저작권법에따른이용자의권리는위의내용에의하여영향을받지않습니다.

Transcription

1 저작자표시 - 비영리 - 변경금지 2.0 대한민국 이용자는아래의조건을따르는경우에한하여자유롭게 이저작물을복제, 배포, 전송, 전시, 공연및방송할수있습니다. 다음과같은조건을따라야합니다 : 저작자표시. 귀하는원저작자를표시하여야합니다. 비영리. 귀하는이저작물을영리목적으로이용할수없습니다. 변경금지. 귀하는이저작물을개작, 변형또는가공할수없습니다. 귀하는, 이저작물의재이용이나배포의경우, 이저작물에적용된이용허락조건을명확하게나타내어야합니다. 저작권자로부터별도의허가를받으면이러한조건들은적용되지않습니다. 저작권법에따른이용자의권리는위의내용에의하여영향을받지않습니다. 이것은이용허락규약 (Legal Code) 을이해하기쉽게요약한것입니다. Disclaimer

2 경제학박사학위논문 Essays on he Opimal Moneary Policy in Small Open Economies 소규모개방경제에서의최적통화정책에대한연구 2015 년 2 월 서울대학교대학원 경제학부경제학전공 호미영

3 Essays on he Opimal Moneary Policy in Small Open Economies 지도교수윤택 이논문을경제학박사학위논문으로제출함 2014 년 10 월 서울대학교대학원 경제학부경제학전공 호미영 호미영의박사학위논문을인준함 2014 년 12 월 위원장김소영 ( 인 ) 부위원장윤택 ( 인 ) 위원정용승 ( 인 ) 위원김진일 ( 인 ) 위원이재원 ( 인 )

4

5 Absrac Essays on he Opimal Moneary Policy in Small Open Economies Mi Young Ho Deparmen of Economics The Graduae School Seoul Naional Universiy In he firs chaper of his hesis, I analyze he opimal policy problem in a canonical DSGE model wih an enforcemen consrain, where he enforcemen consrain akes he form of an inequaliy for value funcions. The main resul in his chaper is ha here is an equivalence relaion in he presence of enforcemen consrain beween he original non-linear opimal policy problem and LQ approximae opimal policy problem. In he second chaper of his hesis, I coninue o analyze he opimal policy problem on he basis of a small-open New Keyensian DSGE model where limied enforcemen and limied spanning inerac o creae an endogenous deb limi. If here are fiscal policy measures o eliminae seady-sae disorions in he case of complee inernaional financial markes, hen enforcemen consrain has no role in he deerminaion of he inflaion of GDP deflaor, real exchange rae, consumpion, and he number of hours worked in each period. In he hird chaper, I show ha he equivalence relaion holds in he small-open New Keyensian DSGE model wih limied enforcemen and limied spanning. Key Words: Endogenous Deb Limi; Capial Conrols; Opimal Moneary Policy; Open Economies Suden Number:

6 Conens 1 Linear-Quadraic Approximaion of Opimal Policy Problems wih Enforcemen Consrains Lagrangian Approach Dynamic Programming Approach Summary of Resuls A Small Open Economy wih Limied Enforcemen: Complee and Incomplee Inernaional Financial Markes Complee Marke and Limied Enforcemen A Bond Model wih Limied Enforcemen Capial Conrol and Moneary Policy in a Bond Model wih Limied Enforcemen The Opimal Moneary Policy in a Small Open Economy wih Endogenous Deb Limi Equivalence beween Non-linear Original Policy Problem and Linear Quadraic Approximaion Numerical Resuls Summary of Resuls

7 Lis of Tables 1 Definiions of Variables Definiion of Variables Calibraion Lis of Figures 1 Truncaed Value Funcion

8 1 Linear-Quadraic Approximaion of Opimal Policy Problems wih Enforcemen Consrains In his chaper, I provide analyical resuls for he equivalence beween he original nonlinear opimal policy and LQ approximae opimal policy problems in he presence of enforcemen consrains. My analysis is inended o show how he mehod of Benigno- Woodford (2012) is applied o models wih enforcemen consrain (i.e. occasionally binding consrain for objecive funcions) based on heir Lagrangian approach. An advanage of his Lagrangian approach helps use a piecewise linear-quadraic approximaion mehod for models wih enforcemen consrain: One is for he reference regime and he oher is for he alernaive regime ha does no ake place along he equilibrium pah. In addiion, I discuss a dynamic programming approach in order o compue a soluion for he opimal policy problem (from a imeless perspecive) wih enforcemen commimen. Before going furher, i should be noed ha here are wo concerns for he validiy of local approximaions. The firs concern is ha a sufficien differeniabiliy for he opimal policy problem may no be guaraneed in he presence of inequaliy consrains. In his regard, i may be useful o disinguish disincions beween a model wih inequaliy consrain for endogenous sae variables and a model wih inequaliy consrain for objecives. In fac, he Lagrangian approach used in his paper helps use a piecewise linear-quadraic approximaion mehod for models wih enforcemen consrain: One is for he reference regime and he oher is for he alernaive regime ha does no ake place along he equilibrium pah. The second concern is he legiimacy of enforcemen consrain under local approximaion. I is hus imporan o check wheher he foreign deb can reach is endogenous limi wih a plausible size of exogenous disurbance under local approximaions. The cumulaive muliplier migh have a uni roo (wihou drif) 1

9 in he linearized model. However, he resuling argeing rule depends on he difference of he cumulaive mulipliers in wo consecuive periods. I begin my analysis by characerizing he opimal policy problem wih limied enforcemen in a general framework. Specifically, he policy auhoriy seeks o deermine he evoluion of an endogenous sae vecor {y } = 0 by maximizing an objecive of he form: E 0 [ β 0 r(y, ξ )] = 0 The evoluion of he endogenous saes mus saisfy sysems of backward-looking and forward-looking srucural equaions F (y, ξ ; y 1 ) = 0, E [G(y, ξ ; y +1 )] = 0, and an inequaliy consrain for he welfare funcion E [ β T r(y T, ξ T )] ṽ (y 1, ξ ) T = ha should hold for each period 0, given he vecor of iniial condiions y 0 1. ṽ (y 1, ξ ) is he value funcion under defaul. The value funcion under defaul can be wrien as follows. The policy auhoriy seeks o deermine he evoluion of an endogenous sae vecor {y +k } k=0 by maximizing an objecive of he form: E [ β k r(y +k, ξ +k )] k=0 I is also necessary o specify srucural changes ha would have occurred under defaul. In each period, all agens agree o anicipae ha hey will face some srucural changes upon defaul. Specifically, hese srucural changes are refleced in he evoluion of he endogenous saes: F (y +k, ξ +k ; y +k 1 ) = 0, E [ G(y +k, ξ +k ; y +k+1 )] = 0 2

