A Macroeconomic Model with Occasional Financial Crises

Transcription

1 FEDERAL RESERVE BANK OF SAN FRANCISCO WORKING PAPER SERIES A Macroeconomic Model wih Occasional Financial Crises Pascal Paul Federal Reserve Bank of San Francisco November 2017 Working Paper hp:// Suggesed ciaion: Pascal Paul A Macroeconomic Model wih Occasional Financial Crises Federal Reserve Bank of San Francisco Working Paper hps://doi.org/ /wp The views in his paper are solely he responsibiliy of he auhors and should no be inerpreed as reflecing he views of he Federal Reserve Bank of San Francisco or he Board of Governors of he Federal Reserve Sysem.

2 A Macroeconomic Model wih Occasional Financial Crises a Pascal Paul Federal Reserve Bank of San Francisco November 2017 Absrac Financial crises are born ou of prolonged credi booms and depressed produciviy. A imes, hey are iniiaed by relaively small shocks. Consisen wih hese empirical observaions, his paper exends a sandard macroeconomic model o include financial inermediaion, long-erm defaulable loans, and occasional financial crises. Wihin his framework, crises are ypically preceded by prolonged boom periods. During such episodes, inermediaries expand heir lending and leverage, hereby building up financial fragiliy. Crises are generally iniiaed by a moderae adverse shock ha pus pressure on inermediaries balance shees, riggering a credior run, a conracion in lending, and ulimaely a deep and persisen recession. Keywords: Financial Crises, Financial Inermediaion, Financial Sabiliy JEL codes: E32, E44, E52, G1, G01, G21 pascal.paul@sf.frb.org; Firs online version: Sepember I am paricularly hankful for deailed commens by Paul Beaudry, Fabrice Collard, Keshav Dogra (discussan), Marin Ellison, Andrea Ferrero, Mark Gerler, Òscar Jordà, Nobuhiro Kiyoaki, and Josef Schroh (discussan). I also hank Joseph Pedke, Michael Tubbs, and Ania Todd for excellen research & ediing assisance and many seminar and conference paricipans for heir insighs a Bank of England, Banque de France, CREi, European Cenral Bank, Federal Reserve Bank of San Francisco, Federal Reserve Board, New York Universiy, Tilburg Universiy, UC Davis, Universiy of Oxford, he 2016 Meeing of he Canadian Macroeconomics Sudy Group, he 2015 European Winer Meeings of he Economeric Sociey, and he 2016 New York Fed - Oxford Moneary Economics Conference. This paper was parly wrien during a visi o New York Universiy, I hank he insiuion and is people for heir hospialiy. A previous version was circulaed wih he ile Financial Crises & Deb Rigidiies. Financial suppor by he German Academic Exchange Service, he German Naional Academic Foundaion, and he David Walon Scholarship during my docoral sudies is graefully acknowledged. All errors are my own. The views expressed herein are solely hose of he auhor and do no necessarily reflec he views of he Federal Reserve Bank of San Francisco or he Federal Reserve Sysem. 1

3 1 Inroducion The Grea Recession revealed he need for macroeconomic models o incorporae connecions beween he financial secor and he macroeconomy ha can amplify economic shocks and lead o occasional deep economic downurns. However, in he vas majoriy of exising models ha accoun for such macro-financial linkages, financial crises are equally likely o occur a any poin in he business cycle and are mosly he resul of large exogenous shocks ha impinge upon he economy unexpecedly (e.g., Gerler and Kiyoaki, 2010). This is in sark conras o he daa. A recen empirical lieraure has shown ha financial crises are ypically preceded by a prolonged buildup of macro-financial imbalances. In his paper, I develop a quaniaive general equilibrium model ha allows for an endogenous increase of financial fragiliy. Based on his model, financial crises display he following feaures: (i) hey occur ou of credi booms, (ii) hey are immediaely surrounded by depressed produciviy, and (iii) hey are ypically iniiaed by a moderae adverse shock. Firs, across a range of empirical sudies, one key deerminan of crises ha has emerged is credi growh (e.g., Schularick and Taylor, 2012). Figure 1 illusraes his finding based on he macrofinancial daa and he crisis daes by Jordà, Schularick, and Taylor (2017). Annual percenage credi growh ypically increases in he years before a crisis, followed by a credi crunch. Second, produciviy growh falls sharply in he years surrounding he oubreak of a crisis eiher being he ulimae rigger of he crisis iself or he resul of a rigger. Figure 1 again illusraes his paern, showing he ypical annual percenage growh of hree indicaors of produciviy around financial crises (based on Paul, 2017, and he daa by Bergeaud, Cee, and Leca, 2016). Third, he ulimae riggers of financial crises can be relaively small. For example, wih respec o he Grea Recession, Goron and Ordoñez (2014) argue ha he losses from morgage-backed securiies he relevan shock for he financial secor around his ime were acually quie modes (see also Ospina and Uhlig, 2016). Thus, a model of financial crises should allow for he possibiliy ha relaively small shocks can have large effecs a imes and rigger a crisis. A he hear of he model ha accouns for hese empirical observaions is he ineracion beween he financial secor, is crediors (households), and is borrowers (nonfinancial firms). The model has hree crucial elemens: long-erm defaulable loans, an agency problem ha resuls in procyclical lending, and occasional credior runs. In he model, inermediaries borrow shor-erm deb from households and issue long-erm and defaulable loans o enrepreneurs resuling in a mauriy mismach. Since lending is long-erm, only a fracion of i maures each period. Nonmaured loans exis for muliple periods and canno be resrucured credi is rigid. The model herefore disinguishes beween ousanding and newly issued loans. 2

4 Figure 1: Macroeconomic Trends around Financial Crises. Annual percenage growh of aggregae bank credi (Source: Jordà e al., 2017) and various produciviy measures (Source: Bergeaud e al., 2016, and Paul, 2017, for addiional calculaions) around financial crises (year zero denoes sar of a crisis based on he daes in Jordà e al., 2017). Median, 33 rd, and 66 h perceniles are shown. Empirical evidence suggess ha new lending is procyclical boh in erms of volume and lending sandards, which ighen during recessions and ease in boom periods. 1 To reconcile he model wih his evidence, I inroduce an agency problem beween inermediaries and enrepreneurs ha deermines lending volume and sandards over he business cycle. Enrepreneurs may wan o risk-shif and inves in projecs wih low expeced reurns bu poenially high upsides ha imply a negaive ne presen value invesmen for financial inermediaries. Credi is raioned o ensure ha enrepreneurs do no inves in such projecs (similar o Sigliz and Weiss, 1981). This agency problem becomes more severe in recessions as incenives o risk-shif increase. As a resul, he quaniy of lending decreases and lending sandards ighen in recessions, while he opposie holds for booms. The mauriy mismach of he financial secor and is procyclical lending give rise o boom-bus episodes around financial crises. These are characerized by he following sequence of evens. During a prolonged boom period, inermediaries expand heir balance shees. They increase heir lending, since incenives o risk-shif decrease in good imes and enrepreneurs are less likely o defaul. Old mauring loans are replaced by new loans ha are backed by less collaeral. The financial secor finances he new loans by increasing is liabiliies. This expansion is simulaed by a decline in he shor-erm real ineres rae, as a resul of he household s desire o smooh consumpion and save during he boom. 1 See Figures 15 and 16 in Appendix A.3.1 for empirical evidence in his regard. 3

5 While inermediaries profi in good imes from increasing leverage and lending, hey also risk sharp losses from a credior run. If he inermediary s marke leverage exceeds a cerain hreshold, hen crediors doub ha hey will evenually be repaid and choose o run. The increase of leverage in good imes moves inermediaries closer o his hreshold, hereby building up financial fragiliy. Evenually, an adverse shock pushes inermediaries over he edge. Tha is because an unfavorable shock decreases he value of loans immediaely due o higher curren and fuure defaul raes. In urn, his leads o a sharp increase in he inermediary s marke leverage which riggers a credior run. Due o he mauriy mismach, he financial secor is lef wih a porfolio of ousanding nonperforming loans and is unable o quickly reduce is increased deb burden from he boom or raise new equiy o lower is leverage. When he financial secor experiences a credior run, is available funding drops disconinuously. In his sense, crises are exis from financial secor deb. In he model, a resricion of available funding o he financial secor is ransmied o he res of he economy hrough a conracion of new lending a credi crunch. Since credior runs only occur if he inermediary s leverage exceeds a cerain hreshold, hey are linked o he inermediary s balance shee. In paricular, hey are no self-confirming equilibria in a seing wih muliple equilibria as in Diamond and Dybvig (1983). Goron (1988) gives empirical evidence for his feaure. He shows ha banking panics during he U.S. naional banking era were sysemaic responses by deposiors o changing percepions of risk based on he arrival of new informaion raher han random evens. Why do inermediaries increase heir lending and leverage in good imes and hereby expose hemselves o he risk of credior runs? The financial secor follows his invesmen sraegy because i is profiable. Ofen, credi booms do no end in crises bu simply die ou, and inermediaries profi from he balance shee expansion. However, some credi booms do end in crises, wih sharp losses for he financial secor, and such episodes are associaed wih declines in produciviy. Goron and Ordoñez (2016) show ha hese paerns are presen in he daa. Credi booms ha are followed by a financial crisis ( bad booms ) are associaed wih lower produciviy growh compared wih credi booms ha do no end in a crisis ( good booms ). Taken ogeher, he model includes sandard business cycle dynamics, a realisic represenaion of he financial secor s balance shee and is lending behavior, and endogenous financial crises. Capuring all hese feaures in one model is imporan o be able o sudy he condiions under which financial crises are mos likely o occur. Given a calibraed version of he model ha maches he frequency and severiy of crises in he daa, I find ha he ypical buildup pah leading o a crisis is characerized by a prolonged credi boom, followed by decreases in produciviy, and a sudden bus riggered by a relaively moderae adverse shock. 4

