Monetary and Macroprudential Policies under Fixed and Variable. Interest Rates

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1 Moneary and Macroprudenial Policies under Fixed and Variable Ineres Raes Margaria Rubio Universiy of Noingham Augus 2015 Absrac In his paper, I analyze he abiliy of moneary policy o sabilize boh he macroeconomy and financial markes under wo differen scenarios: fixed and variable-rae morgages. I develop and solve a New Keynesian dynamic sochasic general equilibrium model ha feaures a housing marke and a group of consrained individuals who need housing collaeral o obain loans. A given share of consrained households borrows a a variable rae, while he res borrows a a fixed rae. I consider wo alernaive ways of inroducing a macroprudenial approach o enhance financial sabiliy: one in which moneary policy, using he ineres rae as an insrumen, responds o credi growh; and a second one in which he macroprudenial insrumen is insead he loan-o-value raio (LTV). Resuls show ha when raes are variable, a counercyclical LTV rule performs beer o sabilize financial markes han moneary policy. However, when hey are fixed, even hough moneary policy is less effecive o sabilize he macroeconomy, i does a good job o promoe financial sabiliy. Keywords: Fixed/Variable-rae morgages, moneary policy, macroprudenial policy, LTV, housing marke, collaeral consrain JEL Codes: E32, E44, E52 Universiy of Noingham, Sir Clive Granger Building, Noingham. margaria.rubio@noingham.ac.uk. The auhor would like o hank Maeo Iacoviello, Fabio Ghironi and Peer Ireland for heir suppor and help in he modelling framework. Special hanks o José A. Carrasco-Gallego for his very useful commens. Par of his projec was underaken while he auhor was visiing he Naional Bank of Poland, for which she acknowledges heir hospialiy. All errors are mine. 1

2 " The explici incorporaion of macroprudenial consideraions in he naion s framework for financial oversigh represens a major innovaion in our hinking abou financial regulaion [...] This new direcion is consrucive and necessary, I believe, bu i also poses considerable concepual and operaional challenges in is implemenaion.". Ben Bernanke, May 5, Inroducion In recen years, especially during he period of he Grea Moderaion, moneary policy had been seen as a very powerful ool o sabilize he economy. However, in he afermah of he crisis, new experiences have revealed ha his saemen is no rue for all cases nor under all circumsances. Firs of all, he effeciveness of moneary policy may depend on srucural facors in he economy. In paricular, here may be insiuional feaures, especially in housing markes, ha are counry-specific and ha may affec he opimal conduc of policies. One source of heerogeneiy, which can be crucial, is he srucure of morgage conracs. Morgage conracs in an economy can be fixed or variable rae, and he proporion of each ype of morgages varies from counry o counry. The link beween he policy rae and fixed raes is weaker, since he laer are more conneced o longer-erm raes, and hus, in his case, moneary policy is less effecive. 1 On he oher hand, wih he crisis, he cener of policy and academic discussions has been how o ensure a more sable financial sysem: a macroprudenial approach o preven he economy from siuaions in which problems in he financial secor are ransmied o he real secor and vice-versa. I is debaable ha moneary policy alone can achieve his goal; i may need he help of oher ools ha help avoiding excessive credi growh. The remaining quesion is he following: Does he morgage srucure of he economy affec he abiliy of moneary policy o enhance financial sabiliy? In his paper, I ry o shed some ligh on his issue. I analyze he abiliy of moneary policy o sabilize financial markes under wo differen scenarios: when he prevalen morgage rae in he economy is variable and when i is fixed. Recen lieraure shows ha he effeciveness of moneary policy o sabilize he macroeconomy is reduced when raes are fixed. Neverheless, he lieraure is silen abou wheher his feaure has an impac on he poenial of moneary policy o promoe financial sabiliy. There is evidence of differen cross-counry morgage conracs. While fixed rae-morgages predominae in he US, he majoriy of consumers borrow a a variable rae in Canada or Ausralia. Wihin 1 See Rubio (2011) or Brzoza-Brzezina (2014) for heoreical models ha show ha fixed rae conracs imply less effecive moneary policy. 2

3 European counries, we have sriking differences such ha he Unied Kingdom or Spain, in which he vas majoriy of consumers have variable-rae morgages, as opposed o Germany or France in which morgages are fixed rae (See Table A1 in he Appendix). Rubio (2011) and Calza e al (2013) show ha he srucure of morgage conracs has imporan implicaions for he ransmission of moneary policy in he sense ha policy rae changes are less effecive when he morgage rae is fixed. However, hese papers do no ouch upon he issue of wheher having fixed- or variable-rae morgages may also affec financial sabiliy and he opimal design of macroprudenial policies. In his paper, I build a new Keynesian dynamic sochasic general equilibrium model wih housing, collaeral consrains, and fixed and variable-rae morgages o sudy how he morgage srucure in an economy affecs he opimal design of boh moneary and macroprudenial policies. In he model, here are hree ypes of consumers; savers, variable-rae borrowers and fixed-rae borrowers. Borrowers need collaeral in order o access credi markes which are more or less igh depending on he loan-ovalue raio (LTV). Moneary policy is conduced by he cenral bank. For he macroprudenial policies I consider wo opions; one in which i is conduced by he cenral bank wih he ineres rae as an insrumen, and a second one in which here is a macroprudenial regulaor ha uses a counercyclical rule for he LTV as a macroprudenial ool. Since his is a microfounded model, i is suiable o sudy welfare-relaed issues. In his seing, here are several channels ha affec welfare and ha are dependen on morgage conracs. In new Keynesian models wih collaeral consrains, here are wo ypes of disorions: sicky prices and credi fricions. Savers prefer policies ha alleviae he firs disorion, since hey own he firms. They are beer off in a scenario wih price sabiliy, he goal of moneary policy. On he conrary, borrowers welfare increases when he credi fricion disorion is minimized. Then, borrowers may prefer siuaions ha generae inflaion or policies ha enhance financial sabiliy, namely macroprudenial. However, hese mechanisms may differ depending on wheher he prevalen morgage conrac in he economy is fixed or variable rae. In he variable-rae scenario, moneary policy is more sabilizing because here is a one-for-one link beween he policy rae and he borrowing rae. Wih respec o macroprudenial policies, heir effeciveness will also depend on wheher he economy has fixed or variable raes, since heir ineracion wih moneary policy will have an effec on financial sabiliy. Wih he purpose of undersanding he dynamics of he baseline model and as a moivaion for my sudy, as a firs sep I presen impulse responses o a echnology shock, for he case in which here are no macroprudenial policies. I find ha having variable or fixed-rae morgages does no only affec 3

