Optimal Monetary Policy Under Financial Sector Risk *

Transcription

1 Federal Reserve Bank of Dallas Globalizaion and Moneary Policy Insiue Working Paper No. 85 hp:// Opimal Moneary Policy Under Financial Secor Risk * Sco Davis Federal Reserve Bank of Dallas Kevin X.D. Huang Vanderbil Universiy June 2 Absrac We consider wheher or no a cenral bank should respond direcly o financial marke condiions when seing moneary policy. Specifically, should a cenral bank pu weigh on inerbank lending spreads in is Taylor rule policy funcion? Using a model wih risk and balance shee effecs in boh he real and financial secors (Davis, "The Adverse Feedback Loop and he Effecs of Risk in he boh he Real and Financial Secors" Federal Reserve Bank of Dallas, Globalizaion and Moneary Policy Insiue Working Paper No. 66, November 2) we find ha when he convenional parameers in he Taylor rule (he coefficiens on he lagged ineres rae, inflaion, and he oupu gap) are opimally chosen, he cenral bank should no pu any weigh on endogenous flucuaions in he inerbank lending spread. However, he cenral bank should adjus he risk free rae in response o flucuaions in he spread ha occur because of exogenous financial shocks, bu we find ha he cenral bank should no be oo aggressive in is easing policy. Opimal policy calls for a wo-hirds of a percenage poin cu in he risk free rae in response o a financial shock ha causes a one percenage poin increase in inerbank lending spreads. JEL codes: E32; E44; E5; F4; G * Sco Davis, Globalizaion and Moneary Policy Insiue, Federal Reserve Bank of Dallas, 22 N. Pearl Sree, Dallas, TX 752. Sco.davis@dal.frb.org Kevin X.D. Huang, Vanderbil Universiy, VA Saion B #3589, 23 Vanderbil Place, Nashville, TN Kevin.huang@vanderbil.edu The views in his paper are hose of he auhors and do no necessarily reflec he views of he Federal Reserve Bank of Dallas or he Federal Reserve Sysem.

2 Inroducion The pas few years have winessed how nancial secor risks could be spread across counries borders o cause a nancial and economic calamiy worldwide. This overwhelming episode has simulaed a surged ineres in invesigaing wheher and how moneary policy should respond o variaions in nancial marke condiions. A popular view is ha he cenral bank should include some inerbank credi spreads in a Taylor ype moneary policy rule. In his esimony on February 26, 28 before he Commiee on Financial Services of he U.S. House of Represenaives, John B. Taylor argued ha he inercep erm in a Taylor ype rule for moneary policy, ha is, he naural rae of ineres, should be adjused downward in proporion o observed increase in he spread beween he erm Libor rae a hree monh mauriy and an index of overnigh federal funds raes expeced for he same period. Similar views have been expressed by ohers, including Goodfriend and McCallum (27), De Fiore and Trisani (27), McCulley and Toloui (28), Meyer and Sack (28), Curdia and Woodford (29 and 2), Woodford (2), and Mishkin (2a and 2b). In his paper, we examine wheher and how a Taylor ype moneary policy rule should be modi ed in an environmen feauring nancial fricions and nancial secor risks. A de ning feaure of our model is a careful disincion beween exogenous and endogenous changes in he nancial marke condiions, which we capure by an inerbank lending spread. A robus nding in he paper is ha boh he naural rae of ineres and he risk free policy rae should be adjused direcly o an exogenous variaion in he spread, which we call a nancial secor shock, bu neiher should respond direcly o any endogenous movemen in he spread. Our paper is a sudy of opimal simple rules for cenral banks in an open-economy moneary model ha feaures muliple sources of fricions and risks in boh real and nancial secors. Esimaed simple moneary policy rules ypically ake he form of a policy rae as a funcion of in aion and he oupu gap, as well as he lagged policy rae. Whereas sabilizing he variabiliy in in- aion and he oupu gap are he basic characerisics of he celebraed Taylor rule (e.g., Taylor, 993), subsequen sudies reveal subsanial evidence of ineres rae smoohing in moneary policy pracice (e.g., Rudebusch, 995; Clarida, Gali, and Gerler, 998 and 2; Orphanides, 2). Argumens in favor of such policy rules have been made using srucural models where he cen- 2

3 ral bank s loss funcion consiss of variaions in in aion, he oupu gap, and ineres rae (e.g., Woodford, 23a). Desirabiliy of such policy rules or heir varians have been shown in models wih sicky prices in one secor (e.g., Clarida, Gali, and Gerler, 999; Goodfriend and King, 2; Aoki, 2), in muliple secors (e.g., Mankiw and Reis, 23; Huang and Liu, 25), in muliple counries (e.g., Benigno, 24; Clarida, Gali, and Gerler, 22), and wih boh sicky prices and sicky wages (e.g., Erceg, Henderson, and Levin, 2; Amao and Laubach, 23). The imporance of ineres rae smoohing is especially emphasized by Woodford (23b). The robusness of such simple policy rules have been shown in various modeling environmens (e.g., Levin and Williams, 23; Levin, Wieland, and Williams, 999 and 23; Levin, Onaski, Williams, and Williams, 25; see Taylor and Williams, 29b for a survey). These models all feaure a fricionless nancial world. In his paper, we examine how such simple moneary policy rules migh need o be modi ed in an environmen feauring nancial fricions and nancial secor risks. Our model akes is roo in he classic nancial acceleraor lieraure pioneered by Gerler (988), Bernanke and Gerler (989), Carlsrom and Fuers (997), Kiyoaki and Moore (997), and Bernanke, Gerler, and Gilchris (999), among ohers. These papers incorporae fricions in nance for manufacuring rms, bu hey do no model nancial inermediaries. Some recen sudies incorporae nancial inermediaries in modeling, bu hey absrac from fricions or risks in he nancial secor. For insance, Chrisiano, Moo, and Rosagno (28) model banks, bu hey do no allow for fricions or risks wihin he banking secor. A few recen papers like Meh and Moran (2), Gerler and Karadi (29), Gerler and Kiyoaki (2), and Dib (2) model nancial fricions wihin he nancial secor, in he form of collaeral consrains, bu hey do no allow for nancial secor risks. All of hese sudies focus on he mechanism of nancial fricions in ransmiing real or moneary shocks. Our model builds on Davis (2) bu is concerned abou he design of opimal simple moneary policy rules in boh a closed and an open-economy moneary environmen ha feaures muliple sources of fricions and risks, in boh real and nancial secors. The model has four imporan feaures. Firs, i incorporaes nancial inermediaries, and fricions and risks in boh he manufacuring indusries and he nancial secor. Thus, he model feaures balance shee e ecs on boh he demand and supply sides of he credi marke. Second, i incorporaes sicky wages, in addiion o sicky prices, so moneary policy faces a nonrivial rade-o beween di eren componens of he 3

