Online Appendix to Fiscal Consolidation in an Open Economy with Sovereign Premia and without Monetary Policy Independence

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1 Online Appendix o Fisal Consolidaion in an Open Eonomy wih Sovereign Premia and wihou Moneary Poliy Independene Aposolis Philippopoulos, a,b Peros Varhaliis, and Vanghelis Vassilaos a a Ahens Universiy of Eonomis and Business b CESifo Eonomi and Soial Researh Insiue, Triniy College Dublin Appendix. ouseholds This appendix presens and solves he problem of he household in some deail. There are i =, 2,..., N idenial domesi households ha a ompeiively. ousehold s Problem Eah household i maximizes expeed lifeime uiliy given by E 0 =0 β U ( i,,n i,,g ), () where i, is i s onsumpion bundle, n i, is i s hours of work, g is per apia publi spending, 0 <β< is he ime preferene rae, and E 0 is he raional expeaions operaor ondiional on he informaion se. The onsumpion bundle, i,, is defined as ( i, ) ν ( F i, ) ν i, = ν ν, (2) ( ν) ν Corresponding auhor: Aposolis Philippopoulos, Deparmen of Eonomis, Ahens Universiy of Eonomis and Business, Ahens 0434, Greee. Tel: aphil@aueb.gr.

2 2 Inernaional Journal of Cenral Banking Deember 207 where i, and F i, denoe he omposie domesi good and he omposie foreign good, respeively, and ν is he degree of preferene for he domesi good. We define he wo omposies, i, and F i,,as i, = [ N h= f= [ i,(h)] φ φ N F i, = [ F i,(f)] φ φ ] φ φ φ φ (3), (4) where φ>0 is he elasiiy of subsiuion aross goods produed in he domesi ounry. The period budge onsrain of eah i expressed in real erms is (see, e.g., Benigno and Thoenissen 2008, for similar modeling) [ P ( + τ ) i, + P F ] F i, + P x i, + b i, + S P fi, h P P P + φh 2 ( S P P = ( τ k ) [ r k fi, h SP P f h ) 2 P P k i, + ω i, P S P + R b i, + Q P P P ] +( τ n ) w n i, P P fi, h τi,, l (5) where P is he onsumer prie index (CPI), P is he prie index of home radables, P F is he prie index of foreign radables (expressed in domesi urreny), x i, is i s domesi invesmen, b i, is i s endof-period real domesi governmen bonds, fi, h is i s end-of-period real inernaionally raded asses denominaed in foreign urreny, r k is he real reurn o inheried domesi apial, k i,, ω i, is i s real dividends reeived by domesi firms, w is he real wage rae, R is he gross nominal reurn o domesi governmen bonds beween and, Q is he gross nominal reurn o inernaional asses beween and, τi, l are real lump-sum axes if posiive (or ransfers if negaive) o eah household, and

3 Vol. 3 No. 4 Fisal Consolidaion in an Open Eonomy 3 τ, τ k, τ n are ax raes on onsumpion, apial inome, and labor inome, respeively. The parameer φ h 0 apures adjusmen oss relaed o privae foreign asses, where variables wihou ime subsrips denoe seady-sae values. Eah household i also faes he budge onsrains P i, = P i, + P F F i, (6) P i, = P F F i, = N P (h) i,(h) (7) h= N P F (f) F i,(f), (8) f= where P (h) is he prie of eah variey h produed a home and P F (f) is he prie of eah variey f produed abroad (expressed in domesi urreny). Finally, he law of moion of physial apial for household i is k i, =( δ)k i, + x i, ξ 2 ( ) 2 ki, k i,, (9) k i, where 0 <δ< is he depreiaion rae of apial and ξ 0isa parameer apuring adjusmen oss relaed o physial apial. For our numerial soluions, we use he usual funional form: u i, ( i,,n i,,m i,,g )= σ i, σ χ n n +η i, where χ n, χ g, σ, η, ζ are preferene parameers. ousehold s Opimaliy Condiions g ζ +η + χ g ζ, (0) Eah household i as ompeiively, aking pries and poliy as given. Following he lieraure, o solve he household s problem, we follow a wo-sep proedure. We firs suppose ha he household deermines is desired onsumpion of omposie goods, i, and F i,, and, in urn, hooses how o disribue is purhases of individual varieies, i, (h) and F i, (f).

4 4 Inernaional Journal of Cenral Banking Deember 207 The firs-order ondiions of eah i inlude he budge onsrains wrien above and also u i, i, P i, i, P ( + τ ) = βe u i,+ i,+ i,+ i,+ u i, i, i, i, P + P S P P ( + τ ) P P + P ( )R +τ () + P + [ ( +φ h S P )] f h SP P P f h = βe u i,+ i,+ i,+ u i, i, i, i, ( + τ ) + { +ξ P + P + ( +τ + )Q S + P + ( ki, k i, )} P + u i,+ i,+ = βe ( ) i,+ i,+ +τ + ( 2 ( δ) ξ ki,+ 2 k i, ) + ( ) ki,+ ki,+ ξ k i, k i, + ( τ+) k r k + P P + (2) (3) u i, n i, = u i, i, i, i, P P ( τ n ) w ( + τ ) (4) i, F i, = ν ν P F P (5) [ ] P φ i,(h) = (h) i, (6) P [ ] P F F φ i,(f) = (f) F i,. (7) P F Equaions () (3) are, respeively, he Euler equaions for domesi bonds, foreign asses, and domesi apial, and (4) is he opimaliy ondiion for work hours. Finally, (5) shows he opimal alloaion beween domesi and foreign goods, while (6) and (7) show he opimal demand for eah variey of domesi and