10 ha should hold for k = 0, 1,,, given he vecor of iniial condiions y 1. There is a leas one endogenous variable ha should be fixed a a consan value upon defaul. y = b. 0 y =. y ny y ny The resuling opimal policy leads o he value funcion under defaul ṽ (y 1, ξ ). I is also imporan o make a precise inerpreaion of enforcemen consrain for he following reason. In his paper, each individual agen does no ake ino accoun he naional-wide enforcemen consrain. No enforcemen consrains are imposed on individual agens. The reason why I care abou his inerpreaion is he disincion beween naional defaul risk versus residen defaul risk. In paricular, Kehoe and Perri (2002) assume ha imperfec risk sharing can ake place endogenously because of he limied abiliy o enforce inernaional credi arrangemens beween sovereign naions. In conras, Wrigh (2006) argues ha capial flow subsidies are poenially Pareo-improving in he presence of residen defaul risk, while capial conrols are poenially Pareo-improving in models of naional defaul risk. The reason for his resul is ha privae agens end o over-borrow in he case of naional defaul risk and privae capial flows are inefficienly low facing he risk of residen defaul on inernaional borrowing. 1.1 Lagrangian Approach I will highligh a Lagrangian approach for an opimal policy problem in a model wih limied enforcemen where he limied enforcemen gives rise o a curren-value consrain: E [ β T r(y T, ξ T )] ṽ (y 1, ξ ). T = 3

11 Table 1: Definiions of Variables Variables Definiion M φ r(y, ξ ) ξ ṽ 0 (y 1, ξ ) Cumulaive Muliplier for Enforcemen Consrain Muliplier for Enforcemen Consrain One Period Reurn Funcion Vecor of Exogenous Variables Value Funcion under Defaul The policy auhoriy obains he opimal evoluion of an endogenous sae vecor {y } = 0 by maximizing an objecive of he form: E 0 [ β 0 (M r(y, ξ ) φ ṽ (y 1, ξ ))] = 0 The evoluion of he endogenous saes mus saisfy a sysem of backward-looking srucural equaions F (y, ξ ; y 1 ) = 0 and a sysem of forward-looking srucural equaions E [G(y, ξ ; y +1 )] = 0 ha should hold for each period 0, given he vecor of iniial condiions y 0 1. In addiion, a se of resricions can be summarized as follows. r(y, ξ ) is concave in y and coninuously wice differeniable in y and ξ. 0 < β < 1. n F + n G < n y in each period = 0, 1,,. 4

12 M = M 1 + φ wih M > 0 and φ 0 ṽ 0 (y 1, ξ ) is concave in y 1 and coninuously wice differeniable in y 1. I will move ono he recursive represenaion by using he imeless perspecive approach developed in Benigno and Woodford (2012). In each period, he policy auhoriy chooses a value of he vecor y and sae-coningen one-period-ahead pre-commimens Ḡ +1 (ξ +1 ) o maximize M r(y, ξ ) φ ṽ 0 (y 1, ξ ) + βe [V (Ḡ+1; y, ξ +1, ξ )] subjec o he following consrains: F (y, ξ ; y 1 ) = 0 G(y 1, ξ 1 ; y ) = Ḡ E [Ḡ+1] = 0 given he values of Ḡ, y 1, ξ 1, and ξ. In addiion, M and φ are aken as given. The required iniial pre-commimens are G(y 0 1, ξ 0 1; y 0 ) = Ḡ 0 In his represenaion, i is essenial o make a self-consisen iniial pre-commimen. Specifically, he policy ha is chosen subjec o he iniial pre-commimens would saisfy exacly he same form in all subsequen periods as well. This requiremen makes he opimal policy ime consisen. Under his requiremen, i is also possible o have ime-invarian coefficiens of he approximae quadraic objecive and approximae linear consrains. While he unconsrained Ramsey problem does no have such self-consisen pre-commimens, he opimal policy from a imeless perspecive leads o he same deerminisic seady sae as he unconsrained Ramsey problem does. The opimal condiions of he opimal policy wih enforcemen consrain can be obained by using a Lagrangian approach. The Lagrangian of he opimal policy problem 5

13 from a imeless perspecive is given by L 0 = V 0 + E 0 [ β 0 {λ F (y, ξ ; y 1 ) + ϕ G(y 1, ξ 1 ; y )}] = 0 where V 0 is defined as V 0 = E 0 [ β 0 (M r(y, ξ ) φ ṽ 0 (y 1, ξ ))] = 0 The opimal condiions in he presence of enforcemen consrain can be summarized as follows. M D y r(y, ξ ) βe [φ +1 D y ṽ +1 (y, ξ +1 )] +λ D y F (y, ξ ; y 1 ) + βe [λ +1 D 3F (y +1, ξ +1 ; y )] +E [ϕ D y G(y, ξ ; y +1 )] + β 1 ϕ 1 D 3G(y 1, ξ 1 ; y ) = 0 In he absence of enforcemen consrain, he opimal condiions can be wrien as D y r(y, ξ ) + λ D y F (y, ξ ; y 1 ) + βe [λ +1 D 3F (y +1, ξ +1 ; y )] +E [ϕ D y G(y, ξ ; y +1 )] + β 1 ϕ 1 D 3G(y 1, ξ 1 ; y ) = 0 Hence, o he exen which he muliplier for enforcemen consrain is always zero, one can find ha he opimal condiions are exacly he same form used in Benigno and Woodford (2012). The opimal opimal seady sae also is he same irrespecive of enforcemen consrains. For example, he opimal seady sae for he wo opimal condiions shown above can be described by a se of vecors {ȳ, λ, ϕ} saisfying D y r(ȳ, 0) + λ D y F (ȳ, 0; ȳ) + β λ D 3 F (ȳ, 0; ȳ) + ϕ D y G(ȳ, 0; ȳ) + β 1 ϕ D 3 G(ȳ, 0; ȳ) = 0 F (ȳ, 0; ȳ) = 0 G(ȳ, 0; ȳ) = 0. The presence of enforcemen consrain herefore has no impac on he seady sae opimal allocaion ha is defined as he soluion o he sysem of equaions shown above. The opimal condiion is approximaed around he opimal seady sae defined above. Before going furher, i would be helpful o make a parameerizaion of exogenous disur- 6

14 bances by making he following assumpion: ξ = ɛu where ɛ is a real number and {u } =0 is a bounded vecor sochasic process. The local approximaion is hen defined as an approximae characerizaion of he soluion near he opimal seady sae in he case of any small enough value of ɛ. Under he local approximaion, he equilibrium evoluion of he endogenous variables saisfies y (ɛ) = ȳ + O(ɛ) a all imes. I is also imporan o noe ha he muliplier φ can be non-differeniable a a poin of endogenous sae variables given values of exogenous disurbances in each period. The sochasic process of his muliplier is assumed o saisfy E 0 [ β 0 φ 2 ] <. = 0 The equilibrium evoluion of his muliplier saisfies φ = O(ɛ) a all imes. The approximae law of moion for cumulaive muliplier is m = m 1 + φ. The policy funcions of endogenous sae variables can change around poins where his muliplier is no differeniable. One migh wan o overcome his siuaion by using piecewise approximaions when linear soluions should be provided. However, i should be noed ha he purpose of my analysis a his poin is o provide approximae condiions ha reflec his siuaion, bu no a specific soluion procedure ha leads o numerical soluions. 7