4 cenral bank s objecive even when he nancial fricions and risks are mued, while in he baseline case he policy rade-o is muli-dimensional. Third, we consider boh a closed and open-economy seing o ake ino accoun fricions and risks in no only domesic bu inernaional inerbank lending markes for shor-erm unsecured loans. Las, i disinguishes beween endogenous and exogenous ucuaions in inerbank lending spreads, he spread beween he bank s cos of capial and he risk free rae. As we show below, exogenous and endogenous ucuaions in he inerbank spreads play fundamenally di eren roles in deermining an opimal simple moneary policy rule. Our resuls are easy o summarize. Saring wih he sandard case ha absracs from nancial fricions and nancial secor risks, we derive a simple opimal rule in erms of he respecive responsiveness of policy rae o he variabiliy in in aion and he oupu gap as well as lagged policy rae, much in line wih he sandard lieraure on opimal moneary policy. We hen nd ha he presence of nancial fricions calls for a greaer degree of gradualism in he ineres rae rule. This nding suppors he classic view of Goodfriend (987) and Cukierman (99) in suppor of ineres rae smoohing from he perspecive of nancial sabiliy. In addiion, policy also shifs weigh from oupu gap sabilizaion o in aion sabilizaion in he presence of nancial fricions. This is consisen wih he nding by Lee (2) who models nancial fricions on he households side, whereas we model nancial fricions and risks on he sides of rms and banks. Bu, in conras o he sandard models ha absracs from nancial fricions and nancial secor risks, our model feaures more han one ineres rae. Time varying spreads in our model have an allocaive role. Issing (26) and Goodhar (27) sugges ha such spreads may have implicaions for moneary policy pracice. I is on his end we nd i imporan o disinguish beween wha we erm endogenous ucuaions in he spread from exogenous ucuaions. Endogenous ucuaions in he spread are he essence of a nancial acceleraor model. In his model wih nancial fricions in he banking secor, he inerbank lending spread ucuaes counercyclically in response o endogenous ucuaions in a bank s capial srucure and loan loss raios. In his paper s mos ineresing nding, we nd ha when he convenional parameers of he Taylor rule, he coe ciens on he lagged ineres rae, he in aion rae, and he oupu gap, are chosen opimally, he cenral bank should ignore hese endogenous ucuaions in he Recen welfare-based moneary policy evaluaions in models wih nancial fricions on he households side also include Iacoviello (25) and Monacelli (29). 4

5 spread. Since hese ucuaions are he endogenous reacions o oher macroeconomic variables in he model, hese ucuaions in he spread provide no new informaion ha is no already conained in measures of he oupu gap and in aion. Since he convenional Taylor rule parameers were chosen opimally, he cenral bank has already found he opimal weighing of he informaion conained in he in aion rae and he oupu gap. Therefore puing any weigh on a new erm ha conains no new informaion would be sub-opimal. However his may no be rue for exogenous ucuaions in he spread. We de ne exogenous ucuaions in he spread as occurring because of some exogenous nancial secor shock, for insance, a sudden ighening of he credi marke ha occurs because of a sudden increase in nancial secor risk or uncerainy. This ype of shock is documened by Taylor and Williams (29a) who describe he sudden increase in inerbank lending spreads a he beginning of he nancial crisis in Augus 27. Bordo and Haubrich (2) documen hisorical insances of hese credi marke shocks going back o 875. Helbling e al. (2) and Gilchris, Yankov and Zakrajsek (29) use economeric echniques he single ou hese credi shocks and demonsrae heir imporance in explaining he ucuaions in broader macro aggregaes. Wihin he framework of a nancial acceleraor model, a number of recen papers, like Aah-Mensah and Dib (28), Chrisiano e al. (23 and 28), Nolan and Thoenissen (29), Jermann and Quadrini (29), and Gilchris, Oriz, and Zakrajsek (29) have inroduced credi shocks ino a DSGE model. We nd ha ucuaions in he spread ha are caused by hese exogenous credi shocks may conain new informaion ha is no already found in measures of he oupu gap and he in aion rae. Thus he cenral bank may wan o reduce boh he naural rae of ineres and he risk free policy rae in response o an exogenous increase in he inerbank spread ha is caused by a credi marke shock. Furhermore, in he open economy versions of he model, he cenral bank will wan o reac o exogenous ucuaions in boh home and foreign inerbank lending spreads. While he cenral bank will wan o employ an accommodaive moneary policy in response o a credi marke shock, we nd ha he cenral bank will no wan o fully accommodae he shock. To fully accommodae he shock would imply ha he cenral bank would lower he risk free rae by % in response o a % exogenous increase in he spread. We nd ha his degree of accommodaion is oo exreme. Opimal policy is for he cenral bank o reduce he risk free rae by abou.7% in response o a % exogenous increase in he spread. 5