5 Vol. 3 No. 4 Fisal Consolidaion in an Open Eonomy 5 foreign goods, respeively; hese ondiions are used in he firm s opimizaion problem below. Impliaions for Prie Bundles Equaions (5), (6), and (7), ombined wih he household s budge onsrains, imply ha he hree prie indexes are Appendix 2. Firms P =(P ) ν (P F ) ν (8) [ N ] φ P = [P (h)] φ (9) h= N P F = [P F (f)] φ f= φ. (20) This appendix presens and solves he problem of he firm in some deail. There are h =, 2,...,N domesi firms. Eah firm h produes a differeniaed good of variey h under monopolisi ompeiion in is own produ marke faing Calvo-ype nominal prie fixiies. Demand for he Firm s Produ Demand for firm h s produ, y (h), omes from domesi households onsumpion and invesmen, C (h) and X (h), where C (h) N i= i, (h) and X (h) N i= x i,(h), from he domesi governmen, G (h), and from foreign households onsumpion, C F (h) N i= F i, (h). Thus, he demand for eah domesi firm s produ is y (h) =C (h)+x (h)+g (h)+c F (h). (2) Using demand funions like hose derived by solving he household s problem above, we have [ ] P φ i,(h) = (h) i, (22) P

6 6 Inernaional Journal of Cenral Banking Deember 207 [ ] P φ x i, (h) = (h) x i, (23) P [ ] P φ G (h) = (h) G (24) F i, (h) = P [ P F P F (h) where, using he law of one prie, we have in (25) P F (h) P F = P S (h) P S = ] φ F i,, (25) P P (h). (26) Sine, a he eonomy level, aggregae demand is Y = C + X + G + C F, (27) he above equaions imply ha he demand for eah firm s produ is [ ] P y φ (h) = (h) Y. (28) The Firm s Problem P The real profi of eah firm h is (see, e.g., Benigno and Thoenissen 2008 for similar modeling) ω (h) = P (h) y (h) P r k k (h) w n (h). (29) P P The produion ehnology is (h) =A [k (h)] α [n (h)] α, (30) where A is an exogenous sohasi oal faor produiviy proess whose moion is defined below. Profi maximizaion is also subje o he demand for is produ as derived above: [ ] P y φ (h) = (h) Y. (3) P

7 Vol. 3 No. 4 Fisal Consolidaion in an Open Eonomy 7 In addiion, firms hoose heir pries faing a Calvo-ype nominal fixiy. In eah period, firm h faes an exogenous probabiliy θ of no being able o rese is prie. A firm h, whih is able o rese is prie, hooses is prie P # (h) o maximize he sum of disouned expeed nominal profis for he nex k periods in whih i may have o keep is prie fixed. This is modeled nex. Firm s Opimaliy Condiions To solve he firm s problem, we follow a wo-sep proedure. We firs solve a os-minimizaion problem, where eah firm h minimizes is os by hoosing faor inpus given ehnology and pries. The soluion will give a minimum nominal os funion, whih is a funion of faor pries and oupu produed by he firm. In urn, given his os funion, eah firm, whih is able o rese is prie, solves a maximizaion problem by hoosing is prie. The soluion o he os-minimizaion problem gives he inpu demand funions: w = m (h)( a) y (h) n (h) (32) P r k = m (h)a y (h) P k (h), (33) where m (h) = Ψ (h) P denoes he real marginal os. Then, he firm hooses is prie o maximize nominal profis wrien as (see also, e.g., Rabanal 2009 for similar modeling) E k=0 θ k Ξ,+k {P # (h) +k (h) Ψ +k ( y +k (h) )}, where Ξ,+k is a disoun faor aken as given by he firm, [ ] y+k (h) = P # φ (h) P Y +k +k and Ψ (h) denoes he minimum nominal os funion for produing y (h) a so ha Ψ (h) ishe assoiaed marginal os.

8 8 Inernaional Journal of Cenral Banking Deember 207 E The firs-order ondiion gives k=0 θ k Ξ,+k [ P # P+k (h) Dividing by P,wehave [ ] E θ k P Ξ # φ { (h),+k Y+k k=0 =0. P +k ] φ { Y+k P # (h) φ } φ Ψ +k(h) =0. P # P (h) (34) } φ φ m +k(h) P +k P (35) Summing up, he behavior of eah firm h is summarized by (32), (33), and (35). A reursive expression of he equaion above is presened below. Noie, finally, ha eah firm h, whih an rese is prie in period, solves an idenial problem, so P # (h) =P # is independen of h, and eah firm h, whih anno rese is prie, jus ses is previous period prie P (h) =P (h). Thus, he evoluion of he aggregae prie level is given by ( ) P φ ( ) = θ P φ ( ) φ +( θ) P #. (36) Appendix 3. Governmen Budge Consrain This appendix presens in some deail he governmen budge onsrain and he menu of fisal poliy insrumens. Reall ha we have assumed a ashless eonomy for simpliiy. Then, he period budge onsrain of he governmen wrien in aggregae and nominal quaniies is B + S F g φ g ( S F g = P 2 P ) 2 SFg + R B + Q S F g P + P G τ (P C + P F C F ) τ k (r k P K + P Ω ) τ n W Ñ T l, (37)

9 Vol. 3 No. 4 Fisal Consolidaion in an Open Eonomy 9 where B is he end-of-period nominal domesi publi deb and F g is he end-of-period nominal foreign publi deb expressed in foreign urreny so i is muliplied by he exhange rae, S. The parameer φ g 0 apures adjusmen oss relaed o publi foreign deb. The res of he noaion is as above. Noe ha B = N i= B i,, C = N i= i,, CF = N i= F i,, K = N i= k i,, Ω N i= ω i,, Ñ N i= n i,, and T l denoes he nomimal value of lump-sum axes. Le D = B + S F g denoe he oal nominal publi deb a he end of he period. This an be held boh by domesi privae agens, λ D, where in equilibrium B λ D, and by foreign privae agens, S F g ( λ ) D, where 0 λ. Then, dividing by he prie level and he number of households, he budge onsrain in real and per apia erms is d = φg 2 [( λ ) d ( λ) d] 2 P + R λ d + Q S P P + P g τ P τ k P P ( P P ( r k P k + ω P P P P S ( λ ) d + P F ) F P ) τ n w n τ, l (38) where, as said in he main ex, in eah period, one of he poliy insrumens (τ, τ k, τ n, g, τ l, λ, d ) follows residually o saisfy he governmen budge onsrain. Appendix 4. Deenralized Equilibrium (DE) This appendix presens in some deail he DE sysem. Following he relaed lieraure, we work in seps. Equilibrium Equaions The DE is summarized by he following equaions (quaniies are in per apia erms):