6 As in he daa, risk premia srongly increase during financial crises (Muir, 2017) and credi spreads predic well he severiy of crises (Krishnamurhy and Muir, 2017). Financial recessions are ypically deeper and more persisen han nonfinancial recessions, confirming exising empirical evidence (e.g., Jordà e al., 2013). The behavior of he economy around nonfinancial recessions is differen, since hey are no preceded by an expansion of he inermediary s balance shee, a credi boom, or a buildup of leverage. I check he model s performance in wo more ways. Firs, I es i wih a hisorical series of srucural produciviy shocks for he U.S. economy; he model is able o predic and replicae he Grea Recession. Second, I show ha he model s mechanism based on he leveraging behavior of financial and nonfinancial firms is suppored by he daa: book and marke leverage of boh ypes of firms behave as heir empirical counerpars around crises and over he business cycle. Since he model maches key empirical characerisics of crises, i is a suiable laboraory o sudy policy inervenions aimed o reduce financial insabiliy. I find ha financial regulaion ha lowers inermediary leverage can address several exernaliies joinly and improve economy-wide welfare. Las, I also model long-erm defaulable deb in a novel and racable way. Firms inves in longerm projecs financed by long-erm loans ha maure sochasically. Firms deb, capial, leverage, and chance of defaul differ depending on when a loan was iniially issued. 2 This feaure implies a rich ime dependence, such ha he whole hisory of loans maers and deermines he overall qualiy of he financial secor s asse porfolio. I also allows he model o mach he empirical evidence by Demyanyk and Hemer (2011) who show ha he qualiy of loans in he U.S. deerioraed for several consecuive years before he Grea Recession. Relaed Lieraure. This paper builds on a vas lieraure abou financial fricions wihin macroeconomic seings. 3 This lieraure has quickly progressed recenly and sared o inroduce financial inermediaries and occasional crises explicily. 4 A relaed paper is Boissay, Collard, and Smes (2016), in which crises follow credi booms and are iniiaed by moderae adverse shocks. In heir paper, households accumulae bank deb during a boom, giving banks incenives o engage in risky aciviies, which can resul in a collapse of he inerbank marke. Goron and Ordoñez (2014, 2016) build on he idea ha deb is informaionally insensiive during booms, bu can suddenly urn informaionally sensiive even afer small shocks and herefore lead o a conracion in lending. Compared o hese conribuions, my paper presens a novel mechanism for boom-bus cycles o occur. Here, i is he ineracion beween leverage, procyclical lending, and he mauriy mismach of inermediaries ha occasionally drives he economy ino financial crises. 2 Alernaively, in Gomes, Jermann, and Schmid (2016), firms are idenical a he beginning of each period, bu may defaul on a fracion of heir long-erm deb due o idiosyncraic shocks. 3 Among he seminal conribuions in his field are Bernanke and Gerler (1989), Kiyoaki and Moore (1997), Carlsrom and Fuers (1997), and Bernanke, Gerler, and Gilchris (1999). 4 Wihou providing a full overview, conribuions o his liaure include Adrian and Boyarchenko (2012), Akinci and Queralo (2017), Bianchi (2011), Bocola (2016), Brunnermeier and Sannikov (2014), Elenev, Landvoig, and Van Nieuwerburgh (2017), Faria-e-Casro (2017), Gerler and Kiyoaki (2015), He and Krishnamurhy (2014), Marinez- Miera and Suarez (2012), and Mendoza (2010). 5

7 2 Model 2.1 Household Following Greenwood, Hercowiz, and Huffman (1988), he represenaive household values consumpion C and dislikes labor H, capured by he flow uiliy U (C, H ) = log ( ) C χ H1+φ 1 + φ where φ represens he inverse Frisch elasiciy of labor supply. The household chooses coningen plans for consumpion, labor supply, and savings in he form of shor-erm and riskless bonds B H, so as o maximize expeced lifeime uiliy. The household discouns he fuure a he rae β H. Taking prices, wages, and ineres raes as given, he household solves he problem subjec o V H ( B H 1, S ) = max C,H,B H { U(C, H ) + β H E [V ( )]} H B H, S +1, C + BH R +1 w H + B H 1 + T S +1 = Γ(S ), where w is he real wage and R +1 is he real ineres rae on shor-erm bonds beween period and + 1, known in period. Γ(.) denoes he law of moion for he aggregae sae variables S. The expecaion E [.] is formed condiional on he informaion se S a ime. Moreover, he household receives lump-sum ransfers T from firms, described in more deail below. The soluion o he above problem gives he iner- and inraemporal opimaliy condiions where 1 = E [Λ,+1 ] R +1 w = χh φ, Λ,+1 = β H is he household s sochasic discoun facor. C χ H1+φ 1+φ C +1 χ H1+φ +1 1+φ 2.2 Good Producion A represenaive good producer operaes according o a Cobb-Douglas producion funcion Y = A K α 1 H1 α, (1) 6

8 combining labor H supplied by he household wih aggregae capial K 1, supplied by enrepreneurs in period 1, o produce he good Y. The only source of aggregae risk in he model eners via he echnology level A A = e a, a = ρ a a 1 + ɛ a, ɛ a N(0, σ 2 a ), where ɛ a is ermed he echnology shock. Facor markes are compeiive. As a resul, all available facors are employed and pay heir marginal producs. These are given by w = (1 α) Y H, r K = α Y K 1, where r K is he renal rae per uni of capial. The role of financial inermediaion is o ransform he household s shor-erm savings B H ino invesmen in he economy s aggregae capial sock K. The wo ypes of agens ha fulfill his role he financial inermediary and enrepreneurs are a he hear of he model and described in deail nex. 2.3 Enrepreneurs Enrepreneurs inves in long-erm projecs. They ake on long-erm deb o finance hese projecs. The model is herefore in line wih he observaion ha firms change heir capial sock and borrowing infrequenly (e.g., Cooper and Haliwanger, 2006). Firs, he sochasic mauriy of longerm deb and he enrepreneurs defaul decision are inroduced for a given loan. Nex, I describe he agency problem beween enrepreneurs and he financial inermediary, which deermines he amoun and collaeral of newly issued loans Sochasic Mauriy and Defaul An enrepreneur who acquires a loan in period is ermed a new enrepreneur in ha period highlighed by he superscrip new. There is a uni mass of new enrepreneurs. Since all new enrepreneurs urn ou o be idenical, I omi enrepreneur-specific noaion. A new enrepreneur has ne worh N available. Addiionally, he enrepreneur acquires a long-erm, collaeralized, and defaulable loan Q L new deermined below. Combining N and Q L new from he financial inermediary. Boh ne worh and loan value are Q K K new, he enrepreneur purchases K new = N + Q L new, unis of capial where Q K denoes he price of capial and Q he price of long-erm loans. The face value of he loan is L new and he underlying collaeral are he unis of capial K new. The capial is len o he good producer and he reurns on he enrepreneurs invesmens are risky, since he amoun of 7

9 capial is chosen a leas one period in advance. Long-erm deb is inroduced as follows. Each loan has an exogenous sochasic mauriy. I maures wih probabiliy 1 γ in he nex period. An enrepreneur wih a nonmauring loan daing from period receives he renal rae r K +k per uni of capial in period + k for k {1, 2,..., } from he good producer. These profis are ransferred lump-sum o he household. The underlying collaeral K new consan as he household refurbishes depreciaed capial δk new, where δ is he rae of depreciaion. 5 Moreover, he face value L new a paymen o he financial inermediary if he loan does no maure. 6 of a nonmauring loan says remains unchanged as an enrepreneur does no have o make When a loan maures in period, ha was originally issued in period + k, hen he enrepreneur s projec ends as well. In ha period, he enrepreneur receives he renal rae r+k K per uni of capial and sells he remaining capial for he price Q+k K. Addiionally, he profis are hi by an idiosyncraic shock ω. This shock is drawn from a uniform disribuion, ω U[0, 2], independen across ime and enrepreneurs, and normalized o have mean and widh suppor uniy. 7 profis are The overall ωr K +k QK +k 1 Knew, (2) where R+k K (QK +k (1 δ)+rk +k ). When a loan maures, hen he enrepreneur can eiher choose o Q+k 1 K repay i or o defaul on his obligaion. An enrepreneur compares he profis in (2) o he face value of deb L new and decides wheher o defaul. An enrepreneur sraegically defauls if L new is larger han he enrepreneur s profis in (2). Or saed differenly, if he idiosyncraic shock ω is lower han a paricular hreshold ω +k in period + k, defined as follows. Defaul Decision. An enrepreneur from period whose loan maures in + k where k {1, 2,..., } defauls iff he realizaion of ω is such ha ω < ω +k, (3) where ω +k L new R K +k QK +k 1 Knew. (4) Equaion (4) shows ha he model incorporaes vinage-specific defaul hresholds ω +k. These depend on he loan-o-collaeral raio Lnew K new for a loan issued in period and he aggregae profis o capial R+k K QK +k 1 in period + k. Loans issued under disinc saes of he economy can herefore have differen defaul hresholds when mauring in he same period. This feaure of he model is illusraed by he following simple example. 5 This assumpion ensures ha he probabiliy of defaul of very old enrepreneurs does no approach one since heir underlying capial and herefore heir profis do no vanish in he long run. 6 This assumpion implies no defaul consideraions before a loan maures, which makes he problem racable and aggregaion feasible as shown in Appendix A The mean uniy ensures ha he idiosyncraic shock does no change profis in he aggregae. The widh suppor of one gives a zero lower bound on ω. This implies ha a fracion of loans always defauls in equilibrium as visible from equaions (3) and (4). 8

