Appendix to Monetary and Macroprudential Policy in an Estimated DSGE Model of the Euro Area. January 8, 2014

Transcription

1 Appendix o Moneary and Macroprudenial Policy in an Esimaed DSGE Model of he Euro Area Dominic Quin Pau Rabanal January 8, 24

2 Inroducion This appendix conains furher deails on he heoreical model and is esimaion. Secion 2 describes he derivaion of he model in greaer deail. Secion 3 derives he seady sae of he model, while Secion 4 provides addiional deails on he properies of log-normal disribuions ha are needed o log-linearizing he model. Secion 5 provides addiional robusness resuls on he Bayesian esimaion of he model. 2 The Model The model can be summarized as follows: Two-counry model of he euro area, wih a home counry H of size n (he core) and foreign counry F of size n (he periphery). In each counry here are wo ypes of agens: savers (of mass ) and borrowers (of mass ). There are wo secors in each counry: non-durable and durable goods. Boh ypes of goods are produced under monopolisic compeiion and nominal rigidiies. The producion funcion is linear in labor in all secors. Non-durable consumpion goods are raded across counries, while durable goods are nonradable and used o increase he housing sock. In each counry, savers and borrowers consume non-durable goods, purchase durable goods and provide labor o boh secors. Borrowers are more impaien han savers and have preference for early consumpion, which creaes he condiion for credi o occur in equilibrium. Borrowers use heir housing sock as collaeral o gain access o credi. We adap he mechanism of Bernanke, Gerler and Gilchris (999), henceforh BGG, o he household side and o residenial invesmen. Shocks o he valuaion of housing a ec he balance shee of borrowers, which in urn a ec he defaul rae on morgages and he lending-deposi spread. Inernaional nancial inermediaries channel funds from one counry o he oher. Savings and (residenial) invesmen need no o be balanced a he counry level period by period. 2

3 Moneary policy is conduced by a cenral bank ha arges he union-wide CPI in aion rae, and also reacs o ucuaions in he union-wide real GDP growh. Macropudenial policy in uences credi marke condiions by a ecing he fracion of liabiliies (deposis and loans) ha banks can lend. This insrumen can be hough of as addiional capial requiremens, liquidiy raios, reserve requiremens or loan-loss provisions ha reduce he amoun of loanable funds by nancial inermediaries and increase credi spreads. Macroprudenial policy in uences credi marke condiions above and beyond curren regulaions. I arges credi spreads by a ecing he fracion of liabiliies (deposis and bonds) ha nancial inermediaries can lend. Spreads can be increased by imposing e.g. addiional capial surcharges, liquidiy raios, loan-loss provisions, or reserve requiremens, whereas he direc provision of liquidiy o he banking secor (eiher hrough convenional or unconvenional policies) can decrease spreads. This could be achieved via measures such as widening of collaeral sandards, he Funding for Lending Scheme launched by he Bank of England in 22, or even liquidiy provision o he real economy as in Gerler and Karadi (2). In wha follows, we presen he home counry block of he model, by describing he domesic and inernaional credi markes, households, and rms. The foreign counry block has a similar srucure and, o save space, is no presened. Unless speci ed, all shocks follow zero-mean AR() processes in logs. 2. Credi Markes We adap he BGG nancial acceleraor idea o he housing marke, by inroducing defaul risk in he morgage marke, and a lending-deposi spread ha depends on housing marke condiions. There are wo main di erences wih respec o he BGG mechanism. Firs, here are no agency problems or asymmeric informaion in he model, and borrowers will only defaul if hey nd hemselves underwaer: ha is, when he value of heir ousanding deb is higher han he value of he house hey own. Second, unlike he BGG seup, we assume ha he one-period lending rae is pre-deermined and does no depend on he sae of he economy, which seems o 3

4 be a more realisic assumpion. 2.. Domesic Inermediaries Domesic nancial inermediaries collec deposis from savers S, for which hey pay a deposi rae R, and exend loans o borrowers S B for which hey charge he lending rae R L. Credi graned o borrowers is backed by he value of he housing sock ha hey own (P D D B ), where P D are nominal house prices and D B is he level of he housing sock owned by borrowers. We inroduce risk in he credi and housing markes by assuming ha each borrower (indexed by j) is subjec o an idiosyncraic qualiy shock o he value of her housing sock,! j, ha is log-normally disribued wih CDF F (!). We choose he mean and sandard deviaion so ha E! j = : There is idiosyncraic risk bu no aggregae risk in he housing marke. This assumpion implies ha log(! j ) N( 2!; ; 2 2!;), wih!; being he sandard deviaion characerizing he qualiy shock. This sandard deviaion is ime-varying, and follows an AR() process in logs: log(!; ) = (! ) log(! ) +! log(!; ) + u!; ; wih u!; N(; u! ). The suppor of he log-normal disribuion is (; ), meaning ha! j canno become negaive. Figure () plos he log-normal disribuion wih di eren values of! (.25 and.33). An increase in!; is mean-preserving, raising only he skewness of he disribuion of! j. Thus, wih a higher sandard deviaion, more mass of he disribuion is concenraed on he lef and lower values for! j become more likely. The qualiy shock! j can lead o morgage defauls and a ecs he spread beween lending and deposi raes. Borrowers use heir housing sock as collaeral o gain access o credis. The value of collaeral is a eced by qualiy shocks and he realizaion of hese shocks is known a he end of he period (afer credis have already been graned and he loan rae has been se). Hence, he value of he housing! j P D D B migh no be su cien o fully repay he loan. Wih high realizaions of! j, he value of he housing sock is higher han he ousanding deb (! j P D D B > R L S B ) and households repay he full amoun of heir ousanding loan R L S B. Realizaions of! j ha are so low ha! j P D D B < R L S B, A similar approach is aken by Suh (22) and Zhang (29). 4