4 he macroeconomic dynamics bu also he financial side of he economy. The fixed-rae economy, has a less powerful moneary policy ool bu borrowers are more exposed o changes in inflaion and house prices, affecing heir financial capaciy. Therefore, he srucure of morgage conracs should have clear implicaions, no only for moneary policy reacion bu also for macroprudenial policies, which care abou financial sabiliy. Then, i is relevan o sudy he opimal implemenaion of boh moneary and macroprudenial policies in he conex of variable and fixed-rae morgages. Then, I analyze how he opimaliy of moneary and macroprudenial policies changes depending on he morgage srucure of he economy. I define opimal policy as he one ha maximizes oal welfare. As menioned, I consider wo alernaive ways of inroducing macroprudenial policies; firs, a simple and auomaic rule on he LTV. Following his rule, he LTV would be he insrumen of he macroprudenial regulaor and would reac o credi growh. In his way, if he economy is, for insance, enering a credi boom, he LTV will be cu, hus resricing credi in he economy and avoiding excessive credi growh. This rule, which resembles a Taylor rule for moneary policy, serves as a proxy for he macroprudenial insrumens ha have been used by some insiuions. 2 Alernaively, I consider including credi growh in he ineres-rae rule of he cenral bank. In his way, he moneary auhoriy would have one insrumen, he ineres rae, o ake care of wo objecives; macroeconomic and financial sabiliy. Resuls show ha macroprudenial policies increase welfare regardless of he morgage srucure prevalen in he economy. Neverheless, when morgages are variable rae, an LTV rule combined wih moneary policy is preferable o including credi variables in he ineres-rae rule. When raes are fixed, using he ineres rae as an insrumen boh for moneary and macroprudenial policy delivers higher welfare and sabiliy han having wo separae insrumens. Ineresingly for he fixed rae case, in which moneary policy is less effecive o sabilize he macroeconomy, i is a more powerful ool o sabilize he financial sysem. This paper relaes o differen srands of he lieraure. Firs, i inroduces heerogeneiy in morgage conracs in he spiri of Rubio (2011), Calza e al. (2013), and Garriga e al. (2013). However, hose sudies resric hemselves o he effecs of he morgage srucure on business cycles and moneary policy, wihou analyzing he implicaions for macroprudenial policies. Second, i is close o he recen macroprudenial lieraure. On he one hand, i relaes wih papers in which macroprudenial policies 2 LTV rules have become paricularly popular. See for insance, Gruss and Sgherri (2009) which analyses he welfare effecs of procyclical loan-o-value raios in a real business cycle model wih borrowing consrains. Funke and Paez (2012) uses a non-linear rule on he LTV and finds ha i can help reduce he ransmission of house price cycles o he real economy. In a similar way, Kannan, Rabanal and Sco (2012) examines a moneary policy rule ha reacs o prices, oupu and changes in collaeral values wih a macroprudenial insrumen based on he LTV. 4

5 inerac wih moneary policy as in Kannan e al. (2012), Rubio and Carrasco-Gallego (2014), and Angelini e al. (2014). However, none of he above menioned papers sudies how fixed and variable-rae morgages affec he implemenaion of macroprudenial policies nor affec financial sabiliy. On he oher hand, my paper also explores he opic of wheher moneary and macroprudenial policies should be conduced by he same regulaor using only one insrumen and wo objecives or wo regulaors wih wo differen insrumens. Following he same line, Beau e al. (2012) claims ha i is preferable o have a combinaion of separae objecives for moneary and macroprudenial policies. Rubio and Carrasco- Gallego (2015) also finds ha moneary policy should focus on price sabiliy while macroprudenial policy should have financial sabiliy as an insrumen. Kannan e al. (2012) experimens wih an augmened Taylor rule and an LTV rule as well and finds ha resuls depend on he source of he shock considered. In my paper, I find ha having wo separae insrumens is preferred in he case of variable-rae morgages bu he augmened Taylor rule delivers higher welfare when raes are fixed. The paper coninues as follows. Secion 2 explains he basic model I build for he analysis and is dynamics. Secion 3 shows he modelling of he macroprudenial policies. Secion 4 analyzes he normaive implicaions of inroducing macroprudenial policies and displays he opimal moneary and macroprudenial policy mix. Secion 5 presens he conclusions. The Appendix conains graphs and ables on he empirical evidence menioned above and model derivaions. 2 The Baseline Model I consider an infinie-horizon economy in which households consume, work and demand real esae. There is a represenaive financial inermediary ha provides morgages and acceps deposis from consumers. Firms se prices subjec o Calvo (1983)-Yun (1996) nominal rigidiy. The moneary auhoriy ses ineres raes endogenously, in response o inflaion and oupu, following a Taylor rule. 2.1 The Consumer s Problem There are hree ypes of consumers: unconsrained consumers, consrained consumers who borrow a a variable rae, and consrained consumers who borrow a a fixed rae. Consrained individuals need o collaeralize heir deb repaymens in order o borrow from he financial inermediary. Ineres paymens for boh morgages and loans canno exceed a proporion of he fuure value of he curren house sock. In his way, he financial inermediary ensures ha borrowers are going o be able o 5

