9 θ θ + C p θ θ p + C C C + + θ h h m m C p C m m m θ + θ θ θpm ph + θ θ θ + + θ θ θ + pm + (4) As already menioned, p h and p m are, respecively, he curren relaive prices of nonraded and impor goods in erms of epors. The consumpion-based price inde P measures he leas ependiure needed o buy a uni of he consumpion inde, on which insananeous uiliy depends. The inde is given by: where κ θ θ θ. θ C m θ + + m h θ P κp + p 2 θ + pm θ + θ The Non-raded Goods Marke Equilibrium Equilibrium in he non-raded goods marke requires ha domesic demand for such goods equals is supply. Using he epression for P in (5), he consumpion demand for he impored good can be wrien as: P + C (5) Now subsiuing his epression ino he firs equaion in (4), he non-raded goods marke equilibrium condiion can be obained as: 23 p h + θ θ p θ Z k N θ + θ h h h θ θ + m Labor demand in he non-raded producion is given by: pm θ + θ + + c P (6) 23 Here we de-rend consumpion and capial sock variables by he labor-augmening echnology componen, and use lower-case leers o denoe he de-rended per-capia variables. This is done here for convenience ahead of ime for laer use. See Appendi B for inra-emporal opimizaion and labor marke equilibrium condiions.

11 N h phzhkh (7) W where W is he economy-wide wage rae in erms of he epor good. The assumpion ha labor can migrae insananeously beween secors ensures a unique economy-wide wage rae: (8) h h h W Z k + p Z k Now one can subsiue (8) ino (7), and use he resuling equaion along wih (5) and (6) o solve for he equilibrium price inde, he relaive price of non-raded goods, and he wage rae. Wih he inraemporal allocaion rules in (5) and (4), he consumer-invesor s ineremporal decision can be analyzed enirely in erms of composie consumpion, C, invesmens in he wo producion secors, I i, (i, h), and he price-inde, P. The behavior of he inra- and iner-emporal opimizing price inde P can be compared wih he behavior of he acual real echange rae. 24 Real echange rae s key ineremporal link in he cone of his small open-economy is as follows. Shocks o echnology or o erms-of-rade cause flucuaions in he relaive price of he nonraded good and, hus, in he consumpion-based price inde. This, in urn, affecs he consumer s effecive real ineres rae. The represenaive consumer-invesor hen adjuss her consumpioninvesmen decisions in order o smooh consumpion ineremporally. We now have all he inpus needed o eplore he model s ineremporal opimizaion condiions. The ne sep is o solve he model o obain is recursive equilibrium laws of moion. These will be used o eamine he implicaions of boh erms of rade and echnology shocks for he real echange rae and oher key macro variables. 2.5 Ineremporal Opimizaion The consumer-invesor s ineremporal opimizaion problem involves he maimizaion of he following Lagrangian by choosing C, K +, K h+, F, I, and I h : In empirical sudies, he real echange rae is ypically defined as he raio of a domesic and foreign aggregae price inde boh measured in unis of he same currency. Given he absence of any nominal echange rae in he presen * model, we can define an analogous measure of he real echange rae: e pmp P, where P* is he foreign counerpar of P, he consumpion-based price inde. Noice ha e gives he price of a uni of he foreign consumpion inde in erms of unis of he domesic consumpion inde. 25 We coninue o follow he convenion ha he ime subscrip indicaes he level of a variable a ime, while he absence of any such ime subscrip indicaes he seady sae level of he variable.