5 2.8 σ ω =.25 σ ω = Figure : E ec of an Increase in!; on he Probabiliy Disribuion of! j however force he household o defaul on her loan in period. Afer he household defauls on is loan, he bank calls a deb collecion agency ha forces he household o repay he value of he housing sock afer he shock has realized,! j P D D B. Afer paying his amoun, he household keeps her house. These deb-collecion agencies charge banks a fracion of he value of he house. The pro s of hese agencies are ransferred o savers, who own hem. The value of he idiosyncraic shock is common knowledge, so ha households will only defaul when hey are underwaer. 2 When graning credi, nancial inermediaries also do no know he hreshold! which de nes he cu-o value of hose households ha defaul and hose who do no. The ex-ane hreshold value expeced by banks is given by:! a E P D + D B + = R L S B : () Thus, he hreshold! a is he value of! j a which borrowers are expeced o be indi eren beween repaying and defauling. Noice ha! a is increasing in he 2 BGG originally assume an agency problem: To observe he nal qualiy of he collaeral! j P D D B, nancial inermediaries mus pay a monioring cos proporional o he collaeral! j P D D B. Under our assumpion however, no fracion of he housing sock is desroyed during he foreclosure process. If, as in BGG, a fracion of he collaeral was los during foreclosure, risk shocks migh have unrealisic expansionary e ecs on housing and residenial invesmen. See Forlai and Lamberini (2). Suh (22) also assumes ha households ha defaul on heir loan pay he value of heir house and ge o keep i. 5

6 expeced loan-o-value (LTV) raio S B =E P D + D B +. Given he ex-ane hreshold, we can now use he CDF of he qualiy shock o de ne he fracion of loans which nancial inermediaries expec o be underwaer in he nex period + : F (! a ;!; ) = Z! a and he fracion of loans which are expeced o be repaid: Nex, we de ne [ F (! a ;!; )] = G (! a ;!; ) Z Z! a df (!;!; )d!; (2)! a df (!;!; )d!: (3)!dF (!;!; ) as he mean value of he qualiy shock condiional on he shock being less han he hreshold! a. We can now also denoe he mean value of he housing sock, which nancial inermediaries expec o be underwaer and will be urned over by borrowers: Z! a G (! a ;!; ) P+D D + B =!P+D D +df B (!;!; ): (4) We inroduce a macroprudenial insrumen ha in uences credi marke condiions by a ecing he fracion of liabiliies ha banks can lend. The balance shee of nancial inermediaries is: n (S B ) = n ( ) S B ; (5) where B are claims on nancial inermediaries in he foreign counry. We assume ha his insrumen is imposed above and beyond curren regulaions. Hence, we assume ha = in he esimaed version of he model, and ha i varies counercyclically in he welfare analysis secion. We can hink of he macroprudenial insrumen as addiional capial surcharges, loan-loss provisions, or reserve requiremens ha resric he amoun of loanable funds and a ec he spread direcly. The macroprudenial insrumen may also ake values smaller han one. In his case, he cenral bank aims a lowering he spread. This could be implemened e.g. by unconvenional moneary policies in he spiri of Gerler and Karadi (2). As S ; B and S B are per-capia quaniies, we need o muliply hem by populaion size n and n ( ). Inermediaries require he expeced reurn from graning 6

7 credi o be equal o he funding rae of banks, which equals he deposi rae R : nr (S B ) = n ( ) E ( ) Z! a!df (!;!; )P D +D B + + [ F (! a ;!; )] R L S B = n ( ) E ( )G (! a ;!; ) P D +D B + + [ F (! a ;!; )] R L S B : (6) Equaion (6) describes he paricipaion consrain of nancial inermediaries. I ensures ha heir obligaions o debors (lef-hand side) are equal o he expeced repaymen by crediors, which is given by he expeced foreclosure selemen (he rs erm of he righ hand side of equaion 6) and he expeced repaymen of households wih higher housing values (he second erm). Due o he fees paid o deb-collecion agencies o make defauling households pay heir debs, nancial inermediaries only receive a fracion ( ) of he morgage selemen. We can use he marke clearing condiion (5) o rewrie he paricipaion consrain as: R = E ( )G (! a ;!; ) P +D D + B + [ F (! a S B ;!; )] R L : (7) For a given demand of credi from borrowers, observed values of risk!;, expeced values of he housing sock, and a given macroprudenial policy sance, inermediaries passively se he lending rae R L and he expeced (ex-ane) hreshold! a so ha equaion () and he paricipaion consrain (7) are ful lled. Unlike he original BGG se-up, he one-period lending rae R L is deermined a ime, and does no depend on he sae of he economy a +. This means ha he paricipaion consrain of nancial inermediaries delivers ex-ane zero pro s. However, i is possible ha, ex-pos, hey make pro s or losses. We assume ha savers collec pro s or recapialize nancial inermediaries as needed. The paricipaion consrain delivers a posiive relaionship beween he LTV raio S B =E P D + D B + and he spread beween he funding and he lending rae, due o he probabiliy of defaul. This becomes obvious, when we rewrie he paricipaion consrain (7) as: R L R 8 < = E : ( )G(! a ;!;)! a 9 = + [ F (! a ;!; )]; : (8) Le s rs assume ha = = so ha no macroprudenial policies are in place 7

8 and, in case of defaul, he nancial inermediary recovers nohing from he defauled loan. According o equaion (), he higher is he LTV raio, he higher is he hreshold! a ha leads o defaul. This shrinks he area of no-defaul [ F (! a ;!; )], and herefore increases he spread beween R L and R. Similarly, an increase in he sandard deviaion!; increases he spread beween he lending and he deposi raes. When!; rises, i leads o a mean-preserving spread for he disribuion of! j : he ails of he disribuion become faer while he mean remains unchanged (as in Figure ). As a resul, lower realizaions of! j are more likely so ha more borrowers will defaul on heir loans. More generally, when he nancial inermediary is able o recover a fracion ( ) of he collaeral value, i can be shown (using he properies of he lognormal disribuion when E [! ] = ) ha he denominaor in he spread equaion (8) is always declining in! a ; and hence he spread is always an increasing funcion of he LTV. Furhermore, a ighening of credi condiions due o macroprudenial measures, re eced in a higher, will increase he spread faced by borrowers. As nancial inermediaries canno use he full amoun of heir liabiliies o gran credi bu only a fracion =, hey will pass hese coss o heir cusomers. Finally, we assume ha he deposi rae in he home counry equals he risk-free rae se by he cenral bank. In he foreign counry, domesic nancial inermediaries behave he same way. In heir case, hey face a deposi rae R and a lending rae R L, and he spread is deermined in an analogous way o equaion (7), including a macroprudenial insrumen. We explain below how he deposi rae in he foreign counry R is deermined Inernaional Inermediaries Inernaional nancial inermediaries buy and sell bonds issued by domesic inermediaries in boh counries. For insance, if he home counry domesic inermediaries have an excess B of loanable funds, hey will sell hem o he inernaional inermediaries, who will lend an amoun B o foreign counry domesic inermediaries. Inernaional inermediaries apply he following formula o he spread hey charge beween bonds in he home counry (issued a an ineres rae R ) and he foreign 8

