COLLATERAL CONSTRAINTS AND MACROECONOMIC ADJUSTMENT IN AN OPEN ECONOMY. Philip L. Brock

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1 January 29 COLLATERAL CONSTRAINTS AND MACROECONOMIC ADJUSTMENT IN AN OPEN ECONOMY Phlp L. Broc Deparmen of Economcs Unversy of Washngon Seale, Washngon Absrac Ths paper analyzes a small open economy Ramsey growh model wh convex nvesmen coss and a collaeral consran on borrowng. Opmal conrol mehods are used o characerze he dynamcs of nvesmen, consumpon, and deb. The analyss demonsraes ha he economy s adjusmen speed depends on he fracon of he capal soc ha can be used as collaeral. In he presence of non-convexes, a hgher loan-o-value of he capal soc may produce a bfurcaon n he dynamcs by ncreasng he economy s adjusmen speed. In conras o he canoncal small open economy model wh convex nvesmen coss, domesc and foregn savngs are growh-rae complemens due o he neracon beween domesc savngs, he prce of capal, and he borrowng consran. The sandard closed economy Ramsey model, he Cohen-Sachs deb repudaon model, and he canoncal small open economy model wh adjusmen coss are shown o be specal cases of he analyss.

2 1 I. Inroducon Durng he lae 197s and early 198s economss began a concered effor o develop opmzng neremporal models of open economes 1. As par of hs push varous aemps were made o open he Ramsey growh model o capal flows. Smply openng he sandard Ramsey model produces nsananeous adjusmen of he capal soc, snce domesc savng s no longer a consran on capal formaon. In order o slow down he adjusmen of he capal soc wo prncpal approaches were aen. The frs opened he Uzawa (1961, 1964) wosecor growh model and made he nvesmen goods secor nonradable. 2 The second approach ncorporaed convex nvesmen coss no he basc Ramsey model so ha he cos of nvesmen ncreased wh he rae of nvesmen. Boh approaches slowed he speed of capal soc adjusmen and creaed a relave prce of nvesmen goods o consumpon goods ha could be nerpreed as a real exchange rae. The adopon of he second approach was nally slowed by he dffculy of negrang he mcroeconomcs leraure on frm adjusmen coss, such as Lucas and Presco (1973), no a Ramsey model wh a represenave prce-ang agen. Hayash (1982) showed ha lnearly homogenous convex adjusmen coss were requred for he specfcaon of a Ramsey model wh prce-ang frms. 3 Hayash s paper proved o be nsrumenal n he developmen of he open economy Ramsey model. In shor order papers by Blanchard (1983), Lpon and Sachs (1983), and Gavazz and Wyplosz (1984) ncorporaed lnearly 1 See, e.g., Sachs 1981 and Svensson and Razn 1983 for early models. 2 Papers layng he groundwor for hs approach nclude Fscher and Frenel (1972) and Bruno (1976). See Murphy (1986), Broc (1988), and Obsfeld (1989) for early examples of explcly neremporal opmzng open economy models wh nonraded nvesmen 3 Whou a lnearly homogeneous nsallaon cos funcon, frms are prce maers ha generae monopoly rens. Hayash (1982) characerzed he problem as one of deermnng he assumpons necessary o mae he margnal q valuaon of he capal soc he same as he Tobn s q average value of he capal soc.

3 2 homogeneous nvesmen coss no open economy versons of he Ramsey model, and many oher papers soon followed. 4 By he 199s he open economy Ramsey model wh convex nvesmen coss had been ncorporaed no graduae exboos and became one of he canoncal models of nernaonal macroeconomcs. 5 Neverheless, here has remaned he queson of how o close hs model wh a borrowng consran. Any nfne-horzon Ramsey model mus mpose a no-ponzgame condon on he represenave agen, whch n he presence of complee mares s equvalen o a solvency condon requrng ha he value of deb canno exceed he presen value of he economy s ne capal ncome plus wage ncome. 6 The solvency condon generaes an upper bound on deb and a lower bound of zero on consumpon. For sandard uly funcons ha mply posve consumpon (e.g., ones ha sasfy he Inada condons), assumpons n addon o he solvency condon mus be ncorporaed o solve for he model s seady sae equlbrum. As s well nown, f boh he rae of me preference ( ρ ) and he world neres rae (r) are aen as consans, hen he exsence of a seady sae for he small open economy model requres, n addon o he solvency condon, ha ρ = r. Ths n urn gves rse o a zero roo n he dynamcs, so ha he seady sae depends on nal deb and consumpon equals 4 See, e.g., Masuyama (1987), Broc (1988), and Turnovsy and Sen (1991). 5 See, e.g., Blanchard and Fscher (1989), Obsfeld and Rogoff (1995), Turnovsy (1997), and Barro and Saleh-- Marn (24). Convex nvesmen coss also have an ncreasngly mporan role n dynamc sochasc general equlbrum models such as Chrsano, Echenbaum and Evans (25) and Adolfson e. al (27). 6 See Levne and Zame (1996). Ths condon can be expressed algebracally as r( s ) b q + w( s) e ds where b s he soc of deb, q s he value of he capal soc, w ( ) s he real wage, and r s he world neres rae. Wh convex, lnearly homogenous nvesmen coss, he value of he capal soc equals he presen dscouned value of fuure renals on capal ne of adjusmen coss.