12 E E β β * C η λ φ hqh Ih + RF + qh Ih + 2 K Q K I δ K Q η φ q I I PC + q I + 2 K K + h K λ h + I X δ K h h h h 2 K F Z K X N Ih K K p Z K X N h 2 h h h h h The specificaion of labor-augmening echnological progress a he gross raeγ implies a mean seady sae growh a he same rae. I will herefore be convenien o ransform he variables ino saionary ones by rescaling hem as c C/X, k K/X, i I/X, ec. Wih he rescaled variables, he Lagrangian can be re-wrien as: η η c η 2 φ q i i Pc + qi + γ 2 k k φ q i i h h h h + + Z k phzhkh qhih 2 k kh γ + δ γ + δ k + R f f Q k k i Q k k i h h h h h 2 k h where γ is he gross rae of growh in he labor-augmening echnological progress as defined in (3), and β * η βγ is he consumer s effecive discoun facor. 26 The firs-order efficiency condiions are obained by differeniaing he Lagrangian wih respec o he choice variables c, f, i, i h, k +, k h+ and he mulipliers, λ, Q, and Q h a each ime. This is done fully in Appendi C. (9) 26 Noe ha in above formulaion we assume ha he consumer-invesor akes he inraemporally opimal fracions of he labour force employed in he wo producion secors as given, and eclude hem from he ineremporal opimizaion problem. Given ha we do no rely on he ineremporal subsiuion of labour o eplain economic flucuaions, inclusion of labour is no cenral o any ineremporal adjusmen process in he presen model.

13 2 3. The Soluion of he Model 3. Log-linearizaion A look a he firs-order condiions associaed wih he ineremporal opimizaion problem derived in Appendi C indicaes ha hese condiions are nonlinear in he decision variables, and, in general, do no lend hemselves o an eac soluion. A sandard approimae soluion procedure is o compue he nonsochasic seady sae values of he variables of he sysem, and hen loglinearize he firs-order condiions, as well as oher equaions, around he nonsochasic seady sae values. The model hen becomes a sysem of log-linear epecaional difference equaions in he capial sock, he foreign deb, and he eogenous variables driving he economy. The resuling sysem of equaions can be solved using he mehod of undeermined coefficiens for he laws of moion for he sae variables k, k h, and f, as well as linear relaions connecing oher variables such as P, p h, c, ec. 27 The model is log-linearized in some deail in Appendi C. Here we jus presen he resuls of his eercise in Bo in an order ha is convenien for he applicaion of he mehod of undeermined coefficiens. In log-linearizing he model, we use a shor-cu proposed by Uhlig (997) ha mus be noed here. 28 Le ~ denoe he percenage deviaion of any variable from is seady-sae value : ~ log log. This can be usefully re-wrien as e ~ + ~ + ~. Thus, if ~ In.4, hen is approimaely 4 percen above is seady sae value. Bo, equaion (.) presens he feasibiliy consrain facing he represenaive consumerinvesor. Equaions (.2) hrough (.5) represen he producion funcions and he capial accumulaion equaions for he epor good and he nonraded good secors, respecively. Equaions (.6) hrough (.8), respecively, capure he price inde, he non-raded goods marke equilibrium condiion consisen wih labor marke equilibrium, and he real ineres rae. 29 The epecaional equaions (.9) hrough (.) represen he Euler equaions for firs consumpion and hen invesmen for nonraded and epor good secors. Equaion (.2) is derived by equaing he gross rae of reurn on invesmen in he epor good secor o he gross finance cos of invesmen. 3 Equaions (.3) hrough (.5), respecively, give he model s eogenous shock processes wih respec o echnology shocks in he raded, and nonraded secors, and erms-of-rade shocks. Bo 2 presens he coefficiens associaed wih he variables in Bo. 27 The mehod has been presened in eensive deails by Uhlig (997), Chrisiano (998), and Campbell (994). There are some alernaive mehods as well. Taylor and Uhlig (99) provide a good review and comparison of hese alernaive mehods. 28 Noe ha we use lowercase leers o indicae per-capia levels of variables de-rended by he growh componen X. Now ildes are used o denoe percenage deviaions of hese de-rended per-capia variables from heir seady sae levels. There are, however, some ecepions. We use P, p h, and p m, respecively, for he price inde, he relaive price of nonraded o epor goods, and he erms-of-rade. The absence of ime subscrip wih any variable is used o indicae he seady sae level of he variable. We use r for he real rae of ineres, while ~r is used o denoe he logarihmic deviaion of he gross rae of ineres from is seady sae value: r ~ log R log R. 29 The wo raios τ and τ τ ha appear in he coefficiens a 2, a 3, and a 4 (see Bo 2) in equaion (7) in Bo depend on he seady-sae oupu in he epor- and he non-raded good secors. They are defined in equaion (C.5) in Appendi C. 3 Noe ha he gross rae of reurn includes he marginal produc of capial in producion and he marginal reducion in he cos of insalling a given flow of capial. See equaion (C.7) in Appendi C (and he associaed discussion) for he gross reurn on a one-period invesmen and he finance cos of invesmen in erms of he epor good.