9 counry (issued a R ): R = R + # exp B B P C Y C : (9) The spread depends on he raio of real ne foreign asses B =P C o seady sae non-durable GDP Y C in he home counry (o be de ned below). When home counry domesic inermediaries have an excess of funds ha hey wish o lend o he foreign counry domesic inermediaries, hen B > : Hence, he foreign counry inermediaries will pay a higher ineres rae R > R. The parameer B denoes he risk premium elasiciy and # is a risk premium shock, which increases he wedge beween he domesic and he foreign deposi raes. Inernaional inermediaries are owned by savers in each counry and opimaliy condiions will ensure ha he ne foreign asse posiion of boh counries is saionary. 3 pro s (R They always make posiive R ) B, which are equally spli across savers of boh counries. 2.2 Households In each counry a fracion of agens are savers, while he res are borrowers Savers Savers indexed by j 2 [; ] in he home counry maximize he following uiliy funcion: ( X E " C log(c j "C ) + ( ) D log(d j ) = L j +' + ' #) ; () where C j, D j, and L j represen he consumpion of he ow of non-durable goods, he sock of durable goods (housing) and he labor disuiliy of agen j. We assume exernal habis in non-durable consumpion, wih " measuring he in uence of pas aggregae non-durable consumpion C on he curren uiliy level. The uiliy funcion is hi by wo preference shocks, a ecing he marginal uiliy of eiher non-durable consumpion ( C ) or housing ( D ). The parameer sands for he 3 Hence, he assumpion ha inernaional inermediaries rade unconingen bonds amouns o he same case as allowing savers o rade hese bonds. Under marke incompleeness, a risk premium funcion of he ype assumed in equaion (9) is required for he exisence of a well-de ned seady sae and saionariy of he ne foreign asse posiion. See Schmi-Grohé and Uribe (23). 9

10 discoun facor of savers, measures he share of non-durable consumpion in he uiliy funcion, and ' denoes he inverse elasiciy of labor supply. Moreover, non-durable consumpion is an index composed of home (C j H; ) and foreign (Cj F; ) goods: C j = C C j C C H; + ( ) C C j C C F; C C ; () wih 2 [; ] governing he preference for domesic over foreign goods and C > being he elasiciy of subsiuion beween hese wo ypes of goods. In seady sae will be he fracion of domesically produced non-durables a home, while denoes he fracion of impored consumpion goods. Goods produced in he home and foreign counry are only imperfecly subsiuable, only for C! hey become perfec subsiues. Similarly, we inroduce imperfec subsiuabiliy beween he labor supply o he durable and non-durable secor o explain comovemen of hours worked a he secor level: L j = L L C;j +L + ( ) L L D;j +L + L : (2) The labor disuiliy index consiss of hours worked in he non-durable secor L C;j and durable secor L D;j, wih denoing he seady sae share of employmen in he non-durable secor. Reallocaing labor across secors is cosly, and is governed by parameer L. Noe ha when L = he aggregaor is linear in hours worked in each secor and here are no coss of swiching beween secors. Wages are exible and se o equal he marginal rae of subsiuion beween consumpion and labor in each secor. 4 The budge consrain of savers in nominal erms reads: P C C j + P D I j + S j R S j + W C L C;j + W D L D;j + j ; (3) where P C and P D are he price indices of non-durable and durable goods, respecively, which are de ned below. Nominal wages paid in he wo secors are denoed by W C and W D. Savers allocae heir expendiures beween non-durable consumpion C j and residenial invesmen I j. They have access o deposis in he domesic nancial sysem S j, ha pay he deposi ineres rae R. In addiion, savers also receive pro s j from inermediae goods producers in he durable and he non- 4 When L >, wages can di er across secors. Only if L = and he elasiciy of subsiuion beween he supply of labor o he wo secors becomes in nie, wages are he same in boh secors.

11 durable secor, from domesic and inernaional nancial inermediaries, and from deb-collecion agencies ha charge fees o domesic nancial inermediaries o make defauling households pay heir debs. Purchases of durable goods (which is he same as residenial invesmen, I j ) are used o increase he housing sock D j wih a lag, according o he following law of moion: D j = ( )D j + " z Ij I j 2!# I j ; (4) where denoes he depreciaion rae and z () re ecs an adjusmen cos funcion. This cos funcion can help he model o replicae hump-shaped responses of residenial invesmen o shocks, and reduce residenial invesmen volailiy. To do so, z () is a convex funcion, which in seady sae mees he following crieria: z = z = and z >. We discuss below, in he maximizaion problem of borrowers, he reason why we inroduce a lag in he law of moion (4). The household decision can be separaed in wo sages. On he rs sage, households decide on he allocaion of heir spending beween non-durable and durable goods and he labor supply o he non-durable and durable secor. In a second sep, households decide on he allocaion of non-durable consumpion expendiures beween home and foreign goods aking he following budge consrain ino accoun: P C C = P H; C H; + P F; C F; ; where P H; sands for he price charged for home non-durabable goods C H; and P F; denoes he price for foreign non-durabable goods C F;. Solving he uiliy maximizaion problem of savers we ge a sandard Euler equaion for he consumpion of non-durable goods: 5 = R E " P C P+ C C + ogeher wih he demand for durable goods: C C "C C + "C # ; (5) ( ) D D = % ( ) E % + ; (6) where % is he Lagrange muliplier associaed wih he law of moion for he housing 5 Since all savers behave he same way, we henceforh drop he j subscrip.