4 3 permanen ncome. 7 For many purposes modelers may wsh o have seady saes ha are nvaran o nal levels of deb. One alernave way o close he model s o pn down longrun consumpon ( c ) by assumng ha he rae of me preference s ncreasng n consumpon (e.g. Obsfeld 1981). A second alernave s o specfy he borrowng rae as an ncreasng funcon of he amoun borrowed where he seady-sae level of deb (b ) s assumed o sasfy he solvency consran (see, e.g., Schmd-Grohé and Urbe 23). The assumpon I mae n hs paper s ha deb mus be collaeralzed by a fracon a of he value of he capal soc. Curren and fuure labor ncome canno be pledged agans repaymen of a loan. 8 Evans and Jovanovc (1989) were among he frs o use a collaeral requremen on capal n an opmzng model. 9 Cohen and Sachs (1986) were he frs o nroduce a collaeral consran n an open economy Ramsey model wh convex nvesmen coss, bu her model proved o be analycally racable only for lnear producon funcons. Mendoza (28) has recenly used a collaeral consran, bu focuses on wheher or no he consran s 7 Three papers n he early 198s ha developed echnques whch permed he analycal characerzaon of open economy models wh zero roos are Blanchard and Kahn (198), Buer (1984), and Gavazz and Wyplosz (1985). 8 The four alernave closure assumpons generae he followng alernave seady-sae borrowng consrans, where ρ s he rae of me preference, r s he world neres rae, and b,, and c are seady-sae values of deb, capal, and consumpon:. ρ = r w ( ) cb ( ) b = + r. ρ( c) = r w ( ) c( r) b = + r. ρ = r( b% ) w ( ) c b ( ρ) = + r v. b aq b a a 1 Gven he seady-sae capal soc ( ), he frs closure assumpon deermnes he seady-sae level of deb as a funcon of he nal soc of deb ( b ). The second assumpon deermnes seady-sae deb by pnnng down seady-sae consumpon va he exogenous world neres rae, whle he hrd assumpon pns down seady-sae deb by equang he exogenous rae of me preference wh he endogenous deb-deermned neres rae. If a = 1 and he collaeral consran bnds, consumpon wll equal he real wage n equlbrum. 9 The paper by Ayagar and Gerler (1999) s also an early closed-economy model ha employs a collaeral consran of he ype used n hs paper.

5 4 bndng raher han on he effec of fraconal changes n he collaeral requremen when he consran s bndng. Barro, Manw and Sala--Marn (1995) and Lane (21) modfy he Cohen-Sachs model by replacng convex nvesmen coss wh an addonal facor of producon ha requres nonraded nvesmen. Caballero and Krshnamurhy (21), Devereux and Poon (24), Char, Kehoe, and McGraan (25), and Braggon, Chrsano, and Roldós (27) also employ collaeral consrans, bu le Mendoza (28) resrc her analyses o he change n model dynamcs when here s a swch from a bndng o a nonbndng consran. Several papers movae he collaeral consran by he ably of foregn debors o seze he capal soc n he case of defaul (see, e.g., Barro e. al. 1995, Lane 21, and Char e. al. 25). In hs paper, le ha of Cohen and Sachs, nernaonal lenders canno seze he capal soc due o adjusmen coss ha fx capal n he shor run, so he borrowng consran refers o he amoun of presen and fuure capal ncome ha can be pledged agans he deb. Ths lm on pledgeable ncome could reflec he ably of foregn credors o mpose a penaly equal o a fracon of he economy s oupu n he case of defaul, as n Cohen and Sachs, or could reflec he nably of agens o pledge wage ncome because of he nalenably of human capal as n Kyoa and Moore (1997). 1 Followng Masuyama (28) and Mendoza (28), I assume ha he collaeral requremen capures he oucome of unmodeled agency problems or legal resrcons ha preven he represenave agen from pledgng more han a fracon of capal ncome when ncurrng deb. 1 There are, of course, oher reasons relaed o adverse selecon and moral hazard. Smlarly, Holmsrom and Trole (1998) pos a prvae benef ha canno be pledge by an enrepreneur o credors.

6 5 2. The Model The open economy model wh convex nvesmen coss s well-nown and s exposed by a number of graduae-level exboos, such as Blanchard and Fscher (1989, 58-69) and Turnovsy (1997, 57-77). The canoncal form of he model s he followng, where f ( ) s a neoclasscal producon funcon, (, ) represens convex nvesmen coss, b s deb, r s he world neres rae, ρ s he rae of me preference, and uc () s a concave uly funcon: ρ max u( c) e d subjec o, b and c, b& = c + (, ) + rb f( ) & = δ (1) Wh he rae of me preference se equal o he neres rae, he model s separable n producon and consumpon decsons. Wh a lnearly homogenous nvesmen coss, he value of he capal soc (q) s equal o he sum of he dscouned value of renal ncome from capal ne of nvesmen coss: r( s ) [ ( ) (, )] q = f e ds s s s s (2) Consumpon s consan and s equal o permanen ncome: r( s ) r( s ) [ ( ) (, )] ( o ) ( ) s c= rb + r f e ds= r q b + r w e d (3) Consumpon s consan because he agen can use he world capal mare o creae an annuy from curren deb and fuure ncome.

7 6 In hs paper he agen has more lmed access o he world capal mare. Ths lmed access aes he form of a borrowng consran: b aq (4) The agen can pledge only a fracon a of he capal soc as collaeral and no par of wage ncome. The capal soc canno lerally serve as collaeral snce adjusmen coss preven s drec pledgng o credors. Snce he value of he capal soc equals he presen value of capal ncome ne of nvesmen coss, he beer nerpreaon of he borrowng consran s ha a mos a fracon a of presen and fuure ne capal ncome can be pledged o lenders. Ths secon loos a he effec of he consran when all ne capal ncome can be pledged ( a = 1). Secons 5 and 6 wll generalze he model o allow a o ae on fraconal values. The assumpon a = 1 s he assumpon made by Kyoa and Moore (1997). I sgnfes ha capal ncome can be pledged o lenders, bu ha borrowers canno pledge her wage ncome. The borrowng consran b q s hen added o he maxmzaon problem gven by (1). The presen value Hamlonan assocaed wh he maxmzaon problem) s: [ ] ρ e H = u() c + λ f() rb c (, ) + q*( δ) + γ *( q b ) (5) where λ s he shadow value of deb, q * s he shadow value of nsalled capal, and γ * s he shadow value aached o he borrowng consran. The frs-order necessary condons are: u () c = λ (6) (, ) = q (7) where q = q* λ s he mare value of capal. The Euler equaon for consumpon s:

8 7 c c& = ( r+ γ ρ) (8) σ where σ = cu () c u () c and γ = γ * λ. The evoluon of he prce of capal s gven by q& = f ( ) + (, ) + ( r+ δ ) q (9) The sae equaon for capal s derved from he frs order condon (, ) = q(equaon 7): & = ( q, ) δ (1) These laer wo equaons are dencal o he canoncal model. The consumpon equaon dffers from he canoncal open economy Ramsey model by he erm γ n he consumpon Euler equaon. Thus, when a =1 he borrowng consran of hs model alers he Euler equaon whle leavng he dynamcs of capal soc adjusmen unchanged from he canoncal model. Capal accumulaon wh he borrowng consran speed of adjusmen as he canoncal model: a = 1 s governed by he same e µ 1 = + ( ) where µ 1,2 2 r r f ( ) = m (11) 2 2 are he characersc roos of equaons (9) and (1). The adjusmen dynamcs are he same for he canoncal and borrowng-consraned economy because wh a = 1 borrowng for nvesmen s compleely collaeralzed by fuure capal ncome. Wh a bndng consran (b=q) he curren accoun equaon (1) mples ha consumpon n he borrowng-consraned economy s equal o he real wage: b& = c+ (, ) + rq f( ) = q & + q& c= f( ) f '( ) = w( ) (12)

9 8 The borrowng consran wll bnd n equlbrum f γ = ρ r > where γ s he shadow value of he borrowng consran n seady-sae equlbrum. As dsnc from he canoncal model, where ρ = r s requred for a saonary seady sae, he borrowng consraned model wll have a saonary seady sae provded ha r ρ. In essence, when he auarchc neres rae ( ρ ) exceeds he world neres rae (r), mposng a borrowng consran s an alernave assumpon o mposng equaly beween he world neres rae and he rae of me preference n he seady sae, ncludng he assumpons ha he neres rae s an ncreasng funcon of he soc of deb or ha he rae of me preference s a decreasng funcon of consumpon. In summary, wh a bndng collaeral consran b =, he capal soc s fnanced q by borrowng from foregners. Agens canno borrow agans fuure wage ncome so ha consumpon s equal o he real wage raher han o a consan value as n he canoncal model. 3. Nonbndng borrowng consran The borrowng consran may be non-bndng ( b < ) under several condons. Frs, f he q nal capal soc ( ) s suffcenly large, he agen can borrow agans o perfecly smooh consumpon, assumng ha ρ = r. Second, f he nal capal soc s greaer han he long-run capal soc, labor ncome wll declne over me. In hs case he agen wll accumulae foregn asses o smooh consumpon so ha he borrowng consran wll no bnd. Once agan, a saonary seady sae wll exs provded ha ρ = r. In hese wo cases, consumpon wll equal permanen ncome and he model s dynamcs wll be he same as hose of he canoncal model.

10 9 In addon, here s a hrd possbly. The nal capal soc may no be large enough o allow he agen o borrow agans o perfecly smooh consumpon, bu may be large enough o allow he agen o emporarly smooh consumpon by borrowng agans. In hs case, he dynamcs are descrbed by a emporary perod n whch consumpon exceeds wage ncome before he borrowng consran becomes bndng and consumpon equals wage ncome. 11 The canoncal model s descrbed by four dynamc equaons n, q, b, and c. 12 The frs wo egenvalues, whch correspond o capal and he prce of capal, are gven by equaon (11). Deb accumulaon and consumpon dynamcs are governed by he hrd and fourh egenvalues, µ =, µ = r 3 4 Durng he emporary perod n whch he borrowng consran s no bndng, he rajecory of consumpon wll be governed by he hrd and fourh egenvalues as follows: c = c + Ae + Ae µ 3 µ = c + A + Ae 1 2 r where and A are consans ha are deermned by he model. If he borrowng consran A1 2 s never bndng, adjusmen s saddlepah so ha c = c + Ae µ See Turnovsy (1997), pp for a dscusson and several mahemacal examples of hs nd of emporary dynamcs. 12 The lnearzed sysem (wh ρ = r ) can be wren as: q& r f ( ) q q & 1 = b& 1 f ( ) r 1 b b c c c &

11 1 whch, wh ρ = r mples ha A 1 =. Therefore, durng he perod of a emporarly nonbndng budge consran, consumpon evolves accordng o: c = c + Ae 2 r Snce he nal seady-sae (borrowng-consraned) consumpon s gven by c = w( ), hen A = c w( ). 2 Ths mples ha durng he emporary perod n whch b< q, consumpon wll exceed wage ncome w ( ) due o he agen s ably o borrow agans he nal capal soc: c c = w( ) + ( c w( )) e r (12) mplyng ha he growh rae of consumpon durng he perod before he borrowng consran becomes bndng s: c& c r = w ( ) 1+ c w( ) r e (13) Throughou he ranson, he growh rae of consumpon s connuous and, wh ρ = r, s gven from equaon (8) by: c& c γ =. σ Durng he perod afer he consran becomes bndng, consumpon equals he real wage. Leng T denoe he me a whch he borrowng consran becomes bndng ( c = w( T T ), hen c& w ( ) & & = = α c w( ) T T T T T T T T T, (14)

12 11 where he laer equaly assumes a neoclasscal producon funcon y = α. Evaluang equaons (11) and (12) a me T and equang he growh rae of consumpon from equaon (13) wh he growh rae of he real wage from equaon (14) a me T gves hree equaons ha deermne, c, and T as funcons of he nal capal soc : T e µ 1T T = + ( ) (15) [ ] rt c = w( ) + w( ) w( ) e (16) T T w ( T ) w ( ) αµ 1 = r T w( T) (17) where µ 1 s he negave egenvalue from equaon (11). Gven he nal capal soc ( ), equaon (17) deermnes T, he sze of he capal soc when he borrowng consran becomes bndng. Gven T, equaon (15) deermnes T, he me a whch he consran becomes bndng. Gven T and T, equaon (16) hen deermnes he nal level of consumpon ( c ). In parcular, can be shown ha T T > c > > < <, =, = (18) The ambguous effec of an ncrease n he nal capal soc on he me a whch he borrowng consran becomes bndng s due o wo offseng facors. A hgher nal capal soc rases he me T capal soc (equaon 17). 13 Ths ncrease n rases T (equaon 15) T whle he ncrease n lowers T (also equaon 15). Smlarly, wh an ncreased and T he wealh effec assocaed wh a larger nal capal soc (va a hgher wage) may be offse by T 13 Dfferenaon of equaon (13) demonsraes ha dt d > for a sandard neoclasscal producon funcon.