14 3 Bo The Log-linearized Model.) yy ~ + py~ p + py~ y Pcp ~ Pcc ~ α ~ + αp ~ ~ ~ ~ ~ iqi iqi h h h Rf γr + f + ff 2.) ~ ~ ~ y z k 3.) ~ ~ ~ yh zh kh ~ ~ ~ 4.) γk + δ k γ + δ i ~ ~ ~ 5.) γk δ k γ + δ i h h h h h h m h + h h h h + +.) ~ α ~ 6 P p ~ 2 pm ph ph 7.) ap ~ + ~ + ~ + ~ m a a3ph ap 6 c ~ + ~ ~ ak az ak + az ~ 2 4 h 4 h 5.) ~ ~ 8 r ωf 9.) E ηc~ ηc~ + + ~ p ~ p+ + r~ + ~ ~ * ~ * ~ γ + +. ) E q λ γ q λ βα i βα k * + ~ ~ ~ ~ β ap γap + α k i * + βα~ z * + βα ~ h + h 3 h + 4 h + 5 h + 5 h + 8 m + 7 m 3 h h ~ ~ + ~ ~ γ + + ~ + ~. ) E q λ γ q λ β α i β α k β α z β a p γap ~ ~ ~ + α k i * * * * m + 9 m ~ ~ ~ ~ α 7k + + Rα 6 γ k i + α 6i + + α ~ 8z + + a ~ p + Ra ~ 9p 2. ) E Rq ~ r+ 3. ) ~ z ψ ~ z + ε 4. ) ~ z ψ ~ h hzh + εh 5. ) ~ p ψ ~ p + ε m m m m 6 m m Wih he model log-linearized, we now have a sysem wih an endogenous sae variable ~ ~ ~ vecor S of size 4, a vecor E of 8 oher endogenous variables, and a vecor Z of 3 eogenous sochasic processes. 3 The objecive in applying he mehod of undeermined coefficiens is o obain he marices P, V, A, and B for he sysem s recursive equilibrium laws of moion: 3 ~ ~ In solving he model, we ake he co-sae variable λ as a sae variable in addiion o k, k and h ~. f

15 4 Bo 2 The Log-linearized Model: Coefficiens θ α ϑ α ϑ α θ κ + ih hqh, i q 2 pm, θ θ κ θ θ, α γq φ γ + δ, α p y k α, 3 h h h 4 h h h 3 α py k, α γqφ γ + δ, α y k α, 5 h h h * η α y k, β βγ, a + a a + θ, a τ a τ, a τ τ, a τ, a a, a α i, a a δ, a α i, a a δ, τ 6 7 h 8 7 h 9 9 Zk + pzk h h h Zk phzk h h τ. Zk + pzk h h h ~ ~ ~ S PS + VZ ~ ~ ~ E AS + BZ (2) The sabiliy of he sysem requires ha he equilibrium described by he rules in (2) is sable. As i urns ou, he equilibrium will be sable if and only if all eigenvalues of P are smaller han uniy in absolue value. The soluion process requires solving numerically a mari quadraic equaion. The resuls from his process are described in Secion 4. Before hese resuls can be obained, however, we mus parameerize he model. 3.2 Parameerizaion of he Model 3.2. Seady sae in he model and sochasic rend in he daa One mus eercise cauion when parameerizing a model ha requires log-linearizaion around he seady sae of he economy. In he parameerizaion of dynamic models, researchers ofen use momens calculaed wih hisorical daa. In he models calibraed o he U.S. daa, parameer values are ofen so se ha he models seady saes mach he average poswar behavior of he U.S. economy. Thus, King e al. (988), and Baer and Crucini (995), for eample, assign he value for β, he discoun facor, o yield he seady sae annual real ineres rae of 6.5 percen, which is he average real reurn o equiy from The use of long hisorical momens o characerize economy s seady-sae and hen viewing he behavior of aggregaes as deviaions from heir seadysae value implies a seady sae ha has remained unchanged for a long period, even in he face of consan changes in ases and echnology, and in he srucure of he economy. The average real rae

16 5 Goods Producing Indusries Goods Indusries Services Indusries Services Producing Indusries Year 58 Figure Srucural Changes in Canadian Economy, Conribuions as a percenage of GDP of Goods and Services producing indusries of reurn on long-erm governmen bonds in Canada was 6.2 percen for he period The average is 7.3 percen for he period , and 9.8 percen for Figure plos percenage conribuions of goods and services producing indusries o Canadian GDP for he period Beween he period , he conribuion of services producing indusries o GDP increased by eigh percen while he conribuion of he goods producing indusries