10 0 Inernaional Journal of Cenral Banking Deember 207 u u ( + τ ) P P P = βe u ( +φ p ( S P ( ) P + +τ + S P ( + τ ) P f h SP P P P = β u + + ( ) P + S + P+ P + + +τ + P+ Q P + P+ { ( )} u k ( + τ +ξ ) k u + + = βe ( ) + + +τ + ( ) 2 ( δ) ξ k+ 2 k ( ) +ξ k+ k k+ k + ( τ+) k r k + P P+ R P + )) f h (39) (40) (4) u n = u F P P P F P = ν ν ( τ n ) w ( + τ ) (42) (43) E θ k k=0 ( k k =( δ)k + x ξ 2 ) 2 k k (44) ( ) ν ( ) F ν ν ν ( ν) ν (45) w = m ( a)a k n a a (46) P r k = m aa k a P (47) ω = P y P [ P Ξ #,+k P +k P r k k w n (48) P { } P # φ P φ m P +k +k =0 P ] φ +k (49)

11 Vol. 3 No. 4 Fisal Consolidaion in an Open Eonomy = [ P ] φ A k n a a (50) P P S P d = R λ d + Q P P + P ( P g τ P P + P F P F P P ) τ k P P S ( λ ) d ( r k P ) k + ω P τ n w n τ l + φg 2 [( λ ) d ( λ) d] 2 (5) = + x + g + F (52) P F + P F F P P ( + φp S P f h SP 2 P P f h ( λ ) d S P P + φg 2 [( λ ) d ( λ) d] 2 ) 2 S P Q P f h P P =( λ ) d S P P f h (53) ) φ (54) ( ) P φ = θp φ (P +( θ) # P = ( P ) ν ( ) P F ν (55) P F ( P P =(P ) ν S ( ) φ ( ) φ ( P = θ P +( θ) Q = Q + ψ e = S P (56) d P P Y ) ν d P # (57) ) φ (58) (59) l R S λ d + Q + S ( λ )d, (60) P P

12 2 Inernaional Journal of Cenral Banking Deember 207 where [ N ( P P f g ( λ )d S P, Ξ,+k β k P ] φ φ, and P ( N h= [y (h)] φ φ ) φ is a measure of prie dispersion. σ +k σ P P +k τ τ, +k = h= [P (h)] φ) φ. Thus, We hus have weny-wo equaions in weny-wo variables, {y,,, F, n, x, k, f h, P F, P, P, P #, P, w, m, ω, r k, Q, d, P, R, l } =0. This is given he independenly se moneary and fisal poliy insrumens, {S, τ, τ k, τ n, g, τ, l λ } =0; he resof-he-world variables, { F,Q,P } =0; ehnology, {A } =0; and iniial ondiions for he sae variables. In wha follows, we shall ransform he above equilibrium ondiions. In pariular, following he relaed lieraure, we rewrie hem firs, by expressing prie levels in inflaion raes, seondly, by wriing he firm s opimaliy ondiions in reursive form, and, hirdly, by inroduing a new equaion ha helps us o ompue expeed disouned lifeime uiliy. Finally, we will presen he final ransformed sysem ha is solved numerially. Variables Expressed in Raios We firs express pries in rae form. We define seven new variables, whih are he gross domesi CPI inflaion rae, Π P P ; he gross foreign CPI inflaion rae, Π P P ; he gross domesi goods inflaion rae, Π P P prie dispersion index, Δ ; he auxiliary variable, Θ P # ; he P ] φ; he gross rae of exhange [ P P rae depreiaion, ɛ S S ; and he erms of rade, TT P F = P S P. In wha follows, we use Π P, Π, Π, Θ, Δ,ɛ, TT insead of P,P,P,P #, P,S, P F, respeively. Also, for onveniene and omparison wih he daa, we express fisal and publi finane variables as shares of nominal oupu, Thus, TT TT = S P S P P P = ɛ Π. Π

13 Vol. 3 No. 4 Fisal Consolidaion in an Open Eonomy 3 P y. In pariular, using he definiions above, real governmen spending, g, an be wrien as g = s g y, while real governmen ransfers, τ, l an be wrien as τ l = s l y TT ν, where s g and s l denoe, respeively, he oupu shares of governmen spending and governmen ransfers. Equaion (49) Expressed in Reursive Form We now replae equaion (49), from he firm s opimizaion problem, wih an equivalen equaion in reursive form. In pariular, following Shmi-Grohé and Uribe (2007), we look for a reursive represenaion of he form E k=0 θ k Ξ,+k [ P # P+k ] φ { y+k P # } φ (φ ) m +kp +k =0. (6) We define wo auxiliary endogenous variables: [ ] z E θ k P # φ Ξ,+k y+k P # (62) P z 2 E k=0 k=0 θ k Ξ,+k [ P # P+k P +k ] φ +km +k P +k P. (63) Using hese wo auxiliary variables, z and z 2, as well as equaion (6), we ome up wih wo new equaions ha ener he dynami sysem and allow a reursive represenaion of (6). Thus, in wha follows, we replae equaion (49) wih z = φ (φ ) z2, (64) where we also have he wo new equaions (and he wo new endogenous variables, z and z 2 ) z =Θ φ y TT ν + βθe σ + σ +τ +τ + ( Θ Θ + ) φ ( Π + ) φ z + (65)