12 sock (4). The invesmen decision (derivaive wih respec o I ) is given by: C P D C "C P C I = E % + z z I I I I I " # E % +2 z I+ I+ : (7) I I Equaion (6) and (7) deermine he allocaion of spending beween non-durable and durable goods. The decision by savers on how o spli heir labor supply beween he wo secors of he economy is: L L ' L ( ) L L ' L L C L D L C = W C ; C "C L C = W D ; (8) C "C aking boh wages W C and W D as given. Given he oal amoun of non-durable consumpion spending C households decide on he allocaion beween home and foreign goods according o: C H; = PH; P C C F; = ( ) c C ; (9) PF; P C c C ; (2) while he price index for non-durable consumpion akes he following form: P C C = (P H; ) C + ( ) (P F; ) C : (2) Borrowers Borrowers di er from savers along hree main dimensions. Firs, heir preferences are di eren. The discoun facor of borrowers is smaller han he respecive facor of savers ( B < ), and we allow for di eren habi formaion coe ciens " B. Second, borrowers do no earn pro s from inermediae goods producers, nancial inermediaries, or deb-collecion agencies. Finally, as discussed above, borrowers are subjec o a qualiy shock o he value of heir housing sock! j. 6 Since bor- 6 We could also assume ha savers are hi by a housing qualiy shock. Since hey do no borrow and use heir housing sock as collaeral, his qualiy shock would no have any macroeconomic 2

13 rowers are more impaien, in equilibrium, savers are willing o accumulae asses as deposis, and borrowers are willing o pledge heir housing wealh as collaeral o gain access o loans. Analogously o savers, he uiliy funcion for each borrower j 2 [; ] reads: E 8 >< >: X = B; C log(c B;j " B C B ) + ( ) D log(d B;j ) L B;j + ' +' 39 >= 7 5 >; ; where all variables and parameers wih he superscrip B denoe ha hey are speci c o borrowers. The indices of consumpion and hours worked, as well as he law of moion of he housing sock have he same funcional form as in he case of savers: C B;j = L B;j = " C C B;j H; L L C;B;j D B;j = ( )D B;j + C C + ( ) +L + ( " z IB;j I B;j 2 C C B;j F; ) L!# (22) # C C C C ; (23) L D;B;j +L + L ; (24) I B;j : (25) Residenial invesmen I B;j increases he housing sock wih a lag. We make his assumpion because an conemporaneous increase would have unrealisic consequences for defauls: borrowers would inves in housing which is already underwaer. The budge consrain for borrowers di ers beween hose who repay heir loans in full: P C C B;j and hose who defaul: + P D I B;j + R L S B;j S B;j + W C L C;B;j + W D L D;B;j ; (26) P C C B;j + P D I B;j +! j P D D B;j S B;j + W C L C;B;j + W D L D;B;j : (27) Independen of he decision o repay or defaul, borrowers consume non-durables C B;j, inves in he housing sock I B;j, supply labor o boh secors (L C;B;j L D;B;j ), and obain loans S B;j and borrowers are paid he same wages W C impac. and from nancial inermediaries. Furhermore, savers and W D in boh secors, as hiring rms 3

14 are no able o discriminae ypes of labor depending on wheher a household is a saver or a borrower. Borrowers, who decide o repay heir loans from las period, pay R L S B;j wih R L being he lending rae which has been xed in he previous period. On he conrary, hose who defaul pay an amoun! j P D D B;j o he bank, afer being conaced by a deb-collecion agency. This fracion of he housing sock is kep by he households ha defauled on heir loans. We de ne! p as he ex-pos hreshold value for which a borrower is jus willing o repay he loan:! p P D D B = R L S B : (28) The ex-pos hreshold! p is he de faco cu-o value of hose households ha defaul and hose who do no afer aggregae and idiosyncraic shocks have hi he economy. As nancial inermediaries do no know his ex-pos hreshold when graning credi, hey form he expeced ex-ane hreshold! a as de ned by equaion (). As he housing sock D B ogeher wih he lending rae R L are pre-deermined variables and are no a funcion of he sae of he economy, i is possible ha! a and! p di er. Noe, however, ha when he loan is signed,! a = E! p. Given he hreshold! p, we can now use he CDF of he qualiy shock o de ne he de faco fracion of loans which are underwaer: Z! p F! p ;!; = he de faco fracion of loans which are repaid by borrowers: F! p ;!; = Z df (!;!; )d!; (29)! p df (!;!; )d!; (3) ogeher wih he de faco mean value of he housing sock, which borrowers pay o nancial inermediaries afer a deb-collecion agency has inervened: Z! p P D G! p ;!; D B = P D!dF (!;!; )D B : (3) Aggregaing he borrower s budge consrains (26) and (27), using he expressions (28)-(3), and dropping he j superscrips, we obain: P C C B + P D S B + W C L C;B I B + G! p ;!; D B + F! p ;!; R L S B + W D L D;B : (32) 4

15 Before deriving he rs order condiions o he borrowers problem, we rewrie he budge consrain rs by inroducing he average ineres rae of hose borrowers who defaul on heir housing sock: R D = G!p ;!; P D D B S B Noe ha R D is he ex-pos rae of reurn on defauled loans (excluding he fracion nancial inermediaries need o pay o deb-collecion agencies). The iming in R L and R D is hus consisen: he lending rae for hose who fully repay is known in advance and is a conracual obligaion, while he average reurn on hose loans ha defaul is only known a ime. The budge consrain for borrowers nally akes he following form: P C C B S B + W C L B;C + P D I B + R D + F! p ;!; R L S B : + W D L B;D ; (33) leading o an Euler equaion for borrowers of he following form: = B E ( [ F (! p ;!; )] R L + R D + " P C P+ C C + C #) C B " B C B : (34) C B + " B C B The demand for durable goods ogeher wih he invesmen decision are given by: ( ) D D B C P D C B " B C B P C = % B B ( ) E % B +; (35) I = B E % B B + z I z B I B I B I B I B " + B 2 I E % B +2z B + I B 2 # + ; (36) I B I B wih % B being he Lagrange muliplier associaed wih he law of moion for he housing sock of borrowers (25). Impaien households spli heir labor supply according o: L L B ( ) L L B ' L L C;B L = ' L L D;B L = C W C ; C B " B C B C W D : (37) C B " B C B 5