15 Having described he original non-linear opimizaion problem, I now urn o he linear-quadraic approximaion. Firs, a second-order Taylor expansion of one-period reurn funcion around he opimal seady sae can be wrien as M r(y, ξ ) φ ṽ 0 (y 1, ξ ) = m (D y r)ỹ φ (D y ṽ)ỹ 1 + (D y r)ỹ + 1 2ỹ (Dyyr)ỹ 2 + ỹ (D yξ 2 r)ξ +.i.p. + O(ɛ 3 ) where m r(y, ξ) and m (D ξ r)ξ are included in erms independen of policy because hey are no choice variables of he moneary auhoriy. The resuling (naive) linear-quadraic approximaion is given by V 0 = E 0 = 0 β 0 {m (D y r)ỹ φ (D y ṽ)ỹ 1 + (D y r)ỹ + 1 2ỹ (Dyyr)ỹ 2 + ỹ (D yξ 2 r)ξ } +.i.p. + O(ɛ 3 ) Benign and Woodford (2012) show ha he linear erm in his represenaion should be eliminaed in order o consruc he equivalence relaion beween he original nonlinear opimizaion problem and is linear-quadraic approximae problem. The approach developed in Benign and Woodford (2012) can be summarized as follows. 1. The coefficiens of he linear erms in he naive approximaion are firs-order derivaives of one-period reurn funcion evaluaed a he seady sae. 2. The opimal condiions can be summarized as Vecors of Lagrange Mulipliers Firs-order Derivaives of Consrains + Firs-order Derivaives of One-period Reurn Funcion = One can use his fac o find a welfare measure ha is purely quadraic. This welfare measure reflecs variaions ha are consisen wih srucural relaions described by 8

16 consrains. Consruc a linear sum of consrains where he seady-sae value of each consrain s muliplier serves as a weigh o each consrain. Compue he second-order Taylor series expansion of his linear sum of consrains. The linear erms in he second-order approximaion of consrains are used o replace he linear erms of he naive approximaion. The second-order approximaion of consrains for he opimal policy problem can be wrien as follows. β 1 ϕ Ḡ 0 = E 0 = 0 β 0 {Φỹ (ỹ Hỹ + 2ỹ Rỹ 1 + 2ỹ Z(L)ξ +1 )} +.i.p. + O(ɛ 3 ) The definiions of coefficiens are given by Φ λ (D y F + βd 3 F ) + ϕ (D y G + β 1 D 3 G) H N F k=1 λ k (D 2 y3 F k + βd 2 3y F k ) + N G i=1 ϕ i(d 2 y3 Gi + β 1 D 2 3y Gi ) R N F k=1 λ k D 2 y3 F k + β 1 N G i=1 ϕ id 2 3y Gi Z(L) N F k=1 λ k (βd 2 3ξ F k + D 2 yξ F k L) + N G i=1 ϕ i(d 2 yξ Gi L + β 1 D 2 3ξ Gi L 2 ) By subsiuing ou he linear erm in he naive linear-quadraic approximaion, one can obain he correc linear-quadraic approximaion as follows. V 0 = E 0 = 0 β 0 {m (D y r)ỹ φ (D y ṽ)ỹ (ỹ Qỹ + 2ỹ Rỹ 1 + 2ỹ B(L)ξ +1 )} +.i.p. + O(ɛ 3 ) 9

17 where Q and B(L) are defined as Q D 2 yyr + H B(L) Z(L) + (D 2 yξ r)l As a resul, a correc linear quadraic approximaion o he original problem wih enforcemen consrain can be summarized as follows. The policy auhoriy chooses {ỹ } =0 o maximize E 0 [ β 0 {m (D y r)ỹ φ (D y ṽ)ỹ (ỹ A(L)ỹ + 2ỹ B(L)ξ +1 )}] = 0 subjec o he following consrains C(L)ỹ = f E [D(L)ỹ +1 ] = h for all 0 and he addiional iniial consrain ha D(L)ỹ 0 = h 0 where coefficiens are defined as A(L) Q + 2RL C(L) D y F + D 3 F L f D ξ F ξ D(L) D 3 G + D y GL h (D ξ G)ξ The Lagrangian of he LQ problem is h 0 h Ḡ 0 L Q 0 = E 0 [ = 0 β 0 {m (D y r)ỹ φ (D y ṽ)ỹ (ỹ A(L)ỹ + 2ỹ B(L)ξ +1 ) + 2 λ C(L)ỹ + 2β 1 ϕ 1 D(L)ỹ }] The firs-order condiion can be wrien as m (D y r) βe [φ +1 (D y ṽ)] + E [J(L)ỹ +1 ] + E [K(L)ξ +1 ] + E [M(L) λ +1 ] + N(L) ϕ = 0 10

18 where he marix polynomials are defined as J(L) 1 2 (A(L) + A (βl 1 )) K(L) B(L) M(L) C (βl 1 )L N(L) β 1 D (βl 1 )L Proposiion 1.1. To exen which soluions o opimal policy problems wih enforcemen consrain exis, he linearizaion of he firs-order condiions of he original policy problem wih enforcemen consrain around he opimal seady sae leads o he same se of he firs-order condiions ha are derived from he opimizaion problems of is corresponding correc linear-quadraic approximae problem. I suffices o show ha he linearizaion of he firs-order condiions of he original policy problem around he opimal seady sae leads o he same se of difference equaions shown above. I is also necessary o provide necessary and sufficien condiions for he uniqueness of he soluion. Le H be he Hilber space of sochasic processes such ha E 0 [ = 0 β 0ỹ ỹ ] < 0. Le H 1 be he subspace of H whose elemens saisfy C(L)ŷ = 0 E [D(L)ŷ +1 ] = 0 for all 0 wih D(L)ŷ 0 = 0. The necessary and sufficien condiions for he exisence of he soluion can be summarized as follows. There are Lagrange muliplier processes m, φ, ϕ and λ in he Hilber space ha processes {m, φ, ϕ, λ, ỹ } =0 saisfy he firs-order condiions shown above and m = m 1 + φ wih m 1 = 0. The following condiion holds: V Q (ŷ) 0 V Q (ŷ) = E 0 [ β 0 {m (D y r)ŷ φ (D y ṽ)ŷ (ỹ A(L)ŷ )}] = 0 11

19 for all processes ŷ H 1. The uniqueness of he soluion requires ha V Q (ŷ) < 0 for all processes ŷ H 1 ha are non-zero almos surely. 1.2 Dynamic Programming Approach The endogenous deb limi is defined as he counry s deb level a which defaul and non-defaul decisions are indifferen. So he deb limi is obained by comparing value funcions under defaul and non-defaul decisions. In order o formulae he endogenous limi, I specify value funcions ha can be obained by solving LQ dynamic programming problems. Firs, he value funcion in he absence of enforcemen consrain is defined as v Q (z ) = 1 2 z P z where he exended sae vecor is defined as z = [ỹ 1 h ξ ξ 1 ] and he marix P is given by he soluion o a sysem of Riccai-ype equaions. In his represenaion, he law of moion for he exended sae vecor is given by z +1 = Φz + Ψɛ +1 and he law of moion for he exogenous sae vecor is ξ +1 = Γξ + ɛ +1. Second, he value funcion under defaul can be wrien as ṽ Q (z a, ) = 1 2 z a, P z a,. where he law of moion for he exended sae vecor under defaul is given by z a,+1 = Φz a, + Ψɛ