14 4 Inernaional Journal of Cenral Banking Deember 207 z 2 =Θ φ y m + βθe σ + +τ ( Θ σ +τ+ Θ + n +η ) φ ( Π + ) φ z 2 +. (66) Lifeime Uiliy Wrien as a Firs-Order Differene Equaion Sine we wan o ompue soial welfare, we follow Shmi-Grohé and Uribe (2007) by defining a new endogenous variable, V, whose moion is given by ( s g ) y ζ V = σ σ χ n +η + χ g + βe V +, (67) ζ where V is household s expeed disouned lifeime uiliy a ime. Thus, in wha follows, we add equaion (67) and he new variable V o he equilibrium sysem. Noe ha, in he welfare numerial soluions repored, we add a onsan number, 2, o eah period s uiliy; his makes he welfare numbers easier o read. Equilibrium Equaions Transformed Using he above, he ransformed DE sysem is summarized by V = σ σ χ n +η ( s g ) n +η + χ y ζ g + βe V + (68) ζ βe σ + ( )R +τ = σ + Π + ( + τ (69) ) βe σ ( )Q +τ TT v +ν + + = σ ( + τ ) TTv β σ + TTν + + +ν Π + ( ) k+ k+ + ξ k = σ [ +φ p ( TT ν +ν f h TT ν +ν f h)] { ( ) δ ξ +τ ( } τ k ) + r k k + [ ( )] k ( + τ +ξ ) k TT ν ( ) 2 k+ k (70) (7)

15 Vol. 3 No. 4 Fisal Consolidaion in an Open Eonomy 5 χ n n η = σ ( τ n ) w ( + τ ) (72) F = ν ν TT (73) k =( δ)k + x ξ ( ) 2 k k (74) 2 k ( ) ν ( ) F ν (ν) ν ( ν) ν (75) w = m ( a)a k n a a (76) TT v r k = m aa k a (77) ω = TT v y z = TT v r k k w n (78) φ (φ ) z2 (79) = Δ A k a n a (80) d = φg 2 [( λ )d ( λ)d] 2 + R Π λ d + Q TT v+v τ ( TT v Π TT v+v + TT v F ) τ k ( λ )d + TT ν s g y ( ) r k TT v k + ω τ n w n TT ν s l (8) = + x + s g + F (82) ( λ )d TT ν +ν f h = TT ν F + TT ν F ( ) + Q TT ν +ν ( λ )d f h Π TT v+v

16 6 Inernaional Journal of Cenral Banking Deember φp 2 + φg ( TT ν +ν f h TT ν +ν f h) 2 2 (( λ )d ( λ)d) 2 (83) ( ) Π φ ( = θ +( θ) Θ Π ) φ (84) z =Θ φ y TT ν z 2 =Θ φ Π Π Π Π ( TT = TT TT = ɛ Π TT Π ( ) ν TT ) ν (85) (86) = (87) TT ( ) Δ = θδ Π φ +( θ)(θ ) φ (88) ( ( ) ) d Q = Q TT + ψ e ν y d (89) + βθe σ + σ y m + βθe σ +τ +τ + ( Θ Θ + + +τ ( Θ σ +τ+ Θ + ) φ ( ) φ ( Π + Π + ) φ z + (90) ) φ z+ 2 (9) l = R s d + Q ɛ + ( λ ) d. (92) TT ν y We hus have weny-five equaions in weny-five variables, {V,y,,, F,n,x,k,f h, TT, Π, Π, Θ, Δ,w,m, ω,r k, Q,d, Π,z,z 2,R, l } =0. This is given he independenly se poliy insrumens, {ɛ,τ, τ k,τ n,s g,s l, λ } =0; he res-of-he-world variables, { F,Q, Π } =0; ehnology, {A } =0; and iniial ondiions for he sae variables. Equaions and Unknowns (Given Feedbak Poliy Coeffiiens) We an now define he final equilibrium sysem. I onsiss of he weny-five equaions of he ransformed DE presened in he previous subseion, he four poliy rules in subseion 2.7 in he main

17 Vol. 3 No. 4 Fisal Consolidaion in an Open Eonomy 7 ex, and he equaion for domesi expors in subseion 2.8 in he main ex. Thus, we have a sysem of hiry equaions. By using wo auxilliary variables, we ransform i o a firs order, 2 so we end up wih hiry-wo equaions in hiry-wo variables, {y,,, F, x, n, f h, d, k, ω, m,π,π,π,θ,δ, TT, w, r k,q, l, z, z 2, V, R ; τ,s g, τ k, τ n ; klead, TTlag, F } =0. This is given he exogenous variables, {Q, Π,A,ɛ, s l, λ } =0, and iniial ondiions for he sae variables. The hiry-wo endogenous variables are disinguished in weny-four onrol variables, {y,,, F,x,n, ω,m, Π, Π, Π, Θ,TT,w,r k,z,z 2,V, klead, F,τ, s g, τ k, τ n } =0, and eigh sae variables, {f,d h,k, Δ,Q,R,l,TTlag } =0. All his is given he values of feedbak poliy oeffiiens in he poliy rules, whih are hosen opimally in he ompuaional par of he paper. Saus Quo Seady Sae In he seady sae of he above model eonomy, if ɛ =, our equaions imply Π =Π =Π = Π. If we also se Π = (his is he exogenous omponen of foreign inflaion), hen Π =Π = Π =Π=;his in urn implies Δ=Θ=. Also, in order o solve he model numerially, we need a value for he exogenous expors, F ; we assume F =.0 F, as i is he ase in he daa. Finally, sine from he Euler for bonds, = βr Π, we residually ge R = β. (Noe ha hese ondiions will also hold in all seady-sae soluions hroughou he paper.) The seady-sae sysem is (in he labor supply ondiion, we use ν TT v =) =β[ δ + ( τ k) r k ] (93) =βq (94) χ n n η σ ν = ) TT v ( + τ ) w (95) 2 In pariular, when we use he Shmi-Grohé and Uribe (2004) MATLAB rouines, we add wo auxiliary endogenous variables, klead and TTlag, o redue he dynami sysem ino a firs-order one.