13 12 a ranson pah of greaer duraon ha ends o lower he nal level of consumpon c (equaon 16). 4. Closed Capal Accoun A closed economy verson of hs model was analyzed by Abel and Blanchard (1983), bu hey d d no characerze he shor-run dynamcs or adjusmen speed of he economy. Ths secon does ha o provde a comparson o he economy wh he capal accoun opened. In a closed economy he ne soc of deb s zero so ha f ( ) = c+ (, ) (19) Mang use of he lnear homogeney of he nvesmen funcon, he growh rae of he capal soc s: f( ) c & = δ (2) The presen value Hamlonan s ρ f( ) c e H = u() c + λ δ (21) By mang use of he frs order condon ( u () c = λ ) and he co-sae equaon & λ f '( ) = ( ρ + δ ) λ (22) he growh rae of consumpon can be expressed as (see Appendx 1): f '( ) ( ρ+ δ) c& = (23) σ + c

14 13 Lnearzng equaons (2) and (23) around he seady sae (wh = 1 and = ) gves σ c& f ( ) + & c = 1 ρ (24) wh correspondng egenvalues gven by: µ 1,2 ρ ρ f ( ) = ± 2 2 σ + c 2 (25) Comparng equaons (11) and (25), he adjusmen speed of he economy s slower when he capal accoun s closed because of he concave uly funcon (whose curvaure s measured byσ c ). Wh an open capal accoun, he adjusmen speed s only lmed by he convex nvesmen coss and becomes nfne as adjusmen coss become lnear ( = ). Fgure 1 shows he consumpon pahs ha correspond o he closed economy, he borrowng-consraned economy (ncludng he ransonally unconsraned one), and he canon cal open economy wh perfec consumpon smoohng. Seady-sae consumpon n he borrowng-consraned economy s less han seady-sae consumpon n he closed economy by he amoun of ne capal ncome, whch s used o mae neres paymens on he deb. 5. Parally Pledgeable Capal Income The analyss of he prevous secons can be generalzed o nclude parally pledgeable ncome: b aq a 1 ( 4 )

15 14 When a = he economy s closed o capal flows and when a = 1 capal ncome s fully pledgeable. In he nermedae cases, he agen can borrow agans a fracon of fuure capal ncome. Ths could be he resul of lmed capably of foregn lenders o punsh he agen n he case of defaul (.e., he Cohen-Sachs argumen), or could arse from moral hazard and legal resrcons ha preven credors from sezng more han a proporon of he agen s collaeral, as emphaszed by Holmsrom and Trole (1998). The opmzaon problem for he represenave agen can be expressed as he followng presen-value Hamlonan: [ ] ρ e H = u() c λ c + (, ) + rb f () + q*( δ) + γ *( aq b) (26) Frs-order necessary condons are: u'( c) = λ (, ) = q whereq= q* λ The co-sae equaons are: & λ λ = σ cc & = ρ r γ (27) [ δ ] q& = f '( ) + (, ) + r+ + (1 a) γ ) q (28) Equaon (28) s he connuous-me analogue of he asse-prcng equaon used by Cohen and Sachs o characerze he borrowng consran wh a lnear producon funcon ( f ( ) = A ). 14 Cohen and Sachs do no provde a soluon for a more general producon 14 Cohen and Sachs wre he borrowng consran as * where he paper solves fo * b h, r h as a funcon of * model parameers. Nong ha along a consraned growh pah where b = h, q& = and cc= & κ (where κ s * a consan) and normalzng =, hen h = aq = aa [(1 a)( ρ r+ σκ ) + r+ δ ]. The cred consran s ncreasng n oal facor producvy A and he pledgeable ncome rao a, and decreasng n boh he rae of me preference and he neres rae (gven he assumpon ρ > r ). These are he resuls of * Cohen and Sachs (see Table 1 and equaon A3.13 of her paper). The endo geney of h n her model reflecs boh he pledgeable ncome rao and he prce of capal (q) ha corresponds o alernave seady-sae growh raes.

16 15 funcon, nong ha once he lnear producon echnology s abandoned, an analycal soluon appears ou of reach. In hs paper s connuous me seng, he more general soluon can derved for a neoclasscal producon funcon f( ) ( f ( ), f ( ) < ). 15 In he general case of parally pledgeable capal ncome (b = aq ), he adjusmen speed of he economy around a seady sae s he followng (see Appendx 1 for he dervaon): ρ ρ f "( ) µ 1 = 2 2 σ + c 2 2 (1 a) (29) As nuon would sugges, fundng nvesmen wh a larger proporon of foregn loans rases he speed of capal accumulaon, snce foregn fundng s suppled perfecly elascally. The dependence of he rae of growh on he loan-o-value rao has also been noed by Cohen and Sachs (1986) and Aghon, Banerjee, and Pey (1999) n AK models, as well as by Aghon, How, and Mayer-Foules (25) n a Schumpeeran growh model. A novel resul shown by equaon (29) s ha he speed of adjusmen depends on he quadrac of he agen s ne worh-o-value consran (1 a ). The effec of an ncrease n he loan-o-value rao (a) on he adjusmen speed s greae s a auarchy. Increasng he loan-ovalue rao has boh he drec effec of relaxng he borrowng consran (b = aq ) as well as he ndrec effec of rasng he valuaon of he capal soc (( b q)( q a). Ths second effec can be vewed as a mulpler ha depends on he elascy of nsallaon coss ( q= (, ) ) wh respec o he borrowng consran, so ha he combned effec s: 15 An advanage of hs paper s connuous me framewor s he ably o express he maxmzaon problem as an opmal conrol problem ha can be solved usng Hamlonan dynamcs, whereas Cohen and Sachs wored n dscree me usng dynamc programmng ha generaed a sngle nonlnear paral dfferenal equaon.