20 As a resul, he endogenous deb limi can be defined as ˆb = max b {b : z P z = z a, P z a, } where ˆb is he endogenous deb limi a period. Once one compues he endogenous deb limi, i is possible o incorporae he endogenous deb limi ino he law of moion for he exended sae vecor as follows: Φz + Ψɛ +1 z +1 = Φẑ + Ψɛ +1 b < ˆb b ˆb One migh wonder if here is any sufficien condiion for he exisence of endogenous deb limi. In order o provide a sufficien condiion for he exisence of endogenous deb limi, i would be helpful o make a ransformaion of he exended sae vecor: z a, = H a z where H a is a m n marix and n m. Given his ransformaion, one can rewrie he value funcion under defaul as follows. ṽ Q (z a, ) = 1 2 z a, P z a, = 1 2 z (H a P H a ) z where he law of moion for z is given by z +1 = (H ah a ) 1 (H a ΦH a ) z + (H ah a ) 1 H a Ψɛ +1. A sufficien condiion for he exisence of he endogenous deb limi is ha here is a range of b and a se of ime periods indexed by such ha he following inequaliy holds z (P (H a P H a ))z 2z (H a P H a ) + (H a P H a ) where = z - z and E 0 [ k= 0 β k 0 k k] <. Before going furher, i should be noed ha here are wo ypes of enforcemen consrain. The firs one is he Kehoe-Perry ype consrain ha can be inerpreed as a curren-value consrain: E r= s r β r u(c(s r ), H(s r )) V a (K(s 1 ), s ) 13

21 where s represens he hisory of saes ha he economy has experienced unil i reaches he curren sae s a period and V a (K(s 1 ), s ) is he value funcion of he social welfare a period under financial auarchy. The second one is he Bai-Zhang ype consrain ha can be inerpreed as a coninuaion-value consrain: W (K(s ), B(s ), s +1 ) V a (K(s ), s +1 ) for all s +1 and where W (K(s ), B(s ), s +1 ) is he value funcion of he social welfare in he absence of defaul a period + 1. I use he dynamic programming approach o solve he opimal policy problem in models wih he second-ype consrain. If one uses he inequaliy of quadraic value funcions direcly, he firs-order condiions for he opimal choice of ỹ in he Bellman equaion are (A A 1L)ỹ + E [B(L)ξ +1 ] + βp 1 E [z +1 ] + C 0 λ + D 0 ψ + βe [ζ +1 (P 1 z +1 ˆP 1 z +1 )] = 0 where ζ +1 represens he Lagrange muliplier for he enforcemen consrain for coninuaion values a a sae in period +1. In his represenaion, he las erm is a second-order erm so ha one canno obain he linear dynamics for he opimal policy. I requires ha he value-funcion consrain should be of he second-order. Given his requiremen, my conjecure abou a correc firs-order condiion for he approximae linear quadraic problem is (A A 1L)ỹ + E [B(L)ξ +1 ] + βp 1 E [z +1 ] + C 0 λ + D 0 ψ + βe [ζ +1 (P 1 ˆP 1 )] = 0 In paricular, if he following condiion can be derived from his firs-order condiion, (A A 1L)ỹ + E [B(L)ξ +1 ] + βp (1) 1 E [z +1 ] + C 0 λ + D 0 ψ = 0, 14

22 hen he firs-order condiions ogeher wih backward-looking and forward-looking consrains lead o he following soluion: My = V z, y = ỹ λ ψ where he n n marix M (n = n y + n G + n F ) is defined as M = A 0 + βp (1) 11 C 0 D 0 C D V 1 = (1/2)A 1 C 1 D 1 V 2 = 0 0 I 1.3 Summary of Resuls The main analysis in his chaper is o consruc he equivalence relaion in he presence of enforcemen consrain beween he original non-linear opimal policy problem and LQ approximae opimal policy problem. In doing so, he mehod of Benigno-Woodford (2012) is applied o models wih enforcemen consrain (i.e. occasionally binding consrain for objecive funcions) based on heir Lagrangian approach. The Lagrangian approach helps use a piecewise linear-quadraic approximaion mehod for models wih enforcemen consrain: One is for he reference regime and he oher is for he alernaive regime ha does no ake place along he equilibrium pah. In addiion, he opimal policy problem in a model wih Bai and Zhang (2010) ype consrain is discussed in he conex of he dynamic programming approach for linear quadraic approximaion of opimal policy problems. The dynamic programming approach helps evaluae he welfare under he opimal policy problem (from a imeless perspecive) wih enforcemen commimen. 15

23 2 A Small Open Economy wih Limied Enforcemen: Complee and Incomplee Inernaional Financial Markes The aim of his chaper is o describe a small-open DSGE model wih limied enforcemen and analyze he effec of fiscal policy (o fix he disorions associaed wih monopolisic compeiion) on he opimal moneary policy problem. 2.1 Complee Marke and Limied Enforcemen In his secion, I briefly highligh he benchmark model of small open economies. There are wo counries, called H (Home) and F (Foreign), in he world where a fracion of agens [0, n) of uni mass lives in counry H and he oher fracion (n, 1] belongs o counry F. A coninuum of differeniaed goods exiss in he world. Each ype of hese radable goods is produced by eiher counry H or counry F, while each counry produces a number of differen brands wih measure equal o populaion size. 1 The preferences a period 0 of he represenaive household in counry H is represened by he following funcion: =0 β E 0 [ C1 σ 1 1 σ ν H1+χ 1 + χ ] where C is he aggregae consumpion index a period, H is he number of hours worked a period, and β is he ime discoun facor. Each household solves wo cos minimizaion problems o derive demand funcions for home and foreign differeniaed goods: min n 0 P i, (z)c i, (z)dz s.. C i, = [n 1/ɛ n 0 C i, (z) ɛ 1 ɛ dz] ɛ ɛ 1 1 The modificaion of a wo-counry model ino a small-open economy model has been widely used in he lieraure such as Gali and Monacelli (2005), De Paoli (2009), and Farhi and Werning (2013a and 2013b). The wo-counry version of his secion model has been used in he analysis of Corsei, Dedola, and Leduc (2010) for opimal moneary policy. 16

24 for i = H and F and where ɛ > 1 and heir sub-price indices are given by n 1 P H, = [n 1 P H, (z) 1 ɛ dz] 1 1 ɛ ; PF, = [(1 n) 1 0 n P F, (z) 1 ɛ dz] 1 1 ɛ In addiion, he consumpion baske in he uiliy funcions of home residens is defined as C = (a 1/θ H C θ 1 θ H, + (1 a H ) 1/θ C θ 1 θ F, ) θ θ 1 The corresponding consumer price index (CPI) is hen given by P = (a H P 1 θ H, + (1 a H )P 1 θ F, ) 1 1 θ The aggregae consumpion index for he res of he world is C = (a H 1/θ (CH,) θ 1 θ + (1 a H) 1/θ (CF,) θ 1 θ θ ) θ 1 I is also assumed ha a H = 1 - (1 n)λ and a H = nλ where λ measures he degree of openness, following Suherland (2005) and De Paoli (2009). The usual echnique employed in he lieraure o urn his model ino a small open economy is o ake he limi for n 0 in he definiions of a H and a H afer deriving boh price indices and consumpion indices. The resuling demand funcion of firm h in he home counry is given by Y (h) = ( P H,(h) ) ɛ Y P H, where Y denoes he aggregae demand a period : Y = ( P H, ) θ ((1 λ)c + λq θ C ) + G P where G is he governmen consumpion of domesic goods. The consumpion price index also urns ou o be The raio of GDP deflaor o CPI is P = ((1 λ)p 1 θ H, + λp 1 θ F, ) 1 1 θ. (P H, /P ) 1 θ = (1 λq 1 θ )/(1 λ) 17