18 8 Inernaional Journal of Cenral Banking Deember 207 F = ν TT ν (96) x = δk (97) ( ) ν ( ) F ν = (ν) ν ( ν) ν (98) w = m( a)ak a n a (99) r k = TT v maak a n a (00) ω = TT v y TT v rk k wn (0) = Ak a n a (02) ( ) d = Qd + TT ν s g τ TT v + TT v F ( ) τ k r k k + ω τ n wn TT ν s l (03) TT v = + x + s g + F (04) ( λ)d TT ν +ν f h = TT ν F + TT ν F ( ) + QT T ν +ν h TT v+v ( λ)d f ) Q = Q + ψ (e ( d TT ν d) (05) (06) z φ = (φ ) z2 (07) z = TT ν + βθz (08) z 2 = m + βθz 2. (09) We hus have seveneen equaions in seveneen variables,,, F, k, x, n,, TT, m, ω, d, z,z 2, Q, f h, r k, w. The numerial soluion of his sysem is in able 2 in he main ex. Appendix 5. The Reformed Eonomy This appendix presens he reformed eonomy as defined in subseion 4. in he main ex. The equaions desribing he reformed

19 Vol. 3 No. 4 Fisal Consolidaion in an Open Eonomy 9 eonomy are as in he previous appendix exep ha now we se Q = Q, so ha he publi deb raio is deermined by he no premium ondiion, d or d dt T ν. Noe also ha d TT ν sine Q = Q,weseβ = Q in he parameerizaion sage. In wha follows, we presen seady-sae soluions of his eonomy. We will sar wih he ase in whih he ounry s ne foreign deb posiion is unresried so ha f ( λ)tt ν d TT ν f h is endogenously deermined. In urn, we will sudy he ase in whih we se he ounry s ne foreign deb posiion equal o zero (meaning ha he rade is balaned) so ha f = 0 in he new reformed seady sae. Seady Sae of he Reformed Eonomy wih Unresried Foreign Deb Now Q = Q so ha d dt T ν. The seady-sae sysem is =β[ δ +( τ k )r k ] (0) χ n n η σ ν = TT ν ) ( + τ ) w () F = ν TT ν (2) x = δk (3) ( ) ν ( ) F ν = (ν) ν ( ν) ν (4) = Ak a n a (5) ω = = + x + s g + F (6) r k = TT v maak a n a (7) w = m( a)ak a n a (8) TT v y TT v rk k wn (9) z φ = (φ ) z2 (20)

20 20 Inernaional Journal of Cenral Banking Deember 207 Table A. Seady-Sae Soluion of he Reformed Eonomy wih Unresried f Seady-Sae Variables Desripion Soluion u Period Uiliy Oupu TT Terms of Trade Q Q Ineres Rae Premium 0 TT ν k TT ν f h TT ν d f ( λ)dt T ν TT ν f h Consumpion as Share of Physial Capial as Share of Privae Foreign Asses as Share of Toal Publi Deb as Share of Toal Foreign Deb as Share of z = ytt ν + βθz (2) z 2 = ym + βθz 2 (22) d dt T ν (23) ) d = Q d + TT ν s g τ ( TT v + TT v F ( ) τ k TT v rk k + ω τ n wn TT ν s l y (24) ( λ)d TT ν +ν f h = TT ν F + TT ν F ( ) + QT T ν +ν h TT v+v ( λ)d f. (25) We hus have sixeen equaions in,, F, k, x, n,, TT, m, ω, d, z,z 2, r k, w, f h. The numerial soluion of his sysem is presened in able A. Noie ha, for he reasons explained in he ex, privae foreign deb, f h > 0, and hene he ounry s ne foreign deb-o- raio, f, are exremely high in his ase.

21 Vol. 3 No. 4 Fisal Consolidaion in an Open Eonomy 2 Table A2. Seady-Sae Soluion of he Reformed Eonomy wih f = 0 when he Residual Fisal Insrumen Is s g Seady-Sae Variables Desripion Soluion u Period Uiliy r k Real Reurn o Physial Capial w Real Wage Rae.2442 n ours Worked Oupu TT Terms of Trade.0 s g Capial Tax Rae Q Q Ineres Rae Premium 0 Consumpion as Share of k Physial Capial as Share of TT ν TT ν f h TT ν d f ( λ)dt T ν TT ν f h Privae Foreign Asses as Share of Toal Publi Deb as Share of Toal Foreign Deb as Share of Seady Sae of he Reformed Eonomy wih Resried Foreign Deb Now Q = Q, so ha d dt T ν as above, and, in addiion, f ( λ)tt ν d TT ν f h = 0, so ha ( λ) dt T ν TT ν f h =0 or f h ν ( λ)dt T. In his ase, he equaion for he balane of TT paymens (25) ν simplifies o TT F = F. (26) Therefore, using his equaion in he plae of (25), we have sixeen equaions in,, F, k, x, n, y, TT, m, ω, d, z,z 2, r k, w, and one of he ax-spending poliy insrumens, s g, τ, τ k, τ n. Noe ha in urn f h follows residually from f h ν ( λ)dt T. The numerial soluion of his sysem under eah publi finaning ν TT insrumen

22 22 Inernaional Journal of Cenral Banking Deember 207 Table A3. Seady-Sae Soluion of he Reformed Eonomy wih f = 0 when he Residual Fisal Insrumen Is τ Seady-Sae Variables Desripion Soluion u Period Uiliy r k Real Reurn o Physial Capial w Real Wage Rae.2442 n ours Worked Oupu TT Terms of Trade.0 τ Capial Tax Rae Q Q Ineres Rae Premium 0 Consumpion as Share of k Physial Capial as Share of TT ν TT ν f h TT ν d f ( λ)dt T ν TT ν f h Privae Foreign Asses as Share of Toal Publi Deb as Share of Toal Foreign Deb as Share of is presened in ables A2 A5, while a summary of hese soluions appears in able 3 in he main ex. Resried Changes in Fisal Poliy Insrumens We now repea he main ompuaions resriing he magniude of feedbak oeffiiens in he poliy rules so ha all ax-spending poliy insrumens anno hange by more han 0 perenage poins from heir averages in he daa (we repor ha he resuls also do no hange when we allow for 5 perenage poins only). Resried resuls for opimal feedbak poliy oeffiiens, he implied saisis, and welfare over various ime horizons are repored in ables A6, A7, and A8, whih orrespond o ables 4, 5, and 6, respeively, in he main ex. Inspeion of he new ables, and omparison o heir ounerpars in he main ex, implies ha he main qualiaive resuls do