10 Example. If he loan-o-capial raios of loans issued in periods 2 and 1 are such ha L new 2 K new 2 > Lnew 1 K new 1, and hese loans are boh mauring in period, hen heir defaul hresholds relae according o ω 2 = L new 2 R K QK 1 Knew 2 > L new 1 R K QK 1 Knew 1 = ω 1. The loan wih a higher loan-o-capial raio also has a higher defaul hreshold. Tha is because boh receive he same aggregae profis R KQK 1 per uni of capial. If an enrepreneur from period defauls in + k, hen he remainder (1 µ)ωr K +k QK +k 1 Knew is recovered by he financial inermediary. The fracion µ is los, reflecing he coss of defaul. If an enrepreneur does no defaul, hen he ousanding credi L new is repaid. The remaining profis ωr+k K QK +k 1 Knew L new are equally spli among new enrepreneurs in period + k. The sharing of profis ensures ha all new enrepreneurs sar wih he same amoun of ne worh. New enrepreneurs consis of all enrepreneurs whose loans maured wheher hey defauled or no such ha he oal number says consan Risk-Shifing Problem Nex, I describe he agency problem ha deermines he amoun of lending in he firs place. When a loan is issued, a new enrepreneur can choose o inves in wo projecs and canno swich projecs for he duraion of he loan conrac. Besides he described projec, which is ermed he good projec, a new enrepreneur can also inves in a second projec, called he bad projec. The wo projecs only differ in erms of he disribuion from which he idiosyncraic shock is drawn. If a loan from vinage maures in + k, hen he profis under he bad projec are denoed ωr +k K QK +k 1 Knew. The shock ω is drawn from a uniform disribuion, ω U[m c, m + c], independen across ime and enrepreneurs. I is assumed ha c > 1 and m < 1, such ha ω has a higher variance and a lower mean compared wih ω. Even hough he bad projec has a lower mean and herefore a lower expeced reurn, a leveraged enrepreneur may sill prefer his projec. Tha is because he bad projec has a higher variance and herefore a higher poenial upside, wihou he need o cover for he lower downside due o he opion o defaul. In conras, lending o enrepreneurs who inves in he bad projec may be a negaive ne presen value invesmen for he financial inermediary. Tha is he case if mean and variance are sufficienly differen beween he wo projecs. If hey are sufficienly differen, hen he financial inermediary would never agree o conracs for which enrepreneurs have he incenive o inves in he bad projec. Any agreemen herefore has o saisfy he incenive compaibiliy (IC) consrain ha demands ha he enrepreneur s objecive funcion under he good 9

11 projec V good is a leas as large as he one under he bad projec V bad, V good V bad. (5) Knowing ha any deb conrac wih he financial inermediary has o saisfy he IC consrain, inequaliy (5) direcly eners as a consrain ino a new enrepreneur s decision problem. In case he IC consrain binds in equilibrium, hen he enrepreneur would prefer o borrow more. However, ha is no feasible. A higher amoun of deb would induce he enrepreneur o swich o he bad projec and credi is herefore raioned. The inersecion of he wo value funcions deermines he loan amoun Q L new ha a new enrepreneur can raise. Figure 2 illusraes his case. Figure 2: Incenive Compaibiliy Consrain. Furher, if he IC consrain binds, hen he enrepreneur s decision problem is largely simplified. While I leave he saemen of he full problem o Appendix A.1.1, Proposiion 1 shows he main equaions of he simplified problem. Proposiion 1. If (26) binds, hen he soluion o he enrepreneur s problem is given by ( ) c m 2c ( K new = (1 γ) L new J + Lnew K new S ) ) ] (m + c)2 J = E [(1 R K+1 4c QK + γj +1 [ (1 S = E 4 1 ) ] 1 + γs +1 4c Q K K new = Q L new + N, R K +1 QK (6) (7) (8) wih he assumpion ha (m + c) 2 < 4c. Proof: See Appendix A

12 Proposiion 1 also shows how lending sandards change over he business cycle. During a recession, when profis o capial are low and defaul hresholds are high, he bad projec becomes relaively more aracive o a new enrepreneur because of he poenially higher upside. The risk-shifing problem herefore aggravaes and credi is raioned. Toal new lending and he loano-collaeral raio on new lending decline. The laer can be shown based on equaions (6) (8) in J S Proposiion 1. Since E[R+k K QK +k 1] > 0 and < 0 for any k {1, 2,..., }, a fall in expeced fuure profis E [ R+k K ] QK L E[R+k K QK +k 1] +k 1 resuls in a decline of he loan-o-collaeral raio new K new on new lending, so ha equaion (6) coninues o hold. The opposie occurs during a boom when expeced fuure profis increase. In wha follows, assume ha any agreemen wih he inermediary has o saisfy he IC consrain, such ha enrepreneurs always inves in he good projec. 2.4 Financial Inermediary The role of he represenaive financial inermediary is o ransform shor-erm and riskless deb B ino long-erm and risky loans L. The inermediary is perfecly diversified and invess in he whole marke porfolio of loans L, defined recursively and encompassing all ousanding loans L = L new + γl 1. I assume ha he inermediary discouns he fuure a he rae β F and values shareholders flow uiliy of real dividends D according o U(D ) = log (D ). The financial inermediary chooses new shor-erm deb B, loans L, and dividends D every period o maximize expeced lifeime shareholder uiliy. Taking prices and ineres raes as given, he financial inermediary solves wha I will refer o as he unconsrained problem, subjec o { [ ]} V FI (S, B 1 R, L 1 ) = max U(D ) + β F E V FI (S +1, B R +1, L ) D,L,B D + Q L + B 1 R B + R L Q 1 L 1 (9) S +1 = Γ(S ), where Q is he price and R L he reurn per loan in period. Consrain (9) is a budge consrain. The righ-hand side saes he available resources: he amoun of new deb B and he profis on las period s loans R L Q 1L 1. These have o cover a leas he payou of dividends D, new loan invesmen Q L, and ousanding deb and ineres B 1 R. The above problem implies ha he inermediary never raises equiy bu always issues a posiive amoun of real dividends. The 11

13 soluion o he inermediary s problem is given by wo ineremporal opimaliy condiions where R L = γq + (1 γ) 1 L 1 j=1 [ ] 1 1 = β F E R +1 (10) D D +1 [ ] 1 R = β F L E +1, (11) D D +1 ( ) 1 Φ(ω j ) γ j 1 L new j + (1 µ) j=1 ω j ωr KQK 1 γj 1 K j newdφ(ω) 0. Q 1 R L capures boh profis from nonmauring loans γq, as well as from mauring loans across all vinages. The laer consis of repaid loans he firs erm in he curly bracke and he recovery from defauled loans he second erm in he curly bracke. Noe ha R L accouns for an infinie L number of vinages, each wih is own vinage-specific defaul hreshold ω j = new j in period. However, i is no necessary o keep rack of he whole disribuion of loans, as R KQK 1 Knew j highlighed by Proposiion 2. 8 Proposiion 2. The aggregae sae variables L = L new + γl 1, x = L new Lnew K new + γx 1, accoun for all ousanding loans and heir vinage-specific loan-o-capial raios. Given R K QK 1 and Q in period, he profis per loan R L Q 1 can be expressed in erms of only hese sae variables, such ha Proof: See Appendix A.1.3. R L Q 1 = γq + (1 γ) { } 1 x 1(1 + µ) 4R KQK 1 L 1. (12) The auxiliary variable x and he simplified expression for R L Q 1 have inuiive forms. The variable x accouns for all vinage-specific loan-o-capial raios since i is updaed wih Lnew K new each period, weighed by he face value L new of newly issued loans. One can herefore inerpre x as a loan risk indicaor. Furher, x allows o derive a weighed defaul hreshold of all ousanding loans ω x 1 R K QK 1 L 1 = γ k 1 L new k ω L k, k=1 1 8 An imporan assumpion ha allows for his aggregaion is ha he idiosyncraic shocks ω and ω follow uniform disribuions, giving convenien expressions for cumulaive disribuion funcions and parial expecaions (see Appendix A.1.1). 12

14 where Lnew k γk 1 L 1 is he remaining fracion of a vinage of loans and ω k is he defaul hreshold for vinage k in. If curren aggregae profis o capial are low, hen he aggregae defaul hreshold ω increases. Using ω, one can furher simplify expression (12) o { } R L (1 + µ) Q 1 = γq + (1 γ) 1 ω 4. (13) Equaion (13) has a simple inerpreaion: curren profis o loans R L Q 1 are higher, he larger he value of ousanding loans Q and he lower he weighed defaul hreshold ω. Taken ogeher, he model implies a rich ime dependence of he inermediary s asse porfolio. Loans ha have been issued in he pas remain on he financial secor s balance shee for some ime. The riskiness of he whole hisory of loans maers and deermines he overall qualiy of he curren porfolio. 2.5 Occasional Financial Crises Nex, I inroduce occasional financial crises ino he above framework. A crisis is defined as a period in which he inermediary s choices under he unconsrained problem are such ha is marke leverage exceeds a paricular hreshold. Given he inermediary s unconsrained choices, if B Q L > κ, (14) hen crediors lose confidence ha hey will evenually be repaid and run on he inermediary. If (14) is saisfied, hen he inermediary has o solve a consrained problem insead. This problem addiionally includes B τ(s ). (15) Hence, when crediors run, he inermediary faces a quaniy resricion on new borrowing. The funcional form of τ(s ) is chosen such ha he inermediary has less deb available compared wih he unconsrained case (described in he calibraion in Secion 3.1). Boh κ and τ(s ) deermine he frequency and he severiy of crises, and are calibraed o mach he daa in his regard (explained in Secion 3.1). In saes in which he inermediary solves he consrained problem, inequaliy (15) changes he opimaliy condiion (10) o [ ] 1 1 λ = β F E R +1, (16) D D +1 where λ is he Lagrange muliplier on he deb consrain in (15). If τ(s ) is sufficienly low, hen λ is posiive and he inermediary is herefore funding-consrained. 9 Noe ha he inermediary s choices under he unconsrained problem ake ino accoun he possibiliy ha a financial crisis may occur in he fuure due o he realizaion of fuure shocks. The soluion mehod described in Appendix A.2.3 accouns for such precauionary behavior. 9 To ensure ha a financial inermediary is no caugh in a spiral of consan credior runs, I furher assume ha credior runs canno occur for wo consecuive periods. This assumpion ensures sabiliy of he model and is in line wih he observaion ha credior runs generally occur wihin a small ime window. 13