25 In addiion, he aggregae resource consrain a period can be wrien as P H, (Y G ) = P (C + NX ) where NX is he real ne expors measured in he uni of consumpion goods. The real ne expors can be hen wrien as a funcion of he consumpion raio beween wo counries and he real exchange rae: NX = λq 1 θ (C 1 λq1 θ Q 2θ 1 C ). 1 λ Alhough asse marke is no complee in he model of sovereign defaul, I begin wih he assumpion ha he asse marke is complee, where a complee se of nominal saeconingen bonds denominaed in he foreign currency is available in he inernaional financial marke for boh foreign and domesic invesors. In his case, he flow budge consrain of an individual household in period can be wrien as S E [Q,+1 B F,+1 ] S B F, + W H P C P H, T + Φ where Q,+1 is he sochasic discoun facor ha can be used o measure he nominal value a period of one uni of foreign currency a period + 1,and B F, denoe holdings of foreign-currency denominaed nominal bonds, C H, and C F, are demands for home and foreign goods wih heir domesic prices denoed by P H, and P F, respecively, W is he nominal wage, H is he number of hours worked a period, T is he real ax in he uni of GDP, Φ is he nominal profi a period. The opimizaion condiions of domesic and foreign invesors for heir bond holdings can be wrien as Q,+1 = β Λ +1P Λ P +1 S +1 S Q,+1 = β Λ +1 P Λ P +1 18

26 where Λ represens he marginal uiliy of consumpion a period for domesic residens, P is he home CPI index, P is he foreign CPI index, and Λ is he marginal uiliy of consumpion for foreign invesors. The opimizaion condiion of labor supply is given by νc σ H χ = W /P. The home governmen issues nominal one-period risk-less securiies, where B H, is he oal ousanding issue of governmen deb a period. Hence, he governmen s flow budge consrain a period is given by B H, 1 + i H, = B H, 1 + S (I I 1) P H, (T G ) where I is he dollar value of inernaional reserves held by he governmen a he end of period, T is he real amoun of lump-sum axes a period, and G is he governmen s real expendiures a period. In order o concenrae on he analysis of moneary policy, he governmen is assumed o follow a Ricardian fiscal policy regime. Hence, he governmen manages is deb o make is budge consrain hold in each period, irrespecive of equilibrium decisions of households and firms. In order o close he model, domesic firms are assumed o se boh heir local and inernaional prices in he uni of he domesic currency (so called producer s currency pricing) according o he Calvo pricing model. In his model, a fracion of firms rese heir price during each period, while he oher fracion do no. Each firm ha reses is price chooses is price by maximizing he following expeced presen-value of profis: (αβ) k E [Λ +k ( P H, ) ɛ Y +k ( P H, W +k )] P H,+k P H,+k A +k P H,+k k=0 where α is he fracion of firms ha do no rese heir prices in each period and P H, is he opimal rese price a period. The firs-order condiions for his profi maximizaion 19

27 can be represened by a se of recursive equaions: L = F = C σ Y + αβe [Π ɛ 1 H,+1 F +1] νɛ ɛ 1 H 1+χ P + αβe [Π ɛ P H,+1L +1 ] H, P H, P H, = L F The GDP deflaor inflaion of domesic goods can be deermined by he following equaion: 1 = (1 α)( P H, P H, ) 1 ɛ + απ ɛ 1 H,. The aggregae producion funcion also can be wrien as Y = A H / where is he relaive price disorion a period : = (1 α)( P H, P H, ) ɛ + απ ɛ H, 1 Having described he benchmark model, I now move ono he opimal moneary policy problem in a limied enforcemen model wih complee marke. In fac, he cenral bank a period 0 seeks a sae-dependen plan by solving he following opimizaion problem. max E 0 β u(c, H ) =0 subjec o a se of implemenabiliy consrains A H = P θ h, ((1 λ)c + λq θ C ) + G (2.1) F = A H C σ + αβe [Π ɛ 1 H,+1 F +1] (2.2) L = νɛ + αβe [Π ɛ ɛ 1 P H,+1L +1 ] (2.3) h, H 1+χ = (1 α)( 1 απɛ 1 H, 1 α ) ɛ ɛ 1 + απ ɛ H, 1 (2.4) 20

28 L = ( 1 απɛ 1 H, 1 α ) 1 1 ɛ F (2.5) 1 = λq 1 θ + (1 λ)p 1 θ h, (2.6) C σ = Q (C ) σ (2.7) β k E [u(c +k, H +k )] V a ( 1, A ) (2.8) k=0 The firs consrain is he social resource consrain. The second, hird, and fifh consrains are profi-maximizaion condiions of firms. The fourh consrain is he evoluion equaion for relaive price disorion. The sixh consrain is he equilibrium relaion beween real exchange rae and erms-of-rade. The sevenh consrain reflecs he equilibrium condiion for sochasic discoun facors ha should hold for domesic and foreign residens in he complee marke for asse ransacions. The final consrain as an enforcemen consrain requires ha he expeced discouned sum of uiliies a period onward is equal o or greaer han he value funcion a period (denoed by V a ( 1, A )) ha would have been obained if he economy were under financial auarchy. The presence of sae-coningen commimen leads o he addiion of lagged Lagrange mulipliers o he firs-order condiions for opimal nominal prices of firms. φ 2 φ 5 ( 1 απ ɛ 1 H, 1 α ) 1 1 ɛ = απ ɛ 1 H, φ 2 1. (2.9) φ 3 + φ 5 = απ ɛ H,φ 3 1 (2.10) The opimizaion condiion for inflaion is given by ( ) 1 1 απ ɛ 1 ɛ 1 H, φ 4 ( 1 Π H, ) = φ 5F 1 α (1 α)ɛ ( 1 απ ɛ 1 H, 1 α ) ɛ ɛ 1 +(1 ɛ 1 )φ 2 1 F +φ 3 1 Π H, L (2.11) The oher firs-order condiions can be summarized as follows. The opimaliy condiion for consumpion is (1 + M )C σ (1 λ)φ 1 P θ A H h + σφ 2 C σ+1 + σφ 7 C σ 1 = 0 (2.12) 21

29 The firs-order condiion for labor is given by ν(1 + M )H χ A + φ A 1 φ 2 C σ φ 3 νɛ(1 + χ)h χ (ɛ 1) P h = 0 (2.13) The opimizaion condiion for he erms-of-rade is θφ 1 P 1 θ h ((1 λ)c + λq θ C ) + vɛφ 3H 1+χ φ 6 (1 λ)(1 θ)p 2 θ (ɛ 1) h = 0 (2.14) The opimizaion condiion for real exchange rae is (1 θ)λφ 6 Q θ + λθφ 1 P θ h Qθ 1 C + φ 7 (C ) σ = 0 (2.15) The opimizaion condiion for relaive price disorion is φ 4 +βe [φ 8+1 V1 a (, A +1 )] = φ 1A H 2 +φ 2 A H C σ 2 The evoluion of a cumulaive muliplier M can be wrien as νɛh 1+χ +φ 3 (ɛ 1)P h, 2 +αβe [Π ɛ H+1φ 4+1 ] (2.16) M = M 1 + φ 8 (2.17) where M 1 = 0 and φ 8 denoes he Lagrange muliplier of he enforcemen consrain. The complemenary slackness condiion associaed wih he enforcemen consrain implies ha he following condiion holds: φ 8 (V c (φ 1, 1, A ) V a ( 1, A )) = 0 (2.18) where V c (φ 1, 1, A ) denoes he value funcion under he opimal commimen plan. Specifically, he value funcion under he opimal commimen plan has a recursive represenaion: V c (φ 1, 1, A ) = u(c, H ) + βe [V c (φ,, A +1 )] (2.19) where φ 1 = 0. 22