17 16 db a q = q 1+ da q a. (3) The model wh paral collaeralzaon of he capal soc encompasses four specal cases. The sandard Ramsey growh model ( a =, = ) and he Abel-Blanchard (1983) model ( a =, > ) are lmng cases wh a closed capal accoun ( a = ). Wh an open accoun, he lmng cases nclude boh he canoncal open economy Ramsey model wh perfec consumpon smoohng ( a = 1, > ) and he model of nsananeous adjusmen of he capal soc ( a = 1, = ). 6. Nonconvexes Openng he economy rases he adjusmen speed of he economy. In he presence of nonconvexes, h e dfference n adjusmen speeds can mae a crcal dfference n he adjusmen dynamcs. Ths secon assumes ha he producon funcon s characerzed by nonconvexes arsng from an exernal effec arsng from publc nowledge (). Ths nowledge s a byproduc of capal accumulaon and s aen as gven by he agen as n Romer (1986). Greaer nowledge resuls n a more durable capal soc so ha he producon funcon can be wren as F (,) = f( ) δ (), δ (). The deprecaon funcon δ () s convex-concave, so ha he rae of mprovemen n he durably of he 16 capal soc nally ncreases before decreasng. Ths allows he possbly of hree 16 I s more common o nroduce he nonconvexy n he producon funcon f(,), as n Romer (1986). The analycal re suls of hs secon would be he same wh ha assumpon, bu he expresson for he egenvalues (equaon 31) would no be as clean.

18 17 equlbra (, ˆ, ), as shown n Fgure 2, where he vercal dsance beween he f ( ) and δ () curves equals ρ a he equlbra. A and, f ( ) δ () < so ha he equlbra are saddlepons. A ˆ, f ( ) δ () > so ha he Hamlonan s convex and he deermnan of he Jacoban s posve, ndcang ha he equlbrum s unsable: µ 1,2 2 ρ δ '() ρ δ '() f ( ) δ '() = 2 m 2 2 σ (31) (1 a) + c The equlbrum a ˆ wll be a node f boh egenvalues are posve and a focus f boh are complex. 17 As shown n equaon (31), a (he rao of pledgeable ncome o deb) s a bfurcang varable. Paral openng of he economy o borrowng ( a = a > ) wll ncrease he sze of he second erm under he radcal. If he equlbrum n auarchy ( a = ) s a node, hen for some posve value of he loan-o-value raon ( a 1) here wll be a bfurcaon n he dynamcs for small enough, a whch value he characersc roos wll swch from beng posve o complex. Fgure 3 graphs he adjusmen pahs of consumpon and he capal soc. Pon he unsable equlbrum. When he capal accoun s closed ( a = ), I assume ha he e 1 s dynamcs are nodal, so ha an nal endowmen of capal > ˆw ll lead o an accumulaon of capal along he dashed pah leadng o pon e 2, whle an nal endowmen of capal < ˆ wll produce a reducon n he capal s oc along he dashed pah leadng o e3. Fgure 17 See Harl e. al. (24) for he use of local properes of he Hamlonan and Jacoban o deermne he sably properes of nvesmen models wh nonconvexes.

19 18 3 also graphs he adjusmen pah assocaed wh focal dynamcs for a large enough loan-o- value rao. Any nal endowmen wll allow he agen o accumulae capal on he % sold adjusmen pah leadng oe 4. Thus, an ncreased loan-o-value rao, by ncreasng he speed of adjusmen, allows he agen o aan he larger long-run capal soc from a lower nal capal soc han wh he nodal dynamcs. An mporan aspec of he adjusmen process nvolves he role of deb. I s common o descrbe nodal dynamcs as beng deermned by hsory (.e., he nal endowmen of capal) whle focal dynamcs are also deermned by expecaons n he spral regon. In Fgure 3 he regon o he rgh of he spral ( > % ) s characerzed by saddlepah dynamcs for any nal endowmen of capal. In he area of he spral ( % ), here are mulple % consumpon choces (and correspondng nvesmen choces) ha are conssen wh saddlepah adjusmen and he ransversaly condon a nfny. 19 In mos models here s nohng n he agen s opmzng behavor ha pns down he adjusmen pah n hs regon. Hence, he dea ha expecaons are crcal o he dynamc behavor of he economy. Wh a collaeral consran, however, he nal soc of bonds ( b ) s an addonal pre-deermned sae varable. Wh an nal collaeral consran gven by b aq, he prce of capal wll be unquely deermned f he consran s bndng. A jump 18 from a lower level of consumpon n he spral regon o a hgher level of consumpon, for example, would reduce nvesmen and lower q, hereby volang he collaeral consran. 18 See Krugman (1991) and Masuyama (1991) for hs nerpreaon. Sba (1978) s he orgnal reference on he dynamcs assocaed wh he Ramsey model wh a convex-concave producon funcon. 19 For example, hgh consumpon and low nvesmen ae place along he op of he spral, whle low consumpon and hgh nvesmen characerze he lower par of he spral. Fgure 3 draws he ouer boundary of he adjusmen pah ha sprals no pon e 1.

20 19 Wh a bndng borrowng consran, herefore, he poenal ndeermnacy of adjusmen pahs n he presence of ncreasng reurns s elmnaed. Hrshlefer (1958), McKnnon (1973), and Krugman (1979) have used a wo-perod Fsheran nvesmen analyss o llusrae he dependence of mulple equlbra on he dscoun rae n he presence of ncreasng reurns o scale. Smlarly, he presence of a node or focus a % also depends on he dscoun rae va s mpac on he speed of adjusmen. To llusrae he mporance of he speed of adjusmen n he presence of nonconvexes, assume (as n Hrshlefer, McKnnon, and Krugman) ha here s a radonal producon echnology ha employs a consan reurns producon echnology over he relevan range (shown by he doed curve gong hrough pon e n Fgure 3). The economy s nally n long-run equlbrum a pon e where f ( ) = ρ + δ. There s also a second echnology characerzed by ncreasng reurns o scale arsng from a declnng deprecaon rae n aggregae capal as oulned above. Wh a closed capal accoun and an nal capal soc, adopng he new echnology wll cause an nal upward jump n consumpon from 2 pon e o he dashed adjusmen pah and a declne n savngs as he economy s capal soc decln es o. Wh an open capal accoun and a suffcenly large fracon of pledgeable capal ncome (so ha a s large enough o produce a bfurcaon n he dynamcs wh a large enough spral) consumpon wll nally fall and he economy wll move along he sold pah oward he long-run equlbrum e 4. 2 The agen may wsh o reduce he capal soc o he long run equlbrum e 3. Unle he closed economy, n whch he neres rae endogenously falls wh he declne n he capal soc along he dashed adjusmen pah, he borrowng consraned open economy can sell capal on he world mare for bonds ha pay he world neres rae. As dscussed n Secon 2, he dynamcs of adjusmen wh a declnng capal soc wll be he same as n he canoncal open economy Ramsey model wh adjusmen coss.