16 The allocaion of non-durable consumpion expendiures beween home and foreign goods is analogous o he decision by savers: C B H; = PH; P C C B F; = ( ) c C B ; (38) c C B ; (39) PF; wih he price index for non-durable consumpion P C being given by equaion (2), which is he Consumer Price Index for he whole counry since i is he same for borrowers and savers. To obain he oal demand for home and foreign non-durable goods CH; T OT and, respecively, we combine he demand funcions (9) wih (38) and (2) wih C T OT F; (39): P C C T OT H; = PH; P C C T OT F; = ( ) c C T OT ; (4) c C T OT ; (4) PF; wih C T OT = C + ( ) C B de ning oal consumpion of non-durable goods in he home counry. The maximizaion problem of savers and borrowers in he foreign counry is similar o he problem of hese agens in he home counry. All funcional forms for preferences are he same across counries, we merely allow he parameer value for governing he preference for domesic over foreign goods o be di eren across counries, i.e. we di ereniae beween and. P C 2.3 Firms, Technology, and Nominal Rigidiies In each counry, homogeneous nal non-durable and durable goods are produced using a coninuum of inermediae goods in each secor (indexed by h 2 [; n] in he home, and by f 2 [n; ] in he foreign counry). Inermediae goods in each secor are imperfec subsiues of each oher, and here is monopolisic compeiion as well as saggered price seing à la Calvo (983). Inermediae goods are no raded across counries and are solely bough by domesic nal goods producers. In he nal goods secor, non-durables are sold o domesic and foreign households. 6

17 Thus, for non-durable consumpion we need o disinguish beween he price level of domesically produced non-durable goods P H;, of non-durable goods produced abroad P F;, and he consumer price index P C, which will be a combinaion of hese wo price levels (as given by equaion 2). 7 Durable goods are solely sold o domesic households, who use hem o increase heir housing sock. Boh nal goods secors are perfecly compeiive, operaing under exible prices Final Goods Producers Final goods producers in boh secors aggregae he inermediae goods hey purchase according o he following producion funcion: Y k " Z n k Y k (h) k k n dh # k k ; for k = C; D; (42) where Y k represens he nal goods produced from inermediae goods Y k (h), while k denoes he price elasiciy of inermediae goods. Final goods producers purchase non-durable inermediae goods a a price of P H (h) and durable inermediae goods a a price P D (h). Pro maximizaion leads o he following demand funcion for individual inermediae goods: Y C (h) = P H P H (h) C Y H ; and Y D (h) = P D P D (h) D Y D : (43) Price levels for domesically produced non-durables P H are obained hrough he usual zero-pro condiion: and durable nal goods P D P H n Z n P H (h) C C dh ; and P D n Z n P D (h) D D dh : (44) 7 The law of one price holds for individual goods and, herefore, P H; and P F; are he same in boh counries. However, he CPI index in he foreign counry di ers from he one in he home counry due o di eren preferences for domesic over foreign goods: P C C = (P F; ) C + ( ) (P H; ) C. 7

18 2.3.2 Inermediae Goods Producers Inermediae goods are produced under monopolisic compeiion wih producers facing saggered price seing in he spiri of Calvo (983). In each period, only a fracion C ( D ) of inermediae goods producers in he non-durable (durable) secor receive a signal o re-opimize heir price. For he remaining fracion C ( D ) we assume ha heir prices are parially indexed o lagged secor-speci c in aion (wih a coe cien C, D in each secor). In boh secors, inermediae goods are produced solely wih labor: Y C (h) = A Z C L C (h); Y D (h) = A Z D L D (h); for all h 2 [; n]: (45) The producion funcions include counry- and secor-speci c saionary echnology shocks Z C and Z D, each of which follows a zero mean AR()-process in logs. In addiion, we inroduce a non-saionary union-wide echnology shock, which follows a uni roo process: log (A ) = log (A ) + " Z : This shock inroduces non-saionariy o he model and gives a model-consisen way of derending he daa by aking logs and rs di erences o he real variables ha inheri he random walk behavior. In addiion, i adds some correlaion of echnology shocks across secors and counries, which is helpful from he empirical poin of view because i allows o explain comovemen of main real variables. Since labor is he only producion inpu, cos minimizaion implies ha real marginal coss in boh secors are given by: MC C =P H; = W C A Z C ; MC D = W D A Z D =P D : (46) Inermediae goods producers in he durable secor face he following maximizaion problem: 82 X >< Max P D (h)e k 6 D ;+k 4 >: k= P D (h) P D D +k P D P D +k MC D +k Y D +k (h) 9 >= >; 8

19 subjec o fuure demand Y D +k (h) = " P D (h) P D P+k D +k P D D # D Y D +k; where ;+k = k +k is he sochasic discoun facor, wih being he marginal uiliy of non-durable consumpion of savers (since hey are he owner of hese rms). The FOC of he opimizaion problem is given by: ^P D (h) P D = D D 2 X ky k k D +k E k= s= 6 4 X ky k k D +k k= (P D +s =P D +s 2) D s= P D +s =P D +s (P D +s =P D +s 2) D P D +s =P D +s! D MC D +k Y D! D Y D +k +k 3 ; (47) 7 5 where ^P D (h) is he opimal price of durables chosen a ime if he producer can reconsider is price in his period. The fracion D of producers, which can opimize heir prices a ime, face he same decision problem and, herefore, choose he same price ^P D (h) = ^P D. Since he remaining fracion D of prices, which are no revised, are parially linked o he pas in aion, he evoluion of he durable secor price level is given by: P D = D ^P D D + ( D )[P D (P D =P D 2) D D ] D : (48) Producers in he non-durable secor face a similar maximizaion problem wih he appropriae change of noaion. 2.4 Closing he Model 2.4. Marke Clearing Condiions For inermediae goods, supply equals demand. We wrie he marke clearing condiions in erms of aggregae quaniies and, hus, muliply per-capia quaniies by populaion size of each counry. In he non-durable secor, producion is equal o domesic demand by savers C H; and borrowers CH; B and expors (consising of 9