6 fulfill heir deb obligaions nex period. As in Iacoviello (2005), I assume ha consrained consumers are more impaien han unconsrained ones. This assumpion ensures ha he borrowing consrain is binding, so ha consrained individuals do no save and wai unil hey have he funds o self-finance heir consumpion. This generaes an economy in which households divide ino borrowers and savers. Furhermore, borrowers are spli ino wo groups, hose who borrow a a fixed rae and hose who borrow a a variable rae. The proporion of each ype of borrower is fixed and exogenous. All households derive uiliy from consumpion, housing services assumed proporional o he housing sock and leisure The Financial Inermediary There is a financial inermediary which acceps deposis from savers, and exends boh fixed and variablerae loans o borrowers. 3 I assume a compeiive framework and hus he inermediary akes he variable ineres rae as given. The profis of he financial inermediary are defined as: F = αr 1 b cv 1 + (1 α) R 1 b cf 1 R 1b u 1, (1) where F represens he profis of he financial inermediary, α is he proporion of variable raes, R 1 is he gross policy rae se by he cenral bank, b cv 1 morgages, respecively. 4 b u 1 represen deposis. and bcf 1 In equilibrium, aggregae borrowing and saving mus be equal, ha is, are one-period variable and fixed-rae αb cv + (1 α) b cf = b u. (2) Subsiuing (2) ino (1),we obain, F = (1 α) b cf 1 ( R 1 R 1 ). (3) In order for he wo ypes of morgage o be offered, he fixed ineres rae has o be such ha he inermediary is indifferen beween lending a a variable or fixed rae. Hence, he expeced discouned profis ha he inermediary obains by lending new deb in a given period a a fixed ineres rae mus be equal o he expeced discouned profis he inermediary would obain by lending i a a variable 3 In counries where FRMs are mos exensively used, financial inermediaries pass on he loans o invesors wih longerm liabiliies (such as pension funds and life-insurance companies). Shor-erm deposis are predominanly used o finance morgages in counries where ARMs are commonly used. These insiuional feaures are ou of he scope of his paper. 4 Noice ha in his model morgages are a flow variable. 6

7 rae: 5 ( C u where Λ,i = C u +i E τ i=τ+1 β i τ Λ τ,i R τ = E τ i=τ+1 β i τ Λ τ,i R i 1, (4) ) is he unconsrained consumer relevan discoun facor. Since he financial inermediary is owned by he savers, heir sochasic discoun facor is applied o he financial inermediary s problem. Noice ha, as saed before, variable-rae deb is one period bu he porion of new deb acquired a a fixed rae is associaed wih a long-erm conrac. Since he agen is infiniely lived, he financial inermediary considers an infiniely lasing mauriy in hese calculaions. 6 We can obain he equilibrium value of he fixed rae in period τ from expression (4) : R τ = E τ i=τ+1 E τ i=τ+1 β i τ Λ τ,i R i 1 β i τ Λ τ,i. (5) Equaion (5) saes ha, for every new deb issued a dae τ, here is a differen fixed ineres rae ha has o be equal o a discouned average of fuure variable ineres raes. Noice ha his is no a condiion on he sock of deb, bu on he new amoun obained in a given period. New deb a a given poin in ime is associaed wih a differen fixed ineres rae. Boh he fixed ineres rae in period τ and he new amoun of deb in period τ are fixed for all fuure periods. However, he fixed ineres rae varies wih he dae he deb was issued, so ha in every period here is a new fixed ineres rae associaed wih new deb in his period. If we consider fixed-rae loans o be long-erm, he financial inermediary obains ineres paymens every period from he whole sock of deb, no only from he new ones. Hence, we can define an aggregae fixed ineres rae ha is he one he financial inermediary effecively charges every period for he whole sock of morgages. This aggregae fixed ineres rae is composed of all pas fixed ineres raes and pas deb, ogeher wih he curren period equilibrium fixed ineres rae and new amoun of deb. Therefore, he effecive fixed ineres rae ha he financial inermediary charges for he sock of fixed-rae deb every period is: 5 The fixed rae loan is priced following his non-arbirage condiion, no by applying he prices of zero-coupon bonds o he fuure cash flows from he new loan. 6 Calza e al. (2010) also have a model in which he financial inermediary offers fixed and variable-rae morgages. However, in heir model, he wo ypes of morgages do no coexis. For hem, he fixed-rae loan is a wo-period conrac while he variable-rae is one period. In my model, I allow for he wo morgages o be offered in order o be able o sudy inermediae cases in which a mix of he wo ypes of conracs are presen in he economy. 7

8 R = ( R 1 b cf 1 +R b cf b cf bcf 1 R 1 if b cf ) if b cf b cf 1 > b cf 1. (6) Equaion (6) saes ha he fixed ineres rae ha he financial inermediary is acually charging oday is an average of wha i charged las period for he previous sock of morgages and wha i charges his period for he new amoun. 7 In he case ha here is no new deb, he fixed ineres rae will be equal o las period s. 8 Then, he same way ha variable raes are revised every period, fixed-raes are revised by including he new opimal fixed ineres rae for he new deb originaed in his period. Imporanly, his assumpion is no crucial for resuls. Boh R τ and R are pracically unaffeced by ineres rae shocks. This assumpion is a way o make he model compaible wih he fac ha fixed-rae loans are no one-period asses bu longer erm ones. 9 As noed above, if any, profis from financial inermediaion are rebaed o he unconsrained consumers every period. Even if he financial inermediary is compeiive and profis are expeced o be zero, if here is a shock a a given poin in ime, he fac ha only he variable ineres rae is direcly affeced can generae non-zero profis Unconsrained Consumers (Savers) Unconsrained consumers maximize: ( max E 0 β ln C u + j ln H u (Lu ) η ), (7) η =0 where he superscrip u sands for "unconsrained", E 0 is he expecaion operaor, β (0, 1) is he discoun facor, and C u, H u and L u are consumpion a, he sock of housing and hours worked, respecively; 1/ (η 1) is he labor supply elasiciy, η > 0 and j > 0 represens he weigh of housing in he uiliy funcion. The budge consrain is: 7 This expression can be inerpreed in a similar way as in Calza e al. (2010). In heir model, he fixed rae loan in repaid in wo periods. Here, he conrac is of infinie mauriy bu I also divide paymens in wo blocks, he new paymens made his period for new loans and he paymens for he old loans se in a compac way. 8 Noice ha, if R > R, remorgaging o a lower R is no allowed in he model. The agen canno repay he mos expensive morgages firs eiher. 9 In he real wold, variable-rae morgages are also long-erm loans. Tha is, boh loans are amorized over a long period of ime. The only difference is ha ineres paymens on adjusable-rae morgages are variable. In he model variable-rae morgages are modeled as one-period loans. 8