21 2 The ncreased speed of adjusmen requres a hgher savngs rae and an nal reducon n consumpon. An ncrease n he domesc savngs rae rases nvesmen and he prce of capal ( q= (, ) ), hereby relaxng he borrowng consran b= aq and ncreasng access o foregn capal. 21 Unle he wo-perod Fsheran analyss, he concavy of he uly funcon and he convexy of he nvesmen funcon also maer for adjusmen n he presence of scale economes because hey affec he speed of adjusmen. Aghon, Banerjee, and Pey (1999) and Aghon, Bacchea, and Banerjee (24) have also consruced models n whch changes n he loan-o-value rao can aler he adjusmen dynamcs. However, her models employ eher lnear (A) or Leonef producon funcons and rely on cobweb dynamcs creaed by prce movemens arsng from he neracon beween savers and enrepreneurs. 22 In addon, hey characerze unsable equlbra as ndcave of negave consequences of capal mare lberalzaon n developng counres. 23 In hs paper s model, n conras, non-convexes n producon dscourage he represenave agen from underang nvesmen when borrowng s prohbed ( a = ). A capal mare lberalzaon ha relaxes he collaeral consran creaes focal (spral) dynamcs ha, raher han creang nsably, offer he possbly of successfully reachng he hgher equlbrum capal soc. The hgher convergence speed assocaed wh a larger loan-o-value rao reduces he presen value of losses ha accompany producon n he convex regon of he producon froner ( & = ). 21 We can hn of a paramerc reducon n σ n equaon (31) as capurng an ncreased propensy o save. 22 See also Masuyama (28) for smlar resuls nvolvng he allocaon of cred beween alernave projecs n an overlappng generaons model. Ao, Bengno, and Kyoa (26) also generae hs ype of resul by employng dual domesc and foregn borrowng consrans n a model wh enrepreneurs and worers. 23 For example, Aghon, Bacchea, and Banerjee (24) wre, The dynamc mpac of a lberalzaon predced by he model [nsably] s also conssen wh he experence of several emergng mare counres ha have lberalzed, n parcular n Souheas Asa and Lan Amerca, bu also n some European counres.

22 21 The dynamcs of consumpon and nvesmen n Fgure 3 are no nconssen wh he resuls of several papers. Jappell and Pagano (1994) fnd ha lqudy consrans can ncrease he growh rae by rasng savng and channelng hese funds o frms. Ths paper s explanaon s conssen wh Jappell and Pagano (1994), bu offers he addonal nsgh ha greaer savng rases he collaeral value of capal, hereby permng greaer borrowng. Prasad, Rajan, and Subraman (26) hypohesze ha fas-growng lower ncome counres canno easly access foregn capal (low a) so ha her curren accoun defcs are smaller han n more developed counres (hgh a) wh smlar growh raes. Ths explanaon s n agreemen wh hs paper s model. In he model of Aghon, Comn, and How (26), fnancal resrcons caused by agency problems are allevaed by ncreased domesc savng, hus rasng foregn nvesmen and growh and hereby creang a complemenary beween domesc and foregn savng. Ths complemenary channel s dsnc from he one n hs paper, bu o he exen ha he borrowng consran n hs paper s mean o capure unmodeled agency problems (as n Masuyama 28), hen he ln beween ncreased savng and he collaeral value of he capal soc s no nconssen wh her model. 7. Concluson Ths paper has aen he canoncal small open economy Ramsey model wh adjusmen coss for nvesmen and added a borrowng consran n he radon of Cohen and Sachs (1986) and Kyoa and Moore (1997). The paper shows ha he borrowng consran, whch can be expressed as a collaeral requremen on he value of he capal soc, provdes an aracve alernave o several sandard borrowng consrans ha have been used n he leraure See he useful summary of hese consrans by Schm-Grohé and Urbe (23).

23 22 The model encompasses he sandard closed economy Ramsey model, he Abel-Blanchard closed-economy Ramsey model wh convex nvesmen coss, and he canoncal small open economy model wh convex nvesmen coss as specal cases. The model can be exended n several drecons. A number of papers (e.g., Char, Kehoe, and McGraan 25 and Mendoza 28) refer o collaeral consrans ha depend on he value of he capal soc ( e.g., b aq ) as endogenous consrans, n conras o fxed lms ( e.g., b b ), whch are referred o as exogenous consrans. A useful exenson o hs model would be he nroducon of a cos-of-collaeralzaon funcon ha would mae he loan-o-value (a) a choce varable and hus provde he collaeral consran wh an addonal degree of endogenzaon. Followng he general heme of Barro, Manw and Sala--Marn (1995), anoher exenson of he model could specfy wo ypes of capal one wh a borrowng consran and one whou a consran. Fnally, followng Broc (29) he model can be exended o nclude fscal varables, where he represenave agen and governmen operae wh dsnc borrowng consrans.

24 23 References Abel, A. and O. Blanchard An neremporal model of savng and nvesmen. Economerca 51, Adolfson, M., S. Laséen, J. Lndé, and V. Maas. 27. Bayesan esmaon of an open economy DSGE model wh ncomplee pass-hrough. Journal of Inernaonal Economcs 72, Aghon, P., P. Bacchea, and A. Banerjee. 24. Fnancal developmen and he nsably of open economes. Journal of Moneary Economcs 51, Aghon, P., A. Banerjee, and T. Pey Dualsm and macroeconomc volaly. Quarerly Journal of Economcs 114, Aghon, P., D. Comn, and P. How. 26. When does domesc savng maer for economc growh? NBER Worng Paper Aghon, P., P. How, and D. Mayer-Foules. 25. The effec of fnancal developmen on convergence: heory and evdence. Quarerly Journal of Economcs 12, Ayagar, S. and M. Gerler Overreacon of asse prces n general equlbrum. Revew of Economc Dynamcs 2, Ao, K., G. Bengno, and N. Kyoa. 26. Adjusng o capal accoun lberalzaon. Worng Paper, Prnceon Unversy. Barro, R., N.G. Manw, and X. Sala--Marn Capal mobly n neoclasscal models of growh. Amercan Economc Revew 85, Barro, R. and X. Sala--Marn. 24. Economc Growh. Cambrdge, MA: MIT Press (second edon). Blanchard, O Deb and he curren accoun defc n Brazl. In P. Armella, R. Dornbusch, and M. Obsfeld, eds., Fnancal polces and he world capal mare: he problem of Lan Amercan counres. Chcago: Unversy of Chcago Press. Blanchard, O. and C. Kahn The soluon of lnear dfference models under raonal expecaons. Economerca 48, Blanchard, O. and S. Fscher Lecures on Macroeconomcs. Cambrdge, MA: MIT Press. Braggon, F., L. Chrsano, and J. Roldós. 27. Opmal moneary polcy n a sudden sop. NBER worng paper

