Technical Appendix: International Business Cycles with Domestic and Foreign Lenders

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1 Technical Appendix: Inernaional Business Ccles wih Domesic and Foreign Lenders Maeo Iacoviello Boson College Raoul Minei Michigan Sae Universi Ma 13, The model As he wo counries are smmeric, we onl describe, wihou loss of generali, agens decisions in he domesic econom Domesic enrepreneurs The producion funcion is Cobb-Douglas in domesicall locaed labor l and real esae owned b enrepreneurs: = A h ν 1l 1 ν (1.1) where producivi follows an exogenous sochasic saionar AR (1) process around a consan mean A. Enrepreneurs maximize heir lifeime uili from he consumpion flow c. Denoing wih E he expecaion operaor condiional on ime informaion and wih γ he enrepreneurs discoun facor, enrepreneurs solve he following problem: subjec o flows and borrowing consrains: max b,bf,,h,l,α E 0 X =0 γ ln c (1.2) A h ν 1l 1 ν + b + b F = c + q (h h 1 )+R 1b 1 + R 1b F F 1 + w l (1.3) R b m α q +1 h (1.4) R F b F q +1 (1 α ) h 1 1 m F (q +1 (1 α ) h ) (1.5) Enrepreneurs choose labor and real esae; how much o borrow from domesic and foreign households; how o allocae shares α of real esae beween domesic and foreign financiers. Define λ and λ F as he ime shadow values of he domesic and foreign borrowing consrain respecivel. The firs-order condiions for an opimum are he consumpion Euler equaions (1.6

2 and 1.7), real esae demand (1.8), choice of α (1.9), and labor demand (1.10): 1 γr = E + λ R (1.6) c c +1 1 γr F = E + λ F R F (1.7) c c +1 ³ γ 1 q = E c +1 ν +1 h + q +1 + λ m α q +1 c +λ F (1 α ) q +1 ³1 2(1 m (1.8) F )(1 α )q +1 h λ m = λ F E 1 2(1 m F )(1 α ) q +1 h (1.9) w l = (1 ν). (1.10) The opimal value of α. Using 1.6, 1.7, and 1.9, we can solve for he opimal α as a funcion of he credi mulipliers and of he value of real esae held b enrepreneurs as follows: α =1 1 λ m λ F (1.11) 1 m F 2q +1 h so ha in sead sae α =1 1 m 2(1 m F ). (1.12) Equaion 1.11 shows how in general he opimal value of α, he share of domesic collaeral, will be posiivel relaed o he average domesic loan-o-value raio (m ) and inversel relaed o average foreign loan-o-value raio (m F ). In a neighborhood of he sead sae, whenever asse values increase, α will increase. Tha is, asse price increases will be associaed wih a swich from foreign owards domesic lenders. Linearizing 1.11 around he sead sae, leing = v, gives: α = 1 1 λ m λ F v (1.13) 1 m F v αbα = 1 λ m λ F v 1 m F v 2 dv = 1 m bv =(1 α) bv 1 m F (1.14) bα = 1 α α bv (1.15) so ha he linearized borrowing consrain for domesic loans is br + b b = bα + bv = 1 α bv (1.16) 2