declined by he same magniude. Noe (see equaions in C.6 in Appendi C) ha seady-sae equilibrium requires βr γ η. Evidenly, wih given preference parameers β and η, seady-sae equilibrium canno be consisen simulaneously wih he hree hisorical averages of he real rae of ineres given above. All his suggess ha resuls based on parameerizaion of dynamic models using momens from long hisorical daa may be unreliable, especially when he model requires loglinearizaion around he seady-sae of he economy. In he parameerizaion of our model, we use annual daa for he period , for which dis-aggregaed secoral daa on oupu and employmen are available, in order o calculae produciviy shocks. 33 Since many macroeconomic ime series possess sochasic rends, he researcher mus remove such rends from he daa before meaningful momens can be compued. The real business cycle modelers almos universally use he linear filer advocaed by Hodrick and Presco (997) o remove he sochasic rend from he daa, and hen compare his filered daa wih 32 All daa, ecep he erms-of-rade series, are from Saisics Canada s CANSIM daabase. Epor and impor price deflaors used for calculaion of he erms-of-rade series are obained from Inernaional Moneary Fund s Inernaional Financial Saisics (IFS). The choice of IFS over CANSIM is mainly for convenience of no having o adjus he daa due o he change in he base year, or disconinuaion of a paricular series. The sandard uni prices of impors and epors repored in he IFS daabase are frequenly used (see, for eample, Sockman and Tesar (995), Mendoza (995)) o calculae erms-of-rade shocks. Appendi D provides complee series codes for all daa used here. 33 Noe ha alhough disaggregaed daa on oupu, capial sock, and invesmen are available for he period since 955, similar employmen daa are available only since 987. While a longer ime period may be desirable, given he above discussion, i may no be paricularly appealing for he parameerizaion of a dynamic model, especially when he model needs log-linearizaion around is seady-sae.

17 he momens produced by he model. 34 The Hodrick-Presco filer (HP-filer) renders saionary any ime series ha is inegraed of order four or less. Technically, he Hodrick-Presco filer compues he smoohed series s of any series y by minimizing he variance of y around s, subjec o a penaly ha consrains he second difference of s: T T 2 2 min y s s s s s + λ + s 2 The penaly parameer λ conrols he smoohness of he series s, such ha he larger he λ, he smooher he series. In he calibraion of he presen model, we firs ake naural logarihms of all he variables, including he forcing variables. Then all he variables, ecep he forcing variables, are de-rended applying he HP-filer wih he smoohing parameer λ, alhough he lieraure sandard is o se λ in case of annual daa. The reasons for our paricular choices are saed below. King and Rebelo (993) emphasize ha he mechanical applicaion of he HP-filer as a derending procedure removes imporan ime series componens ha have radiionally been regarded as represening business cycle phenomena. In paricular, in heir applicaion of he filer o U.S. ime series, he auhors find ha HP filering alers he momen implicaions of he model in a quaniaively imporan manner, and his influence is no consan across series. We use he Solow residuals in he Canadian goods and services producing indusries o calculae he shocks o produciviy in he epor- and he non-raded goods secors, respecively. 35 Augmened Dickey- Fuller uni roo ess applied o hese residuals as well as o he erms-of-rade indicae ha none of he hree forcing variables of our model conains a uni roo over he sample period. 