23 Vol. 3 No. 4 Fisal Consolidaion in an Open Eonomy 23 Table A4. Seady-Sae Soluion of he Reformed Eonomy wih f = 0 when he Residual Fisal Insrumen Is τ k Seady-Sae Variables Desripion Soluion u Period Uiliy 0.93 r k Real Reurn o Physial Capial w Real Wage Rae.2649 n ours Worked Oupu TT Terms of Trade.0 τ k Capial Tax Rae Q Q Ineres Rae Premium 0 Consumpion as Share of k Physial Capial as Share of TT ν TT ν f h TT ν d f ( λ)dt T ν TT ν f h Privae Foreign Asses as Share of Toal Publi Deb as Share of Toal Foreign Deb as Share of no hange. Namely, deb onsolidaion is again preferable o nondeb onsolidaion afer he firs en years. Also, alhough obviously feedbak poliy oeffiiens are now smaller, he bes fisal poliy mix again implies ha we should earmark publi spending for he reduion of publi deb and, a he same ime, u axes o miigae he reessionary effes of deb onsolidaion. The only differene is ha now, sine us in inome axes are resried, we should also u onsumpion axes. Impulse Response Funions (IRFs) wih One Fisal Insrumen a a Time Figures A and A2 ompare deb onsolidaion resuls when he fisal auhoriies an use all insrumens joinly o he ase in whih hey are resried o use one insrumen a a ime only. Resuls are always wih opimized rules. To save on spae, we fous on resuls for he publi deb-o- raio and privae onsumpion

24 24 Inernaional Journal of Cenral Banking Deember 207 Table A5. Seady-Sae Soluion of he Reformed Eonomy wih f = 0 when he Residual Fisal Insrumen Is τ n Seady-Sae Variables Desripion Soluion u Period Uiliy r k Real Reurn o Physial Capial w Real Wage Rae.2442 n ours Worked Oupu TT Terms of Trade.0 τ n Capial Tax Rae Q Q Ineres Rae Premium 0 Consumpion as Share of k Physial Capial as Share of TT ν TT ν f h TT ν d f ( λ)dt T ν TT ν f h Privae Foreign Asses as Share of Toal Publi Deb as Share of Toal Foreign Deb as Share of Table A6. Opimal Reaion o Deb and Oupu wih Deb Consolidaion (resried opimal fisal poliy mix) Fisal Opimal Reaion Opimal Reaion Insrumens o Deb o Oupu s g γ g l =0.4 γy g =0.00 τ γ l = γ y = τ k γ k l = γ k y = τ n γ n l = γ n y =0.95 Noe: V 0 =

25 Vol. 3 No. 4 Fisal Consolidaion in an Open Eonomy 25 Table A7. Saisis Implied by Table A6 Five Ten Tweny Elasiiy Elasiiy Periods Periods Periods Daa o Liabiliies o Oupu Min/Max Average Average Average Average s g 2.06% % 0.30/ τ %.97% 0.352/ τ k % 2.6% 0.243/ τ n %.6% /

26 26 Inernaional Journal of Cenral Banking Deember 207 Table A8. Welfare over Differen Time orizons wih, and wihou, Deb Consolidaion (resried opimal fisal poliy mix) Two Four Ten Tweny Thiry Periods Periods Periods Periods Periods Wih Consolid Wihou Consolid. (.625) (3.79) (7.448) (3.427) (8.904) Welfare Gain/Loss Figure A. IRFs of Publi Deb o (omparison of alernaive fisal poliies) Noe: IRFs are in levels and onverge o he reformed seady sae, while he solid horizonal line indiaes he poin of deparure (saus quo value).

27 Vol. 3 No. 4 Fisal Consolidaion in an Open Eonomy 27 Figure A2. IRFs of Consumpion (omparison of alernaive fisal poliies) Noe: IRFs are in levels and onverge o he reformed seady sae, while he solid horizonal line indiaes he poin of deparure (saus quo value).

28 28 Inernaional Journal of Cenral Banking Deember 207 Appendix 6. Robusness o he Publi Deb Threshold Parameer The Case wih a Lower Publi Deb Threshold We now assume d 0.8. This implies ψ o hi he daa. Then, we have he soluion shown in ables A9 A. Table A9. Saus Quo Seady-Sae Soluion Seady-Sae Variables Desripion Soluion Daa u Period Uiliy r k Real Reurn o Physial Capial w Real Wage Rae.40 n ours Worked Oupu TT Terms of Trade Q Q Ineres Rae Premium Consumpion as Share of k Physial Capial as Share of TT ν TT ν f h TT ν d f ( λ)dt T ν TT ν f h Privae Foreign Asses as Share of Toal Publi Deb as Share of Toal Foreign Deb as Share of

29 Vol. 3 No. 4 Fisal Consolidaion in an Open Eonomy 29 Table A0. Reformed Seady-Sae Soluion Seady-Sae Variables Desripion Soluion u Period Uiliy r k Real Reurn o Physial Capial w Real Wage Rae.2673 n ours Worked Oupu TT Terms of Trade.0 τ k Capial Tax Rae Q Q Ineres Rae Premium 0 Consumpion as Share of k Physial Capial as Share of TT ν TT ν f h TT ν d f ( λ)dt T ν TT ν f h Privae Foreign Asses as Share of Toal Publi Deb as Share of Toal Foreign Deb as Share of Table A. Opimal Reaion o Deb and Oupu wih Deb Consolidaion (opimal fisal poliy mix) Fisal Opimal Reaion Opimal Reaion Insrumens o Deb o Oupu s g γ g l = γy g =0 τ γ l =0 γ y = τ k γ k l =0 γ k y = τ n γ n l =0.000 γ n y =2.36 Noe: V 0 =