3 for foreign loans is, leing s =1 α and q h = v R F b F = s v (1 m F ) v 2 s 2 v (1.17) bdr + Rdb = vds + sdv 1 m F 2v 2 sds +2s 2 vdv (1.18) ³ v Rb br + b = sv (bs + bv ) 2(1 m F ) s (svbs + sbv ) (1.19) ³ Rb br + b = sv (bs + bv ) 2(1 m F ) s (sv)(bs + bv ) (1.20) ³ br + b = sv Rb (bs + bv )(1 2(1 m F ) s) (1.21) in a neighborhood of he sead sae bα = 1 α α bv (1.22) s = 1 α (1.23) bs = α s bα (1.24) ence 1.2. Domesic households bs = α σ 1 α α bv = bv (1.25) br F + b b F =0 (1.26) The household secor (denoed wih a prime) is convenional. In each period, household ener wih real esae holdings h 0 1 (priced a q ) and bonds coming o mauri. The derive uili from consumpion of he final good c 0 and from real esae services proporional o he housing sock h 0. The ren labor l 0 o domesic enrepreneurs a a wage w (labor is immobile across counries), lend b o domesic firms, b F o foreign firms and lend b (or borrow b )o(from) foreign households, while receiving back he amoun len in he previous period imes he agreed ineres raes, respecivel R,R F and R. Formall, households solve: X max E 0 b,bf,h0,b,l =0 β ln c 0 + j ln h 0 τ η lη (1.27) where β is households discoun facor. As we menioned alread, we assume ha γ < β.this ensures ha enrepreneurs borrowing consrains will alwas hold wih he equali sign. The flow of funds for he household is: c 0 + q h 0 h b + b F + b + ψ (b b) 2 = R 2 1b 1 + R 1b F F 1 + R 1 b 1 + w l (1.28) The ψ (b b)2 2 erm reflecs he adjusmen cos of an increase in he household porfolio holding vis-a-vis he res of he world 3

4 Solving his problem ields (choosing b,b,b F,l,h ) 1 1+ψ c 0 b b R = βe c 0 +1 (1.29) R = R F (1.30) R = R 1+ψ b b (1.31) = j q+1 h 0 + βe c 0 +1 Finall, in a sead sae wih consan consumpion, βr =1and 1.3. Foreign econom w = τc 0 l η 1 (1.32). (1.33) q c 0 b = b (1.34) The foreign econom is compleel smmeric o he domesic one and is herefore omied. We jus repor firs order condiions for foreign households involving bond adjusmen cos. The are given b: 1 1 ψ c 0 b b R = βe c 0 (1.35) +1 R = R F (1.36) R = R 1 ψ. (1.37) b b where we denoe foreign variables wih a sar. For smmer, we assume m = m and m F = m F. 2. Bonds and ineres raes The following able summarizes he range of financial asses ha are raded in he econom: smbol bond originaion issuer buer collaeral price b home home firm home household αh R b F home home firm foreign household (1 α) h R F b foreign foreign firm home household (1 α ) h R b F foreign foreign firm foreign household α h R F b (foreign) foreign household home household - R I is eas bu noaionall cumbersome o define an equilibrium. Before doing so, noice ha in equilibrium household will demand he same reurn on domesic versus foreign asses: R = R F = R 1+ψ b b (2.1) 4

5 whereas for foreign households he similar arbirage equaion will be: R = R F R = 1 ψ b b (2.2) Combine 2.1 and 2.2 o ge: 1 ψ b b = R F R and in log-linear erms: using b b = b b : 1 ψ b b = R = R 1+ψ b b = R F 1+ψ b b (2.3) br ψ b b = R F ψ b b = R = R + ψ b b = R F + ψ b b. (2.4) Taken ogeher, hese wo condiions impl ha a each dae a unique ineres rae holds for all bonds if ψ =0. 3. Equilibrium For given bond holdings b 1,b F 1,b 1,b 1, ineres raes R 1,R 1,R 1,R 1 F, domesic real esae holdings h 1,h 1 0, foreign real esae holdings h 1,h 1 0, and echnolog (A,A ), a recursive compeiive equilibrium is characerized b: 1. a pah of asse prices (q,q ); 1,b F 2. a pah of ineres raes R,R,R F,R 3. a pah of wage raes (w,w ); ³ 4. consumpion c,c 0,c,c 0 ; 5. real esae holdings ³ h,h 0,h,h 0 ;,R F ; 6. bond holding dnamics b,b F,b,b F,b ; 7. labour suppl (l,l ); 8. oupu dnamics (, ); 9. mulipliers on he credi consrains (λ,λ ); 10. shares of domesic collaeral (α,α ); such ha: 1. he bond markes clear; 2. he domesic and foreign labor markes clear; 3. he domesic and foreign real esae markes clear; 4. he world final good marke clears. 1,R F 5