6 Our paper is relaed o an emerging lieraure of welfare-based moneary policy evaluaions using models wih nancial fricions, including Moessner (26), Faia and Monacelli (27), De Fiore and Trisani (27), Teranishi (28), Sudo and Teranishi (28), Curdia (28), Curdia and Woodford (29 and 2), Faia and Iliopulos (2), Merola (2), and Kolasa and Lombardo (2), among ohers. Besides our modeling deails, such as he open-economy seup wih boh sicky prices and sicky wages, wha disances our sudy from his lieraure is he incorporaion of nancial secor shocks, fricions in nance in boh real and nancial secors, heir disinguished roles, and he disincion beween exogenous changes of domesic and foreign inerbank spreads and endogenous movemens in hese spreads in deermining simple opimal moneary policy rules. This paper will proceed as follows. Secion 2 presens he model ha is used o assess he desirabiliy of including inerbank spreads in he cenral bank s policy funcion. The model is a muli-counry new Keynesian model, wih nancial fricions inroduced in boh real and nancial secors ha enable he model o move away from he irrelevance of balance shees implied by he Modigliani and Miller (958) heorem. The model is very similar o ha presened in Davis (2), bu unlike he model in Davis (2), he cenral bank s policy funcion is modi ed o give he cenral bank he opion of responding o nancial marke condiions. In addiion, in his paper we inroduce a new ype of shock ha is a direc shock o risk and uncerainy in he nancial secor. This new ype of shock provides he basis for he exogenous ucuaions in he inerbank spread ha become so imporan when discussing opimal simple policy rules. Then he calibraion of he model is discussed in secion 3. The opimal parameers in he Taylor rule as derived from simulaions of he model are presened in secion 4. Firs we discuss how he presence of nancial fricions in he model induces he cenral bank o pu more weigh on ineres rae smoohing in heir Taylor rule funcion. Then we discuss wheher or no he cenral bank will wan o direcly arge inerbank lending spreads. Finally, secion 5 concludes and o ers some suggesions for furher research. 2 Model In he model here are ve ypes of agens: rms, enrepreneurs, capial builders, banks, and households. There is also a cenral bank ha ses he risk free nominal rae of ineres. 6

7 Firms use capial and labor inpus o produce radeable oupu ha is used for consumpion and invesmen. Each rm produces a di ereniaed good and ses prices according o a Calvo (983) syle price seing framework, hus giving rise o nominal price rigidiy. Enrepreneurs own physical capial and ren i o rms. This physical capial is nanced parially hrough deb and parially hrough equiy. In every period, an individual enrepreneur faces an idiosyncraic shock o he value of heir physical capial asses. While hese shocks have no direc aggregae e ecs, hey inroduce heerogeneiy among enrepreneurs. The shock is uninsurable, and a fracion of enrepreneurs may experience an abnormally large shock o he value of heir physical capial sock and be pushed ino bankrupcy, while mos will no. The uncerainy over which enrepreneurs will be pushed ino bankrupcy and which will no is a ype of nancial fricion in he real secor. The raio of deb o equiy on an enrepreneur s balance shee deermines heir abiliy o wihsand a shock o he value of heir capial sock. Crediors use he enrepreneur s deb-equiy raio o deermine he riskiness of lending o he enrepreneurial secor, giving rise o a defaul risk ineres premium ha depends on he deb-equiy raio. 2 Capial builders purchase nal goods from rms for physical capial invesmen. There are diminishing marginal reurns o physical capial invesmen. In periods when invesmen is high, he marginal reurn of ha invesmen in producing new physical capial is low, and vice versa. This gives rise o a procyclical relaive value of physical capial. Banks channel savings from households o rms in he form of working capial loans and o enrepreneurs in he form of physical capial loans. A bank nances is asse porfolio parially hrough equiy and parially hrough deb, which is made up of deposis from domesic and foreign households. Due o bankrupcies in he real secor, a porion of a bank s porfolio of physical capial loans will go ino defaul in any given period. While hese loan losses are no grea enough o push he enire banking secor ino insolvency, here is heerogeneiy among banks wih regards o heir exposure o he se of non-performing loans. A few banks may be over-exposed o he se of bad loans, and hey hemselves may be pushed ino insolvency. The uncerainy abou which banks are over-exposed o he se of non-performing loans and which are no is a ype of nancial fricion in 2 The fac ha his idiosyncraic shock is uninsurable provides he necessary violaion of he complee markes assumpion necessary o overcome he implicaions of he Miller and Modigliani (958) heorem. 7

8 he banking secor. The raio of deb o equiy on a bank s balance shee deermines heir abiliy o absorb loan losses, so he deb-equiy raio deermines he ex-ane riskiness of a paricular bank. This gives rise o an environmen where he spread beween inerbank lending raes and he risk free rae is increasing in he leverage raio of he banking secor. Households supply labor o rms and consume nal oupu. Furhermore hey supply a di ereniaed ype of labor and se wages according o a Calvo-syle wage seing process, giving rise o nominal wage rigidiy. Finally, he cenral bank ries o sabilize oupu and prices by conrolling he risk free nominal rae of ineres. The cenral bank ses policy using a Taylor rule funcion combining he curren period s in aion rae, oupu gap, and he lagged risk free nominal ineres rae. We will also consider he case where he home and foreign inerbank lending spreads are also par of he Taylor rule. When considering he cenral bank s opimal reacion o nancial secor developmens, speci cally we are rying o nd he opimal coe ciens on he spreads in he Taylor rule. The remainder of his secion presens he acual deails of he model. In secion 4 we will examine wo versions of he model, he closed economy, and wo open economies. In wha follows, he key model equaions are wrien for he open economy case. In he model s noaion, he relaive size of he home counry is n and he relaive size of he foreign counry is n. In he case of wo large economies, n = 2. In he case of he closed economy, imagine ha n =. In wha follows, all variables are wrien in per capia erms and foreign variables are disinguished by an aserisk (*). In he open economy version of he model, he wo counries are symmeric, so foreign equaions have been omied for breviy excep where absoluely necessary. 2. Firms In he home counry, inermediae goods producing rms, indexed i 2 [ n], combine capial and labor, k (i) and h (i) o produce a unique inermediae good Y (i). The rm s producion funcion is: Y (i) = A h (i) k (i) () 8

9 where A is an exogenous counry speci c sochasic TFP parameer ha is common o all rms and is a xed cos parameer ha is calibraed o ensure ha rms earn zero pro in he seady sae. The oupu from rm i can be sold o he domesic marke or sold as impors in he foreign marke: Y (i) = y d (i) + y m (i) where y d (i) is oupu from rm i ha is sold domesically and y m (i) is he oupu ha is impored ino he foreign counry. Inermediae goods from domesic and foreign rms are hen combined ino one aggregae nal good. As in Chari, Kehoe, and McGraan (22), domesically supplied and impored inermediae goods are aggregaed by he following: y = " () R n yd (i) R di + f n ym (i) # di (2) where is he elasiciy of subsiuion beween domesic varieies and is he elasiciy of subsiuion beween home and foreign varieies. From his aggregaor funcion he demand in he home counry for he inermediae good from domesic rm i, where i 2 [ n], as a funcion of aggregae demand is: y d (i) = (n) P d (i) P d y (3) P Similarly, he demand in he home counry for he inermediae good from foreign rm i, where i 2 (n ], as a funcion of aggregae demand is: P d n y m (i) = f ( n) P m (i) P m P P m y (4) where P d (i) is he price in he domesic marke for he inermediae good from rm i, P d = R n P d (i) di is a price index of domesically produced inermediae goods, P m = nr n (P m (i)) di is a price index of impored inermediae goods, and he aggregae 9