30 30 Inernaional Journal of Cenral Banking Deember 207 The Case wih a igher Publi Deb Threshold Nex, we assume d. This implies ψ 0.08 o hi he daa. Then, we have he soluion shown in ables A2 A4. Table A2. Saus Quo Seady-Sae Soluion Seady-Sae Variables Desripion Soluion Daa u Period Uiliy r k Real Reurn o Physial Capial w Real Wage Rae.373 n ours Worked Oupu TT Terms of Trade Q Q Ineres Rae Premium Consumpion as Share of k Physial Capial as Share of TT ν TT ν f h TT ν d f ( λ)dt T ν TT ν f h Privae Foreign Asses as Share of Toal Publi Deb as Share of Toal Foreign Deb as Share of

31 Vol. 3 No. 4 Fisal Consolidaion in an Open Eonomy 3 Table A3. Reformed Seady-Sae Soluion Seady-Sae Variables Desripion Soluion u Period Uiliy r k Real Reurn o Physial Capial w Real Wage Rae.2625 n ours Worked Oupu TT Terms of Trade.0 τ k Capial Tax Rae Q Q Ineres Rae Premium 0 Consumpion as Share of k Physial Capial as Share of TT ν TT ν f h TT ν d f ( λ)dt T ν TT ν f h Privae Foreign Asses as Share of Toal Publi Deb as Share of Toal Foreign Deb as Share of Table A4. Opimal Reaion o Deb and Oupu wih Deb Consolidaion (opimal fisal poliy mix) Fisal Opimal Reaion Opimal Reaion Insrumens o Deb o Oupu s g γ g l = γy g =0.007 τ γ l =0 γ y =0.008 τ k γ k l =0.004 γ k y = τ n γ n l =0.306 γ n y =.5629 Noe: V 0 =

32 32 Inernaional Journal of Cenral Banking Deember 207 Appendix 7. Robusness o he Ne Foreign Deb Posiion in he Reformed Seady Sae The Case wih 0. Ne Foreign Deb Raio in he Reformed Seady Sae When we se f ( λ)tt ν d TT ν f h =0. in he reformed seady sae, he resuls are as shown in ables A5 and A6. Table A5. Seady-Sae Soluion of he Reformed Eonomy wih f =0. Seady-Sae Variables Desripion Soluion u Period Uiliy Oupu TT Terms of Trade.0065 Q Q Ineres Rae Premium 0 Consumpion as Share of TT ν k TT ν f h TT ν d f ( λ)dt T ν TT ν f h Physial Capial as Share of Privae Foreign Asses as Share of Toal Publi Deb as Share of Toal Foreign Deb as Share of Table A6. Opimal Reaion o Deb and Oupu wih Deb Consolidaion (opimal fisal poliy mix) Fisal Opimal Reaion Opimal Reaion Insrumens o Deb o Oupu s g γ g l = γy g = τ γl =0 γ y = τ k γl k =0.00 γy k = τ n γl n =0.005 γy n =2.4 Noe: V 0 =

33 Vol. 3 No. 4 Fisal Consolidaion in an Open Eonomy 33 The Case wih Ne Foreign Deb Raio in he Reformed Seady Sae When we se f ( λ)tt ν d TT ν f h =0.209 (as in he daa) in he reformed seady sae, he resuls are as shown in ables A7 and A8. Table A7. Seady-Sae Soluion of he Reformed Eonomy wih f =0.209 Seady-Sae Variables Desripion Soluion u Period Uiliy Oupu TT Terms of Trade.002 Q Q Ineres Rae Premium 0 Consumpion as Share of TT ν k TT ν f h TT ν d f ( λ)dt T ν TT ν f h Physial Capial as Share of Privae Foreign Asses as Share of Toal Publi Deb as Share of Toal Foreign Deb as Share of Table A8. Opimal Reaion o Deb and Oupu wih Deb Consolidaion (opimal fisal poliy mix) Fisal Opimal Reaion Opimal Reaion Insrumens o Deb o Oupu s g γ g l = γy g =0.04 τ γl = γ y =.309 τ k γl k =0.059 γy k =.3759 τ n γl n =0 γy n =2.4 Noe: V 0 =

34 34 Inernaional Journal of Cenral Banking Deember 207 Appendix 8. Robusness o World Ineres Rae Shoks Using he new speifiaion presened in he main ex, he resuls are as shown in ables A9 and A20. Table A9. Opimal Reaion o Deb and Oupu wih Deb Consolidaion (opimal fisal poliy mix) Fisal Opimal Reaion Opimal Reaion Insrumens o Deb o Oupu s g γ g l = γy g =0 τ γl = γ y = τ k γl k =0 γy k = τ n γl n =0 γy n =2.360 Noe: V 0 =

35 Vol. 3 No. 4 Fisal Consolidaion in an Open Eonomy 35 Table A20. Welfare over Differen Time orizons wih, and wihou, Deb Consolidaion (opimal fisal poliy mix) Two Four Ten Tweny Thiry Periods Periods Periods Periods Periods Wih Consolid Wihou Consolid. a (.5236) (2.9533) (6.852) (2.6868) (7.8657) Welfare Gain/Loss a The values of he opimized feedbak poliy oeffiiens when we sar from, and reurn o, he same saus quo seady sae are γ g l =0.009,γ l =0.5206,γk l =0,γ n l =0.3032,γ g y =0.029,γ y =0,γk y =2.2564, and γn y =