15 2.6 Risk and Liquidiy Premia The inermediary s sochasic discoun facor is denoed by M,+1 = β F D +1 and excess reurns by R+1 X = RL +1 R +1. Given he wo Euler equaions in (11) and (16), he expeced excess reurn can hen be divided ino wo componens: a classic risk premium and a liquidiy premium, [ ] E R+1 X = λ D E [M,+1 ] }{{} Liquidiy Premium Cov(M,+1, R X +1 ) E [M,+1 ] }{{} Risk Premium D. (17) If λ > 0, hen he inermediary is funding-consrained and he liquidiy premium is posiive, increasing (17). This is in line wih he observaion ha expeced excess reurns are generally high when an economy experiences a financial crisis during which inermediaries are fundingconsrained (Muir, 2017). 2.7 Capial Producers and Resource Consrain Capial good producers underake real invesmen. Given he price of capial Q K, capial good producers maximize heir profis by choosing he economy-wide unis of invesmen I subjec o { } max Q K I Φ (I, K 1 ) I ( ) I δk 2 1 K 1 Φ(I, K 1 ) = I + ζ 2 K 1 where I = K (1 δ)k 1 and Φ (I, K 1 ) reflec quadraic adjusmen coss ha follow he parsimonious specificaion in He and Krishnamurhy (2014). The above problem gives he inraemporal opimaliy condiion Q K = 1 + ζ ( ) I δ K 1 I complee he descripion of he model by saing he resource consrain Y = C + D + I + ζ ( ) I δk 2 1 K 1 + (1 γ)µ 2 K 1 j=1 0. ω j ωr K Q 1 K γj K j new dφ(ω). This consrain implies ha he final oupu good is used as consumpion, as conversion ino capial goods, and for covering real defaul coss. I consider a compeiive general equilibrium, defined in Appendix A.2.2. The equilibrium condiions of he model are saed in Appendix A.2.1. A nonlinear global soluion is obained using a projecion algorihm, described in Appendix A

16 3 Quaniaive Analysis 3.1 Calibraion Agens Descripion Parameer Value Targe / Source Household Discoun facor β H 0.99 Lieraure Inv. Frisch elasiciy φ 0.5 Lieraure Rel. uiliy weigh χ 1.6 Normalizaion: H 1 in seady sae Good Producer Effecive capial share α 0.3 Lieraure Depreciaion rae δ Lieraure S. dev. echnology shock σ a 0.68% TFP series (Fernald, 2014) Persis. echnology shock ρ a 0.93 TFP series (Fernald, 2014) Inermediary Discoun facor β F Impulse response maching Leverage hreshold κ 0.5 Frequency Crises: 4% Deb consrain τ 0.95 GDP Fin.Rec. Severiy Crises : GDP Ave.Rec Enrepreneurs Mean of ω m 0.9 IC consrain binding Widh suppor of ω c 1.55 Annual Defaul Rae 3% Defaul coss µ 0.12 Bernanke e al. (1999) Sochasic mauriy γ 0.9 Ma. Mismach U.S. Comm. Banks Capial Producer Capial adjusmen cos ζ 3 He and Krishnamurhy (2014) Table 1: Calibraion of Srucural Parameers. The model is calibraed o a quarerly frequency for he U.S. economy. The srucural parameers are lised in Table 1. I discuss he calibraion of he nonsandard ones nex. The persisence and sandard deviaion of he echnology shock are obained by esimaing an AR(1) process o he linearly derended logarihm of he oal facor produciviy (TFP) series by Fernald (2014) for he second half of he pos-wwii period (1980 Q Q4), giving ρ a = 0.93 and σ a = 0.68%. 10 The difference beween he household s discoun facor β H and he inermediary s discoun facor β F deermines wo arges simulaneously. Firs is he level of he inermediary s marke leverage he model s main indicaor of financial sabiliy. Second, by deermining he iniial level of leverage, he impulse response of leverage o he echnology shock is also affeced. 11 In urn, his pins down he ineracion beween aggregae risk and financial sabiliy. I obain empirical evidence 10 The resuls are robus o using a measure of TFP ha is adjused for he uilizaion of capial and labor inpus. 11 To undersand why he shape of he impulse response is affeced, consider wo cases. Firs, he inermediary is fully equiy-funded. In his case, an increase in liabiliies immediaely afer a posiive echnology shock raises leverage. Second, marke leverage of a highly leveraged inermediary iniially falls afer such a shock, since asse prices rise insanly. However, over ime, marke leverage may increase if he inermediary expands is liabiliies. 15

17 Figure 3: Impulse Responses. Impulse responses of marke leverage o a one-sandard-deviaion posiive echnology shock. 95% confidence bands are shown. on boh. Using bank-level daa ses for commercial and invesmen banks, I derive an empirical proxy for he U.S. financial secor s marke leverage (see Appendix A.3.2 for a descripion of he daa and derivaions). Impulse responses of his measure of leverage o a echnology shock are obained using a varian of local projecions (Jordà, 2005). As a proxy for he series of srucural echnology shocks, I use he residual from he esimaed AR(1) process of he TFP series, denoed ˆɛ a, based on he sample 1980 Q Q4. Using he generalized mehod of momens (GMM), I simulaneously esimae he following sysem 12,13 log(tfp) = ρ a log(tfp) 1 + ˆɛ a (18) Lev +k 1 Lev 2 = β k 0 + β k 1 ˆɛa + e +k for k {1, 2,..., 19}, (19) where Lev denoes he marke leverage of he inermediary secor a ime and β k 1 gives he reacion of leverage o a echnology shock a horizon k. Figure 3 shows he resuls. Following a posiive echnology shock, leverage iniially declines and hen rises over ime, urning posiive afer around eigh quarers. By calibraing β F for a given β H, I find ha an asse-o-equiy raio of around 2 gives a model-implied impulse response ha maches well he iniial sign and urning poin of he obained impulse response. This calibraion gives a slighly lower marke leverage han found for commercial and invesmen banks in he colleced daa, wih asse-o-equiy raios above 5 in normal imes. However, i maches well wih oher pars of he inermediary secor such as hedge funds wih raios of around 2 (Ang, Gorovyy, and van Inwegen, 2011). 12 The sysem is esimaed joinly o avoid a generaed regressor problem (Newey and McFadden, 1994). GMM requires he choice of a weighing marix and insrumens. Regarding he weighing marix, a Newey-Wes correcion for heeroskedasiciy and auocorrelaion is used. The insrumens are he regressors in (18) and (19). 13 I consider impulse responses wih respec o leverage a ime 2 o accoun for poenial news shocks a ime 1. 16

18 The leverage hreshold κ governs he frequency of crises in he model. I pick κ such ha crises occur around 4% of he ime in a simulaion of he model, maching he frequency of crises in he macrohisory daa by Jordà, Schularick, and Taylor (2017) (see Appendix A.3.3 for deails). 14 When a credior run occurs, a financial inermediary is resriced in he amoun of new deb B o raise. According o equaion (15), he financial inermediary canno ake on more new deb han τ(s ). I choose he funcional form of τ(s ) o be τ(s ) = τ (D unc + Q unc L new,unc ) + B 1 R R L Q 1 L 1 + Q γl 1, such ha he inermediary is always able o repay is ousanding deb, bu has less new deb B available o finance dividend payous and newly issued loans compared wih is choices under he unconsrained problem (denoed wih he superscrip unc). The parameer 0 < τ < 1 deermines he severiy of a financial crisis. I choose τ such ha he raio beween financial and average recessions in a simulaion of he model maches is empirical counerpar. 15 Based on he macrofinancial daa by Jordà e al. (2017), he change in real GDP from peak o rough is around 31% larger during financial recessions ( 5.31%) han during average recessions ( 4.05%). 16 The following parameers govern he long-erm deb conrac beween he financial inermediary and enrepreneurs. Mean and variance of he idiosyncraic shock ω under he bad projec are chosen o mach wo arges: an annual defaul rae of around 3% in seady sae (as in Bernanke e al., 1999) and a binding IC consrain in seady sae, saisfied by m = 0.9 and c = ,18 In a simulaion of he model, I confirm ha hese choices imply ha lending o enrepreneurs who inves in he bad projec is a negaive ne presen value invesmen from he poin of view of he inermediary. The fracion of profis los in he case of defaul µ is direcly aken from Bernanke e al. (1999) and se o Based on Call Repors for U.S. commercial banks, I find ha he average mauriy of asses is around 3.49 years, and ha of liabiliies is around 0.35 years, wih a raio beween he wo of I normalize he mauriy of shor-erm deb o one quarer and choose γ o give an average mauriy of long-erm deb of 9.97 quarers, giving γ = 0.9. The sochasic seady sae resuling from his calibraion is given in Appendix A.4.4 and he accuracy of he soluion is shown in Appendix A.2.4. Nex, I analyze he dynamics of he model wih respec o he echnology shock. 14 These daa are based on an annual frequency. To ensure consisency, he following definiion is used. A year in which a leas one financial crisis occurs is couned as one financial-crisis-year. 15 In he model simulaion, I deermine business cycle peaks and roughs according o he definiion ha is used by he UK and members of he European Union, among ohers. A business cycle peak is he quarer before oupu falls by wo consecuive quarers. Following a peak, a rough is reached before oupu grows again. I furher resric recessions o occur 14.59% in a simulaion of he model (as in he macrohisory daa by Jordà e al., 2017) by choosing he ones wih larger falls in oupu from peak o rough. If a credior run occurs beween a peak and a rough, hen such an episode is ermed a financial recession. All remaining recessions are called nonfinancial recessions. 16 The model s implied changes in real GDP are very close o hese empirical counerpars: 5.26% wih respec o financial recessions and 3.88% for average recessions. 17 Noe ha he mapping beween hese calibraion arges and he parameers m and c is no unique. However, I find ha for differen parameer pairs ha give he same arges, he impulse responses are much he same. 18 Away from he seady sae, he IC consrain is assumed o bind. 17