9 C u + q H u + b u q H u 1 + w u L u + R 1b u 1 π + F v + S v, (8) where q is he real housing price and w u is he real wage for unconsrained consumers. These can buy houses or sell hem a he curren price q. I assume zero housing depreciaion for simpliciy. As we will see, his group will choose no o borrow a all; hey are he savers in his economy. b u is he amoun hey save. They receive ineres R 1 for heir savings. π is inflaion in period. S and F are lump-sum profis received from he firms and he financial inermediary, respecively. We can hink of hese consumers as he wealhy agens in he economy, who own he firms and he financial inermediary. The firs-order condiions for his unconsrained group are: 1 C u = βe ( R π +1 C u +1 ), (9) w u = (L u ) η 1 C u, (10) j H u = 1 1 C u q βe C+1 u q +1. (11) Equaion (9) is he Euler equaion for consumpion, equaion (10) is he labor-supply condiion, and equaion (11) is he Euler equaion for housing. This saes ha he benefis from consuming housing mus be equal o he coss a he margin Consrained Consumers (Borrowers) Consrained consumers are of wo ypes: hose who borrow a a variable rae and hose who do i a a fixed rae. The difference beween hem is simply he ineres rae hey face. The fixed-rae borrower faces R, se by he financial inermediary, whereas he variable-rae counerpar faces R, se by he cenral bank. The proporion of variable-rae consumers is fixed and exogenous and equal o α [0, 1]. Consrained and unconsrained consumers are differen in he way hey discoun he fuure. Consrained consumers are more impaien han unconsrained ones. I assume ha consrained consumers face a limi on he deb hey can acquire. The maximum amoun hey can borrow is proporional o he value of heir collaeral, in his case he sock of housing. Tha is, he deb repaymen nex period canno exceed a proporion of omorrow s value of oday s sock of housing: 9

10 R E b cv π +1 k E q +1 H cv, (12) E R b cf π +1 k E q +1 H cf, (13) where (12) represens he collaeral consrain for he variable-rae consrained consumer and (13) is he consrain for he fixed-rae one. 10 k represens a proxy for he loan-o-value raio and, as we will see, i is he insrumen of he macroprudenial auhoriy. As we have seen wih he problem of he financial inermediary, R is an aggregae ineres rae ha conains informaion on all he pas fixed ineres raes associaed wih pas deb. Each period, his aggregae ineres rae is updaed wih a new ineres rae linked o he new amoun of deb originaed in ha period. Wihou loss of generaliy, I presen he problem for he variable-rae borrower, since he one for he fixed-rae is symmeric. Variable-rae borrowers maximize heir lifeime uiliy funcion: subjec o he budge consrain: max E 0 =0 β (ln C cv + j ln H cv (Lcv ) η η ), (14) C cv + q H cv + R 1b cv 1 π q H 1 cv + w cv L cv + b cv, (15) and (12), he collaeral consrain. 11 Noice ha variable-rae borrowers repay all deb every period and acquire new one a he curren new ineres rae. This assumpion implies ha he ineres rae on variable rae morgages is revised every period for he whole sock of deb and changed according o he policy rae. 12 In order o make he problem for fixed-rae borrowers symmeric and analogous o he exising models wih borrowing consrains, I assume he same deb-repaymen srucure for his ype of borrowers. Obviously, fixed-rae conracs are no revised every period. However, o make he model more realisic bu sill racable, he fixed ineres rae will be such ha a revised fixed rae will be applied only on new deb, keeping consan he ineres rae applied o exising deb. In his way, I reconcile he srucure of he model wih he fac ha fixed-rae conracs are long erm The superscrip cv sands for "consrained variable" while cf sands for "consrained fixed" 11 We will see from he firm s problem ha w cv = w cf = w c. 12 This assumpion is consisen wih realiy, in which variable ineres raes are revised very frequenly and changed according o an ineres rae index ied o he ineres rae se by he cenral bank. 13 Anoher opion would be o have an overlapping generaion model in which we are able o keep rack of he deb issued 10

11 As noed above, consrained consumers are more impaien han unconsrained ones, so ha β < β. This assumpion is crucial for he borrowing consrain o be binding and herefore, for here o be boh borrowers and savers in he economy. The firs-order condiions for variable-rae consrained consumers are: 1 C cv = βe ( ) R π +1 C+1 cv + λ cv R, (16) w cv = (L cv ) η 1 C cv, (17) j H cv = 1 C cv ( ) q βe 1 C+1 cv q +1 λ cv k E (q +1 π +1 ). (18) These firs-order condiions differ from hose of he unconsrained individuals. In he case of consrained consumers, he Lagrange muliplier on he borrowing consrain (λ cv ) appears in equaions (16) and (18). From he Euler equaion for consumpion of unconsrained consumers, we know ha R = 1/β in seady sae. If we combine his resul wih he Euler equaion for consumpion of consrained individuals we have ha λ cv = β β ( ) /C cv > 0 in seady sae. This means ha he borrowing consrain holds wih equaliy in seady sae. Since we log-linearize he model around he seady sae and assume ha uncerainy is low, we can generalize his resul o off-seady-sae dynamics. Then, he borrowing consrain is always binding, so ha consrained individuals borrow he maximum amoun hey are allowed o and unconsrained consumers are never in deb. 14 Given he borrowing amoun implied by (12) a equaliy, consumpion for variable-rae consrained individuals can be deermined by heir flow of funds: C cv = w cv L cv + b cv ( + q H cv and he firs-order condiion for housing becomes: j H cv = 1 ( C cv q k ) E (q +1 π +1 ) R 1 H cv ) R 1 b cv 1 π, (19) ( ) βe 1 C+1 cv (1 k ) q +1. (20) each period. However, he model would become more complex and less comparable wih he sandard collaeral consrain DSGE models such as Iacoviello (2005). 14 This is a ypical assumpion for his kind of models. See Iacoviello (2005), Appendix C for a deailed analysis of when do consrains bind. 11