10 price level is given by P = h (n) P d + f ( n) (P m ) i. Firm i can discriminae when seing prices for he domesic or foreign marke. Thus hey can se separae prices for he domesic and expor markes. In period, he rm will be able o change is price in he domesic marke wih probabiliy p. If he rm canno change prices hen hey are rese auomaically according o P d (i) = P d (i), where = P P 2. Thus if allowed o change heir domesic price in period, he rm will se a price o maximize: max P E P d(i) = o p + n ;+ P d (i) y+ d (i) MC + y+ d (i) where is he marginal uiliy of income in period. As discussed in his paper s echnical appendix, he rm ha is able o change is domesic price in period will se is price o: P d (i) = P E = P E = ;+ P d p + MC + + P+ d P y+ + ;+ P d p + + ;+ P+ d P y+ + If prices are exible, and hus p =, hen his expression reduces o: P d (i) = MC which says ha he rm will se a price equal o a consan mark-up over marginal cos. Wrie he domesic price se by he rm ha can rese prices in period as ~ P d (i) o denoe ha i is an opimal price. Firms ha can rese prices in period will all rese o he same level, so P ~ d (i) = P ~ d. Subsiue his opimal price ino he price index P d = R n n P d (i) di. Since a rm has a probabiliy of p of being able o change heir price, hen by he law of large numbers in any period p percen of rms will reopimize prices, and he prices of p percen of rms will be auomaically rese using he previous periods in aion rae. Thus he domesic price index, P d, can be wrien as: P d = p ; P d + p P ~ d The full deails of his derivaion as well as he derivaion for prices se for he foreign marke

11 is locaed in he appendix. The rm hires labor and capial inpus, where W is he wage rae paid for labor inpu and R is he capial renal rae, boh of which he rm akes as given. Furhermore he rm mus pay heir wage bill a he beginning of he period, prior o producion. To do so hey borrow b wc (i) = W h (i). The rm s income afer paying for capial and labor inpus is: d f (i) = P d (i) y d (i) + P x (i) y x (i) W h (i) R k (i) r wc b wc (i) (5) where P x (i) is he expor price for he inermediae good from rm i, and r wc on working capial loans. is he ineres rae Since here is no defaul risk from lending working capial o rms, compeiion in he banking secor forces he rae on working capial loans down o he bank s own cos of capial, r wc = r b. The aggregae income from all rms is reurned o households as a lump sum paymen, d f = R n df (i) di. The rm will choose h (i) and k (i) o maximize pro in (5) subjec o he producion funcion in (). The working capial requiremen implies ha he cos of he labor inpu is W ( + r wc ) and he cos of he capial inpu is R. Given hese prices, he rm s demand for labor and capial inpus are: where MC = A W(+r wc ) h (i) = ( ) k (i) = MC Y (i) R R. MC W ( + r wc ) Y (i) (6) 2.2 Enrepreneurs Enrepreneurs, indexed j 2 [ n], buy capial from capial builders and ren i o rms. A he beginning of period, enrepreneur j has a sock of capial, K (j), ha he will ren o rms in period a a renal rae R. In equilibrium, he aggregae sock of capial supplied by all domesic enrepreneurs j is equal o he aggregae sock of capial demanded by all domesic rms

12 i, R n K (j) dj = R n k (i) di. Enrepreneurs nance his sock of capial parially hrough deb. The enrepreneur borrows b e (j) from domesic banks o nance heir capial sock K (j). Thus he marke value of he asses and liabiliies for enrepreneur j a he beginning of period are: Asses: P K K (j) Liabiliies: b e (j) (7) where P K is he price of exising capial. The end of he period he value of he non-depreciaed capial sock for he average enrepreneur is P K ( ) K. However during he period, he individual enrepreneur j receives an idiosyncraic draw ha a ecs he relaive price of heir exising capial, so for enrepreneur j he end of period value of heir non-depreciaed capial sock is:! e (j) P K ( ) K (j) where! e (j) is a i.i.d. draw from a lognormal disribuion on he inerval [ ) wih mean and variance 2 e. Since his draw has a mean, i has no e ec on he aggregae capial sock. I simply inroduces heerogeneiy among enrepreneurs, and in any given period a fracion of enrepreneurs receive a draw ha has a large adverse e ec on he value of heir exising capial (a small! e (j)) and hus a he end of he period, he value of heir liabiliies exceeds he value of heir asses. During he period he enrepreneur rens his capial sock o rms for a renal rae of R. The enrepreneur nances his capial sock wih a loan from he bank wih an ineres rae r e. Thus a he end of he period, afer he realizaion of! e (j), he nominal marke value of enrepreneur j s asses is! e (j) P K ( ) K (j) + R K (j). A he end of he period he nominal value of he enrepreneur s liabiliies is ( + r e ) b e (j). Thus, afer he realizaion of! e (j), enrepreneur j is bankrup if:! e (j) P K ( ) K (j) + R K (j) < ( + r e ) b e (j) (8) 2

13 Thus he hreshold value of! e (j) below which he enrepreneur goes bankrup in period and above which hey coninue operaions is:! e = ( + re ) be (j) K (j) P K ( ) R (9) where DA e (j) = be (j) K (j) is he raio of he book value of deb o he book value of asses on an enrepreneur s balance shee. The hisory of individual enrepreneur j will in uence he level of b e (j) and K (j), bu he raio DA e (j) = be (j) K (j) is equal across all enrepreneurs. This is a key resul for aggregaion, for i implies ha he bankrupcy cuo value! e does no depend on an enrepreneur s hisory. More inuiion behind his resul is presened a he end of his secion and a formal proof is presened in he appendix. When deciding how much o lend o enrepreneurs going ino nex period and a wha rae, banks facor in he fac ha if enrepreneur j does no defaul in period +, crediors receive a reurn of r+ e. If he enrepreneur defauls, crediors receive a share of he enrepreneur s remaining asses, less he bankrupcy cos e. The hreshold value! e + in equaion (9) deermines wheher or no an enrepreneur goes ino defaul nex period. Thus he payo o crediors condiional of he realizaion of he shock! e + (j) is: + r e + b e + (j) if! e + (j)!e + ( e )! e + (j) ( ) P K + K + (j) + R + K + (j) if! e + (j) <!e + () Perfec compeiion in he banking secor implies ha he bank s expeced pro is zero. So he ineres rae he bank charges on physical capial loans, r+ e, is se such ha he expeced reurn, afer facoring in he cos of bankrupcy, is equal o he bank s cos of capial, r b + : + r+ b b e + (j) = Z! e + Z +! e + ( e )! e + (j) ( ) P+K K + (j) + R + K + (j) df! e + + r+ e b e + (j) df! e + where F! e + is he c.d.f. of he lognormal disribuion of! e +. Thus he ineres rae charged by banks for physical capial loans is: 3