19 3.2 Impulse Responses Figures 4 and 5 show impulse responses o a one-sandard-deviaion posiive echnology shock, saring from he sochasic seady sae of he model. For hese responses, a financial crisis does no occur because he inermediary s leverage is sufficienly below he hreshold κ. Following a posiive echnology shock, oupu Y and consumpion C increase. Due o consumpion smoohing and he increase in he real wage, he household s savings increase, he real ineres rae R +1 declines, and he household chooses o work more, such ha H increases. Profis o capial R K QK 1 rise, which decreases he probabiliy of defaul of all ousanding loans as indicaed by he weighed defaul hreshold ω. Since new enrepreneurs are also less likely o risk-shif, hey can borrow more in absolue erms, Q L new value of heir loans L new, in relaive erms o heir collaeral Lnew increases. The loan risk indicaor x herefore rises. K new, and he face Figure 4: Impulse Responses. Impulse responses o a one-sandard-deviaion posiive echnology shock, saring a he sochasic seady sae of he model. Following a posiive echnology shock, he financial inermediary expands is balance shee. Given he reduced incenives o risk-shif and he lower real ineres rae, he inermediary akes on more deb B o increase is new lending. The overall sock of ousanding loans L and he economy s capial sock K herefore rise. In addiion, he profiabiliy of loans, R L Q 1, increases and he inermediary pays ou more dividends D. The inermediary s marke leverage B Q L iniially decreases since he enire loan porfolio rises in value. Over ime, he price of loans Q reurns. The inermediary coninues o ake on more deb o issue new loans, raising leverage. The response of leverage herefore roughly maches he empirical counerpar in Figure 3. 18

20 a Figure 5: Impulse Responses. Impulse responses o a one-sandard-deviaion posiive echnology shock, saring a he sochasic seady sae of he model. 19

21 3.3 Financial Crises In principle, a financial crisis can break ou a any ime if an adverse echnology shock is sufficienly large. Hence, nohing in he model resrics crises o occur ou of booms or recessions. However, financial crises are more likely o happen if cerain condiions are me. To undersand hese condiions, I analyze he ypical behavior of he model around crises. Firs, I simulae he model for 500,000 periods. Then, I collec he sequences of endogenous variables and shocks in a window of 30 quarers before and 20 quarers afer a crisis. Figures 6, 7, and 8 plo period-byperiod he median, 33 rd, and 66 h perceniles across hese sequences for each variable wih respec o windows in which only one financial crisis occurs. In wha follows, he median pah for each variable is referred o as he ypical pah around a crisis. The firs row in Figure 8 shows he ypical behavior of he echnology shock ɛ a and he echnology level a. A ypical buildup period leading o a crisis is characerized by an elevaed echnology level, which reverses wihin a oneyear window before a crisis. The median shock ha riggers a crisis is a negaive 1.49 sandard deviaion shock. Typical financial crises herefore occur ou of slowdowns in produciviy, and he model does no need o rely on exremely large adverse shocks o iniiae financial crises. Figure 6: Typical Financial Crises. Even window around financial crisis a Quarer = 0. Based on a simulaion of 500, 000 periods. Median, 33 rd, and 66 h perceniles are shown. Oupu Y, capial sock K, and hours H all increase in he buildup period and decrease once a crisis is riggered. In his model, financial crises are credi booms gone bus. During he boom, more new loans Q L new wih higher loan-o-capial raios Lnew K new are issued ha are boh srongly decreasing in he bus. The overall sock of loans L and he loan risk indicaor x herefore increase 20

22 in good imes, and boh sharply reverse around he oubreak of a crisis. 19 The inermediary s balance shee and decisions are key o undersanding how crises arise. During he boom, enrepreneurs are less likely o defaul, ω declines, and he inermediary s profis R L Q 1 rise. Due o he higher income, he inermediary pays ou more dividends D. The issuance of new loans during he boom is financed by aking on addiional deb B, leading o a buildup of leverage The higher leverage moves he inermediary closer o he hreshold κ, above which crediors run, hereby building up financial fragiliy. B Q L. Figure 7: Typical Financial Crises. Even window around financial crisis a Quarer = 0. Based on a simulaion of 500, 000 periods. Median, 33 rd, and 66 h perceniles are shown. Given ha inermediaries know abou he he risk of a credior run, why do hey no ry everyhing o avoid i? The simple answer is ha i is profiable for he inermediary o be exposed o a run in good imes. Firs, he ineres on deb R +1 declines as he household saves during he boom o smooh consumpion. Second, defaul raes drop and profis on loans rise. Hence, aking on addiional deb o issue more loans is a profiable invesmen sraegy. No every credi boom ends in a crisis, in fac ofen hey do no ( good booms ). However, some credi booms do end in a crisis ( bad booms ). In he model, ha is he case if produciviy falls afer a credi boom. The produciviy slowdown leads o sharp rises in defaul raes ω, a drop in he value of loans Q, and a spike in inermediary leverage B Q L. The laer pushes he inermediary above he hreshold κ and crediors run. Due o he mauriy mismach, he inermediary is lef 19 Based on he simulaion of he model, Appendix A.4.2 replicaes he baseline regressions in Schularick and Taylor (2012) and confirms heir resul ha credi performs well in predicing financial crises. 21

23 Figure 8: Typical Financial Crises. Even window around financial crisis a Quarer = 0. Based on a simulaion of 500, 000 periods. Median, 33 rd, and 66 h perceniles are shown. wih a porfolio of non-performing loans, an increased deb burden from he boom, and unable o raise equiy o deleverage. Hence, he inermediary canno preven a credior run. While he inermediary is funding-consrained, he expeced excess reurn E [ R+1] X sharply increases. Tha is because of he posiive liquidiy-premium as defined in equaion (17) consisen wih he daa (Muir, 2017). In Appendix A.4.3, I also show ha credi spreads are good predicors of he severiy of crises, following he empirical sraegy by Krishnamurhy and Muir (2017). Taken ogeher, he model is able o mach key empirical characerisics of financial crises. 3.4 Nonfinancial Recessions To undersand how crises differ from normal recessions, I repea he exercise of he las secion around nonfinancial recessions (as defined in foonoe 15). Figures 17, 18, and 19 in Appendix A.4.1 plo he ypical median pahs around nonfinancial recessions and financial crises. Compared wih financial crises, he rise and decline in oupu around nonfinancial recessions is less pronounced. Moreover, nonfinancial recessions are no preceded by srong increases in credi L, he inermediary s deb B, marke leverage B Q L, or he loan risk indicaor x, in conras o he behavior of hese variables in he buildup owards financial crises. During nonfinancial recessions, he weighed defaul hreshold ω, he inermediary s marke leverage, and he expeced excess reurn E [ R X +1] do no increase as srongly as during financial crises. Moreover, he fall in he inermediary s funding B, is dividends D, new lending Q L new, and he value of loans Q are sronger during financial crises. In addiion, nonfinancial recessions are less severe and persisen 22

24 compared wih financial recessions. 20 Overall, his comparison shows ha i is he financial secor s balance shee and is lending behavior ha play crucial roles during he boom-bus cycle around financial crises. In his regard, nonfinancial recessions are differen. 3.5 The Grea Recession How well does he model perform in predicing and replicaing he financial crisis of ? To es he model in his regard, I again use he hisorical series of srucural echnology shocks { ˆɛ a} as discussed in Secion 3.1 and feed he model wih his series. Figure 9: The Grea Recession. Model-implied variables, given series of srucural echnology shocks { ˆɛ a }. The red doed line indicaes he hreshold level of leverage κ, above which a credior run occurs. Gray bars denoe NBER recessions. The resuls are given in Figure 9, showing he pah of he echnology level a, he probabiliy ha a financial crisis will occur in he nex quarer, and he evoluion of he inermediary s marke leverage. During he boom period of 2000 o 2005, he echnology level increases sep-by-sep and so does he inermediary s leverage. This boom period is followed by declines in he echnology level ha iniiae a crisis and lead o a credior run when he inermediary s leverage goes beyond he hreshold κ (indicaed by he red doed line). The probabiliy of a crisis sars o increase around 2006 and jumps close o around 40%. Hence, he model performs well in predicing and replicaing he Grea Recession. For he res of he sample, he probabiliy of a crisis is close o zero, wih he excepions of he early 1980s and he early 1990s, during which a crisis also occurs in he model. 3.6 Leverage The behavior of financial and nonfinancial firm leverage is key o he model s dynamics. Nex, I show ha his behavior is also consisen wih he daa. In his regard, i is useful o disinguish beween book and marke leverage in he model and in he daa. I obain empirical measures for hese wo ypes of leverage for he U.S. financial and nonfinancial secors by combining equiy and balance shee firm-level daa ses (see Appendix A.3.2 for deails). 21 In he model, he financial 20 For example, he median duraion from peak o rough of a financial recession is around 66% longer compared wih a nonfinancial recession in he range of he empirical counerpar of 100% in he macrohisory daa by Jordà e al. (2017). 21 As a proxy for he U.S. financial secor leverage, I use daa for commercial and invesmen banks. 23