30 I would be worhwhile o discuss some issues associaed wih he enforcemen consrain. Following he approach used in Marce and Marimon (2011), Kehoe and Perri (2002), and Bai and Zhang (2010), he cumulaive muliplier (denoed by M ) summarizes he impac of he cenral bank s commimen on consumpion demand and labor supply decisions of households ha should be made in he presence of he enforcemen consrain. Given he definiion of he cumulaive muliplier, he cenral bank s objecive funcion in he Lagrangian formulaion associaed wih he opimal policy problem discussed above urns ou o be β E 0 [(1 + M )u(c, H ) φ 8 V a ( 1, A )]. =0 The cumulaive muliplier (denoed by M ) herefore helps keep rack of impacs of he cenral bank s pas commimens on he curren one-period insananeous uiliy funcion. In addiion, he evoluion of M is affeced by he relaive size of he wo value funcions ha can be obained under he opimal sae-coningen plan and financial auarchy. The complemenary slackness condiion associaed wih he enforcemen consrain can be rewrien as 0 if V c (φ 1, 1, A ) > V a ( 1, A ) M M 1 = φ 8 (> 0) if V c (φ 1, 1, A ) = V a ( 1, A ) (2.20) Having described he cenral bank s opimizaion condiions, I now solve a se of 18 equaions (8 equilibrium condiions (2.1) - (2.7) wih (2.19) and 10 opimizaion condiions (2.9) hrough (2.18)) o pin down decision rules of 18 endogenous variables including he cenral bank s decision variables {C, H, Π H,, P h,,, Q, F, L, V c } and Lagrange mulipliers {φ 1, φ 2, φ 3, φ 4, φ 5, φ 6, φ 7, φ 8, M }, given he value funcion under financial auarchy (denoed by V a ( 1, A )), a Markov process for A, he iniial values of Lagrange mulipliers φ 1, and he iniial relaive price disorion 1. 23

31 The effeciveness of enforcemen consrain depends on he presence of fiscal policy measures o fix seady-sae disorions. In order o show hese resuls, I noe ha opimizaion condiions for consumpion demand and labor supply lead o he following condiions a he seady sae: νc σ H χ = (ɛ 1)P h /ɛ. As discussed in Corsei and Peseni (2001), he inclusion of P h in his equaion implies ha he cenral bank has incenive o raise oupu by making surprise inflaions in models wih inefficien seady-sae deviaions. Hence, if fiscal policy measures are o eliminae seady-sae disorions, one should allow for boh of he wo disorions associaed wih he monopolisic compeiion in goods marke and he erms-of-rade. Proposiion 2.1. Le us suppose ha here are fiscal policy measures o eliminae seadysae disorions. In his case, he enforcemen consrain has no role in he deerminaion of he inflaion of GDP deflaor, real exchange rae, consumpion, and he number of hours worked in each period. Under he assumpion ha fiscal policy is used o eliminae seady-sae disorions as discussed in Woodford (2003) and Yun (2005), Lagrange mulipliers for profi maximizaion condiions should be zero: φ 2 = φ 3 = φ 5 = 0. Specifically, his soluion requires he following condiion o hold 1 Π H, = ( ) 1 1 απ ɛ 1 ɛ 1 H,. 1 α By puing his condiion ino he opimizaion condiions of firms, I can find ha he inflaion of GDP deflaor is given by Π H, = The law of moion for relaive price disorion is 1. (2.21) = 1 (α + (1 α) ɛ 1 1 ) 1/(ɛ 1). (2.22) 24

32 In addiion, he oher opimal condiions can be solved o yield he following condiion: νc σ H χ = A P h, {(1 λ)p 1 θ h, +σλ(q 1 θ +P 1 θ h, Q θ 1/σ )+(σλ 2 /(1 λ))q 1 1/σ } 1 (2.23) Moreover, he aggregae demand equaion and he social resource consrain can be solved o show ha consumpion and labor are deermined as follows: C = Q 1/σ C (2.24) H = P θ A h, ((1 λ)q1/σ + λq θ )C. (2.25) In sum, I have derived 9 equaions (2.6), (2.19), (2.17), (2.18), and (2.21) - (2.25) for 9 variables such as {C, H, Π H,, P h,,, Q, V c, φ 8, M }, given he value funcion under financial auarchy (denoed by V a ( 1, A )), a Markov process for A, he iniial relaive price disorion 1, and he iniial asse holdings B 1. An imporan implicaion of equaions (2.23) - (2.25) is ha he enforcemen consrain has no impac on he deerminaion of real exchange rae, consumpion, and he number of hours worked in each period. Moreover, he evoluion equaion of relaive price disorion and he inflaion of GDP deflaor are no affeced by he Lagrange mulipliers of he enforcemen consrain as shown in equaions (2.21) and (2.22). As a resul, I have proved he saemen of proposiion A Bond Model wih Limied Enforcemen In his secion, I urn o a bond model wih limied enforcemen. In he bond model, non-coningen nominal deb alone is available in inernaional financial markes. The only difference of his secion from he previous secion is he asse marke srucure. Hence, I highligh he opimizaion problem of he represenaive household, while oher equilibrium condiions are idenical o hose of he previous secion. 25

33 The opimizaion problem a period 0 of he represenaive household in counry H can be wrien as follows. max =0 β E 0 [ C1 σ 1 1 σ subjec o a se of one-period flow budge consrains ν H1+χ 1 + χ ] (1 + i F,) 1 S B F,+1 = (1 + τ b )S B F, + W H P C P H, T + Φ for = 0, 1, 2,, and where τ b is a subsidy a period for he household s holdings of foreign asses. The subsidy for holding foreign asses is financed by lump-sum axes denoed by P H, T. Hence, he governmen imposes a ax on capial inflows (subsidy on capial ouflows) in he home counry, while proceeds of capial conrol axes are redisribued o households in he home counry, following he recen lieraure on capial conrols such as Farhi and Werning (2013). In addiion, I absrac from he issue of he home governmen s bonds in order o concenrae on he impac of foreign deb on he opimal design of moneary policy wih he assumpion ha he governmen follows a Ricardian fiscal policy regime. The governmen s one-period flow budge consrain is hus given by τ b S B F, + P H, G = P H, T (2.26) The firs-order condiion of domesic residens for heir bond holdings is given by 1 = β(1 + i F,)E [( C C +1 ) σ S +1 S Π +1 ] The firs-order condiion of foreign households for heir bond holdings is given by 1 = β(1 + i F,)E [( C C+1 ) σ (Π +1) 1 ]. By using he definiion of he real exchange rae, i is possible o rewrie he Euler equaion for he accumulaion of foreign asses as follows: 1 = β(1 + i F,)E [( C ) σ Q +1 C +1 Q Π ] (2.27) +1 26