14 + r+ e = + rb + F! e + h ( e ) R + F! e R + + ( ) P K! e + +! e + df i!e + F! e + b e + (j) K + (j) () where F! e + is he percen of manufacuring rms ha declare bankrupcy. Holding all else equal, his ineres rae, r e +, is increasing in F!e +. If here are nancial fricions in he enrepreneurial secor, F! e + is increasing in! e +.! e + is increasing in he manufacuring rm s deb-asse raio. Thus when here are nancial fricions in he enrepreneurial secor, he ineres rae on physical capial loans is increasing in he level of deb on an enrepreneur s balance shee. The cuo value of! e + (j) in equaion (9) combined wih he ineres rae expression in () demonsraes he feedback loop associaed wih nancial fricions in he enrepreneurial secor. When he price of exising capial, P K + falls, he cuo value!e + rises. This implies ha more rms will receive draws of! e + (j) below his cuo value and be forced ino bankrupcy. When more rms go ino bankrupcy, F! e + increases, and r e + increases as banks now demand a higher ineres rae o compensae for he increased bankrupcy risk. This higher r e + means higher ineres expenses and lower pro for he enrepreneur, which leads o a furher increase in he cuo value! e +. The end of period ne worh for he enrepreneur ha survives is he enrepreneur s pro in ime plus he value of heir non-depreciaed capial sock: ~N e (j) = R K (j) ( + r e ) b e (j) +! e (j) P K ( ) K (j) The enrepreneur will pay a dividend o shareholders of d e (j) and begin he nex period wih ne worh N e + (j) = ~ N e (j) d e (j). Enrepreneurs ha declare bankrupcy in period pay no dividend and drop ou of he marke, hey are replaced wih new enrepreneurs, which are endowed wih sar up capial of N e. Thus he ne worh of he enrepreneurial secor a he beginning of nex period is: 4

15 N e + = Z! e = N e F (! e ) + N e + (j) df (! e ) + Z! e N e + (i) df (! e ) (2) Z r k K ( + r e ) b e d e ( F (! e )) + P K ( ) K! e! e df (! e ) A he beginning of any period, enrepreneurs have di eren levels of ne worh N + (j) ha will depend on he enrepreneur s hisory of idiosyncraic shocks! e (j). The enrepreneur will acquire capial up o he poin where he ineres rae on bank loans is equal o he expeced reurn o holding a uni of capial: r e + = E R + +! e + (j) ( ) P K + P K! Since! e + (j) is i.i.d. and E! e + (j) =, he lef hand side of he above expression is he same across all enrepreneurs j, which implies ha r e + is he same across all enrepreneurs. 2.3 Capial Builders The represenaive capial builder convers nal goods, given by equaion (2), ino he physical capial purchased by enrepreneurs. A he end of period, he non depreciaed physical capial sock is ( ) K, and he physical capial sock a he beginning of he nex period is K +. The evoluion of he physical capial sock is given by: I K + ( ) K = K K where > and < implying ha here are diminishing marginal reurns o physical capial invesmen. Capial builders purchase nal goods for invesmen a a price P and sell exising capial o enrepreneurs a a price P K. Thus he pro s of he represenaive capial builder are given by: d c = P K (K + ( ) K ) P I In a compeiive capial building secor, pro maximizaion implies ha he relaive price of 5

16 exising capial is: Since <, when I K P K = I P K is high, I K is low, so P K P is high. This implies ha during imes of high physical capial invesmen, when he raio of invesmen o he exising capial sock is high, he relaive price of exising capial is high. Since invesmen is highly procyclical, capial adjusmen coss imply ha he relaive price of capial is highly procyclical as well. 2.4 Banks Banks, indexed k 2 [ n] make physical capial loans o domesic enrepreneurs. They nance his loan porfolio parially wih equiy and parially wih borrowing from domesic and foreign households. A he beginning of period, he value of he bank s asses is B e (k), which is he bank s sock of loans o enrepreneurs. The value of he bank s liabiliies is b s (k) + b sf deposis of domesic households and b sf (k) are he deposis of foreign households. 3 (k), where b s (k) are he The bank also makes working capial loans o rms in order o nance he rm s wage bill. This however is no lised as a beginning of period asse for he bank. By assumpion his loan is made afer he beginning of he period and repaid before he end of he period. If he sock of working capial loans were o appear as a asse for he bank a he beginning of period, ha would imply ha he loan was made in period, which implies ha he rm made a decision abou period s labor inpu in period. Bankrupcy in he enrepreneurial secor in period means he bank s asses are worh less a he end of he period. The value of he average bank s asses a he end of he period is ( e ) ( + r e ) B e, where e is he share of he average bank s physical capial loan porfolio ha is los o bankrupcy and liquidaion coss. e represens he share of he average bank s physical capial loan porfolio ha is los o 3 The same sock of bonds ha is a liabiliy o one pary is an asse o anoher. Throughou his paper, when a sock of bonds is an asse, i is wrien wih a capial B, when he sock of bonds is a liabiliy i is wrien wih a lower case b. Thus marke clearing in he bond marke requires ha he sum of physical capial loans across all banks equals he sum of borrowing by enrepreneurs, R n Be (k) dk = R n be (j) dj. 6