25 24 Broc, P Invesmen, he curren accoun, and he relave prce of non-raded goods n a small open economy. Journal of Inernaonal Economcs 24, Broc, P. 29. Capal accoun lberalzaon n an open economy Ramsey model. Economcs Deparmen, Unversy of Washngon. Bruno, M The wo-secor open economy and he real exchange rae. Amercan Economc Revew 66, Buer, W Saddlepon problems n connuous me raonal expecaons models: a general mehod and some macroeconomc examples. Economerca 52, Caballero, R. and A. Krshnamurhy. 21. Inernaonal and domesc collaeral consrans n a model of emergng mare crses. Journal of Moneary Economcs 48, Char, V., P. Kehoe, and E. McGraan. 25. Sudden sops and oupu drops. Amercan Economc Revew papers and proceedngs 95, Chrsano, L., M. Echenbaum, and C. Evans. 25. Nomnal rgdes and he dynamc effecs of a shoc o moneary polcy. Journal of Polcal Economy 113, Cohen, D. and J. Sachs Growh and exernal deb under rs of deb repudaon. European Economc Revew 3, Devereux, M. and D. Poon. 24. A smple model of opmal moneary polcy wh fnancal consrans. Unversy of Brsh Columba, Deparmen of Economcs. Evans, D. and B. Jovanovc An esmaed model of enrepreneural choce under lqudy consrans. Journal of Polcal Economy 97, Fscher, S. and J. Frenel Invesmen, he wo-secor model and rade n deb and capal goods. Journal of Inernaonal Economcs 2, Frenel, J. and C. Rodrguez Porfolo equlbrum and he balance of paymens: a moneary approach. Amercan Economc Revew 65, Gavazz, F. and C. Wyplosz The real exchange rae, he curren accoun, and he speed of adjusmen. In J. Blson and R. Marson, eds., Exchange Rae Theory and Pracce. Chcago: Unversy of Chcago Press. Gavazz, F. and C. Wyplosz A noe on he zero-roo problem : dynamc deermnaon of he saonary equlbrum n lnear models. Revew of Economc Sudes 52, Har, O. and J. Moore A heory of deb based on he nalenably of human capal. Quarerly Journal of Economcs 19,

26 25 Harl, R., P. Kor, G. Fechnger, and F. Wrl. 24. Mulple equlbra and hresholds due o relave nvesmen coss. Journal of Opmzaon Theory and Applcaons 123, Hayash, F Tobn s margnal q and average q: a neoclasscal nerpreaon. Economerca 5, Hrshlefer, J On he heory of opmal nvesmen decson. Journal of Polcal Economy 66, Holmsrom, B. and J. Trole The prvae and publc provson of lqudy. Journal of Polcal Economy. Jappell, T. and M. Pagano Savng, growh, and lqudy consrans. Quarerly Journal of Economcs 19, Kyoa, N. and J. Moore Cred cycles. Journal of Polcal Economy 15, Krugman, P Ineres rae celngs, effcency, and growh: a heorecal analyss. MIT. Mmeograph. Krugman, P Hsory versus expecaons. Quarerly Journal of Economcs 16, Lane, P. 21. Inernaonal rade and economc convergence: he cred channel. Oxford Economc Paper 53, Lpon, D. and J. Sachs Accumulaon and growh n a wo-counry model: a smulaon approach. Journal of nernaonal economcs 15, Lucas, R. and E. Presco Invesmen under uncerany. Economerca 39, Masuyama, K Curren accoun dynamcs n a fne horzon model. Journal of Inernaonal Economcs 23, Masuyama, K Increasng reurns, ndusralzaon, and ndeermnacy of equlbrum. Quarerly Journal of Economcs 16, Masuyama, K. 28. Aggregae mplcaons of cred mare mperfecons. NBER Macroeconomcs Annual 27. Cambrdge: MA. McKnnon, R Money and Capal n Economc Developmen. Washngon, D.C.: Broongs Press. Mendoza, E. 28. Sudden sops, fnancal crses and leverage: a Fsheran deflaon of Tobn s Q. NBER Worng Paper

27 26 Obsfeld, M Macroeconomc Polcy, Exchange-Rae Dynamcs, and Opmal Asse Accumulaon. The Journal of Polcal Economy 89, Obsfeld, M Fscal defcs and relave prces n a growng world economy. Journal of Moneary Economcs 23, Obsfeld, M. and K. Rogoff The neremporal approach o he curren accoun. In G. Grossman and K. Rogoff, Handboo of Inernaonal Economcs, volume 3. New Yor: Norh- Holland. Prasad, E., R. Rajan, and A. Subramanan. Foregn capal and economc growh. 27. NBER worng paper Romer, P Increasng reurns and long-run growh. Journal of Polcal Economy 94, Sachs, Jeffrey The curren accoun and macroeconomc adjusmen n he 197s. Broongs Papers on Economc Acvy, Schm-Grohé, S. and M. Urbe. 23. Closng small open economy models. Journal of nernaonal economcs 61, Sba, A Opmal growh wh a convex-concave producon funcon. Economerca 46, Svensson, Lars and Assaf Razn The erms of rade and he curren accoun: he Harberger-Laursen-Mezler effec. Journal of Polcal Economy 91, Trole, J. 26. The heory of publc fnance. Prnceon: Prnceon Unversy Press. Turnovsy, S Inernaonal macroeconomc dynamcs. Cambrdge, MA: MIT Press. Turnovsy, S. and P. Sen Fscal polcy, capal accumulaon, and deb n an open economy. Oxford Economc Papers 43, Uzawa, H On a wo-secor model of economc growh, I. Revew of Economc Sudes 29, 4-7. Uzawa, H Opmal growh n a wo-secor model of capal accumulaon. Revew of Economc Sudes 31, 1-24.