20 demand by savers CH; and borrowers CB H; from he foreign counry): ny C = n C H; + ( ) CH; B + ( n) CH; + ( ) CH; B : (49) Durable goods are only consumed by domesic households and producion in his secor is equal o residenial invesmen for savers and borrowers: ny D = n I + ( ) I B : (5) In he labor marke oal hours worked has o be equal o he aggregae supply of labor in each secor: Z n L k (h)dh = Z n L k;j dj + ( ) Z n L k;b;j dj; for k = C; D: (5) Credi marke clearing implies ha for domesic credi and inernaional bond markes, he balance shees of nancial inermediaries are sais ed: n(s B )= = n ( ) S B ; (52) nb + ( n) B = : Finally, aggregaing he resource consrains of borrowers and savers, and he marke clearing condiions for goods and nancial inermediaries, we obain he law of moion of bonds issued by he home-counry inernaional nancial inermediaries. This can also be viewed as he evoluion of ne foreign asses (NFA) of he home counry: nb = nr B (53) + ( n) P H; CH; + ( ) CH; B np F; CF; + ( ) CF; B ; which is deermined by he aggregae sock of las period s NFA imes he ineres rae, plus ne expors Moneary Policy and Ineres Raes Moneary policy is conduced a he currency union level by he cenral bank wih an ineres rae rule ha arges union-wide CPI in aion and real oupu growh. The cenral bank ses he deposi rae in he home counry, and he oher raes 2

21 are deermined as described in he model. Le EMU be he seady sae level of union-wide CPI in aion, R he seady sae level of he ineres rae and " m moneary policy shock, he ineres rae rule is given by: R = R P EMU =P EMU EMU Y EMU =Y EMU y an iid R R R exp(" m ): (54) The euro area CPI P EMU and real GDP Y EMU are given by geomeric averages of he home and foreign counry variables, using he counry size as a weigh: P EMU = P C n P C n ; and Y EMU = (Y ) n Y n ; where he naional GDPs are expressed in erms of non-durables: Y = Y C + Y D P D P C ; and Y = Y C + Y D P D P C : Macroprudenial Policy As shown in equaions (8) and (52), he macroprudenial insrumen a ecs he equilibrium in he domesic credi marke and a ecs he lending-deposi spread in each counry. We inerpre his macroprudenial insrumen as being deployed above and beyond curren rules, which are saic o a large degree. Hence, when we esimae he model, we se o a consan value of one. When we conduc an opimal macroprudenial policy exercise, we allow he insrumen o be changed in order o maximize he weighed uiliy of all he ciizens in he moneary union. A ighening of macroprudenial policies will be re eced in a higher, which will ranslae ino a higher lending-deposi spread. Alhough we leave i unspeci ed, his could be implemened via addiional capial surcharges, liquidiy raios, loanloss provisions, or reserve requiremens ha reduce he amoun of loanable funds by nancial inermediaries. We assume ha he insrumen, in principle, can behave symmerically and i can go below one. In ha case, he cenral bank or any oher regulaory agency would provide liquidiy o he banking secor o reduce he lending-deposi spread. This could be achieved via (convenional or unconvenional) measures like a widening of collaeral sandards, he Funding for Lending Scheme launched by he Bank of England in 22, or even a direc provision of liquidiy o he real economy as in Gerler and Karadi (2). 2

22 In he welfare maximizing exercise, we specify he macroprudenial insrumen as reacing o an indicaor variable ( ): = ( ) ; = ( ) : (55) We sudy wo main cases. In each counry he macroprudenial insrumen reacs o: (i) nominal credi growh, or (ii) he credi-o-gdp raio. For boh cases, he parameers and are eiher allowed o be di eren, or are forced o be he same in he moneary union. In all cases, he indicaor reacs o deviaions from seady sae values. 3 Seady Sae We assume a seady sae in aion of zero. The rade balance ogeher wih he ne inernaional posiion of boh economies are zero. Since we calibrae he wo counries symmerically, all relaive prices in all secors equal o one and all percapia quaniies are he same across counries. Therefore, we only need o solve for he per-capia values of he home counry. Given he seady sae cu-o poin for defauling on a loan!, he defaul rae on loans F (!;! ) and he fac ha! = 2 2!, we use he CDF of he log-normal disribuion o obain a value for he sandard deviaion of he qualiy shock (! ). Using! ogeher wih!, we can solve for he mean value of he qualiy shock condiional on he shock being less han he hreshold!: 8 G (!;! ) = Z! 2!F (!;! )d! = 2! ln! ;! wih being he CDF of he sandard normal disribuion. Taking he Euler equaion of savers (5), he lending rae in he currency union is given by he discoun facor of savers: R = : The seady sae leverage raio is deermined by he hreshold value! and he lending rae R L : 8 See also Secion 4 of his Appendix. ~S B D B =! R L ; 22

23 where ~ S B are ousanding loans in real erms (divided by he CPI). We can now use he paricipaion consrain (7) of nancial inermediaries and he fac ha = o ge an expression for he lending rae: R = ( )G (!;! ) RL! + [ F (!;!)] R L : The seady sae average ineres rae of hose who defaul on heir housing sock is: R D = G (!;!) R L :! From he Euler equaion of borrowers (34) we derive he discoun facor of impaien consumers: B = F (!;! ) + G (!;!) R L :! Since in seady sae he adjusmen coss of invesmen are zero, he raio of nondurable o durable consumpion for savers and borrowers is given by: C [ ( )] = ; (56) D ( )( ") C B D = B ( ) B B ( )( " B ) B ; (57) which we obain by combining equaion (6) wih equaion (7) and equaion (35) wih equaion (36). Since he degree of monopolisic compeiion is he same in he durable and non-durable good secor ( C = D = ), we obain from he pricing equaions (47) he level of real wages as: W W C = W D = : (58) Having equal wages across secors, he seady sae supply of labor for savers is: L C = L; L D = ( )L; and analogously for borrowers: L C;B = L B ; L D;B = ( )L B : 23

24 Turning rs o he consumpion expendiures and oal labor supply of borrowers. From he law of moion for he housing sock, we know ha I B = D B so ha he budge consrain (32) can be wrien as: C B + [ + G (!;! )] D B + [ F (!;! )] R L ~ S B = ~ S B + W L B Togeher wih he labor supply (37): we solve for L B : L B ' C B = " B ; " L B = + + G (!;!#!) + F (!;! ) R! +' L : " B B The consumpion of non-durable goods is hen given by: C B = " B " +' + + G (!;!) + F (!;! ) R! L B # ' +' : Knowing C B we can use equaion (56) o solve for he consumpion of durable goods D B. Nex, we solve for he consumpion expendiures and oal labor supply of savers. Using he fac ha I = D ogeher wih he seady sae balance shee ideniy (5) of nancial inermediaries (wih ~ S being deposi holdings in real erms and using he fac ha B = ): ~ S = ( ) ~ S B ; he budge consrain of savers (3) can be expressed as: C + D + ~ S B = R ~ S B + W L + : 24