25 secor s leverage can be defined as Marke Leverage FI B Q L, Book Leverage FI B BA FI, (20) where BA FI sands for book asses of he financial inermediary, defined recursively BA FI = Q L new + γba FI 1, (21) such ha new loans are recorded on he balance shee wih he value a which hey were given ou and held consan unil hey maure. Similarly, define leverage of he nonfinancial secor as where BA NF Marke Leverage NF L Q K K, Book Leverage NF L BA NF sands for book asses of he nonfinancial secor, again defined recursively BA NF = Q K K new + γba NF 1, (22) such ha newly acquired capial is recorded on firms balance shees wih he value a which i was acquired and held consan unil a firm s projec ends. Wih hese definiions a hand, one can compare leverage beween he model and he daa. Firs, consider he behavior of leverage over he business cycle. Saring wih he work of Adrian and Shin (2010), he cyclicaliy of financial insiuions leverage has been discussed as a poenial indicaor of he financial secor s procyclical risk-aking. Adrian and Shin (2010) show ha leverage of cerain financial insiuions is procyclical based on book value daa. These resuls are confirmed in Table (2), showing he correlaion beween he cyclical componen of book leverage, differeniaed by secors, and real indusrial producion. 22 Two samples are considered, one including and one excluding he Grea Recession. For he financial secor, book leverage is procyclical and significan a he 99% confidence level for boh samples. For he nonfinancial secor, book leverage is counercyclical bu only mildly based on he sample ha ends before he Grea Recession. The model s implied correlaion beween oupu Y and book leverage is in line wih his empirical evidence. However, for he financial secor, he correlaion based on he model is slighly larger in magniude., Daa 1980 M M M M12 Model Financial Nonfinancial Table 2: Cyclicaliy of Book Leverage. Correlaion beween cyclical componen of book leverage and real indusrial producion. Noaion: p < 0.01, p < 0.05, p < 0.1. In Table (3), I repea he exercise for marke leverage. Financial secor marke leverage is eiher mildly procyclical or counercyclical in he daa, depending on he sample. Nonfinancial secor 22 The logarihm of real indusrial producion and leverage are derended using a Hodrick-Presco filer (a smoohing parameer of 129,600 for monhly daa is applied following Ravn and Uhlig, 2002). 24

26 marke leverage is counercyclical, for boh samples. The model s implied cyclicaliy of marke leverage is in line wih his empirical evidence. I is close o zero for he financial secor, and srongly negaive for he nonfinancial secor. Daa 1980 M M M M12 Model Financial Nonfinancial Table 3: Cyclicaliy of Marke Leverage. Correlaion beween cyclical componen of book leverage and real indusrial producion. Noaion: p < 0.01, p < 0.05, p < 0.1. Nex, consider he behavior of leverage around crises. Figure 10 shows he evoluion of he U.S. financial secor s leverage around he Grea Recession, comparing i wih he ypical behavior of leverage around crises in he model. In boh model and daa, marke leverage increases mildly before and sharply during a crisis. In conras, book leverage rises in he run-up and decreases during a crisis. The differen behavior of he wo measures is due o he disinc valuaion of loans. Before a crisis, ousanding loans are valued a high marke prices, lowering marke leverage. During a crisis, he value of loans srongly falls, increasing marke leverage. In conras, loan prices are held consan using he measure of book leverage, which herefore evolves differenly. Figure 10: Financial Secor Leverage. Top graphs: Typical behavior of financial secor s leverage based on simulaion of he model as in Secion 3.3. Boom graphs: U.S. financial secor s leverage around he Grea Recession. 25

27 Figure 11 repeas he exercise for nonfinancial firm leverage. Again, model and daa line up, apar from he behavior of book leverage before a crisis, which rises more srongly in he model. Thus, overall, he model performs well in maching he empirical behavior of leverage over he business cycle and around financial crises. However, a cavea is ha he model s calibraion gives smaller levels of leverage and herefore also smaller quaniaive variaions. Figure 11: Nonfinancial Secor Leverage. Top graphs: Typical behavior of nonfinancial secor s leverage based on simulaion of he model as in Secion 3.3. Boom graphs: U.S. nonfinancial secor s leverage around he Grea Recession. 3.7 Financial Regulaion Several exernaliies ha are presen in he model may joinly make he decenralized equilibrium inefficien. Firs, he inermediary akes he value of loans Q as given and does no ake ino accoun how is decisions affec his price. In urn, Q deermines wheher a credior run is riggered hrough is influence on marke leverage (condiion 14), giving rise o a pecuniary exernaliy (similar o Bianchi, 2011). Second, asse prices also ener he defaul consrains of enrepreneurs (equaion 4). Since enerpreneurs are also price-akers, hey do no inernalize how heir borrowing affecs prices, again resuling in a pecuniary exernaliy. Third, he household does no inernalize how is saving decision deermines he real ineres rae R, which affecs he inermediary s borrowing. In good imes, he household saves more o smooh consumpion, lowering R, and encouraging he inermediary o increase is borrowing and leverage, hereby building up financial fragiliy. Similarly, when he boom urns ino a bus, he household saves relaively more o safeguard agains a poenial crisis, again lowering he cos of funds for he inermediary (a saving glu exernaliy similar o Boissay e al., 2016). 26

28 To address hese exernaliies joinly, I analyze wheher financial regulaion can improve welfare in he economy. In paricular, assume ha he financial inermediary has o pay a ax on is liabiliies of ψ 2 B2. However, he financial regulaor rebaes his lump-sum cos back o he inermediary. While he financial secor s budge consrain herefore remains unchanged, he firs-order condiion (10) changes o [ ] 1 1 = β F E (R +1 + ψb ), D D +1 such ha he inermediary perceives an addiional cos of borrowing, and his cos increases in he level of is liabiliies. Due o he higher cos of borrowing, he inermediary s leverage is reduced in seady sae and hereby decreases he likelihood of financial crises. Figure 12 illusraes his relaion beween financial crises and he parameer ψ. In addiion, he figure shows how ψ affecs he household s and he inermediary s welfare. Figure 12: Financial Regulaion and Welfare. Lef: Frequency of financial crises for differen levels of ψ. Righ: Welfare of household, VH VFI T, and welfare of financial inermediary, T, for differen levels of ψ. Boh are based on a simulaion of he model of T = 500, 000 periods. Based on he considered range of ψ, inermediae values raise he welfare of boh agens by reducing leverage and financial inermediaion. In fuure research, i would be ineresing o evaluae differen macro-prudenial policies ha aim o reduce boh he frequency and he severiy of financial crises. 27

29 4 Conclusion Since he Grea Recession, rapid advances have been made o exend macroeconomic models o include financial inermediaion and crises. A he same ime, a quickly growing empirical lieraure has revealed common paerns across a range of hisorical financial crises. However, he vas majoriy of exising models is inconsisen wih his evidence. By conras, in his paper, ypical crises in he model look like ypical crises in he daa. To his end, I augmen a exbook macroeconomic model o include long-erm defaulable loans, financial inermediaion, and occasional credior runs. Wihin my framework, crises occur ou of credi booms, during which financial fragiliy builds up when inermediaries increase heir lending and leverage. They also immediaely follow low produciviy, bu i only akes a moderae adverse shock o iniiae hem. During financial crises, inermediaries face funding consrains, ha cause hem o cu heir new lending, and risk premia sharply rise. The recessions ha follow are deeper and more persisen han nonfinancial recessions, and heir severiy is well prediced by he rise in credi spreads a he sar of recessions. The model behaves differenly around nonfinancial recessions, as hey are no preceded by a srong expansion of he financial secor s balance shee or a credi boom. The model is validaed in wo addiional ways. Firs, when confroned wih a hisorical series of srucural shocks, he model predics and replicaes he occurrence of he Grea Recession. Second, he behavior of leverage of financial and nonfinancial firms is consisen wih he daa, boh around financial crises and over he business cycle. Since he model maches key empirical characerisics of financial crises, i is a suiable laboraory o sudy policy inervenions ha are argeed o reduce financial insabiliy. I find ha financial regulaion ha reduces he financial secor s leverage is welfare-improving. 28

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32 Kiyoaki, N. and J. Moore (1997). Credi cycles. Journal of Poliical Economy 105(2), Krishnamurhy, A. and T. Muir (2017). How credi cycles across a financial crisis. Unpublished manuscrip, Sanford Universiy. López-Salido, D., J. Sein, and E. Zakrajsek (2017). Credi-marke senimen and he business cycle. Quarerly Journal of Economics, forhcoming. Malin, B. A., D. Krueger, and F. Kubler (2011). Solving he muli-counry real business cycle model using a Smolyak-collocaion mehod. Journal of Economic Dynamics and Conrol 35(2), Marinez-Miera, D. and J. Suarez (2012). A macroeconomic model of endogenous sysemic risk aking. CEPR Discussion Papers (9134). Mendoza, E. G. (2010). Sudden sops, financial crises, and leverage. American Economic Review 100(5), Muir, T. (2017). Financial crises and risk premia. The Quarerly Journal of Economics 132(2), Newey, W. K. and D. McFadden (1994). Large sample esimaion and hypohesis esing. Volume 4 of Handbook of Economerics, pp Elsevier. Ospina, J. and H. Uhlig (2016). Morgage-backed securiies and he financial crisis of 2008: a pos morem. Unpublished manuscrip, Universiy of Chicago. Paul, P. (2017). Hisorical paerns of inequaliy and produciviy around financial crises. Federal Reserve Bank of San Francisco Working Paper Ravn, M. O. and H. Uhlig (2002). On adjusing he Hodrick-Presco filer for he frequency of observaions. The Review of Economics and Saisics 84(2), Schmi-Grohe, S. and M. Uribe (2003). Closing small open economy models. Journal of Inernaional Economics 61(1), Schularick, M. and A. M. Taylor (2012). Credi Booms Gone Bus: Moneary Policy, Leverage Cycles, and Financial Crises, American Economic Review 102(2), Sigliz, J. E. and A. Weiss (1981). Credi Raioning in Markes wih Imperfec Informaion. American Economic Review 71(3),