36 I is, herefore, no necessary o pre-filer hese series. Moreover, as Baer (995) noes, i is incorrec o pre-filer variables o obain saionary forcing processes, since he low-frequency and non-saionary componens of he shock processes will generally have implicaions for he behavior of he endogenous variables a all frequencies. The mos common approach o model evaluaion in he real business cycle lieraure has been he comparison of he model s cenral momen predicions wih corresponding momens compued from he daa. We borrow his approach o evaluae he predicions of he presen model. I is, herefore, necessary o remove any non-saionariy presen in he variables of he model. As menioned before, we use he HP-filer wih he smoohing parameer λ. This follows Baer (995), who observes ha higher values of λ in case of annual daa remove oo much of he low- frequency variaion in a series. Table below presens he values of he augmened Dickey-Fuller es saisics applied o he raw (unfilered) daa (all in naural logarihms) as well as o he HPfilered daa wih λ. In he able, all variables, ecep price and ineres rae, are in per-capia erms. In deriving he es saisics presened in Table, we include a consan and a rend in all equaions, ecep in he equaion for he gross rae of ineres, in which case only a consan is 6 34 Hodrick and Presco firs used his filer in a working paper circulaed in he early 98's. The paper was published in 997. King and Rebelo (993) provide a deailed discussion of he Hodrick-Presco filer. 35 Secion 3.3 below describes he measuremen of hese Solow residuals. 36 Calculaed es saisics for he Solow residuals in he epor- and he non-raded goods secors, and for he ermsof-rade, respecively, are -6.64, and -3.94, while he 5% criical value is in all he hree cases.

18 7 Table Augmened Dickey-Fuller Uni-Roo Tess of Variables Variable Unfilered Filered Variable Unfilered Filered Y K h Y h P I F I h C K R Noes: Price (P ) is CPI. The gross rae of ineres (R ) is one plus he real rae of ineres. Filered series are HP-filered wih smoohing parameer λ. The 5% criical value for he es saisic is in all cases, bu he gross rae of ineres, in which case he criical value is Evidenly, HP λ removes nonsaionariy in all cases, ecep for consumpion and he rae of ineres. included. Evidenly, he smoohing parameer λ removes nonsaionariy in all cases, ecep for consumpion and he gross real rae of ineres. One can, however, noe ha he HP-filering increases (in absolue value) he Dickey-Fuller es saisic for consumpion and he real rae of ineres. Given our shor sample period and he generally recognized low power of he uni roo es o rejec a false null hypohesis, we coninue using he smoohing parameer λ, and repor in Table 2 below (see secion 4.3) he sensiiviy of resuls when boh he acual and he simulaed daa are HP-filered wih λ Esimaion and Calibraion of Parameers Recenly some researchers (mos noably, Chrisiano and Eichenbaum (992)) have advocaed he esimaion of he srucural parameers in dynamic macroecomic models using he Generalized Mehod of Momens (GMM). We use he GMM in an aemp o esimae depreciaion and preference parameers. When formal esimaion echniques fail o produce meaningful parameer values, we use he model s resricions o calculae parameers from acual daa. E δ i In esimaing he depreciaion raes, (i, h), we follow Chrisiano and Eichenbaum (992) and impose he following resricion on he laws of moion of capial socks in he wo producion secors: 37 Ki + Ii δ i + Ki Ki (2) 37 This basically re-wries equaion (8) as an epecaional difference equaion.