12 2.2 Firms Final Goods Producers There is a coninuum of idenical final goods producers ha aggregae inermediae goods according o he producion funcion [ 1 Y = 0 Y (z) ε 1 ε ] ε ε 1 dz, (21) where ε > 1 is he elasiciy of subsiuion beween inermediae goods. The final good firm chooses Y (z) o minimize is coss, resuling in demand of inermediae good z: The price index is hen given by: ( ) P (z) ε Y (z) = Y. (22) P [ 1 P = Marke clearing for he final good requires: 0 ] 1 P (z) 1 ε ε 1 dz. (23) Y = C = C u + C c Inermediae Goods Producers The inermediae goods marke is monopolisically compeiive. Following Iacoviello (2005), inermediae goods are produced according o he producion funcion: Y (z) = A L u (z) γ L c (z) (1 γ), (24) where γ [0, 1] measures he relaive size of each group in erms of labor. This Cobb-Douglas producion funcion implies ha labor effors of consrained and unconsrained consumers are no perfec subsiues. This specificaion is analyically racable and allows for closed form soluions for he seady sae of he model. This assumpion can be economically jusified by he fac ha savers are he managers of he firms and heir wage is higher han he one of he borrowers. 15 Experimening wih a producion 15 I could also be inerpreed as he savers being older han he borrowers, herefore more experienced. 12

13 funcion in which hours are subsiues leads o very similar resuls in erms of model dynamics. Under he Cobb-Douglas specificaion each household has mass one. γ is a consan ha represens he laborincome share of he paien household and L u are oal hours worked by he paien household. In he alernaive specificaion, one needs o define he fracion of agens in he populaion, say ω is he fracion of savers. Then, ωl u represens he oal hours worked by he paien household. Therefore, boh specificaions are very similar bu, while γ represens he economic size of savers, ω is is absolue size. 16 A represens echnology and i follows he following auoregressive process: log (A ) = ρ A log (A 1 ) + u A, (25) where ρ A is he auoregressive coeffi cien and u A is a normally disribued shock o echnology. Labor demand is deermined by: w u = 1 γ Y X L u, (26) where X is he markup, or he inverse of marginal cos. 17 w c = 1 (1 γ) Y X L c, (27) The price-seing problem for he inermediae good producers is a sandard Calvo-Yun seing. An inermediae good producer sells is good a price P (z), and 1 θ, [0, 1], is he probabiliy of being able o change he sale price in every period. The opimal rese price P (z) solves: k=0 (θβ) k E { Λ,k [ P (z) P +k The aggregae price level is hen given by: ] } ε/ (ε 1) Y+k X (z) = 0. (28) +k P = [ θp 1 ε 1 + (1 θ) (P ) 1 ε] 1/(1 ε). (29) Using (28) and (29), and log-linearizing, we can obain a sandard forward-looking New Keynesian Phillips curve π = βe π +1 k x +u π, ha relaes inflaion posiively o fuure inflaion and negaively 16 The full derivaion of his alernaive specificaion is available upon reques. 17 Symmery across firms allows us o wrie he demands wihou he index z. 13

14 o he markup ( k (1 θ) (1 βθ) /θ). u π is a normally disribued cos-push shock Aggregae Variables Given he fracion α of variable-rae borrowers, we can define aggregaes across consrained consumers as C c αc cv + (1 α) C cf, L c αl cv + (1 α) L cf, Hc αh cv + (1 α) H cf, b c αb cv + (1 α) b cf. Therefore, economy-wide aggregaes are: C C u + C c, L L u + L, H H u + H c. In his model, aggregae supply of housing is fixed, so ha marke clearing requires: H = H Moneary Policy The model is closed wih a Taylor Rule wih ineres rae smoohing, o describe he conduc of moneary policy by he cenral bank: 20 R = (R 1 ) ρ [ π (1+φπ) (Y /Y 1 ) φy R] 1 ρ εr, (30) where 0 ρ 1 is he parameer associaed wih ineres-rae ineria. φ π, φ y > 0 measure he response of ineres raes o curren inflaion and oupu growh, respecively. R is he seady-sae values of he ineres rae. ε R is a whie noise shock wih zero mean and variance σ 2 ε. 2.5 Dynamics Parameer Values I linearize he equilibrium equaions around he seady sae. Deails are shown in he Appendix. For calibraion, I consider he following parameer values: The discoun facor, β, is se o 0.99 so ha he annual ineres rae is 4% in he seady sae. The discoun facor for borrowers, β, is se o Lawrance (1991) esimaes discoun facors for poor consumers beween 0.95 and 0.98 a quarerly frequency. Resuls are no sensiive o differen values wihin his range. This value of β is low enough o endogenously divide he economy ino borrowers and savers. The weigh of housing on he uiliy funcion, j, is se o 0.1 in order for he raio of housing wealh o GDP in he seady sae o be 18 Variables wih a ha denoe percen deviaions from he seady sae. 19 This assumpion provides an easy way o specify he supply of housing and have variable prices. A wo-secor model wih producion of housing would no generae qualiaively differen resuls. 20 This is a realisic policy benchmark for mos of he indusrialized counries. A more realisic rule would also include oupu bu i complicaes building inuiion abou he workings of he model. Furhermore, esimaions deliver a small response o he oupu gap in he las wo decades (See Clarida, Galí and Gerler (2000)). Neverheless, robusness checks o his specificaion will be performed. 14