34 In addiion, he opimizaion condiion of domesic residens for heir labor supplies can be wrien as νc σ H χ = W /P. I now urn o he analysis of he opimal moneary policy problem in a bond model wih limied enforcemen. Since he real value of foreign bond (measured in he uni of foreign goods) is he cenral bank s choice variable in is opimal policy problem, he oneperiod flow budge consrain shown above can be ransformed ino he following social resource consrain: (1 + i F,) 1 Q B = Q B 1 P h, P h, Π + Y C G P h, where B 1 = B F, /P 1. The cenral bank a period 0 seeks a sae-dependen plan by solving he following opimizaion problem. subjec o max E 0 β u(c, H ) =0 A H = P θ h, ((1 λ)c + λq θ C ) + G (2.28) F = A H C σ + αβe [Π ɛ 1 H,+1 F +1] (2.29) L = νɛ + αβe [Π ɛ ɛ 1 P H,+1L +1 ] (2.30) h, H 1+χ = (1 α)( 1 απɛ 1 H, 1 α ) ɛ ɛ 1 + απ ɛ H, 1 (2.31) L = ( 1 απɛ 1 H, 1 α ) 1 1 ɛ F (2.32) 1 = λq 1 θ + (1 λ)p 1 θ h, (2.33) B A H C G Q ( P h P h 1 + i F, B 1 Π ) = 0 (2.34) E β k u(c +k, H +k ) V a ( 1, A ) (2.35) k=0 27

35 The firs consrain is he social resource consrain. The second, hird, and fifh consrains are profi-maximizaion condiions of firms. The fourh consrain is he evoluion equaion for relaive price disorion. The sixh consrain is he equilibrium relaion beween real exchange rae and erms-of-rade. The sevenh consrain reflecs he equilibrium condiion for sochasic discoun facors ha should hold for domesic and foreign residens in he complee marke for asse ransacions. The final consrain as an enforcemen consrain requires ha he expeced discouned sum of uiliies a period onward is equal o or greaer han he value funcion a period (denoed by V a ( 1, A )) ha would have been obained if he economy were under financial auarchy. The firs-order condiions of he cenral bank s opimal policy problem can be wrien as follows. The opimizaion condiions for he wo variables associaed wih opimal nominal prices of firms can be wrien as follows. ( ) 1 1 απ ɛ 1 1 ɛ H, φ 2 φ 5 1 α = απ ɛ 1 H, φ 2 1. (2.36) φ 3 + φ 5 = απ ɛ H,φ 3 1 (2.37) The opimizaion condiion for inflaion is given by φ 4 ( 1 Π H, ( 1 απ ɛ 1 H, 1 α ) 1 ɛ 1 ) = φ 5F (1 α)ɛ The opimaliy condiion for consumpion is ( ) 1 απ ɛ 1 ɛ ɛ 1 H, +(1 ɛ 1 )φ 2 1 F +φ 3 1 Π H, L 1 α (2.38) (1 + M )C σ φ 1 P θ h (1 λ) + σφ 2 A H C σ+1 φ 7 P h = 0 (2.39) The firs-order condiion for labor is given by v(1 + M )H χ A + A (φ 1 + φ 7 ) φ 2 C σ φ 3 vɛ(1 + χ)h χ (ɛ 1) P h = 0 (2.40) 28

36 The opimizaion condiion for he erms-of-rade is θφ 1 P 1 θ h ((1 λ)c +λq θ C )+ vɛφ 3H 1+χ φ 6 (1 λ)(1 θ)p 2 θ B (ɛ 1) h +φ 7 (C +Q ( 1 + i F, The opimizaion condiion for real exchange rae is B 1 Π )) = 0 (2.41) (1 θ)φ 6 = θφ 1 P θ h Q2θ 1 B C φ 7Q θ ( λp h, 1 + i F, B 1 Π ) (2.42) The opimizaion condiion for relaive price disorion is φ 4 +βe [φ 8+1 V1 a (, A +1 )] = αβe [Π ɛ H,+1φ 4+1 ] (φ 1 + φ 7 )A H The opimizaion condiion for real bond holdings is given by 2 +φ 2 A H C σ 2 νɛh 1+χ +φ 3 (ɛ 1)P h, 2 (2.43) φ 7 Q P h, i F, = βe φ 7+1 Q +1 P h,+1 Π +1 (2.44) The evoluion of a cumulaive muliplier M can be wrien as M = M 1 + φ 8 (2.45) where M 1 = 0 and φ 8 denoes he Lagrange muliplier of he enforcemen consrain. The complemenary slackness condiion associaed wih he enforcemen consrain implies ha he following condiion holds: φ 8 (V c (B 1, φ 1, 1, A ) V a ( 1, A )) = 0 (2.46) where V c (B 1, φ 1, 1, A ) denoes he value funcion under he opimal commimen plan. Specifically, he value funcion under he opimal commimen plan has a recursive represenaion: V c (B 1, φ 1, 1, A ) = u(c, H ) + βe [V c (B, φ,, A +1 )] (2.47) where φ 1 = 0. 29

37 In he same way as is done in he previous secion, i can be shown ha he evoluion of M is affeced by he relaive size of he wo value funcions ha can be obained under he opimal sa-coningen plan and financial auarchy. In addiion, he complemenary slackness condiion associaed wih he enforcemen consrain implies ha he following condiion holds: 0 if V b (B 1, φ 1, 1, A ) > V a ( 1, A ) M M 1 = φ 8 (> 0) if V b (B 1, φ 1, 1, A ) = V a ( 1, A ) Having described he cenral bank s opimizaion condiions, I now solve a se of 19 equaions (8 equilibrium condiions (2.28) - (2.34) wih (2.47) and 11 opimizaion condiions (2.36) hrough (2.46)) o pin down decision rules of 19 endogenous variables including he cenral bank s decision variables {B, C, H, Π H,, P h,,, Q, F, L, V c } and Lagrange mulipliers {φ 1, φ 2, φ 3, φ 4, φ 5, φ 6, φ 7, φ 8, M }, given he value funcion under financial auarchy (denoed by V a ( 1, A )), a Markov process for A, he iniial values of Lagrange mulipliers φ 1, he iniial relaive price disorion 1, and he iniial asse holdings B 1. I now seek a closed-form soluion o he opimizaion condiions shown above under he assumpion ha fiscal policy is used o eliminae seady-sae disorions as discussed in Woodford (2003) and Yun (2005). In his case, Lagrange mulipliers for profi maximizaion condiions should be zero: φ 2 = φ 3 = φ 5 = 0. Specifically, his soluion requires he following condiion o hold 1 Π H, = ( ) 1 1 απ ɛ 1 ɛ 1 H,. 1 α By puing his condiion ino he opimizaion condiions of firms, I can find ha he inflaion of GDP deflaor is given by Π H, = (2.48)

38 The law of moion for relaive price disorion is = 1 (α + (1 α) ɛ 1 1 ) 1/(ɛ 1). (2.49) By combining equaions (2.39) and (2.40), he Lagrange muliplier for he social resource consrain (denoed by φ 7 ) can be wrien as φ 7 = λ 1 C σ Q θ 1 (1 + M )(1 ν C σ H χ P h A ). The subsiuion of his equaion ino he opimal condiion for bond holdings leads o he following wo condiions: 1 = β(1 + i F,)E [( C ) σ Q Q θ (1 + M +1)(1 ν +1 A +1 C+1 σ Hχ +1 ) C +1 Q Π +1 Q θ 1 (1 + M )(1 ν A C σhχ ) ] (2.50) The oher opimizaion condiions are also solved o yield νc σ H χ = A P h 1 κ 1 + (1 λ)κ P 1 θ h (2.51) where κ is defined as κ = θ 1 (1 λ)p 1 θ h Q θ z + λ(q z + C ) λp 1 θ h ((1 λ)c + λq θ C ) + λ(1 λ)p 2(1 θ) h Q 2θ 1 C Moreover, he aggregae demand equaion and he social resource consrain can be solved o show ha consumpion and labor are deermined as follows: where z is defined as H = Q 2θ 1 A C = λ 1 Q θ (λq θ 1 1 λq 1 θ C z ) (2.52) 1 λ ( 1 λq1 θ ) θ 1 θ (C 1 λ λ 1 (1 λ)q 1 θ z ). (2.53) B = (1 + i F,)( B 1 Π + z ). (2.54) In sum, I have derived 11 equaions (2.33), (2.47), (2.45), (2.46), and (2.48) - (2.54) for 11 variables such as {B, z, C, H, Π H,, P h,,, Q, V c, φ 8, M }, given he value funcion 31