19 For he epor good secor, he esimae isδ.986, wih a -raio of and a p-value of. for he -raio. The corresponding esimae for he nonraded secor is δ h.79, wih a -raio of 58.9 and a p-value of. for he -raio. 38 The growh rae of labour-augmening echnological change, γ, is usually se equal o he rae of growh of real per-capia oupu. 39 We se γ.95, which is he (gross) rae of growh of real per-capia oupu for he period The value of.35 is borrowed from Mendoza (995) so ha he elasiciy of subsiuion beween raded and nonraded goods, +, equals θ, he share of domesically produced goods in he consumer s bundle of raded goods, is se a.3 in he benchmark model. 4 Wih hese parameer values se, we aemp o esimae he following consumpion Euler equaion by GMM for he parameers η and β : E η c + P η β R c + γ P + (22) Our esimaion of (22) wih annual daa for he period gives some srange resuls ha are no worh reporing. In order o obain some guidance, however, we ry o esimae he equaion using quarerly daa covering he period 96:-998:4. This gives esimaes of β.979 and η wih -raios 2.38 (p-value.) and -.25 (p-value.8), respecively. When esimaed over he shorer period 987:-998:4, using he same quarerly daa, he esimaes of β andη are,.982 and -.336, respecively, wih -raios 4.83 (p-value.) and -.56 (p-value.87), respecively. While he esimae of β using quarerly daa (over boh ime periods) appears reasonable, i is 8 38 We use he J-saisics from he above wo GMM esimaions o es he validiy of over-idenifying resricions. The J-saisic imes he number of regression observaions is asympoically χ 2 wih degrees of freedom equal o he number of over-idenifying resricions. We use five insrumens including GDP and a consan o esimae he depreciaion parameers in boh secors, and hus here are four over-idenifying resricions in each equaion. The esimaes for our χ 2 saisics are.97 and.362 for he parameers in he epor good and he nonraded good secors, respecively, wih p-values of.998 and.85. Thus, he over-idenifying resricions are saisfied in boh cases. 39 See, for eample, Correia e. al (995), Baer and Crucini (995), Harjes (997). 4 Mendoza esimaed by running a regression of relaive ependiures on relaive prices and per-capia GDP, all in logarihms, as suggesed by Sockman and Tesar (995). The esimae is represenaive of he G-7 counries. Alhough we use Mendoza s esimae for he G-7 in our benchmark model, we eamine he model s sensiiviy o variaions in his parameer laer in sub-secion Alhough his is done informally, he underlying reason for using his value is as follows. Sockman and Tesar (995) se θ.5 in heir model calibraed o he G-7 daa. Macklem (993) pus eporable consumpion a abou 22 percen of radable consumpion ependiure. This parameerizaion is based on evidence from he final demand ables in 986. To accoun for he possibiliy of an increasing conribuion of he impor-compeing domesic non-resource-based manufacuring secor, we se θ.3 as an approimaion. We, however, eamine he model s sensiiviy o variaions in θ in he range beween θ.25 and θ.4. Ecep for he impac effecs of a erms-of-rade shock on consumpion and he wo prices, no considerable change in he resuls in response o he variaions in θ can be noed. We find ha he higher he values of θ are assigned, he lower are he impac effecs of erms-of-rade shocks on consumpion and he price inde, bu he higher are he effecs on he relaive price of he nonraded good.

20 9 Bo 3 Benchmark Parameers β. 94 η..35 δ. 98 δ h φ. φh.3 γ 955. θ. 3 f. 39 τ. 44 ω. Z Zh y pm ψ. 696 ψ h. 726 ψ m. 469 σ. 86 σ h. 75 σ m. 237 σ h,. 93 σ m,. 74 σ hm,. 625 ϑ 3. ϑ 4. h difficul o defend he esimae of η. 42 Given he naure of he GMM esimaion mehodology, one canno accep eiher esimae wih any confidence. We, herefore, resor o he alernaive calibraion mehodology. Presco (986) cies a number of sudies including Hansen and Singleon (983) ha poin o he conclusion ha η is no oo far from. We se η. in our benchmark model. Wih γ.955, all one needs o obain is he esimae of eiher he parameer β or he seady-sae gross rae of ineres R o saisfy he seady-sae equilibrium condiion βr γ η. We use he guidance from he GMM esimae above, se β 94. in he benchmark model, and le R be deermined residually so as o saisfy he seady-sae equilibrium condiion. 43 The producion funcion parameer as well as oher seady-sae raios are se using acual daa for he period Thus, is se equal o.735, which is average labor income as a fracion of he average ne domesic income a facor cos for he period The seady-sae raio τ is se a.44 using average oupus of he goods and services producing indusries for he period Seady-sae deb as a percenage of GDP is se a.39, which is he average foreign deb as a percenage of GDP for above-menioned period. We se he value of ω, he elasiciy of he real ineres rae wih respec o foreign deb, equal o. in he benchmark model, and hen eamine he sensiiviies of resuls o variaions in he values of his parameer. Wih all oher parameer values se, we assign he values φ. and φ h 3. for he capial adjusmen cos parameers in he epor good and he nonraded good producion secors of he economy, respecively, so as o reproduce he observed volailiies of he invesmen series as closely as possible. 45 Bo 3 summarizes our calibraed preference and echnology parameers, various seady-sae 42 Noe ha a negaive value for η implies a negaive ineremporal elasiciy of subsiuion of consumpion! Of course, based on he -raio, one will no be inclined o accep he esimae for η. Bu, ineresingly, he model s over- idenifying resricions are saisfied in ha he J-saisic-based χ 2 value is.3 wih a p-value of.998 (for esimaion over he shorer period). 43 Noe ha R is endogenous in he presen model. We, herefore, do no se i arbirarily a some hisorical mean rae of reurn o equiy or some oher arbirarily chosen asse. In choosing he value of he parameer β, however, we noe ha, given oher parameer values and he mean gross rae of ineres (.88) on long-erm governmen bonds for he period , β is a reasonable approimaion. 44 The raios τ and τ are defined in (C.5) in Appendi C. 45 This follows he approach adoped by Cardia (99).