6 4. Inroducing variable capial Add now variable capial k as a facor of producion. Assuming k is collaeralizable as well, we denoe z hedomesicloanovalueandσ he share of collaeral which is pledged o domesic lenders. The wo borrowing consrains will now be wrien as R b m α q +1 h + z σ k (4.1) R F bf q +1 (1 α ) h 1 (1 m F ) q +1 (1 α ) h (4.2) +(1 σ ) k 1 (1 z F ) (1 σ ) k 2k The firs order condiions for k and σ are: 1 γ = E c z = λf λ c k +1 δ + λ z σ + λ F (1 σ ) 1 (1 z F )(1 σ ) k ) k (1 z F )(1 σ ) k k (4.3) (4.4) Combining he wo expression ields: 1 γ = E c or: 1 c +1 When linearized, he equaion above becomes: bc + c δ + λ z k 1 z γ R = E δ z c +1 k βz 1 βz b R = bc γ 1 δ 1 βz + γz 1 βz ³ b +1 b k (4.5) (4.6) (4.7) 6

7 5. The sead sae The sead sae is described b he following equaions: = Rb Rb F c 0 q (1 h) c 0 = α = 1 1 m 2(1 m F ) (5.1) σ = 1 1 z 2(1 z F ) (5.2) k γ = 1 γ (1 δ) (β γ) z (5.3) γν 1 γ (β γ) m (5.4) = αm + σz k (5.5) = (1 α)(1 (1 m F )(1 α)) +(1 σ)(1 (1 z F )(1 σ)) k (5.6) Rb = (1 β) + RbF +(1 ν) (5.7) j 1 β h 1 h = γν (1 β) j (1 γ (β γ) m ) c0 c = ( + ν) δ k b (R 1) + bf (5.8) (5.9) (5.10) Solving for b and b F gives: Rb = 1 1 m 2(1 m F ) m z k z 2(1 z F ) (5.11) and Rb F = 1 m2 4(1 m F ) + 1 z2 k 4(1 z F ). (5.12) 7

8 6. The log-linear equilibrium 6.1. The model wih variable capial, endogenous α and no quadraic coss of bondholdings ere is our log-linearized model, allowing for variable capial. All he variables wih a subscrip are in % deviaions from heir sead sae levels (indicaed wihou subscrips). For inra-household bonds b, which are close o zero in sead sae, we approximae b /. Normalize sead sae labor suppl so ha income is uni, ha is = =1.Le e k = k (1 δ) k 1. Afer some algebra, some variables can be eliminaed and he equilibrium can be reduced o a dnamic ssem of 18 equaions in 18 unknowns, where he unknowns are: c c 0 h k c c 0 h k b b b F b b F R q q and A and A follows AR (1) he sochasic processes described in Table 1 of he paper. The equaions for home econom are given b: c 0 c 0 + b b + b F b F + b = R b + b R 1 + Rb 1 + Rb b 1 + Rb F b F 1 +(1 ν) + h (6.1) ( + ν) + b b + b F b F = R b + b F R 1 + Rb b 1 + Rb F b F 1 + h + cc + k k e (6.2) c 0 +1 = c 0 + R (6.3) (1 β) h q = βq h h + c 0 βc 0 +1 (6.4) (1 (1 ν ) /η) = A + νh 1 + k 1 ((1 ν ) /η) c 0 (6.5) R + b m =(1/α) m + z k (q z k +1 + h )+(1/σ) m + z k k (6.6) R + b F =0 (6.7) q (γ + m (β γ)) q +1 =(1 βm )(c c +1 )+(1 γ + m (β γ)) ( +1 h ) m βr (6.8) 0=(1 βz )(c c +1 )+(1 γ(1 δ) z (β γ)) ( +1 k ) βz R. (6.9) The equaions for he foreign econom are given b: 8