33 A Appendix A.1 Proofs This secion saes he enrepreneur s decision problem and derives simplified expressions o he soluion of his problem (see Proposiion 1), he reurn on he marke porfolio of loans (see Proposiion 2), and he evoluion of enrepreneurs ne worh (see Appendix A.1.1). A.1.1 The Enrepreneur s Decision Problem Given ne worh N (defined below) and loan price Q, a risk-neural new enrepreneur chooses he amoun of deb L new max K new subjec o,l new V good = E and he unis of capial K new. The complee decision problem is given by γ j 1 j=1 (1 γ) 2 (ωr+j K QK +j 1 Knew ω +k Q K K new = N + Q L new ω +j = L new R K +j QK +j 1 Knew L new )dφ(ω) + γr K +j Knew (23) (24) for j {1, 2,..., } (25) V good V bad (26) S +1 = Γ(S ) where Φ(ω) is he c.d.f. for ω and V bad is defined equivalen o V good, aking ino accoun he capures boh properies of he idiosyncraic shock ω. The enrepreneur s objecive funcion V good he profis from a mauring, non-defauled loan he firs erm in he curly bracke as well as profis received unil mauriy he second erm in he curly bracke. Consrain (24) is he enrepreneur s budge consrain. The consrains summarized in (25) denoe fuure defaul hresholds ω +j and (26) saes he IC consrain. In he above definiion of he objecive funcion, I exclude he household s sochasic discoun facor, since such an exclusion allows for a convenien analyical inerpreaion of Proposiion 1. I found ha he resuls remain much he same if he household s sochasic discoun facor is included insead, as in a previous version of he paper. The ne worh of a new enrepreneur is given by N = (1 γ) 2 γ j 1 j=1 ω j ( ) ωr K Q 1 K Knew j L new j dφ(ω). Appendix A.1.4 shows how o derive a simplified expression for N. 32

34 A.1.2 Incenive Compaibiliy Consrain The cumulaive disribuion funcion and parial expecaion under he uniform disribuion for ω U[0, 2] are Φ(ω +j ) = ω +j 2 2 ( ωdφ(ω) = 1 1 ) 4 ω2 +j ω +j (27). (28) Using expressions (27) and (28), (23) can be wrien as V good = (1 γ)e which can be rearranged o V good = (1 γ)e { γ j 1 R+j K QK +j 1 Knew j=1 ( 1 ω2 +j 4 ) ( L new 1 ω ) } +j + γ 2 1 γ rk +j Knew, (29) { γ j 1 R K (L new +j QK +j 1 Knew ) 2 } + j=1 4R+j K L new QK +j 1 Knew + γ 1 γ rk +j Knew, (30) since Equaion (30) can be simplified o ω +j = V good = (1 γ) L new R K +j QK +j 1 Knew ( K new for j 1. ) 2 K new S g J g Lnew (1 γ) + (Lnew ), where J g and S g are wo forward-looking auxiliary variables, defined as [ ] J g = E R+1 K QK + r+1 K γ 1 γ + γjg +1 [ ] S g 1 1 = E + γs g R K +1 QK Following a similar derivaion, he value funcion under he bad projec V bad = E γ j 1 j=1 (1 γ) m+c (ωr+j K QK +j 1 Knew ω +j L new )dφ(ω) + γr K +j Knew (31) 33

35 can be simplified o V bad = (1 γ) ( K new J b L new (m + c) 2c (1 γ) + (Lnew ) 2 K new S b ), where J b and Sb are wo forward-looking auxiliary variables, defined as [ ] (m + c) J b 2 = E R+1 K 4c QK + r+1 K γ 1 γ + γjg +1 [ ] S b 1 1 = E + γs g 4c +1. R K +1 QK Taking hese represenaions ogeher, he binding IC consrain V good = V bad can be expressed as c m 2c ( K new = (1 γ) L new J + Lnew K new S ), (32) where J and S are again wo forward-looking auxiliary variables, defined as ) ] (m + c)2 J = E [(1 R K+1 4c QK + γj +1 (33) [ (1 S = E 4 1 ) ] 1 + γs +1. (34) 4c R K +1 QK Equaions (32) (34), ogeher wih he budge consrain Q K K new = N + Q L new, give he simplified soluion o he enrepreneur s decision problem as saed in Q.E.D. 34

36 A.1.3 Reurn on Marke Porfolio of Loans The reurn on he marke porfolio of loans is given by R L = γq + (1 γ) 1 L 1 j=1 ( ) 1 Φ(ω j ) γ j 1 L new j + (1 µ) j=1 ω j ωr KQK 1 γj 1 K j newdφ(ω) 0. Q 1 The wo infinie sums in he numeraor can be simplified in he following way. (35) Firs, using he definiion of he cumulaive disribuion funcion in (27), noe ha where ( ) 1 Φ(ω j ) γ j 1 L new j = L j=1 L 1 = L new 1 + γl 2 x 1 = (Lnew 1 )2 K 1 new + γx 2. 1 R K QK 1 Second, using he definiion of he parial expecaion given in (28), rewrie j=1 0 ω j ωr K Q 1 K γj 1 K j new dφ(ω) = R K QK 1 x 1, (36) x 1, (37) Subsiuing (36) and (37) in (35) and rearranging gives R L Q 1 = γq + (1 γ) { } 1 x 1(1 + µ) 4R KQK 1 L 1. This is he expression saed in Proposiion 1, ogeher wih he aggregae sae variables L and x. Q.E.D. 35

37 A.1.4 Enrepreneur s Ne Worh The evoluion of he enrepreneur s ne worh is given by N = (1 γ) 2 γ j 1 j=1 ω j Using expressions (27) and (28), (38) can be wrien as N = (1 γ) j=1 γ j 1 ( R K Q K 1 Knew j ( ( ) ωr K Q 1 K Knew j L new j dφ(ω). (38) 1 ω2 j 4 ) ( L new j 1 ω ) ) j dφ(ω). (39) 2 Noing ha (39) can be wrien as where N = (1 γ) L new j = R K Q K 1 Knew j ω j for j 1, { R K Q K 1 K 1 L x 1 R K QK 1 }, (40) K 1 = K new 1 + γk 2 L 1 = L new 1 + γl 2 x 1 = (Lnew 1 )2 K 1 new + γx 2, are hree aggregae sae variables. Equaion (40) is he expression ha is shown in Secion A.2.1. Q.E.D. Equaion (40) has an inuiive form. The ne worh of new enrepreneurs is higher, he larger he aggregae capial sock and he associaed profis o capial R KQK 1 K 1, he lower aggregae ousanding deb L 1, and he higher he loan risk indicaor x 1. The laer effec allows enrepreneurs o offload he downside risk o he lender, bu his channel is reduced by high aggregae profis o capial. 36

38 A.2 Model Equaions & Soluion A.2.1 Model Equaions Household & Good Producer Enrepreneurs Financial Inermediary E [Λ,+1 R +1 ] = 1 (41) C χ H1+φ 1+φ Λ,+1 = β H C +1 χ H1+φ +1 1+φ (42) (1 α)y = χh 1+φ (43) Y = e a K α 1 H1 α (44) a = ρ a a 1 + ɛ a (45) Q K K new = N + Q L new (46) ( ) ( ) c m K new = (1 γ) 2c L new J + Lnew K new S (47) ) ] (m + c)2 J = E [(1 R K+1 4c QK + γj +1 (48) [ (1 S = E 4 1 ) ] 1 4c R+1 K + γs +1 (49) QK { } N = (1 γ) R K Q K 1 K 1 L x 1 R K QK 1 D + Q L + B 1 R = B + R L Q 1 L 1 (51) [ ] [ ] R+1 1 E + λ = E R+1 L (52) D +1 D +1 (50) L = L new + γl 1 (53) K = K new + γk 1 (54) x = (Lnew ) 2 K new + γx 1 (55) R L Q 1 = γq + (1 γ) { 1 x 1(1 + µ) 4R K QK 1 L 1 } (56) 37

39 a Occasional Financial Crises B If Q L κ or r 1 = 1 λ = 0 If B Q L > κ and r 1 = 0 B = τ(d unc + Q unc Capial Producers and Resource Consrain L new,unc ) + B 1 R R L Q 1 L 1 + Q γl 1 Q K = 1 + ζ( K 1) (57) K 1 Y = C + D + K (1 δ)k 1 + ζ ( ) 2 K 1 K 1 + (58) 2 K 1 ( ) 1 x (1 1 γ)µ 4 R K (59) QK 1 ( R K Q 1 K = Q K (1 δ) + α Y ) (60) K 1 A.2.2 Compeiive General Equilibrium Given he opimaliy condiions o he agens decision problems and he clearing of goods, labor, and deb markes, he equilibrium condiions of he model for = 0, 1,..., are lised in Appendix A.2.1, given an iniial sae S 0. The endogenous variables can be separaed ino a vecor of nonsae variables X and a vecor of sae variables S. X +1 is unknown in period and S = { S, Ŝ } comprises boh exogenous sae variables S and endogenous sae variables Ŝ. S = {ρ a a 1 + ɛ a } includes he echnology shock ɛ a and he probabiliy disribuion of his shock is known o all agens. The realizaion ɛ+1 a is unknown in period. Ŝ = {K 1, L 1, x 1, R B 1, r 1 } collecs he endogenous sae variables and Ŝ +1 is known in period. Overall, he model has six sae variables. One is linked o he shock, hree arise from he defaulable loan conracs, K 1, L 1, and x 1, R B 1 accouns for he ousanding deb of he financial inermediary, and r 1 is an indicaor variable equal o one if here has been a credior run in he previous period and zero oherwise. Definiion 1. A compeiive general equilibrium is a soluion of he model which is given by a se of policy funcions Ŝ +1 = fŝ(s ) and X = f X (S ) ha saisfy he model s equilibrium condiions lised in Appendix A.2.1 for = 0, 1,..., in he relevan sae space. 38