21 raios and facor shares, and he parameers associaed wih he shock processes. In he bo, ψ i andσ i are, respecively, he esimaed auocorrelaion coefficien and he sandard deviaion of he ih shock process, and σ ij is he correlaion coefficien beween he ih and he jh shock processes, for i, j shocks o produciviy in he epor and non-raded good secors, and o he erms-of-rade Esimaion of Shocks o Produciviy and Terms of Trade In he real business cycle lieraure, produciviy shocks are frequenly measured from he Solow residuals. 46 These residuals are measured as he difference beween changes in oupu and he changes in measured inpus (labor and capial) imes heir shares. We follow he same approach here. Using our symbols, he produciviy changes in he epor- and he non-raded goods secors are given by: 47 ln Z lny ln K ln N lnγ (23) ln Z lny ln K ln N lnγ h h h h One ecepion o he definiions provided earlier mus be noed here. In (23), Y i, K i, and N i are, respecively, aggregae oupu, capial sock, and employmen in he ih secor. We use employmen and oupu in he goods and services producing indusries for he employmen and oupu of he raded and non-raded secors, respecively, of he model. However, such aggregaed daa on capial socks in he goods and services producing indusries are no available, alhough disaggregaed daa by indusry caegories are available. We obain he capial sock in he nonraded secor by aggregaing he capial socks of he service-oriened indusries. 48 The capial sock in he epor good secor is hen aken o be he residual from he oal capial sock in he economy. Our model s shock processes are characerized fully by he auo-correlaion coefficiens and sandard deviaions of he erms-of-rade and he Solow residuals described above. These saisics are repored in Bo 3 along wih all oher benchmark parameers for he calibraion of he model. A his poin, i may be useful o remind he reader ha we are working wih an approimae soluion of a highly nonlinear model. The soluion is obained by log-linearizing he firs-order condiions of he model in is seady-sae equilibrium. 49 All variables in he log-linearized sysem are hus epressed as log-deviaions from heir seady-sae values. The linearized model is parameerized and calibraed o Canadian daa for he period Thus, an essenial implici assumpion in he calibraion process of he model is ha he model economy is in is seady-sae 46 In virually all he papers menioned so far ha consider produciviy shocks, hese shocks have been measured from he Solow residuals. Baer and Crucini (995) poin ou some poenial problems when employmen raher han oal hours worked are used for labor inpu. However, due o he pauciy of reliable daa on hours worked, researchers ofen end up using employmen in place of oal hours worked. In he end, Baer and Crucini (995) also use employmen for labour inpu. 47 The las erm arises from specificaion of he growh in labor-augmening echnology: X γ X, wih he assumpion ha X. 48 The indusries considered o be service-oriened are: consrucion, ransporaion, and sorage, communicaion and oher uiliies indusries, wholesale and reail rade, finance and insurance, healh and social service, governmen, business, and oher service. Deailed CANSIM series codes are provided in Appendi D. 49 The seady-sae equilibrium condiions used in he process are presened in (C.7) in he Appendi C. 2