9 c 0 c 0 + b b + b F b F b = R b + b F R 1 Rb 1 + (6.10) +Rb b 1 + Rb F b F 1 +(1 ν) + q h h ( + ν) + b b + b F b F = R b + b F R 1 + (6.11) +Rb b 1 +Rb F b F 1 + q h h + c c + k k e c 0 +1 = c 0 + R (6.12) q = βq+1 (1 β) h + 1 h h + c 0 βc 0 +1 (6.13) (1 (1 ν ) /η) = A + νh 1 + k 1 ((1 ν ) /η) c 0 (6.14) R + b =(1/α m q h ) q m q h + z k +1 + h +(1/σ z k ) m q h + z k k (6.15) R + b F =0 (6.16) q (γ + m (β γ)) q+1 =(1 βm ) c c +1 +(1 γ + m (β γ)) +1 h m βr 0=(1 βz ) c c +1 +(1 γ (1 δ) z (β γ)) ( +1 k ) βz R. (6.17) (6.18) 6.2. The econom wih quadraic coss of adjusing bonds In he bond adjusmen cos model, remember he log-linear relaionship beween ineres raes: R ψb = R F ψb = R = R + ψb = R F + ψb. The mos convenien normalizaion involves expressing all ineres raes as a funcion of he inrahousehold rae. ence we have he following no-arbirage relaionships involving ineres raes: ence he following equaions change: R = R ψb R F = R ψb R = R + ψb R F = R + ψb. c 0 c 0 + b b + b F b F + b = R b + b F + b R 1 + Rb 1 + Rb b 1 + (5.1 ) +Rb F b F 1 + s + h Rb fb 1 +Rb F fb 1 ( + ν) + b b + b F b F = R b + b F R 1 + Rb b 1 + Rb F b F 1+ (5.2 ) + h + cc + k k Rb fb 1 Rb F fb 1 R + b fb =(1/α) c 0 +1 = c 0 + R fb (5.3 ) m m + z k (q z k +1 + h )+(1/σ) m + z k k (5.6 ) 9

10 R + b F fb =0 (5.7 ) q (γ + m (β γ)) q +1 = (1 βm )(c c +1 )+ (5.8 ) +(1 γ + m (β γ)) ( +1 h ) m βr +m βfb 0=(1 βz )(c c +1 )+(1 γ(1 δ) z (β γ)) ( +1 k ) βz R +βz fb. (5.9 ) The modificaions for he foreign econom are: c 0 c 0 + b b + b F b F b = R b + b F b R 1 Rb 1 + (5.10 ) +Rb b 1 + Rb F b F 1 + s + q h h + +Rb fb 1 Rb F fb 1 ( + ν) + b b + b F b F = R b + b F R 1 + Rb b 1 +Rb F b F 1 + (5.11 ) q h h + c c + k k +Rb fb 1 +Rb F fb 1 c 0 +1 = c 0 + R +fb (5.12 ) R + b +fb =(1/α ) m q h q m q h + z k +1 + h +(1/σ ) z k m q h + z k k (5.15 ) R + b F +fb =0 (5.16 ) q (γ + m (β γ)) q+1 =(1 βm ) c c +1 + (5.17 ) +(1 γ + m (β γ)) +1 h m βr m βfb 0=(1 βz ) c c +1 + (5.18 ) (1 γ (1 δ) z (β γ)) ( +1 k ) βz R βz fb The econom wih exogenous α In he exogenous-α econom, he linearized borrowing consrains (equaions 5.6, 5.7, 5.15 and 5.16) become: R + b = R + b F = R + b = R + b F = m m + z k (q z k +1 + h )+ m + z k k (5.6 ) m m + z k (q z k +1 + h )+ m + z k k (5.7 ) m m + z k (q z k +1 + h )+ m + z k k (5.15 ) m m + z k (q z K k +1 + h )+ m + z k k. (5.16 ) 10