40 A.2.3 Soluion Technique Given he definiion of a soluion in Appendix A.2.2, I obain he policy funcions Ŝ +1 = fŝ(s ) and X = f X (S ) using a projecion algorihm. Broadly, his involves hree choices. Firs, one has o choose a grid on which he model is solved. Second, a parameerizaion of he policy funcions has o be deermined. Third, given an iniial parameerizaion, one has o choose an ieraion procedure. These hree choices srucure he descripion below. Grid. The model is solved on a Smolyak sparse grid due o he curse of dimensionaliy (Bellman, 1961). The consrucion of he Smolyak sparse grid works as follows. The grid poins in he space [ 1, 1] for each sae variable are obained by ensor-producs of nesed ses of Chebyshev exrema and he applicaion of he Smolyak rule for a given level of approximaion ha conrols how many of hese ensor-producs are included in he grid (see also Malin, Krueger, and Kubler, 2011). I selec 5 for he level of approximaion, giving 2433 grid poins for he five sae variables (excluding r 1 ). Nex, he grid poins are ransformed from he space [ 1, 1] ino he relevan space given he model s calibraion. In order o deermine he relevan space, I solve he model firs wih a hird-order perurbaion mehod around a deerminisic seady sae (ignoring he occasionally binding consrain), simulae he model for 500, 000 periods, and choose he lower and upper bounds for each sae variable as he associaed ones in he grid (denoed L B and U B ). 23 A linear ransformaion (x + 1) (U B L B ) 2 + L B is used o ransform each grid poin x from [ 1, 1] ino [L B, U B ], giving he full se of grid poins j = 1,..., M in he relevan sae space. Parameerizaion of policy funcions. I parameerize several non-sae variables using hirdorder ordinary polynomials. In paricular, le X P be a parameerized variable where X P {K new 1,, R +1, J, S } D and le r be an indicaor funcion deermining wheher here is a credior run in period, in which case r = 1, or no, such ha r = 0. Then, X P (S ) is parameerized using he piecewise flexible form wih separae coefficiens β X r =0 and βx r =1 X P (S ) = (1 r )β X 0 T(S ) + r β X 1 T(S ), (61) where T(S ) is a vecor collecing he basis funcions. The number of grid poins is larger han he number of coefficiens. Hence, he oulined soluion algorihm does no give an exac soluion on he grid poins such as collocaion mehods (see for example Malin, Krueger, and Kubler, 2011). 23 I slighly perurb he model by adding a small risk premium on deb for he financial inermediary which depends on he level of deb. This gives a unique porfolio choice in a deerminisic seady sae and allows o solve he model wih sandard perurbaion mehods around his deerminisic seady sae (see for example Schmi-Grohe and Uribe (2003) for similar echniques wih respec o small open-economy models). The hird-order perurbaion soluion also serves as a firs guess for coefficiens in he parameerized policy funcions. During he ieraion, he risk premium is sequenially reduced unil i reaches zero (see also chaper 5.9. on Homoopy Mehods in Judd, 1998, in his regard). I hank Fabrice Collard for a discussion on his opic. 39

41 Ieraion. Given an iniial guess for he coefficiens β0 X and βx 1, he ieraion proceeds as follows. 1. Obain he vecor collecing he basis funcions T(S,j ) for a given grid poin j. Assume ha here is no credior run in period and calculae he parameerized variables X P,j (S,j; r = 0) via (61). Subsiue X P,j (S,j; r = 0) ino he se of equaions summarized in Appendix A.2.1. Using he equilibrium condiions, solve for he res of he variables and se he Lagrange muliplier λ = 0. Check wheher he financial inermediary s leverage exceeds he hreshold κ. If so, go back o he beginning, se r = 1, use he parameerizaion X P,j (S,j; r = 1) insead, and add he consrain B = τ(s,j ). This separaes he grid poins ino wo ses: a se of poins for which here is no credior run and a se for which here is a credior run. When here is a credior run, K new is no parameerized, bu given by he equilibrium condiions. Insead, he ineremporal equaion (16) is used o check wheher he Lagrange muliplier λ is posiive. 2. Having solved for all period variables a grid poin j, one obains nex period s endogenous sae variables Ŝ +1,j. To approximae inegrals arising from expecaion operaors in ineremporal equaions, I use nine Hermie-Gaussian quadraure nodes and weighs for period + 1. This gives nex period s exogenous sae variables S +1,j,i and herefore S +1,j,i for each node i {1, 2,..., 9} in period + 1 a each grid poin j. Assume ha here is no credior run in period + 1 a node i for grid poin j. Use he iniial parameerizaion X P +1,j,i (S +1,j,i; r +1 = 0) again and solve for he res of he variables a node i in period + 1. If r = 0, check wheher he financial inermediary s leverage in period + 1 exceeds hreshold κ. If so, use he parameerizaion X P +1,j,i (S +1,j,i; r +1 = 1) insead, and add he consrain B +1 = τ(s +1,j,i ). 3. Having solved for all period + 1 variables, compue he expecaion inegrals in he equilibrium condiions. The ineremporal equaions give an esimae X P,j (S,j) for each of he iniially parameerized variables X P,j (S,j) a each grid poin j. A fixed poin is obained when he iniially assumed value X P,j (S,j) is equal o X P,j (S,j). 4. Unil a fixed poin is reached, ierae over he coefficiens β0 X and βx 1 in he policy funcions (61). Collec he vecors of basis funcions T(S,j ) for all grid poins for which here is no credior run in period, combine hem in a marix T 0 (S ) and projec hese on he obained esimaes X P,j (S,j), giving ˆβ X 0 ( T 0 (S ) T 0 (S ) ) 1 T0 (S ) X P r =0, where X P r is a vecor collecing he obained esimaes XP =0,j (S,j) for which here is no credior run. Repea he same for all grid poins for which here is a credior run in period, resuling in ˆβ X 1 ( T 1 (S ) T 1 (S ) ) 1 T1 (S ) X P r =1. 40

42 5. Compue he coefficiens β X 0 and βx 1 ha are used for he nex ieraion via β X 0 = (1 ξ) ˆβ X 0 + ξβ X 0 and β X 1 = (1 ξ) ˆβ X 1 + ξβ X 1, where 0 < ξ < 1 is a dampening parameer which makes convergence more likely (ξ = 0.1 is used). 6. Check for convergence and end ieraion if M Mb X P A M j=1 X P,j X P,j r =0 X P,j r =0 < η, and Mb X P A M j=1 X P,j X P,j r =1 X P,j r =1 < η, where A = {K new 1, D, R +1, J, S }, A = A\K new, M denoes he number of poins in he sae space and Mb he number of binding grid poins. η = 5 04 is used. 24 Given he policy funcions X P (S ), one can use he model s equaions o obain he full se of policy funcions Ŝ +1 = fŝ(s ) and X = f X (S ) for a soluion of he model as given in Definiion 1. A.2.4 Accuracy Besides having an accurae soluion on he grid poins, I check wheher he soluion is also accurae a oher poins in he sae space. The accuracy of he soluion is confirmed by analyzing he absolue residual equaion errors X P X P X P in a simulaion of he model as suggesed by Judd (1992). X P is obained as described in Appendix A.2.3. I simulae he model for 500, 000 periods and compue he decimal log of he absolue residual equaion errors for each period and for each X P {K new 1, D, R +1, J, S }. In Figure 13, I plo hisograms of hose errors for each parameerized variable, where red lines indicae means. The means all lie around -3. Following (Judd, 1992), I herefore consider he soluion o be accurae. Since I am paricularly ineresed in obaining an accurae soluion a poins for which he occasionally binding consrain holds and a financial crisis occurs, I repor he decimal log of he absolue residual equaion errors a hose poins separaely in Figure 14, ogeher wih he Lagrange muliplier λ. The soluion is sill very accurae in he sae space where financial crises occur, which lies far away from he seady sae of he model. Moreover, he Lagrange muliplier λ on he deb consrain is always posiive during financial crises. I conclude ha he oulined 24 I find ha seing η o a lower value does no srongly increase he accuracy of he soluion away from he grid poins or change any of he resuls, bu resuls in a significan increase in compuaional ime. 41

43 soluion algorihm gives an accurae nonlinear global soluion of he model, boh in he sae space in which sandard business cycles occur, as well as in he region in which he economy experiences a financial crisis. Figure 13: Accuracy. Hisograms for absolue residual equaion errors in decimal log basis, based on a 500,000-period simulaion of he model. Red lines indicae means Figure 14: Accuracy. Hisograms for absolue residual equaion errors in decimal log basis for periods during which he economy experiences a financial crisis, based on a 500,000-period simulaion of he model. The subplo of he Lagrange Muliplier shows he value of λ insead. Red lines indicae means. 42

44 A.3 Daa A.3.1 Empirical Evidence Procyclical Lending Figures 15 and 16 provide empirical evidence on he cyclical behavior of lending volume and sandards. Figure 15 shows he percenage change from a year ago of U.S. commercial banks lending, which srongly declines during recessions and picks up aferwards. Figure 16 shows he ne percenage of senior loan officers reporing a ighening of lending sandards in a survey conduced by he U.S. Federal Reserve. Again, lending sandards ighen during recessions and ease subsequenly. Figure 15: Source: Federal Reserve Bank of S. Louis. Grey bars denoe NBER recessions. Figure 16: Source: Federal Reserve Bank of S. Louis. Grey bars denoe NBER recessions. 43