39 under financial auarchy (denoed by V a ( 1, A )), a Markov process for A, he iniial relaive price disorion 1, and he iniial asse holdings B 1. Comparing his se of opimizaion condiions wih he se of opimizaion condiions used o prove proposiion 3.1, I can see ha he asse marke srucure plays a subsanial role in deermining wheher he enforcemen consrain can have impac on he deerminaion of he inflaion of GDP deflaor, real exchange rae, consumpion, and he number of hours worked in each period. In paricular, equaion (2.50) shows ha he cumulaive lagrange muliplier (denoed by M ) can affec he relaion beween he level of consumpion and he accumulaion of foreign deb. The enforcemen consrain in a bond model wih limied enforcemen can have impac on he deerminaion of he inflaion of GDP deflaor, real exchange rae, consumpion, and he number of hours worked in each period, whereas i does no in he case of complee marke. As a resul, our findings for he role of asse marke srucure on he impac of he enforcemen consrain on he deerminaion of opimal consumpion and hours worked can be summarized as follows. Proposiion 2.2. Le us suppose ha here are fiscal policy measures o eliminae seadysae disorions. In a bond model wih limied enforcemen, he enforcemen consrain can have impac on he deerminaion of he inflaion of GDP deflaor, real exchange rae, consumpion, and he number of hours worked in each period. 2.3 Capial Conrol and Moneary Policy in a Bond Model wih Limied Enforcemen In his secion, I show ha he presence of capial conrol axes helps aain he secondbes allocaion in a decenralized economy. In fac, a compeiive equilibrium canno aain he second-bes allocaion in he absence of capial conrol axes. If his resul holds rue, he inroducion of capial conrol axes raises he social welfare because i 32

40 makes i possible o implemen he second-bes allocaion in a compeiive equilibrium. Definiion 2.1. A compeiive equilibrium under naional defaul risk given a sequence of axes on ineres earnings of foreign asses and lump-sum ransfers {τ b, T } consiss of a se of allocaions {B, C, H }, a se of prices {Π H,, P h,, Q, W /P, F, L }, and a sequence of relaive price disorion { } ha saisfy he following equilibrium condiions: 1. Each household chooses {B, C, H } by maximizing is uiliy subjec o a se of one-period flow budge consrains and no-ponzi consrains. 2. Each firm ses is prices by solving is profi maximizaion condiions (2.29), (2.30), and (2.32). 3. The non-defaul condiion (2.35) is saisfied in each period given a sequence of he value funcion under financial auarchy. 4. The governmen s budge consrain (2.26) is saisfied in each period. In order o see he reason why we need capial conrol axes, we absrac he profi maximizaion condiions of firms and he aggregae demand equaion in he implemenabiliy consrains for a momen. We also assume ha he governmen alone can make defaul decisions on he counry s foreign deb. Hence, here are only naional defaul risks in his case. The opimal condiions of he planner s problem can be wrien as follows. (1 + M )C σ = φ 7 /P h ν(1 + M )H χ = A φ 7 The opimal condiion for bond holdings is given by φ 7 Q P h, i F, = βe [ φ 7+1Q +1 P h,+1 Π ]

41 Hence, subsiuing he consumpion equaion ino he bond-holding equaion leads o he following equaion: i F, = βe [( C C +1 ) σ (1 + M +1)Q +1 (1 + M )Q (Π +1) 1 ]. On he oher hand, when capial conrol axes are imposed in decenralized economies, he Euler equaion urns ou o be i F, = βe [(1 τ b +1)( C C +1 ) σ Q +1 Q (Π +1) 1 ]. Hence, comparing he opimal condiion of he planner s problem and he Euler equaion for bond holdings, we can see ha he size of capial conrol axes should saisfy he following relaion: τ b = φ 8 M 1. We have used his simplified example o demonsrae ha i is necessary o have capial conrol axes if one wans o implemen he second-bes allocaion a a decenralized compeiive equilibrium. In paricular, he inroducion of capial conrol axes in his simplified example raises he social welfare because he implemenaion of he secondbes allocaion can be aained a a compeiive equilibrium. Proposiion 2.3. In a bond model wih limied enforcemen, he second-bes allocaion (defined in proposiion 2.2) can be suppored as a compeiive equilibrium wih naional defaul risk (defined in definiion 2.1) in he presence of axes {τ b, T }. In he absence of axes {τ b, T }, his second-bes allocaion (defined in proposiion 2.2) can be suppored as a compeiive equilibrium wih naional defaul risk. By comparing equaions (2.27) and (2.50), we can compue he opimal level of subsidy on he holdings of foreign asses ha makes he second-bes allocaion decenralized as a 34

42 compeiive equilibrium wih naional defaul risk saisfies he following equaion: τ b = 1 Q θ 1 (1 + M )(1 ν C σ H χ /A ) Q θ 1 1 (1 + M 1)(1 ν 1 C σ 1 Hχ 1 /A 1) (2.55) for = 1, 2,,. In addiion, equaion (2.55) implies ha τ b = 0 onely when Q θ 1 (1 + M )(1 ν C σ H χ /A ) is consan a each sae and dae. However, when he soluion o he cenral bank s opimizaion condiions does no guaranee ha Q θ 1 (1 + M )(1 ν C σ H χ /A ) is consan, he opimal subsidy on holdings of foreign asses should vary over ime. Moreover, a consan value of Q θ 1 (1+M )(1 ν C σ H χ /A ) a each sae and each dae requires an addiional resricion for he second-bes allocaions ha canno be derived from he se of opimal condiions. Hence, he second-bes allocaion (defined in proposiion 2.2) canno be suppored as a compeiive equilibrium wih naional defaul risk in he absence of axes {τ b, T }. An imporan quesion ha migh emerge from his resul is wheher or no he inroducion of capial conrol axes leads o a igher deb limi when he deb limi is endogenously deermined. In order o address his issue, i would be helpful o see how he endogenous deb limi is deermined in he presence of limied enforcemen and limied spanning. As emphasized in Bai and Zhang (2010), boh limied enforcemen and financial incompleeness inerac o creae he endogenous deb limi on non-coningen deb because no lenders are willing o exend heir loans o hose who would defaul for sure in he nex period. Specifically, he endogenous deb limi is defined as he maximum amoun of deb ha can be suppored wihou defaul a all fuure coningencies: B = min A +1 { B : V b (B, φ,, A +1 ) = V a (, A +1 )} where B represens he endogenous deb limi a period. The endogenous deb limi is igh when levels of he social welfare under non-defaul and defaul decisions is on 35