17 bankrupcy and liquidaion coss, however banks don hold fully diversi ed loan porfolios. Some banks may be overexposed o he se of non-performing loans o he enrepreneurial secor. This overexposure may be due o a regional bias in he bank s porfolio, or i may be because a bank has a cerain core compeency and is herefore overexposed o a cerain secor of he economy. 4 The percen of he bank k s loan porfolio ha is los o bankrupcy or liquidaion coss is h i! b (k) e, where! b (k) is an i.i.d. draw from a lognormal disribuion on he inerval e wih mean and sandard deviaion b. If bank k receives a large draw! b (k), i implies ha he bank is overexposed o he se of non-performing loans and may iself face insolvency. The bank is insolven if he end of period value of is asses is less han he end of period value of is liabiliies:! b (k) e ( + r e ) B e (k) < + r b (k) b s (k) + b sf (k) The hreshold value of! b (k) above which bank k is forced o declare bankrupcy and below which he bank will coninue operaions is:! b = ( + re ) + r b (k) bs (k)+bsf (k) B e(k) e ( + r e ) (3) Bank k s hisory of idiosyncraic draws,! b (k), hus is hisory of exposure o non-preforming secors of he economy, will deermine he levels of B e (k), b s (k), and b sf (k). However, a he beginning of he period, all banks will have he same raio of oal deb o oal asses, DA b (k) = b s (k)+bsf (k) B e(k) and will have he same cos of capial, r b (k). This resul is key for he aggregaion of balance shee variables across a coninuum of individual banks, for his implies ha he cuo value! b is common across all banks. The formal proof of his claim is presened in he appendix. When deciding how much o lend o bank k in he nex period and a wha rae, he bank s crediors facor in he fac ha if he bank does no defaul, hey receive a gross ineres rae + r b + (k). If bank k defauls, crediors receive nohing.5 Thus he expeced payo o a bank s 4 Like he banks, many of which are now bankrup or were acquired by healhier rivals, who were overexposed o he subprime secor of he morgage marke during he recen nancial crisis. 5 The assumpion ha crediors receive nohing in he case of bank defaul is because he model is laer calibraed such ha he spread beween he inerbank rae, r b, and he risk free rae, i, in he seady sae of he model is equal o he hisorical average of he spread beween he 3-monh Libor and he 3-monh T-bill. The Libor is an inerbank index rae ha is based on he ineres rae for unsecured lending o banks. 7

18 crediors condiional on he bank s exposure o he se of non-preforming loans is: + r+ b (k) b s + (k) + bsf + (k) if! b + (k)!b + if! b + (k) >!b + (4) Domesic and foreign deposiors will exend bank k credi up o he poin where he expeced reurn, afer facoring in he probabiliy of defaul is equal o he risk free rae: Z! b ( + i + ) b b + (k) + b bf + (k) + = + r+ b (k) b b + (k) + b bf + (k) dg! b +; b + This condiion can be used o solve for he ineres rae on inerbank lending o bank k: + r b + (k) = + i + G! b + ; (5) b + where G! b + ; b + is he c.d.f. of he lognormal disribuion of! b +, and hus measures he proporion of banks ha do no go bankrup in period +. Since DA b + (k) = bs + (k)+bsf + (k) B e + (k) consan across all banks, he inerbank lending rae, and hus banks cos of capial, is consan across all banks. A rs order Taylor series expansion of he expression in (5) highlighs he wo facors, one endogenous and one exogenous, ha can cause ucuaions in he inerbank lending spread, rp b = +r b + +i + : is rp b rp b ss " + g! b +! b ss! b ss! + g 2 b + b ss b ss!# (6) where g + ;b b +! b + G(! b + ;b +) < is he elasiciy of he spread wih respec o changes in he endogenous cuo value! b +, and g 2 + ;b +) b + > is he b + G(! b + ;b +) of he spread wih respec o changes in he exogenous variable describing uncerainy in he inerbank mark, b +. Thus ucuaions in he spread, r^pb = rpb rp b ss, can be dividend ino wo componens, an endogenous componen due o ucuaions in he endogenous cuo value! b +, and an exogenous componen due o change in he exogenous uncerainy variable b +. 8

19 De ne r^p b;endo = g 2 r^p b;exo! = g b +! b ss as he endogenous componen of ucuaions in he spread and! b ss as he exogenous componen. b + b ss b ss The end of period ne worh of he bank ha is no over-exposed o he se of non-preforming loans and is able o coninue operaions is: ~N b =! b (k) e ( + r e ) B e (k) + r b b s (k) + b sf (k) The bank will pay a dividend o shareholders and begin he nex period wih a ne worh N b + (k) = ~ N b (k) d b (k). Banks ha were overexposed o he se of non-preforming loans and hus were forced ino bankrupcy end he period wih no ne worh and drop ou of he marke. They are replaced wih new banks ha are endowed wih sar up capial N b. Thus he ne worh of he enire banking secor a he beginning of nex period is: N b + = Z! b Z! b N+ b (i) dg! b ; b + = N b G! b ; b + ( + r e ) B e Z! b ( + r e ) B e e! b dg! b ; b N+ b (k) dg! b ; b + r b b s + b sf d b G! b ; b (7) 2.5 Households Households, indexed l 2 [ n], supply heerogeneous labor o rms and consume from heir labor income, ineres on savings, and pro income from rms, enrepreneurs, capial builders, and banks. The household maximizes heir uiliy funcion: max P = ln (C (l)) (H (l)) + H H (8) subjec o heir budge consrain: 9

20 P C (l) + B+ s (l) + S B sf + (l) + F (!e ) N e + G! b ; b N b = W (l) H (l) + d f (l) + de (l) + d c (l) + d b (l) + b + r b + b + r b S B sf (l) + e + b b 2 B s (l) S B sf (l) S B sf 2 (9) where C (l) is consumpion by household l in period, H (l) is he household s labor e or in he period, B s (l) is he household s sock of deposis wih domesic banks a he beginning of he period, B sf (l) is he sock of deposis wih foreign banks, W (l) is he wage paid for he household s heerogenous labor supply, b ( b ) represens he small share of deposis o he home (foreign) banking secor ha are los o bankrupcy and liquidaion coss, and d f (l), de (l), d c (l) and d b (l) are he household s share of period pro s from rms, enrepreneurs, capial builders and banks, respecively. 6 The household pays a small quadraic ransacions cos o holding oher han he seady sae level of deposis wih foreign banks, b 2 B sf (l) B sf 2. Each household supplies a di ereniaed ype of labor. The funcion o aggregae he labor supplied by each household ino he aggregae sock of labor employed by domesic rms is: Z n H = H (l) dl (2) where H = R n h (i) di. Since he household supplies a di ereniaed ype of labor, i faces a downward sloping labor demand funcion: W (l) H (l) = H W In any given period, household j faces a probabiliy of w of being able o rese heir wage, oherwise i is rese auomaically according o W (l) = W (l). If household j is allowed o rese heir wages in period hey will se a wage o maximize he 6 Marke clearing in he marke for deposis requires ha he sum of deposis wih domesic banks across all domesic households equals he sum of borrowing from domesic households across all domesic banks, R n R Bs (l) dl = n bs (k) dk, and ha he sum of deposis wih foreign banks across all domesic households equals he sum of borrowing from domesic households across all foreign banks, R n Bsf (l) dl = R n bsf (k) dk. 2