15 consisen wih he daa. This value of j implies a raio of approximaely 1.40, in line wih he Flow of Funds daa. 21 I se η = 2, implying a value of he labor supply elasiciy of For he loan-ovalue raio, I pick k SS = 0.9, consisen wih he evidence ha in he las years borrowing consrained consumers borrowed on average more han 90% of he value of heir house. 23 The labor income share of unconsrained consumers, γ, is se o 0.64, following he esimae in Iacoviello (2005). I pick a value of 6 for ε, he elasiciy of subsiuion beween inermediae goods. This value implies a seady sae markup of 1.2. The probabiliy of no changing prices, θ, is se o 0.75, implying ha prices change every four quarers. For he Taylor Rule parameers I use ρ = 0.8, φ π = 0.5, φ y = 0.5. The firs value reflecs a realisic degree of ineres-rae smoohing. 24 The second and hird ones, are consisen wih he original parameer proposed by Taylor in For α, I consider wo polar cases for comparison. In he firs case, he proporion of variable-rae morgages in he economy is 0, ha is, all consrained consumers in he economy borrow a a fixed rae. In he second case, he proporion of variable-rae morgages is 1. Table 1 shows a summary of he parameer values. 21 See Table B.100. In his model, consumpion is he only componen of GDP. To make he raio comparable wih he daa I muliply i by 0.6, which is approximaely wha nondurable consumpion and services accoun for in he GDP, according o he daa in he NIPA ables. 22 Microeconomic esimaes usually sugges values in he range of 0 and 0.5 (for males). Domeij and Flodén (2006) show ha in he presence of borrowing consrains his esimaes could have a downward bias of 50%. 23 We can idenify consrained consumers wih hose ha borrow more han 80% of heir home. In he US, among hose borrowers, he average LTV raio exceeds 90% for he period See he daa from he Federal Housing Finance Board. 24 See McCallum (2001). 15

16 Table 1: Parameer Values β.99 Discoun Facor for Savers β.98 Discoun Facor for Borrowers j.1 Weigh of Housing in Uiliy Funcion η 2 Parameer associaed wih labor elasiciy k SS.9 Loan-o-value raio γ.64 Labor share for Savers α 0/1 Proporion of variable-rae borrowers X 1.2 Seady-sae markup θ.75 Probabiliy of no changing prices ρ A.9 Technology persisence ρ.8 Ineres-Rae-Smoohing Parameer in Taylor Rule φ π.5 Inflaion Parameer in Taylor Rule Impulse Responses In order o gain some insigh abou he dynamics of he model before sudying macroprudenial policies, figure 1 presens impulse responses o a 1 percen posiive shock o echnology wih 0.9 persisence. We see ha he economy responds slighly more srongly afer a echnology shock when he majoriy of is borrowers have a fixed-rae morgage. In paricular, we can see in figure 1 ha a posiive echnology shock increases oupu and lowers prices. As a reacion, nominal raes decrease. However, for he fixed rae case, inflaion drives he real rae and herefore i increases. House prices, which move inversely wih he ineres rae, increase in he case of he variable-rae economy bu decrease in he fixed-rae one. For variable-rae consumers, he increase in house prices and he decrease in he ineres rae make borrowing increase. And since housing is now a more valuable asse, variable-rae borrowers use his borrowing o increase boh housing and consumpion goods. However, for fixed-rae consumers, he increase in he real rae, ogeher wih he decrease in house prices, makes. borrowing decrease. Fixed-rae borrowers prefer o decrease housing purchases in favor of consumpion goods and his is why oupu ends up increasing slighly by more in he fixed-rae scenario. We can see from he dynamics of he model ha having variable or fixed-rae morgages does no 16

17 1 Oupu 5 Borrowing %dev. seady sae Consumpion Borrowers Inflaion %dev. seady sae Housing Borrowers House Prices Variable Fixed %dev. seady sae quarers quarers Figure 1: Impulse Responses o a Technology Shock only affec he macroeconomy bu also he financial side. The fixed-rae economy, has a less powerful moneary policy ool bu borrowers are more exposed o changes in inflaion and house prices. Therefore, he srucure of morgage conracs has clear implicaions, no only for moneary policy reacion and for macro variables bu also for house prices and borrowing. Thus, i seems clear ha in case of including macroprudenial policies, he morgage conracs ha are prevalen in he economy will affec heir implemenaion, since he macroprudenial regulaor cares abou financial sabiliy. 3 Modelling Macroprudenial Policies For he macroprudenial policy, I will consider wo opions o be compared. The firs one is a rule on he LTV, so ha his variable responds o credi growh. The second one is an exended Taylor rule so ha he ineres rae, apar from responding o inflaion and oupu, also responds o credi growh. The firs case would correspond o a siuaion in which macroprudenial supervision should involve a regulaory agency, differen from he cenral bank or wihin he cenral bank, ha uses a differen insrumen for macroprudenial purposes. The second case represens a world in which macroprudenial and moneary policies are inegraed and assigned o he cenral bank, which uses jus one insrumen, he ineres rae, wo achieve boh macroeconomic and financial sabiliy. In his case, he objecives of moneary policy should be expanded o include financial sabiliy. 17

18 3.1 LTV Rule As an approximaion for a realisic macroprudenial policy, I consider a Taylor-ype rule for he loan-ovalue raio. In sandard models, he LTV raio is a fixed parameer which is no affeced by economic condiions. However, we can hink of regulaions of LTV raios as a way o moderae credi booms. When he LTV raio is high, he collaeral consrain is less igh. And, since he consrain is binding, borrowers will borrow as much as hey are allowed o. Lowering he LTV ighens he consrain and herefore resrics he loans ha borrowers can obain. Recen research on macroprudenial policies has proposed Taylor-ype rules for he LTV raio so ha i reacs inversely o variables such ha he growh raes of GDP, credis, he credi-o-gdp raio or house prices. These rules can be a simple illusraion of how a macroprudenial policy could work in pracice. Here, we assume ha here exiss a macroprudenial Taylor-ype rule for he LTV raio, so ha i responds o credi growh: k = k SS (b /b 1 ) φk b, (31) where k SS is he seady-sae values for he loan-o-value raio. φ k b 0 measures he response of he loan-o-o value o credi growh. This kind of rule would deliver a lower LTV raio in booms, when credi is growing, herefore resricing he credi in he economy and avoiding a credi boom derived from good economic condiions. 3.2 Macroprudenial Taylor Rule Here, I am considering he case in which he cenral bank is adoping a macroprudenial approach and aking care abou credi variables. Then, I exend he Taylor rule o no only respond o inflaion and oupu growh bu also o credi growh. R = (R 1 ) ρ [ π (1+φπ) (Y /Y 1 ) φy (b /b 1 ) φ b R] 1 ρ εr. (32) Thus, we are giving he cenral bank a way o implemen a macroprudenial policy. Noice ha increasing he ineres rae when credi is growing mean resricing credi booms in he economy, since deb repaymens are going up. Therefore, in his case, he goals of he cenral bank are exended o also include financial sabiliy. 18