25 Noe ha aggregae pro s are given by: = Z n P H Y C (h)dh + Z n P D Y D (h)dh Z n W C L C (h) + ( Z n W D L D (h) + ( ) L B;D (h) dh + n ( ) G (!;! ) D B = n(y C + Y D ) n L + ( ) L B + n ( ) G (!;! ) D B L = n + ( ) L B + n ( ) G (!;! ) D B : ) L B;C (h) dh Per capia pro s are hen given by. Plugging his ogeher wih equaion (58) n ino he budge consrain leads o: C + D = (R ) S ~ B + L + + G (!;!) D B = (R ) S ~ B + L + L + L B + LB G (!;!) D B : h Inroducing = (R ) ~ i S B + L B + G (!;! ) D B as a parameer which is consan from he perspecive of savers, we nally arrive a: C + D L = : (59) Bringing ogeher he labor supply of savers (8): L ' C = " and equaion (59) we obain he following expression for L: + " L ' L +' = : For a given value of L, i is sraighforward o obain C and D. To nd a value for we use he marke clearing condiion. The fracion of non-durable producion over oal producion is: Y C Y C + Y D = : In seady sae his has o be equal o he fracion of spending allocaed o non- 25

26 durable consumpion over oal spending: C + ( )C B (C + D) + ( )(C B + D B ) = : Given values for,,,,,, B, ", " B, ',, F () we can solve for he value of. Aggregae expendiures on non-durable consumpion is de ned as: C T OT = C + ( )C B ; so ha aggregae allocaion of expendiures beween home and foreign-produced goods is: C H = C T OT ; C F = ( )C T OT : The marke clearing condiions for nal goods are: Y C = ny C = nc H + ( n) C H; Y D = ny D = n D + ( )D B ; where Y C and Y D are he seady-sae values of Y C and Y D, and Y C and Y D are he seady-sae individual (per capia) producion levels of each rm. Therefore aggregae producion levels are given by: Y C = n L + ( )L B ; Y D = ( )n L + ( )L B : 4 Derivaives of F (!;! ) and G (!;! ) In order o log-linearize he model we need he derivaives of he CDF F (!;! ) and G (!;! ), which denoes he mean value of he qualiy shock condiional on he shock being less han he hreshold. Firs, we use he properies of he qualiy shock! o nd expressions for F () and G (). Then, we deermine he derivaives wih respec o he hreshold and he sandard deviaion. As! follows a log-normal disribuion, E [! ] = e!; + 2 2!; and since we se E [! ] =, he seady sae of he 26

27 mean is given by:! = 2 2!: The CDF of he log-normally disribued qualiy shock in seady sae is de ned as: F (!;! ) = Z! df (!) = = Z! Z! p e (ln!!! 2!! p 2 e!) 2 2 2! d! (ln!+ 2 2!) 2 2 2! d!; which can be used o nd an expression for he derivaive wih respec (!;! =!! p 2 e ( ln!+ 2 2!) 2 2 2! ; and wih respec o he sandard deviaion! (!;! = " F (!;! ) + F (!;! )! " ln! ! = F (!;!)! 2! ln! + 2 2! + ln! + 2 2! 2 2! 2 4! ln! + # 2 2! ; 2 # 2! where we have used he fac = e f(x2) f (x 2 )2x. Nex, we need o nd an expression for he mean value of he qualiy shock condiional on he shock being less han he hreshold G (!;! ). To do so we combine he formula o calculae he expeced value wih he formula for he parial expecaions which are given by: E [!] = E [! j! >!] = Z Z!!dF (!) = e! + 2 2! ;!df (!) = e! + 2 2!! + 2! ln!! ; wih being he CDF of he sandard normal disribuion. We use his o rewrie he expecaion E [!] as: E [!] = Z Z!!dF (!) = = Z!!dF (!) + Z!!dF (!)!df (!) + e! + 2 2!! + 2! ln!! : 27

28 This allows us o nd an express for G (!;! ): G (!;! ) = Z!!dF (!) =E! e! +! !! ln!! =e! +! !! ln! :! Using he fac ha! = 2 2! we can express G (!;! ) as: G (!;! ) = 2 2! ln! :! The derivaive of G () wih respec o he hreshold value!, follows from he de - niion G () = R!!dF (!) so (!;!) Turning now o he derivaive wih respec o he sandard deviaion!. Noe ha he expression for is given by: (x) = p 2 Z x e 2 =2 d: Taking derivaives and evaluaing a x : (x) = p 2 e x2 =2 ; we arrive = 2 2! ln!! ln! + : 2 2! 5 Robusness Resuls on Bayesian Esimaion In secion 3.E of he main ex we discuss di eren model comparison exercises ha we have underaken. Our preferred speci caion is one where here is a common innovaion in he echnology shock of he non-durable secor and in he preference shock of he durable secor across counries. Also, we found ha unlike Chrisiano, Moo and Rosagno (23), anicipaed ("news") shocks in he sandard deviaion of he housing qualiy shock did no improve model. Finally, we esimaed he 28

29 model wih a argeing rule of he ype: P EMU =P EMU + EMU p Y EMU =Y EMU = insead of a Taylor-ype rule as equaion (54). We also esimaed a version of he model where funding coss for nancial inermediaries are he same across counries. None of hese wo exensions improved model so hey were discarded. In he following Table we provide he marginal likelihoods for di eren speci- caions, while in he following subsecions we provide resuls for he Bayesian parameer esimaes. Table : Marginal Likelihoods Baseline Model Di eren AR() Coe ciens No Common Innovaions News shocks in risk, one lag News shocks in risk, wo lags News shocks in risk, hree lags News shocks in risk,four lags Targeing Rule Same funding coss