21 expeced presen value of uiliy from consumpion minus he disuiliy of labor. E P = ( w ) + ;+ W (l) H + (l) (H + (l)) + H H Thus afer echnical deails which are locaed in he appendix, he household ha can rese wages in period will choose a wage: W (l) H + = + H H (W ) H E = P ( w ) P E ( w ) + ;+ = + W + H ;+ W (H+ ) + H H W+ ;+ W H+ If wages are exible, and hus w =, his expression reduces o: W (l) = + H H (H ) H Thus when wages are exible he wage rae is equal o a mark-up, ( ), muliplied by he marginal disuiliy of labor, + H H (H ) H, divided by he marginal uiliy of consumpion,. Wrie he wage rae for he household ha can rese wages in period, W (l), as ~ W (l) o denoe i as an opimal wage. Also noe ha all households ha can rese wages in period will rese o he same wage rae, so ~ W (l) = ~ W. All households face a probabiliy of ( w ) of being able o rese heir wages in a given period, so by he law of large numbers ( w ) of households can rese heir wages in a given period. The wages of he oher w will auomaically rese by he previous periods in aion rae. Subsiue ~ W ino he expression for he average wage rae W = derive an expression for he evoluion of he average wage: W = w ( ; W ) + ( w ) ~W R n W (l) dl, o 2.6 Moneary Policy The moneary policy insrumen is he shor erm risk free rae, i, which is deermined by he cenral bank s Taylor rule funcion: 2

22 i + = i ss + i (i i ss ) + ( i ) p + y ^y + r r^p b + rf r^p b (2) where = P P 4, and ^y = GDP G DP ~, where G ~ DP is he level of GDP a ime in an economy wih he same srucure as he one jus described and subjec o he same shocks, only here are no price or wage fricions, p = w =, and here are no nancial fricions, e = b =. When r = rf =, he cenral bank does no place any weigh on condiions in he inerbank markes and he Taylor rule is simply he convenional Taylor rule wih smoohing. Recall from equaion (6) ha ucuaions in he inerbank lending spread r^p b have an endogenous and exogenous componen, r^p b = r^p b;endo + r^p b;exo. In anoher version of he Taylor rule, we assume ha he cenral bank can disinguish beween hese wo componens and poenially respond di erenly o ucuaions in he spread depending on wheher i is due o endogenous or exogenous facors. In his case he Taylor rule would ake he form: i + = i ss + i (i i ss )+( i ) p + y ^y + endo r r^p b;endo + exo r r^p b;exo + endo rf r^p b;endo + exo rf r^pb;exo (22) 3 Parameer Values The model in he previous secion is solved wih a rs-order approximaion and he resuls are found from simulaions of he calibraed model. This secion will begin by presening he basic parameer values used in his calibraion. Then we will describe he various ypes of exogenous shocks ha will drive he simulaions of he model and he esimaion of hese di eren shock processes. The full lis of he model s parameers and heir values is found in able. The rs eigh parameers: he discoun facor, he capial depreciaion rae, capial s share of income, he elasiciy of subsiuion beween home and foreign goods, he bond adjusmen cos parameer, he labor supply elasiciy, he elasiciy of subsiuion beween goods from di eren rms, and he elasiciy of subsiuion beween labor from di eren households are all se o values ha are commonly found in he lieraure. 22

23 The capial adjusmen cos parameer,, describes he curvaure of he capial adjusmen funcion I K. I is he elasiciy of he relaive price of capial wih respec o changes in he invesmen-capial raio. This parameer preforms he imporan funcions of lowering he relaive volailiy of invesmen and ensuring he procyclicaliy of he price of capial. Empirical esimaes of his parameer vary, bu he value of :25 is in he middle of he range of empirical esimaes and ensures ha he relaive volailiy of invesmen in he model is near wha we see in he daa. The nex wo parameers in he able are he Calvo price and wage sickiness parameers. The wage sickiness parameer is chosen such ha on average a household adjuss heir wages once a year. The price sickiness parameer implies ha prices are a lile more exible han wages and is aken from he DSGE esimaion lieraure (see e.g. Chrisiano e al. 25). The nex four parameers are all deermined so ha he seady sae of he model is able o mach cerain feaures of he daa. The and f parameers from he funcion ha aggregaes home and foreign goods in (2) are se such ha he home counry in he open economy version of he model have a seady sae impor share of 25%. 7 The nex wo parameers, and are he xed cos in he producion of inermediae goods and he weigh on he disuiliy from labor in he household s uiliy funcion, respecively. These are se o ensure ha in he seady sae, inermediae goods rms earn zero economic pro and he household s labor supply is uniy. Finally he las hree parameers in he able relae o he risk of bankrupcy and liquidaion coss in eiher he banking or enrepreneurial secors. The seady sae value of b measures he seady sae level of uncerainy in he nancial secor. This parameer is deermined o ensure ha in he seady sae of he model, when banks have a deb-asse raio of abou :9, here is a 3 basis poin spread beween inerbank raes and he risk free rae, he average quarerly spread beween he 3-monh Libor and he 3-monh T-bill from 984 o 27. The cos of liquidaion and he idiosyncraic bankrupcy risk in he enrepreneurial secor, e and e are joinly deermined. These parameers ensure ha in he seady sae of he model, when rms in he enrepreneurial secor have a deb-asse raio of :5, an enrepreneur faces a 2% probabiliy of bankrupcy and he seady sae spread beween he ineres rae on physical capial 7 From he demand funcions for domesically supplied inermediae inpus and impors, equaions (3) and (4), he seady sae impor share is: m = Z P m (i)ym (i) Z n n Z = f ( n) P d(i)yd (i)+ P m(i)ym (i) (n) + f ( n) n 23