28 27 Appendx 1: Convergence Speed wh Parally Pledgeable Capal Income The represenave agen faces he followng uly maxmzaon problem: max c, uce () ρ subjec o nal sae varables, b and b & = c+ (, ) + rb f( ) (A1) & = +δ (A2) b aq a < 1 (A3) where (, ) s lnearly homogeneous and q s he value of nsalled capal. The opmzaon problem for he represenave agen can be expressed as he followng curren-value Hamlonan: [ ] * ρ e H = u() c λ c + (, ) + rb f () + q ( δ) + γ *( aq b). (A4) Frs-order necessary condons are: u'( c) = λ (, ) q whereq q The co-sae equaons are: * = = λ. & λ c σ ρ r γ λ = & c = (A5) [ δ ] q& = f '( ) + (, ) + r+ + (1 a) γ ) q. (A6) Wh a bndng collaeral consran (b = aq ): b & = aq & + aq & = c + (, ) + raq f ( ) so ha [ ] aq( δ) + a f ( ) + + ( r + δ) q + (1 a) γq = c + (, ) + raq f ( ).

29 28 Nong ha aq + a a( + ) = a(, ) by he lnear homogeney of he nvesmen funcon, and [ γ ] c= f( ) af ( ) (1 a) (, ) a q (A7) & = f( ) af ( ) c+ (1 a) aγ q (1 a) δ (1 a) (A8) for a < 1. Usng equaon (A6), he capal accumulaon consran can be wren as: f ( ) c ( ar ) & + δ = (1 a) + aω. (A9) where ω q = ( ) +.The second erm n he denomnaor equals zero around a seady sae snce by lnear homogeney of he nsallaon cos funcon he prce of capal s 2 2 scale nvaran: ω = δ + =. In addon, around a seady sae ω q = for a quadrac nsallaon cos funcon (see appendx 2), so ha and f( ) c a( r ) & + δ = δ (1 a) (A1) & = f ( ) a( r+ δ) δ (A11) (1 a) around a seady sae. The consraned opmzaon problem can be wren as he followng Hamlonan: ρ f( ) c a( r+ δ) e H = u() c + λ δ (1 a). (A12) The frs-order necessary condon s: u'( c) = λ (1 a) (A13) whle he shadow value of capal evolves accordng o he co-sae equaon

30 29 & λ f ( ) a( r+ δ) = + ρ + δ λ (1 a). (A14) From he economy s resource consran (A1), c = (1 a ), so ha u () c (1 a) u ( c) (1 a) & λ σ & λ c& = + c 2 & = (A15) u ()(1 c a) λ c (1 a) λ cu"( c) where σ. u'( c) Combnng hs wh he co-sae equaon (A14) gves he c& equaon: f ( ) a( r+ δ ) + ( ρ + δ ) (1 a) c& =. (A16) σ + c (1 a) 2 Nong agan from equaon (A6) ha q& = f '( ) + (, ) + [ r+ δ + (1 a) )] from equaon (A5) (wh ρ = r ), around a seady sae he c & equaon s 25 : c& γ q and γ = σ c f ( ) c& + aσ + ( ρ+ δ) c c& = (A17) σ + c (1 a) 2 so ha, by combnng erms, f ( ) + ( ρ+ δ) c& =. (A18) σ (1 a) + c (1 a) 25 The dervaon also maes use of he propery of he nsallaon cos funcon ha a seady sae. 2 2 q = q = around

31 3 Lnearzng around he seady sae and mang use of he seady-sae properes of he nsallaon cos funcon, he dynamc adjusmen equaons around a seady sae ( c, ) are gven by: 26 σ f "( ) (1 a) + c& c (1 a) c c = & 1 ρ 1 a. (A19) The egenvalues of he lnearzed sysem are: µ 1,2 2 ρ ρ f "( ) = ± a < σ (1 a) + c. (A2) The specal case of a closed capal accoun can be deermned by seng a =. The soluon converges o he case wh fully pledgeable capal as a In parcular. = 1, =, and (, ) = (, ) =. I s worh emphaszng ha he dervaon of hese lnearzed dynamc adjusmen equaons depends on he fac ha he long-run asse prce of capal ( q = 1 ) s deermned by he margnal cos of nvesmen, q = (, ).

32 31 Appendx 2. The Invesmen Funcon The nvesmen funcon (, ) s lnearly homogeneous: + = (, ), where = & +δ. 27 general specfcaon seady sae values & (, ) = + φ ( δ, ) = δ / φ, φ >, φ() = & (, ) = 1 + φ ( δ, ) = 1 & 1 (, ) = φ ( δ, ) = φ () > & (, ) = φ ( δ, ) = 2 2 & & δ (, ) = φ + φ (, ) () 2 3 δ = φ > & δ (, ) = φ (, ) 2 δ = φ () < q q= 1 = ( δ, ) δ + ( δ, ) = ( δ, ) δ + ( δ, ) = 2 2 q δ = φ () 2 q= 1 27 & An alernave form of he nvesmen funcon s (, ) = 1+ ϕ. To ensure he convexy of nsallaon coss hs specfcaon requres ha 2ϕ & ϕ & + >, whch s less nuve han he assumpon φ & >. The wo forms of he nvesmen cos funcon generae he same analycal resuls.

33 32 Fgure 1. Growh wh Collaeralzed Borrowng c c & = c & = e 1 e 3 & =, b= rb% = [ f '( ) δ ] c ρ=r c b < q e 2 c b = q c closed T

34 33 Fgure 2 f '( ) ρ δ () ρ ρ ρ f '( ) δ () ˆ

35 34 Fgure 3 c & = c c & = c & = e 2 rb% = ar e 4 & = ( b= ) e 3 e e 1 ˆ % %