19 4 Normaive Analysis In his secion, I inroduce he above menioned macroprudenial policies and sudy heir implicaions for welfare and heir opimal implemenaion. In order o do ha, firs I presen a measure for welfare. Then, using his measure, I analyze he opimaliy of moneary and macroprudenial policies for boh fixed and variable-rae morgage economies and presen impulse-responses using he opimized values. 25 In new Keynesian models wih collaeral consrains, here are wo ypes of disorions: sicky prices and credi fricions. Savers prefer policies ha alleviae he firs disorion, since hey own he firms. They are beer off in a scenario wih price sabiliy, he goal of moneary policy. However, borrowers welfare increases when he credi fricion disorion is minimized. Then, borrowers may prefer siuaions ha generae inflaion, since in his case he collaeral consrain is relaxed hrough lower real deb repaymens. On he one hand, moneary policy has effecs on he consrain, direcly hrough he ineres rae ha borrowers have o pay and indirecly hrough house prices, which make collaeral be more or less valuable. On he oher hand, macroprudenial policies which deliver higher financial sabiliy also lower he negaive effecs of he credi fricion, since hey provide borrowers wih a scenario in which heir consumpion is smooher. However, hese mechanisms differ depending on wheher he prevalen morgage conrac in he economy is fixed or variable rae. In he variable-rae scenario, moneary policy is more sabilizing because here is a direc link beween he policy rae and he borrowing rae. Neverheless, his link is broken for he fixed-rae case. Therefore, an economy wih variable raes will be more effecive o minimize he sicky price disorion, he one ha affecs savers. In he fixed-rae scenario, borrowing is more dependen on inflaion and house prices. Alhough he policy rae does no affec he economy as much as in he variable-rae case, inflaion affecs real raes and herefore borrowing. The policy rae also has an effec on house prices, since as any asse price, hey move inversely wih he ineres rae, and hus has also an effec on credi. Then, borrowers may prefer fixed raes because i creaes a siuaion wih more inflaion and his lowers real deb repaymens, hus relaxing heir collaeral consrain. Wih respec o macroprudenial policies, in he variable-rae case, he combinaion of moneary policy wih an LTV rule would deliver more financial sabiliy because, in a conex of sable inflaion, increasing LTVs in imes of credi growh means conaining credi. However, wih fixed-rae morgages, in which he real borrowing rae basically depends on inflaion, higher inflaion variabiliy may offse 25 I define opimal policy as he one ha maximizes oal welfare. 19

20 he effecs of increasing he LTV and more financial sabiliy may no be achieved. When he macroprudenial policy is included in he Taylor rule, for he variable-rae case, as oher sudies ha include credi variables in he moneary policy rule show, here may be no much gain in erms of welfare. However, for he fixed-rae case, i opens a new mechanism. Making he nominal rae respond o an addiional variable which is more volaile han inflaion and oupu makes house prices reac by more han wih he simple Taylor rule. This increases financial sabiliy hrough he effec of moneary policy on house prices. 4.1 Welfare Measure As discussed in Benigno and Woodford (2008), he wo approaches ha are recenly used for welfare analysis in DSGE models include eiher characerizing he opimal Ramsey policy, or solving he model by using a second-order approximaion o he srucural equaions for given policy and hen evaluaing welfare using his soluion. As in Mendicino and Pescaori (2007), I ake his laer approach o be able o evaluae he welfare of he borrowers and savers separaely and idenify he rade-off which appears beween hem. 26 The individual welfare for savers and he wo ypes of borrowers respecively is defined as follows: ( ) L V u, E β (ln m C+m u + j ln H+m u u η ) +m, (33) η m=0 V cv, E m=0 V cf, E m=0 β m (ln C cv +m + j ln H cv +m ( L cv +m ) η ( L cf β m ln C cf +m + j ln Hcf +m +m η η ) η ), (34), (35) Following Mendicino and Pescaori (2007), I define social welfare as a weighed sum of individual welfare for he differen ypes of households: Borrowers and savers welfare are weighed by ( V = (1 β) V u, + 1 β ) [αv cv, + (1 α) V cf, ]. (36) receive he same level of uiliy from a consan consumpion sream. ( 1 β ) and (1 β) respecively, so ha he wo groups 26 See Monacelli (2006) for an example of he Ramsey approach in a model wih heerogeneous consumers. 20

21 To make he resuls more inuiive, I presen welfare changes in consumpion equivalens, aking as a benchmark he siuaion in which here are no macroprudenial policies Opimal Policy In his subsecion, I sudy wha he mix of macroprudenial and moneary policy ha maximizes welfare is. In paricular, given a grid of possible parameers for he LTV and he Taylor rule (boh he sandard and he macroprudenial one), I perform a search ha maximizes welfare, subjec o deerminacy requiremens. 28 I do he analysis firs for he benchmark case, in which here are no macroprudenial policies, so ha I jus opimize over he parameers of he sandard Taylor rule. I find he parameers boh for he variable and he fixed-rae scenarios. Resuls are presened in Table 2: Table 2: Opimized Taylor Rule (Benchmark) Variable Rae Fixed Rae (1 + φ π) φ y Volailiies σ π σ y σ b Resuls in Table 2 represen he benchmark case, since i does no include macroprudenial policies. We can see he difference in he opimaliy of moneary policy in boh scenarios; fixed versus variable raes. For he variable-rae case, i is opimal for moneary policy o respond aggressively boh agains inflaion and oupu. However, for fixed raes, since he link beween he ineres rae and he macroeconomic variables is weaker, i is no opimal for moneary policy o respond o any of he variables because in any case, he effec of nominal raes on he economy is inexisen, jus real raes maer in his case, and hey are driven by inflaion. Furhermore, he nominal ineres rae, in his case, also affecs house prices and his also affecs borrowing. In erms of sabiliy, we see from he volailiies ha a greaer sabiliy, boh macroeconomic and financial, is achieved wih variable-rae morgages. Macroeconomic 27 I follow Ascari and Ropele (2009). 28 The Taylor Principle also holds in he model wih collaeral consrains, for (1 + φ π) 1, here is indeerminacy. 21