24 loans and he bank s cos of capial is approximaely 7 basis poins Exogenous Shock Processes In his model here are wo ypes of shocks. The rs shock is simply a counry speci c shock o oal facor produciviy (TFP) in (). The second shock is a shock o he uncerainy abou he healh of a bank s asses, b, and is unique o a model wih nancial fricions in he banking secor. Since TFP shocks are no he primary focus of he sudy, we se he exogenous process ha governs TFP shocks o a simple process ha is familiar in he real business cycle lieraure. Shocks o TFP, ^A follow an AR() process wih an auoregressive coe cien of :9. Since he model is solved using a rs order approximaion, we simply normalize he variance of he shocks o TFP o one. To ensure comparabiliy across he closed and open economy versions of he model, we assume ha TFP follows his same process regardless of counry ype, his ensures ha any di erences in opimal policy beween he wo versions of he model are due o he inernal dynamics of he model and no he exogenous shock process. Alernaively we can consider shocks o he nancial secor uncerainy variable, b +. Equaion (6) describes how ucuaions in he inerbank lending spread can be broken down ino wo componens, one due o ucuaions in he endogenous cuo value! b +, and one due o exogenous ucuaions in nancial secor risk, b +. r^p b g! b +! b ss! b ss! + g 2 b + b ss b ss We consruc a ime series of he inerbank spread rp b by aking he di erence beween he 3- monh USD Libor and he 3-monh T-bill from 985:-2:3 (alernaively we consider he period 985:-27:2). A ime series of he cuo value! b + is consruced using daa on U.S. commercial bank deb asse raios, loan delinquency raes, he prime rae, and he 3-monh USD Libor over he same period. In an OLS regression where r^p b, is he dependen variable,!b + is he independen! b ss variable, and g b + b ss 2 is he residual, he esimaed coe cien ^g = :274 wih a sandard b ss error of :79 and an R 2 = :8 (for he 985:-27:2 period ^g =!! b ss :258 wih SE = :53 and 8 The calibraion ha enrepreneurs have a seady sae deb-asse raio of abou :5 and banks have a seady sae deb-asse raio of abou :9 is based on he hisorical average deb-asse raios for U.S. non- nancial and nancial rms as repored in he Federal Reserve s Flow of Funds Accouns. 24

25 R 2 = :2). We use he residual erm g 2 b + b ss b ss o esimae an AR() process for b + and nd ha he nancial secor uncerainy variable has an auoregressive coe cien of :74. When presening he resuls in he nex secion, we will parameerize he exogenous shock process for b + such ha he auoregressive coe cien is :74, bu we will experimen wih he variance of he process and nd ha when ideni ed correcly, he opimal moneary policy is largely invarian o he variance of he exogenous nancial secor uncerainy shocks, b +. 4 Resuls To nd he opimal coe ciens in he cenral bank s Taylor rule and access how hese opimal parameers change as he degree of nancial secor risk in he economy changes, we rs have o de ne he loss funcion ha he cenral bank will aemp o minimize. In order o ensure ha he changes in opimal policy are due o changes in he endogenous srucure of he economy and he ransmission mechanism and no due o changes in he cenral bank s preferences, his loss funcion should remain he same regardless of he degree of nancial secor risk in he economy. When nding he opimal coe ciens in he Taylor rule, he cenral bank will aemp o minimize: L = var ( ) + :5 var (^y ) + : var (i i ) where is he in aion rae, ^y is he oupu gap, and i i is he quarer-over-quarer di erence in he nominal risk free ineres rae. 4. The parameers in he convenional Taylor rule To evaluae how moneary policy should respond o nancial secor risk, we rs idenify he opimal weighs on in aion, he oupu gap, and he lagged ineres rae in he Taylor rule funcion in he model where business cycles are driven by produciviy shocks. We nd hese opimal parameers hree imes, in a model wihou a nancial acceleraor, e = b =, in a model wih a nancial acceleraor only in he enrepreneurial secor, e > and b =, and in a model wih a nancial acceleraor in boh he enrepreneurial and banking secors, e > and b >. 25

26 We nd hese parameers hrough a grid search. We vary p, y, and i unil we nd he combinaion of he hree erms in he cenral bank s Taylor rule funcion ha minimizes he cenral bank s loss funcion. The resuls from his search are presened in able 2. The able presens opimal parameers for boh he closed economy and he open economy versions of he model. In he benchmark parameerizaion of he model, regardless of counry ype, he presence of a nancial acceleraor in he model means ha he cenral bank will wan o pu more weigh on he lagged ineres rae. In he closed economy version of he model, he opimal coe cien on he lagged ineres rae increases from :74 in he version of he model wih no nancial fricions o :76 in he model wih nancial fricions only in he real secor, o :78 in he model wih fricions in boh he real and nancial secors. The resuls for he open economy version of he model show a similar rend. Similarly, as he level of nancial fricions in he model increases, he cenral bank will pu more weigh on in aion and less weigh on he oupu gap. In he closed economy, adding a nancial acceleraor in boh he enrepreneurial and banking secors means ha he cenral bank will wan o increase he weigh on in aion by :5 from :62 o :67, and hey will decrease he weigh on he oupu gap by he same (absolue) amoun from :4 o :35. Thus as he level of nancial fricions in he model increases, he cenral bank will wan o pu more weigh on he lagged ineres rae as opposed o conemporaneous variables, and more weigh on in aion as opposed o oupu. In able 2, nancial fricions lead o endogenous changes in lending spreads (eiher he enrepreneurial lending spread, r e r, b or he inerbank spread, r b i ) in response o ucuaions in balance shees and asse prices ha are ulimaely caused by an exogenous produciviy shocks. Many recen papers have also considered he role of exogenous nancial shocks. In his model, exogenous nancial shocks ake he form of exogenous shocks o he inerbank credi supply schedule. As described in equaion (6), ucuaions in he inerbank lending rae are driven by endogenous ucuaions in! b, he cuo value for insolvency in he banking secor, which is deermined by endogenous variables like deb-asse raios and loan loss raios, and exogenous ucuaions in b, he sochasic variable ha measures he degree of ex-ane uncerainy abou he healh of a paricular bank s asses. De ne as he sandard deviaion of he exogenous ucuaions in he inerbank lending spread. 26