36 36 Inernaional Journal of Cenral Banking Deember 207 Appendix 9. Ramsey Poliy Problem The Ramsey governmen hooses he pahs of poliy insrumens o maximize he household s expeed disouned lifeime uiliy, V, subje o he equaions of he DE. As shown in appendix 4 above, he DE sysem onains weny-four equaions (we do no inlude he equaion for V ). As said in he main ex, we solve for a imeless Ramsey equilibrium, meaning ha he resuling equilibrium ondiions are ime invarian (see also, e.g., Shmi-Grohé and Uribe 2005). If we follow he dual approah o he Ramsey poliy problem, he governmen hooses he pahs of {y,,, F,n,x,k,f h, TT, Π, Π, Θ, Δ,w,m, ω,r k,q,d, Π,z,z 2,R, l } =0, whih are he weny-four endogenous variables of he DE sysem, plus he pahs of he hree ax raes, {τ, τ k,τ n } =0. Noie ha publi deb, {d } =0, is also among he hoie variables. Also noie ha, in solving his opimizaion problem, we ake as given he rae of exhange rae depreiaion, {ɛ } =0; publi spending on goods/servies, {s g } =0; lump-sum governmen ransfers, {s l } =0; and he share of publi deb issued for he domesi marke, {λ } =0. We rea hese variables as exogenous for differen reasons: in a urreny union model, ɛ = all he ime; we ake {s g } =0 as given for simpliiy; we ake {s l } =0 as given beause, if he Ramsey governmen an hoose a lump-sum poliy insrumen, hen is problem beomes rivial; finally, we se λ exogenously and equal o is daa average value (as we have done so far anyway) beause, if i is hosen by he Ramsey governmen, is opimaliy ondiion will be idenial o he household s opimaliy ondiion for finanial asses/deb in a Ramsey seady sae wihou sovereign ineres rae premia, so he Ramsey seady-sae sysem will be indeerminae (in pariular, in he seady sae, he governmen s opimaliy ondiion for λ is redued o = βq, whih is also he household s opimaliy ondiion for foreign asses/deb). For he very same reason (namely, he opimaliy ondiions of he household and he governmen wih respe o f h are boh redued o = βq in he Ramsey seady sae wihou sovereign ineres rae premia), we se f h so as o saisfy f = 0 in he reformed seady sae (as we have done so far anyway in he reformed seady sae). Thus, from f ( λ)dt T ν TT ν f h = 0, i follows f h =( λ) dt T ν.

37 Vol. 3 No. 4 Fisal Consolidaion in an Open Eonomy 37 Table A2. Ramsey Seady-Sae Soluion (unresried) Seady-Sae Variables Desripion Soluion u Period Uiliy.225 Oupu.5676 TT Terms of Trade.0 τ Opimal Consumpion Tax Rae τ k Opimal Capial Tax Rae τ n Opimal Consumpion Tax Rae Q Q Ineres Rae Premium 0 Consumpion as Share of TT ν k TT ν f h TT ν d f ( λ)dt T ν TT ν f h Physial Capial as Share of Privae Foreign Asses as Share of Toal Publi Deb as Share of Toal Foreign Deb as Share of Working in his way, and inluding Ramsey mulipliers, we end up wih a well-defined sysem onsising of fify-one equaions in fifyone endogenous variables (he sysem is available upon reques from he auhors). The numerial soluion of his sysem in he seady sae is presened in able A2 (we do no repor soluions for Ramsey mulipliers). In his soluion, as said before, Q = Q = R = β, and f h = ( λ) dt T ν. (Noe ha nominal variables, like R, do no affe he real alloaion, and hene uiliy, a seady sae.) Noie, as also said in he ex, and as is well esablished in he Ramsey lieraure wih opimally hosen onsumpion axes, ha he soluion in able A2 is haraerized by an exremely large onsumpion ax rae, τ = , and an equally large labor

38 38 Inernaional Journal of Cenral Banking Deember 207 Table A22. Ramsey Seady-Sae Soluion (resried τ n =0) Seady-Sae Variables Desripion Soluion u Period Uiliy.0950 Oupu.622 TT Terms of Trade.0 τ k Opimal Capial Tax Rae τ Opimal Consumpion Tax Rae Q Q Ineres Rae Premium 0 Consumpion as Share of TT ν k TT ν f h TT ν d f ( λ)dt T ν TT ν f h Physial Capial as Share of Privae Foreign Asses as Share of Toal Publi Deb as Share of Toal Foreign Deb as Share of subsidy, τ n = < 0. ene, following he same lieraure, we rule ou a subsidy o labor by seing τ n = 0 a all. This gives he soluion in able A22 (his able is also presened in he main ex, bu we repea i here for he reader s onveniene). Appendix 0. Flexible Exhange Raes The IRFs under flexible exhange raes are shown in figure A3 (he zero lower bound for he nominal ineres rae is no violaed).

39 Vol. 3 No. 4 Fisal Consolidaion in an Open Eonomy 39 Figure A3. IRFs under Flexible Exhange Raes Noe: IRFs are in levels and onverge o he reformed seady sae, while he solid horizonal line indiaes he poin of deparure (saus quo value). Referenes Benigno, G., and C. Thoenissen Consumpion and Real Exhange Raes wih Inomplee Markes and Non-raded Goods. Journal of Inernaional Money and Finane 27 (6): Rabanal, P Inflaion Differenials beween Spain and he EMU: A DSGE Perspeive. Journal of Money, Credi and Banking 4 (6): 4 66.

40 40 Inernaional Journal of Cenral Banking Deember 207 Shmi-Grohé, S., and M. Uribe Opimal Fisal and Moneary Poliy under Siky Pries. Journal of Eonomi Theory 4 (2): Opimal Fisal and Moneary Poliy in a Medium- Sale Maroeonomi Model. In NBER Maroeonomis Annual, ed. M. Gerler and K. Rogoff, Cambridge, MA: MIT Press Opimal Simple and Implemenable Moneary and Fisal Rules. Journal of Moneary Eonomis 54 (6):