22 sabiliy is achieved because moneary policy is more effecive wih variable raes. 29 Wih fixed raes, on he one hand, borrowers are more exposed o changes in house prices. On he oher hand, since he nominal rae is fixed, he real rae mainly depends on inflaion, and his one is more volaile han in he variable-rae case. and real raes are more volaile. All his generaes greaer financial insabiliy. Table 3: Opimized Taylor and LTV Rule Variable Rae Fixed Rae Taylor Rule (1 + φ π) φ y LTV Rule φ k b Welfare Gain Savers Borrowers Toal Volailiies σ 2 π σ 2 y σ 2 b In Table 3, I presen he opimized moneary policy when i ineracs wih an LTV rule. We see ha, in his case, for he variable-rae scenario, he opimal response for moneary policy is subsanially less aggressive han in he benchmark case wihou macroprudenial policies in place. The macroprudenial LTV rule complemens he role of moneary policy and boh ineracing ogeher manage o achieve a more sable macroeconomic and financial scenario. However, a he expense of a slighly larger inflaion volailiy. 30 The increase in inflaion volailiy makes savers worse off because hey care abou he sickyprice fricion. On he conrary, borrowers are beer off for wo reasons; hey like higher inflaion because 29 See Rubio (2011) for a Taylor Curve analysis ha shows ha moneary policy is more effi cien wih variable rae morgages and herefore he economy is always more sable under his scenario. 30 This is a ypical resul found in he lieraure. Resuls are in line, for example, wih Gelain e al (2013) which show ha while macroprudenial policies can sabilize some variables, hey can magnify he volailiy of ohers, especially inflaion. 22

23 hey have o repay heir deb, and hey prefer a more sable financial scenario, since his helps hem smooh heir consumpion. For he fixed-rae case, he opimal reacion of boh moneary and macroprudenial policies is smaller ha in he variable-rae scenario. Moneary policy is sill no effecive, herefore he opimal response is minimal, as in he benchmark case. The inroducion of he LTV rule has similar effecs as in he variable-rae scenario. I reduces he volailiy of oupu bu increases he volailiy of inflaion. As remarked by Lusig (2006) and Rubio (2011), he inflaion channel ha relaxes borrowing consrains should be much more effecive when fixed-rae morgages are predominan, because agens care abou real raes. Therefore, agens are more sensiive o changes in inflaion in a fixed-rae scenario. This is he reason why borrowers welfare gain and savers loss are larger in his case, even hough in he aggregae welfare gains are similar as in he previous case. Welfare gains however, come mainly from he fac ha inflaion is more volaile, bu no from he financial side his ime. Given ha inflaion is less sable, borrowers benefi in erms of deb repaymens, relaxing heir consrain. This offses he consrain ighening ha he LTV rule should impose. As a resul, alhough he economy is beer off, a larger financial sabiliy is no achieved. Table 4: Opimized Macroprudenial Taylor Rule Variable Rae Fixed Rae (1 + φ π) φ y φ b Welfare Gain Savers Borrowers Toal Volailiies σπ σy σb

24 In Table 4, he macroprudenial policy is inroduced direcly in he Taylor rule, by leing he ineres rae respond o credi growh. Resuls show ha, alhough i is opimal o respond o credi growh, he opimal moneary policy is very similar o he case in which he cenral bank only responds o inflaion and oupu. As i is common in he lieraure, for he sandard variable rae case, welfare gains of responding o credi variables are very small. 31 Table 4 shows ha inflaion volailiy is slighly lower han in he benchmark case and financial insabiliy slighly larger. Thus, wih his new opimized Taylor rule, borrowers are slighly worse off wih respec o he case in which credi variables are no included in he rule because inflaion is less volaile, alhough here are no benefis from he financial side. This is offse by he fac ha savers live in a jus slighly more sable world. However, for he fixed-rae case gains are larger, mainly coming from he borrowers side. When he nominal rae responds o credi growh, i reacs more srongly o changes in he economy. Even hough he opimal response is small, credi is a volaile variable and hus he ineres rae responds more srongly han wih he sandard Taylor rule. This has an effec on house prices and he collaeral consrain is affeced hrough his channel. For example, if here is an increase in credi, he ineres rae will increase and his will decrease house prices. The fall in house prices ighens he collaeral consrain and helps achieve more financial sabiliy. A scenario wih greaer financial sabiliy is beneficial for borrowers. 4.3 Impulse Responses Figure 2 presens impulse responses o a echnology shock for a variable-rae economy and for he opimized parameers found in Tables 2-4. The echnology shock increases oupu and decreases inflaion. As a resul, he ineres rae slighly increases, o respond o he increase in oupu, especially in he case in which he ineres rae responds o credi growh. For he case in which he macroprudenial policy is represened by an LTV rule, he ineres rae decreases because he opimal reacion parameers in he Taylor rule are much smaller han in he oher wo cases. House prices move as a mirror image of ineres rae and also responding o he increase in housing demand derived from beer economic condiions. Since house prices increase, borrowing increases. However, he increase in borrowing is sofer in he case in which macroprudenial policies are presen. When he macroprudenial policy is incorporaed in he Taylor rule, he increase in he ineres rae is larger and hen, borrowing increases by less on impac, alhough he effec is dissipaed in subsequen periods. However, borrowing is really conained when 31 See for insance Iacoviello (2005), who shows wih a policy fronier analysis ha lile is gained in erms of inflaion and oupu sabilizaion by responding o asse prices. 24