30 5. Parameer Esimaes, Model wih Di eren AR() Coe ciens parameers prior mean pos. mean conf. inerval prior psdev hea_c bea.5 hea_c_s bea.5 hea_d bea.5 hea_d_s bea.5 phi_c bea.5 phi_c_s bea.5 phi_d bea.5 phi_d_s bea.5 epsilon bea.5 epsilon_borr bea.5 lambda bea.5 phi gamma.5 ioa_c gamma.5 ioa_l gamma.5 psi gamma. kappa_b gamma.2 gamma_pi norm. gamma_r bea.5 gamma_y gamma.5 rho_echc bea. rho_echd bea. rho_echc_s bea. rho_echd_s bea. rho_risk bea. rho_risk_s bea. rho_premium bea. rho_prefc bea. rho_prefd bea. rho_prefc_s bea. rho_prefd_s bea. 3

31 sandard deviaion of shocks prior mean pos. mean conf. inerval prior psdev e_risk_s gamma.25 e_risk gamma.25 e_m gamma.2 e_premium gamma.2 e_ech gamma.2 e_echc gamma.2 e_echc_com gamma.2 e_echd gamma.2 e_echc_s gamma.2 e_echd_s gamma.2 e_prefc gamma.5 e_prefd gamma.5 e_prefc_s gamma.5 e_prefd_s gamma.5 e_prefd_com gamma.5 3

32 5.2 Parameer Esimaes, Model wih No Common Innovaions parameers prior mean pos. mean conf. inerval prior psdev hea_c bea.5 hea_c_s bea.5 hea_d bea.5 hea_d_s bea.5 phi_c bea.5 phi_c_s bea.5 phi_d bea.5 phi_d_s bea.5 epsilon bea.5 epsilon_borr bea.5 lambda bea.5 phi gamma.5 ioa_c gamma.5 ioa_l gamma.5 psi gamma. kappa_b gamma.2 gamma_pi norm. gamma_r bea.5 gamma_y gamma.5 rho_echc bea. rho_echd bea. rho_risk bea. rho_premium bea. rho_prefc bea. rho_prefd bea. 32

33 sandard deviaion of shocks prior mean pos. mean conf. inerval prior psdev e_risk_s gamma.25 e_risk gamma.25 e_m gamma.2 e_premium gamma.2 e_ech gamma.2 e_echc gamma.2 e_echd gamma.2 e_echc_s gamma.2 e_echd_s gamma.2 e_prefc gamma.5 e_prefd gamma.5 e_prefc_s gamma.5 e_prefd_s gamma.5 33

34 5.3 Parameer Esimaes, Model wih News in Risk Shock, One Lag parameers prior mean pos. mean conf. inerval prior psdev hea_c bea.5 hea_c_s bea.5 hea_d bea.5 hea_d_s bea.5 phi_c bea.5 phi_c_s bea.5 phi_d bea.5 phi_d_s bea.5 epsilon bea.5 epsilon_borr bea.5 lambda bea.5 phi gamma.5 ioa_c gamma.5 ioa_l gamma.5 psi gamma. kappa_b gamma.2 gamma_pi norm. gamma_r bea.5 gamma_y gamma.5 rho_echc bea. rho_echd bea. rho_risk bea. rho_premium bea. rho_prefc bea. rho_prefd bea. 34

35 sandard deviaion of shocks prior mean pos. mean conf. inerval prior psdev e_risk_s gamma.25 e_risk gamma.25 e_risk_s gamma.25 e_risk gamma.25 e_m gamma.2 e_premium gamma.2 e_ech gamma.2 e_echc gamma.2 e_echc_com gamma.2 e_echd gamma.2 e_echc_s gamma.2 e_echd_s gamma.2 e_prefc gamma.5 e_prefd gamma.5 e_prefc_s gamma.5 e_prefd_s gamma.5 e_prefd_com gamma.5 35

36 5.4 Parameer Esimaes, Model wih News in Risk Shock, Two Lags parameers prior mean pos. mean conf. inerval prior psdev hea_c bea.5 hea_c_s bea.5 hea_d bea.5 hea_d_s bea.5 phi_c bea.5 phi_c_s bea.5 phi_d bea.5 phi_d_s bea.5 epsilon bea.5 epsilon_borr bea.5 lambda bea.5 phi gamma.5 ioa_c gamma.5 ioa_l gamma.5 psi gamma. kappa_b gamma.2 gamma_pi norm. gamma_r bea.5 gamma_y gamma.5 rho_echc bea. rho_echd bea. rho_risk bea. rho_premium bea. rho_prefc bea. rho_prefd bea. 36

37 sandard deviaion of shocks prior mean pos. mean conf. inerval prior psdev e_risk_s gamma.25 e_risk gamma.25 e_risk_s gamma.25 e_risk gamma.25 e_risk_s gamma.25 e_risk gamma.25 e_m gamma.2 e_premium gamma.2 e_ech gamma.2 e_echc gamma.2 e_echc_com gamma.2 e_echd gamma.2 e_echc_s gamma.2 e_echd_s gamma.2 e_prefc gamma.5 e_prefd gamma.5 e_prefc_s gamma.5 e_prefd_s gamma.5 e_prefd_com gamma.5 37

38 5.5 Parameer Esimaes, Model wih News in Risk Shock, Three Lags parameers prior mean pos. mean conf. inerval prior psdev hea_c bea.5 hea_c_s bea.5 hea_d bea.5 hea_d_s bea.5 phi_c bea.5 phi_c_s bea.5 phi_d bea.5 phi_d_s bea.5 epsilon bea.5 epsilon_borr bea.5 lambda bea.5 phi gamma.5 ioa_c gamma.5 ioa_l gamma.5 psi gamma. kappa_b gamma.2 gamma_pi norm. gamma_r bea.5 gamma_y gamma.5 rho_echc bea. rho_echd bea. rho_risk bea. rho_premium bea. rho_prefc bea. rho_prefd bea. 38

39 sandard deviaion of shocks prior mean pos. mean conf. inerval prior psdev e_risk_s gamma.25 e_risk gamma.25 e_risk_s gamma.25 e_risk gamma.25 e_risk_s gamma.25 e_risk gamma.25 e_risk_s gamma.25 e_risk gamma.25 e_m gamma.2 e_premium gamma.2 e_ech gamma.2 e_echc gamma.2 e_echc_com gamma.2 e_echd gamma.2 e_echc_s gamma.2 e_echd_s gamma.2 e_prefc gamma.5 e_prefd gamma.5 e_prefc_s gamma.5 e_prefd_s gamma.5 e_prefd_com gamma.5 39